Environmental Sciences) (No. 4)

Light Scattering Reviews 4 Single Light Scattering and Radiative Transfer Alexander A. Kokhanovsky (Editor) Light S...

Author: Alexander A. Kokhanovsky

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Light Scattering Reviews 4

Single Light Scattering and Radiative Transfer

Alexander A. Kokhanovsky (Editor)

Light Scattering Reviews 4 Single Light Scattering and Radiative Transfer

Published in association with

Praxis Publishing Chichester, UK

Editor Dr Alexander A. Kokhanovsky Institute of Environmental Physics University of Bremen Bremen Germany

SPRINGER±PRAXIS BOOKS IN ENVIRONMENTAL SCIENCES (LIGHT SCATTERING SUB-SERIES) SUBJECT ADVISORY EDITOR: John Mason B.Sc., M.Sc., Ph.D. EDITORIAL ADVISORY BOARD MEMBER: Dr Alexander A. Kokhanovsky, Ph.D. Institute of Environmental Physics, University of Bremen, Bremen, Germany

ISBN 978-3-540-74275-3 Springer Berlin Heidelberg New York Springer is part of Springer-Science + Business Media (springer.com) Library of Congress Control Number: 2008939874 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. # Praxis Publishing Ltd, Chichester, UK, 2009 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speci®c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Jim Wilkie Project copy editor: Mike Shardlow Author-generated LaTex, processed by EDV-Beratung, Germany Printed on acid-free paper

Contents

List of contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XIII Notes on the contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XXIII . Part I Single Light Scattering 1 Scaled analogue experiments in electromagnetic scattering Bo ˚ A. S. Gustafson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Theoretical basis for scaled electromagnetic experiments . . . . . . . . . . . . . 1.3 Scattering by a few common particle classes . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Mie-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Evaluating the scattering by HCPs and other complex natural particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The scattering problem in the laboratory setting . . . . . . . . . . . . . . . . . . . . 1.4.1 Considerations in designing a high-precision scattering laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Analogue particle materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Scaled analogue scattering laboratories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 The classic laboratories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 The University of Florida laboratory . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Complex interiors scattering experiment example . . . . . . . . . . . . 1.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 14 16 16 20 24 27 28

2 Laboratory measurements of the light scattered by clouds of solid particles by imaging technique Edith Hadamcik, Jean-Baptiste Renard, Anny-Chantal Levasseur-Regourd, Jean-Claude Worms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Astronomical and atmospheric context . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Polarization measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Samples preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 31 32 34 41 41

3 3 4 5 6 10 11

VI

Contents

2.2.2 Levitation techniques, advantages and restrictions . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Phase curves and their parameters . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Optical and physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Solar system dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Atmospheric dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusions and future developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

3 Jones and Mueller matrices: structure, symmetry relations and information content S.N. Savenkov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Basic deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Internal structure of a general Mueller–Jones matrix . . . . . . . . . . . . . . . . . 3.4 Symmetry relations for Mueller–Jones matrix . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Forward scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Backward scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The depolarizing Mueller matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Structure of the depolarizing Mueller matrix . . . . . . . . . . . . . . . . 3.5.2 Matrix models of depolarization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Cloude’s coherency matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Block-diagonal structure of the Mueller matrix . . . . . . . . . . . . . . 3.6 Structure and information content of the Mueller–Jones matrix in continuous medium approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Mueller–Jones matrices of basic types of anisotropy and partial equivalence theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Polar decomposition of Mueller–Jones matrices . . . . . . . . . . . . . . 3.6.3 Generalized matrix equivalence theorem . . . . . . . . . . . . . . . . . . . . 3.6.4 Eigenanalysis of the Jones matrices of dichroic, birefringent, and degenerate media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Green functions for plane wave scattering on single nonspherical particles Tom Rother . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Some basic considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Formulation of the scattering problems . . . . . . . . . . . . . . . . . . . . . 4.2.2 Spherical coordinates and eigensolutions of the vector-wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Dyadics and Green’s theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Dyadic Green functions and light scattering . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Dyadic free-space Green function . . . . . . . . . . . . . . . . . . . . . . . . . .

44 45 45 45 49 53 58 58 62 64 66

71 71 72 76 78 80 81 82 82 83 89 92 98 99 101 103 106 112 114

121 121 122 122 124 128 132 132

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4.3.2 Dyadic Green functions of the scattering problems . . . . . . . . . . . Singular integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Ideal metallic scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Dielectric scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Rayleigh’s hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Symmetry and Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Far-ﬁeld behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 The plane wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 The scattered and total ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4

VII

134 142 142 146 148 151 152 153 158 159 160 164 165

Part II Radiative Transfer 5 Space-time Green functions for diﬀusive radiation transport, in application to active and passive cloud probing Anthony B. Davis, Igor N. Polonsky, Alexander Marshak . . . . . . . . . . . . . . . . . 5.1 Context, motivation, methodology, and overview . . . . . . . . . . . . . . . . . . . . 5.2 Elements of time-dependent three-dimensional radiative transfer . . . . . . 5.2.1 Radiant energy transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Dirac-δ boundary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Remotely observable ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Flux-based spatial and temporal moments . . . . . . . . . . . . . . . . . . . 5.2.5 Vertical variation of scattering coeﬃcient . . . . . . . . . . . . . . . . . . . 5.3 Formulation in the Fourier–Laplace domain . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Temporal Green functions and pulse-stretching problems . . . . . . 5.3.2 Spatial Green functions and pencil-beam problems . . . . . . . . . . . 5.4 Diﬀusion approximation for opaque scattering media . . . . . . . . . . . . . . . . 5.4.1 Derivation from the time-dependent 3D RT equation . . . . . . . . . 5.4.2 Directional and spatial enhancements . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Boundary conditions, including boundary sources . . . . . . . . . . . . 5.4.4 Remote sensing observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Fourier–Laplace transformation for stratiﬁed media . . . . . . . . . . 5.5 Solutions of diﬀusive Green function problems . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Homogeneous cloud with an isotropic boundary point-source . . 5.5.2 Stratiﬁed cloud with an isotropic boundary point-source . . . . . . 5.5.3 Homogeneous cloud with normally incident illumination at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Homogeneous cloud with normally incident illumination at a point from above and a reﬂective surface below . . . . . . . . . . . . . . 5.5.5 Homogeneous cloud with uniform oblique illumination . . . . . . . . 5.6 Inverse Fourier–Laplace transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Uniform clouds with an isotropic boundary point-source in (5.1), using exact boundary conditions in (5.2) . . . . . . . . . . . . . . .

169 169 172 172 174 177 179 184 186 187 187 188 188 191 197 199 200 202 202 206 206 208 209 210 211

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5.6.2

Uniform clouds with an isotropic internal point-source, using extended boundary conditions in (4.34) . . . . . . . . . . . . . . . . . . . . . 5.7 Temporal Green functions applied to in situ cloud lidar . . . . . . . . . . . . . . 5.7.1 Forward model for the radiometric signal . . . . . . . . . . . . . . . . . . . 5.7.2 Illustration with SNR estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Temporal Green functions applied to oxygen A-band spectroscopy of overcast skies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 A-band spectroscopy as observational time-domain RT . . . . . . . . 5.8.2 Path-length moments from below . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3 Path-length moments from above . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Space-time Green functions applied to multiple-scattering cloud lidar (MuSCL) observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Space-based MuSCL systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.2 Ground-based and airborne MuSCL systems . . . . . . . . . . . . . . . . . 5.9.3 Moment-based methods for MuSCL . . . . . . . . . . . . . . . . . . . . . . . . 5.9.4 Deeper mining of MuSCL observations for cloud information . . 5.10 Further applications to passive solar observations of clouds . . . . . . . . . . . 5.10.1 Operational cloud remote sensing in the solar spectrum . . . . . . . 5.10.2 Opacity-driven 3D radiation transport . . . . . . . . . . . . . . . . . . . . . . 5.10.3 The independent pixel approximation for steady/uniform illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.4 The independent pixel approximation for space/time Green functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.5 Landsat-type observations of clouds from space, and the nonlocal IPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.6 Zenith radiance reaching ground, and the nonlocal IPA . . . . . . . 5.10.7 Green functions at work in the adjoint perturbation approach to 3D radiation transport eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˜ A Responses T˜(k) and R(k) for horizontal transport away from an isotropic boundary source in stratiﬁed clouds . . . . . . . . . . . . . . . . . . . . . . . A.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Transmitted light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Reﬂected light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˆ B Responses Tˆ(s) and R(s) for pulse stretching for an isotropic boundary source in stratiﬁed clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Transmitted light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Reﬂected light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˜ C Responses T˜(k) and R(k) for steady illumination by a normally incident pencil-beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Transmitted light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.3 Reﬂected light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˆ D Responses Tˆ(s) and R(s) for pulsed normal or oblique uniform illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

212 220 220 221 223 223 228 232 240 241 242 244 246 247 247 248 248 250 251 255 258 260 265 265 265 265 266 266 266 266 266 266 267 267 268 268

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D.2 Transmitted light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3 Reﬂected light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E Scaling exponents for diﬀusive Green function moments from the random walk approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1 Caveat about photons as ‘particles’ of light . . . . . . . . . . . . . . . . . . E.2 Elements of Brownian motion theory . . . . . . . . . . . . . . . . . . . . . . . E.3 Transmitted light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.4 Reﬂected light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F Scaling exponents for time-domain anomalous diﬀusion by extending the random walk approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1 Anomalous diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.2 Observational validation, and evolution toward anomalous transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Radiative transfer of luminescence light in biological tissue Alexander D. Klose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Light–tissue interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Luminescent imaging probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Fluorescent probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Bioluminescent probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Radiative transfer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Equation of radiative transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Partly-reﬂecting boundary condition . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Partial boundary current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Scattering phase function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Bioluminescence system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Fluorescence system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Rate equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 Steady-state domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Radiative transfer approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Discrete ordinates method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Spherical harmonics method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.3 Simpliﬁed spherical harmonics method . . . . . . . . . . . . . . . . . . . . . . 6.7.4 Diﬀusion method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Finite diﬀerence methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Step-diﬀerencing method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Diamond-diﬀerencing method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.3 Centered-diﬀerencing method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.1 Source iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.2 GMRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.3 Multigrid methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Light propagation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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268 269 269 270 270 271 271 272 272 274 278 293 293 295 297 297 298 300 300 300 302 302 303 304 304 305 306 307 307 308 311 314 317 319 319 320 321 322 322 323 324 324

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6.11 Image reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.1 Optimization techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.2 Algebraic reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Image reconstruction example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Summary and concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Boundary coeﬃcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Coeﬃcients for partial current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The characteristic equation of radiative transfer theory N.N. Rogovtsov, F.N. Borovik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The classical variant of the characteristic equation of radiation transfer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Reducing the classical variant of the characteristic equation of radiative transfer theory to the family of reduced characteristic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Solving the classical variant of the characteristic equation of the radiative transfer theory in an analytical form . . . . . . . . . . . . . . . . . . . . . . 7.3.1 General properties of discrete spectra and eigenfunctions of the reduced characteristic equations of the radiative transfer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Obtaining solutions of inhomogeneous reduced characteristic equations of radiative transfer theory in an analytical form . . . . 7.3.3 Substantiation of the algorithm for calculating discrete spectra of reduced characteristic equations of radiative transfer theory . 7.3.4 On the stability of solutions of reduced characteristic equations and inﬁnite systems of linear algebraic equations . . . . . . . . . . . . . 7.4 Analytical presentation of azimuthal harmonics of Green’s function of the radiative transfer equation for the case of an inﬁnite plane-parallel turbid medium and an arbitrary phase function . . . . . . . . . . . . . . . . . . . . . 7.5 Eﬀective algorithm to calculate the azimuthally averaged reﬂection function and the reﬂection function for a semi-inﬁnite plane-parallel turbid medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 General invariance relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Asymptotical formulas for the azimuthal harmonics of Green’s function of the radiative transfer equation . . . . . . . . . . . . . . . . . . . 7.5.3 Integral equations and formal analytical representations of the azimuthally averaged reﬂection function . . . . . . . . . . . . . . . . . . . . 7.5.4 Numerical results for the case of water cloud C.1 model . . . . . . . 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: General mathematical notations, notions and constructions . . . . Appendix B: Metric spaces and their simplest properties . . . . . . . . . . . . . . . . . . Appendix C: Linear, normed and Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . .

325 328 331 332 333 335 335 336 336 347 347 350 350

354 358

359 371 378 381

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389 389 392 393 403 407 408 410 411

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Appendix D: Elementary notions of spectral theory of operators and linear operator equation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix E: Shturm’s method and the simplest properties of Jacobi’s matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix F: The basic notions of the diﬀerence equation theory. Perron’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix G: Continuous fractions. Van Vleck’s and Pincherle’s theorems . . . Appendix H: The correct algorithm for calculating minimal solutions of three-term recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part III Dynamic and Static Light Scattering: Selected Applications 8 Advances in dynamic light scattering techniques P. Zakharov, F. Scheﬀold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Scattering regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Single scattering limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Multiple scattering limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Intermediate scattering regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 General DLS techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Multi-tau correlation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Binning technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Spectral analysis in DLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Problem of non-ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Pusey and van Megen method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Multi-speckle technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Double-cell technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Echo-technique with sample rotation . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 Echo-technique without sample rotation . . . . . . . . . . . . . . . . . . . . 8.4.6 Combination of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Time-resolved methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Time-resolved spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Time-resolved correlation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Time-resolved structure function analysis . . . . . . . . . . . . . . . . . . . 8.6 Space-resolved methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Space-resolved correlation and structure analysis . . . . . . . . . . . . . 8.6.2 Space-resolved spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Laser speckle imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.4 LSI with active noise reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.5 Dynamic contrast analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.6 Simpliﬁed speckle contrast calculations . . . . . . . . . . . . . . . . . . . . . 8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

433 433 435 435 437 439 441 441 443 444 444 446 446 447 448 449 451 452 452 454 455 456 456 457 457 459 460 461 462 463

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9 Static and dynamic light scattering by aerosols in a controlled environment R.P. Singh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Atmospheric particles and their dispersion . . . . . . . . . . . . . . . . . . 9.1.2 Particle size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Diﬀerent methods for particle sizing . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Optical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Static light scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Laser diﬀraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Dynamic light scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Aerosol size and distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Eﬀect of temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Eﬀect of humidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Eﬀect of concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Aerosols and climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

469 469 470 470 471 472 472 475 476 483 484 485 487 494 494 495

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

List of Contributors

Felix N. Borovik Heat and Mass Transfer Institute National Academy of Sciences of Belarus P. Brovki Street 15 220072 Minsk Belarus [email protected]

Alexander D. Klose Department of Radiology Columbia University ET351 Mudd Building, MC 8904 500 West 120th Street New York, NY 10027 USA [email protected]

Anthony B. Davis Space and Remote Sensing Group Los Alamos National Laboratory P.O. Box 1663 Los Alamos, NM 87545 USA Now at: Jet Propulsion Laboratory California Institute of Technology 4800 Oak Grove Drive Pasadena, CA 91109 USA [email protected]

Anny-Chantal Levasseur-Regourd UPMC Universit´e Paris 06, UMR 7620, SA/CNRS R´eduit de Verri`eres BP3 91391Verri`eres-le-Buisson, France [email protected]

Bo ˚ A. S. Gustafson Department of Astronomy University of Florida Gainesville, FL 32611 USA [email protected]

Edith Hadamcik UPMC Universit´e Paris 06, UMR 7620, SA/CNRS R´eduit de Verri`eres BP3 91371 Verri`eres-le-Buisson France [email protected]

Alexander Marshak Climate and Radiation Branch NASA Goddard Space Flight Center 613.2, Greenbelt, MD 20771 USA [email protected] Jean-Baptiste Renard LPCE/CNRS 3A Avenue de la Recherche Scientiﬁque 45071 Orl´eans-Cedex 2 France [email protected] Igor N. Polonsky Department of Atmospheric Sciences Colorado State University 1371 Campus Delivery Fort Collins, CO 80523-1371 USA [email protected]

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List of Contributors

Nikolai N. Rogovtsov Applied Mathematics Department Belarusian National Technical University Prospect Nezavisimosty 65 220013 Minsk Belarus [email protected] Tom Rother Remote Sensing Technology Institute of the German Aerospace Center Kalkhorstweg 53 D-17237 Neustrelitz Germany [email protected] Sergey N. Savenkov Department of Radiophysics Kiev Taras Shevchenko University Vladimirskaya Street 64 Kiev 01033 Ukraine [email protected] Frank Scheﬀold Department of Physics University of Fribourg Chemin du Mus´ee 3 1700 Fribourg Switzerland [email protected]

Ravindra Pratap Singh Physical Research Laboratory Navrangpura, Ahmedabad – 380009 India [email protected] Jean-Claude Worms ESF & LSIIT-TRIO UMR CNRS 7005 European Science Foundation B.P. 90015 – 1, Quai Lezay-Marn´esia F-67080 Strasbourg Cedex France [email protected] Pavel Zakharov Solianis Monitoring AG Leutschenbachstr. 46 8050 Zurich Switzerland [email protected]

Notes on the contributors

Felix N. Borovik graduated from the Physical Department of the Belarusian State University, Minsk, Belarus, in 1970. His research is directed to the calculation of optical and thermodynamic properties of high-pressure plasma, using the self-consistent ﬁeld equations and taking into account relativistic eﬀects for heavy atoms and ions, and to the construction of wide range equations of state of granite and water, taking into account evaporation, dissociation and ionization. Also he has performed a number of investigations in the ﬁelds of radiative transfer and catalytic growth of nanoﬁbres. He is currently a research worker of Heat and Mass Transfer Institute of National Academy of Science of Belarus, Minsk. F. N. Borovik has published more than seventy papers.

Anthony B. Davis recently joined NASA’s Jet Propulsion Laboratory, coming from an 11-year tenure at Los Alamos National Laboratory, Space & Remote Sensing Group. From 1992 to 1997, he worked at NASA’s Goddard Space Flight Center, Climate & Radiation Branch. He obtained his PhD in physics at McGill University in 1992; his MSc is from Universit´e de Montr´eal and his undergraduate degree from Universit´e Pierre et Marie Curie (Paris 6). His research interests are in the theoretical, computational and observational aspects of three-dimensional radiative transfer (both steady-state and time-dependent), as well as in the mathematics and statistics that apply naturally to the spatial variability of clouds and other turbulent geophysical processes (mostly data analysis and stochastic modelling using wavelets and fractals). He applies this expertise to optical remote sensing of cloud and surface properties using both active and passive techniques. His community involvement includes co-editing a monograph on 3D Radiative Transfer in Cloudy Atmospheres (Springer-Verlag, 2005), sitting on technical committees and organizing sessions

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Notes on the contributors

on specialized topics for the American Geophysical Union (AGU), the American Meteorological Society (AMS), the American Nuclear Society (ANS), NASA’s International 3D Radiation Code (I3RC) project, and the US DOE’s Atmospheric Radiation Measurement (ARM) Program. He has authored or co-authored 49 articles in refereed journals, 48 peerreviewed contributions to edited volumes, 73 conference papers, 7 technical reports, and a number of outreach texts (Eos-AGU Transactions, Laser Focus World, etc.).

Bo ˚ A. S. Gustafson is a Professor of Astronomy at the University of Florida and Director of its Laboratory for Astrophysics. Besides electromagnetic scattering theory and experiments, his research interests include planetary systems formation, the dynamics and physical evolution of cosmic grains, meteoroids, comets and asteroids. A developer of the original concepts for the GIADA instrument on ESA’s Rosetta mission, Gustafson is a CoI of several NASA and ESA space instruments/missions past and present. He served on numerous international panels or expert groups including NASA, UNESCO, and the UN and as president of International Astronomical Union Commission 21. In 1996 the International Astronomical Union named Asteroid 4275 BoGustafson in recognition of his work to develop a model for primitive solar system solids as aggregate structures of evolved interstellar grains, a work only possible because of scaled analogue experiments in the two classical laboratories described in Chapter 1. In 1995 Gustafson designed and built the next generation scaled laboratory facility at the University of Florida. As President of DataGrid Inc. a developer of high-precision Global Navigation Satellite System (GNSS) receivers, land management hardware/software systems and national resource management consulting, as well as a member of the University of Florida’s Center for African Studies, Gustafson’s activities also include the development and introduction of advanced technology and science to address problems in the developing world. Gustafsons contribution to science and technology was recognized through the ID Hall of Fame award at the 2008 ID WORLD International Congress.

Edith Hadamcik is an associated scientist at Service d’A´eronomie (1999–). She has been a high school teacher in physics. She defended her PhD thesis in 1999 for the work ‘Contribution to a classiﬁcation of comets from observations and laboratory simulations’ (UPMC, Universit´e Paris 06). Her main research areas include: (1) studies of physical properties of

Notes on the contributors

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solid particles by the light they scatter; (2) in situ observations with the Optical Probe Experiment (OPE/Giotto experiment on comets 1P/Halley and 26P/GriggSkjellerup); and (3) remote observations of comets, which allow the deﬁning of comae regions and/or comets with diﬀerent physical properties of the dust experimental light scattering simulations, e.g. cosmic dust analogues: cometary particles, asteroidal regoliths and Titan’s aerosols. She is the co-investigator of several experiments such as the PRopri´et´es Optiques des Grains Astronomiques et Atmosph´eriques (PROGRA2 experiment, 1994–), the Interaction in Cosmic and Atmospheric particle System at the International Space Station (ICAPS/ISS experiment, 2001–), the PAMPRE experiment ‘Titan’s aerosols analogues’ (2005–). Dr Hadamcik was a member of the European Space Agency (ESA) topical team (physico-chemistry of ices in space, 2000–2004). She is a member of CLEA (Comit´e de Liaison Enseignant-Astronomes), which is an association to promote astronomy in schools.

Alexander D. Klose graduated from the Technical University of Berlin, Germany, and received a Diploma in Physics in 1997. He earned a PhD in physics from the Free University of Berlin, Germany, in 2001. He was a visiting scientist at Los Alamos National Laboratory, at the State University of New York, and is currently an Assistant Professor at the Department of Radiology of Columbia University in New York. His current research focuses on numerical solutions of light propagation models in biological tissue, and on image reconstruction of ﬂuorescent and bioluminescent sources for optical molecular imaging.

Anny-Chantal Levasseur-Regourd is Professor at UPMC (Universit´e Paris 06). Her research activities are mainly devoted to the physics of comets, asteroids and the interplanetary medium, and to the light scattering properties of solid particles, through remote and in situ observations, as well as through numerical and laboratory simulations. She has been PI for experiments on board a Russian space station and on European rockets and spacecraft, including the OPE experiment for the Giotto mission to comets, which provided the ﬁrst evidence for the evolution of dust particles in cometary comae and resulted in the discovery of their extremely low density. She has obtained the ﬁrst reference tables on zodiacal light and provided a classiﬁcation of comets and asteroids from the physical properties of their dust particles. Professor Levasseur-Regourd now serves actively on science teams for the Rosetta rendezvous mission to comet Churyumov-Gerasimenko,

XVIII Notes on the contributors and coordinates an experiment on light scattering on dust agglomerates, Light Scattering Unit in the Interaction in Cosmic and Atmospheric particle System (LSU-ICAPS) for the International Space Station (ISS). She is the author of about one hundred and ﬁfty papers in refereed journals and books, and convener and editor for various international scientiﬁc meetings; she is also active in the presentation of the science to the public (e.g. being the French point of contact for the International Year of Astronomy 2009). Professor Levasseur-Regourd has been chair of IAU and COSPAR commissions, and of ISSI and ESA working teams. Her contribution to planetary sciences has been recognized with the naming of Asteroid 6170 Levasseur.

Alexander Marshak received his MS in applied mathematics from Tartu University, Estonia, in 1978 and his PhD in numerical analysis in 1983 from the Siberian Branch of the Soviet Academy of Sciences (Novosibirsk, Russia). In 1978, he joined the Institute of Astrophysics and Atmospheric Physics (Tartu, Estonia) and worked there for 11 years. During that time his major research interests were in the numerical solution of transport problems, radiative transfer in vegetation and remote sensing of leaf canopies. In 1989, he received an Alexander von Humboldt fellowship and worked for two years at Goettingen University (Germany). He joined the NASA Goddard Space Flight Center (GSFC), Greenbelt, MD, in 1991, ﬁrst working for SSAI then UMBC/JCET and ﬁnally with the NASA/GSFC, where he has been employed since 2003. He has published over one hundred refereed papers, books, and chapters in edited volumes where the most recent ones are focused on scaling and fractals, cloud structure, 3D radiative transfer in clouds, remote sensing, and interaction between clouds and aerosols. He is a member of several NASA and the DoE Atmospheric Radiation Measurement (ARM) Program science teams.

Igor N. Polonsky received his MS in physics from the Belarussian State University (Minsk, Belarus) in 1983 and his PhD in physics in 1998 from the Institute of Physics (Minsk, Belarus). In 1983 he joined the Institute of Physics in Minsk and worked there for 17 years. His major research interests were in the approximate solution of radiative transport problems, and active and passive remote sensing of the atmosphere–ocean system. In 2004 he received a Los Alamos laboratory director fellowship and worked till 2007 at the laboratory. He joined the Colorado State University in 2007 and currently he is involved in

Notes on the contributors

XIX

CloudSat and OCO projects as an algorithm developer. He has published over thirty-ﬁve refereed papers and chapters in edited volumes where the most recent ones are focused on 3D radiative transfer in clouds, remote sensing of clouds and aerosols, interaction between clouds and aerosols.

Jean-Baptiste Renard received his PhD in 1992 on the optical properties of interplanetary and cometary dust from Paris Universit´e. He is at Laboratoire de Physique et Chimie de l’Environnement (LPCE/CNRS, Orl´eans, France). He is the Principal Investigator (PI) of several instruments onboard stratospheric balloons for the study of the stratospheric chemistry (nitrogen species, bromine species, water vapour) and the physical properties of aerosols. He has also been PI of the PROGRA2 (Propri´et´es Optiques des Grains Astronomiques et Atmosph´eriques) instrument since 2001, and is involved in science teams of GOMOS experiment on the ENVIronmental SATellite (ENVISAT) and of IPE-ICAPS (Interaction in Cosmic and Atmospheric Particle System Precursor Experiment). His research interests concern the development and the data analysis of instruments that perform UV–visible spectroscopy and light scattering measurements for atmospheric and planetary studies. He is the author of more than sixty papers in refereed journals.

Nikolai N. Rogovtsov graduated from the Physical Department of the Belarusian State University, Minsk, Belarus, in 1970. He received his PhD in optics and his Doctor of Sciences degree in physics and mathematics from the B.I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Belarus, in 1976 and 1994, respectively. He is currently a professor of Mathematics at the Applied Mathematics Department of the Belarusian National Technical University. His research interests cover the areas of radiative transfer theory, invariance theory, theory of inﬁnite linear systems of algebraic and diﬀerential equations, and theory of integral and integro-diﬀerential equations. In 1981 he formulated a general invariance principle, giving his algebraic and physical interpretation. During 1976–1983 he developed a general invariance relations reduction method. In 1996 Professor N. Rogovtsov solved in an analytical form the characteristic equation of the radiative transfer theory. Professor N. Rogovtsov is the author of the monograph ‘Properties and principles of invariance. Application to solution of the problems of mathematical physics’ (1999, in Russian). He is the author and co-author of more than one hundred papers.

XX

Notes on the contributors

Tom Rother is a Senior Scientist at the Remote Sensing Technology Institute of the German Aerospace Center (DLR). His research interests are in quantum statistics of charged particle systems, full-wave analysis of planar microwave structures and structures of integrated optics, and electromagnetic scattering. He is the author of over forty peer-reviewed papers and three book contributions. In 1990 he received the Young Scientist Scholarship of the URSI, and in 1991 he received a special grant of the Alexander von Humboldt Foundation. He received his Dr.rer.nat. and Dr. habil. degrees from the University of Greifswald in 1987 and 1998, respectively.

Sergey N. Savenkov received his PhD in 1996 at Kiev Taras Shevchenko University (Ukraine) for studies of the information content of polarization measurements. His research interests are focused on the use of polarization of electromagnetic radiation in remote sensing, crystal optics, biomedical optics and laser/radar technology. His is currently employed at the Department of Radiophysics of Kiev Taras Shevchenko University (Ukraine). Speciﬁc topics are the methods of structural and anisotropy property analysis of objects contained in their Mueller matrices, development of the Mueller matrix measurement methods, optimization of polarimetric sensors for remote sensing applications, and use of polarimetry to improve imaging in turbid media.

Frank Scheﬀold received his PhD in 1998 on the subject of speckle correlations and universal conductance ﬂuctuations of light. The thesis was prepared in the Institute Charles Sadron, Strasbourg, France, and University of Konstanz, Germany. After having worked as a research associate (1999–2002) and senior scientist (2002–2004), he became associate professor in the University of Fribourg, Switzerland. His research interests include physics

Notes on the contributors

XXI

of colloidal dispersions, optical microrheology of complex ﬂuids, static and dynamic light scattering methods, wave propagation in random media, statistical optics and biomedical optical imaging. He is the author of more than thirty papers in leading international journals, an active member of several scientiﬁc working groups and organizer of signiﬁcant scientiﬁc meetings.

Ravindra Pratap Singh is a Senior Scientist at the Physical Research Laboratory of the Department of Space, Government of India. His research interests are in quantum optics, light scattering, optical vortices and nonlinear optics. He is the author of over twenty peerreviewed papers and three book contributions. In 1986 he received the National Research Scholarship of the CSIR. He received his MPhil and PhD degrees from the Jawaharlal Nehru University, New Delhi, in 1988 and 1994, respectively.

Jean-Claude Worms is Head of the space sciences unit of the European Science Foundation, managing all space sciences programmes of the ESF. He holds a PhD in physics from Universit´e Paris 6 and was Assistant Professor in physics and astronomy from 1989 to 1992 (Paris 6 and Versailles). He is associate researcher to the Laboratoire de Physique et Chimie de l’Atmosph`ere, Laboratoire des Sciences de l’Image et de la t´el´ed´etection and Laboratoire des Syst`emes Photoniques and du Service d’A´eronomie. Having worked in radiative transfer in granular media, pre-planetary aggregation and space debris, he was PI on the PROGRA2 facility (polarimetry of dust clouds in microgravity), and the LIBRIS project (in-orbit optical detection of space debris), and Co-Investigator of the ESA ICAPS facility (study of particle systems on the ISS). He has been Main Scientiﬁc Organizer and Editor of solar system sessions in COSPAR Scientiﬁc Assemblies since 1998 and he is a member of the Editorial Board of the International Journal on Nanotechnologies.

XXII

Notes on the contributors

Pavel Zakharov obtained his PhD in 2004 from Saratov State University, Russia, on the subject of optical biophysics and biophotonics. His PhD work was devoted to the light scattering studies of moving interphase boundaries in the porous media. He spent the next 3 years as a postdoctoral fellow in the Soft Condensed Matter group in the University of Fribourg, Switzerland, working on techniques of dynamic light scattering and methods of functional neuroimaging with coherent light. In 2007 he joined Solianis Monitoring AG, a Zurich-based company, developing noninvasive blood glucose measurement systems for medical applications. His scientiﬁc interests cover light scattering methods in application to biomedical problems and soft matter in general, laser speckle imaging and diﬀuse reﬂectance spectroscopy.

Preface

This fourth volume of Light Scattering Reviews is composed of three parts. The ﬁrst part is concerned with theoretical and experimental studies of single light scattering by small nonspherical particles. Light scattering by small particles such as, for instance, droplets in the terrestrial clouds is a well understood area of physical optics. On the other hand, exact theoretical calculations of light scattering patterns for most of nonspherical and irregularly shaped particles can be performed only for the restricted values of the size parameter, which is proportional to the ratio of the characteristic size of the particle to the wavelength λ. For the large nonspherical particles, approximations are used (e.g., ray optics). The exact theoretical techniques such as the T-matrix method cannot be used for extremely large particles, such as those in ice clouds, because then the size parameter in the visible x = 2πa/λ → ∞, where a is the characteristic size (radius for spheres), and the associated numerical codes become unstable and produce wrong answers. Yet another problem is due to the fact that particles in many turbid media (e.g., dust clouds) cannot be characterized by a single shape. Often, refractive indices also vary. Because of problems with theoretical calculations, experimental (i.e., laboratory) investigations are important for the characterization and understanding of the optical properties of such types of particles. The ﬁrst paper in this volume, written by B. Gustafson, is aimed at the description of scaled analogue experiments in electromagnetic scattering. Such experiments for understanding optical properties of small particles (say, of submicrometer size) are based on the fact that light is composed of electromagnetic waves. Electromagnetic scattering (for a given refractive index m) depends not on the size of particles a and the wavelength λ of the incident radiation separately but on their ratio or size parameter x = 2πa/λ. Therefore, one can deduce optical properties of small particles from measurements of microwave scattering by large macroscopic objects, which are much simpler to manipulate as compared to the tiny aerosol particles. The pioneer in this area was the late Professor M. Greenberg, who appears in Fig. 1.1 of Gustafson’s paper setting up one the objects used for microwave scattering experiments. The main problem here is to ﬁnd materials having the same refractive indices in the optical and microwave ranges. This issue is discussed by Gustafson in considerable detail. He also describes several scaled analogue scattering laboratories and reviews results obtained for the elements of the Mueller matrix of the studied objects. The second paper of this volume (E. Hadamcik et al.) deals with the experimental studies of light scattering by collections of nonspherical solid particles with emphasis on the atmospheric and astronomical applications. The experiments were

XXIV Preface

performed on levitated particles at normal atmospheric pressure, and also in conditions of reduced gravity. Measurements at reduced gravity can be made within seconds for any kind of particles without discrimination by weight or composition. The microgravity technique is suitable for particles with characteristic sizes larger than about 10 μm. The measurements are then performed onboard of aircraft operating in parabolic ﬂights. An imaging technique is used to detect the scattered light using, in particular, a CCD sensor with 752 × 582 pixels. Measurements of both intensity and degree of polarization of the scattered light are presented and discussed in this paper. The results are given as functions of the phase angle α = π −θ, where θ is the scattering angle, and also as functions of the size parameter for a vast range of shapes and chemical compositions of particles. This data can be used for the development of realistic models of optical properties of particles that exist in nature. In the third paper of Part I aimed at the description of single scattering eﬀects, S. Savenkov describes the structure, symmetry relations, and information content of Mueller matrices. Mueller matrices describe the transformation of the Stokes parameters of a beam of radiation upon scattering of that beam. An important objective is to extract information on the properties of the scattering medium from the Mueller matrices. A goal of this paper is to present in a systematic way the main properties of Mueller and Jones matrices that are experimentally or numerically derived. Both single scatterers and collections of randomly oriented particles are considered. This paper provides a review of the diﬀerent polarimetric equivalence (decomposition) theorems, and highlights details about their application. Useful applications of the results presented in this paper can be made in lidar studies of ocean and atmosphere. Part I of this volume is concluded by the paper of T. Rother, who describes a quite general technique for theoretical studies of light scattering by nonspherical particles based on the method of Green functions. Waterman’s T-matrix approach has become very popular and widely used in many light scattering applications, where future research is needed to deal with nonspherical particles (e.g., in optics of desert dust and biological media). The author clariﬁes the relationship between the T-matrix and Green function methods introducing the interaction operator describing the interaction of the free-space Green function with the surface of the light scattering object. He shows that the Green function method can be used to examine the symmetry and unitarity of the T-matrix at a purely mathematical level. The consideration is restricted to homogeneous and isotropic scatterers. The study is based on the exclusive use of the spherical coordinate system, although the general case of light scattering by an arbitrarily shaped particle is also considered. The general results presented in the paper can be used for the development of particular approximate and numerical techniques for the solution of a light scattering problem for the case of objects with arbitrary shapes. This study is also of importance for improving our understanding of some general properties of wave scattering phenomena. The second part of the book is aimed at theoretical studies of multiple light scattering phenomena. It starts with the paper by A. Davis et al. describing new techniques for studying light propagation in dense clouds at wavelengths where water droplets are essentially non-absorbing. This survey is entirely based on the

Preface

XXV

theory of space-time Green functions for multiple scattering that can be computed analytically in the diﬀusion (i.e., small mean-free-path) limit of radiative transfer. However, the validity of this popular approximation is constantly checked against accurate numerical (Monte Carlo) solutions of the corresponding time-dependent 3D radiative transfer problems over a broad range of opacities. The localized and/or pulsed sources considered can be internal or on a boundary; receivers can also be inside or, as is naturally required in cloud remote sensing, outside the medium. Techniques that account for the internal variability of the cloud as well as the directionality of the laser or solar source are presented. Moment-based methods are applied to the emerging technologies of high-resolution diﬀerential absorption spectroscopy of sunlight in the oxygen A-band and of multiple-scattering cloud lidar from below, inside or above the cloud layer. In addition, more established techniques in passive cloud remote sensing are revisited productively from a Green-function perspective. Finally, the power of this uniﬁed theory for remarkably diﬀerent observational phenomena is illustrated by invoking possible diagnostics of turbid media beyond clouds. The paper by A. Klose is aimed at studies of luminescence in biological tissue. In most applications of radiative transfer, only the case of monochromatic (elastic) scattering is considered. However, in reality luminescence/ﬂuorescence eﬀects occur in a broad range of materials. One therefore needs to take into account that a scattering medium has the capability of selective absorption of light at one spectral interval with emission at other frequencies. In this case, the scattering matrix describes not only the angular pattern of light scattering by a medium but also must include a description of the frequency change in the inelastic scattering. The exact equations describing these processes are quite complex. However, they can be solved using a range of numerical and approximation techniques discussed by the author. The author considers radiative transfer in ﬂuorescence systems both in steady-state, in time and in frequency domains. In particular, he formulates a system of two separate time-dependent radiative transfer equations for the excitation light at the wavelength λex and the emission at the wavelength λem . Their simultaneous solution leads to a description of radiative transfer phenomena in light scattering media composed of particles capable to produce eﬀects of selective absorption and emission (ﬂuorescence). Various numerical techniques for the solution of the problem at hand are discussed by the author in great detail. The results presented are of considerable interest for both radiative transfer theory, and also for applications (e.g., in the ﬁeld on biomedical optics). In particular, the author describes optical molecular imaging of small animals based on luminescence, instead of intrinsic absorption properties of tissue. Luminescent imaging probes are either administered from outside or genetically expressed inside a small living animal. The results presented in this paper are of considerable importance, e.g., for the development of noninvasive methods in cancer diagnostics. The paper by N. Rogovtsov and F. Borovik is concerned mostly with the mathematics of radiative transfer. They derive an analytical solution of the radiative transfer equation for some speciﬁc types of light scattering media. The techniques they use to solve the characteristic equation of the radiative transfer describing the light ﬁeld deep inside the plane-parallel medium are covered in great detail. In this chapter, the qualitative and constructive mathematical theory of the classical vari-

XXVI Preface

ant of the characteristic equation and reduced characteristic equations of radiative transfer theory is presented. This theory is used as a basis for correct and eﬀective methods for solving characteristic equations and boundary-value problems of radiative transfer theory in an analytical form in the case of arbitrary phase functions (in particular for phase functions highly peaked in the forward direction). The new theory substantially diﬀers from the qualitative mathematical theory of characteristic equations constructed on the basis of ideas and methods of functional analysis and Case’s method. It is based on the extensive use of classical results from mathematical analysis, diﬀerence equation theory and continuous fractions theory. As an application of the new theory, rigorous general analytical expressions for all azimuthal harmonics of Green function of radiative transfer equation are obtained for the case of a plane-parallel inﬁnite turbid medium. In addition, an eﬀective algorithm for calculating the azimuthally averaged reﬂection function and the reﬂection function of a plane-parallel semi-inﬁnite turbid medium is presented. Results of numerical calculations using the algorithm developed here are given for the speciﬁc case of water clouds. The ﬁnal part of the book, Part III, is aimed at the description of dynamic and static light scattering used for studies of turbid media such as suspensions of particles in liquids and air. P. Zakharov and F. Scheﬀold discuss the theoretical foundations and experimental implementations of dynamic light scattering techniques used for diagnostics of the media of various origin – from colloidal suspensions to the moving red blood cells in the living organism. These techniques explore correlation properties of scattered coherent light in order to uncover the dynamical characteristics of underlying systems. Special attention is devoted to the recent developments of the space and time resolved techniques, which extend the range of applications of dynamic scattering methods with the imaging of heterogeneous media and analysis of non-stationary processes. To conclude this book, R. P. Singh gives a comprehensive review of diﬀerent methods to measure properties of aerosols in a controlled environment. Aerosols play many complex roles in environmental dynamics and climate change producing cooling at some places while warming at others, creating clouds that rain and at times making clouds which would not rain. In all these cases, aerosol size is the single most important parameter that determines its dispersion and a host of other properties. The article deals with optical sizing of aerosols using static as well as dynamic light scattering, and compares them with other methods employed for sizing of the particles. A review of light scattering methods used for particle sizing is provided. The author describes experiments designed to investigate aerosols under controlled conditions of humidity, temperature and concentration, and he points out the dearth of work in this area. In summary, the results presented in this volume show that the optics of light scattering media remains an important area of applied research with many new exciting developments ahead. This volume of Light Scattering Reviews commemorates the lives of two outstanding scientists, Gustav Mie (1868–1957) and Peter Debye (1884–1966), who made important contributions to the optics of light scattering media. The light scattering community indeed celebrates now the centennials of the publication in Annalen der Physik of their classic papers on light scattering (G. Mie, 1908:

Preface XXVII

Beitr¨ age zur Optik tr¨ uber Medien, speziell kolloidaler Metall¨ osungen, Annalen der Physik, 330, 377–445; P. Debye, 1909: Der Lichtdruck auf Kugeln von beliebigem Material, Annalen der Physik, 335, 57–136). These papers are of not only historic importance. Even now, 100 years after their publication, they remain relevant and inspirational. Bremen, Germany October, 2008

Alexander A. Kokhanovsky

Part I

Single Light Scattering

1 Scaled analogue experiments in electromagnetic scattering Bo ˚ A. S. Gustafson

1.1 Introduction Bertrand Russell famously contemplated a table: ‘depending upon the angle one has upon looking at it, and the way the light and shadows fall across it, it will appear one way to one observer, another way to another. Since no two people can see it from precisely the same point of view, and since the light falls on it diﬀerently from diﬀerent points of view, it will not look the same to anyone,’ he noted, and the philosopher asked ‘But if that is so, does it make any sense to say what the real shape (or color, or texture) the table REALLY is?’ To the physicist it makes sense to deduce as many properties of an object as possible and to predict what it would look like under diﬀerent observing conditions. These are the inverse respectively direct problems of electromagnetic scattering. It is interesting that it is through experience acquired by observing (and active ‘experimentation’ in part through play) that humankind and other life forms develop the ability to deduct what their observations signify, and interpretation requires no theoretical knowledge. This illustrates that observations and ‘trial-and-error’ experience suﬃce to work out the inverse problem in scattering theory to some level that has practical use, at least in the realms of geometric optics where the physics of interaction simplify. Deductive reasoning combined with systematic observations allowed some of the ﬁrst recorded insights into what we now know as electromagnetic scattering. Aristotle observed how rainbows, solar halos and several other atmospheric phenomena depend on the geometry with respect to the light source and that the same observable features occur in moonlight as in sunlight. He concluded correctly that they result from a sort of reﬂection and that illumination of the reﬂecting object(s) does not depend on the observer. Euclid’s experiments with mirrors and reﬂection demonstrated that light travels in straight lines between reﬂections. His experiments appear to have been conceptionally simple and free from a priori assumptions. Ptolemy established an early de facto laboratory standard as he measured the path of a ray of light from air to water, from air to glass, and from water to glass and tabulated the relationship between the incident and refracted rays. His systematic and quantitative measurements belong to a ﬁrst generation of experiments that took advantage of Euclid’s results although they did not depend on

Bo ˚ A. S. Gustafson

4

insights gained from theoretical models.1 Like Ptolemy, most modern researchers design their experiments using insights gained from earlier experiments and from theory to probe ever further into the unknown. Theoretical computing progress depends on technology and engineering advances similarly to progress in experimental and observation capabilities. Theoretical modeling and reasoning along with experiments and observations developed into interdependent tools in a comprehensive process. The role of experiments has long been to yield the true answer to a speciﬁc question when answers either cannot be obtained from theory or depend on some inadequately tested hypothesis and therefore cannot be relied upon until conﬁrmed by experiment. We also take recourse to the laboratory when, as with the scattering by clouds of particles involving broad ranges of parameter space, the solution may require prohibitive amounts of theoretical calculations. A third class is when the exact parameters to the scattering problem (refractive indices, shapes or other relevant parameters) remain undeﬁned, such as for collected soil samples or for aerosols. Observations are a class when control over the experimental parameters is conﬁned to the observing parameters at best and may naturally ﬁt in as a fourth class in this sequence. However, an important distinction is that observations give answers to questions that may be unknown while experiments are attempts at posing and controlling the questions to varying degree depending on the class. Scaled experiments use particle models speciﬁcally made to represent a particle in the scattering problem posed. The method is therefore applicable to well posed problems involving known particle parameters and is primarily used to test and extend theoretical solutions to address the ﬁrst type of problem listed above.

1.2 Theoretical basis for scaled electromagnetic experiments Scaled electromagnetic experiments build on the ﬁndings of classical electrodynamics. Obtained from experiments and reﬁned throughout the nineteenth century, a set of four partial diﬀerential equations attributed to Gauss, Faraday and Amp`ere describe the properties of the electric and magnetic ﬁelds and relate them to their sources, the charge density and current density. Together they form the basis of classical electrodynamics and are collectively known as Maxwell’s equations. Maxwell used the equations to derive the electromagnetic wave equation and show that light is an electromagnetic wave. The classic electrodynamics concept of electromagnetic radiation and most modern solutions to its propagation and scattering therefore are direct consequences of experimental ﬁndings summarized in the form of Maxwell’s equations. The idea of using scaled models and scaled frequencies to study electromagnetic scattering dates back at least to the late 1940s (Sinclair, 1948). But the scaling in size and frequency is implicit in the formulation of the Mie-solution where particle dimensions are expressed in units of the wavelength using a dimensionless size parameter x instead of absolute units. Use of dimensionless size parameters in classical electrodynamics based formulations as standard practice can be traced back to well before the Mie-solution. This is since Maxwell’s equations themselves only 1

See works by for example Sayili (1939) and Herzberger (1966) for historical accounts.

1 Scaled analogue experiments in electromagnetic scattering

5

contain dimensions in units of the wavelength. Experimental scattering results are also usually reported in terms of a size parameter. This generalizes the response similarly to the way theoretical results are reported and facilitates comparison between works. It follows that the laboratory simulation of any electromagnetic interaction problem can be scaled to any particle size or region of the electromagnetic spectrum and the choice of laboratory wavelength is a matter of convenience. There are, however, a few known complications. One is the requirement that the material of a scaled object must be represented using an analogue material with the same optical properties at the laboratory frequency as those of the replicated material at the original frequency. Analogue materials can often be found or may be produced from a mixture of standard compounds. However, there may be a need for multiple scale models made from a range of analogue materials to study color eﬀects near absorption/emission lines and the experiment may become cumbersome. One may also need to invoke inhomogeneous structures to simulate very short wavelength radiation that experiences matter atom by atom rather than as a continuum. This and statistical eﬀects at low intensities are due to the quantum nature of light and can also be replicated through scaling. Max Planck found that to match the radiation in equilibrium with a blackbody the energy of a monochrome wave must only assume values which are an integral multiple of ωh/2π where h is Planck’s constant and ω the frequency. This quantum property of electromagnetic waves leads to the famous wave–particle duality eﬀect and spawned the ﬁeld of quantum theory. We see that quantum eﬀects occur at all frequencies and that the consequence of changing the frequency is to shift the energy quanta proportionally. This has statistical consequences when the number of detected quanta or photons is small but even this could be replicated by scaling the observing time or intensity so that the number of detected photons remains unchanged.2 Approximately 1019 quanta illuminate the particle during each measurement at the University of Florida facility and the detection limit is close to 108 , which is still statistically large. In practice all measurements considered here therefore correspond to the large number of photons limit so that any statistical uncertainty can be neglected. This also applies to the older laboratories where measurements were made at approximately the same intensity but lower frequencies and longer exposure times so that even larger numbers of photons were collected per measurement. The scaled laboratory measurements should be interpreted as the statistically most likely outcome from a large number of photon-scattering events and are as such entirely valid. Put diﬀerently, the laboratories do not have the sensitivity to detect quantum eﬀects.

1.3 Scattering by a few common particle classes Scaled laboratory experiments are best viewed as complementary to other means of investigating the scattering problem. We therefore consider a few classes of particles that have come to play special roles or may impose special requirements on the investigation method before contemplating the laboratory facilities. 2 The lower the frequency, the shorter the observing time or lower the intensity needed to collect a given number of photons.

Bo ˚ A. S. Gustafson

6

1.3.1 Mie-solutions Known for more than a century (Kerker, 1969), the Mie- or Lorenz–Mie–Debye- analytic solution to Maxwell’s equations in a spherical geometry (Mie, 1908) is often referred to as ‘Mie-theory’. Yet it is not an independent theory but the mathematical application of Maxwell’s equations to an electromagnetic wave illuminating a homogeneous sphere of given size and refractive index. The Mie-solution is therefore a direct consequence of classical electrodynamics, much as Maxwell’s electromagnetic wave equations are the result for empty space. Mie ﬁrst applied his solution to metallic gold spheres but the solution was soon extended to the scattering by homogeneous spheres of arbitrary size and refractive index including magnetic materials. Although the solution involves inﬁnite series, they converge so that it becomes possible to evaluate the scattering to any degree of accuracy. Clearly there is no need to resort to laboratory experiments if the scattering body is a Mie-sphere. However, this allowed the sphere to be used as a convenient laboratory test and calibration case. Mie-spheres remain the de facto test standard for experiments although the high degree of symmetry simpliﬁes scattering (Savenkov, Chapter 3 in this volume) and does not test or calibrate all aspects of the experiment. Similar solutions were developed in cylindrical coordinates and allowed the calculation of scattering by inﬁnite circular cylinders (Rayleigh, 1881; Twersky, 1952; Kerker, 1969). This solution can in principle not be tested in a ﬁnite laboratory. However, the ﬁrst-order eﬀect of truncating an inﬁnite cylinder to ﬁnite size is a diﬀraction broadening of the scattering cone. This eﬀect is clearly seen in laboratory data (Gustafson, 1980, 1983). It appears that long cylinders could be used to probe the eﬀective extent of the ﬁnite wave fronts in the laboratory. Indeed, the diﬀraction due to the ﬁnite ﬁeld generated by the transmitting antenna combined with the corresponding eﬀect due to the receiving antenna3 could be measured conveniently (Gustafson, 1980, 1983). Cross polarization terms4 could not be calibrated until Asano and Yamamoto (1975) derived the solution in spheroidal coordinates. But this requires appropriately oriented spheroids and the advance remains mostly academic due to high sensitivity on geometry compared to routinely achievable orientation accuracy. Spheres, cylinders, spheroids, and other particles with well deﬁned geometries on which Maxwell’s equations may be applied to yield boundary conditions permit solutions to the scattering problem that are sometimes collectively referred to as Mie-solutions. Many of these were soon extended to concentric core mantle geometries and Mie-type solutions now also exist for a limited but growing number of particle geometries. These include structures with oﬀ-center cores, clusters of interacting spheres and cylinders (see contributions in Mishchenko et al., 2002). Further, the Extended Boundary Condition Method (EBCM) and related approaches dramatically relax the shape constraint so we can be optimistic that 3

By time reversal symmetry. The two cross polarization terms are measures of the intensity of radiation polarized (electric vector) in the scattering plane resulting from illumination polarized perpendicular to the scattering plane and scattered radiation polarized perpendicular to the plane resulting from illumination polarized along the plane. 4

1 Scaled analogue experiments in electromagnetic scattering

7

Mie-related solutions will continue to expand the number of geometries for which ‘exact’ solutions exist.5 Does this mean that the scattering problem is about to be solved to the level where it satisﬁes most practical needs? H. C. van de Hulst (2000) addressed this question in the foreword to Light Scattering by Nonspherical Particles (Mishchenko et al., 2000). He wrote: ‘In the majority of applications, the assumption of homogeneous spherical particles is highly unrealistic. That much was clear from the start to all serious research workers.’ His half-page objection is in its essence applicable not only to spheres but to all particles with extensive homogeneous volumes and smooth surfaces. If a grain is exposed to space for any signiﬁcant time there is surely damage from cosmic rays. Amorphous and crystalline mineral mixtures may be crisscrossed by odd-shaped crystals, similar to those forming Widmanst¨ atten patterns, the interleaving of kamacite and taenite ﬁlaments seen in many iron meteorites. Impurities, dents and cracks will dot the particle surfaces as well as the interiors. H.C. van de Hulst concludes: ‘The upshot of these objections is that the eﬀorts of most scientists applying Mie computations to a problem in nature were unwarranted.’ While a step in the right direction, simple nonspherical particle shapes remain unrealistic in most natural settings. Of the particle types discussed in this article, large aggregates of small dissimilar subvolumes may be the morphology that comes closest to representing heterogeneous real particles whose interior may be a mixture of minerals but, as we shall see in section 1.3.1.2 even this is a stretch. 1.3.1.1 Complex inhomogeneous structures It is in principle possible to represent particle geometries with arbitrarily complex interiors using the Coupled Dipole Approximation by Purcell and Pennypacker (1973). The internal ﬁeld is discretized and solved for iteratively using interacting dipoles to numerically calculate the internal ﬁeld and the resulting scattered ﬁeld in a process that has been shown to be equivalent to the Volume Integral Equation Method (Lakhtakia and Mulholland, 1993). How many dipoles are needed to represent the internal ﬁeld? To be ‘safe’ we may want to assign a number of dipoles comparable to the number of atoms in a real particle but in reality there is no practical way to use such large numbers and the discretization of the internal ﬁeld remains intrinsically, as the name indicates, an approximation. How should the dipoles be distributed? Is a speciﬁc lattice geometry appropriate or will it introduce numerical resonances or other disturbances? Many implementations rely on a cubic lattice for computational expediency. These and additional issues such as the polarizability to assign each dipole have been investigated and tested. But they often rely on spheres and related simple geometries with homogeneous and 5

The word ‘exact’ is in quotation marks to acknowledge that the solutions represent materials as a continuum involve inﬁnite series that in practice are truncated and may not necessarily converge except within some parameter ranges. Some like the EBCM can involve additional approximations. For example, Maxwell’s equations degenerate and cannot be solved on a surface with a discontinuous ﬁrst derivate such as the edges and corners of a cube. Solutions can approach these locations but do not include rigorous solutions on the very edges and corners.

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isotropic interiors from section 1.3.1 as test cases. This is hardly making much advance unless the solutions are also tested on more complex structures. Microwave measurements by Gustafson (1980), Zerull et al. (1993) and Gustafson et al. (2002) (discussed in section 1.5.3) illustrate how the scattering at large angles is aﬀected ﬁrst if a homogeneous material is discretized too coarsely. As we model more complex structures convergence issues and other numerical instabilities emerge at increasing rate, especially when interaction between dipoles in complex inhomogeneous materials is locally strong. A comparison of the Coupled Dipole Approximation solution to microwave measurements for even a simple aggregate of spheres illustrates the problem (Xu and Gustafson, 1999). The method is conceptionally attractive and a powerful tool when used within its limits, but the limits of validity are hard to deﬁne and it remains diﬃcult to predict when the approximation breaks down. Undetected erroneous results are likely to emerge more frequently as computing power grows and modeling is pushed toward ever more complex and larger particle models. Veriﬁcation using precise tests in a laboratory of the ﬁrst type or class described in section 1.1 is warranted. 1.3.1.2 Real particles The Rayleigh approximation (Strutt, 1871) applies when the phase is uniform everywhere across the scattering body (van de Hulst, 1957). This requires that the particle dimensions are small compared to the wavelength both outside and inside the particle. Grains meeting the dimensions criterion are so small that any restriction on surface smoothness and internal homogeneity is automatically fulﬁlled. The Rayleigh solution therefore applies to suﬃciently small particles even as they may have been weathered and subject to cosmic rays as long as the polarizability can be assigned. The solution is intrinsically a Volume Integral Equation formulation like the Coupled Dipole Approximation which reduces to Rayleigh scattering in the small particle limit of a single dipole. The Rayleigh approximation is therefore plagued by the same issue of assigning polarizability to the dipole as the Coupled Dipole Approximation. Because of this and because the propagation speed in metals is slow (and therefore the internal wavelength short which can make the Rayleigh requirement very limiting), there may be an arguable need to study the scattering by some very small particles in the laboratory. However, the major laboratory facilities were optimized for the study of larger scatterers. At the other extreme size limit, ray tracing or the geometric optics approximation is said to apply when all particle dimensions are large compared to the wavelength, but the conditions are restrictive and eliminate many naturally occurring structures. The interiors must be homogeneous over distances that are large compared to the wavelength and all surfaces including any internal boundaries must be smooth even on scales that are small compared to the wavelength. The geometric optics approximation is a practical engineering tool for the design of telescopes, microscopes, eyeglasses or spectacles. These are successfully modeled as long as the glass is of good quality (homogeneous) and surfaces are clean and free from scratches and other defects. We just need to consider ‘foggy’ or dirty glasses to see that the deviation might be signiﬁcant when the conditions are not met. How serious is this problem with real particles? Figure 1.1 shows a section

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Fig. 1.1. Cut through the Allende meteorite reproduced from Grossman’s (1980) seminal article in which he described what became known as ‘calcium aluminum rich inclusions’ (CAIs) such as the light-colored object at top center. Having condensed at high temperature (∼ 3000 K) presumably in the cooling inner solar nebula, CAIs are the earliest known surviving solids to form in the solar system. Grossman identiﬁed the two narrow, elongated objects at middle right as ‘ﬁne-grained inclusions’ and the large object at bottom center as an ‘amoeboid olivine aggregate that may include mineral predating the solar system and formed an aggregate of interstellar grains so that the structure is coarse down to the sub micron level’. He also states that a dark clast is barely visible at middle left and that most of the other light-colored objects are chondrules. Reprinted, with permission, c from the Annual Review of Earth and Planetary Sciences, Volume 8; 1980 by Annual Reviews, www.annualreviews.org.

of the Allende meteorite. As all carbonaceous chondrite meteoroids, Allende is a rocky structure in which most chemical bonds formed billions of years ago when partially molten heterogeneous matter accreted and to some extent fused to form asteroids. After being crushed by any overlaying material for billions of years, possibly hydrated (which redistributes minerals), pounded in repeated collisions and probably shattered and reassembled many times over, a meteoroid emerges from the interior of an asteroid as surviving debris from a collision. It only escapes if the collision is of suﬃcient violence to knock the meteoroid away from the gravitational well surrounding the parent. As if that was not tough enough, the meteoroid may have spent the past few thousand years tumbling in space, suﬀering cosmic ray exposure and occasionally colliding with other rocks. The meteorite material seen in Figure 1.1 is believed to be the result of such violent events. We would probably not subject our laboratory optics to similar treatment and expect it to behave even close to an ideal structure. Why then expect it of natural particles that on top of this treatment often started as complex and heterogeneous? Similarly complex inhomogeneous structures that defy simple descriptions using homogeneous macroscopic optical constants are also seen in comet material (Figure 1.2) and may be found across a broad range of dimensions (from nanometerto centimeter-size) in the form of GEMS, chondrules and CAIs (Grossman, 1980),

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Fig. 1.2. Collected IDPs (interplanetary dust particles), that are usually classiﬁed as cometary, have complex interiors that include amorphous silicate materials with scattered inclusions of metals and sulﬁdes (GEMS). Not seen in the TEM (transmission electron microscope) slide is an organic compound that is likely to permeate the particle and further aﬀect the uneven value of the refractive index across these complex particles. Similar structures are found in impact tracks in aerogel returned by NASA’s Stardust mission (John Bradley, LLNL, personal communication; image credit: Bradley).

as individual particles or as inclusions. These scales are of relevance in optical and infrared studies since the wavelength is not large compared to all inclusions6 in any waveband. It is true that particle geometries to which approximations apply can be found but can particles to which the approximations apply be found in nature? Yes, if they are small compared to the wavelength but larger solid bodies, unless they are young crystals (e.g., hail or snow) or liquid (e.g., rain drops) tend to feature imperfections and be better viewed as compact heterogeneous compound particles (HCPs) to which geometric optics does not apply. Mie-type solutions are also not forthcoming and both compact and porous structures of this type need to be the main targets of scaled laboratory studies at facilities like that at the University of Florida. 1.3.2 Evaluating the scattering by HCPs and other complex natural particles The main problem with natural particles is their diversity. We are many times faced with a set of scattering curves and hope to conclude something about the particles that produced them (the ‘inverse problem’). We may try any number of more or less specialized computer codes and chances are good that the number of 6 Note that for this purpose empty cavities can also be viewed as inclusions (often irregular) characterized by the refractive index 1.

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free parameters is so large that a reasonable ﬁt can be found. However, this is no guarantee that the solution is unique. To the contrary, if anything can be said with certainty, it is usually that other solutions also ﬁt. It is for example possible to ﬁnd a good ﬁt to the zodiacal light using physically reasonable dust models but without the need for the level of complexity that the actual particles likely have. In addition, a ﬁt with entirely unreasonable materials (optical constants that only deviate minutely from those of vacuum) and large spherical shapes7 can ﬁt most and possibly all observations including silicate emission features in the infrared. Commercial optical-scattering-based particle sizers/counters where particles are counted and binned in size ranges are examples of non-unique classiﬁcation. The proper interpretation is not that the particles counted are in the indicated size ranges. The instrument could simply not distinguish them from spheres of that size and bins according to a ‘sphere-equivalent’ size. To achieve this level we may have to indicate the particle composition (refractive index) and even so, counters using dissimilar optics generally yield conﬂicting results. Systematic probing of scattering response to the change of some deﬁned parameter such as packing in an aggregate was used to devise an in situ Planetary Aerosol Monitor or Interplanetary Dust Analyzer (Gustafson et al., 2000) where the scattering by an arbitrary single particle can be classiﬁed according to multiple parameters. This may be the direction of future high-end particle counters/characterizers. However, most observations of natural particles are limited and the particles often form clouds with a distribution of particle properties that wash out speciﬁcs and allow multiple interpretations.

1.4 The scattering problem in the laboratory setting The most fundamental role of an experiment is to reveal under given conditions the state of a set of parameters that may be otherwise unknown. A well-designed and well operating experiment provides a truthful answer but the enduring diﬃculty is usually not so much in the answer as in the question. That is, in the control of the experimental conditions. What experimental electromagnetic scattering conditions need to be created and controlled? The answer depends on the speciﬁc scattering problem. We assume that the objective is to measure all properties of independent scattering in the far ﬁeld.8 Further, we adopt the deﬁnitions given by Mishchenko et al. (2000, p. 4) and other restrictions given there, as is usual in the electromagnetic scattering literature. Interaction between particles (dependent scattering) can also be investigated using this same apparatus by considering the interacting particles as an ensemble making up one scattering body. Scattering parameters are assigned to this ensemble rather than to its components. Any aspect of a light scattering solution subject to these restrictions should be testable in the laboratory so that knowledge about scattering can be extended 7 Large porous aggregates approach these values if they are represented using Mietheory and eﬀective medium theory and may in a sense be an optical equivalent although physically diﬀerent. The approximation is known to work well for thermal emission and is also an often usable scattering approximation when the optical constants are low and averages are taken over many orientations and sizes. 8 Called the distant ﬁeld by van de Hulst (1957).

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to particle parameters beyond the reach of theoretical solutions. This means measurements should yield the same type and format scattering information as theory usually yields and numerical computations can produce. These can then ‘plug-in’ to existing codes to address a broad range of practical scattering problems through the use of the measured quantities in the same way as calculated ones. We therefore seek to ﬁrst reduce the description of an arbitrary illumination to its simplest terms and to similarly describe radiation resulting from the scattering process. The scattering process is then fully described through the transformation of one into the other. Scattering in the near ﬁeld is not contemplated here. Stokes (1852) reduced the description of the radiance (intensity) and state of polarization of fully overlapping beams (radiation traveling in the same direction and passing through the same points in space) to a set of only four real elements known as the Stokes parameters. We use the classical notation (I, Q, U, V ) where the Stokes parameters are written as a vector. The ﬁrst Stokes parameter I is the net monochromatic energy ﬂux. Q and U describe the state of linear polarization and V is a phase term to describe the state of circular polarization. H.C.van de Hulst (1957) points out that Stokes parameters make up a complete set of quantities characterizing the intensity and state of polarization of an incoherent monochrome beam of light.9 Naturally the direction of propagation and a set of spatial coordinates it passes through are also needed for completeness. A description of an arbitrary monochrome electromagnetic radiation ﬁeld can be a vector ﬁeld describing the direction of propagation and the associated Stokes vectors while the multi-beam panchromatic generalization is simply a set of such ﬁelds. The most common formulation of the scattering problem is that of an inﬁnite and ﬂat (uniform) incident wavefront of wavelength λ radiation that may be uniquely described by the Stokes vector (I0 , Q0 , U0 , V0 ). The scattered radiation is evaluated at a suﬃcient distance r from a ﬁnite size scattering body so that the scattered spherical ﬁeld can be locally approximated by a ﬂat wavefront. The scattered ﬁeld subject to this far ﬁeld condition can then be expressed as (I, Q, U, V ) =

1 k2 r2

F · (I0 , Q0 , U0 , V0 )

where F is a 16-element transformation matrix and k = 2π/λ. The F-matrix for any speciﬁc particle depends only on the wavelength, particle orientation and scattering angle Θ. The far ﬁeld scattering resulting from illumination by any combination of incoherent distant light sources can be calculated by combining (adding) the Stokes vectors representing the scattered ﬁeld. Calculating and adding ﬁeld amplitudes (using the Stokes vector) instead of adding Stokes parameters trivially makes the generalization to coherent illuminating sources or coherently scattering bodies. Most scattering problems only require the far ﬁeld solution. Generalization to arbitrary distances requires near ﬁeld data. It is understood that some scattering problems are conﬁned to speciﬁc scattering geometries. For example the interstellar grains that Greenberg et al. studied (see section 1.5.1) are mostly seen in extinction. To measure just extinction Greenberg’s original laboratory at Rensselaer Polytechnic Institute (RPI) did not need to include the possibility to measure side scattering. In addition, all scattering problems 9

Alternatives to Stokes’s original formulation are also found in the literature.

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do not in practice require determination of the entire Stokes vector transformation from the incident to scattered light and most scattering experiments have been set up to quantify a limited aspect of the scattering process. Investigations of scattering of natural incoherent light such as sunlight or scattering of any light by clouds of moving particles to yield average scattering functions usually deal with intensity and polarization but not phase. Only the four elements in the upper left quarter of the transformation matrix known as the Mueller matrix are relevant in these cases. To see this, we note that when particles are illuminated by ‘natural light’ Q0 = U0 = V0 = 0 so there is no deﬁnite phase relation between the polarization components. Many experiments are more limited than the example of Greenberg’s original laboratory. In particle sizing it is common to measure the scattered linearly polarized intensity integrated across some scattering angle range while ignoring the phase and the polarization, requiring only one transformation element. Experiments may use monochrome or broad-spectrum sources to obtain averages across some spectral range. 1.4.1 Considerations in designing a high-precision scattering laboratory A minimum of four pairs of incident and scattered Stokes vectors are needed to calculate all elements of the F-matrix (Gustafson, 2000). This means control over the illumination characterized by the incident Stokes vector (I0 , Q0 , U0 , V0 ) and ability to measure the scattered Stokes vector (I, Q, U, V ) in accurately known geometries. This is similar in content and format to results obtained from most theoretical solutions. The laboratory should be capable of retrieving the full Fmatrix at as many scattering angles as possible. All particle parameters including the orientation should be known and preferably be controlled. In practice the polarization state of the incident beam and of the sampled part of the scattered radiation can be controlled using polarizers. A transmission loss of 0.15 dB and rejection of the orthogonal polarization state in the 50 dB range is now reachable in the w-band. Similar or better polarizer performance is available in most wavebands and in the optics range. The polarization direction must be related to the scattering plane and is therefore part of the antenna and target alignment. The measured intensity can be given in units of the incident ﬁeld. Here, the phase diﬀerences between the illuminating and scattered waves are relevant, not the absolute values (Bohren and Huﬀman, 1983). It is therefore good practice to siphon oﬀ a signal from the emitted beam and make all measurements relative this reference. The reference signal needs to travel the same optical length as the scattered radiation to ensure that both signals refer to the same combination of wave trains. Although the theoretician usually considers idealized perfect monochromatic, ﬂat and inﬁnite beams, a real beam consists of many ‘simple waves’ with independent phase in succession. It also is ﬁnite and cannot have a ﬂat wavefront everywhere. A real beam is in addition never fully monochromatic. The wave train length is related to the spectral purity so the ‘cleaner’ (more narrow and phase stable) the radiation source is, the longer the wave trains are. Coherent wave trains of duration 10−8 seconds (a few meters in length) were already achievable in the microwave

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range as Greenberg developed his laboratory. This is comparable to the coherence time of modern stabilized helium-neon lasers although the laser has a much longer train in units of wavelengths. The advent of radar opened the possibility to achieve the requirements described above at microwave frequencies. Lasers enabled measurement of F-matrices throughout most of the spectrum including the visual and IR decades later. Demands from the communication industry drive instrument development across the spectrum and suitable equipment is now often available oﬀ-the-shelf. Modern millimeter-wave stabilized oscillators have very high spectral purity and therefore very long wave trains so this issue is no longer a practical concern. However, alignment and dimensional stability is important at any wavelength and since all relevant dimensions are in units of the wavelength there is great advantage gained from use of the microwave range over the optical range. In addition, particle control is best achieved with models of dimensions in the millimeter to decimeter range. The millimeter or microwave spectral range therefore continues to be preferred for high-precision scattering measurements and will probably always remain so. 1.4.2 Analogue particle materials Macroscopic material properties in optical applications are usually described using the complex refractive index m = mr + imi or equivalently, in electrodynamics, √ through the dielectric constant ε = ε + iε , where i = −1. The relations ε = m2r − m2i and ε = 2mr mi hold for nonmagnetic materials. Although m and ε are known as ‘optical constants’, they depend on the angular frequency, ω, so that an analogue material must usually be substituted as part of the scaling process. To see this, it is useful to note that the material properties could also be represented using a combination of harmonic oscillators of strength proportional to (ω02 −ω 2 )−1 , where the resonance frequency ω0 is higher than microwave frequencies and often closer to the visual range (Bohren and Huﬀman, 1983). Based on this representation, optical properties vary only slowly across the microwave range since it is far removed in frequency from ω0 while the infrared and especially the visual range usually are much closer to resonances so that the oscillator strength may vary across the range. The oscillator strength and loss are related through the Kramers–Kronig relations (Bohren and Huﬀman, 1983) so that the loss as well as the oscillator strength depends on the frequency. Put diﬀerently, most materials have vivid colors in the visual, dull in the far infrared and are shades of grey in the microwave range. There are some exceptions, silicates have their most vivid colors in the infrared with resonances near 9.7 μm produced by Si-O stretching and 18 μm due to O-Si-O bending modes. This makes ‘rocks’ grey at visual as well as microwave frequency ranges. Silicates typically have refractive indices in the 1.55 + i0.0001 to 1.7 + i0.03 range in the visible. They reach extreme values of the order 1.1 + i1 near the resonances in the infrared and then stabilize around 3.5 + i0.03 in the millimeter microwave range and at longer wavelengths. Similarly, water ice has m ≈ 1.31 + i3 × 10−9 in the visible, however, the real part dips below unity near λ = 3 μm, where the imaginary part peaks around mi = 0.6. The real refractive index approaches an asymptotic value close to 1.78 with the imaginary part in the 10−2 to 10−4 range at wavelengths exceeding a few hundred micrometers.

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It is advisable to shift scaled scattering experiments away from resonances if one intends to operate at multiple wavelengths. Millimeter wavelengths and longer are typically far from any resonance so that any detected color eﬀect is due to the scattering body’s morphology, not to wavelength dependence in the material it is made from. As we have seen, many minerals (rocks) are grey in the visual so the analogue material should apply across the microwave frequencies. Water ice has no major resonances in the visual and can for many applications also be represented in the microwave range using a single analogue material. Cosmic organic refractory materials (e.g., Greenberg et al., 1995) are brown, yellow or red. Like most organics, it and other colored materials typically need to be matched using several analogue materials depending on the frequency range to be represented. If the objective is to study color eﬀects near a resonance, it is necessary to use a full-scale laboratory setup unless the wavelength dependence is precisely known and one is prepared to develop matched model materials at each wavelength to be simulated. Conversely material restrictions would make the use of scaled frequencies in the visual or infrared ranges more diﬃcult, at least for color investigations. However, the original material can often be used when the scaling is from lower frequencies to the microwave range. Refractive indices were typically measured in the Greenberg et al. laboratory using the slotted waveguide method by Roberts and von Hippel (1946). This approach is based on measuring the change in a standing wave pattern as a sample of the material is introduced inside a waveguide. Giese et al. ﬁtted the measured angular distribution of scattering by a sphere made from the material with unknown refractive index using the Mie-solution. Both methods appear to be similar in accuracy. Using von Hippel’s method Schuerman et al. (1981) found m ≈ 1.61 + i0.004 at 9.417 GHz (λ = 3.18 cm) for the acrylic resin polymethyl methacrylate, also known under the trade name ‘Lucite’. Gustafson (2000) obtained 1.605 + i0.003 for the same material at 85 equally spaced frequencies across the 75 to 110 GHz interval (2.7 mm < λ < 4 mm) using a ﬁt to Mie-calculations. Zerull et al. (1993) similarly obtained m ≈ 1.735 + i0.007 for nylon at 35 GHz (λ = 8 mm). Gustafson obtained m ≈ 1.740 + i0.005 across the 75 to 110 GHz interval. We conclude that these plastics are remarkably consistent in their refractive index and that there is indeed no measurable frequency dependence. These easy to machine common plastic compounds, including Delrin (a polyacetal marketed by DuPont) with m = 1.655 + i0.00, are convenient analogues for silicates. Use of eﬀective medium theory allows control over the complex refractive index. Many plastics can be expanded or, better, diluted with ‘microspheres’ hollow glass or plastic spheres with a priori known sizes small enough to safely satisfy the conditions of eﬀective medium theory. Or they can be mixed with pigments and/or absorbing compounds (carbon) to represent practically any material in the visual. See Kolokolova and Gustafson (2001) for a modern summary and test of the eﬀective medium approximation. We have developed plastic compounds to produce any refractive index in the range mr = 1.195 to 1.7 and mi = 0.002 to 0.08 with relative ease at the University of Florida. We have also made individual samples with imaginary parts as high as 0.2 and real parts near 2. It is usually practical to use commercially available compounds for testing theoretical solutions and to approach a desired refractive index even if it is not a perfect match. In this way dis-

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crete and highly reproducible values can be obtained including m ≈ 2.517 + i0.017 for BK7 glass or m ≈ 3.08 + i0.001 for alumina glass. The metals aluminum, steel, and copper can be used to reach very high real and imaginary refractive indices. We note that use of analogue materials sometimes has great advantages; for example, water ice and many other volatiles can be represented with ease by plastic compounds that can be machined and that do not melt at room temperatures. It also is possible to make harmless plastic models of bacteria and other hazardous compounds.

1.5 Scaled analogue scattering laboratories We have seen that scaled experiments depend on insights gained from the now very extensively tested and well-established theoretical foundation of classic electrodynamics. We can be conﬁdent that scaling is permitted and introduces no approximation as long as classic electrodynamics applies and the particle material can be described using macroscopic optical constants. Although special considerations apply to the scaling of quantum eﬀects they too could be studied in scaled experiments. 1.5.1 The classic laboratories By mid-twentieth century, the Mie-solution had long been known but solutions for nonspherical particle shapes were rather limited. Most were approximations that make use of simpliﬁcations arising when some particle parameter reaches an extreme value and the complex structures of real particles were mostly beyond contemplation (van de Hulst, 1957, 2000). In addition, the world of the experimentalist was crude in the eyes of the theoretician who contemplates idealized problems, usually involving perfect inﬁnite uniform and coherent wavefronts of monochrome radiation. We have seen that it was the emergence of coherent sources in the microwave range following rapid development of radar technology around the Second World War that opened up the possibility of generating coherent radiation. Although the laboratory version wavefront will always be ﬁnite in extent, it became possible to make precise narrow band microwave measurements of scattered amplitude and phase that were not reachable at other wavelengths. Realizing the potential to circumvent limitations imposed by the state of theoretical solutions for nonspherical particles J.Mayo Greenberg et al. developed a pioneering analogue laboratory (Greenberg, 1960) at RPI in 1960 and applied it to astrophysical problems. Greenberg et al. used X-band equipment operating at 3.18 cm wavelength and centimeter-sized plastic models (Figure 1.3) as the analogue to micrometer and submicrometer-sized silicate and ice interstellar grains scattering starlight at visual wavelengths. Retrieving all elements of the scattering matrix, the apparatus was optimized to study the forward and back scattering by any shape and refractive index particle in the notoriously diﬃcult resonance scattering size range where the size parameter x = (2πa)/λ of a radius a particle illuminated by wavelength λ radiation is of the order of unity and most approximate solutions fail. X-band radar

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Fig. 1.3. J. Mayo Greenberg in the SUNY incarnation of his microwave analogue laboratory circa 1978. The solution for spheroids was just becoming available and this particle model was used in attempts to check the cross polarization coeﬃcients in angular and forward scattering. Absorbers line the anechoic chamber and thin nylon strings (ﬁsh line) support the target and control its orientation using computer-controlled servomotors mounted in the ceiling and on the ﬂoor. (Photograph by D. E. Beeson.)

is designed to place airplanes and ships that radar was developed for in the geometric optics range. Classic size interstellar grains scattering visual light (Greenberg’s main interest) are in the 1 < x < 5 size range so that analogue interstellar grain diameters translate approximately to the convenient 1 cm to 5 cm range. Greenberg et al. were lucky too. Figure 1.4 illustrates the laboratory conﬁguration for angular scattering measurements as it was implemented at the State University of New York at Albany (SUNYA) and at the University of Florida (UF) toward the end of its use. Also shown is the forward scattering antenna which was covered by an absorber while angular measurements were made. When illuminated from the transmitting antenna to the left, the ﬁxed forward scattering and movable receiving antennas to the right intercept not only the scattering by the analogue particle model in the center but also unwanted direct radiation as well as scattered background radiation despite use of ‘anechoic’ materials on the chamber walls. The wanted scattering signal from the model particle was often dwarfed by the unwanted signal. Operation

Bo ˚ A. S. Gustafson

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Fig. 1.4. Layout of the microwave laboratory at the State University of New York at Albany (SUNYA) is shown as used when conﬁgured for angular scattering measurements. Reproduced from Gustafson (1980).

of the laboratory involved creation of conditions in which the weak wanted signal could be separated from the unwanted signal and accurately measured. To achieve this, the stable oscillator to the lower left in the diagram feeds not only the transmitting antenna but also a ‘compensation’ or ‘nulling’ waveguide signal which is mixed with the received signal. By removing the scattering body and manually adjusting the amplitude and phase of the compensation signal it became possible to achieve nulling destructive interference at the waveguide-mounted square law detector. As the analogue model is introduced into the beam an oﬀ-balance signal arises which is a highly accurate measure of the scattering produced by the particle and intercepted by the receiving antenna. This is a type of interferometry creating a condition in which unwanted signals cancel10 and are prevented from interfering with the much weaker scattered radiation. Measurements were only accepted if the null condition returned when the particle model was again removed from the illuminating beam. The most sensitive measurements were carried out in early morning hours when stability was at its best.11 The phase and amplitude of the oﬀ-balance signal could in principle be obtained by adjusting a calibrated set of variable attenuator and phase shifter to again obtain a null through destructive interference. The change in attenuation is a measure of the intensity in units of the incident radiation intensity. The phase 10

The unwanted signal is the sum of multiple signals each with their own Stokes vector. As we have seen their sum is also described by a single Stokes vector. The compensation signal has this same Stokes vector except that the phase element is shifted by the factor π radians. 11 Each component of the signal was subject to drift due to uncompensated power and temperature ﬂuctuations, even trembles due to passing trucks on a nearby road had a noticeable eﬀect.

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shifter measures the phase diﬀerence between the incident and scattered waves. With four polarization combinations at each scattering angle12 this would allow calculation of the entire F-matrix without need for calibration. Instability of the null condition unfortunately made the measurement time window too short to allow use of this method. The intensity of the scattered radiation was instead ampliﬁed and measured using a lock-in ampliﬁer system. The measurement therefore needed calibration using a known standard at the time of each measurement. A circuitry used in the forward scattering setup measured the amplitude and phase of the oﬀ-balance signal on separate branches using ampliﬁers that also needed continual calibration (Wang and Greenberg, 1976). The full F-matrix could be calculated from the forward scattering measurements but since phase measurements were not done in side scattering, only the four elements of the Mueller matrix in the upper left corner of the F-matrix related to intensities were obtained at other scattering angles. Schuerman, Wang and Gustafson ran Greenberg’s setup at SUNYA in Albany through the late 1970s and at the UF in Gainesville from 1980 through the early 1990s.

Fig. 1.5. The Bochum Ka-band laboratory as it appeared shortly before it was dismantled. The ﬁxed transmitting antenna is in the center and computer movable receiving antenna to the left. To the right is a mobile screen to limit the illuminating beam and reduce unwanted radiation. 12

All nulling adjustments must be repeated for each polarization and angle combination.

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Richard Giese et al. performed similar experiments at the Eberhard Karls University Tbingen (Giese and Siedentopf, 1962) and until 1990 at the Ruhr University Bochum (Zerull et al., 1977). Having a primary interest in interplanetary particles, Giese and his team operated their facility (Figure 1.5) at 35 GHz, which eventually became known as the Ka-band. The ∼ 8 millimeter wavelength and larger beam size (in terms of wavelengths) at the Bochum facility allowed study of particle sizes up to approximately x = 40 in the far ﬁeld condition. This corresponds to 5 to 8 micrometer diameter in the visual depending on the wavelength. The angular scattering measurement strategy employed in the Giese et al. laboratory diﬀered primarily from that of Greenberg et al. in that the particle model under study remained illuminated throughout the angular measurement sequence so that scattering plus background in principle were measured. This was only possible because of a low background level. The very last set of angular scattering results (Zerull et al., 1993) included phase measurements using an approach with separate branches for intensity and phase similar to the forward scattering arrangement. Both facilities normally used a sphere for calibration. However, cross polarization vanishes for spheres so they cannot be used to calibrate the oﬀ-diagonal cross polarization elements. It was usually assumed that an arithmetic average of the two obtainable calibration factors (polarization of both transmitter and receiver perpendicular respective parallel to the scattering plane) was a good approximation to the remaining two cross polarization calibration factors. Spheroid based calibrations were never suﬃciently accurate to conﬁrm this assumption because of the uncertainties in the precise orientation of an elongated particle. We note that cross polarization calibration factors primarily depend on loss across a rotary waveguide joint and that the calibration errors are likely to remain small. It is also notable that due to the fact that the orientation of a sphere is degenerate the orientation control is not part of tests commonly used to demonstrate the accuracy of a laboratory apparatus. Inadequate control over orientation quickly leads to errors especially for large highly elongated particles. Both laboratory facilities used similar means of particle support and orientation using nylon string (ﬁsh-line). There was never observable evidence for uncompensated perturbations due to the strings. The teams collaborated to obtain measurements at both wavelengths for some models until both laboratories were discontinued in the early 1990s. While other laboratories also existed, Greenberg’s and Giese’s were probably the main facilities of these early times. 1.5.2 The University of Florida laboratory A modern millimeter multi-wavelength broadband facility at the University of Florida (Figure 1.6) which is described in detail elsewhere (Gustafson, 1996, 2000) can measure phase as well as intensity to generate all elements of the scattering matrix from forward scattering to 163 degrees near backscattering. Synthesizers automatically step through 512 discrete wavelengths across a waveband as broad as the sensitivity range of the human eye and simulate colors as seen through standard color ﬁlters. Like Greenberg’s laboratory it has the sensitivity to measure angular scattering from simulated interstellar grains in the visual but the laboratory is optimized for larger grains and interstellar ones are near the sensitivity limit. Its large

1 Scaled analogue experiments in electromagnetic scattering

21

Fig. 1.6. The multi-wavelength w-band full bandwidth laboratory at the University Florida. The scale model in the center is illuminated by a ﬁxed transmitting antenna on a vibration dampened and temperature stabilized titanium and aluminum mount. A mobile but otherwise similarly mounted receiving antenna to the left is identical to the transmitting antenna. The aperture is covered by a polarizing copper wire grid and a lens made of acetal homopolymer resin (Delrin). (Photograph by the author.)

beam can accommodate particles several times larger than in Giese’s laboratory reaching x ∼ 200 or 250 (depending on the required ﬁdelity). The laboratory was designed to study especially the eﬀect of complex internal structure (e.g., Figs. 1.1 and 1.2) that may exist in compact and ﬂuﬀy aggregates. Particle control considerations and the size range dictate use of the millimeter wavelength range of the w-band (75 to 110 GHz or 2.7 to 4 mm). Objectives include validation of theoretical solutions (e.g., Xu and Gustafson, 1997, 1999; Waldemarsson and Gustafson, 2003), probe the validity range of approximations such as eﬀective medium theory (e.g., Kolokolova and Gustafson, 2001) and to probe sensitivity on otherwise not modeled parameters (Kolokolova et al., 2001). As van de Hulst (1957) points out, space is traversed by the incident and the scattered wave and the two wave systems cannot be separated in a mathematical point. However, using direction sensitive antennas, we can distinguish the two. The combination of an antenna aperture of the order of 100 wavelengths and corrective lenses generate a ﬂat wavefront at the distance of the scattering body approximately 700 wavelengths away. By time reversal symmetry, the identical receiving antenna

Bo ˚ A. S. Gustafson

22

collects not the spherical wave of the scattering body but a similarly ﬂat wavefront taken out of that spherical wave. This helps assure that the measured scattering corresponds to the far ﬁeld. Equivalently, the spherical wave can be viewed as a set of superposed ﬂat wavefronts that travel in all directions and decompose in these ﬂat wavefronts at inﬁnity. Space can then be viewed as being traversed by all these waves plus the incident wave and in practice several unwanted waves scattered by the support of the particle model and by background structures. The direction sensitivity of the antenna helps separate these waves and collect the wanted scattered far ﬁeld signal. The separation is not perfect and other methods including anechoic mats are used to further reduce the unwanted background. Mechanical control as well as frequency and other drift mitigation measures allow the unwanted signals to be measured in the absence of the scattering body and to be mathematically subtracted from the corresponding measurement with the model present in the beam. This is only possible when the system is stable and phase as well as amplitude is measured to allow vector subtraction. The ability to reach a high level of separation of the wanted scattered radiation from the incident radiation and scattered stray radiation is a key feature that sets the microwave analogue method apart from its optical counterparts. While it is possible to derive the scattering components based on the ratio of the scattered intensity to the intensity measured in the unobstructed direct beam, we found that higher accuracy can be obtained using a sphere and the Mie-solution for calibration similarly to praxis in the preceding laboratory facilities. Scattering quantities that we refer to as ‘measured’ are thus not obtained directly but are processed, ﬁrst through the subtraction of the background signal, and then through calibration. Although the scattered signal can be separated out, captured and calibrated, the idealized conditions of light scattering that are commonly assumed in theoretical works cannot be fully reproduced in any laboratory. For example, the incident wavefront is necessarily ﬁnite and suﬀers from imperfections, the distance to other objects is also ﬁnite and the scattering sample must be supported against gravity so some supporting mechanism is typically nearby. Gustafson (1996) describes how the Florida facility is designed for stability with its thermal and mechanical layout,13 how its geometric conﬁguration minimizes the studied model’s interaction with the support, and how the use of lenses compensates for the ﬁnite beam width to produce a ﬂat wavefront in the central part of the beam. When, as in the Florida facility, the change in both intensity and phase suﬀered by a scattered wave can be deduced in the four combinations of polarization along the scattering plane and perpendicular to the plane, we characterize the far ﬁeld scattering process completely (Gustafson, 2000). With both the incident and the scattered waves described using their respective Stokes parameters, the scattering 13

The eﬀective sensitivity is limited by the fraction of the power in the illuminating beam that emerges as unwanted ‘background’ due to direct exposure to the beam or stray radiation even in an anechoic chamber. With the transmitted power in the 0.1 W range (limited by safety concerns), contribution from thermal radiation and thermal noise generated in the electronic circuitry are usually less important than drift in the background due to frequency drift or thermal expansion/contraction which are the most important factors to control.

1 Scaled analogue experiments in electromagnetic scattering

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can be described by a linear transformation, mathematically the 4 × 4 matrix F of section 1.4. We have seen in section 1.4.2 that because the resonant frequencies of most materials are far removed from microwave frequencies, their refractive index changes only slowly across the microwave range and is usually constant across a single microwave band. The broadband facility in Florida thus allows us to study the dependency on frequency while the refractive index remains constant. This allows the separation of color induced by the particle geometry or morphology from the intrinsic color of the material since it is neutral, or gray, by deﬁnition when the refractive index is constant. Practically any number and combination of frequencies can be used at the Florida facility from 75 to 110 GHz (approximately 4 mm to 2.7 mm wavelength) to represent the illuminating spectrum. To give a quantitative measure of the intrinsic color in intensity and in polarization we may integrate the total intensity across the waveband which is represented by the sum over intensities normalized to the width of the band, I(Θ)Δω Δω where Δω is the separation between discrete frequencies. If for example, the laboratory illumination is to simulate sunlight, we may represent the integral of matrix elements illuminated by the solar spectrum across the [ωa , ωb ] interval using ωb

i[ωa ,ωb ] =

I0,freq ifreq Δω

freq=ωa ωb

, I0,freq Δω

freq=ωa

where the indices indicating polarization have been dropped and I0,freq is the solar illumination integrated across the small frequency interval corresponding to one frequency step Δω from ω − 12 Δω to ω + 12 Δω. This mimics the usual format of reported observations where the numerator represents the observed ﬂux across the waveband and the denominator provides normalization to the solar spectrum. The matrix elements are iﬁlter where the ﬁlter may be standard red and blue ﬁlters used in astronomy. The averages Icolor and P color can now be obtained by substituting the corresponding averages to compute the intensity and polarization (van de Hulst, 1957). As an example, we may quantify color through the ratio C(Θ) =

Ired − Iblue . Ired + Iblue

While the intervals [ωa , ωb ] for either red or blue colors can be chosen to simulate a variety of ﬁlters, we simulate the narrow-band continuum ﬁlters 443 and 642 nm using the frequency bands 75–85 GHz and 100–110 GHz to compare laboratory colors with colors obtained from modern comet observations. A single particle orientation requires a few hours of measurement to obtain 1◦ angular resolution from 5◦ to 165◦ at 85 frequencies and four combinations of polarization, with equal time needed for a background measurement. The background

24

Bo ˚ A. S. Gustafson

measurement includes the scattering by the support which does not have perfect symmetry so that a background measurement is made for each orientation of the support. The background usually does not need to be re-measured except when the target support is replaced. The repeatability and accuracy of angular measurements usually are comparable to errors introduced through uncertainties in the target parameters including the refractive index, shape and orientation (Gustafson, 1996). Scattering at angles from 168◦ to 180◦ cannot be measured because the antennas would overlap in this interval. However, it is possible to add a backscattering measurement capability through the addition of a directional coupler and receiving circuitry in the transmitting antenna. The forward scattering amplitude and phase are, in principle, measured in a manner similar to those at other scattering angles. However, due to strong contamination from the illuminating beam, extra care must be taken so that the illumination remains stable between the times of the measurements of the scattered signal plus background and the background alone. This is because the intensity of the scattered signal may constitute a small fraction of the direct beam background signal: ∼ 10−3 for an x ≈ 10 sphere. The measurement error is, to a ﬁrst approximation, proportional to the measured signal strength, which in forward scattering is dominated by direct illumination. This means that the error is expected to be practically independent of the particle’s cross-section so that the forward scattering by large scatterers can be measured with much higher relative accuracy than that by small particles. The large background signal is a direct consequence of the large beam size which, at about 100 wavelengths across, is approximately a factor of 10 wider (∼ 100 times larger in cross-section) than those of the two preceding laboratories. This compromise was made to accommodate particle size parameters up to x = 200 or 250 in the beam. The 0◦ to 5◦ interval is so badly plagued by direct illumination of the receiver antenna by way of the ﬁrst side lobes that it requires a tedious measurement procedure similar to that at 0◦ . 1.5.3 Complex interiors scattering experiment example We have seen that analogue laboratory models can in principle be arbitrary. Microwave and laser technology now exists that allow laboratory measurements at practically any size parameter. Using mostly plastic compounds with pigments and other inclusions to adjust the refractive index, the material and particle internal morphologies as well as shape can be arbitrary. The original motivation dating back to the works of Greenberg and Giese was primarily to study the eﬀects of nonspherical shape. But internal structure may be even more important. Figure 1.7 shows the scattering from a set of 125 x = 0.47 ﬁnite 4:1 aspect ratio prolate circular cylinders averaged over rotation about an axis normal to the scattering plane. All cylinder axes are aligned with the rotation axis and the cylinders randomly distributed in an 80% porosity (20% material) spherical aggregate. A particle with the same amount of scattering material evenly distributed across the volume of the sphere can be represented by a Mie-sphere with the same total polarizability as the aggregate. We see that the deviation, labeled ‘Mie’, is not large in the forward scattering direction where Fraunhofer diﬀraction (van de Hulst, 1957) dominates. The scattering gradually deviates from that for the uniform interior as the scattering angle increases and eventually joins with

1 Scaled analogue experiments in electromagnetic scattering

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Fig. 1.7. Measured (circles) angular distribution of polarized scattered intensities, perpendicular to the scattering plane, resulting from similarly polarized illumination of a 125 aggregate of simulated silicate (m ≈ 1.618 − 0.005i) cylinders aligned perpendicular to the scattering plane (upper left element of the Mueller S-matrix). The scattering is averaged over a rotation axis normal to the scattering plane. All cylinders were 4:1 prolate elongation ratio circular cylinders of size x = 0.47 randomly located in an approximately spherical aggregate at 20% packing factor (80% void). Comparison curves are a Mie calculation representing the homogeneous distribution approximation for the interior using eﬀective medium theory, a Rayleigh–Gans (van de Hulst, 1957) -type approximation labeled ‘Interference’ and a non-interacting and non-interfering approximation labeled ‘independent’ in which the single-scattering solution for each cylinder was added incoherently. Reproduced from Gustafson (1980).

the curve representing the coherent interference pattern from independently scattering cylinders labeled ‘interference’. (The incoherent scattering by the cylinders labeled ‘independent’ is also shown for comparison.) The homogeneous interior is a gross underestimate in the backscattering hemisphere where most of the radiation results from a few instances where scattering centers interfere constructively (Figure 1.8), which is supported by the ‘interference’ calculation. The body ‘glitters’ in backscattering. Coherent scattering allows most of the incident radiation to propagate undisturbed past the particle generating a diﬀerent forward diﬀraction pattern than from a homogeneous body where a signiﬁcant portion of the incident radiation is shifted in phase and/or attenuated. A homogeneous interior can therefore not be used to represent the scattering from a particle with internal structure.

26

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Fig. 1.8. Measured variation of the scattering element shown in Fig. 1.7 (upper left corner of the Mueller S-matrix) before averaging over rotation. The rotation angle χ is about an axis normal to the scattering plane. The pattern is dominated by interference which is seen to shift as the geometry changes with rotation. This fully explains the increasing sensitivity to orientation with increasing scattering angle Θ. Reproduced from Gustafson (1980).

While this is an extreme example of the eﬀect of internal distribution of matter, it is illustrative. One can generally expect that the Fraunhofer diﬀraction dominated forward scattering is aﬀected the least by an inhomogeneous interior, although not unaﬀected, and that there is a transition with increasing scattering angle toward scattering that depends on interference and (more generally) interaction between inhomogeneities or identiﬁable scattering centers. The scattering angles where these transitions occur depend strongly on the scattering volume (Zerull et al., 1993; Gustafson et al., 2002) and the eﬀect reduces to polarizability strength anisotropies in the Rayleigh limit. Interaction between inhomogeneities is low in this speciﬁc case, which makes it easier to identify the overall eﬀect of inhomogenieties. The particle model is one of the so-called ‘bird’s-nest’ structures made to represent aggregates of interstellar grains. It was part of a series of models used to test a theory about the nature of comet material and the process of comet formation in the solar nebula and consequent evolution ending with interplanetary dust (Gustafson, 1980; Greenberg and Gustafson, 1981). Measurements are from the SUNYA version of the Greenberg et al. laboratory and illustrate use of the laboratory to ﬁll in where other solutions do not exist. While eﬀects of irregular internal structures were studied, the relatively small size parameter < 5 reachable in the SUNYA laboratory was a limiting factor. The slightly larger ‘ﬂuﬀy’ particle structures (x < ∼ 40) studied in Bochum (Giese et al., 1978) often had complex shape on the wavelength-scale with inhomogeneities

1 Scaled analogue experiments in electromagnetic scattering

27

on scales that are not small compared to the wavelength so that eﬀective medium theory does not apply and geometric optics fails. Systematic investigations were limited by the diﬃculty of accurately describing the ‘ﬂuﬀy’ particle structures. The scattering by larger porous aggregates and some compact ones continue at the University of Florida facility (Gustafson et al., 2001, 2009; Thomas-Osip et al., 2005) with emphasis on complex structures that could be reasonably well documented. Investigation of the color and polarimetric color eﬀects of aggregation geometry (cluster-cluster or ballistic) as bodies grow in nature are currently the emphasis. Problems in deﬁning the complex geometry of aggregates remain but the emergence of 3D scanners and 3D printers are likely to herald a new era as far as particle documentation and the ability to build complex structures of predeﬁned geometry.

1.6 Discussion Microwave analogue measurements ﬁll a special need because of their versatility, accuracy, and the capability of systematic explorations of electromagnetic scattering at any wavelength. The high degree of control over the scattering experiment allows its use as a guide in theory development; for example, we have tested partial solutions to the scattering by large dielectric cubes by covering faces of the cube with absorbing or reﬂecting materials at will (e.g., Waldemarsson, 2001: Fig. 29). It is the only resort in the cases where no theoretical solution exists or when theoretical solutions are unreliable. It is also useful in the many cases when numerical computing is even more demanding than the experiment. Scaled analogue measurements require the construction of scale models. Because of this and because of the large parameter space to explore, microwave measurements are tedious. Given the level of eﬀort and the many tasks for which an accurate microwave laboratory is uniquely suited, it may be hard to justify its use in the exploration of scattering by broad varieties of natural particles that can be studied using direct optical techniques and clouds of particles. Multiparticle fullscale scattering measurements such as described by Mu˜ noz and Volten (2006) and Hadamcik et al. (Chapter 2 in this volume) are likely more suitable in such instances. There are also a broad range of systematic surveys that do not necessarily require the high accuracy of our w-band facility and that could best be done using a microwave analogue facility that trades some of the accuracy for higher data log rates. In the ‘direct’ problem the particle properties are speciﬁed and the task is to predict the scattering properties. The analogue laboratory is an ideal complement to theoretical computations of the direct problem and will probably remain so for a long time as practically any speciﬁed particle can be machined, molded, or possibly printed using ‘rapid prototyping’ technology. Acknowledgments Operating the microwave laboratory is a team eﬀort and the author is especially grateful to A. Espy, L. Kolokolova, J. Thomas-Osip, T. Waldemarsson, and Y.-l. Xu for their collaboration. I have also proﬁted greatly from many years of collaboration

28

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at various times with R. H. Giese, J. M. Greenberg, R. T. Wang, and R. H. Zerull who all helped introduce me to the microwave analogue method of light scattering measurements. This work was supported in part by NASA’s Planetary Atmospheres Program through grant NAGW-2482. Dedication This paper is dedicated to the pioneers of scaled analogue experiments in electromagnetic scattering, Richard Heinrich Giese 1931-1988 and Jerome Mayo Greenberg 1922–2001.

References Asano, S., and G. Yamamoto, 1975: Light scattering by a spheroidal particle. Applied Optics, 14, 29–49. Bohren, C. F., and D. R. Huﬀman, 1983: Absorption and Scattering of Light by Small Particles, New York: Wiley–Interscience. Giese, R. H. and Z. Siedentopf, 1962: Z. Naturforschung, 17a, 817. Giese, R. H., K. Weiss, R. H. Zerull, and T. Ono, 1978: Large ﬂuﬀy particles: a possible explanation of the optical properties of interplanetary dust. Astron. Astrophys., 65, 265-272. Greenberg, J. M., 1960: Rensselaer Research, 5, No. 1. Greenberg, J. M., and B. ˚ A. S. Gustafson, 1981: A comet fragment model for Zodiacal light particles, Astron. Astrophys., 93, 35–42. Greenberg, J. M., A. Li, C. X. Mendoza-G´ omez, W. A. Schutte, P. A. Gerakines, and M. de Groot, 1995: Approaching the Interstellar Grain Organic Refractory Component, Ap. J., 455, L177. Grossman, L., 1980: Refractory inclusions in the Allende meteorite, Ann. Rev. Earth Planet. Sci., 8, 559–608. Gustafson, B. ˚ A. S., 1980: Scattering by ensembles of small particles, experiment, theory, and application, Reports from the Observatory of Lund, 17, PhD thesis, Lund University. Gustafson, B. ˚ A. S., 1983: Dominant particle parameters in side scattering by some aggregates of cylinders. In Proc. 1982 CSL Scientiﬁc Conference on Obscuration and Aerosol Research (Kohl, R. H., Ed.), ARCSL-SP-83011, Aberdeen: US Army CSL, 281–292. Gustafson, B. ˚ A. S., 1996: Microwave analog to light scattering measurements: a modern implementation of a proven method to achieve precise control. J. Quant. Spectrosc. Radiat. Transfer, 55(5), 663-672. Gustafson, B. ˚ A. S., 2000: Microwave analog to light scattering measurements. In Light Scattering by Nonspherical Particles (Mishchenko et al., Eds), London: Academic Press, 367–390. Gustafson, B. ˚ A. S., F. Giovane, T. Waldemarsson, L. Kolokolova, Y.-l. Xu, and J. McKisson, 2000: Planetary Aerosol Monitor / Interplanetary Dust Analyzer. In Light Scattering by Nonspherical Particles: Halifax Contributions (Videen, G., et al., Eds), Adelphi, Maryland, Army Research Laboratory, 270–274. Gustafson, B. ˚ A. S., J. M. Greenberg, L. Kolokolova, R. Stognienko, and Y.-l. Xu, 2001: Scattering theory and laboratory simulations. In Interplanetary Dust (Gr¨ un et al., Eds) Heidelberg: Springer Verlag, 509–567.

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˚. S., L. Kolokolova, Y.-l. Xu, and T. Waldemarsson, 2002: The microwave Gustafson, B. A analogue method and its main results. In Electromagnetics and Light Scattering by Nonspherical Particles (Gustafson et al., eds.), Adelphi Maryland: Army Research Laboratory Press, 89–92. Gustafson, B. ˚ A. S., L. Kolokolova, and A. Espy, 2009: Evidence of an aggregate’s formation history from its scattering of light, in preparation. Herzberger, M., 1966: Optics from Euclid to Huygens. Applied Optics, 5, 1383–1393. van de Hulst, C. H., 2000: Hints from history: a foreword. In Light Scattering by Nonspherical Particles (Mishchenko et al., Eds), London: Academic Press, xxv–xxx. van de Hulst, C. H., 1957: Light Scattering by Small Particles, New York: John Wiley & Sons. Kerker, M., 1969: The Scattering of Light and Other Electromagnetic Radiation, New York: Academic Press. Kolokolova, L., and B. ˚ A. S. Gustafson, 2001: Microwave analogue experiments comparisons to eﬀective medium theories. J. Quant. Spectrosc. Radiat. Transfer, 70, 611–625. Kolokolova, L., B. ˚ A. S. Gustafson, K. Jockers, and G. Lichtenberg, 2001: Properties and evolution of cometary dust in near-nucleus coma from their color and polarization, J. Geophys. Res., 106, 10113–10128. Lakhtakia, A., and G. W. Mulholland, 1993: On two numerical techniques for light scattering by dielectric agglomerated structures. J. Res. Natl. Inst. Stand. Technol., 98, 699–716. Mie, G. 1908: Contribution to the optical properties of turbid media, in particular of colloidal suspensions of metals. Ann. Phys. (Leipzig), 25, 377-452. Mishchenko, M. I., J. W. Hovenier, and L. D. Travis, 2000: Light Scattering by Nonspherical Particles, London: Academic Press. Mishchenko, M. I., L. D. Travis, and A. A. Lacis, 2002: Scattering, Absorption and Emission of Light by Small Particles, Cambridge: Cambridge University Press. Mu˜ noz, O., and H. Volten, 2006: Experimental light scattering matrices from the Amsterdam Light Scattering Database. In Light Scattering Reviews: Single and Multiple Light Scattering (Kokhanovsky, A. A., Ed.), London: Springer Verlag. Purcell, E. M., and C. R. Pennypacker, 1973: Scattering and absorption of light by nonspherical dielectric grains. Ap. J., 186, 705–714. Rayleigh, Lord, 1881: On the electromagnetic theory of light. Philos. Mag., S. 5, 12, 81. Roberts, S., and A. von Hippel, 1946: A new method for measuring dielectric constant and loss in the range of centimeter waves, J. Appl. Phys., 17, 610–616. Sayili, A. M., 1939: The Aristotelian explanation of the rainbow. Isis, 30, 65–83. Schuerman, D. W., R. T. Wang, B. ˚ A. S. Gustafson, and R. W. Schaefer, 1981: A systematic study of light scattering: I. Particle shape. Applied Optics, 20, 4039–4050. Sinclair, G., 1948: Theory of models of electromagnetic systems, Proc. IRE, 36(11), 1364– 1370. Stokes, G. C. 1852: Trans. Cambr. Phil. Soc., 9, 399. Strutt, J. W. (later Lord Rayleigh), 1871: On the light from the sky, its polarization and colour. Philos. Mag., 41, 107–120, 274–279. Thomas-Osip, J., B. ˚ A. S. Gustafson, L. Kolokolova, Y.-l. Xu, 2005: An investigation of Titan’s aerosols using microwave analog measurements and radiative transfer modeling, Icarus, 179, 511–516. Twersky, V. 1952: Multiple scattering of radiation by an arbitrary conﬁguration of parallel cylinders. J. Acoust. Soc. Am., 24, 42–46. Waldemarsson, K. W. T., 2001: Electromagnetic scattering by ﬂakes of arbitrary refractive index: theory, experiment and applications. PhD thesis, University of Florida.

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˚. S. Gustafson, 2003: Interference eﬀects in the light Waldemarsson, K. W. T., and B. A scattering by transparent plates. J. Quant. Spectrosc. Radiat. Transfer, 79–80, 11111119. Wang, R. T., and J. M. Greenberg, 1976: Scattering by spheres with nonisotropic refractive indices, Appl. Opt., 15, 1212. Xu, Y.-l., and B. ˚ A. S. Gustafson, 1997: Experimental and theoretical results of light scattering by aggregates of spheres, Applied Optics, 36(30), 8026–8030. Xu, Y.-l., and B. ˚ A. S. Gustafson, 1999: Comparison between multisphere light-scattering calculations: (I) Rigorous solution and (II) DDA, Ap. J., 513, 896–911. Zerull, R. H., R. H. Giese, and K. Weiss, 1977: Scattering functions of nonspherical dielectric and absorbing particles VS Mie theory. Applied Optics, 16, 777. Zerull, R. H., B. ˚ A. S. Gustafson, K. Schulz, and E. Thiele-Corbach, 1993: Scattering by aggregates with and without an absorbing mantle: microwave analog experiments. Applied Optics, 32(21), 4088–4100.

2 Laboratory measurements of the light scattered by clouds of solid particles by imaging technique Edith Hadamcik, Jean-Baptiste Renard, Anny-Chantal Levasseur-Regourd, Jean-Claude Worms

2.1 Introduction 2.1.1 Astronomical and atmospheric context Clouds of solid particles are found in numerous regions in the solar system: in comets (comae and tails), in planetary and satellite atmospheres (Earth, Mars, Titan), in the interplanetary dust complex built up from dust released by comets and by asteroidal collisions (Gr¨ un et al., 2001; Levasseur-Regourd et al., 1990; Levasseur-Regourd, 1999; Levasseur-Regourd and Hadamcik, 2003). Space missions are rare and many objects cannot be visited (e.g. new comets). Although most dust clouds have very low number densities, they may be remotely detected by the light they scatter. Their properties are known by in situ observations, remote observations or by the interplanetary dust particles (IDPs) collected in the atmosphere of the Earth (Hanner and Bradley, 2004). Asteroidal and cometary nuclei surfaces are layers of loosely connected grains made of fragmentary debris produced by, for example, meteoritic impacts (regolith) or by gaseous species evaporation (Sullivan et al., 2002; Levasseur-Regourd et al., 2006). When released, these particles are irregular, compact or aggregated. Cometary particles, as conﬁrmed by, for example, the particles captured by Stardust, are mainly made of silicates and carbonaceous compounds with aggregates and compact grains in a large range of sizes from a tenth up to hundreds of micrometres (H¨ ortz et al., 2006; Zolensky et al., 2006). Titan solid aerosols, as observed by the Cassini–Huygens in situ space probe, are made of aggregates of submicrometre-sized organic grains (Tomasko et al., 2005). As far as Earth atmospheric solid particles are concerned, ice crystals, soot produced by biomass burning, industrial or aircraft combustions, sands lifted by winds, volcano ashes or extraterrestrial particles have been identiﬁed (Renard et al., 2003). Light scattering and polarization measurements are one of the tools used to study such particles and to access their physical properties (Herman et al., 1986; Santer et al., 1988; Gayet et al., 2002; Brogniez et al., 2003). Light scattering by clouds of particles can be studied using diﬀerent techniques such as in a steady-state gas ﬂow or in jet streams (West et al., 1997; Mu˜ noz et al., 2004; Volten et al., 2007). These techniques are suitable for submicrometre-sized or micrometre-sized particles but not for large ones (hundreds of micrometers). Microwave analogue experiments on individual or aggregated grains (Gustafson and

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E. Hadamcik, J.-B. Renard, A.-C. Levasseur-Regourd, J.-C. Worms

Kolokolova, 1999) have validated numerous numerical model although the particles are purpose-built and clouds of particles cannot be obtained. Reduced gravity can be a sensible way to achieve conditions close to those prevailing in space for the laboratory experiment measurements which are requested to relate remote or in situ observations to physical parameters (Worms et al., 1996, 1999a,b; LevasseurRegourd, 2003). The choice of parabolic ﬂights to achieve these conditions is determined by the fact that light scattering measurements can be made within seconds for any kind of particles without discrimination by weight or composition. The PROGRA2 experiment (PRopri´et´es Optiques des Grains Astronomiques et Atmosph´eriques) has been developed to study the light scattered by realistic ‘natural’ (opposed to purpose-built) dust clouds with the PROGRA2 -vis instrument (‘vis’ for visible). The second instrument PROGRA2 -surf (‘surf’ for surface) is used for comparison with the light scattered by the same grains deposited on layers on the ground and for the study of regolith analogues. Samples are chosen for two main reasons: (1) astronomical analogues (e.g. lunar or Martian, Titan’s aerosols, cometary, asteroidal) or purpose studies for atmospheric applications (soot, sands); (2) to have diﬀerent samples with only one change of parameter (e.g. size of grains, of particles, structure, absorption). 2.1.2 Polarization measurements 2.1.2.1 Deﬁnitions The linear polarization value depends on the phase angle, the wavelength and the physical properties of the particles. The phase angle (180◦ -scattering angle) is the angle between the direction of the light source and the observer’s line of sight at the scattering particle, in the scattering plane (Fig. 2.1). The plane of observations is perpendicular to the line of sight at the scattering object.

Fig. 2.1. Geometry of observations. Scattering plane deﬁned by line of sight and illumination directions. Plane of observations perpendicular to the line of sight. α = phase angle. R = light source to object distance (Sun-object distance in case of observations), Δ = object to detectors distance (object-Earth distance in the case of observations).

2 Laboratory measurements of the light scattered by clouds of solid particles ...

33

2.1.2.2 Equations The degree of linear polarization P of the scattered light and the intensity I can be written: I⊥ − I|| P = , (2.1) I⊥ + I|| I = I⊥ + I|| ,

(2.2)

where I⊥ and I|| are the polarized components respectively perpendicular and parallel to the scattering plane (they are measured by two separate images). P as a dimensionless ratio does not need any normalization with the distance or the number of particles. 2.1.2.3 Phase curves for comets and asteroids Solar system small bodies present phase curves typical of irregular particles with a positive branch and a maximum polarization Pmax between 90◦ and 120◦ , a small negative branch of 1% or 2% and an inversion angle close to 20◦ . In Fig. 2.2 synthetic phase curves are presented for comets and asteroids of S and C types at green and red wavelengths, and for the interplanetary dust (Levasseur-Regourd et al., 2001). In the positive branch, the polarization increases with wavelength for comets and C-type asteroids. The opposite trend is observed for S-type asteroids, for the circumnucleus region of comets, close to the nucleus (Levasseur-Regourd et al., 2005; Harrington et al., 2007) and for two comets (e.g. Kiselev et al., 2000). For the interplanetary dust, the polarization spectral eﬀect is diﬃcult to measure; it seems to be neutral in the visible domain (Leinert et al., 1998).

Fig. 2.2. (Updated from Hadamcik et al., 2007; Levasseur-Regourd et al., 2001). Synthetic ﬁts of the observed phase curves for comets (HP = high Pmax comets, LP = low Pmax comets), for asteroids of C and S types and for the interplanetary dust (ID = interplanetary dust in the symmetry surface, at 1.5 AU from the Sun). Left: Green wavelength domain. Right: Red wavelength domain. The ID component is assumed to have the same scattering properties as in the green domain.

34

E. Hadamcik, J.-B. Renard, A.-C. Levasseur-Regourd, J.-C. Worms

In cometary comae diﬀerent regions are observed such as jets and a circumnucleus halo (Renard et al., 1996; Hadamcik and Levasseur-Regourd, 2003a). The polarization is higher in the jets and generally in regions following fresh dust emission, e.g. during an outburst. The synthetic phase curve for jets seems to only present a positive branch (or a very small negative branch if it exists at all), indicating diﬀerences in the physical properties of the particles ﬁlling up these regions. After a description of the operating principles of the two instruments, we mainly concentrate on the PROGRA2 -vis instrument and on results for lifted particles but do not neglect the comparison with results obtained on layers (with non-coherent multiple scattering). We also present some comparisons between numerical models and experiments and ﬁnally some applications to interpret astronomical and atmospheric observations. 2.1.3 Instruments Two instruments are used: PROGRA2 -vis for levitating particles studied in the visible spectral domain and PROGRA2 -surf for particles deposited on a surface in the same spectral domain. The average phase angle range is [6–150]◦ with a minimum value of 5◦ and a maximum value of 170◦ . 2.1.3.1 Description Measurement system A randomly polarized light source illuminates lifted or deposited particles. An optical ﬁbre carries the light to the vial in which the particles are lifted or to the collimator lenses to illuminate the deposited sample. The samples scatter the light. A polarizing beam splitter cube splits the scattered light at a given phase angle in its parallel and perpendicular components to the scattering plane. The detection system (photodiodes or cameras) enables measurements (Worms et al. (1999a, b) for the ﬁrst system and Renard et al. (2002, 2005a) for the imaging system). For lifted particles, the incident laser beam and vial rotate to change the phase angle, the detection system being in a ﬁxed position (Fig. 2.3(a)). For deposited particles, two arms can rotate, one with the incident beam, the other one with the detection system (Fig. 2.3(b)).

Fig. 2.3. (Updated from Hadamcik et al., 2002a.) Operating principle of the two instruments. (a) PROGRA2 -vis for lifted particles with measurements in the visible wavelength domain. (b) PROGRA2 -surf for particles deposited on surfaces.

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35

Light source Randomly polarized He-Ne lasers are used at two wavelengths (543.5 nm and 632.8 nm). The lack of residual polarization is checked in the laboratory as a function of the position in the beam (Worms et al., 1999a, b), and as a function of time. For this purpose, it is necessary to enlarge the beam and to reduce the aperture to avoid saturation (measurements in the forward direction). After the laser is switched on, an important increase of intensity in the ﬁrst 20 min is noticed. An oscillation of the polarization value with a period of a few seconds and a small amplitude of a few tenths of one per cent is detected. The whole system (laser and detection) produces an error bar of ±1% on the polarization value. Measurements of the scattered light by transparent and absorbing particles (silica and carbon-black with nanometre-sized grains) at a 10◦ ﬁxed phase angle with the PROGRA2 -surf instrument are also conducted with the two lasers (red and green). In the ﬁrst 40 min, the polarized intensities increase by about 0.12% min−1 ; the increase is about 0.1% min−1 during the next 20 min and only of 0.03% min−1 during the next hour. These results are the same for all the lasers used and perfectly correlated to the direct measurements made with the PROGRA2 -vis instrument. When the steady-state temperature is reached (after about 60 min) in the laboratory conditions, the signals and polarization are stable within an error bar of 0.5% for the second hour of measurements and 0.05% for the third hour (on layers of transparent particles). If the samples are absorbing, their temperature can slightly increase and the intensity slowly change depending on the size of the grains. Although negligible, this problem can be avoided if the beam impact point changes on the surface. Levitating particles are moving and the increase of temperature for absorbing particles is negligible. Sample container for lifted particles The optical ﬁbre head was set at about 35 cm from the sample. In the new version, it is close to the entrance of the vial. The laser beam diameter lies between 2 and 3 mm (4 mm in the ﬁrst version). The samples are introduced in a container. The vial containing the dust particles is cylindrical with a diameter of about 40 mm and is made of glass. The beam enters the vial through a plane-parallel glass window ﬁxed on a glass tube (length about 12 mm, external diameter 8 mm and internal diameter about 6 mm). The unscattered light is trapped in a curved light trap with an external diameter 12 mm where it connects to the vial (Fig. 2.4(a)). The light input cylinder and the light trap are covered with a black paint and their diameters are minimal in order to have a better access to the backward and forward scattering regions. For studies in reduced gravity, the pressure inside the vial is reduced and the vial is hermetically sealed. For particles lifted by air-draught, nitrogen is introduced in the vial by an opening in the cap of the vial (Fig. 2.4(b)). The gas then ﬂows along the vial wall with a slow speed of injection, lifts the particles close to the axis and leaves the vial through the centre cap. This allows, if necessary, a partial cleaning of the vial walls by blowing oﬀ the dust.

36

E. Hadamcik, J.-B. Renard, A.-C. Levasseur-Regourd, J.-C. Worms

Fig. 2.4. Sample containers: (a) Vial for microgravity conditions. (b) Vial for air-draught injection on the ground with its cap allowing to the injection of nitrogen and its circulation around the inner wall.

Beam splitter cube A broadband (450–700 nm) polarizing beam splitter cube separates the parallel and perpendicular polarized components to the scattering plane, corresponding to the reﬂected and transmitted beams (at 90◦ from each other). The refractive index of the substrate material is 1.64 in the visible domain. All external surfaces are antireﬂection-coated. At wavelengths of 543.5 nm and 632.8 nm the transmitted and reﬂected beam eﬃciency are theoretically the same, and as checked in the experiment (if necessary one way is weighted by the measured coeﬃcient). Detectors In the ﬁrst version of the instrument, pin-photodiodes were used, integrating the scattered light on a ﬁeld of 2 cm diameter at about 35 cm from the detectors (such conﬁguration is still available for the deposited particles system). Since 2000, two cameras are used. The cameras (Leutron Vision LV65CE) have a 752 × 582 pixels CCD sensor. To minimize the time of lecture of the CCD, the ﬁeld of view is reduced to 740 × 400 pixels. Telephoto lenses allow imaging with a 10 μm per pixel resolution. At 90◦ phase angle, the horizontal ﬁeld of view is about 6.5 mm and the vertical ﬁeld corresponds to the diameter of the beam (about 2 mm). As the phase angle changes, the ﬁeld of view increases theoretically as 6.5/ sin α. In practice, it is limited by the ﬁeld depth and we only keep the central part of the image; for phase angles smaller than 40◦ and larger than 140◦ , this gives a maximum ﬁeld of view of about 10 mm (and not the 40 mm of the vial). The geometry of the measurements induces an error bar of about 1◦ on the phase angles. The signals are digitized in real time on 10 bits. The cameras record 12.5 images per second. To increase the dynamics, exposure times are between 1/50 s and 1/10 000 s; the choice mainly depends on the particles albedo. If the exposure time is too long the images of the particles are signiﬁcantly elongated. Diﬀerent diaphragm-stops positioned in front of the cameras allow the manual adjustment of the ﬁeld depth or the increase of the ﬂux for very dark samples (thus obtaining less accurate images).

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37

A third camera at a ﬁxed 90◦ phase angle is used to normalize the intensity (reference camera). It is synchronized to the two main cameras. Its resolution is about 18 μm per pixel. This camera allows also a partial 3D view of the particles. Instrument container The whole instrument is installed in a case, whose inside walls were painted black (Fig. 2.5a). The external dimensions are 1.05 × 0.85 m with a height of 0.54 m. The command and acquisition systems are located in an external speciﬁc rack (Fig. 2.5b).

Fig. 2.5. PROGRA2 -vis instrument and container. Left: Instrument inside the container. Right: Container in the A300-Zero-G plane.

2.1.3.2 Measurements process Without particles To control and adjust the optical alignment before each series of measurements (on the ground or during a parabolic ﬂight campaign) a sighting-mark is positioned along the axis of the rotating tray. The vertical and horizontal alignments of the two images are ﬁrst controlled visually. The two images of the sighting-mark are then recorded and the shift parameters between the two images measured automatically. At each phase angle and without any particles in the ﬁeld of view, one (or more) images are recorded to estimate the electronic oﬀset and obtain a so-called ‘oﬀsetimage’. The oﬀset-image also allows subtracting the eventual faint stray light that may be present at small (α < 20◦ ) or large (α > 150◦ ) phase angles (produced mainly by grains stuck in the light-trap or in the input cylinder). With levitating particles Two techniques are used to lift the particles: levitation under reduced gravity conditions or injection of a draught of nitrogen in ground-based conditions (so-called ‘air-draught’). The microgravity technique is suitable for compact particles with

38

E. Hadamcik, J.-B. Renard, A.-C. Levasseur-Regourd, J.-C. Worms

an average size larger than 10 μm, or for relatively compact particles. For ﬂuﬀy particles or for particles with sizes smaller than 10 μm, the air-draught technique is generally used. Novespace operates the parabolic ﬂights with the A300-zeroG dedicated aircraft. The typical proﬁle of a parabola is as follows: an entry pull-up phase from the horizontal normal ﬂight and during which the gravity is about 1.8g (for about 20 s), a transitory injection phase during which the gravity falls down from 1.8g to 0g (less than 5 s), the microgravity phase (±5 × 10−2 g) for some 22 s and the pull-out phase (essentially symmetrical to the pull-up phase) ending by a 2-min period between two consecutive parabolas. The pull-up and pull-out phases may increase the amount of particles sticking to the vial. A mechanical device is used to un-stick the particles at the beginning of each parabola (and sometimes during the parabola). During the 22 s of the microgravity phase, the particles freely ﬂoat in the vial with random movements; the phase angle is constant and the images are continuously recorded. The tray rotates between the parabolas by 5◦ or 10◦ depending on the phase angle range, at the convenience of the operator. Exceptionally, the operator can initiate a two-angle procedure during the same parabola, with two recordings of 9 s each (rotation of the tray 5◦ ). The number of recorded images being lower in the two angles procedure, the consequence can be a loss of precision for the polarization values. A full campaign (three ﬂights with 31 parabolas each), allows studying about six samples at one wavelength. Twenty to thirty series of measurements (phase angles) are necessary to retrieve a whole phase curve. One to three campaigns take place every year (with the French Space agency CNES or the European Space agency ESA). When the particles are lifted by air-draught, they progressively fall down with gravity but, depending on the samples, they may ﬂoat with random movements for 1s up to tens of seconds (sometimes minutes). The images are recorded during the time when particles are in the ﬁeld of view (10 s on average). To achieve better accuracy, many injections can be necessary for each phase angle. When the scatter of the data is too large, an important number of measurements are necessary to retrieve a representative value of the polarization. 2.1.3.3 Data reduction Pre-processing An iterative procedure with a 0.25 pixel step precisely aligns the images of the sighting-mark; the parameters are applied to all pairs of recordings of each series. The ‘oﬀset-image’ containing the electronic oﬀset and the eventual stray-light is ﬁrst subtracted for each series of images corresponding to the same phase angle. For some samples (mainly for large agglomerates or for crystals), some pixels may be saturated. To avoid some bias in the polarization results by these pixels, or by values in the nonlinear part of the curve of the detectors, an automatic procedure suppresses all the pixels, the level of which exceeds 80% of the saturation level. To achieve the best accuracy on the photometry of the individual particles, the images are ﬁltered and smoothed by a mean smoothing IDL procedure. Nevertheless, some noise is still present on the images and to prevent any confusion with

2 Laboratory measurements of the light scattered by clouds of solid particles ...

39

real particles in the ﬁeld of view, all pixels with a digital level smaller than 5 are excluded from the analysis. Polarization maps Polarization and intensity maps are obtained by computing the equations (2.1) and (2.2) for each pixel of the polarized component images. The particles in the ﬁeld of view may be of diﬀerent kinds. Large compact grains (20 μm < size < 300 μm) appear as single particles in the ﬁeld of view, but when lifted in microgravity they can be found to be agglomerates of grains in contact with each other by small surfaces; their large sizes in the ﬁeld of view generally make it possible to locate them on the polarization map. Aggregates of submicrometre-sized grains can be also agglomerated in larger particles. When lifted by an air-draught the average size of the ﬂuﬀy agglomerates is [50–100] μm with a lower cut-oﬀ of 10 μm and some large particles. When levitating in microgravity, the agglomerates of ﬂuﬀy aggregates are generally huge, up to millimetres. When micrometer-sized grains are studied, they are lifted by the air-draught technique. If some large agglomerates are lifted they quickly fall down; some seconds after, the beam seems about continuous in the ﬁeld of view, individual grains and small aggregates of a few number of grains remain present and scatter the light. Figure 2.6 shows a polarization map, the corresponding intensity map and the reference intensity map for lunar analogue particles at a 120◦ phase angle. The number of particles in the ﬁeld of view is relatively large and this type of image is generally not kept for single scattering studies.

Fig. 2.6. Lunar analogue PROGRA2 images at 120◦ phase angle. Top: intensity maps; middle: polarization maps; bottom: intensity maps from the reference camera.

40

E. Hadamcik, J.-B. Renard, A.-C. Levasseur-Regourd, J.-C. Worms

Single and multiple scattering The imaging method enables measurements of both the polarization of light produced from individual particles and that from thick (or dense) clouds of particles. In practice, when single scattering of a sample is studied, images containing strong multiple-scattering eﬀects arising from particles too close from one another are rejected. Inside a particle, optically interacting grains can be present and the variations of the polarization with the size of the particles may be an indication of such interactions. Another indication of multiple scattering comes from important diﬀerences in polarization and/or intensity values in the image as compared to the average values from other images at the same phase angle for the same sample. The behaviour of the polarization is monitored when the number density increases: from optically thin to optically thick clouds with multiple scattering between non-touching particles, loose agglomerates with a few touching grains, loose deposited very large agglomerates, sifted layers and ﬁnally packed layers. Polarization values for clouds of particles All pixels with a digital level higher than the cut-oﬀ level are added for each I|| and I⊥ images. To calculate the intensity normalized over the surface, the resulting sum is divided by the number of used pixels. Then, the parallel and perpendicular components of all the images are summed separately. Finally the polarization value is calculated with equation (2.1). If enough images are available (typically more than 10), the resulting polarization value is the same than when directly calculated on each image and averaged over the whole series. Intensity values The intensity function, in relative units, is obtained as Ip /Ir where Ip is the sum of the surface-normalized polarized components as in equation (2.2) and Ir is the surface-normalized intensity recorded by the reference camera. At small and large phase angles, the ﬁeld of view on the two main cameras and on the reference camera diﬀer markedly; hence the division Ip /Ir can give inaccurate results, except when only one or two particles are in the ﬁeld of view with non-overlapping images. Size distribution of particles The imaging technique allows us to measure the polarization of the light scattered by large single particles (compact or not). The polarization maps (for phase angles between 30◦ and 150◦ ) are automatically scanned; the output is the projected surface (S) of each particle independently of its shape. Two adjoining pixels with values higher than the cut-oﬀ have to be present to be identiﬁed as a particle. The equivalent diameter (in pixels) is 4S . (2.3) d= π

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41

Size ranges are deﬁned (steps of some tens of micrometres) and a size distribution given. For each size range a polarization and a surface-normalized intensity value are calculated. To have a signiﬁcant variation of the polarization as a function of the particles’ size, it is necessary to measure the polarization for numerous (usually more than one thousand) particles. Polarization values for clouds of particles selected by size can be obtained; variations of these values if any give an indication on the structure (compact or ﬂuﬀy) of the particles. In the case of ﬂuﬀy particles, it may be interesting to study the variation of the polarization as a function of the surface-normalized intensity, which can give some indication of the porosity of the aggregates. Due to the large number of images for a session of measurements (some tens of thousands of images), the processing is automatic but manual sorting can be done after the processing if some images give results completely diﬀerent from the average. New parameters (such as ﬂux and polarization as a function of time or polarization as a function of equivalent diameter) may also be deﬁned (see, for example, Hadamcik et al., 2002b).

2.2 Samples For light scattering studies, the main physical properties are linked to the real and imaginary parts of the refractive index, which are related to the material, the size of the grains, their shape (spheres, cubes, irregular particles with rounded or sharp edges), their structure (single grains, aggregates with diﬀerent porosities). In the case of aggregates or agglomerates, diﬀerent sizes are involved. Figure 2.7a presents SEM and TEM images of silica and carbon samples in diﬀerent size ranges. The size distribution of the grains is given by the provider (industrial or laboratory) and controlled on the images. Figures 2.6 and 2.7(b) display some polarization maps used to measure the size distribution of the particles in the ﬁeld of view. For compact particles this size is compared to those estimated from microscope images and to those sometimes given by the provider. Nevertheless, some discrepancy can exist between these measurements mainly due to grain shape (e.g. elongated grains have a smaller size than those measured on the polarization maps). Table 2.1 gives the characteristics of the samples presented in this chapter, the sizes are those given by the provider or measured on the SEM images. To have access to some physical properties from the observed phase curves (inverse problem) it is necessary to know the inﬂuence of each parameter and to disentangle these inﬂuences. Systematic studies are made by changing one parameter at a time. 2.2.1 Samples preparation A small quantity of materials is introduced in the vial (typically less than 0.5 g for compact particles and 0.05 g for ﬂuﬀy aggregates). It is diﬃcult to image together small and large, compact grains. When the size distribution of the grains is too large the smaller grains are not detected on the polarization maps. Small grains can eventually stick to large ones producing irregularities on the particles. The small

Mantle 0 .05 ± 0 .01

Sicastar-black

NaCl

Sodium chloride

Quartz Quartz Silica Carbon Carbon Graphite Mg-silicates

SiC (99.4) SiO2 (50), Al2 O3 (16), FeO (10), CaO (10) SiO2 SiO2 SiO2 C (amorphous) C (amorphous) C (100) MgSiO3 (50 crystals) Mg2 SiO4 (50 crystals)

Silicon carbide Basalt

500 ± 150

200 ± 100 30 ± 20 (0.1 − 6) 150 ± 100 20 ± 10 (1 − 10) × (0.1 − 2) 300 ± 150 (aggregates with 15 ± 12 crystals)

88 ± 5 13 ± 2 9±1 88 ± 8 110 ± 15

100 ± 50

100 ± 25 1.45 ± 0.06 Core 1.45 ± 0.06 Mantle 0 .04 ± 0 .01 Core 20 ± 6

Mean diameter (μm)

SiO2 (71), Na2 O (14) SiO2 SiO2 + Sicastar-black SiO2 +

Main components (%)

IRREGULAR COMPACT Lunar analogue (JSC-1) SiO2 (43.5), Al2 O3 (23.3), Fe2 O3 (15.6) Boron carbide B4 C (crystals)

SPHERES Glass beads Silica Coated silica

Samples

A. Compact particles

Stochastically rough Some sharp edges Some sharp edges Stochastically rough Sharp edges Plates Polyhedral and aggregates of μm-sized crystals Rounded cubes and some aggregates of cubes. Crystals

Rounded edges, some concave parts Polyhedral, sharp edges Idem Idem Polyhedral, sharp edges Polyhedral

Spheres < 4% broken or spheroids

Spheres < 1% spheroids Spheres < 2% spheroids Spheres < 2% spheroids

Shape

White White White Black Black Black White

White

1.54 + i4 .5 × 10 −4

0.065 ± 0.01 (black) 0.045 ± 0.01 (black) 0.045 ± 0.01 (black) Dark grey 0.04 (Dark grey)

Grey

—

White White — Black —

Albedo and colour

(1.4 − 1.5) (1.4 − 1.5) (1.4 − 1.5) 2 2 (2 ± 0.5) + i(1 ± 0.8) 1

2 + i0 .04 Idem Idem (2.67 ± 0.3) 1.6

1.6

Mantle 2 .4 + i0 .4

1.5 (1.51) 1.5 ( 1 .48 + i10 −4 ) Core 1.5 ( 1 .48 +i10 −4 ) Mantle 2 .4 + i0 .4 Core 1.5 ( 1 .48 +i10 −4 )

Re(m) + i I(m)

Table 2.1. Summary of the properties of the samples presented here. Name, chemical formula (components with ratios <10% are not indicated), size range (diameters are those given by the providers and veriﬁed on SEM images ‘in italic deduced from numerical simulations’), shape (HP = high porosity), refractive index (from literature, ‘in italic deduced from numerical simulations’), albedo measured at 8◦ on a sifted deposited sample (by comparison to a same white compact sample) and colour.

42 E. Hadamcik, J.-B. Renard, A.-C. Levasseur-Regourd, J.-C. Worms

Main components (%)

C

C

C (95–99)

Mgx Siy Oz mixtures

Carbon-black

Carbon-black

Soot (e.g. kerosene)

Mg-Silicates (5 = samples) Fe-silicates Titan’s analogues

Fex Siy Oz mixture C x Hy N z 6 = samples

SiO2

Silica

FLUFFY AGGREGATES Alumina Al2 O3

Samples

Table 2.2. Continued. B. Fluﬀy particles

0.013 ± 0.002 (grains) 10 ± 5 (aggregates) 0.040 ± 0.020 (grains) 8 ± 6 (aggregates) 0.095 ± 0.020 (grains) 5 ± 4 (aggregates) 0.014 ± 0.005 (grains) 7 ± 3 (aggregates) [0.01 − 0.1] grains [0.1 ± 5] aggregates [0.01 − 100] grains Average 0.050 0.040 ± 0.02 1.27 ± 0.5, 0.68 ± 0.35, 0.27 ± 0.15, 0.30 ± 0.18 0.39 ± 0.1, 0.19 ± 0.06

Mean diameter (μm)

White 0.87 ± 0.03 (white) 0.0025 ± 0.001 (black) 0.001 ± 0.001 (black) 0.008 ± 0.003 (black) Brown Dark brown From clear to dark brown

1.45 2 2 2 1.6 1.6 [1.6?1.7]

HP aggregates, amorphous HPaggregates, amorphous HP aggregates, amorphous HP aggregates, amorphous HP aggregates, crystals and amorphous grains HP aggregates HP aggregates amorphous Radial growing

Albedo and colour

1.7

Re(m) + i I(m)

HP aggregates, crystals

Shape

2 Laboratory measurements of the light scattered by clouds of solid particles ... 43

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E. Hadamcik, J.-B. Renard, A.-C. Levasseur-Regourd, J.-C. Worms

Fig. 2.7. (a) Silica and carbon samples in diﬀerent size ranges (C spheres = Sicastarblack-coated silica spheres): sub-μm-sized, μm-sized and large as compared to the wavelength. (b) Images of compact and ﬂuﬀy particles on the polarization maps.

grains may also be agglomerated into large ﬂuﬀy particles (that can eventually be detected by the ﬂux per pixel on the intensity images). To relate the optical properties of particles to their physical properties it is necessary to study narrow size distributions mainly for compact particles. Diﬀerent size ranges are usually obtained by crushing and sieving; it is then necessary to clean the sample and eliminate the ﬁne grains, for example, by sedimentation in a liquid. To prepare mixtures, it is necessary to homogenize them, ﬁrst by vigorous manual shaking eventually complemented by a 15 min shaking in an ultrasonic device. Small compact grains well mixed with ﬂuﬀy particles made of submicrometre-sized grains are thus included in the ﬂuﬀy agglomerates. 2.2.2 Levitation techniques, advantages and restrictions The advantages and restrictions of the two levitation methods, which are used in the experiment, are summarized below. Microgravity avoids sedimentation due to weight and allows the study of clouds of freely ﬂoating mixtures of irregular particles of any shape, size, and size distribution. The number of parabolas is limited and it can be necessary to complete

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45

the phase curves at the next campaign. Owing to ﬂuctuations of the microgravity level (±5 × 10−2 g), the particles move inside the vial with in average random speeds and directions. In the vial, the pressure is generally reduced also avoiding the presence of humidity. Nevertheless, the pull-up and pull-out phases (with 1.8g gravity) favour the sticking of the small grains and aggregates of small grains to the bottom of the vial. When they are shaken oﬀ by the kick-oﬀ device, they are lifted in large agglomerates with low number densities. Micrometre-sized grains and ﬂuﬀy aggregates can be lifted from the ground by a low speed air-draught (avoiding orientation of possibly elongated particles); they then ﬂoat by convection and gravity-driven buoyancy with random speeds for several seconds before settling down. It is possible to adjust the injection and have small or large agglomerates of grains. Some samples, with average sizes of tens of micrometres may be studied if their particles have homogeneous densities; when that is not the case the polarization becomes time-dependent. If the statistics are not suﬃcient to retrieve a representative average phase curve, a small quantity of material can be introduced again in the vial, then it is possible to start again the measurements. A limit of the two methods used to lift the particles is the diﬃculty of studying single scattering for grains smaller than 20 μm. The advantage is the possibility of studying large particles compact or ﬂuﬀy as found in, for example, cometary comae and to make measurements on optically thin and thick clouds of particles.

2.3 Results 2.3.1 Calibrations Calibration experiments are ﬁrst carried out with samples for which the polarization values can be easily calculated. With PROGRA2 -surf, specular reﬂection on a black plane glass surface allows to control that the reﬂected light follows the Fresnel laws (Worms et al., 1999a, b). With PROGRA2 -vis, the phase curves (intensity and polarization) obtained for the scattered light by 100 μm diameter glass spheres are compared to Mie curves (Worms et al., 1999a, b; Renard et al., 2002; Hadamcik et al., 2003). A good adequacy between theory and experiment in the two wavelengths is found if about 20 measurements are made for each phase angles. The results at 543.5 nm are presented in Fig. 2.8. If the real part of the refractive index is known, the imaginary part is a free parameter of the ﬁt. 2.3.2 Phase curves and their parameters 2.3.2.1 Transparent and absorbing materials in diﬀerent size ranges The variation of the phase curve characteristics will be presented on series of measurements for silica (transparent) and carbon (black). Their properties can be found in Fig. 2.7 and Table 2.1. For each material, the parameters are the size of the grains and the structure of the particles. The size range is compared to the wavelength range. In Fig. 2.9(a), the maximum polarization (Pmax ) diﬀerence is small between transparent silica with average grain size of (12 ± 3) × 10−3 μm and dark carbon

46

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Fig. 2.8. Polarization (left) and intensity (right) phase curves for 100 μm glass spheres as compared to Mie curves. The Mie curves take into accounts the aperture of the detectors (which produces a 1◦ uncertainty on the phase angle) and the size distribution of the spheres (measured on SEM images and controlled on the polarization maps). λ = 543.5 nm.

with grain size of (14 ± 5) × 10−3 μm. For larger sizes than the wavelength, Pmax increases when the size increases (Fig. 2.9b, c, d); this increase is more important for dark materials. More generally, Pmax is higher for absorbing than for transparent materials. The polarization in the negative branch depends on diﬀerent parameters, the absorption, the morphology and the shape of the grains. Figures 2.9(e, f, g, h) present the phase curves for layers of the same grains. The main diﬀerence is the systematic decrease of Pmax for layers of transparent particles as compared to lifted particles because of multiple scattering between the transparent grains. The other important diﬀerence is the presence of the negative branch for all the samples in layers. For dark particles, the diﬀerence of Pmax between lifted particles and layers is smaller than for transparent particles and Pmax may be higher for deposited particles than for lifted ones. 2.3.2.2 Mixtures of materials Fluﬀy (porosity larger than 90%) particles made of mixtures of silica and carbon aggregates with submicrometre-sized grains have been studied. Figure 2.10(a) presents the phase curves for silica (40 ± 20) × 10−3 μm and carbon (14 ± 5) × 10−3 μm and (95 ± 20) × 10−3 μm in red and green wavelengths. Figure 2.10(b) presents the phase curves for mixtures of 50% silica and 50% carbon (in mass) in two size ranges of the carbon samples. Pmax increases when the average size of the grains in the mixture decreases (Hadamcik et al., 2006a). A negative branch is always found for the mixtures. If micrometre-sized transparent compact particles are mixed to ﬂuﬀy particles of the same material, Pmax decreases (Fig. 2.10(c)).

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Fig. 2.9. (a, b, c, d) Polarization phase curves for particles in levitation: silica and carbon in diﬀerent size ranges and structures. λ = 632.8 nm. (e, f, g, h) Same grains but in layers.

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Fig. 2.10. (Updated from Hadamcik et al., 2006a.) R = red (λ = 632.8 nm) and G = green (λ = 543.5 nm). (a) Phase curves for silica and carbon. (b) Mixtures of silica with carbon. (c) Mixture of ﬂuﬀy silica (0.04 μm) with 1.5 μm size compact spheres.

2.3.2.3 Micrometer-sized spheres Partly for numerical simulation purposes, spheres are used to try and better understand the internal interactions inside the agglomerates with increasing sizes (and in a future phase to make a comparison with irregular grains in regolith). In space as suggested by Greenberg and Hage (1990) grains may be covered by black organic compounds and organized in ‘bird-nest’ structures. Transparent and black-coated spheres are studied in levitation as individual grains and/or small agglomerates of grains (size smaller than 5 μm), as large agglomerates in the [50–100] μm size range and as deposited very large agglomerates (diameter 2 cm, thickness 3 mm). The transparent spheres are made of compact silica; the coated spheres have a silica R core and a thin organic black mantle ( Sicastar-black). Figure 2.11 presents the polarization phase curves. For small agglomerates made of a few spheres only, the oscillations of the Mie scattering are present (Fig. 2.11(a)). The phase curve for the large agglomerates of transparent spheres presents a possible small negative branch at phase angles smaller than 20◦ , a polarization spike around (25–30)◦ , a bell-shaped branch up to 130◦ and an increase of polarization (Fig. 2.11(b)). The internal interactions between the transparent grains seem to decrease the negative branch at small phase angles, and smooth the phase curve between 40◦ and 130◦ . For deposited very large agglomerates (Fig. 2.11(c)), the interactions between the transparent grains become important and the polarization value for phase angles between 30◦ and 120◦ is close to zero with small amplitudes oscillations (Hadamcik et al., 2006b). For agglomerates of coated spheres with a black mantle, the oscillations decrease as compared to transparent spheres (Fig. 2.11(d)) and disappear for large agglomerates; the phase curve is smooth with a shape similar to those obtained for irregular particles (Fig. 2.11(e)). Surprisingly though, for the very large deposited agglomerates of coated spheres, the phase curve present oscillations with maxima and minima corresponding to those obtained for the small agglomerates or single spheres (Fig. 2.11(f)). The amplitude of the oscillations decreases as the phase an-

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Fig. 2.11. Polarization phase curves for μm-sized spheres in agglomerates. λ = 632.8 nm. On the left-hand side, transparent silica spheres (diameter 1.45 μm). On the right-hand side, black-coated spheres (silica core 1.45 μm, black coating 0.04 μm). Top curves: Individual spheres and small agglomerates (less than 5 μm) in levitation. Middle curves: Large agglomerates (tens of μm) in levitation. Bottom curves: Huge agglomerates (cm), deposited.

gle increases, it may come from surface irregularities. The polarization is mainly dominated by single scattering for the coated very large deposited agglomerates (Hadamcik et al., 2007a). 2.3.3 Optical and physical properties 2.3.3.1 Maximum polarization as a function of sizes Size of the particles The maximum polarization is studied as a function of the size of the particles as measured on the polarization maps for compact particles and agglomerates, and of the constituent grains for ﬂuﬀy aggregates (Hadamcik et al., 2006b).

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For ﬂuﬀy aggregates (more than 90% porosity), Pmax generally decreases when the size of the agglomerates increases except for constituent grains smaller than about 0.015 μm for which it does not change with the size of the particles (Fig. 2.12(a)). For relatively compact aggregates (less than 50% porosity) such as the silicates (S) in Fig. 2.12(a), the variation of Pmax with size follows the same trend than for compact particles. For compact particles, which are large compared to the wavelength, Pmax increases up to a value for which the polarization remains stable (all the refracted light being absorbed); Pmax also increases with the absorption (Fig. 2.12(b)). In Fig. 2.12(c), the same trend is observed for deposited particles and for levitating compact particles (Hadamcik et al., 2003a; Levasseur-Regourd et al., 2006).

Fig. 2.12. (Updated from Hadamcik et al., 2006b.) (a) Pmax vs particles size for white samples (ﬂuﬀy and compact). (b) Pmax vs particles size for compact particles with increasing absorption from quartz to CB4 (c) The same as (b) for deposited compact particles (updated from Hadamcik et al., 1996).

Size of the constituent grains in aggregates When the grain size is smaller than about 0.03 μm, the linear decrease of Pmax with the increasing size of the grains is similar for the transparent and the absorbing materials as can be seen in Fig. 2.13 (to allow a comparison with other data, the size parameter X = πaλ−1 is used on the horizontal axis, ‘a’ being the average diameter of the grains and ‘λ’ the wavelength). For larger grains of absorbing materials, the slope decreases and Pmax seems to have a minimum value of about 40%. For transparent materials, the lack of data for grains larger than 0.04 μm in ﬂuﬀy aggregates prevents providing such a limit. Nevertheless, for small aggregates of irregular silica particles with a size of the grains in the range [0.1–6] μm, Pmax is about 20% (Hadamcik et al., 2007a) and can be a limit for grains smaller than the wavelength in silica ﬂuﬀy aggregates. The variations for red and green light are similar; the wavelength eﬀect is only a relative size eﬀect (Fig. 2.13).

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Fig. 2.13. (Updated from Hadamcik et al., 2006a.) Pmax vs grain size for ﬂuﬀy aggregates (SiO2 and C).

2.3.3.2 Maximum polarization as a function of albedo The albedo is the geometric albedo measured at 8◦ phase angle (incident angle 4◦ ) on layers of diﬀerent materials relative to the same compact white sample (MgO) in the same geometric conditions. The albedo depends on the refractive index (mainly on the absorption); it also depends on the grain size and on the porosity of the deposited samples. The measurements are made with sifted samples without any compression. For layers of particles, log(Pmax ) is linearly dependent on log(albedo), with diﬀerent slopes for ﬂuﬀy particles and for compact ones (Hadamcik et al., 2002a; Levasseur-Regourd et al., 2006). To change the albedo, it is convenient to make mixtures of transparent particles with absorbing ones. Nevertheless as shown in Fig. 2.14(a) for levitating ﬂuﬀy particles, Pmax increases when the average submicron size of the grains in the mixture decreases. Nevertheless as expected for single materials (SiO2 or C), log(Pmax ) increases linearly when log(albedo) decreases but this result can also be interpreted as a size eﬀect. For these ﬂuﬀy aggregates made of submicrometre-sized grains, when the size increases, the albedo increases (Fig. 2.14(b) and Hadamcik et al., 2006).

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Fig. 2.14. (Updated from Hadamcik et al., 2006a.) Pmax vs geometric albedo (in log-log scale) for diﬀerent mixtures of SiO2 and C. Linear ﬁts for each set of data points. (1) SiO2 (40 nm) + C (95 nm), (2) SiO2 (40 nm) + C (14 nm), (3) SiO2 (12 nm) + C (14 nm).

2.3.3.3 Maximum polarization as a function of number density of the grains The number density of the grains has been estimated for two samples: one made of large irregular compact silicon carbide grains (88 μm) and the other one made of ﬂuﬀy alumina agglomerates of submicrometre-sized grains (0.013 μm). An average volume for an equivalent sphere diameter for the grains of each sample is calculated. The mass of the sample is measured, the density of the material known and the number of grains calculated. The grains are supposed to be homogeneously distributed in the vial volume. Similar estimation has been made for the ﬂuﬀy sample, the non-homogeneous distribution of the grains being ignored in a ﬁrst approximation. The measurements have been made with the ﬁrst photodiodes system and the number density is supposed to be proportional to the mass in the vial. With the imaging system, we have veriﬁed that this hypothesis generally gives a correct approximation. For the particles deposited in a cylindrical cup, the volume of the cup is measured. The cup is either ﬁlled with sifted grains or the grains are packed in the cup with a glass plate to have a sample with grains as closely packed as possible. The mass is measured in each case and the number density of the grains is calculated. The variation of Pmax as a function of the number density (ND) of the grains decreases when the ND increases for lifted grains and increases for deposited grains (Fig. 2.15). The increase of Pmax with increasing ND for deposited samples was observed for all samples with sifted or packed grains, when the size of the grains is smaller than about 20 μm (it is easier to pack small grains than large ones). To interpret the decrease of Pmax with the increase of ND for lifted grains, multiple scattering between the grains can be suggested. To interpret the increase of Pmax between sifted and packed grains other interpretations have to be found. The ori-

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Fig. 2.15. (Updated from Worms et al., 1999b.) (a) Pmax as a function of number density of the grains for 88 μm SiC. (b) Pmax as a function of number density of the grains for 0.013 μm Al2 O3 .

entation of packed grains may be suggested but the increase is also observed for iso-shaped grains (e.g. spheres in the case of ﬂuﬀy silica or carbon black). 2.3.4 Numerical models The ﬁrst step in such a work on real particles in a 3D volume is to make a model to represent the samples by equations: particles shape, average size and size distribution using mainly ﬂat images from microscopes. The second step is to build a light scattering code corresponding to the experiment and investigate the inﬂuence of shape, surface irregularities, size distribution and the optical properties to interpret the observations. 2.3.4.1 Compact particles, large as compared to the wavelength Penttil¨ a et al. (2003) ﬁrst described the shapes of the boron carbide particles by stochastic polyhedra. They used optical microscope images of 125 particles and chose a convex polyhedral model (Figure 2.16(a)). The same model is used for the three size distributions (9 μm, 13 μm, 88 μm). The particles are large as compared to the red wavelength (632.8 nm) and a ray-tracing code (Macke et al., 1996) is applied. The complex refractive index of the material is not known and diﬀerent combinations are tried. A Fourier ﬁt is made for 20 realizations of the particles model, 1000 rays of the 100 orientations per particle. The goodness-of-ﬁt between measurements and model data is chosen to be the root mean square (rms) between measured data points and the continuous ray-tracing model curve. The best-joined ﬁts, where the rms values of ﬁts are simultaneously minimized for the three sizes, are achieved with a refractive index of (2 + 0.04i). The eﬀect of concavities is also studied but it does not produce any signiﬁcant change in the best refractive index obtained there from. Figure 2.16(b) compares the experimental data to the best ﬁts. Mikrenska et al., (2006) used a Monte Carlo model for direct simulation of polarization at single scattering by rounded NaCl crystals (cubes). The method

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Fig. 2.16. (Update from Penttil¨ a et al., 2003a.) (a) Left side: boron carbide grain with six corners and its maximum diameter d. Right side: model particle viewed from three orthogonal directions. (b) Comparison of the best ﬁts in polarization to the experimental data for the three size ranges of B4 C particles. λ = 632.8 nm.

was ﬁrst described by Stefanov et al., (2002) and applied to glass spheres. Particle orientation is assumed to be random. Modelling is done in geometric optics approximation. First, by numerical approach, the inﬂuence of the diﬀerent parameters is studied: macro and micro shapes, crystals size distribution, refractive indices, single scattering by an ensemble of randomly oriented crystals. Analysis of the SEM images yields deﬁning ratios of shape irregularities and a mean size of the crystals. In Figure 2.17(a), a real NaCl particle and the rounded cube model are presented. The real part of the refractive index is known (1.54) but the imaginary part is a free parameter. In the studied sample 35% cubic, 10% slightly rounded cubic (mean roundness 0.72), 10% highly rounded cubes (roundness degree 0.94) and 45% round spherical shapes are estimated to be present. The degree of roughness is respectively 25% and 40% for the cubic and the spherical part of the surfaces. The output is the imaginary part of the refractive index, which was found to be 4× 10−5 by the best ﬁt. Figure 2.17(b) compares the experimental data to the best ﬁts in polarization and intensity. Considering the wide variety of the grain irregularities (e.g. presence of a few multi-cubes), the comparison between calculated and measured data is quite satisfactory. The Direct Monte Carlo Simulation model is suitable for non-homogeneous particles and/or with complex shapes, and for relatively dense media without or with multiple scattering between the particles (Hadamcik et al., 2003).

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Fig. 2.17. (Updated from Mikrenska et al., 2006.) (a) Left side: real NaCl crystal; right side: model rounded cube. (b) Comparison of the best ﬁts to the experimental data for NaCl crystals. Left side: polarization; right side: intensity. λ = 632.8 nm.

Lasue et al., (2007a) studied the light scattered by transparent spheres covered by diﬀerent coatings. All the diameters are large compared to the wavelengths (632.8 nm and 543.5 nm for some of the samples). The code is adapted from a stratiﬁed sphere model (Toon and Ackerman 1981) to take into account the size distribution as well as the speciﬁc parameters of the experiment. In this work we present the results for Sicastar-black-coated spheres (silica core and organic black mantle). The refractive index of the core is known but not that of the black coating. The diameter of the spheres is about 20 mm and the size distribution of the coated spheres was measured using SEM images. The coating thickness is in the [0.01–0.05] μm range as indicated by the provider. The real and imaginary parts of the refractive index and the thickness of the mantle are free parameters. In order to estimate the conﬁdence level of the ﬁt from the numerical model and the data points, the root mean square (rms) is calculated for the polarization phase curve. The best ﬁt of the measurements is obtained for a thickness of 0.05 μm and a refractive index (2.4+i0.4) for the coating; it is presented with the data in Fig. 2.18. The discrepancy between the data and the ﬁt that is observed at the maximum of

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Fig. 2.18. (Updated from Lasue et al., 2007a.) Comparison between the data points and the best ﬁts for 20 μm diameter silica spheres covered by an organic black mantle: m = 2.4 + i0.4, thickness 0.05 μm. λ = 632.8 nm.

the polarization curve could correspond to some small agglomerates of grains and to some shape irregularities on the surface or among the grains and/or to some uncertainties in the size of the core. 2.3.4.2 Micrometre-sized spheres, single or in small agglomerates, bare or coated Hadamcik et al. (2007a) have compared the phase curves for particles made of micrometre-sized (diameter about 1.5 μm) constituent grains. The particles size is smaller than 5 μm. Single grains and small agglomerates are present in the ﬁeld of view in a statistically similar ratio at each phase angle. The constituent grains are either compact bare silica spheres or the same spheres coated by an organic R mantle (Sicastar-black ). The experimental results are compared to numerical simulations. The size distribution of the spheres was measured using SEM images. The codes used for the numerical model are adapted from Toon and Ackerman (1981) and Lasue and Levasseur-Regourd (2006). The refractive index for the black mantle is taken from Lasue et al. (2007a). The inﬂuences of the size distribution of the grains, of the thickness of the mantle for the coated spheres, and of the number of grains in the agglomerates are considered. Figure 2.19(a) compares the experimental polarization data to the Mie curves obtained for single spheres made of bare silica and of supposed plain spheres made of the mantle material for two diﬀerent sizes (1.45 μm and 1.54 μm). Figure 2.19(b) compares the experimental polarization data for bare silica spheres and for the coated spheres to the best ﬁts for agglomerates of eight spheres with a size distribution (σ = 0.05) of the constituent grains and a 0.04 μm thickness mantle. Figure 2.19(c) compares the

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intensity experimental data to the ﬁts obtained for eight spheres agglomerates. The position of the maxima and minima of the polarization phase curves mainly depends on the size of the grains and not on the refractive index of the external material, which only produces a small shift of the phase angle values (Fig. 2.19(a)). The presence of agglomerates produces a decrease of the amplitude of the oscillations

Fig. 2.19. (Updated from Hadamcik et al., 2007a.) Bare silica (left column) and coremantle (right column) 1.5 μm diameter spheres. λ = 632.8 nm. (a) Experimental polarization phase curves for bare silica spheres and coated silica spheres compared to Mie curves of plain spheres (pure silica and pure Sicastar- black). (b) Experimental polarization phase curves for bare silica spheres and coated silica spheres compared to the best ﬁts obtained for aggregates of eight spheres with a size distribution of the spheres. (c) Experimental intensity phase curves obtained for bare silica spheres and coated spheres compared to the best ﬁts obtained for agglomerates of eight spheres with a size distribution of the spheres.

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and of the negative branch at small phase angles. The oscillations for coated spheres are reduced as compared to bare spheres. In future numerical work, mixtures of diﬀerent size of agglomerates and larger agglomerates will be tested to obtain the best ﬁts between the model and the data points.

2.4 Applications The results of the light scattering experiments are used to interpret remote observations of dust either in the solar system or in the Earth’s atmosphere. 2.4.1 Solar system dust 2.4.1.1 Analogues The relevant samples for astronomical applications are powdered meteorites and analogues (Moon, Mars, regoliths, Titan’s aerosols, cometary comae). Deposited in layers, the lunar and Martian soils analogues (JSC1) polarization phase curves are compatible with phase curves obtained by remote observations (Worms et al., 2000). Very large agglomerates are considered as protoplanetary analogues or as regolith analogues; they are produced by random ballistic deposition (Blum and Schraepler, 2004; Blum et al., 2006), their volume-ﬁlling factor is in the [0.1–0.2] range depending on the sample. All of them are compared to the same levitating grains in agglomerates or not (Worms et al., 2000; Hadamcik et al., 2007a); the main purpose is to underline the optical interactions between grains and ﬁnally compare with asteroids. Some clouds of large agglomerates of submicrometre-sized constituent grains may also be considered as cometary dust analogues (Hadamcik et al., 2006a, 2007b) and Titan’s aerosols analogues (Hadamcik et al., 2009). 2.4.1.2 Interpretation of cometary observations The variations of the polarization values as a function of the diﬀerent physical properties such as the sizes, the porosity or the albedo for the diﬀerent kinds of particles (compact or ﬂuﬀy, transparent or absorbing) are used to interpret some remote sensing observations. For example, the dust in comet C/1995 O1 Hale-Bopp and more generally in active comets and their jets shows a higher polarization than in other regions or comets (Hadamcik et al., 2003a,b). Conﬁrmed by other diagnosis such as colour in intensity or infrared emission features by silicates, the particles seem to be made of mixtures of ﬂuﬀy aggregates with submicrometre-sized constituent grains of diﬀerent materials. Such particles were also found in the dust ejected by Deep Impact from comet 9P/Tempel 1 nucleus subsurface (Harker et al., 2005, Hadamcik et al., 2007c). When particles break up in a coma, the polarization evolution may give indications on the structure of the particles. Large, relatively compact fragmenting particles are suggested by the important decrease of polarization with increasing aperture, correlated to important intensity gradient around a nucleus or its fragment. Such trends were observed around comet C/2000 WM1 LINEAR nuclei and

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around the main debris in comet C/1999 S4 LINEAR (Hadamcik and LevasseurRegourd, 2003a,c). For suﬃciently large apertures including all the main cometary structures, the polarization phase curves (Fig. 2.2) and mainly their positive branches allow deﬁning of three classes of comets (high Pmax comets, low Pmax comets); the third class consist only of comet Hale-Bopp, the polarization of which was the highest ever measured for comets (Hadamcik and Levasseur-Regourd, 2003b). When a comet presents important jet activity, the polarization increases due to a size distribution of the grains shifted towards smaller grains with the presence of Mg-silicates crystals (Hanner, 2002). Cometary analogues with Mg-silicates, Fe-silicates provided from the Goddard Space Center (Nuth et al., 2002) and carbonaceous compounds in ﬂuﬀy agglomerates with average sizes in the [50–100] μm range have been studied with the PROGRA2 -vis instrument. The average size of the constituent grains is 0.05 μm. The polarization phase curves have the same characteristics as for remote observations. To have a maximum polarization in the 30% range, it is necessary to add some compact particles to the mixture (Fig. 2.20), as found in particles captured from 81P/Wild 2 coma by Stardust (Burchell et al., 2008). To have a positive spectral gradient, the presence of carbonaceous compounds is necessary (Mg-silicates spectral gradient is very small, if at all existing, and the spectral gradient of any iron component, e.g. Fe-silicates, is negative). Grains sintered in chains in the aggregates seem to play a role on the presence of the negative branch just as does the random mixture of transparent and absorbing materials (Hadamcik et al., 2006a, 2007b). Numerical simulations with mixtures of aggregates and compact particles give comparable results (Lasue and Levasseur-Regourd, 2006; Levasseur-Regourd et al., 2007; Lasue et al., 2007a). Lasue et al., (2009) compare the phase curves in diﬀerent wavelengths for comets C/1995 O1 Hale-Bopp and 1P/Halley. They ﬁnd an average composition of 50% Mg-silicates and 50% carbonaceous compounds and an average size distribution of [0.25–40] μm range. More precisely, as expected, the average size is found to be smaller for comet Hale-Bopp than for comet Halley, and the ratio of silicates over carbonaceous compounds is found to be higher. In the computations, the numerical code limits the number of grains in the aggregates to 512 grains. The size distribution is deﬁned by the equivalent volume (i.e. the material is supposed to be melted in a sphere), the largest equivalent particles sizes are for compact particles, the smallest ones are for ﬂuﬀy particles and the size of aggregates is dependent on the size of their constituent grains. But very large aggregates [50–500] μm with grains in the [0.05–0.1] μm range cannot be ruled out if the experimental results are considered (with a porosity of [90–95] % for the aggregates, the equivalent size of these aggregates is in the [10–100] μm range. Some tracks made by ﬂuﬀy particles in the aerogel of the Stardust mission, as well as craters on the aluminium foils (H¨ orz et al., 2006) seem to be made by large ﬂuﬀy aggregates. It agrees well with the size range of [10–500] μm and a density of 100 kg m−3 (porosity about 95%), deduced from Giotto in situ observations of 1P/Halley (Levasseur-Regourd et al., 1999; Fulle et al., 2000). Combining numerical and experimental simulations creates synergy to their mutual beneﬁt. The numerical simulations can give quantitative results but the

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Fig. 2.20. (Updated from Hadamcik et al., 2007b.) Phase curves for cometary analogues (mixtures of ﬂuﬀy Mg-silicates, Fe-silicates and dark carbonaceous compounds and same mixture with compact Mg-silicate aggregates). Red (λ = 632.8 nm) and green (λ = 543.5 nm).

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modelling limits have to be given; the results obtained by experimental simulations are usually more qualitative due to the diﬃculty of separating the diﬀerent parameters in real particles. Using real particles is essential to make a complete analysis and, if complemented by numerical simulations, this makes possible discrimination of the involved physical properties parameters. 2.4.1.3 Interpretation of interplanetary dust results As interplanetary dust is concerned, some evolution of the particles may be forecasted as they approach the Sun (Levasseur-Regourd et al., 1990). The temperature of the particles increases and the abundance of the carbonaceous compounds seems to decrease increasing the albedo (Dumont and Levasseur-Regourd, 1988). When less dark carbonaceous compounds are present among the silicates, the polarization spectral gradient becomes neutral or slightly negative (Fig. 2.20). Lasue et al., (2007) have found a similar result by numerical simulations of the scattered light and of thermal evolution. 2.4.1.4 Interpretation of Titan’s aerosols results Observations of the scattered light and its linear polarization by Titan’s aerosols were conducted onboard diﬀerent spacecraft such as Pioneer and Voyager (Tomasko and Smith, 1982; West et al., 1983) for the integrated measurements (Pmax = 50% in red and 54% in green, close to 90◦ phase angle). The polarization values measured in 2005 by DISR/Huygens during its descent in the atmosphere are smaller when the altitude decreases. With a numerical model of fractal aggregates and less than 256 constituent grains, Tomasko et al., (2005) deduce a size for the grains in the [0.05–0.1] μm range and an aggregate size smaller than 1 μm; nevertheless, they do not exclude the possibility of larger aggregates. Analogues of Titan’s aerosols are produced in the gas phase by a radio-frequency plasma technique in methane–nitrogen mixtures by the PAMPRE experiment (Szopa et al., 2006). Depending on the experimental conditions the particles produced there from have diﬀerent physical properties. One important diﬀerence between these particles is the average size of the grains in the range [0.2–1.4] μm. The details of the samples and production process may be found in Hadamcik et al. (2009). The produced samples are studied with the PROGRA2 -vis experiment by the air-draught technique. The lifted particles are ﬂuﬀy agglomerates with an average size of [50–100] μm. In Fig. 2.21, the PROGRA2 -vis measurements, close to 90◦ phase angle, are compared to the results of in situ measurements with grain sizes from numerical simulations (Tomasko et al., 2005). With an extrapolation of our results to 0.075 μm grains in diameter, the observations and laboratory measurements yield comparable results, but in the experiment the size of the particles reaches hundreds of micrometres. The spectral gradient from outside the atmosphere (West, 1983) in polarization is negative for the analogues; it is similar to the observed one and can be due to a higher absorption in green than in red (evaluated by geometric albedo measurements for the analogues and observed for Titan’s aerosols).

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Fig. 2.21. Comparison between observations and experimental simulations with PAMPRE experiment analogues. R = red (λ = 632.8 nm) and G = green (λ = 543.5 nm).

2.4.2 Atmospheric dust Even if the most important components of the aerosols in the Earth’s atmosphere are droplets, solid particles are also present in the upper troposphere and in the stratosphere. They are probably residues of combustions such as soot of various origins (natural ﬁres, industrial activities and in a lesser proportion from aircraft). Their concentration is of a few particles per cm−3 (Renard et al., 2008) even in the middle stratosphere (at altitudes lower than 30 km). Some volcanic ashes can also be present, for example after a strong eruption event such as the Pinatubo in June 1991 (e.g. Russell et al., 1996; Deshler et al., 1997). Interplanetary micrometresized dust and meteorites are also found but with a concentration of about 10−4 particles per cm−3 (Hunten et al., 1980; Renard et al., 2005a). Some sand and other particles lifted by wind are also found in the (lower) troposphere. The composition, size distribution and morphology of some of these solid particles are not yet well known, in particular in the stratosphere. Their small concentration and localization need remote observations of their optical properties from satellites or balloon-borne instruments. Among them, the balloon-borne MicroRadibal radiometer instrument allows us to retrieve the brightness and polarization functions of light scattered by the aerosols (Brogniez et al., 2003; Renard et al., 2008). Just as for astronomical observations, it is necessary to make comparisons with well-documented laboratory measurements and numerical simulations to interpret the observations. Two main materials are systematically studied: soot and sands (sands measurements can be used for future analysis of remote sensing observations in the lower atmosphere). The results for interplanetary particles (powdered meteorites, cometary analogues and asteroidal analogues) can also be taken into account for the analysis of observations of solid particles in the upper atmosphere where such particles could be the main population.

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This work is still in progress, since ﬂights with stratospheric balloons of the improved version of MicroRadibal will probably start in 2009. Some preliminary results for solid particles thought to be found in the atmosphere are presented below. 2.4.2.1 Soot in the atmosphere Soot presents typical aggregates of submicrometre-sized grains and important absorptions in the whole visible spectrum. The individual grains are usually roughly spherical with sizes in the [0.01–0.1] μm range. Obviously due to their ﬂuﬀy morphology, it is impossible to model the light scattered by the particles with Mie curves and to measure the physical parameters such as size, size distribution, and refractive index (Renard et al., 2001, 2005b). The polarization phase curves for almost all carbonaceous soot, independently of their origin, are similar with Pmax in the (75–85)% range and a slight positive spectral gradient (Fig. 2.22). More generally, this kind of spectral variation in polarization was observed for ﬂuﬀy aggregates of submicrometre-sized grains made of grey materials (Hadamcik et al., 2002a). Extinction measurements in the visible domain [400–700] nm seem to indicate a slightly positive gradient for the absorption (Renard et al., 2001), which may also contribute to the polarization increase with wavelength. A small negative branch at phase angles smaller than 20◦ is observed for all the samples. The maximum in polarization increases when the size of the agglomerates increases. This trend may be the result of a smaller number density of the largest agglomerates.

Fig. 2.22. (Updated from Renard et al., 2005b.) Synthetic phase curves for soot. Red (λ = 632.8 nm) and Green (λ = 543.5 nm).

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Phase curves measurements will be systematically extended to other soots of diﬀerent origins together with spectroscopic analysis to characterize the soot by its chemical composition and to have precise measurements of the absorption. Measurements on soot made of carbon nanotubes fall within the scope of our studies as well as soot containing a large amount of non-carbonaceous compounds. Work is in progress to numerically model the morphological and physical properties of soot (Moulin, 2007). Such a novel database will be used for the analysis of remotesensing measurements and a best characterization of the various natures and thus the origin of soot that will be observed with the MicroRadibal radiometer. 2.4.2.2 Sand particles Sand particles of three origins, in a [10–400] μm size-range, have been studied by three levitation techniques, microgravity, air-draught and air-ﬂow (Daugeron et al., 2006). With the air-draught technique, important sedimentation was observed and diﬀerent porosities found in the sand. With the Laboratoire de M´et´eorologie Physique (LaMP) nephelometer, a powder generator with an airﬂow speed between 2 m s−1 and 20 m s−1 ejects the particles downward vertically; a considerable speed is needed to have a continuous ﬂow of hundreds of micrometres particles. For elongated particles with velocities higher than about 5 m s−1 , the polarization phase curves are completely diﬀerent than those obtained in microgravity conditions. The unique and valuable technique to study such large and compact irregular particles is by using microgravity. Nevertheless, in the case of remote observations, the differences in polarization phase curves could provide fundamental indications on the particles shape and wind speeds and/or sedimentation that can orient the particles during their transport. In microgravity conditions, the phase curves follow the same trend for large particles as compared to the wavelength with a bell-shaped curve for phase angles larger than 20◦ . As for other irregular particles, the maximum polarization depends on the average grain size and on the absorption. An extensive database is progressively built with sands of diﬀerent compositions, from diﬀerent origins (e.g. coastal and desert sands); the absorption in the two wavelengths is appreciated using the geometric albedo. The diﬀerent sands will also be studied when deposited in layers to compare to observations made from satellites or balloons and from a theoretical point of view to analyse the eﬀect of multiple scattering at the diﬀerent phase angles. The size studies have to be extended to smaller grains and the diﬀerent parameters involved have to be extensively studied.

2.5 Conclusions and future developments The PROGRA2 experiment results are of interest for the interpretation of astronomical and atmospheric light scattering observations in terms of physical properties of the particles and more generally in terms of their formation. The PROGRA2 -vis measurements on levitating particles have already made possible the interpretation of some of the physical properties of cometary and of

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interplanetary dust particles, as well as the diﬀerences existing between diﬀerent comets and diﬀerent cometary regions. The experimental work is indeed a complement to numerical simulations, which are limited by computer time and by the size of the aggregates in the codes to simulate the particles. The experiment enlarges the size range of studied particles toward large agglomerates and it permits the use of particles of any shapes. Comparison between numerical simulations and our experimental results has validated the experiment protocol and the numerical models making it possible, for instance, to measure the complex refractive index and to better understand the inﬂuence of the diﬀerent irregularities on light scattering of the shape of a complex particle. Measurements with the MicroRadibal instrument compared to brightness measurements on soot (PROGRA2 and the LaMP nephelometer) have proved the presence of soot up to 30 km altitude in the stratosphere. Their detection and characterization is important for climate evolution. A wide spectral structure in extinction produced by aerosols was detected by balloon-borne and satellite instruments; it was interpreted as originating from the interplanetary medium. Polarization measurements associated with other techniques are used to interpret them in order to best document the various origins and nature of the grains. For such purpose, an extended and well-documented database is mandatory and is still in progress. The PROGRA2 -surf instrument was updated in 2008, with cameras replacing the photodiodes, in order to avoid the contribution of surroundings in the measured ﬂux for large phase angles. Up to now, the wavelength dependence was limited to the visible domain with green and red light. A new instrument for levitating particles was built in 2008, for the near-infrared domain at about 1500 nm (PROGRA2 -IR for infrared) and another instrument for deposited particles at the same wavelength is expected to start operating in 2010–2011 (PROGRA2 -SIR, for surface infrared). This wavelength range is particularly important for cosmic dust studies and to interpret the MicroRadibal balloon-borne results. A new set-up (PROGRA2 -aero) is under development for micrometre-sized grains, including an injection system with a turbulent airﬂow, which can de-agglomerate the grains. Photodiodes will be used, the resolution of the cameras being too small for this size range. The samples to be studied by the ICAPS programme (Interactions in Cosmic and Atmospheric particles System) and its precursor experiment IPE onboard the international Space Station (see e.g. Levasseur-Regourd, 2003; Lasue and Levasseur-Regourd, 2007), are currently tested with PROGRA2 -vis. To prevent damage on the Rosetta spacecraft in the innermost coma of comet 67P/Churyumov-Gerasimenko in 2014 (when the lander will be released), it is necessary to have the most precise knowledge on the physical properties of the dust particles (nuclei surface and cometary particles). Experimental (e.g. with PROGRA2 ) and numerical simulations on good candidates for realistic particles are necessary to interpret the preliminary remote observations. The database is available, through a web-connection in 2008 (www.icare.univlille1.fr/PROGRA2), for researchers who want to make comparisons of models, or to use it to interpret remote observations. The physical characteristics of the samples are progressively completed and new results are being added.

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Acknowledgements This research was supported by the French Space Agency (CNES), the French National Program of Atmospheric Chemistry (PNCA), the Programme National de Plan´etologie (PNP) and the ICARE database program. The microgravity ﬂights are operated by Novespace and funded by CNES and ESA. The authors thank S. Borensztajn (UPMC Universit´e Paris 06, UPR 15) for the SEM images. We thank all the members of the technical team (B. Cout´e, B. Gaubicher, V. Guerrini, Th. Lemaire, C. Robert) who help us. We thank all the collaborators who participate on the microgravity ﬂights and specially D. Daugeron who is developing the PROGRA2 -aero instrument. We thank all those who have given samples and cosmic dust analogues. We do not forget the numerous students helping us to develop the instruments, making experiments or organizing the database.

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Russell P.B., J.M. Livingston, R.F. Pueschel, J.J. Bauman, J.B. Pollack, S.L. Brooks, P. Hamill, L.W. Thomason, L.L. Stowe, T. Deshler, E.G. Duttton. R.W. Bergstrom, 1996: Global to microscale evolution of the Pinatubo volcanic aerosol derived from diverse measurements and analyses J. Geophys. Res. 101, 18745–18763. Santer R., M. Herman, D. Tanre, J. Lenoble, 1988: Characterization of stratospheric aerosol from polarization measurements. J. Geophys. Res. 93, 14209–14221. Stefanov S., P. Koulev, J.C. Worms, J.-B. Renard, E. Hadamcik, 2002: Numerical simulation of the scattering of a laser beam introduced in an ensemble of suspended particles. Proceedings 9th Congr. Theor. Appl. Mech. 2, 219–224. Sullivan R.J., P.C. Thomas, S.L. Murchie, M.S. Robinson, 2002: Asteroid geology from Galileo and NEAR Shoemaker data, In: Asteroids III, Bottke, W.F., Cellino A, Paolicchi P., Binzel R. (Eds), University of Arizona Press, Arizona, pp. 331–350. Szopa C., G. Cernogora, L. Boufendi, J.J. Correia, P. Coll, 2006: PAMPRE: A dusty plasma experiment for Titan’s tholins production and study. Planet. Space Sci. 54, 394–404. Tomasko M.G. and P.H. Smith, 1982: Photometry and polarimetry of Titan: Pioneer 11 observations and their implications for aerosols properties. Icarus 51, 65–85. Tomasko M.G., B. Archinal, T. Becker, B. B´ezard, M. Bushroe, M. Combes, D. Cook, A. Coustenis, C. de Bergh, L.E. Dafoe, L. Doose, S. Dout´e, A. Eibl, S. Engel, F. Gliem, B. Grieger, K. Holso, E. Howington-Kraus, E. Karkoshka, H.U. Keller, R. Kirk, R. Kramm, M. K¨ uppers, P. Lanagan, E. Lellouch, M. Lemmon, J. Lunine, E. McFarlane, J. Moores, G.M. Prout, B. Rizk, M. Rosiek, P. Rueﬀer, S.E. Schr¨oder, B. Schmitt, C. See, P. Smith, L. Soderblom, N. Thomas, R. West, 2005: Rain, winds and haze during Huygens probe’s descent to Titan’s surface. Nature 438, 8, 765–778. Toon O.B. and T.P. Ackerman, 1981: Algorithms for calculation of scattering by stratiﬁed spheres, Appl. Opt. 20, 3657–3660. Volten H., O. Mu˜ noz, J.W. Hovenier, F.J.M. Rietmeijer, J.A. Nuth, B.F.M. Waters, W.J. van des Zande, 2007: Experimental light scattering by ﬂuﬀy aggregates of magnesiosilica, ferrosilica, and alumina cosmic dust analogs. Astron. Astrophys. 470, 377–286. West R.A., H. Hart, K.E. Simmons, C.W. Hord, L.W. Esposito, A.L. Lane, R.B. Pomphrey, D.L. Coﬀeen, M. Sato, 1983: Voyager 2 photopolarimeter observations of Titan. J. Geophys. Res. 88, 8699–8708. West R.A., L.R. Doose, A.M. Eibl, G.T. Martin, M. Mishchenko, 1997: Laboratory measurements of mineral dust scattering phase function and linear polarization. J. Geophys. Res. 102, D14, 16871–16881. Worms J.C., A.C. Levasseur-Regourd, E. Hadamcik, E. Bourras, J.-B. Renard, 1996: Scattering properties of dust particles in weightlessness. In: Physics, Chemistry and Dynamics of Interplanetary Dust. Gustafson B.A.S. and Hanner M.S. (Eds), ASP conf. series 104, pp. 415–418. Worms J.C., J.-B. Renard, A.C. Levasseur-Regourd, E. Hadamcik, 1999a: Light scattering by dust particles in microgravity: the PROGRA2 achievements and results. Adv. Space Res. 23, 1257–1266. Worms J.C., J.-B. Renard, E. Hadamcik, A.C. Levasseur-Regourd, J.F. Gayet, 1999b: Results of the PROGRA2 experiment: an experimental study in microgravity of scattered polarized light by dust particles with large size parameter. Icarus 142, 281–297. Worms J.C., J.-B. Renard, E. Hadamcik, N. Brun-Huret, A.C. Levasseur-Regourd, 2000: Light scattering by dust particles with the PROGRA2 instrument: comparative measurements between clouds under microgravity and layers on the ground. Planet. Space Sci. 48, 493–505. Zolensky M.E. and 74 colleagues, 2006: mineralogy and petrology of Comet 81P/Wild 2 nucleus samples. Science 314, 1735–1739.

3 Jones and Mueller matrices: structure, symmetry relations and information content S.N. Savenkov

3.1 Introduction When light propagates through a linear medium, its polarization properties can be described either by the Stokes vector or, under deﬁnite conditions, by Jones vector formalisms. The eﬀect of medium on the light is then to transform the Stokes or Jones vector, so that the medium can be represented by a transformation matrix. This transformation is usually known as the Mueller matrix (Bohren and Huﬀman, 1983; Brosseau; 1998) when it acts on the four-dimensional Stokes vector or as the Jones matrix (Azzam and Bashara, 1987; Collett, 1993; Shurcliﬀ, 1962) when it acts on Jones vector. Information contained in the Mueller matrix has many useful applications in such diverse ﬁelds as interaction with various optical systems (Shurcliﬀ, 1962; Azzam and Bashara, 1977; Collett, 1993; Brosseau, 1998), cloud diagnostics (van de Hulst, 1957; Bohren and Huﬀman, 1983; Mishchenko et al., 2000, 2002; Kokhanovsky, 2003b), remote sensing in the ocean, atmosphere and terrestrial (Boerner, 1992; Kokhanovsky, 2001, 2003a, 2003b; Muttiah, 2002), and tissue optics (Priezzhev et al., 1989; Tuchin, 2002, 2004). Taking into account that the Mueller matrix allows one to determine everything optically about the medium at the incident frequency and, in that sense, is a matrix model of the medium, the main aim of polarimetry lies in the physical interpretation of the information that provides the 16 elements of the Mueller matrix. The information that can be derivable from the Mueller matrix is determined by model of the medium considered. If one considers the medium as a single scatterer or system of scatterers (discrete medium), the Mueller matrix contains information related to the optical properties, size, shape and composition of scatterers (van de Hulst, 1957; Bohren and Huﬀman, 1983). If one considers the model (approximation) of continuous medium, as, say, it is in crystal optics, than the Mueller matrix contains information related to the time and spatial dispersion, which determine, in the general case, the character of anisotropy of medium, namely, linear amplitude and phase, and circular amplitude and phase anisotropies (Landau et al., 1984; Berry and Dennis, 2003). In both the above cases under certain conditions the Mueller matrix can contain information related to depolarization (Chipman, 1995; Mishchenko and Travis, 2000; Gil, 2007). The methods of extraction of this information from the Mueller matrices have been developed by many authors (Hurwitz

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and Jones, 1941; Whitney, 1971; Cloude, 1986; Gil and Bernabeu, 1987; Lu and Chipman, 1994, 1996; Mar’enko and Savenkov, 1994; Savenkov and Yushtin, 2005; Savenkov et al., 2006, 2007b). In this chapter we consider the structure, symmetry relations and information content of the Jones and Stokes vectors transformation matrices for both approaches to medium modeling: system of scatterers and continuous medium approximation. Bibliography presented in this chapter does not pretend to be exhaustive and is merely aimed to direct the reader in the case of his/her further interest.

3.2 Basic deﬁnitions Let us recollect some basic deﬁnitions. The Jones calculus (Jones, 1941; Hurwitz and Jones, 1941; Jones, 1942, 1947, 1948, 1956) is the adequate way to describe the coherent superposition of polarized light because it operates on amplitudes rather than on intensities. However, Jones vectors and matrices can only describe completely polarized light because a monochromatic wave is always completely polarized. The electric vector of a monochromatic light wave traveling along the z axis of a right-handed coordinate system can be decomposed into its x and y components Ex and Ey , which are in general complex quantities with an amplitude and a phase. The Jones vector contains the complex components of electric vector of light in the form Ex . (3.1) E= Ey Note that amplitudes are not observed directly by detectors in the optical wavelength range. Therefore, observables always depend on products of Jones vector components such as |E|2 . The transfer of completely polarized light through an optical medium is described by 2 × 2 matrix T: sct inc Ex Ex t11 t12 t 1 t4 = T where T = or , (3.2) Eysct Eyinc t21 t22 t 3 t2 inc sct represent the electric vector components of a incident light; Ex,y relate where Ex,y to a scattered light that is not necessarily traveling in the same direction as the incident light. The matrix T is called a Jones matrix; its elements are in general complex. In the context of light scattering by small particles, T is frequently called the amplitude matrix (van de Hulst, 1957), while the scattering plane then usually acts as the common reference plane for the incident and scattered beams. Combined Jones matrix describing a series of optical elements is equal to the matrix product of the individual Jones matrices. Examples of various Jones matrices are given in the following sections. Extensive lists have been presented by several authors (Shurcliﬀ, 1962; Kliger et al., 1990; or Gerrard and Burch, 1975). The rotation of Jones matrices is given by

T = RT (−γ)TRT (γ) ,

(3.3)

3 Jones and Mueller matrices

73

where the rotation matrix RT RT =

cos γ sin γ . − sin γ cos γ

(3.4)

Mueller matrices (classic contributions here are Soleillet, 1929; Perrin, 1942; Mueller, 1948; and Parke, 1948, 1949) describe the linear transformation between Stokes vectors (formed by grouping the four Stokes parameters into a single vector) associated with optical elements and media, i.e. Ssct = MSinc

(3.5)

where the Stokes vector (here superscripts ‘sct’ and ‘inc’ are the same as in Eq. (3.5)) consists of the following Stokes parameters ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 2 2 I s1

|Ex |2 + |Ey |2 ⎜Q⎟ ⎜s2 ⎟ ⎜ |Ex | − |Ey | ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ S=⎜ (3.6) ⎝U ⎠ = ⎝s3 ⎠ = ⎝ Ex∗ Ey + Ex Ey∗ ⎠ .

∗ ∗ V s4 i E x Ey + Ex E y The Stokes parameter I is proportional to the total energy ﬂux of the light beam. Stokes parameters Q and U represent the diﬀerences between two components of the ﬂux for which the electric vectors oscillate in orthogonal directions. The Stokes parameter V is the diﬀerence between two oppositely circularly polarized components of the ﬂux. As indicated by the angle brackets, , the Stokes parameter si are ensemble averages (or time averages in case of ergodic, stationary processes). Therefore no coherence eﬀects are considered. A normalized Mueller matrix is obtained by scaling the matrix such that the upper left element is equal to one. When a beam of light passes through N optical elements, each described by a Mueller matrix Mi , the combined Mueller matrix MC of the whole assembly is given by MC = MN MN −1 . . . M2 M1 .

(3.7)

Rotation of Mueller matrices is given by equation which is analogous to that for Jones matrix. In particular, it follows that: M = RM (−γ)TRM (γ) , ⎛

where RM

⎞ 1 0 0 0 ⎜0 cos 2γ sin 2γ 0⎟ ⎟ =⎜ ⎝0 − sin 2γ cos 2γ 0⎠ . 0 0 0 1

(3.8)

(3.9)

However, this rule for rotating Mueller matrices cannot be applied formally. In particular, one has to remember that this rule assumes the same coordinate system is kept for the incoming and outgoing beams. However, when scattering or reﬂections are considered, the convention is to change the coordinate system for the scattered beam as compared to the incident beam of light.

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Stokes vectors and Mueller matrices operate on intensities and their diﬀerences, i.e. incoherent superpositions of light; they are not adequate to describe neither interference nor diﬀraction eﬀects. However, they are suited to describe partially polarized and unpolarized light. The Stokes parameters obey the inequality s21 ≥ s22 + s23 + s24 .

(3.10)

This inequality is called the Stokes–Verdet criterion and is a consequence of Schwartz’s (or Couchy-Buniakovski’s) theorem. The degree of polarization p is deﬁned by s22 + s23 + s24 p= . (3.11) s1 In Eq. (3.10) the equality holds for completely (pure) polarized light. For this case p = 1. Another limiting case p = 0 has occurred when s22 + s23 + s24 = 0, i.e., when the electric vector of light vibrates in all directions with no preferential orientation. The intermediate case 0 < p < 1 implies that light contains both polarized and depolarized components and is called ‘partially polarized’. Inequality (3.10) plays important role in polarimetry because it allows the classiﬁcation of the character of light-medium interaction. We consider input light completely polarized. In this case, if for output light condition Eq. (3.10) is an equality, then the medium is non-depolarizing. It is important to note that the terms ‘nondepolarizing’ and ‘deterministic’ or ‘pure’ are not generally identical. The term ‘deterministic’ means that the Mueller matrix describing such a medium can be derivable from the correspondent Jones matrix (Simon, 1982; Gil, and Bernabeu, 1985; Anderson and Barakat, 1994; Gopala Rao et al., 1998b). This condition is more rigid than the condition of non-depolarizing (Savenkov and Yushtin, 2000). To emphasize the fact that the Mueller matrix of a homogeneous anisotropic medium can be derivable from the correspondent Jones matrix we will hereinafter call this class of matrices ‘Mueller–Jones’ matrices. If for output light we have an inequality in Eq. (3.10) then this medium is neither deterministic nor non-depolarizing. If, in addition, transformation matrix in Eq. (3.5) can be deﬁned as a parallel combination (convex sum) of Mueller–Jones matrices (Cloude, 1986; Gil, 2007), this is a depolarizing Mueller matrix (hereinafter a ‘Mueller’ matrix), otherwise a Stokes transformation matrix, i.e. the transformation matrix ensures the fulﬁllment of the Stokes–Verdet criterion only. The properties of matrices transforming Stokes vectors into Stokes vectors, i.e., satisfying the Stokes–Verdet criterion, have been studied by many authors (Xing, 1992; van der Mee, 1993; van der Mee and Hovenier, 1992; Sridhar and Simon, 1994; Nagirner, 1993; Givens and Kostinski, 1993; Gopala Rao et al., 1998a). Any Jones matrix can be transformed into the corresponding Mueller–Jones matrix, M, using the following relation (Parke, 1949; Azzam and Bashara, 1977): M = A T ⊗ T∗ )A−1 , (3.12) where ∗ indicates the complex conjugate,

3 Jones and Mueller matrices

⎛

1 ⎜1 A=⎜ ⎝0 0

0 0 1 i

0 0 1 −i

75

⎞

1 −1⎟ ⎟, 0⎠ 0

(3.13)

and ⊗ is the tensor (Kronecker) product, or in detailed form: M= ⎛ 1 ⎜ ⎝

(S11 + S22 + S33 + S44 ) 2 1 (−S11 + S22 + S33 − S44 ) 2 S42 + S31

1 (−S11 + S22 − S33 + S44 ) 2 1 (S11 + S22 − S33 − S44 ) 2

S42 − S31 −D42 − D31

−D42 + D31

⎞

S32 + S41 D32 − D41 S32 − S41 D32 + D41 ⎟ , S21 + S43 −D21 − D43 ⎠ D21 − D43 S21 − S43

(3.14) where the following auxiliary quantities Skj = Dkj =

1 ∗ 2 i(tk tj 1 ∗ 2 i(tk tj

+ tj t∗k ) , − tj t∗k ) ,

(3.15)

are used. From Eq. (3.14) it can be in particular seen that m211 − m221 − m231 − m241 = det(T) ;

(3.16)

here mij are elements of the Mueller–Jones matrix M. Since element m11 is a gain for unpolarized incident light it must hold the following inequality: m11 > 0 . (3.17) Besides, elements of the Mueller matrix have to obey the conditions: m11 ≥ |mij | ,

(3.18)

Tr (M) ≥ 0 ,

(3.19)

μT ∼ |μ| M ,

(3.20)

2

where Tr is the trace operation; μ is an arbitrary real or complex constant. Note that last relation concerns the condition for the elements of the Jones and Mueller matrices to represent a ‘physically realizable’ medium (Lu and Chipman, 1994; Anderson and Barakat, 1994; Gil, 2007) which is the result of the physical restriction that the ratio, g, between the intensities of the emerging and incident light beams (the gain or intensity transmittance) must always be in the interval 0 ≤ g ≤ 1. This condition is called the gain or transmittance condition and can be written in function of the elements of T as follows (Barakat, 1987): 12 2 1 + + + Tr (T T) + Tr (T T) + 4 det(T T) ≤ 1, (3.21) 2 or, as a function of the elements of the Mueller matrix, 1 m11 + m212 + m213 + m214 2 ≤ 1 , 1 m11 + m221 + m231 + m241 2 ≤ 1 .

(3.22)

Taking into account Eqs (3.17) and (3.18), the condition 0 ≤ g is satisﬁed.

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3.3 Internal structure of a general Mueller–Jones matrix While the Jones matrix has eight independent parameters Eq. (3.2), the absolute phase information is lost in the Mueller–Jones matrix Eq. (3.12), leading to only seven independent matrix elements for a Mueller matrix derived from a Jones matrix. This results evidently in existence of interrelations for the elements of a general Mueller–Jones matrix. For the ﬁrst time this fact was pointed out, though without derivation of their explicit form, by van de Hulst, (1957). Since then this subject has been studied by many authors (Abhyankar and Fymat, 1969; Fry and Kattawar, 1981; Hovenier et al., 1986). In most complete and reﬁned form these interrelations presented in (Hovenier, 1994) as two following sets of equations. (1) The ﬁrst set of seven equations for the squares of the elements of general Mueller–Jones matrix can be obtained from the matrix ⎤ ⎡ 2 m11 −m212 −m213 −m214 ⎢ −m221 m222 m223 m224 ⎥ ⎥ (3.23) MS = ⎢ ⎣ −m231 m232 m233 m234 ⎦ , −m241 m242 m243 m244 requiring that all sums of the four elements of a row or column of above matrix are equal. That is, one has m211 − m221 − m231 − m241 = −m212 + m222 + m232 + m242 = = −m213 + m223 + m233 + m243 = −m214 + m224 + m234 + m244 = = m211 − m212 − m213 − m214 = −m221 + m222 + m223 + m224 = = −m231 + m232 + m233 + m234 = −m241 + m242 + m243 + m244

(3.24)

(2) Second set of equations includes thirty interrelations that involve products of diﬀerent elements of M, which can be obtained by means of a graphical code presented in Fig. 3.1. For example, the pictogram in the upper left corner of Fig. 3.1(a) means m11 m12 − m21 m22 − m31 m32 − m41 m42 = 0

(3.25)

and the pictogram in the upper left corner of Fig. 3.1(b) stands for m11 m22 − m12 m21 − m33 m44 + m34 m43 = 0 .

(3.26)

From structure of Mueller–Jones matrix presented in Eq. (3.24) and Fig. 3.1 nine expressions connecting the matrix elements can be derived. For that end we consider the following relations: e f g h

= = = =

m11 + m22 − m12 − m21 , m11 + m22 + m12 + m21 , m11 − m22 − m12 + m21 , m11 − m22 + m12 − m21 .

(3.27)

These relations determine the values of amplitude of the Jones matrix elements (Gerrard and Burch, 1975), and, hence, they are non-negative. If they equal to

3 Jones and Mueller matrices

77

a)

b) Fig. 3.1. The 16 dots in each pictogram represent the elements of a Mueller–Jones matrix. A solid line connecting two elements represents a positive product, and a dashed line represents a negative product. In each pictogram, the sum of all positive and negative products vanishes. (a) Twelve pictograms that represent equations that carry corresponding products of any two chosen rows and columns. (b) Eighteen pictograms that demonstrate that the sum or diﬀerence of any chosen pair of complementary subdeterminants vanishes (after Hovenier, 1994).

zero simultaneously then one has a trivial case of null matrix. So, at least one of the above relations has to exceed zero. In each case we have a certain set of nine relations to deal with. In particular, for example, in the case of e > 0 we have, as shown by Hovenier et al. (1986), the following nine relations: (m11 + m22 )2 − (m12 + m21 )2 = (m33 + m44 )2 + (m34 − m43 )2 , (m11 − m12 )2 − (m21 − m22 )2 = (m31 − m32 )2 + (m41 − m42 )2 , (m11 − m21 )2 − (m12 − m22 )2 = (m13 − m23 )2 + (m14 − m24 )2 , e(m13 + m23 ) = (m31 − m32 )(m33 + m44 ) − (m41 − m42 )(m34 − m43 ) , e(m34 + m43 ) = (m31 − m32 )(m14 − m24 ) + (m41 − m42 )(m13 − m23 ) , e(m33 − m44 ) = (m31 − m32 )(m13 − m23 ) − (m41 − m42 )(m14 − m24 ) , e(m14 + m24 ) = (m31 − m32 )(m34 − m43 ) + (m41 − m42 )(m33 + m44 ) , e(m31 + m32 ) = (m33 + m44 )(m13 − m23 ) + (m34 − m43 )(m14 − m24 ) , e(m41 + m42 ) = (m33 + m44 )(m14 − m24 ) − (m34 − m43 )(m13 − m23 ) .

(3.28)

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From Eq. (3.28) one more important equation for elements of general Mueller–Jones matrix can be obtained 4 4 m2ij = 4m211 . (3.29) i=1 j=1

Firstly, this equation has been obtained by Fry and Kattawar (1981) and there it was shown that this equation is satisﬁed by any Mueller–Jones matrix. However, the question whether this is a suﬃcient condition for M to be a Mueller–Jones matrix has been the subject of discussion in many papers (see, for example, Simon, 1982, 1987; Hovenier, 1994; Kim et al., 1987; Kostinski, 1992; Kostinski et al., 1993; Gil and Bernabeu, 1985; Anderson and Barakat, 1994; Brosseau et al., 1990, 1993). Under the premise that a given Mueller matrix can be represented as a convex sum of Mueller–Jones matrices, this equation is both necessary and suﬃcient condition for M to be a Mueller–Jones matrix (Gil, 2007). In addition to the equalities presented above, the following series of inequalities can be derived to characterize the structure of Mueller–Jones matrix (Hovenier et al., 1986): m11 + m22 + m12 + m21 ≥ 0 , m11 − m22 − m12 + m21 ≥ 0 , m11 + m22 − m12 − m21 ≥ 0 , m11 − m22 + m12 − m21 ≥ 0 , (3.30) m11 + m22 + m33 + m44 ≥ 0 , m11 + m22 − m33 − m44 ≥ 0 , m11 − m22 + m33 − m44 ≥ 0 , m11 − m22 − m33 + m44 ≥ 0 . It is important to note that although the analysis of the internal structure of a general Mueller–Jones matrix have been carried out historically in the framework of light scattering by a single particle, the results obtained can also be used in the more general context of properties of a Mueller–Jones matrix.

3.4 Symmetry relations for Mueller–Jones matrix Now Eqs (3.12) and (3.14) can be used to derive relations between Jones matrices and corresponding Mueller–Jones matrices. We proceed from the form of Jones matrix in Eq. (3.2). (i) transposition of the Jones matrix gives ⎛ m11 ⎜ t t m 12 TT = 1 3 ↔ ⎜ ⎝ m13 t 4 t2 −m14

⎞ m21 m31 −m41 m22 m32 −m42 ⎟ ⎟; m23 m33 −m43 ⎠ −m24 −m34 m44

(3.31)

(ii) Hermitian conjugation of the Jones matrix gives simple transposition of corresponding Mueller–Jones matrix:

3 Jones and Mueller matrices

t∗1 t∗3 t∗4 t∗2

⎛

m11 ⎜m12 ↔⎜ ⎝m13 m14

m21 m22 m23 m24

m31 m32 m33 m34

79

⎞

m41 m42 ⎟ ⎟; m43 ⎠ m44

(iii) sign reversal for oﬀ-diagonal elements: ⎛ ⎞ m11 m12 −m13 −m14 ⎜ m21 m22 −m23 −m24 ⎟ t1 −t4 ⎟ ↔⎜ ⎝−m31 −m32 m33 m34 ⎠ ; −t3 t2 −m41 −m42 m43 m44 (iv) interchange of diagonal elements ⎞ ⎛ m11 −m21 m31 m41 ⎜−m12 m22 −m32 −m42 ⎟ t2 t4 ⎟ ↔⎜ ⎝ m13 −m23 m33 m43 ⎠ ; t3 t1 m14 −m24 m34 m44 (v) successive application of (i) and (iii) ⎛ ⎞ m11 m21 −m31 m41 ⎜ m12 m22 −m32 m42 ⎟ t1 −t3 ⎟ ↔⎜ ⎝−m13 −m23 m33 −m43 ⎠ . −t4 t2 m14 m24 −m34 m44

(3.32)

(3.33)

(3.34)

(3.35)

Physical reasons for the above relations are directly clear. For instance, relation Eq. (3.31) and Eq. (3.35) originates from the operation of interchanging the incident and emerging light beams and the principle of reciprocity (Saxon, 1955; Sekera, 1966; Vansteenkiste et al., 1993; Potton, 2004); relation Eq. (3.33) originates from mirror symmetry (Hovenier, 1969, 1970). In the framework of light scattering by a single particle, Jones matrices Eqs (3.2), (3.31), (3.33) and (3.35) correspond to two arbitrary particles and two positions of its mirror particles (van de Hulst, 1957). The eﬀect of the symmetry of a collection of scatterers on the structure (number of independent parameters) of a Mueller matrix has been considered by van de Hulst (1957), in particular: (i) The collection contains one kind of particles. For each particle in one position there is a particle in the reciprocal position (Eq. (3.2) + Eq. (3.35)): ⎛ ⎞ m11 m12 m13 m14 ⎜ m12 m22 m23 m24 ⎟ ⎜ ⎟ (3.36) ⎝−m13 −m23 m33 m34 ⎠ (10 parameters) , m14 m24 −m34 m44 with relation: m11 − m22 + m33 − m44 = 0 (ii) The collection contains equal number of particles and their mirror particles (Eq. (3.2) + Eq. (3.33)): ⎛ ⎞ m11 m12 0 0 ⎜m21 m22 0 0 ⎟ ⎜ ⎟ (8 parameters) . (3.37) ⎝ 0 0 m33 m34 ⎠ 0 0 m43 m44

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If, in addition, t3 = t4 , then m22 = m11 , m33 = m44 , m34 = −m43 , and m211 − m212 − m233 − m234 = 0. If in this case for any particle of Eq. (3.2) there is a particle in position Eq. (3.31), then the Mueller–Jones matrix has a form: ⎛ ⎞ m11 m12 m13 m14 ⎜ m12 m22 m23 m24 ⎟ ⎜ ⎟ (3.38) ⎝ m13 m23 m33 m34 ⎠ (10 parameters) . −m14 −m24 −m34 m44 (iii) The collection contains equal number of particles in positions Eq. (3.2), Eq. (3.35), Eq. (3.33) and Eq. (3.31): ⎛ ⎞ m11 m12 0 0 ⎜m12 m22 0 0 ⎟ ⎜ ⎟ (6 parameters) . (3.39) ⎝ 0 0 m33 m34 ⎠ 0 0 −m34 m44 The scattering angles 0◦ and 180◦ deserve separate attention in the literature through their practical importance. It was van de Hulst (1957) who ﬁrst derived the general form of Mueller matrices for these scattering angles. Hu et al. (1987) presented a comprehensive study of forward and backward scattering by an individual particle in a ﬁxed orientation. For forward scattering they distinguished sixteen diﬀerent symmetry shapes which were classiﬁed into ﬁve symmetry classes and for backward scattering four diﬀerent symmetry shapes which were classiﬁed into two symmetry classes. A large number of relations were derived in this way. Below we present structures of Mueller matrices for various collections of particles in the cases of forward and backward scattering (van de Hulst, 1957; Hovenier and Mackowski, 1998). 3.4.1 Forward scattering (i) There is one kind of asymmetric particles in the collection. The assumption of rotational symmetry is made: ⎛ ⎞ m11 0 0 m14 ⎜ 0 m22 m23 0 ⎟ ⎜ ⎟ (3.40) ⎝ 0 −m23 m22 0 ⎠ (6 parameters) . 0 0 m44 m41 (ii) There is one kind of asymmetric particles. The assumption of rotational symmetry combined with that equal numbers of particles occur in the reciprocal position is made: ⎞ ⎛ 0 0 m14 m11 ⎜ 0 m22 m23 0 ⎟ ⎟ ⎜ (3.41) ⎝ 0 −m23 m22 0 ⎠ (5 parameters) . m14 0 0 m44

3 Jones and Mueller matrices

81

(iii) Particles and mirror particles occur in equal numbers. The assumption of rotational symmetry combined with that for any particle there is a mirror symmetric particle in any plane through the axis is made: ⎞ ⎛ 0 0 m11 0 ⎜ 0 m22 0 0 ⎟ ⎟ (3 parameters) . ⎜ (3.42) ⎝ 0 0 m22 0 ⎠ 0 0 0 m44 with m44 ≥ 2m22 − m11 . For the special case of homogeneous, optically inactive, and spherical particles m11 = m22 = m44 (Hovenier and Mackowski, 1998). (iv) Particles and mirror particles occur in equal numbers. Rotational symmetry is combined with the assumption of mirror symmetry with respect to a plane perpendicular to the axis. ⎞ ⎛ 0 0 m14 m11 ⎜ 0 m22 0 0 ⎟ ⎟ (4 parameters) . ⎜ (3.43) ⎝ 0 0 m22 0 ⎠ 0 m44 −m14 0 3.4.2 Backward scattering (i) No assumption about kind of ⎛ m11 m12 ⎜ m12 m22 ⎜ ⎝−m13 −m23 m14 m24

particles is made: ⎞ m13 m14 m23 m24 ⎟ ⎟ (10 parameters) . m33 m34 ⎠ −m34 m44

(3.44)

(ii) There is one kind of asymmetric particle in the collection. The assumption of rotational symmetry is made: ⎛ ⎞ m11 0 0 m14 ⎜ 0 m22 0 0 ⎟ ⎜ ⎟ (4 parameters) . (3.45) ⎝ 0 0 −m22 0 ⎠ 0 m44 m14 0 Also, m11 ≥ m22 ≥ 0; m44 = m11 − 2m22 , and m22 − m11 ≤ m44 ≤ m11 − m22 . (iii) The collection contains particles and mirror particles occurring in equal numbers. Rotational symmetry is combined with the assumption that any particle has its mirror particle with respect to any plane through the axis. Then it follows: ⎞ ⎛ m11 0 0 0 ⎜ 0 m22 0 0 ⎟ ⎟ (3 parameters) ⎜ (3.46) ⎝ 0 0 −m22 0 ⎠ 0 0 0 m44 For special case of homogeneous, optically inactive, and spherical particles: m11 = m22 = −m44 (Hovenier and Mackowski, 1998).

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3.5 The depolarizing Mueller matrix When the incident light is fully polarized and for output light in Eq. (3.10) we have an inequality, then equalities on matrix elements (Eq. (3.24), Fig. 3.1, Eqs (3.26)– (3.29)) determining the structure of the Mueller matrix as a Mueller–Jones matrix are lost. In this case the output light is composed of several incoherent contributions, and the medium cannot be represented by means of a Jones matrix. However, the medium can be considered as a parallel set of deterministic media, each one is described by a well-deﬁned Jones matrix, in such a way that the light beam is shared among these diﬀerent media. It is important to point out that the same result could be obtained by considering the medium as an ensemble (Kim et al., 1987) so that each realization i, characterized by a well-deﬁned Jones matrix Ti , occurs with a probability pi . If the Mueller matrix can be presented as a weighted sum of the Mueller–Jones matrices (Cloude, 1986; Simon, 1987; Cloude and Pottier, 1995; Gil, 2007) then it is called a depolarizing Mueller matrix. It is important to note that this class of matrices does not coincide with the class of matrices, named Stokes matrices, satisfying the Stokes–Verdet criterion (see section 3.2), i.e., transforming Stokes vectors into Stokes vectors. Any physical Mueller matrix is a Stokes matrix; however, the converse statement is not in general true (Gil, 2007). Furthermore, no method has been quoted to physically realize a Stokes matrix being non-derivable as a weighted sum of Mueller–Jones matrices. 3.5.1 Structure of the depolarizing Mueller matrix Linear inequalities for elements of a Mueller–Jones matrix are also valid for a depolarizing Mueller matrix, Ms (Hovenier and van der Mee, 2000), in particular: ms11 ≥ 0;

ms11 ≥ |msij | ,

ms11 + ms22 + ms12 + ms21 ≥ 0 , ms11 + ms22 − ms12 − ms21 ≥ 0 , ms11 ms11

− −

ms22 ms22

+ −

ms12 ms12

− +

ms21 ms21

(3.47)

≥ 0, ≥ 0.

In this case the following quadratic inequalities are valid as well (Fry and Kattawar, 1981): 2 2 s m11 + ms12 − ms21 + ms22 s 2 2 m11 − ms12 − ms21 − ms22 s 2 2 m11 + ms21 − ms12 + ms22 s 2 2 m11 − ms21 − ms12 − ms22 s 2 2 m11 + m222 − ms12 + ms21 s 2 2 m11 − ms22 − ms12 − ms21

2 2 ≥ ms31 + ms32 + ms41 + ms42 , 2 2 ≥ ms31 − ms32 + ms41 − ms42 , 2 2 ≥ ms13 + ms23 + ms14 + ms24 , 2 2 ≥ ms13 − ms23 + ms14 − ms24 , 2 2 ≥ ms33 + m244 + ms34 − ms43 , 2 2 ≥ ms33 − ms44 + ms34 + ms43 ,

and Eq. (3.29) becomes an inequality as well:

(3.48)

3 Jones and Mueller matrices 4 4

msij

2

83

2 ≤ 4 ms11 .

i=1 j=1

The following relations on elements of the Mueller matrix have to satisfy the condition Eq. (3.10), i.e., be the Stokes vectors (Gil, 2007): S1 = ms11 + m221 S2 = ms11 − m221 S3 = m211 + m212 S4 = m211 − m212 S5 = m211 + ms22 S6 = m211 − ms22

ms12 + ms22 ms12 ms21 ms21 ms21 ms21

− + − + −

ms22 ms22 ms22 ms12 ms12

ms13 + m223 ms13 ms31 ms31 ms33 ms33

− + − + −

m223 ms32 ms32 ms44 ms44

ms14 + ms24 ms14 ms41 ms41 ms34 ms34

− + − − +

T

T ms24 T ms42 T ms42 T ms43 T ms43

, , , ,

(3.49)

, ,

As can be seen, the ﬁrst four expressions can be obtained as results of the following transformations: S1 = Msr (1 1 0 0)T , S2 = Msr (1 − 1 0 0)T , S3 = Ms (1 1 0 0)T ,

(3.50)

S4 = Ms (1 − 1 0 0)T , where Mr = diag (1 1 1 − 1)MT diag (1 1 1 − 1) . The last two expressions in Eq. (3.49) are peculiarities of structure of the Mueller matrix and cannot be derived as results of Stokes vector transformations. 3.5.2 Matrix models of depolarization One of the key points of scattering media optics is the study of one more phenomenon originating in light–medium interaction which is called depolarization. The study and characterization of depolarization is of considerable importance by reason of the fact that depolarization phenomena is involved in many theoretical and experimental applications in polarimetry of discrete random media and medium with bulk and surface inhomogeneities. However, note that light–medium interaction with depolarization is heretofore studied in considerably less degree than the problem described by Mueller–Jones matrices discussed in previous sections. Depolarization is the result of decorrelation of the phases and the amplitudes of the electric vector and selective absorption of polarization states (Brosseau, 1998). Depolarization can be observed in both single and multiple light scattering regimes and depends on the geometrical and physical characteristics of the scattering particles: shape, morphology, refractive index, size parameter (ratio of the particle circumference to the wavelength of the incident light), and orientation with respect to the coordinate system (Mishchenko and Travis, 2000). Furthermore, multiple scattering results in depolarization of the output light, even in case of collections of spherical particles, and often reinforces depolarization caused by particle nonsphericity. The purpose of this section is to consider the depolarization

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phenomenon using Mueller matrix formalism; in particular, we intend to discuss some single-number metrics and Mueller matrix models of depolarization in the literature. In addition to the degree of polarization Eq. (3.11) it is common practice in scattering optics to describe depolarization of light scattered by random and inhomogeneous media in terms of linear, δL , and circular, δC , depolarization ratios (Mishchenko and Hovenier, 1995; Mishchenko and Travis, 2000) δL =

s1 − s2 s1 + s2

(3.51)

δC =

s1 − s4 s1 + s4

(3.52)

The interest of these parameters is explained by the fact that they are susceptible to particle nonsphericity. Indeed, for spherical particles both ratios are equal to zero whereas, for nonspherical scatterers, both δL and δC can substantially be diﬀerent from zero. As pointed out by Mishchenko and Hovenier (1995) for the backscattering direction the depolarization ratios have several interesting properties. In the case of backscattering by a small-volume element comprising arbitrary particles in random orientation, the Mueller matrix has the form of Eq. (3.45). For isotropic spheres the Mueller matrix takes the form of Eq. (3.46), i.e., in this case both ratios vanish. This means that, if the incident light is linearly polarized, the backscattered light is completely linearly polarized in the same plane, whereas, if the incident light is circularly polarized, the backscattered light is completely circularly polarized in the opposite sense. For nonspherical particles this is generally not the case. Combining inequalities on matrix elements in Eq. (3.45) we get

m22 −1 m11

m14 ≤ ≤ m11

1−

m22 m11

2 12 (3.53)

This inequality and the areas in which δC > δL and δC < δL are presented in Fig. 3.2. Based on and in addition to the degree of polarization, p, Eq. (3.11) for a more detailed description of depolarization properties of light the degrees of linear and circular polarization can be determined (Chipman, 1995): s22 + s23 , (3.54) DoLP = s1 DoCP =

s4 . s1

(3.55)

These parameters are very useful for applications in meteorology, astronomy, ophthalmology, optical ﬁbers etc. (Bueno, 2001, and references therein). Quantities Eqs (3.51), (3.52), (3.54) and (3.55) refer to the intrinsic depolarization properties of light; however, in many cases it is important to know about the depolarization features of a medium itself. Depolarization metrics provide a single scalar number that varies from zero relating to totally depolarized output light to a

3 Jones and Mueller matrices

85

m14 m11

m22 m11 Fig. 3.2. Areas of possible combinations of m14 /m11 and m22 /m11 for backscattering by randomly oriented particles (after Mishchenko and Hovenier, 1995)

certain positive number relating to totally polarized output light. All intermediate values are associated to partial polarization. The depolarization index DI(M) is introduced by Gil and Bernabeu (1985, 1986): ⎞ 12 ⎛ 4 ⎝ m2ij − m211 ⎠ DI(M) =

i,j=1

√

3 m11

(3.56)

The bounds on the depolarization index is 0 ≤ DI(M) ≤ 1. Boundary values of DI(M) associate with the case of unpolarized and totally polarized output light respectively. In a sense an ‘analog’ of the degree of linear polarization Eq. (3.54) in terms of Mueller matrix elements, the index of linear polarization was suggested in (Bueno, 2001): 12 √ 4 1 2 3 2 2 2 GL = m2i + m3i (3.57) m21 + m31 + 2m11 3 i=1 It can be seen that GL is a ratio of the mean of sum of the squares of the matrix elements corresponding to linear polarization of the output light and the value of the corresponding averaged intensity normalized by the√maximum value of this ratio which occurs for the linear polarizer (GL )max = 2/ 3. The former provides the range of variation 0 ≤ GL ≤ 1. The average degree of polarization AverageDoP is deﬁned as follows (Chipman, 2005): π π/2 1 p[MS(ε, ζ)] cos(ε) dε dζ (3.58) AverageDoP (M) = 4π 0 −π/2

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The term cos(ε) dε dζ scans the incident polarization state over the Poincar´e sphere, with the latitude ε and longitude ζ. The Stokes vector S(ε, ζ) is here a function of ellipticity and the azimuth of orientation of the polarization ellipse of light: ⎛ ⎞ 1 ⎜cos(2ε) cos(2ζ)⎟ ⎟ S(ε, ζ) = ⎜ (3.59) ⎝ cos(2ε) sin(2ζ) ⎠ sin(2ε) The Q(M) metrics is (Espinosa-Luna and Bernabeu, 2007): 4

Q(M) =

m2ij

i=2 j=1

4

= m21j

3[DI(M)]2 − [D(M)]2 , 1 + [D(M)]2

(3.60)

j=1

m212 + m213 + m214 is the diattenuation parameter and 0 ≤ where D(M) = D(M) ≤ 1. The bounds on the metric Q(M) are 0 ≤ Q(M) ≤ 3. Q(M) = 0 is for a totally depolarizing medium; 0 < Q(M) < 1 is for a partially depolarizing medium; 1 ≤ Q(M) < 3 represents a partially depolarizing medium if, in addition, 0 < DI(M) < 1, otherwise, represents a non-depolarizing diattenuating medium; Q(M) = 3 is for a non-depolarizing non-diattenuating medium. Thus, depolarization metrics provide a summary of the depolarizing property of a medium in a single number. The depolarization index DI(M) and Q(M) metrics are directly related to the Mueller matrix elements only and, in contrast to the average degree of polarization AverageDoP, no need to scan the whole Poincar sphere of input polarizations. Thus, Q(M) provides more detailed information about the depolarization properties of a medium. Since the polarization properties of a medium can be very diﬀerent if the direction of the incident and output light is exchanged, it is appropriate, in addition to the ‘direct’ depolarization metrics, to introduce the depolarization metrics for the reverse direction of light. The Mueller matrix for the reverse direction of light, MR , can be written in the form (Sekera, 1966; Schonhoﬀer and Kuball, 1987): MR = Q MT Q−1 ,

(3.61)

where Q = diag (1 1 1 − 1) . The expression Eq. (3.61) is of importance in a number of problems both in crystalline (Potton, 2004) and in scattering media optics (Mishchenko et al., 2006). As an example of depolarization metrics for the reverse direction of light the reverse index of linear polarization can be pointed out (Bueno, 2001): √

GLR

3 = 2m11

+

m213

12

1 2 mi2 + m2i3 + 3 i=1 4

m212

(3.62)

3 Jones and Mueller matrices

87

It can be directly seen from Eqs (3.57) and (3.62) that the condition of GLR = GL is m2ij = m2ji . Some media depolarize all polarization states equally. Other depolarizing media partially depolarize most polarization states but may not depolarize one or two particular incident states. Nevertheless, a single-number metric for depolarization providing a summary of depolarization of a medium cannot give detailed information about all features of depolarization. Such information can be obtained only from Mueller matrix models of depolarization. The case when for all polarizations of input light the degree of polarization p of output light is the same is called isotropic depolarization. When the degree of polarization of output light is a function of parameters of input polarization, then this case is called anisotropic depolarization. There seems to be a consensus about the form of Mueller matrix model describing the isotropic depolarization (Brosseau, 1998; Chipman, 1999): ⎛ ⎞ 1 0 0 0 ⎜0 p 0 0 ⎟ ⎜ ⎟ (3.63) ⎝0 0 p 0 ⎠ 0 0 0 p It can be seen that properties of this type of depolarization are the following: (i) the transmittance is equal for all polarizations of incident light; (ii) p of output light is equal for all input polarizations. At the same time, in the literature there exists no agreement concerning the Mueller matrix for the case of anisotropic depolarization. Seemingly, one of the most accepted forms of the Mueller matrix describing the dependence of p of output light on incident polarization is the following (Shindo, 1995; Brosseau, 1998; Chipman, 1999): ⎛ ⎞ 1 0 0 0 ⎜0 a 0 0⎟ ⎜ ⎟ (3.64) ⎝0 0 b 0⎠ . 0 0 0 c The physical meaning of the elements a, b and c are interpreted in the following manner: a and b are the degrees of linear depolarization; c is the degree of circular depolarization. When a = b = c = 0 the Mueller matrix is a matrix of the ideal depolarizer. Bicout et al., (1994) discussed the depolarization arising in multiple scattering of light by spherical scatterers in a Rayleigh regime given by the matrix Eq. (3.64) with a = b. One more form of the Mueller matrix describing depolarization is (Williams, 1986): ⎛ ⎞ 1 A1 A2 A3 ⎜0 P1 0 0 ⎟ ⎜ ⎟ (3.65) ⎝0 0 P2 0 ⎠ . 0 0 0 P3 This matrix, as the author stated, shrinks the unit vectors along the three coordinate axes without changing their directions. The diagonal matrix elements P1 , P2 , and P3 are average lengths of the transformed axes. The top row elements A1 ,

88

S.N. Savenkov

A2 , and A3 represent asymmetry in the amounts of depolarization of the oppositely directed input vectors. The terms A1 and P1 are deﬁned by the following equations: A1 =

(LS − LP ) , (LS + LP )

(3.66)

P1 =

2LP LS . (LS + LP )

(3.67)

The terms LP and LS represent the lengths of transformed unit input vectors (1 1 0 0)T , the p wave, and (1 − 1 0 0)T , the s wave. Similar expressions deﬁne A2 , A3 , P2 , and P3 . Thus, the Mueller matrix Eq. (3.9) will transform unit T input vectors (1 1 0 0)T and (1 − 1 0 0)T to 2LS / LP + LS 1 LP 0 0 T and 2LP / LP + LS 1 LS 0 0 . Note that this Mueller matrix contains six degrees of freedom (leaving out of account the degree of freedom connected with transmittance for unpolarized radiation). The most general expression for the Mueller matrix describing depolarization was suggested by Lu and Chipman (1996): 1 OT (3.68) ; mTΔ = mΔ , PΔ m Δ where PΔ denotes the so-called polarizance vector. The polarizance vector describes the state of polarization produced by this Mueller matrix from an unpolarized incident light. The Mueller matrix Eq. (3.68) has nine degrees of freedom and this situation is of interest because this matrix with the generalized deterministic Mueller matrix are jointly characterized by 16 degrees of freedom. This means that one, in that way, has the generalized Mueller matrix of an arbitrary medium, which has 16 degrees of freedom and linearly interacts with polarized light. It is important to note that describing the dependence of p of output light on input polarization, the Mueller matrices Eqs (3.64), (3.65), and (3.68) describe the changes of polarization of incident light as well. Because of that, the question of what exact additional mechanisms of polarization transformation all these Mueller matrices contain is absolutely relevant. Although Gil (2007) suggested one of the possible approaches to gaining insight into the retarding, polarizing and depolarizing properties of the Mueller matrix Eq. (3.68), nevertheless, the problem is far from being completely solved. In Savenkov (2002a) the following Mueller matrix for anisotropic depolarization has been suggested: ⎛ ⎞ 1 m12 m13 m14 ⎜0 m 0 0 ⎟ ⎜ ⎟. (3.69) ⎝0 0 m 0 ⎠ 0 0 0 m Formally, this Mueller matrix diﬀers from the matrix Eq. (3.65) only by the equality of its diagonal elements; however, this distinction results in a substantial implication. This implication consists in the fact that, describing the dependence of p of output light on incident polarization, this Mueller matrix preserves the state of polarization of input light.

3 Jones and Mueller matrices

89

3.5.3 Cloude’s coherency matrix In this section we consider an additive matrix model to present the Mueller matrix based on Cloude’s coherency matrix (Cloude, 1986; Cloude and Pottier, 1995). The coherency matrix J is derived from the corresponding Mueller matrix as follows: J11 = 41 (m11 + m22 + m33 + m44 ) J13 =

1 4 (m13

+ m31 + im24 − im42 )

J12 = 14 (m12 + m21 − im34 + im43 ) J14 = 14 (m14 − im23 + im32 + m41 )

J21 = 14 (m12 + m21 + im34 − im43 )

J22 = 14 (m11 + m22 − m33 − m44 )

J23 = 14 (im14 + m23 + m32 − im41 )

J24 = 14 (−im13 + im31 + m24 + m42 )

J31 = 14 (m13 + m31 − im24 + im42 )

J32 = 14 (−im14 + m23 + m32 + im41 )

J33 = 41 (m11 − m22 + m33 − m44 )

J34 = 14 (im12 − im21 + m34 + m43 )

J41 = 14 (m14 + im23 − im32 + m41 )

J42 = 14 (im13 − im31 + m24 + m42 )

J44 = 14 (m11 − m22 − m33 + m44 ) (3.70) Thus, J depends linearly on M and the linear relation between them is given by four sets of interrelations presented in Fig. 3.3. J43 =

1 4 (−im12

+ im21 + m34 + m43 )

Fig. 3.3. Relation between structures of the Mueller matrix M and corresponding Cloude’s coherency matrix J (after Hovenier and van der Mee, 1996).

It can be seen that coherence matrix J is positive semideﬁnite Hermitian and, hence, has always four real eigenvalues. The eigenvalues of the coherency matrix, λi , can be combined to form a quantity that is a measure of the depolarization of the studied medium. This quantity is called entropy and is deﬁned as: ⎛ ⎞ ⎛ ⎞ S=−

N ⎜ λi ⎟ ⎜ λi ⎟ ⎜ ⎟ logN ⎜ ⎟. ⎝ ⎠ ⎝ ⎠ λ λ j j i=1 j

(3.71)

j

Given eigenvalues λi of coherency matrix J, we have for initial Mueller matrix: M=

4 k=1

λk MkD ;

MkD ⇔ Tk ,

(3.72)

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S.N. Savenkov

here MkD are the Mueller–Jones matrices obtained from the Jones matrices by Eq. (3.12). The Jones matrix, T, in turn, is obtained in the following manner: (k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

t11 = Ψ1 + Ψ2 , t12 = Ψ3 − iΨ4 (k)

(k)

t21 = Ψ3 + iΨ4 , t22 = Ψ1 − Ψ2

k = 1, 4 ,

(3.73)

where Ψ(k) = (Ψ1 Ψ2 Ψ3 Ψ4 )Tk is kth eigenvector of coherency matrix J. Thus, the substance of the Cloude’s coherency matrix concept, which, in essence, is an additive matrix model of depolarizing Mueller matrix representing the initial depolarizing Mueller matrix as a weighted convex sum of four Mueller–Jones matrices, is presented in Fig. 3.4. If three of the eigenvalues of J vanish, then the initial matrix M is a deterministic Mueller–Jones matrix. Indeed, we note above that Fry and Kattawar (1981) showed that the inequality Eq. (3.48) is a necessary condition satisﬁed by any Mueller matrix, which became an equality for Mueller–Jones matrices. From Eq. (3.70) it can be shown that (Simon, 1987; Gil, 2000): Tr (MT M) = 4 Tr (J2 ) = 4

4

λ2i ,

(3.74)

.

(3.75)

i=1

(Tr (J))2 = m211 =

4

2 λi

i=1

Taking into account that λi are non-negative, we can write Deterministic matrices §x ¨ Cloude decomposition O 1 ¨ xx ¨x © Measured matrix §x ¨x ¨x ¨x ©

x x x x

x x x x

x· x¸ x¸ x ¸¹

x x x x

x x x x

+ §x O2 ¨x ¨x ¨x ©

x x x x

x x x x

x· x¸ x¸ x ¸¹

x x x x

§x O4 ¨x ¨x ¨x ©

x x x x

x x x x

x· x¸ x¸ x ¸¹

+ x x x x

M CA Anisotropy parameters ( R , P , J , ' ,D, M ) §x x x x ¨x x ©

Max( O i ) ¨¨ xx

+ §x O3 ¨x ¨x ¨x ©

Generalized equivalence theorem decomposition

x· x¸ x¸ x ¸¹

x· x¸ x¸ x ¸¹

x x x x

x· x¸ x¸ x ¸¹

M LA MLP

MCP

Fig. 3.4. Additive model of the Mueller matrix based on Cloude’s coherency matrix.

3 Jones and Mueller matrices 4

λ2i ≤

4

i=1

91

2 λi

,

(3.76)

i=1

It can be seen that Eq. (3.76) becomes an equality if, and only if, only one eigenvalue is nonzero. If all four eigenvalues of J are not equal to zero and S ≤ 0.5, then the Mueller– Jones matrix, which corresponds to the maximal eigenvalue, is the dominant type of deterministic polarization transformation of the studied medium. So, this theorem on the one hand makes it possible to study the anisotropy properties of depolarizing media and, on the other hand, is a necessary and suﬃcient criterion for a given 4 × 4 real matrix to be a Mueller matrix (the case when all four eigenvalues of J are non-negative) and a Mueller–Jones matrix (the case when three of the eigenvalues vanish) (Hovenier and van der Mee, 1996; Volten et al., 1999). Aiello and Woerdman (2005) have studied the relation between entropy S and depolarization index DI(M). They derived the analytical expression which interconnects S and DI(M): S(n, f ) = − [(1 − nf ) log4 (1 − nf ) + nf log4 (f )] where

1 3n+1 f± = 1± 1− (1 − (DI(M))2 ) n+1 4 n

(3.77)

(3.78)

In the S − DI(M) plane expression Eq. (3.77) determines domain presented in Fig. 3.5. The boundary of this domain is the curves Cij (i, j = 1, . . . , 4), joining the points (1 → 4). The links between the functions S(n, f ) and the curves Cij are given in Table 3.1. S13 is deﬁned as S13 = −(1 − μ) log4 ((1 − μ)/2) − μ log4 (μ/2). The curve C14 sets an upper bound for the value of entropy for any light scattering medium. The entropy on this curve is approximated by

1 2

S

3 4

DI (M) Fig. 3.5. Domain in the S − DI(M) plane corresponding to all physically realizable light scattering media (after Aiello and Woerdman, 2005).

92

S.N. Savenkov

Table 3.1. List of curves Cij (Fig. 3.5) for scattering media determined by eigenvalues of coherency matrix J. Curve

Generating equation

C12 C23 C34 C14 C13 C24

S(3, f+ ) S(2, f+ ) S(1, f± ) S(3, f− ) S13 S(2, f− )

Eigenvalues of J ` ´ `λ μ μ μ´ `λ μ μ 0´ `λ μ 0 0´ `λ μ μ μ´ `λ λ μ μ´ λ μ μ 0

κ S14 ∼ 1 − (DI(M))2

(3.79)

where κ ≈ 0.862. For all other scattering media must be satisﬁed the condition S ≤ S13 . Thus, various light scattering medium described by diﬀerent Mueller matrices can be characterized by corresponding curves and subdomains in Fig. 3.5. 3.5.4 Block-diagonal structure of the Mueller matrix The Mueller matrix of the form Eq. (3.37) plays a key role in many light scattering problems. Certain of them we discuss in this section. The structure Eq. (3.37) can be caused by symmetry of a single particle and a collection of particles in single and multiple scattering (van de Hulst, 1957; Mishchenko, and Travis, 2000) and by illumination-observation geometry for backward (Zubko et al., 2004) and forward (Savenkov et al., 2007a) scattering. A model of the medium described by the Mueller matrix Eq. (3.37) was used in a study (to present only a few examples) of the optical characteristics of oceanic water (Voss and Fry, 1984; Kokhanovsky, 2003d); of an ensemble of identical, but randomly oriented fractal particles (Kokhanovsky, 2003c); of dense spherical particle suspensions in multiple-scattering case (Kaplan et al., 2001); of the multiple scattering of light by an ice cloud consisting of nonspherical ice crystals (Lawless et al., 2006); of polydisperse, randomly oriented ice crystals modeled by ﬁnite circular cylinders with diﬀerent size distributions (Xu et al., 2002); for characterizing cylindrically shaped radially inhomogeneous particles (Manickavasagam and Menguc, 1998); small spherical particles (diameters range from 0.2 to 1.5 μm) sparsely seeded on a surface of crystalline silicon c-Si wafer (Kaplan and Drevillon, 2002); for measurements of the complex refractive index of isotropic materials as matrices of isotropic and ideal metal mirror reﬂections (Deibler and Smith, 2001); in developing a symmetric three-term product decomposition of a Mueller–Jones matrix (Ossikovski, 2008). Cloude’s coherence matrix method enables to derive the conditions on blockdiagonal matrix Eq. (3.37) (van der Mee, 1993) to be representable as a convex sum of the Mueller–Jones: "1 ! (m33 − m44 )2 + (m34 + m43 )2 + (m12 − m21 )2 2 ≤ m11 − m22 ! "1 (m33 + m44 )2 + (m34 − m43 )2 + (m12 + m21 )2 2 ≤ m11 + m22 (3.80)

3 Jones and Mueller matrices

93

Analogously one can see that conditions on matrix elements presented in Eq. (3.37) are the conditions on the matrix to be derivable from corresponding Jones matrix. Note that, in contrast to the case of spherical particles when relations m22 = m11 and m33 = m44 presented in Eq. (3.37) are held, i.e., when the Mueller matrix of the structure Eq. (3.37) is a Mueller–Jones matrix and can be derivable from the corresponding Jones matrix, in many cases this Mueller matrix can contain information on depolarization. In Kokhanovsky (2003c) in examining whether the Mueller matrix of the structure Eq. (3.37) can produce depolarized light, the following parameter # q=

(ms12 )2 + (ms33 )2 + (ms34 )2

(3.81)

was suggested. It can be seen that for normalized Mueller matrix of spherical particles q is equal to 1. To illustrate the physical meaning of the parameter q, consider the illumination of a scattering medium with the Mueller matrix Eq. (3.37) by a linearly polarized light with the Stokes vector S inc = (1 0 1 0)T , which corresponds to the linear polarization at the angle 45◦ to a reference plane. Then for output light we have S out = (1 ms12 ms33 −ms34 )T . For a completely polarized output light the condition Eq. (3.10) is an equality and for this case it can be presented in the form (ms11 )2 = (ms12 )2 +(ms22 )2 +(ms34 )2 , which is equivalent to q = 1. Therefore, in the case q = 1, the depolarized light is not produced in a light scattering event. The ratio (−ms12 /ms11 ) is called (Mishchenko and Travis, 2000; Kokhanovsky, 2003c) the degree of polarization of initially unpolarized incident light. It is negative for most of the scattering angles. This means that direction of an electric vector is predominantly in the plane perpendicular to the scattering plane. For single spheres the ratio (−ms12 /ms11 ) at scattering angles 0◦ and π is identically equal to 1. Mishchenko and Travis (2000) have pointed out that the most prominent feature of nonspherical scattering is the bridge of positive linear polarization at scattering angles near 120◦ . The ratio ms22 /ms11 is equal to unity for spherical particles at any scattering angles. So, the diﬀerence of ms22 /ms11 from unity can be used as a measure of the particle’s nonsphericity, which is sensitive to particle size and aspect ratio (Quinby-Hunt, 2000). The ratio ms34 /ms11 describes the transformation of linear polarized incident light to circular polarized output light. Generally, ms34 /ms11 decreases with the value of refractive index of particles. The value of ratio ms44 /ms11 describes the reduction of the degree of circular polarization for circularly polarized incident light. Much as ms22 /ms11 , the behavior of element ratios ms33 /ms11 and ms44 /ms11 also represents asymmetric particle shapes, i.e., deviation of ms44 /ms11 from ms33 /ms11 indicates nonspherical symmetry of particles (Xu et al., 2002). Figure 3.6 shows the example of dependencies of ratios $ % π of matrix elements discussed above and phase function deﬁned as 2ms11 (θ) 0 ms11 (θ ) sin θ dθ on scattering angle for a power law size distribution of spheres and randomly oriented spheroids in the case of single scattering (Mishchenko and Travis, 2000). Multiple scattering can signiﬁcantly aﬀect the process of polarization (the occurrence of at least one more nonzero Stokes parameter other than overall intensity for unpolarized input Stokes vector) and depolarization. In particular, polarization can be considerably decreased by multiple scattering. At the same time, multi-

m33/m11 (%)

S.N. Savenkov

m34/m11 (%)

- m12/m11 (%)

m44/m11 (%)

m22/m11 (%)

94

Fig. 3.6. Phase function and normalized Mueller matrix elements for power law size distribution spheres and randomly oriented spheroids (after Mishchenko and Travis, 2000).

ple scattering causes nonzero depolarization values even for spherical particles and often reinforce depolarization caused by particle nonsphericity. Figure 3.7 (Mishchenko and Travis, 2000) demonstrates this by the example of randomly oriented bispheres with an index of refraction of 1.5 + 0.005i. The size parameter of the component spheres is 5 and the distance D between the sphere centers. Solid curves show the results of Lorenz–Mie computations for a single sphere with the same size parameter 5. It can be seen that the magnitude of oscillations, which are attributed to manifestation of the interference structure typical for monodisperse spheres, decreases as the component spheres become closer and is minimal for the bisphere with touching components. In particular, the degree of linear polarization (−ms12 /ms11 ) becomes more neutral and less indicative of the physical characteristics of the component spheres. Using the coherency matrix concept (see section 3.5.3) consider the structure of the Mueller matrix for multiple scattering in the exact backscattering direction by optically thick discrete random media (Cloude and Pottier, 1995). In this case backscattering of light is aﬀected by a phenomenon called coherent backscattering or weak photon localization (van Albada et al., 1988; MacKintosh et al., 1989). Mishchenko (1992) have shown that the Mueller matrix for this problem has the structure given by Eq. (3.37) and can be presented as follows: M = M1 + ML + MC

(3.82)

The ﬁrst term, M1 , describes single scattering contribution. The second, ML , describes multiple incoherent scattering contribution. Depending on particle symme-

m33/m11 (%)

95

m34/m11 (%)

- m12/m11 (%)

m44/m11 (%)

m22/m11 (%)

3 Jones and Mueller matrices

Fig. 3.7. Phase function and normalized Mueller matrix elements for randomly oriented bispheres with touching and separated components (after Mishchenko and Travis, 2000).

try this contribution may have high or low entropy S and, hence, support polarization information (MacKintosh et al., 1989). This term can be calculated using classical vector radiative transfer equation (Chandrasekhar, 1950). The third term, MC , represents the contribution results from coherence in the exact backscattering direction between the path and the time-reversed path of light. The third term is diﬃcult to calculate for a general random medium; however, using the vector reciprocity theorem Mishchenko (1992) derived the relationship for given scattering geometry between coherent and incoherent multiple scattering contributions: L mC ij = mij ,

i = j

1 L L L L mC 11 = 2 (m11 + m22 − m33 + m44 ) 1 L L L L mC 22 = 2 (m11 + m22 + m33 − m44 ) 1 L L L L mC 33 = 2 (−m11 + m22 + m33 + m44 ) 1 L L L L mC 44 = 2 (m11 − m22 + m33 + m44 )

(3.83)

Consequently, both matrices ML and MC can be found by solving the vector radiative transfer equation. Since there is a one-to-one correspondence between Mueller matrix M and coherency matrix J, from Eq. (3.82) one can write for overall coherency matrix: J = J1 + JL + JC (3.84)

96

S.N. Savenkov

From Eqs (3.70) and (3.84) it can be seen that: ⎛ ⎞ J11 J12 0 J14 ⎜J21 J22 0 J24 ⎟ ⎟ JL = ⎜ ⎝ 0 0 J33 0 ⎠ J41 J42 0 J44 ⎛ ⎞ J11 J12 0 J14 ⎜J21 J22 0 J24 ⎟ ⎟ JC = ⎜ ⎝ 0 0 −J33 0 ⎠ J41 J42 0 J44

(3.85)

(3.86)

The overall entropy is a sum of the following contributions: the ﬁrst relates with the form of a Mueller matrix that describes a single scattering event for array of particles and the second relates with JL , so that it is determined by size, shape and angular distribution of particles. If these two components have similar eigenvectors of corresponding coherency matrices Eq. (3.84) then the problem has low entropy, even for multiple scattering. The high value of entropy associates with the case when eigenvectors are nearly orthogonal. As an example of the case when the Mueller matrix has the structure of Eq. (3.37) and contains information on depolarization and depolarization depends pronouncedly on the polarization state of input light, consider the exactly forward scattering of polarized light by a slab of inhomogeneous linear birefringent medium (Savenkov et al., 2007a). Inhomogeneity is due to the output side (where radiation exits) of the slab has random roughness and insures the single scattering regime. When inhomogeneity σh is large compared with wavelength λ, i.e., the following condition: σh λ/(n1 − n2 ) (3.87) is satisﬁed, then Mueller matrix Eq. (3.37) ⎛ 1 ms12 ⎜ms12 1 ⎜ ⎝ 0 0 0 0

takes the form ⎞ 0 0 0 0⎟ ⎟. 0 0⎠ 0 0

(3.88)

The value of matrix element m12 is found as follows ms12 =

(n2 − 1)2 − (n1 − 1)2 , (n2 − 1)2 + (n1 − 1)2

(3.89)

where ni are refraction indexes of birefringent medium. Condition Eq. (3.87) is not very hard to fulﬁll and, say, for calcite CaCO3 at λ = 0.63 μm it is already satisﬁed for σh ≈ 8 ÷ 10 μm. For the Mueller matrix Eq. (3.88) Fig. 3.8 presents the theoretical and experimental dependences of polarization degree, p, and intensity, I, of output light for linear polarized input light with azimuth of polarization scanning over 0 ÷ 180◦ region (Savenkov et al., 2005). p takes all physically acceptable values in light scattering by such medium 0 ≤ p ≤ 1. In that way, the eﬀect of so-called polarization memory is observed: the medium ‘keeps in mind’ the input linear polarizations

3 Jones and Mueller matrices

97

p, I 1,2 1,0 0,8 0,6 0,4

p (theor) I (theor) I (exper) p (exper)

0,2 0,0 -0,2 -20

0

20

40

60

80

T,

100 120 140 160 180 200 0

Fig. 3.8. Theoretical and experimental dependences of the intensity, I, and the polarization degree, p, of output light versus azimuth θ of incident linearly polarized light. σh = 20 μm (after Savenkov et al., 2005).

with some azimuths (0◦ and 90◦ ) perfectly (light remains completely polarized) and ‘forgets’ completely the input linear polarizations with some other azimuths (approximately 60◦ and 120◦ ) (light becomes unpolarized). It can be seen that the presence of surface inhomogeneity results in dependence of intensity I of output light on azimuth of input linear polarization as well. Hence, for exact forward scattering direction an inhomogeneous linear birefringent medium exhibits partial dichroism. At that, the ratio of minimal and maximal values of intensity is a function of refraction indexes ni (see Eq. (3.89)) and determined by 1 − ms12 (n1 − 1)2 Imin = = ≈ 0.4 Imax 1 + ms12 (n2 − 1)2

(3.90)

Note that in scope of additive matrix model based on the Cloude’s coherency matrix Eq. (3.72) and Fig. 3.4 the Mueller matrix Eq. (3.88) for the case of inhomogeneous linear birefringent medium is represented as weighed sum of two matrices of ideal (singular) linear polarizers with crossed planes of transmittance. Depolarization depends signiﬁcantly on polarization state of input light in multiple scattering as well (Bicout et al., 1994; Rojas-Ochoa et al., 2004; Kim et al., 2006, and references therein). In particular, Bicout et al. (1994) studied numerically and experimentally how depolarization proceeds for linear and circular input polarizations as the size of the particles increases from very small (Rayleigh regime) to large (Mie regime) for forward scattering geometry. They considered a planeparallel slab of ﬁnite thickness d composed of uncorrelated spherical particles of radius a. For the degree of polarization of the light transmitted by the scattering medium for linearly, pL , and circularly, pC , polarized input light they derived: l d 2d sinh exp − (3.91) pL,C l ξL,C ξL,C

98

S.N. Savenkov

where l is a mean free path and ξL,C are terms deﬁne the characteristic lengths of depolarization for linear (L) and circular (C) input polarizations in the slab geometry. The beam of light (λ = 0.67 μm) normally impinges on one side of the slab (3 mm thickness) and the scattered light transmitted through back side of the slab was detected within a collection angle of 2◦ . The samples, which ensure multiple scattering of input light, are suspensions of polystyrene spheres with mean diameters of 0.22 and 1.05 μm in distilled water at diﬀerent concentrations. Figure 3.9 shows the dependencies of ratio of linear to circular degree of polarization pL /pC as a function of d/l∗ . Dashed lines are ﬁt to Eq. (3.91). Symbols indicate experimental data corresponding to measurements on suspensions of polystyrene latex spheres in water: 0.22 μm (circle); 1.05 μm (square). It can be seen that pL /pC increases in the qa 1 (Rayleigh) regime, decreases in qa > 1 (Mie) regime and is constant in the intermediate regime.

Fig. 3.9. Dependencies of the ratio of the degrees of polarization pL /pC as a function of d/l∗ for three values of size parameters qa (after Bicout et al., 1994).

The character of the dependences of pL /pC the authors explain by the fact that for various values of size parameters qa a various character of scattering is observed: in the Mie regime scattering is predominantly in the forward direction while in the Rayleigh regime forward and backward scattering directions are equally presented. Linear polarization is not aﬀected by backscattering regardless of particle size, whereas backscattering for circular polarization results in helicity ﬂip.

3.6 Structure and information content of the Mueller–Jones matrix in continuous medium approximation The increased interest in the Jones and Mueller–Jones matrices is conditioned by the following circumstances. First, these matrices describe the interaction between polarized light and medium without depolarization, i.e., the problem which is a subject of crystal optics. Second, as can be seen from the discussion of section 3.5.3, the Jones and the Mueller–Jones matrices are the key concept of the additive matrix model of a scattering medium being described by a depolarizing Mueller matrix,

3 Jones and Mueller matrices

99

Fig. 3.4. Indeed, if the entropy S is low (typically <0.5), the medium is weakly depolarizing and the Jones matrix in Eq. (3.72) associated with the eigenvector corresponding to the largest eigenvalue of the coherency matrix is dominant and describes the anisotropy properties of a given medium. If the entropy S is high (typically >0.5), the medium is strongly depolarizing and in this case all four Jones matrices, Fig. 3.4, consist of information about the medium (Cloude and Pottier, 1995). In this section we consider the structures and information content of Jones and Mueller–Jones matrices of homogeneous anisotropic (deterministic) media (Shurcliﬀ, 1962; Azzam and Bashara, 1977; Brosseau, 1998). Note that the results for the internal structure of the general Mueller–Jones matrix and symmetry relations caused by interchanging the incident and emerging light beams and the principle of reciprocity presented in the preceding sections and obtained in the framework of light scattering by particles are relevant for the Jones and Mueller–Jones matrices of homogeneous anisotropic (deterministic) media in continuous medium approximation. 3.6.1 Mueller–Jones matrices of basic types of anisotropy and partial equivalence theorems Polarization of light changes if either amplitudes and phases of components of electric vector E taken separately or both of them simultaneously change (Azzam and Bashara, 1977; Shurcliﬀ, 1962; Brosseau, 1998). It is therefore accepted to distinguish between corresponding classes of anisotropic media: dichroic (or possessing amplitude anisotropy), which inﬂuence only the amplitudes; birefringent (or possessing phase anisotropy), inﬂuencing only the phases; and ‘all other’ (possessing amplitude and phase anisotropy) which inﬂuence both the amplitudes and the phases of the components of the electric vector simultaneously. Among these classes, four types of anisotropic mechanisms are recognized as basic (or primitive) (Jones, 1941; Hurwitz and Jones, 1941): linear and circular phase and linear and circular amplitude anisotropies. The Jones and Mueller–Jones matrices of linear phase anisotropy are 2 cos α + sin2 α exp(−iΔ) cos α sin α[1 − exp(−iΔ)] TLP = , cos α sin α[1 − exp(−iΔ)] sin2 α + cos2 α exp(−iΔ) ⎞ ⎛ LP

M

1

0

0

0

0

sin(2α) sin(Δ)

− cos(2α) sin(Δ)

cos(Δ)

⎜0 cos2 (2α) + sin2 (2α) cos(Δ) cos(2α) sin(2α)[1 − cos(Δ)] − sin(2α) sin(Δ)⎟ =⎝ 0 cos(2α) sin(2α)[1 − cos(Δ)] sin2 (2α) + cos2 (2α) cos(Δ) cos(2α) sin(Δ) ⎠ (3.92)

where Δ is a value, i.e., phase shift between two orthogonal linear components of the electric vector, and α is an azimuth of the anisotropy.

100

S.N. Savenkov

Linear amplitude anisotropy has the matrices cos2 γ + P sin2 γ cos γ sin γ(1 − P ) LA , = T cos γ sin γ(1 − P ) sin2 γ + P cos2 γ ⎛ 1+P (1 − P ) cos(2γ) √ 2 2 ⎜ (1 − P ) cos(2γ) cos (2γ)(1 + P ) + 2 sin√ (2γ) P LA ⎜ M =⎝ (1 − P ) sin(2γ) cos(2γ) sin(2γ)(1 − P )2 0 0 ⎞ (1 − P ) sin(2γ)√ 0 0 ⎟ cos(2γ) sin(2γ)(1 − P )2√ ⎟ 2 2 0 ⎠ sin (2γ)(1 + P ) + 2 cos (2γ) P √ 0 2 P

(3.93)

where P is a value (relative absorption of two linear orthogonal components of the electric vector) and γ is an azimuth of the anisotropy. Circular phase anisotropy has the matrices: cos ϕ sin ϕ , TCP = − sin ϕ cos ϕ ⎛ ⎞ 1 0 0 0 ⎜0 cos(2ϕ) sin(2ϕ) 0⎟ ⎟ MCP = ⎜ (3.94) ⎝0 − sin(2ϕ) cos(2ϕ) 0⎠ , 0 0 0 1 where ϕ is a phase shift introduced for two orthogonal circular components of the electric vector. Finally, the Jones and Mueller–Jones matrices of circular amplitude anisotropy are 1 −iR TCA = , iR 1 ⎛ ⎞ 1 + R2 0 0 2R ⎜ 0 0 0 ⎟ 1 − R2 ⎟, MCA = ⎜ (3.95) 2 ⎝ 0 0 ⎠ 0 1−R 2R 0 0 1 + R2 where R is a value of the anisotropy, i.e., relative absorption of two orthogonal circular components of the electric vector. Six quantities α, Δ, P , γ, ϕ, and R are called anisotropy parameters. The ranges of their physically realizable values are 0 ≤ ϕ ≤ 2π 0 ≤ Δ ≤ 2π −π/2 ≤ α ≤ π/2 0≤P ≤1 −π/2 ≤ γ ≤ π/2 −1 ≤ R ≤ 1

(3.96)

3 Jones and Mueller matrices

101

It can be seen that elements of matrices of linear Eq. (3.92) and circular Eq. (3.94) phase anisotropy satisfy t11 = t∗22 ;

t12 = −t∗21 ,

| det(T )| = 1 ,

(3.97)

i.e., these matrices have structures corresponding to class of unitary matrices (in case of matrices with real elements – orthogonal matrices). Diagonal elements of the matrices of linear Eq. (3.93) and circular Eq. (3.95) amplitude anisotropy are real and non-diagonal: t12 = t∗21 , (3.98) i.e., these matrices belong to the class of Hermitian matrices (in case of matrices with real elements – symmetric matrices). It is evident that, in general, the arbitrary medium can be characterized by these four types of anisotropy simultaneously. The history of matrix models of anisotropic media goes back to the early works of Jones, who in 1941 introduced the formalism of 2 × 2 matrix (Jones, 1941) describing the response of anisotropic media to the incident polarized light. Relying on this approach, Hurwitz and Jones proved three so-called equivalence theorems (Hurwitz and Jones, 1941) providing the basis of the matrix models for a number of classes of homogeneous anisotropic media. According to the ﬁrst theorem, any combination of polarization elements with circular and linear phase anisotropy (i.e., rotators and retardation plates) can be presented by an optical system consisting of only two elements, one with circular and the other with linear phase anisotropy. The Jones matrix of the medium corresponding to the ﬁrst Jones equivalence theorem has the form: cos ϕ cos2 α + sin2 α exp(−iΔ) + 12 sin ϕ sin 2α 1 − exp(−iΔ) CP LP T T = − sin ϕ cos2 α + sin2 α exp(−iΔ) + 12 cos ϕ sin 2α 1 − exp(−iΔ) sin ϕ sin2 α + cos2 α exp(−iΔ) + 12 cos ϕ sin 2α1 − exp(−iΔ) cos ϕ sin2 α + cos2 α exp(−iΔ) − 21 sin ϕ sin 2α 1 − exp(−iΔ) (3.99) Note that the Jones matrix Eq. (3.99) is a matrix model of an arbitrary linear crystalline medium in the transmission band. It can be seen that structure of the matrix Eq. (3.99) satisﬁes the conditions Eq. (3.97) and, hence, is unitary. The second theorem is analogous to the ﬁrst and deals with combinations of elements with circular phase and amplitude anisotropy (rotators and partial polarizers). Finally, the third theorem states that any combination of elements with amplitude and phase anisotropy is equivalent to an optical system containing only four elements: two with linear phase, one with circular phase, and one with linear amplitude anisotropy. Some other partial equivalence theorems can be found elsewhere (Whitney, 1971; Barakat, 1998). 3.6.2 Polar decomposition of Mueller–Jones matrices The development of matrix descriptions of media properties motivated further research in the ﬁeld based on the methods of linear algebra and matrix analysis. In particular, Barakat (1998) gave an analytical proof of the third Jones equivalence

102

S.N. Savenkov

theorem in terms of singular decomposition (Lancaster and Tismenetsky, 1985). Another approach to modeling homogeneous anisotropic media, alternative to that of Jones, is based on the polar decomposition theorem (Lancaster and Tismenetsky, 1985). According to this theorem, an arbitrary matrix T can be presented by a product T = TP TR or T = TR TP (3.100) where TP , and TP are Hermitian matrices and TR is a unitary one. The Hermitian matrix is associated with amplitude anisotropy, and the unitary matrix with phase anisotropy (Whitney, 1971). The matrices TP and TR are called dichroic and phase Jones polar forms (Whitney, 1971; Gil and Bernabeu, 1987; Lu and Chipman, 1996). Polar decomposition for polarization theory was ﬁrst employed in Whitney (1971) without, however, ﬁnding explicit expressions for TP and TR . They were obtained later independently by Gil and Bernabeu (1987) and Lu and Chipman (1996). Alternatively, dichroic and phase polar forms may be presented relying on the spectral problem of linear algebra (Azzam and Bashara, 1977). The models of anisotropic media based on polar decomposition contain six independent parameters, three for the phase TR and three for the dichroic TP polar forms, for four complex elements of the Jones matrix. Two additional degrees of freedom are associated with isotropic changes of the phase and amplitude of the light propagating in the medium. Analogously to Eq. (3.100), polar decomposition can be realized in terms of Mueller–Jones polar forms MP and MR . Phase polar form MR (using notations from Lu and Chipman (1996)) has the form: 1 0T (3.101) MR = 0 mR 3 εijk ak sin(R) , (mR )ij = δij cos(R) + ai aj 1 − cos(R) +

i, j = 1, 3

k=1

ˆ T )T is the normalized Stokes where 0 is a 3 × 1 zero vector, (1 a1 a2 a3 )T = (1 R vector for the fast axis of MR , δij is the Kronecker delta, εijk is the Levi–Civita permutation symbol, mR is a 3 × 3 submatrix of matrix MR obtained by striking out the ﬁrst row and the ﬁrst column of MR and R is the retardance and can be obtained as: Tr (MR ) −1 (3.102) R = arccos 2 The dichroic polar form MP is: MP = Tu mP =

1 DT D mP

,

' & ˆD ˆT , 1 − D2 I + 1 − 1 − D2 D

(3.103) (3.104)

ˆ = D/|D| denotes the unit vector along where I is the 3 × 3 identity matrix, D the diattenuation vector D, and Tu is the transmittance for unpolarized light; the

3 Jones and Mueller matrices

value of diattenuation can be obtained as 1/2 4| det(T)|2 D = 1− [Tr (T∗ T)]2

103

(3.105)

So, we can see that the Mueller–Jones phase polar form is a unitary (orthogonal) matrix and the Mueller–Jones dichroic polar form is a Hermitian (symmetric) matrix. Note that unitarity (orthogonality) of the Mueller–Jones phase polar form, Eq. (3.101) is in complete accordance with the ﬁrst Jones equivalence theorem, Eq. (3.99), and is a general model of birefringence medium. As we see below, the situation with dichroic polar is more complex. This originates from the fact that, in contrast to unitary matrices, the product of Hermitian matrices is generally not a Hermitian matrix (Lancaster and Tismenetsky, 1985). The product of Mueller matrices of polar forms Eqs (3.101) and (3.103) and the depolarizing Mueller matrix Eq. (3.68) is called the generalized polar decomposition and multiplicative matrix model of an arbitrary Mueller matrix (Lu and Chipman, 1996; Gil, 2007). The eﬀect of the order of multiplication of matrices on the results of decomposition has been studied by Morio and Goudail (2004) and Ossikovski et al. (2007). 3.6.3 Generalized matrix equivalence theorem A direct consequence of the generality of polar and singular decompositions is that they can be employed for representing an arbitrary deterministic medium. One has, however, to pay the price for using these formal mathematical approaches by losing in physical interpretability of the decomposition results (Savenkov et al., 2005). The most signiﬁcant issue for any model of media would be its physical validity (Boerner, 1992). Evidently, only those matrix models that simultaneously take into account both the inertia of medium properties and the nonlocality of the medium response to light (i.e., time and spatial dispersion) can be accepted as rigorous and adequate. It is known that these properties of a crystalline medium determine, in the general case, the character of its anisotropy, namely, linear amplitude and phase, and circular amplitude and phase anisotropies (see, for example, Landau et al., 1984; Berry and Dennis, 2003). The subject of this section is an analysis of the structure of the Jones matrix of arbitrary anisotropic medium. The Jones matrix of an arbitrary anisotropic medium is derived on the basis of the so-called generalized equivalence theorem (Savenkov et al., 2006), which is a direct generalization of the ﬁrst and second Jones equivalence theorems. Accepting the style of Jones (Hurwitz and Jones, 1941), in (Savenkov et al., 2006) the following theorem is formulated: Any combination of elements with linear and circular phase and linear and circular amplitude anisotropy is equivalent to an optical system containing only one element of each kind in the order TGen = TCP TLP TCA TLA

(3.106)

Formulated in the form of Eq. (3.106) the generalized equivalence theorem is free from the drawbacks of both polar and singular decompositions. First, the results of

104

S.N. Savenkov

the suggested decomposition can be directly used for an analysis of the anisotropy of a medium, which is based on physically meaningful and realizable parameters. Second, in contrast to singular decomposition (or the third Jones equivalence theorem), Eq. (3.106) includes circular amplitude anisotropy. Therefore, it provides a more physically adequate and complete description of the anisotropic properties of the general case of media possessing all four kinds of anisotropy mechanisms. The fact is that, although circular amplitude anisotropy can operationally result in transformations of light polarization which are analogous to some forms of linear phase anisotropy (half-wave plate), nevertheless, physically circular amplitude anisotropy diﬀers in principle from linear phase anisotropy and, therefore, must be presented in a general model of a medium. An explicit form of the elements of the matrix TGen Eq. (3.106) follows: ˆ ´˜ ˆ ˜ ` ´ ` t11 = sϕ cα sα 1 − exp(−iΔ) + cϕ c2α + s2α exp(−iΔ) c2γ + s2γ P − iRcγ sγ (1 − P ) ˆ ` ´ ` ´˜ ` ´˜ ˆ + sϕ s2α + c2α exp(−iΔ) + cϕ cα sα 1 − exp(−iΔ) cγ sγ (1 − P ) + iR c2γ + s2γ P ˆ ´˜ ˆ ˜ ` ´ ` 2 t21 = cϕ cα sα 1 − exp(−iΔ) + sϕ cα + s2α exp(−iΔ) c2γ + s2γ P − iRcγ sγ (1 − P ) ˆ ` ´ ` ´˜ ` ´˜ ˆ + cϕ s2α + c2α exp(−iΔ) + sϕ cα sα 1 − exp(−iΔ) cγ sγ (1 − P ) + iR c2γ + s2γ P ˆ ´˜ ˆ ` ´˜ ` ´ ` t12 = sϕ cα sα 1 − exp(−iΔ) − cϕ c2α + s2α exp(−iΔ) cγ sγ (1 − P ) − iR s2γ + c2γ P ˆ ` ´ ˜ ` ´˜ ˆ + sϕ s2α + c2α exp(−iΔ) − cϕ cα sα 1 − exp(−iΔ) s2γ + c2γ P + iRcγ sγ (1 − P ) ˆ ´˜ ˆ ` ´˜ ` ´ ` t22 = cϕ cα sα 1 − exp(−iΔ) − sϕ c2α + s2α exp(−iΔ) cγ sγ (1 − P ) − iR s2γ + c2γ P ˆ ` ´ ˜ ` ´˜ ˆ + cϕ s2α + c2α exp(−iΔ) − sϕ cα sα 1 − exp(−iΔ) s2γ + c2γ P + iRcγ sγ (1 − P ) (3.107)

where sϕ = sin(ϕ), cϕ = cos(ϕ); sα = sin(α), cα = cos(α), sγ = sin(γ), cγ = cos(γ). It is important to note that matrix Eq. (3.107) generally does not satisfy either conditions Eq. (3.97) or conditions Eq. (3.98). Assuming that det(TGen ) = 0, from Eq. (3.107) we derive the expressions for parameters of anisotropy: for the azimuth of linear amplitude anisotropy γ: 2 Re (t22 t∗21 + t11 t∗12 ) 1 , γ = − arctan 2 2 |t12 | + |t22 |2 − |t11 |2 − |t21 |2 for the value of linear amplitude anisotropy P : ( |t12 cos γ − t11 sin γ| + |t22 cos γ − t21 sin γ| , P = |t12 sin γ + t11 cos γ| + |t22 sin γ + t21 cos γ|

(3.108)

(3.109)

for the circular amplitude anisotropy R: R= ` ´ q 2 ∗ − |t11 |2 +|t21 |2 +|t12 |2 +|t22 |2 ± (|t11 |2 +|t21 |2 +|t12 |2 +|t22 |2 )2 −4[Im (t∗ 11 t12 +t22 t21 )] ∗ −2Im (t∗ 11 t12 + t22 t21 )

(3.110)

where tij are the elements of the following matrix

3 Jones and Mueller matrices

T = TGen (TLA )−1 = TCP TLP TCA .

105

(3.111)

The parameters ϕ, α, and Δ can then be directly determined from the following equations: t12 − t21 , (3.112) ϕ = arctg t11 + t22 t21 + t12 1 α= + ϕ , (3.113) arctg 2 t11 − t22 (t + t21 ) cos(ϕ) Δ = −2 arctg i 12 , (3.114) (t11 + t22 ) sin(ϕ − 2α) where tij are the elements of the following matrix T = T (TCA )−1 = TCP TLP .

(3.115)

Obviously, a similar theorem can be formulated for the Mueller–Jones matrices as well (Savenkov et al., 2006). The fact that the parameters used in this theorem characterize physically realizable anisotropy mechanisms permits not only clear physical interpretation of the anisotropic properties of a medium but also synthesis of the systems (polarization elements) with arbitrary predetermined polarization properties. Matrix Eq. (3.107) describes the generalized eﬀect operator of anisotropic media on the polarized light in terms of the Jones and Mueller matrix formalisms, respectively. Therefore, Eq. (3.107) may be regarded as generalized polarimetric matrix model of an arbitrary deterministic medium or, in other words, as the basic relation of algebraic (or operator) optics of anisotropic media (Rozenberg, 1977). The anisotropy parameters Eqs (3.108)–(3.115) are then the general solution of the inverse problem of crystal optics based on the model Eqs (3.107). In addition, Eq. (3.107) determines the general matrix form of the polarization transfer equation (Brosseau, 1995) for homogeneous anisotropic media. Measurement of the Mueller matrices and determination of the anisotropy parameters for a corresponding number of directions in the studied medium can then be considered as a content of the method of Mueller tomography (Azzam and Bashara, 1977; Le Roy-Brehonnet and Le Jeune, 1997) for the given class of media. Furthermore, the generalized equivalence theorem clariﬁes the matter and physical interpretation of the non-uniqueness of the solutions of the inverse problem as well as the invariance of describing the anisotropic properties from an electrodynamic point of view. Non-uniqueness of the inverse problem is a result of the fact that the matrices of the primitive anisotropic mechanisms Eqs (3.92)–(3.95) are not interchangeable in Eq. (3.106), the latter gives several diﬀerent ways in which the matrices Eqs (3.92)–(3.95) can be multiplied (Lu and Chipman, 1996; Morio and Goudail, 2004; Ossikovski et al., 2007). The importance of the formulated problems was addressed in a number of earlier publications (Rozenberg, 1973, 1977; Gil and Bernabeu, 1987; Lu and Chipman, 1996; Morio and Goudail, 2004; Ossikovski et al., 2007). The models of the homogeneous anisotropic media available at the moment did not, however, allow their further development. The proven theorem provides a basis for a systematic investigation of these problems.

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3.6.4 Eigenanalysis of the Jones matrices of dichroic, birefringent, and degenerate media Eigenpolarizations and eigenvalues (Lancaster and Tismenetsky, 1985) completely characterize the optical medium (Landau et al., 1984) and determine the structure and inner symmetry of the corresponding matrix models (Azzam and Bashara, 1977; Shurcliﬀ, 1962; Jones, 1941). We use eigenanalysis to derive and analyze the Jones matrix of dichroic, birefringence media and media with degenerate anisotropy. Depending on eigenvalues and eigenpolarizations, the media and associated Jones matrices are classiﬁed as homogeneous and inhomogeneous (Shurcliﬀ, 1962). To avoid the confusion in using the terminology ‘homogeneous and inhomogeneous medium (polarization element)’ accepted in laser polarimetry (Lu and Chipman, 1994, 1996; Sudha and Gopala Rao, 2001) with that used in scattering media optics we will use here the terms ‘homogeneous and inhomogeneous Jones and Mueller– Jones matrices.’ The homogeneous Jones matrix has two orthogonal eigenpolarizations: E+ e1 Ee2 = 0 ,

(3.116)

where + denotes the Hermitian conjugate. The inhomogeneous Jones matrix has nonorthogonal eigenpolarizations. The case when eigenpolarizations of the Jones matrix are coincident is called degenerate anisotropy (Pancharatnam, 1955a, 1955b; de Lang, 1966; Meira-Belo and Leitao, 2000; Mosino et al., 2002). The Jones matrices describing phase and dichroic polar forms Eqs (3.101) and (3.103) discussed in section 3.6.2 are considered as either generalized elliptical retarders (medium with phase anisotropy), or generalized elliptical dichroic medium (medium with amplitude anisotropy). Hence, generalized media with phase and amplitude anisotropy, each taken separately, have been heretofore assumed to possess orthogonal eigenpolarizations (Azzam and Bashara, 1977) and a single ‘origin’ of nonorthogonality of eigenpolarizations of medium is simultaneous presence of phase and amplitude anisotropy. From the Jones’ equivalence theorem (Hurwitz and Jones, 1941) for the medium with general phase anisotropy, this is indeed true. The case for a medium with general amplitude anisotropy turns out to be more complicated and is addressed in this subsection. Making only the most general assumptions on properties of crystal media, below we analyze the structure of the generalized Jones matrices for birefringent and dichroic media. It is convenient to describe polarization state in this case by the complex variable deﬁned by (Azzam and Bashara, 1977): χ = |Ey |/|Ex | exp[j(δy − δx )]

(3.117)

The connection between incident and output polarizations can be written in terms of complex variable Eq. (3.117) in the form: χ0 =

t21 + t22 χi t11 + t12 χi

(3.118)

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If a medium is described by the Jones matrix, T, its eigenvectors Ee1 and Ee2 satisfy the following equations (Lancaster and Tismenetsky, 1985): TEe1 = V1 Ee1 ,

TEe2 = V2 Ee2

(3.119)

where, V1 , V2 are the eigenvalues of T, which are in general complex values. Eigenvectors Ee1 and Ee2 are presented in terms of complex variable Eq. (3.117) by: 1 Ee1,2 = . (3.120) χe1,2 For the Jones matrix, the eigenpolarizations and eigenvalues are, respectively (Azzam and Bashara, 1977): 1 t22 − t11 ± (t22 − t11 )2 + 4t12 t21 , (3.121) χe1,2 = 2 t12 ' 1& (3.122) t22 + t11 ± (t22 − t11 )2 + 4t12 t21 V1,2 = 2 If eigenpolarizations and eigenvectors do not coincide, then the Jones matrix elements can be obtained from Eqs (3.121) and (3.122): 1 V2 χe1 − V1 χe2 V1 − V2 T= (3.123) χe1 − χe2 −(V1 − V2 )χe1 χe2 V1 χe1 − V2 χe2 The structure of the matrix Eq. (3.123) illustrates once more the fact, which we pointed out above for matrix Eq. (3.107), that the Jones matrix describing general homogeneous anisotropic medium is neither unitary nor Hermitian. 3.6.4.1 Medium characterized by phase anisotropy In the most general case the birefringent medium, i.e., the medium which is characterized only by changes to the phases of the components of the electric vector E and by the isotropic changes of the amplitude of the electric vector of incident light (Lu and Chipman, 1994), is completely characterized by the following conditions (Savenkov et al., 2005): ) ) ) V1 ) ) )=1 (3.124) ) V2 ) χe1 χ∗e2 = −1 It is important to note that the orthogonality of eigenpolarizations of the birefringent medium is a consequence of the deﬁnition of birefringence rather than a part of this deﬁnition. Eq. (3.124) results in the following form of the generalized Jones matrix for birefringent media: |V | T= |χe |2 + 1 ⎛ ⎞ & ' & ' φ φ φ φ 2 exp i exp(−iψ)⎟ + |χe | exp −i |χe | exp i 2 − exp −i 2 ⎜ ⎝ & ' 2 & ' ⎠ &2 ' & ' |χe | exp i φ2 − exp −i φ2 exp(iψ) |χe |2 exp i φ2 + exp −i φ2 (3.125)

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where φ and ψ are the phases of eigenvalues and eigenpolarizations respectively. It can be seen that structure of the generalized Jones matrix of birefringent medium is characterized by relations Eq. (3.97). Constant |V | representing the degree of isotropic absorption is present in Eq. (3.125) as a general multiplier and it, obviously, does not inﬂuence the polarization properties of an element. Thus, having the eigenvalues and the eigenpolarizations for a given realization of the Jones matrix, T, using Eq. (3.124), it is possible to specify whether the medium, described by this matrix, belongs to the class characterized by phase anisotropy. If, however, the aim is to analyze the anisotropy of birefringent media, or to synthesize a polarization element with given properties, then it needs to have the conditions analogous to Eq. (3.124) in terms of anisotropy parameters Eq. (3.96). This can be done by applying the model of Eq. (3.106) to the deﬁnitions in Eqs (3.121), (3.122) and (3.124). It can be shown (Savenkov et al., 2007b) that a medium has birefringent eigenvalues, Eq. (3.124), when Δ Δ sin ϕ − (P − 1) sin cos ϕ − 2(α − γ) = 0 R(P + 1) cos 2 2 2 Δ Δ cos ϕ + R(P − 1) sin sin ϕ − 2(α − γ) − 4P (1 − R2 ) < 0 (P + 1) cos 2 2 (3.126) The eigenvalues are then given by Δ Δ Δ 1 (1 + P ) cos cos ϕ − R(1 − P ) sin sin ϕ − 2(α − γ) V1,2 = exp −i 2 2 2 2 ⎞ ( 2 Δ Δ ± i 4P (1−R2 )− (1 + P ) cos cos ϕ−R(1−P ) sin sin(ϕ−2(α−γ)) ⎠ 2 2 (3.127) Eigenpolarizations of a birefringent medium are orthogonal, as determined by the second equation in Eq. (3.124). Analysis of the orthogonality for eigenpolarizations of the generalized Jones matrix Eq. (3.106) together with Eq. (3.126) leads to P = 1 and R = 0. This means that a general birefringent medium consists of only two basic elements: a retarder and rotator; dichroic components (linear and circular diattenuators) are absent. The Jones matrix, for which Eq. (3.126) holds, can be considered as the polar form in the polar decomposition theorem (Whitney, 1971; Gil and Bernabeu, 1987; Lu and Chipman, 1994, 1996). 3.6.4.2 Medium characterized by amplitude anisotropy The anisotropic properties of general dichroic media, despite their signiﬁcance, have been analyzed much less thoroughly. Most studies of dichroic media either rely on symbolic operator approaches or deal with particular examples of these elements (Azzam and Bashara, 1977; Chipman 1995; Savenkov, et al., 2005; de Lang, 1966;

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Lu and Chipman, 1994; Sudha and Gopala Rao, 2001; Tudor, 2003, 2006). It is known, for example, that the simultaneous presence of dichroism and birefringence in a medium may lead to nonorthogonal eigenpolarizations of the latter (Tudor, 2006; Moxon and Renshow, 1990; Savenkov et al., 2006; Tudor and Gheondea, 2007). The questions that arise are: what combinations of anisotropic mechanisms would make a medium demonstrate a dichroic behavior and whether the simultaneous presence of dichroism and birefringence is a necessary or a suﬃcient requirement for nonorthogonal eigenpolarizations? The medium, which is characterized only by changes to the amplitudes of the components of the electric vector, E, is called dichroic. It is also assumed that the isotropic changes to the phase of components of the electric vector of incident light are present (Lu and Chipman, 1994). Note that, unlike intensity, phase-shift cannot be the same for any polarization basis. Dichroic media preserve relative phase (or, equivalently, make isotropic phase-shifts) only in a speciﬁc basis, i.e., in the eigenbasis. Therefore, in general, the dichroic medium is completely characterized by the following conditions (Savenkov et al., 2005): V1 Im =0 V2 (3.128) V1 Re ≥0 V2 Then, taking into account Eqs (3.123) and (3.128), the corresponding generalized Jones matrix of the dichroic medium has a form: exp(iφ) |A| − |B| |B|χe1 − |A|χe2 . (3.129) T= χe1 − χe2 −χe1 χe2 (|A| − |B|) |A|χe1 − |B|χe2 Thus, Eq. (3.129) presents the structure of the generalized Jones matrix of the dichroic medium and this Jones matrix in contrast to the dichroic polar form Eq. (3.103) is not Hermitian. The fact that the generalized dichroic Jones matrix, Eq. (3.129), is not a non-negative deﬁnite Hermitian matrix allows us to assert that in the case when medium is characterized by presence of phase and amplitude anisotropy simultaneously, the Hermitian polar form by itself, which is derived in polar decomposition theorem, cannot not be an adequate and exhaustive description of the dichroic properties of the medium. The generalized Jones matrix of the dichroic medium Eq. (3.129) can be considered as new general equivalence theorem, namely (in terms of the Jones equivalence theorems (Hurwitz and Jones, 1941)): ‘any combinations of the polarization elements with linear and circular amplitude anisotropy will have a resulting matrix of the form’ Eq. (3.129). Indeed, the eigenvalues of any combination of polarization elements with linear and circular amplitude anisotropy have to satisfy the conditions Eq. (3.128), hence, the resulting Jones matrix of the combination of polarization elements will have the form Eq. (3.129). If we have the case when the determinant of the Jones matrix is zero, one of the eigenvalues equals zero. In which case, the output polarization is the same for all incident polarizations. Phase shifts can be regarded as isotropic and, therefore, polarization elements with one zero eigenvalue are still dichroic.

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From Eq. (3.128) and taking into account Eq. (3.106) for dichroic eigenvalues we get Δ Δ R(1 + P ) cos sin ϕ + (1 − P ) sin cos[ϕ − 2(α − γ)] = 0 2 2 2 Δ Δ cos ϕ − R(1 − P ) sin sin[ϕ − 2(α − γ)] − 4P (1 − R2 ) > 0 , (1 + P ) cos 2 2 (3.130) The corresponding eigenvalues are 1 Δ Δ Δ (1 + P ) cos cos ϕ − R(1 − P ) sin sin(ϕ − 2(α − γ)) V1,2 = exp −i 2 2 2 2 ⎤ ( 2 Δ Δ ± (1+P ) cos cos ϕ−R(1−P ) sin sin[ϕ−2(α−γ)] −4P (1−R2 )⎦ 2 2 (3.131) As can be seen from Eq. (3.128), a general dichroic medium is completely determined by the eigenvalues of its Jones matrix; in contrast to the birefringent medium, Eq. (3.124), there are no restrictions for the corresponding eigenpolarizations. The case of orthogonal eigenpolarizations, however, usually receives increased attention in the literature, and it has been a milestone for classifying media and studying their properties (Azzam and Bashara, 1977; Chipman 1995; Savenkov et al., 2005; Sudha and Gopala Rao, 2001). Here, therefore, we discuss the properties of arbitrary dichroic medium with orthogonal eigenpolarizations. From Eqs (3.106) and (3.121) and taking into account Eq. (3.128), eigenpolarizations of dichroic elements are orthogonal when (Savenkov et al., 2007b): ϕ = 0, π; α = π/4 + γ Δ 1−P =− R, tan 2 1+P ϕ = 0, π; α = −π/4 + γ Δ 1−P tan = R. 2 1+P

(3.132)

(3.133)

The two possibilities, Eqs (3.132) and (3.133), diﬀer in the relative orientation of the linear birefringence, α, and the linear dichroism, γ, and in the sign of the phase-shift Δ. Note that Eqs (3.132) and (3.133) are written for the birefringent parameters Δ, α, and ϕ, because, obviously, no restrictions could be placed on the dichroic parameters, P , R, and γ. Then, from Eq. (3.106), the generalized Jones matrix of dichroic elements with orthogonal eigenpolarizations is of the form Δ 1 H, (3.134) T = ± exp −i 2 2

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where the upper sign should be taken for ϕ = 0 and the lower for ϕ = π in Eqs (3.132) and (3.133). H is a Hermitian matrix given by 1 Q1 + Q2 cos 2γ Q2 sin 2γ − 4iP R , (3.135) H= √ Q1 Q2 sin 2γ + 4iP R Q1 − Q2 cos 2γ where Q1 = (1 + P )2 + (1 − P )2 R2 ,

(3.136)

Q2 = (1 − P 2 )(1 + R2 ) .

(3.137)

3.6.4.3 Medium characterized by degenerate anisotropy Heretofore, we have assumed that eigenvalues and eigenpolarizations be not equal. In this section we will discuss the case when eigenpolarizations of a given medium are coincident (degenerate anisotropy) (Savenkov et al., 2007b). This occurs when: (t22 − t11 )2 + 4t12 t21 = 0 . (3.138) The practical examples of media and polarization elements with degenerate anisotropy that are available in the literature refer mostly to the singular case (Lu and Chipman, 1994; Tudor, 2003; Berry and Dennis, 2004). It is, however, known that nonsingular realizations should also be available (Sudha and Gopala Rao, 2001). So, what combinations of anisotropic mechanisms would make a medium be a degenerate one? Note that the impossibility of constructing an eigenbasis for a medium with degenerate anisotropy implies the medium cannot be dichroic. The structure of the Jones matrix of a medium with degenerate anisotropy is then determined: t11 t∗ (3.139) t21 = − ∗ 12 , t22 |t11 |2 = |t22 |2 ,

(3.140)

|t12 | = |t21 | .

(3.141)

2

2

Eqs (3.138) and (3.139) yields: |t212 | =

(t11 − t22 )2 t∗22 4t11

(3.142)

In terms of parameters Δ, α, P , γ, ϕ, and R, of the general model Eq. (3.106), a degenerate medium is given by Δ Δ sin ϕ + (1 − P ) sin cos[ϕ − 2(α − γ)] = 0 R(1 + P ) cos 2 2 2 Δ Δ cos ϕ − R(1 − P ) sin sin[ϕ − 2(α − γ)] − 4P (1 − R2 ) = 0 , (1 + P ) cos 2 2 (3.143) The corresponding eigenvalues are then of the form Δ Δ 1 V1 = V2 = (1 + P ) cos cos ϕ − R(1 − P ) sin sin[ϕ − 2(α − γ)] 2 2 2 Δ · exp −i . (3.144) 2

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3.6.4.4 Transition of eigenvalues In this subsection we intend to consider the interesting symmetry in expressions for the eigenvalues of dichroic, Eq. (3.130), birefringent, Eq. (3.126), and degenerate, Eq. (3.143), media. Indeed, comparing the expressions Eq. (3.126), Eq. (3.130), and Eq. (3.143), one can see that they diﬀer only in the sign of the lower expression: it is negative for birefringent, positive for dichroic, and zero for degenerate media. Hence, it is possible, without changing the arrangement of the constituting parts in the model, Eq. (3.106), to transit eigenvalues of a medium from dichroic to birefringent and to degenerate by changing just a single parameter in Eq. (3.106). We demonstrate this by the following example. To satisfy the ﬁrst expression in Eqs (3.126), (3.130), and (3.143), which is the same for birefringent, dichroic, and degenerate media, we chose R, Δ, α, γ = 0. The medium then is characterized by a combination of a linear dichroism and circular birefringence. The eigenvalues of the corresponding Jones matrix can be readily found as ( 1+P 4P cos ϕ ± cos2 ϕ − V1/2 = . (3.145) 2 (1 + P )2 The sign of the radicand in Eq. (3.145) can be changed by changing the value of circular birefringence, ϕ. The ratio of the eigenvalues, V1 /V2 , is shown in Fig. 3.10 in the complex plane for a medium with P = 0.5, whose parameter ϕ changes in the whole range of physically realizable values, 0 ≤ ϕ ≤ 2π. As can be seen in Fig. 3.10, all possible values of V1 /V2 lie either on a circle with a unitary radius (dashed curve) or on a straight line that corresponds to a zero imaginary part (solid line). The former represents birefringent and the latter is for dichroic media. The point at which the line and the circle intersect characterizes a medium with degenerate anisotropy for which √ 2 P cos ϕ = ± , (3.146) 1+P giving for the chosen value of P four values of ϕ: ϕ = 0.11π; ϕ = 0.89π; ϕ = 1.11π; and ϕ = 1.89π. 3.6.5 Conclusions The principal objective of this chapter has been to consider the main properties of the central concept of mathematical description of scattering process – Jones and Mueller matrices. On the assumption that the Jones and Mueller matrices are the models of corresponding classes of media we have analyzed the structure, symmetry relations and information content of the Jones and Mueller matrices for both approaches to medium modeling: system of scatterers and continuous medium approximation. It is interesting to note that characterizing the information content of the Mueller matrix Bohren and Huﬀman (1983) have pointed out that the complete Mueller matrix contains so much information that only recently have the properties of all its elements begun to be investigated.

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Fig. 3.10. Evolution in the complex plane of V1 /V2 with the change of rotation angle ϕ. The eigenvalues transit from dichroic to degenerate and birefringent depending on the rotation angle (after Savenkov et al., 2007b).

These studies provide a better understanding of the physical information and speciﬁcity provided by deﬁned groups of elements and by each of the Mueller matrix elements individually. Thus, the results discussed in this chapter make it possible to determine the corresponding classes of the inverse problems for the models of inhomogeneous and homogeneous random and dispersive media described by the appropriate incomplete Mueller matrices (Jellison and Modine, 1997; Tang, and Kwok, 2001; Savenkov, 2002b). The former, in its turn, will guide the experimentalist in developing and using the Mueller-polarimeter for which characteristic matrix is mostly appropriate for the measurement the structure of incomplete Mueller matrix describing the properties of a given medium. In particular, this is also evidently of current importance in developing the imaging polarimeters (Tyo, 2002; Tyo et al., 2006). It can be seen that the main properties of the Jones and Mueller–Jones matrices (see sections 3.3, 3.4 and 3.6) are today studied in more detail than those of the depolarizing Mueller matrix (section 3.5). In this connection it is interesting to note that van de Hulst in his milestone book (1957) has pointed out that the Jones matrix of general form characterized by eight independent parameters: b c d a is a most general combination of linear and circular birefringence and linear and circular dichroism. In this chapter we in particular have shown that the answer to the question, i.e., in what exact way the information on birefringence and dichroism consists in the Jones matrix in the most general case, is determined by the generalized matrix equivalence theorem (section 3.6.3).

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At present, perhaps the main challenge associated with depolarizing Mueller matrices is the development of an adequate Mueller matrix model of anisotropic depolarization and studying the physical content of its elements. Acknowledgments The author is obliged to Dr Alexander Kokhanovsky for his valuable advice and encouragement in writing this paper.

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4 Green functions for plane wave scattering on single nonspherical particles Tom Rother

4.1 Introduction Elastic plane wave scattering of electromagnetic waves on three-dimensional structures is a basic physical interaction process which is of great importance in many practical applications. One may think of measurements in technical and medical diagnostics as well as in remote sensing of our Earth’s atmosphere, for example. During the last decades, not at least due to the possibilities of modern computers, there is a growing interest in modelling more and more realistic scattering scenarios which goes beyond the conventional Mie theory for spherical scatterers. A large variety of rigorous numerical methods (i.e., methods which start from the underlying physical equations without any approximations) for solving the problem of plane wave scattering on single but nonspherical particles have been developed in the past. Among those methods Waterman’s T-matrix approach as well as several boundary and volume integral equation approaches have become very popular and widely used in many applications (for an overview see [1], for example, and the literature cited therein). Diﬀering in concept and execution these methods start from the common assumption that the well-known separation of variables method cannot be applied if the scatterer geometry deviates from that of a spherical surface (see [2] chapter 16.11 therein, and [3], for example). A critical discussion of this assumption is one goal of the paper at hand. On the other hand, there is still an ongoing discussion in the literature concerned with conceptual and technical diﬀerences as well as advantages and disadvantages of those standard methods mentioned above. In the paper at hand this discussion is considered from the point of view of Green functions belonging to the scattering problem. An essential step for this purpose is the introduction of an interaction operator describing the interaction of the free-space Green function with the scatterer surface. It is discussed afterwards that several numerical methods can be traced back to certain approximations of this interaction operator. But the main focus of our considerations here is to clarify the relation between Green functions and T-matrices. A consequent Green’s function approach of electromagnetic boundary-value problems has already been given in the well-known book of Tai [4]. Unfortunately, Tai considered only boundary-value problems with boundaries along constant coordinate lines (i.e. spherical boundaries in spherical coordinates, for example). This paper can be con-

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sidered as a generalization of his concept to nonspherical boundaries. The results of this paper reveal the conceptual similarity of the Green’s function concept and Waterman’s T-matrix approach [5]. Surprisingly, this was noticed neither by Tai nor by Waterman even though Tai’s concept (according to a remark in the ﬁrst edition of his book which was released in 1971) has existed since the 1950’s, and even though Waterman’s T-matrix exhibits in some aspects already the nature of a Green’s function. It should be also emphasized that much of the work presented in this paper was motivated by the papers of Kleinman, Roach, and Stroem [6, 7]. In these papers they introduced the so-called ‘modiﬁed Green’s function’ related to the scalar Helmholtz equation in conjunction with the homogeneous Dirichlet and Neumann problem by use of a least-squares approach. By using the Green’s function formalism presented herein, boundary symmetries can be treated in a very comprehensive way, as already demonstrated in [8]. Concerning the diﬀerence between the T-matrix approach and singular integral equations for solving the scattering problem the so-called Rayleigh hypothesis is of special importance. This hypothesis has a quite interesting history, and although it is still an unsolved problem, the Green’s function point of view may add some interesting aspects to it. Another advantage of using Green functions is the ability to consider the symmetry and unitarity property of the T-matrix solely on a mathematical level without employing reciprocity and energy conservation as necessary physical preconditions. Starting from the general representations of the relevant Green functions we will ﬁnally discuss their far-ﬁeld behaviour, the relation between the scattering dyadic and the interaction operator, and resulting but well-known properties of the scattered and total ﬁeld.

4.2 Some basic considerations 4.2.1 Formulation of the scattering problems Plane wave scattering on ideal metallic or dielectric particles can be described mathematically by the following two boundary-value problems (for the scattering conﬁguration see Fig. 4.1): –

solving the inhomogeneous vector-wave equation for the total electric ﬁeld et in the outer space Γ+ in conjunction with the homogeneous Dirichlet condition at the scatterer surface ∂Γ (this corresponds to scattering on an ideal metallic scatterer), i.e., ∇ × ∇ × − k02 et (x) = ρ(x) ; x ∈ Γ+ (1) n ˆ − × et (x) = 0 ; x ∈ ∂Γ . (2)

–

solving the inhomogeneous vector-wave equation for the total electric ﬁeld et in the outer space Γ+ as well as the homogeneous vector-wave equation for the internal ﬁeld eint in conjunction with the transmission conditions at the scatterer surface ∂Γ (this corresponds to scattering on a dielectric scatterer), i.e.,

4 Green functions for plane wave scattering

123

n ˆ− k0 , Γ+ S∞ C scatterer n ˆ− k, Γ− ∂Γ ρ(x)

Fig. 4.1. Scattering conﬁguration of an isotropic, homogeneous, and nonspherical scatterer with boundary surface ∂Γ . The outer space Γ+ represents a single-connected area, bounded by ∂Γ , S∞ , and cut C. The point of origin is located somewhere inside the scatterer.

∇ × ∇ × − k02 et (x) ∇ × ∇ × − k 2 eint (x) n ˆ − × et (x) n ˆ − × ∇ × et (x)

= ρ(x)

; x ∈ Γ+

(3)

= 0 ; x ∈ Γ− =n ˆ − × eint (x)

(4) ; x ∈ ∂Γ

=n ˆ − × ∇ × eint (x)

; x ∈ ∂Γ

(5) (6)

In both cases the total electric ﬁeld et in Γ+ is the sum of the primary incident wave einc and the scattered wave es with the former being a solution of ∇ × ∇ × − k02 einc (x) = ρ(x) ; x ∈ Γ+ . (7) Because of (1) or (3) es has to fulﬁl the homogeneous vector wave equation in Γ+ . In the case of an ideal metallic scatterer the homogeneous Dirichlet condition (2) can be replaced by the inhomogeneous Dirichlet condition n ˆ − × es (x) = − n ˆ − × einc (x)

; x ∈ ∂Γ

(8)

for the scattered ﬁeld. Generally, the primary incident ﬁeld is considered to be given. ∇ denotes the ‘Nabla’-operator, and x represents the space vector (r, θ, φ) in spherical coordinates. k0 and k are the wave numbers of the outer and inner space of the scatterer if scattering on a dielectric scatterer is considered. These quantities are given by (2π) (9) k0 = ω 2 · 0 · μ0 = λ

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and k=

√

r · k0

(10)

with λ and r being the free-space wavelength and the permittivity of the scatterer. n ˆ − is the outward directed unit normal vector related to the inner boundary of Γ+ . To make the solutions unique we need an additional condition for the scattered ﬁeld at the outer but nonlocal boundary S∞ – the far-ﬁeld of Γ+ . This is the so-called radiation condition 1 x . (11) lim ∇ × − ik0 × es (x) = 0 |x| |x| |x|→∞ This condition must be fulﬁlled uniformly for any direction x/|x|. Please, note that the plane wave does not obey this condition! We will see later why this is important for the scattering problem. From the physical point of view we will also require the regularity of the internal ﬁeld inside a dielectric scatterer and the smoothness of the scattered ﬁeld outside. 4.2.2 Spherical coordinates and eigensolutions of the vector-wave equation Cartesian variables can be expressed by spherical variables according to x = r · sin θ · cos φ y = r · sin θ · sin φ z = r · cos θ

(12)

where the angle θ is within the interval [0, π], and the angle φ is within [0, 2π]. The corresponding relations between the unit vectors in Cartesian and spherical coordinates can be seen from Table 4.1. The nabla-operator reads in spherical coordinates 1 ∂ ˆ 1 ∂ ∂ ˆ · rˆ + ·θ + ·φ. (13) ∇= ∂r r ∂θ r sin θ ∂φ For the boundary surface ∂Γ = S of the scatterer we assume the parameter representation r = r(θ, φ) · rˆ . (14) According to this representation we get for the surface element ) ) ) ∂r ∂r )) ) × dθ dφ dS = ) ∂θ ∂φ ) with

∂r ∂r ˆ ∂r ˆ ∂r × = r2 sin θ · rˆ − r sin θ · ·θ−r· ·φ. ∂θ ∂φ ∂θ ∂φ

(15)

(16)

For a spherical surface with radius a this results in dS = a2 sin θ dθ dφ ,

(17)

4 Green functions for plane wave scattering

125

for example. The unit normal vector at the surface (14) pointing into Γ+ is given by ∂r ∂r × ∂θ ∂φ ). (18) n ˆ = )) ) ) ∂r × ∂r ) ) ∂θ ∂φ ) n. Please, note that we have n ˆ − = −ˆ Table 4.1. Relations between the unit vectors in Cartesian and spherical coordinates

rˆ θˆ φˆ

x ˆ

yˆ

zˆ

sin θ cos φ cos θ cos φ − sin φ

sin θ sin φ cos θ sin φ cos φ

cos θ − sin θ 0

The relevant vectorial eigensolutions of the vector-wave equation in spherical coordinates can be calculated from the eigensolutions of the scalar Helmholtz equation and are explicitly given by ψ l,n,1 (r, θ, φ) = γl,n · jn (kr) · cl,n (θ, φ) ψ l,n,2 (r, θ, φ) = γl,n ·

n(n + 1) · jn (kr) · pl,n (θ, φ)+ kr 1 ∂ (rjn (kr)) · bl,n (θ, φ) kr ∂r

ϕl,n,1 (r, θ, φ) = γl,n · h(1) n (kr) · cl,n (θ, φ) ϕl,n,2 (r, θ, φ) = γl,n ·

n(n + 1) (1) · hn (kr) · pl,n (θ, φ)+ kr ' 1 ∂ & (1) rhn (kr) · bl,n (θ, φ) kr ∂r

χl,n,1 (r, θ, φ) = γl,n · h(2) n (kr) · cl,n (θ, φ) χl,n,2 (r, θ, φ) = γl,n ·

where

n(n + 1) (2) · hn (kr) · pl,n (θ, φ)+ kr ' 1 ∂ & (2) rhn (kr) · bl,n (θ, φ) kr ∂r

(19)

(20) (21)

(22) (23)

(24)

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pl,n (θ, φ) := rˆ Pnl (cos θ) · e ilφ il dPnl (cos θ) l ˆ ˆ cl,n (θ, φ) := θ · Pn (cos θ) − φ · e ilφ sin θ dθ il dP l (cos θ) bl,n (θ, φ) := θˆ n + φˆ · Pnl (cos θ) · e ilφ , dθ sin θ (

and γl,n :=

2n + 1 (n − l)! . 4πn(n + 1) (n + l)!

(25) (26) (27)

(28)

Pnl (cos θ) are the associated Legendre polynomials, and the radial dependent func(1) (2) tions jn (kr), hn (kr), and hn (kr) are the spherical Bessel functions and spherical Hankel functions of ﬁrst and second kind. It should be mentioned that only the eigensolutions (21) and (22) fulﬁl the radiation condition (11). Therefore, they are called ‘outgoing’ or ‘radiating’ eigensolutions. (19) and (20) are the ‘regular’ eigensolutions, and (23) and (24) are the ‘incoming’ eigensolutions. For the outgoing eigensolutions we know the far-ﬁeld behaviour (−i)n+1 eikr · γl,n · cl,n (θ, φ) r→∞ kr (−i)n eikr · γl,n · bl,n (θ, φ) . lim ϕl,n,2 (r, θ, φ) = r→∞ kr lim ϕl,n,1 (r, θ, φ) =

(n + l)! · c−l,n (θ, φ) (n − l)! (n + l)! · b−l,n (θ, φ) b∗l,n (θ, φ) = (−1)l · (n − l)! c∗l,n (θ, φ) = (−1)l ·

(29) (30)

(31) (32)

and cl,n (π − θ, φ ± π) = (−1)n · cl,n (θ, φ) bl,n (π − θ, φ ± π) = (−1)n+1 · bl,n (θ, φ) .

(33) (34)

are frequently used relations of the vector functions cl,n and bl,n . Furthermore, they fulﬁl the orthogonality relations 2π π dφ dθ sin θ b∗l,n (θ, φ) · cl,n (θ, φ) =

0 2π

0 π

dφ

0 2π

0 π

dφ 0

dθ sin θ b∗l,n (θ, φ) · pl,n (θ, φ) = dθ sin θ c∗l,n (θ, φ) · pl,n (θ, φ) = 0

2π

dφ

2π

dφ 0

(35)

0

0

0 π

π

dθ sin θ b∗l,n (θ, φ) · bl,n (θ, φ) =

0

dθ sin θ c∗l,n (θ, φ) · cl,n (θ, φ) =

1 2 γl,n

· δll δnn

(36)

4 Green functions for plane wave scattering

2π

π

dφ 0

dθ sin θ p∗l,n (θ, φ) · pl,n (θ, φ) =

0

1 2 · δll δnn . n(n + 1) · γl,n

127

(37)

One summation index i is customarily used instead of the two indices l, n. This common index is deﬁned by i = n(n + 1) + l .

(38)

However, it can be resolved into the two indices l, n by the equations & ' √ 1 −1 + 1 + 4i n(i) = nint 2 l(i) = i − n(i) · [n(i) + 1]

(39) (40)

with function nint(a) being the integer number nearest to the real number a. The regular eigensolutions can be expressed in terms of outgoing and incoming eigensolutions by means of ψ i,τ (r, θ, φ) =

" 1 ! · ϕi,τ (r, θ, φ) + χi,τ (r, θ, φ) . 2

(41)

For later purposes we introduce also the vector functions ˜i,τ (r, θ, φ) = ψ ˜l,n,τ (r, θ, φ) := (−1)l · ψ −l,n,τ (r, θ, φ) ψ

(42)

˜l,n,τ (r, θ, φ) := (−1)l · ϕ−l,n,τ (r, θ, φ) ϕ ˜i,τ (r, θ, φ) = ϕ

(43)

˜l,n,τ (r, θ, φ) := (−1) · χ−l,n,τ (r, θ, φ) , χ ˜i,τ (r, θ, φ) = ϕ

(44)

l

n ˆ

ψ i,τ− (r, θ, φ) := n ˆ − × ψ i,τ (r, θ, φ) ; r ∈ ∂Γ

(45)

n ˆ ϕi,τ− (r, θ, φ) n ˆ χi,τ− (r, θ, φ)

:= n ˆ − × ϕi,τ (r, θ, φ) ; r ∈ ∂Γ

(46)

:= n ˆ − × χi,τ (r, θ, φ) ; r ∈ ∂Γ ,

(47)

and ˆ − × ∇ × ψ i,τ (r, θ, φ) ; r ∈ ∂Γ ∂ nˆ − ψ i,τ (r, θ, φ) := n ∂ nˆ − ϕi,τ (r, θ, φ) := n ˆ − × ∇ × ϕi,τ (r, θ, φ) ; r ∈ ∂Γ

(48) (49)

where τ = 1, 2. If k is a real number we have ˜i,τ (r, θ, φ) = ψ ∗i,τ (r, θ, φ) . ψ

(50)

The vector functions (45)–(47) are the projections onto the tangential planes at the scatterer surface. Please note also that the curl operation in Eqs (48) and (49) must be performed before the projection onto the tangential plane. On the surface of a sphere with radius r = a we have the following orthogonality relations: 2π π ∗ n ˆ n ˆ (ϕ,ϕ) dφ dθ r2 sin θ ϕi,τ− (κa, θ, φ) · ϕj,τ− (κ a, θ, φ) = a2 · di,τ (κ, κ ) · δij δτ τ 0

0

(51)

128

Tom Rother

2π

π

dφ 0

0

2π

π

dθ r2 sin θ

dφ 0

∗

∗

n ˆ ψ i,τ− (κa, θ, φ)

(52) ·

n ˆ ψ j,τ− (κ a, θ, φ)

= a2 ·

(ψ,ψ) di,τ (κ, κ )

· δij δτ τ

0

2π

π

n ˆ

dθ r2 sin θ ψ i,τ− (κa, θ, φ)

dφ 0

∗ n ˆ n ˆ (ϕ,ψ) dθ r2 sin θ ϕi,τ− (κa, θ, φ) · ψ j,τ− (κ a, θ, φ) = a2 · di,τ (κ, κ ) · δij δτ τ

(53) · ϕj,τ− (κ a, θ, φ) = a2 · di,τ n ˆ

(ψ,ϕ)

(κ, κ ) · δij δτ τ .

0

(54) The coeﬃcients di,τ are given by (ϕ,ϕ)

di,1

& '∗ (1) (1) (κ, κ ) = hn(i) (κa) · hn(i) (κ a)

(55)

& '∗ (1) (κ, κ ) = hn(i) (κa) · jn(i) (κ a)

(56)

∗ (κ, κ ) = jn(i) (κa) · jn(i) (κ a)

(57)

(ϕ,ψ)

di,1

(ψ,ψ)

di,1

∗ (κ, κ ) = jn(i) (κa) · hn(i) (κ a) & '∗ ' ∂ & 1 ∂ (ϕ,ϕ) (1) di,2 (κ, κ ) = ∗ 2 r · h(1) r · h (κr) · (κ r) ni ni κ κa ∂r ∂r r=a r=a ∗ & ' 1 ∂ ∂ (ϕ,ψ) r · h(1) (r · jni (κ r)) · di,2 (κ, κ ) = ∗ 2 ni (κr) κ κa ∂r ∂r r=a r=a ∗ ∂ 1 ∂ (ψ,ψ) (r · jni (κr)) (r · jni (κ r)) · di,2 (κ, κ ) = ∗ 2 κ κa ∂r ∂r r=a r=a ∗ & ' 1 ∂ ∂ (ψ,ϕ) (r · jni (κr)) r · h(1) · . di,2 (κ, κ ) = ∗ 2 ni (κ r) κ κa ∂r ∂r r=a r=a (ψ,ϕ)

di,1

(1)

(58) (59) (60) (61) (62)

4.2.3 Dyadics and Green’s theorems 4.2.3.1 Dyadics A vector source given in Cartesian coordinates by ρ = ρx x ˆ + ρy yˆ + ρz zˆ

(63)

e = ex x ˆ + ey yˆ + ez zˆ .

(64)

produces an electric ﬁeld

Between the components of the source and the ﬁeld we assume the linear relations ex = φxx · ρx + φxy · ρy + φxz · ρz ey = φyx · ρx + φyy · ρy + φyz · ρz ez = φzx · ρx + φzy · ρy + φzz · ρz ,

(65)

4 Green functions for plane wave scattering

129

i.e., each component of the source contributes to all components of the ﬁeld. To bring this relation into the simpler form e = ex x ˆ + ey yˆ + ez zˆ = Φ · ρ

(66)

we deﬁne the dyadic Φ according to Φ := x ˆ φx + yˆ φy + zˆ φz

(67)

with φx , φy , φz given by φx = φxx x ˆ + φxy yˆ + φxz zˆ φy = φyx x ˆ + φyy yˆ + φyz zˆ φz = φzx x ˆ + φzy yˆ + φzz zˆ

(68)

(Please note that a dyadic quantity will be denoted by bold capital letters to distinguish it from a vector. The latter will be denoted throughout this paper by bold lower case letters. For conventional matrices we will also use bold capital letters. But from the context it should become clear whether dyadics or matrices are meant.) The symbol ‘’ in (67) denotes the dyadic product of the vectors. The dyadic Φ reads explicitly Φ := φxx x ˆx ˆ + φxy x ˆ yˆ + φxz x ˆ zˆ + φyx yˆ x ˆ + φyy yˆ yˆ + φyz yˆ zˆ + φzx zˆ x ˆ + φzy zˆ yˆ + φzz zˆ zˆ .

(69)

Φ is symmetric if Φ = Φtp

(70)

with Φtp := φx x ˆ + φy yˆ + φz zˆ (71) ' & as the transpose dyadic. The scalar product of a dyadic φ (1) φ (2) with an arbitrary vector f is deﬁned according to ' & ' & (72) φ (1) φ (2) · f := φ (1) · φ (2) · f . From this deﬁnition (66) follows immediately. It follows also that Φ · f = f · Φ holds, in general. The dyadic Φ of (67)–(69) can be represented by the matrix ⎛ ⎞ φxx φxy φxz ⎝ φyx φyy φyz ⎠ . (73) φzx φzy φzz We can represent correspondingly the transpose dyadic by the transpose of this matrix. For the vector product of a vector with a dyadic we have the similar deﬁnition & ' & ' φ (1) φ (2) × f := φ (1) φ (2) × f . (74)

130

Tom Rother

As for the scalar product Φ × f = f × Φ holds, in general. The sum of the dyadic ˆ i represents a special dyadic, the so-called ‘idem products of the unit vectors x factor’ I, 3 ˆi . ˆi x (75) x I := i=1

The idem factor is characterized by its feature I · f (x) = f (x) · I = f (x) .

(76)

Taking the gradient of a vector f will also produce a dyadic. In Cartesian coordinates this operation provides ∇f = x ˆ

∂f ∂f ∂f + yˆ + zˆ , ∂x ∂y ∂z

(77)

where f = fx · x ˆ + fy · yˆ + fz · zˆ. In spherical coordinates, if the vector f is given ˆ we have correspondingly by f = fr · rˆ + fθ · θˆ + fφ · φ, ∇f = rˆ

1 ∂f 1 ∂f ∂f + θˆ + φˆ . ∂r r ∂θ r sin θ ∂φ

(78)

It must be taken into account that in spherical coordinates not only the components fr , fθ , fφ but also the unit vectors must be diﬀerentiated with respect to θ und φ. According to Table 4.1 we have ∂ rˆ ˆ ∂ˆ r =0, = θ and ∂r ∂θ

∂ rˆ = sin θ φˆ , ∂φ

(79)

for example. Thus the gradient of the radial unit vector produces the dyadic ' 1 &ˆ ˆ ∇ˆ r= θ θ + φˆ φˆ . (80) r Furthermore, taking the gradient of the product of a scalar function f with a vector φ provides the dyadic ∇(f φ) = (∇f ) φ + f · ∇(φ) . The curl of the dyadic Φ in (67) is deﬁned according to " ! ∇ × Φ := x ˆ [∇ × φx ] + yˆ ∇ × φy + zˆ [∇ × φz ] .

(81)

(82)

In spherical coordinates this becomes " ! ∇ × Φ := rˆ [∇ × φr ] + θˆ [∇ × φθ ] + φˆ ∇ × φφ

(83)

with ∇ × φr , ∇ × φθ , and ∇ × φφ being the curl of the vector functions φr , φθ , and φφ in spherical coordinates. The following identities are of some importance for the ongoing considerations: a · [b × C] = − b · [a × C] = [a × b] · C

(84)

4 Green functions for plane wave scattering

a · B = Btp · a

(85)

a × B = − [Btp × a] tp C · [a × B] = − [a × C] tp

tp

(86) ·B

[A × b] · C = A · [b × C] − [a × C ]

131

(87) (88)

=C×a

(89)

[A · B] tp = Btp · Atp .

(90)

tp tp

4.2.3.2 Green’s theorems Green’s theorem for the two arbitrary vector functions ψ and φ is given in Γ+ by [ψ(x) · ∇ × ∇ × φ(x) − φ(x) · ∇ × ∇ × ψ(x)] dV (x) = Γ+ * n ˆ − · {φ(x) × [∇ × ψ(x)] − ψ(x) × [∇ × φ(x)]} dS(x) . (91) ∂Γ ∪ S∞

Obviously, this theorem relates two scalar quantities. To get a corresponding theorem which relates vector functions we have to introduce dyadic quantities into this theorem. This can be achieved by taking the scalar product of a dyadic with a constant vector into account. Since this operation yields a vector according to definition (72) we can use such a scalar product within (91). The constant vector can be removed afterwards. By means of this procedure we can derive the vector-dyadic Green theorem {[∇ × ∇ × ψ(x)] · Q(x, x ) − ψ(x) · [∇x × ∇x × Q(x, x )]} dV (x) = Γ+ * n ˆ − · {ψ(x) × [∇x × Q(x, x )] + [∇ × ψ(x)] × Q(x, x )} dS(x) (92) ∂Γ ∪ S∞

as well as the dyadic-dyadic Green theorem + tp [∇x × ∇x × Q(x, x )] · P(x, x ) − Γ+

* ∂Γ ∪ S∞

+

, Qtp (x, x ) · [∇x × ∇x × P(x, x )] dV (x) = [ˆ n− × Q(x, x )] · [∇x × P(x, x )] − tp

tp [∇x × Q(x, x )] · [ˆ n− × P(x, x )] dS(x)

(93)

valid in Γ+ . Similar theorems can be derived in Γ− which diﬀers only in the surface integral on the right-hand side of (92) and (93) (in Γ− we must only consider the surface ∂Γ ), and in the use of the unit normal vector n ˆ instead of n ˆ − . However, for the free space without any scatterer, we have to consider only the surface integral over S∞ .

132

Tom Rother

4.3 Dyadic Green functions and light scattering Since their introduction in the 19th century Green’s functions have become a powerful mathematical tool for solving boundary-value problems. In [9] Freeman Dyson compared the importance of these functions with the innovation of modern computers in science in our days. What makes these functions so useful for our purposes? Mathematically seen, with the help of these functions we are able to transform the formulation of the scattering problem in terms of partial diﬀerential equations into an equivalent formulation in terms of integral equations. From a physical point of view these functions describe the ﬁeld in space point x caused by a unit source located in a diﬀerent space point x . A given source, on the other hand, can be considered as a distribution of unit sources within a located area in space. The ﬁeld of this given source is simply the sum of the ﬁelds of all unit sources. In this way Green’s functions are able to decouple the properties of the source from the properties of the space where the latter is characterized by the properties (i.e., by the geometry and permittivity) of the scatterer. Thus, if only the given source changes, the Green function remains still the same and must be calculated only once. Last but not least, Green functions can be used to express the fundamental Huygens principle in terms of an integral equation which serves as a powerful starting point for solving scattering problems, as we will demonstrate in this paper. 4.3.1 Dyadic free-space Green function We will start with the well-known Green function of the free and unstructured space Γ = Γ− ∪ Γ+ . This free-space Green function is the most important prerequisite for our further considerations. It is a solution of the inhomogeneous vector-wave equation (94) [ ∇x × ∇x × − k02 ] G0 (x, x ) = Iδ(x − x ) with I being the idem factor deﬁned in (75). We require additionally the fulﬁlment of the radiation condition (11) at S∞ with respect to variable x. Levine und Schwinger could show that the solution of this boundary-value problem can be expressed in terms of the scalar free-space Green function

G0 (x, x ) =

eik0 |x−x | 4π|x − x |

(95)

which is the solution of the corresponding boundary-value problem of the scalar Helmholtz equation. They found 1 (96) G0 (x, x ) = I + 2 ∇x ∇x G0 (x, x ) . k0 for the dyadic free-space Green function [10]. Because the nabla operator operates twice on the scalar G0 the dyadic G0 has a pole of order 3 at x = x and exhibits therefore a singularity of higher order than G0 in the same point. But because of ∇x × G0 (x, x ) = ∇x × IG0 (x, x ) = ∇x G0 (x, x ) × I

(97)

4 Green functions for plane wave scattering

133

the curl of this function exhibits only a pole of the order 2. We will come back to this aspect later on when dealing with the application of a singular surface integral equation for solving the scattering problem of an ideal metallic scatterer. From ∇x ∇x G0 (x, x ) = ∇x ∇x G0 (x, x )

(98)

and Green’s theorem (93) the symmetry relation Gtp 0 (x , x ) = G0 (x , x ) = G0 (x , x ) .

(99)

follows for the dyadic free-space Green function. Another important property is its behaviour for large distances between source and ﬁeld points. It is given by

eik0 |x | −ik0 eˆx ·x e ] ·e lim G (x, x ) = [ I − e ˆ ˆ . 0 x x 4π|x | |x |→∞

(100)

r in spherical coordinates. eˆx corresponds to the unit vector ˆ Now, looking for the solution of Eq. (7), we get by use of Green’s theorem (92) if applied with ψ(x) = einc (x) and Q(x, x ) = G0 (x, x ) einc (x) = G0 (x, x ) · ρ(x ) dV (x ) ; x, x ∈ Γ+ . (101) Γ+

einc (x) obeys therefore also the radiation condition (11). In deriving (101) we have already applied the symmetry relation (99). For example, if we have the speciﬁc source vector ρ(x ) = 4π|x | · e−ik0 |x | · δ(x − xq ) · e0 (102) with

|xq | |x| ,

(103)

k = −k0 eˆxq

(104)

and if we denote we obtain from (100) and (101) the plane wave einc (x) = e0 · eik·x

(105)

with polarization e0 in space point x. The importance of the plane wave which does not fulﬁl the radiation condition at S∞ is discussed in more detail in the last section of this paper. For many electromagnetic wave problems it is convenient to split a general vector ﬁeld into two parts according to f = ft + fl ,

(106)

i.e., into a transverse or solenoidal part f t with ∇ · f t = 0, and into a longitudinal part f l with ∇ × f l = 0. The latter part is only of interest if we ask for the ﬁeld in a source region with free charges. The same splitting can be done with the dyadic delta-distribution Iδ which is deﬁned according to Iδ(x − x ) · f (x ) dV (x ) := f (x) . (107) Γ

134

Tom Rother

Their transverse and longitudinal part, if applied to a general vector function f (x), provides the transverse or longitudinal part of this vector function, respectively. In doing so we can represent the dyadic free-space Green function by G0 (x, x ) = Gt (x, x ) −

1 · Dl (x − x ) k02

(108)

where Gt is its transverse part which solves the equation [ ∇x × ∇x × − k02 ] Gt (x, x ) = Dt (x − x ) ,

(109)

and Dl (x − x ) in (108) is the longitudinal part of Iδ. For the transverse part of the dyadic free-space Green function we know the following expansions in terms of the eigensolutions of the vector-wave equation: ⎧ ˜i,τ (k0 , x ) ∞ ⎨ ϕi,τ (k0 , x) ψ 2 ; |x| > |x | Gt (x, x ) = (ik0 ) · ⎩ τ =1 i=0 ˜i,τ (k0 , x ) ; |x| < |x | ψ i,τ (k0 , x) ϕ (110) [3]. If |x| > |x | is valid we will denote the corresponding expansion with Gt> (x, x ), and if |x| < |x | holds we will use the notation Gt< (x, x ). Both approximations Gt> and Gt< are solutions of the homogeneous vector-wave equation (94), and Gt> fulﬁls additionally the radiation condition at S∞ . Using the expansion Gt< in conjunction with (101) and the vector source (102) results in the well-known expansion of the plane wave (105) which is given in spherical coordinates by (see [11], for example) einc (k0 r, θ, φ, θi , φi ) =

2

cl,n,τ · ψ l,n,τ (r, θ, φ)

(111)

τ =1 l,n

with in (2n + 1) · e0 · c−l,n (θi , φi ) · γl,n n(n + 1) in−1 (2n + 1) · e0 · b−l,n (θi , φi ) . = (−1)l · · γl,n n(n + 1)

cl,n,1 = (−1)l ·

(112)

cl,n,2

(113)

This can be proven by taking (29) and (30) as well as (33) and (34) into account. θi and φi are the angles related to the direction of incidence of the plane wave. 4.3.2 Dyadic Green functions of the scattering problems Now we will consider the two Green functions which can be related to the two scattering problems of our interest. As in the case of the dyadic free-space Green function the Green function GΓ+ belonging to the ideal metallic scatterer as well (d) as the Green function GΓ+ belonging to the dielectric scatterer are solutions of the inhomogeneous vector-wave equations

4 Green functions for plane wave scattering

!

135

" 2

∇x × ∇x × − k0 GΓ+ (x, x ) = Iδ(x − x ) ! " (d) ∇x × ∇x × − k02 GΓ+ (x, x ) = Iδ(x − x )

; x, x ∈ Γ+ . (114)

Both Green functions must also obey the radiation condition (11) with respect to x. But for the ideal metallic scatterer we require the additional boundary condition n ˆ × GΓ+ (x, x ) = 0

(115)

at the scatterer surface ∂Γ . On the other hand, if looking at a dielectric scatterer, we have to take into account the additional transmission conditions n ˆ − × GΓ+ (x, x ) = n ˆ − × G (−/+) (x, x ) (d) n ˆ − × ∇ × GΓ+ (x, x ) = n ˆ − × ∇ × G (−/+) (x, x ) (d)

(116) (117)

at the scatterer surface. The dyadic Green function G (−/+) solves the homogeneous vector-wave equation " ! ∇x × ∇x × − k 2 G (−/+) (x, x ) = 0 ; x ∈ Γ− ; x ∈ Γ+ (118) inside the scatterer and must be regular with respect to x. The primary source of the incident ﬁeld is assumed to be located in Γ+ but outside the smallest sphere circumscribing the scatterer. This assumption holds throughout this paper! By use of Green’s theorem (93) we can derive the symmetry relations !

"tp

GΓ+ (x, x )

and

tp

GΓ+ (x, x ) (d)

= GΓ+ (x , x) . = GΓ+ (x , x) (d)

(119)

(120)

for both Green functions. The total ﬁeld outside the scatterer can be calculated from the relations et (x) = GΓ+ (x, x ) · ρ(x ) dV (x ) (121) Γ+

or

GΓ+ (x, x ) · ρ(x ) dV (x ) (d)

et (x) = Γ+

(122)

once we have found appropriate expressions for these Green functions. This is, of course, the crucial question we will try to answer now. For this we introduce the so-called dyadic interaction operator W∂Γ+ by the deﬁnition GΓ+ (x, x ) := G0 (x, x ) + * Gt> (x, x ¯) · W∂Γ+ (¯ x, x ˜) · Gt< (˜ x, x ) dS(¯ x) dS(˜ x) . ∂Γ

(123)

136

Tom Rother

if the ideal metallic scatterer is considered. In the case of dielectric scatterers we (d) (d) introduce the two dyadic interaction operators W∂Γ+ and W∂Γ− by the deﬁnitions GΓ+ (x, x ) := G0 (x, x ) + * (d) Gt> (x, x ¯) · W∂Γ+ (¯ x, x ˜) · Gt< (˜ x, x ) dS(¯ x) dS(˜ x) , (d)

(124)

∂Γ

and G

(−/+)

*

Gt<s (x, x ¯) · W∂Γ− (¯ x, x ˜) · Gt< (˜ x, x ) dS(¯ x) dS(˜ x) . (d)

(x, x ) := ∂Γ

(125)

These deﬁning equations can be considered as expressions of Huygens’ principle in terms of Green functions where the interaction operator describes the interaction of the transverse part of the dyadic free-space Green function with the scatterer surface. Gt<s in Eq. (125) corresponds to the lower expansion given in (110) but with k0 replaced by k in the vector functions ψ i,τ as well as in the pre-factor (ik0 ). It solves the homogeneous vector-wave equation (118) in Γ− and is also regular everywhere inside the scatterer. Thus expression (125) obeys equation (118) together with the requirement of regularity. Similarly, since Gt> is a solution of the homogeneous vector-wave equation in Γ+ , and since it fulﬁls the radiation condition with respect to x, expressions (123) and (124) obey equations (114) together with the radiation condition. The remaining boundary conditions at the scatterer surface can be used to approximate the interaction operators, and, thus, the Green functions we are seeking for. This will be done for each scattering problem separately in the next two subsections. 4.3.2.1 Ideal metallic scatterer Applying boundary condition (115) to the representation (123) results in * " ! ¯) · W∂Γ+ (¯ x, x ˜) · Gt< (˜ x, x ) dS(¯ x) dS(˜ x) = n ˆ − × Gt> (x, x ∂Γ ! " − n ˆ − × Gt< (x, x ) ; x ∈ ∂Γ ; x ∈ Γ+ .

(126)

In the next step we approximate Gt> and Gt< by use of the expansions given in (110), but with a ﬁnite number N of expansion terms. We will denote this approximation further on with a superscript ‘(N)’ on the relevant Green’s function. This provides − (ik0 )

2 N +

-

˜k,τ (k0 , x ) ψ k,τ− (k0 , x) ϕ n ˆ

=

τ =1 k=0

(ik0 ) ·

2 N N 2 !

W∂Γ+

τ =1 i=0 τ =1 k=0

"τ,τ + nˆ − · ϕ (k , x) ϕ ˜ (k0 , x ) 0 k,τ i,τ i,k

(127)

with the vector functions given in (19)–(22) as well as in (45) and (46), and with the matrix elements of the interaction operator deﬁned according to

4 Green functions for plane wave scattering

!

W∂Γ+

137

*

"τ,τ

:= (ik0 ) ·

i,k

˜i,τ (k0 , x ψ ¯) · W∂Γ+ (¯ x, x ˜) · ψ k,τ (k0 , x ˜) dS(¯ x) dS(˜ x) . ∂Γ

(128) For the Green function GΓ+ this results in the approximation GΓ+ (x, x ) = G0 (x, x ) + (N)

(ik0 ) ·

N 2 N 2 !

W∂Γ+

τ =1 i=0 τ =1 k=0

"τ,τ 1 , · ϕi,τ (k0 , x) ϕ ˜k,τ (k0 , x ) . i,k

(129)

This is the dyadic form of the ‘modiﬁed Green function’ discussed by Kleinman, Roach, and Stroem in [7]. Explicit expressions for the matrix elements of the interaction operator W∂Γ+ can be derived from (127) if we change on the left-hand side of this equation in virtue of the transformation n ˆ

ψ k,τ− (k0 , x) =

2 N

T˜∂Γ

τ =1 i=0

τ ,τ k,i

n ˆ

· ϕi,τ− (k0 , x)

; x ∈ ∂Γ

n ˆ

(130)

n ˆ

from the vector functions ψ k,τ− (k0 , x) to the vector functions ϕi,τ− (k0 , x). (130) reads in matrix notation tp ! " ˜ ∂Γ · ϕ nˆ − (k0 , x) tp . ψ nˆ − (k0 , x) =T (131) Equation (127) is obviously met if 2 τ,τ τ ,τ τ,τ 3tp (N ) τ,τ = − T˜∂Γ =− T˜∂Γ = − [T∂Γ ]i,k W∂Γ+ i,k

(132)

i,k

k,i

holds. The transformation matrix itself is given by the product of the two matrices (see [12], for example) τ,τ

[T∂Γ ]i,k =

2 N

ˆ − −1 (g,ϕn )

A∂Γ

ˆ− (g,ϕn )

A∂Γ and

ˆ− (g,ψ n ) B∂Γ

τ,τ i,k

τ,τ i,k

(133)

l,k

i,l

τ¯=1 l=0

where

τ,¯τ τ¯,τ ˆ− (g,ψ n ) · B∂Γ ,

4 5 n ˆ = g i,τ (x) | ϕk,τ− (x)

(134)

∂Γ

4 5 n ˆ = g i,τ (x) | ψ k,τ− (x)

.

(135)

∂Γ

The scalar product denoted by the ‘bra’ and ‘ket’ symbols is deﬁned in terms of surface integrals along the boundary surface ∂Γ according to *

∗ g i,τ (x) · f k,τ (x) dS(x) . (136) g i,τ (x) | f k,τ (x) ∂Γ := ∂Γ

g ∗ denotes the conjugate-complex of g. {g i,τ }N i=0 represents a system of as yet not speciﬁed weighting vector functions thus providing an additional degree of freedom

138

Tom Rother

in determining the transformation matrix. This freedom can be used to match the numerical procedure to a certain scattering geometry. (133) reads in matrix notation −1 ˆ− ˆ− (g,ϕn ) (g,ψ n ) · B∂Γ (137) T∂Γ = A∂Γ ˆ− (g,ϕn )

ˆ− (g,ψ n )

, and B∂Γ being 2 × 2 block matrices, due to the τ, τ with T∂Γ , A∂Γ summation. Please, note also that the indices i, k, l are all combined summation indices as introduced in (38). Of course, one has to ensure the existence of the ˆ− (g,ϕn )

inverse of matrix A∂Γ . For certain scatterer geometries and the restriction that the wave number k does not belong to an internal resonance problem the linear n ˆ n ˆ independence of the vector wave functions ψ k,τ− and ϕk,τ− is known mathematically (for more details see Doicu et al. in [13], for example). Thus, if using one of those ! (g,ϕnˆ − ) "−1 sets as weighting functions the existence of A∂Γ is known in advance for such geometries. For other geometries one can take up a pragmatic position and prove this numerically. Please, note also that in deriving Eq. (129) it was tacitly assumed that integration and summation can be interchanged. The transformation character of the T-matrix employed above to determine the matrix elements of the interaction operator has already been observed by Ursell (see [7], and the remark therein). The above given result runs into that of an ideal metallic sphere with radius r = a if the weighting vector functions n ˆ

g i,τ (x) = ψ i,τ− (k0 , x)

(138)

are used. From the orthogonality relations given in (51)–(62) it follows ˆ− ˆ − −1 τ,τ 1 1 (ψ n ,ϕn ) = δτ,τ δi,k · 2 · A∂Γ a d(ψnˆ − ,ϕnˆ − ) i,k

(139)

i,τ

and

ˆ− ˆ− (ψ n ,ψ n )

B∂Γ

τ,τ i,k

ˆ− ˆ− (ψ n ,ψ n )

= δτ,τ δi,k · a2 · di,τ

(140)

for the relevant matrix elements. The d-values are given by ˆ− ˆ− (ψ n ,ϕn )

ˆ− ˆ− (ψ n ,ϕn )

di,2

∗ = jn(i) (k0 a) · hn(i) (k0 a) (141) di,1 2 ∗ & 3 ' 1 ∂ ∂ (1) r · jn(i) (k0 r) r · hn(i) (k0 r) = 2 2· · (142) k0 a ∂r ∂r r=a r=a (1)

ˆ− ˆ− (ψ n ,ψ n )

∗ = jn(i) (k0 a) · jn(i) (k0 a) (143) di,1 2 ∗ 3 ˆ− ˆ− 1 ∂ ∂ (ψ n ,ψ n ) . (144) r · jn(i) (k0 r) r · jn(i) (k0 r) = 2 2· · di,2 k0 a ∂r ∂r r=a r=a Hence, for the spherical geometry the dyadic Green function becomes

GΓ+ (x, x ) = G0 (x, x ) − ik0 · (N)

N 1

˜i,1 (k0 , x ) + ai,1 · ϕi,1 (k0 , x) ϕ

i=0

,

˜i,2 (k0 , x ) ai,2 · ϕi,2 (k0 , x) ϕ

.

(145)

4 Green functions for plane wave scattering

with

jn(i) (k0 a)

ai,1 =

139

(146)

(1)

hn(i) (k0 a)

and

" ∂ ! r · jn(i) (k0 r) r=a . ai,2 = ∂r ∂ (1) r · hn(i) (k0 r) ∂r r=a This result agrees exactly with that one presented in [4].

(147)

4.3.2.2 Dielectric scatterer We can proceed in a similar way for the dielectric scatterer. If using again ﬁnite expansions for Gt> , Gt<s , as well as Gt< in (124) and (125) we get with the deﬁnitions * τ,τ (d) (d) ˜i,τ (k0 , x W∂Γ+ ψ := (ik0 ) ¯) · W∂Γ+ (¯ x, x ˜) · ψ k,τ (k0 , x ˜) dS(¯ x) dS(˜ x) (148) i,k

∂Γ

and * τ,τ (d) := (ik) W∂Γ− i,k

∂Γ

(d)

ϕ ˜i,τ (k0 , x ¯) · W∂Γ− (¯ x, x ˜) · ψ k,τ (k0 , x ˜) dS(¯ x) dS(˜ x) (149) (d)

(d)

of the matrix elements of the interaction operators W∂Γ+ and W∂Γ− the approximations (d,N)

GΓ+

(ik0 ) ·

(x, x ) = G0 (x, x ) +

2 N N 2

(d)

W∂Γ+

τ =1 i=0 τ =1 k=0

τ,τ 1 , · ϕi,τ (k0 , x) ϕ ˜k,τ (k0 , x )

(150)

τ,τ 1 , · ψ i,τ (k, x) ϕ ˜k,τ (k0 , x )

(151)

i,k

and G(−/+,N) (x, x ) = (ik0 ) ·

2 N N 2 τ =1 i=0 τ =1 k=0

(d)

W∂Γ−

i,k

(d)

for the two Green functions GΓ+ and G(−/+) . τ,τ (d) of the interaction operaNext, to calculate the matrix elements W∂Γ+ i,k

(d)

tor W∂Γ+ , we consider the following transformation problem. We ask for the two n ˆ

matrices Tϕ and Tψ which allows us to express the eigenvectors ψ i,τ− (k0 r, θ, φ) and ∂ nˆ − ψ i,τ (k0 r, θ, φ) deﬁned in (45) and (48) in terms of the eigenvectors n ˆ n ˆ ϕi,τ− (k0 r, θ, φ) and ∂ nˆ − ϕi,τ (k0 r, θ, φ) as well as ψ i,τ− (kr, θ, φ) and ∂ nˆ − ψ i,τ (kr, θ, φ) according to

140

Tom Rother

ψ nˆ − (k0 r, θ, φ) = ϕ nˆ − (k0 r, θ, φ) · Tϕ + ψ nˆ − (kr, θ, φ) · Tψ ∂ nˆ − ψ(k0 r, θ, φ) = ∂ nˆ − ϕ(k0 r, θ, φ) · Tϕ + ∂ nˆ − ψ(kr, θ, φ) · Tψ

(152) (153)

at the scatterer surface. Please note that both of these equations are represented in a condensed noatation, i.e., all the eigenvectors are already summarized in corresponding block vectors consisting of two blocks according to the index τ . Within each block we have the usual eigenvectors as components. For example, ψ nˆ − (k0 r, θ, φ) in (152) represents the block vector & ' n ˆ n ˆ (154) ψ nˆ − (k0 r, θ, φ) = ψ i,1− (k0 r, θ, φ), ψ i,2− (k0 r, θ, φ) with i being again the combined index introduced in (38). Correspondingly, both matrices Tϕ and Tψ are 2 × 2 block matrices where each block is a single square matrix, due to the combined indices i, k. Thus the matrix elements of Tϕ are denoted by τ,τ (155) [Tϕ ]i,k for example. This condensed notation will ease the following derivation. Now, it is τ,τ quite interesting to realize that the negative matrix elements − [Tϕ ]i,k are identical τ,τ (d) (d) of the interaction operator W∂Γ+ we are with the matrix elements W∂Γ+ i,k

seeking for. To see this we have only to apply the transmission conditions (116) and (117) to the approximations (150) and (151) (see [14], for example). Since this is a straight forward analysis without further diﬃculties the reader may verify this by himself. To determine Tϕ from the equation system (152)/(153) we apply a scalar product as deﬁned in (136) with the weighting vector functions g to Eq. (152), and with diﬀerent weighting vector functions h to Eq. (153). Both sets of weighting vector functions are block vectors as shown in (154). But they are again not yet speciﬁed to have an additional degree of freedom. From this procedure it follows the matrix equation system (g,ψ

C∂Γ

n ˆ−

)

(h,∂n ˆ ψ) C∂Γ −

ˆ− (g,ϕ n )

· Tψ = − A∂Γ · Tψ = −

(h,∂n ˆ ϕ) A∂Γ −

n ˆ−

(g,ψ0

· Tϕ + B∂Γ · Tϕ +

)

(h,∂n ˆ ψ0 ) B∂Γ −

(156) .

(157)

The matrix elements of these matrices are deﬁned by the scalar products

ˆ− (g,ϕ n )

A∂Γ

i,k

n ˆ−

(g,ψ B∂Γ 0

τ,τ

τ,τ ) i,k

(g,ψ

C∂Γ

n ˆ−

)

τ,τ i,k

n ˆ := g i,τ (x) | ϕk,τ− (k0 , x) ∂Γ

(158)

n ˆ := g i,τ (x) | ψ k,τ− (k0 , x) ∂Γ

(159)

n ˆ := g i,τ (x) | ψ k,τ− (k, x) ∂Γ

(160)

4 Green functions for plane wave scattering

τ,τ (h,∂n ˆ − ϕ)

A∂Γ

i,k

τ,τ (h,∂n ˆ − ψ0 )

B∂Γ

i,k

τ,τ (h,∂n ˆ − ψ)

C∂Γ

i,k

141

:= hi,τ (x) | ∂nˆ − ϕk,τ (k0 , x) ∂Γ

(161)

:= hi,τ (x) | ∂nˆ − ψ k,τ (k0 , x) ∂Γ

(162)

:= hi,τ (x) | ∂nˆ − ψ k,τ (k, x) ∂Γ .

(163)

Resolving both Eqs (156) and (157) for matrix Tψ is one way to determine the matrix Tϕ . This results in −1 −1 ˆ − −1 ˆ− (h,∂n (h,∂n ˆ ψ) ˆ ϕ) (g,ψ n ) (g,ϕ n ) · A∂Γ − C∂Γ − · A∂Γ − × Tϕ = C∂Γ n ˆ −1 ˆ − −1 (h,∂n (h,∂n ˆ ψ) ˆ ψ0 ) (g,ψ − ) (g,ψ n ) . C∂Γ · B∂Γ 0 − C∂Γ − · B∂Γ −

(164)

In [15], for example, it was shown that for a speciﬁc choice of the weighting vector functions g i,τ and hi,τ the matrix elements of the interaction operator runs exactly into Waterman’s T-matrix derived in [5]. Another possibility consists in resolving both Eqs (156) and (157) for matrix Tϕ at ﬁrst. Thus we get Tψ =

ˆ − −1 (g,ϕ n ) A∂Γ

·

(g,ψ C∂Γ

ˆ − −1 (g,ϕ n )

A∂Γ

n ˆ−

)

− n ˆ−

(g,ψ0

· B∂Γ

−1 (h,∂n ˆ ϕ) A∂Γ −

)

·

(h,∂n ˆ ψ) C∂Γ −

−1 (h,∂n ˆ − ϕ)

− A∂Γ

−1 ×

(h,∂n ˆ − ψ0 )

· B∂Γ

(165)

for the matrix Tψ . Next, in close analogy to (130), we can change from the eigenvectors ψ(k0 , x) to the eigenvectors ψ(k, x) according to ψ(k, x) = ψ(k0 , x) · Tψ0 /ψ .

(166)

at the scatterer surface. Finally, using (130) and (166), we can formulate Eq. (152) in terms of the eigenvectors ϕ(k0 , x) only. From this we get " ! (167) Tϕ = T∂Γ · E − Tψ0 /ψ · Tψ with T∂Γ being the T-matrix given in (137). Obviously, if we chose Tψ = 0 we get from (167) the known result for the ideal metallic scatterer. We will ﬁnish this subsection with the remark that the same result would have been obtained if the deﬁning volume integral equations GΓ+ (x, x ) := G0 (x, x ) + (d) Gt> (x, x ¯) · W∂Γ+ (¯ x, x ˜) · Gt< (˜ x, x ) dV (¯ x) dV (˜ x) (d)

(168)

Γ−

and G (−/+) (x, x ) :=

Gt<s (x, x ¯) · W∂Γ− (¯ x, x ˜) · Gt< (˜ x, x ) dV (¯ x) dV (˜ x) . (d)

Γ−

(169)

142

Tom Rother

instead of the surface integral Eqs. (124) and (125) would have been used in the course of the above given derivation. This choice aﬀects only the deﬁnitions (148) and (149) of the matrix elements of the interaction operators, i.e., then we get τ,τ (d) (d) ˜i,τ (k0 , x W∂Γ+ ψ := (ik0 ) ¯)·W∂Γ+ (¯ x, x ˜)·ψ k,τ (k0 , x ˜) dV (¯ x) dV (˜ x) (170) i,k

Γ−

and τ,τ (d) W∂Γ− := (ik) i,k

Γ−

(d)

ϕ ˜i,τ (k0 , x ¯) · W∂Γ− (¯ x, x ˜) · ψ k,τ (k0 , x ˜) dV (¯ x) dV (˜ x) . (171)

These diﬀerent options in the deﬁnition of interaction operators are related to the equivalence principles which allows us to replace the scatterer by equivalent currents on its surface or within its volume (see [16], for example). We will come back to these latter deﬁnitions of the interaction operators in the next section when considering singular volume integral equations for dielectric scatterers.

4.4 Singular integral equations In this section we will demonstrate how a small but important change in the deﬁning equations of the interaction operators will lead us to the conventional singular integral equation approaches. The singular surface integral equation for the Green function of an ideal metallic scatterer related to the magnetic ﬁeld is considered ﬁrst. Once this ‘magnetic Green function’ is known the ‘electric Green function’ GΓ+ which is of our primary interest here can be calculated by a simple operation. This subsection ends with the derivation of the so-called Lippmann–Schwinger equation which can serve as a starting point for iterative solutions. In the next subsection we will derive the singular volume integral equation known for the dielectric scatterer which already exhibits the nature of a Lippmann–Schwinger equation. This will be achieved by the same small change in the relevant deﬁning equation of the interaction operator. Finally, in the last subsection of this section we will discuss why the abovementioned ‘small changes’ are strongly related to Rayleigh’s hypothesis. Additionally, we will justify a pragmatic position concerning the validity of this hypothesis. 4.4.1 Ideal metallic scatter 4.4.1.1 Singular surface integral equation (h)

We can deﬁne the magnetic Green function GΓ+ according to GΓ+ (x, x ) := ∇x × GΓ+ (h)

(172)

with GΓ+ being the electric Green function introduced in section 4.3.2. Instead of (123) we will now employ the deﬁning equation

4 Green functions for plane wave scattering

GΓ+ (x, x ) := G0 (x, x ) + * ˆ ∂Γ+ (¯ x, x ˜) · G0 (˜ x, x ) dS(¯ x) dS(˜ x) . G0 (x, x ¯) · W

143

(173)

∂Γ

for the interaction operator. Unlike (123), where the quantities Gt> and Gt< have been used within the surface integral, we have now changed to the full dyadic freespace Green function G0 . But the replacement of Gt< by G0 is not really a change because we generally assumed throughout this paper that the source of the primary ﬁeld is located outside the smallest sphere circumscribing the scatterer. Thus, the most signiﬁcant change is the replacement of Gt> by G0 . Please note that we used ˆ ∂Γ+ in (173) to distinguish it from the interaction operator deﬁned the notation W (h) by (123). From (172) we can see that GΓ+ is a solution of !

" (h) ∇x × ∇x × − k02 GΓ+ (x, x ) = ∇x × Iδ(x − x ) .

(174)

Furthermore, if looking at (97), we deﬁne the magnetic free-space Green function according to G0 (x, x ) := ∇x × G0 (x, x ) = ∇x G0 (x, x ) × I . (h)

(175)

Applying (172) to (173) results therefore in GΓ+ (x, x ) = G0 (x, x ) + * (h) ˆ ∂Γ+ (¯ G0 (x, x ¯) · W x, x ˜) · G0 (˜ x, x ) dS(¯ x) dS(˜ x) . (h)

(h)

(176)

∂Γ

(h)

The necessary boundary condition GΓ+ has to fulﬁl at the scatterer surface (beside the radiation condition at inﬁnity with respect to x, of course) is given by * (h) ˆ ∂Γ+ (x, x ˜) · G0 (˜ x, x ) dS(˜ x) . (177) n ˆ × GΓ+ (x, x ) = W ∂Γ

Once we know this magnetic Green function we obtain the corresponding magnetic ﬁelds from the relations 1 (h) · G (x, x ) · ρ(x ) dV (x ) (178) hg (x) = iωμ0 Γ+ Γ+ and

1 hinc (x) = · iωμ0

G0 (x, x ) · ρ(x ) dV (x ) , (h)

Γ+

(179)

respectively. This is a consequence of Maxwell’s equations ∇ × e(x) ∇ × h(x) ∇ · d(x) ∇ · b(x)

= = = =

iωb(x) − iωd(x) + ρ(x) 0 0

(180) (181) (182) (183)

144

Tom Rother

valid for time-harmonic ﬁelds, the linear material equations d(x) = e(x) = 0 r e(x)

(184)

b(x) = μ0 h(x) ,

(185)

and the required boundary condition n ˆ × h(x) = j ∂Γ (x)

; x ∈ ∂Γ

(186)

at the surface of the ideal metallic scatterer. j ∂Γ is the induced surface current. This unknown surface current is related to the unknown interaction operator by (see [14], for example) * i ˆ ∂Γ+ (¯ j ∂Γ (¯ x) = − · x, x ˜) · einc (˜ x) dS(˜ x) . (187) W ωμ0 ∂Γ The prefactor −i/(ωμ0 ) ensures that the current will have the correct dimension ˆ ∂Γ+ from the [A/m]. We are then able to determine the interaction operator W surface integral equation * 1 ˆ ∂Γ+ (x, x · ˜) · G0 (˜ x, x ) dS(˜ x) − W 2 ∂Γ * (h) ˆ p.v. n ˆ× G0 (x, x ¯) · W∂Γ+ (¯ x, x ˜) · G0 (˜ x, x ) dS(¯ x) dS(˜ x) = ∂Γ

n ˆ × G0 (x, x ) (h)

; x ∈ ∂Γ

(188)

which results from the application of boundary condition (177) in conjunction with (176). The factor 1/2 of the ﬁrst contribution on the left-hand side results from (h) treating the singularity of G0 at the scatterer surface appropriately (for the details of this procedure see [17] especially sections 12 to 14, and section 77 therein!). The ‘principal value’ integration on the right-hand side of this equation means * · · · dS(¯ x) := lim · · · dS(¯ x) , (189) p.v. ∂Γ

∂Γδ →0

∂Γ \∂Γδ

i.e., a small surface element ∂Γδ containing the singularity is excluded from the surface integration [17]. (188) is the Green’s function analogue to the conventional surface integral equation for the induced surface current (see [17,19], for example). ˆ ∂Γ+ the magnetic Green function can be calculated according to Once knowing W (176). By taking the curl of the magnetic Green function we get together with (114) GΓ+ (x, x ) =

1 1 (h) · ∇x × GΓ+ (x, x ) − 2 · Iδ(x − x ) . k02 k0

(190)

This electric Green function obeys the homogeneous Dirichlet condition as demonstrated for the related total electric ﬁeld in [17]. That is the way we can choose to circumvent the strong singularity of the dyadic free-space Green function G0 mentioned in section 4.3.1. Another advantage of using the interaction operator instead of the induced surface current is the independence of the former from the direction of the incident ﬁeld. This will ease the orientation averaging process if necessary. The induced surface current, on the other hand, depends on the incident ﬁeld as we can see from Eq. (187). This has been considered to be an essential drawback of the boundary integral equations, so far [1].

4 Green functions for plane wave scattering

145

4.4.1.2 Lippmann-Schwinger equations First, to derive the Lippmann–Schwinger equations for the ideal metallic scatterer, we apply Green’s theorem (93) in Γ+ with the two Green functions Q(¯ x, x) = G0 (¯ x, x)

(191)

P(¯ x, x ) = GΓ+ (¯ x, x ) .

(192)

Taking the symmetry relation (99) into account this provides GΓ+ (x, x ) = G0 (x, x ) + * ! " tp ! " ˆ n ¯ − × G0 (¯ x, x) · ∇x¯ × GΓ+ (¯ x, x ) dS(¯ x) .

(193)

∂Γ

This expression can be reformulated by use of identity (87). We get GΓ+ (x, x ) = G0 (x, x ) − * ! " ˆ G0 (x, x ¯) · n ¯ − × ∇x¯ × GΓ+ (¯ x, x ) dS(¯ x) .

(194)

∂Γ

In close analogy to the ‘classical’ dyadic delta distribution (107) one can introduce a dyadic surface delta distribution via the deﬁnition * D∂Γ (x − x) · f (x ) dS(x ) := f (x) ; x, x ∈ ∂Γ . (195) ∂Γ

But because of the boundary conditions (2) and (8) we should be more interested in a corresponding delta distribution which is restricted to the tangential planes at the scatterer surface ∂Γ . This is realized by the deﬁnition * (ˆ n) D∂Γ (x − x) · f nˆ (x ) dS(x ) := ∂Γ

f nˆ (x)

; x , x ∈ ∂Γ

(196)

where f nˆ (x) is the tangential projection of the vector function f (x) at the scatterer surface [12]. This special delta distribution allows us to deﬁne the operator ˆ ˆ (n ¯) (n ¯) ˆ˜ × ∇x˜ × I U∂Γ (¯ x, x ˜) := − D∂Γ (˜ x−x ¯) · n (197) so that *

ˆ (n ¯ )

ˆ¯ − × ∇x¯ × GΓ+ (¯ U∂Γ− (¯ x, x ˜) · GΓ+ (˜ x, x ) dS(˜ x) = − n x, x )

(198)

∂Γ

holds. Hence, (194) can be further reformulated to become GΓ+ (x, x ) = G0 (x, x ) + * ˆ (n ¯ ) G0 (x, x ¯) · U∂Γ− (¯ x, x ˜) · GΓ+ (˜ x, x ) dS(˜ x) dS(¯ x) ∂Γ

(199)

146

Tom Rother

or, in operator notation, ˆ (n ¯ )

GΓ+ (x, x ) = G0 (x, x ) + G0 (x, x ¯) ◦ U∂Γ− (¯ x, x ˜) ◦ GΓ+ (˜ x, x ) .

(200)

This is the Lippmann–Schwinger equation of GΓ+ related to the scattering problem of an ideal metallic scatterer. The lowest and ﬁrst-order iterations of this equation are given by GΓ+ (x, x ) = G0 (x, x ) (0)

(201)

(this is the unperturbed problem, i.e., if there is no scatterer) and ˆ (n ¯ )

¯) ◦ U∂Γ− (¯ x, x ˜) ◦ G0 (˜ x, x ) GΓ+ (x, x ) = G0 (x, x ) + G0 (x, x (1)

(202)

or GΓ+ (x, x ) = G0 (x, x ) + * ˆ (n ¯ ) G0 (x, x ¯) · U∂Γ− (¯ x, x ˜) · G0 (˜ x, x ) dS(˜ x) dS(¯ x) = ∂Γ * ! " ˆ G0 (x, x ) − G0 (x, x ¯) · n ¯ − × ∇x¯ × G0 (¯ x, x ) dS(¯ x) (1)

(203)

∂Γ

(this is the ﬁrst deviation from the unperturbed problem). These two iterations are not aﬀected by the singularity of G0 , but each higher-order iteration. The Lippmann–Schwinger equation of the corresponding interaction operator can be derived from the deﬁnition (173) which reads in operator notation ˆ ∂Γ+ (¯ GΓ+ (x, x ) := G0 (x, x ) + G0 (x, x ¯) ◦ W x, x ˜) ◦ G0 (˜ x, x ) ,

(204)

An intercomparison of (204) with (200) results in ˆ ∂Γ+ (˜ ˆ ∂Γ+ (x, x ) := U (nˆ¯− ) (x, x ) + U (nˆ¯− ) (x, x ¯) ◦ G0 (¯ x, x ˜) ◦ W x, x ) . (205) W ∂Γ ∂Γ Obviously, the lowest-order iteration of this Lippmann–Schwinger equation provides the ﬁrst-order iteration of the corresponding Green function GΓ+ . 4.4.2 Dielectric scatter 4.4.2.1 Singular volume integral equation To derive the singular volume integral equation for the Green function related to the dielectric scatterer Green’s theorem must be applied twice. First we use (93) in Γ+ with the two dyadics Q(x, x ) = G0 (x, x )

P(x, x ) =

(d) GΓ+ (x, x )

; x ∈ Γ+

; x, x ∈ Γ+

(206) (207)

(case 1). There are two possibilities concerning the location of x . It can be located either in Γ+ (case 1a) or in Γ− (case 1b). Case 1a results in

4 Green functions for plane wave scattering

GΓ+ (x , x ) = G0 tp (x , x ) + + tp (d) [ˆ n− × G0 (x, x )] · ∇x × GΓ+ (x, x ) − ∂Γ tp (d) [∇x × G0 (x, x )] · n ˆ − × GΓ+ (x, x ) dS(x) ,

147

(d)

*

and case 1b correspondingly in * + tp (d) 0 = G0 tp (x , x ) + [ˆ n− × G0 (x, x )] · ∇x × GΓ+ (x, x ) − ∂Γ tp (d) [∇x × G0 (x, x )] · n ˆ − × GΓ+ (x, x ) dS(x) .

(208)

(209)

Next we apply (93) in Γ− with the two dyadics Q(x, x ) = G0 (x, x ) ; x ∈ Γ+ P(x, x ) = G (−/+) (x, x ) ; x ∈ Γ− , x ∈ Γ+

(210) (211)

Here we can again distinguish the two possible locations of x . x ∈ Γ+ (case 2a) provides κ2d · G0 tp (x, x ) · G (−/+) (x, x ) dV (x) = Γ− * + tp [ˆ n × G0 (x, x )] · ∇x × G (−/+) (x, x ) − − ∂Γ tp [∇x × G0 (x, x )] · n ˆ × G (−/+) (x, x ) dS(x) . (212) x ∈ Γ− (case 2b) provides on the other hand − G (−/+) (x , x ) + 2 κd · G0 tp (x, x ) · G (−/+) (x, x ) dV (x) = Γ− * + tp [ˆ n × G0 (x, x )] · ∇x × G (−/+) (x, x ) − − ∂Γ tp [∇x × G0 (x, x )] · n ˆ × G (−/+) (x, x ) dS(x) . κ2d = k 2 − k02

(213) (214)

denotes the diﬀerence in the wave numbers of the scatterer material and the assumed free space outside the scatterer. If one combines the two cases 1a/2a as well as 1b/2b, if taking symmetry relation (99), identity (88), and the transmission conditions (116) and (117) into account, and if one renames the variables afterwards we get from the equality of the surface integrals in all of the above cases ﬁnally the two volume integral equations (d) 2 GΓ+ (x, x ) = G0 (x, x ) + κd G0 (x, x ¯) · G (−/+) (¯ x, x ) dV (¯ x) Γ−

x, x ∈ Γ+

(215)

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Tom Rother

and G (−/+) (x, x ) = G0 (x, x ) + κ2d

G0 (x, x ¯) · G (−/+) (¯ x, x ) dV (¯ x)

Γ−

x ∈ Γ+ , x ∈ Γ− .

(216)

The essential step is the solution of Eq. (216). But the strong singularity of G0 inside the scatterer must be considered very carefully (see [18, 20], for example). Once G (−/+) is known we can calculate the Green function of our primary interest from Eq. (215). Both of these ﬁnal equations serve as a starting point for diﬀerent numerical approaches like the Discrete Dipole Approximation (DDA) or the diﬀerent Method of Moment schemes, for example [1, 16]. 4.4.2.2 Lippmann–Schwinger equation Equation (216) exhibits already the nature of a Lippmann–Schwinger equation. With the deﬁnition GΓ+ (x, x ) := G0 (x, x ) + ˆ (d) (¯ G0 (x, x ¯) W ˜) G0 (˜ x, x ) dV (¯ x) dV (˜ x) , Γ− x, x (d)

(217)

Γ−

in which we have again replaced Gt> (x, x ¯) by G0 (x, x ¯) if compared to (124), we get after intercomparison with (215) and (216) the Lippmann–Schwinger equation (d) 2 ˆ (d) (ˆ ˆ (¯ x, x ˜) = κd · Iδ(¯ x−x ˜) + G0 (¯ x, x ˆ) · W x, x ˜) dV (ˆ x) (218) W Γ−

Γ−

Γ−

of the interaction operator related to the dielectric scatterer. Its lowest-order iteration, after insertion into (217), provides the ﬁrst-order iteration of the Green function which deviates from the unperturbed problem. Each higher-order iteration has to consider again the strong singularity of G0 inside the scatterer. 4.4.3 Rayleigh’s hypothesis Let us bring to our mind again the diﬀerences of sections 4.3 and 4.4. In section 4.3 we started from the deﬁnitions (123), and (124) or (168), respectively. In conjunction with the required boundary conditions at the scatterer surface we were then able to derive appropriate equations for the matrix elements of the interaction operators. By use of these matrix elements we could approximate the relevant Green functions by ﬁnite series expansions which are in accordance not only with the boundary conditions at the scatterer surface but also with the radiation condition and the vector-wave equation. In the course of these derivations it was tacitly assumed that appropriate expressions for the transverse parts Gt< and Gt> of the dyadic free-space Green function are known. Such expressions were given in Eq. (110). In section 4.4, in contrast to what was done in section 4.3, we started from Eqs (173) and (217), respectively, where the quantity Gt> (x, x ¯) was replaced by the full

4 Green functions for plane wave scattering

149

dyadic free-space Green function G0 . Applying again the boundary conditions we could derive singular surface or volume integral equations to determine the interaction operators explicitly. But as a consequence of using G0 instead of Gt> we had to consider the strong singularity of this Green function appropriately. Exceptions were only the lowest-order iterations of the Lippmann–Schwinger equations. How is this diﬀerence related to the so-called Rayleigh hypothesis? Before we will answer this question it should be noted that it is suﬃcient to restrict the following discussion to the ideal metallic scatterer since the dielectric scatterer provides no additional aspects to it. In 1907, surprisingly one year before Mie’s wellknown paper [21] on light scattering on spherical particles, Lord Rayleigh published a paper in which he solved the scattering problem of a plane wave incident on a sinusoidal surface (a grating) [22]. But this is a surface which does not coincide with a constant coordinate line in Cartesian coordinates. Thus the conventional separation of variables method was not supposed to become applicable to this problem. By contrast, Mie’s problem of light scattering on a spherical scatterer is appropriate for this method if applied in spherical coordinates. And that is exactly what Mie did. To solve the scattering problem of the grating Rayleigh used an expansion of the scattered ﬁeld in terms of plane waves which propagate solely away from the grating, i.e., which uses solely outgoing solutions of the homogeneous wave equation. It became clear later on that Rayleigh’s approach is nothing but an appropriate generalization of the separation of variables method to scattering problems of nonspherical scatterers (see [15,23], for example). Moreover, Rayleigh’s approach produces the same equation system as the application of Waterman’s Tmatrix approach will do [24]. We can state therefore that Rayleigh’s approach coincides with the methodology presented in section 4.3 herein. More precisely, Eq. (123) with Gt> (x, x ¯) in the surface integral coincides with Rayleigh’s assumption of using solely outgoing solutions to represent the scattered ﬁeld everywhere outside an ideal metallic scatterer. But why is this called a hypothesis? In 1953 Lippmann published a very short comment on Rayleigh’s approach [25]. He argued that outside the grating but in its valleys the usage of solely outgoing plane waves is not correct. It is just an assumption (a hypothesis) which results in an incomplete solution scheme. In these regions incoming plane waves must additionally be taken into account. Concerning our scattering problem of nonspherical particles treated in spherical coordinates this means that outside the scatterer but within the smallest sphere Smin circumscribing the scatterer (see Fig. 4.2) the usage ¯) is not complete because this quantity contains only outgoing eigenof Gt> (x, x vectors with respect to x, according to Eq. (110). Especially in this intermediate region we have points x = x1 which obviously violates the restriction x>x ¯ ; x ¯ ∈ ∂Γ

(219)

we ﬁxed for Gt> in (110). In his comment Lippmann stated furthermore that this problem does not appear if one uses the full free-space Green function as we did in section 4.4. In other words: it is questionable whether Eq. (123) is able to produce correct results especially outside the scatterer but inside Smin . Equation (173), on the other hand, is assumed to produce correct results everywhere outside the scatterer since the corresponding singular integral equations consider the full dyadic free-space Green function which contains both sets of eigenfunctions, outgoing and

150

Tom Rother k0 , Γ+

x2 scatterer x1

k, Γ− Smin

Fig. 4.2. The region outside the scatterer but inside Smin is the crucial region for the discussion of Rayleigh’s hypothesis.

incoming. This is the reason why in the presentation of the T-matrix approach in the recent literature the near-ﬁeld region outside the scatterer but inside Smin is excluded from the considerations (see [11], for example). Within our Green’s function formalism Rayleigh’s hypothesis is therefore equivalent to the questions whether (123) or (173) or both equations will produce correct results for the Green function of an ideal metallic but nonspherical scatterer. There is no deﬁnite answer, so far, but it is quite interesting to study the diﬀerent answers given in the past. But since such a detailed treatment is outside the scope of this paper we will solely justify a pragmatic position we take up with respect to this question, based on numerical experiences and a few intuitive arguments. We believe that both equations (123) and (173) produce correct results! The numerical experience which supports our believe are the following: –

–

–

Scattering calculations for spheroidal particles based on the T-matrix approach in spherical as well as in spheroidal coordinates produce the same results (see [26], for example). So far, no inconsistencies in such calculations have been observed [27–29]. Treating spheroidal particles in spheroidal coordinates by use of the T-matrix approach is not inﬂuenced by Rayleigh’s hypothesis, as it happens also for the sphere in spherical coordinates. Recent intercomparisons with Rayleigh’s approach and a geometrical optics approach if applied to random rough surfaces support the correctness of Rayleigh’s approach [30]. Very recently we solved the scattering problem of Rayleigh’s original paper by use of a T-matrix approach based on (123), and by use of a singular surface integral equation based on (173). The obtained results of both approaches agree very well even in the near-ﬁeld. These results are published in [31].

The intuitive arguments are: –

For a certain nonspherical geometry and a certain T-matrix approach based on (123) Dallas could prove mathematically the least-squares convergence of the scattering results [32].

4 Green functions for plane wave scattering

151

The requirement x > x ¯ for Gt> is only a necessary requirement if we intend to approximate the dyadic free-space Green function by an appropriate series expansion. But in the course of deriving the T-matrix from Eq. (123) we need only the feature that Gt> is a solution of the homogeneous vector-wave equation which obeys the radiation condition with respect to x. And that is what is warranted by use of Gt> , independent of whether it approximates the dyadic free-space Green function correctly or not. For the determination of the matrix elements of the interaction operator Gt> has ‘merely’ the meaning of an auxiliary function. – We have no additional boundary condition on the surface Smin to split the scattering problem into a near-ﬁeld and far-ﬁeld part. The boundary value problem as formulated in section 4.2 seems to be complete from a mathematical point of view.

–

Therefore we will fall back on the results of section 4.3 for the ongoing considerations. It should be also stated in this context that the various T-matrix approaches developed so far have been proved to produce very accurate and highly reliable results also if compared to experiments, not at least due to the improvements and manifold applications made and performed by Mishchenko et al. [11]. The alternative singular surface integral equations, for example, are much less practicable. Even in the limiting case of an ideal metallic sphere it is not simple to reproduce the results of the Mie series (which is nothing but a special T-matrix approach) with high accuracy (see [19], for example).

4.5 Symmetry and Unitarity Summarizing the results of section 4.3.2 we can state that both Green functions related to the scattering problems of our interest can be approximated by the expansion GΓ+ (x, x ) = G0 (x, x ) + (N)

(ik0 )

2

N

, τ,τ 1 [W ]i,k · ϕi,τ (k0 , x) ϕ ˜k,τ (k0 , x )

τ,τ =1 i,k=0

x, x ∈ Γ+ τ,τ

(220)

with [W ]i,k being the matrix elements of the matrix −T∂Γ according to (137) in the case of an ideal metallic scatterer, and −Tϕ according to (164) or (167) in the case of a dielectric scatterer. Approximation (220) can be reformulated by introducing the so-called ‘scattering matrix’ (S-matrix). This was done ﬁrst by Wheeler and Heisenberg in the context of quantum mechanical scattering but later on also for electromagnetic wave scattering [33–35]. The symmetry and unitarity properties of this matrix are of special importance because they can be related to the physical principles of reciprocity and energy conservation which must be fulﬁlled under certain circumstances. It is quite interesting to note that in the literature these physical principles, which are essentially based on our physical experience, are used as preconditions to prove the symmetry and unitarity properties of the

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Tom Rother

S-matrix. This holds also for the quantum mechanical considerations. But it seems strange to use principles of our physical experience to prove mathematical properties of a matrix. That is, since there exists no proof for energy conservation, for example, this would mean that we are never able to prove the unitarity property of the S-matrix. But we should be able to prove the mathematical properties of symmetry and unitarity of a matrix on a mathematical level without employing reciprocity and energy conservation as preconditions. Furthermore, in the recent literature these considerations are restricted to the far ﬁeld only. But such a restriction is not justiﬁed from a physical point of view. Therefore we want to show now how these properties can be related to the mathematical properties of the Green functions. 4.5.1 Symmetry To consider the symmetry property we write down Eq. (220) once again but with all indices: GΓ+ (x, x ) = G0 (x, x ) + (N)

2

(ik0 )

1 , τ,τ (−1)l · [W ]l,n;l n · ϕl,n,τ (k0 , x) ϕ−l ,n ,τ (k0 , x )

τ,τ =1 l,n;l ,n

x, x ∈ Γ+

(221)

For both cases, the ideal metallic and the dielectric scatterer, the Green functions must fulﬁl the same symmetry relation (119)/(120). Applying this relation to (221) requires the equality of 2

1 , τ,τ (−1)l · [W ]l,n;l n · ϕl,n,τ (k0 , x) ϕ−l ,n ,τ (k0 , x ) =

τ,τ =1 l,n;l ,n 2

1 , τ,τ (−1)l · [W ]l,n;l n · ϕ−l ,n ,τ (k0 , x) ϕl,n,τ (k0 , x ) . (222)

τ,τ =1 l,n;l ,n

The four steps – – – –

insert the identity factor 1 = (−1)l−l on the right-hand side of (222) interchange the notation n, l, τ and n , l , τ on the right-hand side of (222) replace l by −l, and l by −l on the right-hand side of (222) compare the thus converted right-hand side with the left-hand side of (222)

result in the general symmetry relation τ,τ

τ ,τ

[W ]l,n;l ,n = (−1)l+l · [W ]−l ,n ;−l,n

(223)

for the matrix elements of the interaction operator. This corresponds with Eq. (5.34) in [11], for example, which was derived therein on the basis of reciprocity. It should

4 Green functions for plane wave scattering

153

be mentioned that in Waterman’s original paper [5] the simpler symmetry relation τ,τ

τ ,τ

[W ]l,n;l ,n = [W ]l ,n ;l,n

(224)

was given. The diﬀerence between (223) and (224) results from the fact that we use the complex φ-dependence e ilφ in our eigenvectors instead of the two real functions sin φ and cos φ as Waterman did in his paper. Furthermore, since the symmetry relation (119)/(120) holds for the exact Green’s function one may expect that (223) holds only for inﬁnite matrices, in general. Symmetry relation (223) can therefore be used as a measure of numerical convergence for nonspherical scatterers, for example. It should be also mentioned that (223) is independent of whether the scatterer is absorbing or non-absorbing. 4.5.2 Unitarity We assume that the source of the primary incident wave (which need not necessarily be a plane wave!) is located somewhere outside the outer boundary ∂Γa of the now considered volume Γ˜ (see Fig. 4.3). ∂Γa is not restricted to be a spherical surface in the far ﬁeld, i.e., to be identical with S∞ even though it can be S∞ ! The inner boundary of Γ˜ is given by the surface ∂Γ of the scatterer. Furthermore, we assume that |x| < |x | holds for every point x ∈ ∂Γa . This gives us the justiﬁcation to use the lower expansion Gt< of (110) instead of G0 (x, x ) everywhere inside Γ˜ as well as on its outer surface. Thus (220) reads

˜ k0 , Γ ∂Γa scatterer

k, Γ− ∂Γ

Fig. 4.3. Scattering conﬁguration to consider unitarity. The outer space Γ˜ is assumed to be source-free.

154

Tom Rother (<,N)

GΓ+

(x, x ) = (ik0 )

2 N 1

,

˜i,τ (k0 , x ) ψ i,τ (k0 , x) ϕ

+

τ =1 i=0

(ik0 )

2 N

, τ,τ 1 [W ]i,k · ϕi,τ (k0 , x) ϕ ˜k,τ (k0 , x )

τ,τ i,k=0

x ∈ Γ˜ , x ∈ Γ+

.

(225)

In view of relation (41) this expression can be converted into an expression which contains only incoming and outgoing eigensolutions as expansion vectors, (<,N)

GΓ+

(x, x ) = (ik0 )

with

2 N 1 1 · δi,k δτ,τ · χi,τ (k0 , x) + 2 τ,τ i,k=0 τ,τ ˜k,τ (k0 , x ) , [S]i,k · ϕi,τ (k0 , x) ϕ

τ,τ

τ,τ

[S]i,k = δi,k δτ,τ + 2 · [W ]i,k

(226)

(227)

being the matrix elements of the abovementioned S-matrix. (227) reads in matrix notation S=E + 2·W . (228) We can state therefore that the elements of the S-matrix possess the same symmetry property (223) we derived already for the matrix elements of the interaction operator. As a consequence of the above formulated precondition concerning the location of the source of the primary incident wave the Green function GΓ<+ (x, x ) as well as its approximation (226) are solutions of the homogeneous vector-wave equation with respect to x everyhere in Γ˜ . We assume further that the outer space as well as the scatterer are non-absorbing, i.e., the parameters k0 and k are both ∗

real quantities. Then the conjugate-complex Green functions GΓ<+ (x, x ) and ! −/+ "∗ G (x, x ) are also solutions of the corresponding homogeneous vector-wave equations. With these preconditions we are now able to discuss the unitarity property of the S-matrix. For this pupose we introduce the functional * ∂Γa

+

{P(x, x ), Q(x, x )}∂Γa := [ˆ na × P(x, x )]

∗ tp

· [∇x × Q(x, x )] − ∗ tp [∇x × P(x, x )] · [ˆ na × Q(x, x )] dS(x)

(229)

for at ﬁrst two arbitrary dyadics P and Q with n ˆ a being the outward-directed unit normal vector at ∂Γa , and x being an arbitrary but ﬁxed source point outside ∂Γa . Due to the above formulated preconditions it can now be shown that + GΓ<+ (x, x ), GΓ<+ (x, x ) =0 (230) ∂Γa

4 Green functions for plane wave scattering

155

holds for the Green function related to the ideal metallic as well as the dielectric scatterer. The proof for the ideal metallic scatterer is straightforward because of the homogeneous Dirichlet condition (115) which holds also for the conjugate-complex Green function. We simply have to apply Green’s theorem (93) within Γ˜ with ∗

GΓ<+ (x, x ) and GΓ<+ (x, x ) . To prove (230) for the dielectric scatterer we have to apply additionally Green’s theorem (93) within the scatterer (i.e., within Γ− ) ! "∗ with G −/+ (x, x ) and G −/+ (x, x ) , and by taking the transmission conditions into account which must hold also for the conjugate-complex Green functions. In close analogy to (229) we introduce the additional functional * 1 ∗ [ˆ na × ψ(x)] · [∇x × φ(x)] − {ψ(x), φ(x)}∂Γa := ∂Γa ∗, [ˆ na × φ(x)] · [∇x × ψ(x)] dS(x) (231) for at ﬁrst two arbitrary vector functions ψ and φ. Next, by use of Green’s theorem (91) the follwing identities for the eigenvector functions of the vector-wave equation can be proven: 1 , 1 , χi,τ (k0 , x), ϕi,τ (k0 , x) ∂Γ = ϕi,τ (k0 , x), χi,τ (k0 , x) ∂Γ = 0 , (232) a

a

1 , 1 , ϕi,τ (k0 , x), ϕj,τ (k0 , x) ∂Γ = χi,τ (k0 , x), χj,τ (k0 , x) ∂Γ = a ,a 1 χi,τ (k0 , x), ϕj,τ (k0 , x) ∂Γ = 0

(233)

if i = j and τ = τ , and 1 , 1 , χi,τ (k0 , x), χi,τ (k0 , x) ∂Γ = − ϕi,τ (k0 , x), ϕi,τ (k0 , x) ∂Γ = c

(234)

a

a

a

for all i = 0, · · · , N and τ = 1, 2. (232) is a direct consequence of a real k0 and the resulting identity ∗ (2) (k r) = h (k r) (235) h(1) 0 0 n n valid for the Hankel functions of ﬁrst and second order. To prove (233) we apply Green’s theorem (91) within the volume included by the two surfaces ∂Γa and S∞ with ϕi,τ (k0 , x) and ϕ∗j,τ (k0 , x), χi,τ (k0 , x) and χ∗j,τ (k0 , x), and ϕi,τ (k0 , x) and χ∗j,τ (k0 , x), respectively. Since for a real k0 follows 1 , 1 , f i,τ (k0 , x), g j,τ (k0 , x) ∂Γ = − f i,τ (k0 , x), g j,τ (k0 , x) S (236) a

∞

we get (233) from the orthogonality of the eigenvectors on the spherical surface S∞ . Finally, (234) is a consequence of (236) in conjunction with the far-ﬁeld behaviour (29) and (30). The constant c in (234) must not even be known to prove the unitarity. Now, using the approximations (<,N)

GΓ+

(x, x ) = (ik0 )

2 N 1 1 · δp,q δκ,κ · χp,κ (k0 , x) + 2 κ,κ p,q=0 κ,κ ˜q,κ (k0 , x ) . [S]p,q · ϕp,κ (k0 , x) ϕ

(237)

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Tom Rother

and

(<,N)

GΓ+

∗ tp

(x, x )

=

2 N 1 ∗ , 1 ! (ik0 ) · δi,k δτ,τ · ϕ ˜k,τ (k0 , x ) χ∗i,τ (k0 , x) + 2 τ,τ i,k=0 , τ,τ 1 ∗ ˜k,τ (k0 , x ) ϕ∗i,τ (k0 , x) [S ∗ ]i,k · ϕ

(238)

in (230) we get from the identities (233)–(234) + (<,N) (<,N) GΓ+ (x, x ), GΓ+ (x, x ) = ∂Γa

⎡ ⎣

, k02 1 · χi,τ (k0 , x), χi,τ (k0 , x) ∂Γ · a 4 i,τ

δi,k δτ,τ · ϕ ˜∗k,τ (k0 , x )

δi,q δτ,κ · ϕ ˜q,κ (k0 , x ) −

κ ,q

τ ,k

[S ∗ ]i,k · ϕ ˜k,τ (k0 , x ) τ,τ

⎤

[S]i,q · ϕ ˜q,κ (k0 , x )⎦ = 0 . τ,κ

(239)

κ ,q

τ ,k

Because of (234) this can be transformed into !1 , k02 ·c· ϕ ˜∗k,τ (k0 , x ) ϕ ˜k,τ (k0 , x ) − 4 k,τ

1

⎤

,

τ,τ τ,κ [S ∗ ]i,k · [S]i,q · ϕ ˜∗k,τ (k0 , x ) ϕ ˜q,κ (k0 , x ) ⎦ = 0 .

(240)

i,τ,q,κ

The expression inside the square brackets will vanish obviously if τ,τ τ,κ [S ∗ ]i,k · [S]i,q = δk,q δτ ,κ

(241)

i,τ

holds. But this is nothing but the expression of unitarity of the matrix elements of the S-matrix. That is the way we can go to prove the unitarity of the S-matrix independent of energy conservation and far-ﬁeld restrictions as long as the initially formulated preconditions hold. But, as in the case of symmetry, we may expect that the unitarity condition (241) is valid only for inﬁnite matrices, in general. Therefore, beside the symmetry relation (223), we can also use the unitarity condition as a measure of convergence of approximation (225). If we consider only the ideal metallic scatterer there exists another but quite interesting possibility to prove the unitarity of the S-matrix directly by employing "τ,τ ! the transformation character (131)/(132) of the matrix elements W∂Γ+ i,k of the interaction operator at the scatterer surface ∂Γ . For this we deﬁne the scalar product

4 Green functions for plane wave scattering

v nˆ − , u nˆ −

∂Γ

:=

2 N * τ =1 i=0

!

v nˆ − (x)

∂Γ

"∗ i,τ

n ˆ

· ui,τ− (x) dS(x) .

157

(242)

of at ﬁrst two arbitrary block vectors v nˆ − and u nˆ − in which each component of the block vectors is a three-dimensional vector by itself. For the scalar product inside the surface integral on the right-hand side of (242) we have in spherical coordinates "∗ "∗ "∗ ! ! ! ! nˆ − " ∗ n ˆ n ˆ n ˆ n ˆ · ui,τ− = v nˆ − i,τ · ui,τ−r + v nˆ − i,τ · ui,τ−θ + v nˆ − i,τ · ui,τ−φ , (243) v i,τ r

θ

φ

for example, with i = 0, · · · , N and τ = 1, 2. Next we consider the two special block vectors ˆ − × ϕ0,1 (k0 r, θ, φ), · · · , n ˆ − × ϕN,1 (k0 r, θ, φ), φ(k0 , x) = n ˆ − × ϕN,2 (k0 r, θ, φ) (244) n ˆ − × ϕ0,2 (k0 r, θ, φ), · · · , n ˆ − × χ0,1 (k0 r, θ, φ), · · · , n ˆ − × χN,1 (k0 r, θ, φ), χ(k0 , x) = n ˆ − × χN,2 (k0 r, θ, φ) (245) n ˆ − × χ0,2 (k0 r, θ, φ), · · · , n which consist of the tangential projections of the outgoing and incoming eigenvectors of the vector-wave equation at the scatterer surface. By use of (131)/(132) and (41) it can be shown that the S-matrix transforms the block vector φ into the block vector χ according to χ tp = − S tp · φ tp

(246)

χ=−φ·S.

(247)

or, alternatively, Here we want to emphasize again that the S-matrix is a 2 × 2 block matrix. Now, since we have assumed a real k0 , we get the identity (χ, χ)∂Γ = (φ, φ)∂Γ for the two block vectors χ and φ. Together with (246) this results in φ, φ · S† · S ∂Γ = (φ, φ)∂Γ

(248)

(249)

where S† is the conjugate-complex and transpose of S. From this we can infer the unitarity condition S† · S = E (250) of the S-matrix directly. This way makes again clear that (250) holds only for inﬁnite matrices, in general, because the equals sign in (246) is in fact not strictly valid for block vectors with a ﬁnite number of components. The only exception is for the spherical scatterer as one can see from the deﬁnitions of the eigenvectors χ and φ. In this case every ﬁnite S-matrix is already unitary. But this means that this feature is not an appropriate measure of the accuracy of approximation (225) for spherical scatterer geometries. Furthermore, it was shown by Waterman in [5] that every ﬁnite S-matrix (or T-matrix, as well) belonging to a nonspherical scatterer can be treated to become unitary to improve the convergence behaviour of the approximation in certain cases. It should also be mentioned that, unfortunately, we could not ﬁnd a similar way to prove the unitarity of the S-matrix on the basis of the transformation (152) and (153) for dielectric scatterers, so far.

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4.6 Far-ﬁeld behaviour If we speak about scattering in electromagnetic theory it is tacitly meant that we consider scattering of plane waves on spatial structures characterized by certain dielctric properties. A corresponding scattering theory interrelates the asymptotic free states before and after the interaction of the plane wave with the scattering structure, whereas the incoming plane wave expresses the asymptotic free state before the interaction [36]. If we know the asymptotic state after the interaction we can derive and deﬁne quantities which are appropriate to measure. Thus, from scattering measurements we are able to infer information about the physical nature of the scatterer itself. If we consider the steady state of scattering with an assumed time-dependence of exp(−iωt) then the asymptotic states correspond to spatial distances large compared to a characteristic distance of the scattering structure, i.e., we look at the free states in the far ﬁeld. From this understanding of scattering the far ﬁeld becomes an important region, i.e., we must ask for the behaviour of the relevant ﬁelds in this region. All the following far-ﬁeld considerations are based on the general representation GΓ+ (x, x ) = G0 (x, x ) + * G0 (x, x ¯) · W(¯ x, x ˜) · G0 (˜ x, x ) dS(¯ x) dS(˜ x)

(251)

∂Γ

of the Green function for both the ideal metallic and dielectric scatterer. In the (d) x, x ˜) according to (123), and in the latter case W∂Γ+ (¯ x, x ˜) former case W∂Γ+ (¯ according to (124) must be used in the right-hand side of (251). The quantities Gt> and Gt< which appeared in Eqs (123) and (124) can be replaced by G0 since both variables x and x are always outside the smallest sphere circumscribing the scatterer. Gt will be identical with G0 because of (108), and if using the analytical expression (96) there is no need to distinguish between Gt> and Gt< . In Fig. 4.1 we have shown the general scattering conﬁguration. The far ﬁeld is represented by the spherical surface S∞ although it is a nonlocal condition valid for an inﬁnite distance from the scatterer, i.e., for lim|x|→∞ . Its practical importance stems from our experience that the far-ﬁeld behaviour of a certain scattering quantity can be suﬃciently approximated within a ﬁnite distance from the scatterer in a real experiment. As discussed in section 4.3 the dyadic free-space Green function G0 (x, x ) solves the inhomogeneous vector-wave equation (94) and obeys the radiation condition (11) with respect to x. The primary incident ﬁeld, generated by the source distribution (102) is then be given by Eq. (101). To force this primary incident ﬁeld to become a plane wave at the scatterer location we applied the additional condition (103). This allowed us to use the asymptotic expression (100) for G0 in (101). It means that the source of the primary incident ﬁeld must located somewhere in the far ﬁeld of the scatterer. But this statement concerning the location of this source is not suﬃcient to deﬁne scattering quantities in an appropriate way. The above-discussed understanding of scattering implies that the plane wave nature of the primary incident ﬁeld still remains in the far ﬁeld. There it will be interfere with the scattered ﬁeld to a total ﬁeld where the scattered ﬁeld results from the

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interaction of the primary incident plane wave with the scatterer. This linear superposition is expressed by the two contributions on the right-hand side of (251). This becomes problematic since a plane wave is a solution of the homogeneous vector-wave equation which does not fulﬁl the radiation condition in the far ﬁeld. Moreover, this is in contradiction to the solution of (101). Thus, the plane wave seems to contradict somehow the concept of Green’s function because the latter are essentially based on sources. The ideal plane wave represents ‘smoke without ﬁre’, so to speak. Anyway, to generate the required plane wave within a Green’s function technique we have to place the corresponding source behind the conventional far-ﬁeld S∞ , whatever this means theoretically. But it is not suﬃcient to place the source only somewhere behind the conventional far ﬁeld but in the far ﬁeld of the conventional far-ﬁeld, just to generate a plane wave in the conventional XXL (the enlarged far ﬁeld. Therefore we have to introduce a second far ﬁeld S∞ far ﬁeld) with respect to the variable x . Only this conﬁguration will allow us to generate a plane wave at the location of the scatterer as well as in the far-ﬁeld S∞ by use of the free-space Green function. This is the somehow strange situation concerning the plane wave we already mentioned in section 4.3.1. Now we will go into the details. 4.6.1 The plane wave We have already said that the plane wave einc (x) = e0 · eik·x = e0 · e−ik0 ·|x|·ˆex ·ˆex is a solution of the homogeneous vector-wave equation ∇ × ∇ × − k02 einc (x) = 0

(252)

(253)

in Γ+ . eˆx is the unit vector pointing toward the source of the primary plane wave located in the enlarged far-ﬁeld SXXL . eˆx , on the other hand, is pointing toward the point of measurement. These unit vectors will be denoted further on with n ˆ i = − eˆx n ˆ s = eˆx

(254) (255)

and should not be confused with the unit vectors n ˆ or n ˆ − deﬁned at the boundary surfaces. For the plane wave Maxwell’s equations read k × einc (x) = ωμ0 hinc (x) k × hinc (x) = − ω0 einc (x) k · einc (x) = 0 k · hinc (x) = 0

(256) (257) (258) (259)

since the ∇-operator, if applied to eik·x , provides simply ik. From these equations (253) follows immediately. We can see furthermore that the polarizations e0 und h0 are always perpendicular to each other and to the wave-vector k. k · x = const.

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are points of constant phase, and the wave-vector k is always perpendicular to the planes of constant phases. These are the important features of a plane wave. In the far-ﬁeld S∞ the plane wave enjoys an interesting representation. In (252) there appears the term eik0 ·|x|·ˆni ·ˆns . This expression can be rewritten in terms of incoming and outgoing spherical waves according to 2πi e−ik0 |x| eik0 |x| eik0 ·|x|·ˆni ·ˆns = − δ(ˆ ni − n (260) · δ(ˆ ni + n ˆs) ˆs) k0 |x| |x| with ˆ s ) = δ(cos θi ± cos θs ) · δ(φi ± φs ) . δ(ˆ ni ± n

(261)

From this we get einc (x) =

2πi e−ik0 |x| eik0 |x| − δ(ˆ ni − n · e0 · δ(ˆ ni + n ˆs) ˆs) k0 |x| |x|

(262)

for the plane wave in the far ﬁeld (see [11], for example). Obviously, if looking into ni ) we are not able to distinguish the direction of the primary source (i.e., if n ˆ s = −ˆ between an outgoing spherical wave and the plane wave. But the more important aspect of (262) becomes clear if we consider this representation together with the scattered ﬁeld. 4.6.2 The scattered and total ﬁeld The total ﬁeld in the far-ﬁeld S∞ follows from (121) and (122), respectively, in conjunction with the Green function (251). Together with the source (102) and XXL the asymptotic behaviour (100) of G0 (this can be used because of our S∞ construction!) we obtain from the ﬁrst part of the right-hand side of (251) just the above discussed plane wave (262) as the primary incident ﬁeld. The scattered ﬁeld follows from the second term on the right-hand side of (251). Here we can use again the asymptotic expression (100) resulting in ns , n ˆ i ) · e0 · es (x) = A(ˆ where ˆi) = A(ˆ ns , n

1 · 4π

* (ˆ ns )

It

eik0 |x| |x|

· e−ik0 nˆ s x¯ · W(¯ x, x ˜ ) · It

(ˆ ni )

(263)

· eik0 nˆ i x˜ dS(¯ x) dS(˜ x) . (264)

∂Γ

denotes the dyadic scattering amplitude. Because of (ˆ ns )

It

(ˆ n ) It i

= θˆs θˆs + φˆs φˆs = θˆi θˆi + φˆi φˆi

n ˆ s · A(ˆ ns , n ˆ i ) = A(ˆ ns , n ˆi) · n ˆi = 0

(265) (266) (267)

ˆ i )· holds for this amplitude. It should be mentioned that the second relation A(ˆ ns , n n ˆ i = 0 is usually an additional requirement in the existing literature (see [11, 35],

4 Green functions for plane wave scattering

161

for example). Here it is a consequence of representation (251) and must not be additionally required. The corresponding expression for the total ﬁeld reads now 2πi e−ik0 |x| eik0 |x| − δ(ˆ ni − n + · e0 · δ(ˆ ni + n ˆs) ˆs) et (x) = k0 |x| |x| A(ˆ ns , n ˆ i ) · e0 ·

eik0 |x| . |x|

(268)

From this we can see that any other direction n ˆ s = −ˆ ni will register only the ni we observe the interferscattered ﬁeld. Only in the forward direction n ˆ s = −ˆ ence of the plane and scattered wave. However, if we had assumed a primary ﬁeld generated from a source located within a ﬁnite distance from the scatterer (this corresponds to the general radiation problem in electrodynamics) then we would observe in the far-ﬁeld S∞ the interference of two outgoing spherical waves (one from the primary source, and the other from the scatterer) for every direction n ˆs. It is exactly the former behaviour which makes the plane wave especially suited for scattering experiments and their interpretation. Every measurement for which n ˆ s = −ˆ ni holds can be exclusively related to the scattered ﬁeld, and, thus, to the interaction of the given plane wave with the scatterer we want to analyse with a scattering experiment. However, the convenience of plane wave scattering in the far ﬁeld is justiﬁed only by our experimental experience. Consequently, the applicability of both models ‘plane wave’ and ‘far ﬁeld’ in a real experiment must be ensured from the beginning. But sometimes even this is not a simple task. Equation (268) can be split into incoming and outgoing spherical waves according to ns ) · et (x) = f 1 (ˆ

eik0 |x| e−ik0 |x| + f 2 (ˆ ns ) · |x| |x|

(269)

with the amplitude vectors 2πi (ˆ n ) · δ(ˆ ni + n ˆ s ) · It i · e0 k0 2πi · S(ˆ ns , n ˆ i ) · e0 , f 2 (ˆ ns ) = − k0 f 1 (ˆ ns ) =

(270) (271)

and the dyadic far-ﬁeld scattering function S(ˆ ns , n ˆi) =

ik0 (ˆ n ) · A(ˆ ns , n ˆ i ) + δ(ˆ ni − n ˆ s ) · It i . 2π

(272)

This function can now be used as the kernel of the following dyadic far-ﬁeld scattering operator: ˆ:f →g S (273) *

with

ˆ S(n ¯, n ˆ ) · f (ˆ n) dˆ n.

ˆ g(n ¯ ) := S∞

6

‘

S∞

· · · dˆ n’ denotes the integral

(274)

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2π

π

· · · sin θ dθ dφ . 0

(275)

0

This operator allows us to map the above given amplitude vector f 1 (ˆ n) of the incoming wave according to * ns ) = S(ˆ ns , n ˆ ) · [− f 1 (−ˆ n)] dˆ n (276) f 2 (ˆ S∞

to the amplitude vector f 2 (ˆ ns ) of the outgoing wave. The dyadic scattering amplitude as well as the dyadic far-ﬁeld scattering function obey the symmetry relations A(ˆ ns , n ˆ i ) = Atp (−ˆ ni , −ˆ ns )

(277)

S(ˆ ns , n ˆ i ) = S tp (−ˆ ni , −ˆ ns ) .

(278)

and The former is a consequence of the symmetry relations (99) and (119)/(120). Applied to (251), and by taking identity (90) into account we obtain at ﬁrst the symmetry relation W(¯ x, x ˜) = Wtp (˜ x, x ¯) . (279) From this relation and from (264), (277) follows in a straightforward way. The consequential relation of (272) is (278). * τ,τ [S]l,n;l ,n = y ∗l,n,τ (ˆ ns ) · S(ˆ ns , n ˆ i ) · y l ,n ,τ (ˆ ni ) dˆ ns dˆ ni (280) S∞

ˆ The transversal vector functions y l,n,τ are the matrix elements of the operator S. are related to the vector functions cl,n and bl,n given in (26) and (27) according to y l,n,1 (ˆ n) := in · γl,n · cl,n (θ, φ) (n−1)

y l,n,2 (ˆ n) := i

· γl,n · bl,n (θ, φ) .

(281) (282)

Because of the orthogonality of (35) and (36) the transversal vector functions form an orthonormal system on S∞ , i.e., * y ∗l,n,1 (ˆ n) · y l ,n ,1 (ˆ n) dˆ n = δl,l δn,n (283) S∞ * y ∗l,n,2 (ˆ n) · y l ,n ,2 (ˆ n) dˆ n = δl,l δn,n (284) S∞

and

* S∞

y ∗l,n,1 (ˆ n) · y l ,n ,2 (ˆ n) dˆ n=

*

y ∗l,n,2 (ˆ n) · y l ,n ,1 (ˆ n) dˆ n=0

(285)

S∞

hold. The second contribution on the right-hand side of (272) provides therefore the matrix elements δτ,τ δl,l δn,n . (286)

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To determine the matrix elements of the ﬁrst contribution on the right-hand side (ˆ n ) of (272) we have to go back to (264). First we note that the expression It i · eik0 nˆ i x˜ is nothing but the plane wave (105) if multiplied with the polarization vector e0 . But for such a plane wave we already know the expansion (111)–(113). From this and relations (31) and (32) it follows that we can use (ˆ ni )

It

· eik0 nˆ i x˜ = 4π ·

2 ∞ n

˜ φ) ˜ y ∗ (ˆ ψ l,n,τ (k0 r˜, θ, l,n,τ ni )

(287)

τ =1 n=0 l=−n (ˆ ni )

as an appropriate expansion for the dyadic It get (ˆ ns )

It

· e−ik0 nˆ s x¯ = 4π ·

2 ∞ n

· eik0 nˆ i x˜ . Along the same lines we

¯ φ) ¯ y l,n,τ (ˆ ns ) ψ ∗l,n,τ (k0 r¯, θ,

(288)

τ =1 n=0 l=−n

for the dyadic It s · e−ik0 nˆ s x¯ . Now, if we use both expansions in Eq. (264) and we take (50) into account we obtain the matrix elements * ik0 · y ∗l,n,τ (ˆ ns ) · A(ˆ ns , n ˆ i ) · y l ,n ,τ (ˆ ni ) dˆ ns dˆ ni = 2π S∞ * ˜l,n,τ (k0 , x 2 (ik0 ) ψ ¯) · W(¯ x, x ˜) · ψ l ,n ,τ (k0 , x ˜) dS(¯ x) dS(˜ x) . (289) (ˆ n )

∂Γ

But the integral on the right-hand side of this relation is nothing but (128) or (148), respectively. Thus we have ﬁnally τ,τ

τ,τ

[S]l,n;l ,n = δτ,τ δl,l δn,n + 2 · [W ]l,n;l ,n .

(290)

This is identical with (228). Thus we have the interesting result that the matrix elements of the S-matrix introduced in section 4.4.2 are identical with the matrix ˆ Furthermore, since we have elements of the above deﬁned far-ﬁeld operator S. already proven the unitarity of the former in the case of a lossless scatterer, we have thus shown the unitarity of the far-ﬁeld operator for a lossless scatterer without assuming energy conservation. The unitarity property of this operator reads in operator notation * S † (ˆ n, n ˆ ) · S(ˆ n, n ˆ ) dˆ n = S∞ * (ˆ n ) S ∗ (ˆ n , n ˆ ) · S(ˆ n, n ˆ ) dˆ n = It · δ(ˆ n − n ˆ ) . (291) S∞

Consequently, if we introduce the scalar product *

u ∗ (ˆ n) · v(ˆ n) dˆ n u(ˆ n) | v(ˆ n) S∞ := S∞

for two arbitrary transversal vector functions u(ˆ n) and v(ˆ n)

(292)

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f 2 (ˆ n) | f 2 (ˆ n)

S∞

= f 1 (ˆ n) | f 1 (ˆ n) S∞

(293)

holds for the two special amplitude vectors f 1 and f 2 in (269), due to relation (276) and the unitarity of the S-operator for lossless scatterer. According to our initially formulated understanding we can state now that (231), if applied to the total electric ﬁeld by using ψ(x) = et (x) and φ(x) = et (x), is an appropriate quantity to express energy conservation for lossless scatterer. Especially in the farﬁeld region, i.e., if ∂Γa is identical with the far-ﬁeld surface S∞ , and if the far-ﬁeld behaviour (269) of the total electric ﬁeld is considered, it can be shown that (293) is a consequence of (294) {et (x), et (x)}S∞ = 0 (see [35], for example). A further consequence of the unitarity property of the S-operator is the so-called ‘generalized optical theorem’ of the dyadic scattering amplitude. To derive this theorem we must simply insert (272) into (291). This provides * " 2π ! A † (ˆ n, n ˆ ) · A(ˆ n, n ˆ ) dˆ n= · A(ˆ n , n ˆ ) − A † (ˆ n , n ˆ) = ik0 S∞ tp 2π · A(ˆ n , n ˆ ) − A ∗ (ˆ n , n ˆ) . (295) ik0 The conventional optical theorem can be derived from this expression, as also discussed already in [35].

4.7 Concluding remarks The present paper aimed at providing a common methodical Green’s function basis for diﬀerent rigorous approaches developed so far for solving the problem of plane wave scattering on a single, homogeneous, isotropic but nonspherical particle. The considerations have been further restricted to the spherical coordinate system and to ideal metallic and dielectric scatterers. It was shown that diﬀerent deﬁnitions of so-called interaction operators can be linked to diﬀerent approaches. The wellknown T-matrix approach, originally developed by Waterman, could be derived from deﬁnition (123) in the case of an ideal metallic scatterer, and from deﬁnitions (124) or (168) if the dielectric scatterer is considered. Once the matrix elements of the interaction operators are determined from the boundary conditions at the scatterer surface we are able to approximate the corresponding Green functions by ﬁnite series expansions. The derived matrix elements of the interaction operators are identical with the matrix elements of Waterman’s T-matrix if the needed weighting vector functions are appropriately chosen. It was shown afterwards that the conventional singular surface and volume integral equations are a result of a small but important replacement in the deﬁning equations. We ‘simply’ replaced Gt> by the full dyadic free-space Green function G0 . The diﬀerent possibilities of the deﬁnition of interaction operators are the crucial aspect of Rayleigh’s hypothesis. Although there is no deﬁnite answer to this hypothesis, so far, we gave some arguments for our position that these diﬀerent possibilities to deﬁne the interaction operators can be considered to be equivalent.

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In a further chapter we discussed the symmetry and unitarity property of the Tmatrix (or, equivalently, the matrix elements of the interaction operators) by use of the corresponding Green functions but without using the physical experience of reciprocity and energy conservation as a necessary prerequisite. Finally, the well-known far-ﬁeld behaviour of the scattered and total ﬁeld was discussed as a consequence of the Green function representation. Acknowledgements Special thanks are given to Dr K. Schmidt and Dr J. Wauer for valuable discussions, and to one of the anonymous reviewers for helpful suggestions.

References 1. M. Kahnert. Numerical methods in electromagnetic scattering theory: J. Quant. Spectrosc. Radiat. Transfer 79-80, 775, 2003. 2. H.C. van de Hulst. Light Scattering by Small Particles, Dover Publications, New York, 1957. 3. P.M. Morse, H. Feshbach. Methods of Theoretical Physics, McGraw-Hill, New York, 1953. 4. C. Tai. Dyadic Green Functions in Electromagnetic Theory, 2nd edn, IEEE Press, New York, 1994. 5. P.C. Waterman. Symmetry, unitarity, and geometry in electromagnetic scattering: Phys. Rev. D 3, 825, 1971. 6. R.E. Kleinman, G.F. Roach. On modiﬁed Green functions in exterior problems for the Helmholtz equation: Proc. R. Soc. Lond. A 383, 313, 1982. 7. R.E. Kleinman, G.F. Roach, S.E.G. Stroem. The null ﬁeld method and modiﬁed Green functions: Proc. R. Soc. Lond. A 394, 121, 1984. 8. M. Kahnert: Light scattering by particles with boundary symmetries. In Light Scattering Reviews, Vol. 3, edited by A. Kokhanovsky, Springer-Praxis, Chichester, 2008. 9. F. Dyson. George Green and physics: Physics World 6, 33, 1993. 10. H. Levine, J. Schwinger. On the theory of electromagnetic wave diﬀraction by an aperture in an inﬁnite plane conducting screen: Appl. Math. 3, 355, 1950. 11. M.I. Mishchenko, L.D. Travis, A. Lacis: Scattering, Absorption, and Emission of Light by Small Particles, Cambridge University Press, Cambridge, 2002. 12. T. Rother. Self consistent Green’s function formalism for acoustic and light scattering, Part 2: Dyadic notation: Opt. Commun. 251, 270, 2005. 13. A. Doicu, Y. Eremin, T. Wriedt: Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources, Academic Press, New York, 2000. 14. T. Rother. Scalar Green’s function for penetrable or dielectric scatterers: Opt. Commun. 274, 15, 2007. 15. K. Schmidt, T. Rother, J. Wauer. The equivalence of applying the Extended Boundary Condition and the continuity conditions for solving electromagnetic scattering problems: Opt. Commun. 150, 1, 1998. 16. J.J.H. Wang: Generalized Moment Methods in Electromagnetics, Wiley, New York, 1991. 17. H. Hoenl, A.W. Maue, K. Westphal: Theorie der Beugung. In Handbuch der Physik, Vol. 25/1, edited by S. Fluegge, Springer, Berlin, pp. 218–584, 1961.

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18. J. Van Bladel: Singular Elelctromagnetic Fields and Sources, Clarendon Press, Oxford, 1991. 19. J.G. Fikioris, A.N. Magoulas. Scattering from axisymmetric scatterers: a hybrid method of solving Maue’s equation: PIER 25, 131, 2000. 20. J.G. Fikioris. Singular integrals in the source region: J. Electromagn. Waves Appl. 18, 1505, 2004. 21. G. Mie. Beitraege zur Optik trueber Medien, speziell kolloidaler Metalloesungen: Ann. Phys. 25, 377, 1908. 22. Lord Rayleigh. On the dynamical theory of gratings: Proc. R. Soc. Lond. A 79, 399, 1907. 23. W.C. Chew: Waves and Fields in Inhomogeneous Media, IEEE Press, New York, 1995. 24. M.L. Burrows. Equivalence of the Rayleigh solution and the Extended-BoundaryCondition solution for scattering problems: Electr. Lett. 5, 277, 1969. 25. B.A. Lippmann. Note on the theory of gratings: J. Opt. Soc. Amer. 43, 408, 1953. 26. T. Rother, K. Schmidt. The discretized Mie-formalism for plane wave scattering on dielectric objects with non-separable geometries: J. Quant. Spectrosc. Radiat. Transfer 55, 615, 1996. 27. J.P. Barton, D.R. Alexander. Electromagnetic ﬁelds for an irregularly shaped, nearspherical particle illuminated by a focused laser beam: J. Appl. Phys. 69, 7973, 1991. 28. J.P. Barton. Electromagnetic ﬁeld calculations for irregularly shaped, axisymmetric layered particles with focused illumination: Appl. Opt. 35, 532, 1996. 29. J.P. Barton. Electromagnetic ﬁeld calculations for a sphere illuminated by a higherorder Gaussian beam. I. Internal and near-ﬁeld eﬀects: Appl. Opt. 36, 1303, 1997. 30. T. Elfouhaily, T. Hahn. Rayleigh’s hypothesis and the Geometrical Optics limit: Phys. Rev. Lett. 97, 120404, 2006. 31. J. Wauer, T. Rother: Considerations to Rayleighs hypothesis. Opt. Commun. 282, 339, 2009. 32. A.G. Dallas: On the convergence and numerical stability of the second Waterman scheme for approximation of the acoustic ﬁeld scattered by a hard object. In Technical Report, no. 2000-7, Dept. of Mathematical Science, Univ. Delaware, 2000. 33. J.A. Wheeler. On the mathematical description of light nuclei by the method of resonating group structure: Phys. Rev. 52, 1107, 1937. 34. W. Heisenberg. Die beobachtbaren Groessen in der Theorie der Elementarteilchen: Z. Phys. 120, 513, 1943. 35. D.S. Saxon. Tensor scattering matrix for the electromagnetic ﬁeld: Phys. Rev. 100, 1771, 1955. 36. M. Reed, B. Simon: Methods of Modern Mathematical Physics, Vol. 3, Academic Press, San Diego, CA, 1979.

Part II

Radiative Transfer

5 Space-time Green functions for diﬀusive radiation transport, in application to active and passive cloud probing Anthony B. Davis, Igor N. Polonsky, Alexander Marshak

5.1 Context, motivation, methodology, and overview Clouds are a feast for the eye but, when contemplating their ﬂuid beauty, it is important – at least for scientists – to bear in mind that they are also key elements of the Earth’s climate system. They are indeed the ﬁrst-order regulators of the intake in solar energy: What portion goes back to space? What reaches the surface (then warms the ground, drives photosynthesis, etc.)? Clouds also contribute strongly to the vertical distribution of solar heating and, from there, the thermal balance of the atmosphere. These are well-known and relatively well-understood/modeled climate roles of clouds, as can be expected for such naturally occurring components of the atmosphere. We note that these roles involve radiative transfer across the electromagnetic spectrum. What is far less understood about clouds is how they interact microphysically, chemically and thermo-hydrodynamically, with other natural and anthropogenic constituents, especially aerosols. These are known as cloud feedback mechanisms in the parlance of climate science, and have been identiﬁed as the single most resilient roadblock in the way of reducing uncertainty in future climate prediction [1], an enterprise that relies heavily on models to explore various scenarios in global greenhouse gas emissions. The ﬁrst order of business in addressing cloud feedback issues is therefore to improve statistics and accuracy in cloud observation, which is prerequisite to the improvement of cloud process models, which are in turn expected to enable the progress we so desperately need in global and regional climate modeling. Policymakers need and deserve the absolute best climate science deliverables if they are to propose in some ways painful regulations for greenhouse gas emissions in accord with future international treaties. Because cloud systems have regional inﬂuence and strong diurnal cycles, communities engaged in the meteorology, air-quality and surface hydrology are also stakeholders in improved cloud science, both in observations and in models. As soon as we realize that direct airborne probing of clouds is prohibitively expensive on a per datum basis, it follows that improving cloud observations globally is primarily a charge to the cloud remote sensing community. Whether from ground or from space – and these perspectives are in many ways complementary

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– we need better coverage (number of satellites, and their swaths),1 better sampling (spatial and angular resolution), and better retrieval techniques (exploitation of spectral and polarization measurements). This picture is overly simplistic. The notion that we can infer all the cloud properties that matter using only spectral and, with increasing appeal and availability, polarization data follows from the prevailing ‘1D thinking’ behind all current cloud remote sensing products delivered by space agencies world-wide. The operational assumption is indeed that clouds can be represented, for the purposes of the unavoidable radiative transfer, as horizontally inﬁnite plane-parallel slabs at the pixel-scale, irrespective of the size of the pixel. We thus ignore the pixel-to-pixel variations that make satellite cloud imagery so interesting, not to mention the likely presence of sub-pixel variability. This assumption may in fact be justiﬁed in view of other uncertainties: there is more than just radiative transfer modeling error to worry about, and a relatively fool-proof way of minimizing its impact is to steer away from cloud boundaries. This makes cloudy- versus clear-column discrimination an important preliminary task. However, those interesting interactions between clouds and aerosols happen in the (often only partially [2]) cloudy pixels, as well as in those pixels at the cloudyclear interface [3] – precisely where we are sure that neither cloud nor aerosol retrievals are accurate due to unaccounted 3D radiative transfer eﬀects. The emerging paradigm in cloud and aerosol remote sensing is therefore 3D and integrative: multi-spectral and multi-polarization methods are merged with multi-pixel and multi-angle ones and, ultimately, synergies across very diﬀerent kinds of instrumentation are used to optimize cloud property retrievals. No one ever said that cloud remote sensing is easy! This review covers in detail the theory of radiation transport Green functions, as it applies to optical probing of cloud structure by remote observers on both sides of the clouds, as well as in situ observers. We are only interested in scattering – and indeed multiple scattering – phenomenology based primarily, but not exclusively, on solar and laser sources. Our focus will be on the more opaque clouds that form in the Earth’s atmosphere, since we favor analytical methods based on mathematically tractable problems that arise in the diﬀusion (small mean-free-path) limit of transport theory at large.2 Finally, we will make the standard assumption of horizontally extended stratiform cloud geometry.3 In spite of this cascade of restrictions, the modeling framework we present sheds new light on a wide variety of radiometric modalities: some active and some passive; some advanced concepts and some well-established ones; some from ground, 1

From geostationary platforms, one sees the whole visible face of the Earth; however, their distance limits the achievable spatial resolution. 2 We are thus building on the shoulders of the giants that founded and developed statistical physics: kinetic theory (going at least back to L. Boltzmann), neutron transport theory (going back to J. von Neumann, N. Metropolis, R. Peierls, and others), stellar astrophysics (going back to A. Eddington, A. Milne, V.A. Ambartsumian, V.V. Sobolev, S. Chandrasekhar, and others), and even turbulence (going back to L.F. Richardson, A.N. Kolmogorov, G.I. Taylor, and others). 3 However, the space-time Green function is a 3D radiative property, and we will revisit the challenging problem of practical (i.e., eﬃcient and targeted) 3D cloud radiative transfer armed with said Green functions in various parts of the paper.

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some from space, and yet others from aircraft. The application ﬁeld is so vast that we have opted to survey the relevant theoretical and observational literatures as we encounter diﬀerent topics. Our main message is that they all branch away from the same radiation transport theory, namely, the unifying framework of Green functions. We present this body of theory up-front in a hierarchical manner: starting with a very detailed incarnation of radiative transfer4 in three spatial dimensions and with time dependence, implemented numerically mostly with slow-but-accurate Monte Carlo methods, and ending with simple scaling arguments, implemented with the proverbial back-of-the-envelope algebra. In our individual experiences, it is this multi-tiered modeling toolkit that has enabled the pursuit of instrument development ... even though our training is in applied theoretical and computational physics and mathematics. This solid theoretical background also enabled us to propose new ways to use under-exploited components of existing radiometric signals, some of them considered previously as contamination (for example, 3D adjacency eﬀects in remote sensing at high spatial resolutions) or even as noise (for example, solar background in lidar). Although no real-world data is used in this paper, we point the reader to numerous publications by ourselves and others that are all about data analysis. They are all success stories that, as a whole, have shored up our now ﬁrm belief in balance – as well as intense interaction – between (i) theory, (ii) computation, and (iii) observation/experimentation. Stakes are often much higher for engineering projects than for cloud remote sensing: we can think, for instance, of aircraft or nuclear reactor safety and reliability. It is therefore not surprising to learn that engineering scientists have developed a quite formal choreography for the interaction between these three elements. In the process, the notions of ‘veriﬁcation’ and ‘validation’ have been given precise meanings that Roache [6] has distilled into two fundamental questions: Are the equations solved right? (Veriﬁcation) Are they the right equations? (Validation). The ﬁrst V in ‘V&V’ leads in particular to codeto-code comparisons, while the second invariably leads to model-to-measurement comparisons. This framework can be adapted to the study of atmospheric radiation transport in the presence of clouds [7, 8]. We believe many other communities, remote sensing included, have much to gain by adopting – or at least taking inspiration from – this tested framework. In the next section, we survey the necessary prerequisites in time-domain 3D radiative transfer theory. In section 5.3, Fourier–Laplace transformation is introduced, which will prove useful in several ways. Sections 5.4 and 5.5 are the core material of the paper: formulation and solution of the diﬀusive Green function problem. In other words, how do we obtain analytic expressions for the Green function in time and/or space, or for its marginal or conditional moments, for light either reﬂected or transmitted by clouds? Appendices provide the required technical details as well as heuristic approaches to Green functions, in standard as well as anomalous diﬀusion regimes, using random walk theory. A comprehensive list of symbols and abbreviations is provided for reference at the end of the paper. Since the analytical expressions are delivered in Fourier and/or Laplace spaces, section 5.6 presents 4

Radiative transfer has itself been connected to Huygens’ scalar wave theory [4], and only quite recently in ﬁnal form to Maxwell’s classic electromagnetic wave theory [5], through rigorous methods of statistical optics.

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special cases where inverse transforms can be performed in closed-form. The short section 5.7 on (airborne) in situ cloud ‘lidar’, a new probe of optically thick clouds from the inside, is followed by two longer sections devoted to emerging technologies in cloud remote sensing per se. Both time-domain radiative transfer via O2 A-band spectroscopy (section 5.8) and multiple-scattering cloud lidar (section 5.9) can be implemented in ground-, aircraft- or space-based conﬁgurations. We show in particular that there is additional information to harvest in the spatial domain by lidars if the stand-oﬀ distance is not too large. A selection of other theoretical and observational applications of Green functions are described in section 5.10, for the most part based on well-established instrumentation. We oﬀer some closing remarks in section 5.11 with an emphasis on future developments. Notes on reproducibility and validation The bulk of the analytic diﬀusion theoretical computations described in sections 5.5 to 5.10 were performed using a commercially available symbolic mathematics package. We believe that any such product can be used at the reader’s convenience for replicating, generalizing and/or applying these results. Although it is possible to derive them by hand, the exercise can prove extremely tedious. We did it ourselves only in the simplest of cases. Furthermore, diﬀusion is in essence an asymptotic limit of radiative transfer. Before trusting our diﬀusion results to give us insights about radiation transport processes in real clouds, we compare them on a regular basis using a numerical time-dependent radiative transfer solver, generally a straightforward Monte Carlo scheme. Is this comparison of model outputs veriﬁcation and validation? In our view, it would be the former if the very same equations were solved, but that is not the case. One model is higher in the accuracy-based hierarchy than the other, so Monte Carlo can be used to validate diﬀusion. However, because of their role in validation, our Monte Carlo simulations were designed to follow as closely as possible the assumptions of the diﬀusion problem: single-parameter phase functions, predictions for ﬂuxes, and so on. This way, we are informed about the range of cloud parameters where a speciﬁc diﬀusion model can be applied, and about the magnitude of associated modeling error.

5.2 Elements of time-dependent three-dimensional radiative transfer 5.2.1 Radiant energy transport Let G(t, x, Ω) denote radiance at instant t and position x in 3D space propagating into direction Ω (units are W/m2 /sr). In view of the focus of the present study on Dirac δ-sources introduced in the next subsection, we denote radiances here by G, the usual mnemonic for Green functions, and reserve I for ‘at-detector’ radiances discussed in various parts of the paper. At-the-detector is indeed where we quite literally leave the realm of radiative transfer and enter that of radiometry.

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The ﬂow of Green function radiance, G(t, x, Ω), in its (1+3+2)-dimensional space is determined by the monochromatic time-dependent 3D radiative transfer (RT) equation [9], written succinctly as LG = SG + Q.

(2.1)

On the l.-h. side, we identify the propagation operator L = c−1

∂ + Ω · ∇ + σ(x) ∂t

(2.2)

where σ(x) is the extinction coeﬃcient (in 1/m), assumed to depend only on position. The two last terms in L represent losses for the intensity of the light beam as it crosses an elementary volume aligned with Ω, respectively to advection across boundaries and extinction.5 We caution here that = 1/σ(x) may have a popular interpretation as the local value of the mean-free-path (MFP); however, it will generally diﬀer from the actual MFP, except in strictly uniform media [10]. On the r.-h. side of (2.1), we ﬁnd the gains of the elementary volume aligned with Ω. First comes the in-scattering operator (2.3) S = σs (x) p(Ω · Ω)[·] dΩ 4π

where σs (x) is the scattering coeﬃcient. It is also assumed to depend only on position and, for simplicity, in such a way that the single scattering albedo (SSA), 0 =

σs (x) σ(x)

(2.4)

remains constant. The scattering phase function (expressed here in 1/sr) is denoted p(Ωin · Ωout ); it is also assumed, for simplicity, uniform in space as well as axisymmetric around the incident direction Ωin . The local source term, denoted by Q(t, x, Ω) in (2.1), is another net gain for the elementary volume (expressed in W/m3 /sr). Apart from boundaries, to be examined momentarily, the net losses of radiant energy for the whole medium are determined locally by the absorption coeﬃcient σa (x) = σ(x) − σs (x) = (1 − 0 )σ(x). The general RT equation spelled out in (2.1)–(2.3) is usually derived from purely phenomenological considerations grounded in radiant energy conservation [11] or, more formally, by analogy with particle kinetic theory leading to Boltzmann’s transport equation, which is narrowed to the case of ‘photons’ [12], often viewed simply as neutrons without the possibility of multiplication events (that cause the SSA to exceed unity). However, when dealing with electromagnetic radiation, one cannot be satisﬁed with these derivations since, in particular, they do not make clear the conditions of validity of the transport model. Progress towards a wave-theoretical foundation for radiative transfer was achieved steadily using a scalar wave approach 5 As far as we know, extinction σ can only be negative inside a lasing medium due to stimulated emission under conditions far from thermal equilibrium.

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to physical optics [4]. We refer the interested reader to Mishchenko’s recent review [13, and several key references therein], to gain understanding of how the RT equation – without temporal or spatial variability, but plus polarization – follows from Maxwell’s equations through the rigorous methods of statistical optics. The three most important lessons from this deﬁnitive microscopic foundation for RT theory are probably: 1. that the mesoscopic 3D RT equation is valid in all ‘dilute’ optical media, i.e., where inter-particle distances are much larger than the wavelengths of interest; 2. that multiple scattering paths ending in propagation exactly backwards toward a source have a residual wave-theoretical signature in the interesting phenomenon of ‘weak localization’ (resulting in an enhancement of radiance very near the backscattering direction); and 3. that, in view of the purely classical derivation, ‘photons’ have nothing to do with the story in spite of the often-used analogy with particle transport, especially by Monte Carlo (MC) practitioners. As an illustration of the last point, we should probably talk about e-folding distances rather than MFPs, a notion inherited from kinetic theory, which is patently about particulate material. In the following, we will continue to depend on the microscopic description of radiation-matter interactions (Maxwell’s equations and, as needed, their quantum mechanical counterparts) only to provide us with transport coeﬃcients (σ’s) and phase functions; see, for example, the monographs by Goody and Yung [14] for molecules, Bohren and Huﬀman [15] for spherical particles, and Mishchenko et al. [16] for nonspherical ones. We will actually be taking a step in the opposite direction: from the above mesoscopic 3D RT equation to the macroscopic picture where scattering dominates to such a degree that angular details are smoothed down to just two spherical harmonics. That is the ‘diﬀusion’ (a.k.a. ‘P1 ’) limit of RT theory. In this asymptotic (small MFP) approximation to RT, the analogy between light (highly scattered) and particles (in Brownian motion) regains some usefulness, bearing in mind the caveat in the above lesson #3; cf. Appendix E. Strictly speaking, the new microphysical derivation of the RT equation [5] is for spatially uniform, although not necessarily plane-parallel, media under steady and uniform illumination by a collimated beam. Mishchenko [17] extended his derivation to media with ﬂuctuations at scales that are small with respect to the MFP; specifically, one neglects the contributions of electromagnetic wave interactions between particles inside the same (randomly placed) ‘inclusion’. However, extensions of this rigorous framework to pulsed and/or narrow beams and/or larger-scale ﬂuctuations – all topics of interest here – are considered open problems [13]. Pending its derivation from ﬁrst principles (Maxwell’s equations), we will use the time-dependent 3D RT equation in (2.1)–(2.3) as the ‘exact’ numerical benchmark against which analytical diﬀusion-based approximations are assessed. 5.2.2 Dirac-δ boundary sources In view of the potential for spatial variability of σ(x), no generality is lost by bounding the cloudy medium by two horizontal planes at z = 0 and z = H. If we

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exclude from the deﬁnition of G the un-collided part of the radiance impinging on the medium from an external source, then we can subject the 3D RT equation to homogeneous boundary conditions (BCs): G(t, x, y, 0, Ω) = 0, μ > 0, G(t, x, y, H, Ω) = 0, μ < 0, where we denote direction cosines as

(2.5) (2.6)

⎛

⎞ η cos φ Ω(θ, φ) = ⎝ η sin φ ⎠ μ

in Cartesian coordinates using polar angles, with μ = cos θ and η = 1 − μ2 . In this case, nontrivial solutions of the RT equation require a non-vanishing source term Q in (2.1). For boundary-source Green functions, we use Q(t, x, Ω) = δ(t − z/c)δ(x)δ(y) σs (0, 0, z)p(μ) ⎞ ⎛ z × exp ⎝− σ(0, 0, z ) dz ⎠

(2.7)

0

where we assume, for the moment, vertical beam alignment (normal to the boundary at z = 0). Note from the writing of the ﬁrst δ-function that the instant t = 0 is when the the laser pulse hits the cloud at z = 0, precisely at x = y = 0 for convenience. Lasers are indeed physical sources that approximate Dirac-δ’s extremely well in all the variables that matter here: time, location and directionality. In other applications, their δ-in-wavelength and δ-in-polarization qualities also come in handy. Alternatively, we can set Q(t, x, Ω) ≡ 0 in the RT equation and model the source in a revised statement of the BC (2.5) at z = 0: ˆ), Ωz = μ > 0, G(t, x, y, 0, Ω) = δ(t)δ(x)δ(y)δ(Ω − z

(2.8)

ˆ = (0, 0, 1)T orients the positive z-axis. In this case, the radiance ﬁeld where z contains both direct and diﬀuse components. Apart from this interpretation of what is contained in G(t, x, Ω) or not, the two ways of modeling the normally incident Dirac-δ source at a boundary are equivalent. Figure 5.1 oﬀers a comprehensive view of the spatial Green function excited by a steady narrow beam normal to the upper boundary of a uniform non-absorbing cloud assumed to be between altitudes 1 km and 2.2 km. Mean radiance, averaged over all directions, is plotted. We immediately notice that the multiple-scattering Green function permeates the whole optically thick (τ = 36) cloud. In view of the logarithmic scale, we notice the exponential-type decay of the light ﬁeld with distance from the beam inside the cloud. This contrasts with the much slower decay of overall light levels with distance to the cloud boundaries (viewed as sources for remote observers), which is based on a 1/r2 law in the absence of any signiﬁcant scattering or absorption. Let us assume we have a single laser pulse with ∼1018 photons (for example, a 532 nm solid-state device with 5 W in cw power pulsing

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Fig. 5.1. Mean radiance for the search-beam problem in a ﬁnite homogeneous slab. The steady-state version of the RT problem deﬁned in (2.1)–(2.7) was solved with the Spherical Harmonic – Discrete Ordinate Method (SHDOM) [18, 19] for a uniform non-absorbing ( 0 = 1) cloud with thickness H = 1.2 km and uniform extinction σ = 30 km−1 , hence optical thickness τ = σH = 36. The phase function is for a ‘C1’ distribution of droplet size [20] in a Mie scattering computation [15] for λ = 532 nm, which yields asymmetry factor g ≈ 0.85 in (2.21). Mean radiance, J/4π from (4.5), is plotted for a domain larger than the cloud itself. The ‘rays’ emanating from the source region near the top of the cloud are an artifact of the discrete ordinates scheme (in this case, Nμ = 12 and Nφ = 24). This result was graciously contributed by Dr K. Franklin Evans (University of Colorado).

at 10 Hz), and a modest (but highly eﬃcient) detector with a modest aperture of ∼1 mm2 and a 1-sr ﬁeld-of-view (FOV) corresponding to ≈66◦ from side to side. A typical number of photons detected per pulse by such a sensor in the light shaded zone (for example, ground level, right below the source) is then ∼ 1018 ×10−6 /m2 /sr ×1 sr ×(10−3 )2 m2 = 106 , which appears to be enough to spread over, say, several 1000s of bins in space (direction) and/or time before the signalto-noise ratio (SNR) falls below 106 /1000s ∼ 10s for shot noise alone. This leaves plenty of room for a diﬀerent FOV, reduced optical throughput, quantum eﬃciency, and so on.6 For oblique illumination along Ω0 (θ0 , φ0 ), the internal source formulation with (2.7) becomes z z z/c Q(t, x, Ω) = δ t − δ x − η0 cos φ0 δ y − η0 sin φ0 σs (x, y, z) μ0 μ0 μ0 ⎤ ⎡ z dz z z ⎦ , (2.9) × p(Ω0 · Ω) exp ⎣− σ η0 cos φ0 , η0 sin φ0 , z μ0 μ0 μ0 0

while the BC source model in (2.8) becomes simply G(t, x, y, 0, Ω) = δ(t)δ(x)δ(y)δ(Ω − Ω0 ),

(2.10)

6 If it dominates the shot noise, background noise (sun- or moonlight) can be dealt with by appropriate ﬁltering and time integration [21].

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where μ, μ0 > 0. Although we will focus primarily on uniform and slant (solar) or pointwise and normal (lidar) illumination geometries, the above model for the boundary source is general enough to cover all possible spatial, temporal and angular distributions of sources by using straightforward space-time translations and linear superposition (i.e., convolutions). A ﬁnal type of unitary source we are highly interested in is isotropic boundarypoint-sources, which can be expressed as G(t, x, y, 0, Ω) = δ(t)δ(x)δ(y)/π,

(2.11)

for all μ > 0 in the BC (2.5) at z = 0. All of the primary sources of radiation, in the bulk or at the boundary of the optical medium, have now been accounted for. In the same way that the inscattering operator in (2.3) produces a secondary source of radiance for a given beam, boundaries can become secondary sources via partial reﬂection, as opposed to the systematic absorption/escape described in (2.5)–(2.6). This process of boundary scattering is formalized in a revised r.-h. side for the BC (2.6) at z = H for incoming radiance G(t, x, y, H, Ω) with μ < 0. Speciﬁcally, we introduce in analogy with (2.3), Ss = αH (x, y) ps (Ω → Ω)[·] dΩ , (2.12) μ >0

which is applied to the local G; αH (x, y) is the local albedo of the boundary and ps (Ω → Ω) is the surface phase function (in 1/sr).7 If necessary, we could similarly deﬁne a reﬂective BC for the illuminated boundary at z = 0; this would only mean a change of sign in μ and μ . Two contrasting examples of surface scattering/reﬂection are the isotropic (a.k.a. Lambertian) case, ps (Ω → Ω) = |μ |/π, and the specular (a.k.a. Fresnel) case, ps (Ω → Ω) = δ(μ + μ)δ(φ − φ). We have now deﬁned all the components of the RT equation, its BCs, and the boundary δ-sources of primary interest here. This completes the determination of the multiple-scattering Green function in the spirit of Bell and Glasstone [23], who introduced it as a powerful modeling tool in nuclear reactor design and analysis. The remainder of this section narrows our interest to outgoing boundary radiances and their properties. 5.2.3 Remotely observable ﬁelds In remote sensing applications, we use detectors outside the medium. We can therefore access only the outgoing radiance ﬁelds at cloud boundaries, which echo the boundary conditions in (2.5)–(2.6) but with z-axis direction cosines of opposite sign: → → G(t, − ρ , 0, Ω), when Ωz = μ < 0, for reﬂection; G(t, − ρ , H, Ω), when Ωz = μ > 0, for transmission. We denote here − → ρ x → − . ρ = , hence x = z y 7 To connect with the popular ‘bidirectional reﬂection distribution function’ (BRDF) [22]: ρ(Ω → Ω) = αH ps (Ω → Ω)/|μ |.

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More precisely, we assume an imaging detector is measuring this radiance at some ﬁnite stand-oﬀ distance dobs > 0 from the cloud boundary of interest: the sensor is → − → − thus positioned at xobs = ( 0 , −dobs )T or ( 0 , H +dobs )T . See schematic in Fig. 5.2.

Fig. 5.2. Schematic of boundary-source/boundary-detector Green function problem. We illustrate the case of normal incidence and an observer along the incident beam.

Notice that we assume geometric alignment with the normally incident collimated beam. For reﬂected light, this is tantamount to modeling a monostatic lidar (i.e., transmitter and receiver collocated). For transmitted light, the detector is looking straight at the point-source on the opposite boundary of the optical medium, a typical conﬁguration in imaging and visibility studies. In the latter case, the proportion of the emitted light directly transmitted from the collimated source → − in (2.7) or (2.8) to the detector is exp[−τ ( 0 )]. More generally, we deﬁne − τ (→ ρ)=

H

− σ(→ ρ , z) dz

(2.13)

0

→ as the optical thickness of the medium at horizontal position − ρ . The interesting questions, however, are about the diﬀuse component generated by one or more scatterings. We are thus restricting ourselves to coaxial source–detector geometry → − → → (− ρ0 =− ρ obs = 0 ), simply because of the applications treated in the present study; → − → if necessary, generalization to − ρ obs = 0 is straightforward. We denote the radiance recorded by the time-resolving/imaging detector as Iobs (tobs , Ωobs ). From this vantage point, we just sub-sample the Green func→ → tion for boundary illumination, G(t, − ρ , 0, Ω) or G(t, − ρ , H, Ω) is equated with

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→ − G(t, − ρ , ·, Ωobs (→ ρ )). In the time-domain, t refers to the time interval from entry at the z = 0 boundary to escape from anywhere; however, unless the point of escape is the closest to the detector, an observer at ﬁnite distance will see an extra delay illustrated in Fig. 5.2. So we have 1 dobs . (2.14) −1 t = tobs − cos θobs c In the case of monostatic lidar observations (in reﬂection, when the detector is near the source), one might use tround-trip = tobs + 2dobs /c. To equate ‘at detector’ and ‘at cloud’ radiances in the spatial domain, we must also factor into the independent variables the ﬁnite distance to the observation point: ρ(θobs ) = x2 + y 2 = dobs tan θobs , hence, and

θobs (ρ) = tan−1 (ρ/dobs ), ⎞ cos φobs sin θobs (ρ) → Ωobs (− ρ ) = ∓ ⎝ sin φobs sin θobs (ρ) ⎠ , cos θobs (ρ) ⎛

(2.15)

where − is for reﬂection and + is for transmission. In the limit dobs → ∞, a reasonable approximation for an orbital detector, the connections in (2.14)–(2.15) still make sense by taking simultaneously the limit θobs → 0, but keeping ρ constant. We thus denote the detector response as Iobs (t, ρ), after accounting for the large but ﬁnite time delay; the last connection in (2.15) z. simpliﬁes to Ωobs (ρ) = (0, 0, ∓1)T = ∓ˆ 5.2.4 Flux-based spatial and temporal moments To summarize, we are interested in computing and measuring the time-dependent equivalent reﬂectance (or albedo) ﬁeld → → → Robs (t, − ρ ) = πG(t, − ρ , 0, Ωobs (− ρ ))/μ0

(2.16)

normalized by total source energy, and its counterpart in transmittance → → → ρ ) = πG(t, − ρ , H, Ωobs (− ρ ))/μ0 . Tobs (t, −

(2.17)

These are just alternatives to boundary-leaving radiance using 1. a Lambertian assumption about outgoing directions other than Ωobs (hence the factor π), and 2. normalization by total incoming ﬂux integrated over space, time and direction (hence the 1/μ0 factor that accounts for the reduction in ﬂux when illumination is unitary but oblique).

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Temporarily ignoring angular sampling and truncation issues in real measurements, we deﬁne 1 − R(t, → ρ) = μ0 1 − T (t, → ρ) = μ0

+π π

→ | cos θ|G(t, − ρ , 0, Ω(θ, φ)) sin θ dθ dφ,

(2.18)

−π π/2

+π π/2

− cos θ G(t, → ρ , H, Ω(θ, φ)) sin θ dθ dφ,

(2.19)

−π 0

as the local time-dependent reﬂected and transmitted ﬂux ﬁelds, respectively. The equivalent and actual reﬂectance and transmittance ﬁelds displayed above are by deﬁnition ratios of outgoing-to-incoming hemispherical ﬂuxes, estimated or actual. Lastly, we note that (2.19) is either diﬀuse or total transmittance depending on the adopted formulation of the 3D RT equation. If we only have the diﬀuse transmittance, then the total one is obtained by adding exp[−τ (Ω0 )]/μ0 where H dz z z τ (θ0 , φ0 ) = σ η0 cos φ0 , η0 sin φ0 ,z μ0 μ0 μ0 0

is the optical path across the whole medium along the incident beam. It coincides → − → ρ = 0 there. with (2.13) only when μ0 = 1 here and −

Fig. 5.3. Space-time boundary Green functions of a uniform cloud under diﬀuse pointwise illumination for reﬂection (left) and transmission (right). These responses in ﬂux were estimated numerically using a MC simulation with 109 histories that resulted in reﬂection with probability R = 0.557 and transmission with probability T = 1 − R. The logarithmic grayscale is the same for both panels. Lateral transport distance ρ runs vertically from 0 to 1.5 km while in-cloud path ct runs horizontally from 0 to 3 km. Reproduced from Ref. [24] with permission. More details in text.

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Figure 5.3 displays numerical (hence binned) estimates of R(ct, ρ) and T (ct, ρ) for a uniform cloud, using the convenient units of in-cloud path ct rather than transit time per se. The pointwise illumination is isotropic: boundary condition at z = 0 from (2.11) and Q ≡ 0 in (2.1). The cloud has H = 0.3 km and τ = 16; there is no absorption (0 = 1) and scattering is according to the Henyey–Greenstein (H–G) model [25]: 1 − g2 1 p(μs ) = , (2.20) 4π (1 + g 2 − 2gμs )3/2 where μs is the cosine of the scattering angle, i.e., Ω · Ω. The new parameter g, the well-known ‘asymmetry factor’, is critical to this study of diﬀusion regimes. In general, it is deﬁned as g= 4π

+1 Ω · Ωp(Ω · Ω) dΩ = 2π μs p(μs ) dμs .

(2.21)

−1

In this case, we used8 g = 0.85. The most striking diﬀerence between the two radiative responses in Fig. 5.3 for a localized and pulsed excitation is that reﬂectance happens immediately, thanks to low orders of scattering, while transmittance occurs only for ct ≥ H, with ‘=’ being very unlikely since it calls for a direct transmission (at the e−16 ≈ 10−7 probability level). Supports of both responses are inside the causality cone (ρ ≤ ct) dictated by shallow quasi-ballistic lateral propagation away from the isotropic source followed by a reﬂection. At large ct and ρ, the two responses are indistinguishable since after a large number of scatterings escape is equally probable through either boundary. Also, for ﬁxed ct H, we see that the value of ρ that maximizes the Green function (where the tangent to the isophote is √ vertical in the panels of Fig. 5.3) follows a roughly parabolic trend (ρmax (ct) ∼ ct). This is typical of diﬀusive radiation transport, as we will see further on. Figure 5.4 focuses on spatial Green functions. The l.-h. panels are the axisym→ metric ﬁelds F (− ρ ) ≡ F (ρ), F = R, T , from (2.22) for a uniform cloud with τ = 13 and conservative g = 0.85 H–G scattering under normal collimated illumination by a narrow beam. As in Fig. 5.3, the main diﬀerences between reﬂection and transmis→ sion are in the near-axis region. The top and r.-h. panels are F (− ρ ), F = R, T , for a randomly variable cloud with long-range spatial correlations. The ﬂuctuating values of the local optical depth in (2.13) have the same mean optical depth τ = 13; vertical structure is assumed uniform.9 The boundary ﬂux ﬁelds for this scaling (a.k.a. ‘fractal’) stratocumulus (Sc) cloud model of course have no axial symmetry. Closer examination shows that there is less overall reﬂection and correspondingly more transmission; in both cases, we see a slight increase in the horizontal dispersion → quantiﬁed by the root-mean-square (RMS) value of − ρ. 8 It is notable that phase functions for observed droplet-size distributions in boundarylayer clouds yield g ≈ 0.85 with remarkably small variability [26]. 9 This stochastic cloud model was generated with a so-called ‘bounded cascade’ model [27] tuned to have a Kolmogorov-type wavenumber spectrum in k−5/3 and an amplitude of variation similar to those observed in marine stratocumulus [28]. More details on this fractal cloud model are provided in section 5.10.6.

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Fig. 5.4. Spatial Green functions of homogeneous (left) and fractal (right) clouds with the same mean optical depth under steady normal illumination by a narrow beam. These are the outcome of two MC simulations, each with 108 histories; these events resulted in reﬂection (b,b’) with probability R ≈ 0.5 in the uniform case and ≈ 0.3 in the fractal case, or in transmission (c,c’) with probability T = 1 − R. A 128×128 grid was used to bin the boundary ﬂuxes F ( ρ) with F = R, T . In the fractal case, we notice the systematically larger spread towards rows with high rank, which correspond to the more tenuous region in the top graphic (a) showing τ (x, y) from (2.13). Reproduced from Ref. with permission. More details in text.

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Partly to improve the SNR and partly for conceptual simplicity, we like → use spatial and/or temporal integrals of the observed ﬂux ﬁeld F (t, − ρ ), for F R, T . We are particularly interested in statistical moments when it is viewed a probability density function (PDF) for escape in reﬂection or transmission. order to normalize the PDFs, we start by estimating ∞ F =

+∞

F (t) dt =

− → F (→ ρ ) d− ρ (x, y) =

−∞

0

∞

+∞

dt

→ → F (t, − ρ ) d− ρ (x, y),

to = as In

(2.22)

−∞

0

the cloud’s albedo (F = R) or transmittance (F = T ) for steady and uniform illumination, either collimated or isotropic. We can then proceed to compute 1 t F = F

∞

q

1 t F (t) dt = F

∞

q

0

+∞ q

t dt

→ → F (t, − ρ ) d− ρ (x, y)

(2.23)

→ → ρ2 F (t, − ρ ) d− ρ (x, y).

(2.24)

−∞

0

for q = 1, 2, or more, and 1 ρ F = F

+∞

2

1 − → ρ F (→ ρ ) d− ρ (x, y) = F

∞

2

−∞

+∞

dt 0

−∞

Angular brackets will always denote averages over space and/or time in cloud radiative responses while an overscore denotes an average over spatial disorder, i.e., 1/2 cloud structure. Examples of ρ2 F (RMS horizontal transport) for normal illumination are rendered graphically with double-headed arrows in Fig. 5.4. The above are ‘marginal’ moments; ‘joint’ moments tq ρp F and ‘conditional’ moments can also be estimated. Of particular interest in this last class is 1 ρ F (t) = F (t)

+∞

2

→ → ρ2 F (t, − ρ ) d− ρ (x, y),

(2.25)

−∞

the mean-square horizontal transport at a ﬁxed time, where F (t) follows from (2.22) without the time integral. Note that the moment estimations in (2.24)–(2.23) are immune to uncertainties → in a multiplicative constant for F (t, − ρ ). From an observational standpoint, and in sharp contrast with the estimation of cloud albedo or transmittance based on (2.22), absolute calibration is not required. The easier task of ﬂat-ﬁelding of the imager’s focal-plane array is, however, still necessary. Of course, real-world observations give us no information on the boundary Green → → → ρ )), hence on R(t, − ρ ), outside of the receiver’s FOV, i.e., function G(t, − ρ , 0, Ωobs (− → the actual limits of all the above integrals over − ρ (x, y) are ﬁnite when F = R, and → − similarly for T . Moreover, for each value of ρ we only get one value of θ and φ in (2.15). The latter problem is resolved by using an angular model to convert an observed radiance into a boundary ﬂux. The former problem is best addressed by designing instruments with the widest possible FOV, such that it contains at least a couple of the Green function’s e-folding distances away from the axis. We can then assume quite safely that the residual truncation errors in both numerator and denominator in (2.24)–(2.23) do not bias the estimates too much.

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5.2.5 Vertical variation of scattering coeﬃcient From this point on, we restrict our interest to stratiform clouds that can be reasonably well represented by plane-parallel slabs of uniform thickness H, M(H) = {x ∈ R3 ; 0 < z < H}, still allowing for some degree of internal structure. In the above, we examined cloud models with fractal behavior in the horizontal plane but uniformity in the vertical. In the following, we will consider two complementary kinds of variability: smooth (convection/radiation-driven) stratiﬁcation along the z-axis and random (turbulence-driven) 3D ﬂuctuations. Discussion of the latter is postponed until section 5.4.2.3, leaving us until then with σ(x) ≡ σ(z), and similarly for σs and σa . The parameters of primary interest for a purely stratiﬁed cloud are its physical thickness H and optical depth τ from (2.13), but without the dependence on → − ρ (x, y). Basic cloud physics informs us that stratus clouds are expected to be strongly stratiﬁed. For instance, liquid water content (LWC) is predicted and widely observed [30] to follow the adiabatic gradient in their ‘convective cores’, i.e., a linear trend in z over the vertical extent of the cloud.10 This classic result from the baseline ‘rising parcel’ theory in cloud microphysics (number density assumed constant) leads to a 2/3 power-law in extinction, from straightforward dimensional analysis. Formally, and depending on what side of the cloud is being illuminated by the δ-source, we can write this as σ0 (γ; z) = σ × (1 + γ) (z/H)γ , or σH (γ; z) = σ × (1 + γ) (1 − z/H)γ ,

(2.26)

with γ ≥ 0 (in this case, 2/3) and σ denoting the mean extinction (obtained, say, from cloud optical depth τ = σH). Instead of the power-law model, it is advantageous to use a linear (constant gradient) model, (2.27) σΔ (z) = σ × [1 + Δ (z/H − 1/2)], where |Δ| ≤ 2 is the relative diﬀerence in extinction at the two cloud boundaries with respect to its mean value (invariably crossed at z = H/2). To put the linear and power-law models in one-to-one correspondence, we propose to minimize their diﬀerence squared (distance in L2 ). This exercise leads to γ+1 Δ(γ) = ±6 × 2 −1 (2.28) γ+2 where + is mapped to σ0 (γ; z) and − to σH (γ; z). Values of special interest are Δ = ±3/2 since they approximate γ = 2/3, the abovementioned expectations based on parcel theory for a cloud illuminated from below (+) and above (−). Conversely, we have γ(Δ) = 2|Δ|/(6 − |Δ|). (2.29) 10 In a broader view of liquid water clouds, the maximum value of LWC is typically reached at a depth of ≈H/3 from the top.

5 Space-time Green functions for diﬀusive radiation transport

185

Fig. 5.5. The impact of internal stratiﬁcation on space-time cloud responses in ﬂux to an isotropic point-source (top) and a narrow collimated beam (bottom). These results are from a number of MC simulations (2 · 108 histories) with H–G scattering (2.20). For simplicity, the maximum cross-section algorithm was used to account for internal structure; see Refs [19, 29]. Moments deﬁned in (2.23)–(2.24) were computed, using their square-root for second-order moments, and normalized to H = 1. We used τ = 15 with g = 0.46 in the top panel, and τ = 20 with g = 0.85 in the lower panel. In spite of the larger optical depth, we notice the smaller spread of results in the latter case (half the vertical range) due to more forward-peaked scattering and a more anisotropic source. More discussion in the main text.

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Stratiﬁcation in σ(z), as parameterized in the above, will directly aﬀect the spatial (2.24) and temporal (2.23) observables, even if it has no eﬀect whatsoever on the cloud’s albedo (F = R) or transmittance (F = T ) in (2.22). Indeed, the local value of the MFP will be diﬀerent at the top and bottom of the cloud and, physically, this means that the random walk representing the diﬀusing light propagation is scaled up (near cloud base) or down (near cloud top). Active and passive instruments are already probing clouds from both sides and will continue to do so; it is therefore imperative to quantify the eﬀect of stratiﬁcation on the observables. Figure 5.5 shows numerical MC results for ρ2 F /H, for ctF /H and for (ct)2 F /H (F = R, T ) using both stratiﬁcation models. The constant-gradient model in (2.27) is sampled at 0.5 intervals from Δ = −2 to +2 for the prescribed clouds and sources. The power-law model in (2.26) degenerates to the linear case when γ = 1 (Δ = ±2), and of course when γ = 0 (Δ = 0), but it is interesting to seek diﬀerences between the γ = 2/3 cases and the associated values of Δ = ±3/2 using (2.28). They are very small compared to the overall eﬀect of stratiﬁcation. In turn, these eﬀects are signiﬁcantly larger for reﬂection than for transmission, especially in the time domain. We also note in the top panel that, when illumination is pointwise but isotropic and the response is for boundary ﬂuxes, the Green functions for transmission depend only on |Δ| due to source–detector reciprocity and a mirror-symmetry around z = H/2. The qualitative diﬀerences between boundary-ﬂux responses R and T we have uncovered can be traced to the fact that reﬂected light is a balanced mix of lowand high-order scattering. The low orders in the observed signal come almost surely from near the source, hence from the illuminated side of the cloud, while the high orders come from radiation that has permeated the whole cloud. In contrast, radiation transmitted by optically thick clouds is made almost entirely of highly scattered light.

5.3 Formulation in the Fourier–Laplace domain Moment integrals in (2.23)–(2.24) are easy to compute by manipulation of transforms in Fourier–Laplace space. In probability theory, the Fourier or Laplace transform of a PDF is called its ‘characteristic’ function or, more tellingly, ‘momentgenerating’ function [31]. Which transform is used depends on the support of the PDF. In our application, we need both Laplace for time t ∈ [0, ∞) and 2D Fourier → for position − ρ in the z = 0, H planes. We are thus interested in the behavior of → − F˜ (s, k ) =

∞

+∞

dt 0

→ → − → − exp(−st + i k · − ρ ) F (t, − ρ ) d→ ρ (x, y)

−∞

→ → − = F × exp(−st + i k · − ρ )F , F = R, T. We will apply a similar transformation to other ‘3+1D’ quantities as needed.

(3.1)

5 Space-time Green functions for diﬀusive radiation transport

187

5.3.1 Temporal Green functions and pulse-stretching problems → − → − Consider the case of uniform, but still pulsed, illumination ( k ≡ 0 ). One can → − show [32] from (3.1) that coeﬃcients of the Taylor expansion of F˜ (s, 0 ) at s = 0 can be used to estimate temporal moments in (2.23). Speciﬁcally, we compute → − F = F˜ (0, 0 ), and then q ) ) ∂ 1 q t F = − . (3.2) F˜ )) F ∂s s=0,k=0 These numbers describe quantitatively how the incoming pulse is stretched out in the responses of the scattering medium. There is an interesting interpretation of the Laplace conjugate variable s in terms of absorption by a uniformly distributed gas in the otherwise purely scattering medium. Taking the Laplace transform of the general 3+1D RT equation, boundary conditions, and source term in (2.1)–(2.7), we ﬁnd notable changes, on ˆ = SG ˆ+Q ˆ where the one hand, in LˆG Lˆ = Ω · ∇ + σ(x) + s/c

(3.3)

and, on the other hand, in ˆ x, Ω) = δ(x)δ(y) σs (0, 0, z)p(μ) Q(s, ⎞ ⎛ z s × exp ⎝− σ(0, 0, z ) dz − z ⎠ , c

(3.4)

0

and similarly using (2.9) if μ0 < 1. All is therefore as if the extinction coeﬃcient is boosted everywhere by s/c, but the scattering coeﬃcient is unchanged. This reading of (3.3)–(3.4) is known as the ‘equivalence theorem’ that attracted considerable attention in the 1960s and early 1970s [33–39]. It clearly separates absorption and scattering processes in the general RT problem, and states that radiance at an absorbing wavelength can be calculated from the radiance at a nonabsorbing one and the attenuation along all possible paths from the sources to the point/direction of interest. It is interesting to note that the earliest numerical investigations (known to the authors) of time-domain RT with multiple scattering (MC-based of course) were performed in the same time period; see Ref. [40]. Most of these early studies of pulse stretching, with or without the Laplace transform, were focused on the F = R scenario. We will exploit it computationally and observationally further on for both F = R and F = T , this balance being a recurring theme in this review. We can thus gain access to temporal moments of sunlight ... even though the source is steady and the detection system is passive (cf. section 5.8). 5.3.2 Spatial Green functions and pencil-beam problems The rotational symmetry acquired in section 5.2.5 for the cloud under already normal/collimated or isotropic illumination conditions in direction space carries → − over from physical to Fourier space; we therefore have F˜ (s, k ) ≡ F˜ (s, k).

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Consider now the case of steady illumination (s ≡ 0). One can easily show from (3.1) that the Taylor coeﬃcients of F˜ (0, k) at k = 0 can be used to estimate spatial moments in (2.24). Speciﬁcally, we compute F = F˜ (0, 0), and then11 ) −2 ∂ 2 F˜ )) 2 ρ F = . (3.5) ) F ∂k 2 ) s=0,k=0

RT problems, generally in uniform semi-inﬁnite (H → ∞) media, with localized narrow (collimated) steady beams, are known in the literature as ‘pencil-beam’ or ‘search-light’ problems (cf. Figs 5.1 and 5.4). They have attracted quite a lot of attention ever since the very earliest investigations of 3D RT we are aware of [41–47]. Lately, such problems have been proposed as analytical benchmarks for numerical transport code veriﬁcation [48, 49]. There is not an exact equivalence between horizontal wavenumber k > 0 with an eﬀective absorption process, except if we remove the dependence of the source on Ω (that controls the direction of the derivative in L) and consequently forgo all but isotropic scattering. At any rate, more eﬀort and creativity is required than in the time-only problem.

5.4 Diﬀusion approximation for opaque scattering media In the above, we have established a complete formalism for computing space-time Green functions for scattering media illuminated and observed at a boundary. However, it invariably leads to a numerical implementation, for example, a MC algorithm. This is ﬁne for case studies but impractical for applications in remote sensing. Our goal now is to establish a physically reasonable theory leading to F (t, ρ) or, equivalently, F˜ (s, k) in closed form (section 5.5). We can then use the above deﬁnitions and relations that predict analytically the spatial, temporal, joint or conditional moments of the Green functions (sections 5.8 and 5.9), even derive closed-form expressions for the Green functions themselves (sections 5.6 and 5.7). 5.4.1 Derivation from the time-dependent 3D RT equation Consider dense clouds, say, through which one cannot detect the silhouette and maybe not even the general direction of the sun in the transmitted radiance ﬁeld. According to Bohren et al. [50], this translates to optical thickness 9 ± 1 (geometrical thickness 9 ± 1 MFPs). We can then safely assume that the transmitted light at least is transported via diﬀusion, the well-known approximation to RT per se. That statement should carry over to reﬂected light as long as we focus on higher orders of scattering; in space-time Green function studies, that translates to later times and further distances from the source. In other words, while bearing in mind the caveat in section 5.2.1 about thinking about RT as ‘photon’ transport, all is as if the radiative ﬂuxes measured in transmission or, for the most part, in reﬂection 11

The factor of 2 originates, when the axial symmetry is applied, from the 2D dyadic →− − → → − tensor k k used in the multivariate Taylor expansion of F˜ (0, k ) in (double-dot) combi−→−→ nation with the second-order diﬀerential tensor operator ∇k ∇k .

5 Space-time Green functions for diﬀusive radiation transport

189

were currents of particles executing long convoluted random walks starting at the localized and/or collimated source and ending at a cloud boundary. Equations for this simpliﬁed transport theory can be derived by integrating (2.1)–(2.3) angularly term-by-term over 4π, once directly and once after multiplication by Ω. This leads to [51]: ∂J + ∇ · F = −σa (x)J + qJ (t, x), ∂t ∂F + ∇ · K = −σt (x)F + q F (t, x). c−1 ∂t c−1

(4.1) (4.2)

In (4.2), an important new coeﬃcient appears: the ‘transport’ extinction, σt (x) = (1 − g)σs (x) + σa (x),

(4.3)

equivalently (1 − 0 g)σ(x), with 0 from (2.4) and g from (2.21). The associated (local) transport MFP [4, 52] is denoted t (x) = 1/σt (x).

(4.4)

There was been some controversy over the past 15 years about the exact role of σa (i.e., absorption) in σt , hence in the diﬀusivity D, which we deﬁne further on as ct /3. In the mid-1990s, the rising importance of biomedical imaging applications of ‘photon migration’ (diﬀusion) theory motivated several authors [53–58] to revisit and conﬁrm the idea originated by Furutsu [59] in 1980 that σa is not present in (4.3)–(4.4). Rather, the eﬀect of absorption is added to J after the fact with a multiplicative term in exp(−σa ct), assuming a uniform medium. This makes sense from the standpoint of the equivalence theorem discussed in section 5.3.1. Subsequent investigations [60–63], all based on steady-state transport, argued convincingly that absorption does impact σt , although more weakly than in (4.3), for example, Aronson and Korngold [62] give D = 1/3(σs +σa /5) when g = 0. Cai et al. [64] re-ignited the debate by framing D as dependent on time rather than absorption. We believe that Pierrat et al. [65] have resolved the controversy by distinguishing between the dynamical diﬀusion constant without σa , and steady-state one with it. All this is for uniform media where one can gain insights by way of analytical manipulations, even solutions, of the RT (not just diﬀusion) equation. In the presence of spatial variability, we further need to distinguish absorption by a uniform interstitial gas and by the clumped material that produces the scattering. The former case can be treated with the equivalence theorem as stated, but not the latter unless it is extended to include a discrete sum over successive orders of scattering [66]. We also introduced here the zeroth-, ﬁrst-, and second-order angular moments of Green-function radiance as J(t, x) = G(t, x, Ω) dΩ, 4π

ΩG(t, x, Ω) dΩ,

F (t, x) = 4π

K(t, x) =

ΩΩG(t, x, Ω) dΩ, 4π

(4.5)

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respectively, the scalar- (a.k.a. actinic-), vector- and (dyadic) tensor-ﬂuxes. We similarly deﬁne qJ (t, x) = Q(t, x, Ω) dΩ, (4.6) 4π

q F (t, x) =

ΩQ(t, x, Ω) dΩ.

(4.7)

4π

For instance, based on (2.7), we have &

qJ (t, x) = δ t −

z' c

⎛ δ(x)δ(y) σs (z) exp ⎝−

z

⎞ σ(z ) dz ⎠ ,

(4.8)

0

ˆ. q F (t, x) = qJ (t, x) g z

(4.9)

More generally, (2.9) yields here z z z/c δ x − η0 cos φ0 δ y − η0 sin φ0 qJ (t, x) = δ t − μ0 μ0 μ0 ⎞ ⎛ z dz ⎠ × σs (z) exp ⎝− σ(z ) , (4.10) μ0 0

q F (t, x) = qJ (t, x) g Ω0 ,

(4.11)

noting that in (2.9) we had not yet assumed that extinction is at most dependent on z; here, the horizontal variations are driven only by the pointwise source distribution. In analogy with particle transport theory, (4.1)–(4.2) are local expressions of the conservation of radiant energy and momentum respectively [12]. These are the basic ingredients of a macroscopic theory of radiation transport where the 3D RT equation describes mesoscopic processes involving directional details while Maxwell’s equations describe microscopic details involving wave phenomena. As stated, the ‘continuity’ (or conservation) equations for energy (4.1) and momentum (4.2) are exact. The diﬀusion approximation follows from making two simplifying assumptions about (4.2): ﬁrst, the time-derivative is assumed negligible and, second, a natural closure is introduced. The closure statement is that the radiation pressure tensor K/c is isotropic, i.e., oﬀ-diagonal components vanish and on-diagonal components are equipartitioned (each one is equal to 1/3 of the radiant energy density J/c) [51]. This is indeed the expectation when radiation – viewed as a monokinetic gas – is in local equilibrium with a dense highly scattering medium. Small deviations from isotropy are then entirely captured by two spherical harmonics, namely, a monopole and a dipole. Speciﬁcally, we have12 G(t, x, Ω) ≈ [J(t, x) + 3Ω · F (t, x)] /4π 12

(4.12)

We refer to the key paper by King, Radke, and Hobbs [67] for empirical evidence of this representation of in-cloud radiance collected in extensive marine boundary-layer stratocumulus clouds, which are of considerable interest here.

5 Space-time Green functions for diﬀusive radiation transport

and, accordingly,

p(Ω · Ω) ≈ [1 + 3gΩ · Ω] /4π

191

(4.13)

for the phase function. These two expressions are often used as the point of departure in diﬀusion (a.k.a. ‘P1 ’) theory. We notice right away that (4.13) is a rather poor representation of the phase function of real distributions of cloud droplet size [20], most notably, the forward diﬀraction-induced peak is absent. By the same token, (4.12) is a poor representation of radiance anywhere near the highly collimated beam illuminating the medium, whether localized like a laser source or spread out like the solar source. The latter problem is mitigated by separating the un-collided beam from the diﬀuse ﬁeld and thus using internal source terms rather than a source in the boundary conditions. The former problem is addressed in the following subsection. No matter how one derives the diﬀusion transport model from the RT equation,13 we obtain the ‘constitutive’ equation: ∇J/3 = −σt (x)F + q F (t, x),

(4.14)

a.k.a. Fick’s law, especially when the source term is absent (F = −∇J/3σt ). This deﬁnes the diﬀusion approximation as a ﬁrst-order closure of the hierarchy of transport equations started in (4.1)–(4.5). In the applications to come, we will focus on wavelengths where the condensed water in cloud particles has negligible absorption: σa (x) ≡ 0, hence 0 = 1 and σs (x) = σ(x). The continuity equation (4.1) then simpliﬁes to ∂J c−1 + ∇ · F = qJ (t, x). (4.15) ∂t These coupled partial diﬀerential equations (PDEs) encapsulate the diﬀusion transport model we will exploit in the remainder of this paper. 5.4.2 Directional and spatial enhancements Diﬀusion theory can easily be improved in highly relevant ways for cloud remote sensing applications. Among many possibilities, we discuss one well-known approach in previously raised issues in direction space, and two quite recent developments germane to position space. 5.4.2.1 δ-Eddington rescaling for the impact of the diﬀraction peak in the phase function As previously mentioned, an inherent weakness of diﬀusion-based radiation transport modeling is the smooth one-parameter phase function in (4.13) whereas realworld phase functions have prominent forward peaks. We can partially mitigate this 13 A more contemporary derivation of the 3D diﬀusion approximation would use asymptotic analysis of the general RT equation where the magnitude of the local MFP 1/σ(x) is small and only large terms are kept; see Refs [68, 69]. Yet another notable derivation [70] makes use of a special limiting case of the discrete-angle phase function in the ‘six-ﬂux’ model for 3D RT [71].

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disconnect by applying the classic δ-Eddington rescaling [72]. The phase function is recast as a combination of a δ-function in the forward direction and a complementary term with two spherical harmonics. In the absence of absorption, this results in a rescaling given by σ (x) = (1 − f )σ(x), (1 − g ) = (1 − g)/(1 − f ),

(4.16)

where f is the fraction of ‘δ-scattering’ (physically, just prolonged ballistic propagation). This operation decreases σ ≡ σs (increases the MFP), but leaves σt invariant in (4.3). We will see in the next section that this scaling therefore leaves a large class of diﬀusion models with isotropic boundary sources unaﬀected. A popular choice is f = g 2 because it ﬁts the second as well as ﬁrst spherical harmonic coeﬃcients of the H–G model phase function, hence g = (g − f )/(1 − f ) = g/(1 + g).

(4.17)

For liquid water clouds, where g ≈ 0.85, we get f ≈ 0.72, hence σ ≈ 0.28σ and g ≈ 0.46. Alternatively, the whole diﬀraction peak – half of the scattered energy for particles with very large size parameters (Babinet’s principle) – can be recast as prolonged propagation in the original direction: f = 0.5, hence σ ≈ 0.5σ and g ≈ 0.7. 5.4.2.2 Eﬀects on bulk radiation transport of turbulence-driven random 3D variability of extinction, general considerations Although not our preferred approach in Green function calculation, F can be eliminated between (4.14) and (4.15), leading to [∂t − ∇ · (D∇)] J = cqJ − D∇ · q F

(4.18)

D(x) = c/3σt (x) = ct (x)/3

(4.19)

where is the (local) radiative diﬀusivity.14 In an arbitrary 3D optical medium with neither sources nor sinks (absorption), apart from boundaries, the local expression for conservation of radiant energy is [∂t − ∇D · ∇ − D∇2 ]J = 0. In the Fokker–Planck interpretation of this boundary-value diﬀusion problem, −∇D is an eﬀective drift velocity. Radiant energy thus ﬂows naturally from low to high diﬀusivity (high to low extinction) regions. Dwelling on the no-source/no-sink media, we can use (4.18) to equate the classic operator for diﬀusion in a uniform medium with a speciﬁc variability term: [∂t − D∇2 ]J = ∇D · ∇J. In the vector-ﬂux picture, this new term on the r.-h. side can be written as ∇ ln D · F , and it formally acts as a pseudo-source/sink term for the mean ﬂow in the hypothetical uniform medium. Davis and Marshak [73] investigated the steady-state (∂t J ≡ 0) problem in detail, showing systematic 14

We recall that transport coeﬃcients are the product of particle density and crosssection for scattering, absorption, or both (extinction), per particle. Although there is no compelling rationale for this, we tend to assign the random spatial ﬂuctuations to the particle density and hold the optical properties constant.

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eﬀects of any given 3D structure on ﬂux line geometry and, from there, domainaverage boundary ﬂux (T = 1 − R). The transport mechanisms behind systematic 3D eﬀects observed in absorbing media and time-dependent deterministic ﬂows remain open to investigation. For our present purposes, we now tap into a statistical approach developed for steady sources to account for 3D eﬀects, be they for temporal characteristics – a conjecture that will eventually call for at least numerical veriﬁcation. Barker and Davis [74] survey the two broad classes of models in 3D RT that target large-scale eﬀects of unresolved small-scale variability in cloud structure, which is invariably assumed random in nature. Members of one class of such ‘mean ﬁeld’ theories lead to new transport equations to solve. Members of the other class pursue homogenization: redeﬁne coeﬃcients in 1D RT so that the known solutions of that problem capture the main 3D eﬀects, which is clearly the path of least resistance. Among these ‘eﬀective medium’ approaches to random 3D variability, we favor the rescaling techniques proposed by Cairns et al. [75] and by Larsen [76]. Although one-parameter solutions, they stem from careful treatments of both both 1- and 2-point statistics, i.e., the PDF of σ(x) and its autocorrelation function respectively. 5.4.2.3 Homogenization via Cairns’ rescaling Starting with the δ-rescaled (primed) quantities in (4.16) that account for the problematic forward-scattering peak in the phase function, Cairns’ renormalization theory leads to σ (x) = (1 − )σ (x), (1 − g ) =

1 − 2 (1 − g ), 1−

(4.20)

in the case of conservative scattering, where is the new variability parameter. We see immediately that 1/2 is a strict upper limit for , and that it is probably best to not approach too closely in practice, and especially not in diﬀusion modeling. While δ-Eddington rescaling leaves the product (1 − g)σ invariant, it decreases here both through σ and through 1 − g as increases (since g > g ). For diﬀusion models with strict similarity (dependent only on σt = (1 − g)σ = (1 − g )σ ), we have (4.21) σt (x) = (1 − 2)σt (x). About the dependence of σ (or σt ) on x, we should bear in mind here that, as soon as we apply this rescaling, we have taken care of all the arguably random small-scale variability – up to a few MFPs (see below). So, implicitly, we are now interested only in cloud scale variations of the extinction, such as the stratiﬁcation trends discussed in section 5.2.5. How does one compute ? Recalling that overscores denote averages over the spatial variability, Cairns et al. show speciﬁcally that, for moderate-amplitude ﬂuctuations, the 3D RT eﬀects are captured with = a − a2 − v 2 (4.22) where

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Fig. 5.6. Cairns’ scaling factor used in (4.20). The parameter is plotted as function 1/2

of σ 2 /σ and σlc using (4.22)–(4.23). Values up to ∼1/3 can be used with some conﬁdence (this divides 1 − g at most by 2). Therefore at most moderate 1-point variability ratio (RMS/mean for σ) can be considered (only slightly more than unity), unless the correlations are very short range vis-` a-vis the MFP (deﬁned here as 1/σ, even though this is known to be an underestimation [10]). More discussion in main text.

1 1 1+ a= , v= 2 σlc

( σ2 − 1. σ2

(4.23)

Parameter v is the standard deviation to mean ratio, itself expressed with the RMS-to-mean ratio, for σ and we denote here the characteristic correlation scale of the spatial variability by lc . We see that for small-scale ﬂuctuations (i.e., when lc MFP ≈ 1/σ), we anticipate little eﬀect since ≈ (v/a)2 /2 ≪ 1 (irrespective of v) as a becomes very large; – for ﬂuctuations at larger scales (i.e., when σlc 1), we can have a strong impact ( 1/2) although this scenario clearly stretches the validity of the model, in particular, amplitude is then limited to cases where v 2 a − 1/4 (σ 2 /σ 2 5/4 + 1/σlc ); – for ﬂuctuations at the largest scales (σlc 1, hence a ≈ 1/2 and v 1/2), one should average over macro-scale responses rather than try to ﬁnd a single eﬀective medium to account for micro-scale variability eﬀects.

–

Figure 5.6 illustrates this analysis of . In the last (‘slow’) variability regime, the large-scale averaging of radiative responses can be computed locally using a strong uniformity assumption, which is the essence of the independent pixel approximation (IPA). See Ref. [74], references therein, and the penultimate section of this paper for applications of the IPA to solar RT (section 5.10.3) and to a related time-dependent problem (section 5.10.4). The above scale-by-scale breakdown of spatial variability impacts is consistent with the ﬁrst-principles analysis by two of the present authors [10] who, incidentally,

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show that the actual MFP is ≈1/σ in a broad class of variable media with longrange correlations, including clouds. Moreover, that estimate always exceeds 1/σ (they are equal only when σ is uniform). This is a direct consequence of Jensen’s inequality [77] in probability theory concerning averages of functions with deﬁnite convexity (in this case, the exponential). Those authors come to the same scale-based classiﬁcation of variability eﬀects in RT from the standpoint of steady-state 3D diﬀusion theory [73]. The only diﬀerence is that the transport MFP in (4.4) replaces the usual MFP used in the present arguments based on propagation between successive scatterings (or, for example, an emission or an absorption). 5.4.2.4 Homogenization via Larsen’s rescaling As competition for Cairns’ model, we highlight another notable development in homogenization theory by Larsen [76], who includes a diﬀusion limit of immediate interest here. It builds on the ideas of Kostinski [78], Davis and Marshak [10] who predict non-exponential step distributions between scatterings in randomly variable media: sub-exponential distributions if spatial correlations are positive, super-exponential otherwise [79]. Positive correlations are the norm in clouds, so we anticipate the higher-order moments of to exceed the exponential-based prediction. Let P (s) be the distribution of steps in the Markovian propagation process of multiply scattered light. Recall that = s is the well-known MFP. Then the exponential case is completely deﬁned, P (s) = exp(−s/)/, and it leads to sq = Γ (q)q (for any real q > −1). The sub-exponential laws of interest here will therefore have sq > Γ (q)q for q > 1 because of the longer tail (slower decay). Dwelling on moments of lowest integer order, we can use r=

s2 >1 2s2

(4.24)

to measure the deviation of P (s) from the exponential case. By taking a careful asymptotic limit, Larsen ﬁnds (in diﬀerent notations) that, in the case where 0 = 1, homogenized diﬀusivity is Deﬀ = Dexp × [(1 − g)r + g],

(4.25)

which exceeds the classic value of Dexp = c/3(1 − g) if r > 1, for all |g| < 1. Interesting things happen to radiation transport when s2 = ∞ leading to ‘anomalous’ diﬀusion, which is out of the scope of this article; they are brieﬂy discussed in Appendix F and in Refs [80, 81]. To implement this statistical 3D RT correction, we can assign the boost Deﬀ /Dexp in (4.25) to an eﬀective reduction of σt in our diﬀusion models with isotropic boundary or internal sources, hence strict similarity. Before application to models with non-isotropic internal sources, the derivation in Ref. [76] would have to be revisited. In analogy with Cairns’ rescaling in (4.21), we can use 1 1 L = 1− , (4.26) 2 r − (r − 1)g

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Fig. 5.7. Larsen’s scaling factor L (g, r) from (4.26). Parameter L is to be used in lieu of Cairns’ counterpart in (4.21). It is plotted here as function of asymmetry factor g and the ratio r of the second-order moment of P (s) and its predicted value based on the exponential distribution. As expected from the deﬁnition in (4.21), L does not exceed 1/2. We believe that values up to ∼1/3 can be used with conﬁdence, which would at most divide σt = (1 − g)σ by 2, hence for 1 < r 10 based on g = 0.85.

which of course does not exceed 1/2 for r ≥ 1 and |g| < 1. This homogenization factor is plotted in Fig. 5.7 for the relevant range of its arguments (g, r). Larsen’s L depends on one local optical parameter, g, and only one variability parameter, r. The later in turn depends however on both 1-point/PDF and 2-point/correlation statistics in ways that are not yet well understood. Indeed, it can be shown that if correlations are only very short range compared to the MFP, then P (s) is very near exponential [10, 17]. Correlations that are very long by the same standard of comparison are better treated by the above-mentioned IPA methods, leaving correlations on the same scales as the MFP as the ones15 where homogenization theories are truly helpful. In short, exponential free-path PDFs prevail only in homogeneous optical media. Random-but-correlated media have wider step PDFs, in particular, in the sense of the moment inequality in (4.24). Consequently, they can sustain in steady-state systematically larger ﬂuxes thanks to greater eﬀective diﬀusivity for the bulk of the medium. Guided by the numerical results displayed in Fig. 5.4 for ρ2 F (F = R, T ), we speculate that the spatial variability will also lead to increased horizontal transport away from a localized source. Although their derivations may need to be revisited for time-dependent transport, we will allow ourselves in the latter sections 15 The plural is deliberate since, in stochastic optical media, the MFP is itself a random variable dependent on position, direction, and realization. Although a mean MFP s can be deﬁned and investigated, it is no longer the only moment of interest in the step PDF P (s) for the spatial-, directional- and ensemble-average [10]. So scales ‘commensurate with the MFP’ could cover a wide range, at least going from s to s2 /2s ≥ s (where ‘=’ is for the exponential case).

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of this paper to use Larsen’s or Cairns’ homogenizations of σt (x) for time-dependent diﬀusion. 5.4.3 Boundary conditions, including boundary sources Bearing in mind that the local optical properties that matter in diﬀusion {σ(x), 0 , g} can be rescaled to account for unavoidable phase function attributes and 3D random variability, we now need to assign boundary conditions (BCs) to the coupled ﬁrst-order PDE problem at hand. If the opaque (radiatively diﬀusive) cloud is reasonably stratiform, then we can model it with slab geometry {x ∈ R3 ; 0 < z < H}. We furthermore recall that, in this plane-parallel framework, cloud optical depth τ is the integral of σ(z) from 0 to H. BCs for the above coupled PDEs for J and F must then express hemispherical ﬂuxes crossing a constant-z plane in the ± directions, namely, +π F± = −π

±1 J/2 ± Fz , dφ G(·, Ω)μ dμ = 2

(4.27)

0

obtained from (4.12). The no-incoming-radiance (a.k.a. ‘absorbing’) BCs for radiance bring us ﬂux-based BCs 4F+ (t, x, y, 0) = J(t, x, y, 0) + 2Fz (t, x, y, 0) = 0, 4F− (t, x, y, H) = J(t, x, y, H) − 2Fz (t, x, y, H) = 0,

(4.28) (4.29)

for all x, y, and t. If we have partial reﬂectivity at the z = H boundary, which relates outgoing (μ > 0) and incoming (μ < 0) radiances, as described in (2.12), then the associated BC becomes [1 − αH (x, y)] J(t, x, y, H) − 2 [1 + αH (x, y)] Fz (t, x, y, H) = 0.

(4.30)

The precise kind of reﬂection (for example, Lambertian vs. specular) is of course inconsequential since only ﬂuxes are modeled in diﬀusion theory. Alternatively – but no longer equivalently – in the diﬀusion approximation, one can put the pulsed source in the BC at z = 0. In that case, we set qJ = qF ≡ 0 in the r.-h. sides of (4.14)–(4.15) and require J(t, x, y, 0) + 2Fz (t, x, y, 0) = 4q0 (t, x, y)

(4.31)

for a general distribution of isotropic boundary sources, in lieu of (4.28). The postulated boundary source term q0 (t, x, y) is the μ-weighted angular integral over μ > 0 of the incoming radiance ﬁeld. The radiance BC at z = 0 in (2.8) or in (2.10), irrespective of Ω0 , then yields q0 (t, x, y) = δ(t)δ(x)δ(y),

(4.32)

as does the expression in (2.11) for an isotropic boundary source; all this while the homogeneous BC at z = H in (4.29) or (4.30) is unchanged. Since ﬂux alone

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tells us nothing about directionality, we are now eﬀectively modeling the source as pointwise and pulsed but isotropic in the μ > 0 hemisphere. Equations (4.28)–(4.29), and their generalizations in (4.30) and (4.31) respectively for secondary and primary sources, express the least usual ‘3rd-type’ of BCs that occur in generic applications of diﬀusion-type PDE problems, both timedependent (parabolic) or steady-state (elliptical). They involve the density J at the boundary and the boundary-crossing current Fz , equivalently, J and its normal derivative from (4.14). These BCs can be expressed as a variable mixture of Dirichlet/ﬁrst-type (ﬁx J) and Neumann/2nd-type (ﬁx Fz ) BCs: → → → ρ , 0) = 4q0 (t, − ρ) J(t, − ρ , 0) + 3χFz (t, − → − → − → − → − [1 − αH ( ρ )] J(t, ρ , H) − 3χ [1 + αH ( ρ )] Fz (t, ρ , H) = 0. (4.33) Although often referred to as ‘mixed’ BCs,16 these are known technically as ‘Robin’ BCs [82]. At any rate, they are the most general BCs we will need to consider in the following applications of diﬀusion theory to cloud remote sensing. → When q0 (t, − ρ ) does not vanish, the BC mixing factor χ can diﬀer from its 2/3 value used in (4.31), but typically not very much (at least in the most common transport applications). This is basically a tuning parameter that was introduced by early neutron transport theorists to help diﬀusion models reproduce high-precision solutions of the transport equations in critical applications [52]; this boost in accuracy is naturally applied where diﬀusion is at its weakest, namely, boundaries. The physical interpretation of χ is that of an ‘extrapolation length’ measured in transport MFPs. Indeed, in the absence of anisotropic internal sources, Fick’s law → in (4.14) tells us that Fz (t, − ρ , 0) = −[∂z J/3σt (z)]z=0 , and similarly at z = H. By substitution into (4.33), the l.-h. side reads as a linear extrapolation formula for J, given its derivative along the z-axis, over a distance χ/σt (0) into the z < 0 region; we have a similar reading of the BC at z =√H, going into the z > H region. Some values for χ found in the literature are: 1/ 3 (‘S2 ’ model [11]), 2/3 (Marshak ﬂux BCs [51]), 0.7104 · · · (Milne half-space problem [52]), 1/n (nD ‘discrete angle’ RT model [70], including 3D ‘six-ﬂux’ theory [71]), 4/3 (optically thin limit [52]). This realization in fact opens the possibility of recasting the above Robin BCs as ‘extrapolated’ Dirichlet BCs: χ → → J t, − ρ ,− − ρ) (4.34) = 4q0 (t, − σt (→ ρ , 0) → 1 + αH (− ρ) χ → J t, − ρ ,H + = 0. → → σt (− ρ , H) 1 − αH (− ρ) → ρ , z) in the bulk and the We need to assume here that the transport extinction σt (− → − surface albedo αH ( ρ ) vary at most rather slowly (almost everywhere ﬁnite gradients). The support of the coupled PDE problem at hand has thus been formally extended from 0 < z < H to → χ ρ) 1 + αH (− χ
t

H

16 In mathematically correct terminology, ‘mixed BCs’ refers to problems with somewhere Dirichlet and elsewhere Neumann BCs.

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which may be between two wavy boundaries. This Dirichlet-type approximation of the exact Robin BCs normally required for diﬀusion theory is rarely used in atmospheric radiation applications, a notable exception being work by E.P. Zege et al. [83, and references therein]. As demonstrated below, these approximate BCs lead to simpler expressions that in turn enable deeper results, albeit at a cost in accuracy and/or reduction of the applicable parameter space. Finally, we need to reconsider our parameterizations of internal stratiﬁcation from section 5.2.5 from a diﬀusion standpoint. Because g is assumed constant, σt (z) will have the same behavior as σ0,H (γ; z) for the presumably superior power-law model. However, the vanishing σt (z) at either z = 0 (source below cloud) or z = H (source above cloud) is problematic for the diﬀusion model. Indeed, the BCs in → ρ , z) in (4.14) for z = 0 and z = H, (4.33) make necessary the evaluation of Fz (t, − one of which contains a division by σt (z) = 0; this problem is even more obvious in the extrapolated BCs in (4.34). Physically, the local transport MFP is divergent at one of the cloud boundaries. Diﬀusion, as an approximation to RT, is already known to deteriorate near boundaries. However, if the associated extrapolation length is inﬁnite, then the failure is likely to be catastrophic. This is the main reason why we introduced the linear gradient model as a surrogate. 5.4.4 Remote sensing observables The quantities of interest in cloud remote sensing are local/instantaneous reﬂectivity and transmittivity, in other words, the outgoing ﬂuxes normalized by total incident energy. This total energy is the space-time integral of qJ (t, x) always assumed unitary in Green function analyses. Speciﬁcally, we seek: J(t, x, y, 0)/2 − Fz (t, x, y, 0) F− (t, x, y, 0) = , F+ (0) 2μ0 F+ (t, x, y, H) J(t, x, y, H)/2 + Fz (t, x, y, H) T (t, x, y) = = , F+ (0) 2μ0

R(t, x, y) =

(4.35) (4.36)

where F+ (0) is the incoming ﬂux (in the hemisphere with μ > 0) integrated over time and the illuminated cloud boundary. Allowing for the possibility of oblique illumination, we have F+ (0) = μ0 ≤ 1. Invoking the BC at z = 0 in (4.28), we can express these basic cloud responses simply as R(t, x, y) = J(t, x, y, 0)/2μ0 , T (t, x, y) = J(t, x, y, H)/2μ0 ;

(4.37) (4.38)

we recall that when, as is the case here, the source is speciﬁed internally, then the diﬀuse transmittance in (4.38) does not include the un-collided ﬂux contained in direct transmittance. If the isotropic boundary-source model in (4.33) is used for the BCs, then J and F necessarily contain the incident ﬂux. We must therefore compute the required space-time reﬂectivity and transmittivity ﬁelds in (4.35)–(4.36) from F− (t, x, y, 0) = J(t, x, y, 0)/2 − q0 (t, x, y), F+ (0) F+ (t, x, y, H) T (t, x, y) = = J(t, x, y, H)/2. F+ (0) R(t, x, y) =

(4.39) (4.40)

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The outcome in the latter case will contain the contribution of directly transmitted ﬂux, although that should not be a signiﬁcant contribution in diﬀusion regimes. If the extrapolated Dirichlet BCs in (4.34) are used then, in principle, reﬂectivity R(t, x, y) should be computed as the backward hemispherical ﬂux at z = 0 (now a point inside the extended domain) combining J and Fz according to (4.27). However, the gain in accuracy is likely to be small compared to the loss incurred by simplifying the BCs; so, for simplicity, it can still be obtained from (4.39). The same remark applies to transmittivity T (t, x, y) from (4.38). Finally, we recall that diﬀusion theory only predicts ﬂuxes at cloud boundaries (and, for that matter, elsewhere). A zeroth-order estimate of cloud-leaving radiance is given by R(t, x, y)/π, a Lambertian assumption that is not unreasonable for highly scattered light. A ﬁrst-order angular model would make use of (4.12). This radiance-to-ﬂux conversion can be done with better angular models, and should be for actual cloud remote sensing applications; see, for example, Ref. [84]. We have now completed the modeling framework for predicting remote sensing signals originating from internally variable stratiform clouds in the spatial and temporal domains using diﬀusion theory. Moreover, several options are available to control the degree of ﬁdelity in the model’s representation of collimated sources. 5.4.5 Fourier–Laplace transformation for stratiﬁed media In the case of constant coeﬃcients, or simple-enough vertical variability models, Fourier–Laplace transformation of the PDE system in (4.1) and (4.14), with the appropriate boundary conditions, leads to a class of analytically tractable problems for our representations of pulsed laser or solar sources. Equipped with an eﬀective medium approach such as Cairns’ or Larsen’s for small-scale random 3D variability, we can now restrict ourselves to cloud structure that unfolds only along the z-axis according to the previously introduced stratiﬁcation models. We therefore deﬁne → ˜ − J(s, k ; z) =

∞

+∞

dt 0

→ → − → → exp(−st + i k · − ρ ) J(t, − ρ , z) d− ρ (x, y).

(4.41)

−∞

→ → ˜ (s, − We similarly transform all the components of F (t, − ρ , z), yielding F k ; z). We can now think of (s, k) as parameters rather than independent variables, hence the deliberate insertion of the ‘;’ separator. → − Furthermore, we let F = ( F h , Fz )T , similarly for q F , and we recall that ∇ = → − → (∂/∂ − ρ , ∂/∂z)T transforms to (i k , d/ dz)T . Our PDE system in (4.1) and (4.14) then becomes a system of three ordinary diﬀerential equations (ODEs): → → − − (s/c)J˜ + i k · F˜ h + F˜z = −σa (z)J˜ + q˜J → − →˜ − → − i k J/3 = −σt (z) F˜ h + q˜ F h , J˜ /3 = −σt (z)F˜z + q˜F z .

(4.42)

→ − → − Note that q˜ F h ≡ 0 in all of our laser source models by axial symmetry, although this is an assumption we can relax when dealing with extended uniform (solar)

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→ − − → sources since it corresponds to the k = 0 case. From (4.8)–(4.9), Fourier–Laplace transformed internal source terms are Rz

q˜J (s) = σs (z) e−(s/c)z−

0

σ(z ) dz

, q˜F z (s) = g × q˜J (s),

(4.43)

independent of k in the case of normal laser-like incidence. → − Under these conditions, elimination of F˜ h between the ﬁrst and second equations leads to k2 s + + σa (z) J˜ + q˜J . (4.44) F˜z = − c 3σt (z) This last ODE is an expression of energy conservation (with transport) along the z-axis where local time variation and horizontal divergence of J are recast as ‘effective’ absorption processes: σa(e) (s, k; z) = s/c + k 2 /3σt (z).

(4.45)

This is a key coeﬃcient that, in general, is stratiﬁed diﬀerently than σx (z) (with x = s,a,t), which all vary together (since 0 and g are assumed constant). The general boundary conditions in (4.33) become ˜ k; 0) + 3χF˜z (s, k; 0) = 4˜ J(s, q0 (s, k), ˜ k; H) − 3χ(1 + αH )F˜z (s, k; H) = 0, (1 − αH )J(s,

(4.46) (4.47)

where q˜0 (s, k) ≡ 0 and χ = 2/3 if the distributed internal source model in (4.43) is used. If the boundary point-source model is used instead, q0 (t, x, y) in (4.32) leads to q˜0 (s, k) ≡ 1 in (4.46). For simplicity, the optional surface albedo at z = H is assumed uniform; otherwise, we would need to interpret the product of two func→ − → tions of − ρ as a convolution in k -space. Under the same simplifying assumption, treatment of the extrapolated boundary conditions in (4.34) is straightforward. We recall ﬁnally that in remote sensing our interests are limited to ˜ k) = J(s, ˜ k; 0)/2μ0 , R(s, ˜ k; H)/2μ0 , T˜(s, k) = J(s,

(4.48) (4.49)

for collimated (possibly oblique) illumination, depending on what side of the cloud the observation is performed. We must bear in mind that the latter expression is restricted to the diﬀuse (scattered) component of transmission; for total transmittance, we need to add the un-collided component exp[−(σ + s/c)H/μ0 ], which includes the Laplace-space signature of the time-delay going from z = 0 to the point of escape. When using the (necessarily isotropic) boundary source option, we instead use ˜ k) = J(s, ˜ k; 0)/2 − 1, R(s, ˜ ˜ T (s, k) = J(s, k; H)/2, and recall that, in this case, T stands for total transmission.

(4.50) (4.51)

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5.5 Solutions of diﬀusive Green function problems Not all of our best representations of the source and of the cloud structure and of the boundary conditions can be used at the same time, even when using computer-assisted algebra. The mathematical complexity becomes intractable. However, boundary-source Green functions can be computed in closed-form for useful regions of the ﬂexible parameter space we have set up in the above diﬀusiontheoretical framework. In the following, we will move through the models adding and removing capability with speciﬁc applications in mind. These applications will be called out here, but described in full detail in the remaining sections of the paper. In particular, we restrict ourselves here to Fourier–Laplace expressions, and move on to PDFs and space-time moments in the two following sections on cloud probing applications, by remote sensing or by in situ radiometry. Throughout, we assume conservative scattering (σa (z) ≡ 0, 0 = 1), hence σs (z) ≡ σ(z) and σt (z) = (1 − g)σ(z). 5.5.1 Homogeneous cloud with an isotropic boundary point-source In this ﬁrst approach to boundary-source Green function calculus, we do not bother with internal stratiﬁcation. Moreover, we do not attempt to capture the collimation property of laser beams nor source anisotropy induced by the forward-peaked phase function of cloud droplets (beyond the classic scaling of σ by 1 − g in σt ). This is the entry-level model used by Davis, Love and co-workers in their 1999 [85] and 2001 [86] proof-of-concept papers on multiple-scattering cloud lidar (reﬂected laser light), and by Davis and Marshak in their 2002 paper [24] on transmitted solar light, primarily with ground-based O2 A-band observations in mind. The resulting boundary-value problem is quite simple: ˜ J˜ /3 = −σt F˜z , F˜z = −σa(e) (s, k) J, subject to boundary conditions ) ) J˜ + 3χF˜z )

z=0

) ) = 4, J˜ − 3χF˜z )

= 0,

(5.1)

(5.2)

z=H

leaving χ as an unspeciﬁed tuning parameter. By inspection, we see that nondimensional cloud responses will depend only on the transport or ‘scaled’ optical depth (5.3) τt = H/t = σt H = (1 − g)τ, and of course χ. This means in particular that the δ-Eddington rescaling in (4.16) for forward scattering has no eﬀect on this model. Furthermore, we will have similar behavior between s/c and k 2 /3σt since they are interchangeable in the constant (e) coeﬃcient σa (s, k) in (4.45). This coupled ODE problem can be formally mapped to the simplest version of the well-known two-stream model for solar transport in plane-parallel clouds, with a ﬁxed amount of scattering and a variable amount of absorption; see Ref. [87]. This (e) identiﬁcation is purely formal because, in spite of their connection in (4.3), σa varies from 0 to ∞ independently of σt . In practice, this means that the eﬀective

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0 varies from 1 to −∞. Nonetheless, the analogy can be used if necessary to obtain the solution of (5.1)–(5.2). We start with the s/c = k = 0 case, corresponding to steady/uniform illumi(e) nation, hence for σa = 0. It is easy to show, by direct integration of (5.1)–(5.2), that radiation density decreases only linearly through the medium: ˜ 0; z) = 2 (1 + R) + (1 + R − T ) z (5.4) J(0, H where we use cloud transmittance T and albedo R given by T (τt /2χ) =

1 , R(τt /2χ) = 1 − T (τt /2χ). 1 + τt /2χ

(5.5)

These are the classic results for the two-stream/boundary source model in solar radiation transport at non-absorbing wavelengths, going back to Schuster’s landmark 1905 paper [88] about atmospheric RT in the presence of scattering.17 Letting (e) (5.6) LD (s, k) = (3σa(e) σt )−1/2 = 1/ k 2 + 3σt s/c be the so-called diﬀusion length (in this case, an ‘eﬀective’ one), the solution we seek is ˜ k; z) = 4 J(s, ±(1 ± X) e±(1−z/H)Y / ±(1 ± X)2 e±Y {±}

{±}

sinh[(1 − z/H)Y ] + X cosh[(1 − z/H)Y ] . = 4 (1 + X 2 ) sinh Y + 2X cosh Y

(5.7)

(e)

We have introduced here the length-scale ratios X = χt /LD = (2χ/τt )Y where (e) Y = H/LD . Expressions for the required boundary ﬂuxes are then T˜(s, k; H, τt /2χ) =

2X cosech Y 1 + X 2 + 2X coth Y

(5.8)

˜ k; H, τt /2χ) = R(s,

1 − X2 1 + X 2 + 2X coth Y

(5.9)

from (4.51), and

from (4.50). L’Hˆ opital’s rule can be used to retrieve the above baseline limit where (e) s, k (hence σa ) → 0, as do X, Y . Figure 5.8 illustrates this Fourier-space solution, F˜ (0, k; H, τt /2χ)/F (τt /2χ) (F = T, R) from (5.8)–(5.5), therefore with a focus on steady sources (s = 0). For transmission through a diﬀusive cloud, we have (kH/ξ) × (1 + ξ) T˜(kH; ξ) = , 2 T (ξ) [1 + (kH/ξ) /4] sinh kH + (kH/ξ) cosh kH

(5.10)

17 For the record, the remarkable result in (5.4)–(5.5) was in fact derived in two earlier but lesser-known papers by Lommel [89] and Chowlson [90].

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Fig. 5.8. Flux-based MTFs for similar diﬀusing clouds. Top: Fourier space ﬁlters T˜(k; H, τt /2χ)/T (τt /2χ) are plotted against non-dimensionalized wavenumber kH for ﬁxed H and χ = 2/3; they determine the loss of deﬁnition in imaging (with local ﬂux, not √ radiance) through clouds with increasing optical depth, from bottom to top: τt = 1, 10 ≈ 3.16, 10, and the somewhat academic case of ∞ (where there is no light left to transmit). The curvature of the MTF at k = 0 determines the variance of the lateral transport in transmitted light ρ2 T . Middle: Same as top panel but for reﬂection, and τt = 31.6 is used instead of ∞. Bottom: Same as top panel but plotting the reﬂection MTF versus kt rather than kH; ordering of optical depth (modulated here with H) refers to r.-h. side, but we note the increasing curvature on the l.-h. side. More discussion in main text.

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where we deﬁne ξ = τt /2χ = R/T,

(5.11)

using (5.5). This normalized response plotted in the top panel of Fig. 5.8 as a function of kH using χ = 2/3. This is basically a poor-person’s solution of the pencil-beam problem, with no attempt at accounting for collimation, only localization. It can also be recognized as the ﬂux-based modulation transfer function (MTF) of the cloud for an isotropic point-source. The MTF has immediate applications in imaging theory as the spatial Fourier ﬁlter that the medium applies to any pattern viewed by a sensor through the cloud. We refer to Weinmann and Masutani [91] for an interesting atmospheric application: viewing cities through clouds at night from space. Notice how the more tenuous clouds have a narrower band-pass: the resulting image is brighter but more blurry. This is counterintuitive and results from the propagation away from the assumed isotropic point-source in a medium where the transport MFP t is increasing, thus promoting further horizontal transport. However, we see that the MTF is not very sensitive to τt as it becomes large: it rapidly approaches the limiting case, kH/ sinh kH, for ξ ∝ τt → ∞. Since the negative curvature of T˜(kH; ξ)/T (ξ) at the origin becomes asymptotically constant, (3.5) tells us immediately that ρ2 T will be ∝ H 2 (and somewhat larger at ﬁnite τt ). This last prediction will be veriﬁed and made quantitative in the next section; see also Appendix E. For reﬂection from a diﬀusive cloud, we have ˜ (1 − (kH/ξ)2 /4) × (1 + ξ)/ξ R(kH; ξ) = , R(ξ) [1 + (kH/ξ)2 /4] + (kH/ξ) coth kH

(5.12)

with the same deﬁnition for ξ in (5.11). This normalized response is plotted in the middle and lower panels with the same value for χ. As in the top one, the middle panel uses kH as the non-dimensional independent variable. By comparison with the case of transmission, we see that curvature at the origin decreases as τt increases, and so will ρ2 R ; the size of the reﬂected spot on a very opaque/reﬂective nonabsorbing medium (such as this sheet of paper!) is hardly bigger than the laser beam (as is easy to verify with a laser pointer). Another asymptotic limit of interest is when ξ ∝ τt ∝ H → ∞ with H/ξ = 2χt held constant; the limiting form of the reﬂected counterpart of the MTF is ˜ ξ)/R(ξ) versus wavenum(1−χ|k|t )/(1+χ|k|t ). The bottom panel re-plots R(kH; ber non-dimensionalized as kt . Recalling that we are dealing with a necessarily → symmetric function of k (R(− ρ ) is axisymmetric), we see that the curvature at the origin is inﬁnite. In other words, ρ2 R for semi-inﬁnite non-absorbing media is inﬁnite.

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5.5.2 Stratiﬁed cloud with an isotropic boundary point-source The above bare-bones model for a pulsed point-source was recently generalized [92] to include stratiﬁcation according to the constant gradient parameterization from section 5.2.5.18 We thus wish to solve both the space-domain (s = 0) problem, ˜ J˜ /3 = −σt,Δ (z) F˜z , F˜z = −[k 2 /3σt,Δ (z)] J,

(5.13)

and its time-domain (k = 0) counterpart, ˜ J˜ /3 = −σt,Δ (z) F˜z , F˜z = −(s/c) J,

(5.14)

in both cases, subject to boundary conditions in (5.2) with χ as an ﬂoating parameter. By inspection, we see that non-dimensional cloud responses can only depend on Δ and τt = σ t H (and χ). This again implies that δ-Eddington rescaling will have no eﬀect on the results. In contrast with the previous Δ = 0 case, however, we will not have similarity-induced behavior between s/c and k 2 /3σt since they are (e) interchangeable in σa (s, k) only when σt is constant. ˜ k; z) is solvable in the The spatial diﬀusion problem in (5.13) and (5.2) for J(0, Fourier domain in terms of Bessel functions of the second kind (Yn (x), n = 1, 2) and modiﬁed Bessel functions of the second kind (In (x), n = 1, 2); see Appendix A for ˜ 0; z) is solvable details. The temporal diﬀusion problem in (5.14) and (5.2) for J(s, in the Laplace domain in terms of Airy functions and their derivatives, which are in turn related to modiﬁed Bessel functions with 1/3-integer orders. In both cases, the expressions are too complex to be reproduced here from the computer-assisted symbolic math tool and, at any rate, they are only used after setting z = 0 or z = H. This leads to somewhat simpler expressions for Fˆ (s) = F˜ (s, 0), F = R, T , in terms of (regularized) conﬂuent hypergeometric functions 0 F1 (a, x)/Γ (a); see Appendix B for details. Boundary ﬂuxes F˜ (s, 0) and F˜ (0, k) are obtained as above for F = R, T and, at zeroth order in both k and s, we retrieve (again using L’Hˆ opital’s rule) the standard result in (5.5) for cloud transmittance F = T and albedo F = R. As expected, they are insensitive to internal structure since optical properties 0 and g are held constant with respect to z. 5.5.3 Homogeneous cloud with normally incident illumination at a point Rather than an isotropic point-source speciﬁed in the boundary conditions, we now move to a more accurate representation of the pulsed laser beam formalized in (4.8)– (4.9), hence (4.43) in Fourier–Laplace variables, as an exponential distribution of anisotropic internal sources along the z-axis. However, to achieve analytical results, we must abandon the cloud-scale internal structure we just addressed; we can still apply the Cairns rescaling to correct for the eﬀect of small-scale turbulence. 18

For another solvable diﬀusion model (with a steady isotropic point-source) that features exponential stratiﬁcation, we refer to Section 6.2.2 in Zege et al.’s monograph [83]. Like here, Bessel functions arise; see Appendix A.

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Following Ref. [92], we solve this space-domain (s = 0) problem, F˜z = −[k 2 /3(1 − g)σ] J˜ + σ e−σz , J˜ /3 = −(1 − g)σ F˜z + gσ e−σz ,

(5.15)

and its time-domain (k = 0) counterpart, F˜z = −(s/c) J˜ + σ e−(s/c+σ)z , J˜ /3 = −(1 − g)σ F˜z + gσ e−(s/c+σ)z , subject to the Fourier–Laplace version of (4.28)–(4.29): ) ) ) ) J˜ + 2F˜z ) = 0, J˜ − 2F˜z ) = 0. z=0

(5.16)

(5.17)

z=H

We anticipate here that non-dimensionalized responses will depend on both τ and g, not just on (1−g)τ . Consequently, δ-Eddington rescaling can improve the accuracy of the model by reducing the impact of forward-peaked phase functions, which we recall is detrimental to diﬀusion. Also, we notice that s/c enters the exponential source term, as an eﬀective σa should since it participates in overall extinction, but k 2 /3(1 − g)σ does not. This transformed 3D time-dependent problem is therefore not formally identical to any known solar two-stream problem, at least when k = 0. ˜ k; z) of (5.15), with (5.17), can be expressed as a The desired solution J(0, lengthy rational function of k, e±kH , e±kz and e±σz with coeﬃcients containing σ, g, and e±σH . The diﬀuse boundary ﬂuxes are now computed from (4.48)–(4.49) ˜ 0; z) for with μ0 = 1; details are provided in Appendix C. The counterpart J(s, (5.16), with (5.17), can similarly be expressed as an even lengthier rational function √ √ ± 3s/c(1−g)σH ± 3s/c(1−g)σz ±(s/c)H ±(σ+s/c)z , e , e and e with coeﬃcients of s/c, e containing σ, g, and e±σH ; details are given in Appendix D with μ0 = 1. The zeroth-order result in both k and s, yet again calling for L’Hˆ opital’s rule, yields the known expressions [87] for total (direct plus diﬀuse) cloud transmittance, T =

5 − e−τ , 3(1 − g)τ + 4

(5.18)

and its albedo R = 1 − T , leading to R=

3(1 − g)τ − 1 + e−τ , 3(1 − g)τ + 4

(5.19)

for normal solar incidence, this is as expected when the source becomes steady (s = 0) and uniform (k = 0). Following Meador and Weaver [87], we note that for very small optical depths T can slightly exceed unity (hence R becomes slightly negative) if g > 2/3. Although we do not expect to use diﬀusion theory for such optically thin clouds, this underscores the need to use δ-Eddington rescaling in (4.16), which maps g = 0.85 to g = 0.46 (thus crossing the critical 2/3 threshold for obtaining physical values of R and T for all values of τ ).19

19 Since Cairns’ rescaling in (4.20) goes in the wrong direction of larger g-values, it is better not to stretch the diﬀusion model toward media with insuﬃcient opacity.

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5.5.4 Homogeneous cloud with normally incident illumination at a point from above and a reﬂective surface below This scenario could apply to a down-looking lidar probing fog, low-level arctic clouds, or turbid coastal water. However, to represent accurately the later situation one would need to add: (i) a ﬂexible combination of collimated (normal) and diﬀuse (isotropic) illumination to account for roughness of the air–water interface; (ii) partial reﬂection at the illuminated boundary determined by total internal reﬂection; (iii) the related conservation of ﬂux when crossing a discontinuity in index of refraction, another Brewster-angle eﬀect [93]; and (iv) some level of absorption in the optical medium. We explore here the simple version. In this case, we activate the boundary albedo αH in the boundary condition at z = H. We thus need to solve the space- and time-domain problems in (5.15) and (5.16) subject to (4.46)–(4.47) with q0 = 0 and χ = 2/3, speciﬁcally, ) ) 1 + αH ˜ )) ) = 0, J˜ − 2 = 0, (5.20) J˜ + 2 F˜z ) Fz ) 1 − αH z=0 z=H assuming 0 < αH < 1 (and F˜z (H) = 0, a Neumann boundary condition, when αH = 1). There is no new or fundamental diﬃculty in this enhanced version of the previous problem. The analytical expression of the solution will of course be more complex. Rather, we have here the opportunity to demonstrate a superposition → − principle of 3+1D RT in (s, k )-space using a physical argument. If αH = 0, we already know what happens: some light is reﬂected, some is transmitted spread over space and time, and we know the Laplace and Fourier transforms of these distributions. Since the surface at z = H is partially reﬂective, each point on it becomes a secondary source for the medium with a space- and time-dependent intensity determined by the problem with αH = 0. We know from section 5.5.1 how the homogeneous medium responds to each isotropic point-source on a boundary. Let F˜col and F˜iso (F = R, T ) be the boundary ﬂuxes for the two problems in → − ˜ k), hence R(t, ρ), (s, k )-space and let us focus on the remote sensing signal R(s, accessible from above the scene. The zeroth-order contribution, in the sense of lower surface reﬂections, is ˜ col (s, k). The ﬁrst-order contribution to the signal is T˜col (s, k)αH T˜iso (s, k). InR deed, if we think of t and ρ as random variables with PDFs PF (t, ρ) = F (t, ρ)/F , then the associated P˜F (s, k) = F˜ (s, k)/F˜ (0, 0) are their characteristic functions. In probability theory, addition of two independent random variables calls for a convolution of their PDFs, but just a simple product of characteristic functions. Consequently, we can read T˜col (s, k)αH T˜iso (s, k) = Tcol P˜T,col (s, k) × αH × Tiso P˜T,col (s, k) as the properly composed probability of three sequential but independent random events: (1) transmission from z = 0 to z = H and dispersion according to PT,col (t, ρ); (2) isotropic reﬂection by the uniform surface; (3) re-transmission back to z = 0 and dispersion according to PT,iso (t, ρ). Generalization to the next order is straightforward. Rather than transmission back to z = 0 the once surface reﬂected light is reﬂected back to the surface by

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209

˜ iso αH T˜iso . By the cloud/fog, surface reﬂected again and then transmitted: T˜col αH R ˜ iso ]n−1 αH T˜iso . induction, the contribution for n ≥ 1 surface reﬂections is T˜col [αH R By summing up all the contributions, we have for the requested ﬂux ˜ k) = R ˜ col (s, k) + T˜col (s, k) × R(s,

∞

˜ iso (s, k) αH R

n

× αH T˜iso (s, k),

n=0

hence ˜ k) = R ˜ col (s, k) + T˜col (s, k) R(s,

αH T˜iso (s, k) . ˜ iso (s, k) 1 − αH R

(5.21)

When working in a software environment for symbolic math, it is easy to implement this expression once we have solved the two coupled ODEs problems with diﬀerent types of source and stored the results. 5.5.5 Homogeneous cloud with uniform oblique illumination We do not foresee the immediate need to investigate the axially asymmetric spatial patterns of reﬂected (and even less transmitted) light that would result from oblique illumination by a narrow laser beam.20 However, it is of interest to understand the impact of slant illumination on the temporal properties (path-length distributions and moments) of reﬂected and diﬀusely transmitted light. This is largely because of the emerging capability of measuring such properties from ground and space using sunlight and diﬀerential absorption spectroscopy in the oxygen A-band (to be discussed in detail in section 5.8). As always with diﬀusion, we only obtain responses in ﬂux, at the cloud boundaries in particular. However, one can invoke optical reciprocity to ﬁnd new applications. In this case, we have spatially and angularly integrated but time-resolved observation of a pulsed collimated source. This situation can be transposed to a large-scale single-direction (i.e., radiance) observation of an isotropic burst of light, which may or may not be localized at a single point (there is no attempt at imaging). This scenario is directly applicable to light-curve analysis, for example, for optical detection of cloud-to-ground lighting [95–98]. Referring back to (4.10)–(4.11) as needed, with φ0 = 0, it is straightforward to generalize to μ0 < 1 the source terms in the time-domain problem in (5.16) for normal (μ0 = 1) illumination. We thus need to solve Fˆz = −(s/c) Jˆ + σ e−(s/c+σ)z/μ0 , J /3 = −(1 − g)σ Fˆz + μ0 gσ e−(s/c+σ)z/μ0 , ˆ

(5.22)

subject to boundary conditions similar to those in (5.17). However, this is only for Laplace-transformed variables, so we use ˆ· rather than ˜· symbols, referring back ˆ z) is a nontrivial generalization of the to section 5.3.1. The sought solution J(s; ˜ already complex expression for J(s, 0; z) obtained in section 5.5.3; see Appendix D for details. 20

See Section 6.3 in Zege et al.’s monograph [83] and the paper cited therein by Zege, Polonsky and Chaikovskaya [94] where this problem is addressed in steady state to obtain the non-axisymmetric MTF.

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ˆ z) we derive R(s) ˆ From J(s; and Tˆdif (s) using (4.48) and (4.49), ignoring the horizontal wavenumber k. When s → 0, along with the eﬀective absorption, we retrieve classic formulas from, for example, Ref. [87], for total cloud transmittance, T = Tdif + e−τ /μ0 = and albedo R=1−T =

(2 + 3μ0 ) + (2 − 3μ0 ) e−τ /μ0 , 3(1 − g)τ + 4

3(1 − g)τ + (2 − 3μ0 )(1 − e−τ /μ0 ) , 3(1 − g)τ + 4

(5.23)

(5.24)

resulting from slant illumination by the (steady) solar beam. Here also there are caveats about unphysical outcomes if g > 2/3μ0 and τ is too small. The same remedy as above is recommended, namely, to use δ-Eddington rescaling and also to steer away from grazing solar zenith angles (SZAs) if τ is small. As demonstrated in the previous case, we can easily add surface reﬂectivity to the model without going through the whole derivation. Indeed, since each term in (5.21) has a physical meaning, it is not hard to transpose it to a diﬀerent problem. For instance, what if we were to remotely observe an isotropic point-source on a reﬂective surface (0 ≤ α0 ≤ 1) with a non-imaging but time-resolving radiometer through a cloud layer? This could be a cloud-to-ground lightning stroke viewed from satellite. Either way, we need to partition what time-dependence belongs to the source and what ‘pulse stretching’ was added by the intervening scattering medium. The Laplace transform of the radiance time evolution for a unitary δsource viewed at an angle θ = cos−1 μ ≥ 0 (away from nadir) is predicted to be Tˆcol (s; μ) 1 , (5.25) Iˆobs (s) ≈ × ˆ iso (s) π 1 − α0 R where we have used reciprocity and emphasized that the Tˆ(s) computed here is for a slant illumination (μ0 < 1) by a uniform collimated beam or, by reciprocity, remote observation (μ < 1) with a large footprint. As in all of the previous cases, we postpone discussion of higher-order terms ˆ in the Taylor expansion of R(s) and Tˆ(s) around s = 0 (in this case, moments of path-length ct) until we address speciﬁc applications to cloud remote sensing in the following sections. In the interim, we describe what can be done about the challenge of inverse Fourier–Laplace transformation.

5.6 Inverse Fourier–Laplace transformation We now face the diﬃcult task of performing inverse Fourier and/or Laplace transforms of analytical (but not simple) expressions to obtain explicit expressions for cloud remote sensing signals. Or else we need to look for other ways of exploiting the diﬀusion models in remote sensing applications, a route we explore in sections 5.8 and 5.9. Since none of the above models have known analytical inverse transforms, we show here (1) the outcome of a numerical approach and (2) how a somewhat degraded representation of the laser beam source enables the inverse transforms to be performed analytically.

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5.6.1 Uniform clouds with an isotropic boundary point-source in (5.1), using exact boundary conditions in (5.2) It is highly desirable to validate the diﬀusion model for Green functions using numerical solutions of the 3D RT equation, say, by way of Monte Carlo techniques. It is possible to compute Fourier and/or Laplace transforms of radiative responses using MC, i.e., characteristic functions of the MC random variables. However, it is more compelling to see the diﬀusion predictions and MC benchmarks in physical space. Unfortunately, not even the simplest of the above models, isotropic boundary point-sources in (5.1)–(5.2), has analytical inverse transforms, so we need to implement them numerically. Figure 5.9 shows an example using the normalized spatial Green function for transmission PT (ρ) obtained from P˜T (k) = T˜(0, k)/T in (5.10), itself plotted in Fig. 5.8(a). In imaging terminology, we are deriving the cloud’s point spread function (PSF) from its MTF via inverse 2D Fourier transformation. In axial symmetry, the 2D Fourier transform in (4.41) morphs into the Hankel transform: ∞ P˜T (k) = 2π 0

1 PT (ρ)J0 (kρ)ρ dρ ↔ PT (ρ) = 2π

∞ P˜T (k)J0 (kρ)k dk,

(6.1)

0

Fig. 5.9. Normalized spatial Green function for transmitted light according to analytical diﬀusion theory and numerical transport theory. The cloud optical depth was set to τ = 16 and illumination was isotropic from a point; the outcome is the PSF for this diﬀusive medium. The extrapolation length parameter was set to χ = 0.7 for the diﬀusion expression in Fourier space, which was inverted numerically. Other details about the MC transport simulation are as in Fig. 5.3, which was replicated here for isotropic as well as forward (H–G) scattering. Adapted from Ref. [24].

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where J0 (x) is the zeroth-order Bessel function of the ﬁrst kind. Agreement between transport (MC results) and diﬀusion theories is excellent for the prescribed τ = 16 cloud, especially for isotropic (g = 0) scattering, when χ = 0.7. As is appropriate for a validation exercise, the MC scheme mimicked the isotropic boundary source used in the diﬀusion model. The only noticeable diﬀerence is for the near-axis ﬁeld when g = 0.85. 5.6.2 Uniform clouds with an isotropic internal point-source, using extended boundary conditions in (4.34) The above numerical approach to inverse Fourier or Laplace transformation is viable for case studies, but not for routine application to remote sensing signal analysis. So we still need an analytical predictor for space-time Green functions, especially for the multiple-scattering cloud lidar application where both dimensions are exploited. Since none of the models in the previous section are amenable to analytic inversion, we ﬁrst need to modify the diﬀusion model. Polonsky and Davis [99] revisited the way the pulsed laser-type sources for the Green function are represented. Following the path blazed by Zege, Katsev and Sherbaf [100] for weakly absorbing semi-inﬁnite media and Bushmakova, Zege and Katsev [101] for ﬁnite media, they proposed to compute the radiative Green function at z = 0 (the illuminated boundary) and z = H (the opposite boundary) using an isotropic source concentrated at a single point inside the cloud, but judiciously positioned. For the moment, however, we will denote this roaming position as z0 ∈ (0, H). With the 3D RT equation in mind, we write this as21 Q(t, x, Ω) = δ(t)δ(x)δ(y)δ(z − z0 )/4π

(6.2)

in the RT equation, which translates to qJ (t, x) = δ(t)δ(x)δ(y)δ(z − z0 ), q F (t, x) = 0,

(6.3)

for the diﬀusion model. In Fourier–Laplace space, this translates simply to q˜J (z) = δ(z − z0 ) and q˜F z = 0 (hence no attempt at capturing internal source anisotropy in this particular diﬀusion model). We now need to solve22 F˜z = −σa(e) (s, k) J˜ + δ(z − z0 ), J˜ /3 = −σt F˜z ,

(6.4)

which we subject to homogeneous extended Dirichlet boundary conditions, ˜ ˜ J(−χ/σ t ) = 0, J(H + χ/σt ) = 0.

(6.5)

This is a textbook boundary-value problem [103] leading to 21 For lidar applications at least, causality dictates that we also shift the δ-in-time by z0 /c. However, this only modulates the Laplace-transformed source q˜J (z) by a constant, exp(isz0 /c), and the shift itself can be implemented simply in the end-result anyway. By omitting it here, there is thus no loss in generality, but some gain in simplicity. 22 Incidentally, this model found an application in one of the earliest papers in 3D RT known to the present authors: Richards’ 1956 study of point-sources in plane clouds [102].

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213

˜ k; z0 , z) = cosh[m(τt + 2χ − σt |z − z0 |)] − cosh[m(τt − σt (z + z0 ))] , J(s, χm sinh[m(τt + 2χ)]

(6.6)

where we recognize τt = σt H from (5.3) and deﬁne23 (e)

m(s/c, k)2 =

3σa (s, k) s/c =3 + σt σt

k σt

2 =

s/D + k 2 , σt2

(6.7)

recalling that D = c/3σt . This Green function reﬂects reciprocity, i.e., z0 z symmetry between the positions of the isotropic source and detector (in this case, a ‘J-meter’). It also has space-time similarity in the sense that k 2 is interchangeable with s/D in the expression for m2 ; this will translate in physical space to ρ2 ↔ Dt maps in, for example, the marginal moments. Most importantly, the analytical expression contained in (6.6)–(6.7) can be inverse Fourier–Laplace transformed. Details of the inverse 2D Fourier–Laplace (actually, Hankel-Laplace) transfor˜ mation of J(s/c, k; z, z0 ) can be found in Ref. [99]. The most interesting aspect is that the straightforward expansion of the solution into a series (invertible term-byterm) that converges slowly at large times and distances from the source – precisely the regions we are most interested in. Application of Poisson’s sum-rule resolves this issue and delivers a series with reasonably fast convergence in the regime of interest. The end-result is: 2 1 e−ρ /4Dt 2cσt J(t, ρ, z; z0 ) = × τt + 2χ π 4Dt ∞ σ t z0 + χ σt z + χ × sin πn sin πn τt + 2χ τt + 2χ n=1 2 πn σt2 Dt . (6.8) exp − τt + 2χ A closely related quantity of interest is net ﬂux in the vertical, from Fick’s law: 1 ∂J D ∂J =− , c ∂z 3σt ∂z 2 e−ρ /4Dt 2πcσt 1 × =− 3(τt + 2χ)2 π 4Dt ∞ σ t z0 + χ σt z + χ × sin πn n cos πn τt + 2χ τt + 2χ n=1 2 πn σt2 Dt . exp − τt + 2χ

Fz (t, ρ, z; z0 ) = −

(6.9)

In particular, the two above expressions can be combined according to (4.27) to obtain the hemispherical ﬂuxes F± that are directly measured with standard (2π FOV) radiometers. 23 In the case of real absorption in steady-state asymptotic 1D RT, this key nondimensional ratio is known as the ‘similarity’ parameter [104].

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At this point, a well-known physical defect of diﬀusive transport theory becomes apparent: one obtains J(t, ρ, z; z0 ) > 0 for all t > 0, and not just t > ρ2 + (z − z0 )2 /c. For instance, although it is in very small amounts, light reaches the opposite cloud boundary without delay, thus violating basic causality and in sharp contrast with the MC simulation outcome in Fig. 5.3(b). This artifact can be traced back to the choice of neglecting in (4.2) the ∂t F term. Not dropping that term leaves us with the so-called telegrapher’s equation (system). In some applications, it is desirable to enforce causality in signal modeling [105–107]. In our present application, however, we prefer to commit to not use the diﬀusion results in regions of space-time where the theory has known defects, over and beyond the present issue with inﬁnite propagation velocity. Avoidance of the problematic short times/paths and distances from the source is especially easy when one has an explicit representation in space and/or time. We discuss another advantage at the close of section 5.9. In principle, the response to the correct distribution of internal sources in can now be reconstructed by linear superposition (integrating over z0 ).24 However, to preserve the relative simplicity of the above expression, we propose to select a single representative value of z0 rather than integrate over all the sources, i.e., exponentially weight and integrate25 over z0 between 0 and H (even if we can safely assume here that the upper limit is ∞). A physical argument for positioning such an ‘eﬀective’ isotropic point-source goes as follows. If we were to collapse the axial exponentially decaying distribution of anisotropic point-sources in (5.15) or (5.16) into a single isotropic point, we would likely place it at z0 ∼ t , hence σt z0 ∼ O(1), recalling that χ ∼ O(1) as well.26 Indeed, the basic idea of the transport MFP is to prolong ballistic propagation just far enough for the memory of the original direction to be erased by multiple forward-biased scatterings [108]. In essence, t is the medium’s eﬀective MPF for an isotropic scattering [80] and, for the same physical reasons, deﬁnes the depth of the radiative boundary-layer in the cloudy medium. 5.6.2.1 Transmission properties For transmission, we evaluate ˜ k; H, z0 ) cosh[m(σt z0 + 2χ)] − cosh[mσt z0 ] J(s, = , T˜(s, k) = 2 2χm × sinh[m(τt + 2χ)]

(6.10)

and we note that the combined numerator and the 2χm factor in the denominator are well-approximated by cosh [m(σt z0 + χ)] = sinh[m(σt z0 + χ)] if 2χm 1. In 24 There is, however, no q˜F z = g q˜J term here, hence no account of ﬁrst-scattering source anisotropy. 25 This superposition of point-source solutions would change the second sine term in (6.8)–(6.9) into a combination of two trigonometric functions, four if the upper bound is set to H < ∞. 26 A closer look at the spatial Green function plotted in Fig. 5.1 reinforces this choice. The assumed cloud has τt = H/t = 5.4. We can see overall light levels decreasing radially with distance from a point at ∼1–2 transport MFPs t = 1/σt ≈ 0.2 km below the upper (illuminated) boundary.

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this case, the desired asymptotic (τt χ) behavior when m → 0 is Tts (τt ) ≈

2χ , τt + 2χ

(6.11)

from basic two-stream theory (5.5). It is obtained when we set σt z0 = χ. It can be shown that this choice also gives the correct leading terms for ρ2 T and ctT because the leading term in m2 has the ‘correct’ pre-factor for both k 2 and s/c, using as a benchmark the solutions of (5.1)–(5.2). By enforcing correct small (s, k) behavior, we are sure to obtain the correct behavior for large (t, ρ) after inverse transformation. Rather than using (6.10) in (s, k)-space, we can thus use the somewhat simpler model27 sinh[m(2χ)] . (6.12) T˜(s, k) ≈ sinh[m(τt + 2χ)] Figure 5.10 demonstrates how close this model is to (6.10), with σt z0 = χ, and how well these internal point-source models approximate the boundary-source model in (5.8) where it matters, i.e., as k → 0.

Fig. 5.10. Three models for T˜(0, k). We compare in semi-log axes the ‘exact’ diﬀusion model in (5.8) for an isotropic boundary source expressed in Robin BCs, its counterpart in (6.10) based a judiciously positioned internal point-source (σt z0 = χ) with extended Dirichlet BCs, and the approximation in (6.12). The ﬁrst of these MTFs is also plotted in Fig. 5.8 (top panel), but here we use τt = (1 − g)τ = 8.1 and χ = 2/3. Notice the osculating curvatures at k = 0, meaning that the predicted value for ρ2 T will be the same; cf. (3.5).

27 Viewed, like in Fig. 5.10, as a model for the MTF, T˜(0, k), this expression is adapted from Bushmakova et al.’s 1972 article [46], generalized by the same authors in 1974 to space-time Green functions. However, those studies use σt z0 = (5/4)χ in our notations to achieve slightly better accuracy for a collimated beam at normal incidence. See also Katsev and Zege’s 1986 paper [109] and the monograph by Zege et al. [83, Section 4.3.3].

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Adapting the Fourier–Laplace inversion procedure in [106], we ﬁnd 2 2 3τt e−3τt ρ /4Hct 1 2π Rts × T (ct, ρ) ≈ 3H τt π 4Hct ∞ 2 1 ct × . n sin(πnRts ) exp − (πnRts ) 3τt H n=1

(6.13)

where Rts (τt ) is given by 1−Tts (τt ) from (6.11), equivalently, the classic two-stream model outcome in (5.5). We have somewhat rewritten here the Green function to emphasize the two main cloud remote-sensing unknowns, H and τt = (1 − g)τ . As expected, we see in particular that the outcome of (6.13) does not change, for a given value of H, within a class of similar clouds, i.e., where (1 − g)τ is constant. We recognize in (6.13) a Gaussian spatial proﬁle of ﬂux with an increasing value of the variance ρ2 (t) of the lateral transport at each instant t (or path ct); speciﬁcally, 4c ρ2 (t) = t. (6.14) 3σt This is as expected for ρ2 = x2 + y 2 recalling that, for boundary-free isotropic diﬀusion away from a point-source, we have r2 = x2 + y 2 + y 2 = 6Dt with D = c/3σt [103]. Detailed time-only dependence is obtained from (6.13) by multiplying both sides by 2πρdρ and integrating from 0 to ∞: ∞ 2 π Rts 2 1 ct . × n sin(πnRts ) exp − (πnRts ) T (ct) ≈ 2H 3τt n=1 3τt H

(6.15)

This result can be applied to the monitoring of cloud-to-ground lightning [97] and other rapidly varying time-dependent sources, through dense clouds from space. We see in (6.15) a linear superposition of exponential decays in time, the dominant (slowest, n = 1) rate being [101] 3 τt ct = H × , (6.16) × π2 Rts (τt )2 where we continue to emphasize the invariance within similarity classes (constant τt ). Integrating (6.13) over path ct from 0 to ∞ yields the spatial response to a steady source as observed with an imaging detector:28 T (ρ) ≈

Rts H

2 ×

& ρ' . n sin(πnRts ) K0 πnRts H n=1 ∞

(6.17)

where K0 (·) is the zeroth-order modiﬁed Bessel function of the second kind. This would apply to the monitoring cloudy regions from space for steady localized sources using imaging sensors; missiles during their boost phase are possible targets. 28

Use identity

R∞ 0

√ exp[−(at + b/t)] dt/t = 2K0 (2 ab).

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Although deﬁned above as a sum of Gaussians, the dominant (n = 1) term decays radially as an exponential approached from above with a relatively slow power law [110]: exp(−ρ/ρ ) T (ρ) ∼ , (6.18) √ ρ where the e-folding radius is ρ = H/πRts .

(6.19)

As expected from k ↔ s/cD similarity (D = c/3σt ), examination of (6.16) shows that we have (ρ )2 = (c/3σt ) t . This connection serves as a reminder that the key √ space-time variability parameter m deﬁned in (6.7) is ∝ k when s = 0, but ∝ s when k = 0. Consequently, for this class of models cunningly designed to delivery explicit expressions in space end/or time, moment predictions will be accurate for ρ2 and ct, but not (ct)2 . 2

5.6.2.2 Reﬂection properties With reﬂection in mind, we propose to set σt z0 ≈ 1, this time irrespective of χ (although we also have σt z0 ≈ (3/2)χ since χ ≈ 2/3). This leads to ˜ ˜ k) = J(s, k; 0, (3/2)χ/σt ) = cosh[m(τt + χ/2)] − cosh[m(τt − (3/2)χ)] . R(s, 2 2χm × sinh[m(τt + 2χ)] (6.20) In the asymptotic regime where τt σt z0 ∼ χ ∼ O(1), one notes that cosh[m(τt + χ/2)]− cosh[m(τt −(3/2)χ)] will be very close to 2χm sinh[m(τt −χ/2)] if 2χm 1. The boundary ﬂux at z = 0 can therefore be written even more simply as [106] ˜ k) ≈ sinh[m(τt − χ/2)] . R(s, sinh[m(τt + 2χ)]

(6.21)

The above expression still gives us in the limit m → 0 the correct asymptotic (τt χ) form of cloud albedo in (5.19) for the collimated-beam model when the short-lived exponential terms are omitted: R≈

τt /2χ − 1/4 . 1 + τt /2χ

(6.22)

Here also, we get the correct leading terms for ρ2 R and ctR since the leading term in m2 has the correct pre-factor for both k 2 and s/c. Figure 5.11 shows how this coarsest of all representations of the pulsed laser source (in terms of directionality and spatial distribution) compares with our best model for the pencil-beam problem in reﬂection. Although there are interesting deviations at large wavenumbers,29 there is no apparent diﬀerence at small k (nor 29

˜ k) We note that the exact diﬀusion model in Fig. 5.11 leads to negative values of R(0, ˜ if g > 0; the behavior of this model for k H, not plotted here, is R(0, k) ∼ −3g/2k (following a single negative minimum). This is unphysical since characteristic functions (Fourier transforms of PDFs) are everywhere non-negative if the PDF is axisymmetric. In this sense, the proposed approximation is more useful than the exact theory in applications ˜ k) must make physical sense over the full range of k. In our case, its utility is where R(0, to yield tractable inverse Fourier–Laplace transforms.

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˜ k). We plot as functions of k (in units of 1/H) results Fig. 5.11. Two models for R(0, for both ‘exact’ diﬀusion theory that models a collimated incident beam that excites anisotropic sources (details in Appendix C) and a more practical approximation in (6.21) that uses a single judiciously placed isotropic point-source. Two cases with (1 − g)τ = (1 − g )τ = 8.1 are examined. When working with moments and/or long-path and/or far-ﬁeld properties, only the behavior near k = 0 matters. In this case, we have identical behavior up to O(k2 ), having set χ = 2/3 in the approximate model.

would there be for small s), where it matters for spatial (temporal) moment estimation as well as long-path and far-ﬁeld trend analyses. The end-result of the inverse Fourier–Laplace transform of (6.21) is [106] 2 2 3τt e−3τt ρ /4Hct 1 2π Rts × R(ct, ρ) ≈ 3H τt π 4Hct ∞ 5 2 1 ct × πnTts exp − (πnRts ) , (6.23) n sin 4 3τt H n=1 where Tts = 1 − Rts is from (6.11). This result can be applied directly to imaging cloud lidar systems with a wide ﬁeld-of-view, but maybe not wide enough for robust moment estimation [84]. Integration over space yields ∞ 2 5 π Rts 2 1 ct πnTts exp − (πnRts ) . (6.24) × n sin R(ct) ≈ 2H 3τt n=1 4 3τt H This result can be applied directly to non-imaging (for example, space-based) lidar systems with wide-enough footprints [111]. Figure 5.12 shows how well the model in (6.24) performs with respect to the usual ‘gold standard’ of Monte Carlo simulation. In this case, no eﬀort was made to mimic the actual source used in the diﬀusion model, namely, an isotropic pointsource at about one transport MFP from the observed cloud boundary; as in real cloud lidar, a collimated source was used. So deviations at early times are expected and observed, but diﬀusion-based models are not to be used in this regime anyway.

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Fig. 5.12. Diﬀusion theoretical and MC estimates of the temporal Green function for reﬂected ﬂuxes. We have plotted the closed-form expression in (6.24) for the diﬀusionbased model and two numerical solutions of the basic radiative transfer problem for slab geometry in (2.1)–(2.7). The inset shows the noise level for the adopted MC scheme with 2 · 108 histories. Cloud thickness is H = 1 km. The diﬀusion prediction is the same for all similar clouds, i.e., with the same scaled optical depth (1 − g)τ = 8.1, while slightly diﬀerent answers are obtained numerically with the RT model (only at early times). All results are normalized to their respective predictions for steady-state albedo, which are quoted to the accuracy of the MC simulations; only ≈2% of the signal lies beyond the cutoﬀ at ct = 10H. We see that the diﬀusion model is an accurate representation of the radiation transport when ct H for the similar cases of H–G scattering with g = 0.85 (τ = 54) and g = 0.46 (τ = 15).

At long times, the agreement is excellent. The vertical double-headed arrow indicates the range (path ct/2) where the single scattering contribution to the signal vanishes. All the signal beyond this point is from multiple scattering only but none of it is in the standard model for backscatter lidar signals. It would be interpreted in classic lidar as a spurious distribution of scattering particles on the opposite side of the cloud. Integrating (6.23) over path ct yields: R(ρ) ≈

Rts H

2 ×

∞ n=1

n sin

& 5 ρ' πnTts K0 πnRts . 4 H

(6.25)

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This spatial response would be observable with an imaging detector for a steady narrow-beam source such as a cw laser. An interesting laboratory application is described in Refs [112, 113] targeting oxygen levels in blood. As noted already for Fig. 5.3, there is no diﬀerence in the far-ﬁeld behavior between reﬂected and transmitted light in either space or time. So the relations between observables and cloud parameters in (6.14), (6.16) and (6.19) apply to both sides of the cloud [114], and has proven useful in remote sensing applications mentioned previously, but that we now discuss in detail.

5.7 Temporal Green functions applied to in situ cloud lidar 5.7.1 Forward model for the radiometric signal The bulk of this review is about ﬂuxes emerging from cloud boundaries since they are the only ones accessible by remote sensing methods. This section is the exception. We discuss here an interesting new application of forward models for internal radiative properties: in situ cloud lidar.30 In their feasibility study for in situ (airborne) cloud lidar, Evans et al. [21] studied the special case z = z0 = H/2 in (6.8) as a ﬁrst-cut model for a proposed timeresolving wide-FOV radiometry. Notably, they solved the Green function problem directly in the space-time domain using the method of images to satisfy extended Dirichlet BCs (with χ = 2/3). In this highly symmetric situation, the two sine terms reduce to sin2 (nπ/2) = 1 if n is odd, 0 if even. Thus 2 e−ρ /4Dt 2cσt 1 × J(t, ρ, H/2; H/2) = τt + 2χ π 4Dt 2 ∞ π(2j + 1) σt ct × . (7.1) exp − τ + 4/3 3 t j=0 where we recall that diﬀusivity D = c/3σt = cτt /3H. The authors also argue that, for short times, one can use the classic diﬀusion Green function for an isotropic pulse at the origin in the absence of boundaries: 2 ρ + z2 c , (7.2) exp − J(t, ρ, z) = 4Dt (4πDt)3/2 where z is now reckoned from the position of the isotropic point-source. For simplicity, Evans et al. assume a monostatic conﬁguration where source and detector are essentially collocated, hence ρ = 0 in (7.1) and ρ = z = 0 in (7.2). From a cloud characterization perspective, early times can be used to determine a volumeaveraged estimate of opacity σ = σt /(1 − g) from D, knowing g ≈ 0.85, and later times to determine H = τt /σt . 30

Lidar stands for LIght raDAR, or LIght Detection And Ranging, but there is obviously no cloud ranging to be done in this case. The present time-domain signal is entirely about multiple scattering while the conventional ranging application in Lidar assumes necessarily a single scattering or reﬂection.

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This model for in situ lidar signals can easily be improved by setting the detector at a horizontal distance ρ ≈ t = 1/σt from the eﬀective isotropic point-source of light pulses. Instead, Evans et al. moved to qualitatively better forward modeling by using a 3+1D MC code in conjunction with many realizations of a realistic (datadriven) 3D stochastic cloud model [115]. They used this extensive forward modeling to train a neural network to solve the inverse problem at hand: determine from the temporal signal the cloud unknowns {σt , H}. Notably, in situ lidar retrieval methods must also consider the distance to cloud base (z in our present notation) as an unknown, and its determination calls for estimates of both down- and upwelling ﬂuxes that can be modeled (for arbitrary z) using (6.8) and (6.9) combined according to (4.27). The more recent paper by the same authors [116] concludes the proof-of-concept with the successful exploitation of real cloud data gathered from a LearJet operated by the Stratton Park Engineering Company (SPEC), Inc., in Boulder (Colorado). 5.7.2 Illustration with SNR estimation Figure 5.13 shows F± (ct) = J(ct, t , · · · )/4, based on (4.27) and (7.1) in this special situation were Fz ≡ 0 in (6.9). We upgraded the model only for the radial oﬀset of the eﬀective isotropic source (to the side of the aircraft the laser is pointing away from) and the time-resolving radiometer (looking up or down somewhere on the aircraft). We factored into the result a hypothetical path-bin width of Δct = 20 m, which is easily achievable with laser and detector technology currently used in airborne lidar. The two panels illustrate the sensitivity of this observable as a function of ct for (a) varying H (equivalently, σ) at ﬁxed τ and (b) viceversa. We have high sensitivity to σ at all times. Sensitivity to H increases at later times, which is explained immediately by the dependencies of ct in (6.16). Increasing optical depth (equivalently, extinction) for ﬁxed cloud thickness of course increases the general level of the signal. However, we may not want to depend on absolute calibration of the radiometer as well as accurate monitoring of the laser power; so we should rely only on the shapes of the radiative responses in Fig. 5.13, and not their relative positions on the ﬂux scale. The positions of the plots in Fig. 5.13 along the vertical (ﬂux) axis can, however, be used for a rough but informative SNR computation for the airborne in situ cloud lidar. Lasers used in airborne lidar studies can easily reach 5 W of equivalent cw radiant power, translating to ≈1018 photons/pulse at 532 nm (assuming here a frequency-doubled Nd:YAG solid-state laser with a typical 10 Hz rep rate). The lowest major tick of both plots corresponds to a 10−15 probability of detection per laser photon by a (non-imaging) sensor with a reasonable 1 cm2 aperture. At the 100s of m/s air speed of a jet aircraft, one should not integrate temporally for more than a few pulses, or else cloud structure will not be captured at the natural resolution of multiple-scattering cloud lidar, as determined by ρ in (6.19). This still leaves Nγ ≈ 103 multiply-scattered laser photons available for detection. Ignoring background (solar/lunar) and electronic noises, this Poisson count rate yields an estimated SNR of Nγ ≈ 30: enough leeway for losses by optical throughput, quantum eﬃciency, an aperture less than 1 cm2 and/or a FOV reduced from 2π sr. We therefore anticipate all the signals predicted in Fig. 5.13 to be measurable with

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Fig. 5.13. Sensitivities of in situ lidar signal to changes in cloud thickness H and optical depth τ . As a surrogate for ﬂux captured by a large (but still practical) FOV, we use here hemispherical ﬂuxes estimated at mid-cloud. Both H and τ are varied over one order-ofmagnitude around a typical value: H = 1 km in upper panel (a) and τ = 31.6 in lower panel (b). Other parameters used to estimate F± = J/4 from (7.1) are χ = 2/3 and g = 0.85 to compute τt , hence σt and D; we also assume a ﬁnite path bin Δct of 20 m. The presence of a maximum signal at ﬁnite time reﬂects the oﬀset of the detector from the eﬀective origin of the pulse, which is represented here by a eﬀective isotropic point-source. The late-time exponential decay and even the details of the approach to this asymptotic behavior are known to be accurate in the present diﬀusion model for dense enough clouds, cf. Fig. 5.12.

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current lidar technology. Evans et al. [116] did just that (with a detector that had an eﬀective aperture of ≈20 cm2 and a quantum eﬃciency estimated to be ≈0.1.) and they successfully inferred cloud thickness and volume-averaged extinction. Validation was conveniently done by executing a ‘porpoising’ ﬂight pattern where the aircraft goes in and out of cloud base and cloud top. With variable vertical positions (z, z0 ) restored, as in (6.8), plus the possibility of ρ > 0 already exercised in Fig. 5.13, this model can be applied to a conceptually simple cloud observation system based on miniaturized (for example, eﬃcient ﬁberlaser) sources and detectors ﬂying inside the cloud, but now on two or more separate unmanned aerial vehicles (UAVs). This would enable suﬃcient sampling of the cloud’s Green function, in both t and ρ, to retrieve the standard set of in situ lidar → unknowns {z, (1−g)σ, H}, given (− ρ , z0 −z) from accurate relative GPS positioning; cloud base altitude would of course follow from absolute GPS estimation of the altitude of the various aircraft. We anticipate that Fz = F+ − F− , obtained by diﬀerencing signals from up- and down-looking radiometers, will be sensitive to z (assuming z0 ≈ z); Fz ≈ 0 would mean near the center of the cloud (z ≈ H/2), and proximity to the z = 0 and z = H boundaries signaled by diﬀerent signs. Last but not least, these slow-moving platforms would enable longer time-integrations, hence improved SNRs; we indeed expect integration will compensate for the diminished transmitted and received power due to size/weight limitations. Another possibility is to outﬁt UAVs with stand-alone high-resolution spectrometers dedicated to the A-band of O2 (dispersive devices can be made very compact). We now show that this is indeed a passive (no-laser-required) approach to timedomain information. Imagine a uniform pulsed source incident at cloud top with some zenith angle cos−1 μ0 ≥ 0. The basic theory for the absorption spectrum (i.e., uniform gas with variable absorption) is laid out in section 5.5.5 and the associated Appendix D. However, we now broaden our interest from internal ﬂuxes to those at the cloud boundaries, speciﬁcally about how they depend on cloud properties.

5.8 Temporal Green functions applied to oxygen A-band spectroscopy of overcast skies 5.8.1 A-band spectroscopy as observational time-domain RT Oxygen A-band (759–771 nm) spectroscopy has attracted interest in atmospheric remote sensing for at least four and a half decades. The earliest discussions [117–119] were already about using it systematically for cloud observations from space, and the earliest A-band cloud data was indeed collected with a hand-held camera aboard Gemini 5 [120,121]. Multiple scattering eﬀects were considered from very early on, especially in the Former Soviet Union [34, 35], ﬁrst in modeling and eventually in data analysis. Both airborne instruments and spectral models progressed steadily in the United States [122, 123], Europe [124, 125] Japan [126, 127], and FSU [128–147]. The oxygen B-band (c. 687 nm) was also investigated, for example, in Ref. [123]. We are dealing here with a typical diﬀerential absorption spectroscopy that tells us how much O2 was cumulated along the optical path of the light dispersed across

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wavelengths. The lower panel of Fig. 5.14 illustrates the ﬁne spectral structure of the A-band with its absorption optical depth across an assumed low-level cloud layer. The upper panel shows examples of simulated spectra for typical clear and cloudy conditions over a dark ocean surface as observed in reﬂection, normal viewing with a 60◦ solar zenith angle (SZA); the inset gives the values of the corresponding nadir radiance in the continuum.31 What can one do with knowledge of exactly how much of a major gaseous constituent is in the optical path? Some early proposals were to determine atmospheric pressure in remote locales, especially over oceans where there is a dearth of surface stations and radiosonde data, the driving application being numerical weather prediction.32 It quickly became clear that atmospheric scattering was a wild card in this proposition. The optical path is not always straight down from the sun through an oblique airmass followed by another airmass (say) straight up to a spectral detector. There are both longer and shorter paths because of atmospheric scattering. Details of the line proﬁles thus depend on the amounts of aerosol, cloud, etc. It was soon realized that this nuisance could be turned around and exploited to study the scattering components of the atmosphere. One can possibly quantify and qualify the aerosol load, from ground or from space. Concerning clouds, Aband spectroscopy was ﬁrst thought to be a means of deriving cloud-top pressure using the same logic as for the original idea of measuring surface pressure. This concept also proved problematic because clouds are not optically ‘hard’ targets. In fact, they are optically speaking very ‘soft’ in that incident light can permeate the whole cloud before being reﬂected, i.e., returned from possibly very deep inside the cloud back to the illuminated boundary. Single- and multiple-scattering are of course the mechanism for reﬂection, as well as for diﬀuse transmission. Current modeling of reﬂected A-band signals fully and accurately accounts for multiple scattering [150–155], and more and more includes 3D RT eﬀects [156–158]. As will soon become obvious, the standard retrievals from A-band data are cloud-top pressure and cloud (pressure) thickness. In this study, we take the standpoint that O2 A-band spectroscopy at high resolution is a portal to time-domain radiative transfer in clouds, but using steady sources. This powerful connection is encapsulated in the so-called equivalence theorem already discussed in diﬀerential form in section 5.3.1. This relation states that, if we know the time-dependent radiance ﬁeld G(t, x, Ω) resulting from a sudden burst of light in an arbitrary scattering medium, then we can predict the radiance that would be observed at the same point in the presence of a uniform gas with a variable absorption coeﬃcient κν > 0.

31 We note that the A-band sits at the peak of the solar spectrum when reckoned in photons/s/m2 (as opposed to W/m2 ), which is optimal for the SNR. 32 The competition for cloud-top height retrieval from remote sensing data at large comes (1) from signatures in thermal bands (sometimes called ‘CO2 slicing’) [148], but the atmosphere’s thermal structure varies far more than its pressure proﬁle, and (2) from direct geometric method using multi-angle imaging [149], which is foolproof as long as the clouds have enough ‘texture’ for automated stereo matching. Because of the time-delay between the observations of a given cloud at diﬀerent angles (as the satellite moves along its orbit), the latter method furthermore provides mean wind at the cloud-top altitude.

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Fig. 5.14. Simulated O2 A-band spectra in reﬂection. Bottom: Fine structure of the Aband displayed using, as a relevant example, the O2 optical thickness across a layer from 860 to 911 hPa (c. altitudes 0.85 and 1.3 km), where one could ﬁnd a typical low-level cloud. Top: The reference spectrum is for a background aerosol atmosphere above an ocean surface (Cox-Munk BRDF for 5 m/s wind speed). The other is for the same situation plus a liquid water (Mie scattering) cloud between 911 and 860 hPa with optical depth 64. Lineby-line computations were coarsened to the 0.0146 nm resolution of the Orbiting Carbon Observatory (OCO) spectrometer. Both spectra were normalized to maximum radiance (given, for reference, in the inset). Computations were kindly provided by Dr Hartmut B¨ osch (University of Leicester, Dept. of Physics & Astronomy, Earth Observation Science, Space Research Centre).

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Formally, we let L denote the a priori random path that the light has followed in the medium. Then the optical path for gaseous absorption is κν L and Beer’s law of exponential extinction states that light with path L is absorbed with probability exp(−κν L). In summary, for a ﬁxed κν , radiance is33 1 I(κν , x, Ω) = c

∞

G(L/c, x, Ω) e−κν L dL.

(8.1)

0

The diﬀerent absorption values are realized at diﬀerent wavelengths λ, equivalently, frequency c/λ or (more conventionally) wavenumber ν = 107 /λ in cm−1 when wavelength is expressed in nm. We have thus assumed up front that the coeﬃcient κν is a function of wavenumber ν, in compliance with spectroscopic tradition. At any rate, we recognize in (8.1) the Laplace side of (3.1) with radiance rather than ﬂux and identifying κν with s/c. So, as in (3.2), we can compute path moments using the spectrometry: ) q ) ∂ 1 − I(κν , · · · ))) , (8.2) Lq = I(0, · · · ) ∂κν κν =0 where the normalizing factor is simply radiance in the continuum near the A-band. Conveniently, all of our diﬀusive temporal Green function theory has ended up in Laplace space, or at least used it as a stepping stone. We can therefore obtain, after the usual radiance-to-ﬂux conversion, empirical estimates of Fˆ (s) = F˜ (s, 0). Identiﬁcation of observed and theoretical estimates of radiative properties is of course an opportunity for cloud remote sensing: retrieval of cloud parameters, for example, {H, τ }, by standard ﬁtting procedures. Once we have a theoretical ﬁt, one can derive other cloud radiative properties such as the temporal/path-length moments in (8.2). Another possibility is to use the observed Fˆ (s) = I(κν , · · · )/I(0, · · · )|κν =s/c in (8.2) to derive the path-length moments directly. In practice, both groups currently engaged in high-resolution A-band spectroscopy from ground (SUNY-Albany [160–163] and University of Heidelberg [159, 164, 165]) use a convenient parameterization of Fˆ (s) using the ﬁrst two moments of L, namely,34 Fˆ (s) ≈

1 1 . , where a = 2 a (1 + Ls/ca) L /L2 − 1

Both of these approaches avoid the problem of inverse Laplace transforming noisy data from instruments, a numerical process known to be highly unstable. A third group, from NOAA, has performed ground-based spectroscopy of oxygen under cloudy skies at somewhat lower resolution and focusing on the γ-band of 33

We assume in (8.1), purely monochromatic RT. No forward model for A-band spectroradiometry is complete without a convolution of this expression with the spectrometer’s ‘slit function’. This issue of spectral resolution, as well as out-of-band rejection, is absolutely critical to the a priori estimation of how many pieces of path-length/cloud information can be extracted robustly from the radiometric data [159, 160]. 34 This leads back to a Gamma distribution of path-lengths, cf. (10.1)–(10.2) with τ replaced by L. We owe to van de Hulst [166] the original idea of approximating the temporal Green functions with Gamma distributions.

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O2 (c. 628 nm) [167]. Following previous work on low-resolution A-band observations from space, they preferred to directly compare their spectroscopy to a forward 1D RT model (adapted from [168]) with an assumed cloud, which they varied to ﬁt the data. Another interesting development from the NOAA group was to adapt their instrumentation and RT model to work with the spectrally much smoother absorption features of cloud droplets in the near-IR (0.9 to 1.7 μm). Because absorption cross-section (where it exists) is ∝ droplet volume, Daniel et al. [169] were able to retrieve liquid water path (LWP). It is important to bear in mind that a forward model for A-band spectroscopy will have more than just the component for in-cloud multiple scattering we have just described. However, the other contributions are essentially deterministic. Consider a narrow FOV ground-based instrument measuring zenith radiance. The spectrum is formed, on the one hand, by the cloud’s temporal multiple-scattering Green function by identifying s with cκν (a pressure-weighted average over the cloud layer) in the Laplace transform for transmitted light; on the other hand, this contribution must be convolved with two degenerate (zero-variance) distributions of optical path (a) (b) through the absorbing gas, namely, τν /μ0 and τν , the absorption optical depths of the above- and below-cloud layers respectively. Now, in Laplace/κν -space, the convolutions translate to simple products and the two δ-like path distributions (a) (a) (b) (b) transform to constants, Tν = exp(−τν /μ0 ) and Tν = exp(−τν ). In summary, the forward model for the spectroscopy is (a)

I(κν ) = I(0) × Tˆ(cκν ) × e−(τν

/μ0 +τν(b) )

.

For light reﬂected back to an instrument on an aircraft or satellite with oblique viewing capability, the forward model is ˆ ν ) × e−(1/μ0 +1/μ)τν(a) . I(κν ) = I(0) × R(cκ Substitution of these models into the expression (8.2) for path moments will of course yield the moment based on Fˆ (s), for F = T or R, but oﬀset by a known constant. Alternatively, one can correct the data for the spectral oﬀsets before computing the path moments or retrieving cloud properties directly.35 In the remainder of this section, we will exploit a subset of our explicit Laplacespace solutions (from section 5.5) for the temporal Green function to assess the cloud information content of O2 A-band data. Speciﬁcally, we will formally identify the random light path L through the absorbing gas with ct, the path covered between emission from a δ-in-time boundary source and escape in transmission (F = T ) or in reﬂection (F = R). We will then determine the dependence of path moments (ct)q F (F = T, R, q = 1, 2) on cloud properties, which we have not yet done in framework of diﬀusion, and we will continue to validate the diﬀusion model with MC simulations. These are in essence the observables in O2 A-band spectroscopy. Spatial moments ρ2 F also play a role. Insights from this kind of a priori cloud information content analysis are useful in planning for remote sensing mission objectives as well as in the development of retrieval algorithms. 35

The contributions of multiple reﬂections between the cloud and the surface can also be added, recalling that green vegetation can have a relatively high albedo at A-band wavelengths; see Scholl et al. [165].

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5.8.2 Path-length moments from below 5.8.2.1 Isotropic boundary point-source Using the same assumptions as in our 2002 paper [24] investigating light transmitted by the cloud to a ground-based sensor, we start with the simple diﬀusion model based on an isotropic boundary point-source. Applying (3.5) and (3.2) to (5.8) with all ancillary deﬁnitions, we ﬁnd 1 (1) τt × 1 + CT,ct (τt /2χ) , 2 7 2 (2) τt × 1 + CT,ct (τt /2χ) , (ct)2 T /H 2 = 20 2 (2) ρ2 T /H 2 = × 1 + CT,ρ (τt /2χ) , 3 ctT /H =

(8.3) (8.4) (8.5)

for three low-order moments of interest, all normalized by H. We have highlighted here the asymptotic trends. The pre-asymptotic correction terms are given by (1)

(2)

CT,ct (ξ) = CT,ρ (ξ) = (4ξ + 3)/2ξ(ξ + 1), (2)

CT,ct (ξ) = (56ξ 3 + (166ξ 2 + 15(10ξ + 3)))/14ξ 2 (ξ + 1)2 , recalling that ξ = τt /2χ. The leading terms made explicit in (8.3) and (8.5) make sense from the random-walk perspective on diﬀusion exposed in Appendix E wherein the scaling exponents of τt are derived. However, in all cases, pre-factors and pre-asymtotic corrections terms in (8.3)–(8.5) call for the PDE-based approach promoted throughout this review. It is worthwhile to compare these results with those provided by Zege et al. [83]. In their Section 4.8 on pulsed sources, their Eq. (4.8.38) gives, in our notations, the same asymptotic trends for τt 1 and 1 for mean optical path σctT as in (8.3): respectively, we ﬁnd (1 − g)τ 2 /2 (irrespective of χ) and (3χ/2)τ (= τ when χ = 2/3).36 Zege et al.’s same equation gives the trend of the st.dev. σ (ct)2 T − ct2T of the optical path when τt 1; in our notations, it yields a pre-factor of 11/20 rather than 7/20 in (8.4). We interpret this discrepancy as the result of requiring only second-order accuracy in m(s, k) for (6.21), on which their temporal moment computations are predicated; to√obtain the correct trend for (σct)2 T , one needs fourth-order accuracy in m ∼ s when k = 0. Finally, Zege et al.’s counterpart of (8.5) is in their Table 4.1 and follows from Ref. [46]. That paper by Bushmakova et al. generalizes to anisotropic scattering the studies by Romanova in Refs [44, 45] that focused on the case of g = 0. In our notations, the early prediction is (σρ)2 T ≈ τ /3 when τt 1, irrespective of g and χ. We ﬁnd twice that value. It is not clear where this systematic diﬀerence comes from 36 The relation ctT ≈ H = τ /σ for vanishing σ (hence τ ) makes intuitive sense: most of the light goes straight through the medium in direct transmission. However, a straightforward ﬂux -based estimate of mean path for light transmitted through an optical vacuum of thickness H under isotropic illumination is 2H (and, interestingly, the RMS path is divergent). This serves as a reminder (cf. section 5.4.3) that the eﬀective value of χ for optically thin media (τt 1) is 4/3.

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229

but we will see further on that it occurs again with (σρ)2 R , for reﬂected light, and that our MC simulations agree with our diﬀusion estimates for both transmission (Fig. 5.15) and reﬂection (Figs 5.18 and 5.21). We suspect that the factor of 2 in (3.5), which is nontrivial, may have been omitted. Figure 5.15 shows the low-order moment predictions in (8.3)–(8.5), making RMS values out of second-order moments; this suite is augmented with the zeroth-order transmission term T in (5.5) for the classic two-stream model corresponding to the s/c, k → 0 limit of this case. MC simulations of the same transport problem, including the isotropic boundary point-source, using H–G and isotropic phase functions show excellent agreement with the diﬀusion model, as long as the proper χ-value is used. We note however that χ is not present in the leading terms of the moments in (8.3)–(8.5).

Fig. 5.15. Cloud responses to a pulsed isotropic point-source in transmission. Diﬀusion predictions from (8.3)–(8.5), with correction terms, and (5.5) are in solid lines; MC validation data are plotted with symbols (two values of χ used to reproduce the MC benchmarks). The circled points identify parameters used for the space-only case studies in Fig. 5.9. We note the striking similarity between the two temporal moments. Adapted from Ref. [24].

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What do these moments tell us about O2 A-band spectroscopy from ground? The fact that we have (ct)2 T ≈ (7/5)ct2T over the whole range of τt , i.e., that one moment predicts the other one, is not good news for cloud property remote sensing with stand-alone O2 A-band sensors. If we can somehow merge the time-domain A-band data with a spatial methodology that yields ρ2 T (cf. section 5.10.6) or conventional radiometry with absolute calibration that yields T , then we are in a far better position. Indeed, the fact that we have diﬀerent asymptotic scaling exponents for all three observables (including a null exponent for ρ2 T ) makes it at least theoretically possible to determine τt , say, from T and then H from the measured value of, say, ctT and the non-dimensional ratio in (8.3). Even if there is no direct empirical estimate of ρ2 T because the sun is a uniform source, the utility of (8.5) is to inform other retrievals (A-band or radiometric) about their minimum eﬀective spatial resolution. It is indeed the fundamental radiative smoothing scale [170] for the transmitted light ﬁeld [24], below which spatial details are lost for the remote observer due to horizontal radiative transport caused by multiple scattering. In ground-based cloud remote sensing, spatial sampling is determined by temporal sampling Δt and the mean wind v at cloud level – Taylor’s classic ‘frozen turbulence’ hypothesis. For instance, if vΔt < ρ2 T ∼ H, then there is redundancy in the spatial sampling.37 This redundancy can be viewed as desirable or not, depending on circumstances. A case were spatial redundancy is counter-indicated is when compiling statistics that target spatial correlations. 5.8.2.2 Stratiﬁcation eﬀects We showed in section 5.5.2 that the above model could be generalized to clouds endowed with internal stratiﬁcation according to the linear (constant gradient) model in (2.27). Should we worry about this issue in ground-based O2 A-band spectroscopy? We are here in the case of −2 < Δ < 0 in the upper panel of Fig. 5.5 (gray curves only), which plots MC results for an isotropic point-source at z = 0 and distributed ﬂux detectors at z = H, In passing, we note that the converse (0 < Δ < +2) yields the same answer, by optical reciprocity. We see that, for the prescribed cloud (τ = 15, g = 0.46), the spatial statistic in (8.5) is essentially unaﬀected by stratiﬁcation, but there is a ∼10% eﬀect in the time-domain properties of primary interest here in (8.3)–(8.4). In many modeling applications, this level of accuracy is worth pursuing. Further computer-assisted algebra based on the solution presented in section 5.5.2 (and Appendix A), following the same guidance as above, leads to

37 This estimate of the smoothing scale is for ground-based narrow FOV radiometry. That is indeed how one can access radiance, hence ﬂux, at the lower cloud boundary. In ground-based radiometry with a wide FOV (for example, 2π sr, hemispherical ﬂux sensors), the same rule applies but ρ2 T must account for the extensive horizontal transport in the sub-cloud layer. Being about a fundamentally non-diﬀusive transport process, this problem is out of scope for the present review.

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Fig. 5.16. Eﬀects of stratiﬁcation on Green functions for transmission. Diﬀusion predictions for the prescribed cloud (τt = 8.1) are in solid lines; they were obtained from (8.6)–(8.8), including correction terms, with χ = 0.71. MC validation data are plotted with symbols; they are extracted from Fig. 5.5, gray curves in upper panel. We note the nearly ﬂat behavior of ρ2 T away from |Δ| = 0 and up to the onset of the logarithmic singularity at |Δ| = 2 in (8.8), which is discussed in the main text. √ This is because the prescribed cloud, with τt /2χ = 5.7, happens to be very close to (5+ 35)/2 = 5.46 · · · , the value that cancels the coeﬃcient of Δ2 in (8.9). For larger values of τt /2χ, the diﬀusion model for stratiﬁcation will improve with respect to MC results, and for smaller values it is expected to worsen.

1 Δ2 (1) − τt × 1 + CT,ct (ξ, Δ2 ) , (8.6) 2 40 ξ=τt /2χ Δ2 (30 − Δ2 ) 7 (2) − τt2 × 1 + CT,ct (ξ, Δ2 ) , (8.7) (ct)2 T /H 2 = 20 800 ξ=τt /2χ ⎛ ⎞ ( 2 2 2 2 2 1 Δ 1 − Δ/2 ⎠ ln + 1+ ρ2 T /H 2 = ⎝ −1 4 Δ 2 1 + Δ/2 Δ (2) × 1 + CT,ρ (ξ, Δ2 ) , (8.8)

ctT /H =

ξ=τt /2χ

which are all symmetric in Δ and lead back to (8.3)–(8.5) in the limit |Δ| → 0. Figure 5.16 shows how well the full expressions reproduce the MC results from the gray curves in the upper panel of Fig. 5.5. Overall, the diﬀusion results are just slightly oﬀset from their MC counterparts. The most remarkable diﬀerence between the diﬀusion and MC predictions in Fig. 5.16 is the logarithmic divergence of ρ2 T at |Δ| → 2 that is manifest in (8.8). All is as if the eﬀective diﬀusivity constant ρ2 T /tT , as observed at the cloud boundary (in this case, opposite the δ-source), becomes inﬁnite with the value of the transport MFP at z = 0 or z = H, for example, t (0) = 1/σ t (1 − Δ/2).

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Diverging t is clearly a challenge for diﬀusion models since trajectories become more ballistic within a transport MFP of cloud boundaries.38 However, this spatial statistic is only of marginal importance in O2 A-band observations. We therefore recommend using the uniform cloud (Δ = 0) estimate of ρ2 T when and where needed. If more accuracy is needed, one can expand (8.8) in a short series with respect to Δ: 2 ξ(ξ + 3) + 3/2 2ξ(ξ − 5) − 5 2 2 2 − Δ + O(Δ4 ), (8.9) ρ T /H = 3 ξ(ξ + 1) 60ξ(ξ + 1) recalling that ξ = τt /2χ. The full variation of ρ2 T in diﬀusive clouds is quite small anyway, cf. MC results in Fig. 5.15. Although there is less and less light to speak of, diﬀusion becomes quasi-exact deep inside the medium when τt 1. In this limit, ρ2 T is ≈2/3 when Δ = 0 and using (8.8), with correction term, one can show that this moment becomes ≈1/2 when |Δ| = 2 in the same limit. For a start, the two terms in (8.9) yield 8/15 when ξ → ∞. The range in RMS ρ is even smaller. 5.8.2.3 Uniform oblique collimated illumination It is of interest to go beyond isotropic sources in A-band transmission studies since solar illumination comes at a speciﬁc incidence depending on geographic location, season, and time of day. This can be done for uniform sources using the model introduced in section 5.5.5 (supplemented by Appendix D). ˆ z= The same steps as in the previous section are followed: obtain Tˆ(s) from J(s, H)/2μ0 ; expand into a short Taylor series at s = 0; estimate low-order moments from coeﬃcients of sq . We ﬁnd that the scaling, including pre-factors, for τt = (1 − g)τ in (8.3)–(8.4) is unchanged, but that the pre-asymptotic correction terms (q) are aﬀected. In other words, the simple expression for CT,ct ((1 − g)τ /2χ) becomes (q)

a rather complex expression CT,ct (τ, g, μ0 ) for q = 1, 2. This SZA-independence of the leading terms is not too surprising since we are characterizing light that has ﬁltered through an optically thick medium, and is therefore almost always highly scattered. Recall that memory of the initial direction of propagation is all but forgotten after ∼1–2 transport MFPs. Clouds with diﬀusive RT regimes have H 1 transport MFP, so only a subtle dependence on μ0 is expected in the transmitted radiation emerging at z = H. Figure 5.17 illustrates for g = 0.46 the eﬀects of SZA on the ﬁrst two moments of path ct. These plots show us where in {τ, μ0 } parameter space and what is the magnitude of the diﬀerence with the isotropic source prediction. 5.8.3 Path-length moments from above To summarize so far, we have established that ground-based O2 A-band spectroscopy is poised to become a valuable asset in cloud remote sensing as long as 38 This problem could probably be ﬁxed by introducing an ad hoc parameterization χ(Δ) where χ → 0 as |Δ| → 2.

5 Space-time Green functions for diﬀusive radiation transport

ct col 1.5 ct iso 1 0.5

233

1.5

0.2

0.4 Μ0 0.6

0.5 0.8

1 0

1.5 RMS ct 7 1.4 1.3 mean ct 5 1.2 1.1 1 0 0.5 log10 Τ

1 log10 Τ

1 0.8 0.6 0.4 Μ0 0.2

1 1.5

Fig. 5.17. SZA dependence of temporal moments for light transmitted through diﬀusive clouds. Rescaled optical depth τ for such clouds must exceed unity using the rescaled g = 0.46 (original τ , for g = 0.85, is then 3.6× more); we explore up to τ = 101.5 (≈114 for g = 0.85). The upper panel addresses predictions for mean path ctT and ranges from 0.5 to 1.8; we see very little diﬀerence between the collimated and isotropic sources for any μ0 when τ 3 (11 for g = 0.85), less if μ0 ≈ 1/2. The lower panel compares the RMS/mean ratio for ct with the quasi-constant prediction for isotropic illumination, p namely, 7/5 = 1.183 · · · (cf. Fig. 5.15). The variation is only from 1 to 1.5, and we see deviations only when τ 3 and μ0 1/2 (SZA less than about 60◦ ). The ratio for collimated beams slightly exceeds the isotropic value for all but a small area of low τ under near-grazing incidence angles.

cloud morphology is approximately plane-parallel. When cloud geometry is far from plane-parallel, O2 A-band spectroscopy can be used in cloud-radiation diagnostics that probe the complex (and highly climate-relevant) process of gaseous absorption in the presence of 3D clouds [161, 163–165]; see Appendix F for a brief description. At the time of writing, a satellite mission with high-resolution oxygen A-band capability is being prepared for launch in early 2009: the Orbital Carbon Observatory (OCO) [171]. Space-based precursors of OCO with A-band coverage at lower resolutions – but suﬃcient to initiate cloud studies – are POLDER/Parasol [172], GOME [173], and SCIAMACHY/ENVISAT [174]. There are also opportunities for airborne O2 A-band at high resolution from above the clouds, for example, the Langley Airborne A-Band Spectrometer (LAABS) instrument. It is therefore timely to hone our predictive skills for A-band products for clouds when observed from above, namely, path-length moments (ct)q R (q = 1, 2, . . . ).

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We will start our investigation of reﬂected light, as we did for transmission, with the simple diﬀusion model for isotropic illumination of a uniform plane-parallel cloud. We will then examine the eﬀects of internal stratiﬁcation. While the emphasis is on temporal responses, we will keep a tangential interest in spatial properties. Finally, we turn to SZA eﬀects strictly in the time-domain. 5.8.3.1 Isotropic boundary point-source A bare-bones model for radiative Green functions was introduced in section 5.5.1 using an isotropic pulsed source at a boundary point. Davis et al. [85] and Love et al. [86] developed this model with multiple-scattering cloud lidar in mind – an application to discussed in the next section. However, some of the time-domain signals of this active remote sensing technology and O2 A-band cloud products are indistinguishable from the Green function perspective. We are thinking here of low-order moments of in-cloud path-lengths for reﬂected light. The simple diﬀusion model predicts the following dependencies on cloud properties: (1) ctR /H = 2χ × 1 + CR,ct (τt /2χ) , (8.10) 4χ (2) τt × 1 + CR,ct (τt /2χ) , (8.11) (ct)2 R /H 2 = 5 4 2χ (2) × 1 + CR,ρ (τt /2χ) , (8.12) ρ2 R /H 2 = 3 τt where (1)

(2)

CR,ct (ξ) = CR,ρ (ξ) = (ξ + 3/2)/2ξ(ξ + 1), (2)

CR,ct (ξ) = (8ξ 3 + 41ξ 2 /2 + 75ξ/4 + 1/8)/2ξ 2 (ξ + 1)2 . We use here the same deﬁnition for ξ as in (5.11). Figure 5.18 shows the outcome of the above model, in excellent agreement with MC validation data in the regime that matters (τt > 1). Recalling that Monte Carlo simulation is grounded in random walk theory, it is in fact possible to perform a very simple form of MC simulation analytically. We can thus derive the various scaling exponents of the leading terms in (8.10)–(8.12) from heuristic arguments based on random walk (a.k.a. Brownian motion) theory; see Appendix E. In contrast with the corresponding expressions for transmitted light in (8.3)– (8.5), we see that the idiosyncratic extrapolation length parameter χ in diﬀusion theory now aﬀects the leading terms directly. We interpret this diﬀerence as a reminder that, in the case of optically thick (diﬀusive) cloud, the main physical diﬀerence between reﬂected and transmitted light is that the latter is almost always highly scattered while the former is a balanced mixture of low- and high-order scatterings. Light that has suﬀered only a few scatterings originates necessarily from the radiative boundary layer near the source. The present incarnation of diﬀusion theory uses χ to mitigate its known weakness in the boundary layer. Unsurprisingly, this ‘signature’ boundary condition parameter of diﬀusion theory appears in the leading terms of reﬂected characteristics, but not those of transmitted ones.

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Fig. 5.18. Cloud responses to a pulsed isotropic point-source in reﬂection. Diﬀusion predictions from (8.10)–(8.12), with correction terms, and (5.5) are in solid lines; MC√validation data are plotted with symbols. The best overall ﬁt was obtained for χ = 1/ 3 = 0.577 · · · . In contrast with the corresponding Fig. 5.15 for transmitted light, we see that the ﬁrst- and second-order moments of path-length scale diﬀerently with τt (cf. Appendix E). This turns out to be an opportunity for active as well as passive cloud remote sensing. Adapted from Ref. [86].

As made clear in Appendix E, another consequence of the balanced mixture of low- and high-order scatterings in reﬂected light is the diﬀerent scaling expo1/2 nents for ctR and (ct)2 R with respect to τt . This gives a unique advantage to the cloud remote sensing application of O2 A-band spectroscopy from above the observed cloud layer. Indeed, without any need for radiometric calibration, a stand-alone O2 A-band spectrometer can now deliver two pieces of information from the two path-length moments in reﬂected light. The ratio (ct)2 R /ct2R is now a sensitive function of τt alone, which can be unambiguously inverted; from there, the known value of τt and the observed value of ctR can be used in (8.10) to infer cloud thickness H. As was the case for transmitted light, O2 A-band spectroscopy alone does not give us access to information on the horizontal transport captured in ρ2 R . However, the predicted value in (8.12) gives us insight into the spatial resolution of the A-band spectroscopic cloud remote sensing process. We notice that, again because of the non-negligible contribution of lower orders of scattering to reﬂected light, ρ2 R contracts as τt increases while ρ2 T in (8.5) remains essentially constant, on the order of H 2 . On the other hand, since reﬂected light also has non-negligible contributions from light that has diﬀused almost down to the non-illuminated boundary and back, the volume of cloud being sampled is eﬀectively ≈H 3 . We

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will therefore retain H 2 as a reasonable estimate of the eﬀective footprint of the A-band approach to cloud property retrieval. Finally, we notice in Fig. 5.18 that the diﬀusion prediction for mean path exceeds the one for RMS path for scaled optical depths less than about unity (more precisely, τt /2χ < 0.798); this is a clear violation of Schwartz’s inequality, as it applies to statistics, and is of course not observed in the corresponding MC results. This anomaly results ultimately from the fact that solutions of diﬀusion equations are not constrained by causality; probabilistically speaking, they need not have all the deﬁning properties of characteristic (moment-generating) functions, even though we have used them as reasonable approximations thereof throughout this paper. Although we have not found predictions for the temporal moments in (8.10)– (8.11) for reﬂected light in the early literature, we did ﬁnd one for the variance of horizontal transport distance ρ2 R in Zege et al.’s [83] Table 4.1. That table is compiled from 1972 results by Bushmakova et al. [46] that generalize to cases with g = 0 results from Romanova’s 1971 papers [44, 45] based on isotropic scattering. In our notations, they predict that (σρ)2 R ≈ (8/9)τ /(1 − g) when absorption is weak and τt 1. We ﬁnd precisely twice39 that value when we set χ = 2/3 in (8.12), 16/9 rather than 8/9. The origin of this factor-of-2 discrepancy is not clear, but our estimate is validated by high-precision MC simulations in Fig. 5.18. Since the same factor-of-2 diﬀerence was found for transmitted light in section 5.8.2, we speculate that it originates at a higher level, possibly with the nontrivial factor of 2 in (3.5). 5.8.3.2 Stratiﬁcation eﬀects Looking back at the black curves in Fig. 5.5(top), we see that internal stratiﬁcation matters for reﬂected light, more than for its transmitted counterpart. In particular, the impact of positive and negative gradients in the extinction (equivalently here, scattering) coeﬃcient away from the illuminated boundary have opposite eﬀects on all the space-time moments. This is suﬃcient numerical evidence for attempting to capture these eﬀects in the diﬀusion framework. Moreover, in this observation geometry, we note that the stratiﬁcation aﬀects multiple-scattering cloud lidar (spatial moment) as well as O2 A-band (temporal moments only). That is indeed what ﬁrst motivated two of the present authors to develop in [92] the diﬀusion model described in section 5.5.2 that generalizes the above results to media with a linear gradient in extinction.

39 We show further on that the pre-factor of τt is actually 20/9 for a collimated narrow beam normally incident on the cloud, as opposed to an isotropic boundary point-source, cf. (9.4) and Fig. 5.21. This extra oﬀset is traceable to the term we incorporate in (4.14), hence in (5.15)–(5.16), for the anisotropy of the ﬁrst-scattering source, i.e., the P1 in the truncated phase function in (4.13).

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Fig. 5.19. Eﬀects of stratiﬁcation on Green functions for reﬂection. Diﬀusion predictions for the prescribed cloud (τt = 8.1) are in solid lines; they were obtained from (8.13)–(8.15), including correction terms, with χ = 2/3. MC validation data are plotted with symbols; they are extracted from Fig. 5.5, black curves in upper panel. We note the presence of a logarithmic singularity in ρ2 R at Δ = +2, which is manifest in the leading term written explicitly in (8.15) and discussed in the main text. Adapted from Ref. [92].

Following the usual procedure, we ﬁnd Δ(10 + Δ) (2) × 1 + CR,ct (ξ, Δ) , ctR /H = 2χ 1 + 40 ξ=τt /2χ Δ(80 − Δ(20 + Δ(10 + Δ))) 4χ (ct)2 R /H 2 = 1+ τt 5 360 (2) × 1 + CR,ct (ξ, Δ) ,

(8.13)

(8.14)

ξ=τt /2χ

in the time domain, and in the spatial domain, ( 4 2 2 2 2χ 1 Δ 1 + Δ/2 +1 ln − 1+2 2+ ρ2 R /H 2 = 4 Δ 2 1 − Δ/2 Δ Δ τt (2) × 1 + CR,ρ (ξ, Δ) . (8.15) ξ=τt /2χ

These all lead back to (8.10)–(8.12) in the limit Δ → 0. Figure 5.19 compares the above predictions with the MC results from the black curves in the upper panel of Fig. 5.5. The analytical results for the temporal responses are in very good agreement with the numerical benchmarks. As was the case for transmission, the most notable discrepancy is the logarithmic divergence in (8.15). However, this now happens only at Δ → +2, i.e., when

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the extinction vanishes at the illuminated boundary. We attribute this to the same physical cause we assign to the appearance of χ in all the pre-factors: the extrapolation length (outside the cloud) – as well as the estimated depth of the dreaded radiative boundary layer (inside the cloud) – goes to ∞ as 1/σ(z) ∝ 1/z when Δ = +2 in (2.27). This is a clear harbinger of catastrophic failure of the diﬀusion model. In his discussion of multiple-scattering cloud lidar applications, Davis [92] notices that the ‘true’ MC values for the logarithms of ctR and ρ2 R are almost linear in Δ. He therefore suggests using the diﬀusion model only to predict the response at Δ = 0 from (8.10) and (8.12), or something better, and then use the new model with stratiﬁcation capability only to estimate the logarithmic partial derivatives ∂ lnctR /∂Δ and ∂ lnρ2 R /∂Δ at Δ = 0; then a log-linear extrapolation is performed that will closely follow the MC validation data. For the spatial response in (8.15), one can use the short series expansion 2 4 ξ(ξ + 3/2) + 3/4 2 2 ρ R /H = + Δ + O(Δ2 ), (8.16) 3ξ ξ(ξ + 1) 3ξ recalling that ξ = τt /2χ, to make this estimation. As noted in Ref. [92], the logarithmic derivative for ρ2 R at Δ = 0, namely, ξ(ξ + 1)/(4ξ 2 + 6ξ + 3), is the same as for ctR (cf. Fig. 5.19). Moreover, we note that the useful range for this quantity is rather small: ≈2/13 to 1/4 for 1 ξ ≤ ∞. 5.8.3.3 Uniform oblique collimated illumination Oxygen A-band spectroscopic observations of clouds from above need to account for the fact that sunlight is collimated and incident under all possible angles. The diffusion model for uniform-but-possibly-slant illumination presented in section 5.5.5 can again be brought to bear, this time, for reﬂected light. The usual procedure in computer-assisted calculus and algebra leads to 1 (1) (8.17) ctR /H = (2 + 3μ0 ) × 1 + CR,ct (τ, g, μ0 ) , 3 2 (2) (2 + 3μ0 ) (1 − g)τ × 1 + CR,ct (τ, g, μ0 ) , (8.18) (ct)2 R /H 2 = 15 4 (3) (2 + 3μ0 ) [(1 − g)τ ]2 × 1 + CR,ct (τ, g, μ0 ) . (8.19) (ct)3 R /H 3 = 35 We notice that μ0 ﬁnds its way into the pre-factor of the dominant power-law relation for each moment. This is a strong dependence in comparison with counterparts for transmission described in the previous subsection, where μ0 only inﬂuences the pre-asymptotic correction terms. Revisiting Fig. 5.17 for quantiﬁcation of the subtle SZA eﬀects in transmission, using the isotropic source predictions as reference. This diﬀerence in sensitivity to SZA between transmitted and reﬂected ﬂuxes in the time-domain is once again attributable to the fact that reﬂected light is a mixture of low and high orders of scattering ns ≈ σct. Indeed, all the light observed in reﬂection originates from some last scattering event within the upper radiative boundary layer of the cloud (i.e., between z = 0 and z ≈ t ); in this mix, the least scattered light will certainly have a memory of its original direction of incidence.

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2.5 ct R 2 H 1.5 1

1 0.8 0.6 0.4 Μ0

0 0.5 log10 Τ

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0.2

1 1.5

log10

t2 R 12 0.4 t R 0.2 0

1.5 0.2

0.4 Μ0 0.6 0.8 08

1 log Τ 0.5 10 1 0

Fig. 5.20. SZA dependence of temporal moments for light reﬂected from diﬀusive clouds. Upper panel: (8.17) is plotted for 0 < μ0 ≤ 1 and 1 ≤ τ ≤ 101.5 with g = 0.46, corresponding to 3.6 ≤ τ 114 for the original g = 0.85 if we interpret these values as δ-rescaled (cf. section 5.4.2.1); the vertical range is between 2/3 (limit for τ → ∞ when μ0 → 0) and 2.5. Lower panel: (1/2) log10 [(ct)2 R /ct2R )] is plotted for the same range of parameters; the vertical axis ranges only from 0 (log10 1, the minimum theoretical value) to 0.544 (log10 3.5). Further discussion in the main text.

Figure 5.20 illustrates SZA eﬀects in reﬂected light, which are easy to show in their own right. (Unlike transmitted light, there is no need here to refer back to the isotropic source model as a baseline.) In the upper panel, (8.17) is plotted over the relevant range of parameters, namely, 0 < μ0 ≤ 1 and 1 ≤ τ ≤ 101.5 for the δ-Eddington rescaled g = 0.46 (3.6 ≤ τ 114 for the original g = 0.85). We see that in-cloud path increases signiﬁcantly with μ0 at any given τ , which is to be expected since the light is injected deeper into the cloudy medium as μ0 approaches unity. The lower panel plots the log of the RMS-to-mean path ratio over the same region of parameter space. We see a monotonic dependence of the non-dimensional moment ratio with τ , which conﬁrms our above prediction that cloud optical depth can be retrieved unambiguously from the two observable moments for any given SZA.

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5.9 Space-time Green functions applied to multiple-scattering cloud lidar (MuSCL) observations We now turn to a new active technology for cloud remote sensing in the diﬀusive regime: Multiple-Scattering Cloud Lidar (MuSCL),40 that we ﬁrst proposed in 1996 on purely theoretical grounds [176]. Although developed independently, this cloud-probing technique is a direct extension of Multiple-FOV (MFOV) lidar, which targets aerosols [177–181] as well as clouds [182–184]. MFOV lidar itself extends the standard design of elastic backscatter lidar implemented, for instance, in the eye-safe design of micro-pulse lidars [185]. In the classic lidar equation, the assumption is single scattering through 180◦ for the most common conﬁguration, so-called ‘monostatic’ systems, where the transmitter and receiver are collocated. This is indeed the opposite asymptotic limit in RT theory to the diﬀusion/P1 approximation used throughout this paper: it is valid only in optically thin media. In essence, MFOV lidar capitalizes on the strong forward peaked of the scattering phase function for cloud droplets and most aerosol types. To this eﬀect, the small-angle approximation to time-dependent RT with multiple scattering is invoked, and it is known to work well close to the incident beam [83, 186, 187, among others]. Although the MFOV lidar project brought it to a high level of sophistication and application, the time-dependent41 small-angle approximation was applied to pulse propagation through turbid media at least going back to Belyantsev et al. in 1967 [193], Romanova in 1970 [194], and Weinman and Shipley in 1972 [195] who all considered a one-way transit, ending in transmission.42 Investigations of reﬂected laser pulses in the same approximation soon followed [196–199]. Because of the necessarily narrow FOV used in classic lidar, which only needs to contain the volume illuminated by the highly-collimated laser beam, there is little penetration into opaque media such as the clouds. Even if we could separate the single-scattering signal from contamination by multiple-scattering, which we cannot, there would be no photons left to count after two-way propagation to optical depths in the 10–100 range. MFOV does vastly better by indeed exploiting the multiple scattering signal coming from relatively near the laser beam. Optical depths up to 8–10 can be reached; after that wide-angle scattering starts to dominate the signal and we therefore enter the realm of MuSCL. MuSCL signal physics are the same as in the in situ cloud lidar discussed in section 5.7 and, to a large extent, also those of the reﬂected O2 A-band spec40 This obvious and compelling acronym is closely related to, but not to be confused with, MUSCLE (MUltiple-SCattering Lidar Experiments), an on-going series of international workshops; see Ref. [175] for a special section of Applied Physics – B that followed from one of them. 41 In steady-state, application of the small-angle approximation to the pencil-beam problem goes back much further. In charged particle transport, it has been known as the Fermi–Eyges/Fokker–Planck approximation since c. 1950. This modeling framework has recently regained considerable interest, largely motivated by emerging applications to the accurate computation of minimal-yet-eﬀective dosimetry in electron-beam radiation therapy [188–192]. 42 Formally, this is an isotropic diﬀusion process in the non-Euclidean space of directions, starting from a given initial position and slowly but surely ﬁlling the whole unit sphere, one scatter at a time [81, 108].

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troscopy covered in the previous section. The main diﬀerence is that, having a pulsed laser as a source and observing the Green function from a signiﬁcant standoﬀ distance to the cloud, MuSCL can access and exploit the signal in the spatial domain . . . although not always easily, as we now show, by starting our discussion with past and future space-based MuSCL systems. 5.9.1 Space-based MuSCL systems If the lidar instrument is at a very great distance to the cloud, as in space-based systems, then there will be limited, if any, information about the horizontal radiation transport away from the beam. We have estimated this span of the spatial Green function to be ∼H (∼km scales). Although imaging technology exists for much smaller scales, it would soon be ‘photon-starved’ if also required to resolve the rapid time evolution. Furthermore, the laser beam itself will be spread out to a signiﬁcant diameter, thus diluting the spatial Green function (by convolution). That said about the present and near-future, systems may prove far more capable in the longer term. Following Miller and Stephens [200], we make the safe assumption that we only have access to time-domain information. But this is exactly what the proposed cloud remote sensing using reﬂected oxygen A-band spectroscopy is about. The diﬀerence is that a lidar system will access the temporal signal directly with fast radiometry, while the spectroscopic technique delivers the path-length moments as ‘products’ derived from the diﬀerential absorption spectroscopy. The only other diﬀerence is the in down-looking lidar we have μ0 = 1 while passive A-band observation can have any incidence angle. So we are in fact looking at a special case of (8.17)–(8.19): 5 (1) × 1 + CR,ct (τ, g, 1) , 3 2 (2) 2 2 (ct) R /H = (1 − g)τ × 1 + CR,ct (τ, g, 1) , and 3 4 (3) 3 3 (ct) R /H = [(1 − g)τ ]2 × 1 + CR,ct (τ, g, 1) , 7 ctR /H =

(9.1) (9.2) (9.3)

where, following Ref. [92], we have returned to the model for normal incidence in section 5.5.3. Figure 5.21 illustrates (9.1)–(9.3) along with MC validation data and the predicted asymptotes using both g = 0.85 and its δ-rescaled counterpart g = 0.45. As expected, the later case is more accurate with respect to the MC simulation benchmarks. We see that the higher the order of the moment, the longer it takes to approach the asymptote; this underscores the practical importance of (q) knowing the CR,ct terms with high accuracy. At present, the prime application of this limited set of results is for the analysis of the nighttime orbit #135 of the Lidar-In-space Technology Experiment (LITE), which ﬂew on Space Shuttle Discovery (STS-64 mission), September 9–20, 1994 [111]. The LITE payload was in essence a standard research lidar system: 5 W laser transmitting at 532 nm (the popular ‘doubled Nd:YAG’ solid-state technology) with a pulse rep-rate of 10 Hz and a diﬀraction-limited beam divergence leading to ≈0.3 km diameter at cloud level); receiver composed of a ≈1-m telescope feeding

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Fig. 5.21. Spatial and temporal moments of the reﬂected Green functionpfor normal 3 3 illumination by a pulsed narrow beam. From the top p pdown, we have plotted (ct) /H, 2 2 (ct) R /H, and ctR /H from (9.1)–(9.3) and ρ R /H from (9.4) as functions of rescaled optical depth. We note the diﬀerent results for the canonical g = 0.85 value for liquid water clouds and its rescaled counterpart g = 0.45. Indeed, when the un-collided and diﬀuse components of the radiance ﬁeld are treated separately, we no longer have exact similarity in (1 − g)τ . Adapted from Ref. [92].

a high-eﬃciency photon-counting detector with a deliberately large FOV, namely, 0.2◦ (footprint at cloud level with ≈0.9 km diameter (at least during nighttime orbit #135). That data collection targeted a dense marine Sc layer and geometry tells us that the measured light was transported up to 1.2 km horizontally. Since the anticipated spatial Green function for such clouds has RMS-ρ ∼ H ≈ 0.3 km, we can safely assume that very high orders of scattering were present in the LITE signal. We refer the reader to the studies by Davis et al. [114, 201] for quantitative analyses of temporal moments and other time-domain characteristics of the smallbut-interesting subset of non-saturated LITE returns from this orbit. It suﬃces to say here that the inferred cloud properties, i.e., {H, τ } pairs, are consistent with what we know about such clouds. 5.9.2 Ground-based and airborne MuSCL systems We now assume that the MuSCL system is at a ﬁnite distance from the cloud. In other words, going back to notation from section 5.2.3 for the distance of the

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observer to the closest cloud boundary, we have dobs H, H, but not H (a case covered in the previous subsection) nor43 H. In short, platforms include those on the ground [84] or in an aircraft (including blimps) ﬂying above the cloud top [202], up to the ≈20 km limit for high-altitude aircraft; as described in the provided references, both kinds of MuSCL system have been deployed and demonstrated. We can then easily measure spatial characteristics such as 20 1 (2) × 1 + CR,ρ (τ, g, 1) , (9.4) ρ2 R /H 2 = 9 (1 − g)τ given here for the same normal/narrow-beam illumination used for the temporal moments in (9.1)–(9.3). The RMS value for ρ obtained from the above is also plotted in Fig. 5.21 with its MC validation data and estimated asymptote. Note (2) that, for consistency, we have reserved in CR,ρ (τ, g, 1) a slot for the possibility of μ0 < 1 although there is no longer axial symmetry in that case, and one would need to modify the diﬀusion model to capture other spatial statistics than the RMS horizontal transport. At present, however, we have no practical reason to take the diﬀusion transport modeling to that level of complexity. But that is only because of the limited scope of our applications; we refer the interested reader to Zege et al. in Ref. [83, Section 6.3, and citations therein] who address this steady-state problem in order to derive the non-axisymmetric MTF that controls oﬀ-axis imaging through a turbid medium. The random-but-correlated variability of extinction in clouds driven by turbulence aﬀects all aspects of the 3+1D RT, and we have promoted in section 5.4.2.3 a straightforward homogenization approach developed by Cairns et al. [75] to cope with this issue. We recall that in homogenization theory we simply use new values of optical properties σ and g dependent on original values – possibly already δ-rescaled (to account for the anisotropic source and phase function) – and variability parameters. Although all the time-domain responses in (9.1)–(9.3) are aﬀected, we choose to illustrate with the RMS ρ. Figure 5.22shows the relative eﬀect (in %-diﬀerence) of Cairns rescaling for up to 0.4 on ρ2 R . It is systematic and positive: turbulence makes the RMS horizontal transport larger. For a more quantitative validation of this prediction, we can turn back to Fig. 5.4 where we see that, going from the uniform to the fractal stratus cloud, the RMS ρ increases by ≈15–20%. That simulation was for a mean cloud optical depth τ ≈ 13, hence τ ≈ 3.6 after δ-Eddington rescaling (log10 τ ≈ 0.56). This is about in the middle of Fig. 5.22 where we see that a 15–20% eﬀect is obtained for (σ 2 /σ 2 , σ lc ) ≈ 0.1–0.15. Looking at Fig. 5.6, we see that this magnitude is 2 delivered by a variety of values of the one-point PDF parameter σ 2 /σ 2 = σ 2 /σ and of the two-point correlation parameter lc (knowing that σ = τ /H ≈ 3.6/0.3 = 12 km−1 ). Fractal clouds have, by deﬁnition, long-range correlations: horizontally, the variability spectrum observed for marine Sc goes from tens of meters to tens of 43 For dobs = 0, the detector is at the cloud boundary, so no imaging (discrimination between diﬀerent values of ρ > 0) is possible, at least in monostatic systems where source and detector are collocated. This is in fact a limiting case of in situ cloud lidar covered in section 5.7, corresponding to the occurrence of dense fog for a ground-based system. Recalling the source/detector geometry for that technique, it may prove useful to decouple the axis of the receiver FOV from the laser beam.

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1/2

Fig. 5.22. The impact of random internal variability on ρ2 R in %. We have plotted 100 × ([ρ2 R ()/ρ2 R (0)]1/2 − 1) where , obtained from (4.22)–(4.23), is used in (4.20) to change both σ and g . In turn, this rescaling of optical properties changes the outcome of (9.4) for ρ2 R , as an example. We see that over the useful range of the parameterized impact of turbulence-driven variability is signiﬁcant, but only weakly dependent on optical thickness.

kilometers [203–205]. However, we are only interested here in the spatial correlations over the horizontal extent of the spatial Green function, which is ∼H; so we can take σ lc ∼ τ ≈ 3.6. Similarly, we are only interested in the relative one-point variability, as captured by the RMS/mean ratio, over the horizontal extent of the 2 spatial Green function; this implies that σ 2 /σ is only slightly more than unity. For 2 this combination of (σ 2 /σ , σ lc ), we indeed ﬁnd that has the right magnitude in Fig. 5.6. In summary, we have presented upfront (Fig. 5.4) numerical evidence that, for the moderate variability found in Sc cloud layers at scales ∼H, we can expect a boost in ρ2 R on the order of ≈15–20% with respect to strictly uniform clouds with the same (domain-average) optical depth. Furthermore, this observation is consistent with our implementation of Cairns’ parameterization for small-scale variability eﬀects in the spatial Green function. Finally, it is important to compare quantitatively the impacts of random turbulence and of deterministic stratiﬁcation (sections 5.2.5, 5.5.2 and 5.8.3) on the characteristics of reﬂected light. We summarized the latter eﬀect by estimating the 1/2 relative eﬀect of a change in relative gradient |Δ| of unity on either ρ2 R or ctR : it varies between 2/13 (≈15%) and 1/4 (25%) as (1 − g)τ goes from ≈1 to 1. We therefore retain that, under typical circumstances, the two eﬀects are of the same order of magnitude. 5.9.3 Moment-based methods for MuSCL Figure 5.23 illustrates how a hypothetical MuSCL retrieval of cloud thickness H and optical depth τ would proceed, including error propagation. We assume here an observational technology that can deliver one spatial and two temporal moments.

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Fig. 5.23. A typical moment-based cloud remote sensing algorithm for MuSCL data processing. The lower curves show the non-dimensional ratio of observable moments ρ2 R /(ct)2 R as a function of rescaled cloud optical depth (1 − g)τ for g = 0.85 and the related value of g = 0.46. Although not very diﬀerent, the curve for g is favored, and gives us τ , via scaled optical depth τ (upper axis). We then use the corresponding prediction for H/ct in the upper curves to determine cloud thickness H. The numerical example uses rescaled g = 0.46, which is expected to be the more accurate. A ratio of second-order moments of (3.0 ± 0.5) 10−2 is assumed, and yields τ ≈ 34 ± 4 (12% uncertainty) and, from there, H ≈ (0.560 ± 0.005)× the mean in-cloud path ct (whatever that may be). Note that the observational error on this ﬁrst-order moment will very likely overwhelm the ≈1% error on the multiplier (associated with ≈12% on τ , resulting itself from the hypothetical ≈17% on the ratio of second-order moments). Adapted from Ref. [92].

As already suggested in our discussion of cloud remote sensing based on A-band spectroscopy, the non-dimensional property τ is determined from a non-dimensional ratio of moments, and then the dimensional cloud property H is determined from any one of the dimensional observables. Here, we opt for the mean in-cloud path. Hogan and Battaglia [107] recently developed a time-dependent two-stream model, which is akin to the diﬀusion model, but with better performance at early times (in terms of causality in particular). Their model furthermore features an improved representation of the source of diﬀuse radiation by embedding a propagation-with-dispersion model by Hogan [187] for the pulsed laser beam (in the small-angle approximation). This idea was pioneered by Dolin [206] and Zege et al. [207] 20–25 years ago with analytical methods, but Hogan and Battaglia designed

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their model from the start as a numerical tool. In particular, arbitrary vertical structure can be accommodated. In view of the alternation between physical-space modeling (requiring inverse Fourier and Laplace transforms) and moment-based methods in the present study, they oﬀer an interesting middle road. The temporal signal is modeled in detail (as straightforward outcome of the numerical PDE solution) and, for space, they predict the conditional moment ρ2 R (ct) (basically by assuming, as in diﬀusion theory, that the instantaneous proﬁle is Gaussian). This framework should prove very fruitful for all future MuSCL-type probes, and especially those in space from whence information about horizontal transport will necessarily be limited. 5.9.4 Deeper mining of MuSCL observations for cloud information Can we use Cairns’ parameterization of internal variability from section 5.4.2.3 in cloud remote sensing with MuSL? When using moment-based retrievals, there is certainly no harm in improving the ﬁdelity of the forward model by prescribing a value of (σ 2 /σ 2 , σlc ) from the climatology of cloud variability; see Fig. 5.22 for the impact in the ρ-domain. However, if we wish to determine (hence a metric of internal cloud variability) empirically, we would have to start with at least three moments. We can thus form two or more dimensionless ratios and we include here, along with moments, cloud albedo R; albedo is indeed the zeroth-order moment, but also the ratio of overall reﬂected ﬂux to incoming ﬂux. These data in principle enable us to determine simultaneously the twice-rescaled τ and g using the full nonlinear dependence on both parameters (rather than simply prescribe the asymmetry factor); see Fig. 5.24 for an illustration. Then one can derive = 1 − 1/[2 − (1 − g )/(1 − g )] from (4.20) for known g = 0.46; from there, we work back to τ and τ . Knowing τ (and τ ) and having prescribed g (and g ), any predicted moment from (9.1)–(9.4) continues to give us H along the way by comparison with the corresponding observed value. In the same spirit, a three-moment retrieval scheme can be devised that targets Δ (on top of H and τ ) using the parameterizations of stratiﬁcation impact in (8.13)–(8.15). In principle, a four-moment scheme could target all four cloud parameters. That, however, may be overextending MuSCL capability in cloud remote sensing. Only practice in modeling and in the ﬁeld with speciﬁc implementations will determine how much reliable cloud information can be extracted. Of course, beyond its ‘validation-by-intercomparison’ phase of development [84, 202], MuSCL can eventually be combined with other cloud remote sensing instruments in optimal multi-modality approaches. We anticipate particularly fruitful integration with millimeter cloud radar and multi-channel passive microwave radiometers. Natural complementarity with thermal and solar radiometers should also be examined carefully and thoroughly. Proximity of the observer to the cloud will naturally increase the SNR [92]. However, a serious problem arises when the ﬁnite FOV of the instrument prevents one from estimating moments reliably – the far-ﬁeld tail of the spatial Green function is truncated! Although it may not be as obvious, the temporal Green function will be truncated if the spatial one is. If the receiver is a true imager, then it can be tilted with respect to the laser beam, thus gaining theoretically (in the absence of

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Fig. 5.24. Feasibility of joint retrieval of the eﬀective τ and g, hence of a metric of random variability, with MuSCL. Gray-scale plots of cloud albedo R (left) and the nondimensional moment ratio ρ2 R /ct2R (right) as functions of log10 τ and g , the twice rescaled cloud parameters in (4.20). Observational determination of R, which requires absolute calibration, ctR and ρ2 R can lead to physical and optical thickness as well as the internal variability parameter. As explained in the main text, (g , g ), τ (g , g , τ ) and τ (g, g , τ ) are determined in that order, assuming we can take g = 0.85 and the associated g = 0.46 for granted. However, visual inspection of the two panels reveals that joint retrievals of τ and g (hence Cairns’ ) – the cloud’s physical thickness H being determined later – will not be easy in general because isophotes and ‘iso-moment-ratio’ lines are more parallel than perpendicular. The most favorable region is where τ is low (reinforcing the importance of δ-Eddington rescaling as an intermediate step) especially when g (hence ) is large.

noise) up to a factor of 2 in oﬀ-beam distance at the focal plane. A radical solution is, however, to return to the models that express the Green function in space and time, i.e., the PDFs that lead to the moments, cf. section 5.6 and Refs [84, 99].

5.10 Further applications to passive solar observations of clouds Having covered in detail the application of radiative Green functions to two emerging cloud remote sensing technologies in the previous sections, we now survey brieﬂy their application to more tested approaches. 5.10.1 Operational cloud remote sensing in the solar spectrum Reﬂected and transmitted sunlight has always been a resource in cloud remote sensing in the solar spectrum, as long as absolute radiometric calibration is maintained. The standard model for retrieving cloud properties is, like here, the plane-

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parallel slab treated at various levels of accuracy in the RT ranging from a twostream/diﬀusion model (with separation of direct and diﬀuse components) from section 5.5.5 to a full multi-stream 1D RT model such as DISORT [168]. In the latter case, the computational burden is heavy enough that the forward modeling is done ahead of time and used to generate extensive look-up tables used to map cloud properties to radiances and vice versa [208–210]. Excluding O2 A-band, cloud thickness H is not accessible by passive remote sensing since all one has is steady-state reﬂected (R) or transmitted (T ) radiances44 at one or more wavelengths, both being functions of τ, 0 and45 g, as well as of μ0 and the viewing angle coordinates μ, φ − φ0 . The targeted cloud properties are invariably optical depth τ and the eﬀective radius of the droplet size distribution re = r3 /r2 . The latter is obtained indirectly, via an estimate of the SSA, 0 (re ), at a wavelength where liquid water has non-negligible absorption. This joint retrieval is based on the fact that, to a ﬁrst approximation, σ ∼ r2 and σa ∼ r3 , hence 1 − 0 ∼ re for the co-SSA. 5.10.2 Opacity-driven 3D radiation transport The question of how applicable 1D RT is to remote sensing of real (3D) cloud has been investigated quite thoroughly; see Ref. [211] for a recent survey. Since 1D RT is applied irrespective of the pixel scale, one must distinguish two qualitatively diﬀerent kinds of modeling error: 1. if the pixel size is somewhat greater than the characteristic scale of the spatial Green function (namely, ∼H), then there is a high probability that there is signiﬁcant sub-pixel variability and, because of the nonlinear dependence of radiance on cloud properties (τ and re ), the retrieved values will be biased with respect to their mean values; 2. if the pixel size is than H, then the observed radiance is surely ‘contaminated’ (in the 1D RT sense) by horizontal ﬂuxes coming from adjacent pixels. In the former case (problem #1), so-called ‘plane-parallel’ biases occur, which are generally systematic in sign [212]; in the latter case (problem #2), so-called ‘independent pixel approximation’ biases occur, which are of both signs (depending on structural details) [213–215]. Solutions adapted to both of these situations are discussed in the remainder of this section. 5.10.3 The independent pixel approximation for steady/uniform illumination The above problem #1 (sub-pixel variability) can be addressed simply by averaging (linearly mixing) the radiances predicted in 1D RT over an assumed variability in cloud properties. We can anticipate nontrivial results because radiances are 44 It is conventional to normalize the observed radiance by estimating the eﬀective Lambertian reﬂectivity, R(μ, φ; μ0 , φ0 ) = πI(μ, φ; μ0 , φ0 )/μ0 F0 , where F0 is the spectral solar ﬂux integrated over the band of interest, and similarly for T . 45 In this context of multi-stream 1D RT, we understand ‘g’ to symbolize the whole phase function representation, in whatever space that may be (Ω or Pn ).

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nonlinear functions of all the cloud optical properties. This procedure is known in 3D RT as the independent pixel approximation (IPA) [212]. It is natural and convenient to choose a model for the variability that leads to analytical treatment. In stratiform clouds, τ is by far the most variable property in comparison with H, g and re (or 0 ). The preferred variability model for τ has been the twoparameter Gamma distribution: Pa (τ ) =

1 & a 'a a−1 −aτ /τ τ e , Γ (a) τ

where a=

1 τ 2 /τ 2

−1

.

(10.1)

(10.2)

Its popularity follows from the ease of integrating rational functions over arbitrary combinations of power laws and exponentials, resulting at most in exponential integral functions and/or incomplete Gamma functions (possibly inﬁnite series thereof). In the context of radiation budget parameterization for the large-scale domainaverages required in GCMs, Barker [216] worked out the integrals ∞ F a (τ , 0 , g; μ0 ) =

Pa (τ )F (τ ; 0 , g, μ0 ) dτ

(10.3)

0

for boundary ﬂuxes F = R, T (hence also cloud absorptance A = 1−R−T ) resulting from uniform collimated illumination in the Eddington/diﬀusion-based version of the two-stream approximation [87]. This ‘Gamma-Weighted Two-Stream’ model was later generalized to multiple partially cloudy layers by Oreopoulos and Barker [217], thus calling for the same computations for the R and T responses to a uniform isotropic source, which depend only on 1 − 0 and τt = (1 − 0 g)τ . For instance, at non-absorbing wavelengths (0 = 1), they use the simple expression for transmittance in (5.5) and obtain ∞ T a (τ t ) =

) 1 Pa (τt ) dτt = X eX Ea (X))X=2χa/τ , t 1 + τt /2χ

(10.4)

0

although in a diﬀerent notations; here, Ea (X) is the exponential integral of (any real) order a > 0, and we note that X = a/ξ from (5.11). This leads to the systematic positive bias observed in Fig. 5.25(a) of T a (τ t ) with respect to T ∞ (τ t ) = 1/(1 + τ t /2χ). This is a well-known result in 3D RT: structured clouds transmit more than their homogeneous counterparts with the same mean τ . This is an immediate consequence of Jensen’s [77] inequality in the case of a convex function like T (τt /2χ). Kokhanovsky [218, 219] performed similar computations, with satellite remote sensing in mind, using the asymptotic 1D RT [166,220] expressions for reﬂected radiances I(τ, 0 , g; μ0 , μ), which become Ia (τ t , 0 , g; μ0 , μ).

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5.10.4 The independent pixel approximation for space/time Green functions There is no reason why the same averaging procedure cannot be performed on the spatial or temporal Green functions or associated moments obtained in the previous sections, as need be. This computation would apply, as for the above steady-state RT, to solar O2 A-band spectroscopy at coarse scale. In that respect, it complements the homogenization approaches based on Cairns (section 5.4.2.3) or Larsen (section 5.4.2.4) rescaling, which applies primarily to small-scale variability. At any rate, that is precisely how Davis and Marshak [24] preemptively approached the problem of spatial variability for the spatial dimension in transmitted light, without considering homogenization. They used (10.3) with τ = 10 and a = 4.5, typical values found by Barker et al. [221] for stratocumulus, to average T × ρ2 T from (5.5) and (8.5), and then normalized by T in (10.4). Davis and Marshak thus captured quantitatively the ≈10–15% diﬀerence in RMS ρ clearly visible in Fig. 5.4 between uniform and fractal plane-parallel clouds. Here is another example, this one in the time-domain hence directly applicable to passive (solar) remote sensing of clouds based on O2 A-band spectroscopy. We apply (10.3) with F (τ ; 0 , g, μ0 ) replaced by ctT from (8.3) times the corresponding T (τt ) from (5.5). After dividing by T from (10.4), we obtain for the mean path in variable clouds: ctT /H =

")) T ctT χ! 1 + a + X + (2 − X)/T a (X) ) , = 2 X=2χa/τ t T ×H

(10.5)

Fig. 5.25. Transmittance and mean path-length for a Gamma-weighted two-stream model. (a) The expression in (10.4) is plotted versus τ t for selected values of a in log-log axes; we note the increasing transmission as the unresolved variability increases (a decreases) at ﬁxed τ t . (b) Mean path-length, in units of H, from (10.5) versus τ t for the same values of a as in panel (a); we note that paths decrease on average as variability increases (a decreases) at ﬁxed τ t . See text for more explanation.

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where T a (X) is expressed in (10.4). Figure 5.25(b) shows ctT /H for representative values of a over a relevant range of τ t . We see that, for a given H, the unresolved variability reduces the observed mean path with respect to the prediction for a uniform cloud with the same optical depth. The physical reason for this outcome is the same dominance of high transmission values (low τt ) that affect T in Fig. 5.25(a). Indeed, the whole variation of ﬂux-weighted mean path, T × ctT = χ(τt2 + 6χτt + 6χ2 )/(τt + 2χ)2 , is between χ 1 and 3χ/2 ≈ 1 for all choices of χ near its canonical value46 of 2/3. We note that, if the application is ground-based O2 A-band spectroscopy of clouds, then the Gamma- or otherwise-weighted averaging should actually be performed on the prediction of the forward RT model in Laplace space, namely, F˜ (s, 0, τt , χ) from (5.8), or (5.9) for space-based, since that is the quantity observed at a scale so coarse that sub-pixel variability needs to be accounted for. That is a harder computation yielding the same expressions for the path-length moments, but it can be used directly with A-band observations. 5.10.5 Landsat-type observations of clouds from space, and the nonlocal IPA How can we address the above problem #2 of pixel ‘adjacency’ eﬀects? In this case, the pixel scale is too small for 1D RT to be realistic. More precisely, the pixel footprint is so small that, even if it were internally homogeneous, net horizontal ﬂuxes coming from denser or more tenuous neighboring pixels would aﬀect the observed radiance at cloud top. With their 30 m pixels, NASA’s series of Landsat/ThematicMappers are by far the most popular assets delivering imagery that fall in this category. To the best of our knowledge, the ﬁrst systematic attempt to go beyond quantiﬁcation and actually attempt to mitigate this inescapable 3D RT eﬀect was by Marshak et al. [222] who proposed the ‘nonlocal IPA’ (NIPA). NIPA is based on the intuitive idea that multiple scattering processes cause an apparent smoothing of the cloud structure, as observed in the remotely sensed radiance ﬁeld [170, 223–225]. Rather than run a full 3D RT simulation with an expensive MC code, or even a more eﬃcient grid-based solver such as SHDOM [18], one can simply apply a low-pass ﬁlter (smoothing kernel) to the IPA prediction. This approximate 3D RT method works well, at least for stratiform clouds under near-normal illumination. At more oblique illumination, brightening/shadowing effects produce a radiative roughening in the sense of enhanced amplitudes in Fourier space [226, 227] at scales H. → − We note that what is required here is the Fourier transform P˜ ( k ) of the smooth− ing kernel P (→ ρ ) since we wish to perform the convolution product of P and the → 2D IPA-derived radiance ﬁeld IIPA (− ρ ): − INIPA (→ ρ)=

+∞

→ → → → P (− ρ )IIPA (− ρ −− ρ ) d− ρ ,

(10.6)

−∞ 46

We note that ctT /H = 3χ/2 at τt → 0(X → ∞) in (10.5), which is unity when χ = 2/3, for all values of a (cf. Fig. 5.25(b)). As remarked earlier (in section 5.8.2), this limit should yield 2 (the mean μ-weighted value of 1/μ) and, accordingly, we should be using χ = 4/3 for optically thin media, as suggested in section 5.4.3.

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→ − → − → − which becomes a simple product in Fourier space, I˜NIPA ( k ) = P˜ ( k )I˜IPA ( k ). Any one of the Fourier-space reﬂected Green functions computed in section 5.5 for illumination can be used. They depend parametrically on H, τt and g or Δ. Depending on whether local albedo or nadir radiance is targeted, we would choose a spatial Green function for isotropic or normal illumination. Marshak et al. [222] had an even more empirical approach, however. Eschewing normalized solutions from section 5.5 in Fourier space, they used a convenient two-parameter expression like (10.1) but for ρ instead of τ and averages based on cloud radiative Green functions (in ·’s) instead of averages over cloud structural disorder (indicated with ·). The 2D Fourier–Hankel transform in (6.1) of a Gammashaped radial Green function (normalized with the appropriate 2πρ weighting) can be expressed as an Euler hypergeometric function: 2 2 + a ρk 1 + a , , 1; − . (10.7) P˜ (k) = 2 F1 2 2 a+1 However, the authors did their proof-of-concept computations with cloud models having optical depth variability in a single horizontal direction, say, x. The required 1D Fourier transform of (10.1), with τ → |x| and division by 2 (to cover the new support, all of R), is a simpler expression: cos a tan−1 ( |x| k ) a P˜ (k) = . (10.8) '2 a/2 & |x| k 1+ a Just like the diﬀusion-based MTFs for reﬂections presented in section 5.5.1 (Fig. 5.8), these smoothing kernels act in Fourier space as low-pass ﬁlters; however, they feature gentle power-law cutoﬀs in k −a at wavenumber kc ≈ 1/ρR (2D) or 1/|x|R (1D).47 Power-law tails in P˜ (k) are a natural choice to reconcile the spatial correlations observed in satellite images of extensive stratocumulus [225] with those observed with airborne in situ probes [204] for the same type of cloud system. The latter have scale-invariant (power-law) internal structure (obviously driven by turbulence); speciﬁcally, one ﬁnds extinction (actually, liquid water content) ﬂuctuations in k −5/3 , typically over scales from ∼tens of kilometers down to ∼tens of meters. Satellite (nadir-looking) radiances also have this trend, which follows from the IPA (a nonlinear but one-to-one mapping of local τ to local radiance), but only down to a scale found by numerical simulation to be ≈ ρ2 R [170]. Above the associated cut-oﬀ wavenumber, a trend approaching k −3 is found, which translates to a function at least once diﬀerentiable.48 Noting that the low-pass ﬁltered NIPA radiance goes as k −(5/3+a) when k → ∞. This sets a to a value 4/3. Only slightly smaller 47 Recall that this substitution of Gamma-type functions for Green functions derived from transport physics was also done in the time-domain, in particular, for practical data exploitation in O2 A-band spectroscopy of cloudy skies [159, 160]. This is largely because, as for Fourier transformation, it has a closed-form Laplace transform. 48 This level of smoothness is quite remarkable since the 3D RT equation puts no constraints on gradients perpendicular to the beams (which, in this situation, are vertical).

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Fig. 5.26. A simple and convenient stochastic cloud model for radiative smoothing studies. Left: In the ﬁrst step, a fraction f1 of cloud ‘mass’ is transferred in a random horizontal direction from one half to the other. For every step after that, a fraction fn is similarly transferred at scale 1/2n such that fn /fn−1 = · · · = f2 /f1 . Right: The outcome for one realization at n = 14 when the parameters are f1 = 1/4 and f1 /f2 = 21/3 . This so-called ‘bounded cascade’ model [27, 228] has been tuned to yield a 1-point st.dev. of 1/3 of the (unit) mean and 2-point correlations reﬂecting a Fourier spectrum in 1/k −5/3 . These values are typical of the observed variability of local optical thickness for real marine stratocumulus clouds [28]. Both panels are reproduced with permission from Ref. [229].

values (a 1) are required for consistency between the Landsat observations [225] and the near-ﬁeld behavior of simulated [170] and observed [85] spatial Green functions for reﬂected laser light. In short, the anticipated range for a is quite narrow and, in any event, its precise value is not as important as that of |x| (or ρ) that determines the spatial extent of the running average. Figures 5.26 and 5.27 illustrate the NIPA procedure for a fractal cloud model. The stochastic ‘bounded cascade’ model used to generate horizontal cloud structure for a stratocumulus is explained graphically in the l.-h. panel of Fig. 5.26 while the resulting transect of cloud variability normalized to yield a unit mean is plotted in the r.-h. panel; its wavenumber spectrum is prescribed to be ∝ k −5/3 , as observed in real stratocumulus layers. This model is illustrated for 1D and used here as such, but it is readily generalized to 2D [230]. Figure 5.27 exempliﬁes the diﬀerences between MC, IPA and NIPA. The upper panel shows, on the one hand, τ (x) for a 2-km portion of the synthetic fractal cloud that extends to 12.8 km (and is periodically replicated beyond that). On the other hand, both MC and IPA predictions are plotted for the local albedo: we see how the IPA responds immediately to the fractal variability while the MC results are much like a running mean over several pixels. The lower panel shows MC, IPA and NIPA predictions for the local value of nadir radiance over the same portion of cloud. By comparing the two registered panels, we see that the MC radiance ﬁeld is not as smooth as its counterpart for albedo, patently because there is no longer angular averaging. The NIPA computation used the smoothing kernel in (10.8) with a = 0.5, |x| = 0.1 km (eight pixels). Finally, we see how much the prediction error with respect to MC ‘truth’ is reduced by going from the IPA to the NIPA. That completes the description of the forward NIPA where we improve the realism of the IPA by introducing scale-speciﬁc smoothness. The inverse NIPA

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Fig. 5.27. Comparison of simulated reﬂectivity ﬁelds using MC, IPA and NIPA for a portion of a 1D fractal stratoculmulus cloud. Top: On the l.-h. axis, we read the 1D horizontal variation in x of the local optical depth τ (x) (lower curve); the vertically uniform cloud is generated with a 10-step bounded cascade process from Fig. 5.26 with τ = 13 and H = 0.3 km (pixel/grid-scale = 12.5 m). The upper curves (r.-h. axis) in the same panel show the associated ﬂuctuations of albedo R(x), the normalized up-welling ﬂux in (2.18) for steady and uniform illumination, using both IPA and MC schemes; SZA is 22.5◦ and scattering is according to a Deirmendjian C1 phase function for a red wavelength for simplicity (both water- and land-surface albedo is negligibly small). Bottom: The r.-h. axis is the same as in the top panel but for normalized nadir radiance in (2.16) rather than hemisphericial ﬂux, under the same conditions of spatially uniform and steady illumination, and the computational NIPA scheme is added. The lower curves (l.-h. axis) highlights the reduced error with respect to MC when NIPA is used instead of IPA. Both panels are reproduced, with permission, from Ref. [229].

consists in taking actual cloud radiances and applying the corresponding roughening ﬁlter to restore the IPA and, from there, perform straightforward retrievals of (say) the cloud optical depth ﬁeld. Formally, that amounts to solving (10.6), → → viewed as an integral equation, for IIPA (− ρ ) knowing INIPA (− ρ ) from observation or 3D RT computation. In an ideal (inﬁnite-accuracy, noiseless) world, one only needs to perform the → − → − → − → − inverse FFT of I˜IPA ( k ) = I˜NIPA ( k )/P˜ ( k ). However, 1/P˜ ( k ) is a high-pass ﬁlter that will amplify any noise or small-scale numerical error. This is a classic ill-posed

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(i.e., numerically unstable) inverse problem. Marshak et al. [222] demonstrate on ‘observations’ obtained with a MC code (where the ‘truth’ is known), that careful → ρ ) even in the Tikhonov-type regularization [231] can be used to estimate IIPA (− presence of considerable noise from the MC scheme itself and, from there, obtain reasonable estimates of the local value of τ from a pre-computed inverse map of τ to nadir radiance from 1D RT. While the inverse NIPA can be applied to inject more realism into retrieved values of τ for stratocumulus-type clouds, the prerequisite determination of the critical wavenumber kc where the scale break occurs is an opportunity for cloud remote sensing in its own right. Speciﬁcally, it can be used to infer H, knowing the mean value of τ . Indeed, the reﬂected MTF used to inverse ﬁlter the quasi-nadir radiance data can only be a function of kH and dimensionless cloud or modeling parameters (τ , g, maybe χ, maybe a for Gamma-weighting, etc.). So, if we know kc in 1/km from observations and the suite of cloud parameters, we can estimate H in km.49 This idea is pursued using physical- rather than Fourier-space methods in the next example. 5.10.6 Zenith radiance reaching ground, and the nonlocal IPA Ground-based narrow-FOV radiometers that capture down-welling radiance from zenith have proven at once quite simple to build/maintain and extremely useful in cloud probing. One monospectral approach uses the solar background from the sensitive and well-calibrated detector in an operational ground-based micro-pulse lidar system [185] to infer optical depth of stratiform clouds by comparison with the predictions of 1D RT models for spectral zenith radiance at the laser wavelength [232].50 From the 3D RT standpoint, narrow-FOV observations from ground (or space) complement the more classic ﬂux-based measurements from ground (and at least moderately large pixels in MODIS-type instruments): the latter have the above 3D problem #1 while the former present #2. Another application of zenith cloud radiance measurements proposed by Marshak et al. [235] works around the 3D RT eﬀects by using a bi-spectral technique for inference of cloud optical depth above green vegetation; the method works even for a ﬁeld of broken clouds and, moreover, does not require absolute calibration. Viewed as a time-series, rapidly sampled zenith radiance from an overcast sky contains information about the spatial correlations in the radiance ﬁeld in essentially the same way high-resolution satellite imagery does. The only diﬀerence is 49 The simplest approach is to equate 1/kc with the RMS value for ρ, which appears parametrically in Gamma-approximated MTFs, since the ratio of ρ2 R and H 2 is a known function of known dimensionless cloud parameters. 50 This same calibration-based technique was proposed almost a decade ago [233] for the ultra-narrow FOVs in space-based lidar systems, but just recently applied to the Geophysical Lidar and Altimeter System (GLAS) lidar onboard the current ICESat (Ice, Cloud, and land Elevation Satellite) mission. In this time conﬁguration, a comparison with 1D RT predictions for nadir reﬂected radiance at the laser wavelength is performed [234].

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the nontrivial but standard time-to-space conversion using Taylor’s frozen turbulence hypothesis: just use the mean wind at cloud altitude to account for advection across the instrument’s FOV. In this context, the transmitted spatial Green function can be used in both forward and inverse NIPA, applied respectively –

to approximate a rigorous 3D RT computation of zenith radiance, given the cloud’s detailed structure, and – to derive the column optical thickness from a time-series of zenith radiance observations. Recall that the transmitted spatial Green function required for NIPA work has the same exponential decay in the far-ﬁeld as its reﬂected counterpart, but it is ﬂat rather than quasi-singular in the near-beam region. That implies in particular that the only reasonable Gamma-based parameterization of this Green function is for the special value of a = 1, yielding a simple exponential for the radial proﬁle√in (10.1); this choice is equivalent to stating that the RMS-to-mean ratio for ρ is 2. There is, however, the possibility of directly exploiting the outcome of the spatial correlation analysis to derive a key cloud property, namely, the physical thickness H of the (unbroken) layer. Beyond the Fourier wavenumber spectrum invoked in

Fig. 5.28. Simulated structure functions for local transmittance and zenith radiance. Scale breaks between smooth (h ≈ 1) and turbulent (h ≈ 1/3) behaviors are clearly visible, respectively at ≈ H and ≈ H/2. The ensemble of 10 simulated realizations of cloud optical depth were generated using a 10-step 1D bounded cascade model from Fig. 5.26 with the same parameters. Other cloud parameters of interest are mean optical depth τ = 13 and the constant physical thickness H = 0.3 km (indicated with a vertical dashdotted line). Scattering follows the usual H–G model (2.20) with g = 0.85. SZA is 30◦ . This plot is adapted from Ref. [24].

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the previous subsection, a popular statistical characterization of spatial correlation is the (second-order) ‘structure function’: SF(r) = [Izen (x + r) − Izen (x)]2 for a one-dimensional horizontal transect of zenith radiance, assumed to be along the x axis; r is a given scale parameter and the average is over a large interval in x. As for the Fourier spectrum, one naturally seeks power-law behaviors in r:

SF(r) = [Izen (x + r) − Izen (x)]2

1/2

∼ rh ,

(10.9)

where h is the ‘Hurst’ exponent (a.k.a. the global H¨older–Lipschitz exponent).51 Recognizable scaling behaviors are h = 0 for all stationary processes (i.e., that are decorrelated over the associated range of r), h = 1/3 for turbulence-like variability (corresponds to a Fourier spectrum in k −5/3 , as for the fractal model in Fig. 5.26), and h = 1 for all smooth (i.e., diﬀerentiable) ﬁelds. One can of course ﬁnd diﬀerent values of h in (10.9) over diﬀerent ranges in r. Indeed, von Savigny et al. [236,237] found that the time-averaged structure function of ground-based zenith radiance at a non-absorbing (red) wavelength went from h ≈ 1 scaling to h = 1/3 and ended with h = 0 at very long time lags, in excess of 1/2 to 1 hour or so (translating to 5–10 km for a nominal 5 m/s wind speed). Their sampling rate was 2 Hz and the longest records covered ≈4 hours. The authors found the predicted transition from smooth behavior to turbulence-like behavior at time lags that translated (via Taylor’s hypothesis) to scales commensurate with the thickness of the cloud deck (known through collocated mm-wavelength radar). Another way to ‘calibrate’ this simple method of estimating H from zenith radiance records (without a cloud radar present) is to use 3D RT simulations for stochastic cloud models tuned to reproduce the amplitude and scaling of realworld stratus. Figures 5.26(a) and (b) showed how to generate such a model in one horizontal direction and the outcome for one realization. Figure 5.28 shows SFs for both the simulations of zenith radiance ﬁeld and of the local ﬂux ﬁeld transmitted at cloud base. Recall from (8.5) and Fig. 5.15 that the RMS radius of the Green function T (ρ) for transmitted steady-state ﬂux is ≈H over the full range of interest in transport optical depths. We see in Fig. 5.28 that the scale break for ﬂux is, as expected, at ≈H. In contrast, the numerics show on the same ﬁgure that the scale-break for zenith radiance – the remotely observable quantity – is at ≈H/2. As was just noted for reﬂected light, this smaller value (favoring roughness) is understandable since, unlike oblique views (let alone the angular integration for ﬂux), a zenith radiance characteristic (i.e., a vertical beam) does not average by propagation over spatial variability unfolding in the horizontal. This is clearly an opportunity for a remote-sensing retrieval of H using very simple ground-based instrumentation described in Ref. [236]. Only a single nonabsorbing wavelength is required; it could be from a few spectral pixels in the continuum of an O2 A-band spectrometer. Knowing H, one can look back at the cloud information contained in the main cloud product anticipated from groundbased A-band spectrometers, namely, the mean in-cloud path-length ctT . The ratio with the estimated values of ctT and H can then be used to infer τ from 51

A variant of the Weiner–Khintchin theorem for non-stationary processes with stationary increments relates the Fourier spectrum and the second-order structure function. In particular, if the spectrum scales as k−β with 1 < β < 3 then β = 2h + 1.

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a consistently randomized diﬀusion-theoretical result, such as (10.5), given g, χ and52 a, or a homogenization approach. 5.10.7 Green functions at work in the adjoint perturbation approach to 3D radiation transport eﬀects Finally, we brieﬂy describe here another approach altogether to the above problem #2 (traced to pixel adjacency eﬀects) that is also grounded in Green function formalism. Adjoint perturbation theory is very general and can therefore be applied to many aspects of atmospheric RT [238–241], estimation of 3D transport eﬀects is just one example [238, 242–244]. Focusing on steady sources, we start by recasting the 3D RT problem in formal operator language: ΛI = Q where Λ = L − S,

(10.10)

is the linear transport (i.e., propagation and scattering) operator; it is obtained from (2.2), without the time derivative, and (2.3). In this application, we think of Q as a general distribution of sources over (x, Ω). We also deﬁne the adjoint 3D RT problem Λ+ I + = Q+ where Λ+ = L+ − S + .

(10.11)

The general deﬁnition of adjoint transport system is that, for all ‘reasonable’ test functions g and h of x and Ω, we have (g, Λh) = (Λ+ g, h) where (f1 , f2 ) = f1 (x, Ω)f2 (x, Ω) dx dΩ (10.12) M(H) 4π

is the scalar product in the function space of interest. In cases of interest here, the optical medium is M(H) = {x ∈ R3 ; 0 < z < H}. In particular, we require that the response of a detector with a response function D(x, Ω) is the functional E = (D, I) = (I + , Q). (10.13) With these deﬁnitions, it can be shown [245] that – L+ is L with Ω → −Ω, – S + is S with Ω → −Ω and Ω → −Ω, – Q+ (x, Ω) is D(x, Ω). Furthermore, the applicable boundary conditions for (10.11) at z = 0, H express that no adjoint radiance I + escapes the medium M. In other words, when going from the direct to adjoint transport problems, light sources and detectors reverse their roles and, correspondingly, the direction 52 The Gamma-weighting parameter a = 9 for the model in Fig. 5.26, generally less in real Sc.

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of propagation is reversed in space and in scattering. While the response in (10.13) describes how an arbitrary sensing device samples the radiance ﬁeld, it also describes how the adjoint radiance samples an arbitrary distribution of sources. Consequently, adjoint radiance is often called ‘importance’ (of any localized source for a given detector). Now suppose we are interested in the nadir radiance generated by reﬂected → sunlight, as can be observed from space, at a horizontal position − ρ obs . We then → − → − have Q(x, Ω) = F0 δ(z)δ(Ω − Ω0 ) and D(x, Ω) = δ(z)δ( ρ − ρ obs )δ(Ω − ˆz), noting that both distributions at z = 0 can also be expressed in the boundary conditions. This means that (10.11) is the adjoint counterpart of the deﬁning RT equation for the spatial Green function. Therefore, I + ≡ G+ in the present problem. Adjoint perturbation theory seeks to determine the deviation δE of E in (10.13) with respect to some known ‘base case’ when Λ goes from Λb to Λb + δΛ (and similarly for the adjoints). In the present problem of 3D RT eﬀect quantiﬁcation, we naturally take Λb as the uniform plane-parallel case, while δΛ captures deviations from uniformity in the extinction and scattering coeﬃcients and the ensuing horizontal gradients. Assuming uniform scattering properties (only extinction varies), this operator perturbation is ⎛ ⎞ ∂ + δσ(x) ⎝1 − 0 p(Ω · Ω)[·] dΩ ⎠ , δΛ = η − (10.14) ∂→ ρ 4π

where δσ(x) = σ(x) − σ. A general result from perturbation theory is that [241] δE = −(Ib+ , δΛIb )

(10.15)

to a ﬁrst-order approximation. For the present problem in remote sensing of heterogeneous clouds, we have Ib+ = G+ b , the adjoint Green function for an adjoint source at a roaming point on the illuminated boundary. The above expression (10.15) therefore reads in (x, y)-space as the convolution product of G+ b with δΛIb . We note immediately that the horizontal gradient term in (10.14) contributes nothing to δΛIb since the base-case radiance ﬁeld is invariant under arbitrary horizontal translation. Such convolutions are of course best done in Fourier space, which is precisely where we can obtain closed-form expressions for Green functions in the diﬀusion/P1 approximation. Assuming a non-absorbing wavelength in the example of the nadirviewing satellite imager probing an heterogeneous cloud layer, we could determine G+ b in Fourier space from the following system of ODEs

subject to

−F˜z+ = −[k 2 /3(1 − g)σ] J˜+ + σ e−σz , −J˜+ /3 = −(1 − g)σ F˜z+ + gσ e−σz ,

(10.16)

) ) J˜+ − 2F˜z+ )

(10.17)

z=0

) ) = 0, J˜+ + 2F˜z+ )

= 0. z=H

Note the changes in sign with respect to (5.15) and (5.17) that are dictated by the time-reversal implicit in adjoint transport. The solution of this problem has about the same complexity as the one presented in Appendix C.

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Since the receivers in the direct problem in section 5.5.3 measure radiation density J inside the medium and hemispherical boundary ﬂuxes F±z , the easiest sources to model here are isotropic, either internal or at a boundary. The expression for Ib has to take this into account. It can of course also be estimated within the same diﬀusion approximation. In that case, we would turn to the solution of (5.1), (e) when σa = 0, subject to (5.2), with χ = 2/3, for a uniform boundary source at z = 0; for an internal source distributed evenly over a horizontal plane at z = z0 , we would use (6.4)–(6.5), with the same restrictions.53 Recalling that the angular integral implicit in (10.15) is easily carried out in the diﬀusion limit, cf. (4.12)–(4.13), the beginning-to-end computation of the 3D RT eﬀect captured by δE can be done analytically. This is unique in the 3D RT literature, which is dominated by numerical techniques. For a glance at computational recipes for higher-order perturbations in 3D RT, we refer to Box et al. [243]. It is instantly clear that Green functions play a central role, much like propagators in perturbation expansions used in quantum mechanics. Incidentally, for domain-average 3D RT eﬀects, the ﬁrst-order estimate in (10.15) is zero because the integrated contribution of the second term in (10.14) vanishes identically. Thus, according to adjoint perturbation theory, large-scale 3D RT eﬀects are second-order at best. This is corroborated by countless numerical experiments: the domain-average impact of 3D RT is accurately captured by the IPA. Physically, the local horizontal ﬂuxes will indeed lead to both positive and negative deviations from the 1D RT prediction that tend to cancel upon spatial integration. This leads us to the idea of using the IPA as the base case instead of the strictly uniform plane-parallel medium. Details for this approach are out of the scope of the present review; we refer to Polonsky et al. [244]. However, the three panels of Fig. 5.29 illustrate the power of the IPA-based adjoint perturbation approach to 3D RT using a cloud model adapted from the two-dimensional ‘Case 2’ stratus cloud of the Intercomparison of 3D Radiation Codes (I3RC) project [246]. Rather than the boundary-leaving radiance of interest in cloud remote sensing, this computation was for the redistribution of the solar heating inside the cloud due to 3D RT, namely, J(x, z). Both perturbation and 3D RT results are reduced to the uniform planeparallel result based on the mean optical depth. Being based on an implementation in code of closed-form expressions, the perturbation estimate is instantaneous in comparison with the full 3D RT estimate. At the same time, most of the 3D RT eﬀect is reproduced in the right locations.

5.11 Summary and outlook Our primary goal in this review was to establish the sweeping utility of radiative Green functions in both passive and active cloud remote sensing, extended to incloud radiometry and used to retrieve cloud properties. We naturally pursue those 53

More sophistication is required to account for slant illumination, not for the base-case radiance (where the diﬀusion solution for a slant uniform collimated beam is well-known), but for the adjoint Green function (where a superposition of spatially distributed isotropic and anisotropic detectors along a narrow line is in order).

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Fig. 5.29. Adjoint perturbation approach to 3D RT. Top: Cloud model structure varies only in one horizontal (x) direction, from x = 0 to x = 32 km (periodic replication beyond and before); the local value of the extinction plotted here is uniform in the vertical (z) direction, from z = 1 km to z = 2.2 km hence H = 1.2 km (the rest of the 0 < z < 2.5 km domain is empty). Middle: J(x, z) − J1D (z) is plotted for the SHDOM estimate. Bottom: Same as above, but for the adjoint perturbation estimate. These panels are reproduced with permission from Ref. [244].

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properties of importance in climate science. The parameters of an opaque stratiform cloud of primary interest here are therefore, beyond the cloud’s altitude, its physical thickness H and optical thickness τ (equivalently, its volume-averaged extinction coeﬃcient σ = τ /H). Properties of secondary interest address internal variability of the cloudy medium, both cloud-scale stratiﬁcation and small-scale random ﬂuctuations driven respectively by radiation/convection processes and inescapable turbulence. For the remote sensing of cloud particle size, also an essential quantity in climate and precipitation studies, we point the interested readers to the appropriate passive and active techniques. These methods capitalize either on polarization [247] or wavelengths where condensed water has non-negligible absorption (sensitive to particle eﬀective radius) [210], primarily in passive approaches, or on the multiplescattering lidar signal coming from short times/ranges and, therefore, from very near the laser beam. In the latter active approaches, that early/near-beam signal is indeed dominated by the phase function, especially the forward diﬀraction peak (itself sensitive to the particle size distribution) [184]. Neither polarization (a means of selecting low orders of scattering), nor strong particle absorption processes, nor small-angle scattering are amenable to diﬀusion-based modeling presented here for conservative scattering. They are all serviced, however, by Green function theory in the broader framework of radiative transfer, including diﬀusion in the presence of weak absorption; see, for example, [248] on polarization, [249] on absorption, [250] on forward scattering. Our focus has been exclusively on optical wavelengths in the visible and nearIR where one can conﬁdently assume that absorption by cloud particles is small and often negligible. Scattering therefore dominates the radiation transport and, consequently, the physics of radiometry signals. Indeed we soon made the safe assumption for suﬃciently opaque clouds – when it is all but impossible to see where the sun comes from in transmission (τ 9) – that diﬀusion theory can be used to model the multiple scattering. The main beneﬁt of this classic approximation is that, in many relevant situations, the Green functions can be computed analytically; either in closed-form for direct comparison with observations, or else their space-time moments can be expressed analytically and compared with observational estimates. In this respect, we have simply opened new space-time dimensions in radiative transfer models that have already served the GCM community very well for uniform and steady sources, namely, the solar ﬂux. Another one of our goals was to show that the diﬀusion-theoretical approach to Green function estimation provides a powerful signal modeling framework, built largely with computer-assisted symbolic math, that uniﬁes very diverse modalities in the remote sensing of clouds. There are at least two emerging technologies that directly target the Green functions in space and/or time: (1) multiple-scattering cloud lidar with a very wide FOV, and (2) high-resolution diﬀerential absorption spectroscopy in the oxygen A-band. We have discussed implementations of these techniques from ground, space, and aircraft, each platform having its own particularities. In airborne systems, we include those designed to work from inside the cloud itself. The signal from so-called ‘in situ’ cloud lidar – and probably in situ O2 A-band spectroscopy as well – contains information to glean about the macroscopic cloud structure, thus complementing the usual microphysics instrumentation car-

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ried by aircraft into the depths of clouds to sample particle sizes, composition, and so on. It is not common, at least in atmospheric science, that the kind of advanced radiative transfer theory surveyed herein inﬂuences major thrusts in instrument development. Yet that is precisely what happened in the case of multiple-scattering cloud lidar, as well as for the cloud remote sensing application of oxygen A-band spectroscopy. We wish to reinforce this theory-driven approach to innovation in the design of actual instruments and/or algorithms for data exploitation. To this eﬀect, we have (1) revisited better-established methodologies in cloud remote sensing, showing them to rely implicitly on Green function formalism, and (2) pointed out throughout this review a number of avenues yet to be explored. Some of these proposed optical observation techniques target unknown sources below or imbedded in clouds, for example, lightning strokes. Others simply look for cloud-like media in Nature: turbid coastal waters, snow packs, sea-ice ﬂoes [251], dense vegetation canopies [252], etc. We have no doubt that, some day, lidars with multiple scattering capability will probe Europa’s ice cover, Titan’s thick haze, and worlds beyond. Meanwhile, back on planet Earth, there are applications of atmospheric optics that are not driven by meteorology or climate science. Visibility studies come to mind. It has been shown that a large proportion of small aircraft crashes – too often with fatalities – are due to pilot error that could be avoided with better knowledge of the prevailing low-visibility conditions. Multiple-scattering lidars are not expensive compared to both the cost of even a single accident and that of the competing technology in active cloud probing, mm-wave radars.54 So small airports, including aircraft carriers, will eventually be outﬁtted with such instruments that reliably deliver immediately actionable diagnostics on low clouds and fog. First responders on all-weather rescue missions can also beneﬁt from real-time quantitative knowledge of reduced visibility conditions. And what if that approaching aerosol cloud is transporting a toxin? For this potentially life-saving diagnostic, we can take a clue from recent developments in medical optics. Multiple-scattering cloud lidar was developed simultaneously and independently of optical tomography (OT), although they share a large amount of signal physics. OT [255] uses light diﬀusely transmitted through soft tissue, which is highly scattering in the near-IR, to locate and gauge anomalous (absorbing or vacuous) inclusions indicative of pathology, for example, aneurysms and tumors. This is made possible by fast and widely available numerical solvers for Laplace’s equation with a given, although nontrivial, outer geometry and arbitrary internal boundaries; the latter are varied until the forward model ﬁts the radiometric (Green function) data from as many source and sensor positions as necessary.55 OT has made tremendous progress as a low-resolution but inherently 54

The strategy in mm-wave cloud radar is to retain the single-scattering radar/lidar equation, and therefore to seek a wavelength in the electromagnetic spectrum where its assumptions are valid [148,253]. This leads to ∼mm wavelengths, with the extra burden on theory to connect radar reﬂectivities from cloud droplets (∝ r6 ) to properties of interest in the radiation budget (r2 ) and hydrology (r3 ) [254]. 55 In this picture, multiple-scattering cloud lidar determines the unknown distance of the absorbing cloud boundary opposite the laser source, as well as the unknown opacity of the cloud in between. With very restricted Green function sampling by OT standards,

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non-invasive medical imaging technique. A relatively recent development, reviewed in this volume by Klose [256], is inspirational for our problem of detecting toxic material in an optically thick cloud of scattering particles from a safe stand-oﬀ distance: we can excite ﬂuorescence with the laser light scattered throughout the cloud, and tune the detectors to the tell-tale ﬂuorescence wavelengths. Finally, the scope of this survey was limited to ‘normal’ diﬀusion modeling, i.e., amenable to classic PDEs and closely related to standard random-walk theory (used, for example, in Appendix E to derive a coarse but insightful characterization of Green functions based on their asymptotic scaling behavior). Furthermore, cloud structure was limited to horizontal plane-parallel slabs with random 3D opacity ﬂuctuations around a deterministic stratiﬁcation in the vertical. Although the climatically and hydrologically important class of single-layer stratocumulus clouds can be represented in this framework, many other cloud types require full-blown 3D radiative transfer with time-dependence. Yet Green functions can still be brought to bear on this more complex cloudiness, and analytical results (supported by realworld observations) are achievable using ‘anomalous’ diﬀusion; a brief overview is provided in Appendix F. In conclusion, we strongly advocate systematic exploitation of multiple-scattering Green functions, particularly for boundary sources and boundary ﬂuxes, which apply most directly to remote sensing. In the case of the dense clouds that have so far been our foremost concern, they have proven useful far beyond the eﬀort invested in a refresher in mathematical physics. We are conﬁdent that many other insights and applications will follow. We also advocate hierarchical modeling frameworks in any application area. In our case, the fact that we have approaches to Green function computation that range from back-of-envelope estimations to detailed numerical simulations of radiation transport physics has enabled us to advance with conﬁdence into quite foreign territory, for instance, completely new instrument concepts. Acknowledgments This research was supported largely by the Oﬃce of Biological and Environmental Research of the U.S. Department of Energy as part of the Atmospheric Radiation Measurement Program. The authors also acknowledge ﬁnancial support from NASA’s Radiation Sciences Program and from Los Alamos National Laboratory’s Lab-Directed Research & Development (LDRD) Programs. Publication of this paper was supported by the JPL, Caltech, under a contract with the National Aeronautics and Space Administration. We thank Howard Barker, Luc Bissonnette, Craig Bohren, Hartmut B¨ osch, Michael Box, Bob Cahalan, Brian Cairns, Christine Chiu, Dave Crisp, Ed Eloranta, Frank Evans, Nicolas Ferlay, Philip Gabriel, Barry Ganapol, Mike Hall, Lee Harrison, Robin Hogan, Yongxiang Hu, Yuri Knyazikhin, Alexander Kokhanovsky, Ed Larsen, Paul Lawson, Jim Liljegren, Steve Love, Matt McGill, Qilong Min, Michael Mishchenko, Dennis O’Brien, Lazaros Oreopoulos, Klaus Pfeilsticker, Gilles Roy, Christian von Savigny, Jim Spinhirne, Graeme Stephens, Tamas V´ arnai, Jim Weinman, Dave Winker, Warren Wiscombe, Yuekui Yang, and Eleonora Zege for many fruitful discussions. one has to make assumptions about inner and outer cloud structure, as described in the main text.

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˜ A Responses T˜(k) and R(k) for horizontal transport away from an isotropic boundary source in stratiﬁed clouds A.1 Deﬁnitions We recall from Bessel function theory [110] that In (x) = i−n Jn (+ix), and we deﬁne by analogy Xn (x) = i+n Yn (−ix), as an alternative modiﬁed Bessel function of the second kind; like In (x), it is a real-valued function for real-valued x. We also deﬁne f (x, y; z) = xI0 (z) + yI1 (z), g(x, y; z) = xX0 (z) + yX1 (z). Finally, we deﬁne DF (kH; τt /2χ, Δ) = f (+kH, (2 + Δ)τt /2χ; (2/Δ + 1)kH/2) × g(−kH, (2 − Δ)τt /2χ; (2/Δ − 1)kH/2) + f (−kH, (2 − Δ)τt /2χ; (2/Δ − 1)kH/2) × g(−kH, (2 + Δ)τt /2χ; (2/Δ + 1)kH/2). A.2 Transmitted light As required in section 5.5.2, the deﬁnition in (4.51) leads to Δ τt /2χ 8 . T˜(kH; τt /2χ, Δ) = × π DF (kH; τt /2χ, Δ)

(A.1)

A.3 Reﬂected light Similarly, the deﬁnition in (4.50) leads to NR (kH; τt /2χ, Δ) ˜ , R(kH; τt /2χ, Δ) = DF (kH; τt /2χ, Δ) where NR (kH; τt /2χ, Δ) = f (−kH, (2 + Δ)τt /2χ; (2/Δ + 1)kH/2) × g(+kH, (2 − Δ)τt /2χ; (2/Δ − 1)kH/2) + f (+kH, (2 − Δ)τt /2χ; (2/Δ − 1)kH/2) × g(+kH, (2 + Δ)τt /2χ; (2/Δ + 1)kH/2).

(A.2)

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ˆ B Responses Tˆ(s) and R(s) for pulse stretching for an isotropic boundary source in stratiﬁed clouds B.1 Deﬁnitions For a given cloud (τt , Δ) and χ, we deﬁne the following eight functions of s = s/c, i.e., the Laplace conjugate variable for path ct (expressed in 1/m): A± n (s H; τt , Δ) =

1 n τt s H (1 ± Δ)3 ), n = 1, 2, 4, 5, 0 F1 ( ; Γ (n/3) 3 3Δ2

where Γ (a) is Euler’s Gamma function and 0 F1 (a, x) is the conﬂuent hypergeometric function. We also deﬁne − DF (s H; τt , χ, Δ) = A− 1 − [χs H(1 − Δ/2)/Δ]A4 + 2 × A+ 2 + [τt (1 + Δ/2) /3χΔ]A5 + + A+ 1 + [χs H(1 + Δ/2)/Δ]A4 − 2 × A− 2 − [τt (1 − Δ/2) /3χΔ]A5 . B.2 Transmitted light As required in section 5.5.2, the deﬁnition in (4.51) leads to √ 3 1 ˆ T (s H; τt , χ, Δ) = × . π DF (s H; τt , χ, Δ)

(B.1)

B.3 Reﬂected light Similarly, the deﬁnition in (4.51) leads to ˆ H; τt , χ, Δ) = NR (s H; τt , χ, Δ) , R(s DF (s H; τt , χ, Δ)

(B.2)

where − NR (s H; τt , χ, Δ) = A− 1 + [χs H(1 − Δ/2)/Δ]A4 + 2 × A+ 2 − [τt (1 + Δ/2) /3χΔ]A5 + + A+ 1 − [χs H(1 + Δ/2)/Δ]A4 − 2 × A− 2 + [τt (1 − Δ/2) /3χΔ]A5 .

˜ C Responses T˜(k) and R(k) for steady illumination by a normally incident pencil-beam C.1 Deﬁnitions It is of interest to compare the system of ODEs at hand in (5.15) with the classic two-stream model (cf. Appendix D). Indeed, if one can ﬁnd an analogous choice

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of parameters, even allowing for an alternative choice of g and (more formally) for μ0 = 1, then one would have a solution. However, no such choice of ODE (e) parameters exists because our eﬀective absorption coeﬃcient, σa = k 2 /[3(1−g)σ], contributes neither to the inverse of the diﬀusivity constant, which multiplies Fz in the constitutive equation (the lower one), nor to the extinction coeﬃcient in the exponential source terms. Thus, there are too many constraints, and we therefore need solve the stated ODEs directly. We are brought to deﬁne " ! DF (kH; τ, g) = τ 2 − (kH)2 " ! × (3(1 − g)τ + 2kH)2 − (3(1 − g)τ − 2kH)2 e−2kH . C.2 Transmitted light Deﬁne NT (kH; τ, g) = τ × − (τ + kH)(3(1 − g)τ + 2kH)(3(1 − g)τ + gkH) + 2kH(3(5 − g(7 − 2g))τ 2 + 2(kH)2 ) e−kH

+ (τ − kH)(3(1 − g)τ − 2kH)(3(1 − g)τ − gkH) e−(τ +kH) . Applying the deﬁnition in (4.49), as required in section 5.5.3, with μ0 = 1 (and s = 0), for diﬀuse transmittance then leads to NT (kH; τ, g) T˜(kH; τ, g) = e−τ + DF (kH; τ, g)

(C.1)

for total (direct + diﬀuse) transmittance. C.3 Reﬂected light Deﬁne NR (kH; τ, g) = τ × − (τ − kH)(3(1 − g)τ − 2kH)(3(1 − g)τ + gkH) + (τ + kH)(3(1 − g)τ − 2kH)(3(1 − g)τ − gkH) e−kH + 2kH(3(1 − g(3 − 2g))τ 2 − 2g(kH)2 ) e−(τ +kH) . The deﬁnition in (4.48), as required in section 5.5.3, with μ0 = 1 (and s = 0), then leads to NR (kH; τ, g) ˜ (C.2) R(kH; τ, g) = DF (kH; τ, g) for reﬂectance.

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ˆ D Responses Tˆ(s) and R(s) for pulsed normal or oblique uniform illumination D.1 Deﬁnitions The ‘Eddington’ (diﬀusion-based) version of the classic two-stream model for steady-state solar radiation transport in uniform plane-parallel clouds is, at ﬁrst glance at least, a resource. This is especially attractive since we have easy access to Meador and Weaver’s [87] deﬁnitive treatment. In our notations, this problem is described by this system of ﬁrst-order ODEs: Fz = −σa J + σs e−(σs +σa )z/μ0 , J = −3[(1 − g)σs + σa ]Fz + 3μ0 gσs e−(σs +σa )z/μ0 . They are subjected to the same boundary conditions in (5.17) as our model from section 5.5.3, bearing in mind that Meador and Weaver’s cloud optical depth τ = τ + σa H. The case of normal incidence (μ0 = 1) is of particular interest in the application covered in section 5.9. Comparison of the above classic two-stream problem and the system of present interest in (5.22) shows that there is no simple analogy to be made because s = s/c, the eﬀective absorption coeﬃcient, does not contribute to the inverse of diﬀusivity that multiplies Fz in the constitutive equation (the lower one). A formal analogy can nonetheless be made, but at the cost of recasting the asymmetry factor as g in Meador and Weaver’s expressions. We would then use σa = s and σs = σ, hence SSA (in Meador and Weaver’s notation) ω0 = 1/(1 + s /σ) ≤ 1, with g = g/(1 − gs /σ) ≥ g (its magnitude can exceed unity!) and τ = τ + s H. We can exploit this approach, or solve the stated ODEs directly. In the end, we deﬁne DF (s H; τ, g, μ0 ) = 3(1 − g)τ s H × τ 2 + τ s H(2 − 3(1 − g)μ20 ) + (s H)2 √ × (3(1 − g)τ + 4s H)(1 − e−2 3(1−g)τ s H ) √ + 4 3(1 − g)τ s H(1 + e−2 3(1−g)τ s H ) . D.2 Transmitted light Applying the deﬁnition in (4.49), as required in section 5.5.5, with μ0 ≤ 1 (and k = 0), for diﬀuse transmittance then leads to NT (s H; τ, g, μ0 ) Tˆ(s H; τ, g, μ0 ) = e−τ /μ0 + DF (s H; τ, g, μ0 )

for total (direct + diﬀuse) transmittance, where

(D.1)

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NT (s H; τ, g, μ0 ) = 3τ s H × ((1 + 2μ0 − 3gμ20 )(1 − g)τ − (2gμ0 )s H − (1 − g)τ 2 ) √ × 3(1 − g)τ s H(1 − e−2 3(1−g)τ s H ) e−(τ +s H)/μ0 − [(2 + 3μ0 )τ + (2 + 3gμ0 (1 + 2μ0 ))s H] × (1 − g)τ √ √ × ((1 + e−2 3(1−g)τ s H ) e−(τ +s H)/μ0 − 2 e− 3(1−g)τ s H ) . D.3 Reﬂected light The deﬁnition in (4.48), as required in section 5.5.5, with μ0 ≤ 1 (and k = 0), leads to ˆ H; τ, g, μ0 ) = NR (s H; τ, g, μ0 ) R(s (D.2) DF (s H; τ, g, μ0 ) for reﬂectance, where NR (s H; τ, g, μ0 ) = 3τ × 3(1 − g)τ s H[(1 − g)τ 2 − gμ0 s H] − τ s H[(1 − g)τ (3μ0 − 2) + 3(1 − g)τ s H(2μ0 − (1 − g)(1 + 3gμ20 ))] & ' √ × 1 + e−2 3(1−g)τ s H − (1 − g)τ (s H)2 [2 + 3gμ0 (2μ0 − 1)] & ' √ × 1 − e−2 3(1−g)τ s H − 2(1 − g)τ s H[2 − 3gμ0 (2μ0 − 1) + (2 − 3μ0 )τ ] √ × e− 3(1−g)τ s H .

E Scaling exponents for diﬀusive Green function moments from the random walk approach In this extensive survey of diﬀusion theory in application to multiple-scattering Green function estimation, we have systematically used Monte Carlo simulation to validate numerically the radiative transfer approximation leading to convenient closed-form results. It is informative to go to the other extreme of this hierarchy of Green function modeling and perform a highly simpliﬁed version of MC simulation analytically, namely, estimate statistical properties of random (a.k.a. drunkard’s) walks. This approach reveals the physical essence of the problem of transport in dense clouds. In particular, one can derive the scaling exponents for τt = (1 − g)τ in all the dominant terms of the expressions we have derived for Green function moments, viz. (8.3)–(8.5) for transmitted light and (8.10)–(8.12) for reﬂected light. The same scaling appears in all other such groupings based on more sophistication in the cloud or source representation: the novelty aﬀects only pre-factors and preasymptotic corrections. We note that in random walk theory τt is the ratio of the only two scales that matter:

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– H, the outer scale (size of the domain bounding the stochastic process); – t , the inner scale (one MFP for an eﬀectively isotropic scattering). The latter scale deﬁnes diﬀusivity D = ct /3 in three spatial dimensions. E.1 Caveat about photons as ‘particles’ of light The term ‘photon’ was coined by Gilbert Lewis in 1926 to describe the quantum of the electromagnetic ﬁeld, of which light is a prime example. Even if this so-called ‘second quantiﬁcation’ assigns energy E = hν, momentum p = Ωh/λ, and spin S = ±h to photons, it is fundamentally incorrect to think of them as either classic or quantic particles traveling through space-time at velocity c. For instance, by deﬁnition, it is not the same photon that is incident and re-emitted by a scattering entity. Photons can populate energy levels in, for example, thermal sources and laser cavities; they can also be detected using photoelectric materials. In between, it is light – not photons – that propagates in optical media according to the laws of radiative transfer theory, which is a nontrivial construct from statistical optics [5]. The radiance ﬁeld predicted by the radiative transfer equation, and associated boundary conditions, is only a probability of detecting a photon (per photon emitted at the source) with a roaming virtual instrument. In Monte Carlo computation, it is very tempting to talk about the ‘photons’ launched in a simulation. This should be avoided, proper terminology is ‘histories’ or ‘trajectories’ or ‘realizations’. Recall that Monte Carlo is only a random quadrature approach for estimating integrals over radiances. The random walk theory presented here is basically a poor person’s Monte Carlo: too poor to own a computer, and only has some elements of probability theory to work with. So, although strongly reminiscent of wandering particles, we are dealing with light intensities, to be interpreted as probability densities for detection events. E.2 Elements of Brownian motion theory In boundary-free homogeneous 3D space, an isotropic source at x = y = z = 0 emits a diﬀusing ‘wavefront’ of particles propagating at a decreasing √ ‘velocity’ such that the mean distance from the origin, ≈ r 2 , grows only as Dt. This is just a reading of the classic unbounded diﬀusion relation [257] r 2 = 6Dt,

(E.1)

which results directly from the well-known Green function for the basic diﬀusion equation for particle density n(t, r) = J(t, r)/c: [∂t +D∇2 ]n = δ(t)δ(r) ⇔ n(t, r) = 2 e−r /4Dt /(π4Dt)3/2 , as stated in (7.2) for the short-time/near-ﬁeld in situ cloud lidar signal. In the statistical physics of Brownian motion, a lesser known but extremely useful result is the ‘law of ﬁrst returns’ [258]. Focusing, for simplicity on 1D random walks (where D = ct ) along the z-axis, we seek the PDF of t > 0, the random epoch when the coordinate of Brownian particle (that left z = 0 at t = 0) ﬁrst changes sign. It can be shown [31, 259], that

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3/2

t c e−t /2ct dt πt ct dt (E.2) ∼ 3/2 , t if we ignore the exponential cutoﬀ at early times. This is an interesting PDF associated with the gambler’s ruin problem: How long does it take a person who comes to the roulette table with $1, and always bets on ‘red’, to walk away with nothing. There is no mean time – it is divergent – and that may go a long way in explaining why gambling is addictive, and why casinos should never close. Before loosing everything in time with probability one to this casino with an inﬁnite reserve, gains can be considerable. The corresponding RT problem in this review is that of reﬂection from a semiinﬁnite (H → ∞) non-absorbing medium, where ct is indeed inﬁnite, as are all higher moments. Fractional-order moments of order q < 1/2 are, however, ﬁnite. Pr{t, dt} = √

E.3 Transmitted light Now r 2 = x2 + y 2 + z 2 and, by symmetry, all three components are equal in magnitude on average. Therefore, since z = H where ρ2 T is computed, ρ2 = x2 + y 2 ≈ H 2 . This immediately explains the independence of that moment in (8.5) with respect to cloud opacity, i.e., optical depth. As soon as the diﬀusing wavefront reaches the opposite boundary, i.e., t ≈ 3H 2 /D ∼ H 2 /ct = (H/c) × τt based on (E.1), we will detect the transmitted Green function. In other words, reinterpreting the ﬁxed t in (E.1) as a random variable, we can anticipate ctT /H ∼ τt . This expectation is conﬁrmed by ‘exact’ (PDE-based) diﬀusion theory in (8.3). There is no simple argument for the scaling of the second-order moment in time in (8.4). The fact that it goes as ct2T tells us that the distribution of arrival times at the boundary opposite the source of particles is relatively narrow. It is interesting that we can estimate at least the scaling of Green function moments in transmission without knowledge of the overall probability of transmission for given H and D = ct /3 (alternatively, τt = H/t ). This calls for the law of ﬁrst returns in (E.2). Real clouds have ﬁnite physical and optical thicknesses and real casinos have ﬁnite banks. We can approximate the probability of transmission – breaking the casino’s bank – by truncating the PDF in (E.2) at tT ∼ H 2 /ct . This leads to ∞ Pr{t, dt}, (E.3) T ≈ Pr{t > tT } = t T

which scales as t /H = 1/τt . That is indeed the asymptotic behavior for T (τt ) in (5.5) and all other transmission laws we have come across for 0 = 1; see, for example, Fig. 5.15. E.4 Reﬂected light Temporal/path moments for reﬂected light can be estimated for a ﬁnite domain, namely, 0 < z < H, by deﬁning a truncated (and renormalized) version of the

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PDF in (E.2) for the ﬁrst-return process. Allowing time for the particle to return to z = 0 from almost being transmitted at z = H, we compute speciﬁcally 2t T

tq Pr{t, dt},

Iq = 0

tq R ≈

Iq ∼ I0

t c

1/2

H ct

q−1/2 ,

(E.4)

where we have neglected the diﬀerence between I0 and unity, namely, T in (E.3). 1/q Recalling that H/t = τt , this leads to (ct)q R ∼ H × (τt )1−1/q , as was found in the limit τt → ∞ for (8.10)–(8.11) and all other reﬂection laws. It is remarkable that the moments (ct)q R all scale diﬀerently with τt whereas we fully expect that (ct)q T ∼ ctqT , for q ≥ 2. We can trace this property to the mixture, made clear in (E.4), of short and long paths. We have emphasized several times in the main text how much this helps the cloud remote sensing enterprise. As we did for the spatial Green function in transmission, we can roughly estimate the RMS value of ρ for reﬂection from (E.1), with D ∼ ct and (E.4) for q = 1. We obtain ρ2 R ∼ DctR ∼ Ht . In other words, the RMS ρ for reﬂected light goes as the harmonic mean of t and H, the inner and outer scales of the diﬀusion problem at hand.

F Scaling exponents for time-domain anomalous diﬀusion by extending the random walk approach F.1 Anomalous diﬀusion At ﬁrst glance, the problem of 3D RT through an atmospheric column populated with broken and/or multiple cloud layers seems intractable, except maybe with heavy-duty numerical methods. Depending on what radiative properties are targeted, that ﬁrst impression may be quite inaccurate. For instance, Pincus, Barker, et al. [260, 261] developed (with GCMs in mind) the McICA model, a numerical but eﬃcient and unbiased method of estimating large-scale boundary ﬂuxes and ﬂux-divergence proﬁles, hence radiative heating/cooling rates. McICA creatively merges the concepts – –

of MC, viewed as a robust method of random quadrature (rather than a numerical solution of the 3D RT equation), and of IPA, appropriately renamed ICA (for independent column approximation) in this context of radiation energy budget computation where no pixels exist.

Moreover, two of the present authors have explored the alternate theory of ‘anomalous’ diﬀusion, which is still far from being well developed, for estimating largedomain/ensemble average ﬂuxes (currently, only at the boundaries). In their original paper, Davis and Marshak [80] generalized the random-walk approach used in Appendix E to situations where steps are usually small (inside

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clouds) but not infrequently very large (between clouds). They assumed distributions of step size s with power-law tails, ∼ 1/s1+b , such that all moments of an order greater than b > 1 are divergent; it indeed seems natural to require that the MFP (average value of s) be ﬁnite. We continue to use here the transport MFP t = s/(1−g) as the eﬀective MFP for an isotropic scattering. Davis and Marshak then addressed ﬁnite cloudy media with slab geometry (thickness H), showing −α/2

1. that transmittance Tα scales as τt , and 2. that mean path for transmitted light ctT goes as H × τtα−1 , where τt = H/t is the total scaled optical depth of the variable cloudy layer, and α = min{b, 2}. These scaling laws revert to our present ﬁndings for any b ≥ α = 2 (cf. Appendix E). Succinct derivations of these generalized scaling laws are as follows: –

First, we need to consider L´evy’s generalizations [31, 262, 263] of the central limit theorem. The standard result is that the variance of a sum of independent random variables is the sum of their variances, and it becomes normally distributed as the length of the sum increases without bound. But what if the variances are inﬁnite? Then other cumulants than variance are additive, and the PDF of the (normalized) sum becomes asymptotically close to a class of distributions known as ‘L´evy-stable’. These PDFs are parameterized in particular by the L´evy ‘index’ α < 2, which is the order of the smallest diverging moment. Let zn be the coordinate of a 1D random-walking L´evy particle after n isotropic scatterings, starting at z0 = 0. In the absence of boundaries, it obeys (zn /t )α ∼ n.

(F.1)

where α = min{b, 2} ∈ (1, 2) (restricted here to cases where the MFP t is ﬁnite). The angular brackets have a somewhat diﬀerent meaning here: the above relation can be interpreted as an attempt to estimate the lowest (logarithmically) diverging moment of the symmetric (randomly oriented) steps ±s when n b < 2 (step variance is then ∞), viz. zn /n = i=1 ±si /n. This replaces zn2 = (variance of ±s) ×n in classic diﬀusion, the 1D equivalent in discrete time of z(t)2 = r(t)2 /3 = 2t ct from (E.1). Noting that n ≈ ct/t if suﬃciently large, we can now obtain from (F.1) the scaling of ctT for a ﬁnite medium of thickness α−1 H and scaled optical depth τt : H α ∼ α . t × (ctT /t ), hence ctT /H ∼ τt QED. – Second, we need to update the continuous law of ﬁrst returns in (E.2) for path ct in a semi-inﬁnite domain with a discrete version for n: Pr{n ≥ N } ∼ N −1/2 ,

(F.2)

where n is the number of scatterings suﬀered by a light beam before crossing the z = 0 plane where it departed from. This expression is far more general than (E.2), which assumes Gaussian steps: (F.2) only only requires that the distribution of algebraic steps (±s) to be symmetric, i.e., ± with equal probability [264, 265]. By deﬁnition, transmittance is then Tα ∼ Pr{n ctT /t }, −α/2 . QED. which equals τt

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It is interesting to note that, in order to estimate a steady-state transport property like Tα , we need to go through the framework of time-dependent transport. Transport unfolds in time, even for steady sources. F.2 Observational validation, and evolution toward anomalous transport Surprisingly – or maybe not – empirical evidence has cumulated over the past decade that supports anomalous diﬀusion theory [161, 164, 165]. It is based on estimates of ctT from ground-based O2 A-band spectroradiometry collected under all kinds of cloudy skies; see section 5.3.1 and, for the special case of uniform clouds, section 5.8.2. Although that theory was ﬁrst inspired by research in contemporary statistical physics [266], it was later justiﬁed by investigations of light propagation in random-but-correlated media56 [10, 78, 79, 267, 268]. Further justiﬁcation of the P (s) ∼ 1/s1+b ansatz for RT in cloudy atmospheric columns comes in retrospect from the 1996 paper by Barker et al. [221]. Their ﬁnding of Gamma distributions for the optical depth in a wide variety of Landsat cloud ﬁelds can indeed be interpreted as an observation of Gamma-distributed optical paths across a ﬁxed distance that happens to be H and happens to be along the vertical. Recall from section 5.10.3 that the Gamma distribution’s parameters are the mean τ and the variability parameter a = 1/(τ 2 /τ 2 − 1). We then ﬁnd for mean direct transmittance ∞ Tdir (τ , a) = Tdir (τ ) = exp(−τ ) Pr{τ , a; τ, dτ } 0

1 = . (1 + τ /a)a It is easy to verify that Tdir (τ , a) → exp(−τ ) as a → ∞. The transmission law in (F.3) can in turn be interpreted as a new propagation kernel with a power-law tail that can be used in a mean-ﬁeld 1D RT model. As shown by the ﬁrst author in Ref. [81] using the more recent observational evidence [165], this anomalous transport model supersedes the older anomalous diﬀusion model where the steps are eﬀectively L´evy stable; it is however, only a numerical recipe at present. Interestingly, Barker et al.’s original goal was to motivate Barker’s [216] Gamma-weighted two-stream model, which develops in the opposite logic: ﬁrst solve the multiple-scattering 1D RT problem for cloud optical depth τ , then randomize the result and determine the domain average (cf. section 5.10.3). Here, we start by averaging the propagation kernel over the spatial disorder, then formulate 56

Three of these papers were by Kostinski, Shaw, and Lanterman [78, 79, 267] who start from very general considerations in discrete-point statistics; these authors challenge the ability of 3D radiative transfer, a theory grounded in a continuum representation of optical media, to account for the likely deviations from Poissonian behavior that they speculate about. The last one in the chronological series is by two of the present authors [10] who pick up the challenge, and leave only media with the intriguing possibility of ‘super-homogeneity’ (i.e., negatively correlated particle positions discussed in Ref. [79]), as an open frontier for radiative transfer theory.

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and solve the resulting new 1D RT equation that targets domain average ﬂuxes, radiances, etc. As competition for the anomalous diﬀusion/transport models in explaining the emerging cloudy-sky climatology of ctT /H from O2 A-band (it decreases as variability increases), we oﬀer the Gamma-weighted diﬀusion model described in section 5.10.4. Equations (10.4) and (10.5) in the main text yield respectively − min{a,1}

T ∝ τt ctT ∝

and

min{a,1} τt

in the limit of asymptotically large τt . Only future simulations and observations can help decide which is the more accurate representation of Nature’s way of propagating solar radiation in the Earth’s cloudy atmosphere. Finally, the importance of pre-factors and pre-asymptotic behavior was amply demonstrated in the present survey. Since it is based entirely on classic diﬀusion theory, we are of course curious about how to predict them for its anomalous counterpart. The Gamma-weighted time-dependent diﬀusion model from section 5.10.4 delivers the desired pre-factors and pre-asymptotic corrections; see Fig. 5.25. Scholl et al. [165] propose an ad hoc hybrid of the scaling results from Davis and Marshak [80] and the details they obtained for homogeneous clouds [24]. At present, all we know [81, 269] is that a rigorous approach to anomalous transport theory will involve pseudo-diﬀerential equations (fractional-order PDEs [270]) that can be cast as integral equations with singular kernels. This program remains to be implemented. List of abbreviations nD 3+1D BC BRDF cw FFT FOV GCM GPS H–G IPA LWC

n-dimensional (n = 1, 2, 3) three-plus-one-dimensional (i.e., space-time) boundary condition bi-directional reﬂection distribution function continuous-wave (describes steady-source lasers) Fast Fourier transform ﬁeld-of-view Global Climate Model Global Positioning System Henyey–Greenstein (scattering or phase function) from Ref. [25] independent pixel approximation [kg/m3 ] (cloud) liquid water content, (4π/3)r3 × the density of water (103 kg/m3 = 1 g/cm3 ) × droplet concentration (1/m3 ) LWP [cm] (cloud) liquid water path, LWC ×H ≈ (2/3)τ re in the limit of large size parameters (2πr/λ) MC Monte Carlo MFOV multiple ﬁeld-of-view (lidar) MFP mean-free-path MTF modulation transfer function MuSCL multiple-scattering cloud lidar NIPA nonlocal independent pixel approximation

276

ODE OT PDE PDF PSF RMS RT Sc

SF SNR SSA SZA UAV

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ordinary diﬀerential equation optical tomography partial diﬀerential equation probability density function point spread function root-mean-square radiative transfer stratocumulus, a frequent cloud formation (especially in the marine boundary layer) with relatively ﬂat top and bottom, but often with quite variable opacity in between (observed to have long-range/fractal-like correlation structures). second-order structure function signal-to-noise ratio single scattering albedo solar zenith angle unmanned aerial vehicle

List of notations a b

c ct dobs D f F F g G H I J − → k K t m p ps P q

[-] characteristic exponent in Gamma PDF, inverse of reduced variance [-] generic exponent for PDFs with ‘fat’ tails, i.e., decaying in a power law with exponent 1 + b (hence b for the cumulative probability of the random variable exceeding a given value) [m/s] speed of light in vacuum [m] path-length of light since emission from pulsed source [m] distance from remote observer to nearest cloud boundary [m2 /s] radiative diﬀusivity, ct /3 [-] fraction of forward vs. P1 scattering in δ-Eddington rescaling [J/s/m2 ] generic for a hemispherical radiative ﬂux, possibly space- or timeintegrated [J/s/m2 ] radiative vector (net) ﬂux [-] asymmetry factor of phase function [J/s/m2 /sr] radiative transfer Green function, almost but not quite always for a boundary source and boundary observation [m] physical thickness of (horizontal) plane-parallel cloud [J/s/m2 /sr] radiance at detector [J/s/m2 ] radiative scalar ﬂux (radiant energy density ×c) → [rad/m] horizontal wavenumber (2D Fourier–Hankel conjugate of − ρ) 2 [J/s/m ] radiative tensor ﬂux (radiative pressure ×c) [m] mean-free-path (MFP), 1/σ [m] transport MFP, /(1 − 0 g) = 1/σt [-] non-dimensional eﬀective similarity ratio in space-time Green function estimation, (k/σt )2 + 3s/cσt [1/sr] (volume) scattering phase function [1/sr] surface scattering phase function, its relation to the BRDF being ps (Ω → Ω) = |μ |ρ(Ω → Ω)/α0 [1/(random variable units)] generic for probability density functions [-] integer or fractional order of a statistical (non-centered) moment

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qF qJ q0 Q r re R s s/c t T x

[J/s/m3 ] anisotropic volume source term (for F ) [J/s/m3 ] isotropic volume source term (for J) [J/s/m2 ] isotropic boundary source term in diﬀusion theory [J/s/m3 /sr] radiative transfer volume source term [μm] cloud droplet radius (hence moments rq over the size distribution) [μm] eﬀective cloud droplet radius, r3 /r2 [1/s/m2 ] local/instantaneous reﬂectance, or integrals thereof [1/s] Laplace conjugate variable of t [1/m] Laplace conjugate variable of ct [s] time, typically starting at the release of a radiant energy pulse [1/s/m2 ] local/instantaneous transmittance, or integrals thereof [m] position in 3D space (x, y, z)T , where zˆ is vertical (oriented along the direction of source-beam propagation)

α

[-] L´evy index, relating to sums of independent identically distributed (iid) random variables, inﬁmum of 2 and the order of smallest diverging moment of the PDF decaying as a power law with exponent −(1 + α) [-] surface albedo, ground at z = 0, H [-] characteristic exponent of power-law wavenumber spectrum in turbulent media, which decays in 1/k β [-] exponent for power-law model of internal stratiﬁcation [-] parameter for linear gradient model of internal stratiﬁcation [-] rescaling parameter for Cairns’ or Larsen’s homogenization theories of small-scale 3D RT eﬀects [-] sin θ = 1 − μ2 [-] sin θ0 = 1 − μ20 [◦ or rad] zenith angle for propagation direction [◦ or rad] zenith angle for solar source [1/m] absorption coeﬃcient for a gas [nm] wavelength [-] vertical direction cosine, Ωz = cos θ [-] μ for solar source, cos θ0 [cm−1 ] wavenumber = 107 /λ, when wavelength is in nm [-] reserved for τt /2χ in strictly similar diﬀusion models [-] single scattering albedo, σs /σ [m] position in horizontal plane, typically reckoned from a normally incident laser beam [1/m] local extinction coeﬃcient, cross-section (in m2 ) × droplet concentration (in 1/m3 ), where cross-section ≈ 2πr2 in the limit of large size parameters (2πr/λ) [1/m] local absorption coeﬃcient [1/m] local scattering coeﬃcient [1/m] local transport extinction coeﬃcient [-] cloud optical depth [-] rescaled or transport optical depth, (1 − 0 g)τ [◦ or rad] azimuthal angle for propagation direction [◦ or rad] azimuthal angle for solar source

α0,H β γ Δ η η0 θ θ0 κν λ μ μ0 ν ξ 0 − → ρ σ

σa σs σt τ τt φ φ0

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[-] extrapolation length in units of transport MFPs t [-] direction of propagation [sr] inﬁnitesimal solid angle, dμ dφ [-] propagation direction for source [-] direction of observation (opposite of propagation toward instrument)

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6 Radiative transfer of luminescence light in biological tissue Alexander D. Klose

6.1 Introduction Radiative transfer of light in biological tissue plays a vital part in in vivo biomedical optics. Tissue constitutes a highly scattering and absorbing medium for light at wavelengths of the visible and near-infrared spectrum, and its light-tissue interaction can be exploited not just for therapeutic but also for diagnostic purposes [1–3]. In biomedical optics, light sources are placed on top of the tissue surface, either in contact or non-contact, and illuminate the tissue. Optical images of the reﬂected or trans-illuminated light are taken, which subsequently provide valuable biomedical information about macroscopic tissue changes. These optically detected tissue changes are mostly based on the absorption contrast that is caused by intrinsic tissue chromophores. The optical contrast of the detected images is, however, relatively poor due to the multiple scattered light inside tissue. Optical imaging has already been utilized in clinical and pre-clinical practice in order to detect these macroscopic tissue changes. Advances have been made in, for example, studying brain function [4,5], optical mammography and detecting breast cancer [6,7], small animal (for example rodents) imaging [8–11], or in detecting the inﬂammatory progression of rheumatoid arthritis in ﬁnger joints [12]. Optical molecular imaging of small animals utilizes luminescent imaging probes as imaging contrast, instead of intrinsic absorption properties of tissue. These probes emit light that can be utilized for studying gene expressions, protein–protein interactions, and drug eﬀects at a cellular and molecular level prior to the appearance of macroscopic tissue changes [13–21]. Images of the luminescence light intensities on the tissue surface can directly be taken, but they do not reveal the actual location and strength of the sources deep inside tissue due to strong light scattering. Luminescence tomography, on the other hand, measures the light intensities on the tissue surface and, subsequently, an image reconstruction algorithm recovers the spatial location of the unknown luminescent probe distribution [20]. In this regard, radiative transfer models play an important role for image reconstruction by predicting the luminescence light intensities on the tissue boundary. The correct choice of a radiative transfer approximation impacts the accuracy of the ﬁnal image reconstructions and the speed of computation.

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Solving the radiative transfer model for a given luminescent probe distribution is also referred to as solving the forward problem of the image reconstruction task. Whereas, ﬁnding the unknown luminescent probe distribution from the measured luminescence light intensities on the tissue surface is considered as solving the inverse problem. Both entities are closely intertwined within an image reconstruction algorithm [1]. A major diﬃculty in solving the inverse problem is imposed by strong light scattering of tissue, which limits the application of well-established image reconstruction algorithms as used in x-ray computed tomography (CT) [22]; see Table 6.1 for the abbreviations used. CT only deals with non-scattered or single-scattered high-energy photons and radiative transfer can be approximated by a solution of an integral equation. Inversion methods, such as the inverse Radon transform or the back-projection technique, yield the spatial distribution of the x-ray attenuation coeﬃcient in tissue. Tomographic imaging of luminescent probes, however, requires solving an inverse problem based on an integro-diﬀerential equation, namely the equation of radiative transfer (ERT). Solving the inverse source problem based on the ERT for tissue with nonuniform optical parameter distributions and complex geometries still remains a challenging task. Table 6.1. Abbreviations ART BLT CCD CT DD DE DOT EGFP ERT FD FMT FV GMRES GTLS Hb HbO2 ICG LED mfp PN RFP SD SI SPN SOR SN tmfp TSVD

algebraic reconstruction technique bioluminescence tomography charged-coupled-device x-ray computed tomography diamond-diﬀerencing delta-Eddington diﬀuse optical tomography enhanced green ﬂuorescent protein equation of radiative transfer ﬁnite diﬀerence ﬂuorescence molecular tomography ﬁnite volume generalized minimal residual gaseous tritium light source hemoglobin oxyhemoglobin indocyanine green light-emitting diode mean free path spherical harmonics red ﬂuorescent protein step-diﬀerencing source iteration simpliﬁed spherical harmonics successive overrelaxation discrete ordinates transport mean free path truncated singular value decomposition

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This paper will provide an overview about the ERT in steady-state, time, and frequency domain for modeling ﬂuorescence and bioluminescence light propagation. We will discuss in detail diﬀerent radiative transfer approximations to the ERT that are currently used in optical molecular imaging of luminescent in small animal tissue. These approximations are the methods of discrete ordinates, the spherical harmonics method, the simpliﬁed spherical harmonics method, and the diﬀusion method. Last, we will give some examples of numerical techniques for solving the forward and the inverse luminescent source problem of radiative transfer.

6.2 Light–tissue interaction Light–tissue interaction is governed by the optical tissue parameters, i.e. the scattering coeﬃcient, μs , the absorption coeﬃcient, μa , the anisotropy factor, g, and the refractive index, nm [23, 24]. The scattering and absorption coeﬃcients, in units of cm−1 , are the product of the microscopic scattering and absorption cross-sections and the particle density of scatterers and absorbers. All optical parameters are wavelength-dependent. Table 6.2 shows some examples of optical tissue parameters. Absorption of light in tissue is caused by chromophores, such as hemoglobin, oxyhemoglobin, NADH, and cytochrome oxidase [3]. The absorption coeﬃcient is the product of the extinction coeﬃcient, , in units of cm−1 M−1 , and the concentration, C, with unit M, of the chromophore. Light absorption of hemoglobin can vary over several orders of magnitude for wavelengths of the visible and near-infrared spectrum (Figure 6.1). Typical absorption parameters are in the range of 0.5 cm−1 to 5 cm−1 at wavelengths λ < 600 nm. In the red and near-infrared regions with λ > 600 nm, the absorption coeﬃcient varies between 0.01 cm−1 and 0.5 cm−1 . Scattering of light is caused by optical interfaces with diﬀerent refractive indices. Mitochondria and nuclei in tissue cells have been found responsible for most scattering events in tissue [25–28]. The scattering coeﬃcient varies only slightly as a function of wavelength and typical values can be found between 10 to 200 cm−1 for diﬀerent tissue types [29]. Light scattering events are strongly forward-peaked and are well-described by the Henyey–Greenstein scattering phase function with the mean scattering cosine g (also termed anisotropy factor) [30]. The anisotropy factor varies typically between 0.5 to 0.98 depending on the tissue type. The reduced scattering coeﬃcient is deﬁned as μs = (1 − g)μs . Typical reduced scattering coeﬃcients of most tissue types are between 5 and 50 cm−1 . The mean free path (mfp) in radiative transfer is the length (μs + μa )−1 , whereas the transport mean free path (tmfp) is deﬁned as the length (μs + μa )−1 , which plays an important role in diﬀusion theory (section 6.7.4). The index of refraction nm of bulk tissue is mostly due to the predominant water content of tissue (nm = 1.33). Its values in tissue approximately range from 1.3 to 1.5 [31–34]. The refractive index mismatch at the tissue–air interface is responsible for partial reﬂection of light that tries to escape through the tissue boundary. Moreover, a refractive index nm > 1 has an impact on the speed of light inside tissue and is, thus, important for time and frequency domain radiative transfer. We refer the reader to [23] and [29] for a comprehensive review of optical tissue parameters.

296

Alexander D. Klose Table 6.2. Examples of optical tissue properties in vitro

Tissue

aorta (human) bladder (human) brain (human): – white matter – gray matter liver (human ) liver (bovine) liver (rat) liver (rat) liver (rat) lung (human) muscle (chicken) muscle (rabbit) skin (human) skin (rat) skin (rat) skin (rat)

Wavelength

Scattering coeﬃcient

Reduced scattering coeﬃcient μs −1 [cm ]

Absorption coeﬃcient

μs [cm−1 ]

Anisotropy factor g [n.u.]

λ [nm]

Reference

633 633

31 29.3

0.9 0.91

3.1 2.64

2.3 1.4

[35] [29]

633 633 630 633 500 633 750 635 633 630 633 500 633 750

51 60.2 414 — — — — 324 229 110 187 — — —

0.96 0.88 0.95 — — — — 0.75 0.965 0.846 0.81 — — —

2.04 7.22 — 5.23 13 7.5 6.5 — 8 16.5 35.5 45 23 17

1.58 2.63 3.2 3.21 12 3 1.3 8.1 0.12 1.4 2.7 0.48 0.22 0.25

[29] [29] [29] [29] [23] [23] [23] [29] [29] [23] [29] [23] [23] [23]

μa [cm−1 ]

Fig. 6.1. Extinction coeﬃcient of hemoglobin (Hb) and oxyhemoglobin (HbO2 ) as a function of wavelength. Light absorption signiﬁcantly increases towards wavelengths of the visible spectrum. (Data taken from http://omlc.ogi.edu/spectra/hemoglobin/index.html, Dr Scott Prahl, Oregon Medical Laser Center, Portland.)

6 Radiative transfer of luminescence light in biological tissue

297

6.3 Luminescent imaging probes Luminescent imaging probes, also termed contrast agents or , provide the optical imaging signal originating from a speciﬁc molecular or cellular event. These imaging probes are either administered from outside or are genetically expressed inside a living small animal. Luminescent probes are mainly divided into two large groups depending on the mechanism of their light emission activation. A ﬂuorescent probe requires an external energy source for light emission activation. The excitation energy is usually provided by a light-emitting diode (LED) or a laser, which illuminate the tissue surface. In contrast, a bioluminescent probe does not require an external energy source for activation, but instead draws its energy for light emission from a chemiluminescent reaction inside tissue [36]. 6.3.1 Fluorescent probes The simplest ﬂuorescence imaging probe is indocyanine green (ICG) [37]. This non-speciﬁc probe, however, is not targeted to any speciﬁc biological processes at the cellular level and, hence, is mainly used for vascular imaging [38]. Fluorescent probes that report only on a speciﬁc molecular or cellular event are generally characterized as active , activatable , or reporter proteins [20,21,39]. Active and activatable are administered to the animal from outside, whereas ﬂuorescent reporter proteins are genetically expressed at the target site inside the animal. 6.3.1.1 Active Active consist of an aﬃnity and a signaling component. The signaling component is a ﬂuorophore, for example a ﬂuorescent dye or quantum dot, that constantly emits ﬂuorescence light upon photonic activation. The aﬃnity component is a ligand (for example peptide, protein, antibody) that binds to target molecules, such as to speciﬁc cell-surface receptors [17, 40–45]. Hence, active accumulate at target sites, while most nonbound active probes are cleared. The remaining nonbound probes inside tissue, however, are a limiting factor for detecting the speciﬁc ﬂuorescent target signal and they contribute to the background noise of ﬂuorescence light. 6.3.1.2 Activatable Activatable , also termed smart probes, have been designed for overcoming the limitations of active by partially eliminating the ﬂuorescent background signal of nonbound probes. Their signaling component, usually two identical ﬂuorophores in close proximity being in a quenched state, is activated by a speciﬁc target enzyme. This enzyme cleaves the peptide of the probe molecule, which kept the two ﬂuorophores in close proximity. Therefore, activatable are used for imaging of enzyme activity in tissue [46, 47].

298

Alexander D. Klose

6.3.1.3 Fluorescent proteins A ﬂuorescent protein is an intrinsic ﬂuorophore that has initially been isolated from the jellyﬁsh Aequorea victoria [39]. Diﬀerent mutants, for example, enhanced green ﬂuorescent protein (EGFP) or red ﬂuorescent protein (RFP), are nowadays available for molecular imaging. The diﬀerence between a ﬂuorescent protein and a ﬂuorescent dye or quantum dot is such that the protein is expressed by a reporter gene inside the cell instead of being administered from outside. Hence, the ﬂuorescent protein is an intrinsically produced reporter at the site of transcription of the reporter gene. Fusing the reporter gene to a gene of interest allows for imaging of almost any protein inside living small animals. Therefore, ﬂuorescent proteins are mostly used for studying gene expression and gene regulation (Fig. 6.2) [14, 20, 39, 48]. 6.3.1.4 Fluorophores The imaging contrast of a reporter probe in tissue is impacted by the optical properties of the ﬂuorophore. Most common ﬂuorophores in ﬂuorescence molecular imaging are ﬂuorescent dyes, quantum dots, and ﬂuorescent proteins [21,39,49–51]. A ﬂuorophore, after being administered as part of the reporter probe or being expressed, changes the light absorption and ﬂuorescence properties of the targeted tissue site to a diﬀerent extent relative to the surrounding normal tissue. The choice of a ﬂuorophore with speciﬁc optical properties determines whether its ﬂuorescence light is detectable on the tissue surface, thus, providing a suﬃciently large image contrast. First, a ﬂuorophore should preferably possess an excitation and emission spectrum with maxima between 600 nm and 900 nm. This spectral constraint is derived from the necessity that light absorption by endogenous tissue chromophores (for example, hemoglobin, NADH) and autoﬂuorescence of tissue is minimal. Second, the ﬂuorophore should have a high Stokes shift that exhibits the spectral diﬀerence between excitation and emission wavelengths. Third, a prerequisite for high sensitivity detection is a large extinction coeﬃcient and a high quantumyield [50]. A list of diﬀerent ﬂuorescent dyes and proteins is given in Table 6.3. 6.3.2 Bioluminescent probes Bioluminescent follow a similar reporting strategy as ﬂuorescent proteins, except that bioluminescent probes are not intrinsically luminescent but instead require an additional substrate for chemiluminescent activation. This substrate needs to be administered to the animal. External light sources for photonic activation are not required [19, 60–62]. In imaging of bioluminescent , cells of biological interest are transfected with a luciferase gene, which expresses an enzyme luciferase [19, 61, 63]. For example, ﬁreﬂy luciferase (reporter gene: Fluc) catalyzes the emission of visible light from a substrate luciferin, when ATP and oxygen are present. The emission peak is at 560 nm. Renilla luciferase (reporter gene: RLuc) catalyzes light emission at shorter wavelengths from a substrate coelenterazine without the aid of ATP. The emission peak is at 482 nm. Both bioluminescent reporter systems have relatively

6 Radiative transfer of luminescence light in biological tissue

299

Table 6.3. Optical properties of diﬀerent ﬂuorophores: ICG, ﬂuorescein, CyDye family, AlexaFluor family, and ﬂuorescent proteins. Fluorophore

ICG ﬂuorescein Cy5 Cy5.5 Cy7 AlexaFluor488 AlexaFluor700 AlexaFluor750 EGFP DsRed mRFP1 mCherry

Excitation wavelength λx [nm]

Emission wavelength λm [nm]

Extinction coeﬃcient [cm−1 M−1 ]

Quantumyield η [n.u.]

Lifetime τ [ns]

Reference

780 492 649 675 743 495 702 749 488 558 584 587

830 515 670 694 767 519 723 775 509 583 607 610

41,000 81,000 250,000 190,000 200,000 71,000 192,000 240,000 63,000 57,000 44,000 78,000

0.012 0.97 0.28 0.23 0.29 0.92 0.25 0.12 0.65 0.79 0.25 0.22

0.52 4 1 1 < 0.3 4.1 1 0.7 2.3 3.6 2.05 1.4

[37, 52] [53, 54] [18] [18] [18] [53] [18, 53] [18, 53] [55] [57, 58] [57, 59] [56]

Fig. 6.2. Luminescence light distribution on top of tissue surface of a small animal. Light originates from a dual reporter with location in proximity to spine. The dual reporter is a fused red ﬂuorescent protein and a ﬁreﬂy luciferase reporter. Left: ﬂuorescence image. Right: bioluminescence image. (With courtesy from Dr Inna Serganova, Memorial SloanKettering Cancer Center, New York City.)

broad emission spectra. Luciferase-substrate systems function as light-emitting for molecular events occurring within cells, such as levels of gene expression under diﬀerent conditions or in response to speciﬁc stimuli. Moreover, high detection sensitivities can be achieved in bioluminescence imaging due to low background light and the absence of autoﬂuorescence (Fig. 6.2) [62, 64].

300

Alexander D. Klose

6.4 Radiative transfer model In general, propagation of light through a scattering host medium, such as biological tissue, is termed radiative transfer [65–67]. The goal of a radiative transfer model is to determine the distribution of light in the host medium, taking account of the light interaction with the host medium. In particular, the radiative transfer model of luminescence light in tissue describes the propagation of light that originates from luminescent sources inside small animal tissue. The amount of light that leaks through the tissue boundary is a quantity that is subsequently used in image reconstruction of luminescent in small animals. 6.4.1 Equation of radiative transfer The basic equation (see Table 6.4 for notation) for radiative transfer of luminescence light in small animal tissue is the time-dependent ERT [65–67]. It is a balance equation for the radiance or angular photon ﬂux ψ(r, Ω, t) at spatial location r, direction Ω, and time t. The physical units assigned to ψ are either Wcm−2 sr−1 or photons s−1 cm−2 sr−1 . The spatial position and direction of ψ are deﬁned by means of two coordinate systems. First, a laboratory coordinate system describes the spatial geometry of the host medium, i.e. the small animal tissue geometry, and a spatial point inside tissue is given by r = (x, y, z). Second, a local coordinate system at point r describes the local scattering process of light into the diﬀerential solid angle dΩ around the direction Ω with spherical coordinates (ϕ, ϑ). The direction Ω can also be expressed by the coordinates of the laboratory coordinate system Ω = (Ωx , Ωy , Ωz ) = (sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ). Thus, the ERT is an equation with six independent variables (x, y, z, ϑ, ϕ, t) and is given by: Q(r, t) 1 ∂ + Ω · ∇ + μt (r) ψ(r, Ω, t) = μs (r) p(Ω · Ω )ψ(r, Ω , t) dΩ + . v ∂t 4π 4π

(1) The attenuation coeﬃcient, μt , with units of cm−1 , is the sum of the absorption, μa , and scattering, μs , coeﬃcients. The phase function p(Ω · Ω ) is the distribution function for photons anisotropically scattering from direction Ω into the diﬀerential solid angle dΩ around direction Ω. The term Q(r, t) describes the timedependent luminescent source power density of a luminescent probe. This source term will diﬀer for bioluminescence and ﬂuorescence radiative transfer, as well as for light transfer in steady-state, time, and frequency domain. The speed of light, v, in tissue is given by the ratio v = c n−1 m of the speed of light in vacuum and the refractive index of tissue. The ﬂuence or photon ﬂux φ(r, t) in units of W cm−2 or photons s−1 cm−2 is deﬁned by: (2) φ(r, t) = ψ(r, Ω, t) dΩ . 4π

6.4.2 Partly-reﬂecting boundary condition A signiﬁcant diﬀerence between radiative transfer of luminescence light in tissue and other particle transport problems (for example neutron transport in reactor

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939 *>.67, 7B7:+ G *JG]V#Ve ]J # 'AJDaEAG1X*1G*1 XeX]1E- # ;aJV1X*1G*1 XeX]1E *JGXAX]X J6 ]cJ X]#=1X, MKN 1d*A]#]AJG J6 ;aJVJL?JV1X M6JV 1d#ELD1- ;aJV1X*1G] .e1X- ;aJV1X*1G] LVJ]1AGX- JV Ra#G]aE .J]XN AGXA.1 ]AXXa1- #G. M`N Xa'X1Ra1G] 1EAXXAJG J6 ;aJV1X*1G*1 DA=?]P G 1d]1VG#D DA=?] XJaV*1- H - ADDaEAG#]1X ]?1 ]AXXa1 XaV6#*1 cA]? DA=?] #] c#b1D1G=]? 9H P ?1 ;aJVJL?JV1X AG ]?1AV L?J]JGA* =VJaG. X]#]1 L#V]A#DDe #'XJV' ]?1 DA=?] 6VJE ]?1 1d*A]#]AJG XJaV*1 #G. #V1 Xa'X1Ra1G]De 1D1b#]1. AG]J #G 1d*A]1. L?J]JGA* X]#]1P ?1 #EJaG] J6 #'XJV'1. DA=?] AX LVJLJV]AJG#D ]J ]?1 1d]AG*]AJG *J18*A1G]- -- #G. ]J ]?1 *JG*1G]V#]AJG- - J6 ]?1 ;aJVJL?JV1XP JE1 6V#*]AJG J6 ]?1 1d*A]1. ;aJVJL?JV1X- .19G1. 'e ]?1 Ra#G]aEeA1D. .- 1EA]X #=#AG DA=?] #] # DJG=1V c#b1D1G=]?- 9/ 2 9H - #G. V1]aVGX ]J A]X =VJaG. X]#]1P ?1V16JV1- ]?1 ;aJVJL?JV1X *JGX]A]a]1 #G AXJ]VJLA* DA=?] XJaV*1 cA]? X]V1G=]? '1AG= LVJLJV]AJG#D ]J .- ]?1 ;aJVJL?JV1 #'XJVL]AJG =H/ 4 - #] ]?1 1d*A]#]AJG c#b1D1G=]? 9H - #G. ]J ]?1 1d*A]#]AJG 1G1V=e =Ab1G 'e ]?1 L?J]JG ;ad EH #] 9H (2K)P C
MK`N

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A._C+.Q K AES.

C
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Yb/

A._C+.Q K AES.

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A._C+.Q K AES.

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YF9

A._C+.Q K AES.

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YFV

A._C+.Q K AES.

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M[ZN

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YF/

A._C+.Q K AES.

( A MN 4

/ ( / K gD: Q ; Q T ET MN % ET MN D < ^=; MN

M[HN

?1 .A7aXAJG #LLVJdAE#]AJG AX cA.1De #LLDA1. AG G1#V@AG6V#V1. *"=9 01<*- <0) .0'71(A MN ]J ]AXXa1X cA]? V1D#]Ab1De D#V=1 =1JE1]VA1X- Xa*? #X ?aE#G 'V#AG #G. 'V1#X] ]AXXa1 (^)P DXJ- EJX] V#.A#]Ab1 ]V#GX61V EJ.1DX AG ]JEJ=V#L?A* AE#=AG= J6 DaEAG1X*1G] LVJ'1X AG XE#DD #GAE#D ]AXXa1 #V1 #LLVJdAE#]1. 'e ]?1 .A7aXAJG 1Ra#]AJG- 6JV 1d#ELD1 AG ]JEJ=V#L?e J6 ;aJV1X*1G] XJaV*1X (K^H3K
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I2A;35 I235 CI D C[ [

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326

Alexander D. Klose

Fig. 6.4. Calculated partial boundary current J + on top of tissue surface of a nude mouse. The source location (GTLS cylinder in animal rectum) was known from CT coregistration. The absorption coeﬃcient of tissue varies; from left to right: 0.1, 0.2, 0.5, 1, and 2 cm−1 . The light distribution becomes less diﬀuse for large absorption coeﬃcients (far right). All data are normalized to their maximum light intensity value.

for given optical tissue properties, for example scattering and absorption coeﬃcients, and luminescent probes with deﬁned concentration, extinction coeﬃcient, quantumyield, and lifetime. Generally, we will refer to the source term Q when also speaking of individual source properties such as concentration, lifetime, or quantumyield, since a direct relationship between Q and these physical quantities was shown in sections 6.5 and 6.6. A functional relation F between the given and sought variables is given by the forward problem J + = F (Q, μs , μa ). In optical molecular imaging, however, scientists are interested in the quantiﬁcation of the luminescent probes instead of the directly observed light intensities on the tissue surface. As already outlined in section 6.3, luminescent probes reveal information about biological processes in living small animal models of human disease, for example, cell-traﬃcking, protein interaction, or tissue oxygenation. These probes, however, cannot directly be observed with an optical imaging system because of multiple scattering of luminescence light in tissue, and no information about the luminescent probe’s properties is directly accessible. Therefore, optical image reconstruction methods have been developed, which reconstruct the luminescent source Q or its properties (for example concentration, lifetime, quantumyield) as a function of location inside tissue from the measured luminescence light intensities on the tissue boundary. In opposite to the forward model F of luminescence light transfer, the inverse source problem is now deﬁned as Q = F −1 (J + , μs , μa ). It is an ill-posed and underdetermined inverse problem that will be further discussed now [148]. The instrumentation, necessary for performing light intensity measurements on the tissue boundary, in conjunction with image reconstruction techniques, which transform the measured light intensity data into spatial maps of luminescent probe properties, is also termed optical luminescence tomography. The reconstructed ¯ = (Q1 , . . . , Qm , . . . , QM ) are displayed in three-dimensional imsource maps of Q ages with voxels Qm . Diﬀerent experimental instrumentation and imaging protocols

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1V1 ]?1 XeE'JD MP P P -P P P N E1#GX # X*#D#V LVJ.a*]Y *?. AX ]?1 VJG1*C1V XeE'JDP 1] #XXJ*A#]1. 1=1G.V1 6aG*]AJGX '1 .1GJ]1. 'e 1/ M=NP ?1X1 LJDeGJEA#DX #V1 .19G1. 'e ]?1 V1D#]AJGX ' .==> .=AM ' 1= M=N 4 K =M ( K g( K( D D D ( > ( > T ( = K( K ( 1 M=N '. ' . T c?1V1 1T M=N 4 1 M=N 4 1 M=NP ] X?JaD. '1 GJ]1. ]?#] ]?1 .19GA]AJG J6 ]?1X1 6aG*]AJGX =Ab1G AG (K^[) .A71VX 6VJE ]?1 .19GA]AJG LJAG]1. Ja] #'Jb1 'e #

U <. &<Q&X.Q>SX>& .M\X>EC E3 Q+>X>]. XQCS3.Q X<.EQ`

Y77

EaD]ALDA1V MKN= P ?1 #XXJ*A#]1. 1=1G.V1 6aG*]AJGX 6JVE #G AG9GA]1 JV]?J=JG#D XeX]1E J6 6aG*]AJGX JG (K- K) #G. X#]AX6e ]?1 6JDDJcAG= JV]?J=JG#DA]e *JG.A]AJGX, ' . # ; ;QK 5 ` ' . *1. D 1= M=N.= M=N .= 4 MK`N `; Q K ; K 5 ; G #..A]AJG- ]?1e #V1 # L#V]A*aD#V XJDa]AJG J6 ]?1 ]?V11@]1VE XeX]1E J6 V1*aVV1G*1 V1D#]AJGX J6 ]?1 6JVE ' . ' . ' . ; K Q K .A;= M=N `; Q K =.= M=N Q ; Q K .;= M=N 4 g( ; T ( K g( K( `( D D D ( ; ( MK^N .=AM ' . = ' `K 5P` K5 c?1V1 .= K( K V1D#]AJGX =;= M=N 4 g- == M=N 4 K =M ?JD.P "3@1 8 :B
A : ' ?)T

. K `M Q K /? ? M=N( /? 4 `

#

;

' . ' . ? = / = .= ( = K( K

MK
;

#G. ]?1 X1VA1X *JGb1V=1X AG E1#G #] D1#X]P "3@1 H :B
' . # . MK 5 ' . K ; ' . . /?= ?= M=N( /?= 4 `M Q K ' / = ?= = .= ` ; MQK 5

A : ' ?)=

MK:N

?JD. #G. ]?1 X1VA1X J6 MK:N *JGb1V=1X AG E1#G #] D1#X]P ] X?JaD. '1 GJ]1. ]?#] cA]?Ja] #..A]AJG#D *JGX]V#AG]X AELJX1. JG ]?1 6aG*]AJG / M=N ]?1 X1VA1X AG MK
Y7V

KK E:E]XSE]* KK EQE]>@

'

. = (

. . =

#

'

4 #

M:

4

. =

. =( F

#

;

. =( F .

'

.F T

' . =

;

. =

. =( F

' . =

. =( F .=

' . . ; ; Q K; 5 ' `G ' . K Q *3= *. . *= = ( MKZN 4 `;K Q K ;; K; 5 c?1V1 ;; ( ;M T K; ;; ( D D D ( K( g( K( D D D ( ;; KM ;M ( D D D ( K( g( K( D D D ( ;M P ?1G cVA]AG= MKZN A] c#X ]#C1G AG]J #**JaG] ]?#] #G AG9GA]1XAE#D 1D1E1G] J6 # XJDA. #G=D1 . *#G '1 LV1X1G]1. AG ]?1 6JVE . 4 XAG S .S .F 4 .= .FP JV ]?1 X*#D#V b#VA#G] J6 ]?1 #..A]AJG ]?1JV1E AX J6 =V1#] AELJV]#G*1P G #**JV.#G*1 cA]? ]?AX ]?1JV1E ]?1 6JDDJcAG= V1D#]AJGX ' . . : ' . ' . ;K 5 ' . ` ' . .= =( F .= = ( F 4 . % 4 K Q *T= ; Q K 5 =). 4 . M=N. M= N Q `MK *3. N

. : M; KN5 =)T

M; Q KN5

.= M=N.= M= N *JX KMF F N MK2N

X #G. #V1 XL1*A91. 'e #G=aD#V *JJV.A@ ?JD.- c?1V1 ' . ; 'T #G. .]?1 aGA] b1*]JVX G#]1X S( F #G. S ( F M*JX S 4 =- *JX S 4 = NP "3@1 F :B
' .=

. =( F (

MKHN

.)T =).

.= 4

K . ' `G K Q *3=

#

' .

' .=

. = ( F .

M`gN

?JD.P 1V1 ]?1 X1VA1X AG MKHN *JGb1V=1X AG E1#G #] D1#X]P 1] aX cVA]1 M2N AG ]?1 6JVE # ' ' . . ' . ' . . ' AT B % ?( = ( F . Q 0 ?( =( F D M`KN K 4?= ?( =( F 4
1V1 ?( =( F 4 ?( % 4 ' . *JX F F Q== Y ( Y =( F #G. = ( F #V1 #G=aD#V *JJV.AG#]1X *JVV1XLJG.AG= X #G. ]J ]?1 aGA] b1*]JVX AG ]?1 XL?1VA*#D XeX]1E J6 *JJV.AG#]1X .19G1. #'Jb1P 1] ]?1 L#V#E1]1V ? AG AG]1=V#D 1Ra#]AJG M`KN GJ] '1DJG= ]J # X1] 4 M4( 4) ' . - (4( Q4NP G.1V ]?AX *JGX]V#AG] ]?1 EaD]ALDA1V '16JV1 ]?1 6aG*]AJG ?( =( F AG ]?1 XA.1 J6 M`KN cADD GJ] b#GAX?P aLLJX1 ' D16]@?#G. . ' ]?1. L?#X1 6aG*@ ]AJG BM=N M K( K P VJE ]?AX *JG.A]AJG A] 6JDDJcX BM=N ; K( K P aV]?1V c1 ' . ' . ' . X M P #DXJ #XXaE1 ]?#] 0 ?( =( F 4 0 ?(

U <. &<Q&X.Q>SX>& .M\X>EC E3 Q+>X>]. XQCS3.Q X<.EQ`

Y7U

JV ]?1 XaLLJXA]AJGX LJAG]1. Ja] #'Jb1 A] AX V1#XJG#'D1 ]J 9G. # XJDa]AJG J6 ' . 'M`KN AG .]?1 *D#XX' M .P XAG= 1E#VCX K MZP`P`N #G. ^ MZP`P`N ]?1 6aG*]AJGX BM=N- ?( =( F - #G. 0 ?( =( F *#G '1 V1LV1X1G]1. AG ]?1 6JVE J6 ]?1 6JDDJcAG= X1VA1X BM=N 4

A : ' ?)T

. K `M Q K /? ? M=N( /? 4 `

#

;

BM=N ? M=N .=( = K( K Y

M``N

;

. A : ' . M; <N5 ' . : ./ M?N `; Q K ?( =( F 4 M; Q <N5

' ./

. =( F Y

M`^N

.)T /).

0M?( =( FN 4

. A : :

M`; Q KN

.)T /).

M; <N5 0./ M?N M; Q <N5

./ M=( FN(

M=( FN (K( K !(g( `G) D

M`
; A : : ' . K ' . ' . ' . 0 ?( =( F 4 0T ?( = Q 0= ?( = *JX KF Q 0= ?( = XAG KF( ` =) =); ' . =( F K( K ! g( `G (

M`:N

M`[N

c?1V1 ' .@ K = ?( = @= T 4 G 4

K G

'

.@ K 0= ?( = @= T 4 G K 4 G

# #

M:

' .@ ' . ?( =( F *JX KF .F( = ?( = @=;

M:

' . ?( =( F XAG KF .FY

T

M`ZN

T

# #

M:

' .@ ' . 0 ?( =( F *JX KF .F( 0= ?( = @=;

M:

' . 0 ?( =( F XAG KF .FD

T

M`2N

T

VJE 6JVEaD#X MK[N- M`^N- M`
Y7/

KK E:E]XSE]* KK EQE]>@

A ' .@ .: ' . M; KN5 ' . .= M?N.= M=N ( `; Q K ' = ?( = @= 4 K Q *3= ; Q K 5 .)= A ' .@ .: ' . M; KN5 ' . 0.= M?N.= M=N( = K( K ( M`HN `; Q K ' 0= ?( = @= 4 K Q *3= ; Q K 5 .)= ..; $ ; ' . ..; ' ' ' ' .= M=N= ?( = .=-0.= M?N 4 ` K Q *3= c?1V1 .= M?N 4 ` K Q *3= ; ' . $; M=N0= ?( = .=- " 4 D D D ( `( K( g( K( `( D D D D ; .= ' . G AG.AbA.a#D AG]1=V#D 1Ra#]AJG *#G '1 J']#AG1. 6JV 1#*? J6 6aG*]AJGX = ?( = .K "P JV ]?AX 1Ra#]AJG M`KN X?JaD. '1 EaD]ALDA1. 'e *JX KF 6JV K D D D ( `( K( g JV 'e XAG KF 6JV K P ?1G ]?1 1dLV1XXAJG J']#AG1. X?JaD. '1 AG]1=V#]1. AG F Jb1V ]?1 X1=E1G] g( `G #G. 6JVEaD#X MK2N- M``N- M`:N3M`2N #G. ]?1 JV]?J=JG#DA]e J6 ]?1 '#XA* XeX]1E J6 ]VA=JGJE1]VA* 6aG*]AJGX AG ]?1 X1=E1G] X?JaD. '1 aX1.P X # V1XaD] ]?1 6JDDJcAG= AG]1=V#D 1Ra#]AJGX #V1 J']#AG1., # ' . ' . AT ; K 4?= = ?( = 4 = M=( = N= M?( = N .= Q 0= M?( =N( = (K( K)( K "D ` ; M^gN ?1 C1VG1D J6 ]?1 AG]1=V#D 1Ra#]AJGX M^gN AX J6 ]?1 6JVE

= M=( = N 4

A : ?)=

M`M Q KN/?

MM KN5 ?= M=N?= M= N( MM Q KN5

M=( = N (K( K) ! (K( K)( K T D

M^KN

?1 AG]1=V#D 1Ra#]AJGX AGbJDbAG= AG ]?1 6#EADe J6 X M^gN 6aV]?1V cADD '1 *#DD1. ]?1 V1.a*1. *?#V#*]1VAX]A* 1Ra#]AJGX MN ' J6. ]?1 V#.A#]Ab1 ]V#GX61V ]?1JVeP G ]?1X1 1Ra#]AJGX ]?1 V1RaAV1. 6aG*]AJGX = ?( = JGDe .1L1G. JG JG1 #G=aD#V b#VA@ #'D1 =P VJE 6JVEaD# M`:N A] 6JDDJcX ]?#] ]?1 XJDa]AJG J6 M`KN *#G '1 J'@ ]#AG1. JG ]?1 '#XAX J6 ]?1 CGJcG XJDa]AJGX 'J6 6#EADA1X J6 M^gNP G ]aVG 6VJE . M`HN A] 6JDDJcX ]?#] ]J 9G. ]?1 6aG*]AJG ?( = 6JV #Ge 9d1. = K " A] AX Xa8@ *A1G] ]J 9G. # X1Ra1G*1 J6 b#Da1X == M?N( =A;= M?N( D D D c?A*? AX # XJDa]AJG J6 ]?1 AG9GA]1 XeX]1E J6 DAG1#V #D=1'V#A* 1Ra#]AJGXP 1*?GARa1X 6JV XJDbAG= ]?AX XeX]1E cADD '1 LV1X1G]1. AG ]?1 G1d]' X1*]AJGP . ] X?JaD. '1 GJ]1. ]?#] ]?1 #f@ AEa]?#D ?#VEJGA*X J6 ]?1 6aG*]AJG ?( =( F #V1 .AV1*]De 1dLV1XX1. AG ]1VEX J6 = ' . ]?1 b#Da1X = ?( = XL1*A91. 'e 6JVEaD#X M`ZNP fAEa]?#D ?#VEJGA*X J6 ]?1 6aG*]AJG ' . . $ M: ' ?( =( F #V1 =Ab1G 'e b#Da1X G ; T ?( =( F *JX KF .F 6JV K D D D ( `( K( g . $ M: ' #G. G ; T ?( =( F XAG KF .F 6JV K P ?1 E#]?1E#]A*#D Xa'X]#G]A#]AJG J6 ]?1 *JVV1*] .1VAb#]AJG J6 ' 6VJE. 'M`KN M#DXJ M2NN c?1V1 ]?1 L?#X1 6aG*]AJG . BM=N '1DJG=X ]J ]?1 *D#XX ; K( K JV M K( K MX11 LL1G.Ad N- c#X =Ab1G AG (:^- :Z- :2)P

83; .*?(,% :& *77(* ?6(,: . :& &6:6(7:( 4>:(., . :& 6(:(? :6,76 :&.6B (, , ,*B:(* .6+ ?1 9VX] XJDa]AJGX J6 ]?1 ?JEJ=1G1JaX AG]1=V#D 1Ra#]AJGX *JVV1XLJG.AG= ]J 1Ra#@ ]AJGX M2N- M`KN #G. M^gN c1V1 J']#AG1. cA]? ]?1 ?1DL J6 #X1UX E1]?J. (`^- `<- K^2)

U <. &<Q&X.Q>SX>& .M\X>EC E3 Q+>X>]. XQCS3.Q X<.EQ`

Y7D

#XXaEAG= ]?1 L?#X1 6aG*]AJG ]J '1 # LJDeGJEA#DP G ]?AX *#X1 6JV .M 2 > c?1V1 > - ]?1 *J18*A1G]X /? AG 1dL#GXAJG M``N #V1 1Ra#D ]J f1VJP XAG= #X1UX E1]?J. #G. ]?1 AG]1VV1D#]AJG '1]c11G ]?1 XJDa]AJGX J6 #G. MX11 Xa'X1*]AJG ZP`PKN A] AX GJ] .A8*aD] ]J 9G. XJDa]AJGX J6 AG?JEJ=1G1JaX M2N- M`KN #G. M^gNP JVE#D XJDa]AJGX J6 M2N- M`KN #G. M^gN AG ]?1 *#X1 J6 GJG@LJDeGJEA#D L?#X1 6aG*]AJGX ]?#] *#G '1 LV1X1G]1. AG ]?1 6JVE M``N- #DXJ *#G '1 J']#AG1. aXAG= ]?AX E1]?J.P

Jc1b1V- ]?1 9VX] E#]?1E#]A*#DDe *JVV1*] XJDa]AJG J6 M^gN 6JV ]?1 *#X1 J6 K 4 g c#X J']#AG1. AG (:Z)P ?AX V1XaD] c#X 6JaG. aXAG= GJ]AJGX #G. *JGX]Va*]AJGX J6 6aG*@ ]AJG#D #G#DeXAX (Kg<- KK^- K^^)P DXJ AG (:Z) A]' c#X. XaLLJX1. ]?#] ]?1 6aG*]AJG BM=N X#]AX91X ]?1 JD.1V *JG.A]AJG J6 AG.1d # g( K JG ]?1 X1=E1G] (K- K) MX11 ]?1 .19GA]AJG J6 ]?1 JD.1V 6aG*]AJG *D#XX AG LL1G.Ad NP X A] c#X GJ]1. AG (:Z- :2) ]?1 XJDa]AJGX J6 M^gN #DXJ *#G '1 J']#AG1. 6JV #DD K #XXaEAG= ]?#] ]?1 .1VAb#@ ]Ab1 B M=N '1DJG=X ]J ]?1 JD.1V *D#XX 6aG*]AJGX JG ]?1 X1=E1G] (K- K)P Jc1b1V#DD ]?1 XJDa]AJGX J6 M2N- M`KN- M^gN J']#AG1. 'e E1#GX J6 ]?1 E1]?J. LJAG]1. Ja] #'Jb1 *#GGJ] '1 171*]Ab1De aX1. AG *#VVeAG= Ja] ]?1 *#D*aD#]AJGX J6 V#.A#]Ab1 91D. *?#V#*]1VAX]A*X AG V1#D ]aV'A. =1JL?eXA*#D E1.A#- c?A*? #V1 *?#V#*]1VAf1. 'e L?#X1 6aG*]AJGX ?A=?De L1#C1. AG ]?1 6JVc#V. X*#]]1VAG= .AV1*]AJGP ?1 9VX] #G#De]A*#D XJDa]AJGX J6 M^gN 6JV ]?1 *#X1 J6 K 4 g #G. #DDJcAG= 171*]Ab1 GaE1VA*#D V1#DAf#]AJG LV#*]A*#DDe 6JV #Ge L?#X1 6aG*]AJGX c1V1 =Ab1G AG ([<[:)P ?1 XJDa]AJG J6 AG]1=V#D 1Ra#]AJG M^gN 6JV #DD K T c#X =Ab1G AG ([K) AG #G #G#De]A*#D 6JVEP ?1 E#]?1E#]A*#D Xa'X]#G]A#]AJG J6 ]1*?GARa1X 6JV *#D*aD#]AG= ]?AX XJDa]AJG 6JV ]?1 *#X1 J6 #V'A]V#Ve L?#X1 6aG*]AJGX X#]AX6eAG= JD.1VUX *JG.A]AJG JG (K- K)- c#X #DXJ =Ab1G AG ([K)P G ([K- [<- [:) A] c#X XaLLJX1. ]?#] ]?1 L#V#E1]1V ? P P .AX]AG*]Ab1 61#]aV1 J6 ]?1 #LLVJ#*? Xa==1X]1. AG ([K- [<- [:) AX ]?1 cA.1 aX1 J6 ]?1JVA1X J6 .A71V1G*1 1Ra#]AJGX- *JG]AGa1. 6V#*]AJGX- #G#De]A*#D 6aG*]AJGX ]?1JVe #G. JaVA1V ]V#GX6JVE#]AJGP DXJ ]?1 *JGX]Va*]AJGX J6 #X1UX E1]?J. #V1 GJ] aX1. #Gec?1V1 AG #G 1dLDA*A] 6JVEP G *JGX]Va*]AG= ]?AX #LLVJ#*? A] c#X 6JaG. ]?#] EJX] J6 ]?1 LVJL1V]A1X J6 #G. AG 6#*] c1V1 *JG]#AG1. AG #G AELDA*A] 6JVE AG *D#XXA*#D ?1JV1EX J6 ]?1 *JG]AGa1. 6V#*]AJG ]?1JVe #G. ]?1 .A71V1G*1 1Ra#]AJG ]?1JVeP ?1X1 ?1JV1EX c1V1 X]#]1. #] ]?1 1G. J6 ]?1 GAG1]11G]? *1G]aVe #G. #] ]?1 '1=AGGAG= J6 ]?1 ]c1G]A1]? *1G]aVeP ?AX 6#*] #DDJcX aX ]J LV1X1G] ]?1 ]?1JVe J6 AG]1=V#D 1Ra#]AJGX M2N- M`KN- M^gN EJV1 *JGX]Va*]Ab1De #X *JEL#V1. ]J ]?1 ]?1JVe .1b1DJL1. JG ]?1 '#XAX J6 6aG*]AJG#D #G#DeXAX ]1*?GARa1X (:^- :2)P Jc1b1VAG 6JVEaD#]AG= 9G#D V1XaD]X c1 X?#DD cA.1De aX1 =1G1V#D GJ]#]AJGX- ]1VEAGJDJ=e #G. A.1#X .1b1DJL1. AG 6aG*]AJG#D #G#DeXAX #G. AG A]X #LLDA*#]AJGXP B
#

;

' . ' . = =( = = ?( = .= ( = K( K ( K "D

M^`N

;

AG*1 ]?1 C1VG J6 M^`N .J1X GJ] .1L1G. JG # XA=G J6 ]?1 AG.1d K JG1 *#G V1X]VA*] ?AX *JGXA.1V#]AJG ]J ' . ]?1 6#EADe J6 M^`N 6JV K T P X X?JcG AG (:^- :Z- :2) ]?1 6aG*]AJG = =( = .19G1. 'e 6JVEaD# M^KN *#G '1 LV1X1G]1. AG ]?1 6JVE

YVb

KK E:E]XSE]* KK EQE]>@

'

= =( =

.

K 4 `G

#

M:

5' .' . . ' K =M K =M *JX F *JX KF .F( B = = Q

M^^N

T

' . ' . A6 ]?1 6aG*]AJG BM=N ; K( K JV BM=N M K( K P ?1 .AX*V1]1 XL1*]VaE J6 M2N- M`KN AX .1GJ]1. 'e P ?1 .AX*V1]1 XL1*]VaE J6 ?JEJ=1G1JaX M^`N *JVV1XLJG.AG= ]J K T AX .1GJ]1. 'e = V1XL1*]Ab1DeP G (:^- :Z- :2) A] c#X LVJb1. ]?#] ]?1 V1D#]AJG 4 - = J**aVX- AP1P ]?1 .AX*V1]1 =

XL1*]VaE J6 AX # aGAJG J6 ]?1 .AX*V1]1 XL1*]V# J6 P G bAV]a1 J6 ]?AX 6#*] A] AX Xa8*A1G] ]J 9G. #DD X1]X = c?1G K T P X # .AX*V1]1 XL1*]VaE J6 #Ge M^`N A] cADD '1 # X1] J6 b#Da1X J6 ]?1'L#V#E1]1V ? MJV L#V#E1]1V 6 4 4?N 6JV c?A*? . M^`N ?#X ]?1 1A=1G6aG*]AJGX = ?( = c?JX1 GJVEX #V1 9GA]1 AG XJE1 *D#XX J6 aXa#D 6aG*]AJGXP XAG= V1XaD]X LV1X1G]1. AG Xa'X1*]AJG ZP`P` #G. ]#CAG= AG]J #**JaG] AX1X3 AX?1VUX ?1JV1E #G. #VX1b#DUX 1Ra#DA]e (KK^) ]?1 6JDDJcAG= ?1JV1E *#G '1 LVJb1.P -"7@"3 8' :B
U <. &<Q&X.Q>SX>& .M\X>EC E3 Q+>X>]. XQCS3.Q X<.EQ`

' . ' . ' . 7? MKN ' . _? MKN $?A; ?Y K 4 V $? ?Y K $?; ?Y K ( $; ?Y K 4 g( M T ,?A; ,?A;

YVF

M^:N

' .; c?1V1 V 4 4? P eX]1E M^:N cA]?Ja] ]?1 9VX] 1Ra#]AJG AX 6JVE#DDe A.1G]A*#D ]J ]?V11@]1VE V1*aVV1G*1 V1D#]AJG XeX]1E MK`[N MX11 LL1G.Ad N ]?#] AX # DAG1#V .A71V@ 1G*1 1Ra#]AJG J6 ]?1 X1*JG. JV.1V XAEaD]#G1JaXDeP **JV.AG= ]J ]?1 DAG1#V .A71V1G*1 1Ra#]AJG ]?1JVe MX11 LL1G.Ad N XeX]1E M^:N cA]?Ja] ]?1 9VX] 1Ra#]AJG LJXX1XX1X ]cJ DAG1#V AG.1L1G.1G] XJDa]AJGXP VJE ]?1 .19GA]AJGX J6 ]?1 b#Da1X ,? ( _? MKN( 7? MKN #G. 6VJE 1VVJGUX ?1JV1E MX11 LL1G.Ad N A] 6JDDJcX ]?#] ]?1 *?#V#*]1VAX]A* 1Ra#]AJG J6 XeX]1E M^:N AX J6 ]?1 6JVE M `V Q K 4 g( V K( K D M^[N J 1X]AE#]1 ]?1 EJ.aD1X J6. ]cJ VJJ]X ' ' . ; #G. M J6 1Ra#]AJG M^[N c1 cVA]1 A] AG ; ; Q 4 P VJE #G#De]A*#D LVJL1V]A1X J6 ]?1 6aG*]AJG ]?1 6JVE V 4 ` @ @ ' . @@ : K M N JG]J A] 6JDDJcX ]?#] A] E#LX 'AB1*]Ab1De ]?1 JL1G *AV*D1 @ @ @ @ ]?1 1d]1VAJV @ @.? @ @ J6 ]?AX J6 ]?1 *DJX1. *AV*D1 K #G. *JGb1VX1DeP G bAV]a1 K( K ]?1V1 1dAX] ]cJ VJJ]X ; - M J6 1Ra#]AJG M^[N 6JV c?A*? @; @ : K- @M @ 2 KP VJE 1VVJGUX ?1JV1E A] 6JDDJcX ]?#] ]?1 =1G1V#D XJDa]AJG cA]?Ja] ' . J6 XeX]1E ' M^:N . ' . ]?1 9VX] 1Ra#]AJG *#G '1 LV1X1G]1. AG ]?1 6JVE' $.? ?Y'K . 4 ; W_; M Q M W_M M c?1V1 ; #G. M #V1 #V'A]V#Ve *JGX]#G]X- #G. W_; M - W_M M #V1 DAG1#V AG.1L1G.1G] XeX]1EX J6 XJDa]AJGX J6 M^:N cA]?Ja] ]?1 9VX] 1Ra#]AJGP G #..A]AJG #**JV.AG= ]J ]?AX ?1JV1E ]?1 1Ra#DA]A1X ' ' . ' .. ' ' . ' .. DAE W_; M Q K P_ W; M 4 ; ( DAE W_M M Q K P_ W M M 4 M M^ZN ?A

?A

?JD.P @ @ Jc X1D1*] Xa*? LJXA]Ab1 @ @ GaE'1VX ; #G. M 6JV c?A*? ]?1 AG1Ra#DA]A1X @; @ Q ; 4 : K- @M @ M 4 2 K ]#C1 LD#*1P ?1G 6VJE M^ZN A] ; M c?1V1 M AX # Xa8*A1G] D#V=1 G#]aV#D GaE'1V- ]?1 AG@ 6JDDJcX ]?#] 6JV M M T T @ ' .@ @ ' .@ ' .?? ' .?? @ @ @ @ #G. W_M M &_M ?JD.P 1V1 &_; #G. 1Ra#DA]A1X W_; M &_; ; M LJXA]Ab1 *JGX]#G]XP ?1 X1*JG. AG1Ra#DA]e #G. ]?1 6#*] ]?#] ]?1 X1@ &_M #V1 XJE1 ' . Ra1G*1 $? ?Y K '1DJG=X ]J ]?1 *D#XX ;M MKN D1#. ]J &M 4 g MJ]?1VcAX1 ]?1 X1VA1X A . .' ' . .@ ' .@M ' . ;' ' ` M Q K Q K M5P M Q `K 5 @$? ?Y K @ cADD '1 .Ab1V=1G] #G. $? ?Y K cADD '1 ?)T ' . ' . GJ] '1DJG= ]J ]?1 *D#XX ;M MKNNP VJE #6JV1X#A. A] 6JDDJcX ]?#] $? WY K 4 ; W_; M #G. ' . W_; M ' . 4 gD DAE ?A W _M M ] E1#GX ]?#] _ W; MMN AX ]?1 EAGAE#D XJDa]AJG J6 XeX]1E M^:N cA]?Ja] ]?1 9VX] 1Ra#]AJGP VJE AG*?1VD1UX ?1JV1E MX11 LL1G.Ad N #G. 6VJE ]?1 V1D#]AJG V 4 ' .; ]?1V1 6JDDJcX #G 1Ra#DA]e 4? . ' . ' . ' . ' . JT MKN? M J; MKN? M ( (DDD D WM g 4 47T MKNP,; ? gY $; ?Y K P$T ?Y K 4 W_; K P_ K K '

M^2N

VJE M^2N #G. 6VJE ]?1 9VX] 1Ra#]AJG J6 XeX]1E M^:N c1 9G. ]?1 1Ra#]AJG c?JX1 VJJ] EaX] '1 ]?1 L#V#E1]1V ?P ?AX 1Ra#]AJG AX J6 ]?1 6JVE

YV[

KK E:E]XSE]* KK EQE]>@

( / JT MKN? M J; MKN? M K JT MKN? M J; MKN? M K K Q gY ( (DDD 4 KY ( (DDD ? K K ? K K K ' M . 4 ZT ? Y K 4 gD M^HN ? . VJE M^HN A] 6JDDJcX ]?#] 6JV AT g( K ]?1 b#Da1 ? 4 g .J1X GJ] X#]AX6e ]?AX 1Ra#]AJGP 6V#*]AJG ' . 1G*1 ]?1 L#V#E1]1V ? EaX] '1 # f1VJ J6 ]?1 AG9GA]1 *JG]AGa1. ZT ? M Y K P ] 6JDDJcX ]?#] ]?1 L#V#E1]1V ? AG bAV]a1 J6 V K( K P ?1 G1*1XXA]e AX LVJb1.P I'/"5M< Jc #XXaE1 ]?#]' 6JV .XJE1 K T ]?1V1 1dAX]X # f1VJ ? 4 ?T J6 ]?1 AG9GA]1 *JG]AGa1. 6V#*]AJG ? M Y K #G. ]?AX f1VJ '1DJG=X ]J ]?1 X1] P G T . ' Z M bAV]a1 J6 ]?AX ]?1 6V#*]AJG ZT ? Y K *JGb1V=1XP e AG*?1VD1UX ?1JV1E ]?1 ]?V11@ ]1VE V1*aVV1G*1 V1D#]AJG M^:N cA]?Ja] ]?1 9VX] 1Ra#]AJG ?#X ]?1 EAGAE#D XJDa]AJG ' ' .; . . A ' 6JV ? 4 ?T V 4 4?T A7 ]?1 AG9GA]1 *JG]AGa1. 6V#*]AJG "? P0? c?1V1 "? 4 . .. ?); ' ' ' "? MKN 4 _? MKNP,?A; - 0? 4 0? MKN 4 7? MKNP 4?T ,?A; *JGb1V=1XP e E1#GX J6 1D1E1G]#Ve ]V#GX6JVE#]AJGX ]?AX 6V#*]AJG *#G '1 V1.a*1. ]J ]?1 6JVE A ' "? . 47T MKN JT MKN?TM J; MKN?TM ( (DDD D 4 gY

,; ? T K K ?); 0? AG*1 ]?1 1Ra#DA]e '

ZT ?TM Y K

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?AX E1#GX ]?#] ]?1 =Ab1G AG9GA]1 *JG]AGa1. 6V#*]AJG *JGb1V=1XP ?aX ]?1 ]?V11@ ]1VE V1*aVV1G*1 V1D#]AJG M^:N cA]?Ja] ]?1 9VX] 1Ra#]AJG 6JV ? 4 ?T ?#X # EAGAE#D XJDa]AJG MX11 AG*?1VD1UX ]?1JV1ENP G LL1G.Ad A] c#X X?JcG ]?#] ]?1 X1] J6 #DD ]?1 EAGAE#D XJDa]AJGX AX ]?1 [email protected]#D Xa'XL#*1 J6 ]?1 ][email protected]#D XL#*1 J6 #DD XJDa]AJGX J6 # XeX]1E J6 ]eL1 MK`[N MX11 LL1G.Ad NP eX]1E M^:N cA]?Ja] ]?1 9VX] 1Ra#]AJG AX BaX] Xa*? # XeX]1EP J A] AX Xa8*A1G] ]J LJAG] Ja] JGDe JG1 ' .; EAGAE#D ?#X ]cJ VJJ]X ; #G. @ @ XJDa]AJGP @ X @ ?T 1Ra#]AJG M^[N 6JV V 4 4?T @ @ @ @ AG' EJ.aD1XP G LVJbAG= ]?1 G1*1XXA]e M M ; : K M 2 KN- AP1P ]?1e #V1 ' .A71V1G] . . A] c#X X?JcG ]?#] ]?1 XJDa]AJG $? ?T Y K 4 &; W_; M X?JaD. '1 ]#C1G #X # EAGAE#D XJDa]AJGP G #..A]AJG- ]?1 b#DA.A]e J6 ]?1 1Ra#DA]e ' ' . . $; ? T Y K W_; K 47T MKN JT MKN?TM J; MKN?TM ' .4 ' .4 ( (DDD gY ,; ? T K K $T ? T Y K W_M g ?#X '11G LVJb1.P VJE ]?1X1 V1D#]AJGX #G. 6VJE ]?1 9VX] 1Ra#]AJG J6 XeX]1E M^:N c1 J']#AG1. ]?#] ' . ' . 47T MKN 7T MKN JT MKN?TM J; MKN?TM $; ?T Y K P$T ?T Y K 4 ( (DDD 4 D gY ,; ? T K K 4?T ,;

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a] ]?AX 1Ra#DA]e V1#DDe ?JD.X XAG*1 AG bAV]a1 J6 ]?1 X]#V]AG= #XXaEL]AJG ZT ? M Y K 4 gP 1G*1 ]?1 EAGAE#D XJDa]AJG X1D1*]1. X#]AX91X #DD 1Ra#]AJGX J6 ]?1 XeX]1E @ ' M^:NP .@ ?#] ]?AX XJDa]AJG '1DJG=X ]J ]?1 *D#XX ;M MKN 6JDDJcX 6VJE #G 1X]AE#]AJG @W_; M @ ' .?? &_; aX1. AG LVJbAG= ]?1 G1*1XXA]eP ?1 Xa8*A1G*e AX LVJb1.P ?1JV1E ` ; MZP^PKN AX LVJb1.P e ?1JV1E ` MZP^PKN ]?1 .AX*V1]1 XL1*]VaE J6 ?JEJ=1G1JaX M^`N *JVV1@ XLJG.AG= ]J K T *JAG*A.1X cA]? ]?1 VJJ] X1] J6 #G 1Ra#]AJG ' . ZT ? M Y K 4 K Q

JT MKN? M KQ

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cA]? V1XL1*] ]J ? P ] X?JaD. '1 GJ]1. ]?#] 1Ra#]AJG M
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' . Jc *JGXA.1V # Xa'XeX]1E *JG]#AGAG= MT Q K 1Ra#]AJGX J6 XeX]1E M<`N #G. cVA]1 .JcG ]?AX Xa'XeX]1E AG # E#]VAd 6JVE M<
1V1 4 "./ AX #*J'eUX ' 9GA]1 . E#]VAd J6 ]?1 .AE1GXAJG#DA]e MT Q K ! MT Q K ]?#] AX 'aAD] J6 ]?1 9VX] MT Q K VJcX #G. *JDaEGX J6 ]?1 AG9GA]1 E#]VAd J6 XeX]1E ' ' ' . ' . .. M<`NY 4 ]T ?T Y K ( ]; ?T Y K ( D D D ( ]? ?T Y K M AX # ]V#GXLJXA]AJG JL1V#]AJGNY T

' ' ' ' ..;AM .. 4 g( g( D D D ( g( ,? A; _? A; MKNP 7? MKN7? A; MKN ]? A; ?T Y K D

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A;

_? A; MKNPM7? MKN7? A; MKNN

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?1 XeE'JD & AG M<:N E1#GX # *JELD1d *JGBa=#]AJG JL1V#]AJGP G bAV]a1 J6 E#]VAd V1#DA]e #G. XeEE1]Ve ]?1 D16] L#V] J6 M<:N AX # V1#D GaE'1VP ?1 XaE ? @ ' .@ ; @]. ?T Y K @M AX #DXJ # V1#D GaE'1VP XAG= ]?1 1dLDA*A] 6JVE J6 #*J'AUX E#]VAd .)T

]?1 XaE J6 ]?1 D16] L#V] J6 M<:N *#G '1 LV1X1G]1. AG ]?1 6JVE 6 ? : ' ' ' . ' . . ' .. ,. _. MKN ].; ?T Y K ]. ?T Y K Q ].; ?T Y K ]. ?T Y K ( MT D ? 4 7.; MKN7. MKN .); M<[N DAEA] G bAV]a1 J6 M<^N #G. #G AG1Ra#DA]e g : ; : K ]?1V1 1dAX]X # 9GA]1@ V1#D .@M ;A ' @ @ J6 ]?1 b#Da1 ? 6JV MT QP JV XAEAD#V V1#XJGX ]?1 X1VA1X .)T ]. ?T Y K *JGb1V=1X #G. A]X XaE AX GJ] 1Ra#D ]J f1VJ #X ]?1 EAGAE#D XJDa]AJG J6 XeX]1E M^:N cA]?Ja] ]?1 9VX] 1Ra#]AJG AX .A71V1G] 6VJE ]?1 ]VAbA#D XJDa]AJGP #CAG= # DAEA] MT Q AG M<:N #G. ]#CAG= AG]J #**JaG] M<^N c1 J']#AG DAE ? 4 VT

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1G*1 A] 6JDDJcX ]?#] ?T 4( 4 P G bAV]a1 J6 ]?1 #V'A]V#VAG1XX J6 ]?1 .AX*V1]1 VJJ] ?T X1D1*]AJG ]?1 b#DA.A]e J6 ]?1 #XX1V]AJG MKN J6 ]?1 ?1JV1E 6JDDJcX 6VJE ' ]?1. #6JV1X#A.P ?1 b#DA.A]e J6 #XX1V]AJG M`N 6JDDJcX 6VJE ]?#] ? #LL1#VX AG ZT ? M Y K M^N .AV1*]De 6JDDJcX 6VJE ]?1 6JVE J6 ]?1 AG9GA]1 #X ? M P ?1 b#DA.A]e J6' #XX1V]AJG . *JG]AGa1. 6V#*]AJG ZT ? M Y K #G. #DXJ 6VJE ]?1 .19GA]AJG J6 ]?1 b#Da1X J? MKN #G.

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M<2N

?1 b#DA.A]e J6 ]?AX V1E#VC 6JDDJcX 6VJE ?1JV1EX ` MZP^PKN #G. ^ MZP^PKN #G. 6VJE 1VVJGUX ]?1JV1EP "3@1 F :B 4

M 4 g( K M
' . c?1V1 .; ]?1 AG9GA]1 *JG]AGa1. 6V#*]AJGX Z. ? M Y K #V1 .19G1. 'e V1D#]AJGX ' . J. MKN? M J.A; MKN? M ( (DDD D Z. ? M Y K 4 KY K K

M:gN

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4? 7@722@M +<8 :B
#

;

' . == M=N= &46MKN( = .= 4 KD

M:`N

;

' . M`N 6 *JG.A]AJGX M:`N ?JD. ]?1 1A=1G6aG*]AJGX = &46MKN( = *#G '1 LV1X1G]1. .= K( K AG Xa*? 6JVEX ([K) A ' . K : . ' . ' = &46MKNY = 4 # M( K ? &46MKNY K ?A== M=N( ` ?)T

M:^N

A ' . ' .; : ' . ' . # M( K /?A= ? &46MKNY K ?A== M=ND M:
M^N 6 M:`N ?JD.X ]?1 GJVE#DAf#]AJG AG]1=V#DX ([K) #

; ;

' . =M= &46MKN( = .= 4

$; ;

' . =M= &46MKNY = .= #V1 1Ra#D ]J

A . ' . K : ' .' # M( K K AT /?A= M &46MKN Y K D M::N `6MKN )T

?1 b#DA.A]e J6 ?1JV1E : MZP^PKN 6JDDJcX 6VJE LV1X1G]#]AJG M^KN- 1Ra#]AJG M^`N?1JV1EX K MZP^PKN 3 ^ MZP^PKN- 1E#VCX ` MZP^PKN- < MZP^PKN #G. V1D#]AJGX # ; ' . ' . ?A== M=N = &46MKN( = .=( $? &46MKNY K 4 ;

' . *JGG1*]AG= ]?1 cA]? ]?1 6aG*]AJGX ' . 1D1E1G]X J6 ]?1 X1Ra1G*1X $? &46MKNY K = &46MKN( = P ' . G L1V6JVEAG= V1#D *#D*aD#]AJGX J6 ]?1 1A=1G6aG*]AJGX = &46MKN( = AG ]?1 *#X1 J6 ?A=?De #GAXJ]VJLA* L?#X1 6aG*]AJGX A] AX aX16aD ]J ]#C1 AG]J #**JaG] ]?1 6JDDJcAG= #XX1V]AJGP

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K .' MQK MQKQ 6 K( >; ( ; @$1 &46MKNY K @ ` ?)1

@ ' .@ ' ' ..; AT M @$1 &46MKNY K @ K 6 K( >M

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MQKQ

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@' ' .= @ .. K .' @/?A= @ 6 K( >M ?1 ( MQK `

A6 AG *#D*aD#]AG= ]?1 AG9GA]1 X1VA1X AG M:^N #G. M: #..1G.X M> K #G. > KN V1XL1*]Ab1DeP

1V1 6 K( > 4 *JG]#AGAG= > ; M ; M ' . ' . ' . ' . 6MKN Q , K( > - , K( > g( K 6MKN #G. DAE , K( > 4 g DAE ; 4 g 1A

DAE

1 A

1 A

M 4 gP

G AELJV]#G] Ra1X]AJG ]?#] *#G '1 #GXc1V1. cA]? ]?1 Ra#DA]#]Ab1 E#]?1E#]A*#D ]?1JVe J6 #G. AX ]?1 Ra1X]AJG J6 ]?1 GaE'1V J6 1D1E1G]X *JG]#AG1. AG ]?1 X1]X #G. ? c?1V1 M T P ] X?JaD. '1 GJ]1. ]?#]- cA]?AG ]?1 6V#E1cJVC J6 ]?1 #X1UX E1]?J. (`^- `<- K^2- K^H) #G. =1G1V#D *JGX]Va*]AJGX J6 ' 6aG*]AJG#D . #G#DeXAX (:<- :Z- :2)- A] c#X AG 6#*] LVJb1. ]?#] ]?1 X1] 4 4( 4 - 4( Q4 *JAG*A.1X cA]? 1XX1G]A#D XL1*]V# J6 M2N- M`KN- M^`N cA]? V1XL1*] ]J ]?1 L#V#E1]1V ' .; ]?1 1XX1G]A#D XL1*]VaE J6 MHN ? McA]? V1XL1*] ]J ]?1 L#V#E1]1V V 4 4? *JAG*A.1X cA]? ]?1 X1=E1G] (K( K)NP AG=aD#V M=1G1V#DAf1.N 6aG*]AJGX ]?#] #V1 GJ] GJVE#DAf1. AG ]?1 aXa#D X1GX1 M]?1 1A=1G6aG*]AJGX *JVV1XLJG.AG= ]J .AX*V1]1 XL1*]V# J6 #G. LJXX1XX 9GA]1 GJVEX AG ]?1 aXa#D 6aG*]AJG XL#*1XN *JVV1XLJG. ]J ]?1 1XX1G]A#D XL1*]V# J6 LJAG]1. Ja] #'Jb1P GJ]?1V .AX]AG*]Ab1 61#]aV1 J6 ]?1 1XX1G]A#D XL1*]VaE MAG *JG]V#X] cA]? .AX*V1]1 XL1*]V#N AX A]X AG.1L1G.1G*1 J6 ]?1 6aG*]AJG#D 6JVE J6 # L?#X1 6aG*]AJGP ] X?JaD. '1 1XL1*A#DDe GJ]1. ]?#] ]?1 *JGX]Va*]Ab1 aX1 J6 ]?1 *JELD1]1G1XX ?1JV1EX (`^- `<- :<- :Z- :2) M#G.- =1G1V#DDe- ]?1 #X1UX E1]?J.N ]J XJDb1 *JG*V1]1 'JaG.#Ve@b#Da1 LVJ'D1EX J6 AX AELJXXA'D1 A6 ]?1 .AX*V1]1 XL1*]V# J6 E1G@ ]AJG1. #'Jb1 #V1 AG9GA]1 JV 9GA]1 X1]X #G. DJ*#]AJGX J6 #DD 1D1E1G]X J6 ]?1X1 X1]X JG ]?1 *JELD1d LD#G1 #V1 AELJXXA'D1 ]J *#D*aD#]1P 1] # L?#X1 6aG*]AJG BM=N '1 # LJDeGJEA#D J6 ]?1 >]? JV.1V c?1V1 > T P JV ]?AX XAELD1 *#X1 A] c#X X?JcG AG (`<- K^2- K^H) ]?#] .AX*V1]1 XL1*]V# J6 #G. *JG]#AG # 9GA]1 GaE'1V J6 1D1E1G]XP G #..A]AJG- .K g( K( D D D ( > ]?1 XL1*]VaE J6 *JVV1XLJG.AG= ]J KX1D1*]1. X1] = *JAG*A.1X cA]? ]?1 .AX*V1]1 ' . #G. *JG]#AGX GJ EJV1 ]?#G ` > K Q K 1D1E1G]X M.K > Q K( > Q `( D D D ]?1 X1] = 4 NP ?1 XA]a#]AJG AX *JGXA.1V#'De EJV1 *JELD1d 6JV GJG@LJDeGJEA#D L?#X1 6aG*]AJGXP GaE'1V J6 Xa8*A1G] *JG.A]AJGX ?#b1 '11G 6JVEaD#]1. XJ 6#V #G. A6 ]?1e ?JD. .AX*V1]1 XL1*]V# J6 #G. *JG]#AG 1A]?1V # 9GA]1 JV AG9GA]1 GaE'1V J6 1D1E1G]XP G1 J6 ]?1 9VX] V1XaD]X 6JV ]?1 *#X1 J6 GJG@LJDeGJEA#D L?#X1 6aG*]AJG AX =Ab1G AG (K
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V1#D L#V]X J6 #DD *JELD1d GaE'1VX *JVV1XLJG.AG= ]J ]?1 LJAG]X J6 ]?1X1 *JG]JaVX X?JaD. '1 'JaG.1. AG EJ.aD1X 'e XJE1 LJXA]Ab1 GaE'1VP ] X?JaD. '1 1EL?#XAf1. ]?#] 6JV R R; 2 g A] AX G1*1XX#Ve ]J X1D1*] # *JG]JaV XA]a#]1. AG ]?1 DJc1V ?#D6@LD#G1 J6 ]?1 *JELD1d LD#G1P 6 R R; : g # *JVV1XLJG.AG= *JG]JaV X?JaD. '1 XA]a#]1. AG ]?1 aLL1V ?#D6@LD#G1 J6 ]?1 *JELD1d LD#G1P ?1 aX1 J6 *JG]JaV AG]1=V#]AJG @ @ ]1*?@ GARa1 #DDJcX JG1 ]J ]V#GX6JVE MaG.1V ]?1 6aD9DDE1G] J6 # *JG.A]AJG @R R; @ #4 gN ]?1 AG]1=V#DX AG 1dLV1XXAJG MHgN AG]J AG]1=V#DX AG c?A*? AG]1=V#G. 6aG*]AJGX cADD *JG@ ]#AG 6aG*]AJGX .1*V1#XAG= 1dLJG1G]A#DDe A6 ? AG*V1#X1XP ?AX *AV*aEX]#G*1 1XX1G@ _ G= MR( =Y R; ( =; ( F; ( AT NP DXJ ]A#DDe XAELDA91X *#D*aD#]AG= ]?1 #fAEa]?#D ?#VEJGA*X A] X?JaD. '1 GJ]1. ]?#] ]?1 *JG]JaV AG]1=V#]AJG ]1*?GARa1 #DDJcX JG1 AG #G 1dLDA*A] 6JVE ]J AXJD#]1 ]?1 *JG]VA'a]AJGX ]J V11GUX 6aG*]AJG c?A*? #VAX1 6VJE ]?1 .AX*V1]1 XL1*]VaE MA6 A] 1dAX]XN J6 M2HN MX11 #DXJ M^gNNP ?1 *#X1 c?1G R R; 4 &g AX ]?1 EJX] *JELD1dP aXAG= 1dLDA*A] #G#De]A*#D LV1X1G]#]AJGX J6 #DD ]1VEX J6 ' Jc1b1V- . ]?1 X1Ra1G*1 $? ?Y KY =; Y AT MA] AX GJ] .A8*aD] ]J J']#AG ]?1E 'e 6JVEaD#X M[HNN #G. #XeEL]J]A* '1?#bAJV ' . ' J6 ]?1X1 . ]1VEX 6JV ? Q M6JV #Ge M( K T #XeEL@ ]J]A*X $? ?Y KY =; 4 ?; 6JV ? Q ?JD.N #DD ]?1 #fAEa]?#D ?#VEJGA*X _ MD D DN *#G '1 *#D*aD#]1. cA]? ]?1 ?1DL J6 6JVEaD# MHgNP ] J6 V11GUX 6aG*]AJG X?JaD. '1 GJ]1. ]?#] 6JV R R; 4 &g #G. =; #4 g #DD ]?1 AG]1=V#G. 6aG*]AJGX AG AG]1=V#DX J6 MHgN .1*V1#X1 6JV ? Q AG EJ.aD1X GJ XDJc1V ]?#G # 6aG*]AJG ?M 6JV ? QP VJE ]?AX 6#*] A] 6JDDJcX ]?#] #DD ]?1 AG]1=V#DX LJAG]1. Ja] #'Jb1 #V1 *JGb1V=1G]P

83" :(? *%.6(:&+ :. *>*: :& C(+>:&**B ?6% 6#:(., >,:(., , :& 6#:(., >,:(., .6 7+('(,!,(: 0*,'06*** :>6( +(>+ G ]?AX X1*]AJG ]?1 171*]Ab1G1XX J6 ]?1 XAEaD]#G1JaX aX1 J6 =1G1V#D AGb#VA#G*1 V1D#@ ]AJGX (2- [[3[H- Z`) #G. E1]?J.X J6 J']#AGAG= XJDa]AJGX J6 M^gN #G. #fAEa]?#D ?#VEJGA*X J6 V11GUX 6aG*]AJGX J6 ]?1 'JaG.#Ve@b#Da1 LVJ'D1E M2KN- M2`N cADD '1 .1EJGX]V#]1.- aXAG= #X #G 1d#ELD1 ]?1 *D#XXA*#D LVJ'D1E J6 .1]1VEAGAG= ]?1 #f@ AEa]?#DDe #b1V#=1. V1;1*]AJG 6aG*]AJG 6JV ]?1 *#X1 J6 # L?#X1 6aG*]AJG ?A=?De L1#C1. AG ]?1 6JVc#V. .AV1*]AJGP

B<)<8 "5"@2 /5J@/5" @"2E/75A AVX] c1 cADD LV1X1G] #G. 1dLD#AG AG 'VA16 ]?1 X1GX1 J6 JG1 =1G1V#D AGb#VA#G*1 V1D#]AJG ]?#] AX # L#V]A*aD#V *#X1 J6 V1D#]AJGX J6 Xa*? ]eL1 J']#AG1. AG (2- [[3[H) JG ]?1 '#XAX J6 EJ.1D L?eXA*#D *JGXA.1V#]AJGX JV 6JVE#D E#]?1E#]A*#D ]V#GX6JVE#]AJGXP VAE#VADe A] X?JaD. '1 GJ]1. ]?#] ]?1 =1G1V#D AGb#VA#G*1 V1D#]AJGX MXN *JGG1*] XJDa]AJGX J6 'JaG.#Ve@b#Da1 LVJ'D1EX J6 J6 ]?1 X#E1 ]eL1 JV .A71V1G*1 LVJ'D1EXP ?1e *#G '1 *JGXA.1V1. #X *JGX1Ra1G*1X J6 ]?1 AGb#VA#G*1 J6 *?#V#*]1VAX]A* J6 V#.A#]Ab1 91D.X AG ]aV'A. E1.A# J6 .A71V1G*1 *JG9=aV#]AJGX cA]? V1XL1*] ]J *1V]#AG X1]X J6 X1EA@=VJaLX J6 JL1V#]AJGX M]?1AV X1GX1 AX 1dLD#AG1. AG .1]#AD AG (2- [[3[2)NP

YDb

KK E:E]XSE]* KK EQE]>@

JGXA.1V # E#*VJX*JLA*#DDe ?JEJ=1G1JaX AXJ]VJLA* ]aV'A. E1.AaE MD#e1VN J6 LD#G1@L#V#DD1D *JG9=aV#]AJG MX11 X1*]AJG ZPK- G]VJ.a*]AJGNP 1] # D#e1V aG.1V *JG@ XA.1V#]AJG ?#b1 #G JL]A*#D ]?A*CG1XX RT MRT 4 7%QJ ^T c?1V1 7%QJ #G. ^T #V1 #G _ _ !_ AX # 1d]AG*]AJG *J18*A1G] #G. # =1JE1]VA*#D ]?A*CG1XX J6 ]?1 D#e1VNP 1] _ _ J6 ]?AX .AE1GXAJGD1XX VA=?]@?#G.1. #V]1XA#G *JJV.AG#]1 XeX]1E- ]?1 LD#G1 XeX]1E AX XA]a#]1. JG ]?1 9VX] 'JaG.#Ve J6 ]?1 D#e1V- #G. ]?1 #dAX !_ MA] AX #G #dAX J6 JL]A*#D .1L]?XN AX .AV1*]1. ]Jc#V.X A]X X1*JG. 'JaG.#VeP 1] aX .1GJ]1 ]?1 AE#=1 _ _ !_ 'e _TD #G. .1GJ]1 ]?1 *JVV1XLJG.1G*1 AE@ J6 ]?AX E1.AaE AG ]?1 XeX]1E _ _ !_ ]?VJa=? #=1X J6 ]?1 9VX] #G. X1*JG. 'JaG.#VA1X J6 ]?1 D#e1V AG ]?1 XeX]1E _ _ X X - X ; .19G@ ;; #G. M; P J .1LA*] ]?1 #G=D1 *JJV.AG#]1X J6 ]?1 aGA] b1*]JVX - AG= .AV1*]AJGX J6 LVJL#=#]AJG JV 1EAXXAJG J6 V#.A#]AJG c1 cADD aX1 ]?1 GJ]#]AJGX AG]VJ.a*1. AG Xa'X1*]AJG ZP`P` #G. X1*]AJG ZP

T M:

Q #

# .F

T M:

T

# .F

; ;

; ; ;

T

' . ' . _ L R( =( FY g( = ( F ( AT _TD Qg( = ( F Y AT .= = L

' . ' . _ L R( =( FY RT ( = ( F ( AT _TD RT g( = ( F Y AT .= D = L

MHKN

. ' . '

1V1 ' (g( RT ) AX # *?#V#*]1VAX]A* 6aG*]AJG J6 # X1=E1G] (g( RT ) M' (g( RT ) 4 K ' . ' . P (g( RT )NY _TD R( =( F( AT AX # XJDa]AJG A6 R (g( RT ) #G. ' (g( RT ) 4 g A6 R J6 # .AE1GXAJGD1XX aG.1V ]?1 *JG.A]AJGX =Ab1G #'Jb1 #G. V1#DAf1. AG ]?1 _ L MR( =( FY R; ( =; ( F; ( AT N AX V11GUX 6aG*]AJG J6 ]?1 .AE1GXAJGD1XX D#e1V _TD Y L c?A*? *JAG*A.1X cA]? 1Ra#]AJG M2KN AG c?A*? ]?1 XeE'JD AX V1LD#*1. 'e

U <. &<Q&X.Q>SX>& .M\X>EC E3 Q+>X>]. XQCS3.Q X<.EQ`

YDF

$!/ 3/+/ <. :.EB.XQ` E3 X<. GQE#A.BK

' . ]?1 XeE'JD (_ "( _$) #G. A] AX #XXaE1. ]?#] R( R; " _( _$ #G. 'JaG.#Ve *JG.A]AJGX _ L M_$( =( FY R; ( =; ( F; ( AT N 4 g ]#C1 LD#*1P _ L M_ L "( =( FY R; ( =; ( F; ( AT N 4 g- L VJE MHKN A] 6JDDJcX ]?#] 6JV #DD K " ]?1 6JDDJcAG= V1D#]AJGX ' ' . $; ' . ' . . $ D T _ L ' (g( RT ) _TD G= R( =( AT 4 ; .= T L G= R( =Y R ( = ( AT 0= R ( = .R Q ' . ' . $ ; _ L R( =Y g( = ( AT _TD G= Qg( = ( AT .= Q ; = . ' . $ ; L G= ' _ _ L ; = L G= R( =Y RT ( = ( AT TD G= RT g( = ( AT .= MH`N ' . ' . ' . $ M: ?JD.P 1V1 b#Da1X _TD G= R( =( AT 4 G ; T _TD R( =( F( AT Y= F .F #G. ' . . ' . $ M: ' 0= R( = 4 G ; T 0 R( =( F Y= F .F #V1 ]?1 K]? #fAEa]?#D ?#VEJGA*X J6 6aG*@ ' . ' . ]AJGX _TD R( =( F( AT #G. 0 R( =( F V1XL1*]Ab1De M]?1 .19GA]AJG J6 ]?1 6aG*@ ' . ' . _ L ]AJG Y= F AX =Ab1G #6]1V 6JVEaD# M2:NNY L G= R( =Y R; ( =; ( AT AX V11GUX 6aG*@ ]AJG J6 ]?1 .AE1GXAJGD1XX' . *JAG*A.AG= AG ]?1 6JVE ' cA]? 1Ra#]AJG. M2ZN AG _; _ L c?A*? ]?1 6aG*]AJG D D D AX Xa'X]A]a]1. 'e L G= R( =Y R; ( =; ( AT #G. 1d@ G= ' .; ' . ' . ' . ' . LV1XXAJG AT
YD[

KK E:E]XSE]* KK EQE]>@

' . ' . ' . $ _ L R( =( FY R; ( =; ( F; ( AT .F _ L R( =Y R; ( =; ( F; ( AT 4 G ; M: Y= F 1Ra#DA]e L G= L T ' . ' . _ L 4 G ; Y= F; L G= R( =Y R; ( =; ( AT ?JD.XP 1D#]AJGX MH`N E#C1 X1GX1 J6 'a] 6JV ]?1 #fAEa]?#D ?#VEJGA*X J6 ]?1 V1RaAV1. 6aG*]AJGXP MHKN #G. X J6 (2- [[3ZK) *#G '1 *JGXA.1V1. #X '#D#G*1 V1D#]AJGXP ?1X1 X c1V1 J']#AG1. aXAG= AGb#VA#G*1 LVJL1V]A1X (2- [[3ZK) #G. b#VAJaX .1*JELJXA@ ]AJG ]eL1X J6 *JG]VA'a]AJGX J6 L?J]JGX c?A*? AG # V#G.JE E#GG1V c#G.1V AG ]?1 AGA]A#D ]aV'A. E1.AaE #G. #GJ]?1V ]aV'A. E1.AaE AG]J c?A*? ]?1 AG]1VG#D L#V] J6 ]?1 AGA]A#D E1.AaE AX 1E'1..1.P X .A71V AG =1G1V#D 6VJE AGb#VA#G*1 V1D#]AJGX J']#AG1. 1#VDA1V- 6JV 1d#ELD1 AG (K:<) MX11 #DXJ V161V1G*1X AG A]NP G X AG ]?1 =1G1V#D *#X1 ]?1 AG]1=V#]AJG AX L1V6JVE1. Jb1V ]?1 *JELD1]1 aGA] XL?1V1 P ?1X1 V1D#]AJGX AG L#V]A*aD#V *#X1X ]aVG AG]J AGb#VA#G*1 V1D#]AJGX J6 ]?1 9VX] #G. D#X] AG@ ]1VX1*]AJGX (K:<) *JG]#AGAG= ]?1 AG]1=V#]AJG 1A]?1V Jb1V ]?1 ?1EAXL?1V1 A JV Jb1V ]?1 ?1EAXL?1V1 M 4 A - Y ]?1X1 ?1EAXL?1V1X #V1 *?#V#*]1VAf1. 'e ]?1 XJDA. #G=D1 `GNP B<)
' . ' . _ G= R( =Y R; ( =; ( AT AX ?1X1 6JVEaD#X ?JD. 6JV R R; & V1XL1*]Ab1DeP 1V1 V11GUX 6aG*]AJG J6 .AE1GXAJGD1XX M2ZN 6JV ]?1 *#X1 J6 #G AG9GA]1 LD#G1@L#V#DD1D _ _ !_ AG]J _ N #G. ]?1 LV1X1G*1 J6 ]aV'A. E1.AaE MA] .AX'E#LL1. . AG ]?1 XeX]1E ' . ' &46 MKN( = #V1 1A=1G6aG*]AJGX J6 M^`N T# XJaV*1U * R R; * = =; AG A]Y . ' =. GJVE#DAf1. 'e *JG.A]AJG M:`NY '&. AT.Y K AX GJVE#DAf#]AJG 6#*]JVP G (^g- [<) 6JV #DD ; K( `( D D D ( < ]?1 b#Da1 &. 'AT Y K. c#X 1dLV1XX1. ]?VJa=? ]?1 .1VAb#]Ab1 J6 ]?1 *JG]AGaJaX 6V#*]AJG ZT ^Y K cA]? V1XL1*] ]J ]?1 #V=aE1G] ' . . ^ #] ]?1' LJAG]. 'AG9GA]1 6.M MKN P 1G1V#D 1dLV1XXAJGX 6JV ]?1 GJVE#DAf#]AJG 6#*]JVX &; AT Y K ( D D D ( &/ AT Y K #DXJ *#G '1 J']#AG1. aXAG= ]?1 JV]?J=JG#DA]e J6 b#VAJaX 1A=1G6aG*]AJGX MaXa#D #G. XAG=aD#VN J6 M^`N #G. *JELD1]1G1XX ]?1JV1EX (`<- :<- :Z- :2) aG.1VDeAG= #X1UX E1]?J.P ?aX ]?1X1 GJVE#DAf#]AJG 6#*]JVX *#G '1 1dLV1XX1. ]?VJa=? ]?1 GJVE#D@ Af#]AJG AG]1=V#DX J6 ]?1 D16]@?#G. XA.1 J6 1dLV1XXAJGX M::NP X #G 1d#ELD1 J6 171*]Ab1G1XX J6 aX1 J6 X A] cADD '1 X?JcG ]?#] ]?1 GJVE#DAf#]AJG 6#*]JVX AG MH^N *#G '1 AEE1.A#]1De J']#AG1. 6VJE X #G. ]?1 6JVE J6 .#XeEL]J]A*#D ' ' . 6JVEaD#X MH^N A]X1D6P J .J ]?AX c1 #XXaE1 ]?#] T# XJaV*1U * R R; * = =; AX XA]a#]1. AG _ P G bAV]a1 J6 ]?1 ]V#GXD#]AJG XeEE1]Ve J6 ' . _ G= R( =Y R; ( =; ( AT A] *#G '1 #XXaE1. ]?#] R; 4 gP 1] aX X1D1*] # ]?1 6aG*]AJG

U <. &<Q&X.Q>SX>& .M\X>EC E3 Q+>X>]. XQCS3.Q X<.EQ`

YDY

?#D6@XL#*1 AG _ cA]? ]?1 ?1DL J6 #G AG1Ra#DA]e R RT c?1V1 RT 2 gP ?1 AE@ #=1 _D A J6 ]?1 X1EA@AG9GA]1 LD#G1@L#V#DD1D ]aV'A. E1.AaE *JVV1XLJG.X ]J ]?AX _ _ !P _ ?1 'JaG.#Ve J6 ]?AX E1.AaE AX .19G1. 'e #G ?#D6@XL#*1 AG ]?1 XeX]1E 1Ra#]AJG R 4 ^_ 4 RT AG ]?AX XeX]1EP 1] #G TJ'X1Vb#]AJGU LJAG] '1 DJ*#]1. AG _D A #] #G JL]A*#D .1L]? `RT P A]?AG ]?1 6V#E1cJVC J6 ]?1 #XXaEL]AJGX #**1L]1. MH`N AX J6 ]?1 6JVE ' . _ G= `RT ( =Y g( =; ( AT 4

#

;

' . ' . _ G= RT ( = Y g( =; ( AT .= D _ G= RT ( =Y g( = ( AT =

;

MH
.

#

;

&. A T Y K 4

= ;

M=

'

.

46. MKN( = .=

;

D

MH:N

VJE ]?1 #6JV1X#A. A] 6JDDJcX ]?#] 6JVEaD# MH:N 6JV ]?1 GJVE#DAf#]AJG 6#*]JVX J6 #XeEL]J]A*#D 6JVEaD#X MH^N #V1 *JGX1Ra1G*1X J6 ]?1 =1G1V#D AGb#VA#G*1 V1D#]AJGX #G. ]?1e *#G '1 .1]1VEAG1. cA]?Ja] aXAG= #X1UX E1]?J.P ] X?JaD. '1 GJ]1. ]?#] ]?1 *#D*aD#]AJG J6 ]?1X1 6#*]JVX LV#*]A*#DDe 6JV #Ge L?#X1 6aG*]AJGX *#G '1 L1V6JVE1. 171*]Ab1De aXAG= 6JVEaD#X M::NP B<)
YD9

KK E:E]XSE]* KK EQE]>@

V1X1G]#]AJG MH[N *#G '1 V#]?1V XAELDe J']#AG1. 6VJE 1dLV1XXAJG MHgN 6JV K 4 g ]#CAG= AG]J #**JaG] ]?1 GJ]#]AJGX aX1. AG 6JVEaD#X MH^NP GaE'1V < AG MH[N X?JaD. '1 *?JX1G ]#CAG= AG]J #**JaG] . *JGb1GA1G*1 *JGXA.1V#]AJGX #G. LVJL1V]A1X J6 ' _ R( =Y g( =; ( AT Y < P ]?1 6aG*]AJG GT 1] # X1EA@AG9GA]1 ]aV'A. E1.AaE TA > '1 1dLJX1. ]J # aGA.AV1*]AJG#D AG@ 9GA]1De cA.1 '1#E J6 1d]1VG#D V#.A#]AJGP XXaE1 ]?#] ]?1 .1GXA]e J6 ]?1 AG*A@ .1G] DA=?] ;ad JG ]?1 #V1# L1VL1G.A*aD#V ]J ]?1 '1#E AX 1Ra#D ]J #] ]?1 ]JL J6 ]?1 ' ]aV'A. . ' E1.AaEP . 1G*1 ]?1 AG]1GXA]e J6 ]?1 1d]1VG#D V#.A#]AJG AX 1Ra#D ]J * = =; * F F; P ] AX G1*1XX#Ve ]J J']#AG ]?1 f1VJ]? #fAEa]?#D ?#VEJGA* J6 AG]1GXA]e J6 DA=?] *JEAG= 6VJE TA P #D*aD#]AG= ]?AX 6aG*]AJG AX V1.a*1. ]J XJDb@ AG= ]?1 6JDDJcAG= 'JaG.#Ve@b#Da1 LVJ'D1E, ' . _ ?L C ' . TA >GT R( =Y =; ( AT _ ?L = 4 TA >GT R( =Y =; ( AT CR # ; ' . ?L ' . AT _ T =( = Q TA >GT R( = Y =; ( AT .= ( ` ; ' . ' . ' . ? L _ TA >GT Qg( =Y =; ( AT 4 * = =; Y R g( Q Y ' . = K( K ( =; g( K D MHZN ' . _ ?L ?1 6aG*]AJG G ; TA >GT R( =Y =; ( AT AX ]?1 f1VJ]? #fAEa]?#D ?#VEJGA* J6 ]?1 XaV@ 6#*1 V11GUX 6aG*]AJG (`<) J6 ]?1 .AE1GXAJGD1XX P G MHZN # .AE1GXAJGD1XX XeX]1E _ _ !_ AX aX1.P JV ]?AX XeX]1E ]?1 *JJV.AG#]1 LD#G1 _ _ AX DJ*#]1. JG ]?1 'JaG.@ _ _ _ #Ve J6 ]?1 E1.AaE TA > #G. ]?1 #dAX ! AX #G JL]A*#D .1L]? #dAX'#G. AX .AV1*]1.. .11L AG]J ]?AX E1.AaEP ?1 f1VJ]? #fAEa]?#D ?#VEJGA* TA >GT g( =Y =; ( AT J6 ]?1 AG]1GXA]e JG ]?1 'JaG.#Ve _ AX .AV1*]De 1dLV1XX1. ]?VJa=? ]?1 XJDa]AJG J6 'JaG.#Ve@b#Da1 LVJ'D1E MHZN, ' . ' . _ ?L TA >GT g( =Y =; ( AT 4 G ; TA >GT g( =Y =; ( AT ' . 4 G ; T =( =; ( AT =; ' . 4 `G ; T =( =; ( AT =; Y =; Mg( K)D MH2N ' . . $ M: '

1V1 T =( =; ( AT 4 G ; T '=( FY =; ( F; ( AT ..F AX ]?1 f1VJ]? #fAEa]?#D ?#V@ EJGA* J6 ]?1 V1;1*]AJG 6aG*]AJG =( FY =; ( F; ( AT (<- :- Z- ' . T ; =( = 4 ` =( = NP ] X?JaD. '1 GJ]1. ]?#] ( A ( A ]AJG (
U <. &<Q&X.Q>SX>& .M\X>EC E3 Q+>X>]. XQCS3.Q X<.EQ`

YD7

' c?1V1 = g( K ( =; g( K D VJE MH[N A] *#G '1 X11G ]?#] ]?1 C1VG1D _ GT MQg( =Y g( = ( AT N= J6 AG]1=V#D 1Ra#]AJG MHHN *#G '1 cVA]]1G #X # XaE J6 ' . _ .1=1G1V#]1 C1VG1D #G. #..A]AJG#D #..1G.X XL1*A91. 'e ]?1 6aG*]AJGX = GT D D D ' . _ D D D P G.1V *1V]#AG *JG.A]AJGX c?A*? #V1 V1#DAf1. AG ]?1 *#X1 J6 AXJ]VJLA* = GT #G. L1#C1.@AG@]?1@6JVc#[email protected]*]AJG L?#X1 6aG*]AJGX- GJVEX J6 ]?1X1 #..A]AJG#D #.@ .1G.X cADD '1 Ea*? D1XX ]?#G aGA]e AG XJE1 *JELD1]1 XL#*1XP JG]VA'a]AJGX ]J # XJDa]AJG J6 MHHN .a1 ]J ]?1 .1=1G1V#]1 C1VG1D *#G '1 AXJD#]1. AG ]?1 1dLDA*A] 6JVEP ?1X1 *JG]VA'a]AJGX *JG]#AG # GaE'1V J6 AG.1]1VEAG#]1 *JGX]#G]X M]?1 GaE'1V J6 ]?1X1 *JGX]#G]X *#G '1 1Ra#D ]J K JV ]J
1V1 ]?1 6aG*]AJG ' . ^ U T = ; ( K 4 =; Q ` <

#

; T

' . ^ =M T =( =; ( K .= 4 =; Q <

#

;

' . =M T =( =; ( K .=

T

' . $; AX GJVE#DAf1. 'e ]?1 *JG.A]AJG ` T =UT =( K .= 4 K (:) #G. .19G1X ]?1 #G=aD#V .1L1G.1G*1 J6 ]?1 ADG1UX LVJ'D1E XJDa]AJG (:3Z) JG ]?1 'JaG.#Ve J6 ]?1 X1EA@ AG9GA]1 LD#G1@L#V#DD1D *JGX1Vb#]Ab1De X*#]]1VAG= E1.AaEP JGXA.1V # DAG1#V AG]1=V#D JL1V#]JV # ; . ' . ' ' . !

=( = / = .= ( = g( K ( MKgKN

/ M=N 4 T

. ' ' .. ' _ Qg( =Y g( = ( K c?1V1 'e =( = AX E1#G] # 6aG*]AJG ; M=( = N 4 = GT ' .. ' _ Qg( =Y g( = ( KY < P 6 < 4 K ]?1X1 6aG*]AJGX #V1 1Ra#D JV # 6aG*]AJG = GT GJ 61c1V ]?#G ]J 1#*? J]?1V 'e .19GA]AJGP 6 ]?1 X1] T 6JV AT 4 K LJXX1XX1X . ' _ ]cJ GJGf1VJ 1D1E1G]X #G. < `- ]?1G #G 1Ra#DA]e GT Qg( =Y g( = ( K 4 / ' . ' ' . . ' ' . ' . . ; _ Qg( =Y g( = ( KY < &. KY g T 46. g ( = T 46. g ( = Q GT .)M

YDV

KK E:E]XSE]* KK EQE]>@

. ' ?JD.XP e ]?1 6aG*]AJG =( = X?JaD. '1 E1#G] ]?1 6aG*]AJG ; M=( = N A6 ]?1 X1] T M6JV AT 4 KN .J1X GJ] *JG]#AG GJGf1VJ 1D1E1G]X MXa*? # XA]a#]AJG ]#C1X LD#*1 ! . .1GJ]1 ]?1 ;]? LJc1V 6JV 1d#ELD1 6JV AXJ]VJLA* #G. DAG1#V L?#X1 6aG*]AJGXNP 1] ' . ' .; ' .. . ! ! ! ! / M=N J6 ]?1 JL1V#]JV P ?AX LJc1V AX .19G1. 'e # V1D#]AJG / M=N 4 T ! AX E1#G] ]?1 A.1G]A]e JL1V#]JV MAP1P c?1V1 ; #G. = g( K P e ]?1 XeE'JD ' . ! T / M=N 4 / M=NNP 1] ]?1 AG]1=V#D JL1V#]JV ! '1 # *JG]V#*]AJG JL1V#]JV (Kg<- KK^ K^^) AG XJE1 GJVE#DAf1. XL#*1 J6 6aG*]AJGX .19G1. JG ]?1 X1=E1G] g( K P ] AX @@ ' .@@ @@ @@ ! / @@ J @@/ @@ ?JD.X c?1V1 J AX XJE1 E1#G] ]?#] ./ M=N #G AG1Ra#DA]e @@ LJXA]Ab1 *JGX]#G] D1XX ]?#G aGA]eP 1] aX cVA]1 .JcG # 6JVE#D XJDa]AJG J6 MKggN 6JV < `P A]?AG ]?1 6V#E1cJVC J6 ]?1 *JG.A]AJGX #XXaE1. #'Jb1 #G. ]#CAG= AG]J #**JaG] ]?1 .19GA]AJG J6 ]?1 ! MX11 MKggNN # 6JVE#D XJDa]AJG J6 MKggN *#G '1 LV1X1G]1. AG ]?1 6JVE JL1V#]JV ' ' . . T =( =; ( K 4 `T =( =; ( K / ' ' . : ' . ' . ' . . ' . 4 `UT =; ( K T = Q ; =( =; &. KY g . = \. =; D .)M

MKg`N ' . ' . ' ;A ! ? ' .' . ;A

1V1 = g( K -=; g( K Y T = 4 ?)T K = - ; =( =; 4 ?)T A ' .' . ' . .' . ' . ; ! ?' ? _ !

GT = Y 6JV #DD ; 4 `( < . = 4

TG. = D #G. \. =; 4 ?)T ' . . ' . $; ' _ = T 46. g ( = T = ( =; ( K .= P e ]?1 XeE'JDX TG. #G. GT X?JaD. T ' ' . . '1 E1#G] V1XL1*]Ab1De ]?1 6aG*]AJG T 46. g ( =' #G. ]?1 X1*JG. . ' . ]1VE ' AG . ]?1 VA=?]@?#G. XA.1 J6 MKggNP DD ]?1 6aG*]AJGX T = - ; =( =; - M = ' . D D D -/ = AG 1dLV1XXAJG MKg`N #V1 DAEA]X J6 ]?1 *JVV1XLJG.AG= X1Ra1G*1X '1 M=N #G. ]?1 ]1VEX J6 ]?1X1 X1Ra1G*1X . ''1..1]1VEAG1. 'e E1#GX J6 # V1*aVV1G*1 $ ; ' X?JaD. 6JVEaD# '1A; M=N 4 0 T =( = '1 = .= c?1V1 > T #G. = g( K #G. 0 AX '# =Ab1G AG .1]1VEAGAG= ]?1 6aG*@ . '6aG*]AJGP . X' #G. AGA]A#D #LLVJdAE#]AJG ' . ]AJGX T = - ; =( =; -M = -D D D -/ = JG1 *#G ]#C1 # 6aG*]AJG 0P e 0 AX ' ' ' . . . _ Qg( =Y g( =; ( K - T 46M g ( = ( D D D ( E1#G]- V1XL1*]Ab1De- 6aG*]AJGX K- GT ' ' . . T 46/ g ( = P $; Jc c1 aX1 ]?1 GJVE#DAf#]AJG *JG.A]AJG T M=( = ( KN= .= 4 T ' $; T ` T M=( = ( KN= .= 4 K ]?#] AX b#DA. 6JV #DD =; g( K P XAG= ]?AX *JG.A]AJG #G. 6JVEaD#X MKg`N ]?1 6JDDJcAG= 1dLV1XXAJG AX J']#AG1., / : UT M=; ( KN 4 M`NT N; $ M=; N Q &, MKY gNN, \, M=; N (

MKg^N

,)M

$; c?1V1 6JV 8 4 g #G. 6JV #DD 8 4 `( < ]?1 b#Da1X N, 4 T =, M=N .=( $ M=; N 4 $; K T =; M=( =; N .=D XAG= MKg^N 6VJE MKg`N ]?1 6JDDJcAG= 1dLV1XXAJG AX J']#AG1. / ' ' ' . : ' . ' . ' . . . &. KY g \. =; . = D MKg
U <. &<Q&X.Q>SX>& .M\X>EC E3 Q+>X>]. XQCS3.Q X<.EQ`

YDU

' . ' . ' . ' . '

1V1 = g( K ( =;' . g( K - ; =('=; . 4 MN'T N;.$ =; T = Q ; =('=; .( 6JV #DD '; 4.`( < , = 4 MN, PNT NT = , = P GCGJcG 6aG*]AJGX \M =; D D D -\/ =; LV1X1G] AG 1dLV1XXAJG MKg
1V1 c1 =Ab1 # GaE'1V J6 1dLD#G#]AJGX c?A*? X?JaD. '1 ]#C1G AG]J #**JaG] AG L1V6JVEAG= *JG*V1]1 *#D*aD#]AJGX cA]? ]?1 ?1DL J6 ]?1 #D=JVA]?E LV1X1G]1. #'Jb1P G1 J6 ]?1 EJX] AELJV]#G] AX ]J *#D*aD#]1 ]?1 6aG*]AJGX . . ' X]#=1X J6 ]?AX #D=JVA]?E ' _ Qg( =Y g( =; ( KY < N #G. _ ; M=( =; N MJV =; Qg( =Y g( =; ( K 4 GT GT =; ; M M=( =; NNP Jc1b1V- ]J *#D*aD#]1 ]?1X1 6aG*]AJGX ]?1 6JDDJcAG= AG9GA]1 XeX@ ]1E X?JaD. '1 XJDb1.

^MK /; N$; M?Y gY =; ( KN Q `4?$M M?Y gY =; ( KN 4 ^/; ; M=; N 4?$T M?Y gY =; ( KN( `4?$; M?Y gY =; ( KN :MK /M N$M M?Y gY =; ( KN Q ^4?$K M?Y gY =; ( KN 4 :/M M M=; N( %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M4?$?; M?Y gY =; ( KN M`M Q KNMK /? N$? M?Y gY =; ( KN Q MM Q KN4?$?A; M?Y gY =; ( KN 4 M`M Q KN/? ? M=; N( %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M 4 K( `( ^( D D D ( ? M - T ND MKg:N G ]?AX XeX]1E ]?1 b#Da1X $T M?Y gY =; ( KN #G. $; M?Y gY =; ( KN #V1 *JGXA.1V1. ]J '1 CGJcG b#Da1X #G. ]?1e *#G '1 LV1X1G]1. AG ]?1 6JVE A

$T M?Y gY =; ( KN 4

: ^MK /; N M Z M? Y gN M`5 Q KN/) ) M?Y gN) M=; N( ; ?M ))T

$; M?Y gY =; ( KN 4 M4?N; D

MKg[N

1V1 ]?1 b#Da1X Z; M? M Y gN #G. ) M?Y gN #V1 .19G1. 'e 6JVEaD#X M:gN #G. M[HN #D=1'V#A* 1Ra#]AJGX AG c?A*? K 4 g #G. AT 4 KP ?1 AG9GA]1 XeX]1E J6 DAG1#V @ MKg:N c#X J']#AG1. 6VJE XeX]1E M[`N 6JV K 4 g- 0?T M?N@?

4 M`M Q KN/? ? M=; N #G. ? M - T N c?1G AT K MA6 AT 4 K ]?1G g T NP 1] ]?1 XJDa]AJG $T M?Y gY =; ( KN( $; M?Y gY =; ( KN( $M M?Y gY =; ( KN( D D D J6 XeX]1E MKg:N AG ]?1 *D#XX 8M MgN M]?1 .19GA]AJG J6 ]?1 *D#XX 8M MgN AX =Ab1G AG LL1G.Ad N AX J']#AG1.P ?1 1d#*] XJDa]AJG J6 XeX]1E MKg:N *#G '1 J']#AG1. 'e L#XX#=1 ]J ]?1 DAEA] AT K AG 6JVEaD#X M[HNP 6 ]?AX XJDa]AJG AX CGJcG ]?1 6JDDJcAG= #G#De]A*#D LV1X1G]#]AJGX J6 ; M=( =; N #G. M M=( =; N *#G '1 J']#AG1. aXAG= J6 6JVEaD#X MHgN #G. MH[N #G. ]?1 *JG]JaV AG]1=V#]AJG E1]?J.

YD/

KK E:E]XSE]* KK EQE]>@ A

; M=( =; N 4

=; : M`M Q KN/? ? M=N
#

$? M?Y gY =; ( KN )? MK Q 4?=NMK Q 4?=; N

MKgZN

=; T M=( =; N Q `M= Q =; N # A $? M?Y gY =; ( KN =; : Q )?D MKg2N M`M Q KN/? ? M=N
M M=( =; N 4

1V1 ]?1 6aG*]AJG T M=( =; N AX .19G1. 'e 6JVEaD# M^KN 6JV K 4 g #G. ]?1 .AXLJXA]AJG #G. ]?1 L#]?@]V#*AG= .AV1*]AJG J6 *JG]JaV JG ]?1 *JELD1d LD#G1 #V1 X?JcG AG A=P ZP`P 6 ]?1 LJXA]Ab1 VJJ] 6M MgN XA]a#]1. AG ]?1 AG]1Vb#D Mg-KN J6 ]?1 1Ra#]AJG ZT M6 M Y gN 4 g 1dAX]X MX11 1Ra#]AJG M
$ ; ' _ Qg( =Y g( = ( KY < .= : K M]?1 XeE'JD XaL ('M=; N) XaL =; T ; GT 0 =T;

0 =T;

$!/ 3/7/ <. +>SGES>X>EC C+ X<. GX<=XQ&>C: +>Q.&X>EC E3 &ECXE\Q EC X<. &EBGA._ GAC.K

U <. &<Q&X.Q>SX>& .M\X>EC E3 Q+>X>]. XQCS3.Q X<.EQ`

YDD

AX E1#GAG=6aD ]J #G 1d#*] ]JL XA.1 J6 X1] J6 b#Da1X c?A*? #V1 #**1L]1. cA]? 6aG*]AJG 'M=; N JG ]?1 ?#D6@AG]1Vb#D Mg-K)NP ?1 *?1*C J6 6aD9DDE1G] J6 #Ge J6 ]?1X1 AG1Ra#DA]A1X X?JaD. '1 *#VVA1. Ja] 6JV 1#*? L?#X1 6aG*]AJG X1D1*]1.P . ' *JG*V1]1 *#D*aD#]AJGX J6 b#Da1X J6 ]?1 6aG*]AJGX T =( =; ( K ( 'G L1V6JVEAG= . UT =; ( K ;6JV ]?1 L?#X1 6aG*]AJG BM=N X1D1*]1. A]X LJDeGJEA#D #LLVJdAE#]AJG 1 B M=N 4 ?)T M`M Q KN/? ? M=N MX11 Xa'X1*]AJG ZP^P
^ M Z M? N ?M

*#G '1 J']#AG1. c?1V1 J; ? M JM ? M Z M? N 4 KY ( (DDD K K

M

AX #G AG9GA]1 *JG]AGaJaX 6V#*]AJG MX11 LL1G.Ad N #G. 6JV #DD M J? 4

MM Q KNM D M`M Q KNM`M Q ^N

1] ]?1 L?#X1 6aG*]AJG '1 # DAG1#V 6aG*]AJG cA]? V1XL1*] ]J =- AP1P /T 4 K( /; #4 g#G. 6JV #DD M `( ^( <( D D D /? 4 gP ?1G ]J *#D*aD#]1 ]?1 6aG*]AJGX ; M=( =; NM M=( =; N JGDe ]?1 b#Da1X $T M?Y gY =; ( KN- $; M?Y gY =; ( KN 6JV #DD ? X?JaD. '1 CGJcGP VJE 6JVEaD#X MKg[N A] 6JDDJcX ]?#] $T M?Y gY =; ( KN 4 c?1V1

. ^MK /; N ' K Q 4/; ?=; MK /; N; ;G; M? M N ;G; M? M N( M ?

J ? M JM ? M JK ? M J. ? M ( ( ( DDD ;G; M? N 4 KY ; K K K K

MKgHN

M

M P ?1 b#Da1 $; M?Y gY =; ( KN AX #G AG9GA]1 *JG]AGaJaX 6V#*]AJG AG c?A*? J; 4 K=;#

>; #**JV.AG= ]J MKg[N AX 1Ra#D ]J $; M?Y gY =; ( KN 4 M4?N P JV XA]a#]AJGX c?A*? V1#DAf1 AG =1JL?eXA*#D ]aV'A. E1.A# # LJDeGJEA#D 6aG*]AJG ;1 B M=N 4 ?)T M`M Q KN/? ? M=N #LLVJdAE#]AG= ]?1 L?#X1 6aG*]AJG 1d#*]De 1GJa=? EaX] *JG]#AG X1b1V#D ?aG.V1.X JV ]?JaX#G.X J6 #..1G.XP ?1V16JV1 c1 *JGXA.1V # =1G1V#D *#X1 c?1G > `P G bAV]a1 J6 6JVEaD#X MKgZN- MKg2N 6JV ]?AX XA]a#]AJG ]J J']#AG ]?1 6aG*]AJGX ; M=( =; N- M M=( =; N A] AX G1*1XX#Ve ]J *#D*aD#]1 ]?1 b#Da1X $T M?Y gY =; ( KN- $; M?Y gY =; ( KN( D D D ( $1 M?Y gY =; ( KN 6JV #DD ? P VJE ?1JV1E [ MZP^P`N #G. ]?1 6JVE J6 AG9GA]1 XeX]1E MKg:N A] 6JDDJcX ]?#] ]?1X1 b#Da1X #V1 ]?1 aGARa1 XJDa]AJG J6 ]?1 6JDDJcAG= 9GA]1 XeX]1E J6 DAG1#V #D=1'V#A* 1Ra#]AJGX

9bb

KK E:E]XSE]* KK EQE]>@

2 Y$ H0 I)2 H2T bT 0 FI YHF $ I H2T bT 0 FIL[)2 H2T bT 0 FI [)2 H2T bT 0 FI7HF$ I H2T bT 0 FILY)2 H2T bT 0 FI2 7$ H0 I H1 FI)2 H2T bT 0 FIH[1 FIHF $ I H2T bT 0 FI L 1)2 H2T bT 0 FI 2 H[1 FI$ H0 I 1)2 H2T bT 0 FI H2 I H2T bT 0 FI 2 H[1 L FI$ H0 I6 HFFbI

1V1 %M? M N 4 M`> Q KNMK /1 N Q #G. 6JV #DD 8

M> Q KNM ? M M M`> Q ^NZ 1A; M? N

J, ? M J,A; ? M ( (DDD 4 KY K K

M Z , M? N

AX #G AG9GA]1 *JG]AGaJaX 6V#*]AJGP G #..A]AJG ]?1 b#Da1X $T M?Y gY =; ( KN- #G. $; M?Y gY =; ( KN J6 XeX]1E MKKgN *#G '1 *JGXA.1V1. ]J '1 CGJcGP ?1e #V1 .19G1. 6JV #DD > `( ^( D D D 'e ]?1 6JDDJcAG= 1dLV1XXAJGX ( / 1 : ^MK /; N M $T M?Y gY =; ( KN 4 ;G1 M? N K Q M`5 Q KN/) )G1 M?N) M=; N ( ?M )); $; M?Y gY =; ( KN 4 M4?N; D

MKKKN

?1 b#Da1X #LL1#VAG= AG ]?1 9VX] J6 6JVEaD#X MKKKN ?#b1 ]?1 6JDDJcAG= E1#GAG=, 6JV #DD 8 K( `( D D D ( > K ]?1 b#Da1 J1; MgN? M J1 ? M J, MgN? M M ,G1 M? N 4 KY (DDD( ( Y K K Z1A; M? M N M> Q KNM Y M`> Q KNM`> Q ^NMK /1 N . ' M ; 1G1 M? M N 4 K Q J1 ? M Z Y 6JV #DD 5 K( `( D D D ( > ]?1 b#Da1 1A; M? N J1 4

)G1 M?N 4 M4?N)

) 3 ,);

8 M`8 Q KNMK /, N,G1 M? M N

?1JV1E ` MZP^PKNNP M]?1 .19GA]AJG J6 ]?1 b#Da1 J? MgN AX =Ab1G AG ]?1 6JVEaD#]AJG ;J6 1 J]A*1 ]?#] AG aXAG= # LJDeGJEA#D #LLVJdAE#]AJG B M=N 4 ?)T M`M Q KN/? ? M=N J6 ]?1 L?#X1 6aG*]AJG BM=N ]?1 X1VA1X AG 6JVEaD#X MKgZN- MKg2N X?JaD. '1 Xa'X]A]a]1. 'e 9GA]1 XaEX AG c?A*? ]?1 XaEE#]AJG AX *#VVA1. Ja] 6VJE f1VJ ]J > M> `NP G #..A]AJG- AG MKg2N 'e ]?1 6aG*]AJG T M=( =; N X?JaD. '1 E1#G] ]?1 XaE ; 1 ?)T M`M Q KN/? ? M=N? M=; NP Jc c1 X?#DD =Ab1 XJE1 GJ]1X JG ]?1 ]1*?GARa1 J6 *#D*aD#]AG= ]?1 b#Da1X $T M?Y gY =; ( KN- $; M?Y gY =; ( KN( $M M?Y gY =; ( KN( D D D ( $1 M?Y gY =; ( KN 6JV .A6@ 61V1G] b#Da1X ? P A71V1G] E1]?J.X 6JV *#D*aD#]AG= ]?1 b#Da1X LJAG]1. Ja] #'Jb1 X?JaD. '1 aX1. .1L1G.AG= JG c?#] L#V] J6 ]?1 *JG]JaV ]?1 LJAG] #XXJ*A#]1. cA]? ]?1 GaE'1V ? AX XA]a#]1.P AG*1 c1 ?#b1 ]?1 1d#*] XJDa]AJGX J6 ]?1 AG9GA]1 XeX]1E MKg:N #G. ]?1 9GA]1 XeX]1E MKKgN A] AX #Dc#eX LJXXA'D1 ]J 1b#Da#]1 ]?1 1d#*]G1XX J6

7 The characteristic equation of radiative transfer theory

401

the solution obtained with any known numerical method for solving a ﬁnite system of linear algebraic equations. In particular when ν ∈ Γ and |ν| is not more to" than some positive number ς0 (it is of the order of several tens) it is useful ! apply the right sweep method (see, for example, [105]). If ν ∈ Γ and |ν| ∈ ς0 , ξ0 where ξ0 is of the order of hundreds or thousands then to calculate the values b2 (ν; 0; μ1 1), . . . , bn (ν; 0; μ1 1) the straight recursion can be used (it is not diﬃcult to write the straight recursion using the equations of system (110) itself). It should be noted that the selection of concrete numbers ς0 , ξ0 depends on values μ1 and a phase function model selected. In performing calculations it is necessary to take into account that for all ν ∈ Γ the values b0 (ν; 0; μ1 , 1) and b1 (ν; 0; μ1 , 1) are given analytically with formulas (111). When the absolute value of ν situated on Γ is more then ξ0 it is useful to apply asymptotic formulas for the values b0 (ν; 0; μ1 , 1), b2 (ν; 0; μ1 1), . . . , bn (ν; 0; μ1 1) required (they are valid for |ν| → ∞). The technique for obtaining these asymptotic formulas was given in [65]. Once the values b0 (ν; 0; μ1 , 1), b1 (ν; 0; μ1 , 1), b2 (ν; 0; μ1 1), . . . , bn (ν; 0; μ1 1) on the contour Γ (or Γ0;m ) have been obtained (within the framework of the polynomial approximation of the phase function) the values of the functions Q1 (|μ|, |μ1 |) ˜ ∗∗ +0, −|μ|; 0, −|μ1 |, 1; m and Q2 (|μ|, |μ1 |) can be calculated using for(or |μ1 |G ∞;0 mulas (107), (108) (proper modiﬁcations and simpliﬁcations pointed out above should be introduced into these formulas). Knowing the values of these functions it is possible to calculate the values of the required function R0 (|μ|, |μ1 |, 1), u0 (|μ1 |, 1) using formulas (103), (104) and the iteration algorithm described in this subsection. In Table 7.1 the relations and the equalities which allow one to calculate the function R0 (|μ|, |μ1 |, 1), u0 (|μ1 |, 1) within the framework of the simplest variant Table 7.1. The simplest variant of the iteration algorithm to calculate functions R0 (|μ|, |μ1 |, 1), u0 (|μ1 |, 1) No.

The required values

The relations and equalities which allow one to calculate these values

1.

Υ0 (|μ|)

Υ0 (|μ|) = lim χs (|μ|), s→+∞ R1 χs+1 (|μ|) = 1 − 0 Q1 (|μ|, μ )χs (μ ) dμ , χ0 (μ ) ≡ 1.

2.

Υ1 (|μ|, |μ1 |)

Υ1 (|μ|, |μ1 |) = lim χs (|μ|, |μ1 |), s→+∞

3.

σ0

4.

b∗ (|μ1 |)

χs+1 (|μ|, |μ1 |) = |μ1 |−1 Q2 (|μ|, |μ1 |)− R1 − 0 Q1 (|μ|, μ )χs (μ , |μ1 |) dμ , χ0 (|μ|, |μ1 |) = |μ1 |−1 Q2 (|μ|, |μ1 |). R1 σ0 = 0 μΥ0 (μ) dμ. R1 b∗ (|μ1 |) = 1 − 0 μΥ1 (μ, |μ1 |) dμ.

5.

u0 (|μ1 |, 1)

u0 (|μ1 |, 1) = (2σ0 )−1 b∗ (|μ1 |).

6.

W1 (|μ|, |μ1 |)

W1 (|μ|, |μ1 |) = σ0−1 b∗ (|μ1 |)Υ0 (|μ|) + Υ1 (|μ|, |μ1 |).

7.

R0 (|μ|, |μ1 |, 1)

R0 (|μ|, |μ1 |, 1) = 12 W1 (|μ|, |μ1 |).

402

N.N. Rogovtsov, F.N. Borovik

of this algorithm (i.e. when the function K(μ, μ ) in the integral operator (101) is substituted by the function Q1 (|μ|, |μ1 |)) are presented. In conclusion of this subsection we give a block scheme (Scheme 7.1) of the simplest variant of the algorithm to calculate the functions R0 (|μ|, |μ1 |, 1), u0 (|μ1 |, 1)

Finding out the solution of the system (105) (or of the system (110)) on the contour

The function is calculated using the formula (107)

The function is calculated using the formula (108)

The function is calculated using the iteration procedure No. 7.1 from Table 7.1

The function is calculated using the iteration procedure No. 2 from Table 7.1

The value is calculated with the formula No. 3 from Table 7.1

The function is calculated with the formula No. 4 from Table 7.1

The function is calculated with the formula No. 6 from Table 7.1

The function is calculated with the formula No. 5 from Table 7.1

The function is calculated with the formula No. 7 from Table 7.1 Scheme 7.1.

7 The characteristic equation of radiative transfer theory

403

7.5.4 Numerical results for the case of water cloud C.1 model The method presented above for calculating the azimuthally averaged reﬂection function R0 |μ|, |μ1 |, 1 and the reﬂection function R0 1, |μ1 |, 1 is eﬀective for any phase functions (both are isotropic and peaked in the forward direction). Of special interest is this method for the case of phase functions highly peaked in forward directions which are inherent in geophysical turbid media. The reason is that the integral operator (101) for the case of such phase functions has a norm (for example, in the space of absolutely integrable functions on [0, 1], i.e. in the space L1 (0, 1)) much less than unity.Such a situation takes place even when the ˜∗ function |μ |G∞;0 +0, −|μ|; 0, −|μ |, 1 is used as a kernel K(|μ|, |μ |) of integral operator (101). If the degenerative part (it is generated by the presence of discrete roots of equation (40) for r = 0 not equal to zero) is extracted from this function then the corresponding norm of integral operator (101) will be still smaller. Further to illustrate the aforesaid, a number of results obtained numerically using the algorithm of this chapter for the case of water cloud C.1 model (λ = 0.6457 μm; λ is a wavelength) [155] is presented. For this model the single scattering albedo ω0 is equal to 1. In performing the concrete calculations the polynomial approximation of the phase function was used, n being equal to 471. In Fig. 7.3 the dependence of the phase function p(μ) of the argument μ is shown. In Table 7.2 the values of roots of equation ℘0 (−k 2 , 0) = 0 (see equation (40)) situated on [0, 0.99999] and written in ascending order are presented. Recall that the parameter k is connected with the parameter ν of IE (32) by the relation ν = ± ik. In this table the normalization constants cl (1; 0)of relations (95) (l ≥ 2) corresponding to these roots is also given. ) ) ) ) )μ), )μ1 ) = of the functions Q 1 ) )In ∗ Figs 7.4 and 7.5 ) ) ) )the dependences ) ) ) ) ) ) )μ1 ) G∞;0 (+0, −|μ|; 0, −)μ1 ), 1) and Q2 )μ), )μ1 ) = )μ1 ) G∗∞;0 (+0, −|μ|; 0, )μ1 ), 1) ) ) ) ) of the arguments )μ) and )μ1 ) are shown. The ﬁrst of these functions deﬁnes the kernel K(μ, μ1 ) of IE (100). The second term of the right-hand side of IE (100) is expressed through the second function.

Fig. 7.3. The dependence of the phase function p(μ) for the case of water cloud C.1 model of the argument μ.

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Table 7.2. The roots of the equation ℘0 (−k2 , 0) = 0 and their normalization constants. kl (0)

0

0.41270

0.55819

0.60515

0.63992

cl (1; 0)

–

0.68721

0.12242

0.06567

0.05227

kl (0)

0.67043

0.69839

0.72451

0.74914

0.77251

cl (1; 0)

0.04498

0.03997

0.03616

0.03305

0.03040

kl (0)

0.79472

0.81585

0.83595

0.85502

0.87307

cl (1; 0)

0.02808

0.02599

0.02408

0.02229

0.02059

kl (0)

0.89007

0.90602

0.92089

0.93464

0.94724

cl (1; 0)

0.01897

0.01741

0.01589

0.01441

0.01295

kl (0)

0.95865

0.96882

0.97771

0.98525

0.99138

cl (1; 0)

0.01149

0.01005

0.00858

0.00709

0.00554

kl (0)

0.99600

0.99897

cl (1; 0)

0.00391

0.00211

˛ ˛ ˛ ˛ `˛ ˛ ˛ ˛´ Fig. 7.4. The dependence of the function Q1 ˛μ˛, ˛μ1 ˛ of the arguments ˛μ˛ and ˛μ1 ˛; 1 – |μ| = 0.0003125, 2 – 0.003125, 3 – 0.0175, 4 – 0.045, 5 – 0.095, 6 – 0.19, 7 – 0.39, 8 – 0.69, 9 – 1.

) ) ) ) ) ) ) ) It is seen that the function Q1 )μ), )μ1 ) for all )μ), )μ1 ) ∈ (0, 1] has values less than unity. In addition, the norm of the integral operator in IE (100) in the space ) ) %1 L1 (0, 1) (see Appendix A) is not more than a number sup Q1 (μ, )μ1 )) dμ that 0 μ1 ∈(0,1]

is equal about to 0.1055. In )Fig. ) ) azimuthally averaged reﬂection function ) )7.6 the dependences ) of) the ) R0 )μ), )μ1 ), 1 of the arguments )μ), )μ1 ) for the semi-inﬁnite plane-parallel conservatively scattering turbid medium are shown. The results of ) ) ) calculations ) numerical satisfy general constraints imposing on the function R0 )μ), )μ1)), 1) . This ) )function is non-negative, symmetrical with respect to the arguments )μ) and )μ1 ) and

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˛ ˛ ˛ ˛ `˛ ˛ ˛ ˛´ Fig. 7.5. The dependence of the function Q2 ˛μ˛, ˛μ1 ˛ of the arguments ˛μ˛ and ˛μ1 ˛; 1 –|μ|=0.0003125, 2 – 0.045, 3 – 0.095, 4 – 0.29, 5 – 0.69, 6 – 1

Fig. 7.6. The dependence of the function R0 (|μ|, |μ1 |, 1) of the arguments |μ| and |μ1 |; 1 – |μ| = 0.0475, 2 – 0.19, 3 – 0.39, 4 – 0.59, 5 – 0.89, 6 – 1.

) ) %1 satisﬁes the normalization condition 2 0 μR0 (μ, μ1 , 1) dμ = 1 for all )μ1 ) ∈ (0, 1] and [7]. In Fig. 7.7 the dependence of the function ) de Hulst–Sobolev )relation ) ) van u0 )μ), 1 of the argument )μ) is shown. ) ) In Table 7.3 values of the reﬂection function R0 1, )μ1 ), 1 calculated with the algorithm described above are given. ) ) 0 ) ) In )Fig. ) argu) 7.8 the dependence of the reﬂection function R 1, μ1 , 1 of ) the ment )μ1 ) is shown. In addition, in this ﬁgure values of the function R0 1, )μ1 ), 1 calculated with the algorithm presented in [156] at points situated on the segment

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Fig. 7.7. The dependence of the function u0 (|μ|, 1) of the argument |μ|. ` ˛ ˛ ´ Table 7.3. Values of the reﬂection function R0 1, ˛μ1 ˛, 1 μ1

R0 (1, μ1 , 1)

μ1

R0 (1, μ1 , 1)

μ1

R0 (1, μ1 , 1)

0.00000 0.00125 0.00250 0.00375 0.00500 0.00625 0.00750 0.00875 0.01000 0.02500 0.05000 0.07500 0.10000 0.12500 0.15000 0.17500 0.20000

0.29151 0.30331 0.31369 0.32259 0.33055 0.33779 0.34447 0.35067 0.35647 0.40664 0.45710 0.49567 0.52982 0.56143 0.59116 0.61937 0.64631

0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625

0.67210 0.69688 0.72069 0.74375 0.76607 0.78771 0.80879 0.82924 0.84913 0.86833 0.88682 0.90467 0.92173 0.93817 0.95406 0.96966 0.98541

0.6500 0.6750 0.7000 0.7250 0.7500 0.7750 0.8000 0.8250 0.8500 0.8750 0.9000 0.9250 0.9500 0.9750 0.9950 0.9975 1.0000

1.0018 1.0197 1.0398 1.0628 1.0880 1.1116 1.1270 1.1299 1.1264 1.1286 1.1368 1.1449 1.1525 1.1631 1.1937 1.2192 1.2607

[0.01, 1] are marked by black circles. The maximal relative deviation of reﬂection function values obtained with the algorithms mentioned above is less than 0.004% at these points.

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˛ ˛ ` ˛ ˛ ´ Fig. 7.8. The dependence of the reﬂection function R0 1, ˛μ1 ˛, 1 of the argument ˛μ1 ˛: the straight line is the reﬂection function calculated with the algorithm of this work, black circles are the reﬂection function calculated with the technique of [156].

7.6 Conclusions We have presented in close detail the qualitative and constructive mathematical theory of the classical variant of the characteristic equation of radiative transfer theory. This theory has been used as a basis of correct and eﬀective methods for solving CVCE, RCE and boundary-value problems of RTT. The theory pointed out above substantially diﬀers from the qualitative mathematical theory of CVCE constructed on the basis of ideas and methods of functional analysis and Case’s method [23, 24, 53–60]. In fact this theory is based on wide use of classical results of mathematical analysis, diﬀerence equation theory and continuous fraction theory. In virtue of this fact, in presenting the results obtained, it was necessary to give and formulate initial statements and important theorems of diﬀerence equation theory and continuous fraction theory. Also, to make easier the perception of this chapter, results concerning functional analysis (in particular, elements of linear operator theory and linear operator equation) were included. On the base of the developed theory of CVCE we have succeeded in obtaining solutions of CVCE, RCE in an analytical form in the case of arbitrary phase functions (in particular, for phase functions highly peaked in the forward direction). In addition, with the help of this theory, rigorous general analytical expressions for all azimuthal harmonics of Green’s functions of the radiative transfer equation (RTE) were obtained for the case of a plane-parallel inﬁnite turbid medium. As, in general, phase functions corresponding to real turbid media cannot be speciﬁed exactly in numerical algorithms, particular attention has been given to stability analysis of solutions of CVCE, RCE (and also expressions involved in them). It has been shown that all supposed algorithms for obtaining these solutions and expressions are stable and allow a convenient numerical realization for the case of arbitrary phase functions. In section 7.5 the eﬀective algorithm for calculating the azimuthally averaged reﬂection function and the reﬂection function for the case of a plane-parallel semi-inﬁnite

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turbid medium has been presented. In constructing this algorithm the usefulness of simultaneous use of the CVCE theory presented above and also general invariance relations [8, 66–72] which allow simple physical interpretation have been shown. It should be noted that this algorithm is eﬀective both for isotropic phase functions and for phase functions peaked in the forward direction at arbitrary values of albedo single scattering. It should be noted that the ﬁrst attempts at constructive use of the simplest notions of the continuous fraction theory to solve RCE were undertaken in [162, 163]. In generalizing the results of this chapter for the case of the vector characteristic equation the information given in [125, 164] can be useful. The results given in section 7.5 for the case of the water cloud C.I model (λ = 0.6457 μm) show that the approach presented allows not only one to ﬁnd rigorous (or numerical) solutions but also to obtain highly accurate approximate analytical solutions of RTT boundary-value problems. It should be noted that, on the basis of the approach presented in this chapter, boundary-value problems of RTT can be solved for the case of turbid media of non-concave forms (see, for example [8, 66– 90]). Also the presented results can be used to obtain new analytical approximate solutions of RTE for the case of plane-parallel turbid media of any optical thickness. Acknowledgments This work was supported by the National Academy of Sciences of Belarus and the Education Ministry of Belarus within the framework of the Fundamental Investigation Programs ‘Wave’ and ‘Coherence’. The authors are thankful to M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, and N. T. Zakharova for providing the code for calculations of the reﬂection function of semi-inﬁnite media. The authors are grateful to Alexander A. Kokhanovsky for valuable and constructive discussions of problems touched upon in this chapter. The authors are also grateful to Tatiana Karpuk for the help in typing the manuscript.

Appendix A: General mathematical notations, notions and constructions A set is one of major general mathematical notions. If A is a ﬁnite set it is written , 1 in the form A = a1 , a2 , . . . , an where a1 , a2 , . . . , an are its elements, and n is the number of elements belonging to A. The relation of belonging of an element b to a set B is denoted by a symbol ∈ (i.e. 1 b) ∈ B). , A general way to write any set M is the following presentation: M = m)S(m) where S(m) is some condition (or their collections) to which the elements of this set should satisfy. If the condition S(m) does not hold (or S(m) is a false assertion) we write m ∈ / S(m), i.e. m does not belong to M . In set theory the symbols ⊂, ∪, ∩, \ are used. A notation A ⊂ B means that any element of the set A belongs to the set B, i.e. A is a subset of the set B. Sets A and B are considered to be equal, i.e. A = B if relations A ⊂ B, B ⊂ A hold simultaneously. Further, for brevity, a symbol ∀ will be used instead of the term ‘for all’ in writing mathematical expressions. By a symbol 1 ) , ∪ is meant the union operation of sets. It means that A∪ B = x )x ∈ A 1or )x ∈ B . An expression , A ∩ B deﬁnes an intersection of the sets, i.e. A ∩ B = x )x ∈ A and x ∈ B . A

7 The characteristic equation of radiative transfer theory

409

1 ) , construction A\B means a diﬀerence of sets, i.e. A\B = x)x ∈ A and x ∈ / b . An empty set ∅ is thought to be a set containing no elements. Another major general mathematical notion is the notion of a map (function, operator) of a set A into a set B. Symbolically a map f is written as f : A → B. The symbol f is thought to be a rule (law, algorithm) that ∀a ∈ D(f ) (D(f ) ⊂ A) points out even if one element b ∈ B. The symbol D(f ) is thought to be the domain of the map f . For all a ∈ D(f ) the map f deﬁnes one or a collection of elements of the set B. If b = f (a) where a ∈ D(f ) and b ∈ B then a is a prototype of the element b and b is an image of the element 1 ) a. An image Im f of the set A into the set B, is referred to as a set Im f = b) b is the image of the element of the set D(f ) . The set Im f is the range of the map f : A → B. The map f : A → B is referred to as simple if ∀a ∈ D(f ) there exists the unique image b = f (a) ∈ B. A map f : A → B is said to be injective if ∀b ∈ Im f there exists only one prototype a ∈ D(f ) (b = f (a)). If a map f : A → B is injective, D(f ) = A and Im f = B a map is said to be a bijective map of the set A onto the set B. A map f : A → B generates the inverse map. By deﬁnition the inverse map f −1 : B → A to the map f : A → B is referred to as such a map in which an element a ∈ A is an image of an element b ∈ B iﬀ f (a) = b (in addition b ∈ Im f ). A map that is inverse to a simple map is not always simple. But a map that is inverse to a bijection map is always simple. Let us denote sets {1, 2, 3, . . .}, {0, 1, 2, 3, . . .}, {. . . , −2, −1, 0, 1, 1 2, ,. . .} and the set of the real numbers by N, N0 , Z and R respectively (N0 = N ∪ 0 ). A set A is called denumerable if there exists a bijection map of this set onto the natural number set N (or the set of the integer numbers Z). Finite and denumerable sets are referred to as discrete sets. The open set of the complex numbers is denoted by a symbol C and by a symbol i is meant the imaginary unit (i2 = −1). If z ∈ C then z = x + i y, where x, y ∈ R. By a complex-valued function is meant a map f : A → C. If D(f ) ⊂ A ⊂ C the function w = f (z) is a complex-valued function of the complex variable z. To isolate single-valued branches of multi-valued analytical functions it is necessary to make cuts of the complex plane C with the help of curves or straight " ! lines. In theory CVCE and RCE a set C\! is used where ! = −i∞, −i ∪ i, +i∞ is the union of two non-crossing rays lying on the imaginary axis of the complex plane C and coming from the points z1 = −i and z2 = +i. 1 ) , A set C = (a, b) ) a ∈ A and b ∈ B where (a, b) is an ordered pair is called a Cartesian product C = A × B of sets A and B. By a set C = A1 × A2 × · · · × An is meant the set of all ordered 1 ,collections of elements (a1 , a2 , .1. . , an ) where , a1 ∈ A1 ,. . . , an ∈ An (n ∈ N \ 1 ). Let Ar = R ∀r = 1, n = 1, 2, . . . , n . Introduce operations of addition and multiplication in the set R × R × · · · × R consisting of all ordered collections of n real numbers. These operations are deﬁned in the following way: (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn ) = (x1 + y1 , x2 + y2 , . . . , xn + yn ); α(x1 , x2 , . . . , xn ) = (αx1 , αx2 , . . . , αxn ) for any collections of real numbers x1 , x2 , . . . , xn and y1 , y2 , . . . , yn and ∀α ∈ R. Denote this set by Rn . It is a particular case of an n-dimensional real linear space [102]. If a scalar product (!x, !y ) = !x · !y for any vectors !x = (x1 , x2 , . . . , xn ) and !y = (y1 , y2 , . . . , yn ) is introduced into "1/2 !n then Rn will be an Rn with the help of an expression (!x, !y ) = j=1 xj yj n n n-dimensional Euclidean space E . Two vectors !x, !y ∈ E are called orthogonal if

410

N.N. Rogovtsov, F.N. Borovik

) ) ! "1/2 (!x, !y ) = !x · !y = 0. A value )!x) = !x, !x is called a length of a vector !x ∈ E n . n In the space E n an equation j=1 x2j = 1 deﬁnes a unit sphere. By a symbol Ω is meant a unit sphere in the space E 3 . A unit sphere is formally deﬁned by an ) ) ! 2 1 ) "1/2 , ! )! equality Ω = Ω Ω = (Ω1 , Ω2 , Ω3 ), )! Ω) = Ω1 + Ω22 + Ω32 =1 . Consider a number of classes of functions used in theory CVCE, RCE. By C(Ω) denote the class of all continuous complex-valued functions deﬁned on the unit sphere Ω. Any of these functions is a map of a type f : Ω → C. By a symbol L2 (Ω) is meant a class of all complex-valued square-integrable on Ω functions. Any func)2 ! " % ) ! ! ∈ Ω). By C −1, 1 ) ) tion of this class satisﬁes a condition Ω f (Ω) dΩ < +∞ (Ω and L2 (−1, 1) are meant respectively classes of all complex-valued continuous and square-integrable on segment [−1, 1] functions. In mathematics classes of all absolutely and square integrable on a set X complex-valued functions are denoted by L1 (X) and L2 (X) respectively. ! " Let f (μ) be a complex-valued function deﬁned on [−1, 1]. If ∀μ1 , μ2 ∈ −1, 1 ) ) ) )γ an inequality )f (μ1 ) − f (μ2 )) < a)μ1 − μ2 ) holds where a, γ ∈ 0, +∞ , then the function ! " f (μ) is said to satisfy Holder’s condition of index γ. If γ > 1, then ∀μ ∈ −1, 1 f (μ) = const. Therefore it is further assumed that γ ∈(0, 1]. A class of all complex-valued functions satisfying Holder’s condition of index γ is denoted by Hγ [−1, 1]. 1 , A class of all numerical sequences bs = b0 , b1 , b2 , . . . , bn , . . . satisfying a ) )2 +∞ condition s=0 2 s + r + 1 s! / s + 2r ! )bs ) < ∞ is denoted by l2 (r) where r ∈ N0 . Such sequence each is a map f : N0 → C, i.e. it is a complex-valued function of a discrete argument s ∈ N0 .

Appendix B: Metric spaces and their simplest properties A set X is called a metric space if the unique non-negative number ρ x, y corresponds to each ordered pair x, y ∈ X×X.This number is called a distance (or metric) ofthe space X and satisﬁes the following condition: (1) ρ x, y = 0 iﬀ x =y; (2) ∀ x, y ∈ X × X ρ x, y = ρ y, x ; (3) ∀x, y, z ∈ X ρ x, y + ρ y, z ≥ ρ x, z . If X is n-dimensional Euclidean space E n the distance ρ x, y between any of its vectors (elements) !x and !y is deﬁned by a formula n 2 xj − yj ρ !x, !y =

1/2

j=1

where !x = x1 , . . . , xn , !y = y1 , . . . , yn . Not only is E n a metric space but so ! and f2 (Ω) ! are any are sets C(Ω), L2 (Ω), C[−1, 1], L2 (−1, 1). If, for example, f1 (Ω) ) ) ! ! ) ) functions of the class C(Ω) a value ρ f1 , f2 = → max f1 (Ω) − f2 (Ω) should be used − Ω ∈Ω ) " !% ) ! f2 (Ω) ! ∈ L2 (Ω) a value ρ f1 , f2 = ! − f2 (Ω) ! )2 dΩ 1/2 )f (Ω) as a metric. If f1 (Ω), Ω 1 should be used as a metric. Let X is space. A set of all elements (points) x ∈ X satisfying a a metric condition ρ x, x0 < r (r = const ∈ 0, +∞ ) is called an open sphere of a radius

7 The characteristic equation of radiative transfer theory

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r of the center at the point x0 ∈ X. A set of all points of X satisfying a condition ρ x, x0 ≤ r is called a closed sphere. A set M ∈ Xis called bounded if it belongs to some open or closed sphere of a ﬁnite radius. Any open sphere of the center x0 is called a neighborhood Or x0 of the point x0 ∈ X. A point x0 is an internal point of the set M if there exists a neighborhood Or x0 of this point which entirely belongs to M . A set 1 is, called 1 an open, set or region if all its points are internal. An inﬁnite sequence xn = x1 , x2 , . . . of elements of space X is said to the metric converge to an element (point) x0 ∈ X if lim ρ xn , x0 = 0 and in this case a n→+∞

notation

lim xn = x0 will be used. A point x0 ∈ X is called a limit point of a

n→+∞

set M ∈ X if any neighborhood of this point contains even if one point of the set M diﬀerent from x0 . Let us denote a subset of the metric space X that contains all points of the subset M and all its limit points by M . If the subset M contains all its limit points it is called closed. In this case M = M . Let A and B are subsets of the metric space , subset A is called dense in B if B ⊆ A. 1 X. The , x , . . . is called a fundamental sequence if ∀p ∈ N an equality A sequence x 1 2 lim ρ xn+p , xn = 0 holds. If in the metric space X any fundamental sequence n→+∞

whose all terms belong to X, converges to an element of X, then X is a complete space. For example, the spaces C[−1, 1] and L2 (−1, 1) are complete spaces. Examples of complete spaces are the spaces l2 (r) ∀r ∈ N0 . A subset M of the metric space 1 X is,called compact if convergence subsequences can be extracted of any sequence xn of elements of this subset. But the limits of these sequences do not necessary belong to M .

Appendix C: Linear, normalized and Banach spaces A linear space X is called a set in which a sum x + y and a product αx ∀x, y ∈ X and ∀α ∈ C (or R) are deﬁned unambiguously. In addition, (x + y) ∈ X and αx ∈ X. These operations of addition and multiplication should possess the algebraic properties identical to the properties in the space Rn . If elements (vectors) of X can be multiplied only by real numbers, then X is a real linear space. Otherwise X is a complex linear space. Let X be a real linear space. By a scalar product of vectors of X is meant a real function !x, !y deﬁned for anypairs of vectors !x, !y ∈ X and possessing the following properties: (1) ∀! x , ! y ∈ X ! x , ! y = ! y , ! x ; (2) ∀! x , ! y , ! y ∈ X 1 2 !x, !y1 + !y2 = !x, !y1 + !x, !y2 ; (3) ∀α ∈ R and ∀x, y ∈ X α!x, !y = α !x, !y : (4) !x, !x ≥ 0 and ! ! !x, !x = 0 iﬀ !x = 0(0 is the zero vector). A linear real space X is called Euclidean if a scalar product is deﬁned in this space. In the Euclidean space a notion of vector ) ) 1/2 length is deﬁned by a formula )!x) = !x, !x . The properties of vector lengths of the Euclidean space can be taken as a basis to deﬁne a normalized space. By a normalized space 7 7 X is meant a linear space in which ∀!x ∈ X the unique number 7!x7 ∈ [0, +∞ corresponds to !x. In addition, 7 7 the following conditions (1) 7!x7 = 0 iﬀ 7!x = !0; (2) 7 7 should ) ) 7 hold: 7 7 for7 any 7 number 7 7 α and ∀!x ∈ 7X 7 7α!x7 = )α) 7!x7; (3) ∀!x1 , !x2 ∈ X 7!x1 + !x2 7 ≤ 7!x1 7 + 7!x2 7. The number 7!x7 is called a norm of the vector !x ∈ X. A normalized space X is a

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7 7 metric space with respect to a metric ρ !x, !y = 7!x − !y 7. By a convergence in norm is meant a convergence in this metric. Spaces C[−1, 1], L2 (Ω), l2 (r) (∀r ∈ N0 ) are ) 7 7 !% ) " )f (!x))2 dX 1/2 is examples of normalized spaces. A convergence in norm 7f 7 = X

called a mean convergence. A normalized space X is called Banach if X is complete with respect to the convergence in norm. Spaces L2 (Ω), C[-1, 1], L2 (-1, 1), L1 (-1, 1) are examples of Banach spaces.

Appendix D: Elementary notions of spectral theory of operators and linear operator equation theory It is impossible to construct CVCE and RCE theory in practice without using basic notions and constructions of operator spectral theory and linear operator equation theory [104, 113, 144, 157]. Therefore further notions and constructions of the theories mentioned above will be given brieﬂy and their connection with classical variants of characteristic integral equations RTT will be pointed out since CVCE and RCE are such. Further notations and deﬁnitions presented in Appendices A, B, C are used. Let F8 : X → Y be a single-valued operator (mapping) whose domain is D F8 ⊂ X and range Im F8 ⊂ Y (X and Y are normalized spaces). Sometimes to simplify 8 notation this operator will be denoted by symbol F . An operator is called a bounded 8 operator if it maps any set of D F bounded in space X into a set bounded in space Y. Otherwise the operator mentioned above is calledunbounded. An operator F8 : X → Y is called continuous at the point x0 ∈ D F8 ⊂ X, if an equality )) )) )) )) ))F8 x − F8 x0 )) = 0 holds where x ∈ D F8 . Here by ))x − x0 )) and lim ||x−x ||→0 )) 0 )) ))F8 x − F8 x0 )) mean the norms of the elements (x − x0 ) and F8 x − F8 x0 8 respectively in the spaces X and Y. The operator F :X → Y is called continuous 8 if it is continuous at any point x ∈ D F , i.e. ∀x ∈ D F8 . A single-valued operator F8 : X → Y, where X,Y are Banach spaces is called 8 linear if D F is a linear subspace of the space X and ∀x1 , x2 ∈ D F8 an equality F8 αx1 + βx2 = αF8 x1 + β F8 x2 holds. Here α, β are any numbers of R (or C) where R is the set of the real numbers and C is the set of the complex numbers. A linear operator F8 deﬁned on X is called completely continuous if it maps any bounded subset of Banach space X into a compact subset of Banach space Y. An important value for applications has notions of a spectrum and a resolvent set of the linear single-valued operator F8 : X → X when D F8 = X and X is a Banach space. This notion is directly connected with the existence and properties 8 −1 : X → X, where E 8 is the identity operator of the inversion operator F8 − η E 8 x = x) and η ∈ C. The spectrum of the operator F8 consists of a point (∀x ∈ X E (discrete), essential and residual spectra. The point (discrete) spectrum of the operator F8 is referred to as the set 1 , Pσ F8 = η | η ∈ C and F8 x = ηx for some nonzero element x ∈ X . In this 8 : X → X has no inversion operator. case the operator F8 − η E

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The essential spectrum of the operator F8 is referred to as the set Cσ F8 = 1 8 : X → X has the unbounded inversion η | η ∈ C and the operator F8 − η E operator whose domain is dense in X}. A number η belongs to Cσ F8 if there 8 8 does not exist such a nonzero element x ∈ X that F − η E x =)) 0))but ∀ε > 0 there exists an approximate eigenvector x = x ε whose norm ))x)) = 1 and )) )) 8 x )) < ε. ))F8(x) − η E The residual spectrum of the operator F8 is referred to as the set Rσ F8 = 1 8 −1 : X → X η | η ∈ C and there exists a bounded or unbounded operator F8 − η E which has no domain dense in X}. For most operators used in applications the residual spectrum is an empty set. An important exclusion is the generation operator used in the quantum theory [144]. The resolvent set of the operator F8 is referred to as the set ρ(F8 ) = {η|η ∈ C 8 : X → X possesses the bounded inversion operator with and the operator F8 − η E a domain dense in X}. Let F8 : X → X be a linear operator. Consider a linear operator equation F8 x = y (112) where y is a given element of Banach space Y, x is a required element of D F8 ⊂ X, where X is a Banach space (X and Y do not necessarily coincide with each other). In the integral equation theory [31, 133, 144] the equation (112) is usually referred to as the integral equation of the ﬁrst kind. The set of all y ∈ Im F8 for which the equation (112) is solvable is a linear subspace of the space Y. The set of all solutions of the homogeneous equation F8 x = 0 corresponding to equation (112) (0 is the zero element of the space Y) is a linear subspace of Banach space X and is referred to as the Kern F8 of the operator F8 : X → X. The equation (112) is referred to as uniquely solvable on Im F8 if the homogeneous equation F8 x = 0 1 , has the unique solution, i.e. if Kern F8 = 0 ∈ D F8 . In this case ∀y ∈ Im F8 there exists the unique solution of equation (112). Then the operator F8 possesses the inversion operator F8 −1 : Im F8 → D F8 on Im F8 and the solution of the equation (112) is given by the formula x = F8−1 y where y ∈ Im F8 . Now consider a linear operator equation x = η1 F8 x + g1 , (113) where x is a required element of Banach space X,η1 ∈ C,F8 : X → X is a linear operator and g1 is a given element of the space X. In the integral equation theory [31, 144, 157] the equations of the type (113) are usually referred to as the integral equation of the second kind. Transform equation (113) to a form 8 x = g, F8 − η E (114) −1 −1 , g = − η1 g1 . If η ∈ ρ F8 (ρ F8 is the resolvent set of the where η = η1 8 8 −1 g (x ∈ operator F ) equation (114) possesses the unique solution x = F8 − η E 8 ) ∀g ∈ Im F8 − η E 8 . This solution will be also the unique solution of D F8 − η E equation (113) if η1 = η −1 and g1 = −η1 g.

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The classical variant of the characteristic equation and the reduced characteristic equations (CVCE and RCE) are particular cases of the following linear operator equation 8 − η1 M 8 x =F 91 x + g0 . E (115) Here η1 ∈ C, x and g0 are respectively required and given elements of Banach 8 and F 91 are linear operators mapping elements of the space X into space X; M 8 = X and there exists the inthe same space. Here it is supposed that D M −1 8−F 91 : X → X whose domain satisﬁes the condition version operator E −1 8−F 91 −1 equation (115) is transformed 8 8 9 . If g0 ∈ D E Im M ⊂ D E − F1 into the form of equation (113) in which the operator F8 should be substituted by 8−F 91 −1 M 8 (it is a product of the operators) and the element g1 should be substiE 8−F 91 −1 g0 . In turn this equation is reduced to the standard equation tuted by E 8−F 91 −1 M, 8 η1 = η1 −1 of the form (114) in which it should be assumed F8 = E −1 −1 −1 8−F 91 8−F 91 8−F 91 8 − ηE 8 and η g0 . If E g0 ∈ D E and g = −η E M −1 −1 8 8 9 8 the 8 9 M of the operator E − F1 M belongs to the resolving set ρ E − F1 formal solution of equation (115) can be presented in the form −1 8−F 91 −1 M 8−F 91 −1 g0 = R 9η g0 8 − η1 −1 E 8 −1 E x = − η1 E

(116)

The linear operator equations (113), (114), (115) are called homogeneous equations if for them the elements g1 , g, g0 are equal to the zero elements of the space X. Otherwise these equations are called inhomogeneous. In this chapter the mathematically substantiated and eﬀective method for ob9η and the discrete spectra of the operators of the type taining the resolvent R −1 8 8 9 8 F = E − F1 M that correspond to the integral equations CVCE and RCE is given.

Appendix E: Shturm’s method and the simplest properties of Jacobi’s matrices Let hn x be a polynomial of the nth order (n ∈ N \{1} = {2, 3, 4, . . .}) in a real variable x. We assume that the polynomial coeﬃcients are real and the algebraic equation hn (x) = 0 only possess roots of multiplicity one. The problem of ﬁnding the number of the roots of such an equation in the interval (α, β) ∈ (−1, 1) is very important in the theory of CVCE and RCE. Shturm’s method [146, 147] is one of the classical methods to solve this problem. The substance of this method is the following. Let an ordered ﬁnite system of real numbers q1 , q2 , . . . qn n ≥ 2 be given. The pair of elements placed alongside each other is said to possess a sign change if an inequality qs qs+1 < 0 holds. If an inequality qs qs+1 > 0 holds the pair qs , qs+1 is said to have no sign change. The total number of sign changes in all pairs of the adjacent elements qs , qs+1 s = 1, 2, . . . , n − 1 of the system q1 , q2 , . . . qn is called the number ofsign changes system of real of the given system. A ﬁnite ordered functions f0 x , f1 x , f2 x , . . . , fs x where f0 x = hn x and s ≥ 2 is called

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Shturm’s system of the algebraic equation in the interval α, β if the following conditions hold: (1) The function fs x has no real roots (i.e. it is unchangeable in sign in segment α, β ); (2)For no value x two adjacent functions together are not equal to zero; (3) f−1 γ and f+1 γ possess the diﬀerence signs if γ x 1 ≤ ≤ s − 1 of the is a real root of one of intermediate functions f system f0 x , f1 x , f2 x ,. . . , fs x . (4) If γ is a real root of the polynomial hn x the product f0 x f1 x changes the sign from minus to plus when x increasing passes through the point x = γ. Let W x be the total number of sign changes in Shturm’s system. The calculation of the number of real roots of the polynomial can be carried out using Sturm’s theorem [146, 147]. Theorem E). An inequality W α ≥ W β holds and the 1 (Appendix diﬀerence W α − W β is equal to the number of real roots of the polynomial hn x contained in the interval α, β if the real numbers α and β (α < β) are not roots of the polynomial one. 1 roots of multiplicity , possessing only 1, n) and n ∈ N is . . . , n (i.e. , j = A matrix A = aj where , j ∈) 1, 2, ) ) − j ) > 1. If the notations a = a = 1, n called Jacobi’s matrix if aj = 0 for and b = −a+1 , c = −a+1 = 1, n − 1; n ≥ 2 are introduced the matrix can be written in the form ⎤ ⎡ a1 −b1 0 0 0 ....... 0 0 0 ⎥ ⎢ −c1 a2 −b2 0 0 ....... 0 0 0 ⎥ ⎢ ⎥ ⎢0 a −b 0 . . . . . . . 0 0 0 −c 2 3 3 ⎥ ⎢ (117) ⎢ ................................................. ⎥. ⎥ ⎢ ⎣0 0 0 0 0 . . . . . . . −cn−2 an−1 −bn−1 ⎦ 0 0 0 0 0 ....... 0 −cn−1 an ) ) Let dn η = )a j − ηδ j ) be the determinant of the matrix A − ηE where E = δj is the identity matrix, δj is the Kronecker symbol, η ∈ C. The determinant dn η is the characteristic polynomial of the 1matrix A. Jacobi’s matrix (117) is , called normal if ∀ = 1, n − 1 where n ∈ N \ 1 the inequalities bl > 0, c > 0 hold. The following theorem takes place [158]: Theorem 2 (Appendix E). The characteristic polynomial dn η of the normal Jacobi’s matrix (117) possesses n real simple roots. In presenting the methods of solving the integral equations CVCE and RCE we shall consider also the properties of inﬁnite Jacobi’s matrices and some of their properties will be made manifest using Theorem 2 (Appendix E).

Appendix F: The basic notions of the diﬀerence equation theory. Perron’s theorem The ﬁnite-diﬀerence techniques are one of the most eﬀective methods of solving various problems of the mathematical physics. In using this technique the continuous range of arguments of the required functions is substituted by a discrete set of points J. The approximate calculation of the required functions is reduced to

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solving linear or nonlinear algebraic equations that are referred to as the grid equations. The solutions of this equations are grid functions, given on the set J. The diﬀerence equations are a particular case of the grid equations. A number of general properties of these equations will be used to construct the eﬀective algorithm of solving CVCE and RCE. Let y = y be a grid function of the discrete argument ∀ ∈ N0 = N ∪{0}. Its domain is the set N , and the range is a sequence {y0 , y1 , y2 , . . .} where ∀ ∈ N0 , a1 , . . . , am (m ∈ N ) and yl ∈ C. We assume that on the grid functions a 0 g = g are deﬁned. We assume also that ∀ ∈ N0 all these functions have ﬁnite values and the inequalities a0 = 0 am = 0 hold. Consider an equation a0 y + a1 y+1 + · · · + am y+m = g , ∈ N0 (118) where m is a ﬁxed positive integer. Equation (118) is called a linear diﬀerence equation of the m-th order with respect to the grid function y = y . The diﬀerence equation (118) is called homogeneous if ∀ ∈ N0 g = 0. Otherwise the equation (118) is called inhomogeneous. By a solution of equation (118) is meant any grid function y = y that is such that the substitution of values of this function in (118) transforms it into the identity ∀ ∈ N0 . Proceed to considering the linear homogeneous diﬀerence equation corresponding to (118). Write down it in the form m

as y+s = 0, ∈ N0 , m ∈ N.

(119)

s=0

Let v1 , v2 , . . . , vp be the solutions of homogeneous equation (119). Then the grid function y = y = C1 v1 + C2 v2 + · · · + Cp vp (120) is a solution of the equation (119) (C1 , C2 , . . . , Cp are arbitrary constants). By deﬁnition the system of the grid functions v1 (l), v2 (l), . . . , vm (l) are called the linearly independent system solutions of equations (119) if these functions assume only ﬁnite values and the right-hand side of expression (120) for p = m does not vanish although for one value l ∈ N0 and for any constants C1 , C2 , . . . , Cm which are not equal to zero simultaneously. The general solution of equation (119) can be presented in the form of the expression (120) for p = m in which by the system of the grid functions should be meant a linear independent system of grid functions, each of these functions being a solution of equation (119). Let y× = y × be a particular solution of the linear inhomogeneous diﬀerence equation (118). Then the general solution corresponding to this particular solution can be written in the form y = y× + C1 v1 + C2 v2 + · · · + Cm vm , ∈ N0 , m ∈ N (121)

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The algorithms the linear independent system of the grid func for constructing tion v1 , v2 , . . . , vm and the particular solution y× involved in (121) are (for example) presented in [105]. In the theory of linear diﬀerence equations (118) that is directly connected with the theory of inﬁnite system of linear algebraic equation involving inﬁnite Jacobi’s matrices, Perron’s theorem is very important [150–152]. Theorem 1 (Appendix F). Perron’s1theorem. Let ,the following conditions hold: (1) ∀ ∈ N = 1. (2) ∀s ∈ 0, 1, . . . , m − 1 there exist the ﬁnite a 0 m limit lim as = a× . s (3) all roots Λ1 , Λ2 , . . . , Λm of the characteristic equation →+∞

× × 2 × m−1 a× + Λm = 0 are diﬀerent in module; (4) ∀ ∈ N0 0 +a1 Λ + a2 Λ + · · · + am−1 Λ diﬀerence equation (119) can be a0 = 0. Then any solution of homogeneous = C v v presented in the form y + · · · + C where C1 , . . . , Cm ∈ C and 1 1 m m v1 , . . . , vm is a linear independent solution system of equation (119) and the 1 , functions of this system ∀s ∈ 1, 2, . . . , m − 1 satisfy the relation vs + 1 = Λs lim (122) →+∞ vs

Perron’s theorem is a generalization of Poincar´e’s theorem [106, 150–152].

Appendix G: Continuous fractions. Van Vleck’s and Pincherle’s theorems To investigate the properties of the integral equations CVCE and RCE and to construct their solutions in an analytical form a number of basic notions and constructions of the continuous fraction theory [108–111, 115] are used. It should be noted that along with the serious and inﬁnite product notions constructions involving the continuous fractions as the constituents allow one to obtain analytically and numerically solutions of various equations eﬀectively. By a ﬁnite continued fraction is meant the following expression: b1 b 1 b2 bn (123) = a ; , , . . . , a0 + 0 b2 a1 a2 an a1 + b3 a2 + . a3 + . . + abnn where values a0 , a1 , . . . , an and b1 , . . . , bn can be numbers or functions. In (123) the elements bs and as are called the sth partial numerators and denominators. To denote a ﬁnite continued fraction the following notations are widely used: n n bs b1 b2 bn bs b1 b2 bn = a0 ; = a0 + ... = a0 + K a0 ; , , . . . , a1 a2 an as 1 a a + a + + a s=1 s 1 2 n

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By an inﬁnite continued fraction is meant the following construction ∞ +∞ bs b1 bs = a0 + K = a0 + (124) a0 ; b2 as 1 s=1 as a1 + b3 a2 + a + ... 3 1 , 1 , By a0 , a1 , a2 , . . . and b1 , b2 , . . . in (124) are meant sequences of numbers or functions. If for some s0 ∈ N bs0 = 0 inﬁnite continued fraction (124) is transformed into the ﬁnite continued fraction. By deﬁnition values a0 b1 b2 bn An Wn = a0 ; , , . . . , (n ∈ N) = W0 = 1 a1 a2 an Bn are called the zeroth and nth approximants of inﬁnite continued fraction (124). Here it is supposed that the fraction An /Bn is directly calculated of b 1 b2 bn a0 ; , , . . . , a1 a2 an without using any cancellations. According to Euler [159, 160] it is usually assumed that A0 = a0 , B0 = 1 and A−1 = 1, B−1 = 0. The following theorem holds [111, 159–161]. Theorem 1 (Appendix G). Values As and Bs where s = −1, 0, 1, 2, . . . deﬁned by the recurrence formulas As = as As−1 + bs As−2

Bs = as Bs−1 + bs Bs−2 ,

(125)

where A−1 = 1, B−1 = 0, A0 = a0 , B0 = 1 are numerators and denominators of the approximants of inﬁnite continued fraction (124). (They are often called canonical numerators and denominators of the approximants.) In the continued fraction theory a continued fraction 1 (124) is said,to converge if the ﬁnite limit of the approximant sequence W0 , W1 , W2 , . . . exists. If lim Ws = ∞ an inﬁnite continued fraction is said to possess inessential dis→+∞

vergence. A continued fraction is assumed to be divergent if this limit does not exist. In the continued fraction theory a number of convergent criterions for inﬁnite continued fraction are stated [108–111]. Of special interest are , the 1 convergent , 1 b1 , b2 , b3, . . . criteria in which all the elements of the sequences a0 , a1 , a2 , . . . are constants or analytic functions in variable z ∈ C since such criteria allow one to construct solutions of CVCE and RCE in an analytical form. One of such a convergent criterion is stated by the theorem proved by Van Vleck in 1904 [110]. Theorem 2 (Appendix G). (Van Vleck). Let the following condition hold: ∞ (1) K as z/1 is an inﬁnite continued fraction for which there exists the limit s=1 as 1 )) ) , lim = a, a = 0 and a ∈ C; 2)Ma = z ) ) arg az + (1/4) ) < π . Then the

s→+∞

following assertions hold: (1) if G is a compact subset of a set Ma there exist such a region DG that G ⊂ DG ⊂ Ma and there exists such a positive integer number nG that an inﬁnite continued fraction anG z anG +1 z anG +2 z ... 1 + 1 + 1 +

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converges uniformly on each compact subset in the region DG to a holomorphic ∞ function in DG ; (2) the inﬁnite continued fraction K as z/1 converges to a funcs=1 tion f z that is either meromorphic on Ma or equal inﬁnite identically. A lot of special function sequences can be calculated by means of three-term recurrent relations governing these functions. On the other hand these relations can be considered as particular cases of inﬁnite systems of linear algebraic equations involved Jacobi matrices. In turn solving this system is in fact solving linear diﬀerence equations of the second order which are subject to certain boundary conditions. For some sequences (called minimal solutions) it is impossible to obtain results with fair accuracy if three-term recurrence relations are used directly to calculate special functions involving large values of indexes through the functions of the same kind involving lesser indexes. The reason for this is the numerical instability of algorithms in using forward recursion. In 1967 it was Gautchi [107, 111] who was the ﬁrst to attract attention to this problem and point out ways of overcoming this diﬃcult. The techniques developed by Gautchi are based on using backward recursion and the theorem proved by Pincherle in 1894 [111]. Some definitions and explanations will be given before Pincherle’s theorem is formulated. Consider a system of three-term recurrence relations yn+1 = bn yn + an yn−1 , n ∈ N; ∀n ∈ N an = 0, an , bn ∈ C. (126) 1 , The set of all the solutions1 yn, of (126) forms a linear space L2 , having 1 ,dimensionality 2. A sequence hn is called a minimal solution of (126) if hn is nontrivial (i.e. hn = 0 for some n ∈ N ) and there exist another solution , 1 , of (126) 1 pn satisfying the condition lim (hn /pn ) = 0. For any solution yn of (126) n→+∞ 1 , not proportional to hn it is easily shown that lim (hn /pn ) = 0. If the set of n→+∞ 1 , all solutions of hn of the type of (126) is not empty, then it is a one-dimensional subspace of L2 . A solution of (126) which is not minimal is called dominant. In general a system of recurrence relations (126) may or may not have a minimal solution. Theorem 3 (Appendix G). (Pincherle). Let ∀n ∈ N be elements an , bn ∈ C and an = 0. Then the following assertions hold: 1 (1) , the system of three-term recurrence relations (126) has a minimal solution hn , if the continued fraction +∞ K an /bn converges (to a ﬁnite value or to inﬁnity); (2) suppose that (126) has n=1 1 , a minimal solution hn . Then ∀m ∈ N +∞ an am hm am+1 am+2 . (127) ··· = − K =− hm−1 bm + bm+1 + bm+2 + bn n=m By (127) we mean the following: If hm−1 = 0, then hm = 0 and the continued fraction (127) converges to (hm /hm−1 ) = ∞. If hm−1 = 0 then (127) converges to the ﬁnite value (hm /hm−1 ) ∈ C.

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Appendix H: The correct algorithm for calculating minimal solutions of three-term recurrence relations Consider the problem of correct calculation of minimal solutions of relations (126) which are linear homogeneous diﬀerence equations of the second order. We shall assume that ﬁnite nonzero limits lim an = a, lim bn = b exist and an inn→+∞

n→+∞

equality b2 = −4a holds. Under conditions pointed out above the characteristic equation (126) have equation Λ2 − bΛ − a = 0 of the linear homogeneous ) ) ) diﬀerence ) roots Λ1 and Λ2 not equal in modules (0 < )Λ1 ) < )Λ2 )). On the base of Pincherle’s theorem (see Appendix G) it follows that system 1 , ∞ (126) has a minimal solution hn if the continued fraction K cn /1 converges n=1 (to a ﬁnite value or to inﬁnity) where ∀n ∈ N an element cn = (an /bn−1 bn )−1 and b0 = 1. This continued fraction satisﬁes the conditions of Van Vleck’s theorem for z = 1 (see Appendix G). Hence it1converges in the sense pointed out. Therefore , system (126) has a minimal solution hn . As was pointed out in Appendix G, the space of the solutions of system (126) is two-dimensional and the subspace of its 1 , solution minimal solutions is one-dimensional. Further un will denote a dominant of system (126) which by deﬁnition satisﬁes the condition lim hn /un = 0. n→+∞

By Pincherle’s theorem it follows that the roots Λ1 and Λ2 of the characteristic equation correspond to the minimal and dominant solutions respectively. The general solution of system (126) is of the form yn = αhn + βun , n ∈ N0 ,

(128)

where α and β are arbitrary numbers. By Perron’s theorem ∀ε1 > 0 there exist a number ) such−1 ) n1 ∈ N that ∀n ≥ n1 + 1 will be held an inequality ) hn hn−1 − Λ1 ) < ε1 . Similarly ∀ε2 > 0 there exists such a number n2 ∈ N ) ) −1 ) −Λ that ∀n ≥ n2 + 1 will be held an in inequality ) un un−1 ) ε)2 . Se) )2 < ) ) − )Λ1 ) , a way as to hold inequalities ε < (1/2) Λ lect ε1 and)ε2 )in such 1 2 ) ) 1 , ε2 < (1/2) )Λ2 ) − )Λ1 ) . Then ∀n ≥ max n1 + 1, n2 + 1 = n0 an inequality ) ) ) un un−1 −1 − Λ1 ) > 0 will hold. Now let n ≥ n0 . To obtain a minimal solution it is necessary to take such a solution of system (126) which will satisfy −1 = Λ1 . On the base of this condition and an additional condition yn0 yn0 −1 −1 in which (128) one obtains a relation β = α Λ1 hn0 −1 − hn0 un0 − Λ1 un0 −1 un0 − Λ1 un0 −1 = 0. Substituting this relation in (128) the particular solution of system (126) ! −1 " un , n ∈ N0 yn = α hn − (hn0 −1 /un0 −1 ) (hn0 /hn0 −1 ) − Λ1 (un0 /un0 −1 ) − Λ1 (129) is obtained. From formula (129) and the deﬁnitions of minimal and dominant solutions it follows that one should proceed in such a way to obtain the set of minimal solutions: (1) it is necessary to select a fair large natural number n0 , for which the elements an and bn will be fair close to their limits a and b; (2) it is necessary −1 = Λ1 where Λ1 to take some nonzero yn0 and impose a condition yn0 yn0 −1 is the minimal in modulus root of the characteristic equation Λ2 − bΛ − a = 0 (a

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particular value yn0 deﬁnes normalization of the minimal solution); (3) on the basis −1 yn0 by means of backward recursion yn−2 = of the given yn0 and yn0 −1 = Λ1 −1 yn − bn−1 yn−1 one can obtain elements yn0 −2 , yn0 −3 , . . . , y2 , y1 , y0 ; (4) an−1 in carrying out concrete calculations it is necessary to take into account that the , 1 elements yn0 will be fair close to the elements of the minimal solution αhn of system (126) for a ﬁxed n if a condition n n0 only holds. The algorithm presented above can be used in calculating discrete spectra and eigenfunctions of CVRCE corresponding to these spectra.

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Part III

Dynamic and Static Light Scattering: Selected Applications

8 Advances in dynamic light scattering techniques P. Zakharov, F. Scheﬀold

8.1 Introduction Recent developments in processing techniques and detection hardware have opened new horizons for the application of light scattering methods based on the dynamic analysis of coherent scattered light. The increased computational power of modern microprocessors allows real-time data evaluation on standard desktop computers. The continuous improvement of detector arrays, such as cameras based on CCD or CMOS technologies, facilitate space-resolved detection of scattering intensities, which can be used to boost the statistical weight accumulated in a single experiment. New methods and improved accuracy on the other hand also provide answers to questions concerning the quantitative data interpretation which were only partially addressed in some of the earlier work. Since the early days dynamic light scattering (DLS) methods have been applied to a wide variety of systems and therefore the further development continued under diﬀerent names such as DLS, laser Doppler, photon correlation spectroscopy, quasielastic light scattering and the family of laser speckle methods. All these techniques rely on the same phenomena, but look at it from diﬀerent perspectives. Although the importance of exchanging ideas and concepts was realized relatively early [1,2], researchers still prefer to play on their home ﬁelds. Here we make an attempt to adopt a slightly more general point of view. Monochromatic light scattered by a set of particles of mesoscopic size produces a random interference pattern on a screen. This pattern in a general case has the form of irregularly spaced and sized bright spots called speckles. The scattered light intensity pattern remains unchanged as long as the particles do not change their position. Particle motion naturally leads to a temporal evolution of the scattered speckle ﬁeld since one interference pattern is continuously replaced by another. In a single speckle spot this evolution is observed as strong temporal ﬂuctuations of the intensity, with a certain well deﬁned temporal correlation. The time frame of correlated intensity observed in the single spot is called temporal speckle. The intensity ﬂuctuations are inherently linked to the scatterers’ dynamics and therefore the temporal correlation function depends on the particles displacement. Thus the measurable correlation properties of light can be linked to the dynamical proper-

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Fig. 8.1. Illustration of the intensity ﬂuctuations of scattered light and the corresponding autocorrelation function.

ties of particles, which in turn can provide information about ﬂow velocity and direction, particles size, density of moving scatterers, etc. In the scope of DLS temporal ﬂuctuations are usually analyzed by means of the intensity or photon autocorrelation function (ACF; see Fig. 8.1). In the time domain analysis the correlation function usually decays starting from zero delay time (also called the lag time), faster dynamics leads to faster decorrelation of scattered intensity trace. It can be shown that for a random process the intensity ACF is the Fourier transform of the power spectrum [3] and therefore the DLS measurements can be equally well performed in the spectral domain. In fact the DLS experiment was initially discussed in terms of broadening of the spectrum peak of monochromatic light due to Doppler shifts experienced by the propagating light waves scattered by moving particles [4]. Faster particle dynamics leads to a broader spectrum, thus, by measuring the width, dynamical properties can be explored. The eﬀect of light scattering by moving particles is technically easier to analyze in the temporal domain when the scattering events are assumed to produce phase shifts rather than frequency shifts. The intensity trace can be statistically analyzed and its correlation properties can be used to study particles dynamics.1 The data interpretation is usually diﬀerent for the intensity ﬂuctuations of light scattered only once (single scattering limit) and for the case of multiple light scattering (diﬀusing wave spectroscopy or DWS). The intermediate limit represents the most diﬃcult case since no rigorous theory can be applied. After a brief introduction to the theory of single and multiple dynamic light scattering we discuss the experimental techniques that have been developed recently to extract the single-scattering signal from the detected light in the intermediate regime. Next, we address the recent advances of DLS experimental methods with respect to temporal and spatial resolution opening a wealth of new possibilities 1

For the case of random motion Doppler broadening and loss of correlation can be used interchangeably. For a stationary random process the power spectral density is the Fourier transform of the corresponding autocorrelation function, a relation known as the Wiener–Khinchin theorem

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for studies in physics, colloid science, material and process characterization and medicine.

8.2 Scattering regimes 8.2.1 Single scattering limit In the DLS experiment laser light with a wave vector k0 (|k0 | = 2πn/λ where λ is the wavelength in vacuum and n is the solvent refractive index) is scattered by the weakly scattering medium and detected at the scattering angle θ. The light is considered to be (quasi-)elastically scattered, which implies that the magnitude of the wave vector after scattering |ks | is essentially equal to the initial: |ks | = |k0 |. The wave vectors deﬁne the scattering vector q = ks − k0 (see Fig. 8.2), with |q| = 2|k| sin (θ/2). All the scattering sites with centers of mass located at the positions rj relative to a common reference point are illuminated by light with the same ﬁeld amplitude E0 and frequency w0 : E(rj ) = E0 exp i [k0 rj − w0 t],

(8.1)

which produces a spherical wave of the same frequency emitted by the induced dipole, which is detected at the position r Es ∝ exp (ik0 rj )

exp (i|k0 ||r − rj |) |r − rj |

(8.2)

For a far-ﬁeld detection (|r| |rj |) one can write: ks + ··· . |r − rj | ∼ = r − rj |ks |

(8.3)

As a result the electric ﬁeld amplitude detected from N independent scattering particles at the position r is: exp ik0 r exp ik0 r exp [i (k0 − ks ) rj (t)] = exp [iqrj (τ )] (8.4) Es (q, t) ∝ r r j=1 j=1 N

N

θ

Fig. 8.2. Typical single scattering experiment: incident light E0 is scattered by particles. The scattered electric ﬁeld amplitude Es is detected at angle θ and used for the analysis.

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Particle movement produces a continuous change of qrj and thus the frequency of the scattered light is shifted from the reference frequency w0 . Combination of frequency shifts from all the scatters results in the frequency distribution which can be detected with the correlator or spectrum analyzer. When a non-scattered light is blocked or detection is performed at suﬃciently large angles (outside of the primary beam) a self-beating of scattered light is observed (homodyne detection scheme) whereas inside the primary beam the signal interferes with the unscattered beam (heterodyne detection). Here we will restrict our analysis to the homodyne case. For random uncorrelated movement

of thescattering particles with Gaussian statistics of displacements and a mean Δr2 (τ ) the normalized Elementary Linear Correlator (FCF) g1 (q, τ ) can be expressed in the following way using Eq. (8.4): g1 (q, τ ) =

! " E(q, 0)E ∗ (q, τ ) = exp iq [ri (0) − rj (τ )] = exp −q 2 Δr2 (τ ) /6 2 |E(q, 0)| i,j

(8.5) For the Brownian motion of colloidal particles the mean square displacement is

2 Δr (τ ) = 6D0 τ , where D0 is the particle diﬀusion coeﬃcient deﬁned by the Stokes–Einstein relation: kT0 , (8.6) D0 = 6πηR with η being the solvent viscosity, T0 the sample temperature, and R the particle radius. In this case the normalized Elementary Linear Correlator can be written as [5]: g1 (q, τ ) = exp −D0 q 2 τ = exp [−τ /τc (q)] , (8.7) where τc (q) = 1/D0 q 2 is the relaxation time. Eq. (8.7) is widely used in dynamic light scattering for the sizing of small particles [5]. If the particles are in random Gaussian ballistic motion with isotropic

veloc ity distribution with mean ΔV 2 , displacement equals Δr2 (τ ) = ΔV 2 τ 2 . Substitution to Eq. (8.5) leads to (8.8) g1 (q, τ ) = exp − ΔV 2 q 2 τ 2 . Thus for the case of Brownian motion correlation function decays as a simple negative exponent of time; meanwhile for the random ballistic motion it decays as a negative exponent of the square of time. Thermal motionin complex ﬂuids can display more complicated patterns. The analysis of Δr2 (τ ) can thus provide valuable information about the microscopic dynamic properties. In a typical DLS experiment the intensity ﬂuctuations are monitored and analyzed by means of the normalized ACF: g2 (q, τ ) =

I(q, 0)I(q, τ )t , I(q, t)2t

(8.9)

where · · · t denotes time averaging. The measurable quantity g2 (q, τ ) can be linked to the normalized ﬁeld autocorrelation function g1 (q, τ ) via the Siegert relation [5]: 2

g2 (q, τ ) = 1 + β |g1 (q, τ )| ,

(8.10)

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where the coeﬃcient β is usually referred to as coherence factor and depends on the detection optics. In the ideal case, when only a single speckle mode is detected, β is approaching a value of 1. 8.2.2 Multiple scattering limit The single-scattering approach of DLS is only applicable to systems where the probability of double, triple or multiple scattering is negligible which strictly speaking implies the transmission in line of sight to be more than 95%. Nevertheless for systems with high turbidity when the light reaching the detector has been scattered a signiﬁcant number of times, the ﬂuctuations of scattered light intensity can also yield valuable information about the sample dynamics. In this case the light propagation can be considered as a diﬀusion process which in turn can be described by the diﬀusion equation. This is the basis of an approach known as diﬀusing wave spectroscopy (DWS) [6, 7]. Since its development DWS has become established as a key method for characterization of many diﬀerent kinds of turbid systems [8–16]. The physical picture of the single scattering is the same as considered in DLS and the only diﬀerence is the large number of consecutive scattering events. In this process the angular-dependent scattering information is lost and q is replaced with the mean square of the scattering vector q 2 . The scattered photons (or partial waves) accumulate phase shifts along scattering paths. The magnitude of these phase shifts can easily be estimated in the frame of a generalized DLS theory: N N : 1 2 1 2 N 2 exp − qi Δr (τ ) = exp − qi Δr2 (τ ) g1 (τ ) = 6 6 i=0 i=0 1 (8.11) = exp − N q 2 Δr2 (τ ) , 6 where g1N (τ ) is the FCF of the single photon after N scattering events with the sequence of momentum transfers qi , where i = 1, . . . , N . The last equality in Eq. (8.11) holds for the case of a large number of N . Assuming the statistical independence of photon paths the total FCF can be analyzed as the mean of all possible photon paths, weighted by the normalized distribution of number of scatterings ρ(N ): g1 (τ ) = ρ (N ) g1N (τ ) (8.12) N

Using the mean distance between the scattering events l, the sum in Eq. (8.12) can be replaced by an integration over pathlengths s, since N = s/l: ∞ ∞ 1 2 s s 2 g1 (τ ) = (8.13) ρ(s)g1 (τ ) = ρ(s) exp − q Δr (τ ) , 6 l 0 0 where g1s (τ ) is the FCF corresponding to a path of length s. The square mean of the momentum transfer can be written as q 2 = 4k02 sin2 (θ/2) = 2k02 (1 − cos θ). Here cos θ is the mean cosine of the scattering angle, which is often denoted as the scattering anisotropy parameter g. Let us introduce the transport mean free path l∗ = l/(1 − g), which has a physical

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Fig. 8.3. In the multiple scattering limit every scattering event eﬀectively changes the direction of propagation. A statistical measure for this change is the the average cosine of the scattering angle cos θ = g. The transport mean free path l∗ = l/(1−g) is the distance that photon has to travel before it completely loses memory of the original direction.

meaning of the mean distance a photon travels inside the scattering medium before its direction is completely randomized [17] (see Fig. 8.3 for illustration). When substituting it to Eq. (8.13) we obtain ∞ s 1 2 2 (8.14) g1 (τ ) = exp − k0 Δr (τ ) ∗ ρ (s) ds 3 l 0 The diﬀusion equation can be solved for various geometries and source–detector locations and in some of the most important cases analytical solutions can be found which allow for a quantitative interpretation of experimental data [17]. Due to the strong multiple scattering DWS oﬀers more ﬂexibility in the experimental design and higher sensitivity since the contribution of small displacements is enhanced by the long scattering paths. For the backscattering geometry with plane-wave illumination one obtains the following result for the intensity ACF [6]: (8.15) g1 (τ ) = exp −γk0 Δr2 (τ ) , where γ is a constant of order γ ≈ 2. For the case of simple Brownian motion we ﬁnd: (8.16) g1brn (τ ) = exp −γk0 6D0 τ = exp −γ 6τ /τ0 , where τ0 = 1/D0 k 20 is the DWS relaxation time. For random ballistic motion the FCF can be found as g1rnd (τ ) = exp −γk0 τ ΔV 2 .

(8.17)

At the same time, however, the DWS experiment cannot provide q-resolved information since it is lost in the multiple-scattering process and thus directional characteristics of particle movement cannot be studied. Another complication originates from the limitations of diﬀusion approximation. It is only applicable in the strong multiple scattering limit, when the optical pathlength is large compared to the transport mean free path l∗ . This assumption does not hold under some measurement conditions even for turbid samples, notably in back-reﬂection [18, 19].

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8.2.3 Intermediate scattering regime In the crossover regime the interpretation of the intensity correlation function is signiﬁcantly complicated. Incident light experiences single, double, triple scattering and so on. The eﬀective momentum gradually transfers from q to q. Theoretical modeling for this complicated situation is very diﬃcult and sample-dependent. Therefore many experimental techniques focus instead on the suppression of nonsingly scattered light instead. Several experimental approaches can be used to reduce the inﬂuence of multiple scattering. One can dilute the sample, reduce scattering eﬃciency by refractive-index matching [20] or decrease the cell size. The ﬁrst two approaches require a change of the sample properties, which is very often impossible or undesirable. Reducing the size of the container has its own limitations which are set by the optical quality of the sample cell and by boundary eﬀects. For cylindrical cells minimal diameters of typically 1–3 mm are used whereas in ﬂat or rectangular containers even smaller photon path lengths can be achieved [21, 22]. To maximally reduce the photon path lengths one can use ﬁber optical probes [23] directly immersed in the liquid sample. This approach known as Fiber Optical Quasi Elastic Light Scattering (FOQELS) has been applied in a number of recent studies (see, for example, Refs [24–26]). The application of FOQELS is, however, limited to backscattering angles around 180◦ and the interpretation of the data is often complicated due to the incomplete suppression of multiple scattering. Along with eﬀorts to eliminate the contribution of multiple scattering one can try to use diﬀerent schemes of signal processing in order to extract the signal of interest. One of the most eﬃcient approaches is to study spatial correlations of the detected ﬁeld attributed to the volume of single scattering. One can carry out two simultaneous light scattering experiments with exactly the same scattering vectors in the same scattering volume and analyze the time cross-correlation function of intensities. It has been clearly shown that under suitable conditions (see Refs [27– 29]) the cross-correlation function equals the autocorrelation function for single scattering within the experimentally accessible range (the experimental intercept β of the cross-correlation system can be signiﬁcantly smaller than one). In the 3D cross-correlation scheme multiple scattering suppression can be achieved by performing two scattering experiments simultaneously on the same scattering volume (with two laser beams, initial wave vectors ki1 and ki2 , and two detectors positioned at ﬁnal wave vectors kf 1 and kf 2 ). The signal seen by the two detectors is correlated only for the case of single light scattering where both detectors are set to have the same scattering vector q. Therefore for the case of single scattering, each detector sees the same spatial Fourier component of the sample [21, 30]. This approach has been implemented as a commercially available system [31]. Instead of a tilting two laser beams out of the scattering plane it is also possible to employ two lasers of diﬀerent wavelengths [29, 30] Another cross-correlation approach can be implemented using a single-beam two-detector conﬁguration [33, 34]. This approach is based on the so-called van Cittert–Zernike theorem [35], which states that intensity correlations in an observation region are closely related to the Fourier transform of the intensity distribution across the source. This means that a small region dominated by single scattering (for example, the volume of a focused beam) will produce large correlated areas (speckles), whereas a comparably large halo of multiple scattered photons will give

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(a)

(b)

Fig. 8.4. (a) Illustration of the van Cittert–Zernike theorem: small coherently illuminated areas (such as a focused laser beam) produce large areas of correlated intensity (speckles) and vice versa: large coherence areas (halo from multiple scattering) produce small speckles. (b) Suggested suppression scheme using cross-correlation processing: (I) intensity values measured within the same speckle are correlated; (II) two diﬀerent speckles are uncorrelated; (III) for a superposition of small and large speckles intensities detected at a certain distance will be correlated only due to the larger speckles (reproduced from [32] with permission of OSA).

rise to small speckles (see Fig. 8.4(a)). This is reﬂected in the expression for the speckle size S in the far-ﬁeld geometry: S = λz/D, where z is the distance from the light emitting object to the detector and D is the lateral extension of the object along one chosen direction [35]. In other words, spatially resolved detection of the scattered intensity carries selective information about the spatial distribution of light in the scattering volume. The single-scattering volume D1 is deﬁned by the beam cross-section and the size of the halo D2 is determined by the scattering mean free path l. Let us consider the two points geometrically separated by distance Δx chosen between the speckle sizes produced by beam and halo: λ·z λ·z Δx ≤ , D2 D1

(8.18)

In these points the intensity ﬂuctuations are correlated mostly due to the single scattering as shown in Fig. 8.4(a). The normalized intensity cross correlation function (CCF) of the intensity signals measured at positions separated by Δx: g2Δx (q, τ ) =

I(q, t, 0)I(q, t + τ, Δx)t I(q, t, 0)t I(q, t, Δx)t

(8.19)

provides the proper estimate of autocorrelation function of single-scattered intensity. Such an approach has already been successfully demonstrated by Meyer et al. [33] with a scheme based on cross-correlation of scattered intensities detected by two spatially separated optical ﬁbers. While the underlying principle is highly plausible, it is more complicated to put forward a detailed theoretical description since it requires modeling of the low-order

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scattering processes. Such a treatment has been derived by Lock [36] for the case of double scattering. He ﬁnds the multiple scattering suppression eﬃciency to be approximately proportional to the speckle size ratio S1 /S2 = D2 /D1 if the detectors are placed at the distance Δx ≈ S1 = λ · z/D1 . The suppression ratio can be raised by choosing a larger separation Δx < S1 albeit at the cost of a decreased signal. Choosing a large distance on the other hand might prove unnecessary for small amounts of multiple scattering. It is due to these practical diﬃculties, that the technically simpler single-beam cross-correlation geometry is often considered inferior to the two-beam realization, where a sample independent accurate theoretical description is available. The recently proposed single-beam cross-correlation camera-based method allows one to overcome this shortcoming [32]. Using a digital camera as a detector one can analyze speckle correlations and adapt Δx accordingly, assuring singlescattering detection with high accuracy. This allows for more ﬂexibility and in combination with the inherently high statistical accuracy of multi-speckle detection leads to an improved performance of the single-beam two-detector conﬁguration while preserving its technical simplicity. Several novel processing techniques are essential for the camera-based suppression of non-singly scattered light: multi-speckle and multi-tau averaging and speckle binning. Multi-speckle indicates an improved measurement statistics by additional averaging over set of speckles in space. This approach is discussed in detail in the following section on averaging of non-ergodic signals. Conceptually related, multitau and binning algorithms perform low-pass ﬁltering of the speckle ﬁeld in the time and space domain, correspondingly, in order to improve the signal-to-noise ratio and decrease the computational load at the same time.

8.3 General DLS techniques 8.3.1 Multi-tau correlation scheme The multi-tau correlation scheme was originally proposed by Sch¨ atzel [37, 38] to increase the accuracy of existing correlators for large lag times and is now implemented in the majority of modern correlator hardware [39]. Recently it has been also adopted for the software-based correlators [32,40,41]. The multi-tau algorithm assumes, that in most cases the linear spacing of lag times is not required for the analysis of intensity correlation functions (ICFs) produced from light scattering in colloidal systems. The spacing can be increased for large lag times, thus saving valuable correlator memory and processing time. Similarly, the minimal sampling time is not required for the calculation of the high-order lag times. Instead the sampling time can be increased proportionally to the explored lag time. The correlation function measured with multi-tau scheme is smoothed with a triangular kernel due to the nonzero integration time T > 0 of intensity trace[37,40]. The intensity IT (t) produced by integration with sampling time T from an original intensity trace I(t) can be written in the following way: rect(t/T ) 1 t+T /2 , (8.20) I(ξ) dξ = I(t) ∗ IT (t) = T t−T /2 T

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where rect(x) is equal to 1 for x ≤ 0.5 and zero elsewhere. The autocorrelation function as calculated from the sampled intensity trace is modiﬁed in the following way [37, 40]: t+T 1 Λ(τ /T ) . (8.21) g2 (ξ) [T − |ξ − τ |] dξ = g2 (τ ) ∗ g2T (τ ) = 2 T t−T T The kernel Λ(x) denotes triangular averaging deﬁned as Λ(x) = 1 − |x| for |x| ≤ 1 and zero elsewhere. It can be shown that for a single exponential decay the absolute accuracy error of the ICF can be as low as 10−3 for lag times τ ≥ 8T [37, 40]. In other words, high accuracy can be achieved with a multi-tau scheme if the lag time is at least eight times larger than the sampling time of the intensity. This naturally leads to a processing scheme where the sampling time T is adjusted for the calculation of the correlation value at the lag time τ by resampling of the original intensity trace, involving integration of two or more sequential intensity values, which furthermore improves statistical accuracy. In the study [32] the multi-tau scheme has been applied for the processing of the multi-detector data from a CCD camera. The data was provided with an initial exposure time T with 1/T Hz frame rate (the images were acquired without delays). The doubling of the exposure time to 2T is realized by integrating two sequential intensity values with exposure time T . In a similar way the eﬀective exposure time increased to 4T and so on as is shown in Fig. 8.5(a). From the obtained time series of intensity ﬂuctuations with diﬀerent exposure times the correlation coeﬃcients can be calculated with a simple linear scale processing. In software terms this scheme can be elegantly implemented via an objectoriented approach. The Elementary Linear Correlator (ELC) shown in Fig. 8.5(b) at every cycle i receives the intensity data point I(i), multiplies this value with previously accumulated data points I(i − k) for diﬀerent linearly spaced integer delays k = 0, 1, 2, . . . and updates the mean values of the corresponding products I(i)I(i−k). Every second cycle the correlator estimates the mean integrated value of intensity for two sequential time steps [I(2n)+I(2n+1)]/2 (where n = 0, 1, 2, . . . ) and delivers it to the input of the next ELC, operating in the similar manner and providing the integrated value to the next ELC and so on. This creates a cascading

〈Ι〉

(a)

(b)

Fig. 8.5. (a) Principles of the multi-tau correlation scheme. Sequential intensity values I(t) are integrated for the computation of larger lag times. (b) Simpliﬁed object scheme to illustrate the cascading of linear correlators used for the realization of the multi-tau correlation scheme. (Modiﬁed ﬁgures from [32] reproduced with permission of OSA.)

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line of ELC. Since the data output rate of ECL is half of the input data rate, the evaluation period doubles at every correlator. With this method extra precautions have to be taken to optimize the cycles with respect to limitations in computational power. The ELC used in the study [32] performed calculations of the cross-correlation coeﬃcients as well as autocorrelation coeﬃcients. ACF coeﬃcients were calculated on every pixel individually to obtain local values of I(x, t)I(x, t+τ )t and I(x, t)t . For the CCF with separation Δx the following averages were obtained: I(x, t)I(x+ Δx, t+τ )t , I(x, t)t and I(x+Δx, t)t . The values of products were accumulated by ECL for further spatial averaging. This scheme described here makes it possible to evaluate the data in real time on a standard desktop computer and therefore does not limit the duration of measurement, since no raw speckle data is recorded. 8.3.2 Binning technique The cross-correlation technique for multiple scattering suppression usually involves extraction of small single-scattering signals from the dominating background [29, 33]. This requires light detectors with a high dynamic range, like modern photodiodes or photomultipliers. Unfortunately, currently available and reasonably priced CCD or CMOS cameras are limited in either sensitivity or frame rate and thus their application to this task is challenging. Nevertheless, their effective dynamic range can be partially extended with a pixel binning technique. When the size of a single scattering speckle exceeds the area of several pixels, its intensity can be represented by the mean intensity of these pixels. Let us call the bin or meta-pixel the area of Sx × Sy pixels size represented by a single intensity value obtained by integration (or ﬂoating-point averaging) of pixel intensities within this area. Due to multiple sampling the noise of a single speckle measurements can be reduced by factor Sx Sy and thus the dynamic range deﬁned as DR = 20 log10 SN R dB, where SN R is a signal-to-noise ratio, will increase on 10 log10 Sx Sy dB. The loss of spatial resolution reduces the statistical accuracy and intercept β only if the binning area Sx × Sy is comparable √ or larger than the coherence area S1 × S2 . Binning also results in partial suppression of multiple scattering due to ﬁltering out speckles smaller than the bin. Binning introduces a two-dimensional ﬁltering with a box kernel function which reduces more eﬃciently the high-frequency spatial ﬂuctuations (smaller speckles from multiple scattering volume) than the lower-frequency ﬂuctuations (larger speckles from single-scattering beam volume). Binning also helps to dramatically reduce computational and memory load, thereby preserving statistically signiﬁcant information of the original signal in a way similar to the multi-tau scheme. Experimental results demonstrate that binning alone can yield a considerable reduction of the multiple-scattering part in the signal (Fig. 8.6) [32]; however, the crosscorrelation processing is still required in order to achieve the maximal suppression eﬃciency.

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1 300

g2(τ) í 1

D, nm

0.8

0.6

0.4

0.2

0

250 200 150

ACF 1x2 ACF 5x2 ACF 20x2 ACF 80x2 CCF í1

10

100 0 2 10 10 binning size, pixels

0

10 τ, s

1

10

Fig. 8.6. Autocorrelation functions obtained with diﬀerent binning areas are shown together with the cross-correlation function. As the vertical size of the binning area increases, the ACF approaches the CCF, indicating partial but not suﬃcient suppression of multiple scattering. The inset shows the estimated particle size (◦) as a function of binning size. The actual size is indicated by a dashed line (reproduced from [32] with permission of OSA).

8.3.3 Spectral analysis in DLS The power spectrum S (w) can be equally well employed for the analysis of the intensity autocorrelation function g2 (τ ) since both are related via a Fourier transformation [42, 43]: 1 ∞ S (w) = cos wt [g2 (τ ) − 1] dt . (8.22) π 0 It was demonstrated [44, 45] that the moment of the zero order of the power spectrum is proportional to the number of dynamic scatterers Cdyn (e.g. red blood cells in the case of biomedical signal) while the ﬁrst moment is related to the mean velocity of particles V multiplied by Cdyn . Thus the ratio of both integrals, representing the mean frequency w∗ provides the velocity-dependent signal, inversely proportional to the correlation time [42]: %∞ wS(w) dw ∗ . (8.23) V ∝ 1/τc ∝ w = %0 ∞ S(w) dw 0

8.4 Problem of non-ergodicity In the case of solid-like or highly viscous media, full statistical averaging is not possible in a single coherence area (speckle spot) or it requires extremely long data-collection times. In the past researchers have reported on experiments with

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a measurement time of hours and days (for example, [46, 47]). For such materials the statistics of the spatial variation of intensity is diﬀerent from the statistics accumulated in time. This situation is often referred to as the non-ergodicity problem (NEP), since ergodicity implies interchangeability of temporal and ensemble (spatial) statistics. The NEP may arise when the motion of scattering particles is restricted, like in gels or colloidal glasses, or when the system dynamics is macroscopically heterogeneous (e.g. in areas of frozen motion or completely static regions, like bone in a biological measurements). System with ergodic dynamics in the general sense can demonstrate apparent non-ergodic properties on the ﬁnite time scales of an experiment. In this quasi-ergodic case proper time averaging is theoretically possible, but requires measurement times not accessible or limited by the sample stability (for example, aging, sedimentation). Thus for in vivo biomedical measurements minimization of data collection time is one of the most important requirements for reliable measurement due to the natural time evolution of the biological system. Scattering media undergoing continuous slow changes, such as colloidal systems out of equilibrium (gelation, foam coarsening, sedimentation, ﬁlm drying and so on), can be also considered as the NEP systems. Let us consider the simple case of non-ergodicity, when the ﬂuctuating dynamic signal is mixed with a constant scattered ﬁeld. The electric ﬁeld amplitude of detected light E(t) is composed of dynamic Ed (t) and static Es (t) contributions: E(t) = Ed (t) + Es (t),

(8.24)

The Elementary Linear Correlator for this case can be written in a following way: g1 (τ ) =

E(0)E ∗ (τ ) = (1 − ρ) |g1 d (τ )| + ρ . |E(0)|2

(8.25)

where ρ = Is /(Id + Is ) is the static part in the detected intensity (Is = Es Es∗ and Id = Ed Ed∗ are the intensities of static and dynamic parts, correspondingly) and g1 d (τ ) = Ed (0)Ed∗ (τ )/|Ed (0)|2 is the Elementary Linear Correlator of the dynamic part. For lag times τ signiﬁcantly larger than the characteristic relaxation time of g1 d (τ ) the total FCF can be written simply as g1 (τ → ∞) = ρ. When the measurement of such signal is performed with the inﬁnite time averaging in a single speckle spot the ICF [43] assumes the following form: 2 (8.26) g2 (τ ) − 1 = β(1 − ρ)2 |g1 d (τ )| + 2 βρ (1 − ρ) |g1 d (τ )| . When the ensemble averaging is performed using spatial as well as temporal averaging the ICF has the following ergodic form: 2

g2 (τ ) − 1 = β(1 − ρ)2 |g1 d (τ )| + 2βρ (1 − ρ) |g1 d (τ )| + βρ2 .

(8.27)

Equations (8.26) and (8.27) describe the measured correlation functions when the time-averaged intensity It equals the intensity obtained with ergodic averaging Ierg . This, however, might not be the case if the averaging time is insuﬃcient and FCF does not decay to ρ at the maximal accessible lag time τmax : g1 (τmax ) = g1 (∞) = ρ. If we introduce the ratio of time-averaged and ergodic intensities Y = It /Ierg this diﬀerence can be quantiﬁed as following: g1 (τmax ) − ρ = 1 − Y [48].

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The dynamic part of correlation function is deﬁned in the following way: g1 d (τ ) = [g1 (τ ) − g1 (τmax )] / [1 − g1 (τmax )], thus Eq. (8.26) can be rewritten for the full-coherence case β = 1 in the following form: 2 2 g1 (τ ) − g1 (τmax ) It − Is g2 t (τ ) − 1 = IT 1 − g1 (τmax ) Is It − Is g1 (τ ) − g1 (τmax ) . +2 IT It 1 − g1 (τmax )

When Is is substituted, Eq. (8.28) turns to: ! " g2 t (τ ) − 1 = Y 2 g12 (τ ) − g12 (τmax ) + 2Y (1 − Y ) [g1 (τ ) − g1 (τmax )] .

(8.28)

(8.29)

There are two main eﬀects due to time-averaging reﬂected in Eq. (8.29). First of all the experimental correlation function is distorted by the diﬀerence of timeaveraged mean intensity from the ensemble-averaged one (expressed in Y ). This aﬀects the intercept of the correlation function as well as the shape, due to the introduction of a linear term in g1 (τ ). Next, the term g1 (∞) which is the correlation value for the longest accessible lag time τmax is subtracted from the experimental ICF, which leads to the shift of the ICF, in such a way that ICF g2 t (τ )−1 decays to zero at the maximal lag time, even for a properly estimated mean intensity value. The correct averaging of the ICF and the extraction of the FCF is nevertheless possible for non-ergodic samples with methods described below. 8.4.1 Pusey and van Megen method Existing hardware correlators are usually used in combination with a single-point detectors and they are therefore unable to perform spatial averaging. The timeaveraged ICF g2 t (τ ) always decreases to 1 for the maximal lag time τmax , even if the underlying FCF g1 (τ ) does not decay to zero. Thus the normalized temporallyaveraged ICF g2 t (τ ) should be post-processed in order to correct the artifacts of non-ergodic averaging. This was successfully performed by several groups (e.g. [48, 49] for single-scattering DLS and [50] for DWS). If the ergodic intensity Ierg is measured in the separate experiment and the β factor of the setup is known, the FCF g1 (τ ) can be extracted from the experimentally measured g2 t (τ ) using Eq. (8.29). In order to estimate the ergodic intensity the sample has to be translated or rotated to access a large number of independent speckles [15, 48, 49, 51]. 8.4.2 Multi-speckle technique One can also use the multi-speckle detection schemes in order to perform averaging over a large number of speckles. DLS with a digital (CCD or CMOS) camera as a detector oﬀers the possibility of performing simultaneously a large number of independent light scattering experiments, thus achieving ensemble averages in real time [32, 41, 52–54]. In a typical camera-based DLS experiment the speckle ﬁeld is detected with spatial resolution, which allows one to monitor the intensity ﬂuctuations of multiple speckles simultaneously. Using a large number of these

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speckles the NEP can be solved. In this case the averaging of the intensity products and the mean intensity is based on the speckle ensemble statistics and intensity history: I(t)I(t + τ )ms,t g2ms (τ ) = (8.30) I(t)2ms,t . where . . .ms,t = . . .ms t denotes the averaging along the entire two-dimensional camera matrix ms and time t. Sch¨ atzel suggested the use of the so-called symmetric normalization of the correlation function [37, 38] by normalizing with the product of mean intensities I(t)ms,t ·I(t+τ )ms,t instead of the square of the mean intensity. This reduces the noise in the estimates of the normalized correlation function by diminishing the inﬂuence of drifts. It is of special importance for multi-speckle processing, since ﬂuctuations of unaccounted background light, camera matrix sensitivity or the sample properties usually aﬀect all pixels on the matrix in a similar way and thus cannot be reduced with multi-speckle processing. The multi-speckle averaged ICF with symmetric normalization is deﬁned as: g2ms (τ ) =

I(t)I(t + τ )ms,t I(t)ms,t · I(t + τ )ms,t

(8.31)

For the camera system the speckle size should be adjusted to ﬁt the pixel size in order to optimize the signal-to-noise ratio by maximizing the number of monitored speckles and keep a reasonable coherence factor β. Due to sampling times of typically 1–10 ms, digital-camera-based detection is restricted to rather long correlation times. Thus, in many cases, photon correlation spectroscopy has to be performed in a separate experiment to access the full range of correlation times. Another solution might be ﬁber-based multispeckle detection using avalanche photodiodes. As it was recently demonstrated by Dietsche et al. [55] such a system can measure FCF with a decay time of tenths of microseconds with a time resolution of 26 milliseconds. However, with the rapid improvement of semiconductor technologies, the implementation of DLS systems with cameras on the detection side is probably more promising. Recent implementations of the smart-pixels technologies with on-site processing of the detected intensity already realized for some tasks [56] might lead in the future to fast camera-based DLS systems. 8.4.3 Double-cell technique An alternative approach might be the active alteration of scattered ﬁeld in order to achieve the necessary statistical averaging. This can be performed in a DWS experiment by introducing a second cell in the optical path [14, 57] with slow ergodic dynamics and moderate turbidity. Light transmitted through the ﬁrst cell of thickness L1 has to pass through the second cell of thickness L2 . The second cell contains a slowly evolving sample with known correlation function g2 (τ, L2 ). In this conﬁguration the resulting intensity correlation function g2 (L1 , L2 , τ ) of the light transmitted through both cells can be found according to multiplication rule [14]: (8.32) g2 (L1 , L2 , τ ) − 1 = [g2 (τ, L1 ) − 1] · [g2 (τ, L2 ) − 1]

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Fig. 8.7. Two-cell DWS setup. Light transmitted diﬀusively from the sample cell is imaged via a lens onto a second cell (containing a highly viscous colloidal suspension of moderate optical density). Subsequently the light is detected with a single-mode ﬁber and analyzed digitally (correlator and PC). Element from the system setup in [14].

In the original implementation of the double-cell system the sandwich-like arrangement of the glass cells with equal thicknesses separated by a thin wall has been employed [14,57]. In order to apply the multiplication rule formalism the amount of light making loops between the two cells should be negligible. This can be achieved by introducing an absorbing layer between cells. Nowadays spatially separated cells like the one presented in Fig. 8.7 have been used in most experimental realizations [54]. The stability of the second-cell can be improved by replacing it with a ground glass ﬁxed on a stepper motor. When rotated slowly this system provides the same performance as the liquid second cell but beneﬁts from higher stability and better control. In the latter case the system is not temperature-dependent and the correlation function can be easily tuned by simply changing the motor rotation speed. The importance of this approach has been recognized by the scientiﬁc communities and commercial light scattering setups are available [31]. Although the two-cell technique can provide the proper statistical averaging for the non-ergodic sample, it does not solve the problem imposed by the timeaveraging problem and the associated long measurement times. 8.4.4 Echo-technique with sample rotation An alternative approach is the so-called correlation echo scheme. The original method, as proposed by Pham et al. [58] for DLS, uses precise sample rotation in order to explore diﬀerent scattering conﬁgurations. For a solid sample the same speckle pattern reappears after a full sample revolution. If the sample is slowly evolving the speckle pattern changes due to the internal dynamics. A contribution for each sample position is eﬀectively recorded which allows one to reconstruct an ensemble-averaged signal. In practice the sample rotation causes rapid changes of the measured intensity signal. The correlation function in this case displays narrow echo peaks separated by time intervals equal to the rotation period. The shape of the peak is deﬁned by the sample rotation speed, but the height follows the correlation function of sample internal dynamics. A limitation of this technique is the loss of information about the sample correlation function between the echo peaks and the limited rotation

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speed. Increasing the rotation speed to more than 0.1–1 Hz in such an arrangement is diﬃcult due to possible perturbations of the sample. 8.4.5 Echo-technique without sample rotation For DWS measurements the echo method can be implemented using a rotating ground glass, in a way similar to the double-cell method. With this conﬁguration the sample is kept at rest and illuminated with a light ﬁeld scrambled by the rotating ground glass (Fig. 8.8). Rotation continuously modiﬁes the illumination geometry as well as the scattered speckle ﬁeld detected in the far ﬁeld. After a full revolution of the glass the speckle pattern reappears and generates the echo peak in the measured ICF. Since there is no direct inﬂuence of the sample cell, the motor rotation speed is limited by the mechanical stability of the system itself and detection time of the measuring system. Therefore the gap between echoes can be signiﬁcantly narrowed. In the ﬁrst implementation of this system [61] the motor was operated at about 40 Hz which generated the ﬁrst echo at around 25 ms. With the most recent version of the echo setup at the University of Fribourg, motor frequencies as high as 100–200 Hz can be achieved.

Fig. 8.8. Setup for echo-based multispeckle DWS measurements. Light from a laser is dispersed by ground glass rotated by the motor and, collected by the lens, illuminates the sample. Scattered light is detected both in transmission and reﬂection for further processing. (Reproduced from [59] with permission of OSA.)

8.4.5.1 Processing of echo speckle data In order to resolve the narrow correlation echoes special emphasis has to be given to the data analysis of the echo speckle signal. Since the echo width can be as much as three orders of magnitude smaller than the echo period, very precise measurements are required. In a traditional linear correlator the ICF is obtained by averaging products of photon counts in a certain sampling time interval T separated by lag time τ (channel) [5]. Over several decades of lag times the linear channel

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layout becomes impractical and the computation time increases rapidly. The multitau scheme discussed above computes the ICF with the highest resolution for the shortest lag times. Resolution is reduced proportionally for longer lag times. Such loss of resolution is not acceptable for the echo signal since all echoes are expected to have the same width, even at large lag times. On the other hand only a small number of correlation channels is needed to resolve the ICF in the vicinity of the correlation echoes, meanwhile the ICF in the inter-echo intervals usually equals 1 and does not contribute. Furthermore not all echoes need to be resolved to cover a given range of lag times. Pham et al. proposed the modiﬁed multi-tau scheme [58] optimized for the echo signal which integrated intensity values separated by the echo period. Fast computation of the correlation function can be also performed with the fast Fourier transform (FFT). According to the Wiener–Khinchin theorem the autocorrelation function of the signal can be determined as an inverse Fourier transform of its power spectrum which is the square of the Fourier amplitudes [60, 62]. For a given sampling time T FFT provides the correlation function for all available channels much faster than any other intensity correlation technique. However, in order to resolve the echo shape the sampling time has to be minimized, which requires large data arrays and long processing time. Another disadvantage is that FFT requires the complete intensity history and is therefore not suitable for the online processing. A fundamentally diﬀerent approach to calculate the ICF has been introduced by Chopra and Mandel [63] as the photon correlation function or photon time-ofarrival correlator. It estimates the distribution of time intervals between photons arrivals which is proportional to the ICF. This method does not require data resampling and therefore does not depend on the sampling time T . This approach is signiﬁcantly faster if high resolution is required for the ICF and if the data is provided in the corresponding format. The latter is the case for one of the main commercial producers of hardware correlators (www.correlator.com). The method has been used for the calculation of ICFs in Fig. 8.9(a) at the highest available resolution of 12.5 ns. The performance of diﬀerent data analysis algorithms has been compared in Ref. [60]. The data of a 12-second measurement at an average count rate of 210 kHz has been analyzed under identical conditions with the linear correlator, FFT and the time-of-arrival correlator. Correlation echoes were explored up to echo number 240, corresponding to the lag time τ = 6 seconds with the time delay between echoes doubling after each linear block of 16, resulting in a total number of 60 echoes. The time resolution of time-of-arrival correlator is determined by a 12.5-ns hardware time step of 80 MHz. For a computer with a 1.7 GHz Intel Xeon processor with an echo integration window of 6 μs the computation time is about 14 seconds for the time-of-arrival correlator at 12.5-ns resolution. The FFT correlation function contains all available lag times and computation time is inversely proportional to the sampling time T . For the same data it takes 1.7 seconds with 12 μs sampling time and less than one second for 24 μs. The whole analysis with the linear correlator takes approximately 1.5 seconds if a single channel for each calculated echo with a sampling time of 12 μs is chosen. Most of this time is needed to convert the photons arrival history to the intensity trace, which could be easily done on-the-ﬂy

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1

Time-averaged Echo: delay-based processing Echo: sampling time 8 mcs

0.6

0.6

0.4 0.2

g2 (τ) - 1

g2 (τ) - 1

g2 (τ) - 1

0.8

0

0.4

451

-4 -2 0 2 4

10-1

Δτ [μs]

0.2

10-2

0 0.1

1

τ [s]

(a)

10

0

0.2

0.4

0.6 τ

1/2

0.8 1/2

[s

1

1.2

1.4

]

(b)

Fig. 8.9. (a) Intensity correlation function detected in backscattering from TiO2 particles in glycerol as detected with the echo method. Inset shows the shape of the ﬁrst echo. (b) Sample correlation function reconstructed from echo data using photon time-of-arrival correlator with 12 ns resolution () and linear intensity correlation calculation with 8 μs sampling time (), as compared to the traditional single-point-averaged ICF over 20 minutes (solid line). Echo statistics have been accumulated for 12 seconds. (Reproduced from [32] and [60] with permission of OSA.)

during data recording. Thus time required for the data processing with optimized algorithms is smaller than the measurement time and does not signiﬁcantly increase the total duration of the experiment. 8.4.6 Combination of methods The double-cell and echo methods using the rotating ground glass can be implemented with the same experimental arrangement utilizing one motor, and thus it is possible to switch between these schemes even during a single experiment by simply changing the motor rotation speed and processing algorithms [61]. In the double-cell mode the ground glass is rotated slowly in order to record the sample correlation function for short times. The sample correlation function can be recovered from the multi-run average by the Pusey–van Megen method Eq. (8.29). When the experiment is switched to the echo-mode it measures the correlation values at lag times separated by the rotation period. By merging the acquired short time and long time parts of the recorded sample ICF, an extremely wide range of correlation times can be explored with a single experimental setup consisting of the detector, the photon counter and the computer-controlled stepper motor with the ground glass [61]. Fig. 8.10 shows the results obtained with such technique from a slow-evolving solution of polystyrene particles in gelatine. The dynamics gradually slows down as the system gels. At some point the particle motion becomes spatially restricted and the system becomes non-ergodic. From this point on the measured correlation function does not decay to zero at the maximal lag time of 10 seconds. Echo DWS also makes it possible to follow in time the evolution of the correlation coeﬃcient in the arrested state and thus enables us to characterize changes in a sample dynamics (see inset in Fig. 8.10).

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Fig. 8.10. Time evolution of the ICF for a gelling mixture of 4 wt.% gelatine and 1 wt.% polystyrene tracer spheres (diameter 720 nm) in water kept at 22 ◦ C. Data taken at 1, 41, 57, 68 and 138 minutes. (◦), Two-cell DWS based on a hardware multi-tau analysis, and (•), echo analysis of 15 seconds long measurements. Dotted line, ICF determined according to [48] with motor at rest. Inset: time evolution of the echo correlation function at τ = 0.1 s. (Reproduced from [61] with permission of APS.)

8.5 Time-resolved methods The multi-speckle methods described above signiﬁcantly reduce the measurement time by substituting temporal speckle averaging with spatial averaging. It has been pointed out that this can be of extraordinary importance for the characterization of systems undergoing continuous or intermittent changes such as colloidal systems out of equilibrium, biological systems or any other systems in evolution. Thus, the techniques solving the NEP lead to a reduced measurement time which in turn enables time-resolved measurements. 8.5.1 Time-resolved spectral analysis A straightforward step towards time-resolved analysis with a spectral approach is by dividing the intensity time trace in equal intervals and processing them separately using multi-speckle averaging [64, 65]. With the camera operating in the line-scan mode the speckle data can be collected with a single pixel row and accumulated line-by-line into a single image, where the columns of the sensor matrix represent time traces of the intensity ﬂuctuations at the single pixel. In this way processing of the single image can provide the time-resolved multi-speckle average of the intensity correlation function. The spectral processing can be performed, for example, with a windowed Fourier transform (WFT) method or a continuous wavelet transform (CWT). These methods were recently applied for a study of

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the time evolution of a rough front during the evaporation of ﬂuid phase from the porous media and non-stationary dynamics has been clearly observed in the time-resolved measurements [66]. The time-resolved spectral information can be obtained separately from the intensity trace of every pixel when the results are averaged over the whole pixel ensemble. The CWT and WFT estimation of spectral statistics from non-stationary time traces are well known methods, which are extensively covered in the literature on data processing [62, 67] and thus will not be discussed here. Another method, making use of characteristic multi-speckle data, can be realized through the conversion of the instantaneous intensity to the instantaneous frequency and the estimation using available spatial speckle statistics [68]. Such ¯ a conversion can be performed through the complex analytical signal I(t) = (r) (r) I (t) + jI (t) calculated from the intensity trace using a Hilbert transform: +∞ (r) I ¯ = I (r) (t) + j dξ. (8.33) I(t) π −∞ ξ − t It can also be obtained through inverse Fourier transform u ¯(t) 2F −1 [(1 + sgn w) F {I}] ,

(8.34)

where F and F −1 are operators of forward and inverse! Fourier transforms, corre" spondingly. In the next step, the phase φ(t) = arctg I (i) (t)/I (r) (t) and instantaneous frequency wh (t) = dφ(t)/dt can be easily calculated from the analytical signal. Since in the multi-speckle scheme a large set of intensity traces is available at each time, many instantaneous frequencies can be found. The distribution of these frequencies is related to the power spectrum and the mean of instantaneous frequencies wh (t). For a Gaussian process the latter corresponds to the width of the instantaneous spectrum w∗ (t) which in turn provides an estimate of the intensity ﬂuctuations. The spectrum width can also be estimated through fractional diﬀerentiation of intensity ﬂuctuations. The expression (8.23) linking the spectral moments of intensity ﬂuctuation to the underlying dynamics can be evaluated in the time domain by a separate calculation of denominator and numerator. Actually, the denominator is the energy of the signal and can be easily calculated in the time domain: Δt ∞ 1 ∞ S(w) dw = S(w) dw = π I 2 (t) dt . (8.35) 2 0 −∞ 0 The numerator, on the other hand, can be calculated in the time domain only by means of a half-integer diﬀerentiation: ∞ Δt 2 ⏐ 1 ∞⏐ ⏐ ⏐2 D1/2 {I(t)} , wS(w) dw = (8.36) ⏐F {I} jw⏐ dw = π 2 −∞ 0 0 where D1/2 {·} is an operator of half-integer diﬀerentiation. For a non-stationary signal of intensity ﬂuctuations the spectrum width w∗ can be calculated for a moment of time t inside the averaging window Δt:

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"2 D1/2 {I(ξ)} t−Δt/2 % t+Δt/2 I 2 (ξ) dξ t−Δt/2

% t+Δt/2 ! wf∗ d =

dξ .

(8.37)

The statistics of intensity ﬂuctuations can be estimated with fractional derivation using digital ﬁltering techniques [66, 68] which allows for spectral width estimation in the real-time and provides an extremely simple, ﬂexible and fast way to compute the spectrum width in the time domain. 8.5.2 Time-resolved correlation analysis The multi-speckle calculation of the correlation function can be performed with no temporal averaging other than the ﬁnite exposure time of the digital camera. The latter is typically has the order of milliseconds. The Time-Resolved Correlation (TRC) technique proposed by Cipelletti et al. [69] uses spatial statistics of the farﬁeld speckles to study time-dependent processes. In the original implementation two sequential camera frames are cross-correlated and the products are averaged and normalized with the mean intensities using the symmetric normalization: g2T RC (t, τ ) − 1 =

[I (t) I (t + τ )]ms , I (t)ms I (t + τ )ms

(8.38)

where multi-speckle averaging . . . ms is carried over all the pixels of the matrix or its part. The lag times τ can be adjusted by the step of camera inter-frame time Δt: τ = nΔt. In this way evaluation of the correlation coeﬃcients can be studied in time domain which allows measurements of non-stationary and intermittent processes with high temporal resolution [69–72]. One can also use echo-based multi-speckle methods to obtain time-resolved information. With this method statistics of the spatial speckles is supplemented with the speckles obtained on diﬀerent echo phases. An individual echo frame has an eﬀective ‘exposure time’ given by the short time average which deﬁnes the lower limit of the time resolution. This time will typically be two or three orders larger compared to the exposure time of an individual picture since the ground glass motion has to be resolved for a signiﬁcant set of statistically uncorrelated images. This additional averaging step, on the other hand, makes it possible to further decrease the noise level. If TRC is extended by the averaging over diﬀerent time phases φ of the motor revolution, assuming that the motor period T = τ and thus 0 < φ < τ: g2T RCecho (t, τ ) =

[I (t + T φ/2π, r) · I (t + T φ/2π + τ, r)]φ , I (t)φ I (t + τ )φ

(8.39)

where the averaging . . . φ is performed over the ground glass phases φ and corresponding times T φ/2π, where T in this case is the motor rotation period. The lag time τ can take on the values of nT , where n is the echo order. The echo-based time-resolved measurements can be performed with the commonly used single-point detectors and do not require cameras for the multi-pixel averaging. However, the quality of the data is limited by the number of independent speckles produced by

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the ground glass which is usually smaller than the number of independent spatial speckles that can be detected simultaneously with the camera (usually deﬁned by the number of pixels, if the speckle size is minimized to the pixel size). Thus the camera multi-speckle technique easily outperforms the echo technique for timeresolved measurements in the far ﬁeld. 8.5.3 Time-resolved structure function analysis Along with the frequently used intensity correlation function less well known statistical measures of the temporal decorrelation can be used for the processing of intensity ﬂuctuations. One example is the structure function of the intensity ﬂuctuations [73–75]. The normalized intensity structure function (ISF) is deﬁned in the following way: 2 [I (t) − I (t + τ )] . (8.40) d2 (τ ) = I2 The ISF can be directly related to the ICF g2 (τ ) [73] for the stationary ergodic case via the following relation: d2 (τ ) = 2 [g2 (0) − g2 (τ )] .

(8.41)

It was shown theoretically [73] and experimentally [74] that the structure function is much less sensitive to low-frequency noise or drifts in the intensity and outperforms the ICF in all cases when the photon count rate is high. In addition, the structure function estimator signiﬁcantly suppresses the impact of the static scattered light on the measured statistics of ﬂuctuations. We can estimate time-resolved structure function (TRS) similar to time-resolved ICF deﬁned in [69]: 2 [I (t) − I (t + τ )] ms dT2 RS (t, τ ) = , (8.42) I (t)ms I (t + τ )ms where multi-speckle averaging is performed over all the pixels of the matrix or its part. The dT2 RS (t, τ ) quantity characterizes the magnitude of speckle ﬁeld changes inside the multi-speckle averaging region in the period from t to t + τ measured as the normalized mean square diﬀerence in the detected intensity values. This type of statistical measure can also be called speckle deviation. As follows from Eq. (8.41) the meaning of d2 (τ ) and g2 (τ ) are inverted and thus faster dynamics corresponds to larger values of structure function while the opposite is true for g2 (τ ). Fig. 8.11 shows a comparison of the time-resolved correlation function estimate g2 (τ ) − 1 to the ISF d2 (τ )/2 during the intermittent process of drying paint with camera-based multi-speckle processing. The structure function clearly indicates the rearrangement events which are diﬃcult to distinguish in the history of the correlation coeﬃcient with the same lag time. When compared to the time trace of ACF intercept g2 (0) − 1 presented in the same Fig. 8.11 one can see that the noise in g2 (τ ) − 1 is dominated by the ﬂuctuations of the intercept estimate. The calculation of structure function does not require that many independent speckle

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τ

τ

0 τ

Fig. 8.11. Comparison of TRC and TRS analysis of the drying colloidal ﬁlm [75]. The rearrangement of water droplets inside the ﬁlm is seen as intermittent alterations of the speckle ﬁeld which can be detected by the peaks of the TRS signal (d2 (τ )/2). They can be also observed on the TRC plot (g2 (τ ) − 1) as the deviations from the time-resolved estimate of intercept (g2 (0) − 1). However, the TRC signal is dominated by the noise of the intercept.

realizations as the estimation of ICF intercept and thus time-resolved ISF is more suitable for the monitoring of the intermittent processes [75].

8.6 Space-resolved methods 8.6.1 Space-resolved correlation and structure analysis The DWS experiment can be equally well realized in the imaging geometry [75, 76], where each point in the image plane corresponds to a spatial position on the sample surface. Spatially resolved local dynamics can be studied with the suitable processing techniques. The straightforward approach would be to use the camera-based imaging scheme and estimate the speckle statistics (ACF) pixelwise with minimal spatial averaging. Thus the characteristics of intensity dynamics can be mapped on the sample geometry and the sample dynamics can be studied with full spatial resolution. The quantiﬁcation of characteristic relaxation time can be performed with a cumulant analysis [5, 77]. The dynamics evaluated with cumulants can be used to create a two-dimensional image representing the sample dynamics. This approach has been used to study temporal and spatial dynamics of the imbibition front propagation in a vertically mounted paper sheet [66, 76]. This technique makes it possible to qualitatively distinguish the dynamic areas of the moving interphase boundary from a static background and track the front propagation. To some extent data can be quantitatively analyzed, but it is still sensitive to the noise introduced by the limited averaging time. Moreover, for the partially wet areas, when the dynamic part in the overall detected intensity is low, the measurements are suﬀering from a non-ergodicity problem which manifests itself as increased noise on the dynamic

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maps. This problem can be reduced only by increasing the temporal or spatial averaging windows, which would lead to unavoidable loss of resolution. A possibility of overcoming these limitations is to combine the digital-camera-based approach and the echo technique in the image plane. An implementation of such an approach is currently in progress at the University of Fribourg (Switzerland) [75, 78]. 8.6.2 Space-resolved spectral analysis In recent years Doppler-based methods for measuring perfusion blood ﬂow have made a signiﬁcant progress both in hardware and processing algorithms. The technique of Laser Perfusion Imaging [44,45] has been introduced, which involves beam scanning or full-ﬁeld acquisition [56,79], when speckle data is collected with a camera and a mapping of the dynamics is performed with pixelwise spectral processing. Such a processing scheme advances Doppler-based methods to the level of timeand space-resolved DLS techniques for biomedical imaging which in the past has been limited to laser speckle imaging (LSI) approaches. Doppler-based methods oﬀer some fundamental advantages compared to LSI, since they allow to distinguish diﬀerent time-rates of dynamics as well as diﬀerent types. Nevertheless, they suﬀer from the non-ergodicity problem if the multi-speckle averaging is not implemented in the processing algorithm. This leads to the known diﬃculty of quantitative data interpretation and the appearance of the speckle noise [80] in the image, as soon as contributions with suﬃciently slow relaxation rates are present in the detected signal. In general, the formalism developed for the temporally averaged correlation function in subsection 8.4.1 can be equally well applied to the Doppler signal. Obviously, the imaging performance of Doppler methods can beneﬁt from the application of multi-pixel detection in terms of noise reduction. 8.6.3 Laser speckle imaging In both TRC and TRS methods the diﬀerence in the acquired image frames is analyzed and the detector exposure time is minimized in order to limit smearing induced by the time averaging during the exposure. However, the overall limited number of detected photons does not allow one to decrease the exposure time T below a certain threshold, typically of the order of milliseconds for most standard applications. The minimum time resolution is then given by the maximum frame rate of the order of 1000 fps or less. In this way, the time resolution of TRC, TRS methods is typically limited to 1 ms or more. On the other hand, if the exposure time T is of the order of decorrelation time the speckle pattern evolves while the camera is acquiring a single frame. This leads to the blurring of the speckle and decreases the contrast of speckle image as detected by the camera. Contrast as a function of exposure time T is usually deﬁned as the standard deviation of the intensity estimated along pixels of detector normalized with mean intensity: # K(T ) = σT (I)/Ims = IT2 ms /I2ms − 1. (8.43)

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The idea of Fercher and Briers [81, 82] was to use the amount of blurring in order to estimate the correlation time of intensity ﬂuctuations. The approach has been coined Laser Speckle Contrast Analysis (LASCA) by the inventors. The contrast K of the time-integrated speckle ﬂuctuations can be related to the correlation function in the following way [37, 83] (in order to keep the notation simple we will discuss the properties of square of contrast K 2 (T ) in the following part): K 2 (T ) =

2 T

T

g2 (τ ) (1 − τ /T ) dτ = 0

2β T

T

|g1 (τ )|2 (1 − τ /T ) dτ,

(8.44)

0

It is important to note that this equation is diﬀerent from that originally introduced by Fercher and Briers [84]. As reaﬃrmed by Bandyopadhyay and coworkers[83], the correct expression, Eq. (8.44), has to take into account the eﬀect of triangular averaging of the correlation function according to Eq. (8.21) [37]. Essentially, the contrast K 2 (T ) is an intercept of the time-integrated intensity correlation function: K 2 (T ) = g2T (0) − 1,

(8.45)

where g2T (τ ) is deﬁned by Eq. (8.21). The correlation function of the particular form can be substituted in Eq. (8.44) in order to ﬁnd contrast dependence on the correlation time. The results for some important cases can be found in [83]. For the DWS case, when the FCF is deﬁned by Eq. (8.16): ! √ " √ K 2 = β 3 + 6 x + 4x exp −2 x − 3 + 2x / 2x2 , (8.46) where x = 6γ 2 T /τ0 . For single-scattering case, when the FCF is deﬁned by Eq. (8.7): T T τc2 2 −1+2 , (8.47) K = β 2 exp −2 2T τc τc LASCA was originally developed for biomedical imaging, where the image of the speckle surface of the object has been formed in the detector plane with the objective lens [84, 85]. It can be equally well employed in the far-ﬁeld geometry [83, 86] to study non-stationary processes in complex media, where the spatial information is not of the major interest. The contrast analysis based on the spatial speckle statistics ensures ergodic averaging, since the speckle size on the detector is adjusted to the speckle size and calculation within LASCA window N × N performs averaging over approximately N 2 statistically independent speckles. The technical arrangement for the speckle contrast experiment is relatively simple, as no online processing of the images is required. A quantitative interpretation of the contrast values requires some assumptions concerning the particular shape of the correlation function and the impact of a static component in the detected light [87]. In addition, the contrast has the maximal sensitivity (with respect to the underlying relaxation processes) if the correlation time is of the order of the exposure time [88], and it is signiﬁcantly reduced if the exposure time is much smaller or much larger than the correlation time. In this limit a conversion of the contrast to the correlation values can be performed only with signiﬁcant error.

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Since in biomedical imaging absolute values are not always needed for data interpretation (like functional imaging), LASCA received signiﬁcant attention [85, 89] and commercial devices are already available on the market [90]. 8.6.4 LSI with active noise reduction The improvement of LASCA technique has been recently proposed by V¨ olker et al. [91] as an active noise-reduction scheme which is closely related to a double-cell approach described in subsection 8.4.3. The sample is illuminated with the laser beam expanded by slowly rotating ground glass. The decay time of the intensity correlation function due to the ground glass rotation is larger than the decay time of the sample dynamics, and thus the full speckle statistics can be explored in time (provided the speckle size is smaller than the scattering length l∗ ). In order to minimize the necessary data transfer, sequential frames have to be captured with the period larger than the correlation time due to the rotation. In such a way obtained speckle images are statistically independent. To study the eﬃciency of noise reduction V¨ olker et al. [91] used a biomedical phantom consisting of the Teﬂon block with a cylindrical liquid inclusion containing latex particles. The optical properties of the solution were equal to the Teﬂon and no static scattering diﬀerences could be observed. The camera has been imaging the bottom of the Teﬂon block (Fig. 8.12(a)). In this way the liquid inclusion has been separated by the static Teﬂon volume (Fig. 8.12(b)). The sample was illuminated through the rotating ground glass. The rotation speed and the camera frame rate were set in such a way that the sequential speckle images were completely

(a)

(b)

Fig. 8.12. (a) Imaging setup. A laser beam (785 nm) is incident on a ground glass mounted on a motor with 1 Hz rotation speed. Light passing the ground glass is moderately divergent and illuminates the sample surface as an expanded light spot. The illuminated surface is imaged via a beam-splitter by the camera objective onto the CCD chip of the digital camera. (b) Heterogeneous sample obtained by milling a cylindrical void (diameter D0 = 3 mm) into a solid block of Teﬂon. The inclusion is ﬁlled with an aqueous suspension of 710 nm polystyrene latex spheres that matches the optical properties and creates dynamic contrast. A layer of thickness d = 0.45 mm separates the inclusion from the imaging surface. (Reproduced from [91] with permission of OSA.)

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uncorrelated. The authors could demonstrate that temporal and spatial speckles have the same statistical properties and that the contrast can be calculated by a time-space average [91]. In addition, spatial resolution can be easily increased at the expense of temporal resolution and vice versa, which allows one to adapt the data processing to the speciﬁc application. 8.6.5 Dynamic contrast analysis There exist a number of practically important experimental situations when the scattering ﬁelds of interest are mixed with the non-ﬂuctuating electric ﬁelds scattered from static regions. A prominent example is a measurement with a strong static background or dynamical imaging of a volume covered by a static scattering surface. This is the case, for example, for a layered medium with dynamics localized in a buried layer (neuroimaging through the skull). These cases present a problem for the LASCA method [87, 91], but slight modiﬁcation of the LASCA technique allows one to correct the contrast signal for the static background. The original LASCA algorithm performs the spatial averaging over a number of speckle and ensures the multi-speckles averaging of the signal. If the static (nonﬂuctuating) component ρ is present in the detected signal, the measured ACF is deﬁned by Eq. (8.27). The resulting measured contrast value Km can be found by substituting Eq. (8.27) into Eq. (8.44): 2 Km

2β = T

T 2

[(1 − ρ) g1d (τ ) + ρ] (1 − τ /T ) dτ 0 2

= β (1 − ρ) K22 d + 2βρ (1 − ρ) K12 d + βρ2 , where we introduce the following quantities: 2 T 2 g (τ ) (1 − τ /T ) dτ K22 d = T 0 1d 2 T g1 d (τ ) (1 − τ /T ) dτ. K12 d = T 0

(8.48)

(8.49)

K2 d is the contrast of the dynamic part of the scattered intensity and K1 d is the contrast of dynamic part of the scattered ﬁeld. The two terms deﬁne a mixed dynamic contrast: 2

2 2 2 K12 d = β (1 − ρ) K2 d + 2βρ (1 − ρ) K1 d ,

(8.50)

2 thus the measured contrast Km is composed of K12 d and the static part: 2 2 2 Km = K12 d + βρ .

(8.51)

Contributions of the static and dynamic components to the contrast cannot be distinguished in a traditional LASCA experiment[82, 84, 92]. Nevertheless, the quantity ρ can be obtained with an additional processing step which does not involve any modiﬁcations in setup and data collection procedure. Usually in an LSI experiment the exposure time T is large compared to the relaxation time related to

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blood ﬂow τ0 , and sequential frames are acquired with a period Δt which is larger than T , hence τ0 < T < Δt. Two frames separated by maximal accessible lag time tmax Δt can be considered uncorrelated with respect to the dynamic component of interest, as the ratio of static intensity ρ is expected to remain the same. From Eq. 8.25 we ﬁnd g1 (tmax ) = ρ and the ICF as deﬁned in Eq. (8.27) reduces to g2 (tmax ) − 1 = βρ2 . The ρ can be determined from two separated frames using the multi-speckle averaged correlation function: 1 ms 1 I (0) I (tmax )ms 2 ρ = [g2 (tmax ) − 1] = −1 . (8.52) β β I2ms Finally, with the knowledge of ρ and a model for the short time relaxation a quantitative interpretation of the measured contrast values is possible. 8.6.6 Simpliﬁed speckle contrast calculations 8.6.6.1 Modiﬁed laser speckle imaging This approach is essentially a temporal analog of the original LASCA method, motivated by the desire to improve spatial resolution of the speckle images (see Fig. 8.13). The implementation is quite obvious: instead of calculating contrast in space one can perform the same type of processing with a stack of sequentially obtained images without utilization of spatial statistics[93–96]: Kt2 =

IT2 t − 1. I2t

(8.53)

This method provides the images with improved spatial resolution and also qualitatively suppresses the impact of static component, produced by the skull, for example[96]. The suppression is facilitated by the fact that the temporal averaging

(a)

(b)

Fig. 8.13. Pseudo-color-coded images of internal dynamics in the open rat brain. Spatial resolution of two laser speckle imaging methods is compared: (a) spatial contrast (b) temporal contrast. (Images are from Bruno Weber, Zurich University Hospital.)

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of correlation function leads to the loss of static part ρ2 in the measured contrast 2 . In this case the dynamic range of the contrast variations is covered by the Km dynamic contrast components: Kt2 = β 1 − ρ2 K22 d + 2 βρ (1 − ρ) K12 d . (8.54) 2 The Kt2 is similar to√K12 d deﬁned by Eq. (8.50), except for the multiplication of the second term by β instead of β. Along with obvious similarity with dynamic contrast the temporal contrast is not multi-speckle averaged in modiﬁed LSI and cannot guarantee proper averaging of the non-ergodic signal. Temporal contrast should be used for qualitative imaging when the maximal spatial resolution is crucial, meanwhile for the quantitative mapping the spatial statistics should be included in the processing to ensure the multi-speckle averaging and to suppress speckle noise.

8.6.6.2 Speckle deviation analysis The eﬀect of static background can be signiﬁcantly reduced in the LSI signal with structure function processing of time-integrated intensity. According to Eq. (8.41) the structure function is closely related to the correlation function and according to Eq. (8.45) the intercept of the latter deﬁnes the contrast. If the lag time τ used for the ISF dT2 (τ ) calculation is much larger than the correlation time of the dynamic signal under study (τ τc ) the ISF can be written in the following way: ! " ! 2 " 2 − βρ2 = 2K12 (8.55) dT2 (τ ) = 2 g2T (0) − g2T (τ ) = 2 Km d. Eq. (8.55) indicates that the diﬀerence between two uncorrelated time-integrated 2 speckle images is twice the square of the mixed dynamic contrast K12 d . This provides an easy way to estimate the dynamic contrast and suppress the static component. Since multi-speckle averaging is used in this scheme the dynamic contrast does not suﬀer from the multiplicative noise associated with modiﬁed laser speckle analysis, nevertheless it relies on the assumption that the correlation time of the process under study cannot exceed the lag time τ used for analysis. Without multispeckle averaging the contrast values cannot be unambiguously converted to the correlation times τc for the signal with a static component. In general, the knowledge of ρ is required for the data interpretation and estimation of τc .

8.7 Summary The current article presents a brief overview of some recent developments in dynamic light scattering methods with a particular emphasis on advanced processing schemes allowing for faster and more reliable signal averaging. It also demonstrates that improved averaging schemes make possible dynamic light scattering measurements with temporal and spatial resolution. Thanks to the advances in the image sensors and fast data processing hardware, sophisticated analysis algorithms can be implemented in real time. Nevertheless, one should always mind the limitations of the newly introduced algorithms and avoid losing the quantitative information.

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If the signal averaging is performed by the analysis of temporal ﬂuctuations, the non-ergodicity problem might arise. This can potentially lead to a signiﬁcant error in the estimated statistical characteristics. The artifacts of temporal averaging, however, can be reduced if the multi-speckle averaging is performed with additional spatial speckle statistics or with active speckle sampling as discussed in the paper.

Acknowledgments The authors acknowledge discussions with Andreas V¨ olker, Bruno Weber, Mathias Wyss, Alfred Buck. P.Z. is grateful to Irina Lebedeva for help with the manuscript preparation. Financial support from the Swiss National Science Foundation (project no. 200020-111824) is gratefully acknowledged.

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9 Static and dynamic light scattering by aerosols in a controlled environment R.P. Singh

9.1 Introduction Aerosols are deﬁned as particles suspended in air or any other gaseous medium. These particles may be solid (dust, smoke, fume etc.) or liquid (mist, fog etc.). They range in size from 0.01 micrometers to several tens of micrometers, for example cigarette smoke particles are in the middle of this size range and typical cloud drops are 10 or more micrometers in diameter. Because of their particle nature they are also called ‘particulate matter’ and in literature one often encounters terms like PM2.5 and PM10 , which are used for particles smaller than 2.5 micrometers and 10 micrometers respectively. These particles can be classiﬁed under two headings: (i) dispersion aerosols, formed by atomization of solids and liquids, and (ii) condensation aerosols, formed by condensation of supersaturated vapor or due to chemical reaction in the gas phase. The particles, which serve as nuclei for condensation are termed as condensation nuclei or cloud condensation nuclei. All aerosol particles are formed either by mechanical processes or by condensation of gases and vapors. Mechanical processes producing aerosols may be natural, like windblown dust, sea salt spray, and volcanoes, or man-made, such as industrial activities, burning of fossil fuels, and the alteration of natural surface cover. The particles produced by mechanical processes in general are coarse with a size of a few micrometers while ﬁne submicrometer particles are formed by condensation. Aerosol particles larger than about 1 micrometer in size are produced by windblown dust and sea salt from sea spray and bursting bubbles. Aerosols smaller than 1 micrometer are mostly formed by condensation of sulfur dioxide (SO2) gas to sulfate particles and by formation of soot and smoke during burning processes. The books by Friedlander (1977), Dennis (1975), Reist (1986, 1993) and Seinfeld (1986) are highly recommended for details. Even after decades of research into the ﬁeld still there is much to learn about the way aerosols aﬀect regional and global climate. When it comes to quantifying the relative impacts of natural aerosols and those that are man-made on the climate or on human health, we cannot yet be very accurate. Also, we do not know the regions of the planet where amount of atmospheric aerosols is increasing, decreasing or remaining approximately constant. Several types of aerosols have been identiﬁed and models have been developed about the amount of aerosols to be found in diﬀerent seasons and locations (Forster et al., 2007, Third Assessment

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Report, IPCC). However, key details about the amount and properties of aerosols to calculate their eﬀect on surface temperatures are still far from complete. 9.1.1 Atmospheric particles and their dispersion Size and shape of the dispersed particles determine the properties and behavior of aerosols. In general, aerosols are polydisperse, i.e., the particles have diﬀerent sizes. Particles having the same size are termed monodisperse aerosols. Particles constituting the aerosols may be in the size range of 0.1–100 micrometers. Particles with sizes greater than 100 micrometers settle down fast and therefore are not of much interest. Particles of sizes >10 micrometers aﬀect the visibility and consequently the biological energy conversion (e.g. photosynthesis). Particles <5– 10 micrometers size can aﬀect health and are of physiological interest (Cohen et al., 1979; Ferron, 1994). These particles, once formed, are mixed and transported by atmospheric motions and are primarily removed by cloud and precipitation processes. Since particle size is the single most important physical parameter that determines the dispersion of aerosols and hence their eﬀect on health, visibility and climate, a great deal of attention has been paid to studies pertaining to particle size measurements (Hinds, 1982; Barth and co-workers, 1985, 1987, 1989, 1991, 1993, 1995; Etzler and co-workers, 1995, 1997, 2004; Jillavenkatesa et al., 2001; Spurny, 1986). Several studies have shown that there is a strong association between the airﬂow pattern in the buildings governing the dispersion of airborne bacteria and the spread of infectious diseases (Li et al., 2007). Besides bio-aerosols, smoke particles can increase respiratory irritation and aggravation of existing respiratory or cardiovascular disease. For example, prolonged exposure to asbestos ﬁber increases the risk of lung cancer and mining aerosols can be carcinogenic or mutagenic. In the present global scenario one cannot deny the possibility of terrorist attacks via airborne release of chemical or biological aerosols and the extent of damage can be found out only by studying their dispersion. Looking at various possibilities a number of dispersion models have been developed taking into account size factor, dynamics, chemical composition for aerosols and also the topography of the immediate environment which can be the human respiratory system, buildings, the urban environment, forest or open terrain (Chen et al., 2006; Nardell et al., 1986; Mangili and Gendreau, 2005; Wallace, 1996, Mistra, 1980; Egan, 1984; Pielke, 1984; Yamada, 1985; Enger, 1986, 1990a,b,c). 9.1.2 Particle size Before we go for particle sizing using light scattering or other methods it would be worthwhile to understand the meaning of particle size. Suppose one wants to tell if the particle size has increased or decreased with increase in temperature or decrease in humidity, one needs to give a single number, which is possible only for a spherical particle that can be deﬁned by a single number – the diameter of the sphere. But particles would not be spherical in general and in such cases one resorts to the concept of an equivalent sphere. One measures some property of the particle and assumes this refers to a sphere made up of the same material. For example, the weight of the particle, which is a unique number, assigned to a sphere

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of same material density could be used to get the size of the particle. However, one should have no doubt that this is the size of a sphere with the same weight as the particle, and if someone else considers a sphere with the same surface area as the particle, the size may not come out to be same. Similarly, one can have a sphere of the same minimum length, a sphere of the same maximum length, a sphere of the same sedimentation rate and sphere passing through the same sieve. The determination of particle size on the same sample by all these methods and others which are not mentioned will yield diﬀerent results for mean size, modal size, and size distribution in most cases. Thus several diﬀerent sizes could be obtained for a single, nonspherical particle due to various methods of measurement. In fact, in most of the cases particle size is obtained by measurement of characteristics other than the complete geometry. These characteristics are associated with a physical phenomenon in which the particle is involved, for example, sedimentation velocity of the particle in a ﬂuid or pattern of light scattered by the particle. The observed sedimentation velocity and the scattered light can be used to ﬁnd out the geometry or the size of the particle using an appropriate theory or model. However, if the particles under test are of irregular shape, then most likely the results on size will be diﬀerent because two particles which settle with the same velocity can scatter light diﬀerently. The mathematical complexities introduced by nonspherical geometry make it diﬃcult to develop models for such particles and also prevent assessment of the extent of error. The eﬀect may be negligible or may be severe, depending on the sizing technique and the shape of the particles, and often cannot be predicted reliably. 9.1.3 Diﬀerent methods for particle sizing The interest in particle size emanates from diﬀerent areas of research and commercial applications – color appeal of a paint or an emulsion, particle size of the powdered drug or the size of inhaled aerosols in pharmaceutical preparations, dispersion of radioactive particles for the nuclear power industry, health hazard of atmospheric aerosols, atmospheric chemistry, cloud formation, and of course the radiation budget and climate change – it could be any of these. There are various particle sizing techniques, which could be employed depending on the application. There is no single technique that can claim to be superior to other techniques in all the applications (Allen, 1997; McMurry, 2000). Some of the popular techniques are sieves, sedimentation, electrozone sensing, microscopy and imaging, and light scattering. Sieves are cheap and readily available. They are suited for large particles which can be separated into some size bands, usually four to ﬁve, and are thus a low-resolution technique. Sedimentation uses Stoke’s law for the settling velocity of a spherical particle in a ﬂuid medium. This is a traditional method and employs equipment as simple as the Andreason pipette or as complex as centrifuges. The technique has a limited range: one faces problems below 2 μm and above 50 μm. Electrozone sensing measures the change in resistivity of the electrolyte displaced by the particle passing through an aperture. This change in resistivity is used to ﬁnd out the size of the particle. Porous particles and dense materials create problems in the use of this technique. Microscopy and imaging is an excellent technique and allows direct examination of the particles in question. However, since relatively

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few particles are examined, there is a real danger of unrepresentative sampling. Techniques based on light scattering measurements, which are often nondestructive and nonperturbative to the medium investigated, have become most popular for particle size instrumentation and will be discussed in detail in the next section.

9.2 Optical Methods The use of light scattering by particles to infer their properties is a well established and important technique. As a diagnostic technique it has become common in research and industry. Combustion of fuel sprays, production and characterization of pharmaceuticals, biological cell analysis, smoke formation and cloud physics represent a few areas in which optical particle sizing methods have become indispensable diagnostic tools. Prompted by many important applications, researchers have directed considerable attention towards optical particle sizing (van de Hulst, 1957, 1981; Penndorf, 1958; Chu et al., 1987; Gulari et al., 1979, 1985; Chowdhury et al., 1984; King et al., 1983; Gouesbet and Grehan, 1988). An early historical perspective on light scattering can be obtained in the book by Kerker (1969) while Bohren and Huﬀman (1983) point out important books and articles on light scattering under ‘notes and comments’ in the ﬁrst chapter of their book. Most of these deal with classical light scattering (CLS) that is time-averaged or static light scattering. Laser diﬀraction and Mie scattering come into this class of scattering. However, there is another class of scattering which measures the time correlation of scattered light, the change in scattering with time, and rightly called dynamic light scattering (DLS) or photon correlation spectroscopy (PCS). Over the years PCS has become a widely used particle sizing tool in the submicrometer range. We will discuss each of them one by one. 9.2.1 Static light scattering The scientiﬁc study of light scattering may be said to have commenced with the work of Tyndall (1869a,b) on aerosols. It was in 1871 that Lord Rayleigh ﬁrst explained the observed color and polarization of the sunlight scattered in the atmosphere even though he considered light as mechanical vibrations and based his treatment on the old elastic theory of light. In deriving his expressions he used density ﬂuctuations. Rayleigh’s most inventive paper on light scattering was published in 1881. Here, he used Maxwell’s equations in deriving the volume integral for the scattered ﬁeld as observed from Green’s theorem. This integral can be represented as an expansion in Δε, the diﬀerence between the dielectric constant of the particle and the medium. The solution is perfectly general with no need to maintain any restriction either on the particle size or shape or on Δε provided that the indicated integrations which are far from trivial can be performed. The restrictions on size and shape could be relaxed when Δε is not too large. Later in 1899 Rayleigh deduced that for non-interacting, non-absorbing and optically isotropic particles having sizes very small compared with the wavelength of the incident light amount of the scattering should be proportional to the reciprocal fourth power of the wavelength λ, now known as Rayleigh’s law. Rayleigh’s approach has since

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been elaborated to cover absorbing and anisotropic particles having sizes comparable to the wavelength of the incident radiation. Up to this point Rayleigh has used the Green’s theorem method to express the scattered ﬁeld as the volume integral over the ﬁeld within the particle. This is a completely general formulation, but its solution by expansion into a power series in Δε involves great computational diﬃculty except for the leading term which gives the Rayleigh–Gans–Debye–Born approximation. In the last part of the 1881 paper Rayleigh returns to scattering by a cylinder which he formulates as a boundary-value problem. There are no limitations on particle size, dielectric constant or magnetic permeability. The ﬁrst application of this boundary-value method to spheres was by L. Lorenz in 1890, and is now generically termed Mie scattering following the detailed discussions of its applicability to the colors of gold sols by G. Mie (1908). A large number of more or less independent boundary-value treatments of the sphere of arbitrary size were made by Lorenz (1890), Thomson (1893a,b), Love (1899), Walker (1900), Bateman (1915) and Bromwich (1919, 1920). Later Oster (1948) reviewed the scattering of light for application to chemistry and van de Hulst (1957) gave a good treatment for light scattering by small particles. After this a number of calculations have been performed adding the refractive indices and absorption coeﬃcients appropriate to diﬀerent aerosols. For the latest developments one can go through the book by Mishchenko et al. (2002). The scattered intensity distribution for a spherical particle of radius ‘r’ depends on size parameter α (= 2πr/λ, λ being the wavelength of incident light). For spherical particles relevant to atmospheric scattering like dust, Penndorf (1960), Deirmendjian et al. (1961) made the scattering calculations. Penndorf (1957–1962) did extensive work on Mie scattering and described an approximate method of calculating the total scattering coeﬃcient for transparent spheres, which is valid for all particle sizes when refractive index n ≤ 2. Ellison and Peetz (1959) calculated the intensity of scattered light in the forward direction using an approximate method applicable to large particles, good enough to be treated by geometrical optics. Measurements of 90◦ scattering by plastic latex spheres using polarized light were made by Heller and Tabibian (1962). They evaluated the depolarization ratio. The results obtained using Mie theory agreed well with direct size measurements by electron microscopy. Hodkinson (1966) has given a good account of particle sizing by means of a forward scattering lobe and listed all the principal published Mie theory computations of the angular distributions of the scattered intensity for medium and large transparent and moderately absorbing spheres. The list includes refractive index from 1.05 to 2.105 and the size parameter α from 0.1 to 159 with varying increments. Experiments on suspended particles (polystyrene latex in water) were performed by Bateman et al. (1959). They measured speciﬁc extinction using a spectrophotometer and calculated the particle size from its variation with the wavelength of light. Angular distribution function based on Mie coeﬃcients were deﬁned by Chu and Churchill (1955b) and Clarke et al. (1957), which were easier to handle than the usual form involving Mie functions. Angular distribution up to α = 30 can be calculated with these functions. Intensity of light scattered by aerosol droplet of diameters between 1.4–3 μm suspended freely between two charged condenser plates have also been measured by Gucker and his associates (1960, 1961). For an-

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gles between 40◦ and 140◦ from the forward direction, there was good agreement with theoretical intensities calculated from Mie theory. If the light scattered by one particle is intercepted by other particles, it is called multiple scattering (Chandrasekhar, 1960; Woodward, 1964). A theory for this type of scattering was developed by Churchill and his associates (Chu and Churchill 1955a,b; Churchill et al., 1960). With dense suspension of monodisperse latex particles (0.8–1.7 μm) satisfactory results were obtained experimentally. However, multiple light scattering involves a lot of complexities which can lead to erroneous results in many cases (Czerwinski et al., 2001). Kokhanovsky has done a good amount of work on the subject (2001, 2002). For a detailed treatment one can consult the book by Mishchenko et al. (2006). In the above mentioned studies the particles were assumed to be spherical and of uniform size. However, many naturally occurring aerosols are of irregular shape and variable size. Light scattering by such irregular-shaped particles has been dealt in a monograph by Mishchenko et al. (2000). Stevenson et al. (1961) considered the eﬀect of the particles being unequal in size. It was found that a unimodal and positively skewed size distribution can be deﬁned from measurements of the depolarization ratio. It would not be out of place to mention that Ghosh et al. (2003a,b, 2004, 2006) have extensively used the depolarization ratio to study turbid media. Methods for ﬁnding the size distribution from turbidity measurements at diﬀerent wavelengths were developed by Wallach et al. (1961) and Curcio (1961). Chin et al. (1955, 1955a,b) gave a method of ﬁnding size distribution based on the angular variation of the scattered light at very small angles. Experimental data on scattering of fogs has been summarized by Spencer (1960). He developed a scattering function which is valid over a wide range of conditions from thin to dense fogs. At about the same time Went (1960) showed that it is scattering by aerosols of sizes 0.1 μm or less consisting of aggregates of condensed molecules which is responsible for blue atmospheric haze. Experiments were carried out with quartz dust suspended in liquid. Ellison (1957) found that the forward scattered light for the quartz dust was approximately the same as scattering calculated for spheres of the same refractive index and size distribution. Due to the assumption of sphericity and transparency of quartz, his results could not be applied immediately to opaque and nonspherical airborne dust. Later Berry (1962) studied the scattering by non-spherical particles, like silver bromide crystals (0.1–1 μm). A number of studies followed to determine the sizes of nonspherical particles based on Mie theory: Holland and Gagne (1970), Pinnick et al. (1976), Ch´ ylek (1977a), Zerull et al. (1977), Perry et al. (1978), and Janzen (1980). Jaggard et al. (1981) compared the experimental data for latex spheres of regular shape and ammonium sulfate and soil dust particles of irregular shape with theoretical Mie calculations. Mie theory was in complete agreement for latex spheres, also applicable to small and resonant sized ammonium sulfate particles – not spherical but regular in shape. They concluded that as the particles become larger or more irregular, the diﬀerence between Mie theory calculations and experimental data could go on increasing. It must be noted that with the increase in particle size, Rayleigh’s λ−4 law ceases to be valid, dispersion becomes less selective of λ, and Mie scattering theory, which is more general and contains Rayleigh scattering and geometrical optics as limiting cases, should be used (van de Hulst, 1957). However, Mie scattering works basically for

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spherical particles and for suﬃciently large particles the dispersion of radiation approaches a 1/λ dependence leading to diﬀuse reﬂection. Scattering functions, i.e., the scattered intensity as a function of angle, have been obtained to include the eﬀect of nonspherical shape of the particles (Koepke and Hess, 1988). It is observed that with increase in wavelength the inﬂuence of nonspherical particles decreases. The methods for simulating the scattering of light by various nonspherical shapes have rapidly evolved over the last two decades. However, the majority of aerosol remote-sensing retrievals still rely on the Mie theory because retrievals accounting for particle nonsphericity are not as well deﬁned methodologically. Dubovik et al. (2002) have proposed a method for the retrieval of the optical properties of nonspherical aerosols based on the model of a shape mixture of randomly oriented polydisperse spheroids that shows a signiﬁcant improvement in dust-particle phase functions, size distributions, and refractive indices as compared to Mie-based retrievals. It has been observed that the amount of large particles can be underestimated when nonspherical particles are substituted with Mie spheres (Kocifaj and Horvath, 2005). 9.2.2 Laser diﬀraction Laser diﬀraction, also called Low Angle Laser Light Scattering (LALLS), has become one of the most widely used techniques in many industries for particle size analysis (Kippax, 2005). It relies on the fact that particles passing through a laser beam will scatter light at an angle that is inversely proportional to their size, the scattered intensity being proportional to the particle volume. This means larger particles will scatter at low angles with high intensity while smaller ones will scatter with low intensity but at high angles. The instruments based on laser diﬀraction use this fact to determine the size of particles. The dynamic range of the instrument is directly related to the angular range of the scattering measurement that varies from 0.02 degrees to 140 degrees and beyond – no more Low Angle Laser Light Scattering! The wavelength of light used for the measurement is also important: smaller wavelengths like blue laser light provide improved sensitivity to submicrometer particles. For calculating particle size distribution using laser diﬀraction, the sample’s scattering pattern is compared with a suitable theoretical model. Older instruments and some existing instruments rely on Fraunhofer approximation (Kouzelis et al., 1987; Ma et al., 2000), which assumes: particles are much larger than the wavelength of light used; the particles being measured are opaque and scatter light at small angles; all sizes of particle scatter with equal eﬃciency. Consequently, it is more suited to large particles and will give an incorrect result when applied to ﬁne particles. However, the latest instruments use full Mie theory (Veihelmann et al., 2006) that completely solves the equations for interaction of light with the particle and provides a more rigorous solution for the calculation of particle size distributions from light scattering data. It calculates scattering intensities for all particles, small or large, transparent or opaque. This allows correct assessment over a large size range (0.02–2000 μm). Mie theory assumes the volume of the particle, as opposed to Fraunhofer that gives the projected area. The penalty for the accuracy over a wide range of sizes is that the refractive indices for the material and the medium must be known since Mie theory allows for primary scattering from the surface of the particle, with the intensity predicted by

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the diﬀerence of refractive indices between the particle and the dispersion medium. However, in most cases this does not pose a problem as the values of refractive indices are either known or can be measured, which is not required for Fraunhofer approximation. 9.2.3 Dynamic light scattering Dynamic light scattering (DLS), also known as quasi-elastic light scattering (QELS) or photon correlation spectroscopy (PCS), is a convenient and frequently used technique to determine the size and size distribution of particles suspended in a ﬂuid. Since the particles perform Brownian motion, the total scattered ﬁeld at any point in space resulting from superposition of all wavelets scattered from individual particles ﬂuctuate with time. The analysis of these ﬂuctuations yields the lateral diffusion coeﬃcient of the particles and by means of the Stokes–Einstein relation, the hydrodynamic radius for the case of the spherical particles. The term quasi-elastic in QELS refers to the fact that the frequency of the scattered light is slightly shifted due to particle motion. The small frequency broadening in the light scattered from dilute solutions of macromolecules or particles is used to derive information about diﬀusion and also to study critical phenomena (Pike and Stanely, 1981; Kramer and Frederick, 1971, 1972). The experimental realization of these ideas depends not only on directly measuring the small frequency changes involved by use of monochromators but the use of ‘optical beating’ technique (Bertolotti et al., 1967; DiPorto et al., 1969; Jakeman and Pike, 1968a). It is convenient to think of scattered light ﬁelds from each beam as suﬀering a constant frequency shift due to a constant rate of phase change of the scattered light seen by the detector while the particle moves through the scattering volume. Since the scattering volume is ﬁnite, the number of Doppler beat cycles measured is limited which gives rise to a broadening of the spectrum that is usually referred to as the Doppler ambiguity. Alternatively, the particles give rise to a small fraction of a Doppler cycle before they change direction by collision and therefore the concept of a frequency shift is not directly applicable in such cases. In this case, as in the case of Rayleigh scattering, it is perhaps useful to think of the scattered light as a random diﬀraction pattern varying on a time scale governed by the internal changes occurring in the scattering volume. In all cases, therefore, the detector measures the changes in the light intensity, either as regular Doppler beats or as randomly varying diﬀraction lobes. One can relate these ﬂuctuations either implicitly or explicitly to the spectrum of the scattered light arising from the temporal behavior of the sample. That is why the technique is known as light beating or intensity ﬂuctuation spectroscopy or, when the scattered light ﬂux is measured by processing individual photons, photon correlation spectroscopy (PCS). PCS is characterized by a ‘digital’ technique for measuring intensity ﬂuctuations in which the number of photons arriving at the detector at the set time interval is repeatedly counted and its time autocorrelation function (ACF) computed. Digital correlation techniques greatly improve the signal-to-noise ratio. There are a number of books and review articles that deal with PCS and its applications (Jacobs et al., 1975; Berne and Pecora, 1976; Cummins and Pike, 1974, 1977; Chu, 1970, 1974; Pecora, 1985; Schmitz, 1990; Brown, 1993). In condensed media or whenever the scatterers are close to one another, a detailed computation of the induced electromagnetic ﬁeld surrounding a parti-

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cle becomes very complex because inter-particle interactions have to be taken into account. Einstein (1910) was able to bypass the above diﬃculties inherent in Rayleigh’s analysis as applied to a collection of interacting particles. He assumed that local density ﬂuctuations in the neighboring volume elements could be independent of one another and carried out a quantitative calculation of the mean square amplitude of those ﬂuctuations from a statistical mechanics approach. Although Einstein’s theory was able to explain the scattering from pure liquids and to predict the enormous increase in the scattering as the liquid gas critical point was approached, the so-called critical opalescence, it failed to account for angular dissymmetry of the strong scattered intensity in critically opalescent systems. This idea of local density ﬂuctuations and its connection with diﬀusion became the key element in extracting information about the system using the PCS. Later Brillouin showed that ﬂuctuations that propagated with a velocity give rise to ‘doublets’ that were frequency shifted by an amount proportional to ±v from the frequency of the incident light (1914, 1922). Gross (1930, 1932) experimentally observed this doublet and, in addition, a central peak of unshifted frequency. Even though there was a wealth of information to be obtained from the spectral proﬁles of scattered light, the intrinsic linewidth of the radiation was too broad to allow meaningful information contained in the relatively small frequency shifts to be gained except under most unusual circumstances. This situation was changed in the 1960s with the invention of laser. Pecora (1964) showed that the frequency proﬁle of a scattered electric ﬁeld had broadened by the diﬀusion processes of macromolecules. The ﬁrst experimental report using lasers as the source of incident radiation for the study of macromolecular solutions (polystyrene) was by Cummins et al. (1964) while that for pure ﬂuids near the critical point was by Ford and Benedek (1965). In 1964, the ﬁrst observation of the Brillouin–Mandelstham lines were reported using lasers coupled with ultra-high-resolution Fabry-Perot etalons (Chiao et al., 1964, Chiao and Stoicheﬀ, 1964) and grating spectrometers (Benedek et al., 1964). However, even the highest resolving power Fabry-Perot etalons fall short of providing sufﬁcient resolution to measure the narrowing of the central component in critically opalescent systems. Usually the frequency shift produced by scatterers is very small in comparison to the frequency of light. The resulting spectral linewidth can be as narrow as few kilohertz and the resolution required to decipher it is 1012 , which calls for high-resolution technique while a conventional optical spectroscopy fails above a resolution of 107 . The optical mixing spectrometers were devised to measure very narrow line widths (Ford and Benedek, 1965; Cummins et al., 1964, DuBois and Berge, 1971). The nonlinear detector or mixer in an optical mixing spectrometer is a photoelectric device such as a photomultiplier tube or photodiode, which produces a photocurrent proportional to the square of the total electric ﬁeld falling on the photosensitive surface of the device. Optical mixing spectroscopy utilized a wave (spectrum) analyzer in which a scanning electrical ﬁlter was used to analyze the spectrum of photocurrent ﬂuctuations from a square law detector, such as a photomultiplier tube. The equivalent information, namely the photocurrent correlation function which is a Fourier transform of the power spectrum, was measured using a signal correlator. Unfortunately, until the end of 1960s, most commercial signal correlators took only analog signals even though the internal processes of

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these correlators were often ‘digital’ (Hanbury Brown and Twiss, 1956). The most promising technique which recognized and took advantages of the digital nature of photon statistics of scattered laser light was that of photocount autocorrelation (Arecchi et al., 1966; Cantrell, 1970). Pike and co-workers did pioneering work in this ﬁeld (Jakeman and Pike, 1968a,b, 1969; Jakeman et al., 1968a,b, 1970a,b; Ford et al., 1971) that was followed by Chopra and Mandel (1972). Asch and Ford (1973) constructed a correlator approaching ideality by increasing the bit-multiplication capacity of each channel. When combined with lasers, photoelectric mixing is particularly suitable for quasi-elastic light scattering (QELS) resulting from various time-dependent non-propagating local thermodynamic ﬂuctuations. Several closely related experimental approaches have been developed around two schemes of detection: heterodyne detection and homodyne detection. Early applications of DLS methods focused on the successful determination of molecular weight and shapes. The credibility of this technique was accepted by the scientiﬁc community as a whole in the mid-1970s. As a large group of researchers began to use these methods, new information about speciﬁc systems began to emerge. Technological advances in the instrumentation led to more precise and accurate data. Minor discrepancies between theory and experiments soon became apparent. These discrepancies led to more sophisticated theories and methods of analysis, and to the development of new techniques for preparation and handling of samples. Following the introduction of light-beating spectroscopy by Benedek and clipped photocount autocorrelation by Jakeman and Pike several extensive studies on the statistical accuracy of spectral linewidth in optical mixing spectroscopy were reported (Jakeman, Pike and co-workers, 1970c, 1971a,c; Bertolotti et al., 1967; Cummins and Swinney, 1970; Chen et al., 1972, 1973; Hughes et al., 1973). Benedek (1968) discussed the experimental problem of using a detector area much larger than the coherence area of light. Haus (1969), Cummins and Swinney (1970), Degiorgio and Lastovka (1971) presented the statistical errors inherent in intensity correlation spectroscopy owing to the stochastic nature of both, the scattering and the photoemission processes. They determined the statistical errors on the opticalintensity correlation function as measured by two photocounting digital correlator models and on the intensity spectrum as measured by a ‘self-beat’ optical mixing spectrometer. From these errors and a generalized least-mean-squares ﬁtting procedure they calculated the uncertainty on the measured correlation time (linewidth) for the case of a Gaussian optical ﬁeld with an exponential intensity correlation function (a Lorentzian spectrum). Independently Jakeman et al. (1971b) investigated the statistical errors (caused by ﬁnite duration of electronic resolving time and ﬁnite receiving apertures) in the intensity autocorrelation function and in the spectral linewidth of Gaussian–Lorentzian light (Lorentzian broadened light with Gaussian statistics). Analysis of Gaussian light by clipped photocount autocorrelation has also been extended to include the eﬀect of ﬁnite duration sampling intervals and incomplete spatial coherence by Koppel (1971). The details of these lengthy developments can best be obtained by reading the original papers, especially the work of Degiorgio and Lastovka (1971) on intensity correlation spectroscopy and of Jakeman et al. (1971b) on statistical accuracy in digital autocorrelation of photoncounting ﬂuctuations. While the analysis of data using three variables generally leads to larger errors, the two-parameter approach of Pike and his co-workers (1981)

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for single exponential decays represents a more realistic treatment of this complex problem. The ﬁrst major variation in light scattering technique occurred in 1971 when Ware and Flygare (1971, 1972) reported dynamic laser light scattering (DLS) on Bovine Serum Albumin (BSA) in the presence of static electric ﬁeld E0 . The eﬀect of E0 is to superimpose a constant drift velocity, proportional to the electrophoretic mobility of the species, on the random Brownian motion of the charged particles. The resulting spectral density is thus composed of peaks that are Doppler-shifted from central position. This technique is referred to as Doppler-shifted spectroscopy (DSS) or electrophoretic light scattering (ELS). Experiments such as laser light near threshold (Jakeman et al., 1970b), motions of particles in turbulent ﬂuids (Diporto et al., 1969; Bourke et al., 1970), and light scattering from small number of particles undergoing Brownian motions (Schaefer and Berne, 1972; Schaefer and Pusey, 1972; Adrian, 1972) follow non-Gaussian statistics. Schaefer and Berne (1972) showed that non-Gaussian concentration ﬂuctuations arise at low concentrations because the particles in the scattering volume have a Poisson distribution. In this case, the homodyne intensity autocorrelation function decays on two widely diﬀerent time scales. The fast process is the usual Brownian motion of the individual particles, while the slow process is related to the time-dependence of the ﬂuctuations in the total number of particles in the scattering volume. The slow process does not appear in the heterodyne correlation function. Thus, the homodyne technique can be used to probe fundamental properties of colloid statistics. In many cases the standard procedure of particle size determination by eliminating inter-particle interactions with dilution is undesirable due to change of particle conformation upon dilution or a need to identify the nature of interaction and particle motions under the inﬂuence of neighboring particles. A way of extracting size information from interacting particles is of practical importance. Pusey and Tough (1982) brieﬂy reviewed the application of PCS to study of interacting Brownian particle dynamics. The importance of structure relaxation time scale, as well as the range of wave vector, to interpretation of results was emphasized. Tsang and Tang (1982) treated the problem as radial diﬀusion in an eﬀective potential of the mean force. Medine-Moyola (1982) presented a theory based on description of the coupled diﬀusion of macromolecules. The short-time and the long-time features of correlation function could be predicted. The wave vector dependence of the effective diﬀusion coeﬃcient from the ﬁrst cumulant was examined theoretically by Wills (1981) and Boon and Yip (1982). Lakkerkerker and Dhont (1984) discussed and compared two diﬀerent ways of calculating the self-diﬀusion coeﬃcient of the interacting Brownian particles as a function of concentration. By means of an interacting hard sphere model without hydrodynamic interaction, the two approaches – the so-called memory eﬀect and relaxation eﬀect – yielded identical results. The diﬃcult problem of DLS by concentrated suspensions of polydisperse hard spheres was explored theoretically by Pusey et al. (1982). The authors concluded that even for narrow distributions the correlation function should be composed of two independent modes with well separated decay times when the volume fractions were above 0.15. The fast and the slow modes were related to the collective diﬀusion and the self-diﬀusion of the hard spheres. The applications and testing of these theo-

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retical endeavors were carried out in micro-emulsions, latex dispersions, colloidal suspensions and protein solutions (Bohidar et al., 1980; Bohidar and Chopra, 1981; Kaler and Prager, 1982; Cintre, 1986). Konak et al. (1991) have used a nonlinear regularization method for an inverse Laplace transformation called REPES developed by group member Jakes for analyzing the autocorrelation curve obtained as a result of multiple scattering by monodisperse polystyrene latex particles. Itoh and Takahasi (1991) have considered the eﬀects of non-Gaussian concentration ﬂuctuation in real-time PCS for measurement of aerosol particles by dynamic light scattering. All synthetic macromolecules are more or less polydisperse. The eﬀect of polydispersity on the spectrum of light scattered by macromolecules in solution has been examined by Dubin et al. (1967). They considered a mixture of equal numbers of molecules of the same molecular weight but of diﬀerent diﬀusion coeﬃcients. His results showed that only for large macromolecules with quite broad molecular weight distributions, can we consider light scattering as a tool to learn something about polydispersity. For narrower molecular weight distributions, we simply obtain an averaged diﬀusion coeﬃcient. As a rule very precise measurements are required to examine the eﬀects of polydispersity. Thompson (1971a,b) has studied the bimodal distribution with sharp peaks using suspensions of polystyrene latex spheres in water as his test system. He observed that the self-beat method is quite sensitive to small amounts of large particles in the presence of large number of small particles. Analysis of intensity correlation spectra of a bimodal distribution of polystyrene latex suspensions has been presented by Bargeron (1973, 1974) and Chen et al. (1974) using a straightforward least-squares method and by Lee and Chu (1974) using an integration least-squares method. Pecora and Tagami (1969) have used the two-parameter unimodal Schulz molecular weight distribution to compute the light scattering spectra from a solution of rigid rods and ﬂexible coils. Their calculations showed that polydispersity eﬀects could produce small deviations from a single Lorentzian ﬁt. Intramolecular interference eﬀects on the Rayleigh line spectrum of macromolecules in solution also become observable when the scattering species are comparable in size with the incident wavelength. For randomcoil polymers such large sizes can only be achieved at high molecular weights where the polydispersity also increases. The polydispersity eﬀects in random-coils were investigated by Frederick et al. (1971) and Kramer and Frederick (1971, 1972) also. Thompson (1971b) considered the spectrum of light scattered from polydisperse polymer solutions and developed a graphical technique to characterize particle average size and standard deviation of particles obeying the lognormal distribution. Major diﬃculties in data analysis arise from multiexponential (or multi-Lorentzian) signals. Extreme care must be exercised in ﬁtting the experimental data with a sum of exponentials consisting of arbitrary decay times and amplitudes although computer programs exist. On the other hand it is advantageous to be able to detect deviations from a single decay time. Koppel (1972) developed a technique based on the method of cumulants which is still widely used for multiexponentials. The application of PCS for the analysis of particle size distribution continues to be an area of active research. The mathematical treatment of the data in general could be classiﬁed into two approaches. One method directly deals with the so-called ill-conditioned Laplace transform involved in the extraction of size dis-

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tribution from the correlation function. The other one is based on empirical data reduction such as moment expansion, various parameterized distributions, and the histogram and higher order spline-function decomposition techniques. Pike and coworkers (1981, 1983) used the direct inversion approach and argued in favor of a linear eigenfunction analysis. Provencher (1982a,b) and others made use of regularization constraints to ensure a smooth solution of the Laplace transform. The histogram analysis, a zeroth-order spline method, was persued by Chu and DiNapoli (1983, 1984). Goll and Stock (1983) employed the ﬁrst-order spline functions for the determination of particle size distribution. Lavery and Earnshaw (1984) chose to work with the cubic B-spline function due to its numerical stability and linear relation to ﬁeld autocorrelation function. The histogram method with exponential sampling was described by Fletcher and Ramsay (1983) for continuous and broad distribution of particles. Nash and King (1983) discussed the analysis of decay curves by the ﬁtting of sums of positive exponentials by the S-exponential sum method of Cantor and Evans. It was shown that this method had advantages over previous methods for analyzing multiexponential decay functions. An interesting article was presented by Pusey and Van Megan (1984) for detecting small degrees of polydispersity with PCS. The eﬀective diﬀusion coeﬃcient obtained from the ﬁrst cumulant of the correlation function was examined theoretically as a function of scattering angle. When the Rayleigh–Gans–Debye approximation was satisﬁed, the eﬀective diﬀusion coeﬃcient exhibited a characteristic variation with the wave vector which was very sensitive to the polydispersity, provided that the mean particle radius was greater than 170 nm. In principle, the polydispersity deﬁned as the standard deviation divided by the mean size could be detected at levels as low as 0.01. Weinear and Tscharnuter (1987) have demonstrated various advantages and disadvantages of PCS taking diﬀerent samples: NBS SRM 1691, a crosslinked polystyrene divinylbenzene, a fractionated sample of polyvinyl chloride, ‘dusty’ samples of reference latex, a bimodal mixture of rubber particles, and a 2:1 aggregate for particle sizing. Vaidya et al. (1987) compared various methods available for determining macromolecular polydispersity from DLS data with their own GEX function ﬁt method. Bott (1987) has shown that weight distributions from singleangle PCS measurements can have huge errors and suggested that complementary information obtained by measurement at an additional angle allow a good measure of the weight-averaged size distribution. Bertero and Pike (1983) and Bertero et al. (1989) have also successfully used PCS data to extract polydispersity information. Since the design of a correlator by Asch and Ford (1973) a number of instruments have been developed for DLS. Ford (1983), and Elings and Nicoli (1984) reviewed the current state of the art of instrumentation and techniques. Oliver (1981) critically examined the parameters involved in spectrum analysis and photodetection experiments. Variations of typical light scattering experiments were explored by Phillies (1983) who discussed the potential advantages of multi-detector and multi-beam experimental approaches. Two-beam, two-detector homodyne coincidence spectroscopy was applied to subtract the eﬀect of multiple scattering on the diﬀusion measurement of the turbid sample. In contrast, the multiple scattering problem was otherwise dealt with, either theoretically (Dhont and de Kruif, 1983) or experimentally using novel short path length scattering cells (Cummins and Staples, 1981). Kam and Rigler (1982) used cross-correlation of scattered in-

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tensity at diﬀerent angles between two detectors to eliminate contributions from concentration ﬂuctuations. As a result, scattering from individual particles was enhanced. They applied this approach to study the rotation dynamics of tobacco mosaic virus, DNA and tysozome. Hutchins and Mazumdar (1983) demonstrated the feasibility of the use of a dual-beam, frequency-biased laser-Doppler velocimetry for measuring diﬀusion diameter distributions on a single-particle basis. This technique was particularly attractive for measuring aerosol particles in the size range of 0.05–0.5 μm. In recent years an entirely new technology base for PCS has been considered – laser diodes in place of gas lasers (Brown and Grant, 1987), optical ﬁbers and micro-lenses in place of bulkier optical systems (Brown et al., 1986; Brown, 1987), and software and/or custom-made electronic cards and chips in place of hard-wired boxes of correlation electronics. These developments have led to miniaturization of the PCS system for particle size analysis (Brown et al., 1990; Brown, 2001). Van der Meeren et al. (1993) reviewed ﬁber-optic quasi-elastic light scattering (FOQELS) as a fast and reproducible characterization method. They discussed proper operating conditions and limitations of the technique. PCS was used with a variety of other techniques to determine the nature of materials in solution. By using a combination of static and dynamic laser light scattering and transient electric birefringence Xu et al. (1987) were able to determine structural characteristics and size distributions of polydisperse disk-shaped particles (bentonite) in suspensions. Gulari et al. (1987) have compared PCS with turbidity spectra and Fraunhofer diﬀraction pattern analysis in the overlap region of 0.5 μm to 4 μm for bimodal and trimodal distributions of latex spheres. PCS was found to be a better method for mean size determination but not so good for distribution measurements. Chowdhury et al. (1984) studied the application of PCS to a system of randomly diﬀusing particles suspended in a ﬂuid undergoing uniform translational motion relative to the optical scattering volume. For this they derived theoretical expressions for both the homodyne and the heterodyne correlation functions in dilute as well as non-dilute particle limits and tested these results with experiments on a ﬂowing system and found good agreement. In an interesting experiment Bronk et al. (1993) used the technique to measure submicrometer particles held in micrometers-sized droplets of saturated sodium chloride solution, which were captured in an electrodynamic levitator and maintained at constant diameter for several days at a time. A ﬂowing aerosol was analyzed for concentration, aperture, velocity, and particle-size eﬀects by Weber and co-workers (1993). Weber also discussed experiences in the determination of particle size in the nanometer range using PCS. Ross and Dimas (1993) considered noise and distortion in correlation data. Ruf (1993) discussed data accuracy and resolution in particle sizing by PCS. The application of diﬀusing-wave spectroscopy to particle sizing in concentrated dispersions was described by Horne and Davidson (1993). Problems in working with high-concentration submicrometers particle size distributions using correlation techniques were discussed by Trainer et al. (1992). Colloidal suspension of iron oxide particles was studied experimentally by Chu et al. (1987) who combined laser light scattering – angular distribution as well as autocorrelation function – with transient electric birefringence (TEB). Their results were consistent with electron microscope measurements. This technique requires very small amounts of the often-precious samples. It is an advantage as well as a

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limitation of the technique since the necessity for very low particle concentrations to prevent multiple scattering leads to extreme dilution of the sample. However, a very dilute sample also cannot be measured. The detected intensity ﬂuctuations then arise not only from the Brownian motion, but also from the changing number of particles in the scattering volume. This results in an apparent decrease in the diﬀusion coeﬃcient and thus an apparent increase in the particle diameter (Weber et al., 1993). However, if one assumes that the particle ﬂuctuations and the number ﬂuctuations each have their own frequency domain which can be separated, the problem can be solved by introducing an electronic ﬁlter in the standard PCS setup (Willemse et al., 97). The experiments were carried out with an experimental standard pinhole setup on laminar ﬂowing aerosols of the submicrometers particle size range. It is shown that beside local mean particle size and local aerosol velocity, the local particle number concentration may be obtained simultaneously from a single measured autocorrelation function. The proposed procedure does not require calibration. It is pointed out that measurement conditions can be adapted to the properties of the aerosol to be characterized, thus allowing characterization of aerosols over a wide parameter range – it is not restricted to the case of low particle concentration. The experimental results are compared to data from the literature, data from reference measurements and data from a theoretical model, respectively. The method can also be useful for characterization of other ﬂuid–particle systems as hydrosols (Weber and Schweiger, 1999). Finally there is a note of caution for practitioners of Photon Correlation Spectroscopy. It is very important that before taking the observations one takes care of certain practical considerations associated with the PCS setup (Brown and Smart, 1997; Pusey et al., 1983).

9.3 Aerosol size and distribution Generally one derives information regarding particle size distribution from angular scattering measurements (Westwater and Cohen, 1973; Post, 1976; Alger, 1979; Nelson, 1981; Heintzenberg et al., 1981; Hansen, 1980; Hansen and Evans, 1980; Walters, 1980), spectral extinction measurements (Yamamoto and Tanaka, 1969; Kuriyan and Phillips, 1974; Heintzenberg, 1975; King et al., 1978; Michael et al., 1978; Heintzenberg and Welch, 1982, Sun et al., 2007; Roy and Sharma, 2005) or a combination of these two procedures (Thomalla and Quenzel, 1982). Retrieving the size distribution from these measurements is a diﬃcult inverse problem (Dubovik et al., 1995). It requires the inversion of a Fredholm integral equation of the ﬁrst kind (McWhiter and Pike, 1978). This equation gives the measured property as an integral over the size distribution with a kernel determined by scattering theory. However, inversion techniques do not provide a unique solution for the particle size distribution and may cause numerical instabilities (Twomey, 1975; Fymat, 1975; Box and McKeller, 1976). Therefore, some sort of smoothing constraint is generally required, a reason for retrieval schemes to be based upon homogeneous sphere Mie theory solutions. To solve the inverse problem with real data using Mie theory, three routes have been followed in the literature: (1) assume an a priori trial expression for the size distribution, depending on a ﬁnite number of parameters, and search in the this parameter space for the best ﬁt with the given data (Dobbins

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and Jizmagian, 1966; Tarantola and Valette, 1982), or (2) try to obtain an approximation to the size distribution by discretizing the integral equation and then solve a (constrained) matrix inverse problem (Phillips, 1982; Twomey, 1975), or (3) use an approximate scattering kernel and invert the integral equation analytically (Shifrin and Perelman, 1963; Franssens et al., 2000). Although Mie theory assumes homogeneous spherical particles, aerosols are generally nonspherical. In scattering measurements attempts are often made to minimize the eﬀect of nonsphericity by using only forward scattering directions, i.e., at scattering angles ≤20–40◦ . However, it is worthwhile to investigate the eﬀect of nonsphericity on retrieved particle size distribution (Heintzenberg, 1978; Heintzenberg and Welch, 1982). Studies have also been carried out to see the eﬀects of variation in the particle index of refraction and the truncation of angular scattered intensities due to measurement limitations (Heintzenberg and Welch, 1982; Veihelmann et al., 2006). It should be pointed out that, along with Mie theory, diﬀraction as well as T-matrix calculations have also been used for retrieving the size distribution (Veihelmann et al., 2006). 9.3.1 Eﬀect of temperature The early study of the eﬀect of temperature on aerosols was made by Husar and Shu (1975). They studied the physical and chemical properties of Los Angeles smog aerosols as a function of temperature using two in situ detection techniques – nephelometry and optical aerosol counting. Their setup is shown in Fig. 9.1. In this study volatility was studied quantitatively by heating the smog aerosol up to 250◦ C and simultaneously measuring the scattering coeﬃcient using a nephelometer and the change of size distribution by an optical counter. Thermal analyses revealed that 50–80% of submicrometer aerosol mass is volatile at 220◦ C. The shapes of curves for the total light scattering coeﬃcient obtained by nephelometer versus temperature (thermo-nephelograms) were found to be diﬀerent for photochemically formed aerosols than those formed on ‘humid’, hazy days. It was suggested that the thermal analyses may at least in some cases provide the means of online monitoring the origin of the ambient aerosols. Thermal dependence on the size of aerosols was found to be diﬀerent for volatile and nonvolatile aerosols. In the ﬁeld, the aerosol scattering coeﬃcient is usually measured with a nephelometer by heating the ambient aerosol to a low reference relative humidity (∼40%) in order to measure a light scattering coeﬃcient that is intrinsic to an aerosol rather than dependent on atmospheric relative humidity. Bergin et al. (1997) studied the ammonium nitrate aerosols and observed a decrease in light scattering coeﬃcient of the aerosol due to evaporation in a heated nephelometer. They measured the changes in the scattering coeﬃcient of a laboratory-generated ammonium nitrate aerosol as a function of temperature and the mean residence time within the nephelometer sample volume. At the same time, the change in the aerosol size distribution due to ammonium nitrate evaporation was directly measured with a laser particle counter. The change in the aerosol size distribution and the scattering coeﬃcient was modeled as a function of mean residence time and temperature. Model results for the change in the aerosol scattering coeﬃcient due to evaporation were found in agreement with measurement. King et al. (1983) measured the aerosol diﬀusion constants in the intermediate Knudsen regime. The form of the diﬀusion constant of a particle suspended in the

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Fig. 9.1. Schematics of the experimental setup for the aerosol thermal analyses (adapted from Husar and Shu, 1975).

medium is dependent on the Knudsen number Kn = l/r, where l is the mean free path of the gas molecules and r is the radius of the aerosol. Their experiment based on light scattering showed that in the intermediate Knudsen regime, corresponding to aerosol size comparable to the mean free path of the gas molecules, observed diﬀusion constants do not agree with the kinetic-theory calculations. The experiments performed in two very diﬀerent environments, one with the soot particles in ﬂame and the other with monodisperse polystyrene latex spheres, lead consistently to the same conclusion. Formation of aggregates and their fractal nature for soot in ﬂames was also studied by Sorensen et al. (1992). Singh et al. (2006) studied the eﬀect of temperature on the size of cigarette smoke aerosols using dynamic light scattering. They found that the size of smoke aerosols decreases with increase in temperature. It was very much in agreement with the early results of Husar and Shu (1975) who observed a decrease in light scattering with an increase in temperature for cigarette-smoke aerosols. 9.3.2 Eﬀect of humidity Aerosol hydration has important consequences for the eﬀect of aerosols on the Earth’s radiation budget. The size increase of aerosol particles resulting from uptake of water vapor has important implications for the direct scattering of radiation (the ‘direct eﬀect’) (e.g., Hegg et al., 1996; Hansen et al., 1997). It also has a bearing on the ability of these particles to serve as cloud condensation nuclei (CCN) and, under the right circumstances, to form cloud droplets (the ‘indirect eﬀect’) (Twomey, 1974; Eichel et al., 1996; Wulfmeyer and Feingold, 2000; Hansen et al., 1997; Jiang et al., 2002). Light scattering coeﬃcients as a function of %RH have been calculated for the sulfate and nitrate aerosols of atmospheric interest (Tang, 1996). It is important to note that the chemical eﬀect on light scattering is outweighed by the size eﬀect of the aerosols under consideration.

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The traditional method of measuring hydration and consequent growth uses a scattering nephelometer that aspirates an air sample, dries it, and then re-exposes it to varying levels of RH, typically 65%, 75%, and 85% (Charlson et al., 1984; Hegg et al., 1996; McInnes et al., 1998). The growth in total scattering is referred to as f (RH) (Charlson et al., 1992). The humidiﬁed nephelometers cannot expose air samples to RH > 85% without risking condensation on their chilled mirrors and spurious measurements. However, the region of 85% < RH < 100% is of great interest because it is here that particles experience their most dramatic growth. Moreover, the extent of this growth hints at their ability to act as cloud condensation nuclei (CCN). Instead, Feingold and Morley (2003) used backscatter lidar to collect information on the uptake of water vapor by aerosol in a well-mixed, cloudcapped, boundary layer. The lidar method for determining hygroscopic growth has numerous advantages, including the fact that the enhancement in backscatter due to changes in RH is measured under ambient, unperturbed atmospheric conditions and the range of measurement can be extended very close to saturation. A vertically pointing, airborne lidar was used to measure vertical proﬁles of aerosol backscatter beneath a stratocumulus cloud deck. In situ aircraft thermodynamic measurements were used to derive simultaneous proﬁles of relative humidity (RH) under the assumption that the boundary layer is well mixed. The change in backscatter is derived as a function of relative humidity over the range ∼85% RH to ∼98.5% RH. In situ measurements of the aerosol size distribution and composition were used to calculate the expected enhancement in backscatter due to equilibrium uptake of water vapor. Calculations of enhancement in total scatter due to water vapor uptake with enhancement in backscatter suggest that the eﬀects agree to within ∼20% of one another for RH <∼ 95% but they diﬀer signiﬁcantly for RH > 95%. It, therefore, suggested that broad application of backscatter lidars in well-mixed, cloud-capped boundary layers could provide a source of valuable information for addressing the aerosol direct and indirect eﬀects. It should be pointed out that Raman lidar, which also has the ability to simultaneously measure backscatter (or extinction) and water vapor concentration, has also been applied to measurement of hygroscopic growth (Ferrare et al., 1998). In another related work Wulfmeyer and Feingold (2000) showed how a diﬀerential absorption lidar (DIAL) can be used to infer the extent of water vapor uptake by aerosols. Hitzenberger et al. (1997) also studied the humidity-dependent growth of sizesegregated aerosol samples. The collected samples were exposed to elevated relative humidities in a chamber containing an aqueous solution of CaCl2 of speciﬁed concentration. Calculation of optical parameters (i.e., extinction coeﬃcient and single-scattering albedo) showed that extinction (therefore also the optical depth) and single-scattering albedo increased with humidity. From the changes of the single-scattering albedo it could be deduced that for the dry aerosol heating effects probably dominated over cooling eﬀects, while for high humidities cooling predominated. Hygroscopicity of aerosols and their growth with humidity is very important from health point of view (Khan et al., 1988). Along with the particle size and breathing pattern it plays a crucial role in distribution of inhaled aerosols to various parts of the body as well as determining the impact of aerosolized drugs for therapeutic purposes. Therefore, much work on the eﬀect of humidity on the size

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of aerosols has been done in the ﬁeld of medicine. Dautrebande and Walkenhorst (1961) were ﬁrst to demonstrate that dry NaCl aerosol particles grow hygroscopically by as much as six times during respiration. Later experimental and theoretical studies on hygroscopic aerosol particles revealed that the extent of growth depends on a number of important parameters that have been reviewed by Morrow (1986) and Hiller (1991). In a recent work, Hiller et al. (2006) have studied physical properties of two metered-dose bronchodilator aerosols packaged as solutions and two aerosols packaged as ﬁnely ground powders, measured at low and high relative humidity. The aerodynamic size distribution and particle concentration were measured in real time using the single-particle aerodynamic relaxation time analyzer, which can measure the aerodynamic diameter of single suspended particles in the respirable size range. The median aerodynamic diameter, the mass median aerodynamic diameter, the total particles per dose, and the total aerodynamic mass per dose were calculated. Signiﬁcant increases were noted in the median aerodynamic diameter for three aerosols and in the mass median aerodynamic diameter for two aerosols. The number of particles in the measured size range increased 3.6and 4.1-fold for the droplet aerosols and 1.4-fold for the powder preparations. The aerodynamic mass per dose in the measured size range increased 5.7- and 11.4-fold for the droplet aerosols and 3.1- and 1.6-fold for the powder aerosols. These data indicate that all aerosols tested increased in size at high humidity and that aerosols dispensed as droplets may be more unstable than those dispensed as powders. It would be proper to mention at the end of the section that humidity and temperature are quite closely related parameters. In many cases an increase in temperature would translate to a decrease in humidity, while a decrease in temperature will lead to an increase in humidity. 9.3.3 Eﬀect of concentration It is interesting to note that in most of the cases of light scattering by aerosols/ particles, it is the dilution which is desired, not the concentration. This is required to avoid the eﬀect of multiple light scattering and particle interactions. However, attempts are being made to study concentrated solutions and instruments have been designed which give quite good results even with concentrated samples. Thomas and Dimonie (1990) have described measurements of particle size during the course of a latex emulsion polymerization reaction using ﬁber optic dynamic light scattering (FODLS). By collecting the scattered signal at 180◦ and using the proper focusing optics, one avoids the eﬀect of multiple light scattering. The setup is shown in Fig. 9.2. These measurements were compared to results from DLS measurements performed in the usual way on diluted samples. It was observed that the rate of growth of the particle size was faster in a concentrated solution compared to the diluted one, reaching to saturation after sometime, Fig. 9.3. Chung and Dunn-Rankin (1996) while studying sidestream cigarette-smoke aerosols calibrated their light scattering setup, shown in Fig. 9.4 that measured the scattered intensity at two angles and two polarization components at complementary angles. Their calibration showed that multiple scattering in ensemble aerosol systems aﬀect the scattering ratio measurement diﬀerently for diﬀerent particle sizes. They observed that for 0.197 μm hydrosol, multiple scattering has little

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Fig. 9.2. Experimental setup for FODLS measurements (adapted from Thomas and Dimonie, 1990).

Fig. 9.3. Latex particle size as a function of reaction time (, determined by standard (very dilute) DLS measurements; ♦, determined by FODLS measurements directly on an undiluted sample) (adapted from Thomas and Dimonie, 1990).

eﬀect on the measurement; for 0.300 μm hydrosol, higher particle concentration results in a lower scattering ratio; and for 0.596 μm hydrosol, higher concentration produces higher scattering ratio measurements. Singh et al. (2006) not only studied the eﬀect of concentration on the size of smoke aerosols but also did experiments to see the eﬀect of temperature along with concentration on the size of aerosols. The experimental setup used to study the effect of temperature as well as concentration is shown in Fig. 9.5. They used cigarette smoke for the experiment. Change of concentration was made by increasing the number of cigarettes assuming the same rate of combustion for each cigarette of the same brand, which was actually conﬁrmed by measuring the weight of the cigarette initially and after 12 minutes of lighting. The excitation source was a red

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Fig. 9.4. Schematic of the experimental set-up for light scattering ratio measurements and polarization ratio measurements (adapted from Chung and Dunn-Rankin, 1996).

He-Ne laser (Spectra Physics, 10 mW, 633 nm), which was mounted on the ﬁxed arm of the goniometer. A photomultiplier tube (RCA 31034A) was mounted on the other arm. The scattered light from the sample was detected by a photomultiplier tube at right-angles and the photocurrent was suitably ampliﬁed and digitized by ampliﬁer/discriminator (EG&G, PARC, Model 1182) before it could be fed to a photon correlator (Malvern 7032 CN). The correlator gave the autocorrelation function of the scattered light. The data was transferred to a computer for storage of spectra and its subsequent analysis. The whole spectrometer was installed on a

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Fig. 9.5. Schematic diagram of the experimental set-up. P1 and P2 : pump; R1 and R2 : regulator; F1 and F2 : ﬂowmeter; T: thermometer; S: smoke chamber; L: lens; PMT: photomultiplier tube; HV: high voltage power supply; LV: low voltage power supply; A/D: ampliﬁer/discriminator; DC: digital correlator; C: computer (adapted from Singh et al., 2006).

vibration isolation table. Proper lenses and pinholes were used for better signalto-noise ratio. The lens for collecting the scattered light was an f/3 biconvex lens made of BK7. The size of pinhole before the PMT was 0.3 mm that takes care of a single coherence area being detected by the PMT. This is very important since we take the ﬂuctuating scattered electric ﬁeld as Gaussian and in the Gaussian regime signal is dominated by interference ﬂuctuations. Diaphragm pumps were used to regulate the ﬂow of dry ﬁltered air in the smoke chamber. Air ﬂow of 2 l/min was maintained with the help of regulators and ﬂowmeters to keep the cigarettes burning. The smoke chamber was made of glass with dimensions 20 × 20 × 30 cm. The cigarette smoke produced is taken through a tube (internal diameter 1 cm) to scattering region (2 × 4 cm) at the height of 15 cm from the top of the smoke chamber. The method of cumulants (Koppel, 1972) was used to ﬁnd out the mean size of aerosols. Since the interest was in the evolution of size with concentration and temperature they went for mean size only instead of size distribution, although the latter would have been preferred for the polydisperse sample like smoke aerosols. However, it was ensured that the polydispersity remained below a limit to give correct mean size while using the method of cumulants (Taylor et al., 1985). To test the dynamic light scattering experimental setup for particle sizing, the ﬁrst measurements were made with a standard sample of polystyrene latex beads of size 0.32 μm (Bangs Laboratories). The setup gave the same size as provided by the supplier. Observations were made after one minute of lighting the cigarette. The ﬂow rate was kept to the minimum possible so that the system could be treated as al-

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most Brownian having negligible aerosol ﬂow velocity in comparison to movement in horizontal plane. This ensured that correlation time used for ﬁnding the size was not modiﬁed by the ﬂow (Chowdhury et al., 1984). The beam size in the scattering region was 2 mm. For the He-Ne laser (λ = 633 nm) the illuminated volume had a suﬃciently large number of sub-regions of volume smaller than the wavelength, permitting particle motions in one sub-region to be independent of those in another. This justiﬁed the assumption of Gaussian distribution of ﬂuctuating scattered electric ﬁeld. In calculating the size, the refractive index of the cigarette smoke aerosols was taken to be same as soot particles since the experimental setup provided with the cigarette smoke ‘oﬀ the tip’ of the cigarette, not the mixture of mainstream and sidestream smoke from the cigarette. The absorption values of this part of the cigarette smoke (Terhume and Anderson, 1977) suggested it to be carbonaceous matter like soot. However, numerous reported values of the refractive index for soot might lead to uncertainty in particle size (Sorensen et al., 1992). Since the concern was relative change in size, this uncertainty must not aﬀect the conclusions. Table 9.1 presents the estimated decay constants and mean size. The mean size of the smoke aerosols was found to be 0.143 μm at 25◦ C. Table 9.2 shows estimated decay constants and sizes at various concentrations for T = 25◦ C. It was observed that the size increases with the concentration. No sudden increase in the size was observed. The eﬀect of temperature on size at various concentrations is presented in Table 9.3. It was observed that the size decreases with increase in temperature. Table 9.1. Mean size for smoke aerosols at 25◦ C S. no.

Decay constant γ (s−1 ) × 104

Size (μm)

Mean size (μm)

1 2 3

5.21 5.35 5.32

0.145 0.141 0.142

0.143

Table 9.2. Eﬀect of concentration on size of smoke aerosols at 25◦ C S. no.

Concentration (wt%)

Decay constant Γ (s−1 ) × 104

Size (μm)

1 2 3 4 5

0.01 0.05 0.1 0.5 1

5.32 3.59 1.99 1.29 1.06

0.142 0.210 0.378 0.586 0.712

An examination of the nature of size versus concentration curve (Fig. 9.6) reveals that, although the size increases with concentration, the increment in the mean size is not a multiple of itself. This indicates the stochastic nature of the

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0.01%

0.05%

0.1%

0.5%

1%

20 25 30 35 40

0.154 0.143 0.112 0.089 0.061

0.255 0.210 0.198 0.154 0.097

0.413 0.378 0.315 0.298 0.212

0.622 0.586 0.525 0.516 0.420

0.782 0.712 0.681 0.632 0.613

Fig. 9.6. Change in size with concentration at a ﬁxed temperature (T = 25◦ C) for data in Table 9.2.

coagulation (aggregation) process. Moreover, the increase in size may not be due only to the coagulation. The condensation of gases present in the smoke chamber may also contribute to the increase in size. Fig. 9.7 shows log of size versus log of concentration. The size and concentration values were multiplied by a factor of a hundred before taking the log. It shows a power law behavior indicating the fractal nature of aggregation (Schaefer et al., 1984; Weitz et al., 1985). Fig. 9.8 shows the three-dimensional plot of data in Table 9.3. One can see that an increase in temperature reduces the size as the condensation is reduced and that the size increases with concentration at all temperatures. A number of factors including the mechanism of particle formation, growth, diﬀusion, convection and sedimentation together with coagulation decide the form of size distribution function of aerosols. Therefore, for a better understanding of the evolution of the distributions a comparison of the experimentally observed distributions with those derived from theoretical considerations is needed. One of the methods to achieve this could be setting up a general dynamic equation (GDE) as suggested by Friedlander (1977) and solving it for appropriate initial and boundary conditions relevant to the experiment. Otherwise, various models have been proposed for the mechanism of aggregation phenomenon, which is still an enigma. Few important ones are the Eden model (Eden, 1961), diﬀusion limited aggregation (DLA) (Witten and Sander, 1981), Mole’s labyrinth (Hermann, 1983) and cluster-

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Fig. 9.7. Log of size versus log of concentration for data in Table 9.2.

Fig. 9.8. Plot of data in Table 9.3. Change in size with concentration and temperature.

ing of clusters (Meakin, 1983). Leyvraz (1986) has described a general qualitative theory for aggregation taking into account the various exponents describing the cluster size distribution and its time evolution. A few models have been developed on the basis of stochastic process (Durrett et al., 1999) as well. One can say that the theory of size evolution is still in ﬂux, which makes the explanation of experimental results a diﬃcult task. More experiments taking good care of experimental

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parameters and precautions are required for better understanding of aerosol size distributions.

9.4 Aerosols and climate Kaufman et al. (2002) have described the role of atmospheric aerosols on the climate in a ﬁne review article. The third assessment report of the Intergovernmental Panel on Climate Change (IPCC) by Forster et al. (2007) can also be consulted for a good overview of aerosols and their eﬀect on the climate. It has been established that atmospheric aerosols are intricately linked to the climate system and to the hydrologic cycle of our planet. Aerosols interact both directly and indirectly with the Earth’s radiation budget and therefore the climate. As a direct eﬀect, the aerosols scatter sunlight directly back into space. As an indirect eﬀect, aerosols in the lower atmosphere can modify the size of cloud particles, changing how the clouds reﬂect and absorb sunlight, thereby aﬀecting the Earth’s energy budget (Penner et al., 2006). The net eﬀect of aerosols is to cool the climate system by reﬂecting sunlight (Schulz et al., 2006). Depending on their composition, aerosols can also absorb sunlight in the atmosphere, further cooling the surface but warming the atmosphere in the process (Stier et al., 2007). These eﬀects of aerosols on the temperature proﬁle, along with the role of aerosols as cloud condensation nuclei, impact the hydrologic cycle, through changes in cloud cover, cloud properties and precipitation. Unraveling these feedbacks is particularly diﬃcult because aerosols take a multitude of shapes and forms, ranging from volcanic emissions to desert dust and urban pollution, and because aerosol concentrations vary strongly over time and space (Textor et al., 2006; Ramachandran et al., 2006). The warming eﬀect of the greenhouse gases is expected to take place everywhere, but the cooling eﬀect of the pollution aerosols will be somewhat regionally dependent (Jayaraman et al., 2006; Ganguly et al., 2005), near and downwind of industrial areas. No one knows what the outcome will be of atmospheric warming in some regions and cooling in others. Climate models are still too primitive to provide reliable insight into the possible outcome. Current observations of the buildup are available only for a few locations around the globe and these observations are fragmentary.

9.5 Conclusion We have reviewed the application of static and dynamic light scattering to the study of aerosols. It is observed that the instruments for ﬁeld studies of aerosols use mostly static light scattering for retrieving the data while in laboratory studies dynamic light scattering is used along with Mie scattering. The recent trend shows that use of both methods at the same time yields a better physical picture of the system. It is heartening to note that in the year that celebrates 100 years of the classic paper on light scattering by Mie (1908), it is Mie scattering that comes out on top. In industrial processes involving aerosols, instruments based on laser diﬀraction which combines Mie scattering with Fraunhofer diﬀraction have gained in popularity because of their wide dynamic range and ease of operation.

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In recent years, satellites, lidars and lidar networks, and a network of groundbased sunphotometers have enabled the collection of a huge amount of data related to aerosols (Forster et al., 2007). A number of models have been developed for retrieving the relevant data from these sources, but still there is considerable amount of uncertainty in the prediction of various parameters aﬀecting the climate (Forster et al., 2007). It emphasizes the importance of laboratory studies of various types of aerosols under simulated conditions for validation of the models used in climate studies. The paper shows that not many laboratory studies have been made with this intention. The present study aims to initiate this process. A much better understanding will require multi-wavelength scattering under controlled conditions of humidity, temperature, concentration, airﬂow and the presence of buﬀer gases in laboratory studies. Environmental and atmospheric scientists can take a lead from pharmacy and medicine where it is not uncommon to study the eﬀect of humidity and the airﬂow under controlled conditions to understand the health hazards of inhaled aerosols as well as the therapeutic use of aerosolized drugs. Acknowledgments The author acknowledges the help of Ashok, Jitendra and Pravin for typing the references in the desired format and also checking them through the text.

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Index

Q(M) metrics, 86 3D cross-correlation scheme, 439 absorption, 5, 32, 46, 50, 51, 61, 63, 64, 83, 100, 108, 173, 175, 177, 181, 187–189, 191, 192, 195, 201, 202, 208–210, 213, 223, 224, 226, 227, 233, 236, 241, 248, 262, 267, 268, 277, 293, 295, 296, 298, 300, 301, 304, 305, 308, 312, 318–322, 324–328, 330, 332–334, 473, 486, 491 active noise reduction, 459 adjoint equation method, 328–331, 335 aerosol, 4, 31, 32, 58, 61, 62, 65, 169, 170, 240, 469–475, 477, 479–495, 497, 499, 501, 503, 505, 507, 509 aggregates, 21, 26, 27, 39, 41, 45, 46, 49–51, 57–61, 63, 65, 474, 485 albedo, 36, 51, 52, 58, 61, 64, 173, 177, 179, 183, 186, 198, 201, 203, 206–208, 210, 217, 219, 227, 246, 247, 252–254, 276, 277, 301, 311, 318, 322, 351, 386, 403, 408, 486 algebraic reconstruction, 294, 332, 335 algorithmic diﬀerentiation, 328, 330, 331, 335 amplitude, 12, 16, 18, 19, 22, 24, 35, 48, 57, 71, 72, 76, 83, 99–104, 106–109, 160–162, 164, 181, 193, 194, 251, 257, 435, 445, 450, 477, 480 analogue material, 5, 14–16 anisotropic depolarization, 87, 88, 114 asymmetry factor, 176, 181, 196, 246, 268, 276 backward scattering, 80, 98 bioluminescence tomography, 294, 327 bird’s-nest structures, 26 birefringence, 103, 106, 107, 109, 110, 112, 113, 482 boundary-value problem, 121, 122, 132, 347–349

Cairns rescaling, 206, 243 characteristic equation, 347–355, 357–359, 361, 363, 365, 367, 369, 371, 373, 375, 377–379, 381, 383, 385, 387, 389, 391, 393, 395, 397, 399, 401, 403, 405, 407–409, 411, 413–415, 417, 419–421, 423, 425, 427, 429 cigarette smoke, 469, 485, 488, 490, 491 cloud, 4, 11, 13, 27, 31–33, 35, 37, 39–41, 43–45, 47, 49, 51, 53, 55, 57–59, 61, 63, 65, 67, 69, 71, 92, 169–172, 174–193, 195, 197–212, 214, 216–228, 230–257, 259, 260, 262–264, 266, 268–277, 403, 408, 469–472, 485, 486, 494 cloud optical depth, 184, 197, 211, 239, 243, 245, 254–256, 268, 274, 277 coherence time, 14 coherency matrix, 89, 90, 92, 94, 95, 97, 99 color, 3, 5, 9, 14, 15, 20, 23, 27, 461, 471–473 color eﬀect, 5, 15 cometary comae, 34, 45, 58 comets, 31, 33, 58, 59, 65 compact particles, 37, 38, 41, 44, 46, 49, 50, 59 concentration, 62, 98, 275, 277, 295, 301, 304, 309, 318, 326, 328, 333, 334, 479, 480, 482, 483, 486–488, 490–495 continuous fractions, 349, 373, 374, 376–378, 381, 383–385, 417 continuous medium approximation, 72, 98, 99, 112 convex sum, 74, 78, 90, 92 correlation time, 444, 447, 451, 457–459, 462, 478, 491 cosmic dust, 65, 66 Coupled Dipole Approximation, 7, 8 cross polarization, 6, 17, 20 degenerate anisotropy, 106, 111, 112

512

Index

degree of polarization, 74, 84–87, 93, 97, 98 delta-Eddington approach, 309 method, 249, 268 rescaling, 191–193, 202, 206, 207, 210, 239, 243, 247, 276 depolarization index, 85, 86, 91 depolarization metrics, 86 depolarization ratio, 84, 473, 474 dichroism, 97, 109, 110, 112, 113 dielectric constant, 14, 472, 473 dielectric scatterer, 122–124, 134, 135, 139, 142, 146, 148, 149, 151, 152, 155, 157, 158 diﬀerence equation theory, 353, 359, 361, 370, 407, 415 diﬀraction broadening, 6 diﬀuse optical tomography, 294, 318, 330 diﬀusing wave spectroscopy, 434, 437 diﬀusion coeﬃcient, 436, 476, 479–481, 483 diﬀusion equation, 236, 270, 308, 312, 314, 315, 317, 318, 321, 322, 324, 437, 438 diﬀusion theory, 195, 197–200, 207, 211, 218, 234, 246, 262, 269, 271, 274, 275, 277, 295 direct problem, 3, 27, 260 Dirichlet condition, 122, 123, 144, 155 discrete ordinates, 176, 294, 295, 307–310 discrete spectra, 348, 359, 360, 363, 365, 367–371, 377, 378, 381, 383, 414, 421 dispersion, 71, 103, 181, 208, 245, 469–471, 474–476, 480, 482 dust clouds, 31, 32 dyadic, 122, 129–138, 144–146, 148, 149, 151, 154, 158, 160–164, 188, 190 dyadic delta distribution, 145 dyadic free-space Green function, 132–134, 144, 149, 151, 158, 164 dyadic Green functions, 132, 138 dyadic Green theorem, 131 dyadic interaction operator, 135, 136 dyadic scattering amplitude, 160, 162, 164 dynamic light scattering, 433–437, 439, 441, 443, 445, 447, 449, 451, 453, 455, 457, 459, 461–463, 465, 467, 469, 471–473, 475, 477, 479–481, 483, 485, 487, 489–491, 493–495, 497, 499, 501, 503, 505, 507, 509 echo technique, 455, 457

eigenfunction, 149, 348, 353, 359, 360, 366, 367, 369, 371, 383, 392, 421, 481 eigenpolarizations, 106–111 eigensolutions of the vector-wave equation, 125 eigenvectors, 96, 107, 139–141, 149, 153, 155, 157, 369 electromagnetic waves, 5, 121 Elementary Linear Correlator, 436, 442, 445 energy conservation, 122, 151, 152, 156, 163–165, 173, 201 entropy, 89, 91, 95, 96, 99 equivalence theorem, 99, 101–106, 109, 113, 187, 189, 224 experimental simulations, 59, 61, 62 F-matrix, 12, 13, 19, 25 far ﬁeld, 11, 12, 20, 22, 152, 153, 158–161, 449, 455 Fick’s law, 191, 198, 213 ﬁnite diﬀerence, 294, 319 ﬂuﬀy particles, 38, 41, 44, 46, 51, 59 ﬂuorescence molecular tomography, 294, 327 ﬂux actinic, 190 hemispheric, 180, 197, 200, 213, 222, 230, 254, 260, 276 net, 213, 276 scalar, 190, 276 tensor, 276 vector, 190 forward scattering, 17, 19, 20, 24, 26, 35, 80, 96, 97, 202, 262, 354, 359, 473, 484 Fourier space, 187, 204, 211, 251, 252, 259 fractional derivation, 454 general invariance relations, 348, 349, 389, 393, 408 generalized equivalence theorem, 103, 105 geometric optics, 3, 8, 10, 17, 27, 54 Green function, 121–123, 125, 127, 129, 131–139, 141–149, 151–155, 157–161, 163–165, 169–173, 175, 177–183, 185–189, 191–193, 195, 197, 199, 201–203, 205, 207, 209, 211–217, 219–221, 223, 225–227, 229, 231, 233–235, 237, 239, 241–253, 255–265, 267, 269–273, 275–277, 279, 281, 283, 285, 287, 289, 291

Index Green’s theorems, 128, 131 Henyey–Greenstein, 181, 275, 295, 303, 308, 310 Hermitian conjugation, 78, 364 heterodyne detection, 436, 478 Heterogeneous Compound Particle (HCP), 10 Hilbert transform, 453 homodyne detection, 436, 478 humidity, 45, 470, 484–487, 495 ideal depolarizer, 87 ideal metallic scatterer, 123, 133–136, 141, 144, 146, 149, 155, 156 image reconstruction, 293, 294, 300, 326–328, 332–335 integral equation, 121, 132, 133, 141, 142, 144, 146, 147, 149, 150, 164, 254, 275, 294, 347–349, 351, 352, 356, 358, 359, 375, 395, 398, 412–415, 417, 483, 484 intensity correlation function, 439, 447, 452, 455, 458, 459, 478 intensity cross-correlation function, 499 intensity structure function, 455 interaction operator, 121, 122, 135–144, 146, 148, 149, 151, 152, 154, 156, 164 inverse problem, 3, 10, 41, 105, 113, 221, 255, 294, 326–328, 332, 483, 484 isotropic depolarization, 87, 88, 114 Jones matrix, 71–78, 80, 82, 90–93, 98, 99, 101–103, 106–113 Jones vector, 71, 72 Jones’s equivalence theorems, 114 Laplace space, 171, 186, 212, 226, 251 laser, 14, 24, 34, 35, 106, 170, 175, 191, 200–202, 205, 206, 209, 210, 212, 217, 220, 221, 223, 240, 241, 243, 245, 246, 253, 255, 262–264, 270, 275, 277, 297, 328, 433, 435, 439, 440, 449, 457, 459, 461, 462, 475, 477–479, 482, 484, 489, 491, 495 Laser Perfusion Imaging, 457 Laser Speckle Contrast Analysis (LASCA), 458 light scattering, 11, 22, 28, 32, 41, 53, 58, 64, 65, 72, 78, 79, 83, 91–93, 96, 99, 132, 149, 293, 294, 309, 433–437, 439, 441, 443, 445–449, 451, 453, 455, 457, 459, 461–463, 465, 467, 469–487,

513

489–491, 493–495, 497, 499, 501, 503, 505, 507, 509 Lippmann–Schwinger equation, 142, 145, 146, 148, 149 luminescence, 293–295, 297, 299–301, 303, 305–309, 311–313, 315, 317–319, 321–327, 329, 331–335, 337, 339, 341, 343, 345 mean free path, 98, 170, 173, 275, 276, 278, 294, 295, 301, 306, 437, 438, 440, 485 microgravity, 36–39, 45, 64, 66 Mie, 4, 6, 7, 10, 11, 15, 16, 22, 24, 25, 45, 46, 48, 56, 57, 63, 94, 97, 98, 121, 149, 151, 176, 225, 472–476, 483, 484, 494, 495 minimal solutions, 362, 365, 385, 419, 420 mirror symmetry, 79, 81 mixed dynamic contrast, 460, 462 modiﬁed laser speckle imaging, 462 molecular imaging, 295, 298, 307, 326, 332, 333 Monte Carlo method, 171, 347 Mueller matrix, 13, 19, 71, 73–76, 78, 79, 82–98, 103, 105, 112–114 Mueller–Jones matrix, 74–78, 80, 82, 90–93, 98, 99 multi-speckle averaging, 452, 454, 455, 457, 462, 463 multi-tau averaging, 441 multiple scattering, 34, 40, 46, 52, 54, 64, 83, 87, 92–98, 170, 174, 187, 219, 220, 224, 227, 230, 240, 251, 262, 263, 326, 347, 437–441, 443, 444, 474, 480, 481, 483, 487, 488 multiple scattering suppression, 439, 441, 443 Multiple-Scattering Cloud Lidar, 240 multiplication rule, 447, 448 natural light, 13 near-infrared, 65, 293, 295, 318, 322, 333 non-ergodicity problem, 445, 456, 457, 463 nonspherical particles, 16, 84, 121, 149, 474, 475 optical imaging, 297, 326, 327 optical theorem, 164 optimization, 327, 328, 330, 335 oxygen A-band spectroscopy, 241, 263 parabolic ﬂights, 32, 38

514

Index

partial reﬂection, 177, 208, 295, 306, 307, 313 particle size, 5, 11, 20, 24, 93, 98, 262, 263, 444, 470–475, 479–484, 486–488, 491 particulate matter, 469 phase, 8, 12, 13, 16, 18–20, 22, 24, 25, 71, 72, 76, 83, 93, 99–104, 106–110, 160, 295, 300–303, 308, 310, 311, 349–352, 354, 356, 358, 359, 366–368, 371, 373, 377–379, 381, 383, 386, 388, 389, 393, 395, 396, 399–401, 403, 407, 408, 469, 475, 476 phase curves, 33, 41, 45–49, 56–59, 63, 64 phase function, 93, 172–174, 176, 177, 181, 191–193, 197, 202, 207, 229, 236, 240, 243, 248, 254, 262, 275, 276, 295, 300–303, 308, 310, 311, 349–352, 354, 356, 358, 359, 366–368, 371, 373, 377–379, 381, 383, 386, 388, 389, 393, 395, 396, 399–401, 403, 407, 408, 475 photon correlation spectroscopy, 433, 447, 472, 476 photon time-of-arrival correlator, 450, 451 pixel binning, 443 plane wave, 121, 123–125, 127, 129, 131, 133–135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157–161, 163–165 Poincar´e sphere, 86 polar decomposition, 102, 103, 108, 109 polarimetry, 71, 74, 83, 106 polarization, 6, 12, 13, 17, 19, 20, 22, 23, 31–35, 38–41, 44–46, 48–59, 61–64, 71, 74, 84–88, 91, 93–98, 101, 102, 104–106, 108, 109, 111, 133, 159, 163, 170, 174, 175, 262, 472, 473, 487, 489 polarization memory, 96 polarized light, 34, 72, 74, 88, 93, 96–98, 101, 102, 105, 473 polydispersity, 480, 481, 490 principal value integration, 144

349–355, 357–359, 361, 363, 365, 367, 369, 371, 373, 375, 377–379, 381, 383, 385–387, 389, 391–393, 395, 397, 399, 401, 403, 405, 407, 409, 411, 413, 415, 417, 419, 421, 423, 425, 427, 429 radiative transfer equation, 95, 270, 347, 386, 392, 407 radiative transfer theory, 171, 263, 270, 274, 347, 349, 351–355, 357–359, 361, 363, 365, 367, 369, 371, 373, 375, 377–379, 381, 383, 385, 387, 389, 391, 393, 395, 397, 399, 401, 403, 405, 407, 409, 411, 413, 415, 417, 419, 421, 423, 425, 427, 429 random walk, 171, 186, 189, 234, 269, 270 Rayleigh approximation, 8 Rayleigh’s hypothesis, 142, 148, 150, 164 reciprocity, 79, 95, 99, 122, 151, 152, 165, 186, 209, 210, 213, 230 recurrence relations, 354, 355, 385, 419, 420 reﬂection, 3, 36, 45, 73, 92, 177, 179–183, 186, 188, 197, 204, 205, 208, 209, 217, 220, 224, 225, 227, 229, 235, 237, 238, 252, 271, 272, 275, 295, 302, 306, 307, 311, 313, 348, 349, 389, 393, 394, 403–408, 438, 449, 475 reﬂection function, 349, 389, 393, 394, 403–408 refractive index, 6, 10, 11, 14–16, 23, 24, 36, 41, 45, 51, 53–57, 63, 65, 83, 92, 93, 295, 300–302, 306, 325, 334, 435, 473, 474, 491 remote sensing, 58, 62, 71, 121, 169–172, 177, 188, 191, 198–202, 208, 210, 212, 220, 223, 224, 226, 227, 230, 232, 234, 235, 240, 241, 245–250, 255, 259, 260, 262–264, 272 reporter probes, 295, 297–300, 318, 333, 334 rotational symmetry, 80, 81, 187, 308

quantum nature of light, 5 radiance, 12, 172–179, 183, 187–191, 197, 200, 204, 209, 210, 224–227, 230, 242, 248, 249, 251–260, 270, 275, 276, 300, 301, 303, 307, 308, 311, 312, 319, 321–324, 330, 331 radiative transfer, 95, 169–173, 219, 224, 262–264, 269, 270, 274, 276, 277, 293–295, 300, 303, 305, 307, 308, 312, 314, 318, 322, 325, 327–333, 335, 347,

scattered ﬁeld, 7, 12, 123, 124, 149, 158, 160, 161, 445, 447, 460, 472, 473, 476 scattering, 3–8, 10–22, 24–28, 31–36, 39–41, 45, 46, 48, 49, 52–54, 58, 64, 65, 72, 73, 78–81, 83, 84, 86, 87, 91–99, 106, 112, 121–124, 132–134, 136, 138, 146, 149–151, 158, 160–162, 164, 170, 172–178, 181, 184–190, 192, 193, 195, 202, 203, 210–212, 214, 219–221, 223–225, 227, 228, 230,

Index 234–236, 238, 240, 242, 251, 254, 258, 259, 262–264, 269, 270, 273–277, 293–296, 300–303, 305, 307, 309, 311, 312, 319, 324–326, 330, 333, 347–349, 351, 354, 359, 386, 388, 395, 399, 403, 404, 408, 433–441, 443–446, 448, 458–460, 462, 469–491, 493–495, 497, 499, 501, 503, 505, 507, 509 scattering anisotropy parameter, 437 Shturm’s method, 414 Siegert relation, 436 simpliﬁed spherical harmonics, 294, 295, 308 single scattering, 39, 40, 45, 49, 53, 54, 93, 94, 96, 173, 219, 220, 240, 276, 277, 351, 386, 388, 403, 408, 434, 435, 437, 439, 440, 443 single scattering albedo, 173, 276, 277, 351, 386, 403, 486 singularity, 132, 144, 146, 148, 149, 231, 237 size distribution, 41, 44, 46, 53–57, 59, 62, 63, 92–94, 181, 248, 262, 277, 471, 474–476, 480–484, 486, 487, 490, 492–494 size parameter, 4, 5, 16, 24, 26, 50, 83, 94, 98, 192, 275, 277, 473 slotted waveguide method, 15 small animal models, 326 source iteration, 294, 322, 330, 331 space-time moments, 202, 236, 262 speckle, 433, 437, 439–441, 443–449, 452–463, 513 speckle contrast, 458, 461 speckle deviation, 455 speckle deviation analysis, 462 spectral variations, 63 spherical harmonics, 174, 190, 192, 294, 295, 301, 308, 309, 311, 312, 314 spherical particles, 7, 16, 81, 83, 84, 92–94, 97, 121, 149, 174, 473–476, 484 static light scattering, 472, 494 Stokes parameters, 12, 22, 73, 74 Stokes transformation matrix, 74 Stokes vector, 12, 13, 18, 71–74, 82, 83, 86, 93, 102 Stokes–Verdet criterion, 74, 82 surface albedo, 198, 201, 254, 277

515

surface current, 144 surface integral equation, 133, 142, 144, 150 symmetric normalization, 447, 454 symmetry, 6, 21, 24, 33, 72, 79–81, 92, 93, 95, 99, 106, 112, 122, 133, 135, 145, 147, 151–154, 156, 162, 165, 271, 308–310, 364, 386, 390, 392 T-matrix, 121, 122, 138, 141, 149–151, 157, 164, 165, 484 temperature, 9, 16, 18, 21, 35, 61, 303, 436, 448, 470, 484, 485, 487, 488, 490–495 tensor product, 75 three-dimensional radiative transfer, 290 time-dependent radiative transfer, 172 time-resolved correlation, 455 time-resolved correlation analysis, 454 time-resolved spectral analysis, 452 time-resolved structure function, 455 tissue, 71, 263, 293–303, 305–309, 311, 312, 314–316, 318–322, 324–328, 331–335 Titan’s aerosols, 32, 58, 61 total ﬁeld, 122, 135, 158, 160, 161, 165 transmission, 10, 13, 101, 122, 135, 140, 147, 155, 177, 179–183, 186, 188, 201, 203, 205, 208, 211, 214, 224, 227–229, 231, 232, 234, 237, 238, 240, 250, 251, 262, 271, 272, 274, 348, 437, 449 transmission condition, 122, 135, 140, 147, 155 transmittance, 75, 87, 88, 97, 102, 179–181, 183, 186, 199, 201, 203, 206, 207, 210, 249, 256, 267, 268, 273, 274, 277 transport mean free path, 189, 195, 198, 199, 201, 205, 214, 218, 231, 232, 273, 276, 278, 294, 295, 301, 437, 438 triangular averaging, 442, 458 turbid medium, 243, 347–350, 386, 389, 390, 392–394, 404, 407, 408 unitarity, 103, 122, 151–157, 163–165 vector-wave equation, 122, 124, 125, 132, 134–136, 148, 151, 154, 155, 157–159 volume integral equation, 121, 141, 142, 146, 147, 149 wave trains, 13, 14

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