Advances in Imaging and Electron Physics, Volume 156

Advances in IMAGING AND ELECTRON PHYSICS VOLUME 156 Edited by PETER W. HAWKES CEMES-CNRS, Toulouse, France AMSTERDAM...

Author: Peter W. Hawkes

73 downloads 637 Views 5MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Advances in

IMAGING AND ELECTRON PHYSICS VOLUME

156 Edited by

PETER W. HAWKES CEMES-CNRS, Toulouse, France

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands 84 Theobald’s Road, London WC1X 8RR, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA First edition 2009 Copyright © 2009, Elsevier Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http: //elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material. Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent veriﬁcation of diagnoses and drug dosages should be made. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-374762-4 ISSN: 1076-5670 For information on all Academic Press publications visit our Web site at www.books.elsevier.com Printed in the United States of America 09 10 10 9 8 7 6 5 4 3 2 1

PREFACE

The four chapters in this latest volume of these Advances come from very different worlds. We begin with a survey of photometric stereo by V. Argyriou and M. Petrou, very timely for it is only recently that the recovery of surface orientation and depth has been completely formulated mathematically and translated into applicable language. The authors present not only these important developments but also include an introductory section on radiometry, image formation and the models used to characterize surfaces, which make the chapter accessible to a wide audience. The second chapter, by F. Brackx, N. de Schepper and F. Sommen, describes a most interesting development in Clifford analysis, a subject that has already appeared in this series in a chapter by C. Doran, A. Lasenby, S. Gull, S. Somaroo and A. Challiner (Vol. 95, 1996, 271–386). Clifford analysis, based on Clifford algebra, is encountered in the study of quaternions and in the Dirac and Pauli matrices. It has often been claimed to be the proper way of unifying much of physics and mathematics but scientists remain slow to adopt it, doubtless through lack of familiarity and the fact that it is not taught at an elementary level. I am therefore all the more pleased to include this very scholarly account of the incorporation of Fourier transforms into Clifford analysis. In this long chapter, a short monograph on the subject, the authors recapitulate the basic ideas of Clifford analysis and then explain in detail how the various forms of the Fourier transform are integrated into it. I am convinced that this will soon become the standard presentation of the subject. The third chapter also deals with a subject of great current interest, the use of carbon nanotubes as sources for electron microscopy. This is a very practical chapter, with sections on mounting and preparing the nanotubes, electron microscope studies of such sources and measurement of brightness. There is also a section on the physics of the emission process. Readers will be able form a balanced assessment of the current situation. These sources are in full development and perhaps a successor will appear in these pages in a few years time, conﬁrming N. de Jonge’s forecasts for the future of nanotube electron sources. The ﬁnal long chapter by E. Recami and M. Zamboni-Rached, gives an account of localized waves, or non-diffracting waves as they are often called. These have a vast literature and are of practical importance – optical tweezers are perhaps the best known example. The authors take us through

vii

viii

Preface

the theory with many helpful explanations and provide guidance through the many publications on the subject. In conclusion, I thank all the authors for the trouble they have taken to make their subjects accessible to readers outside their particular ﬁeld. Peter W. Hawkes

CONTRIBUTORS Vasileios Argyriou and Maria Petrou Communications and Signal Processing Group, Electrical and Electronic Engineering Department, Imperial College London SW7 2AZ, United Kingdom

1

Fred Brackx, Nele De Schepper, and Frank Sommen Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Ghent, Belgium

55

Niels de Jonge Oak Ridge National Laboratory, Materials Science and Technology Division, Oak Ridge, TN 37831-6064, USA

203

Vanderbilt University Medical Center, Department of Molecular Physiology and Biophysics, Nashville, 37232-0615, USA Erasmo Recami Faculty of Engineering, University of Bergamo, Bergamo, Italy; and INFN, Sezione di Milano, 24044 Bergamo, 20133 Milan, Italy

235

Michel Zamboni-Rached Center of Natural and Human Sciences, Federal University of ABC, Santo André, SP, Brazil

235

ix

FUTURE CONTRIBUTIONS

S. Ando Gradient operators and edge and corner detection K. Asakura Energy-Filtering x-ray PEEM W. Bacsa Optical interference near surfaces, sub-wavelength microscopy and spectroscopic sensors C. Beeli Structure and microscopy of quasicrystals C. Bobisch and R. Möller Ballistic electron microscopy G. Borgefors Distance transforms Z. Bouchal Non-diffracting optical beams A. Buchau Boundary element or integral equation methods for static and time-dependent problems B. Buchberger Gröbner bases E. Cosgriff, P. D. Nellist, L. J. Allen, A. J. d’Alfonso, S. D. Findlay, and A. I. Kirkland Three-dimensional imaging using aberration-corrected scanning confocal electron microscopy T. Cremer Neutron microscopy P. Dombi (Vol. 158) Ultra-fast monoenergetic electron sources A. N. Evans Area morphology scale-spaces for colour images A. X. Falcão The image foresting transform R. G. Forbes Liquid metal ion sources B. J. Ford (Vol. 158) The earliest microscopical research

xi

xii

Future Contributions

C. Fredembach Eigenregions for image classiﬁcation J. Giesen, Z. Baranczuk, K. Simon, and P. Zolliker Gamut mapping J. Gilles (Vol. 158) Noisy image decomposition A. Gölzhäuser Recent advances in electron holography with point sources M. Haschke Micro-XRF excitation in the scanning electron microscope L. Hermi, M. A. Khabou, and M. B. H. Rhouma Shape recognition based on eigenvalues of the Laplacian M. I. Herrera The development of electron microscopy in Spain J. Isenberg Imaging IR-techniques for the characterization of solar cells K. Ishizuka Contrast transfer and crystal images A. Jacobo Intracavity type II second-harmonic generation for image processing L. Kipp Photon sieves G. Kögel Positron microscopy T. Kohashi Spin-polarized scanning electron microscopy R. Leitgeb Fourier domain and time domain optical coherence tomography B. Lencová Modern developments in electron optical calculations H. Lichte New developments in electron holography M. Mankos High-throughput LEEM M. Matsuya Calculation of aberration coefﬁcients using Lie algebra S. McVitie Microscopy of magnetic specimens

Future Contributions

I. Moreno Soriano and C. Ferreira Fractional Fourier transforms and geometrical optics M. A. O’Keefe Electron image simulation D. Oulton and H. Owens Colorimetric imaging N. Papamarkos and A. Kesidis The inverse Hough transform K. S. Pedersen, A. Lee, and M. Nielsen The scale-space properties of natural images E. Rau Energy analysers for electron microscopes G. Rudenberg The work of R. Rüdenberg R. Shimizu, T. Ikuta, and Y. Takai Defocus image modulation processing in real time S. Shirai CRT gun design methods A. S. Skapin The use of optical and scanning electron microscopy in the study of ancient pigments T. Soma Focus-deﬂection systems and their applications P. Sussner and M. E. Valle Fuzzy morphological associative memories S. Svensson (Vol. 158) The reverse fuzzy distance transform and its applications I. Talmon Study of complex ﬂuids by transmission electron microscopy M. E. Testorf and M. Fiddy Imaging from scattered electromagnetic ﬁelds, investigations into an unsolved problem N. M. Towghi Ip norm optimal ﬁlters E. Twerdowski Defocused acoustic transmission microscopy Y. Uchikawa Electron gun optics

xiii

xiv

Future Contributions

K. Vaeth and G. Rajeswaran Organic light-emitting arrays M. van Droogenbroeck and M. Buckley (Vol. 158) Anchors in mathematical morphology V. Velisavljevic and M. Vetterli Space-frequence quantization using directionlets M. H. F. Wilkinson and G. Ouzounis Second generation connectivity and attribute ﬁlters D. Yang, S. Kumar, and H. Wang (Vol. 158) Time lenses M. Yavor (Vol. 157) Optics of charged particle analysers P. Ye Harmonic holography

CHAPTER

1 Photometric Stereo: An Overview Vasileios Argyriou and Maria Petrou*

Contents

1 2 3 4 5

Introduction Definitions Radiometry and Image Formation Overview Bidirectional Reflectance Distribution Function Reflectance Models 5.1 Lambertian Reflectance Model 5.2 Phong Reflectance Model 5.3 Dichromatic Reflectance Model 6 Surface-Recovering Methods 6.1 Binocular Stereo 6.2 Shape From Shading from a Single Image 6.3 Photometric Stereo 7 Color Photometric Stereo [Assumption (1)] 8 Photometric Stereo with Highlights and Shadows [Assumptions (4) and (5)] 9 Photometric Stereo with an Extended Light Source [Assumption (2)] 10 Dynamic Photometric Stereo and Motion [Assumption (6)] 11 Perspective Photometric Stereo [Assumption (3)] 12 Photometric Stereo Error Analysis 12.1 Errors in the Image Acquisition Stage 12.2 Sensitivity Analysis of Surface Normals 13 Optimal Illumination Configuration for Photometric Stereo 14 Conclusions Acknowledgments References

2 3 6 9 11 12 14 15 17 17 18 20 26 26 28 30 33 38 39 40 46 51 52 52

* Communications and Signal Processing Group, Electrical and Electronic Engineering Department, Imperial College London SW7 2AZ, United Kingdom Advances in Imaging and Electron Physics, Volume 156, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01401-8. Copyright © 2009 Elsevier Inc. All rights reserved.

1

2

Vasileios Argyriou and Maria Petrou

1. INTRODUCTION Although it has long been known that shading in images provides an important depth cue, only relatively recently have surface orientation and depth recovery problems been properly formulated and solved sufﬁciently reliably for commercial applications. The earliest work on the quantitative use of shading information was apparently in the mid-1960s (Hapke, 1963; Orlova, 1956; Rindﬂeisch, 1966) on recovering the shape of parts of the lunar surface in preparation for human exploration of the moon. This chapter presents an overview of the state-of-the-art methods for three-dimensional (3D) shape reconstruction, starting with a tutorial on the basic understanding of the problem with emphasis on photometric stereo technology. Photometric stereo is a method for recovering local surface shape and albedo from a number of images captured under different illumination directions. The photometric stereo method is simple to implement. However, a brief introduction on radiometry, image formation, and surface description models is essential to understanding its underlying principles. Section 2 deﬁnes the imaging geometry and provides some basic definitions of the necessary geometric equations. Section 3 deﬁnes some basic concepts of radiometry and derives the fundamental equation that connects the brightness of an imaged surface patch with the image brightness of the corresponding pixel. The interaction of light with a surface is modeled by the so-called bidirectional reﬂectance distribution function (BRDF), which is discussed in Section 4. Some well-known in computer vision reﬂectance models are introduced in Section 5. Surface recovering methods—either from binocular stereo or a single image (shape from shading)—are presented in Section 6. In Section 6 we also reﬁne the term reﬂectance map, which relates image irradiance to surface orientation, for a given light source direction and surface reﬂectance, since it is fundamental in photometric stereo and image rendering for the description of the shape of an object. Section 7 discusses extensions of photometric stereo for color images. Sections 8 and 9 deal with the relaxation of some of the most restrictive assumptions of photometric stereo: reconstruction methods when highlights and shadows are present (Section 8) or the lights are at a ﬁnite distance from the surface (Section 9). Photometric stereo requires the reconstructed object to remain still during the acquisition stage, and this limitation is discussed in Section 10. Section 11 reviews the solutions proposed for the problems introduced due to the invalid assumption of orthographic projection. In Section 12 the error analysis literature on photometric stereo is reviewed in relation to errors due to acquisition or illumination calibration, and some ways of reducing them are reported. Finally, solutions to the problem of optimal illumination conﬁguration are

3

Photometric Stereo: An Overview

reviewed in Section 13, based on image rendering and the properties of the reﬂectance maps. Conclusions are drawn in Section 14.

2. DEFINITIONS Figure 1 shows a simple imaging system. The viewing direction is aligned with the z-axis of the coordinate system, pointing toward the camera, since a right-handed coordinate system is assumed. The x and y axes are perpendicular to the viewing direction with the origin of the system at the intersection of the z-axis and the observed plane. The shape of a surface may be modeled either by using the height of every point above a reference plane or the gradient vector of the surface at each position. Various active vision methods (e.g., laser scanners or radar-based systems) usually recover a height map for the imaged surface. Conventional stereo also recovers a height (or depth) map using disparity measures. Conversely, photometric stereo recovers the normal vector ﬁeld of the surface, modeling the surface as a collection of ﬂat patches, each corresponding to a pixel of the captured images. The general equation of a 3D plane is given by

Ax + By + Cz + D = 0 ⇒

B D A x+ y+z+ = 0, C C C

(1)

where the ﬁrst partial derivatives of z with respect to x and y correspond to the components of the slope of the surface, pn and qn , respectively, as

Camera Illumination source

z

L (2pl , 2ql , 1)

␪l ␪i

Normal to the average surface

V

N (2pn ,2qn ,1)

␪n

y x

␸l

L9

␸n

N9

is FIGURE 1 Imaging geometry. L is the incident illumination direction vector, and N the normal to the surface vector. V is the viewing direction.

4

Vasileios Argyriou and Maria Petrou

follows:

A ∂z = − = −pn ∂x C B ∂z = − = −qn . ∂y C

(2)

T A B − ,− ,1 = (−pn , −qn , 1)T is normal to the plane C C (Figure 2a). An alternative deﬁnition, which is also useful in this chapter, is based on the representation of a plane by equation r = a + αb + βc, where r is the position vector of any point on the plane, a is the position vector of a speciﬁc known point on the plane, and b and c are two nonparallel vectors lying on the plane (Figure 2b). Then the normal to the plane is given by the cross product b × c. For example, if the nonparallel vectors lying on the plane are chosen as

Vector

b = (1, 0, pn )T c = (0, 1, qn )T ,

(3)

a normal vector to the surface is

b × c = (−pn , −qn , 1)T .

(4)

z

2D/C

N z c

q 2B/C 2D/A

y

y

p N

1 b

a

1

x

(a)

x

(b)

FIGURE 2 (a) A plane and its normal vector. (b) The slope of a surface patch may be expressed in terms of p and q. See text for details.

Photometric Stereo: An Overview

5

We deﬁne the surface normal to be of unit length, so T n , −qn , 1) = (−p N . p2n + q2n + 1

(5)

In terms of azimuth angle ϕn (measured in the xy-plane counterclockwise from the x-axis) and the zenith angle θn (measured between the surface normal and the z-axis), the surface slope components may be written as (Figure 1)

pn = − cos ϕn tan θn qn = − sin ϕn tan θn .

(6)

Furthermore, orthographic projection is assumed; that is, the camera is considered to be far from the surface relative to the size of the surface. The light source is regarded as a point source at inﬁnite distance and therefore constant illumination is assumed over the entire scene. These assumptions are relaxed in Sections 9 through 11, when we review some methods of photometric stereo designed to cope with non-orthographic projection and an illumination source of ﬁnite extent at a ﬁnite distance from the surface. The normal vector of a surface perpendicular to the light rays is used to specify the direction of the illumination source (−pl , −pl , 1). The unit light source direction vector is then given by T l , −ql , 1) = (−p . L 2 2 pl + ql + 1

(7)

The components of this vector in terms of angles ϕl and θl (deﬁned in Figure 1) are

pl = − cos ϕl tan θl ql = − sin ϕl tan θl

(8)

In terms of the zenith and azimuth angles, the unit illumination vector may be written as

= (sin θl cos ϕl , sin θl sin ϕl , cos θl ), L

(9)

where we used Eqs. (7) and (8) and the fact that cos θl = 1/( 1 + p2l + q2l ).

6

Vasileios Argyriou and Maria Petrou

L

V

N

V

L

N Cast-shadow (a)

Self-shadow

(b)

FIGURE 3 Masking (a) and self- and cast-shadows (b). V is the viewing direction, L is is the surface normal. the illumination direction, and N

The effect of obstructions in the path of the reﬂected light is called masking or occlusion, whereas the term shadowing is used to describe obstructions in the path of the incident light, (Figure 3). The masking effects may be neglected only when the viewer is in the zenith of a ﬂat surface. Shadows may be distinguished as self- and cast-shadows. Self-shadows occur when the to the surface facet and the illumination angle between the normal vector N ◦ is larger then 90 . The term cast-shadow is used when the shadow vector L is projected from another facet (Figure 3).

3. RADIOMETRY AND IMAGE FORMATION OVERVIEW The image intensity of a pixel is a function of the orientation of the corresponding surface patch (surface normal), the reﬂectance of the material from which the surface is made or with which the surface is coated, and the spectrum and direction of the illumination used to capture the image. In order to recover the shape and the surface reﬂectance properties of an object, we need to understand how the intensity of each image pixel depends on all these factors. Some concepts relevant to this chapter are introduced here. First, we introduce the concept of the solid angle subtended by a surface patch. A solid angle, subtended by an area dA on the surface of a sphere of radius R, has its vertex at the center of the sphere and is equal to dω steradian (sr) measured as dω = dA /R2 . Thus, the solid angle subtended by a surface patch dA with normal vector forming angle θk with the radius of the sphere is

dω =

dA cos θk , R2

(10)

Photometric Stereo: An Overview

7

N R

␪i

d␻

dA

dA9

FIGURE 4

Solid angle subtended by a surface patch of area dA.

since dA cos θk is the projected area of patch dA on the plane orthogonal to the radius. According to the National Bureau of Standards (Nicodemus et al., 1977), the radiant ﬂux d is deﬁned as the power propagated as optical electromagnetic radiation and is measured in watts (W). The incident light power per unit surface area is called irradiance E and is measured in watts per square meter of surface (Wm−2 )

E=

d , dA

(11)

where d is the light power received by surface dA. Radiance is the amount of light emitted by a particular area within a given solid angle in a speciﬁed direction. Surface radiance (brightness) I, in a direction forming angle θn with respect to the surface normal, is the energy emitted per unit solid angle and per unit area orthogonal to direction θn :

I=

d ⇒ d = IdA cos θn dω. dA cos θn dω

(12)

Surface radiance is measured in watts per square meter per steradian (Wm−2 sr−1 ), (Figure 5). First, we show that the brightness recorded by the sensor used is directly proportional to the brightness of the imaged surface. This then allows use of the captured digital image to draw conclusions about the physical imaged surface. For the purpose of this argument, the E in Eq. (11) will be taken to represent the light that falls on the imaging sensor, while the I in Eq. (12)

8

Vasileios Argyriou and Maria Petrou

dF N

␪n

d␻

dA

dA9

FIGURE 5

Surface radiance (brightness).

will be taken to represent the light that leaves the imaged surface. We shall make this explicit by using the right sufﬁxes in E and I. Assuming that the lens does not absorb any light, the d in Eqs. (11) and (12) will be the same. In order to ﬁnd the relationship between the surface radiance I and image irradiance E, the area of the region in the image dAi and the area of the corresponding patch on the surface dAs must be determined. Based on Eqs. (11) and (12) the irradiance of the image is given by

Eon_image =

Ifrom_surface dAs cos θn dω d = , dAi dAi

(13)

where dω is the solid angle subtended by the lens, as seen from the scene patch. For a surface patch in direction α with respect to the lens axis, the surface of a lens of diameter d orthogonal to the direction of the patch is π( d2 )2 cos α, whereas the distance of the surface patch from the center of the lens is z0 / cos α. Thus, dω is given by

dω =

π

2 d 2

cos α π d 2 ⇒ dω = cos3 α, 2 4 z0

z0 cos α

(14)

where z0 is the distance of the scene patch from the lens along the z-axis, (Figure 6).

9

Photometric Stereo: An Overview

␪n

N

dAs

d␻s

Ri

␣

␣

Rs

d d␻

dAi d␻i

Image

Lens Scene object f

z0

FIGURE 6 The image irradiance is proportional to the scene radiance.

Because the solid angles subtended by the image patch and the scene object patch are equal (Figure 6), we obtain

dAi cos α dAs cos θn dAs cos α dωi = dωs ⇒ = ⇒ = dAi cos θn ( f / cos α)2 (z0 / cos α)2

z0 f

2 . (15)

Substituting from Eqs. (14) and (15) into Eq. (13), we obtain

Eon_image

π = Ifrom_surface 4

2 d cos4 α. f

(16)

Thus, the image irradiance E, which corresponds to our notion of “brightness,” is proportional to the scene radiance I. Therefore, E depends on the amount of light that falls on the imaged surface and the reﬂected fraction of it. It also depends through I on the illumination and viewing directions. Here E is what falls on the ﬁlm and I is what leaves the surface. Therefore, all we do is modeling what happens from the moment the light leaves the surface until the moment it reaches the sensor. The next section discusses what happens to the light from the moment it leaves the light source until the moment it leaves the surface. It is that part of the light’s journey from the light source to the sensor that depends on the physical properties of the imaged surface and the imaging geometry.

4. BIDIRECTIONAL REFLECTANCE DISTRIBUTION FUNCTION The reﬂectance models developed in the ﬁeld of computer vision are classiﬁed into two main categories: physical models and geometrical models. The latter have simple forms and are derived by analyzing the surface and illumination geometry. In contrast, the physical models are based on the

10

Vasileios Argyriou and Maria Petrou

electromagnetic wave theory providing accurate and detailed descriptions, but requiring complex manipulations and signiﬁcant prior information concerning the imaged scene. This makes them inappropriate for computer vision applications. Reﬂectance models are generally presented in terms of the BRDF, relating the incident light energy from the direction of illumination Eon_surface (λ, θl , ϕl ), to the reﬂected brightness Ifrom_surface (λ, θn , ϕn ) in the direction of the viewer:

fBRDF (λ, θl , ϕl , θn , ϕn ) ≡

Ifrom_surface (λ, θn , ϕn ) Eon_surface (λ, θl , ϕl )

.

(17)

Note that fBRDF has units of sr−1 because I is energy per unit area per unit solid angle, measured in W m−2 sr−1 , whereas E is energy per unit area, measured in W m−2 . In this expression, I is the same as the I in Eqs. (12) and (16), but we have made explicit here its dependence on the wavelength λ and the viewing direction with respect to the normal to the surface. E(λ, θl , ϕl ) is different from the E in Eq. (16), and again we have made explicit here its dependence on the wavelength λ and the direction of illumination (θl , ϕl ) (Figure 7). The E in Eq. (16) is the light that falls on the sensor cell, whereas the E in Eq. (17) is the light that falls on the imaged surface. The dependence of fBRDF on λ, θl , ϕl , θn , and ϕn is complicated. As it is difﬁcult to deﬁne models based on the exact physical processes that take place, empirical models of fBRDF are used. Accurate models of the BRDF are N L

V E

␪l

␪n

␸n

I

␸l

y

x

FIGURE 7 Geometry of light reflection. E(θl , ϕl ), irradiance due to source in direction (θl , ϕl ). I(θn , ϕn ), radiance of the surface in direction (θn , ϕn ). V is the vector of the viewing direction, and L is the vector of the direction of the illuminating source.

Photometric Stereo: An Overview

11

generally complex with many parameters. However, various simplifying assumptions may be made. One of the simpliﬁcations is separating the radiometric from the geometric dependence on the right-hand side of Eq. (17), which therefore may be written as

fBRDF (λ, θl , ϕl , θn , ϕn ) = r(λ)

I˜from_surface (θn , ϕn ) . E˜ on_surface (θl , ϕl )

(18)

Here r(λ), also known as surface albedo, is the reﬂectance coefﬁcient that expresses the fraction of light energy reﬂected by the surface as a function of the wavelength of the incident light. We have used a tilde to distinguish between radiometric quantities I and E, that depend on wavelength and ˜ that depend only on the geometry, from radiometric quantities I˜ and E, geometry. Albedo r(λ) is often treated as a constant, independent of λ. In the section on photometric stereo, we shall see how it may be recovered up to a multiplicative factor, for the wavelengths at which the sensors we use have nonzero response. Another simplifying assumption in modeling f BRDF is that each surface facet is isotropic, since for many surfaces the radiance is not altered if the surface is rotated about its normal (He et al., 1991). In that case, the BRDF depends on the difference of the two azimuth angles ϕn − ϕl , and not explicitly on each angle. Based on these simpliﬁcations, many BRDF models have been presented over the past two decades, but only the Lambertian and the Phong models have enjoyed widespread use for modeling smooth surfaces and surfaces with specular and diffuse reﬂections, respectively.

5. REFLECTANCE MODELS The Lambertian and Phong models have been widely used to model surface reﬂectance properties in computer vision. The Lambertian model, often used to model reﬂectances of matte appearance, is simpler and predicts that light incident on a surface is reﬂected equally in all directions. Phong’s model (Phong 1975) has found great acceptance within the computer graphics community and has become the industry standard. In this section, we discuss both models in detail. Finally, we present the dichromatic reﬂectance model, which is a physical model, and we show that both the Lambertian and Phong models are special cases of it (Klinker, Shafer, and Midha, 1988; Shafer, 1985).

12

Vasileios Argyriou and Maria Petrou

5.1. Lambertian Reflectance Model A Lambertian surface appears equally bright from any direction, for any illumination direction. Therefore, its reﬂected intensity is independent of the viewing direction. The BRDF function that can be deduced from this deﬁnition must be equal to a constant: fBRDF (λ, θl , ϕl , θn , ϕn ) = r(λ) × constant. In order to determine the value of this constant, we proceed as follows. An elementary patch dB on the surface of a sphere of radius R subtends a solid angle dω = RdB2 with respect to the center A of the sphere (Figure 8). However, an elementary patch of the same area (dA = dB), θn placed at the center of the sphere, subtends an angle d = dA cos as seen R2 from point B. Thus, we may say d = dω cos θn . In terms of spherical coordinates, dω = sin θn dθn dϕn , so d = sin θn cos θn dθn dϕn . Note that d is the solid angle of the viewer. To estimate how much light point B receives, we must integrate f BRDF over all values of d, since f BRDF expresses the fraction of energy received by the viewer per unit sterad. Under the Lambertian assumption, I˜from_surface (θn , ϕn ) = E˜ on_surface (θl , ϕl ) (i.e., the light the viewer sees is independent of the imaging geometry), and so, from Eq. (18) we have

fBRDF (λ)d = r(λ)

⇒ fBRDF (λ)

π

−π

π/2

0

⇒ 2πfBRDF (λ)

sin θn cos θn dθn dϕn = r(λ)

π/2

0

⇒ πfBRDF (λ)

sin θn cos θn dθn = r(λ)

π/2

0

sin 2θn dθn = r(λ)

⇒ fBRDF (λ) =

r(λ) . π

(19)

N dB ␪n

B

d␻ d⍀

dA A

FIGURE 8

Point B receives the light emitted by all facets of elementary patch dA.

Photometric Stereo: An Overview

13

Thus, using the BRDF function for a Lambertian surface, we obtain from Eq. (17) that the surface radiance is proportional to the surface irradiance

Ifrom_surface (λ) =

r(λ) r(λ) Eon_surface (λ) = Ifrom_source (λ) cos θi for cos θi ≥ 0, π π (20)

where Ifrom_source indicates the source radiance. The dependence on the incident angle θi comes from the geometry between the illuminating source and the surface. If the surface is face-on opposite the illuminating source =L in Figure 7), whatever light energy leaves the source reaches (i.e., N = L, the surface will receive only as much light the surface. If, however, N ⊥ L, the light from the as its effective area is in the path of the light. If N source will just graze the surface (cos θi = 0) and the surface will receive no light energy. In that case, the surface is expected to appear totally black to the viewer. Equation (20) may be combined with Eq. (13) to allow us to say that the brightness observed on the image is directly proportional to the cosine of the angle between the surface normal and the direction of illumination. We wrap up all constants of proportionality that appear in these equations into a single function ρ(λ) and refer to this as the albedo of the imaged surface, all other factors ignored.1 Thus, we may write

· L) I(λ, x, y) = Ifrom_source (λ)ρ(λ) cos θi = ρ(λ)(N 1 + p l pn + q l qn = ρ(λ) 1 + p2n + q2n 1 + p2l + q2l = ρ(λ)

−pn cos ϕl sin θl − qn sin ϕl sin θl + cos θl , p2n + q2n + 1

(21)

where the following apply: • I(x, y) is the image intensity; is the unit vector normal to the surface z(x, y) at point (x, y); • N • pn and qn are partial derivatives of the surface with respect to the x and y coordinates, respectively [see Eq. (2)]; is the unit vector toward the light source (Figure 1); • L • pl and ql are the partial derivatives of a virtual planar surface orthogonal to the direction of the illumination, with respect to the x and the y coordinates, respectively [see Eq. (7)]; 1 We may ignore factors of proportionality in brightness because the gray values of an image have signiﬁ-

cance only in relation to each other (i.e., in relative terms and not in absolute terms). Thus, r(λ) of Eq. (18) and ρ(λ) of Eq. (21) are both the surface albedo, up to a constant of proportionality.

14

Vasileios Argyriou and Maria Petrou

IIIumination source

Directional diffuse Specular

Uniform

FIGURE 9 Components of the Phong reflectance model.

• θl and ϕl are the illumination vector’s zenith and azimuth angles deﬁned in Figure 1; • ρ(λ) is the surface albedo, and • λ is the wavelength of the light.

5.2. Phong Reflectance Model Phong’s model is more complete and realistic compared with the diffuse reﬂectance model. The linear combination of three components—namely, diffuse, specular, and ambient components—constitute the Phong model. The diffuse component is nothing but a Lambertian component. The specular component forms a lobe of reﬂected light that spreads out around the specular direction and is modeled by a cosine function raised to a power. The ambient component accounts for the ambient light caused by interreﬂections. It is this light that causes the shadows not to be black.2 This model is shown schematically in Figure 9 and mathematically expressed as

I(λ, n, β) = Iα (λ)kα (λ) + Ifrom_source (λ)ρ(λ) cos θi + Eon_surface (λ)ks (λ) cosn β · N) = Iα (λ)kα (λ) + Ifrom_source (λ)ρ(λ)(L · V) n, + Eon_surface (λ)ks (λ)(R

(22)

where Iα (λ) is the ambient constant light intensity, kα (λ) is the reﬂectance coefﬁcient of the surface, and ks (λ) is the specular reﬂectance coefﬁcient. β and the perfect reﬂector denotes the angle between the viewing direction V vector R, and n controls the width of the specular lobe. is given by R = 2N( N · L) −L It can be determined that direction R (Figure 10). This computation is time consuming, so angle β in the model is halfway between often is replaced by angle α (Figure 10). Vector H 2 In the Lambertian model, the shadows are black.

Photometric Stereo: An Overview

R N·H

H N a ␪9

␪9

R·V V

15

␤

␪

L

␪

and the viewing direction, and its FIGURE 10 Angle β between the specular R and normal N. approximation by the angle α between the halfway vector H

and V vectors L

= L+V. H 2

(23)

This is the required orientation for a surface to reﬂect light along direction V. lie in the same plane (Blinn, 1977), α is equal to half of If V, L, and N − θ = β2 . Thus, using angle α instead of angle angle β: α = θ − θ = θ+θ+β 2 β in Eq. (22) changes the model. However, because this is an empirical model, exponent n can be adjusted to achieve the desirable effect. As α is smaller than β, cos α is larger than cos β, and to achieve the same effect as when using β, we must increase exponent n to n˜ > n:

· N) I(λ, n, α) = Iα (λ)kα (λ) + Ifrom_source (λ)ρ(λ)(L · H) n˜ . + Eon_surface (λ)ks (λ)(N

(24)

This approximation means that intensity I is solely a function of the surface If we assume the viewer and the light are at inﬁnity, then the halfnormal N. is independent of position and surface curvature. Therefore, angle vector H it can be calculated once for each light source and used for the entire frame, or while the viewpoint and the light remain in the same relative position. depends on In Phong’s original model [Eq. (22)], the reﬂected light vector R the surface curvature and must be calculated for each pixel of the rendered image.

5.3. Dichromatic Reflectance Model The Dichromatic reﬂectance model is a physical model that attempts to model the physical process that occurs when light reaches the surface of an opaque object. According to this theory, the light I, which is reﬂected from a point of the surface of an object, is a mixture of the light Ispec reﬂected by the surface itself (surface, or specular, reﬂection) and the light Ibody reﬂected

16

Vasileios Argyriou and Maria Petrou

by the body of the object (body, or Lambertian, or matte reﬂection):

I(λ, θi , θl , θn ) = Ispec (λ, θi , θl , θn ) + Ibody (λ, θi , θl , θn ) = ms (θi , θl , θn )w(λ) + mb (θi , θl , θn )c(λ).

(25)

Function Ispec (λ, θi , θl , θn ) is assumed separable (Nicodemus et al., 1977) and it may be written as the product of a spectral power distribution, w(λ), and a geometric scale factor, ms (θi , θl , θn ). Similarly, the body reﬂection component Ibody (λ, θi , θl , θn ) is also assumed to be a separable function, and it may be written as the product of a spectral power distribution, c(λ), and a geometric scale factor, mb (θi , θl , θn ). The light reﬂected by the surface has approximately the same spectral power distribution as the light source. The light that is not reﬂected by the surface penetrates into the material body where it is scattered and selectively absorbed. Some fraction of the light arrives again at the surface and is re-emitted. The light traveling through the body is increasingly absorbed at wavelengths that are characteristic of the material of the viewed object. The body reﬂectance provides the characteristic object color (Klinker, 1993). By comparing Eq. (25) with Eq. (21), we may easily infer that the Lambertian model assumes that

w(λ) = 0 mb (θi , θl , θn ) = cos θi c(λ) = Ifrom_source (λ)ρ(λ).

(26)

Note that under the assumption of a white light illumination source, Ifrom_source (λ) = constant, and the color of the object c(λ) is truly the observed color ρ(λ) by the imaging sensor. Under any other light source, we need to know exactly the spectrum of the light Ifrom_source (λ) to work out c(λ) from ρ(λ). Comparing Eq. (25) with Eq. (22), we see that according to the Phong model

w(λ) = Eon_surface (λ)ks (λ) ms (θi , θl , θn ) = (2 cos θi cos θn − cos θl )n c(λ) = Ifrom_source (λ)ρ(λ) mb (θi , θl , θn ) = cos θi .

(27)

The Phong model adds also a term of uniformly distributed ambient light, which could be due to the object itself or to other objects. One might consider Iα (λ) as an extra illuminating source, which, however, is diffuse and

Photometric Stereo: An Overview

17

nondirectional. The effect of such a light on the surface cannot possibly depend on angles θi and θl , because these angles cannot be deﬁned. Thus, this term may be considered another body (Lambertian) component, with geometric factor mbα = 1 and spectral factor cα (λ) = Iα (λ)c(λ).

6. SURFACE-RECOVERING METHODS We may state that we know the shape of a surface if we know the normal or vector (−pn (x, y), −qn (x, y), 1)T of every facet (x, y) that makes vector N up the surface [see Eqs. (4) and (5)]. Alternatively, we may say that we know the shape of a surface if we know the distance of every point (x, y) of the surface from some reference plane. However, a unique facet orientation or point distance cannot be determined from a single image intensity or radiance value observed, since an inﬁnite number of surface orientations can give rise to the same value of image intensity. This is obvious since the recorded brightness of a facet is only one piece of information, whereas the facet orientation (−pn (x, y), −qn (x, y), 1) has two degrees of freedom. Therefore, we need additional information to determine local surface orientation. Several approaches exist for the recovery of surface relief, including binocular stereo, shape from shading, and photometric stereo.

6.1. Binocular Stereo Binocular stereo is used to recover surface distance by identifying corresponding points in two images taken from different viewpoints. Stereoscopic vision has been used successfully in cartography and robot navigation but it has several drawbacks. The difﬁculty of applying binocular stereo arises from reliably determining the corresponding features between two separate images and, if surface detail is the main concern, the problem becomes more evident. In addition, surface depth is recovered rather than surface orientation, as illustrated in Figure 11. This introduces noise and artifacts. The surface depth z is obtained as

z=

f d , x1 − x 2

(28)

where f is the camera focal length, d the distance between the two cameras, and x1 and x2 are the coordinate positions of the images of the same physical point on the two focal planes of the cameras, with respect to coordinate systems that are fully aligned, but each is centered on the axis of the corresponding camera lens.

18

Vasileios Argyriou and Maria Petrou

Scene object (x, y, z )

z

f

d

(x1, y1)

(x2, y2) Left and right image plane

FIGURE 11

Geometry of binocular stereo method.

6.2. Shape From Shading from a Single Image The topic of shape from shading (SFS) deals with the determination of an object’s shape solely from the intensity variation in the image plane. For a Lambertian surface and for a given illumination direction (ϕl , θl ), Eq. (21) may be viewed as a function that relates the brightness of a pixel (x, y) with the orientation (−pn , −qn , 1) of the facet of the surface depicted by the pixel. If we multiply both sides of Eq. (21) with the sensitivity function s(λ) of the imaging sensor we use and integrate over all λ, the left-hand side becomes the brightness of pixel (or sensor element) (x, y) as recorded by the imaging sensor we use. On the right-hand side integration over λ results in yet another version of the albedo of the surface, this time closer to what can actually be measured. Thus, Eq. (21) may eventually be simpliﬁed to

I(x, y) = ρS (x, y)

−pn (x, y) cos ϕl sin θl − qn (x, y) sin ϕl sin θl + cos θl , p2n (x, y) + q2n (x, y) + 1 (29)

where we have made explicit the dependence of pn , qn , and ρS on the coordintates of the facet they refer to, and ρS ≡ λ ρ(λ)s(λ)dλ. Note that ρS depends on the sensor characteristics and I now may be treated as the gray value of the image produced by this sensor. If the sensor sensitivity function s(λ) were a delta function, ρS would have been the value of the true albedo r(λ) of Eq. (18) of the imaged surface at point (x, y), at the wavelength to which the sensor we use is sensitive, times some constant of proportionality. The right-hand side of Eq. (29) is a function of pn (x, y) and

Photometric Stereo: An Overview

19

qn

Iso-brightness contour R (pn, qn) is maximum when (pn, qn) ⫽ (pl, ql) I ⫽ 0.8

Shadow line

I ⫽ 0.9 I ⫽ 1.0 (pl, ql) pn pl ⫹ qn ql ⫹ 1 ⫽ 0

pn I ⫽ 0.7

I ⫽ 0.5

I ⫽ 0.0

I ⫽ 0.2

I ⫽ 0.3

FIGURE 12 Reflectance map R(pn , qn ) for a Lambertian surface illuminated from a direction perpendicular to the point with pl = 0.7 and ql = 0.3 and for ρS = 1. All points on each iso-brightness contour are solutions of Eq. (29) that relates image intensity to surface orientation. The straight line corresponds to the so-called shadow line. Points along this line have gradient vector orthogonal to the illumination direction, so they receive only grazing light. All points with orientations on the left of that line receive no light and so they have I = 0.

qn (x, y) and may be called R(pn (x, y), qn (x, y)). If we plot qn (x, y) versus pn (x, y) for certain values of I(x, y), we produce the so-called reﬂectance map (Figure 12). An analogous analysis may be made for the Phong model, expressed either by Eq. (22) or Eq. (24). In all cases, Eqs. (21), (22), or (24) may be expressed as

I(x, y) = R(pn (x, y), qn (x, y)).

(30)

The above equation has two unknowns, namely, pn (x, y) and qn (x, y), and, therefore, determining object shape from image intensity alone is impossible. For a given reﬂectance model and given imaging geometry, function R(pn (x, y), qn (x, y)) is a function of the two unknowns and it can be plotted in a two-dimensional (2D) (pn (x, y), qn (x, y)) coordinate system. Figure 12

20

Vasileios Argyriou and Maria Petrou

shows the contours of constant value of R as functions of (pn , qn ) for the Lambertian model. Therefore, if we know the brightness of a pixel, its (pn (x, y), qn (x, y)) values could be any point on the contour of that brightness. Additional constraints are required to calculate object shape (i.e., to identify the (pn (x, y), qn (x, y)) values of every pixel in the image of the object). This was one of the ﬁrst areas of study in computer vision. The initial work was done by Horn (Horn, 1975; Horn, 1986; Horn and Brooks, 1989; Horn and Sjoberg, 1979). Horn’s method of solving this problem relies on growing a solution by starting at a single point in the image plane I(x0 , y0 ) where the surface orientation is known. The single-image shape from shading algorithm is still limited even if the exact lighting conditions and surface reﬂectivity are known. The original shape from shading method has been advanced by many researchers in recent years. Most of these methods rely on the use of a regularization term that controls the smoothness of the created surface (e.g., Bors, Hancock, and Wilson, 2003, 2000a,b,c; Worthington and Hancock, 1997; Worthington and Hancock, 1999; Zhang et al., 1999) or the roughness of the created surface when the surface is known to exhibit fractal characteristics (Liao, Petrou, and Zhao, 2008).

6.3. Photometric Stereo Photometric stereo is a reﬂectance map–based technique that uses two or more images to solve the underdetermined problem of recovering surface shape from a single image. Photometric stereo allows estimation of local surface orientation by using several images of the same surface taken from the same viewpoint but under illuminations from different directions (Woodham, 1980). The light sources are ideally point sources, some distance away in different directions, so that in each case there is a welldeﬁned light source direction. The variation of the intensities observed in an image depends on variation in both surface reﬂectance and surface relief. Although the reﬂectance properties are intrinsic to a surface, the surface relief produces a pattern of shadings that depends strongly on the direction of the illumination. The appearance of a 3D surface changes drastically with illumination. An example is shown in Figure 13. The idea is to use this information to recover the intrinsic surface parameters—that is, local surface orientation and reﬂectance—independent of the illumination direction. Woodham (1980) was the ﬁrst to introduce photometric stereo. He proposed a method that was simple and efﬁcient, but it dealt only with Lambertian surfaces and was sensitive to noise. In Woodham’s method, the surface gradient may be recovered by using two photometric images, assuming that the surface albedo is already known for each point on the surface. Coleman and Jain (1982) extended photometric stereo to four light sources, where specular reﬂections were discarded and estimation

Photometric Stereo: An Overview

(a)

21

(b)

FIGURE 13 An example that shows how the appearance of the same 3D surface changes drastically with illumination direction.

of surface shape could be performed by means of diffuse reﬂections and the Lambertian model. Nayar, Ikeuchi, and Kanade (1990) developed the photometric approach, which uses a linear combination of Lambertian and an impulse specular component to obtain the shape and reﬂectance information for a surface. The simplest approach is to take two images of the same surface under different illumination directions. Therefore, two values of image intensity, I 1 (x, y) and I 2 (x, y), at each point (x, y) can be obtained. In general, the image intensity values of each light source correspond to two contours on the reﬂectance map, namely,

I 1 (x, y) = R1 (pn (x, y), qn (x, y)) I 2 (x, y) = R2 (pn (x, y), qn (x, y)).

(31)

This approach gives a unique solution for surface orientation at almost all points in the image, since the (pn (x, y), qn (x, y)) values of a point are determined by the intersection of the iso-brightness contours (reﬂectance functions) on which it lies (Figure 14). In general, Eqs. (31) are nonlinear, so more than one solution is possible (contours may intersect at two points; twofold ambiguity). One solution would be to construct the reﬂectance function for a third illumination vector (Figure 15)

I 3 (x, y) = R3 (pn (x, y), qn (x, y))

(32)

to remove such ambiguities (Figure 16). This also allows estimation of the third surface parameter, ρS (x, y), which is especially useful in cases where a surface is not uniform in its reﬂectance properties.

22

Vasileios Argyriou and Maria Petrou

qn

pn

FIGURE 14 At least two reflectance maps R1 (pn (x, y), qn (x, y)) and R2 ( pn (x, y), qn (x, y)) are required to determine the surface orientation if the reflectance factor ρS (x, y) is known.

L2

N L3

L1

y

x

FIGURE 15

Illustration of photometric stereo geometry with three lighting directions.

Let us consider a Lambertian surface illuminated in turn by three 1, L 2 , and L 3 . In this case, the illumination sources with directions L intensities of the obtained pixels can be expressed as

k · N(x, y), where k = 1, 2, 3. I k (x, y) = ρS (x, y)L

(33)

Photometric Stereo: An Overview

23

qn

A

pn

FIGURE 16 Three views are sufficient to determine both surface orientation and surface albedo uniquely.

We stack the intensities and the illumination vectors to form the pixel intensity vector I(x, y) = (I 1 (x, y), I 2 (x, y), I 3 (x, y)) and the illumination 1, L 2, L 3 )T . Then Eq. (33) may be rewritten in matrix form as matrix [L] = (L

I(x, y) = ρS (x, y)[L]N(x, y).

(34)

k do not lie in the same plane (nonIf the three illumination vectors L coplanar), then the photometric illumination matrix [L] is nonsingular and can be inverted, giving

y), M(x, y) = [L]−1I(x, y) = ρS (x, y)N(x,

(35)

where M(x, y) = (m1 (x, y), m2 (x, y), m3 (x, y))T . The surface gradient components could be obtained from pn (x, y) = −m1 (x, y)/m3 (x, y) and qn (x, y) = −m2 (x, y)/m3 (x, y), and the surface albedo is recovered by calcu ρS (x, y) = m2 (x, y) + m2 (x, y) + m2 (x, y). lating the length of vector M, 1

2

3

24

Vasileios Argyriou and Maria Petrou

Equation system (34) may also be written using the gradient representation of the surface orientation and direction of light source as

1 + pn (x, y)pl1 + qn (x, y)ql1 I 1 (x, y) = ρS (x, y) 1 + p2n (x, y) + q2n (x, y) 1 + p2l + q2l 1

1

1 + pn (x, y)pl2 + qn (x, y)ql2 I 2 (x, y) = ρS (x, y) 1 + p2n (x, y) + q2n (x, y) 1 + p2l + q2l 2

2

1 + pn (x, y)pl3 + qn (x, y)ql3 I 3 (x, y) = ρS (x, y) . 1 + p2n (x, y) + q2n (x, y) 1 + p2l + q2l 3

(36)

3

Hence, the values of pn (x, y) and qn (x, y) may be determined uniquely by solving the above set of three nonlinear equations, given the image intensity values I 1 (x, y), I 2 (x, y), and I 3 (x, y) and the gradients of the three light sources (−pl1 , −ql1 , 1), (−pl2 , −ql2 , 1), and (−pl3 , −ql3 , 1), respectively. We deﬁne A1 ≡ 1 + p2l + q2l , A2 ≡ 1 + p2l + q2l , and A3 ≡ 1 1 2 2 1 + p2l + q2l . By pairwise division by parts, we may eliminate ρS (x, y) 3 3 from Eq. (36). Then

1 + pn (x, y)pl1 + qn (x, y)ql1 I 1 (x, y)A1 = 2 1 + pn (x, y)pl2 + qn (x, y)ql2 I (x, y)A2 1 + pn (x, y)pl2 + qn (x, y)ql2 I 2 (x, y)A2 = 1 + pn (x, y)pl3 + qn (x, y)ql3 I 3 (x, y)A3 1 + pn (x, y)pl3 + qn (x, y)ql3 I 3 (x, y)A3 = . 1 1 + pn (x, y)pl1 + qn (x, y)ql1 I (x, y)A1

(37)

Cross-multiplication and transposition allows us to write the above equations as

pn (I 2 A2 pl1 − I 1 A1 pl2 ) = I 1 A1 − I 2 A2 + qn (ql2 I 1 A1 − ql1 I 2 A2 ) pn (I 3 A3 pl2 − I 2 A2 pl3 ) = I 2 A2 − I 3 A3 + qn (ql3 I 2 A2 − ql2 I 3 A3 ) pn (I 1 A1 pl3 − I 3 A3 pl1 ) = I 3 A3 − I 1 A1 + qn (ql1 I 3 A3 − ql3 I 1 A1 ),

(38)

where for simplicity, we dropped the explicit dependence on (x, y). Any pair of the above set of equations may be used to yield a value of qn by division by parts and rearrangement. We adopt as value of qn the average

25

Photometric Stereo: An Overview

of these three solutions: 1 qn = 3

+

(I 1 A1 − I 2 A2 )(I 3 A3 pl2 − I 2 A2 pl3 ) − (I 2 A2 − I 3 A3 )(I 2 A2 pl1 − I 1 A1 pl2 ) − I 1 A1 pl2 )(I 2 A2 ql3 − I 3 A3 ql2 ) − (I 3 A3 pl2 − I 2 A2 pl3 )(I 1 A1 ql2 − I 2 A2 ql1 )

(I 2 A2 pl1

(I 2 A2 − I 3 A3 )(I 1 A1 pl3 − I 3 A3 pl1 ) − (I 3 A3 − I 1 A1 )(I 3 A3 pl2 − I 2 A2 pl3 ) − I 2 A2 pl3 )(I 3 A3 ql1 − I 1 A1 ql3 ) − (I 1 A1 pl3 − I 3 A3 pl1 )(I 2 A2 ql3 − I 3 A3 ql2 )

(I 3 A3 pl2

(I 3 A3 − I 1 A1 )(I 2 A2 pl1 − I 1 A1 pl2 ) − (I 1 A1 − I 2 A2 )(I 1 A1 pl3 − I 3 A3 pl1 ) . + 1 (I A1 pl3 − I 3 A3 pl1 )(I 1 A1 ql2 − I 2 A2 ql1 ) − (I 2 A2 pl1 − I 1 A1 pl2 )(I 3 A3 ql1 − I 1 A1 ql3 ) (39)

Each of Eqs. (38) may be solved for pn in terms of qn . The value of pn we adopt is the average of these three solutions:

1 pn = 3

I 3 A3 − I 1 A1 + qn (I 3 A3 ql1 − I 1 A1 ql3 ) I 1 A1 pl3 − I 3 A3 pl1

I 1 A1 − I 2 A2 + qn (I 1 A1 ql2 − I 2 A2 ql1 ) I 2 A2 pl1 − I 1 A1 pl2

I 2 A2 − I 3 A3 + qn (I 2 A2 ql3 − I 3 A3 ql2 ) . + I 3 A3 pl2 − I 2 A2 pl3

+

(40)

From the knowledge of pn (x, y) and qn (x, y) then one may derive from Eq. (36) three values of ρS (x, y), which may be averaged to yield an estimate of the albedo of facet (x, y). Clearly, a more elegant solution would be to solve system (38) in the least-square error sense. Section 11 shows how the least-square error solution may be obtained in the more general case of photometric stereo with perspective projection. The above equations were derived under the following simplifying assumptions: The images used are gray. All pixels have the same illumination vector in each image. The object size is much smaller than the viewing distance. The observed object is directly illuminated without the presence of castor self-shadows. 5. The surface of the object is Lambertian. 6. The imaged surface is static during image capture. 1. 2. 3. 4.

In the sections that follow, we examine how each of these assumptions may be relaxed.

26

Vasileios Argyriou and Maria Petrou

7. COLOR PHOTOMETRIC STEREO [ASSUMPTION (1)] Color images typically consist of three bands identiﬁed with the red, green, and blue parts of the electromagnetic spectrum. In general, if more than one type of sensor is used to image the same scene, there will be several gray bands for each illumination direction, one photometric set for each sensor type. Therefore, we shall, have as many sets of Eqs. (36) as different sensors, identiﬁed by index S. We may solve each such set of equations following the method of the previous section, each time recovering the orientation of each facet and its albedo in the band of the corresponding sensor. The orientations derived in this manner may be averaged to yield a single orientation for each facet (Schluns and Wittig, 1993), while the albedos will correspond to the color of the surface in the bands of the sensors we used. The main problem of this approach is that in most cases the color bands have different intensities and the low-intensity bands are much more prone to noise corruption. Therefore, Barsky and Petrou (2001a,b, 2006) suggested an algorithm that simultaneously recovers the optimal gradient and color estimates taking into consideration the relationship between the color bands. This algorithm is based on the fact that for a Lambertian surface patch, the position of a pixel in the 3D color space is shifted along a line as the level of brightness changes. Introduced errors may disturb the collinearity of the positions of a pixel in the different photometric sets, and, therefore, the principal direction corresponding to the chromaticity of the body color is obtained using principal component analysis, (Figure 17). The problem is then reduced to the monochromatic case by projecting all input pixels on the principal color line. The use of the ﬁrst principal component instead of the individual color bands reduces the problem to that of gray image photometric stereo.

8. PHOTOMETRIC STEREO WITH HIGHLIGHTS AND SHADOWS [ASSUMPTIONS (4) AND (5)] Photometric stereo, in order to recover color and gradient information of a surface patch, assumes that neither highlights nor shadows are present in the images used. In case this assumption is not valid, the surface recovery will be affected, with the body color appearing different and the normal leaning more toward the light source that produced the highlight (Figure 18a and b) or away from the source that produced the shadow (Figure 18c). Coleman and Jain (1982), and Solomon and Ikeuchi (1996) proposed a method of determining highlights in the absence of shadows utilizing four images of the same surface. Combining all the recovered albedos from all four possible triplets of pixels, the surface patch was regarded as a highlight if the albedos differed signiﬁcantly from each other. Barsky

27

Photometric Stereo: An Overview

G⬘

G

B⬘ A3

A1 A2

PCL

R⬘

MCL

B

R

FIGURE 17 An example of reducing color photometric stereo to the monochromatic case based on the mean and the principal chromaticity line. Points A1 , A2 , and A3 identify the positions of the same pixel in the red-green-blue (RGB) color space under −−→ −→ the three different illumination directions. Vectors MCL and PCL are the mean and the principal color lines, respectively. The dotted lines indicate the projections of points A1 , A2 , and A3 on the principal line. The R, G, and B axes correspond to the red, green, and blue components of the colored pixels. The R , G , and B axes correspond to the color coordinate system shifted to the center of points A1 , A2 , and A3 . L1

L2 N ␪1

L⬘1

L ⬘2

N⬘ N ␪2

N

V

N⬘

L

Erroneous position

True position

(a)

(b)

(c)

FIGURE 18 The effect of a highlight on the estimated normal of a facet. (a) With no leans toward the light highlights; (b) highlighted facet, where the erroneous normal N 1 source L that produced the highlight; (c) the effect of a shadow on the estimated leans away from the source L, which normal of a facet, where the erroneous normal N produced the shadow.

and Petrou (2001b, 2003) presented an algorithm for separating the local gradient and color information by using four-source color photometric stereo in the presence of highlights and shadows. Based on the fact that any four vectors in a 3D space are linearly dependent, there must exist constants a1 , a2 , a3 , and a4 such that the following equation holds for the four illumination direction vectors:

1 + a2 L 2 + a3 L 3 + a4 L 4 = 0. a1 L

(41)

28

Vasileios Argyriou and Maria Petrou

A similar equation for the four pixel intensities I k may be obtained by multiplying the above equation with the albedo and the normal of the surface patch using vector dot product, as follows:

1 · N) + a2 ρS (L 2 · N) + a3 ρS (L 3 · N) + a4 ρS (L 4 · N) =0⇒ a1 ρS (L a1 I 1 + a2 I 2 + a3 I 3 + a4 I 4 = 0. (42) The problematic pixel quadruples, containing either shadows or highlights, do not satisfy Eq. (42) and therefore this method allows ruling out the majority of quadruples that are not purely Lambertian. The use of color information was suggested to separate shadows from highlights. The color of a highlighted pixel is the combination of “matte” (body) Ibody color, and the color of the illuminant (specular component) Ispec as expressed by Eq. (25). Thus, the chromaticity of the brightest pixel was compared with the chromaticity of the darker pixels, and if the difference exceeded a certain threshold, the pixel was labeled as a highlight. This method cannot provide reliable classiﬁcation if the chromaticity of the surface color is close to the chromaticity of the incident light. In this case, Barsky and Petrou (2003) suggested discarding the brightest pixel and reconstructing the normal using only the darkest three pixels; the pixel was regarded as a highlight if the recovered normal was close to the specular direction of the corre near vector R in Figure 10). In sponding imaging conﬁguration (vector N the rest of the problematic quadruples, the darkest pixel was discarded as a shadow, and the color and the normal were recovered using the brightest three pixels.

9. PHOTOMETRIC STEREO WITH AN EXTENDED LIGHT SOURCE [ASSUMPTION (2)] In conventional photometric stereo, the viewing and lighting positions are assumed to be distant, with the lighting source modeled as a collimated point source. This assumption appears limiting for practical applications. Various authors (Farooq et al., 2005; Lee et al., 2005; Smith, 1999; Smith and Smith, 2005) have made a ﬁrst attempt to solve the problem of the extended light source. This work is in relation to an industrial inspection problem, where strip lights are used to illuminate objects on a conveyor belt. Following their approach, we show here that under certain assumptions, an extended strip light may be replaced by a virtual point source at inﬁnity. The imaged object is on a conveyor belt, illuminated by a strip of light, and viewed by a line scan camera. Thus, the problem may be treated as a one-dimensional (1D) one. The following discussion is in relation to

Photometric Stereo: An Overview

dx P1

Linear illumination x dx D a

r

Pi

a da

rda

a

N

Surface

dx P2

a P2 da rda

a

29

␪

FIGURE 19 Distributed linear illumination.

Figure 19. Let us consider a strip light source with length much longer than the size of the imaged object. Let us consider a point Pi on the imaged object, at distance D from the line of the light. Let us consider two points P1 and P2 on the light source, symmetrically placed on either side of the point of interest Pi , their positions forming an angle α with the normal from point Pi to the linear light source. Let us consider two inﬁnitesimal elements at these points, each of length dx. If the brightness of the light source is I0 per unit length, the light one of these elements produces is I0 dx and at point Pi creates an intensity I0rdx 2 cos α, where r is the distance Pi P1 = Pi P2 and factor cos α appears because the normal of the surface patch at Pi is at angle α with respect to either direction rdα Pi P1 or direction Pi P2 . Figure 19 shows that dx cos α = rdα ⇒ dx = cos α and r cos α = D. So, the light point Pi receives from segments P1 and P2 2I0 2I0 rdα is 2Ir20 cos α cos α = r dα = D cos αdα. The total light, therefore, the surface element at point Pi receives is the integral of this expression over angle α. If the largest value angle α obtains is α0 , the total light at point Pi is 2ID0 sin α0 . Next, let us assume that the imaged surface element is rotated by an angle θ, about the normal of the page, so the facet at point Pi now receives different amounts of light from segments P1 and P2 . To avoid shadowing, θ < (π/2) − α. The intensity at point Pi of the surface is then given by

I0 dx I0 dx cos(α + θ) + 2 cos(α − θ) 2 r r I0 dx = 2 [cos(α + θ) + cos(α − θ)] r I0 dx = 2 [2 cos α cos θ] r I0 dα =2 cos α cos θ. D

(43)

Integrating then over α, we obtain the total illumination point Pi receives: 2I0 D sin α0 cos θ. This result indicates that the strip of light behaves as if

30

Vasileios Argyriou and Maria Petrou

A set of point illumination sources

Equivalent virtual point illumination source

N

FIGURE 20

N

Approximating an extended light source.

a single point source is placed at the zenith of the imaged surface and at distance D from it, with intensity I ≡ 2I0 sin α0 D, so that an elementary part of the surface tilted away from the average surface normal by an angle θ, receives light DI 2 cos θ = 2ID0 sin α0 cos θ. Note that as long as the extent of the imaged object is much smaller than the length of the strip of light, this expression is valid for all points on the same line of the imaged surface parallel to the strip of light. Thus, all imaged points will receive equal intensity of light from a virtual point source that is above each of them and at distance D. This is equivalent to saying that the point source is at inﬁnity with its rays arriving parallel at all imaged points. Note also that different lines on the surface, parallel to the strip of light, are at different distances D from that strip of light and thus receive different amounts of light. The approximation of a linear distributed illumination source (long strip of light) by an equivalent virtual point source has practical interest for spatial multiplexing approaches (Figure 20). In this case, each light source directs the light onto a different area of the acquired surface at the same time, avoiding any photometric overlap. Therefore, the only practical way to use spatially multiplexed photometric stereo is in line scan fashion (Figure 21). The main drawbacks of this method are the need for registration among the interlaced images and the restriction to rigid objects avoiding self-motion between successive line scans.

10. DYNAMIC PHOTOMETRIC STEREO AND MOTION [ASSUMPTION (6)] Many algorithms for shape reconstruction from multiple images of a moving Lambertian object have recently appeared in the literature (Lim, Yang, and Kriegman, 2005; Maki, Watanabe, and Wiles, 2002; Simakov and Basri, 2003; Smith and Smith, 2005; Zhang et al., 2003). Smith and Smith (2005) considered the sixth main constraint of photometric stereo, namely, the limiting assumption that the relative position of the illumination conﬁgurations and the imaged surface must be ﬁxed during image capture. This assumption is violated for rapid or uncontrolled relative movement between the

31

Photometric Stereo: An Overview

Image formation

Object motion

Image 1 Image 2 Image 3 A1 B1 C1 A2 B2 C2 A3 B3 C3

A1 A2 A3

B1 B2 B3

C1 C2 C3

A B C

Interlaced image

Object motion A B C

Visual example Frame 1 Frame 2

Image 1

Image 2

Image 3

A1

B2

C3

A2

B3

C4

A3

B4

C5

Frame 3 Frame 4 Frame 5

FIGURE 21 A schematic representation for spectral multiplexing. The shaded blocks in the top-left depiction of the imaging rig indicate barriers that prevent the light of one source illuminating the surface outside its designating area on the conveyor belt. Camera recordings A1, B1, and C1 are images of the same strip of the imaged surface under three different illumination directions. As this strip of surface moves out of the field of view of the camera, the next strip enters it and creates image segments A2, B2, and C2, which when added to the corresponding previously imaged strips help build the three images for the three different illumination directions.

camera and the surface, since the simultaneous acquisition of multiple images under different lighting conditions is required. Several approaches for achieving dynamic photometric stereo have been proposed. Temporal multiplexing represents an adaptation of conventional photometric stereo to dynamic applications, in which separate lighting conﬁgurations are deployed and images are rapidly acquired at closely spaced intervals in time. This approach is inherently complex, and there are signiﬁcant implications in terms of achievable camera frame rates and the high intensity of lighting needed to realize very short image acquisition periods, particularly if more than three images are required. Spatial multiplexing involves separate images of the same surface location acquired from different points in space. Image capturing occurs simultaneously; therefore, the images must be registered between viewing positions and synchronized with the speed of the moving surface. This case is shown in Figure 21.

32

Vasileios Argyriou and Maria Petrou

Object IR filter 3

IR filter 2

IR filter 1

CCD IR3

CCD IR2

CCD IR1

Lens

CCD RGB

FIGURE 22

Narrow infrared (IR) photometric stereo. CCD, charge-coupled device.

Spectral multiplexing allows multiple images to be captured from a single point in space and time, since the cross-talk of image-illumination conﬁguration can be minimized in terms of light frequency separation. Smith and Smith (2005) suggest a narrow-band infrared photometric stereo technique utilizing narrow frequency channels that are closely spaced. This approach diminishes a number of key limitations that the use of broad-band color photometric stereo introduces. For example, similar components of colored light may be reﬂected in similar proportions, both for a surface of a particular color or inclination (e.g., a red surface with a certain inclination may appear similar to a pink one with a different inclination). Therefore, to reduce the sensitivity to surface color changes and decouple the gradient data from color, medium to long-wave infrared (IR) light was suggested. In addition, if required, a white light source may be simultaneously included to provide fully registered color data (Figure 22). This approach, however, replaces the assumption that the optical albedo of the imaged surface is constant with the assumption that its IR albedo is constant. Any variation in the surface albedo will interfere with the spectral multiplexing of the lights used. In addition, the recovery of color by a synchronized color camera is imprecise, as the color inferred from a single image only is illumination dependent. Another approach followed by many (Lim, Yang, and Kriegman, 2005; Maki, Watanabe, and Wiles, 2002; Simakov and Basri, 2003; Zhang et al., 2003) is to assume an initial surface and then use an algorithm that iteratively estimates a new surface based on the previously reconstructed estimates. The algorithm assumes that the object can be segmented from the background and the scene illumination is modeled by a constant ambient illumination plus a directional source. One camera is used to capture the moving object. A sequence It of F frames, indexed by t = 1, . . . , F and m manually selected scene points, indexed by p = 1, . . . , m, are considered. Let {P1 , . . . , PF } denote the F orthogonal projections of points in 3 to images {I1 , . . . , IF }, respectively. Each pair of (It , Pt ) deﬁnes a mapping

33

Photometric Stereo: An Overview

from 3D to a gray value on the image plane. The object is assumed to be both rigid and Lambertian, and the observed intensity of a point p on the surface at frame t is given by Eq. (22) without the highlight term. Making explicit the application of the equation to a speciﬁc point p, and integrating out the dependence on the wavelength, we may write:

t · N(p). It (p) = Iα + ρ(p)L

(44)

The m manually selected feature points are used to estimate camera projections, light source directions, and an initial piecewise surface. The camera projection matrix Pt and the 3D positions of the feature points with respect to the image plane of the initial frame I1 , can be recovered using the Tomasi– Kanade factorization algorithm for all frames (Tomasi and Kanade, 1992). An initial depth map is calculated by linearly interpolating the depth values of the selected m feature points. Given the initial surface St and its associated depth map zt (x, y), the intensity matrix It (p) = It (Pt (x, y, zt (x, y))) may be computed for all the pixels inside the segmented object region, which initially was not possible because the pixel correspondences across different frames were not known a priori. Since the correspondence problem and the intensity matrix have been is then produced based on calculated, an integrable normal vector ﬁeld N photometric stereo by minimizing the following function: F

t · N p − I α )2 , (It (Pt (x, y, zt (x, y))) − ρL

(45)

t=1 (x,y)∈

where is a region in I1 containing the segmented object. The normals may be recovered in the least-square error sense (see Barsky and Petrou, 2001b, 2003) and they may be integrated to obtain a new surface St+1 and its associated depth map zt+1 (x, y), while the known 3D positions of the m scene points may be used to reﬁne the estimated surface. This process is repeated indeﬁnitely and a sequence of the reconstructed surfaces can be estimated. Furthermore, other methods utilize a coarse-to-ﬁne reﬁnement step using image pyramids to improve the accuracy of the optical ﬂow and the normal vector ﬁeld (Zhang et al., 2003).

11. PERSPECTIVE PHOTOMETRIC STEREO [ASSUMPTION (3)] A common assumption in the ﬁeld of computer vision and photometric stereo is that image projection is orthographic. This assumption is valid only for distant viewing positions. A perspective model should be introduced for positions closer to the observed object. Tankus et al. (Tankus,

34

Vasileios Argyriou and Maria Petrou

Sochen, and Yeshurun, 2003, 2004, 2005; Tankus and Kiryati, 2005) and Prados and Faugeras (2003) reexamined the basic set of equations of photometric stereo under the assumption of perspective projection. In these papers, the scene object is assumed Lambertian and is illuminated from k = (−pl , −ql , 1), where k = 1, 2, 3. The acquired surfaces are directions L k k assumed representable by functions of real-world coordinates (x, y, z(x, y)), as well as of image coordinates (u, v, h(u, v)). Function z(x, y) denotes the depth in a Cartesian coordinate system, the origin of which is on the camera plane. Projecting the real coordinates (x, y, z(x, y)) onto the image point (u, v), with the depth denoted by h(u, v), would result in h(u, v) = z(x, y) by deﬁnition. Let f denote the known focal length of the camera, y) the surface normal, and ρW (x, y) the albedo at point (x, y, z(x, y)), N(x, while ρI (u, v) is the albedo projected onto image point (u, v), with ρW (x, y) = ρI (u, v). The image irradiance equation for a Lambertian surface in its general form is given by Eq. (21). The real-world coordinates (x, y, z(x, y)) under a perspective projection model are related to image coordinates (u, v) by

x=−

u z(x, y) , f

y=−

v z(x, y) . f

(46)

Equation (21) may be expressed with the current notation as

1 + pl zx + ql zy I(u, v) = ρW (x, y) . 2 2 2 2 1 + pl + ql 1 + zx + zy

(47)

In order to derive the perspective image irradiance equation, the authors deﬁne a surface S = {(x, y, z(x, y))} and its projection as S˜ = −uh −vh f , f , h(u, v) . In addition, the surface is assumed differentiable with respect to (u, v) and (x, y). The setup is shown schematically in Figure 23. A curve on the image plane with parameter s is deﬁned as

c(s) = (u(s), v(s), −f )

(48)

and due to the perspective projection the real-world curve may be written as

h(s) u(s)h(s) v(s)h(s) ,− , h(s) = (−u, −v, f ), C(s) = − f f f

(49)

Photometric Stereo: An Overview

35

v c2(v)

y c1(u)

u

P v0

x u0

z0

z

x0

2f

C1(u)

y0 Q

C2(v)

Image plane P

Scene object S

FIGURE 23 The image plane is = {(u, v, −f )}, where f is the focal length. Point = (x0 , y0 , z(x, y)) on the surface S is projected onto point P = (u0 , v0 , −f ) on image Q plane . The curves c1 (u) and c2 (v) are parallel to axes x, y (or u, v). The curves C1 (u), C2 (v) ∈ S are the curves on the object, the perspective projections of which on

are curves c1 (u) and c2 (v), respectively. The tangents to C1 (u) and C2 (v) at point Q are computed from P and the perspective projection equations. The normal to the two tangents is the normal to S at point Q.

The gradient of the real-world curve C(s) is

1 dC(s) = [−us (s)h(s) − u(s)hs (s), −vs (s)h(s) − v(s)hs (s), f hs (s)]. (50) ds f where subscript s indicates differentiation with respect to s. The authors consider two different curves parallel to the x and y axes, = (u0 , v0 , −f). Thus, through the image point P

c1 (u) = (u, v0 , −f ) c2 (v) = (u0 , v, −f ).

(51)

as We may calculate the two tangents at point P

1 dC1 (u) (−h − uh = , −vh , f h ) u u u du u=u0 f u=u0 1 dC2 (v) (−uh = , −h − vh , f h ) . v v v dv v=v0 f v=v0

(52)

36

Vasileios Argyriou and Maria Petrou

The normal to the surface is parallel to the cross-product

dC2 (v) h dC1 (u) × = 2 f hu , f hv , uhu + vhv + h , du u=u0 dv v=v0 f u=u0 , v=v0 (53) and dropping the subscript 0 to make point P generic, we deduce that the unit normal is given by

= (f hu , f hv , uhu + vhv + h) N . (uhu + vhv + h)2 + f 2 (hu2 + hv2 )

(54)

the image irradiance equation For unit illumination direction vector L, thus becomes

·L I(u, v) = ρI (u, v)N (−pl , −ql , 1) · f hu , f hv , uhu + vhv + h = ρI (u, v) 2 2 1 + pl + ql (uhu + vhv + h)2 + f 2 (hu2 + hv2 ) (u − f pl )hu + (v − fql ) hv + h . = ρI (u, v) 2 2 2 2 2 2 1 + pl + ql (uhu + vhv + h) + f (hu + hv )

(55)

The above equation shows direct dependence of the image irradiance on both h(u, v) and its ﬁrst derivatives. Note that in Eq. (55) h(u, v) and its two derivatives explicitly appear in the equation. This makes the problem rather difﬁcult to solve. If, however, ln(h(u, v)) is used instead of h(u, v), the problem becomes simpler, as the irradiance equation is expressed as (u − fpl )˜pn (u, v) + (v − fql )˜qn (u, v) + 1 I(u, v) = ρI (u, v) , 2 2 1 + pl + ql (u˜pn (u, v) + v˜qn (u, v) + 1)2 + f 2 p˜ 2n (u, v) + q˜ 2n (u, v) (56) ln h ln h where p˜ n (u, v) ≡ hhu = ∂ ∂u and q˜ n (u, v) ≡ hhv = ∂ ∂v , and it depends on the derivatives of ln(h(u, v)), but not on ln(h(u, v)) itself. Therefore, the problem of recovering h(u, v) from the image irradiance equation is reduced to the problem of recovering the surface ln(h(u, v)), since the natural logarithm is a bijective mapping and h(u, v) > 0. In the case of photometric stereo, several images I i (u, v) of the same i are acquired and the object under different illumination directions L

Photometric Stereo: An Overview

37

perspective image irradiance equations are as follows:

IIi (u, v) = ρ(u, v)

(u−fpli )˜pn (u, v) + (v − fqli )˜qn (u, v)+1

. i | (u˜pn (u, v)+v˜qn (u, v)+1)2 +f 2 (˜p2 (u, v) + q˜ 2 (u, v)) |L n n (57)

Dividing the ith image by the kth, assuming the latter is nonzero everywhere, yields

k |[(u − fpl )˜pn (u, v) + (v − fql )˜qn (u, v) + 1] |L I i (u, v) i i , = i I k (u, v) |L |[(u − fplk )˜pn (u, v) + (v − fqlk )˜qn (u, v) + 1]

(58)

which may be written as

Aik p˜ n + Bik q˜ n + Cik = 0,

(59)

where

i |(u − fpl ) − I k (u, v)|L k |(u − fpl ) Aik = I i (u, v)|L i k i |(v − fql ) − I k (u, v)|L k |(v − fql ) Bik = I i (u, v)|L i k i | − I k (u, v)|L k |. Cik = I i (u, v)|L

(60)

If we have three images, we will have three equations like Eq. (58), which correspond to Eq. (37) for the orthographic projection:

A01 p˜ n + B01 q˜ n + C01 = 0 A12 p˜ n + B12 q˜ n + C12 = 0 A02 p˜ n + B02 q˜ n + C02 = 0.

(61)

This system of equations may be solved like Eq. (37) (i.e., considering all three pairs of them and solving for q˜ n , averaging the three answers and doing the same for p˜ n ). Alternatively, we may solve the system in the leastsquare error sense. In a matrix form linear system (61) may be written as

⎛

A01 ⎝ A12 A02

⎛ ⎞ ⎞ B01 −C01 ˜ p B12 ⎠ n = ⎝ −C12 ⎠ ⇒ Wx = Y. q˜ n B02 −C02

(62)

38

Vasileios Argyriou and Maria Petrou

Multiplying both sides with the transpose of W, we obtain

WT Wx = WT Y,

(63)

and solving for p˜ n and q˜ n we have

Qx = R ⇒

M1 N1 M2 N 2

p˜ n q˜ n

⇒

=

⎧ ⎪ ⎨ p˜ n =

R1 R2

⇒

N2 R1 −N1 R2 N2 M1 −N1 M2

⎪ ⎩ q˜ = − M2 R1 −M1 R2 , n N2 M1 −N1 M2

(64)

where Q ≡ WT W, R ≡ WT Y, M1 = A201 + A212 + A202 , N2 = B201 + B212 + B202 , and M2 = N1 = A01 B01 + A12 B12 + A02 B02 . Substituting from (60) in the previous equations, we obtain

pn (u, v) =

0 |q21 + I 1 |L 1 |q02 + I 2 |L 2 |q10 I 0 |L l l l 0 |D12 + I 1 |L 1 |D20 + I 2 |L 2 |D01 I 0 |L

qn (u, v) =

0 |p21 + I 1 |L 1 |p02 + I 2 |L 2 |p10 I 0 |L l l l . 0 |D12 + I 1 |L 1 |D20 + I 2 |L 2 |D01 I 0 |L

(65)

ik Here pik l = pli − plk , ql = qli − qlk , and Dik = u(qli − qlk ) + v(plk − pli ) + f(pli qlk − plk qli ) with i, k = 0, 1, 2. Albedo ρI is obtained by substitution of p˜ n and q˜ n into one of the image irradiance equations. The ﬁnal step of the perspective photometric stereo method is taking the exponent of the result, since the derivatives of the natural logarithm of the depth have been calculated rather than the depth itself.

12. PHOTOMETRIC STEREO ERROR ANALYSIS The photometric stereo method is susceptible to systematic errors derived from a number of factors, such as errors in the image acquisition stage, inaccurate measurement of light source orientations, effects of specular reﬂections and shadows, the spatial and spectral distribution of incident light, imaging and illumination geometry, and surface size and material. Jiang and Bunke (1991) and Ray, Birk, and Kelley (1983) analyzed some of the above error sources and performed a sensitivity analysis of surface

Photometric Stereo: An Overview

39

normals computed by the photometric stereo method with respect to the majority of the errors. Furthermore, Barsky and Petrou (2006) presented a detailed performance and error analysis for four-light color photometric stereo. Their error analysis was based on the assumptions that errors may arise due to Gaussian image noise and inaccuracies with which the geometry of the illuminating setup was known.

12.1. Errors in the Image Acquisition Stage Errors that occur during the acquisition stage are due mostly to theoretical assumptions, limitations of the available equipment, and the general system architecture. For example, the assumption that the light rays reaching the surface of the object are parallel requires the distance between the illumination sources and the object to be signiﬁcantly large. Furthermore, the illumination direction vectors should be non-coplanar, and a perfect match for all the light sources in their spatial and spectral properties is required. The reﬂected light of a surface depends on its albedo and the wavelength of the incident light; therefore, the photometric stereo cannot be applied with light sources of different wavelengths, since different albedo factors then are involved in each image. The limitations of the available equipment introduce additional errors. The presence of noise and faults in the CCD arrays alter the intensity values. The focal lens may introduce distortions (e.g., pin cushion and barrel) mainly at the image borders. In order to achieve orthographic projection, which is assumed in photometric stereo, the distance between the acquired object and the viewing point should be much larger than the size of the object itself, with a good approximation being achieved for a ratio of 1/30. In contradiction, by placing the camera close to the object, higher resolution and precision may be achieved. Since the resolution of the reconstructed surface using photometric stereo depends on the camera resolution and its relative position with respect to the acquired object, the optimal distance regarding the desired resolution of the surface may be accurately deﬁned using camera calibration, which also can eliminate the focal lens distortions. Camera calibration consists of the estimation of a model for an uncalibrated camera. The objective is to ﬁnd the external parameters (position and orientation relative to a world coordinate system), and the internal parameters of the camera (principal point or image center, focal length, and distortion coefﬁcients). One of the most commonly used camera calibration techniques is the one proposed by Tsai (1986). During calibration, a target of known geometry is imaged and correspondences between target points and their images are obtained. These form the basic data on which the calibration is based.

40

Vasileios Argyriou and Maria Petrou

12.2. Sensitivity Analysis of Surface Normals Errors in measurement of light source orientations and image intensity result in errors in the estimated surface orientation. The sensitivity of photometric stereo in various input parameters may be analyzed using the gradient representation (pn (x, y), qn (x, y)) of the reﬂectance maps. The accuracy of the computed values of pn and qn depends on the accuracy in measuring the zenith θli and azimuth angles ϕli . Light calibration methods are required to measure the angles of the illumination sources. Therefore, we consider lights placed symmetrically about the optical axis z. Based on Eqs. (7) and (8), in that case it can be observed that all the zenith angles are equal θl = θl1 = θl2 = θl3 with cos θl = 1/ 1 + p2l + q2l . The logical consequence of the above symmetric arrangement is that the acquired image intensities from the three light sources are the same if the observed surface is directly facing the camera (pn (x, y) = qn (x, y) = 0). As shown in Figure 24, the ratio of the distance of the camera from the three light sources dl over the distance between the object and the camera dc corresponds to the tangent value of the zenith angle θl . Hence,

θl = tan−1

dl , dc

0 ≤ θl ≤ 90◦ .

Camera L1 x⬘

dl L⬘1

(66)

L2

␸l1 dl ⬘1 L3 dc

y⬘

␪l1

x

Object

y

FIGURE 24 Points Li indicate the positions of the light sources. L1 is the projection of L1 on the zx-plane, θl1 is its zenith angle, and ϕl1 is the azimuth angle of L1 .

41

Photometric Stereo: An Overview

N3 N2 N1

FIGURE 25 A rectangular parallelepiped placed so that three distinct surfaces are clearly visible from the viewing position simultaneously.

In a similar manner, the azimuth angle may be obtained by measuring the distance between the projection of the light sources on the xz-plane and the center of the axes dl , assuming that the camera is on the z-axis. The i values of ϕli are given by

cos ϕli =

dl

i

dl

and

Li L i sin ϕli = dl

(67)

so that 0 ≤ ϕli < 360◦ . A second method that is often selected for illumination directions calibration requires positioning a rectangular parallelepiped such that three distinct surfaces are clearly visible from the viewing position simultaneously (Figure 25). The normals of the three visible surfaces are known a priori and, using the values of the acquired image intensities for a single illumination direction, we obtain

1 + pn1 (x, y)pli + qn1 (x, y)qli I 1 (x, y) = ρS (x, y) 1 + p2n1 (x, y) + q2n1 (x, y) 1 + p2l + q2l i

i

1 + pn2 (x, y)pli + qn2 (x, y)qli I 2 (x, y) = ρS (x, y) 1 + p2n2 (x, y) + q2n2 (x, y) 1 + p2l + q2l i

i

1 + pn3 (x, y)pli + qn3 (x, y)qli I 3 (x, y) = ρS (x, y) , 1 + p2n3 (x, y) + q2n3 (x, y) 1 + p2l + q2l i

(68)

i

where the surface gradients (pn1 (x, y), qn1 (x, y)), (pn2 (x, y), qn2 (x, y)), and (pn3 (x, y), qn3 (x, y)) correspond to the three visible surfaces. Assuming

42

Vasileios Argyriou and Maria Petrou

diffuse reﬂection and invariant albedo for equivalent point locations, the orientation (pli , qli ) of the ith light source is determined by solving the system of equations in the same manner as system (36). The above procedure is repeated for the remaining light sources to calculate the remaining zenith and azimuth angles, θl , ϕl1 , ϕl2 , and ϕl3 . The sensitivity of the photometric stereo algorithm with respect to the changes in various input parameters can be analyzed based on the gradient representation of the normal vectors and Eqs. (6) and (36). Solving these equations algebraically (Jiang and Bunke, 1991; Ray, Birk, and Kelley, 1983), the surface gradients are given by

pn (x, y) = g(I 1 , I 2 , I 3 , θl , ϕl1 , ϕl2 , ϕl3 )

(69)

qn (x, y) = f (I 1 , I 2 , I 3 , θl , ϕl1 , ϕl2 , ϕl3 ).

(70)

and

In order to estimate the deviation of the normal vector components pn and qn , with respect to small changes in the input parameters, we obtain 3 3 ∂g i ∂g ∂g dI + dθ + dϕli l i ∂θl ∂ϕli ∂I

(71)

3 3 ∂f ∂f i ∂f qn (x, y) = dI + dθl + dϕli . ∂θl ∂ϕli ∂I i

(72)

pn (x, y) =

i=1

i=1

and

i=1

i=1

The expression of pn (x, y) and qn (x, y) in gradient space leads to complicated formulae. For this reason, in Jiang and Bunke (1991) the surface orientation was represented in terms of unit normal vectors, reducing the complexity of the sensitivity analysis process. Based on Eqs. (34) and (35)

= 1 [L]−1I, N ρ

(73)

where [L] is a 3 × 3 matrix, and its inverse is given by

[L]−1 =

1 3 ×L 1 ×L 2 ×L 3 L 1 L 2 ), (L 2 L 3] 1 L [L

(74)

Photometric Stereo: An Overview

43

1 · (L 2 × L 3 ) (i.e., the vol2 L 3 ] representing the triple product L 1 L with [L 1 2 3 , and L ), Eq. (73) may be , L ume of a parallelepiped formed by L written as

1 1 2 ×L 3 ) + I 2 (L 3 ×L 1 ) + I 3 (L 1 ×L 2 )) = (I 1 (L W. 1 2 3 1 2 L 3] ρ[L L L ] ρ[L L (75) 1 2 3 L L ] > 0 and observConsidering, without loss of generality, that [L ing that both N and W point in the same direction, the azimuth ϕn and the and W. Instead of using zenith θn angles are identical for the two vectors N to compute the deviation of the normal vector components, W may be N used, resulting in much simpler sensitivity formulas. is given by The surface normal vector W = N

= (xn , yn , zn ) ≡ I 1 (L 2 ×L 3 ) + I 2 (L 3 ×L 1 ) + I 3 (L 1 ×L 2 ), W

(76)

where

xn = fx (I 1 , I 2 , I 3 , θl1 , θl2 , θl3 , ϕl1 , ϕl2 , ϕl3 ) yn = fy (I 1 , I 2 , I 3 , θl1 , θl2 , θl3 , ϕl1 , ϕl2 , ϕl3 ) zn = fz (I 1 , I 2 , I 3 , θl1 , θl2 , θl3 , ϕl1 , ϕl2 , ϕl3 ).

(77)

with respect to errors To the ﬁrst order, the errors in the component of W in the input parameters may be written as

= (dxn , dyn , dzn ) = dW

3 ∂W i=1

=

3 i=1

∂I i

dI i +

3 ∂W i=1

∂W dI i + ∂I i

3 i=1

∂θi ∂W i ∂L

dθi +

3 ∂W i=1

∂ϕi

dϕi =

i i ∂L ∂L dθi + dϕi , (78) ∂θi ∂ϕi

L i is a matrix consisting of partial derivatives of the where the ratio ∂W/∂ with respect to the components of vector L i = components of vector W (xli , yli , zli ): ⎤ ⎡

∂xn ⎢ ∂xl ⎢ i ⎢ ∂yn ∂W =⎢ ⎢ ∂xl i ∂L ⎢ i ⎣ ∂zn ∂xli

∂xn ∂yli ∂yn ∂yli ∂zn ∂yli

∂xn ∂zli ⎥ ⎥ ∂yn ⎥ ⎥ for i = 1, 2, 3. ∂zli ⎥ ⎥ ∂zn ⎦ ∂zli

(79)

44

Vasileios Argyriou and Maria Petrou

all the partial derivatives in Eq. (78) In order to estimate the errors in W, must be computed. For the ﬁrst part of Eq. (78), we obtain

∂W 2 × L 3 =L ∂I 1 ∂W 3 × L 1 =L ∂I 2 ∂W 1 × L 2. =L ∂I 3

(80)

Using the matrix notation of cross-product

⎡

0 i × L j = ⎣ zl L i −yli

−zli 0 xli

⎤⎡ ⎤ xlj yli ⎦ ⎣ ylj ⎦, −xli 0 zlj

(81)

it can be proven (Jiang and Bunke, 1991) that

⎡ 0 i × L j ∂L = ⎣ zli j ∂L −y

li

−zli 0 xli

⎤ yli −xli ⎦ ≡ Hi , for i = j, i = 1, 2, 3. 0

(82)

Based on the above equation, we obtain

∂W = I 2 H3 − I 3 H2 1 ∂L ∂W = I 3 H1 − I 1 H3 2 ∂L

∂W = I 1 H2 − I 2 H1 . 3 ∂L

(83)

The partial derivatives of the illumination vectors using the azimuth and zenith angles for representation (see Eq. (9)) are given by

i ∂L = (cos θli cos ϕli , cos θli sin ϕli , − sin θli ) ∂θli i ∂L = (− sin θli sin ϕli , sin θli cos ϕli , 0). ∂ϕli

(84)

Photometric Stereo: An Overview

45

Now, substituting all the partial derivatives in Eq. (78), an estimation of can be found. Furthermore, representing the errors in surface normals W surface normal vectors W = (xn , yn , zn ) using the zenith θn and azimuth ϕn angles from Eq. (8), we obtain

θn = tan

−1

xn2 + yn2 −1 2 2 pn (x, y) + qn (x, y) = tan zn

ϕn = tan−1

(85)

pn (x, y) yn = tan−1 . qn (x, y) xn

(86)

and the errors in surface normal vectors are estimated by calculating their derivatives dθn and dϕn with respect to xn , yn , and zn , as 1 dθn = 2 1 + (xn + yn2 )/zn2 1 = 2 xn + yn2 + zn2 1 = 2 xn + yn2 + zn2

xn2 + yn2 dxn + dyn − dzn zn2 zn xn2 + yn2 zn xn2 + yn2 yn

xn

y n zn

xn z n

dxn + dyn − xn2 + yn2 xn2 + yn2

xn z n

yn zn

, xn2 + yn2 xn2 + yn2

1 dϕn = 1 + (yn /xn )2 = =

xn2

xn2 + yn2 dzn

2 2 , − xn + y n · d W

1 yn − 2 dxn + 2 dyn xn xn

(87)

1 −yn dxn + xn dyn 2 + yn

1 −y , x , 0 · dW. n n xn2 + yn2

(88)

Substituting Eq. (78) in the above equations yields the magnitude of the errors in azimuth and zenith angles of the surface normals as functions of the input parameters of the photometric stereo. The most signiﬁcant outcome of the above sensitivity analysis is that, in general, a 1-degree error in the calculation of light source orientation or 1% error in image intensity, results in a 1-degree error in the orientation of the surface normal vectors. However, errors due to highlights or shadows

46

Vasileios Argyriou and Maria Petrou

are not covered by this analysis. A discussion on such errors may be found in Barsky and Petrou (2006).

13. OPTIMAL ILLUMINATION CONFIGURATION FOR PHOTOMETRIC STEREO The inﬂuence of a lighting arrangement on the accuracy of surface reconstruction based on photometric stereo has been considered by Lee and Kuo (1992), Spence and Chantler (2003) and Woodham (1980) and suggestions for optimal illumination conﬁguration in terms of azimuth ϕl and zenith θl angles were made. Woodham (1980), using reﬂectance maps, recommended dense iso-intensity contours to obtain maximum accuracy, since in this case a small change in the surface gradients pn and qn results in a large intensity change. In order to achieve dense iso-intensity contours the zenith angle θl has to be increased, but this results in increasing the presence of shadows, which is undesirable. Regarding the azimuth angles of the light sources, Woodham pointed out that the illumination vectors must not be coplanar, otherwise the illumination matrix in Eq. (34) is not invertible. Lee and Kuo (1992), using two reﬂectance maps for their two-image photometric stereo algorithm, deduced that it is desirable to incorporate reﬂectance maps that compensate for each other’s weaknesses, in order to determine the optimal illumination conﬁguration. Observing that the azimuth angle ϕl determines the orientation of the reﬂectance map around the origin, whereas the zenith angle θl determines the distance between the origin and (pl , ql ) in the gradient space (Figures 26 and 1), the angular difference between two reﬂectance maps would be given by |ϕl1 − ϕl2 |. The above observations can be easily proven using Eq. (8) and calculating the angle between the line passing through the (0, 0) and (pl , ql ) points in the gradient space and the p-axis. Obviously,

φ = arctan

ql = arctan(tan ϕl ) = ϕl , pl

(89)

and the distance d between points (0, 0) and (pl , ql ) is

d=

q2l + p2l = tan θl .

(90)

Angle φ is the same as the azimuth angle ϕl , and the distance from the origin of the gradient space depends on the zenith angle θl .

Photometric Stereo: An Overview

5

q

0

p

x

⫺5 ⫺5

47

x

0

5

FIGURE 26 Contour plots of Lambertian reflectance maps with (ϕl1 , θl1 ) = (45◦ , 45◦ ) and (ϕl2 , θl2 ) = (135◦ , 45◦ ). The x's mark points (pl1 , ql1 ) and (pl2 , ql2 ).

Assuming that the zenith angle is in the range of 30◦ to 60◦ , the reﬂectance map would cover the central region of the gradient space. Therefore, the azimuth angles of the illumination sources deﬁne the optimal lighting condition. Taking into consideration that the reﬂectance map provides good sensitivity along the gradient direction but poor sensitivity along the tangential direction, the optimal illumination directions of two sources are obtained when the gradient directions of one reﬂectance map correspond to the tangential directions of the other reﬂectance map, |ϕl1 − ϕl2 | = 90◦ . To understand this better, the value of qn (x, y) of the reﬂectance map is regarded ﬁxed and equal to qn0 (x, y). Now, viewing the reﬂectance map as a function of pn (x, y), the sensitivity of pn (x, y) with respect to changes in I(x, y) is given by

! ∂pn ∂R(pn (x, y), qn0 (x, y)) −1 ∂I = ∂pn

(91)

and is inversely proportional to the slope of the reﬂectance map at pn (x, y). Thus, for a ﬁxed value of ∂I, the value of ∂pn is smaller where R(pn (x, y), qn0 (x, y)) is steepest and the function changes most rapidly.

48

Vasileios Argyriou and Maria Petrou

Similarly, if pn (x, y) = pn0 (x, y), we obtain

! ∂qn ∂R(pn0 (x, y), qn (x, y)) −1 . ∂I = ∂qn

(92)

For a given point (pn0 (x, y), qn0 (x, y)) and according to the above, the sensitivity is greatest along the direction perpendicular to the contour; that is, ∇R(pn0 (x, y), qn0 (x, y)) = [Rp (pn0 (x, y), qn0 (x, y)), Rq (pn0 (x, y), qn0 (x, y))] and lowest along the tangential direction; that is ∇R(pn0 (x, y), qn0 (x, y)) = [−Rp (pn0 (x, y), qn0 (x, y)), Rq (pn0 (x, y), qn0 (x, y))]. Based on the work by Gullon (2002), Lee and Kuo (1992) further conﬁrmed that the two-light photometric stereo is more sensitive to the azimuth rather than the zenith angle difference and that the optimal value is 90◦ . Gullon moves to the three-image photometric stereo and suggests that distributing the illumination azimuth angles equally through 360◦ is optimal. A theoretical analysis of Gullon’s arrangement was presented by Spence and Chantler (2003, 2006) based on the sensitivity analysis of photometric stereo deriving expressions of each surface normal vector with respect to image intensities as shown in the previous section. The illumination matrix [L], represented using the zenith and azimuth angles (see Eq. (9)), is deﬁned as

⎡

sin θl1 cos ϕl1 [L] = ⎣sin θl2 cos ϕl2 sin θl3 cos ϕl3

sin θl1 sin ϕl1 sin θl2 sin ϕl2 sin θl3 sin ϕl3

⎤ cos θl1 cos θl2 ⎦ cos θl3

(93)

= and depends on six parameters. The scaled surface normals ρN T [xn yn zn ] are given by Eq. (35). Calculating the inverse of matrix [L] and substituting into Eq. (35), the surface normals are obtained as

1 xn = − ((sin ϕl3 cos θl2 sin θl3 − sin ϕl2 sin θl2 cos θl3 )I 1 h + (sin ϕl1 cos θl3 sin θl1 − sin ϕl3 sin θl3 cos θl1 )I 2 + (sin ϕl2 cos θl1 sin θl2 − sin ϕl1 sin θl1 cos θl2 )I 3 )

(94)

1 yn = − ((cos ϕl2 cos θl3 sin θl2 − cos ϕl3 sin θl3 cos θl2 )I 1 h + (cos ϕl3 cos θl1 sin θl3 − cos ϕl1 sin θl1 cos θl3 )I 2 + (cos ϕl1 cos θl2 sin θl1 − cos ϕl2 sin θl2 cos θl1 )I 3 )

(95)

Photometric Stereo: An Overview

zn =

49

1 ((sin(ϕl3 − ϕl2 ) sin θl2 sin θl3 )I 1 h + (sin(ϕl1 − ϕl3 ) sin θl3 sin θl1 )I 2 + (sin(ϕl2 − ϕl1 ) sin θl1 sin θl2 )I 3 ),

(96)

where h ≡ sin(ϕl3 − ϕl2 ) cos θl1 sin θl2 sin θl3 + sin(ϕl1 − ϕl3 ) cos θl2 sin θl3 sin θl1 + sin(ϕl2 − ϕl1 ) cos θl3 sin θl1 sin θl2 . Assuming independent Gaussian noise of variance σi2 in each image, the variance of the noise in the surface normal components is given by

" σx2n σx2n σ¯ i2

=

σ¯ i2 "

=

∂xn ∂I 1

" σy2n σy2n σ¯ i2

=

σ¯ i2 "

=

σz2n σ¯ i2

" =

+

2

+

2

2

+

2

∂zn ∂I 2

2

+

2

+

2

∂yn ∂I 3

⇒

(98)

2

+

∂zn ∂I 3

2

∂zn ∂I 3

⇒

(97)

∂yn ∂I 3

+

2

∂zn ∂I 2

2

+

+

2

2

∂xn ∂I 3

+

+

∂yn ∂I 2

∂xn ∂I 3

∂yn ∂I 2

+

2

2

∂xn ∂I 2

∂xn ∂I 2

2

∂zn ∂I 1

∂zn ∂I 1

2

2

∂yn ∂I 1

∂yn ∂I 1

" σz2n = σ¯ i2

∂xn ∂I 1

⇒

.

(99)

Calculating the derivatives of Eqs. (94)–(96) with respect to the three image intensities, expressions describing the sensitivity of surface normals to errors in image intensity may be obtained:

sin ϕl3 cos θl2 sin θl3 − sin ϕl2 sin θl2 cos θl3 ∂xn =− 1 h ∂I

(100)

50

Vasileios Argyriou and Maria Petrou

sin ϕl1 cos θl3 sin θl1 − sin ϕl3 sin θl3 cos θl1 ∂xn =− 2 h ∂I

(101)

sin ϕl2 cos θl1 sin θl2 − sin ϕl1 sin θl1 cos θl2 ∂xn =− 3 h ∂I

(102)

cos ϕl2 cos θl3 sin θl2 − cos ϕl3 sin θl3 cos θl2 ∂yn =− 1 h ∂I

(103)

cos ϕl3 cos θl1 sin θl3 − cos ϕl1 sin θl1 cos θl3 ∂yn =− 2 h ∂I

(104)

cos ϕl1 cos θl2 sin θl1 − cos ϕl2 sin θl2 cos θl1 ∂yn =− 3 h ∂I

(105)

sin(ϕl3 − ϕl2 ) sin θl2 sin θl3 ∂zn = 1 h ∂I

(106)

sin(ϕl1 − ϕl3 ) sin θl3 sin θl1 ∂zn = 2 h ∂I

(107)

sin(ϕl2 − ϕl1 ) sin θl1 sin θl2 ∂zn . = 3 h ∂I

(108)

Substituting the derivatives in Eqs. (97)–(99) the noise ratio for all the surface normal components is obtained, which depends on the illumination conﬁguration. The optimal illumination arrangement is estimated by minimizing each of the three noise ratios in Eqs. (97)–(99). In order to provide a single objective function, Spence and Chantler (2003, 2006), assumed a Lambertian surface and generated the rendered intensities I r under given illumination directions (θl , ϕl ), as

I r (x, y) = xn cos ϕl sin θl + yn sin ϕl sin θl + zn cos θl .

(109)

This indicates that the rendered intensities are weighted sums of the surface normal components xn , yn , and zn , since the trigonometric terms are scalars. Therefore, the sum of the variances of the normal components is selected

Photometric Stereo: An Overview

51

as the objective function to be minimized, given by

"

F≡

∂xn 2 ∂xn 2 ∂xn 2 + + ∂I 1 ∂I 2 ∂I 3 " ∂yn 2 ∂yn 2 ∂yn 2 + + + ∂I 1 ∂I 2 ∂I 3 " ∂zn 2 ∂zn 2 ∂zn 2 + + + . ∂I 1 ∂I 2 ∂I 3

(110)

Substituting for the partial derivatives, the above equation becomes a function of the azimuth and zenith angles of the light sources and the minimum value of F determines the optimal illumination conﬁguration. It should be noted that the effect of inaccuracies in the measurement of the light source angles was not considered in this theoretical analysis. Spence and Chantler (2003) found that the optimal azimuth and zenith angles cannot be speciﬁed and that the conﬁguration that results in the minimum noise ratio is not unique. Instead, it was determined that an orthogonal arrangement of the illumination vectors (with an angle of 90◦ to each other) is the only restriction to obtaining the optimal conﬁguration. If the common zenith angle was constrained, the optimal values for azimuth angles were estimated and the use of a 120-degree difference in a three-light Lambertian photometric stereo was suggested. This result is in agreement with the work of Gullon (2002), where a uniform distribution of illumination azimuth angles was recommended. The optimal zenith angle in case of equally distributed light sources was found to be around 55◦ , but the angle should be reduced if shadows are present. In contradiction, if the surface is smooth and shadows are not an issue, the zenith angle may be increased. Furthermore, Drbohlav and Chantler (2005) extended the above for n light sources and proved the same optimal zenith angle when the sources are equidistantly spaced in azimuth angle, 360/n degrees apart.

14. CONCLUSIONS This chapter analyzed the basic concepts of radiometry and image formation and discussed the most well-known surface models. A short review of photometric stereo and a literature review of algorithms improving the standard technique were presented. Methods identifying problematic pixels, containing shadows and highlights, or taking into consideration object

52

Vasileios Argyriou and Maria Petrou

motion and perspective projection were analyzed. A sensitivity analysis of photometric stereo with respect to the image acquisition process and the accuracy of the illumination angles was presented. Finally, a theoretical analysis of the optimal illumination conﬁguration was discussed and recommendations were reported. The ﬁeld of photometric stereo has not yet reached its maturity, and its full potential in relation to applications has not been realized. Many advances are expected in the upcoming years with a resultant increased use in applications as diverse as medicine, robotics, and security.

ACKNOWLEDGMENTS This work was supported by EPSRC grant EP/E028659/1 “Face Recognition using photometric Stereo (PhotoFace).”

REFERENCES Barsky, S., and Petrou, M. (2001a). Color photometric stereo: simultaneous reconstruction of local gradient and color of rough textured surfaces. Proc. Int. Conf. Comput. Vis. ICCV 2, 600–605. Barsky, S., and Petrou, M. (2001b). Shadows and highlights detection in 4-source color photometric stereo. Int. Conf. Image Process. 3, 967–970. Barsky, S., and Petrou, M. (2003). The 4-source photometric stereo technique for 3-dimensional surfaces in the presence of highlights and shadows. IEEE Trans. Patt. Anal. Machine Intell. 25(10), 1239–1252. Barsky, S., and Petrou, M. (2006). Design issues for a color photometric stereo system. J. Math. Imaging Vis. 24(1), 143–162. Blinn, J. (1977). Models of light reﬂection for computer synthesized picture. Comput. Graphics 11(2), 192–198. Bors, A., Hancock, E., and Wilson, R. (2000a). 3-D terrain from synthetic aperture radar images. Proc. IEEE Workshop on Computer Vision Beyond the Visible Spectrum: Methods and Applications, pp. 63–72. Bors, A., Hancock, E., and Wilson, R. (2000b). A Bayesian framework for radar shape-fromshading. Proc. IEEE Int. Conf. Comput. Vis. Patt. Recogn. (CVPR ’00) 1, 262–268. Bors, A., Hancock, E., and Wilson, R. (2000c). Terrain modeling in synthetic aperture radar images using shape-from-shading. Proc. Int. Conf. Patt. Recogn. (ICPR ’00) 1, 798–801. Bors, A., Hancock, E., and Wilson, R. (2003). Terrain analysis using radar shape-from shading, IEEE Trans. Pattern Anal. Mach. Intell. 25(8), 974–992. Coleman, E., and Jain, R. (1982). Obtaining 3-dimensional shape of textured and specular surfaces using four-source photometry. Comput. Vis. Graph. Image Process. 18, 309–328. Drbohlav, O., and Chantler, M. (2005). On optimal light conﬁgurations in photometric stereo. IEEE Int. Conf. Comput. Vis. ICCV 2, 1707–1712. Farooq, A., Smith, M., Smith, L., and Midha, P. (2005). Dynamic photometric stereo for on line quality control of ceramic tiles. Computers in Industry (Special edition, Machine Vision) 56(8), 918–934. Gullon, C. (2002). “Height Recovery of Rough Surfaces from Intensity Images.” Doctoral thesis, School of Engineering and Physical Sciences, Heriot-Watt University. Hapke, B. (1963). A theoretical photometric function for the lunar surface. J. Geophys. Res. 68(15), 4571–4586.

Photometric Stereo: An Overview

53

He, X., Torrance, K., Sillon, F., and Greenberg, D. (1991). A comprehensive physical model for light reﬂection. Comput. Graph. 25(4), 175–186. Horn, B. (1975). Obtaining shape from shading information. In “The Psychology of Computer Vision” (P. H. Winston, ed.). McGraw-Hill, New York. Horn, B. (1986). “Robot Vision.” The MIT Press, Cambridge, MA. Horn, B., and Brooks, M. (1989). “Shape from Shading.” Cambridge, MA, The MIT Press. Horn, B., and Sjoberg, R. (1979). Calculating the reﬂectance map. Appl. Optics 18(11), 1770–1779. Jiang, X., and Bunke, H. (1991). On error analysis for surface normals determined by photometric stereo. Signal Proc. 23, 221–226. Klinker, G. (1993). “A Physical Approach to Color Image Understanding.” A K Peters, Ltd, Wellesley, MA. Klinker, G., Shafer, S., and Kanade, T. (1988). The measurement of highlights in color images. Int. J. Comput. Vis. 2, 7–32. Lee, J., Smith, M., Smith, L., and Midha, P. (2005). A mathematical morphology approach to image based 3D particle shape analysis. Machine Vision Appl. 16(5), 282–288. Lee, K., and Kuo, C. (1992). Shape reconstruction from photometric stereo. Comput. Vision Patt. Recogn., pp. 479–484. Liao, Y., Petrou, M., and Zhao, R. (2008). A fractal-based relaxation algorithm for shape from terrain image. Comput. Vis. Image Und. 109(3), 227–243. Lim, J., Yang, J., and Kriegman, D. (2005). Passive photometric stereo from motion. IEEE Int. Conf. Comput. Vis. ICCV 2, 1635–1642. Maki, A., Watanabe, M., and Wiles, C. (2002). Geotensity: combining motion and lighting for 3D surface reconstruction. Int. J. Comput. Vis. 48(2), 75–90. Nayar, S., Ikeuchi, K., and Kanade, T. (1990). Determining shape and reﬂectance of hybrid surfaces by photometric sampling. IEEE Trans. Robot. Automat. 6(4), 418–431. Nicodemus, F., Richmond, J., Hsia, J., Ginsberg, I., and Limperis, T. (1977). “Geometrical Considerations and Nomenclature for Reﬂectance” (Tech. Report NBS Monograph 160). National Bureau of Standards, Gaithersburg, MD. Orlova, N. (1956). Photometric relief of the lunar surface. Astron. Z 33(1), 93–100. Phong, B. T. (1975). Illumination for computer generated images. Comm. ACM 18(6), 311–317. Prados, E., and Faugeras, O. (2003). Perspective shape from shading and viscosity solutions. IEEE int. Conf. Comput. Vis. ICCV 2, 826–831. Ray, R., Birk, J., and Kelley, R. (1983). Error analysis of surface normals determined by radiometry. IEEE Trans. Patt. Anal. Machine Intell. 5(6), 631–641. Rindﬂeisch, T. (1966). Photometric method for lunar topography. Photogramm. Eng. 32(2), 262–277. Schluns, K., and Wittig, O. (1993). Photometric stereo for non-Lambertian surfaces using color information. Proc. 5th Int. Conf. Comput. Anal. Images Patt., pp. 444–451. Shafer, S. (1985). Using color to separate reﬂection components. Color Res. Appl. 10, 210–218. Simakov, D., Frolova, D., and Basri, R. (2003). Dense shape reconstruction of a moving object under arbitrary, unknown lighting. Proc. Ninth ICCV Conf. Comput. Vis. 2, 1202–1209. Smith, M. (1999). The analysis of surface texture using photometric stereo acquisition and gradient space domain mapping. Image Vision Comput. 17(14), 1009–1019. Smith, M., and Smith, L. (2005). Dynamic photometric stereo—a new technique for moving surface analysis. Image Vision Comput. 23, 841–852. Solomon, F., and Ikeuchi, K. (1996). Extracting the shape and roughness of specular lobe objects using four light photometric stereo. IEEE Trans. Patt. Anal. Machine Intell. 18(4), 449–454. Spence, A., and Chantler, M. (2003). Optimal illumination for three-image photometric stereo acquisition of texture. Proceedings of the 3rd International Workshop on Texture Analysis and Synthesis, pp. 89–94.

54

Vasileios Argyriou and Maria Petrou

Spence, A., and Chantler, M. (2006). Optimal illumination for three-image photometric stereo using sensitivity analysis. IEEE Proc. Vis. Image Sign. Proc. 153(2), 149–159. Tankus, A., and Kiryati, N. (2005). Photometric stereo under perspective projection. IEEE Int. Conf. Comput. Vis. ICCV 1, 611–616. Tankus, A. Sochen, N., and Yeshurun, Y. (2005). Shape-from-shading under perspective projection. Int. J. Comput. Vis. 63(1), 21–43. Tankus, A., Sochen N, and Yeshurun Y. (2004). Perspective shape-from-shading by fast marching. IEEE Comput. Soc. Conf. Comput. Vis. Patt. Recogn. CVPR 1, 43–49. Tankus, A., Sochen, N., and Yeshurun, Y. (2003). A new perspective [on] shape-from-shading. Proc. Ninth IEEE Int. Conf. Comput. Vis. 2, 862–869. Tomasi, C., and Kanade, T. (1992). Shape and motion from image streams under orthography: a factorization method. Int. J. Comput. Vis. 9(2), 137–154. Tsai, R. (1986). An efﬁcient and accurate camera calibration technique for 3D machine vision. Proc. IEEE Conf. Comput. Vis. Patt. Recogn., pp. 364–374. Woodham, R. (1980). Photometric method for determining surface orientation from multiple images. Opt. Eng. 19(1), 139–144. Worthington, P., and Hancock, E. (1997). Shape from shading using robust statistics. 13th International Conference on Digital Signal Processing Proceedings 2, 1145–1148. Worthington, P., and Hancock, E. (1999). Data-driven shape-from-shading using curvature consistency. IEEE Comput. Soc. Conf. Comput. Vis. Patt. Recogn. 1, 288–293. Zhang, L., Curless, B., Hertzmann, A., and Seitz, S. (2003). Shape and motion under varying illumination: unifying structure from motion, photometric stereo, and multi-view stereo. Proc. ICCV., pp. 618–625. Zhang, R., Tsai, P., Cryer, J., and Shah, M. (1999). Shape-from-shading: a survey. IEEE Trans. Patt. Anal. Machine Intell. 21(8), 690–706.

CHAPTER

2 The Fourier Transform in Clifford Analysis Fred Brackx, Nele De Schepper, and Frank Sommen*

Contents

1 Introduction 56 2 The Clifford Analysis Toolkit 58 2.1 Clifford Algebra 59 2.2 Clifford Analysis 62 2.3 Some Useful Results Concerning the Fourier Transform and Spherical Harmonics 68 2.4 The Generalized Clifford–Hermite Polynomials 73 2.5 Multidimensional “Analytic Signals” 83 3 The Fractional Fourier Transform 86 3.1 Introduction 87 3.2 The Classical Fractional Fourier Transform 87 3.3 Multidimensional Fractional Fourier Transform: Definition and Operator Exponential Form 90 3.4 The Mehler Formula for the Generalized Clifford–Hermite Polynomials 93 4 The Clifford–Fourier Transform 95 4.1 Introduction 95 4.2 Clifford–Hermite Monogenic Operators 98 4.3 Alternative Representations of the Classical Fourier Transform 113 4.4 Clifford–Fourier Transform: Definition and Properties 115 4.5 The Two-Dimensional Case 122 5 Clifford Filters for Early Vision 143 5.1 Introduction 143 5.2 Generalized Clifford–Hermite Filters 145 5.3 The Two-Dimensional Clifford–Gabor Filters 159 6 The Cylindrical Fourier Transform 167 6.1 Definition 167 6.2 Properties 170 6.3 Cylindrical Fourier Spectrum of the L2 -Basis Consisting of Generalized Clifford–Hermite Functions 175 Acknowledgments 197 References 197

* Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Ghent, Belgium Advances in Imaging and Electron Physics, Volume 156, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01402-x. Copyright © 2009 Elsevier Inc. All rights reserved.

55

56

Fred Brackx et al.

1. INTRODUCTION The Fourier transform is the most important integral transform. Since its introduction by Fourier in the early 1800s, it has remained an indispensable and stimulating mathematical concept at the core of the highly evolved branch of mathematics called Fourier analysis. It has found use in innumerable applications and has become a fundamental tool in engineering sciences, thanks to the generalizations extending the class of Fourier transformable functions and to the development of efﬁcient algorithms for computing the discrete version of it. The Fourier transform provides a representation of functions deﬁned over an inﬁnite interval and showing no particular periodicity in terms of a superposition of periodic, say sinusoidal, functions. More precisely, if f (t) is a real- or complex-valued integrable function of the real variable t, then its Fourier transform, also called Fourier image or frequency contents or spectrum, is the result of the following integral:

1 fˆ (ω) = F [ f ](ω) = √ 2π

+∞ −∞

exp (−iωt) f (t) dt,

which is a continuous and bounded function of the frequency variable ω ∈ ] − ∞, +∞ [. Still more important is that the original function f (t) may be recovered from its spectrum on the condition that it is continuous and bounded and that its Fourier image is integrable; this is the so-called inverse Fourier transform

f (t) = F

−1

1 [ fˆ ](t) = √ 2π

+∞

−∞

exp (itω) fˆ (ω) dω.

The natural habitat of the Fourier transform, however, is the space S of rapidly decreasing functions; these are indeﬁnite continuously differentiable functions decaying at inﬁnity more rapidly than the inverse of any polynomial. On S the Fourier transform is a one-to-one mapping. By a density argument the Fourier transform may then be deﬁned on the space L2 of square integrable functions—the so-called signals of ﬁnite energy—on which it becomes a homeomorphism, that is, a bicontinuous bijection, that preserves the L2 -norm. The latter is expressed by the celebrated Parseval formula: for all f , g in L2 (R) holds

f , g = F [ f ], F [g]. For handling functions of several variables the extension of the Fourier transform to higher dimension is obtained in a straightforward tensorial

The Fourier Transform in Clifford Analysis

57

manner, considering Fourier transforms in each of the Cartesian variables separately:

1 fˆ (ω) = F [ f ](ω) = √ ( 2π)m

Rm

exp (−iω, x) f (x) dV(x),

where ω stands for (ω1 , . . . , ωm ), x for (x1 , . . . , xm ), and ω, x for the traditional scalar product in Euclidean space: ω, x = m j=1 ωj xj . The properties of this standard multidimensional Fourier transform are similar to those of the one-dimensional (1D) transform. The second player in this paper is Clifford analysis. This is a function theory for functions deﬁned in Euclidean space Rm of arbitrary dimension m and taking values in the real Clifford algebra R0,m constructed over Rm . Clifford algebra, named after William Kingdon Clifford (1845– 1879) who himself talked about geometric algebra, is an associative but noncommutative algebra with zero divisors, which combines the algebraic properties of the reals, the complex numbers, and the quaternions with the geometric properties of Grassmann algebra. Clifford algebra has been rediscovered several times, among others as the algebra of Pauli matrices, which is the geometric algebra of the physical space; and the algebra of Dirac matrices, which is the geometric algebra of Minkowski space-time. Much of the recent interest in Clifford algebras can be traced to the works of David Hestenes in the 1960s, who viewed Clifford’s geometric algebra as a unifying language for mathematics and physics. During the past 50 years, Clifford analysis has gradually developed into a comprehensive theory offering a direct, elegant, and powerful generalization to higher dimension of the theory of holomorphic functions in the complex plane. In its most simple but still useful setting, ﬂat m-dimensional Euclidean space, Clifford analysis focuses on monogenic functions, i.e., null solutions of the Clifford vector–valued Dirac operator, ∂x = m j=1 ej ∂xj , where (e1 , . . . , em ) forms an orthogonal basis for the quadratic space R0,m underlying the construction of the real Clifford algebra R0,m . Monogenic functions have a special relationship with harmonic functions of several variables in that they are reﬁning their properties. The reason is that, as does the Cauchy–Riemann operator in the complex plane, the rotation-invariant Dirac operator factorizes the m-dimensional Laplace operator. At the same time, Clifford analysis offers the possibility of generalizing 1D mathematical analysis to higher dimension in a rather natural way by encompassing all dimensions at once, in contrast to the traditional approach, which consists of taking tensor products of 1D phenomena. This last qualiﬁcation of Clifford analysis will be exploited in this paper to construct a genuine multidimensional Fourier transform within the

58

Fred Brackx et al.

context of Clifford analysis. In fact, over the past years several speciﬁc multidimensional Fourier transforms were constructed using Clifford algebra with goals focused mostly in the ﬁeld of signal analysis. An overview of these constructs is given in Section 4 where our Clifford–Fourier theory is developed. We have included an introductory section on Clifford analysis, and each section starts with an introductory situation. The new Clifford–Fourier transform is given in terms of an operator exponential, or alternatively, by a series representation. Particular attention is directed to the two-dimensional (2D) case since then the Clifford–Fourier kernel can be written in a closed form. The main section is preceded by a section on the fractional Fourier transform wherein, surprisingly, it is shown that the traditional and the Clifford analysis approach coincide. Section 5 develops the theory for the Clifford–Hermite and Clifford–Gabor ﬁlters for early vision. The topic in the last section is still in an experimental stage; we try to devise a cylindrical Fourier transform. The idea is the following: For a ﬁxed vector in the image space, the level surfaces of the traditional Fourier kernel are planes perpendicular to that ﬁxed vector. For this Fourier kernel we now substitute a new Clifford–Fourier kernel such that, again for a ﬁxed vector in the image space, its phase is constant on coaxial cylinders w.r.t. that ﬁxed vector. The point is that when restricted to dimension two, this new cylindrical Fourier transform coincides with the Clifford–Fourier transform of Section 4. Thus we are faced with the following situation: In dimension greater than two, we have a ﬁrst Clifford–Fourier transform with elegant properties but no kernel in closed form, and a second cylindrical one with a kernel in closed form but more complicated calculation formulae. In dimension two both transforms coincide. Our aim is to present a consistent theory on a speciﬁc multidimensional Fourier transform in the hope that it might be used in applications.

2. THE CLIFFORD ANALYSIS TOOLKIT Clifford analysis is a well-established mathematical discipline that is closely related but complementary to harmonic analysis. It has gradually developed into a comprehensive theory that offers a direct, elegant, and powerful generalization to higher dimension of the theory of holomorphic functions in the complex plane. In its most simple but still useful setting, it focuses on the null solutions of various special partial differential operators arising naturally within the Clifford algebra language, the most important of which is the so-called Dirac operator, ∂x = m j=1 ej ∂xj . Here (e1 , . . . , em ) forms an orthonormal basis for the quadratic space R0,m underlying the construction of the real Clifford algebra R0,m . Numerous papers, conference proceedings, and books have molded this theory and

The Fourier Transform in Clifford Analysis

59

shown its ability for applications (e.g., Brackx, Delanghe, and Sommen, 1982; Delanghe, Sommen, and Souˇcek, 1992; Gilbert and Murray, 1991; Gürlebeck and Sprößig, 1990; Gürlebeck and Sprößig, 1997; Gürlebeck, Habetha, and Sprößig, 2006; Qian et al., 2004; Ryan, 2004; Ryan, 1996; Ryan, 2000; and Ryan and Struppa, 1998). This section is included to make the chapter self-contained and readable by an audience unacquainted with Clifford analysis. It deals with the basic notions of Clifford algebra (Section 2.1) and Clifford analysis (Section 2.2) that are necessary for our purpose. Moreover, in Section 2.3 we collect some results concerning the standard tensorial Fourier transform and spherical harmonics that are used in the following sections. The Clifford–Hermite functions, forming a generalized basis for the space L2 Rm , dV(x) of square-integrable functions, are shown to be eigenfunctions of the standard Fourier transform (Section 2.4). The last section discusses multidimensional analytic signals.

2.1. Clifford Algebra Clifford algebra may be considered a generalization to higher dimension of the norm division algebras of the real numbers R, the complex numbers C, and the quaternions H. In these traditional algebras, R, C, and H, division by a nonzero number is always possible and in each of these cases there is a norm . such that λ μ = λ μ, for all λ, μ ∈ R, C or H. Such as in the skew ﬁeld of quaternions H the multiplication in a Clifford algebra is noncommutative, but still associative. However, there are zero divisors making division by a nonzero Clifford number impossible in general. One of the most constructive ways to deﬁne a Clifford algebra is as follows. Let the Euclidean space Rm be endowed with a nondegenerate quadratic form of signature (p, q), p + q = m, and let (e1 , . . . , em ) be an orthonormal basis for Rp,q . The noncommutative multiplication in the universal Clifford algebra Rp,q , constructed over Rp,q , is governed by the following rules:

ej2 = 1, j = 1, . . . , p 2 ep+j = −1, j = 1, . . . , q

ej ek + ek ej = 0, j = k, j, k = 1, . . . , m. A canonical basis for Rp,q is obtained by considering for any set A = {j1 , . . . , jh } ⊂ {1, . . . , m} = M, ordered by 1 ≤ j1 < j2 < . . . < jh ≤ m, the element eA = ej1 ej2 . . . ejh . Moreover, for the empty set ∅ one puts e∅ = 1, the latter being the identity element (i.e., the neutral element with respect to multiplication). Note that R0,1 is isomorphic with C via the correspondence

60

Fred Brackx et al.

e1 ←→ i, while R0,2 is isomorphic with H via the correspondence

e1 ←→ i e2 ←→ j e1 e2 ←→ ij = k. Any Clifford number λ in Rp,q may thus be written as λ = A⊂M eA λA , λA ∈ R, or still as λ = m |A|=k eA λA is the so-called k=0 [λ]k , where [λ]k = k-vector part of λ (k = 0, 1, . . . , m). Denoting by Rkp,q the subspace of all k-vectors in Rp,q (i.e., the image of Rp,q under the projection operator [ . ]k ), one has the multivector structure decomposition Rp,q = R0p,q ⊕ R1p,q ⊕ . . . ⊕ Rm p,q , leading to the identiﬁcation of R with the subspace of real scalars R0p,q and of the Euclidean space Rm with the subspace of real Clifford vectors R1p,q . The Clifford number eM = e1 e2 . . . em is usually called the pseudoscalar; depending on the dimension m, the pseudoscalar commutes or anticommutes with the k-vectors and squares to ±1. In the following text we consider the real Clifford algebra R0,m and the complex Clifford algebra Cm , which may be seen as its complexiﬁcation Cm = C ⊗ R0,m = R0,m ⊕ i R0,m (i.e., all coefﬁcients are taken to be complex). An important anti-automorphism of Cm leaving the multivector structure invariant is the Hermitean conjugation, deﬁned by

(λμ)† = μ† λ† † (λA eA )† = λcA eA

ej† = −ej

(A ⊂ M) ( j = 1, . . . , m).

Here λcA denotes the complex conjugate of the complex number λA . In view of the decomposition Cm = R0,m ⊕ i R0,m , any complex Clifford number λ ∈ Cm may also be written as λ = a + ib with a, b ∈ R0,m . Moreover, the restriction of the Hermitean conjugation to R0,m coincides with the usual conjugation in the Clifford algebra R0,m (i.e., the main antiinvolution for which ej = −ej , j = 1, . . . , m). Hence, one may also write λ† = (a + ib)† = a − ib. The Hermitean conjugation leads to a Hermitean inner product and its associated norm on Cm , given respectively, by

(λ, μ) = [λ† μ]0

and |λ|2 = [λ† λ]0 =

A

|λA |2 .

The Fourier Transform in Clifford Analysis

61

For λ, μ ∈ Cm the following properties hold:

|λμ| ≤ 2m |λ| |μ|

|λ + μ| ≤ |λ| + |μ|.

and

(1)

The Euclidean space Rm is embedded in the Clifford algebras R0,m and Cm by identifying variable x the point (x1 , . . . , xm ) with the real vector m+1 is identiﬁed given by x = m e x , whereas the Euclidean space R j=1 j j with R00,m ⊕ R10,m by identifying (x0 , x1 , . . . , xm ) with the real paravector x0 + x . The product of two vectors divides up into a scalar part (the inner product up to a minus sign) and a 2-vector, also called bivector, part (the wedge product):

x y = x . y + x ∧ y, where

x . y = − < x, y > = −

m

xj yj

j=1

and

x∧y =

m m

ei ej (xi yj − xj yi ).

i=1 j=i+1

Note that the square of a vector variable x is scalar valued and equals the norm squared up to a minus sign

x2 = − < x, x > = −|x|2 . This implies that each vector is invertible

x−1 = −

x . |x|2

In addition, each paravector x0 + x is invertible with

(x0 + x)−1 =

x0 − x . |x0 + x|2

The spin group SpinR (m) of the Clifford algebra consists of all products of an even number of unit vectors

SpinR (m) = {s = ω1 . . . ω2 ; ωj ∈ Sm−1 , j = 1, . . . , 2 , ∈ N},

62

Fred Brackx et al.

with Sm−1 the unit sphere in Rm . The spin group doubly covers the rotation group SOR (m): for T ∈ SOR (m) there exists s ∈ SpinR (m) such that T(x) = sxs. But then also T(x) = (−s)x(−s), explaining the double character of this covering. For a detailed study of Clifford algebra, see Porteous (1995).

2.2. Clifford Analysis Clifford analysis offers a function theory that is a higher-dimensional analog of the theory of the holomorphic functions of one complex variable. The functions considered are deﬁned in the Euclidean space Rm or Rm+1 (m > 1) and take their values in Clifford algebra R0,m or in its complexiﬁcation Cm . The central notion in Clifford analysis is the notion of monogenicity, a notion that is the multidimensional counterpart to that of holomorphy in the complex plane. A function F(x1 , . . . , xm ), respectively F(x0 , x1 , . . . , xm ), deﬁned and continuously differentiable in an open region of Rm , respectively Rm+1 , and taking values in R0,m or Cm , is called left monogenic in that region if

∂x [F] = 0,

respectively

(∂x0 + ∂x )[F] = 0.

Here ∂x is the Dirac operator in Rm :

∂x =

m

ej ∂xj .

j=1

This Dirac operator is an elliptic, rotation-invariant, vector differential operator of the ﬁrst order, which may be viewed as the “square root” of the Laplace operator in Rm , since

m = −∂x2 .

(2)

The operator ∂x0 + ∂x is termed the Cauchy–Riemann operator in Rm+1 ; it factorizes the Laplace operator in Rm+1 :

m+1 = (∂x0 + ∂x )(∂x0 + ∂x ) = (∂x0 + ∂x )(∂x0 − ∂x ). The factorization shown in Eq. (2) of the Laplace operator establishes a special relationship between Clifford analysis and harmonic analysis in that monogenic functions reﬁne the properties of harmonic functions. Note, for instance, that each monogenic function is harmonic and that in its turn each harmonic function h(x) can be split as h(x) = f (x) + x g(x) with f and g monogenic, and that a real harmonic function is always the real part of

The Fourier Transform in Clifford Analysis

63

a monogenic one, which need not be the case for a harmonic function of several complex variables. The notion of right monogenicity is deﬁned in a similar manner by letting act the Dirac operator or the Cauchy–Riemann operator from the right. It is easily seen that if a Clifford algebra–valued function F is left monogenic, its Hermitean conjugate F† is right monogenic. Introducing spherical coordinates in Rm by:

x = rω ,

r = |x| ∈ [0, +∞[,

ω ∈ Sm−1 ,

the Dirac operator takes the form

1 ∂x = ω ∂r + , r

where

= −x ∧ ∂x = −

m m

ei ej (xi ∂xj − xj ∂xi )

i=1 j=i+1

is the angular Dirac operator acting only on the angular coordinates. In Subsection 4.5.1 we will also use the angular momentum operators

Lij = xi ∂xj − xj ∂xi ,

i, j = 1, 2, . . . , m.

Another fundamental operator is the Euler operator

E = < x , ∂x > =

m

xi ∂xi = r∂r

i=1

which measures the degree of homogeneity of both Clifford polynomials and Clifford polynomial operators. If P s denotes the space of scalar-valued polynomials in Rm , then a Clifford algebra–valued polynomial (Clifford polynomial for short) is an element of P = P s ⊗ Cm . The inner product on the space P is deﬁned as

< P(x), Q(x) > = P† (∂x ) Q(x) |x=0 , with P(∂x ) the differential operator obtained by substituting for all i, ∂xi for xi in P(x). The subspaces Pk of homogeneous Clifford polynomials of degree k, k ∈ N, are the polynomial eigenspaces of the Euler operator

64

Fred Brackx et al.

(i.e., E[Rk ] = kRk , Rk ∈ Pk ). Obviously, every Clifford polynomial can be decomposed into homogeneous ones. Let End(P s ) be the algebra of endomorphisms of P s ; the elements of End(P s ) ⊗ Cm are called Clifford polynomial operators. A Clifford polynomial operator A transforms a Clifford polynomial P ∈ P into another Clifford polynomial A[P] ∈ P. Like a Clifford polynomial, every Clifford polynomial operator can be decomposed into homogeneous parts:

A=

A ,

(3)

∈Z

with A a homogeneous polynomial operator of degree (i.e., A [Pk ] ⊂ Pk+ , k ∈ N). Moreover, the homogeneous operators A are determined by the commutation relation [E, A ] = EA − A E = A . In Section 4.2 we consider so-called Clifford differential operators with polynomial coefﬁcients. These operators take the form

P(x, ∂x ) =

α

pα (x) ∂x ,

α α

α

with ∂x = ∂x11 . . . ∂xαmm and where pα (x) ∈ P. Let D(Cm ) denote the algebra of Clifford differential operators with polynomial coefﬁcients. Then clearly D(Cm ) ⊂ End(P s ) ⊗ Cm . This algebra D(Cm ) is generated by the basic operators {ej , xj , ∂xj , j = 1, 2, . . . , m}. Examples of such Clifford differential operators with polynomial coefﬁcients are the Dirac operator ∂x , the Euler operator E, the angular Dirac operator , and the left vector multiplication operator f → x f. Many classical theorems from complex analysis in the plane (e.g., Cauchy’s theorem, Cauchy’s integral theorem, Taylor series, Laurent series) have their multidimensional counterpart in Clifford analysis. However, note that due to the noncommutativity of the multiplication in the Clifford algebra, the product of two monogenic functions is, generally no longer monogenic. The natural powers of the vector variable x or the paravector variable x0 + x are not monogenic either. These drawbacks have been overcome in the following way. There is a fundamental method for constructing monogenic functions, the so-called Cauchy– Kowalewskaia (CK) extension procedure, introduced by Sommen (1981). It runs as follows. If ⊂ Rm is open, then an open neighborhood of in Rm+1 is said to be x0 -normal if for each x ∈ the line segment {x + t ; t ∈ R} ∩ is connected and contains exactly one point in (Figure 1). Considering Rm as the hyperplane x0 = 0 in Rm+1 , a real analytic function f (x) in an open connected domain in Rm can be uniquely extended to a monogenic function F(x0 , x) in an open connected and x0 -normal neighbourhood of

The Fourier Transform in Clifford Analysis

65

x

m

{x t ;t }

FIGURE 1

An x0 -normal neighborhood of in Rm+1 .

in Rm+1 . This CK extension of f (x) is given by

F(x0 , x) =

∞ x (−1) 0 ∂x [ f (x)] = exp (−x0 ∂x )[ f (x)]. !

(4)

=0

In particular, the CK extension of the real variables xj , j = 1, . . . , m, are zj = xj − x0 ej , j = 1, . . . , m, the so-called monogenic variables. The CK extension procedure leads to the CK product which, despite the noncommutativity of the Clifford algebra, preserves the monogenicity of the factors; the CK product of two monogenic functions in Rm+1 is the CK extension to Rm+1 of the product of the real analytic restrictions to Rm . For example, the CK product of the monogenic variables zj and zk (k = j) is the CK extension of xj xk , given by

zj zk =

1 (zj zk + zk zj ). 2

The CK products of the monogenic variables are precisely the building blocks of the Taylor series expansion of a monogenic function. In this chapter, the monogenic homogeneous Clifford polynomials and functions, or spherical monogenics, play an important role. A left, respectively right, monogenic homogeneous Clifford polynomial Pk of degree k (k ≥ 0) in Rm is called a left, respectively right, solid inner spherical monogenic of order k. A left, respectively right, monogenic homogeneous function Qk of degree −(k + m − 1) in Rm \ {0} is called a left, respectively right, solid outer spherical monogenic of order k. The set of all left, respectively right, solid inner spherical monogenics of order k will be denoted by M+ (k), respectively Mr+ (k), whereas the set of all

66

Fred Brackx et al.

left, respectively right, solid outer spherical monogenics of order k will be denoted by M− (k), respectively Mr− (k). The dimension of M+ (k) is given by

m+k−2 (m + k − 2)! . dim M+ (k) = = (m − 2)! k! m−2 For Pk ∈ M+ (k) and s ∈ N the following fundamental formula holds:

s

∂x [x Pk ] =

−s xs−1 Pk −(s + 2k + m − 1) xs−1 Pk

for s even for s odd.

(5)

Moreover, the left solid inner spherical monogenics are polynomial eigenfunctions of the angular Dirac operator, i.e., [Pk ] = −kPk , Pk ∈ M+ (k). The set of harmonic homogeneous polynomials Sk of degree k in Rm :

m [Sk (x)] = 0

and

Sk (tx) = tk Sk (x),

usually called solid spherical harmonics, is denoted by H(k). Obviously, we have that

M+ (k) ⊂ H(k)

and Mr+ (k) ⊂ H(k).

Let H(r) be a unitary right Clifford module; that is, H(r) , + is an abelian group and a law ( f , λ) → f λ from H(r) × R0,m into H(r) is deﬁned such that for all λ, μ ∈ R0,m and f , g ∈ H(r)

(i) f (λ + μ) = f λ + f μ (iii) ( f + g)λ = f λ + gλ

(ii) f (λμ) = ( f λ)μ (iv) fe∅ = f .

Note that H(r) becomes a real vector space if R is identiﬁed with Re∅ ⊂ R0,m . Then a function ( . , . ) : H(r) × H(r) → R0,m is said to be an inner product on H(r) if for all f , g, h ∈ H(r) and λ ∈ R0,m

(i) ( f , gλ + h) = ( f , g)λ + ( f , h); (iii) [( f , f )]0 ≥ 0;

(ii) ( f , g) = ( g, f ); (iv) [( f , f )]0 = 0 iff f = 0.

From this R0,m -valued inner product ( . , . ), one can deduce the real inner product

( f , g)R = [( f , g)]0

The Fourier Transform in Clifford Analysis

67

on H(r) , the latter being considered a real vector space. Putting for each f ∈ H(r) : f 2 = [( f , f )]0 , . is a norm on H(r) turning it into a normed right Clifford module. Now, let H(r) be a unitary right Clifford module provided with an inner product ( . , . ). Then it is called a right Hilbert–Clifford module if H(r) considered as a real vector space provided with the real inner product ( . , . )R is a Hilbert space (see Delanghe and Brackx, 1978). Let h be a positive function on Rm . Then we consider the Clifford algebra–valued inner product of the functions f and g deﬁned in Rm and taking values in Clifford algebra Cm

< f,g > =

Rm

h(x) f † (x) g(x) dV(x),

where dV(x) is the Lebesgue measure on Rm , and moreover the associated norm

f 2 = [< f , f >]0 . The unitary right Clifford module of Clifford algebra–valued measurable functions on Rm for which f 2 < ∞ is a right Hilbert–Clifford module, denoted by L2 Rm , h(x) dV(x) . In particular, if we take h(x) ≡ 1, we obtain the right Hilbert–Clifford module of square integrable functions:

m

f : Lebesgue measurable in Rm for which

L2 R , dV(x) =

f 2 =

1/2

Rm

| f (x)|2 dV(x)

<∞ .

A representation of the spin group on this space is the H-representation given by H(s)f (x) = sf (sxs)s, s ∈ SpinR (m). Naturally, L1 Rm , dV(x) denotes the right Clifford module of integrable functions

m

f : Lebesgue measurable in Rm for which

L1 R , dV(x) = f 1 =

Rm

| f (x)| dV(x) < ∞ ,

whereas C0 Rm , L∞ Rm and S Rm denote the right Clifford modules of Cm -valued, respectively, continuous, bounded, and rapidly decreasing functions. A basic integral formula used in Section 6.2 is the Clifford–Stokes formula.

68

Fred Brackx et al.

Let ⊂ Rm be open, let C be a compact orientable m-dimensional manifold with boundary ∂C, and deﬁne the oriented vector-valued surface element dσx on ∂C by the Clifford differential form

dσx =

m (−1)j ej dxM\{j} , j=1

where

dxM\{j} = dx1 ∧ . . . ∧ [dxj ] ∧ . . . ∧ dxm ,

j = 1, 2, . . . , m.

We also introduce the volume element

dxM = dx1 ∧ . . . ∧ dxm . Theorem 2.1 [Clifford–Stokes theorem]. Let f , g ∈ C1 ( ). Then for each C ⊂ , one has

∂C

f (x) dσx g(x) =

C

[ f ∂x g + f (∂x g)] dxM .

2.3. Some Useful Results Concerning the Fourier Transform and Spherical Harmonics In Clifford analysis extensive use is made of the standard tensorial multidimensional Fourier transform given by:

F [ f ](ξ) =

1 (2π)m/2

Rm

exp (−i < x , ξ >) f (x) dV(x).

(6)

Note that the Fourier image inherits its Clifford algebra character only from the original function f (x) itself, because the traditional Fourier kernel is scalar valued. We now list a few fundamental results concerning the Fourier transform.

2 Proposition 2.1 The Gaussian function exp − |x|2 is an eigenfunction of the Fourier transform:

2 |ξ|2 (ξ) = exp − 2 . F exp − |x|2 Theorem 2.2 The Fourier transform F is an isometry of square on the space integrable functions, in other words, for all f , g ∈ L2 Rm , dV(x) the Parseval

The Fourier Transform in Clifford Analysis

69

formula holds

< f , g > = < F [f ], F [g] > . In particular, for each f ∈ L2 Rm , dV(x) one has f 2 = F [f ]2 . Proposition 2.2 The Fourier transform F satisﬁes (i) the multiplication rule:

F [x f (x)](ξ) = i∂ξ [ F [f (x)](ξ)]; (ii) the differentiation rule:

F [∂x [ f (x)]](ξ) = iξ F [ f (x)](ξ). The properties established in Proposition 2.2 also may be expressed by stating that the vector variable x and the Dirac operator ∂x are each other’s Fourier symbol; this property is usually called Fourier duality; in Clifford analysis circles, it is often termed Fischer duality. Next, we formulate the Funk–Hecke theorem (see Hochstadt, 1971), which is the key to the calculation of certain integrals involving spherical harmonics. Theorem 2.3 [Funk–Hecke theorem on the unit sphere]. Let Sk ∈ H(k) be a spherical harmonic of degree k and η a ﬁxed point on the unit sphere Sm−1 . Denote η) = tη for ω ∈ Sm−1 . Then < ω, η > = cos (ω,

Sm−1

f (tη )Sk (ω)dS(ω) = Am−1

1

−1

f (t)(1 − t2 )(m−3)/2 Pk,m (t)dt Sk (η),

where Pk,m (t) denotes the Legendre polynomial of degree k in m-dimensional Euclidean space and

Am−1 =

2 π(m−1)/2

m−1 2

the surface area of the unit sphere Sm−2 in Rm−1 . From the above we can easily deduce a so-called Funk–Hecke theorem in space. Theorem 2.4 [Funk–Hecke theorem in space] Let Sk ∈ H(k) be a spherical harmonic of degree k and η a ﬁxed point on the unit sphere Sm−1 . Denote

70

Fred Brackx et al.

< ω, η > = tη for ω ∈ Sm−1 , then

Rm

g(r) f (tη ) Sk (ω) dV(x)

= Am−1

+∞

g(r) r

m−1

dr

1

−1

0

2 (m−3)/2

f (t) (1 − t )

Pk,m (t) dt Sk (η).

As the Legendre polynomials are even or odd according to the parity of k, we can also state the following corollary, which will be frequently used in Section 6.3. Corollary 2.1 Let Sk ∈ H(k) be a spherical harmonic of degree k and η a ﬁxed point on the unit sphere Sm−1 . Denote < ω, η > = tη for ω ∈ Sm−1 , then the 3D integral

Rm

g(r) f (tη ) Sk (ω) dV(x)

is zero whenever • f is an odd function and k is even; • f is an even function and k is odd. Theorem 2.3 leads to the following result (see Stein and Weiss, 1971). Proposition 2.3 Let Sk ∈ H(k) be a spherical harmonic of degree k, then for r, ρ > 0 and ω, η ∈ Sm−1 , one has

Sm−1

exp (−irρ < ω, η >) Sk (ω) dS(ω)

= (−i)k (2π)m/2 Sk (η) (ρr)1−m/2 Jk+m/2−1 (ρr)

m with Jk+m/2−1 the Bessel function of the ﬁrst kind of order k + − 1 . 2 Proof. Application of the Funk–Hecke theorem on the unit sphere leads to

Sm−1

exp (−irρ < ω, η >) Sk (ω) dS(ω)

= Am−1 Sk (η)

+1

−1

exp (−irρt) (1 − t2 )(m−3)/2 Pk,m (t) dt.

The Fourier Transform in Clifford Analysis

71

As

Pk,m (t) =

k! (m − 3)! (m−2)/2 C (t) (k + m − 3)! k

(7)

and the Gegenbauer polynomials Ckλ satisfy

Ckλ (−x) = (−1)k Ckλ (x), we obtain

Sm−1

exp (−irρ < ω, η >) Sk (ω) dS(ω) k

= (−1) Am−1 Sk (η)

+1 −1

exp (irρt) (1 − t2 )(m−3)/2 Pk,m (t) dt.

(8)

Taking into account (see Hochstadt, 1971)

+1

−1

exp (irρt) (1 − t2 )(m−3)/2 Pk,m (t) dt k

m/2−1

=i 2

√

m−1 π 2

(ρr)1−m/2 Jk+m/2−1 (ρr)

m with Jk+m/2−1 the Bessel function of the ﬁrst kind of order k + − 1 , 2 Eq. (8) becomes

Sm−1

exp (−irρ < ω, η >) Sk (ω) dS(ω) k

m/2−1

= (−i) 2

√

m−1 π 2

Am−1 Sk (η) (ρr)1−m/2 Jk+m/2−1 (ρr).

Expliciting the area Am−1 of the unit sphere Sm−2 in Rm−1 in terms of the Gamma function, we ﬁnally obtain

Sm−1

exp (−irρ < ω, η >) Sk (ω) dS(ω)

= (−i)k (2π)m/2 Sk (η) (ρr)1−m/2 Jk+m/2−1 (ρr).

72

Fred Brackx et al.

By means of this result one can prove the following proposition (see Brackx, De Schepper, and Sommen, 2003). Proposition 2.4 Let Sk ∈ H(k) be a solid spherical harmonic of degree k, then

2 2 = (−1)k Sk (x) exp − |x|2 . Sk (∂x ) exp − |x|2 Proof.

By means of Proposition 2.1 and 2.2, we have

F Sk (x) exp

2 − |x|2

|x|2 (ξ) = Sk (i∂ξ ) F exp − 2 (ξ)

k

= i Sk (∂ξ ) exp

|ξ|2 − 2

or equivalently

Sk (∂ξ ) exp

|ξ|2 − 2

|x|2 (ξ). = (−i) F Sk (x) exp − 2

k

Using spherical coordinates

x = r ω,

ξ = ρ η where

r = |x| ,

ρ = |ξ|,

ω, η ∈ Sm−1 ,

we obtain

2 F Sk (x) exp − |x|2 (ξ) 1 = (2π)m/2 =

1 (2π)m/2

Rm

0

2 exp (−i < x, ξ >) Sk (x) exp − |x|2 dV(x)

+∞

2 rk+m−1 exp − r2 dr

Sm−1

exp (−irρ < ω, η >) Sk (ω) dS(ω).

Next, Proposition 2.3 leads to

2 (ξ) F Sk (x) exp − |x|2 = (−i)k ρ1−m/2 Sk (η)

0

+∞

2 rk+m/2 Jk+m/2−1 (ρr) exp − r2 dr.

The Fourier Transform in Clifford Analysis

73

From the theory of Bessel functions (see Erdélyi et al., 1953b) we know that

+∞

0

2 2 rk+m/2 Jk+m/2−1 (ρr) exp − r2 dr = ρk+m/2−1 exp − ρ2 .

Hence, we ﬁnally obtain

2 2 (ξ) = (−i)k ρk Sk (η) exp − ρ2 F Sk (x) exp − |x|2 |ξ|2 = (−i)k Sk (ξ) exp − 2 , which leads to the desired result.

2.4. The Generalized Clifford–Hermite Polynomials On the real line the

Hermite polynomials associated with the weight x2 function exp − 2 may be deﬁned by the Rodrigues formula

Hen (x) = (−1)n exp

2 2 dn x exp − x2 , 2 dxn

n = 0, 1, 2, . . .

They constitute an orthogonal

basis for the weighted Hilbert space x2 L2 ] − ∞, +∞[ , exp − 2 dx , and satisfy the orthogonality relation

+∞ −∞

x2 exp − 2

Hen (x) Hen (x) dx = n!

√ 2π δn,n

and moreover the recurrence relation

Hen+1 (x) = x Hen (x) −

d [Hen (x)]. dx

Furthermore, Hen (x) is an even or an odd function according to the parity of n; that is, Hen (−x) = (−1)n Hen (x). Sommen (1988) introduced the generalized Clifford–Hermite polynomials, which are a speciﬁc generalization to Clifford analysis of the above Hermite polynomials on the real line.

74

Fred Brackx et al.

2.4.1. Definition In order to a basis for the weighted Hilbert module

obtain |x|2 m L2 R , exp − 2 dV(x) , Sommen (1988) introduced the generalized Clifford–Hermite polynomials. These generalized Clifford–Hermite ∗ polynomials

are deﬁned by the CK-extension G of the weight function

exp − |x|2 order k :

2

Pk (x), with Pk (x) any left solid inner spherical monogenic of

∗

G (x0 , x) = exp

2 − |x|2

∞

x

0

=0

!

H,k (x) Pk (x).

From the monogenicity of G∗ , we obtain

H+1,k (x) Pk (x) = (x − ∂x )[H,k (x) Pk (x)], which in its turn leads to the following recurrence relations:

H2+1,k (x) = (x − ∂x )[H2,k (x)] and

H2+2,k (x) = (x − ∂x )[H2+1,k (x)] − 2k

x H2+1,k (x). |x|2

A straightforward calculation yields

H0,k (x) = 1 H1,k (x) = x H2,k (x) = x2 + 2k + m = −|x|2 + 2k + m H3,k (x) = x3 + (2k + m + 2) x = x (−|x|2 + 2k + m + 2) H4,k (x) = x4 + 2(2k + m + 2) x2 + (2k + m)(2k + m + 2) = |x|4 − 2(2k + m + 2)|x|2 + (2k + m)(2k + m + 2) H5,k (x) = x5 + 2(2k + m + 4) x3 + (2k + m + 4)(2k + m + 2) x = x |x|4 − 2(2k + m + 4)|x|2 + (2k + m + 4)(2k + m + 2) etc.

(9)

75

The Fourier Transform in Clifford Analysis

It is important to remark that the functions H,k (x) do not depend upon the particular choice of the left solid inner spherical monogenic Pk , but only upon the order k. Note that the generalized Clifford–Hermite polynomials H,k are of degree in the variable x with real coefﬁcients depending on k. The polynomial H2,k (x) contains only even powers of x and so is scalar valued, while H2+1,k (x) contains only odd ones and so is vector valued. Moreover, taking into account Eq. (4), it is easily seen that the generalized Clifford–Hermite polynomials satisfy the Rodrigues formula

H,k (x) Pk (x) = exp

|x|2 2

2 (−∂x ) exp − |x|2 Pk (x) .

(10)

Furthermore, they can be expressed in terms of the generalized Laguerre polynomials Lα on the real line given by

Lα (u)

= (−1)s s=0

us ( + α + 1) . ( − s + 1) (α + s + 1) s!

We indeed have:

H2p,k (x) = 2p p!

m/2+k−1 Lp

|x|2 , 2

H2p+1,k (x) = 2p p!

m/2+k Lp

|x|2 2

x, (11)

conﬁrming that H2p,k is scalar valued, while H2p+1,k is vector valued. Moreover, it follows that

H2p+1,k (x) = x H2p,k+1 (x). In Section 6 we will need the so-called Kummer function, also termed conﬂuent hypergeometric function ∞

1 F1 (a; c; u) =

(c) (a + s) us , (a) (c + s) s!

c = 0, −1, −2, . . .

s=0

The fact is that the Laguerre polynomials may be expressed in terms of Kummer’s function:

Lα (u) =

( + α + 1) 1 F1 (−; α + 1; u), ( + 1) (α + 1)

76

Fred Brackx et al.

which leads to the following expression of the generalized Clifford– Hermite polynomials:

m 2 p + + k |x| m 2 1 F1 −p; + k; H2p,k (x) = 2p 2 2 m2 + k p + m2 + k + 1 |x|2 m x 1 F1 −p; + k + 1; . (12) H2p+1,k (x) = 2 2 2 m2 + k + 1 p

It may also be shown that the generalized Clifford–Hermite polynomials are mutually orthogonal with respect to a Gaussian weight function, more precisely

Rm

† 2 H,k1 (x) Pk1 (x) Ht,k2 (x) Pk2 (x) dV(x) = γ,k1 δ,t δk1 ,k2 exp − |x|2 (13)

with

γ2p,k

22p+m/2+k p! πm/2 = m2

m 2

+k+p

and

γ2p+1,k

22p+m/2+k+1 p! πm/2 = m2

m 2

+k+p+1

.

Furthermore, the set

1 (j) Ht,k (x) Pk (x) ; t, k ∈ N, j ≤ dim(M+ (k)) (γt,k )1/2

2 constitutes an orthonormal basis for L2 Rm , exp − |x|2 dV(x) . Here

(j) Pk (x) ; j = 1, 2, . . . , dim M+ (k)

denotes an orthonormal basis of the space M+ (k) of left solid inner spherical monogenics of order k.

The Fourier Transform in Clifford Analysis

77

2.4.2. Differential Equations In this subsection we list a few useful differential equations satisﬁed by the generalized Clifford–Hermite polynomials. Proposition 2.5 The generalized Clifford–Hermite polynomials satisfy (i) the eigenvalue equation

(∂x x − x∂x )[H,k (x) Pk (x)] = a,k H,k (x) Pk (x) (ii) the annihilation equation

∂x [H,k (x) Pk (x)] = −C,k H−1,k (x) Pk (x) (iii) the second-order differential equation

∂x2 [H,k (x) Pk (x)] − x ∂x [H,k (x) Pk (x)] − C,k H,k (x) Pk (x) = 0 with

−(m + 2k) m + 2k − 2

a,k = and

C,k =

− 1 + m + 2k

for even for odd

for even for odd.

Proof. (i) Taking into account Eq. (5), we indeed have for even—that is, = 2p,

∂x [x H2p,k (x) Pk (x)] = ∂x [H2p,k (x)] x Pk (x) + H2p,k (x) ∂x [x Pk (x)] = x ∂x [H2p,k (x)]Pk (x) − (m + 2k) H2p,k (x) Pk (x) = x ∂x [H2p,k (x) Pk (x)] − (m + 2k) H2p,k (x) Pk (x). Now we consider the case odd—that is, = 2p + 1. As H2p+1,k (x) takes the form x f (r) with f a polynomial of degree p in r2 , we obtain

∂x [x H2p+1,k (x) Pk (x)] = ∂x [ f (r) x2 Pk (x)] = ∂x [ f (r)] x2 Pk (x) + f (r) ∂x [x2 Pk (x)] = x ∂x [ f (r)] x Pk (x) − 2f (r) x Pk (x).

78

Fred Brackx et al.

As

∂x [H2p+1,k (x) Pk (x)] = ∂x [f (r)] x Pk (x) − (m + 2k) f (r) Pk (x), we obtain the desired result:

∂x [x H2p+1,k (x) Pk (x)] = x ∂x [H2p+1,k (x) Pk (x)] + (m + 2k − 2) H2p+1,k (x) Pk (x). (ii) We prove the statement by induction. First, for = 0 we have

∂x [H0,k (x) Pk (x)] = ∂x [Pk (x)] = 0, while for = 1 ∂x [H1,k (x) Pk (x)] =∂x [x Pk (x)] = − (m + 2k) Pk (x) = − C1,k H0,k (x) Pk (x). Next, assume that the property holds for − 1; that is,

∂x [H−1,k (x) Pk (x)] = −C−1,k H−2,k (x) Pk (x). Using Eq. (9) and the induction hypothesis, we obtain

∂x [H,k (x) Pk (x)] = ∂x [x H−1,k (x) Pk (x)] − ∂x2 [H−1,k (x) Pk (x)] = ∂x [x H−1,k (x) Pk (x)] + C−1,k ∂x [H−2,k (x) Pk (x)]. By means of (i), the induction hypothesis and Eq. (9), this becomes consecutively

∂x [H,k (x) Pk (x)] = a−1,k H−1,k (x) Pk (x) + x ∂x [H−1,k (x) Pk (x)] + C−1,k ∂x [H−2,k (x) Pk (x)] = a−1,k H−1,k (x) Pk (x) − C−1,k (x − ∂x )[H−2,k (x) Pk (x)] = a−1,k H−1,k (x) Pk (x) − C−1,k H−1,k (x) Pk (x) = − C,k H−1,k (x) Pk (x).

79

The Fourier Transform in Clifford Analysis

(iii) By means of (ii) and Eq. (9), we have consecutively

∂x2 [H,k (x) Pk (x)] − x ∂x [H,k (x) Pk (x)] = −C,k ∂x [H−1,k (x) Pk (x)] + C,k x H−1,k (x) Pk (x) = C,k (x − ∂x )[H−1,k (x) Pk (x)] = C,k H,k (x) Pk (x). Note that formula (iii) of Proposition 2.5 generalizes the differential equation

d d2 [Hen (x)] + n Hen (x) = 0 [Hen (x)] − x 2 dx dx satisﬁed by the classical Hermite polynomials on the real line. Furthermore, combining formula (ii) of Proposition 2.5 and Eq. (9) yields

H+1,k (x) Pk (x) − x H,k (x) Pk (x) − C,k H−1,k (x) Pk (x) = 0,

(14)

which is a generalization of the recurrence relation

Hen+1 (x) − x Hen (x) + n Hen−1 (x) = 0 satisﬁed by the classical Hermite polynomials.

2.4.3. Orthonormal Basis for the Space of Square Integrable Functions By means of the generalized Clifford–Hermite polynomials we now construct an orthonormal basis for the space of square integrable functions, the ﬁnite energy signals. Proposition 2.6 The set

1 (j) |x|2 ; s, k ∈ N, j ≤ dim(M+ (k)) H (x) P (x) exp − s,k 4 k (γs,k )1/2

constitutes an orthonormal basis for L2 Rm , dV(x) . Proof. By means of Eq. (13), the orthonormality of the set is straightforward. Now, take f ∈ L2 Rm , dV(x) . This means that

Rm

f (x)2 dV(x) =

Rm

2 2 |x| |x|2 f (x) exp 4 exp − 2 dV(x) < ∞,

80

Fred Brackx et al.

2

2 in other words, f (x) exp |x|4 ∈ L2 Rm , exp − |x|2 dV(x) . Consequently, there exists a linear combination

N N

dim M+ (k)

s=0 k=0

as,k,j (γs,k

j=1

(j)

)1/2

Hs,k Pk ,

as,k,j ∈ Cm

such that

+ dim M (k) N N as,k,j ( j) f exp |x|2 − Hs,k Pk 4 (γ )1/2 s=0 k=0

|x|2

L2 Rm ,exp − 2

s,k

j=1

dV(x)

tends to zero if N, N → ∞. This also implies that

+ dim M (k) 2 N N 2 as,k,j ( j) |x| f exp − Hs,k Pk 4 1/2 (γ ) s=0 k=0

j=1

|x|2

L2 Rm ,exp − 2

s,k

dV(x)

tends to zero if N, N → ∞ or that

Rm

+ dim M (k) 2 N N

as,k,j 2 (j) f (x) exp |x| − Hs,k (x) Pk (x) 4 1/2 (γ ) s=0 k=0

2 × exp − |x|2 dV(x)

j=1

s,k

tends to zero if N, N → ∞. So ﬁnally we obtain that

Rm

+ dim M (k) N N

2 as,k,j (j) |x|2 f (x) − Hs,k (x) Pk (x) exp − 4 dV(x) (γ )1/2 s=0 k=0

j=1

s,k

tends to zero if N, N → ∞, which proves the statement.

2.4.4. Eigenfunctions of the Fourier Transform The ﬁnal step is to obtain an L2 -basis consisting of eigenfunctions of the Fourier transform. In view of Proposition 2.1, we carry out the

The Fourier Transform in Clifford Analysis

81

√ substitution x → 2 x, which leads to the following orthonormal basis for L2 (Rm , dV(x)):

φs,k,j (x) =

√ 2m/4 (j) √ |x|2 H ( 2x) P ( 2x) exp − s,k 2 ; k (γs,k )1/2 dim(M+ (k))

s, k ∈ N, j ≤

.

(15)

Note that an equivalent orthogonal basis is given by

−p;

1 F1

2 m (j) + k; |x|2 exp − |x|2 Pk (x), 2

1 F1

2 m (j) −p; + k; |x|2 exp − |x|2 x Pk−1 (x) 2

with p, k ∈ N, j ≤ dim(M+ (k)). Proposition 2.7 For each left solid inner spherical monogenic Pk of order k ∈ N and all s ∈ N one has

F exp

2 − |x|2

× exp

|ξ|2 − 2

√ √ Hs,k ( 2x) Pk ( 2x) (ξ) = exp −i(s + k) π2

√ √ Hs,k ( 2ξ) Pk ( 2ξ).

Proof. (by induction). By means of Propositions 2.1, 2.2, and 2.4, one can easily verify that the statement holds for s = 0 and s = 1. Assuming that it holds for s, we now prove it for s + 1. By means of the recurrence relation in Eq. (14) we have

√ √ 2 exp − |x|2 Hs+1,k ( 2x) Pk ( 2x) =

√ √ √ 2 2x exp − |x|2 Hs,k ( 2x) Pk ( 2x)

√ √ 2 + Cs,k exp − |x|2 Hs−1,k ( 2x) Pk ( 2x).

82

Fred Brackx et al.

Taking the Fourier transform yields

F exp

2 − |x|2

√ √ Hs+1,k ( 2x) Pk ( 2x) (ξ)

√ √ √ |x|2 = 2i ∂ξ F exp − 2 Hs,k ( 2x) Pk ( 2x) (ξ)

√ √ 2 + Cs,k F exp − |x|2 Hs−1,k ( 2x) Pk ( 2x) (ξ). In view of the induction hypothesis, this becomes

√ √ 2 F exp − |x|2 Hs+1,k ( 2x) Pk ( 2x) (ξ) =

√ √ √ |ξ|2 2i exp −i(s + k) π2 ∂ξ exp − 2 Hs,k ( 2ξ) Pk ( 2ξ) + Cs,k exp

−i(s − 1 + k) π2

exp

|ξ|2 − 2

√ √ Hs−1,k ( 2ξ) Pk ( 2ξ).

Using formula (ii) of Proposition 2.5, we have

∂ξ exp

|ξ|2 − 2

= −ξ exp × exp

√ √ Hs,k ( 2ξ) Pk ( 2ξ)

|ξ|2 − 2

|ξ|2 − 2

√ √ √ Hs,k ( 2ξ) Pk ( 2ξ) − 2 Cs,k

√ √ Hs−1,k ( 2ξ) Pk ( 2ξ).

Consequently, we ﬁnally obtain

√ √ 2 F exp − |x|2 Hs+1,k ( 2x) Pk ( 2x) (ξ) = exp ×

−i(s + 1 + k) π2

√

exp

|ξ|2 − 2

√ √ √ √ 2ξ Hs,k ( 2ξ)Pk ( 2ξ) + Cs,k Hs−1,k ( 2ξ)Pk ( 2ξ)

√ √ |ξ|2 π = exp −i(s + 1 + k) 2 exp − 2 Hs+1,k ( 2ξ) Pk ( 2ξ).

The Fourier Transform in Clifford Analysis

83

2.5. Multidimensional “Analytic Signals” Let us introduce the fundamental solution E of the Cauchy–Riemann operator in Rm+1 :

E(x0 , x) =

x0 − x . Am+1 |x0 + x|m+1 1

In the complement of the origin it is a strong null solution of the Cauchy– Riemann operator ∂x0 + ∂x , this means a C∞ -smooth function satisfying

(∂x0 + ∂x )[E(x0 , x)] = 0. Moreover, considered as a distribution it satisﬁes

(∂x0 + ∂x )[E(x0 , x)] = δ(x0 , x), with δ(x0 , x) the delta or Dirac distribution supported at the origin. We may now deﬁne for a square integrable function f ∈ L2 (Rm , dV(x)), its Cauchy integral (see, for example, Gilbert and Murray, 1991) in the half spaces

Rm+1 = {(x0 , x) ∈ Rm+1 : x0 > 0} ± <

by

C[ f ](x0 , x) = E(x0 , . ) ∗ f ( . )(x) =

Rm

E(x0 , x − y) f (y) dV(y).

Clearly this Cauchy integral is left monogenic in Rm+1 w.r.t the Cauchy– ± Riemann operator ∂x0 + ∂x and has limit zero for (x0 , x) tending to inﬁnity. Moreover, it is a linear isomorphism between L2 (Rm , dV(x)) and the so-called Hardy spaces H 2 (Rm+1 ± ), deﬁned by

H 2 (Rm+1 ± )

= F(x0 , x) left monogenic in Rm+1 ±

sup x0

> <

0

2

Rm

|F(x0 , x)| dV(x) < +∞ .

such that

84

Fred Brackx et al.

Furthermore, the L2 (Rm , dV(x)) nontangential boundary values for x0 → 0 + and x0 → 0− of the Cauchy integral—these are the Cauchy transforms—take the following form:

C + [ f ](x) := lim C[ f ](x0 , x) = x0 →0 >

1 1 f (x) + H[ f ](x) 2 2

and

1 1 C − [ f ](x) := lim C[ f ](x0 , x) = − f (x) + H[ f ](x). x0 →0 2 2 < Here H[ f ] denotes the Hilbert transform of the function f given by

x 2 Pv m+1 ∗ f Am+1 r x−y 2 = Pv f ( y) dV( y) m+1 Am+1 Rm |x − y| x−y 2 =− lim f (y) dV(y), 0 |x−y|> |x − y|m+1 Am+1 → >

H[ f ](x) =

where Pv denotes the Cauchy principal value. The properties of this Hilbert transform are the following: (i) the Hilbert transform H is a bounded operator on L2 (Rm , dV(x)) (ii) the Hilbert transform H is a unitary operator on L2 (Rm , dV(x)), this means H ∗ H = HH ∗ = I, H ∗ being the adjoint and I the identity operator (iii) the inverse of H is H itself, or equivalently, H 2 = I (iv) the adjoint operator H ∗ is H itself, which means

H[ f ], g = f , H[g], for all f and g in L2 (Rm , dV(x)). Next, we introduce another Hardy space, namely H 2 (Rm ), which is deﬁned as the closure in L2 (Rm , dV(x)) of the space of the nontangential bound2 m ary values for x0 → 0+ of all functions in H 2 (Rm+1 + ). Because H (R ) is a m closed subspace of the Hilbert space L2 (R , dV(x)), we obtain the following orthogonal decomposition:

L2 (Rm , dV(x)) = H 2 (Rm ) ⊕ H 2 (Rm )⊥ .

The Fourier Transform in Clifford Analysis

85

Hence, there exist two projection operators, the so-called Szegö projections, denoted by

P+ : L2 (Rm , dV(x)) → H 2 (Rm )

and

P− : L2 (Rm , dV(x)) → H 2 (Rm )⊥ .

It is clear that these Szegö projections coincide with the Hardy projections or Cauchy transforms C + and −C − mentioned above:

P+ [ f ] = C + [ f ] =

1 1 ( f + H[ f ]) and P− [ f ] = −C − [ f ] = ( f − H[ f ]). 2 2

Moreover, since the Fourier transform of the Hilbert convolution kernel, considered as a tempered distribution, is given by

F

2 Am+1

Pv

x |x|m+1

(ξ) = i

ξ |ξ|

,

the Fourier transform of the Hilbert transform is given by

F [H[ f ]](ξ) = iη F [ f ](ξ), where we have used spherical coordinates in frequency space given by

ξ = ρ η, ρ = |ξ| ∈ [0, +∞[, η ∈ Sm−1 . The orthogonal decomposition of an L2 (Rm , dV(x))-function f :

f = P+ [ f ] + P− [ f ] thus reads in frequency space

F[ f ] =

1 1 (1 + iη) F [ f ] + (1 − iη) F [ f ]. 2 2

Here the so-called Clifford–Heaviside functions

P+ =

1 1 + iη , 2

P− =

1 1 − iη , η ∈ Sm−1 , 2

appear; they were introduced independently by Sommen in 1982 and McIntosh (Li, McIntosh, and Qian, 1994, and McIntosh, 1996). The Clifford–Heaviside functions may be considered the higher-dimensional

86

Fred Brackx et al.

analogs of the Heaviside step-function on the real axis and are a typical feature of Clifford analysis; they are self-adjoint mutually orthogonal primitive idempotents satisfying:

P+ + P− = 1; (P± )† = P± ; P+ P− = P− P+ = 0; (P± )2 = P± . It thus follows that the Hardy components of a ﬁnite energy signal f ∈ L2 (Rm , dV(x)) may be computed as follows:

P+ [ f ] = F −1 P+ F [ f ] ; P− [ f ] = F −1 P− F [ f ] , where F −1 denotes the inverse Fourier transform. It also follows that

P∓ F P± [f ] = 0. If we call a function g (anti-)causal when P± g = 0, then we have shown that “the Fourier spectrum of the Hardy components of a ﬁnite energy signal is (anti-)causal.” This property of the Hardy components, together with their property of being the nontangential boundary values of a monogenic function in the upper, respectively, lower half space, allows us to call

P± [ f ] =

1 f ± H[ f ] 2

an (anti-)analytic or (anti-)monogenic signal, a mathematically sound candidate for the multidimensional counterpart to the well-known notion of (anti-)analytic signal in 1D signal analysis. For further reading on the multidimensional Hilbert transform and Hardy spaces, see Bernstein and Lanzani (2002), Brackx et al., (2006d), Brackx and Van Acker (1992), Delanghe (2002a,b, 2004), Gilbert and Murray (1991), Li, McIntosh, and Qian (1994), McIntosh (1996), Mitrea (1994) and Murray (1985).

3. THE FRACTIONAL FOURIER TRANSFORM A multidimensional fractional Fourier transform is deﬁned in the Clifford analysis context as an operator exponential. It coincides with the tensorial fractional Fourier transform. In this way we are able to prove Mehler’s formula for the generalized Clifford–Hermite polynomials (see Brackx et al., 2007).

The Fourier Transform in Clifford Analysis

87

3.1. Introduction The fractional Fourier transform (abbreviated FrFT) may be considered a fractional power of the classical Fourier transform. It has been intensely studied during the past decade, an attention it may have gained partially because of the vivid interest in time-frequency analysis methods of signal processing (see, e.g., Bultheel and Martinez 2002; McBride and Kerr 1987; Namias 1980). In the 1D case, an integral representation for the FrFT can be obtained by means of Mehler’s formula for the classical Hermite polynomials (see Watson 1933). Here, we proceed in the reverse manner. First, we introduce a multidimensional FrFT in the framework of Clifford analysis using the generalized Clifford–Hermite polynomials introduced in Section 2.4. Then we show that this FrFT coincides with the classical tensorial FrFT in higher dimension. Thus, we are able to prove Mehler’s formula for the generalized Clifford–Hermite polynomials. In Section 3.2 we describe the classical FrFT. Next, we introduce the FrFT in the Clifford analysis setting and show that it can be written as an operator exponential. From this operator exponential form it becomes clear that our FrFT coincides with the classical tensorial higher-dimensional FrFT (Section 3.3). This allows us to derive the so-called Mehler formula for the generalized Clifford–Hermite polynomials (Section 3.4).

3.2. The Classical Fractional Fourier Transform The concept of fractional powers of the Fourier operator appears in the mathematical literature as early as 1929 (see Condon 1937; Kober 1939; Wiener 1929). It has been rediscovered in quantum mechanics, optics, and signal processing. The boom in publications started in the early years of the 1990s and it is still ongoing. A recent state of the art can be found in Ozaktas et al. (2001). The FrFT on the real line is ﬁrst deﬁned on a basis for the space of rapidly decreasing functions S(R). For this basis, one uses a complete orthonormal set of eigenfunctions of the Fourier transform given by

1 F [f ](ξ) = √ 2π

+∞

−∞

exp (−iξx) f (x) dx,

f ∈ S(R).

A possible choice for these eigenfunctions are the normalized Hermite functions:

2 21/4 exp − x2 Hn (x), φn (x) = √ 2n n!

(16)

88

Fred Brackx et al.

where

Hn (x) = (−1)n exp (x2 )

dn [exp (−x2 )] dxn

(17)

are the Hermite polynomials associated with the weight function exp (−x2 ). These eigenfunctions satisfy the orthonormality relation

< φn , φn > = δn,n with respect to the L2 -inner product

1 < f, g > = √ 2π

+∞ −∞

f (x) g(x) dx

and the eigenvalue equation

F [φn ] = exp −in π2 φn .

(18)

The eigenvalue for φn is thus given by λn = λn with λ = exp −i π2 representing a rotation over an angle π2 . It is precisely that concept of rotation that is generalized by the FrFT. Just as the classical Fourier transform corresponds to a rotation in the timefrequency plane over an angle π2 , the FrFT corresponds to a rotation over an arbitrary angle α = a π2 with a ∈ R. Consequently, the FrFT is deﬁned by

F a [φn ] = exp −ina π2 φn = λan φn = λna φn ,

(19)

with λa = exp −ia π2 = exp (−iα). Thus, the classical Fourier transform corresponds to F 1 . Note also that for α = 0 or a = 0, we get the identity operator F 0 = I and for α = π or a = 2 we get the parity operator F 2 [ f ](ξ) = f (−ξ) . The FrFT can be written as an operator exponential F a = exp (−iαH), so that

2 2 exp (−iαH) exp − x2 Hn (x) = exp (−inα) exp − x2 Hn (x).

Differentiating the above relation with respect to α, setting α = 0, and then using the differential equation

d d2 [Hn (x)] + 2n Hn (x) = 0, [Hn (x)] − 2x dx dx2 one can easily verify that the operator H is given by H = − 12

d2 dx2

− x2 +1 .

89

The Fourier Transform in Clifford Analysis

As the set of normalized Hermite functions φn constitutes an orthonormal basis for L2 (R, dx), each function f ∈ L2 (R, dx) can be expanded in terms of these eigenfunctions φn :

f =

∞

an φn ,

n=0

where the coefﬁcients an are given by

1 an = √ 2π

+∞

−∞

φn (x) f (x) dx

1

= √ 2n n! π 2

+∞

−∞

2 Hn (x)exp − x2 f (x) dx.

(20)

Applying the FrFT on this function yields a

F [f] =

∞

an exp −ina π2 φn .

(21)

n=0

The calculation of FrFTs by means of Eq. (21) is usually not practical. In order to obtain the integral representation of the operator F a , a formula provided by Mehler (1866), is used: ∞ 1 exp (−inα) Hn (ξ) Hn (x) n 2 n! n=0

1 2xξ exp = 1 − exp (−2iα)

exp (−iα)−exp (−2iα) (ξ 2 +x2 ) 1−exp (−2iα)

.

Inserting an from Eq. (20) into Eq. (21) and using Mehler’s formula, one obtains 1 F a [ f ](ξ) = √ π 1 − exp (−2iα)

+∞ −∞

exp

2xξ exp (−iα)−exp (−2iα) (ξ 2 +x2 ) 1−exp (−2iα)

2 2 f (x) dx. exp − ξ +x 2

(22)

90

Fred Brackx et al.

Note that for 0 < |α| < π, this expression can also be written as

1 exp − 2i π2 αˆ − α exp 2i ξ 2 cot (α) 2π| sin (α)| +∞

exp −i sinxξ(α) + 2i x2 cot (α) f (x) dx,

F a [ f ](ξ) = √

−∞

where αˆ = sgn(sin (α)). It was previously mentioned that F 0 [ f ](ξ) = f (ξ) and F ±π [ f ](ξ) = f (−ξ). Furthermore, when |α| > π, the deﬁnition is taken modulo 2π and reduced to the interval [−π, π]. The FrFT can be extended to higher dimension by taking tensor products. If Ka (ξ, x) denotes the kernel of the 1D FrFT, that is,

a

F [ f ](ξ) =

+∞

−∞

Ka (ξ, x) f (x) dx,

then one deﬁnes the m-dimensional FrFT as follows:

F a1 ,..., am [ f ](ξ1 , . . . , ξm ) =

+∞ −∞

...

+∞

−∞

Ka1 ,..., am (ξ1 , . . . , ξm ; x1 , . . . , xm )

f (x1 , . . . , xm ) dV(x), where

Ka1 ,..., am (ξ1 , . . . , ξm ; x1 , . . . , xm ) = Ka1 (ξ1 , x1 ) . . . Kam (ξm , xm ).

3.3. Multidimensional Fractional Fourier Transform: Definition and Operator Exponential Form In Section 2.4, we showed that

F [φs, k, j ](ξ) = exp −i(s + k) π2 φs, k, j (ξ). Following the deﬁnition on the real line [see Eq. (19)], we deﬁne the multidimensional FrFT in Clifford analysis by

FCa [φs, k, j ](ξ) = exp −i(s + k)a π2 φs, k, j (ξ); = exp −i(s + k)α φs, k, j (ξ) π with α = a . 2

a∈R

The Fourier Transform in Clifford Analysis

91

We now will show that, similar to the classical case, the FrFT FCa can be written as an operator exponential. Proposition 3.1 The FrFT FCa can be written as an operator exponential

FCa = exp (−iαHC ) = exp −ia π2 HC , where the operator HC is given by

HC =

1 2 1 (∂x − x2 − mI) = − (m − |x|2 + mI) 2 2

with I the identity operator. Proof. First, we note that the operator exponential exp (−iαHC ) is deﬁned as the series ∞ Hn exp (−iαHC ) = (−iα)n C . n! n=0

Differentiating the relation

√ 2 (j) √ exp (−iαHC ) Hs, k ( 2x) Pk ( 2x) exp − |x|2

√ 2 (j) √ = exp −i(s + k)α Hs, k ( 2x) Pk ( 2x) exp − |x|2 with respect to α, and setting α equal to zero, yields

√ 2 (j) √ HC Hs, k ( 2x) Pk ( 2x) exp − |x|2

√ 2 (j) √ = (s + k) Hs, k ( 2x) Pk ( 2x) exp − |x|2 . Now we will verify that the operator HC is indeed given by

HC =

1 2 (∂ − x2 − mI). 2 x

92

Fred Brackx et al.

We have

√ 2 ( j) √ (∂x2 − x2 − mI) exp − |x|2 Hs, k ( 2x) Pk ( 2x)

√ 2 ( j) √ = −exp − |x|2 ∂x [x Hs, k ( 2x) Pk ( 2x)]

√ 2 ( j) √ − x exp − |x|2 ∂x [Hs, k ( 2x) Pk ( 2x)]

(23)

√ 2 ( j) √ + exp − |x|2 ∂x2 [Hs, k ( 2x) Pk ( 2x)]

√ 2 ( j) √ − m exp − |x|2 Hs, k ( 2x) Pk ( 2x). From formula (i) of Proposition 2.5, we readily obtain

√ ( j) √ ∂x [x Hs, k ( 2x) Pk ( 2x)] √ √ ( j) √ ( j) √ = as, k Hs, k ( 2x) Pk ( 2x) + x ∂x [Hs,k ( 2x) Pk ( 2x)]. Consequently, Eq. (23) becomes

(∂x2

√ ( j) √ |x|2 − x − mI) exp − 2 Hs, k ( 2x) Pk ( 2x) 2

√ √ 2 ( j) √ ( j) √ = exp − |x|2 ∂x2 [Hs, k ( 2x) Pk ( 2x)] − 2x ∂x [Hs, k ( 2x) Pk ( 2x)] √ ( j) √ − (as, k + m) Hs, k ( 2x) Pk ( 2x) . Furthermore, formula (iii) of Proposition 2.5 implies

√ √ ( j) √ ( j) √ ∂x2 [Hs, k ( 2x) Pk ( 2x)] − 2x ∂x [Hs, k ( 2x) Pk ( 2x)] √ ( j) √ = 2 Cs, k Hs, k ( 2x) Pk ( 2x),

The Fourier Transform in Clifford Analysis

93

which ﬁnally leads to

√ 1 2 ( j) √ |x|2 2 (∂ − x − mI) exp − 2 Hs, k ( 2x) Pk ( 2x) 2 x

√ 2 1 ( j) √ = 2Cs, k − as, k − m Hs, k ( 2x) Pk ( 2x) exp − |x|2 2

√ 2 ( j) √ = (s + k) Hs, k ( 2x) Pk ( 2x) exp − |x|2 , since for all s

2Cs, k − as, k − m = 2s + 2k. From Proposition 3.1 we observe that, surprisingly, our FrFT coincides with the classical tensorial higher-dimensional FrFT F a1 ,..., am with a1 = a2 = . . . = am = a.

3.4. The Mehler Formula for the Generalized Clifford–Hermite Polynomials A Clifford algebra–valued square integrable function f can be expanded in terms of the eigenfunctions {φs, k, j }: +

f (x) =

∞ ∞ dim(M (k)) s=0 k=0

φs, k, j (x) as, k, j ,

j=1

where the Clifford algebra–valued coefﬁcients as, k, j are given by

as, k, j = < φs, k, j , f > =

Rm

†

φs, k, j (x)

f (x) dV(x).

(24)

By applying the operator FCa , we get +

FCa [ f ](ξ) =

∞ ∞ dim(M (k)) s=0 k=0

j=1 +

=

∞ dim(M ∞ (k)) s=0 k=0

FCa [φs, k, j ](ξ) as, k, j exp −i(s + k)α φs, k, j (ξ) as, k, j .

j=1

We thus have obtained the deﬁnition of the FrFT FCa in the form of a series.

94

Fred Brackx et al.

By replacing as, k, j in the series by their integral expression in Eq. (24) it is turned into +

FCa [ f ](ξ) =

∞ ∞ dim(M (k)) s=0 k=0

exp −i(s + k)α

j=1

√ |ξ| ( j) √ × Hs, k ( 2ξ) Pk ( 2ξ) exp − 2 Rm

2m/4 (γs, k )1/2 2

√ † 2m/4 ( j) √ |x|2 ( 2x) P ( 2x) exp − H f (x) dV(x) s, k 2 k (γs, k )1/2 + ∞ dim(M ∞ (k)) exp −i(s + k)α

m/2

=2

√ ( j) Hs, k ( 2ξ) Pk

γs, k √ √ † |x|2 +|ξ|2 ( j) √ exp − 2 f (x) dV(x). ( 2ξ) Hs, k ( 2x) Pk ( 2x) Rm

s=0 k=0

j=1

(25) Conversely, from the previous section we know that our FrFT FCa coincides with F a,..., a . Consequently, by means of Eq. (22) we have

FCa [ f ](ξ) =

Rm

Ka (ξ1 , x1 ) . . . Ka (ξm , xm ) f (x) dV(x)

=

1

m

√ π 1 − exp (−2iα) 2x1 ξ1 exp (−iα)−exp (−2iα)(ξ12 +x12 ) ξ12 +x12 exp − 2 × exp 1−exp (−2iα) Rm

2 +x2 ) 2xm ξm exp (−iα)−exp (−2iα)(ξm m . . . exp 1−exp (−2iα) 2 2 ξ +x exp − m 2 m f (x) dV(x) m 2<x, ξ>exp (−iα) 1 = √ exp 1−exp (−2iα) π 1 − exp (−2iα) Rm (|x|2 +|ξ|2 ) exp (−2iα) |x|2 +|ξ|2 × exp − 1−exp (−2iα) exp − 2 f (x) dV(x). (26) Comparing Eqs. (25) and (26) yields the following result.

95

The Fourier Transform in Clifford Analysis

Theorem 3.1 The Mehler formula for the generalized Clifford–Hermite polynomials takes the following form: ∞ ∞ exp (−i(s + k)α)

γs, k

s=0 k=0 + dim(M (k))

=

√ Hs,k ( 2ξ)

√ † ( j) √ † ( j) √ Hs, k ( 2x) Pk ( 2ξ) Pk ( 2x)

j=1

1 1 √ 2π 1 − exp (−2iα)

m

exp

2<x, ξ>exp (−iα)−(|x|2 +|ξ|2 ) exp (−2iα) 1−exp (−2iα)

.

4. THE CLIFFORD–FOURIER TRANSFORM Recently several generalizations to higher dimension of the Fourier transform, using Clifford algebra, have been introduced, including our Clifford– Fourier transform, which we deﬁned in Brackx, De Schepper, and Sommen (2005) as an operator exponential with a Clifford algebra–valued kernel. This section provides an overview of all these generalizations. Moreover, an in-depth study of our Clifford–Fourier transform is presented. Particular attention is paid to the 2D situation, since in this case we succeed in ﬁnding a closed form for the integral kernel of the Clifford–Fourier transform leading to further properties, in both the L1 and the L2 context (see Brackx, De Schepper, and Sommen, 2006b).

4.1. Introduction The Fourier transformation is, next to convolution, one of the two robust and frequently used tools for the analysis of scalar ﬁelds and image processing as computer vision. Quite naturally, attempts have been made to extend these methods to analyze 2D and 3D vector ﬁelds and even multivector ﬁelds. This introduction sketches an overview of these generalizations of the Fourier transform. Bülow and Sommer (2001) deﬁne a quaternionic Fourier transform of 2D signals f (x1 , x2 ) taking their values in the algebra H of real quaternions. Recall that the quaternion algebra H is nothing but the Clifford algebra R0,2 where, traditionally, the basis vectors are denoted by i and j, with i2 = j2 = −1, and the bivector by k = ij. In terms of these basis vectors, this quaternionic Fourier transform takes the form

q

F [ f ](u1 , u2 ) =

R2

exp (−2πiu1 x1 ) f (x1 , x2 ) exp (−2πju2 x2 ) dV(x).

96

Fred Brackx et al.

Due to the noncommutativity of the multiplication in H, the convolution theorem for this quaternionic Fourier transform takes a rather complicated form, which is moreover the case for its higher-dimensional analogue, the Clifford–Fourier transform given by

cl

F [ f ](u) =

Rm

f (x) exp (−2πe1 u1 x1 ) . . . exp (−2πem um xm ) dV(x).

Note that for m = 1, this Clifford–Fourier transform reduces to the standard Fourier transform on the real line, whereas for m = 2, the quaternionic Fourier transform is reobtained when restricting to real signals. In addition Bülow and Sommer introduce a commutative hypercomplex Fourier transform given by

F h [ f ](u) =

Rm

˜ f (x) exp −2π m u x e j=1 j j j dV(x),

where the basis vectors (˜e1 , . . . , e˜m ) obey the commutative multiplication rules e˜j e˜k = e˜k e˜j , j, k = 1, . . . , m, while still e˜j2 = −1, j = 1, . . . , m. This commutative hypercomplex Fourier transform offers the advantage that the corresponding convolution theorem has a rather simple outlook. The hypercomplex Fourier transforms F q , F cl , and F h enable Bülow and Sommer to establish a theory of multidimensional signal analysis, and in particular, to introduce the notions of multidimensional analytic signal, Gabor ﬁlter (see the next section), instantaneous and local amplitude and phase, and so on. Using the low-dimensional Clifford algebras R2,0 and R3,0 , Felsberg (2002) deﬁnes his Clifford–Fourier transform as

F fe [ f ](u) =

exp (−2πI < u, x >) f (x) dV(x),

where I denotes the pseudoscalar e1 e2 in the case of 1D signals, or e1 e2 e3 in the case of 2D signals. It is used among others to introduce a concept of 2D analytic signal. Ebling and Scheuermann (2003, 2005) studied convolution and Clifford– Fourier transformation of 2D and 3D signals, using the respective Fourier kernels exp (−e1 e2 (ξ1 x1 + ξ2 x2 )) and exp (−e1 e2 e3 (ξ1 x1 + ξ2 x2 + ξ3 x3 )), where again e1 e2 and e1 e2 e3 are the pseudoscalars in the Clifford algebras R2,0 and R3,0 , respectively. Note that the latter Fourier kernel is also used in Mawardi and Hitzer (2006) to deﬁne a Clifford–Fourier transform of 3D signals. These Clifford–Fourier transforms and the corresponding convolution theorems allow Ebling and Scheuermann for, among others, the analysis of vector-valued patterns in the frequency domain.

The Fourier Transform in Clifford Analysis

97

Note that the above-mentioned Clifford–Fourier kernel of Bülow and Sommer exp (−2πe1 u1 x1 ) . . . exp (−2πem um xm ) was already introduced by Sommen (1981) and Brackx, Delanghe, and Sommen (1982) as a theoretical concept in the framework of Clifford analysis. This generalized Fourier transform was further elaborated by Sommen (1982b, 1983) in connection with similar generalizations of the Cauchy, Hilbert, and Laplace transforms. In this context, the work of Li, McIntosh, and Qian (1994) should also be mentioned; they generalize the standard multidimensional Fourier transform of a function in Rm by extending the Fourier kernel exp (i < ξ, x >) to a function that is holomorphic in Cm and monogenic in Rm+1 . Recall that one of the most fundamental features of Clifford analysis is the factorization of the Laplace operator: m = −∂x2 . Whereas in general the square root of the Laplace operator is only a pseudodifferential operator, by √ embedding Euclidean space into a Clifford algebra, −m can be realized as the Dirac operator ∂x . In the same order of ideas, our ﬁrst purpose is neither to replace nor to improve the classical multidimensional Fourier transform by a Clifford analysis alternative, but rather to reﬁne it within the language of Clifford analysis, in much the same way as the notion of electron spin appears in the Pauli matrix formalism—it is what we call the Clifford–Fourier transform. The key step in its construction is to interpret the standard Fourier transform as an operator exponential (see Section 4.3): ∞ 1 π k k −i F = exp −i π2 H = H , k! 2 k=0

where H is the scalar operator

H=

1 (−m + r2 − m). 2

By means of two commuting operators, O1 and O2 , which are introduced while studying (anti-) monogenic operators in the generalized Clifford– Hermite polynomial setting (see Section 4.2), H can be split into a sum of Clifford algebra–valued second-order operators containing the angular Dirac operator . This leads in a natural way to a pair of transforms FH± , the harmonic average of which is precisely the standard Fourier transform. Moreover, the 2D case of this Clifford–Fourier transform is special in that we succeed in ﬁnding a closed form for the kernel of the integral representation. This closed form enables us to generalize the well-known results for the standard Fourier transform both in the L1 and in the L2 context. Let us end this introductory section with an overview of the content of this chapter. We start with considering operators acting on a subspace M of the space L2 Rm , dV(x) of square integrable functions and, in particular,

98

Fred Brackx et al.

Clifford differential operators with polynomial coefﬁcients. This subspace M is deﬁned as the orthogonal sum of spaces Ms, k of speciﬁc Clifford basis functions of L2 Rm , dV(x) (see subsection 4.2.1). In subsection 4.2.2, we show that these spaces Ms, k are simultaneous eigenspaces of two commuting operators, O1 and O2 . Every Clifford endomorphism of M can be decomposed into Clifford–Hermite monogenic operators. These Clifford– Hermite monogenic operators are characterized in terms of commutation relations involving O1 and O2 and they transform a space Ms, k into a similar space Ms , k (subsection 4.2.3). Hence, once the Clifford–Hermite monogenic decomposition of an operator is obtained, its action on the space M is known. Furthermore, in subsection 4.2.4, the monogenic decomposition of some important Clifford differential operators with polynomial coefﬁcients is studied in detail. In Section 4.3 we then recall two alternative approaches to the classical Fourier transform. Next, we split the scalar-valued kernel operator H by means of the commuting operators, O1 and O2 , which leads to a new Fourier transform in the Clifford analysis setting (subsection 4.4.1). The eigenfunctions of this Clifford–Fourier transform are computed and its relation with the standard Fourier transform is established. Furthermore, we develop an adequate operational calculus in subsection 4.4.2. Next, we thoroughly study the Clifford–Fourier transform in the speciﬁc case of two dimensions. We start with the computation of its integral kernel (subsection 4.5.1). In subsection 4.5.2, we examine the 2D Clifford–Fourier transform as a linear operator in, respectively, the space of integrable functions, the space of rapidly decreasing functions and the space of square integrable functions. Furthermore, in subsection 4.5.3, we give an explicit connection between the 2D Clifford–Fourier transform and the standard tensorial Fourier transform, and a surprising connection with the Clifford–Fourier transform of Ebling and Scheuermann (2005). We end this section by calculating, as an example, the 2D Clifford–Fourier transform of the box function.

4.2. Clifford–Hermite Monogenic Operators In this section we study Clifford–Hermite monogenic operators. To introduce this notion, we ﬁrst explain the already existing notion of monogenic operator in the polynomial framework.

4.2.1. Monogenic Operators Every homogeneous Clifford polynomial Rk of degree k admits a canonical decomposition of the form

Rk (x) =

k s=0

xs Pk−s (x),

Pk−s ∈ M+ (k − s).

The Fourier Transform in Clifford Analysis

99

This monogenic decomposition also yields a monogenic decomposition of the space P of Clifford polynomials

P=

∞ ∞

⊕⊥ Ms, k ,

s=0 k=0

where

Ms, k = {xs Pk (x); Pk ∈ M+ (k)}. Similar to this polynomial setting, the decomposition of Eq. (3) of a Clifford polynomial operator into homogeneous ones can be further reﬁned to a decomposition into monogenic operators A± λ, κ :

A=

− (A+ λ, κ + Aλ, κ ).

(27)

λ, κ

These monogenic operators A± λ, κ transform each space Ms, k into a similar space Ms , k . Hence, once the monogenic decomposition in Eq. (27) of a Clifford polynomial operator is obtained, its action on the space P of Clifford polynomials is known. As the spaces Ms, k are the simultaneous eigenspaces of the operators E and , the monogenic operators are characterized in terms of commutation relations involving E and . Sommen and Van Acker (1992) studied the monogenic decomposition of differential operators acting on Clifford polynomials in detail. In what follows we study the action of operators on the space M given by the algebraic orthogonal sum

M=

∞ ∞

⊕⊥ Ms, k .

s=0 k=0

Each f ∈ M can be decomposed into a ﬁnite sum: f = function + f , where f s, k s, k ∈ Ms, k = span{ψs, k, j (x); j = 1, 2, . . . , dim (M (k))} s k with

√ 2 ( j) √ ψs, k, j (x) = exp − |x|2 Hs, k ( 2x) Pk ( 2x),

s, k ∈ N ∪ {0}, j = 1, 2, . . . , dim(M+ (k)). From Section 2.4, we know that the set

ψs, k, j (x); s, k ∈ N ∪ {0}, j = 1, 2, . . . , dim M+ (k)

(28)

constitutes an orthogonal basis for the space L2 Rm , dV(x) of square integrable functions. Hence, this space is precisely the closure of M: M = L2 Rm , dV(x) .

100

Fred Brackx et al.

As in the polynomial framework, every Clifford endomorphism of M can be decomposed into Clifford–Hermite monogenic (CH-monogenic) operators. These CH-monogenic operators transform a space Ms, k into another such space Ms , k . As the spaces Ms, k are simultaneous eigenspaces of the operators

O1 =

1 (∂x − x)(∂x + x) 2

and

O2 =

1 (∂x + x)(∂x − x), 2

our CH-monogenic operators are characterized in terms of commutation relations involving O1 and O2 .

4.2.2. The Operators O1 and O2 In this subsection we consider the operators O1 and O2 introduced above. They will play a crucial role not only in our study of CH-(anti-)monogenic operators, but also in deﬁning a new Clifford–Fourier transform (see Section 4.4). They satisfy the following properties. Proposition 4.1 One has (i)

O1 =

1 2 m (∂x − x2 ) + − 2 2

O2 =

1 2 m (∂x − x2 ) − − 2 2

(ii)

(iii)

O1 + O2 = ∂x2 − x2 (iv)

m O1 − O 2 = 2 − 2

(v) O1 and O2 are commuting operators (vi)

O1 [ψs, k, j (x)] = Cs, k ψs, k, j (x) (vii)

O2 [ψs, k, j (x)] = Cs+1, k ψs, k, j (x).

The Fourier Transform in Clifford Analysis

101

Proof. (i)(ii) Taking into account that the angular Dirac operator may be written as

1 = − (x ∂x − ∂x x − m), 2 the results follow from a straightforward computation. (iii)(iv) Trivial. (v) As commutes with the Laplace operator m and with the multiplication operator r, we have that

1 1 2 2 2 (∂ − x ), = (−m + r ), = 0 2 x 2

which, in view of (i) and (ii) yields [O1 , O2 ] = 0. (vi)(vii) First, we have that

√ 2 ( j) √ (∂x − x)[ψs, k, j (x)] = exp − |x|2 (∂x − 2x)[Hs, k ( 2x) Pk ( 2x)]. (29) Formula (9) implies that

√ √ √ ( j) √ ( j) √ (2x − ∂x )[Hs,k ( 2x) Pk ( 2x)] = 2 Hs+1,k ( 2x) Pk ( 2x). Consequently, Eq. (29) becomes

√ (∂x − x)[ψs,k,j (x)] = − 2 ψs+1,k,j (x).

(30)

Next, it is immediately veriﬁed that

√ 2 ( j) √ (∂x + x)[ψs,k,j (x)] = exp − |x|2 ∂x [Hs,k ( 2x) Pk ( 2x)]. Moreover, from formula (ii) of Proposition 2.5, we readily obtain that

√ √ √ ( j) √ ( j) √ ∂x [Hs,k ( 2x) Pk ( 2x)] = − 2 Cs,k Hs−1,k ( 2x) Pk ( 2x). Hence, we ﬁnd that

√ (∂x + x)[ψs,k,j (x)] = − 2 Cs,k ψs−1,k,j (x).

(31)

102

Fred Brackx et al.

By combining the results of Eqs. (30) and (31), the basis functions ψs,k,j are found to be eigenfunctions of O1 and O2 . Remark 4.1 Note that (∂x − x) increases the degree of the generalized Clifford– Hermite polynomial, so that it may be qualiﬁed as a creation operator. In the same order of ideas, (∂x + x) is an annihilation operator.

4.2.3. CH-Monogenic Operators: Definition and Properties As mentioned in subsection 4.2.1, the CH-monogenic operators we now introduce will transform elements of a space Ms,k into elements of a space Ms ,k for some s and k . As the spaces Ms,k are simultaneous eigenspaces of the operators O1 and O2 , the CH-monogenicity property is expressed in terms of commutation relations involving these operators. Deﬁnition 4.1 (i) A Clifford endomorphism A of M is called CH-monogenic of degree (λ, κ), + notation: A ∈ χλ,κ , if

[O1 , A] = λA

and

[O2 , A] = κA.

(ii) A Clifford endomorphism B of M is called CH-anti-monogenic of degree − (λ, κ), notation: B ∈ χλ,κ , if

O1 B = BO2 + λB

and

O2 B = BO1 + κB.

Remark 4.2 (i) As the set

√ ( j) √ Hs, k ( 2x) Pk ( 2x); s, k ∈ N, j = 1, 2, . . . , dim M+ (k)

constitutes a basis for the space of Clifford polynomials and as D(Cm ) ⊂ End(P s ) ⊗ Cm , Clifford differential operators with polynomial coefﬁcients belong to the set of endomorphisms of M. (ii) The operators O1 and O2 themselves are CH-monogenic of degree (0, 0). (iii) A Clifford endomorphism A of M is CH-monogenic of degree (λ, κ) if and only if

1 λ+κ 2 (−m + r ), A = A 2 2

and

m λ−κ A. − ,A = 2 2

The Fourier Transform in Clifford Analysis

103

(iv) A Clifford endomorphism B of M is CH-anti-monogenic of degree (λ, κ) if and only if

1 λ+κ 2 (−m + r ), B = B 2 2

and

m λ−κ − ,B = B 2 2

with {A, B} = AB + BA the anti-commutator of two operators. CH-monogenic and CH-anti-monogenic operators are closed under composition as shown in the next proposition. Proposition 4.2 + + + (i) If A ∈ χλ,κ and B ∈ χλ+ ,κ then AB ∈ χλ+λ ,κ+κ and BA ∈ χλ+λ ,κ+κ . − + + (ii) If A ∈ χλ,κ and B ∈ χλ− ,κ then AB ∈ χλ+κ ,λ +κ and BA ∈ χλ +κ,λ+κ . + − − (iii) If A ∈ χλ,κ and B ∈ χλ ,κ then AB ∈ χλ+λ ,κ+κ and BA ∈ χλ− +κ,λ+κ .

Proof. The proof of (ii) is as follows:

O1 AB = (AO2 + λA)B = A(BO1 + κ B) + λAB = ABO1 + (λ + κ )AB and

O2 AB = (AO1 + κA)B = A(BO2 + λ B) + κAB = ABO2 + (λ + κ)AB. Similarly, we ﬁnd for the operator BA:

O1 BA = BAO1 + (λ + κ)BA

and

O2 BA = BAO2 + (λ + κ )BA.

The proofs of (i) and (iii) are similar. We now prove that the CH-monogenic operators indeed transform Ms,k into an Ms ,k . + Proposition 4.3 Let A ∈ χλ,κ and f ∈ Ms, k . Then A[ f ] ∈ Ms ,k for some s and k depending on s, k, λ, κ, and m.

Proof. We start by observing that A[ f ] is a simultaneous eigenfunction of O1 and O2

O1 [A[ f ]] = (AO1 + λA)[ f ] = (Cs, k + λ) A[ f ] O2 [A[ f ]] = (AO2 + κA)[ f ] = (Cs+1,k + κ) A[ f ].

104

Fred Brackx et al.

As A[ f ] belongs to the space M, it can be written as

A[ f ] =

∞ ∞

dim(M+ (t))

j=0 t=0

ψj,t,i (x) aj,t,i ;

aj,t,i ∈ Cm .

i=1

Hence, we also have ∞ ∞

O1 [A[ f ]] =

dim(M+ (t))

j=0 t=0

Cj,t ψj,t,i (x) aj,t,i .

i=1

Comparing the above expression with

O1 [A[ f ]] =

∞ ∞

dim(M+ (t))

j=0 t=0

(Cs,k + λ) ψj,t,i (x) aj,t,i ,

i=1

we obtain that either aj,t,i = 0 or Cj,t = Cs,k + λ. Similarly, comparing ∞ ∞

O2 [A[ f ]] =

dim(M+ (t))

j=0 t=0

Cj+1,t ψj,t,i (x) aj,t,i

i=1

with

O2 [A[ f ]] =

∞ ∞

j=0 t=0

dim(M+ (t))

(Cs+1,k + κ) ψj,t,i (x) aj,t,i ,

i=1

yields that either aj,t,i = 0 or Cj+1,t = Cs+1,k + κ. Consequently, we must prove that at most one pair of indices ( j, t) satisﬁes the set of equations

Cj,t = Cs,k + λ Cj+1,t = Cs+1,k + κ.

(32)

To that end, we must distinguish several cases.

CASE A If s is even, the set of equations (32) becomes

Cj,t = s + λ Cj+1,t = s + m + 2k + κ.

(33)

The Fourier Transform in Clifford Analysis

105

CASE A.1: m odd, λ odd. From the ﬁrst equation, we obtain that j must be odd. For j odd, the set of equations in (33) becomes

j − 1 + m + 2t = s + λ j + 1 = s + m + 2k + κ,

leading to

j = s + m + 2k + κ − 1

and

2t = λ + 2 − 2m − 2k − κ.

In this case, the second equation implies that κ must be odd. As t must be positive, we thus have

A : Ms,k →

0 Ms+m+2k+κ−1,(λ+2−2m−2k−κ)/2

for 2k > λ + 2 − 2m − κ for 2k ≤ λ + 2 − 2m − κ.

CASE A.2: m odd, λ even. Now the ﬁrst equation of (33) implies that j must be even. For j even, the set of equations in (33) becomes

j =s+λ j + m + 2t = s + m + 2k + κ,

which implies

j =s+λ

and

2t = 2k + κ − λ.

Now κ must be even and we have

A : Ms,k →

0 Ms+λ,(2k+κ−λ)/2

for 2k < λ − κ for 2k ≥ λ − κ.

CASE A.3: m even, λ odd. In this case, s + λ is odd. As Cj,t is always even, we have A : Ms,k → 0. CASE A.4: m even, λ even. As to the ﬁrst equation of (33), both j even and j odd are possible. Hence, we need to make a distinction between κ even and κ odd. In the case where κ is even, both j even and j odd are possible for the second equation.

106

Fred Brackx et al.

For j even, the set of equations in (33) becomes

j =s+λ j + m + 2t = s + m + 2k + κ,

and hence,

j =s+λ

and

2t = 2k + κ − λ.

For j odd, we have

j − 1 + m + 2t = s + λ

j + 1 = s + m + 2k + κ, thus,

j = s + m + 2k + κ − 1

and

2t = λ − 2m + 2 − 2k − κ.

As t must be positive, we have that for j even: 2k ≥ λ − κ; while for j odd: 2k ≤ λ − κ − (2m − 2) < λ − κ. This implies

⎧ ⎪ Ms+λ,(2k+κ−λ)/2 for 2k ≥ λ − κ ⎪ ⎪ ⎨0 for λ − κ − (2m − 2) < A : Ms,k → ⎪ 2k < λ − κ ⎪ ⎪ ⎩ Ms+m+2k+κ−1,(λ−2m+2−2k−κ)/2 for 2k ≤ λ − κ − (2m − 2). In the case where κ is odd, the second equation of (33) implies that neither j even, nor j odd is possible. Hence, we have A : Ms,k → 0.

CASE B The case where s is odd is treated in a similar way. In a completely analogous manner, we obtain the following result for the CH-anti-monogenic operators. − Proposition 4.4 Let B ∈ χλ,κ and f ∈ Ms,k . Then B[ f ] ∈ Ms ,k for some s and k depending on s, k, λ, κ, and m.

Now we are able to prove that every Clifford endomorphism of M admits a decomposition into CH-(anti-)monogenic operators, which we call the Clifford–Hermite monogenic decomposition (CHM decomposition).

107

The Fourier Transform in Clifford Analysis

Theorem 4.1 Every Clifford endomorphism of M admits a CHM decomposition. Proof. It is clear that anarbitrary ∞ Clifford endomorphism A of M can be written as follows: A = ∞ s=0 k=0 As,k , where, by deﬁnition,

As,k |Ms,k = A|Ms,k

whenever (s, k) = (s , k ).

and As,k |Ms ,k = 0

As A is an endomorphism of M, we also have As,k [Ms,k ] = A[Ms,k ] ∈ M and hence,

As,k [Ms,k ] =

∞ ∞

Ms ,k, .

s =0 k =0

This implies that every As,k , in its turn, can be decomposed as

As,k =

∞ ∞ s =0 k =0

Ass,k,k ,

with

Ass,k,k : Ms,k → Ms ,k , and Ass,k,k |Ms ,k = 0 whenever (s, k) = (s , k ). Consequently, for the endomorphism A, we obtain A =

s,k

s ,k

Ass,k,k .

It is easily seen that every operator Ass,k,k is both CH-monogenic and CHanti-monogenic. For example, if s and s are even, we have

Ass,k,k ∈ χs+ −s,s −s+2(k −k)

and Ass,k,k ∈ χs− −s−m−2k,s −s+m+2k .

Collecting the CH-monogenic and CH-anti-monogenic operators of the same degree yields a decomposition of A of the form

A=

− (A+ λ,κ + Aλ,κ ) λ,κ

with

± A± λ,κ ∈ χλ,κ .

4.2.4. CHM Decomposition of Clifford Differential Operators With Polynomial Coefficients In this subsection, we study the CHM decomposition of Clifford differential operators with polynomial coefﬁcients. It is sufﬁcient to search for the CHM decomposition of the basic operators {ej , xj , ∂xj ; j = 1, 2, . . . , m}, since they are generating the algebra D(Cm ).

108

Fred Brackx et al.

The operators f → ej f , j = 1, 2, . . . , m For the special case where m = 2, we have that

1 2 2 (∂ − x ), ej = 0 2 x

and

ej = −ej .

Consequently, we ﬁnd

O1 ej =

1 2 1 2 m m 2 2 (∂ − x ) + − ej = ej (∂x − x ) − + − mej 2 x 2 2 2 = ej O2 − mej

and similarly

O2 ej = ej O1 + mej . − Hence, for m = 2, ej ∈ χ−2,2 . For the general case where m > 2, we ﬁrst introduce the operators

τj = − ej + ej ( − m + 2) and δj = [, ej ] = ej − ej ,

for which we prove the following lemma. + − and δj ∈ χ−2,2 . Lemma 4.1 One has τj ∈ χ0,0

Proof. Naturally we have that

1 2 2 (∂ − x ), τj = 0. 2 x

Furthermore, as the Laplace–Beltrami operator ∗ω = (m − 2 − ) is a scalar operator, we ﬁnd

[, τj ] = −2 ej − ej ( − m + 2) + ej + ej ( − m + 2) = −( − m + 2)ej + ej ( − m + 2) = [∗ω , ej ] = 0. This implies

O1 τj = τj O1

and O2 τj = τj O2 .

The operator δj satisﬁes

1 2 2 (∂ − x ), δj = 0 2 x

The Fourier Transform in Clifford Analysis

109

and

δj = 2 ej − ej = (m − 2)ej − ∗ω ej − ej = (m − 2)ej − ej (m − 2 − ) − ej = (m − 2)(ej − ej ) − (ej − ej ) = (m − 2)δj − δj . Hence, we obtain

O1 δj =

1 1 2 m m (∂x − x2 ) + − δj = δj (∂x2 − x2 ) − + − 2δj 2 2 2 2 = δj O2 − 2δj ,

and similarly

O2 δj =

1 2 1 2 m m 2 2 (∂ − x ) − + δj = δj (∂x − x ) + − + 2δj 2 x 2 2 2 = δj O1 + 2δj .

Next we have that τj + δj = ej (m − 2 − 2). The operator (m − 2 − 2) is invertible in the set of endomorphisms of M, since it has eigenvalues 2k + m − 2 (for s even) and −(m + 2k) (for s odd) which, for m > 2, are never zero. Consequently, we can write

ej = (τj + δj )(m − 2 − 2)−1 := E0j + E1j , where

E0j = τj (m − 2 − 2)−1 = − ej + ej ( − m + 2) (m − 2 − 2)−1 and

E1j = δj (m − 2 − 2)−1 = (ej − ej )(m − 2 − 2)−1 . Naturally, the operator (m − 2 − 2) is CH-monogenic of degree (0, 0) and + . hence (m − 2 − 2)−1 ∈ χ0,0

110

Fred Brackx et al.

+ − In view of Proposition 4.2, we obtain that E0j ∈ χ0,0 and E1j ∈ χ−2,2 . Summarizing, the CHM decomposition of ej takes the form

+ − ej = E0j + E1j with E0j ∈ χ0,0 and E1j ∈ χ−2,2 .

Finally, with regard to the action on the spaces Ms,k , we have by means of Propositions 4.3 and 4.4 that

E0j : Ms,k → Ms,k and

E1j : Ms,k

⎧ ⎪ ⎨Ms+1,k−1 → 0 ⎪ ⎩M s−1,k+1

for s even and k ≥ 1 for s even and k = 0 for s odd.

The operators f → x f and f → ∂x f The operators x and ∂x satisfy (see, e.g., Van Acker, 1991)

x = x(m − 1 − ) and ∂x = ∂x (m − 1 − ).

(34)

This result will be combined with the following lemma. Lemma 4.2 One has

Proof.

1 2 (∂x − x2 ), x = −∂x 2

and

1 2 (∂x − x2 ), ∂x = −x. 2

We obtain consecutively

m 1

1 2 1 (∂x − x2 ), x = (∂x2 x − x ∂x2 ) = − 2 2 2 =− =−

1 2

k,j

k=1

⎞ ⎛ m 1 ∂x2k ⎝ xj ej ⎠ − x ∂x2 2 j=1

1 ∂xk (δk,j ej + xj ej ∂xk ) − x ∂x2 2

1 1 (2δk,j ej ∂xk + xj ej ∂x2k ) − x ∂x2 2 2 k,j

1 1 = −∂x + x ∂x2 − x ∂x2 = −∂x 2 2

111

The Fourier Transform in Clifford Analysis

and

⎞ ⎛ m m 1 2 1 ⎝ 1 2 1 2 2 2 2 ej ∂xj ⎠ xk (∂ − x ), ∂x = − (x ∂x − ∂x x ) = − x ∂x − 2 x 2 2 2 j=1

1

1 = − x 2 ∂x − 2 2

k=1

ej (2xk δk,j + xk2 ∂xj )

j,k

1 1 = − x2 ∂x − x + x2 ∂x = −x. 2 2 In view of the above, we now have

O1 x =

1 2 1 2 m m 2 2 (∂ − x ) + − x = x (∂x − x ) − + − x − ∂x 2 x 2 2 2 = xO2 − x − ∂x

and

O2 x =

1 1 2 m m (∂x − x2 ) − + x = x (∂x2 − x2 ) + − + x − ∂x 2 2 2 2 = xO1 + x − ∂x ,

whereas for the operator ∂x , we obtain

O1 ∂x = ∂x O2 − ∂x − x

and

O2 ∂x = ∂x O1 + ∂x − x.

Hence, x and ∂x are neither CH-monogenic nor CH-anti-monogenic. However, we do have the following result: − − and x − ∂x ∈ χ0,2 . Lemma 4.3 One has x + ∂x ∈ χ−2,0

Proof.

Straightforward.

We now readily obtain the CHM decomposition of x and ∂x : x = X 0 + X 1 with

X0 =

1 − (x − ∂x ) ∈ χ0,2 , 2

X1 =

1 − (x + ∂x ) ∈ χ−2,0 2

D1 =

1 − (x + ∂x ) ∈ χ−2,0 . 2

and ∂x = D0 + D1 with

1 − D0 = − (x − ∂x ) ∈ χ0,2 , 2

112

Fred Brackx et al.

The operators f → xj f , j = 1, 2, . . . , m. Once the CHM decomposition of ej , j = 1, 2, . . . , m and x is obtained, the CHM decomposition of xj easily follows from

1 1 0 1 0 1 0 1 0 1 xj = − (xej + ej x) = − (X + X )(Ej + Ej ) + (Ej + Ej )(X + X ) 2 2 = xj0 + xj1 + xj2 + xj3 + xj4 + xj5 ,

with

1 + xj0 = − X 0 E1j ∈ χ2,0 2 1 + xj1 = − E1j X 0 ∈ χ0,2 2 1 + xj2 = − E1j X 1 ∈ χ−2,0 2 1 + xj3 = − X 1 E1j ∈ χ0,−2 2 1 − xj4 = − (X 0 E0j + E0j X 0 ) ∈ χ0,2 2 1 − xj5 = − (X 1 E0j + E0j X 1 ) ∈ χ−2,0 . 2 Here we have used the composition rules for CH-monogenic and CH-antimonogenic operators derived in Proposition 4.2. Finally, for the action on the spaces Ms,k we have

xj0 : Ms,k

xj1 : Ms,k

xj2

: Ms,k

⎧ ⎪ ⎨Ms+2,k−1 → 0 ⎪ ⎩M s,k+1

for s even and k ≥ 1 for s even and k = 0 for s odd.

⎧ ⎪ ⎨Ms,k+1 → Ms+2,k−1 ⎪ ⎩0

for s even for s odd and k ≥ 1 for s odd and k = 0

⎧ ⎪ ⎨Ms−2,k+1 → Ms,k−1 ⎪ ⎩0

for s even for s odd and k ≥ 1 for s odd and k = 0

The Fourier Transform in Clifford Analysis

xj3 : Ms,k

⎧ ⎪ ⎨Ms,k−1 → 0 ⎪ ⎩M s−2,k+1

113

for s even and k ≥ 1 for s even and k = 0 for s odd.

xj4 : Ms,k → Ms+1,k xj5 : Ms,k → Ms−1,k . The operators f → ∂xj f , j = 1, 2, . . . , m. By means of

1 ∂xj = − (∂x ej + ej ∂x ), 2 the CHM decomposition of ∂xj follows at once from the CHM decomposition of ej and ∂x . The results are completely similar to these for the operators considered in the previous subsection.

4.3. Alternative Representations of the Classical Fourier Transform The idea behind the deﬁnition of our Clifford–Fourier transform originates from the operator exponential representation of the classical Fourier transform

F [ f ] = exp

∞ 1 π n n −i [f] = H [ f ], n! 2

−i π2 H

(35)

n=0

with H the scalar-valued differential operator given by

1 2 1 (∂x − x2 − m) = (−m + r2 − m). 2 2 Note that the operators H and exp −i π2 H are Fourier invariant; that is, H=

F [H[ f ]] = H[F [ f ]]

and F exp −i π2 H [ f ] = exp −i π2 H [F [ f ]].

The equivalence of this operator exponential form in Eq. (35) with the traditional integral form in Eq. (6) may be proved rather simply in the framework of Clifford analysis. To that end, we again use the orthogonal basis of Eq. (28) of the space L2 Rm , dV(x) . The basis functions ψs,k,j satisfy the orthogonality

114

Fred Brackx et al.

relation

< ψs, k1 , j1, ψt, k2 , j2 > =

γs, k1 δs, t δk1 , k2 δj1 , j2 . 2m/2

(36)

Hence, a square integrable function f can be expanded as follows

f (x) =

+

M (k) ∞ ∞ dim s=0 k=0

ψs,k,j (x) bs,k,j .

(37)

j=1

The orthogonality relation in Eq. (36) implies that the Clifford algebra– valued coefﬁcients bs,k,j are given by the integrals

bs,k,j =

2m/2 γs,k

Rm

†

ψs,k,j (x)

f (x) dV(x).

(38)

In the foregoing, we have shown that these L2 Rm , dV(x) -basis functions ψs,k,j are simultaneous eigenfunctions of the Fourier transform operator F in integral form and of the kernel operator H. We thus have at the same time (see Proposition 2.7)

1 exp (−i < x , ξ >) ψs,k,j (x) dV(x) (2π)m/2 Rm = exp −i(s + k) π2 ψs,k,j (ξ)

F [ψs,k,j ](ξ) =

and (see the proof of Proposition 3.1)

H[ψs,k,j (x)] = (s + k) ψs,k,j (x). It then follows that ∞ 1 π n n −i exp −i π2 H [ψs,k,j ] = H [ψs,k,j ] n! 2 n=0

∞ 1 π n = −i (s + k)n ψs,k,j n! 2 n=0 = exp −i π2 (s + k) ψs,k,j = F [ψs,k,j ],

which immediately gives rise to the desired equivalence in L2 Rm , dV(x) .

The Fourier Transform in Clifford Analysis

115

Moreover, if the function f ∈ L2 Rm , dV(x) is developed in terms of the basis functions ψs,k,j according to Eq. (37), then its Fourier transform takes the series expansion form +

F [ f ](ξ) =

∞ dim(M ∞ (k)) s=0 k=0

exp −i(s + k) π2 ψs,k,j (ξ) bs,k,j .

j=1

4.4. Clifford–Fourier Transform: Definition and Properties Note that due to the scalar character of the standard Fourier kernel, the Fourier spectrum inherits its Clifford algebra character from the original signal with no interaction with the Fourier kernel. So in order to genuinely introduce the Clifford analysis character in the Fourier transform, it occurred to us to replace in the operator exponential in Eq. (35) the scalarvalued operator H with a Clifford algebra–valued one. To that end, we aim at factorizing the operator H, making use of the factorization of the Laplace operator by the Dirac operator. This leads us to again consider the commuting operators (see Subsection 4.2.2)

O1 =

1 (∂x − x)(∂x + x) 2

and

O2 =

1 (∂x + x)(∂x − x), 2

which turn out to be crucial in our approach.

4.4.1. Definition In view of Proposition 4.1 (vi) and (vii), we deﬁne the Clifford–Fourier transform as the pair of transformations

FH+ = exp −i π2 H+

and

FH− = exp −i π2 H−

with the operators H+ and H− closely linked to the operators O1 and O2 . As we want the classical Fourier transform to be the harmonic average of the Clifford–Fourier transform pair {FH+ , FH− }; that is, F 2 = FH+ FH− with F 2 the parity operator: F 2 [ f ](x) = f (−x), the operators H+ and H− must satisfy

H+ + H− = 2 H = ∂x2 − x2 − m = O1 + O2 − m. This inspires the following deﬁnition of H+ and H− .

116

Fred Brackx et al.

Deﬁnition 4.2 One puts

H+ = O1

H− = O2 − m.

and

Note that the operators H+ and H− contain a scalar part and a bivector part. The following properties are easily proved. Proposition 4.5 One has (i)

H± = H ± . (ii) H+ and H− are Fourier invariant operators. (iii) ± H± [ψs,k,j (x)] = Cs,k ψs,k,j (x) + with Cs,k := Cs,k and

− Cs,k

:= Cs+1,k − m =

s + 2k s+1−m

for s even for s odd.

Proof. (i)(iii) Trivial. (ii) This property follows directly from the Fourier invariance of the operators H and . Corollary 4.1 The basis functions ψs,k,j are eigenfunctions of the Clifford– Fourier transform:

± FH± [ψs,k,j ](ξ) = exp −i π2 Cs,k ψs,k,j (ξ). Now if f ∈ L2 Rm , dV(x) is expanded w.r.t. the basis ψs,k,j (x); s, k ∈ N ∪ {0}, j = 1, . . . , dim M+ (k) , the eigenvalue equation of Corollary 4.1 immediately yields the series representation of the Clifford–Fourier transform as follows: +

FH± [ f ](ξ) =

∞ ∞ dim(M (k)) s=0 k=0

± ψs,k,j (ξ) bs,k,j , exp −i π2 Cs,k

j=1

the coefﬁcients bs,k,j being given by Eq. (38).

The Fourier Transform in Clifford Analysis

117

Moreover, as the orthogonal L2 -basis ψs,k,j (x); s, k ∈ N ∪ {0}, j = + 1, . . . , dim M (k) consists of eigenfunctions of both the operators H and , one can easily verify the following properties. Proposition 4.6 + − (i) The operators m H, , O1 , O2 , H and H are self-adjoint; that is, for all f , g ∈ L2 R , dV(x) and T any of the mentioned operators, one has

< T[ f ], g >=< f , T[ g] > . (ii) The operators H, O1 , and O2 are nonnegative—for each f ∈ L2 Rm , dV(x) and T any of the mentioned operators, one has

[< T[ f ], f >]0 ≥ 0. Next, by means of Proposition 4.5 (i), we obtain in terms of operator exponentials

FH± = exp −i π2 (H ± ) = exp ∓i π2 exp −i π2 H = exp ∓i π2 F .

(39) This establishes the relationship between the classical Fourier transform and the newly introduced Clifford–Fourier transform. Note that the commuting property of the operators H and has been used, so that indeed exp −i π2 (H ± ) = exp −i π2 H exp ∓i π2 = exp ∓i π2 exp −i π2 H . Thus, the Clifford–Fourier transform is obtained as the composition of the classical Fourier transform with the operator exponential ∞ 1 π k k ∓i exp ∓i π2 = . k! 2 k=0

As an immediate consequence, we obtain an integral representation for the Clifford–Fourier transform as follows:

1 FH± [ f ](ξ) = (2π)m/2

Rm

exp ∓i π2 ξ [ exp (−i < x , ξ >)] f (x) dV(x).

Introducing the square root of the Clifford–Fourier transforms, in the sense of the FrFT (see Namias, 1980, and Ozaktas, Zalevsky, and Kutay, 2001), by

FH± = exp −i π4 H± ,

118

Fred Brackx et al.

we also obtain that

FH+ FH− = FH− FH+ = exp −i π4 H+ + H− = exp −i π2 H , leading to the factorization of the standard Fourier transform:

F=

FH+ FH− = FH− FH+ .

Note that each operator that is anti-invariant under the classical Fourier transform and commutes with the angular Dirac operator , is also antiinvariant under the Clifford–Fourier transform. For example, the operators m m − and E + are, respectively, invariant and anti-invariant under the 2 2 classical Fourier transform. Because they both commute with the angular Dirac operator , they show the anti-invariance property w.r.t the Clifford– Fourier transform. For the inversion of the Clifford–Fourier transform, it sufﬁces to observe that

(FH± )−1 = exp i π2 H± = exp ±i π2 F −1 . Finally, using the notation TT = exp −i π2 T , we can draw the following diagram: (see next page)

4.4.2. Operational Calculus As is the case for the classical Fourier transform, an operational calculus may be based on the Clifford–Fourier transform. The operational formulas are derived from Eq. (39) expressing the Clifford–Fourier transform in terms of the classical Fourier transform F . Proposition 4.7 The Clifford–Fourier transform satisﬁes the following: (i) the linearity property

FH± [ f λ + gμ] = FH± [ f ] λ + FH± [ g]μ

for λ, μ ∈ Cm ,

(ii) the change of scale property

ξ 1 FH± [ f (ax)](ξ) = m FH± [ f (x)] a a

for a ∈ R+ ,

119

The Fourier Transform in Clifford Analysis

FH [f ] 1/2

F H

F F

FH

1/2 H [f ]

T

T 1/2

1/2 H

F H

F [f ]

f

F2 [f ]

F

F 1/2

1/2

F H

F H

F

1/2 H [f ]

T

F

T

FH

1/2

H

FH [f ]

Decomposition of the Fourier transform in term of the Clifford–Fourier transforms.

(iii) the multiplication rule

FH± [x f (x)](ξ) = ∓ (∓i)m ∂ξ FH∓ [ f (x)](ξ) and more generally

FH± [x2n f (x)](ξ) = (−1)n ∂ξ2n FH± [ f (x)](ξ) FH± [x2n+1 f (x)](ξ) = ∓ (−1)n (∓i)m ∂ξ2n+1 FH∓ [ f (x)](ξ), (iv) the differentiation rule

FH± [∂x f (x)](ξ) = ∓ (∓i)m ξ FH∓ [ f (x)](ξ) and more generally

FH± [∂x2n f (x)](ξ) = (−1)n ξ 2n FH± [ f (x)](ξ) FH± [∂x2n+1 f (x)](ξ) = ∓ (−1)n (∓i)m ξ 2n+1 FH∓ [ f (x)](ξ),

and

120

Fred Brackx et al.

(v) the mixed product rule

FH± [(x∂x )n f (x)](ξ) = (−1)n (∂ξ ξ)n FH± [ f (x)](ξ) FH± [(∂x x)n f (x)](ξ) = (−1)n (ξ∂ξ )n FH± [ f (x)](ξ).

Proof. (i) Immediate. Note, however, that the Clifford algebra–valued coefﬁcients λ and μ must be at the right of the functions f and g. (ii) As the classical Fourier transform F satisﬁes the change of scale property

ξ 1 , a ∈ R+ , F f (ax) (ξ) = m F f (x) a a we obtain

FH± [f (ax)](ξ) =

ξ 1 π F [f (x)] . exp ∓i ξ 2 m a a

The angular Dirac operator ξ is homogeneous of degree zero, since it commutes with the Euler operator E. Consequently, we have ξ = ξ/a . Hence, the change of scale property also holds for the Clifford-Fourier transform

ξ 1 π FH± [f (ax)](ξ) = m exp ∓i 2 ξ/a F [f (x)] a a ξ 1 = m FH± [f (x)] . a a (iii) From the multiplication rule for the classical Fourier transform (see Proposition 2.2), we obtain at once that

FH+ [xf (x)](ξ) = i exp −i π2 ξ ∂ξ F [f (x)](ξ). Repeated application of the commutation relation [see Eq. (34)]

ξ ∂ξ = ∂ξ (m − 1 − ξ ),

The Fourier Transform in Clifford Analysis

121

yields

kξ ∂ξ = ∂ξ (m − 1 − ξ )k , which enables us to prove that

exp

−i π2 ξ

∞ ∞ 1 π k k 1 π k ∂ξ = −i −i ξ ∂ξ = ∂ξ (m−1−ξ )k k! 2 k! 2 k=0 k=0

= ∂ξ exp −i π2 (m − 1 − ξ ) .

Consequently, we obtain

FH+ [x f (x)](ξ) = i ∂ξ exp −i π2 (m − 1 − ξ ) F [ f (x)](ξ)

= i exp −i π2 m i ∂ξ exp i π2 ξ F [f (x)](ξ) = −(−i)m ∂ξ FH− [f (x)](ξ).

The analogous result for the Clifford–Fourier transform involving the operator H− is derived in a similar way. The more general formulas are now proved by induction. For example, the formula

FH+ [x2n f (x)](ξ) = (−1)n ∂ξ2n FH+ [ f (x)](ξ) holds for n = 1, since FH+ [x2 f (x)](ξ) = −(−i)m ∂ξ FH− [xf (x)](ξ) = −(−i)m im ∂ξ2 FH+ [f (x)](ξ) = −∂ξ2 FH+ [f (x)](ξ). Assuming that it holds for n, we now prove it for n + 1: FH+ [x2(n+1) f (x)](ξ) = FH+ [x2n x2 f (x)](ξ) = (−1)n ∂ξ2n FH+ [x2 f(x)](ξ) 2(n+1)

= (−1)n+1 ∂ξ

FH+ [ f (x)](ξ).

(iv) By means of the differentiation rule for the classical Fourier transform and the commutation relation

ξ ξ = ξ (m − 1 − ξ ),

122

Fred Brackx et al.

a similar argument may be developed as in the proof of the multiplication rule. (v) This result follows by combining the multiplication and differentiation rules. For example, we have FH+ [x∂x f (x)](ξ) = −(−i)m ∂ξ FH− [∂x f (x)](ξ) = −(−i)m im ∂ξ ξFH+ [ f (x)](ξ) = − ∂ξ ξFH+ [ f (x)](ξ) and hence, by repeated use,

FH+ [(x∂x )n f (x)](ξ) = (−1)n (∂ξ ξ)n FH+ [ f (x)](ξ). Because the Fourier transform of a radial function remains radial, and the angular Dirac operator does not affect radial functions, the next result follows readily. Proposition 4.8 For a radial function f ; that is, f only depends on |x| = r, one has FH± [ f ] = F [ f ] and in particular, FH± [δ] = √ 1 m and FH± [1] = 2π √ m 2π δ.

4.5. The Two-Dimensional Case The purpose of this section is twofold: (1) to present an in-depth study of our Clifford–Fourier transform in the speciﬁc case of two dimensions, thus providing a theoretical background for the use of this integral transformation, and (2) to show how our 2D Clifford–Fourier transform ﬁts in the picture of all already existing Clifford–Fourier transforms described in Section 4.1 and in this way to promote our higher-dimensional Clifford– Fourier transform as a possible tool for multidimensional signal analysis. The 2D case of the Clifford–Fourier transform is special in that we are able to obtain a closed form for the kernel of the integral representation.

4.5.1. The Integral Kernel In the sequel, the following Clifford numbers play a crucial role:

P± =

1 (1 ± ie12 ). 2

They are self-adjoint mutually orthogonal idempotents which, by multiplication, transform e12 into the imaginary unit i.

The Fourier Transform in Clifford Analysis

123

Lemma 4.4 The Clifford numbers P+ and P− satisfy the following properties: (i)

P+ + P− = 1 ;

P+ P− = P− P+ = 0 ;

(P± )2 = P±

(ii)

P+ (ie12 ) = P+

or

P− (ie12 ) = −P−

P+ i = P+ (−e12 ) = (−e12 )P+ ; P− i = P− e12 = e12 P−

or

(iii) for k ∈ N:

P+ (ie12 )k = P+

or

P− (ie12 )k = (−1)k P−

P+ (e12 )k = P+ (−i)k ; or

P− (e12 )k = P− ik .

Proof. (i) By a straightforward computation, we ﬁnd

P+ P− =

1 1 (1 + ie12 )(1 − ie12 ) = (1 − ie12 + ie12 − 1) = 0 4 4

(P± )2 =

1 1 (1 ± ie12 )(1 ± ie12 ) = (1 ± 2 ie12 + 1) = P± . 4 4

and

(ii) We easily obtain

P+ (ie12 ) =

1 1 (1 + ie12 )(ie12 ) = (ie12 + 1) = P+ , 2 2

which by multiplication with (−e12 ) yields

P+ i = P+ (−e12 ). The result for P− is derived similarly. (iii) In view of the above, we have

P+ (ie12 )k = P+ (ie12 )(ie12 ) . . . (ie12 ) = P+ (ie12 ) . . . (ie12 ) = . . . = P+

124

Fred Brackx et al.

or, by multiplication with (−i)k ,

P+ (e12 )k = P+ (−i)k . The result for P− is proved in an analogous manner.

A. Calculation of the kernel for FH+ We calculate the kernel of the Clifford–Fourier transform involving the operator H+ . This kernel is given by

exp −i π2 ξ [exp (−i < x , ξ >)]. By means of Lemma 4.4 (i) and = −e12 L12 , we now have

exp −i π2 ξ [exp (−i < x , ξ >)] = P+ exp i π2 e12 L12 [exp (−i < x, ξ >)] + P−exp i π2 e12 L12 ×[exp (−i < x , ξ >)].

(40)

Moreover, again using the properties of the Clifford numbers P+ and P− , we obtain ∞ (ie12 )k π k P+ exp i π2 e12 L12 = P+ (L12 )k k! 2 k=0

=P

+

∞ 1 π k (L12 )k = P+ exp π2 L12 k! 2 k=0

and similarly

P− exp i π2 e12 L12 = P− exp − π2 L12 . Hence, Eq. (40) becomes π exp −i ξ [exp (−i < x , ξ >)] 2

π π + L12 [exp (−i < x , ξ >)] + P− exp − L12 [exp (−i < x, ξ >)]. = P exp 2 2 (41)

We now prove the following intermediate result.

The Fourier Transform in Clifford Analysis

125

Lemma 4.5 Let f be a real-analytic function in R2 , let L12 = ξ1 ∂ξ2 − ξ2 ∂ξ1 be the angular momentum operator, and let R± be the operator exponential given by

R± := exp ± π2 L12 . Then one has

R+ [ f (ξ1 , ξ2 )] = f (−ξ2 , ξ1 )

and

R− [f (ξ1 , ξ2 )] = f (ξ2 , −ξ1 ).

Proof. In terms of polar coordinates

ξ1 = r cos (θ) ξ2 = r sin (θ)

with r = |ξ| ∈ [0, +∞[ and θ ∈ [0, 2π[, the operator exponential R+ takes the form

Rθ+ := exp

π 2

∂θ .

We have + Rθ+ [ f (r, θ)] = Rψ [ f (r, θ + ψ)]ψ=0 =

∞ 1 π k k ∂ψ [ f (r, θ + ψ)]ψ=0 . k! 2 k=0

As we assume f to be real-analytic in R2 , this becomes

π , Rθ+ [ f (r, θ)] = f r, θ + 2 which leads to the desired result. The result for the operator exponential R− is proved in a similar way. Remark 4.3 The operator exponentials R+ and R− may be qualiﬁed, respectively, as an anti-clockwise, and a clockwise rotation by a right angle. Returning to the calculation of the 2D Clifford–Fourier kernel, we obtain the following by applying Lemma 4.5 to Eq. (41):

exp −i π2 ξ exp (−i < x, ξ >) = P+ exp −i(x2 ξ1 − x1 ξ2 ) + P− exp −i(x1 ξ2 − x2 ξ1 ) . This expression for the kernel of FH+ can be further simpliﬁed using the following result.

126

Fred Brackx et al.

Lemma 4.6 One has

P+ exp −i(x2 ξ1 − x1 ξ2 ) = P+ exp (ξ ∧ x) and P− exp −i(x1 ξ2 −x2 ξ1 ) = P− exp (ξ ∧ x). Proof.

By means of Lemma 4.4 (iii), we have consecutively

∞ (−i)k (x2 ξ1 − x1 ξ2 )k P+ exp −i(x2 ξ1 − x1 ξ2 ) = P+ k! k=0

=P

+

∞ (e12 )k k=0

k!

(x2 ξ1 −x1 ξ2 )k

= P+ exp e12 (ξ1 x2 −ξ2 x1 ) = P+ exp (ξ∧x). The second statement is proved similarly. In view of the foregoing lemma, we ﬁnally obtain a closed form for the kernel of FH+ :

exp −i π2 ξ [exp (−i < x , ξ >)] = P+ exp (ξ ∧ x) + P− exp (ξ ∧ x) = exp (ξ ∧ x).

Hence, the Clifford–Fourier transform involving the operator H+ has the following integral representation:

FH+ [f ](ξ) =

1 2π

R2

exp (ξ ∧ x) f (x) dV(x).

B. Calculation of the kernel for FH− The computation of the kernel of the 2D Clifford–Fourier transform involving the operator H− follows the same lines. It is given by

exp i π2 ξ [exp (−i < x , ξ >)] = exp −(ξ ∧ x) = exp (x ∧ ξ). The integral representation for FH− then takes the form

FH− [ f ](ξ) =

1 2π

R2

exp (x ∧ ξ) f (x) dV(x).

The Fourier Transform in Clifford Analysis

127

Remark 4.4 1. Note that the 2D Clifford–Fourier kernels consist of a scalar and a bivector part—they are so-called parabivectors. 2. The Clifford–Fourier transform FH± may be qualiﬁed as a coaxial Fourier transform, since its integral kernel may also be rewritten as

exp ±(ξ ∧ x) = cos (ξ1 x2 − ξ2 x1 ) ± e12 sin (ξ1 x2 − ξ2 x1 ),

(42)

where ξ1 x2 − ξ2 x1 takes constant values on coaxial cylinders, which in two dimensions take the form of two lines parallel and symmetric w.r.t. the ﬁxed vector ξ (see also Section 6). Hence, in terms of the Fourier cosine and the Fourier sine transform,

1 Fcos [ f ](ξ) = 2π Fsin [ f ](ξ) =

1 2π

R2

cos (ξ1 x2 − ξ2 x1 ) f (x) dV(x),

R2

sin (ξ1 x2 − ξ2 x1 ) f (x) dV(x),

the Clifford–Fourier transform FH± takes the form

FH± [ f ](ξ) = Fcos [ f ](ξ) ± e12 Fsin [ f ](ξ).

4.5.2. The Two-Dimensional Clifford–Fourier Transform as a Linear Operator 1. The Clifford–Fourier Transform in L1 R2 , dV(x)

The 2D Clifford–Fourier transform FH+ [ f ] is well deﬁned for each integrable function f ∈ L1 R2 , dV(x) . Indeed, by means of the properties in Eq. (1) of the Clifford norm, we have

R

exp (ξ ∧ x) f (x) dV(x) ≤ 2

R2

≤4

| exp (ξ ∧ x) f (x)| dV(x)

R2

| exp (ξ ∧ x)| | f (x)| dV(x).

Furthermore, using Eq. (42), we obtain:

| exp (ξ ∧ x)| = | cos (ξ1 x2 − ξ2 x1 ) + e12 sin (ξ1 x2 − ξ2 x1 )| ≤ | cos (ξ1 x2 − ξ2 x1 )| + | sin (ξ1 x2 − ξ2 x1 )| ≤ 2.

128

Fred Brackx et al.

Hence, we have

R

exp (ξ ∧ x) f (x) dV(x) ≤ 8 2

R2

f (x) dV(x) < ∞

and thus also

exp (ξ ∧ x) f (x) ∈ L1 R2 , dV(x) . A similar argument holds for the Clifford–Fourier transform involving the operator H− . The above reasoning leads to the following theorem. Theorem 4.2 Let f ∈ L1 R2 , dV(x) . Then FH± [f ] ∈ L∞ R2 ∩ C0 R2 , and moreover,

FH± [ f ]

∞

Proof.

≤

4 f . 1 π

In view of the above, we have

4 1 ≤ f (x) dV(x) exp ±(ξ ∧ x) f (x) dV(x) 2π R2 π R2 4 = f 1 . π

|FH± [ f ](ξ)| =

Moreover, one can easily verify that

|FH± [ f ](ξ) − FH± [ f ](ξ )| ≤

8 f . 1 π

Hence, |ξ − ξ | → 0 implies

|FH± [ f ](ξ) − FH± [ f ](ξ )| → 0; in other words, FH± [ f ] is continuous. In Subsection 4.4.2 some operational formulas (viz., the right Cm linearity, change of scale, multiplication, differentiation, and mixed product rule) were derived for the Clifford–Fourier transform in arbitrary dimension. In the special 2D case, the obtained closed forms for the integral kernels enable some further results for the Clifford–Fourier us to prove transform in L1 R2 , dV(x) .

The Fourier Transform in Clifford Analysis

129

Proposition 4.9 Let f , g ∈ L1 R2 , dV(x) . The 2D Clifford–Fourier transform satisﬁes the following (i) the shift theorem

FH± [τh f (x)](ξ) = exp ±(ξ ∧ h) FH± [ f (x)](ξ), where τh denotes the translation by h , i.e. τh f (x) = f (x − h) (ii) frequency reversion

FH± [ f ](−ξ) = FH∓ [ f ](ξ) (iii) Hermitean conjugation

†

FH+ [ f ](ξ)

†

FH− [ f ](ξ)

=

1 2π

1 = 2π

R2

f † (x) exp (x ∧ ξ) dV(x),

R2

f † (x) exp (ξ ∧ x) dV(x)

(iv) the modulation theorem

FH± exp (x ∧ h) f (x) (ξ) = τ±h FH± [ f (x)](ξ) (v) the transfer formula

R2

†

FH± [ f ](ξ)

g(ξ) dV(ξ) =

R2

f † (ξ) FH± [ g](ξ) dV(ξ)

(vi) the convolution theorem

FH± [ f p ∗ g](ξ) = 2π FH± [ f p ](ξ) FH± [ g](ξ), FH± [ f ∗ g](ξ) = 2π FH± [ f ](ξ) FH∓ [ g](ξ), where ∗ denotes the Clifford convolution given by

( f ∗ g)(x) =

R2

f (x − x ) g(x ) dV(x ),

130

Fred Brackx et al.

and f p , respectively f , denotes, respectively, a parabivector-valued and vector-valued function; that is,

f p (x) = f0 (x) + f12 (x) e12 ;

f = f1 (x) e1 + f2 (x) e2

( f0 , f1 , f2 , f12 : R2 → C). (vii) the multiplication theorem

1 FH± [ f p ] ∗ FH± [ g] (ξ) 2π

FH± [ f p g](ξ) =

1 FH± [ f g](ξ) = FH± [ f ] ∗ FH∓ [ g] (ξ), 2π where again f p , respectively f , denotes, respectively, a parabivector-valued and vector-valued function, which is Fourier invertible. (viii) the rotation rule

FH± [ f (sxs)](ξ) = FH± [ f (x)](sξs), where s ∈ SpinR (m). Proof. We restrict ourselves to the proofs for the Clifford–Fourier transform involving the operator H+ , the proofs for FH− being similar. (i) By means of the substitution u = x − h, we obtain

FH+ [τh f (x)](ξ) =

1 2π

1 = 2π

R2

exp (ξ ∧ x) f (x − h) dV(x)

R2

exp (ξ ∧ u + ξ ∧ h) f (u) dV(u)

1 = exp (ξ ∧ h) 2π

R2

exp (ξ ∧ u) f (u) dV(u)

= exp (ξ ∧ h) FH+ [ f ](ξ). Here we have used the fact that

exp (ξ∧u + ξ∧h) = exp (ξ∧u) exp (ξ∧h) = exp (ξ∧h) exp (ξ∧u), since the exponentials exp (ξ ∧ u) and exp (ξ ∧ h) commute.

The Fourier Transform in Clifford Analysis

131

(ii) This property follows at once from

exp (−ξ) ∧ x = exp (x ∧ ξ). (iii) Taking into account that ξ ∧ x = x ∧ ξ and hence also that

exp (ξ ∧ x) = exp (x ∧ ξ), the result follows immediately. (iv) We have consecutively

1 FH+ exp (x∧h) f (x) (ξ) = exp (ξ∧x) exp (x∧h) f (x) dV(x) 2π R2 1 = exp (ξ − h) ∧ x f (x) dV(x) 2 2π R = FH+ [ f (x)](ξ − h). (v) First note that both integrals in formula (v) are well deﬁned, † since FH± [ f ](ξ) g(ξ) and f † (ξ) FH± [ g](ξ) belong to the space L1 R2 , dV(x) of integrable functions. Moreover, using property (iii) and changing the order of integration yields

†

R2

= =

FH+ [ f ](ξ) 1 2π R2

=

R2

R2

f † (x)

R2

g(ξ) dV(ξ)

f † (x) exp (x ∧ ξ) dV(x) g(ξ) dV(ξ)

1 2π

R2

exp (x ∧ ξ) g(ξ) dV(ξ) dV(x)

f † (x) FH+ [ g](x) dV(x).

(vi) Let us ﬁrst consider the case of a parabivector-valued function f p . Changing the order of integration and applying the substitution u = x − x , yields, consecutively,

FH+ [ f p ∗ g](ξ) 1 p = exp (ξ ∧ x) f (x − x ) g(x ) dV(x ) dV(x) 2π R2 R2

132

Fred Brackx et al.

=

1 2π

1 = 2π

R2

R2

R2

exp (ξ ∧ x) f p (x − x ) dV(x) g(x ) dV(x )

R2

exp (ξ∧u) exp (ξ∧x ) f (u) dV(u) g(x ) dV(x ). p

As a parabivector-valued function f p commutes with the Clifford– Fourier kernels; that is,

exp ±(ξ ∧ x) f p (u) = f p (u) exp ±(ξ ∧ x) , we indeed obtain

FH+ [ f p ∗ g](ξ) 1 = exp (ξ ∧ u)f p (u) dV(u) exp (ξ ∧ x )g(x )dV(x ) R2 2π R2 p = FH+ [ f ](ξ) exp (ξ ∧ x ) g(x ) dV(x ) R2

p

= 2π FH+ [ f ](ξ) FH+ [ g](ξ). On the other hand a vector-valued function f , by commutation, transforms the kernel of FH+ into the kernel of FH− and vice versa; that is,

exp ±(ξ ∧ x) f (u) = f (u) exp ∓(ξ ∧ x) . Hence, for a vector-valued function f we ﬁnd

FH+ [ f ∗ g](ξ) 1 = exp (ξ ∧ u) exp (ξ ∧ x ) f (u) dV(u) g(x ) dV(x ) 2π R2 R2 1 = exp (ξ ∧ u) f (u) dV(u) exp (x ∧ ξ) g(x ) dV(x ) R2 2π R2 = FH+ [ f ](ξ) exp (x ∧ ξ) g(x ) dV(x ) R2

= 2π FH+ [ f ](ξ) FH− [ g](ξ).

The Fourier Transform in Clifford Analysis

133

Note that in case of a general C2 -valued function f = f p + f , the convolution theorem inevitably leads to two terms:

FH± [ f ∗ g](ξ) = 2π FH± [ f p ](ξ) FH± [ g](ξ) + 2π FH± [ f ](ξ) FH∓ [ g](ξ). (vii) First, recall from Subsection 4.4.1 that

(FH± )−1 = exp ±i π2 F −1 or in integral form (FH± )−1 [ f ](ξ) =

1 2π

R2

exp ±i π2 ξ exp (i < x, ξ >) f (x) dV(x).

Similarly, as in Subsection 4.5.1, one can prove that

π exp ±i ξ exp (i < x, ξ >) = exp ±(ξ ∧ x) , 2 from which we obtain that (FH± )−1 = FH± . Hence, we also have that

1 FH+ [ f g](ξ) = 2π

p

R2

−1 p exp (ξ ∧ x) FH + [FH+ [ f ]](x) g(x) dV(x)

1 exp (ξ ∧ x) (2π)2 R2 p × exp (x ∧ u) FH+ [ f ](u) dV(u) g(x) dV(x). =

R2

The Clifford–Fourier spectrum of a parabivector-valued function is again parabivector valued as follows:

FH+ [ f p ](u) = Fcos [ f p ](u) + e12 Fsin [ f p ](u) = Fcos [ f0 ](u) + Fcos [ f12 ](u) e12 + e12 Fsin [ f0 ](u) −Fsin [ f12 ](u),

134

Fred Brackx et al.

and thus commutes with the Clifford–Fourier kernels, which yields

FH+ [ f p g](ξ) 1 p FH+ [ f ](u) exp (ξ − u) ∧ x g(x)dV(x) dV(u) = (2π)2 R2 R2 1 FH+ [ f p ](u) FH+ [ g](ξ − u) dV(u) = 2π R2 =

1 FH+ [ f p ] ∗ FH+ [ g] (ξ). 2π

As the Clifford–Fourier transform of a vector-valued function f is again vector valued,

FH+ [ f ](u) = Fcos [ f1 ](u) e1 + Fcos [ f2 ](u) e2 + e2 Fsin [ f1 ](u) − e1 Fsin [ f2 ](u), the multiplication theorem in case of a vector-valued function f reads as follows:

FH+ [ f g](ξ) 1 exp (ξ∧x) exp (x∧u)FH+ [ f ](u)dV(u) g(x)dV(x) = (2π)2 R2 R2 1 + F [ f ](u) exp x ∧ (ξ − u) g(x)dV(x) dV(u) = H (2π)2 R2 R2 1 FH+ [ f ](u) FH− [ g](ξ − u) dV(u) = 2π R2 =

1 FH+ [ f ] ∗ FH− [ g] (ξ). 2π

Again, in case of a general C2 -valued function f = f p + f , the multiplication theorem divides into two pieces

FH± [ f g](ξ) =

1 FH± [ f p ] ∗ FH± [ g] (ξ) 2π 1 + FH± [ f ] ∗ FH∓ [ g] (ξ). 2π

The Fourier Transform in Clifford Analysis

135

(viii) By means of the substitution u = sxs, we have

1 FH+ [ f (sxs)](ξ) = 2π =

R2

exp (ξ ∧ x) f (sxs) dV(x)

1 2π

R2

exp (ξ ∧ sus) f (u) dV(u).

Taking into account that s(u ∧ ξ)s = (sus) ∧ (sξs) yields

FH+ [ f (sxs)](ξ) =

1 2π

R2

1 = s 2π

exp s(sξs ∧ u)s f (u) dV(u)

R2

exp (sξs ∧ u) s f (u) dV(u).

As s ∈ SpinR (m) takes the form s = ω1 ω2 . . . ω2 with ωj ∈ Sm−1 , j = 1, . . . , 2, and moreover, e12 ωj = −ωj e12 , we ﬁnd that

e12 s = e12 ω1 . . . ω2 = (−1)2 ω1 . . . ω2 e12 = s e12 . Hence, s commutes with the parabivector-valued exponential kernel, which ﬁnally leads to 1 FH+ [ f (sxs)](ξ) = ss 2π

R2

exp (sξs ∧ u) f (u) dV(u) = FH+ [ f ](sξs),

since ss = 1. Remark 4.5 1. Property (ii) implies that it is sufﬁcient to compute one of the Clifford–Fourier transforms. 2. As the Clifford algebra is not commutative, property (iii) cannot be rewritten in the more elegant form

†

†

FH± [ f ](ξ)

= FH∓ [ f † ](ξ).

However, we have

FH± [ f ](ξ)

= [ f † ]FH∓ (ξ),

136

Fred Brackx et al.

where the notation [ f † ]FH∓ means that the Clifford–Fourier transform is now acting from the right on the function f † . 3. If f is rotation invariant (i.e., f (sxs) = f (x) for each s ∈ SpinR (m)), then the rotation rule implies that FH± [ f (x)](sξs) = FH± [ f (x)](ξ) for each s ∈ SpinR (m). Hence, FH± [ f ] is rotation invariant as well.

2. The Clifford–Fourier Transform in S R2 We now investigate the Clifford–Fourier transform in a dense subspace of L1 R2 , dV(x) , viz. the right C2 -module of rapidly decreasing C2 -valued functions. Theorem 4.3 Let ϕ ∈ S R2 . Then FH± [ϕ] ∈ S R2 . Proof. We will show that for every ϕ ∈ S R2 , the following inequality holds: β p∗,k FH± [ϕ] = sup sup sup ξ α ∂ξ [FH± [ϕ](ξ)] ξ∈R2 |α|≤k |β|≤

2 +2

≤ C sup (1 + |x| ) x∈R2

γ ∂x ϕ(x) = C p+2,k (ϕ).

(43)

|γ|≤k

Here {p∗,k ; , k ∈ N} and {p,k ; , k ∈ N} are two equivalent systems α α of seminorms on S R2 . Moreover, we use the notation ξ α = ξ1 1 ξ2 2 and β β β ∂ξ = ∂ξ11 ∂ξ22 . One can easily verify that for every multi-index α ∈ N2 : α

α

∂ξα [FH± [ϕ](ξ)] = (±e12 )α1 (∓e12 )α2 FH± [x1 2 x2 1 ϕ(x)](ξ) and α

α

FH± [∂xα ϕ(x)](ξ) = (±e12 )α1 (∓e12 )α2 ξ1 2 ξ2 1 FH± [ϕ(x)](ξ). For arbitrary ξ ∈ R2 and multi-indices α, β ∈ N2 with |α| ≤ k, and |β| ≤ , we thus have β β β α α ξ α ∂ξ FH± [ϕ](ξ) = (±e12 )β1 (∓e12 )β2 ξ1 1 ξ2 2 FH± x1 2 x2 1 ϕ(x) (ξ) α α β β = (±e12 )β1 +α1 (∓e12 )β2 +α2 FH± ∂x12 ∂x21 x1 2 x2 1 ϕ(x) (ξ).

The Fourier Transform in Clifford Analysis

137

By means of the properties of the Clifford norm and the estimate | exp ±(ξ ∧ x) | ≤ 2 , we obtain 1 α2 α1 β2 β1 α β exp ±(ξ ∧ x) ∂x1 ∂x2 x1 x2 ϕ(x) dV(x) ξ ∂ξ FH± [ϕ](ξ) ≤ 4 2π R2 16 α2 α1 β2 β1 (44) ≤ ∂x1 ∂x2 [x1 x2 ϕ(x)] dV(x). π R2 Furthermore, using the estimate

α2 α1 β2 β1 α −γ α −γ β β γ γ Cγα ∂x12 1 ∂x21 2 [x1 2 x2 1 ] ∂x11 ∂x22 [ϕ(x)] ∂x1 ∂x2 [x1 x2 ϕ(x)] ≤ |γ|≤|α|

≤ C (1 + |x|2 )|β|

γ ∂x ϕ(x), |γ|≤|α|

the inequality in Eq. (44) still for |α| ≤ k and |β| ≤ becomes

γ 16 α β ∂x ϕ(x) dV(x) (1 + |x|2 ) ξ ∂ξ [FH± [ϕ](ξ)] ≤ C π R2 |γ|≤k γ 16 2 +2 ≤C sup (1 + |x| ) ∂x ϕ(x) π x∈R2 |γ|≤k 1 × dV(x) 2 2 R2 (1 + |x| ) γ 2 +2 = C sup (1 + |x| ) ∂x ϕ(x) . x∈R2

|γ|≤k

Hence, we have proved the desired inequality in Eq. (43). Inequality (43) immediately yields the following result, where use we the notation ϕj → ϕ for the convergence of the sequence ϕj in S R2 . S

Corollary 4.2 If ϕj → ϕ, then FH± [ϕj ] → FH± [ϕ] . S

S

Remark 4.6 The 2D Clifford–Fourier transform in S R2 is thus a continuous operator. For the inversion of the Clifford–Fourier transform, one now has the following proposition.

138

Fred Brackx et al.

Proposition 4.10 For each ϕ ∈ S R2 , one has

FH± [FH± [ϕ]] = ϕ, or, in other words, (FH± )2 = IS R2 . By Corollary 4.2 and Proposition 4.10 we ﬁnally obtain the following fundamental result. Theorem 4.4 The 2DClifford–Fourier transform FH± is a homeomorphism of 2 the right C2 -module S R .

3. The Clifford–Fourier Transform in L2 R2 , dV(x) The deﬁnition of the Clifford–Fourier transform in L2 R2 , dV(x) follows classical lines and makes use of the results obtained in the foregoing subsection on the Clifford–Fourier transform in the dense subspace S R2 . We start with the following lemma. Lemma 4.7 For all ϕ, ψ ∈ S R2 , one has < ϕ, ψ > = FH± [ϕ], FH± [ψ] . Proof. Taking into account Proposition 4.10 and the transfer formula (see Proposition 4.9 (v)), we have consecutively

< ϕ, ψ > =

ϕ(x)

R2

=

†

R2

† ψ(x) dV(x) = FH± [FH± [ϕ]](x) ψ(x) dV(x) R2

†

FH± [ϕ](x)

& ' FH± [ψ](x) dV(x) = FH± [ϕ], FH± [ψ] .

In particular we have the following result. Lemma 4.8 For each ϕ ∈ S R2 , one has ϕ2 = FH± [ϕ]2 . This implies that the operator FH± can be uniquely extended to L2 R2 , dV(x) . Indeed, take f ∈ L2 R2 , dV(x) . By means of the density of S R2 in L2 R2 , dV(x) , there exists a sequence (ϕj )j∈N ∈ S R2 that 2 converges in L2 R , dV(x) to f . Hence, (ϕj )j∈N is a Cauchy sequence in L2 R2 , dV(x) . In view of Lemma 4.8, FH± [ϕj ] j∈N is also a Cauchy sequence in L2 R2 , dV(x) and thus convergent in L2 R2 , dV(x) . The limit of FH± [ϕj ] j∈N , which is independent of the chosen sequence (ϕj )j∈N , is called the Clifford–Fourier transform of f . We denote it by FH± [ f ].

The Fourier Transform in Clifford Analysis

139

The above-introduced Clifford–Fourier transform in L2 R2 , dV(x) has the following properties. Proposition 4.11 The 2D Clifford–Fourier transform FH± in L2 R2 , dV(x) satisﬁes the following properties: (i) FH± is right C2 -linear; that is, for all f , g ∈ L2 R2 , dV(x) and for all λ, μ ∈ C2 , one has

FH± [ f λ + gμ] = FH± [ f ] λ + FH± [ g] μ. (ii) The restriction of FH± to S R2 is FH± . 2 (iii) FH± is bounded on L2 R , dV(x) . (iv) The inverse of FH± is precisely FH± , or (FH± )2 = IL R2 ,dV(x) . 2

(v) The adjoint (FH± )∗ is given by (FH± )∗ = FH± = (FH± )−1 .

Proof. (i) Trivial. (ii) Immediate. (iii) Take f ∈ L2 R2 , dV(x) and a sequence ϕj ∈ S R2 that converges in 2 L2 R , dV(x) to f ; that is, ϕj 2 → f 2 . Furthermore, by deﬁnition R FH± [ϕj ] → FH± [ f ]; that is, FH± [ϕj ]2 → FH± [ f ]2 . Taking into L2 R account Lemma 4.8, we thus obtain f 2 = FH± [ f ]2 , which proves the statement. (iv) Take again f ∈ L2 R2 , dV(x) and a sequence ϕj ∈ S R2 that converges in L2 R2 , dV(x) to f . By deﬁnition, we have consecutively

FH± [ϕj ] → FH± [ f ] L2

and

FH± [FH± [ϕj ]] → FH± [FH± [ f ]]. L2

Hence, by means of Proposition 4.10 we ﬁnd

ϕj → FH± [FH± [ f ]], L2

which leads to

f = FH± [FH± [ f ]]. (v) To compute the adjoint (FH± )∗ , we start with f , g ∈ L2 R2 , dV(x) and 2 2 sequences ϕj and ψj in S R that converge in L2 R , dV(x) to f and

140

Fred Brackx et al.

g, respectively. By deﬁnition, we have

FH± [ϕj ] → FH± [ f ]

and

L2

FH± [ψj ] → FH± [ g]. L2

In terms of the inner product on L2 R2 , dV(x) , it follows that < ϕj , ψj >→< f , g > and C2

& ' ' & FH± [ϕj ], FH± [ψj ] → FH± [ f ], FH± [ g] . C2

Moreover, applying Lemma 4.7 gives

& ' < f , g >= FH± [ f ], FH± [ g] and hence also

(

'

)

& ' FH± [ f ], g = FH± [FH± [ f ]], FH± [ g] = f , FH± [ g] .

&

Summarizing we obtain the following fundamental result. Theorem 4.5 The 2D Clifford–Fourier transform FH± is a unitary operator on the right C2 -module L2 R2 , dV(x) .

4.5.3. Connection With the Classical Fourier Transform and the Clifford–Fourier Transform of Ebling and Scheuermann In this subsection we ﬁrst derive an explicit connection between the 2D Clifford–Fourier transform pair {FH+ , FH− } and the classical tensorial Fourier transform F in the plane. By means of Lemma 4.6 we can rewrite the Clifford–Fourier transform FH+ as follows:

FH+ [ f ](ξ) = P+

1 2π

+ P−

R2

1 2π

exp −i(ξ1 x2 − ξ2 x1 ) f (x) dV(x)

R2

exp −i(ξ2 x1 − ξ1 x2 ) f (x) dV(x).

Furthermore, one easily veriﬁes that

1 2π

R2

exp −i(ξ1 x2 − ξ2 x1 ) f (x) dV(x) = F [ f ](e12 ξ)

The Fourier Transform in Clifford Analysis

141

and similarly

1 2π

R2

exp −i(ξ2 x1 − ξ1 x2 ) f (x) dV(x) = F [ f ](−e12 ξ).

This yields the following relation between the Clifford–Fourier transform involving the operator H+ and the standard Fourier transform:

FH+ [ f ](ξ) = P+ F [ f ](e12 ξ) + P− F [ f ](−e12 ξ).

(45)

Similarly, one obtains

FH− [ f ](ξ) = P+ F [ f ](−e12 ξ) + P− F [ f ](e12 ξ).

(46)

Remark 4.7 1. The transformations ξ → e12 ξ and ξ → −e12 ξ represent, respectively, an anti-clockwise and a clockwise, rotation by a right angle. 2. For a radial function f , expressions (45) and (46) reduce to FH± [ f ] = F [ f ], since the Fourier transform of a radial function remains radial. Note that we already obtained this result for the Clifford–Fourier transform in arbitrary dimension (see Proposition 4.8). Moreover, the Clifford–Fourier transform of Ebling and Scheuermann (see Section 4.1)

e

F [ f ](ξ) =

exp (−e12 (x1 ξ1 + x2 ξ2 )) f (x) dV(x)

R2

can be expressed in terms of the Clifford–Fourier transform:

FH± [ f ](ξ) = FH± [ f ](ξ1 , ξ2 ) 1 = 2π

R2

exp ±e12 (ξ1 x2 − ξ2 x1 ) f (x) dV(x).

Indeed, we have

F e [ f ](ξ) = 2π FH± [ f ](∓ξ2 , ±ξ1 ) = 2π FH± [ f ](±e12 ξ),

142

Fred Brackx et al.

taking into account that under the isomorphism between the Clifford algebras R2,0 and R0,2 , both pseudoscalars e12 are isomorphic images of each other. Note that in Ebling and Scheuermann (2005) the Fourier kernel is at the right-hand side of the function f instead of at the left.

4.5.4. Example: The Box Function As an illustration, in this subsection we calculate the Clifford–Fourier transform pair of the box function:

A if a ≤ x1 ≤ b and c ≤ x2 ≤ d 0 if otherwise

f (x1 , x2 ) =

with A a constant. Its classical Fourier transform reads

1 exp (−ibξ1 ) − exp (−iaξ1 ) ξ1 ξ2 × exp (−idξ2 ) − exp (−icξ2 ) .

F [ f ](ξ) = −

A 2π

In view of relation (45), this yields the following expression for the Clifford– Fourier transform involving the operator H+ :

1 exp (ibξ2 ) − exp (iaξ2 ) FH+ [ f ](ξ) = P ξ1 ξ2 1 A × exp (−idξ1 ) − exp (−icξ1 ) + P− 2π ξ1 ξ2 × exp (−ibξ2 ) − exp (−iaξ2 ) exp (idξ1 ) − exp (icξ1 ) . +

A 2π

(47) By means of, inter alia,

P

+

exp (ibξ2 ) = P

+

∞ k ∞ i (−e12 )k k + (bξ2 ) = P (bξ2 )k k! k! k=0

= P+ exp (−e12 bξ2 ),

k=0

The Fourier Transform in Clifford Analysis

143

expression (47) can be simpliﬁed to FH+ [ f ](ξ) 1 A P+ exp (−e12 bξ2 )−exp (−e12 aξ2 ) exp (e12 dξ1 )−exp (e12 cξ1 ) = 2π ξ1 ξ2 + P− exp (−e12 bξ2 )−exp (−e12 aξ2 ) exp (e12 dξ1 ) − exp (e12 cξ1 ) 1 A exp (−e12 bξ2 ) − exp (−e12 aξ2 ) exp (e12 dξ1 )−exp (e12 cξ1 ) . 2π ξ1 ξ2

=

By a similar compution, one obtains

1 A exp (e12 bξ2 ) − exp (e12 aξ2 ) 2π ξ1 ξ2 × exp (−e12 dξ1 ) − exp (−e12 cξ1 ) .

FH− [ f ](ξ) =

5. CLIFFORD FILTERS FOR EARLY VISION Among the mathematical models for the receptive ﬁeld proﬁles of the human visual system, the Gabor model is well known and widely used. Another lesser used model that agrees with the Gaussian derivative model for human vision is the Hermite model. It is based on analysis ﬁlters of the Hermite transform, which was introduced by Martens (1990b), and offers some advantages such as being an orthogonal basis and better matching with experimental physiological data. In this section we expand the ﬁlter functions of the classical Hermite transform into the generalized Clifford–Hermite polynomials. Moreover, we construct a new multidimensional Hermite transform within Clifford analysis using the generalized Clifford–Hermite polynomials and, we compare this newly introduced Clifford–Hermite transform with the Clifford– Hermite continuous wavelet transform. Next, we introduce Gabor ﬁlters in the Clifford analysis setting. These Clifford–Gabor ﬁlters are based on the Clifford–Fourier transform discussed in the previous section. Finally, we present additional properties of the Clifford–Gabor ﬁlters, such as their relationship with other types of Gabor ﬁlters and their localization in the spatial and frequency domain formalized by the uncertainty principle.

5.1. Introduction Image processing has been much inspired by human vision, particularly with regard to early vision. The latter refers to the earliest stage of visual

144

Fred Brackx et al.

processing responsible for the measurement of local structures, such as points, lines, edges, and textures, in order to facilitate subsequent interpretation of these structures in higher stages (known as high-level vision) of the human visual system. According to the Gaussian derivative theory (see Young, 1991), the receptive ﬁeld proﬁles of the human visual system can be approximated quite well by derivatives of Gaussians. Two mathematical models suggested for these receptive ﬁeld proﬁles are the Gabor model and the Hermite model, which is based on analysis ﬁlters of the Hermite transform. The Hermite ﬁlters are derivatives of Gaussians, whereas Gabor ﬁlters, which are deﬁned as harmonic modulations of Gaussians, provide a good approximation to these derivatives. It is important to note that, even if the Gabor model is more widely used than the Hermite model, the latter offers some advantages, such as being an orthogonal basis and better matching with experimental physiological data. In this section we establish the construction of the Hermite and Gabor ﬁlters both in the classical and in the Clifford analysis setting. We begin by describing the classical Hermite transform, in both the 1D and multidimensional case (see Subsection 5.2.1). Next, in Subsection 5.2.2, we expand the ﬁlter functions of the classical multidimensional Hermite transform into the generalized Clifford–Hermite polynomials introduced in Section 2.4. Furthermore, we construct a new higher-dimensional Hermite transform within the framework of Clifford analysis using the generalized Clifford–Hermite polynomials (see Subsection 5.2.3). In Subsection 5.2.4 we compare this Clifford–Hermite transform with the Clifford–Hermite continuous wavelet transform. The topic of Section 5.3 is Gabor ﬁlters, a prominent tool for local spectral image processing and analysis. First, in Subsection 5.3.1, we discuss classical 1D complex and real Gabor ﬁlters. They are closely related to Fourier analysis, since the impulse response of a complex Gabor ﬁlter is the conjugated integral kernel of the complex Fourier transform at a certain frequency, multiplied by a Gaussian. In Subsection 5.3.2 we examine two nonclassical 2D Gabor ﬁlters. First, we brieﬂy describe the so-called quaternionic Gabor ﬁlters of Bülow and Sommer, followed by the Clifford–Gabor ﬁlters of Ebling and Scheuermann. Next, we proceed with developing our 2D Clifford–Gabor ﬁlters (see Subsection 5.3.3). This new types of Gabor ﬁlters arises quite naturally from our study of the 2D Clifford–Fourier transform. Indeed, using the conjugated kernels of the 2D Clifford–Fourier transform modulated by a Gaussian gives rise to the 2D Clifford–Gabor ﬁlters. We also give an explicit connection between these new ﬁlters and the standard complex Gabor ﬁlters and the Clifford–Gabor ﬁlters of Ebling and Scheuermann. An often-cited property of Gabor ﬁlters is their optimal simultaneous localization in the spatial and the frequency domain, which is formalized by the uncertainty principle. This property makes Gabor ﬁlters suitable for local frequency analysis. We end this section by showing that

The Fourier Transform in Clifford Analysis

145

our 2D Clifford–Gabor ﬁlters also exhibit the best possible joint localization in position and frequency space.

5.2. Generalized Clifford–Hermite Filters 5.2.1. The Classical Hermite Transform The Hermite transform was introduced by Martens (1990b) as a signal expansion technique in which a signal is windowed by a Gaussian at equidistant positions and is locally described by a weighted sum of polynomials. In van Dijk and Martens (1997) an image compression scheme based on an orientation-adaptive steered Hermite transform is presented. Comparison with other compression techniques show that the proposed scheme performs very well at high compression ratios, not only in terms of peak signal to noise ratio (the commonly used objective measure for the quality of coded images) but also in terms of perceptual image quality. Moreover, in Martens (1990c), it is demonstrated how the Hermite transform can be used for image coding and analysis. In the image coding application, the relation with existing pyramid coders is described. A new coding scheme, based on local 1D image approximations, is introduced. In the image analysis application, the relation between the Hermite transform and existing line/edge detection schemes is described. An algorithm based on the Hermite transform was applied to astronomical images in Venegas-Martinez, Escalente-Ramirez, and Garcia-Barreto (1997). Hermite transforms have been used in applications such as image deblurring (see Martens, 1990a), noise reduction (see Escalante-Ramirez and Martens, 1992) and also estimation of perceived noise and blur (see Kayargadde and Martens, 1994).

A. The Classical One-Dimensional Hermite Transform The 1D Hermite transform ﬁrst localizes the original signal f (x) by multiplying it by a Gaussian window function

x2 *σ (x) = √1 V exp − 2σ 2 . πσ In order to obtain a complete description of the signal f (x), the localization process should be repeated at a sufﬁcient number of window positions with the spacing between the windows chosen equidistant. In this way, the following expansion of the original signal f (x) is obtained:

f (x) =

1

+∞

* σ (x) W k=−∞

*σ (x − kT), f (x) V

(48)

146

Fred Brackx et al.

where

* σ (x) = W

+∞

*σ (x − kT), V

k=−∞

is the so-called weight function, which is positive for all x. The next step in the Hermite transform is the decomposition of the *σ (x − kT) into a series of orthonormal functions. localized signal f (x)V Fundamental for this expansion are the polynomials Gnσ (x), which are *σ (x))2 ; that is, orthonormal with respect to (V

+∞

−∞

*σ (x))2 Gσ (x) Gσ (x) dx = δn ,n . (V n n

These uniquely determined polynomials have the following form:

Gnσ (x) = √

1 2n n!

Hn

x σ

,

where Hn is the standard Hermite polynomial of order n (n = 0, 1, 2, . . .) associated with the weight function exp (−x2 ) [see Eq. (17)]. Under very general conditions (see Boas and Buck, 1985) for the original signal f (x), we get the following decomposition of the localized signal into *σ (x) Gσ (x): the orthonormal functions Knσ (x) = V n

*σ (x − kT) f (x) = V

∞

cnσ (kT) Knσ (x − kT),

(49)

n=0

where

cnσ (kT)

=

∞

−∞

*σ (x − kT))2 dx f (x) Gnσ (x − kT) (V

(50)

are the Hermite coefﬁcients. As the Hermite transform is locally (i.e., within each window function), a unitary transformation, we have the following generalization of Parseval’s theorem:

+∞

−∞

f (x)

2

*σ (x − kT))2 dx = (V

∞ (cnσ (kT))2 . n=0

The Fourier Transform in Clifford Analysis

147

In other words, the energy of each local signal can be expressed in terms of the Hermite coefﬁcients cnσ (kT) of the expansion. The deﬁning relation in Eq. (50) of the Hermite coefﬁcients cnσ (kT) can be rewritten as the convolution of the original signal f (x) with the functions

x 2 (−1)n 1 *σ (−x))2 = √ exp − σx 2 , Dσn (x) = Gnσ (−x) (V √ Hn σ 2n n! σ π followed by a downsampling by a factor T. The functions Dσn (x) are called the Hermite ﬁlters. By means of the Rodrigues formula Eq. (17) for the classical Hermite polynomials, they can be expressed as Gaussian derivatives

Dσn (x)

2 σn dn 1 =√ √ exp − σx 2 . 2n n! dxn σ π

The Fourier transform of the Hermite ﬁlter Dσn takes the form of a Gaussian modulated by a monomial of degree n

2 2 1 σn F [Dσn ](ξ) = √ √ (iξ)n exp − σ 4ξ . 2n n! 2π The mapping from the original signal f (x) to the Hermite coefﬁcients cnσ (kT) is called the forward Hermite transform. The signal reconstruction from the Hermite coefﬁcients is called the inverse Hermite transform. Combining Eqs. (48) and (49), we get the expansion of the complete signal into the so-called pattern functions Qσn :

f (x) =

∞ +∞

cnσ (kT) Qσn (x − kT),

n=0 k=−∞

where

Qσn (x) =

Knσ (x) . * σ (x) W

This formula implies that the inverse Hermite transform consists of interpolating the Hermite coefﬁcients {cnσ (kT); k integer} with the pattern function Qσn (x) and summing up over all orders n. As mentioned in Section 5.1, the Hermite transform models the information analysis carried out by the visual receptive ﬁelds. Because these receptive ﬁelds vary in size, each ﬁeld is suited for detecting the presence of a speciﬁc spatial frequency. With the Hermite transform, ﬁeld sizes can be modeled by varying the standard deviation σ of the Gaussian envelope,

148

Fred Brackx et al.

while orientation selectivity can be obtained by rotation of the Hermite ﬁlters. In practice, the Hermite transform is often limited to the ﬁrst few terms, which introduces effects of ﬁltering and aliasing. In order for the ﬁnite Hermite transform to describe the signal adequately, σ must be properly selected. On the one hand, σ should be as large as possible since integrating over large areas improves the output signal to noise ratio and the efﬁciency of our signal representation. On the other hand, σ cannot be too large because then the signal cannot be described accurately by the ﬁrst few terms in the Hermite expansion. The important problem of selecting the right value of σ is the main topic of Martens (1990c). Remark 5.1 The Hermite transform provides the connection between the derivatives of Gaussians and the Hermite functions [see also Eq. (16)]. In the Hermite transform, the analysis functions Dσn are the derivatives of Gaussians, whereas the reconstruction functions Knσ are the Hermite functions. The difference between the Hermite function and the derivative of the Gaussian of order n is the scale of the Gaussian in relation to the scale of the Hermite polynomial. In case of the Hermite functions, the Hermite polynomials grow as fast as the exponential decays and hence the maxima of the Hermite functions all have approximately the same height, giving them the shape of a truncated sine/cosine wave. The Hermite functions have two interesting properties. First, they maximize the uncertainty principle (see subsection 5.3.3) and second, as for the Gaussian, their Fourier transform has the same functional form as the function itself [see Eq. (18)].

B. The Classical Multidimensional Hermite Transform The Hermite transform is generalized to higher dimension in a tensorial manner (for the 2D and 3D case, see Martens, 1990b). Let us start from the Gaussian window function *σ (x) = V

1 √ πσ

m

exp

x2 +···+x2 − 1 2σ 2 m

.

*σ (x) = V *σ (x1 ) . . . V *σ (xm ). This window function is separable; that is, V Naturally, the polynomials

1 Giσ1 −i2 ,i2 −i3 ,...,im (x) = 2i1 (i1 − i2 )!(i2 − i3 )! . . . im ! x x x m 1 2 × Hi1 −i2 Hi2 −i3 . . . Him σ σ σ = Giσ1 −i2 (x1 ) Giσ2 −i3 (x2 ) . . . Giσm (xm )

The Fourier Transform in Clifford Analysis

149

*σ (x))2 , that is, are orthonormal with respect to (V

δi1 ,1 . . . δim ,m =

Rm

σ *σ (x))2 Gσ (V i1 −i2 ,...,im (x) G1 −2 ,...,m (x) dV(x)

for i1 , 1 = 0, . . . , ∞; i2 = 0, . . . , i1 ; 2 = 0, . . . , 1 ; . . . ; im = 0, . . . , im−1 ; m = 0, . . . , m−1 . In a manner similar to the 1D case, we obtain the following decomposition of a signal f (x) into the pattern functions Qσi1 −i2 ,...,im (x):

f (x) =

i1 ∞ i1 =0 i2 =0

im−1

...

im =0 (px1 ,...,pxm )∈P

ciσ1 −i2 ,...,im (px1 , . . . , pxm )

× Qσi1 −i2 ,...,im (x1 − px1 , . . . , xm − pxm ), where (px1 , . . . , pxm ) ranges over all coordinates in a square sampling grid P. Thus, the reconstruction of the signal consists again of interpolating the Hermite coefﬁcients ciσ1 −i2 ,...,im (px1 , . . . , pxm ) with the pattern functions

Qσi1 −i2 ,...,im (x) =

*σ (x) Gσ V i1 −i2 ,...,im (x) = Qσi1 −i2 (x1 ) . . . Qσim (xm ), * σ (x) W

where

* σ (x) = W

*σ (x1 − px , . . . , xm − px ) V m 1

(px1 ,...,pxm )∈P

is the positive weight function. The Hermite coefﬁcients ciσ1 −i2 ,...,im (px1 , . . . , pxm ) are again obtained by convolving the original signal f (x) with the ﬁlter functions

*σ (−x))2 Gσ Dσi1 −i2 ,...,im (x) = (V i1 −i2 ,...,im (−x) m x x 1 m 1 . . . Him Hi1 −i2 = √ i σ σ πσ 2 1 (i1 − i2 )! . . . im ! x2 +···+x2 exp − 1 σ 2 m (−1)i1

= Dσi1 −i2 (x1 ) . . . Dσim (xm )

150

Fred Brackx et al.

followed by a downsampling along the grid P. These ﬁlter functions can be written as derivatives of a Gaussian:

Dσi1 −i2 ,...,im (x)

σ i1

= 2i1 (i1 − i2 )! . . . im ! ∂i1 −i2 ∂i2 −i3 i −i2

∂x11

i −i3

∂x22

...

∂im im ∂xm

m 1 √ πσ 2 x12 +···+xm exp − σ 2 .

Now we aim to construct a generating function of these ﬁlter functions. Putting

* Fiσ1 (u; x)

i1 σ 2 1 ∂ ∂ ∂ * (x) V u1 + u2 + · · · + um i1 ! ∂(x1 /σ) ∂(x2 /σ) ∂(xm /σ) m km−1 i1 k2 i −k k −k u11 2 u22 3 . . . ukmm ∂i1 −k2 1 ... = √ (i1 − k2 )!(k2 − k3 )! . . . km ! ∂(x1 /σ)i1 −k2 πσ =

k2 =0 k3 =0

×

∂k2 −k3

∂ km ... ∂(xm /σ)km

∂(x2 /σ)k2 −k3 i1 k2

i1 /2

=2

i −k2

...

exp

x2 +···+x2 − 1 σ2 m

km

k −k3

u22

km−1

k2 =0 k3 =0

× u11

km =0

1 (i1 − k2 )!(k2 − k3 )! . . . km ! =0

. . . ukmm Dσi1 −k2 ,...,km (x),

we obtain

1

Dσi1 −i2 ,...,im (x) =

2i1 /2

√

∂im *σ 1 ∂i1 −i2 . . . [F (u; x)]. im i1 (i1 − i2 )! . . . im ! ∂ui1 −i2 ∂u m 1

Hence, the function

* Fσ (u; x) =

∞ i1 =0

=

1 2i1 /2

i1 ∞ i1 =0 k2 =0

1 i1 !

∂ ∂ u1 + · · · + um ∂(x1 /σ) ∂(xm /σ)

km−1

...

km

i1

2

*σ (x) V

1 i −k u11 2 . . . ukmm Dσi1 −k2 ,...,km (x) (i1 − k2 )! · · · km ! =0

151

The Fourier Transform in Clifford Analysis

generates the ﬁlter functions, since

1 Dσi1 −i2 ,...,im (x) = √ (i1 − i2 )! . . . im !

∂i1 −i2 i −i2

∂u11

∂im σ . . . i [* F (u; x)] ∂umm

. u=0

Moreover, we have that

ciσ1 −i2 ,...,im (t) = ( f ∗ Dσi1 −i2 ,...,im )(t) ∂ im ∂i1 −i2 1 σ . . . i [( f (x) ∗ * F (u; x))(t)] =√ (i1 − i2 )! . . . im ! ∂ui1 −i2 ∂umm 1

.

u=0

So, if we deﬁne

cσ (u; t) = ( f (x) ∗ * Fσ (u; x))(t), we have obtained the generating function of the Hermite coefﬁcients as follows:

* cσ (u; t) =

i1 ∞

im−1

...

i1 =0 i2 =0

×

=

im =0

∂i1 −i2 i −i2

∂u11

i1 ∞ i1 =0 i2 =0

...

1 i −i u 1 2 . . . uimm (i1 − i2 )! . . . im ! 1 ∂im ∂uimm

im−1

...

im =0

√

σ

[c (u; t)] u=0

1 i −i u 1 2 . . . uimm ciσ1 −i2 ,...,im (t) . (i1 − i2 ) . . . im ! 1

5.2.2. Expansion of the Classical Multidimensional Hermite Filters Into the Generalized Clifford–Hermite Polynomials In this subsection we show how the classical ﬁlter functions of the multidimensional Hermite transform can be expressed in terms of Clifford analysis. The starting point is the general term of the deﬁning series of the generating function of the ﬁlter functions (see subsection 5.2.1), which, up to constants, may be rewritten as

Fk (u; x) =

1 < u, ∂x >k [V(x)], k!

152

Fred Brackx et al.

where we have put

2 V(x) = exp − |x|2 . Now the Fischer decomposition leads to the following expansion (see Sommen and Jancewicz, 1997):

1 < u, x >k = us Zk,s (u, x) xs , k! k

s=0

where the functions Zk,s (u, x) are the so-called zonal monogenics. These zonal monogenics are homogeneous of degree k − s in u and x , left monogenic in u and right monogenic in x and have the form

1 Zk−s (u, x), Bs,k−s . . . B1,k−s

Zk,s (u, x) =

with B2s,k = −2s, B2s+1,k = −(2s + 2k + m) and

Zk (u, x) =

( m2 − 1)

(|u||x|)k 2k+1 (k + m2 ) u ∧ x m/2 m/2−1 Ck−1 (t) . × (k + m − 2)Ck (t) + (m − 2) |u||x|

Here Ck denotes the classical Gegenbauer polynomial with variable t = |u||x| . Hence, we obtain that

Fk (u; x) =

k

s

u

Zk,s (u, ∂x ) ∂xs [V(x)]

s=0

=

k

us Zk,s (u, ∂x ) V(x) ∂xs .

s=0

Next, Proposition 2.4 leads to

Fk (u; x) =

k

(−1)k−s us [Zk,s (u, x) V(x)]∂xs .

s=0

Moreover, as Zk,s (u, x) is a homogeneous right monogenic polynomial of degree k − s in x, the Rodrigues formula in Eq. (10) for the generalized

The Fourier Transform in Clifford Analysis

153

Clifford–Hermite polynomials yields

[Zk,s (u, x) V(x)]∂xs = Zk,s (u, x) H s,k−s (x) V(x). Hence, we ﬁnally obtain the decomposition

1 < u, ∂x >k [V(x)] = Fk (u; x) = (−1)k−s us Zk,s (u, x) H s,k−s (x) V(x). k! k

s=0

Substituting

x for x and putting σ

V σ (x) = V

x σ

|x|2 = exp − 2σ 2 ,

we obtain

Fk

k x 1 k 1 k σ u; =σ < u, ∂x > [V (x)] = k (−1)k−s σ s us Zk,s (u, x) σ k! σ s=0

× H s,k−s

x σ

V σ (x).

As

1 πm/2 σ m−i1 √2σ < u, ∂x >i1 [V σ (x)] = (i −m)/2 * Fi1 (u; x), i1 ! 2 1 we have ﬁnally obtained the following decomposition of the classical ﬁlter functions into the generalized Clifford–Hermite polynomials: √

2σ Di1 −i (x) = 2 ,...,im

1 2i1 /2

∂im *√2σ 1 ∂i1 −i2 . . . [Fi1 (u; x)] √ im (i1 − i2 )! . . . im ! ∂ui1 −i2 ∂u m 1

i1 1 1 1 (−1)i1 −s σ s = √ (2π)m/2 σ m+i1 (i1 − i2 )! . . . im ! s=0

∂i1 −i2 i −i2

∂u11

...

∂im ∂uimm

[us Zi1 ,s (u, x)] H s,i1 −s

x σ

V σ (x).

154

Fred Brackx et al.

5.2.3. The Generalized Clifford–Hermite Transform In this subsection we develop a new multidimensional Hermite transform directly in Clifford analysis; we call it the generalized Clifford–Hermite transform. First, using the Gaussian window function

|x|2 V σ (x) = exp − 2σ 2 , we get the following decomposition of the original real-valued signal f (x):

f (x) =

1 f (x) V σ (x − p) W σ (x)

(51)

p∈P

with

W σ (x) =

V σ (x − p)

p∈P

the positive weight function and P a sampling grid in R m . For the decomposition of the ﬁltered localized signal f (x) V σ (x − p) Fil , we use the generalized Clifford–Hermite polynomials Hn,k (x), which satisfy the orthogonality relation [see Eq. (13)]

Rm

Pk† (x)Hn,k

√ 2x σ

√ 2x σ

Hn ,k

Pk (x)(V σ (x))2 dV(x) =

σ 2k+m 2(2k+m)/2 × γn,k δn,n δk,k .

The above orthogonality relation leads to the following decomposition of the ﬁltered localized signal into the orthogonal generalized Clifford– √ σ (x) = V σ (x) H Hermite functions Kn,k n,k

σ

V (x − p) f (x)

Fil

=

∞ ∞

2x σ

Pk (x) (see also Section 3.3):

† σ σ cj, (p) Kj, (x − p) ,

(52)

j=0 =0

where we have put σ cn,k (p) =

=

2(2k+m)/2 σ 2k+m γn,k 2(2k+m)/2 σ 2k+m γn,k

√

Rm

Rm

f (x)Hn,k

2(x−p) σ

Pk (x − p)(V σ (x − p))2 dV(x)

σ f (x) Kn,k (x − p) V σ (x − p) dV(x).

The Fourier Transform in Clifford Analysis

155

σ (p) the generalized Clifford–Hermite coefﬁcients. We call cn,k Combining formulas (51) and (52), we obtain the following decomposition of the ﬁltered signal:

fFil (x) =

∞ ∞

σ cj, (p) Qσj, (x − p),

j=0 =0 p∈P

where we have introduced the pattern functions

†

Qσj, (x)

=

σ (x) Kj,

W σ (x)

=

P† (x) H j,

√ 2x σ

V σ (x)

W σ (x)

.

Note that the generalized Clifford–Hermite coefﬁcients may be expressed as the convolution of the original signal f (x) with the generalized Clifford–Hermite ﬁlter functions Dσn,k :

σ cn,k (p) = f ∗ Dσn,k (p) with

Dσn,k (x) =

√ 2(2k+m)/2 H − σ2x Pk (−x) (V σ (−x))2 n,k 2k+m σ γn,k

=

√

(−1)n+k 2(2k+m)/2 2x |x|2 P H (x) exp − n,k k σ σ2 σ 2k+m γn,k

=

2 2(2k+m−n)/2 σ n−m−2k P (−1)k ∂xn exp − |x| (x) . k σ2 γn,k

(53)

The last expression in Eq. (53) is obtained by using the Rodrigues formula [Eq. (10)] of the generalized Clifford–Hermite polynomials. Note that the parameters of the generalized Clifford–Hermite ﬁlters are the scale σ of the Gaussian, the derivative order n and the order k of the left solid inner spherical monogenic. Moreover, their Fourier transform in spherical coordinates ξ = ρη, ρ = |ξ|, η ∈ Sm−1 , is given by

F [Dσn,k ](ξ) =

2 2 ik+n σ n n n+k η P (η) ρ exp − σ 4ρ . k γn,k 2n/2

156

Fred Brackx et al.

Hence, the generalized Clifford–Hermite ﬁlters are polar separable, this means that their Fourier transform is expressed as the product of a spatial frequency tuning function and an orientation tuning function. Daugman (1983) has demonstrated the importance of polar separable ﬁlters.

5.2.4. Connection With the Clifford–Hermite Continuous Wavelet Transform The continuous wavelet transform (CWT) is a signal analysis technique suitable for nonstationary, inhomogeneous signals for which Fourier analysis is inadequate (see Chui, 1992, and Daubechies, 1992). In the 1D case, it is given by the integral transform

f (x) −→ F(a, b) =

+∞

−∞

c

ψa,b (x)

f (x) dx.

The kernel function of this integral transform is the dilate translate of a mother wavelet ψ:

1 x−b , a > 0, b ∈ R, ψa,b (x) = √ ψ a a where the parameter b indicates the position of the wavelet, whereas the parameter a governs its frequency. The analyzing wavelet function ψ is a quite arbitrary L2 -function that is well localized in both the time domain and the frequency domain. Moreover, it must satisfy the admissibility condition:

Cψ ≡ 2π

+∞

−∞

|F [ψ](ξ)|2 dξ < ∞. |ξ|

The constant Cψ is called the admissibility constant. In the case where ψ is also in L1 , this admissibility condition implies that ψ has “zero momentum,” that is,

+∞

−∞

ψ(x) dx = 0,

which can be fulﬁlled only if ψ is an oscillating function, explaining the terminology “wavelet.” A wavelet is a function that oscillates like a wave in a limited portion of time or space and vanishes outside of it:—it is a wavelike but localized function. Higher-dimensional CWTs, still enjoying the same properties as in the 1D case, traditionally originate as tensor products of 1D phenomena, an

The Fourier Transform in Clifford Analysis

157

exception being the 2D CWT incorporating rotation as well as translation and dilation (see Antoine, Murenzi, and Vandergheynst , 1996). Brackx and Sommen (2001), introduced the so-called generalized Clifford–Hermite CWT. The building blocks in the construction of this multidimensional CWT are the generalized Clifford–Hermite polynomials. The mother wavelets of this transform take the form

2 2 ψn,k (x) = exp − |x|2 Hn,k (x) Pk (x) = (−1)n ∂xn exp − |x|2 Pk (x) , n = 1, 2, . . . Introducing the continuous family of wavelets a,b,s ψn,k (x)

=

1 am/2

s ψn,k

s(x − b)s a

s,

the generalized Clifford–Hermite CWT is deﬁned by

Fn,k (a, b, s)

† a,b,s = ψn,k (x) f (x) dV(x) Rm

=

1 am/2

†

2 s(x−b)s s(x−b)s P s exp − |x−b| H s f (x) dV(x), k n,k a a 2a2

Rm

(54) with f ∈ L2 Rm , dV(x) the signal to be analyzed, a ∈ R+ the dilation parameter, b ∈ Rm the translation parameter, and s ∈ SpinR (m) the spinorrotation parameter. The original signal may be reconstructed from its transform Fn,k (a, b, s) by the inverse transformation,

1 f (x) = Cn,k

SpinR (m) Rm

0

+∞

a,b,s

ψn,k (x) Fn,k (a, b, s)

da am+1

dV(b) ds, (55)

with m

Cn,k = (2π)

(n + k − 1)! 2

the admissibility constant.

Sm−1

|Pk (η)|2 dS(η) < +∞

158

Fred Brackx et al.

√ Putting σ = 2μ in the generalized Clifford–Hermite coefﬁcients of Subsection 5.2.3 σ cn,k (p) =

=

Rm

Dσn,k (p − x) f (x) dV(x)

2(2k+m)/2 σ 2k+m γn,k

Rm

√

|x−p|2 Hn,k σ2 (x − p) exp − σ 2

× Pk x − p f (x) dV(x), we obtain √

2μ cn,k (p)

=

1 μk+m γn,k

|x−p|2

exp − 2μ2 Rm

Hn,k

x−p x−p μ Pk μ f (x) dV(x). (56)

Leaving the spinor-rotations in the generalized Clifford–Hermite CWT out of consideration, by comparing Eqs. (54) and (56), the connection between the generalized Clifford–Hermite transform and the ditto CWT is clear: σ plays the role of the dilation parameter a, p plays the role of the translation parameter b, and the translated ﬁlter functions Dσn,k (p − x) play the role of a,b,s the wavelets ψn,k . However, differences exist between the two transforms. The wavelet translation parameter b is continuous, whereas the parameter p in the Hermite transform is discrete. Furthermore, the reconstruction of the ﬁltered signal in the Hermite transform

fFil (x) =

∞ ∞

σ cj, (p) Qσj, (x − p)

(57)

j=0 =0 p∈P

depends on σ, which is not the case for the CWT. As mentioned previously, σ is an important parameter for practical applications. In the CWT, the quality of the signal reconstruction [Eq. (55)] depends on the mother wavelet ψn,k (x); in other words, it depends on the order of the derivative of the Gaussian modulated by Pk and of the order k of the left inner spherical monogenic. In the inverse Hermite transform [Eq. (57)], we sum over all these orders.

The Fourier Transform in Clifford Analysis

159

5.3. The Two-Dimensional Clifford–Gabor Filters 5.3.1. The Classical One-Dimensional Gabor Filters The Gabor ﬁlter is one of the most prominent tools for local spectral image processing and analysis. Gabor ﬁrst introduced the ﬁlters in the ﬁeld of 1D signal processing in 1946 for a joint time-frequency analysis. Gabor ﬁlters have the primary advantage of being simultaneously optimally localized in the spatial and in the frequency domain. Hence, spatial and frequency properties are optimally analyzed at the same time by Gabor ﬁlters (see also Subsection 5.3.3). Gabor ﬁlters also give access to the local phase of a signal. A close correspondence has been shown between the local structure of a signal and its local phase. Furthermore, certain regions in the human visual cortex can be modeled as Gabor ﬁlters (see Section 5.1). Hence, Gabor ﬁlters conform well to the human visual system’s capabilities. Gabor ﬁlters have been successfully applied to different image processing and analysis tasks such as texture segmentation (see Bülow, 1999; Weldon, Higgins, and Dunn, 1996), edge detection, and local phase and frequency estimation for image matching.

a. One-dimensional complex Gabor filters. In this subsection we consider the classical Fourier transform with the angular frequency in the kernel function 1 F [ f ](u) = √ 2π

+∞

−∞

f (x) exp (−i2πux) dx.

Complex Gabor ﬁlters are closely related to Fourier analysis in the following way. They are linear shift-invariant (LSI) ﬁlters; hence, they can be applied by simply convolving the signal with the impulse response of the ﬁlter. The impulse response h of a complex Gabor ﬁlter is the complex conjugated integral kernel of the classical Fourier transform F of some frequency u∗ multiplied by a Gaussian g centered at the origin; that is,

h(x) = g(x) exp (i2πu∗ x) with

x2 g(x) = N exp − 2σ 2 .

(58)

Whereas Gabor ﬁlters analyze the local spectral properties of a signal, the Fourier transform decomposes a signal into its global spectral components. Hence, the Fourier transform can be considered the basis on which Gabor ﬁlters were introduced. The parameters of the complex Gabor ﬁlter are the normalization constant, N, the center frequency, u∗ , and the variance or standard

160

Fred Brackx et al.

deviation, σ, of the Gaussian. Normally, N is chosen such that the Gaussian is amplitude normalized

g(x) = 1

+∞ −∞

g(x) dx = 1,

which implies N = √ 1 . 2πσ Sometimes other parameterizations of the complex Gabor ﬁlter than the one given above are used, viz.

h(x) = g(x) exp (iξ ∗ x) = g(x) exp

icx σ

.

Here ξ ∗ = 2πu∗ is the angular frequency and c = ξ ∗ σ is the oscillation parameter. The transfer function of a complex Gabor ﬁlter is a shifted Gaussian:

1 H(u) := F [h](u) = √ exp − 2π2 σ 2 (u − u∗ )2 . 2π In terms of the angular frequency, ξ = 2πu, this becomes

2 1 H(ξ) = √ exp − σ2 (ξ − ξ ∗ )2 . 2π Hence, Gabor ﬁlters are bandpass ﬁlters. The majority of energy of the Gabor ﬁlter is centered around the frequency, u∗ , in the positive half of the frequency domain. Analogously, the deﬁnition of the 2D complex Gabor ﬁlters is based on the classical 2D Fourier transform.

b. One-dimensional real Gabor filters. In addition to complex Gabor ﬁlters, real Gabor ﬁlters also appear in the literature (see Rivero-Moreno and Bres, 2003a, b). Naturally they are obtained as the real and imaginary part of the complex Gabor ﬁlters introduced above. Hence, the impulse responses of these real Gabor ﬁlters take the form cx hc (x) = g(x) cos (2πu∗ x) = g(x) cos (ξ ∗ x) = g(x) cos σ and

hs (x) = g(x) sin (2πu∗ x) = g(x) sin (ξ ∗ x) = g(x) sin with g the Gaussian given by Eq. (58).

cx σ

,

161

The Fourier Transform in Clifford Analysis

Their associated transfer functions are

σ exp (−σ 2 2π2 (u + u∗ )2 ) Hc/s (u) := F [hc/s ](u) = N 2 ± exp (−σ 2 2π2 (u − u∗ )2 ) , where the plus sign and the minus sign, correspond, respectively, with the cosine and the sine, Gabor ﬁlter. The above expression can be rewritten in terms of the angular frequency as

σ (ξ+ξ ∗ )2 (ξ−ξ ∗ )2 2 2 Hc/s (ξ) = N ± exp −σ . exp −σ 2 2 2

5.3.2. Different Types of Two-Dimensional Gabor Filters a. Quaternionic Gabor filters. In Bülow (1998, 1999), quaternionic Gabor ﬁlters were constructed and applied to the problems of disparity estimation and texture segmentation. The impulse response hq of a quaternionic Gabor ﬁlter is a Gaussian windowed kernel function of the quaternionic Fourier transform mentioned in Section 4.1: hq (x) = g(x) exp (i2πu∗1 x1 ) exp ( j2πu∗2 x2 ) with

g(x) = N exp

x2 +(x )2 − 1 2σ 2 2

.

(59)

The parameter is the aspect ratio. In the quaternionic frequency domain, these Gabor ﬁlters are shifted Gaussians:

q

q

q

H (u) := F [h ](u) = exp

−2π2 σ 2

(u1 − u∗1 )2

+

(u2 −u∗2 )2 2

.

b. Gabor filters of Ebling and Scheuermann. Ebling and Scheuermann, (2004) introduce 2D and 3D Gabor ﬁlters based on their Clifford–Fourier transforms (see Subsection 4.5.3) and use them to describe local patterns in ﬂow ﬁelds. The impulse responses he of their 2D Gabor ﬁlters take the

162

Fred Brackx et al.

form

he (x) = g(x ) exp (e12 (ξ1∗ x1 + ξ2∗ x2 )), with x a rotated version of x and g the Gaussian given by Eq. (59).

5.3.3. The Two-Dimensional Clifford–Gabor Filters a. Definition. As we dispose of a closed form for the integral kernel of our 2D Clifford–Fourier transform (see Subsection 4.5.1), we are now able to deﬁne a new type of 2D Gabor ﬁlters (see Brackx, De Schepper, and Sommen, 2006a). Deﬁnition 5.1 The 2D Clifford–Gabor ﬁlters G ± are linear shift-invariant ﬁlters with impulse response given by

h± (x) = g(x) exp ±(x ∧ ξ ∗ ) = g(x) cos (x1 ξ2∗ − x2 ξ1∗ ) ± e12 g(x) × sin (x1 ξ2∗ − x2 ξ1∗ ), where g is the Gaussian given by

g(x) =

1 |x|2 . exp − 2 2σ 2πσ 2

The parameters of the Clifford–Gabor ﬁlters are the angular frequency ξ ∗ and the variance σ, which determines the scale of the Gaussian envelope. It turns out that both types of Clifford–Gabor ﬁlters G ± have the same transfer function. Proposition 5.1 The transfer function of the Clifford–Gabor ﬁlter G ± is given by

H ± (ξ) := FH± [h± ](ξ) =

2 1 exp − σ2 |ξ − ξ ∗ |2 . 2π

Proof. By means of the modulation theorem of the 2D Clifford–Fourier transform (see Proposition 4.9 (iv)), we have

H ± (ξ) = FH± exp x ∧ (±ξ ∗ ) g(x) (ξ) = FH± [ g(x)](ξ − ξ ∗ ) = F [ g(x)](ξ − ξ ∗ ), since the Gaussian g(x) is a radial function.

The Fourier Transform in Clifford Analysis

163

The desired result now follows from

1 |ξ|2 2 exp −σ 2 . F [ g(x)](ξ) = 2π Note that, similar to the classical case, the transfer functions H ± (ξ) are shifted Gaussians, which implies that the Clifford–Gabor ﬁlters G ± are bandpass ﬁlters. Remark 5.2 In a similar way, one obtains

FH± [h∓ ](ξ) =

2 1 exp − σ2 |ξ + ξ ∗ |2 . 2π

b. Relationship with other Gabor filters. Using the properties of the Clifford numbers P± introduced in Subsection 4.5.1, we can derive an explicit connection between the 2D Clifford–Gabor ﬁlters G ± and the classical complex Gabor ﬁlters (see Subsection 5.3.1). By a straightforward computation, we obtain h± (x) = P+ h(∓e12 x) + P− h(±e12 x), where

h(x1 , x2 ) = g(x) exp i(ξ1∗ x1 + ξ2∗ x2 ) is the classical 2D Gabor ﬁlter in case of a symmetric Gaussian. Naturally the Clifford–Gabor ﬁlters G ± can also be expressed in terms of classical 1D Gabor ﬁlters which, for the sake of clarity, are now denoted with a superscript specifying the angular frequency: ∗

hξ (x) = g(x) exp (iξ ∗ x). It is easily seen that ∗

∗

∗

∗

h+ (x) = P+ h−ξ2 (x1 ) hξ1 (x2 ) + P− hξ2 (x1 ) h−ξ1 (x2 ). A similar result holds for the impulse response of G − . Finally, let us look for a relationship between the Clifford–Gabor ﬁlters G ± and the Gabor ﬁlters of Ebling and Scheuermann in case of a symmetric Gaussian (see Subsection 5.3.2).

164

Fred Brackx et al.

We have

he (x) = g(x) exp e12 (ξ1∗ x1 + ξ2∗ x2 ) = h± (∓e12 x). c. Localization in the spatial and in the frequency domain. An often-cited property of Gabor ﬁlters is their optimal simultaneous localization in the spatial and the frequency domain. This makes them suitable for local frequency analysis. The notion “optimal simultaneous localization” is formalized by the uncertainty principle, which in its most cited form states that a nonzero function and its Fourier transform cannot both be sharply localized. The uncertainty principle appeared in 1927 under the name Heisenberg inequality in the ﬁeld of quantum mechanics in Heisenberg’s paper (1927). However, it also has a useful interpretation in classical physics; namely, it expresses a limitation on the extent to which a signal can be both timelimited and band-limited. This aspect of the uncertainty principle was already expounded by Wiener in a lecture in Göttingen in 1925. Unfortunately, no written record of this lecture seems to have survived, apart from the nontechnical account in Wiener’s autobiography (1956). The uncertainty principle became really fundamental in the ﬁeld of signal processing after the publication of Gabor’s famous 1946 article. For a 1D complex-valued signal f , the uncertainty principle takes the form x ξ ≥

1 . 2

(60)

Here x denotes the width or spatial uncertainty of f , deﬁned as the square root of the variance of the energy distribution of f

+ +∞

x2 f (x) f c (x) dx . (x) = −∞ + +∞ c (x) dx f (x) f −∞ 2

Analogously, the bandwidth ξ is given by

+ +∞ 2

(ξ) =

c 2 −∞ ξ F [ f ](ξ) F [ f ](ξ) dξ . + +∞ c −∞ F [ f ](ξ) F [ f ](ξ) dξ

The functions that minimize the inequality (60) are the complex Gabor ﬁlters. Hence, depending on the parameter σ, the Gabor ﬁlters are better localized in position or frequency space but they always exhibit the best possible joint localization.

165

The Fourier Transform in Clifford Analysis

Daugman extended the uncertainty principle to 2D complex-valued ﬁlters or signals (see Daugman, 1985):

x1 x2 ξ1 ξ2 ≥

1 , 4

(61)

where x1 is deﬁned by

+

x12 f (x1 , x2 ) f c (x1 , x2 ) dV(x) . c R2 f (x1 , x2 ) f (x1 , x2 ) dV(x)

(x1 ) = R+

2

2

The uncertainties x2 , ξ1 , and ξ2 are deﬁned analogously. It can be shown that 2D complex Gabor ﬁlters achieve the minimum product of uncertainties; that is,

x1 x2 ξ1 ξ2 =

1 . 4

Let us now consider 2D Clifford algebra–valued functions:

f : R2 −→ C2 x = (x1 , x2 ) −→ f (x) = f (x1 , x2 ) = f0 (x) + f1 (x) e1 + f2 (x) e2 + f12 (x) e12 , with fα : R2 −→ C, α = 0, 1, 2, 12. First, we extend the deﬁnition of the uncertainties to these Clifford algebra–valued functions as follows:

+ 2

(x1 ) = and

+ (ξ1 )2 =

x12 [ f (x) f † (x)]0 dV(x) † R2 [ f (x) f (x)]0 dV(x)

R+2

† ξ12 [FH± [ f ](ξ) FH± [ f ](ξ) ]0 dV(ξ) .

† + R2 [FH± [ f ](ξ) FH± [ f ](ξ) ]0 dV(ξ)

R2

Analogous deﬁnitions hold for x2 and ξ2 . For complex-valued signals, the real-valued energy distribution is given by |f |2 = ff c . For Clifford algebra–valued signals, this is given by

| f (x)|2 = [ f (x) f † (x)]0 = | f0 (x)|2 + | f1 (x)|2 + | f2 (x)|2 + | f12 (x)|2 .

166

Fred Brackx et al.

Hence, the uncertainty relation for 2D Clifford algebra–valued signals is identical to Daugman’s relation [Eq. (61)]. In case of the Clifford–Gabor ﬁlters G ± , we have

±

∗

2

∗

|h (x)| = g(x) exp (±(x ∧ ξ )) g(x) exp (∓(x ∧ ξ )) = [(g(x))2 ]0 =

0

2

1 , exp − |x| σ2 4π2 σ 4

where we have used the fact that

exp (±(x ∧ ξ ∗ ))

†

= exp (∓(x ∧ ξ ∗ )).

Hence, for G ± we obtain

2 dV(x) x12 exp − |x| σ2 σ2

= . (x1 )2 = + |x|2 2 exp − dV(x) 2 2 R σ +

R2

Furthermore, we have

FH± [h± ](ξ)2 = 1 exp (−σ 2 |ξ − ξ ∗ |2 ), 4π2 which yields

+

(ξ1 )2 =

2 R2 ξ 1

+

R2

exp (−σ 2 |ξ − ξ ∗ |2 ) dV(ξ)

exp (−σ 2 |ξ

− ξ ∗ |2 )

dV(ξ)

=

1 . 2σ 2

Summarizing, the uncertainties of the Clifford–Gabor ﬁlters G ± are given by

σ x1 = x2 = √ 2

and

1 ξ1 = ξ2 = √ , 2σ

which implies

x1 x2 ξ1 ξ2 =

1 . 4

Hence, as in the classical setting, the Clifford–Gabor ﬁlters G ± are jointly optimally localized in the spatial and in the frequency domain.

The Fourier Transform in Clifford Analysis

167

6. THE CYLINDRICAL FOURIER TRANSFORM In Section 4 we deﬁned the Clifford–Fourier transform as an operator exponential. In only the 2D case, we were able to write it as a standard integral transform with a closed form for the integral kernel function— namely, the exponential of the wedge product of the old and new vector variable. Now taking a generalization of this speciﬁc 2D Clifford–Fourier kernel as a new integral kernel, we are able to introduce a new multidimensional Fourier transform. As the phase of the new integral kernel takes constant values on coaxial cylinders, we call this new Fourier transform the cylindrical Fourier transform.

6.1. Definition The cylindrical Fourier transform is obtained by substituting for the standard inner product in the classical exponential Fourier kernel a wedge product of the old and new vector variable as argument. Deﬁnition 6.1 The cylindrical Fourier transform of a function f is given by

1 Fcyl [ f ](ξ) = (2π)m/2 with exp(x ∧ ξ) =

∞

r=0

Rm

exp (x ∧ ξ) f (x) dV(x)

(x∧ξ)r r! .

Remark 6.1 As is to be expected, the Clifford–Fourier transform and the cylindrical Fourier transform reduce to the same integral transform in the 2D case as follows:

FH± [ f ](ξ) = Fcyl [ f ](∓ξ). In the sequel, we often appeal to the following basic formulas. Lemma 6.1 For all x, t ∈ Rm , one has

(x ∧ t)2 = −|x ∧ t|2 = (< x, t >)2 − |x|2 |t|2 . Proof. First, by deﬁnition of the Clifford norm, we have

|x ∧ t|2 = [(x ∧ t)† (x ∧ t)]0 = −[(x ∧ t)2 ]0 . Next, let us decompose t as t = t + t⊥ , where t , respectively t⊥ , denotes the component of t which is parallel, respectively perpendicular, to x. This

168

Fred Brackx et al.

implies that

x ∧ t = x ∧ t⊥ = xt⊥ = −t⊥ ∧ x = −t⊥ x , which in turn leads to

(x ∧ t)2 = xt⊥ xt⊥ = −t⊥ x2 t⊥ = |x|2 t2⊥ = −|x|2 |t⊥ |2 . We thus have proved that (x ∧ t)2 is scalar valued and consequently we obtain that |x ∧ t|2 = −(x ∧ t)2 . We now prove the second part of the statement, namely,

|x ∧ t|2 = |x|2 |t|2 − (< x, t >)2 . Using spherical coordinates t = |t| η with η ∈ Sm−1 , we ﬁnd |x ∧ t|2 = |t|2 |x ∧ η|2 . Now, we decompose x into its components parallel and perpendicular to η:

x = x + x⊥ = < x, η > η + x⊥ . By a similar reasoning as above, this yields

(x ∧ η)2 = x⊥ η x⊥ η = −x2⊥ η2 = x2⊥ = −|x⊥ |2 . Hence, we obtain

|x ∧ η|2 = −[(x ∧ η)2 ]0 = −(x⊥ )2 . Moreover, we have consecutively

(x⊥ )2 = (x− < x, η > η)(x− < x, η > η) = −|x|2 − < x, η > (ηx + xη) − (< x, η >)2 = −|x|2 − < x, η > (−2 < x, η >) − (< x, η >)2 = −|x|2 + (< x, η >)2 . So, ﬁnally we indeed ﬁnd

|x ∧ t|2 = |t|2 |x|2 − (< x, η >)2 = |x|2 |t|2 − (< x, t >)2 .

The Fourier Transform in Clifford Analysis

169

This result enables us to rewrite the integral kernel of our newly introduced Fourier transform in terms of the sine and cosine function. Proposition 6.1 The kernel of the cylindrical Fourier transform can be rewritten as

exp (x ∧ ξ) = cos (|x ∧ ξ|) + I sin (|x ∧ ξ|) = cos (|x ∧ ξ|) + x ∧ ξ sinc(|x ∧ ξ|), where I =

x∧ξ |x∧ξ|

and sinc(x) :=

sin (x) x

the unnormalized sinc function.

Proof. Splitting the deﬁning series expansion of exp (x ∧ ξ) into its even and odd part yields

exp (x ∧ ξ) =

∞ (x ∧ ξ)r r=0

r!

=

∞ (x ∧ ξ)2 =0

(2)!

+

∞ (x ∧ ξ)2+1 =0

(2 + 1)!

∞ ∞ |x ∧ ξ|2 (x ∧ ξ) |x ∧ ξ|2+1 = + (−1) (−1) (2)! |x ∧ ξ| (2 + 1)! =0

=0

= cos (|x ∧ ξ|) + I sin (|x ∧ ξ|). Remark 6.2 1. From Proposition 6.1 it is clear that the cylindrical Fourier kernel is parabivector valued. 2. As I satisﬁes I 2 = −1, it can be viewed as a kind of imaginary unit. Let us end this subsection by explaining why we have chosen the name cylindrical for our new Fourier transform. To that end, we rewrite Lemma 6.1 as follows:

,ξ)2 |x ∧ ξ|2 = |x|2 |ξ|2 − (< x, ξ >)2 = |x|2 |ξ|2 1 − cos (x, ,ξ) . = |x|2 |ξ|2 sin (x, 2

,ξ) is Hence, for ξ ﬁxed, the “phase” |x ∧ ξ| is constant if and only if |x| sin (x, constant. In other words, for ξ ﬁxed, the phase |x ∧ ξ| is constant on coaxial cylinders w.r.t. ξ (Figure 2). For comparison, in the case of the classical Fourier transform, for ξ ﬁxed ,ξ) is constant if and only if |x| cos (x, ,ξ) the phase < x, ξ > = |x||ξ| cos (x,

170

Fred Brackx et al.

x x sin(x,␰)

␰

FIGURE 2 to ξ.

For fixed ξ, the phase |x ∧ ξ| is constant on coaxial cylinders with regard

x

x cos(x, ␰)

␰

FIGURE 3 For fixed ξ, the phase < x, ξ > is constant on planes perpendicular to ξ.

is constant. Hence, for ξ ﬁxed, the level surfaces of the traditional Fourier kernel are planes perpendicular to that ﬁxed vector (Figure 3).

6.2. Properties Let us start by showing that the cylindrical Fourier transform Fcyl [ f ] is well deﬁned for each integrable function f ∈ L1 Rm , dV(x) .

The Fourier Transform in Clifford Analysis

Theorem 6.1 Let f ∈ L1 Rm , dV(x) . m C0 R , dV(x) and moreover

Fcyl [ f ]

∞

Then

171

Fcyl [ f ] ∈ L∞ Rm , dV(x) ∩

m/2 2 f . ≤2 1 π

Proof. Taking into account Proposition 6.1, the proof is similar to the one of Theorem 4.2. Next, we collect some operational formulas satisﬁed by the cylindrical Fourier transform. Proposition 6.2 Let f , g ∈ L1 Rm , dV(x) . The cylindrical Fourier transform satisﬁes (i) the linearity property

Fcyl [ f λ + gμ] = Fcyl [ f ] λ + Fcyl [ g] μ

λ, μ ∈ Cm

(ii) the reﬂection property

Fcyl [ f (−x)](ξ) = Fcyl [ f (x)](−ξ) (iii) Hermitean conjugation

†

1 = (2π)m/2

Fcyl [ f ](ξ)

Rm

f † (x) exp (ξ ∧ x) dV(x)

(iv) the change of scale property

ξ 1 for a ∈ R+ Fcyl [ f (ax)](ξ) = m Fcyl [ f (x)] a a (v) the differentiation rule

Fcyl [∂x [ f (x)]](ξ) = −ξ Fcyl [ f (x)](−ξ) + (2 − m) ξ × with sinc(x) :=

sin (x) x

Rm

sinc(|x ∧ ξ|) f (x) dV(x)

the unnormalized sinc function.

1 (2π)m/2

172

Fred Brackx et al.

(vi) the multiplication rule

Fcyl [x f (x)](ξ) = −∂ξ [Fcyl [ f (x)](−ξ)] + (2 − m)

1 (2π)m/2

×

Rm

sinc(|x ∧ ξ|) x f (x) dV(x)

(vii) the transfer formula

Rm

†

Fcyl [ f ](ξ)

g(ξ) dV(ξ) =

Rm

f † (ξ) Fcyl [ g](ξ) dV(ξ)

(viii) the rotation rule

Fcyl [ f (sxs)](ξ) = s Fcyl [sf (x)](sξs) with s ∈ SpinR (m). Proof. (i) Trivial. (ii) Straightforward. (iii) This result can be proved taking into account that x ∧ ξ = ξ ∧ x, which implies that

exp (x ∧ ξ) = cos (|x ∧ ξ|) +

ξ∧x |x ∧ ξ|

sin (|x ∧ ξ|) = exp (ξ ∧ x).

(iv) By means of the substitution u = ax, we have

Fcyl [ f (ax)](ξ) =

1 (2π)m/2

Rm

1 1 = m a (2π)m/2

exp (x ∧ ξ) f (ax) dV(x)

Rm

exp u ∧

ξ 1 . = m Fcyl [ f (x)] a a

ξ a

f (u) dV(u)

173

The Fourier Transform in Clifford Analysis

(v) First, by means of the Clifford–Stokes theorem (see Theorem 2.1), we obtain

1 Fcyl [∂x [ f (x)]](ξ) = exp (x ∧ ξ) ∂x [ f (x)] dV(x) (2π)m/2 Rm 1 = exp (x ∧ ξ) dσx f (x) (2π)m/2 ∂Rm 1 − [exp (x ∧ ξ)]∂x f (x) dV(x) (2π)m/2 Rm 1 =− [exp (x ∧ ξ)]∂x f (x) dV(x). (2π)m/2 Rm Next, a straightforward computation yields

x∧ξ sin (|x ∧ ξ|) ∂x [exp (ξ∧x)] = ∂x cos (|x ∧ ξ|) − |x ∧ ξ| = − sin (|x∧ξ|) ∂x [|x∧ξ|]− +

sin (|x∧ξ|) |x∧ξ|2

cos (|x∧ξ|) |x∧ξ|

∂x [|x∧ξ|](x∧ξ) −

∂x [|x∧ξ|](x∧ξ)

sin (|x ∧ ξ|) |x ∧ ξ|

∂x [x ∧ ξ] (62)

and

∂x [x ∧ ξ] = ∂x [xξ+ < x, ξ >] = −mξ + ξ = (1 − m) ξ.

(63)

Furthermore, in view of Lemma 6.1 we also have

∂x [|x ∧ ξ|2 ] = ∂x [|x|2 |ξ|2 − (< x, ξ >)2 ] = 2x |ξ|2 − 2 < x, ξ > ξ = −2ξ (ξx+ < x, ξ >) = −2ξ (ξ ∧ x) = 2ξ (x ∧ ξ). Combining the above result with

∂x [|x ∧ ξ|2 ] = 2 |x ∧ ξ| ∂x [|x ∧ ξ|], we obtain

∂x [|x ∧ ξ|] = ξ

(x ∧ ξ) |x ∧ ξ|

.

(64)

174

Fred Brackx et al.

Inserting Eqs. (63) and (64) in (62) and using (x ∧ ξ)2 = −|x ∧ ξ|2 yields

∂x [exp (ξ ∧ x)] = ξ cos (|x ∧ ξ|) − ξ ×ξ

(x ∧ ξ)

sin (|x ∧ ξ|) − (2 − m)

|x ∧ ξ|

sin (|x ∧ ξ|) |x ∧ ξ|

= ξ exp (ξ ∧ x) − (2 − m) ξ

sin (|x ∧ ξ|) |x ∧ ξ|

.

Taking the Hermitean conjugate of the above result, we obtain

[exp (x ∧ ξ)]∂x = exp (x ∧ ξ) ξ + (m − 2) ξ = ξ exp (ξ ∧ x) + (m − 2) ξ

sin (|x ∧ ξ|) |x ∧ ξ| sin (|x ∧ ξ|) |x ∧ ξ|

.

Hence, we ﬁnally have

1 Fcyl [∂x [ f (x)]](ξ) = −ξ (2π)m/2 + (2 − m) ξ

Rm

exp (ξ ∧ x) f (x) dV(x)

1 (2π)m/2

sinc(|x ∧ ξ|) f (x) dV(x)

Rm

1 = −ξ Fcyl [ f (x)](−ξ) + (2 − m) ξ (2π)m/2 × sinc(|x ∧ ξ|) f (x) dV(x). Rm

(vi) As

∂ξ [exp (ξ ∧ x)] = − exp (x ∧ ξ) x − (m − 2) x

sin (|ξ ∧ x|) |ξ ∧ x|

we indeed obtain

1 Fcyl [xf (x)](ξ) = exp (x ∧ ξ) x f (x) dV(x) (2π)m/2 Rm 1 =− ∂ξ [exp (ξ ∧ x)] f (x) dV(x) (2π)m/2 Rm

,

The Fourier Transform in Clifford Analysis

1 + (2 − m) (2π)m/2

175

Rm

sinc(|ξ ∧ x|) x f (x) dV(x)

1 = −∂ξ [Fcyl [ f (x)](−ξ)] + (2 − m) (2π)m/2 × sinc(|x ∧ ξ|) x f (x) dV(x). Rm

(vii) Similar to the proof of the transfer formula of the 2D Clifford–Fourier transform (see Proposition 4.9). (viii) See the proof of the rotation rule for the 2D Clifford–Fourier transform (Proposition 4.9). Remark 6.3 1. The differentiation and multiplication rules, once more, indicate that the 2D case is special. 2. Note that the shift theorem, the modulation theorem, and the convolution theorem (see Proposition 4.9 for the 2D case) do not hold in the general m-dimensional case, since

(u ∧ ξ) (x ∧ ξ) = ±(x ∧ ξ) (u ∧ ξ) for m ≥ 3.

6.3. Cylindrical Fourier Spectrum of the L2 -Basis Consisting of Generalized Clifford–Hermite Functions In this ﬁnal subsection, we aim at calculating the cylindrical Fourier spectrum of the L2 -basis

φs,k,j (x) =

√ 2m/4 (j) √ |x|2 H ( 2x) P ( 2x) exp − s,k 2 ; k (γs,k )1/2 + s, k ∈ N, j ≤ dim(M (k))

consisting of generalized Clifford–Hermite functions.Note that these basis elements belong to the space S(Rm ) ⊂ L1 Rm , dV(x) . Hence, their cylindrical Fourier image should be a bounded and continuous function (see Theorem 6.1). The calculation method is based on the Funk-Hecke theorem in space (see Theorem 2.4), which needs dimension m > 2. Moreover, introducing spherical coordinates

x = rω,

ξ = ρη,

r = |x|,

ρ = |ξ|,

ω, η ∈ Sm−1 ,

176

Fred Brackx et al.

denoting < ω, η > = tη and taking into account Lemma 6.1 and Proposition 6.1, we can rewrite the cylindrical Fourier kernel as follows:

exp (x ∧ ξ) = cos rρ 1 − tη2 − ξ ∧ x sinc rρ 1 − tη2

= cos rρ 1 − tη2 − rρ tη sinc rρ 1 − tη2

− rρ η ω sinc rρ 1 − tη2 .

(65)

Let us denote these three terms in the decomposition of the kernel by A, B, and C—we put

A = cos rρ 1 − tη2 ,

B = rρ tη sinc rρ 1 − tη2 ,

C = rρ η ω sinc rρ 1 − tη2 .

Note that A and C are even functions in tη , while B is an odd function in tη .

6.3.1. The Cylindrical Fourier Spectrum of the Gaussian (= φ0,0,1 ) By means of the above decomposition of the integral kernel, we have that

Fcyl exp

2 − |x|2

1 (ξ) = (2π)m/2 − −

1 (2π)m/2 1 (2π)m/2

2 A exp − |x|2 dV(x)

Rm

Rm

Rm

2 B exp − |x|2 dV(x)

2 C exp − |x|2 dV(x).

In view of Corollary 2.1 to the Funk-Hecke theorem, the integrals containing the B- and C-term of the kernel decomposition reduce to zero. Furthermore, applying the Funk-Hecke theorem in space (Theorem 2.4), we obtain

2 Fcyl exp − |x|2 (ξ) =

1 (2π)m/2

Rm

2 cos rρ 1 − tη2 exp − r2 dV(x)

The Fourier Transform in Clifford Analysis

=

177

+∞ 2 Am−1 m−1 r exp − dr r 2 (2π)m/2 0 1

2 (m−3)/2 2 × cos rρ 1 − t (1 − t ) dt . −1

Next, taking into account the series expansion of the cosine function

cos (u) =

∞ (−1) =0

(2)!

u2 ,

(66)

this becomes

2 (ξ) Fcyl exp − |x|2 +∞ ∞ 2 Am−1 (−1) 2 2+m−1 r ρ exp − 2 r dr = (2)! (2π)m/2 0 =0 1 2 (2+m−3)/2 (1 − t ) dt × −1

=

Am−1 (2π)m/2

∞ (−1) =0

(2)!

√π 2+m−1

2 m

2+m/2−1 + 2+m 2 2

ρ2

∞ 2+m−1 (−1) ρ2 2

= (2)! 2 m−1 1

2

=0

m−1 1 |ξ|2 = 1 F1 ; ;− . 2 2 2

Recall that 1 F1 (a; c; z) denotes Kummer’s function, also called the conﬂuent hypergeometric function, which is deﬁned by the following inﬁnite series, and all its analytic continuations: ∞ (a) z =0

(c) !

=

∞ (a + ) (c) z =0

(a) (c + ) !

with (α) = α(α + 1) . . . (α + − 1) = also Subsection 2.4.1).

,

(α+) (α)

c = 0, −1, −2, . . . Pochhammer’s symbol (see

Conclusion 6.1 The cylindrical Fourier image of the Gaussian is the radial function given by (see also Figure 4):

178

Fred Brackx et al.

1

m 3 m 4 m 5 m 6

0.8 0.6 0.4 0.2 0 2

4

6

0.2

|␰|

8

10

0.4 0.6

2 FIGURE 4 The cylindrical Fourier spectrum of the Gaussian exp − |x|2 for m = 3, m = 4, m = 5, and m = 6.

Fcyl exp

2 − |x|2

m − 1 1 |ξ|2 (ξ) = 1 F1 ; ;− 2 2 2 m 1 |ξ|2 |ξ|2 exp − 2 . = 1 F1 1 − ; ; 2 2 2

Note that in the case where the dimension m is even and hence 1 − m2 ∈ −N, the 2 m 1 |ξ| Kummer function 1 F1 1 − 2 ; 2 ; 2 reduces to the classical Hermite polyno mials [Eq. (17)] of even degree associated with the weight function exp −u2 , since

1 2 (−1)n (2n)! 1 F1 −n; ; u and H2n (u) = n! 2 3 (−1)n (2n + 1)! 2u 1 F1 −n; ; u2 . H2n+1 (u) = n! 2

6.3.2. The Cylindrical Fourier Spectrum of the Gaussian Multiplied With the Clifford Vector (= φ1,0,1 ) Let us now calculate the cylindrical Fourier transform of the basis function φ1,0,1 (x) which is given, up to constants, by x exp − |x|2 . Now, Corollary 2.1 implies that the integral containing the A-term of the kernel 2

179

The Fourier Transform in Clifford Analysis

decomposition in Eq. (65) reduces to zero. By means of the Funk-Hecke theorem in space, we thus obtain the following:

2 Fcyl exp − |x|2 x (ξ)

1 1 |x|2 =− x dV(x) − B exp − 2 m/2 m (2π) (2π)m/2 R

2 × C exp − |x|2 x dV(x) Rm

2 1 2 r 2 ω dV(x) ρ r exp − sinc rρ 1 − t =− t η η 2 (2π)m/2 Rm 2 1 2 r 2 dV(x) sinc rρ ρ η r exp − 1 − t + η 2 (2π)m/2 Rm +∞ 2 Am−1 m+1 r exp − r2 dr = −ρ (2π)m/2 0 1

2 2 (m−3)/2 2 t sinc rρ 1 − t (1 − t ) dt η × −1

+∞ 2 Am−1 m+1 +ρη r exp − r2 dr (2π)m/2 0 1

2 (m−3)/2 2 sinc rρ 1 − t (1 − t ) dt × −1

+∞ 2 Am−1 m+1 =ξ r exp − r2 dr (2π)m/2 0 1

sinc rρ 1 − t2 (1 − t2 )(m−1)/2 dt , × −1

where we have used the fact that P1,m (t) = t. Moreover, in view of the series expansion of the sinc function

sinc(u) =

∞ (−1) u2 , (2 + 1)!

(67)

=0

this becomes

2 Fcyl exp − |x|2 x (ξ) +∞ ∞ 2 Am−1 (−1) 2 2+m+1 = ξ ρ r exp − r2 dr (2 + 1)! (2π)m/2 0 =0

180

Fred Brackx et al.

×

1 −1

2 (2+m−1)/2

(1 − t )

dt

∞

m Am−1 (−1) 2 +m/2 ρ 2 + +1 = ξ (2 + 1)! 2 (2π)m/2 =0

√π 2+m+1 2

× 2+m+2 2 ∞ 2 + m + 1 (−1) 2

ξ ρ 2 = (2 + 1)! 2 m−1 2

=0

2

= (m − 1) 1 F1

m + 1 3 |ξ|2 ; ;− 2 2 2

ξ.

2 Conclusion 6.2 The cylindrical Fourier spectrum of exp − |x|2 x is the vectorvalued function given by (see also Figure 5):

Fcyl

2 exp − |x|2 x (ξ) = (m − 1) 1 F1

m + 1 3 |ξ|2 ; ;− 2 2 2

m 3 |ξ|2 1− ; ; 2 2 2

= (m − 1) 1 F1

In the even-dimensional case, Kummer’s function 1 F1 1 −

ξ |ξ|2 exp − 2 ξ.

2 m 3 |ξ| 2 ; 2; 2

now yields

classical Hermite polynomials of odd degree. Remark 6.4 We can now easily check the multiplication rule (see Proposition 6.2

(vi)) on the Gaussian exp − |x|2

(2 − m) (2π)m/2

Rm

2

:

2 sinc(|x ∧ ξ|) x exp − |x|2 dV(x)

2 2 (ξ) + ∂ξ Fcyl exp − |x|2 (−ξ) = Fcyl x exp − |x|2

= (m − 1) 1 F1

m + 1 3 |ξ|2 ; ;− 2 2 2

. ξ + ∂ξ

1 F1

m − 1 1 |ξ|2 ; ;− 2 2 2

/ .

181

The Fourier Transform in Clifford Analysis

m 53 m 54 m 55 m 56

1.5

1

0.5

0

2

4

6

8

10

|␰ | 20.5

2 FIGURE 5 The norm of the cylindrical Fourier spectrum of x exp − |x|2 for m = 3, m = 4, m = 5, and m = 6.

As

d dz [1 F1 (a; c; z)]

(2 − m) (2π)m/2

Rm

=

a c 1 F1 (a + 1; c

+ 1; z) (see Magnus et al., 1966), we arrive at

2 sinc(|x ∧ ξ|) x exp − |x|2 dV(x)

m+1 3 |ξ|2 m+1 3 |ξ|2 = (m−1)1 F1 ; ;− ξ − (m−1)1 F1 ; ;− ξ = 0, 2 2 2 2 2 2

2 which was to be expected, since sinc(|x ∧ ξ|) x exp − |x|2 is an odd function in x.

6.3.3. The Cylindrical Fourier Spectrum of the Gaussian Multiplied With Pk (= φ0,k,j )

2 The L2 -basis function φ0,k,j (x) is given, up to constants, by Pk (x) exp − |x|2 with Pk a left solid inner spherical monogenic of order k. From Corollary 2.1 it is clear that we must make a distinction between k even and odd.

a. k even (B- and C-term of kernel decomposition yield zero). In the case where k is even, the integrals containing the B- and C-term of the kernel

182

Fred Brackx et al.

decomposition are zero. Applying the Funk-Hecke theorem and the cosine expansion (66), we arrive at

2 Fcyl exp − |x|2 Pk (x) (ξ) =

1 (2π)m/2

=

Am−1 (2π)m/2

1 −1

Rm

2 exp − r2 rk cos rρ 1 − tη2 Pk (ω) dV(x)

+∞

0

2 exp − r2 rk+m−1 dr

2 (m−3)/2 2 cos rρ 1 − t (1 − t ) Pk,m (t) dt Pk (η) ∞

=

k!(m−3)! Am−1 (−1) 2 ρ (k+m − 3)! (2π)m/2 (2)! =0

1 −1

(m−2)/2

(1 − t2 )(2+m−3)/2 Ck

0

2 exp − r2 r2+k+m−1 dr

+∞

(t) dt Pk (η),

(68)

where we have also used the expression in Eq. (7) of the Legendre polynomials in Rm in terms of the Gegenbauer, also called ultraspherical, polynomials Ckλ (t). As these Gegenbauer polynomials Ckλ are orthogonal

on ] − 1, 1[ w.r.t. the weight function (1 − t2 )λ−1/2 λ > − 12 , it is easily seen that for ≤

1

−1

k 2

− 1 holds (m−2)/2

(1 − t2 ) (1 − t2 )(m−3)/2 Ck

(t) dt = 0.

Moreover, combining the integral formula (see Gradshteyn and Ryzhik, 1980, p. 826, formula 4 with α = β)

1

−1

2 α

(1 − t )

Ckλ (t)

2 22α+1 (α + 1) (k + 2λ) dt = 3 F2 k! (2λ) (2α + 2) 1 × −k, k + 2λ, α + 1; λ + , 2α + 2; 1 , 2

183

The Fourier Transform in Clifford Analysis

valid for Re(α) > −1, with Watson’s theorem (see Erdélyi et al., 1953a)

√ 1 a+b+1 1−a−b+2c π c + 2 2 2 a+b+1

, 2c; 1 = a, b, c; b+1 1−a+2c 1−b+2c 2 a+1 2 2 2 2

3 F2

results in

1

−1

(1 − t2 )α Ckλ (t) dt

2 √ 2α+1 (α + 1) (k + 2λ) α + 32 λ + 12 2α−2λ+3 π2 2

. = k+2λ+1 2α+3+k 2α−2λ+3−k k!(2λ)(2α + 2) −k+1 2 2 2 2 (69) Hence, Eq. (68) becomes

2 Fcyl exp − |x|2 Pk (x) (ξ) ∞ 2π(m−1)/2 (−1) 2 (k+m+2−2)/2 1 k+m+2 k! (m−3)! 2 ρ = (k+m−3)! (2π)m/2 m−1 (2)! 2 2 =k/2

⎛

⎞ 2+m m−1 √ 2+m−2 2+m−1 2 π2 (k + m − 2) ( + 1) 2 2 ⎝

2

⎠ Pk (η). k+m−1 2+m+k 2+2−k k (m − 2)(2 + m − 1) −k+1 2 2 2 2

Taking into account that (see Magnus, Oberhettinger, and Soni, 1966)

(2z) = π

−1/2

1 , (z) z + 2

2

2z−1

the last result can be simpliﬁed to

2 Fcyl exp − |x|2 Pk (x) (ξ)

√ ∞ (−1) 2 ! 2+m−1 2 π

Pk (ξ)

= |ξ|2−k −k+1 k+m−1 2+2−k (2)! =k/2 2 2 2 2 k − 1 + m k + 1 |ξ| ; ;− . = Pk (ξ) 1 F1 2 2 2 2k/2

184

Fred Brackx et al.

b. k odd (A-term of kernel decomposition yields zero). For k odd, the FunkHecke theorem implies that

2 Fcyl exp − |x|2 Pk (x) (ξ)

2 1 k+1 r 2 P (ω) dV(x) =− r ρ exp − t sinc rρ 1 − t η k η 2 (2π)m/2 Rm

2 1 k+1 r 2 ω P (ω) dV(x) − r ρ η exp − sinc rρ 1 − t k η 2 (2π)m/2 Rm +∞ 2 Am−1 = −ρ exp − r2 rk+m dr (2π)m/2 0 1

2 (m−3)/2 2 t sinc rρ 1 − t (1 − t ) Pk,m (t) dt Pk (η) −1

+∞ 2 Am−1 k+m r exp − 2 r dr −ρ η (2π)m/2 0 1

2 (m−3)/2 2 sinc rρ 1 − t (1 − t ) Pk+1,m (t) dt η Pk (η) −1

+∞ 2 Am−1 k+m r Pk (η) exp − 2 r dr =ρ (2π)m/2 0 1

sinc rρ 1 − t2 (1 − t2 )(m−3)/2 Pk+1,m (t) − tPk,m (t) dt . −1

Now, taking into account the Gegenbauer recurrence relation (see Magnus, Oberhettinger, and Soni, 1966) λ+1 λ (k + 2λ) t Ckλ (t) − (k + 1) Ck+1 (t) = 2λ (1 − t2 ) Ck−1 (t),

we obtain

(k + 1)! (m − 3)! (m−2)/2 Ck+1 (t) (k + m − 2)! k! (m − 3)! (m−2)/2 −t C (t) (k + m − 3)! k k! (m − 3)! (m−2)/2 = (k + 1) Ck+1 (t) (k + m − 2)! (m−2)/2 (t) − t (k + m − 2) Ck

Pk+1,m (t) − tPk,m (t) =

=−

k! (m − 2)! m/2 (1 − t2 ) Ck−1 (t), (k + m − 2)!

The Fourier Transform in Clifford Analysis

185

which in its turn yields

2 k! (m − 2)! Am−1 Fcyl exp − |x|2 Pk (x) (ξ) = − ρ Pk (η) (k + m − 2)! (2π)m/2 +∞ 2 k+m r × exp − 2 r dr 0

1

−1

2 (m−1)/2 m/2 2 sinc rρ 1 − t (1 − t ) Ck−1 (t) dt .

Next, applying the series expansion in Eq. (67) the orthogonality of the Gegenbauer polynomials, and expression (69) we ﬁnd consecutively

2 Fcyl exp − |x|2 Pk (x) (ξ) +∞ ∞

k! (m−2)! Am−1 Pk (ξ) (−1) ρ2 2+k+m r2 exp − 2 r dr =− (k+m−2)! (2π)m/2 |ξ|k−1 (2 + 1)! 0 =0

1 −1

=−

2 (2+m−1)/2

(1 − t )

m/2 Ck−1 (t)

⎛

dt

2π(m−1)/2

⎞

Pk (ξ) 1 k! (m − 2)! ⎝

⎠ m/2 m−1 (k + m − 2)! (2π) |ξ|k−1 2 ∞

=(k−1)/2

(−1) ρ2 (2 + 1)!

2(2+k+m−1)/2

2 + k + m + 1 2

⎞

2

2+m+2 m+1 ! π 22+m 2+m+1 (k + m − 1) 2 2 2 ⎟ ⎜

⎠ ⎝ −k+2 k+m 2+m+1+k 2−k+3 (k−1)! (m)(2+m+1) 2 2 2 2 ⎛

√

√ 2π 2k/2

Pk (ξ) = − 2k k+m 2 = −Pk (ξ) 1 F1

(−1) 2 ! 2+m+1 2

|ξ|2−k+1 2−k+3 =(k−1)/2 (2 + 1)! 2 ∞

m + k k + 2 |ξ|2 ; ;− . 2 2 2

186

Fred Brackx et al.

2 Conclusion 6.3 The cylindrical Fourier spectrum of exp − |x|2 Pk (x) is once more expressed in terms of Kummer’s function: a. k even

Fcyl exp

2 − |x|2

Pk (x) (ξ) = 1 F1

k − 1 + m k + 1 |ξ|2 ; ;− 2 2 2

= 1 F1

m k+1 |ξ|2 ; 1− ; 2 2 2

Pk (ξ)

|ξ|2 exp − 2 Pk (ξ)

b. k odd

Fcyl exp

2 − |x|2

Pk (x) (ξ) = − 1 F1

k + m k + 2 |ξ|2 ; ;− Pk (ξ) 2 2 2

m k+2 |ξ|2 ; =−1 F1 1− ; 2 2 2

2 |ξ| exp − 2 Pk (ξ).

6.3.4. The Cylindrical Fourier Spectrum of the Gaussian Multiplied With the Clifford Vector and Pk (= φ1,k,j ) The calculation of the cylindrical Fourier transform

of the basis function |x|2 φ1,k,j , which is given, up to constants, by exp − 2 x Pk (x), runs along similar lines as in the previous subsection. Hence, we restrict ourselves to stating the results.

a. k even (A-term of kernel decomposition yields zero)

2 Fcyl exp − |x|2 x Pk (x) (ξ) (k + m − 1) = 1 F1 (k + 1)

k + m + 1 k + 3 |ξ|2 ; ;− 2 2 2

ξ Pk (ξ)

m k + 3 |ξ|2 (k + m − 1) |ξ|2 ; ; ξ Pk (ξ) exp − 2 . = 1 F1 1 − (k + 1) 2 2 2

The Fourier Transform in Clifford Analysis

187

b. k odd (B- and C-terms of kernel decomposition yield zero)

Fcyl exp

2 − |x|2

k + m k + 2 |ξ|2 x Pk (x) (ξ) = 1 F1 ; ;− ξ Pk (ξ) 2 2 2 2 m k+2 |ξ|2 |ξ| ; ξ Pk (ξ) exp − 2 . =1 F1 1− ; 2 2 2

6.3.5. The Cylindrical Fourier Spectrum of φ2p,k,j Let us now tackle the problem of calculating the cylindrical Fourier transform of the general basis element φ2p,k,j (x), which is, again up to constants, given by

√ 2 H2p,k ( 2x) Pk (x) exp − |x|2 . The starting point of the calculation is very similar to the one performed in subsection 6.3.3. Hence, we will skip the details.

a. k even (B- and C-terms of kernel decomposition yield zero) Expressing the generalized Clifford–Hermite polynomial of even degree in terms of the classical Laguerre polynomial on the real line [see Eq. (11)] and applying moreover the integral formula (see Magnus, Oberhettinger, and Soni, 1966, p. 245) 0

+∞

(α)

exp (−zu) uλ Ln (u) du =

(λ + 1) (α + n + 1) n! (α + 1) × z−λ−1 2 F1 (−n, λ + 1; α + 1; z−1 ) (70)

valid for Re(λ) > −1 and Re(z) > 0, we arrive at

√ 2 (ξ) Fcyl H2p,k ( 2x) Pk (x) exp − |x|2

2 m/2+k−1 = 2p p! Fcyl Lp (|x|2 ) exp − |x|2 Pk (x) (ξ) √ 2p+k/2 π m2 + k + p

= Pk (ξ) k+m−1 m −k+1 + k 2 2 2

∞ (−1) 2 ! 2+m−1 2 k + 2 + m m

2 F1 −p; ; + k; 2 |ξ|2−k . 2+2−k 2 2 (2)! =k/2

2

188

Fred Brackx et al.

By means of the substitution = − 2k , the above result is turned into √

√ 2p+k/2 π m2 +k+p (−2)k/2 |x|2

Fcyl H2p,k ( 2x) Pk (x) exp − 2 (ξ) = Pk (ξ) k+m−1 m2 +k −k+1 2 2

∞ (−2) + k ! 2 +k+m−1 2 2 (2 +k)! !

=0

2 +2k+m m F1 −p, ; +k; 2 |ξ|2 . 2 2 2 (71)

Moreover, when a or b are nonpositive integers, the hypergeometric function 2 F1 (a, b; c; z) represented by the inﬁnite series, and all its analytic continuations 2 F1 (a, b; c; z)

=

∞ (a) (b) z , (c) ! =0

reduce to a polynomial. In fact, taking into account

(−s) =

0

if > s

s! (−1) (s−)!

if ≤ s

,

we ﬁnd 2 F1 (−s, b; c; z) =

s (−1) =0

s

(c)

(b)

z =

s s (b + ) (c) z . (−1) (b) (c + ) =0

Hence, the hypergeometric function 2 F1 −p, 2 +2k+m ; m2 + k; 2 takes the 2 following form:

2 + 2k + m m ; + k; 2 −p, F 2 1 2 2

2 +2k+m+2u p m2 + k 2 p u u

(−1) = 2 . 2 +2k+m m u +k+u u=0

2

2

The Fourier Transform in Clifford Analysis

189

Inserting the above result in Eq. (71) and exchanging the sum and series yields √

√ 2 2p+k/2 π m2 +k+p (−2)k/2

Fcyl H2p,k ( 2x) Pk (x) exp − |x|2 Pk (ξ) (ξ) = k+m−1 −k+1 2 2

p ∞ (−2) + k ! 2+k+m−1 2+2k+2u+m u 2 2 2 p (−2)

|ξ|2 . 2+2k+m u m2 + k + u (2 + k)! ! u=0 =0 2

Fortunately, the series over can now be written in a closed form: ∞ (−2)

+

k 2

!

2+k+m−1 2

2+2k+m 2

2+2k+2u+m 2

|ξ|2

(2 + k)! !

√ 2k+m+2u 2−k π m−1+k 2 2

= 2k+m k+1 2 2 2k + m + 2u m − 1 + k 2k + m k + 1 |ξ|2 ×2 F2 , ; , ;− , 2 2 2 2 2

=0

where 2 F2 (a, b; c, d; z) is the generalized hypergeometric series given by 2 F2 (a, b; c, d; z) =

∞ (a) (b) z . (c) (d) ! =0

Finally, taking into account that 12 + z 12 − z = Oberhettinger, and Soni, 1966, p. 2), we obtain

π cos (πz)

(see Magnus,

√ |x|2 p m + 2k (ξ) = 2 Fcyl H2p,k ( 2x) Pk (x) exp − 2 Pk (ξ) 2 p p p 2k + m + 2u m − 1 + k 2k + m k + 1 |ξ|2 u , ; , ;− . (−2) 2 F2 2 2 2 2 2 u

u=0

(72) Now our aim is to express the above cylindrical Fourier image in terms of Kummer’s function. For that purpose, let us start with the following lemma.

190

Fred Brackx et al.

Lemma 6.2 One has

n n (b)u zu 1 F1 (b + u; c + u; z). u (a)u (c)u

2 F2 (a + n, b; a, c; z) =

(73)

u=0

Proof. We prove this result by induction on n. For n = 0, it is clear that 2 F2 (a, b; a, c; z)

= 1 F1 (b; c; z),

while for n = 1, a straightforward calculation yields 2 F2 (a + 1, b; a, c; z)

= 1 F1 (b; c; z) +

b z 1 F1 (b + 1; c + 1; z). ac

Hence, the proposed sum representation in Eq. (73) holds for the lower order cases n = 0, 1. Assuming that it holds for order n, we now prove the expression (73) in case of order n + 1. First, one can easily verify that 2 F2 (a + n + 1, b; a, c; z)

bz ac × 2 F2 (a + n + 1, b + 1; a + 1, c + 1; z).

= 2 F2 (a + n, b; a, c; z) +

Next, applying the induction hypothesis we arrive at 2 F2 (a + n + 1, b; a, c; z) =

n n (b)u zu 1 F1 (b + u; c + u; z) u (a)u (c)u u=0

n bz n (b + 1)u zu + 1 F1 (b + 1 + u; c + 1 + u; z). u (a + 1)u (c + 1)u ac u=0

Executing in the second sum the substitution u = u + 1 and rearranging the terms indeed yields the desired summation as follows: 2 F2 (a + n + 1, b; a, c; z)

=

n n (b)u zu 1 F1 (b + u; c + u; z) u (a)u (c)u u=0

+

n+1 n (b + 1)u −1 zu −1 bz 1 F1 (b + u ; c + u ; z) u − 1 (a + 1)u −1 (c + 1)u −1 ac u =1

The Fourier Transform in Clifford Analysis

191

n n (b)u b n (b+1)u−1 = 1 F1 (b; c; z) + zu + u (a)u (c)u ac u − 1 (a+1)u−1 (c+1)u−1 u=1

1 F1 (b + u; c

=

+ u; z) +

b (b + 1)n zn+1 1 F1 (b + n + 1; c + n + 1; z) ac (a + 1)n (c + 1)n

n+1 n + 1 (b)u zu 1 F1 (b + u; c + u; z). u (a)u (c)u u=0

We can now easily prove the following proposition. Proposition 6.3 One has p p n=0

n

(−2)n 2 F2 (a + n, b; a, c; z) p

= (−1)

p u=0

p

= (−1)

p u=0

p (b)u zu 2 1 F1 (b + u; c + u; z) u (a)u (c)u u

p (b)u zu 2 1 F1 (c − b; c + u; −z) exp (z). u (a)u (c)u u

Proof. Using the previous lemma, followed by an exchange of the two summations, we obtain p p n=0

=

n

(−2)n 2 F2 (a + n, b; a, c; z)

p p n=0

n n (b)u zu (−2) 1 F1 (b + u; c + u; z) n u (a)u (c)u n

u=0

p

(b)u zu = (a)u (c)u u=0

p

p n (−2) 1 F1 (b + u; c + u; z). n u n=u n

Using the symbolic software Maple (Maplesoft, Waterloo, Ontario, Canada), we ﬁnd p n=u

(−2)n

p n p = 2u (−1)p , n u u

which yields the desired expression.

192

Fred Brackx et al.

By means of the above proposition, we can rewrite the cylindrical Fourier transform in Eq. (72) in terms of Kummer’s function as follows: 2

√ 2 2k + m |ξ| (ξ) = (−2)p Fcyl H2p,k ( 2x)Pk (x)exp − |x|2 Pk (ξ) exp − 2 2 p p p

u

u=0

m−1+k 2

u 2k+m k+1 2 2 u u

2 u

(−|ξ| )

1 F1

|ξ|2 m k+1 1− ; + u; . 2 2 2

Expressing also the generalized Clifford–Hermite polynomial in terms of Kummer’s function [see Eq. (12)], we derive Fcyl

2

m |ξ| |x|2 2 p −p; P (ξ) F (x) exp − = (−1) P (ξ) exp − 2 + k; |x| 1 1 k k 2 2

p p u=0

u

m−1+k 2

u 2k+m k+1 2 2 u u

|ξ|2 m k+1 + u; . 1− ; 2 2 2

(−|ξ|2 )u 1 F1

When k = 0 we can also calculate the cylindrical Fourier spectrum using the integral formula (see Gradshteyn and Ryzhik, 1980, p. 427, formula 4 with u = 1):

1

x 0

2ν−1

2 μ−1

(1 − x )

a2 1 1 (74) cos (ax) dx = B(μ, ν) 1 F2 ν; , ν + μ; − 2 2 4

valid for Re(μ) > 0 and Re(ν) > 0. Indeed, by means of the Funk-Hecke theorem we ﬁnd, as before,

√ 2 (ξ) Fcyl H2p,0 ( 2x)exp − |x|2 Am−1 = 2 p! (2π)m/2

p

1 −1

0

+∞

m/2−1 2 Lp (r )

2 m−1 r exp − 2 r dr

2 (m−3)/2 2 cos (rρ 1 − t ) (1 − t ) dt .

193

The Fourier Transform in Clifford Analysis

Executing the substitution x = we have that

1 −1

1 − t2 and taking into account Eq. (74),

cos (rρ 1 − t2 ) (1 − t2 )(m−3)/2 dt = 2

0

√ =

1

dx cos (rρx) xm−2 1 − x2

π m−1 2 m − 1 1 m r 2 ρ2 ; , ;− , 1 F2 2 2 2 4 m2

which yields

√ 2 2p−m/2+1 p! Fcyl H2p,0 ( 2x) exp − |x|2 (ξ) = m2

+∞ 0

m/2−1 2 Lp (r )

2 − r2

exp

r

m−1

1 F2

m − 1 1 m r2 |ξ|2 ; , ;− 2 2 2 4

dr .

Comparing the above with [see Eq. (72)]

√ 2 (ξ) Fcyl H2p,0 ( 2x) exp − |x|2 = 2p

p 2 p |ξ| + p m − 1 m 1 m + 2u 2 , ; , ;− , (−2)u 2 F2 2 2 2 2 2 u m2 u=0

m

we have thus proved the following integral formula. Proposition 6.4 One has

+∞ 0

m/2−1 2 Lp (r )

exp

2 − r2

r

m−1

1 F2

m − 1 1 m r2 |ξ|2 ; , ;− 2 2 2 4

dr

p 2 |ξ| 2m/2−1 m2 + p p m − 1 m 1 m + u, ; , ;− , (−2)u 2 F2 = p! 2 2 2 2 2 u u=0

where 1 F2 (a; b, c; z) and 2 F2 (a, b; c, d; z) denote generalized hypergeometric series (α) and L the generalized Laguerre polynomial on the real line.

194

Fred Brackx et al.

a. k odd (A-term of kernel decomposition yields zero). Taking into account formula (70), we ﬁnd in a similar way as in Subsection 6.3.3, case b Fcyl

√ 2 H2p,k ( 2x)Pk (x)exp − |x|2 (ξ) =

=(k−1)/2

k + 2+m+1 m

2 F1 −p, ; + k; 2 |ξ|2+1−k . 2 2 (2 + 1)! 2+3−k 2

(−1) 2 !

∞

√ 2π 2p+k/2 m2 + k + p

Pk (ξ) m2 + k − 2k k+m 2

2+m+1 2

Executing the substitution = − k−1 2 , writing the hypergeometric function as a summation, and exchanging the sum and series, we ﬁnd Fcyl

√ 2 H2p,k ( 2x) Pk (x) exp − |x|2 (ξ) =

√ 2π(−1)(k−1)/2 2p+k−1/2 m2 +k+p

Pk (ξ) − 2k k+m 2

p ∞ (−2) + k − 1 ! 2+k+m 2+2k+m+2u 2 2 2 2 p 1

|ξ|2 , (−2)u m u (2 + k)!! 2 +k+u 2+2k+m =0

u=0

2

which can further be simpliﬁed to

√

Fcyl H2p,k ( 2x) Pk (x) exp p p u=0

u

(−2)

u

2 F2

2 − |x|2

(ξ) = −

2p

m

+k+p

2 2k+m 2

Pk (ξ)

2k + m k + m 2k + m k + 2 |ξ|2 + u, ; , ;− . 2 2 2 2 2

Applying Proposition 6.3 we can equivalently state

√ 2 2k + m |ξ|2 (ξ) = −(−2)p Fcyl H2p,k ( 2x)Pk (x)exp − |x|2 Pk (ξ)exp − 2 2 p

k+m (−|ξ|2 )u p 2 m k + 2 + 2u |ξ|2 u

1 F1 1 − ; ; 2k+m k+2 u 2 2 2 p

u=0

2

u

2

u

The Fourier Transform in Clifford Analysis

195

or rewriting also the generalized Clifford–Hermite polynomial in terms of Kummer’s function 2

m |ξ| |x|2 2 p+1 (ξ) = (−1) Fcyl 1 F1 −p; + k; |x| Pk (x) exp − 2 Pk (ξ) exp − 2 2

p k+m (−|ξ|2 )u 2 p m k + 2 + 2u |ξ|2 u

1 F1 1 − ; ; . 2k+m k+2 u 2 2 2 u=0

2

u

2

u

6.3.6. The Cylindrical Fourier Spectrum of φ2p+1,k,j Since the calculations of the cylindrical Fourier spectrum of the basis function φ2p+1,k,j given, up to constants, by

√ 2 H2p+1,k ( 2x) Pk (x) exp − |x|2 are very similar to the ones of the previous subsection, we only give the results.

a. k even (A-term of kernel decomposition yields zero)

√ 2 Fcyl H2p+1,k ( 2x) Pk (x) exp − |x|2 (ξ)

m (k + m − 1) k + + 1 ξ Pk (ξ) k+1 2 p p 2 |ξ| 2k+m+2+2u m+k+1 2k+m+2 3+k p (−2)u F2 , ; , ;− 2 2 2 2 2 u 2

= 2p+1/2

u=0

√ (k + m − 1) m |ξ|2 k + + 1 ξ exp − 2 Pk (ξ) 2 k+1 2 p

p m+k+1 (−|ξ|2 )u |ξ|2 2 m 3+k p u

1 F1 1 − ; + u; 2k+m+2 3+k 2 2 2 u

= (−2)p

u=0

2

u

2

u

or in terms of Kummer’s function [see Eq. (12)],

Fcyl

m |x|2 2 −p; x P + k + 1; |x| F exp − (x) (ξ) = 1 1 k 2 2

(75)

196

Fred Brackx et al.

k+m−1 |ξ|2 ξ exp − 2 k+1

p m+k+1 (−|ξ|2 )u p |ξ|2 2 m 3+k u

1 F1 1 − ; Pk (ξ) + u; . 3+k 2 2 2 u 2k+m+2

(−1)p

u=0

2

u

2

u

b. k odd (B- and C-terms of kernel decomposition yield zero)

√ 2 Fcyl H2p+1,k ( 2x) Pk (x) exp − |x|2 (ξ)

m = 2p+1/2 k + + 1 ξ Pk (ξ) 2 p p 2k+m+2+2u m+k 2k+m+2 2+k |ξ|2 u p , ; , ;− (−2) 2 F2 2 2 2 2 2 u u=0

√ m |ξ|2 = (−2)p 2 k + + 1 ξ exp − 2 Pk (ξ) 2 p

m+k p 2 (−|ξ|2 )u |ξ| 2 p m 2 + k u

1 F1 1 − ; + u; 2k+m+2 2+k u 2 2 2 u=0

2

2

u

u

or in terms of Kummer’s function

m |ξ|2 |x|2 2 p Fcyl 1 F1 −p; +k+1; |x| x exp − 2 Pk (x) (ξ) = (−1) ξ exp − 2 Pk (ξ) 2

m+k p (−|ξ|2 )u |ξ|2 2 p m 2+k u

1 F1 1 − ; + u; . 2k+m+2 2+k u 2 2 2

u=0

2

u

2

u

Taking into account the integral formula (see Gradshteyn and Ryzhik, 1980, p. 426, formula 3 with u = 1):

1

x 0

2ν−1

a 1 (1 − x )μ−1 sin (ax) dx = B μ, ν + 2

2

1 a2 1 3 × 1 F2 ν + ; , μ + ν + ; − 2 2 2 4

2

197

The Fourier Transform in Clifford Analysis

valid for Re(μ) > 0 and Re(ν) > − 12 , we also ﬁnd that

√ 2 (m − 1)

ξ (ξ) = 2p+1/2−m/2 p! Fcyl H2p+1,0 ( 2x) exp − |x|2 m+2 2 +∞ 2 m + 1 3 m + 2 r2 |ξ|2 m/2 2 m+1 r Lp (r ) exp − 2 r ; , ;− dr 1 F2 2 2 2 4 0 which combined with [see Eq. (75)] Fcyl

√ 2 H2p+1,0 ( 2x) exp − |x|2 (ξ) = 2p+1/2 (m − 1)

p u p (−2) 2 F2 u u=0

m

2 + p +1 m2 + 1

ξ

m + 2 + 2u m + 1 m + 2 3 |ξ|2 , ; , ;− 2 2 2 2 2

yields the following result. Proposition 6.5 One has

m + 1 3 m + 2 r2 |ξ|2 ; , ;− dr r exp 1 F2 2 2 2 4 0 p 2m/2 m2 +p+1 m+2+2u m+1 m+2 3 |ξ|2 u p , ; , ;− , (−2) = 2 F2 u p! 2 2 2 2 2

+∞

m/2 Lp (r2 )

2 − r2

m+1

u=0

where 1 F2 (a; b, c; z) and 2 F2 (a, b; c, d; z) denote generalized hypergeometric series (α) and L the generalized Laguerre polynomial on the real line.

ACKNOWLEDGMENTS We are indebted to Wouter Hamelinck for assistance in preparing the ﬁgures.

REFERENCES Antoine, J.-P., Murenzi, R., and Vandergheynst, P. (1996). Two-dimensional directional wavelets in image processing. Int. J. Imag. Syst Technol. 7, 152–165. Bernstein, S., and Lanzani, L. (2002). Szegö projections for Hardy spaces of monogenic functions and applications. Int. J. Math. Math. Sci. 29(10), 613–624. Boas, R., and Buck, R. (1985). “Polynomial Expansions of Analytical Functions.” Springer, New York.

198

Fred Brackx et al.

Brackx, F., Delanghe, R., and Sommen, F. (1982). ”Clifford Analysis.” Pitman, Boston. Brackx, F., De Schepper, N., and Sommen, F. (2003). The higher dimensional Hermite transform: a new approach. Complex Variables and Elliptic Equations 48(3), 189–210. Brackx, F., De Schepper, N., and Sommen, F. (2005). The Clifford–Fourier transform. J. Fourier Anal. Appl. 11(6), 669–681. Brackx, F., De Schepper, N., and Sommen, F. (2006a). Clifford–Hermite and two-dimensional Clifford–Gabor ﬁlters for early vision. In (digital) Proceedings 17 th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering (K. Gürlebeck and C. Könke, eds.), July 12–14, 2006, Bauhaus-Universität Weimar. Brackx, F., De Schepper, N., and Sommen, F. (2006b). The two-dimensional Clifford–Fourier transform. J. Math. Imaging Vis. 26(1–2), 5–18. Brackx, F., De Schepper, N., and Sommen, F. (2006c). Clifford–Hermite-monogenic operators. Czech. Math. J. 56(131), 1301–1322. Brackx, F., De Knock, B., De Schepper, H., and Eelbode, D. (2006d). On the interplay between the Hilbert transform and conjugate harmonic functions. Math. Meth. Appl. Sci. 29(12), 1435–1450. Brackx, F., De Schepper, N., Kou, K. I., and Sommen, F. (2007). The Mehler formula for the generalized Clifford–Hermite polynomials. Acta Math. Sin. 23(4), 697–704. Brackx, F., and Sommen, F. (2001). The generalized Clifford–Hermite continuous wavelet transform. Adv. Appl. Clifford Algebras 11(51) (Special issue: Clifford Analysis, Proceedings of the Clifford Analysis Conference, Cetraro, Italy, October 1998, 219–231. Brackx, F., and Van Acker, N. (1992). Hp spaces of monogenic functions. In “Clifford Algebras and Their Applications in Mathematical Physics” (A. Micali et al. (eds.), pp. 177–188. Kluwer, Dordrecht. Bülow, T. (1999). “Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images.” Doctoral thesis, Christian-Albrechts-Universität, Institut für Informatik und Praktische Mathematik, Kiel. Bülow, T., and Sommer, G. (1998). Quaternionic Gabor ﬁlters for local structure classiﬁcation. In “14th International Conference on Pattern Recognition,” Brisbane, Australia, pp. 808–810. Bülow, T., and Sommer, G. (2001). The hypercomplex signal—a novel approach to the multidimensional analytic signal. IEEE Trans. Signal Process. 49(11), 2844–2852. Bultheel, A., and Martinez, H. (2002). “A Shattered Survey of the Fractional Fourier Transform.” Report TW 337, Department of Computer Science, K.U. Leuven, April 2002. Chui, C. K. (1992). “An Introduction to Wavelets.” Academic Press, San Diego. Condon, E. U. (1937). Immersion of the Fourier transform in a continuous group of functional transformations. Proc. Natl. Acad. Sci. 23, 158–164. Daubechies, I. (1992). “Ten Lectures on Wavelets.” Society for Industrial and Applied Mathematics, Philadelphia. Daugman, J. G. (1983). Six formal properties of two-dimensional anisotropic visual ﬁlters: structural principles and frequency/orientation selectivity. IEEE Trans. Syst, Man. Cybern. 13, 882–887. Daugman, J. G. (1985). Uncertainty relation for resolution in space, spatial frequency, and orientation by two-dimensional visual cortical ﬁlters. J. Opt. Soc. Am. A 2(7), 1160–1169. Delanghe, R., Sommen, F., and Souˇcek, V. (1992). “Clifford Algebra and Spinor-Valued Functions—A Function Theory for the Dirac Operator.” Kluwer, Dordrecht. Delanghe, R. (2002a). Some remarks on the principal value kernel in Rm . Comp. Var. Theory Appl. 47, 653–662. Delanghe, R. (2002b). On the Hardy spaces of harmonic and monogenic functions in the unit ball of Rm+1 . In “Acoustics, Mechanics and the Related Topics of Mathematical Analysis” (A. Wirgin, ed.), pp. 137–142. World Scientiﬁc Publishing, River Edge, NJ. Delanghe, R. (2004). On some properties of the Hilbert transform in Euclidean space. Bull. Belg. Math. Soc.-Sim. 11, 163–180.

The Fourier Transform in Clifford Analysis

199

Delanghe, R., and Brackx, F. (1978). Hypercomplex function theory and Hilbert modules with reproducing kernel. Proc. London Math. Soc. 3(37), 545–576. Ebling, J., and Scheuermann, G. (2003). Clifford convolution and pattern matching on vector ﬁelds. In “Proceedings of IEEE Visualization ’03,” pp. 193–200. IEEE Computer Society, Los Alamitos, CA. Ebling, J., and Scheuermann, G. (2004). Pattern Matching on Vector Fields using Gabor Filter. In “Proceedings of the 14th IASTED International Conference on Visualization, Imaging and Image Processing,” pp. 825–830. Marbella, Spain, September 6–8, 2004. Ebling, J., and Scheuermann, G. (2005). Clifford Fourier transform on vector ﬁelds. IEEE Trans. Vis. Comput. Gr. 11(4), 469–479. Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1953a). “Higher Transcendental Functions 1.” McGraw-Hill, New York. Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1953b). “Higher Transcendental Functions 2.” McGraw-Hill, New York. Escalante-Ramirez, B., and Martens, J. B. (1992). Noise reduction in computerized tomography images by means of polynomial transform. J. Vis. Comm. Image R. 3(3), 272–285. Felsberg, M. (2002). “Low-Level Image Processing with the Structure Multivector.” Doctoral thesis, Christian-Albrechts-Universität, Institut für Informatik und Praktische Mathematik, Kiel. Gabor, D. (1946). Theory of communication. J. Inst. Electr. Eng. 93, 429–457. Gilbert, J., and Murray, M. (1991). “Clifford Algebras and Dirac Operators in Harmonic Analysis.” Cambridge University Press, Cambridge. Gradshteyn, I. S., and Ryzhik, I. M. (1980). “Table of Integrals, Series, and Products.” Academic Press, New York. Gürlebeck, K., Habetha, K., and Sprößig, W. (2006). ”Funktionentheorie in der Ebene und im Raum”. Birkhäuser Verlag, Basel. Gürlebeck, K., and Sprößig, W. (1990). “Quaternionic Analysis and Elliptic Boundary Value Problems.” Birkhäuser Verlag, Basel. Gürlebeck, K., and Sprößig, W. (1997). “Quaternionic and Clifford Calculus for Physicists and Engineers.” J. Wiley & Sons, Chichester. Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematic und Mechanik. Zeit. Physik 43, 172–198. Hestens, D. (1966). “Space-Time Algebra.” Gordon and Breach, New York. Hestens, D. (1968a). “Multivector Calculus.” J. Math. Anal. and Appl. 24, 313–325. Hestens, D. (1968b). “Multivector Functions.” J. Math. Anal. and Appl. 24, 467–473. Hochstadt, H. (1971). The functions of mathematical physics. In “Pure and Applied Mathematics,” Volume 23. Wiley-Interscience, New York. Kayargadde, V., and Martens, J. B. (1994). Estimation of edge parameters and image blur using polynomial transforms. CVGIP: Graphical Models and Image Processing 56(6), 442–461. Kober, H. (1939). Wurzeln aus der Hankel-und Fourier und anderen stetigen Transformationen. Q. J. Math. Oxford Ser. 10, 45–49. Li, C., McIntosh, A., and Qian, T. (1994). Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces. Rev. Mat. Iberoam. 10(3), 665–721. Magnus, W., Oberhettinger, F., and Soni, R. P. (1966). “Formulas and Theorems for the Special Functions of Mathematical Physics.” Springer-Verlag, Berlin. Martens, J. B. (1990a). Application of scale space to image coding. IEEE Trans. Commun. 38(9), 1585–1591. Martens, J. B. (1990b). The Hermite transform—theory. IEEE Trans. Acoust, Speech Signal Process. 38(9), 1595–1606. Martens, J. B. (1990c). The Hermite transform—Applications. IEEE Trans. Acoust. Speech Signal Process. 38(9), 1607–1618. Mawardi, B., and Hitzer, E. (2006). Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl3,0 . Adv. Appl. Cliff. Alg. 16(1), 41–61.

200

Fred Brackx et al.

McBride, A. C., and Kerr, F. H. (1987). On Namias’s fractional Fourier Transforms. IMA J. Appl. Math. 39, 159–175. McIntosh, A. (1996). Clifford algebras, Fourier theory, singular integrals and harmonic functions on Lipschitz domains. In “Clifford Algebras in Analysis and Related Topics” (J. Ryan, ed.), pp. 33–87. CRC Press, Boca Raton, FL. Mitrea, M. (1994). Clifford wavelets, singular integrals and Hardy spaces. In Lecture Notes in Mathematics 1575. Springer-Verlag, Berlin. Murray, M. (1985). The Cauchy integral, Calderon commutation, and conjugation of singular integrals Rn . Trans. AMS 298, 497–518. Namias, V. (1980). The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Math. Applic. 25, 241–265. Ozaktas, H. M., Zalevsky, Z., and Kutay, M. A. (2001). “The Fractional Fourier Transform.” Wiley, Chichester. Porteous, I. R. (1995). “Clifford Algebras and the Classical Groups.” Cambridge University Press, Cambridge. Qian, T., Hempﬂing, Th., McIntosh, A., and Sommen, F. (eds.) (2004). “Advances in Analysis and Geometry: New Developments Using Clifford Algebras.” Birkhäuser Verlag, Basel. Rivero-Moreno, C. J., and Bres, S. (2003). Conditions of similarity between Hermite and Gabor ﬁlters as models of the human visual system. In “Computer Analysis of Images and Patterns, Lecture Notes in Computer Science LNCS 2756,” pp. 762–769. Springer-Verlag, Berlin. Rivero-Moreno, C. J., and Bres, S. (2003). Les Filtres de Hermite et de Gabor Donnentils des Modèles Equivalents du Système Visuel Humain? In “ORASIS ’03, Journées Francophones des Jeunes Chercheurs en Vision par Ordinateur.” Gérardmer, France, 2003, 423–432. Ryan, J. (ed.). (1996). “Clifford Algebras in Analysis and Related Topics” (Studies in Advanced Mathematics). CRC Press, Boca Raton, FL. Ryan, J. (2000). Basic Clifford analysis. Cubo Math. Educ. 2, 226–256. Ryan, J. (2004). Clifford analysis. In “Lectures on Clifford (Geometric) Algebras and Applications” (R. Ablamowicz and G. Sobczyk, eds.), pp. 53–89. Birkhäuser, Boston. Ryan, J., and Struppa, D. (eds.) (1998). “Dirac Operators in Analysis.” Addison Wesley Longman Ltd, Harlow, UK. Sommen, F. (1981). A product and an exponential function in hypercomplex function theory. Appl. Anal. 12, 13–26. Sommen, F. (1982a). Some connections between complex analysis and Clifford analysis. Complex Variables: Theory and Applications 1, 97–118. Sommen, F. (1982b). Hypercomplex Fourier and Laplace transforms I. Ill. J. Math. 26(2), 332–352. Sommen, F. (1983). Hypercomplex Fourier and Laplace transforms II. Complex Variables: Theory and Appplication 1, 209–238. Sommen, F. (1988). Special functions in Clifford analysis and axial symmetry. JMAA 130(1), 110–133. Sommen, F., and Jancewicz, B. (1997). Explicit solutions of the inhomogeneous Dirac equation. J. Anal. Math. 71, 59–74. Sommen, F., and Van Acker, N. (1992). Monogenic differential operators. Results Math. 22, 781–798. Stein, E. M., and Weiss, G. (1971). Introduction to Fourier analysis on Euclidean spaces. In “Princeton Mathematical Series” (P. A. Grifﬁths, M. Morse, and E. M. Stein, eds.). Princeton University Press, Princeton, NJ. Van Acker, N. (1991). “Clifford–differentiaaloperatoren en randwaardentheorie van de nuloplossingen ervan op de sfeer en de Lie-sfeer.” Doctoral thesis, Ghent University, Ghent, Belgium [in Dutch].

The Fourier Transform in Clifford Analysis

201

van Dijk, A. M., and Martens, J. B. (1997). Image representation and compression using steered Hermite transforms. Signal Proc. 56(1), 1–16. Venegas-Martinez, S., Escalante-Ramirez, B., and Garcia-Barreto, J. A. (1997). Applications of polynomial transforms in astronomical image restoration and deconvolution. Rev. Mex. Astron. Astroﬁs. 33, 173–185. von Mehler, F. G. (1866). J. Reine Angew. Math. 66, 161–176. Watson, G. N. (1933). Notes on generating functions of polynomials: (2) Hermite polynomials. J. London Math. Soc. s(1-3)8, 194–199. Weldon, T. P., Higgins, W. E., and Dunn, D. F. (1996). Efﬁcient Gabor ﬁlter design for texture segmentation. Patt. Recogn. 29(12), 2005–2015. Wiener, N. (1929). Hermitian polynomials and Fourier analysis. J. Math. Phys. 8, 70–73. Wiener, N. (1956). “I Am a Mathematician.” MIT Press, Cambridge, MA. Young, R. A. (1991). Oh say, can you see? The physiology of vision. Proc. SPIE 1453, 92–123.

CHAPTER

3 Carbon Nanotube Electron Sources for Electron Microscopes Niels de Jonge*

Contents

1 Introduction 2 Mounting of Individual Carbon Nanotubes with Closed Caps 2.1 Mounting Procedure 2.2 Cap Closing 2.3 Cleaning the Nanotubes and Emission Patterns 2.4 Field Emission Microscopy of the Apex of Carbon Nanotube Electron Sources 3 TEM Investigations of Carbon Nanotube Electron Sources 3.1 In Situ Specimen Holder 3.2 High-Resolution TEM Images of Individual Free-Standing Carbon Nanotubes 3.3 In Situ TEM Investigation of Cap Closing 3.4 In Situ TEM Investigations at High Currents 3.5 Discussion 4 The Electron Emission Process 4.1 The Fowler–Nordheim Model 4.2 Results 5 Figure of Merit of Electron Optical Performance 5.1 Causes of Emission Instability 5.2 Emission Stability Measurements 5.3 Brightness Measurements 5.4 Figure of Merit 6 Evaluation of the Use of Carbon Nanotube Electron Sources for Electron Microscopes 6.1 Noise Level 6.2 The Figure of Merit Under Realistic Conditions 6.3 Lifetime Testing

204 205 205 205 206 207 208 208 209 209 210 211 213 213 214 216 216 217 219 221 222 222 224 224

* Oak Ridge National Laboratory, Materials Science and Technology Division, Oak Ridge, TN 37831-6064, USA; and Vanderbilt University Medical Center, Department of Molecular Physiology and Biophysics, Nashville, 37232-0615, USA Advances in Imaging and Electron Physics, Volume 156, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01403-1. Copyright © 2009 Published by Elsevier Inc.

203

204

Niels de Jonge

6.4 Considerations for a Source Module 6.5 Prototype Carbon Electron Source in SEM 7 Specifications 8 Summary and Outlook Acknowledgments References

225 228 229 230 231 232

1. INTRODUCTION Carbon nanotubes have been investigated as next-generation highbrightness electron point sources for electron microscopes (de Jonge and Bonard, 2004; de Jonge et al., 2002). The carbon nanotube electron source could possibly replace the Schottky emitter for high-resolution imaging and analysis in scanning electron microscopes (SEMs) and scanning transmission electron microscopes (STEMs). It is expected that carbon nanotubes have several advantages over ﬁeld emission sources made from sharp metal tips (typically tungsten or molybdenum) that have been used for four decades (Hainfeld, 1977). The drawback of a cold ﬁeld-emitter of metal is that the current ﬂuctuations are too large; these are caused by surface migration of atoms under the high electric ﬁeld and by ion back sputtering. The carbon nanotube is not a metal but a highly ordered crystalline structure built by covalent bonds. Each carbon atom is bound to three other carbon atoms by a covalent sp2 (sigma) bond. The threshold for the removal of one atom is 17 eV (Crespi et al., 1996), much higher than the activation energy for surface migration of a tungsten atom (3.2 eV) (Hainfeld, 1977). Thus, the carbon nanotube should be much less sensitive to surface migration of the carbon atoms. In addition, carbon nanotubes have a very large Young’s modulus and a high tensile strength. With no (or few) dangling bonds, carbon nanotubes are also chemically inert and react only under extreme conditions or at high temperature with oxygen or hydrogen (Saito, Dresselhaus, and Dresselhaus, 1998). Moreover, carbon has one of the lowest sputter coefﬁcients (Paulmier et al., 2001). Therefore, the current emitted from a carbon nanotube is expected to be highly stable compared with metal emitters, and the carbon nanotube itself is expected to be a robust emitter even at high temperatures (Purcell et al., 2002). We describe the measurement on electron sources made from individual carbon nanotubes with closed caps and carefully cleaned surfaces. Carbon nanotubes grown by chemical vapor deposition (CVD) (Lacerda et al., 2004) and arc discharge (Colbert et al., 1994) were investigated. This chapter gives an overview of the research conducted at Philips Research, the Netherlands.

Carbon Nanotube Electron Sources for Electron Microscopes

205

2. MOUNTING OF INDIVIDUAL CARBON NANOTUBES WITH CLOSED CAPS The use of a carbon nanotube as an electron point source in an electron microscope requires precise mounting of an individual carbon nanotube on a support—in our case, a tungsten tip on a heating ﬁlament. By 1995 electron emitters had already been made. They consisted of single carbon nanotubes with the help of micromanipulators and an optical microscope (Rinzler et al., 1995). This method was improved by using nanomanipulators inside an SEM, thereby avoiding the (undesired) mounting of bundles of carbon nanotubes or multiple carbon nanotubes (de Jonge, Lamy, and Kaiser, 2003).

2.1. Mounting Procedure An individual carbon nanotube was mounted on a tungsten support tip connected to a heating ﬁlament. The tungsten tip (typically with a radius of curvature of 50 nm) was obtained by electrochemical etching. The tip was transferred into the SEM and carefully pierced into carbon tape to apply glue necessary for a ﬁrm attachment of the nanotube. A sample with nanotubes protruding from a sharp edge also placed in the SEM was searched for a long, straight, thin, and free-standing nanotube. This nanotube was brought into contact with the tungsten tip on which it stuck (Figure 1). Then, the nanotube was broken off the nanotube sample by applying a voltage difference over the nanotube, leading to Joule heating by a current of more than 20 μA (corresponding to a current density higher than 6 × 1010 A/m2 for a tube with a radius of 10 nm). The length and the diameter can be selected with this method to a precision of 200 nm and 10 nm, respectively (de Jonge, Lamy, and Kaiser, 2003). Moreover, the contact length and the angle between the nanotube and the support tip can be set.

2.2. Cap Closing For optimal use as an electron source carbon nanotubes with open caps are undesirable because the manifold of dangling bonds from an open cap leads to current ﬂuctuations and may even lead to rapid destruction of the nanotube under the presence of the extremely strong electric ﬁeld needed for electron emission (Rinzler et al., 1995). However, the breaking action in the described mounting procedure results in an open cap. A considerable research effort was invested in ﬁnding a procedure to close the cap, which was predicted to be possible (Dean and Chalamala, 2003; Rinzler et al., 1995). It turned out that carbon nanotubes with up to ﬁve walls close their cap when emitting a current of a few microamperes, as demonstrated by TEM investigations (see Section 3.1). This effect was applied

206

Niels de Jonge

CNTs

Metal tip CNT

Substrate (a) CNTs Metal tip CNT Substrate (b) CNTs gap

Metal tip CNT

Substrate (c) Metal tip CNT (d)

FIGURE 1 Mounting procedure of a carbon nanotube (CNT) electron source as performed inside an SEM. (a) A carbon nanotube protruding from a thin substrate containing many carbon nanotubes is selected and (b) attached to a tungsten support tip. (c) The carbon nanotube is broken by Joule heating. (d) The open tube end is finally closed. Source: de Jonge et al., 2005a.

in the procedure to make high-quality electron sources from individual carbon nanotubes (de Jonge et al., 2005a).

2.3. Cleaning the Nanotubes and Emission Patterns The ﬁnal step in the preparation of the carbon nanotube electron source was cleaning. A newly made electron source was transferred into an ultrahigh vacuum system (10−10 torr) for the characterization of the electron emission properties. The nanotube was ﬁrst heated to the carbonization temperature (Saito, Dresselhaus, and Dresselhaus, 1998) of 700◦ ± 50◦ C for 10 minutes to remove volatile species from the tube (de Jonge, 2004). The samples were heated to a temperature of 300◦ –500◦ C during the emission experiments to continuously clean the tubes.

Carbon Nanotube Electron Sources for Electron Microscopes

207

2.4. Field Emission Microscopy of the Apex of Carbon Nanotube Electron Sources We used a ﬁeld emission microscope (FEM) to investigate the morphology of the caps of twelve CVD-grown carbon nanotubes. FEM images provide information on the electronic structure and the local tunneling probability of an emitting area (Good and Mueller, 1956). Figure 2 panels a–d and f shows almost round patterns obtained for carbon nanotubes #8–#12. The patterns have several local maxima within the emission pattern but no sharp transitions from regions of low to high intensity. We interpret these patterns as typical emission patterns of single-walled or thin multiwalled carbon nanotubes with closed caps (Dean and Chalamala, 2003). The images were highly stable with time for emitted currents up to 1 μA as expected for a closed cap. The emission patterns of ﬁve other carbon nanotubes (#13– #17) showed similar patterns reﬂecting closed caps (data not shown). A different type of emission pattern, showing rings with ﬁvefold and sixfold symmetry and interference fringes, has been observed for bundles containing thick multiwalled carbon nanotubes with highly symmetric caps (Hata, Takakura, and Saito, 2001). Such patterns do not

CNT#9

CNT#8 (a)

CNT#12, initially (e)

CNT#11

CNT#10 (b)

CNT#12

(c)

CNT#19

CNT#18 (f)

(d)

(g)

(h)

FIGURE 2 Field emission microscopy images of carbon nanotube electron sources. The emission patterns have an approximate width of 1 cm of the phosphor screen. Images (a)−(f) correspond to carbon nanotubes #8−#12 showing patterns of closed caps. Image (e) shows the initial emission pattern of carbon nanotube #12, while image (f) presents the emission pattern after additional treatment (see text). Images (g) and (h) for carbon nanotubes #18 and #19 are the emission patterns of open caps. Source: de Jonge et al., 2005a.

208

Niels de Jonge

occur for our carbon nanotubes, as these caps do not have a completely regular and symmetric structure because of the breaking procedure, as seen in Figure 4b and c. Stable emission patterns were not always found. Three patterns of a different nature are shown in Figure 2e, g, and h. The pattern in Figure 2e was the initial emission pattern of carbon nanotube #12 and had a single round spot that changed its position every few minutes. The pattern changed to the pattern of Figure 2f only after repeated heating to 700◦ C and extracting an emission of 10 μA. We believe that the unstable pattern of Figure 2e belongs either to a strongly adsorbed species, amorphous carbon, or an open carbon nanotube. The FEM images of Figure 2g and h are the patterns of the respective carbon nanotubes #18 and #19, displaying separated and uncorrelated spots. These spots changed their position and intensity every few seconds; the pattern of Figure 2h even rotated. Because additional heating to 700◦ C and emission of several microamperes of current did not change this behavior, we interpret these patterns as being caused by open carbon nanotubes, whose wide emission pattern and ﬂuctuating emission are undesirable for ﬁeld emission applications.

3. TEM INVESTIGATIONS OF CARBON NANOTUBE ELECTRON SOURCES The main advantage of the use of carbon nanotubes as an electron source over other materials (commonly metals such as molybdenum or tungsten) is expected to be the stability of the structure under electron emission conditions. TEM experiments were performed to investigate the relation between the structure of carbon nanotubes and their electron emission properties. The nature of the end of the free-standing nanotube (e.g., the cap) was investigated by high-resolution TEM imaging. The behavior of the nanotube under electron emission conditions was investigated in situ in a TEM, enabling the direct imaging of the structural changes of the nanotube during electron emission (Wang et al., 2002). In situ studies were also performed at high currents to determine the limit of operation. It was found that a cap-closing mechanism exists (de Jonge et al., 2005a).

3.1. In Situ Specimen Holder TEM images were taken of free-standing individual carbon nanotubes mounted on tungsten support tips at an electron beam energy of 300 kV in low-dose mode. For the in situ observation of a carbon nanotube while it was emitting electrons, a special TEM specimen holder was constructed with a metal electrode for electron emission measurements at a distance of 2.0 mm from the carbon nanotube (Kaiser et al., 2006) (Figure 3). A large electric ﬁeld was generated at the free-standing end of the nanotube by

Carbon Nanotube Electron Sources for Electron Microscopes

209

13.5

1

FIGURE 3 Schematic drawing of the tip of the TEM specimen holder for in situ electron emission measurements with the grounded tungsten tip with a mounted carbon nanotube, the counter electrode, and the electrical connections to apply the field and measure the emission current.

5 nm

10 nm (a)

10 nm (b)

(c)

FIGURE 4 High-resolution TEM images of free-standing carbon nanotubes mounted individually on tungsten support tips. (a) Image of a multiwalled carbon nanotube with an open cap and eight walls. (b) and (c) Images of two thin carbon nanotubes with closed caps. Source: de Jonge et al., 2005a.

applying a voltage difference between the electrode and the frame of the holder.

3.2. High-Resolution TEM Images of Individual Free-Standing Carbon Nanotubes The caps of six carbon nanotubes were imaged with a TEM. After breaking the carbon nanotubes in the mounting procedure we found that four of the six carbon nanotubes had closed caps at their broken ends (Figure 4). Because all of the carbon nanotubes were broken in the mounting procedure, it was initially expected that all carbon nanotubes would have open caps. Thus, this observation of closed caps shows that a cap-closing mechanism exists for small-diameter carbon nanotubes.

3.3. In Situ TEM Investigation of Cap Closing We next demonstrated the closing of a carbon nanotube in situ in a TEM (de Jonge et al., 2005a). A high-resolution image of the carbon nanotube taken

210

Niels de Jonge

5 nm (a)

10 nm (b)

(c)

(d)

FIGURE 5 In situ electron emission experiment in the TEM for a multiwalled carbon nanotube. (a) High-resolution image showing five walls. (b) Image of the carbon nanotube end taken under the presence of a strong electric field resulting in electron emission of a current of 0.2 μA. (c) Image recorded directly after the carbon nanotube had closed. The carbon nanotube emitted a current of 1.5 μA. The arrows in (b) and (c) indicate a piece of amorphous carbon showing the same distance between the carbon nanotube end and the piece in both images. (d) Image of the closed cap recorded while the carbon nanotube was emitting a current of 0.08 μA. Source: de Jonge et al., 2005a.

at a position close to the tungsten support tip revealed it had ﬁve walls (Figure 5a). Starting with a carbon nanotube with an open cap (Figure 5b), a voltage was applied between the carbon nanotube and the counterelectrode in the TEM specimen holder, thus obtaining electron emission. Running a current of 0.5 μA at 220 V for 3 minutes did not lead to any changes of the open cap. At a voltage of 240 V, the carbon nanotube emitted a current of 1 μA with strong ﬂuctuations of 0.5 μA (∼50%). Over a period of approximately 5 minutes, the current slowly increased to 3 μA, although the same voltage was maintained at the electrode. Then the carbon nanotube suddenly closed (Figure 5c). The length of the carbon nanotube did not change within the experimental error of 0.5 nm during this experiment. The current dropped to 1.5 μA after the closing, as expected for a reduction of the sharpness of the emitter caused by the closing. An image recorded at a lower emission current is shown in Figure 5d.

3.4. In Situ TEM Investigations at High Currents Several carbon nanotube electron sources were operated in the TEM to test their structural changes under electron emission conditions. The ﬁrst important observation in this study is that structural changes of the carbon nanotubes were not observed for emission currents up to a threshold

211

Carbon Nanotube Electron Sources for Electron Microscopes

current of 2–9 μA (different for each nanotube) (Doytcheva, Kaiser, and De Jonge, 2006). These results rule out a gradual change of the structure as observed for most metal ﬁeld emitters and demonstrate the great rigidity of the carbon nanotube. Deﬁning the limits of safe operation of carbon nanotube electron sources is required for their successful application as ﬁeld emission sources. Table I shows that the breaking limit is 5 μA for nanotubes with diameters of 5–13 nm and lengths of 0.7–1 μm, 2 μA for a longer nanotube in the same diameter range, and 9 μA for a thicker nanotube. Setting a safe limit of operation signiﬁcantly below the limit of breaking, our nanotube electron sources can be operated stably up to 1 μA. Sudden changes occurred only above the threshold current and several different processes were observed, including shortening of the nanotubes (Figure 6), splitting of their ends, inner segment removal, as well as closing of their open caps (Figure 5). All observed processes are summarized in Table I.

3.5. Discussion Several mechanisms were proposed to explain the structural changes of carbon nanotube electron sources operated at large currents: Joule heating, Coulomb explosion, and ambient effects. Joule (resistive) heating causes a rise in the emitted current because of the temperature dependence of TABLE I Structural changes at high currents Walls

d(nm)

l (μm)

1

7

20.6 ± 0.4

0.83

2

8

9.8 ± 0.2

3.0

3

14–16

13.0 ± 0.5

1.0

4

6

6.0 ± 0.5

0.91

5

5

5.2 ± 0.2

0.69

Carbon nanotube

Process

Splitting Shortening Splitting Shortening Cap change Splitting Shortening Segment removal Splitting Shortening Shortening Shortening Cap closing

I(μA)

I after (μA)

9 15 2 5 4 6 16 8 5 12 5 8 3

19 7 4 2 4 6 2 3 2 6 2 5 1.5

Data on the structural changes of carbon nanotube electron sources under the inﬂuence of emission at large currents for ﬁve samples. Indicated are the number of walls, the diameter (d) of the nanotube, its initial length (l), the type of process that occurred, the current I at which the change occurred, and the current observed after the event (Iafter ).

212

Niels de Jonge

100 nm

100 nm

(a)

(b)

100 nm

100 nm

(d)

(e)

100 nm (c)

100 nm (f)

FIGURE 6 In situ TEM observation of the changes in the structure of a multiwalled carbon nanotube while emitting electrons. (a) Overview image of carbon nanotube #1. The tungsten tip on which it was mounted is visible at the bottom. (b) Zoomed image of the end of the nanotube, directly before the electron emission was started. (c) Image recorded after the top part of the nanotube was shortened while emitting a current of 12 μA at 190 V on the counter electrode. (d) Image of the end of the nanotube while emitting a current of 9 μA at 190 V. (e) Image recorded after the right-side branch shortened (see arrow). (f) Image of the nanotube after the top part of the nanotube was cut off, 60 seconds after image (e). The emission current dropped to 12 μA at a constant voltage of 190 V. The arrows indicate the split branch at the right side of the nanotube.

ﬁeld emission. An important factor is believed to be the force on the end of the nanotube due to the interaction of the large electric ﬁeld and the electrical dipole it induced in the nanotube. A combined effect of Joule heating and charging may damage a part of the outer shell at once;

Carbon Nanotube Electron Sources for Electron Microscopes

213

this is termed Coulomb explosion. Ambient effects, including ion bombardment, adsorbed molecules, or small ﬂuctuations of the extraction voltage, also could have a signiﬁcant effect on the degradation of the nanotube electron source.

4. THE ELECTRON EMISSION PROCESS 4.1. The Fowler–Nordheim Model Often it is assumed that the emission process of a carbon nanotube is ﬁeld emission as described by the Fowler–Nordheim theory (Fowler and Nordheim, 1928; Good and Mueller, 1956). The tunneling current density J through a potential barrier between a metal surface and vacuum is given by Hawkes and Kasper (1996) as follows:

√ 8π 2mφ3/2 e3 F 2 exp − J= v(y) 3heF 8πhφ t2 (y) F2 φ3/2 = c1 2 exp −c2 v(y) F φ t (y)

(1)

(2)

with workfunction φ, electron mass m, electric ﬁeld F, Planck’s constant h, the electron charge e, and the functions t(y) and v(y). For a triangular surface potential barrier, t(y) and v(y) are unity. A plot of log( J/F2 ) versus 1/F, the so-called Fowler–Nordheim plot, is a linear curve for a triangular barrier. The functions v(y) and t(y) can be approximated by de Jonge et al., 2004; Groening et al., 2000 v(y) = a1 − a2 × y2 = 0.958 − 1.05y2 and t(y) = b1 = 1.05. The function y is expressed as

1 y= φ

√ e3 F F = c3 4πε0 φ

(3)

with the permittivity of free space ε0 . The current density simpliﬁes to

φ3/2 2 1 . J = c1 2 exp a2 c2 c3 √ exp −a1 c2 F φ b1 φ F2

(4)

Of importance is the tunneling parameter, given by

d=

ehF c4 F . =√ √ φ t(y) 4π 2mφ t(y)

(5)

214

Niels de Jonge

The tunneling parameter relates to the energy distribution of the emitted electron beam. The current density as function of the energy is denoted as

J(E) = a

exp(E/d) ; 1 + exp(E/kb T)

(6)

here, kb is the Boltzmann constant, T is the temperature, and a is a constant. The total current I is a function of J and the emitting surface A, which is often taken as a half sphere with radius of curvature R.

I = AJ = 2πR2

(7)

The ﬁeld at a sharp metal tip is

F = βU,

(8)

with the extraction voltage U and the ﬁeld enhancement factor β. The ﬁeld enhancement factor can be calculated numerically from the geometry of the emitter and the extractor. The beta factor also can be readily obtained from the current-voltage characteristics of the emitter. It was found that experimentally obtained values equaled calculated ones within a factor of two (de Jonge et al., 2004).

4.2. Results Thin multiwalled CVD carbon nanotubes (three to ﬁve walls) were characterized by recording their current-voltage characteristics and energy spectra (de Jonge et al., 2004). As shown in Figure 7, the data follow a straight line in the Fowler–Nordheim plot, which indicates a ﬁeld emission process. Fitting the Fowler–Nordheim equation (assuming a workfunction of 5 eV) to this data gave a value of the ﬁeld enhancement factor β = 8.1 × 106 m−1 . The tube radius was extracted from the emitting area and amounted to 4.5 nm. The numerically calculated ﬁeld enhancement factor for a tube with a radius of 5 nm and a length of 25 nm on a support tip is 8.4 × 106 m−1 . Because these values of the radius and the length are typical for our samples, and the values of the ﬁeld enhancement factor correspond, it can be concluded that the Fowler–Norhdeim model applies in ﬁrst order. Advanced modeling accounts for the highly curved surface of the emitter (Edgcombe and de Jonge, 2007). As a second test of the validity of the Fowler–Nordheim model, the energy spectrum of the emitted electron beam of the nanotube was measured with a hemispherical energy analyzer (VSW) (Figure 7c). Fitting the spectrum to Eq. (6) yields the value d = 0.19 eV (the limited resolution of

215

Carbon Nanotube Electron Sources for Electron Microscopes

227

1000

1

228 229

10

0.1 J(E) / a.u.

I/nA

Iog(I/U2)

100

230 231

0.01

232 233

1 500 550 600 650 700 750 U/ V

(a)

234

0.001 0.0014 0.0016 0.0018 0.002 1/U

21.5

21

(b)

20.5

0

0.5

E / eV

(c)

FIGURE 7 Field emission measurements of a CVD-grown carbon nanotube electron source at room temperature. (a) The emitted current as a function of the extraction voltage and a fit of the Fowler−Nordheim theory (line). (b) Fowler−Nordheim plot with a slope of −9.0 × 103 and a linear fit (line). (c) Energy spectrum recorded at an extraction voltage of 552.8 V at room temperature and with an emitted current of 11 nA (dotted line) fit with the Fowler−Nordheim theory (dashed line). Numerically generated energy spectrum taking into account the limited resolution of the spectrometer. Source: de Jonge et al., 2004.

the spectrometer had no inﬂuence on this value). The value of d as calculated with Eq. (5) using β = 8.1 × 106 m−1 , φ = 5.0 eV, and U = 552.8 V equals 0.19 eV, consistent with the measurement. The data of the Fowler– Nordheim plot and the energy spectrum can be combined to determine the value of the workfunction from the ﬁeld emission measurements. The slope of the Fowler–Nordheim plot b = −a1 c2 φ3/2 /β, yielding (Groening et al., 2000)

φ=−

3 b1 bd bd b1 bd =− = −1.64 , a1 c2 c4 V 2 a1 V V

(9)

with d and V from the energy spectrum. For our data this equation yields a value of the workfunction of 5.1 ± 0.1 eV as expected for a carbon nanotube (Gao, Pan, and Wang, 2001) and is equal to the value of graphite. The same result was obtained for another (thinner) nanotube. The fact that the obtained value of the workfunction was equal to the expected value reafﬁrms that the Fowler–Nordheim model applies. An important consequence of the validity of the Fowler–Nordheim model is that the behavior of carbon nanotubes electron sources can now be calculated with the analytical expressions shown in Eqs. (4)–(8). An example is given in Figure 8. Energy spectra were recorded at different currents for two carbon nanotubes. For both nanotubes the obtained values of the tunneling parameter were plotted versus the current. The value of d increases with increasing current due to the necessary increase of the electric ﬁeld. A second important observation is that the value of d for a similar current increases with decreasing radius. This ﬁnding can be

Niels de Jonge

d /eV

216

0.34 0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16

Nanotube 1 Nanotube 2 r 5 5 nm r 51.5 nm

10

100 I /nA

1000

FIGURE 8 Tunneling parameter d as a function of the total emitted current measured for two CVD-grown carbon nanotubes, with respective radii of 5 and 1.5 nm. The lines present the calculated relation for corresponding tip radii.

explained as follows. A thin nanotube has a small emitting area and thus requires a relatively large electric ﬁeld to provide a certain current level, whereas for a larger nanotube a moderate ﬁeld is already sufﬁcient to provide the required current. The radii of both nanotubes were obtained from their emitting areas and were 5 and 1.5 nm, respectively. The relation between d and I was calculated using Eqs. (4), (5), and (8). The trend ﬁts well with the measurements except for the data point at the highest currents. The latter is possibly due to the effect of Coulomb interactions (Kruit, Jansen, and Orloff, 1997) or large effects on the energy spectra of Joule heating (Purcell et al., 2002).

5. FIGURE OF MERIT OF ELECTRON OPTICAL PERFORMANCE The ﬁgure of merit of electron optical performance of an electron source is given by the relation between the reduced brightness and the energy spread in the region of stable emission. A thorough study was performed to determine the stability of the emitted current and the region of stable emission. A method was developed to determine the reduced brightness. Finally, brightness measurements were combined with measurements of the energy spread for a series of nanotubes.

5.1. Causes of Emission Instability Field emitters of sharp metal tips have existed for many decades and are presently the electron sources with the lowest energy spread and high brightness (Hainfeld, 1977). The main drawback of the cold ﬁeld-emitter is that the current ﬂuctuations (I/I) are too large (3–5%). The current ﬂuctations have two major causes: variations of F and variations of φ.

Carbon Nanotube Electron Sources for Electron Microscopes

217

As can be derived from the Fowler–Nordheim equation, the exponential term predominates and J is very sensitive to small ﬂuctuations of F or φ. A change of 1% causes ∼ 10% change in J at typical operating conditions. The workfunction is sensitive to changes in the layer of absorbates on the surface. Even in ultrahigh vacuum (10−12 torr) and with a fully baked-out tip, a layer of absorbates forms after a few hours and a dynamic process of absorbtion, vaporizing, and molecular ﬂow is unavoidable at room temperature. To obtain best performance, the emitter must be ﬂashed (heated shortly) about every 6 hours in a special ﬂash procedure. Local changes of F are likely to occur due to small geometric changes of the tip caused by ion bombardment and surface migration in the very large electric ﬁeld at the tip. At room temperature the surface migration is too problematic because of the activation energy of 2.4 eV/atom. Only ions bombarding the tip can provide sufﬁcient energy for the process to occur. However, especially at the high temperatures (>1000◦ C) needed to evaporate all surface contamination, the tips rapidly (seconds) change shape. It was predicted that the problems of surface migration and ion sputtering could be circumvented by using carbon nanotubes. The primary argument is that the carbon nanotube is not a crystal but a structure built by covalent bonds. Each carbon atom is bound to three other carbon atoms by a (covalent) sp2 bond. The threshold for the displacement of one atom is 17 eV (Crespi et al., 1996). Thus, the energy required to move one atom is a factor of 7 larger than for tungsten. Carbon nanotubes have a very large Young’s modulus and maximal tensile strength. Carbon nanotubes are chemically highly inert and react only under extreme conditions or at high temperature with oxygen or hydrogen (Saito, Dresselhaus, and Dresselhaus, 1998). Finally, carbon has one of the lowest sputter coefﬁcients (Paulmier et al., 2001).

5.2. Emission Stability Measurements The stability of the emitted current of a carbon nanotube was determined by running the source at a constant voltage provided by a high-stability power supply of in-house design, with less than 50 parts per million (ppm) ripple and 5 ppm drift. The power supply provided both the extraction voltage and the ﬁlament-heating current used to heat the carbon nanotube continuously at 800◦ K during emission to avoid contamination. The probe current Ip was collected in a Faraday cup with an opening diameter of 1 mm placed ∼2 cm from the emitter connected with coaxial cable to a precision ampliﬁer (PerkinElmer) as illustrated in Figure 9a. The extractor electrode was grounded and the emitter was biased negatively at high voltage. The current measuring electrode in the Faraday cup was biased positively to prevent current loss by secondary electron emission. Figure 9b shows that the emitted current has a maximal drift of 0.5% over 1 hour. The drift is

218

Niels de Jonge

2.43

118 V

Electrode

Ip /nA

2.42 Faraday cup

2.41 Electron beam

2.4 Electron source

1300 V

0

10

(a)

20

30 40 Time (min)

50

60

(b)

FIGURE 9 Measurements of the emission stability of CVD-grown carbon nanotube. (a) The probe current was measured using a Faraday cup with a biased electrode placed above the electron source. (b) The probe current (Ip) measured as a function of time at U = 270 V, I = 190 nA, T = 800◦ K, and a vacuum level of 2 × 10−10 torr.

1027

DI 2p /I 2pDf (sec)

Urms(V)

1024

1025

1026 0.1

1028

1029

10210 1

f (Hz) (a)

10

0.1

1 f (Hz)

10

(b)

FIGURE 10 Spectral analysis of the probe current of the carbon nanotube of Figure 9. (a) Urms as a function of f . The data present the average of 10 measurements recorded in a period of 10 minutes. The noise of the measuring system (dotted line) and a constant fitted to it (dashed line) are also plotted. (b) The normalized spectral density and a fit with the function y = a/xb (dashed line). Source: de Jonge et al., 2005b.

probably caused by a small change of the temperature. The short-term peak to peak ﬂuctuations varied by less than 0.2%. The frequency characteristic of the emission process was investigated by recording the fast Fourier transformation (FFT) of Ip with a spectrum analyzer (Hewlett-Packard) (de Jonge et al., 2005b). The related root mean square (RMS) voltage Urms is plotted in Figure 10a corrected for the background noise of the measuring system. The signal was normalized on the

Carbon Nanotube Electron Sources for Electron Microscopes

219

average probe current to obtain the normalized spectral density

Sn ( f ) =

Ip2 Ip2 f

,

which gives the spectral noise power of the emitter (Figure 10b). The curve shows that Sn ( f ) is proportional to 1/f. At 25 Hz, Sn ( f ) is almost equal to the shot noise limit = 2e/Ip = 1.3 × 10−10 sec, which is in general the lowest possible noise level. Integrating the measured Sn ( f )

f2 Sn ( f )df = f1

<Ip2> Ip2

= np2

leads to a noise percentage (np) of 0.02% as a measure for the emission stability in the 0.1–25 Hz bandwidth. The noise percentage is the inverse signal to noise ratio. These results show a highly stable emission process for a ﬁeld emitter (Hainfeld, 1977) and the stability is even better than that of the Schottky emitter (Swanson, Schwind, and Jon, 1997; Tuggle, Swanson, and Orloff, 1979).

5.3. Brightness Measurements The most important parameter for the resolution in high-resolution electron beam instruments is the reduced brightness (Hawkes and Kasper, 1996). The reduced brightness Br measures the amount of current that can be focused into a spot of a certain size and from a certain solid angle. It is a function of the radius of the virtual source rv , the angular current density I corresponding to the brightest fraction of the emitted electron beam, and the beam potential U:

Br =

I . πrv2 U

(10)

The virtual source of an electron emitter is the area from which the electrons appear to originate when tracing back their trajectories. To determine the virtual source size, nanotubes were mounted as an electron source in a point-projection electron microscope (Spence, Qian, and Silverman, 1994) in an ultrahigh vacuum system. In this setup, the emitter is positioned at a small distance (a few micrometers) from one of the holes in the carbon ﬁlm of a TEM grid. Electrons emitted by the source generate an image on the screen at a distance of a few tens of centimeters from the source (Figure 11).

220

Niels de Jonge

2 mm

2 mm

(a)

(b)

FIGURE 11 Images generated with the point projection microscope using a multiwalled carbon nanotube as an electron source, recorded with a phosphor screen and a microchannel plate. (a) The Fresnel interference pattern at the edges of the hole in the carbon grid. The electron energy was 55 eV. (b) The fringe pattern of a carbon fiber. The electron energy was 39 eV. Source: de Jonge, 2004.

The radius of the virtual source size varied between 2.1 and 3.0 nm for arcdischarge–grown multiwalled nanotubes (de Jonge et al., 2002; de Jonge 2004). Furthermore, the radius of the virtual source size is almost equal to the real radius of the nanotube, and the radius as found by ﬁtting the Fowler–Nordheim model R; thus,

rv ∼ = R.

(11)

The angular current density I was measured with a Faraday cup. To account for the focusing effect of the beam energy, the angular density must be divided by the beam energy U, thus obtaining the normalized or reduced angular current density Ir . Ir was approximately linear with the total emitted current I. Furthermore, the absolute values did not deviate more than a factor of 2 among eight different nanotubes from the equation

Ir ∼ = 0.02I × I/srV.

(12)

We can now write a simple expression for the reduced brightness (de Jonge et al., 2005c) as follows:

Br =

Ir ∼ 0.02I = 0.04 J(d, φ), = πrv2 πR2

(13)

Carbon Nanotube Electron Sources for Electron Microscopes

221

with J the current density at the emitting surface as a function of the tunneling parameter d (which is a measure for the fraction of the energy spread independent of the temperature) and the workfunction φ. This equation has an important implication. Because φ is constant for a certain material, the reduced brightness is a function of d only. Typical values thus obtained for the brightness were 1 × 109 Am−2 sr−1 V−1 . When the source is applied in an electron microscope the brightness will be somewhat lower due to the effect of Coulomb interactions between the electrons in the emitted beam (Kruit, Jansen, and Orloff, 1997). This effect on the brightness was calculated and was found to be small only in our parameter range (de Jonge, 2004). Current state-of-theart electron sources for electron microscopes are the Schottky emitter and the cold ﬁeld-emission gun (CFEG), with values of the reduced brightness varying between 1 × 107 and 2 × 108 Asr−1 m−2 V−1 . Our measurements show that the carbon nanotube electron emitter outperforms these sources by at least a factor of ten.

5.4. Figure of Merit The new model of the brightness actually gives the ﬁgure of merit. We compared this model with our data containing datasets of both Br and d for a total of ﬁve carbon nanotubes. Note that the presentation of just one of these two parameters has a limited use as high brightness is usually obtained at the expense of a low energy spread, affecting the performance of the source in an electron optical system. Figure 12 shows the data obtained for the carbon nanotubes. The data correspond well with the theory. An important aspect of the model is that it is independent 1011

B r (A/sr m2V)

1010 109 108

CNT CNT CNT CNT CNT

107 106 105 0.15

0.2

0.25

0.3 0.35 d (eV)

1 2 3 4 5

0.4

0.45

FIGURE 12 Reduced brightness (Br ) as a function of the tunneling parameter (d) measured for 5 carbon nanotubes and compared with the calculated curve using Br = 0.04 J (dashed line), within its error range of a factor of 3 as indicated by the dotted lines. Source: de Jonge et al., 2005c.

222

Niels de Jonge

of the actual shape of the carbon nanotube, expressed in R, its length l, and other geometrical factors, such as the geometry of the surrounding electrodes. Our systematic study shows that it is now possible to select the optimal carbon nanotube electron point source from carbon nanotubes with a range of diameters and lengths, because the relation between Br and d, determining the optical performance of the source, does not depend on these parameters. We expect that our result is more generally applicable to ﬁeld emitters of nanometer size, for example, nanometer-sized tungsten tips and metallic nanowires. The upper and lower limits are set by the minimal current required for sufﬁcient signal to noise in the application (e.g., imaging) and the maximum current at which the emitter can still operate properly.

6. EVALUATION OF THE USE OF CARBON NANOTUBE ELECTRON SOURCES FOR ELECTRON MICROSCOPES The previous Sections have shown how an electron source can be made from a carbon nanotube, and have discussed in detail the electron emission properties. The electron optical properties show signiﬁcant advantages compared with state-of-the-art electron sources—for example, a low energy spread, high brightness, and good emission stability. The question is now whether the nanotube electron source also can be applied practically in an electron microscope. This section addresses several aspects of this question. First, we describe the noise level as a function of emission current and vacuum level. Then the ﬁgure of merit under realistic conditions is given and the lifetime is addressed. Some design considerations are discussed with the goal of using the source in an SEM. Finally, we present the testing of a carbon nanotube electron source in an SEM.

6.1. Noise Level The noise characteristics of six carbon nanotube electron sources were recorded (de Jonge et al., 2005b); three were deposited by arc discharge and three from chemical vapor deposition. Measurements also were recorded at different total currents and different vacuum levels. As shown in Figure 13a, the carbon nanotube emitter can be operated at currents up to a few microamperes without losing its stability. The average value of noise percentage was 0.08% with a standard deviation of 0.06% for all seven carbon nanotubes. The worst noise percentage measured was 0.2%, corresponding with signal to noise ratio of 500. For imaging applications, it is desirable to resolve at least 8 bits in image contrast (256 gray levels). Thus, the carbon nanotube electron source meets this requirement.

Carbon Nanotube Electron Sources for Electron Microscopes

223

10

1

np (%)

np (%)

1 0.1

0.1

0.01

1000

100

I (nA) (a)

0.01 10210

1029 1028 Vacuum (torr) (b)

1027

FIGURE 13 Noise percentages of seven carbon nanotubes measured in the frequency range of 0.1−25 Hz with a measuring time of 10 minutes. The probe current was typically 2–10 nA, depending on the total current level and the distance from the carbon nanotube to the Faraday cup. The temperature was between 700◦ and 900◦ K. (a) Noise percentage as a function of the total current for a vacuum level of 10−10 torr. (b) Noise percentage as a function of the vacuum level for total current levels of 90–500 nA. Source: de Jonge et al., 2005b.

The vacuum level is an important factor that inﬂuences the noise observed in the emission current. At lower vacuum levels, the rate of molecule adsorbtion and subsequent desorbtion (because the carbon nanotube is at 800◦ K) becomes higher, thus affecting the emitter stability. We measured noise percentage as a function of vacuum level (Figure 13b). Clearly, at higher levels the noise percentage increases. For imaging applications, an noise percentage above 0.39% (1/256 gray levels) is undesirable. Thus, from Figure 13b we can see that a vacuum level of 2 × 10−8 torr or better is needed. Operation at this vacuum level is beneﬁcial for low-cost electron columns, but further research in this direction is required as large current jumps occasionally took place at higher vacuum levels. The current ﬂuctuation and noise percentages of the carbon nanotubes measured here are extremely low. Metal cold ﬁeld-emitters typically exhibit 3–5% ﬂuctuations of the probe current under similar measurement conditions (Hainfeld, 1977). The noise percentage of value 0.12% is even better than that obtained from the state-of-the-art Schottky emitter of 0.2% at a bandwidth of 10 kHz (Swanson, Schwind, and Jon, 1997; Tuggle, Swanson, and Orloff, 1979). (NB: the Schottky emitter is often chosen in the electron microscopy industry over metal ﬁeld emitters because of its stability.) Thus, the carbon nanotube is a very promising candidate as a stable and low-noise electron source for imaging applications.

224

Niels de Jonge

6.2. The Figure of Merit Under Realistic Conditions At room temperature and at low currents, d is almost equal to the energy spread E, and therefore Figure 12 presents the ﬁgure of merit of the carbon nanotube electron source. However, some heating to 600–800◦ K was required to obtain stable emission, which broadens the energy spectrum. A secondary effect is that Joule heating at larger currents also leads to a broadening of the energy spectrum. The ﬁgure of merit of electron sources under realistic conditions, Br versus E, is shown in Figure 14. For comparison, data of state-of-the-art electron sources, the tungsten CFEG and the Schottky emitter are included (Hainfeld 1977; Swanson, Schwind, and Jon, 1997). As cold ﬁeld-emitters, the carbon nanotube and the CFEG have an almost identical E. However, the CFEG often lacks emission stability, which the carbon nanotube clearly has (see Figure 13.) Note also that the carbon nanotube has a much higher Br than the CFEG at high currents. Our values imply a signiﬁcant improvement with respect to the Schottky emitter, either a lower E at similar Br , or a much higher Br at similar E.

Br (A/sr m2 V)

1010 109 108 CNT Schottky CFEG

107 106 0.2

0.3

0.4

0.5 0.6 DE (eV)

0.7

0.8

FIGURE 14 Reduced-brightness (Br ) as a function of the energy spread (E) measured for several carbon nanotubes at currents of 10–500 nA and temperatures of 600◦ –700◦ K and calculated curve (dashed line) and error range (dotted line). The data of the Schottky emitter and the CFEG are also included. Source: de Jonge et al., 2005c.

6.3. Lifetime Testing A lifetime setup was built and three carbon nanotube electron sources were tested successfully for 2 years, running continuously at 100 nA of emission current, while being heated to 900◦ K in a vacuum of 10−10 torr (Figure 15). One electron source died after 18 months because of a failed power supply. The electronics setup was such that it regulated the emission current to remain a level of 100 nA within a range of 20%. From this experiment it can be concluded that carbon nanotube electron sources have a long lifetime as expected because of their extremely rigid structure. It must be noted that

Carbon Nanotube Electron Sources for Electron Microscopes

225

FIGURE 15 Lifetime setup testing four carbon nanotube electron sources.

the ion back-sputtering does not occur toward the tip of the nanotubes, which would lead to rapid destruction, but rather toward the shaft of the tungsten support tip.

6.4. Considerations for a Source Module A source module for use with a carbon nanotube electron source has to be designed. The extractor of the commonly used Schottky emitter is at a potential of 4.5 kV and forms an essential part of the condenser lens. For the carbon nanotube the extraction voltage requires only a few hundred volts, which demands a redesign of the condenser lens. However, only some minor modiﬁcations are necessary. In the original design, the extractor is positioned 0.5 mm from the tip and the beam-deﬁning aperture (BDA) is at 2.8 mm. For the carbon nanotube source, the BDA can be placed closer to the tip because both the total current and the extraction voltage are much lower than for the Schottky emitter, leading to a reduction of the deposited energy and associated heating of the aperture. The power that

226

Niels de Jonge

must be dissipated by the extractor is P = 4500 V · 200 μA = 0.9 W with the Schottky emitter and only P = 310 V · 1 μA = 0.3 mW with the carbon nanotube (i.e., 3000 times less power). The BDA can be placed directly in front of the tip at a distance of 0.5 mm and serve as extractor (Figure 16). The extraction voltage of the carbon nanotube source is only ∼ 300 V. Because of this low potential a so-called cathode lens can be used as condenser lens. The required condenser voltage is between 300 V and the column potential; the optimal value depends on the current that is required. The optical properties of a cathode lens are superior to other types of electrostatic lenses that can be used as condenser lens (e.g., Einzel lenses). It was debated in literature that the high brightness of the carbon nanotube source would not be a beneﬁt because of problems with the condensor lenses (Kruit, Bezuijen, and Barth, 2006). However, this conclusions was drawn without taking into account that the extractor electrode and gun lens have to be optimized for a cold ﬁeld-emission gun. In fact, the cold emission gun size as the carbon nanotube source, has been successfully used for scanning TEM (STEM) imaging (Crewe, Wall, and Langmore, 1970; Hainfeld, 1977; Lupini et al., 2007) because of its high brightness.

Extractor

Condenser

FE source BDA

(a) Extractor

Condenser

CNT source BDA

(b)

FIGURE 16 The design of the extractor and the objective lens of the Schottky emitter (a) and the modified design for the carbon nanotube (CNT) emitter (b). FE, field emission; BDA, beam-defining aperture.

227

Carbon Nanotube Electron Sources for Electron Microscopes

To demonstrate the beneﬁts of applying the carbon nanotube source we calculated the spot size obtained in an SEM equipped with an immersion lens (Br = 3 × 109 Am−2 sr−1 V−1 , U = 310 V, rv = 2.5 nm, E = 0.3 eV). These calculations were also conducted for the Schottky emitter for comparison (Br = 3 × 108 Am−2 sr−1 V−1 , U = 4500 V, rv = 12 nm, E = 0.8 eV). The calculations were performed numerically with software of local design. The program simulates an electron ray path and calculates the various contributions to the spot size dtot —diffraction ddiff , chromatic aberration dcc , spherical aberration dcs , Coulomb blurring dblur , and the geometric effect of the source size dgeo . The different contributions are combined according to Barth and Kruit (1996) as follows:

2 dtot

=

4 ddiff

4 + dcs

1.3/4

+

2 dgeo

2 + dblur

1.3/2 2/1.3

1/2 2 + dcc

.

(14)

In the spot size calculations we considered only the lens aberrations of the objective lens assuming a parallel incoming beam. This assumption is valid if the magniﬁcation is not too close to 1 (say, M < 0.5) and if the carbon nanotube source is equipped with an optimized condenser lens. Table II compares spot sizes for both type of sources at 10 pA and 10 nA for landing energies of 1 keV and 0.5 keV. A drastic decrease in spot size can be obtained, varying from a factor of 1.7 at 10 pA to a factor of 2.4 at 10 nA. The result is shown graphically in Figure 17. These calculations show that the carbon nanotube source can improve the resolution of SEM columns equipped with an immersion lens compared with the Schottky emitter. The resolution improves by a factor of 1.7 at a current of 10 pA to 1.8 nm and by a factor of 2.4 at 10 nA to 2.6 nm, at a landing energy of 1 keV. This makes the carbon nanotube source interesting for high-resolution applications. Moreover, the carbon nanotube TABLE II Theoretical spot-sizes I [nA] U (landing voltage) dSchottky [nm] [keV]

0.01 10 0.01 10

1.0 1.0 0.5 0.5

3.0 6.3 5.1 10.5

M (magnification)

dCNT [nm]

M (magnification)

0.018 0.29 0.06 0.47

1.8 2.6 3.1 4.2

0.015 0.37 0.024 0.56

Spot sizes calculated with the Schottky emitter (dSchottky ) and the carbon nanotube source (dCNT ) of an SEM immersion lens. The immersion lens at full excitation has a working distance of 1 mm, a Cs (spherical aberration) of 0.6 mm, and a Cc (chromatic aberration) of 0.8 mm.

228

Niels de Jonge

7 6

d (nm)

5 4

3 2 1 0 0.01

0.1

1

10

I (nA)

FIGURE 17 Calculated spot sizes of an SEM with an immersion lens for the carbon nanotube emitter (solid line) and the Schottky emitter (dashed line) at a landing energy of 1 keV.

source would allow recording of high-resolution images at 1000 times faster recording times than with the present SEM.

6.5. Prototype Carbon Electron Source in SEM A prototype electron source was built at Philips Research for an SEM from an individual carbon nanotube. The carbon nanotube was mounted on a standard tungsten ﬁlament (as used also for the commercially available Schottky emitter) with a mounting procedure as described in Section 2. A standard source module in which a ﬁlament is placed in the SEM and containing an extractor electrode, condensor lens, and so on was modiﬁed for the carbon nanotube. The ﬁlament with the carbon nanotube was positioned in this module, and the position of the tungsten tip was aligned with the optical axis with a precision of 5 μm using a turntable and a specially designed holder. The source module with carbon nanotube was then placed in the SEM (Figure 18, left). The standard bake-out procedure of the SEM was used, leading to a vacuum level of 1 × 10−9 torr. The source was ﬁrst heated to 600◦ C to clean it from adsorbed species and then operated while it was heated. With 90-V extraction voltage, a stable current of 100 nA was obtained. The electron beam was aligned in the column and images were taken of Sn spheres. As shown in the right image of Figure 18, a high-resolution image was taken. The resolution was analyzed by drawing the 25–75% edges in the image and plotting a histogram of the measured edge widths (Figure 19). The maximum of counts is at 3.0 nm, which is considered the resolution of the SEM. The commercial Schottky emitter provides 3.0 nm as well at these settings. This shows that the carbon nanotube electron source provides at

229

Carbon Nanotube Electron Sources for Electron Microscopes

Acc.V Spot Magn Det WD 1.00 KV 1.0 1000000x SE 5.0 cnt

50 nm

FIGURE 18 The photo at the left shows the SEM with a carbon nanotube electron source. The image at the right shows Sn spheres using the carbon nanotube electron source, a probe current of 20 pA, a landing voltage of 1 kV, and a working distance of 5 mm.

350 300

Counts

250 200 150 100 50 0

0

1

2

3

4

5

6

7

8

9

Resolution (25%–75% edge)/nm

(a)

(b)

FIGURE 19 (a) 25–75% Edges drawn in the image of Figure 18 of Zn spheres. (b) Statistics for the edges with a maximum at 3.0 nm.

least the same resolution in an SEM with respect to the Schottky emitter. It is expected that the resolution can be further improved because the resolution measurement was limited by vibrations of the sample stage, (visible as horizontal lines in the image), by interferences of the electronics, and by the size of the Sn spheres. It is expected that with simple improvements a resolution improvement of a factor of 2 can be obtained.

7. SPECIFICATIONS Table III summarizes the speciﬁcations of an electron source made from a thin (up to ﬁve walls) multiwalled carbon nanotube with a closed cap and cleaned surface of high quality (little defects).

230

Niels de Jonge

TABLE III Specifications of the Carbon nanotube Electron Source

Virtual source size Typical operating current Typical extraction voltage Reduced angular current density Beam proﬁle Noise percentage Brightness versus energy spread

Lifetime

2–3 nm 100 nA 100 V @ 0.2 mm, 400 V @ 20 mm 3 nAsr−1 V−1 @ 100 nA Broad beam, typically 5◦ –10◦ off-axis < 0.2% 1 × 108 Am−2 sr−1 V−1 @ 0.35 eV 1 × 109 Am−2 sr−1 V−1 @ 0.4 eV 1 × 1010 Am−2 sr−1 V−1 @ 0.5 eV >2 years continuous

8. SUMMARY AND OUTLOOK Electron sources were constructed from individual multiwalled carbon nanotubes with closed caps and thoroughly cleaned surfaces. Nanotubes from both chemical vapor deposition growth and arc discharge growth were investigated. These emitters provide a highly stable emission current up to a threshold current of a few microamperes. At too-large currents several processes occurs such as splitting, breaking, and cap closing. The emission process is ﬁeld emission for a workfunction of 5 eV. The electron optical performance is highly beneﬁcial for use of carbon nanotubes as high-brightness point sources in electron microscopes and advantageous with respect to state-of-the-art electron sources. The lifetime is at least 2 years. We have tested the source successfully in an SEM. The future use of the carbon nanotube electron source in electron microscopes will depend on the successful development of a production process. First, an industrial process is needed to mount a closed-cap carbon nanotube inside an electron miroscope so that it provides an electron beam with a cone around the optical axis of the microscope. This requires not only that the carbon nanotube is mounted in a direction parallel to the optical axis, but also that the cap structure is deﬁned such that the electrons are emitted primarily in a forward direction. One approach would be to further develop the process of mounting as used in these studies (de Jonge, Lamy, and Kaiser, 2003). The procedure can be simpliﬁed by using a puriﬁed sample of carbon nanotubes of similar diameter aligned parallel and protruding from a knife edge. Second, the cap closure procedure needs to be optimized under highly controlled conditions so that the cap always closes in the same way. The emitters must be tested by recording their emission patterns. An alternative approach would be to use short (a few micrometers) carbon nanotubes with caps that are already closed and ﬁnd a procedure to place many

Carbon Nanotube Electron Sources for Electron Microscopes

231

of them on a knife edge aligned parallel. These can then simply be picked up and a cap-closing procedure will not be necessary. In this case, the difﬁculties will shift to the production of such carbon nanotubes. Third, carbon nanotubes can be grown directly on the support tip. Promising results were obtained with CVD-grown carbon nanotubes on tungsten tips (Mann, Teo, and Milne, 2006). In this case, however, the nanotubes exhibit many structural defects and do not provide highly stable emitters. The carbon nanotubes must be heated to the carbonization temperature (Saito, Dresselhaus, and Dresselhaus, 1998) to obtain a highly pure structure. Heating, however, may drive the carbon nanotube off the tungsten tip. Perhaps laser heating could be used to heat only the carbon nanotube and not the tungsten tip. The carbon nanotube source would beneﬁt applications that require high current and high resolution—for example, analysis and imaging in an SEM. The carbon nanotube source also would improve the imaging capabilities of an STEM, typically operating at voltages of 100–300 kV. The highest resolution yet obtained with aberration-corrected STEMs approaches 0.5 Å (Nellist et al., 2004) and can be further improved with a source of higher brightness and better stability than the Schottky source (Lupini et al., 2007) or the CFEG. Materials analysis using electron energy loss spectrospcopy can be enhanced by focusing more current in a high-resolution probe, since much more current can be focused in a small spot with a high-brightness source. The beneﬁts will be faster imaging and reduced effects of drift and mechanical instabilities. The high-brightness source also will beneﬁt time-resolved imaging in in situ electron microscopy. Because of the low extraction voltage and the resulting low-power consumption of the carbon nanotube source, it is of particular interest to make micrometer-sized SEMs with these sources (Saini et al., 2006). High-brightness sources are also needed for other applications, such as for interferometry in strain measurements (Hytch et al., 2008) and for the generation of quantum-degenerate electron beams (Zolotorev, Commins, and Sannibale, 2007). The ﬁgure of merit of the reduced brightness versus the energy spread perhaps can be improved by reducing the work function by doping, for example, with cesium (Zhao et al., 2006), if this can be done in a manner that preserves emission stability.

ACKNOWLEDGMENTS I thank M. Allioux, P. Bachmann, M. Buijs, B. Buijsse, A.M. Calvosa, M.A. Doytcheva, E.C. Heeres, M. Kaiser, Y. Lamy, S.A.M. Mentink, M. Meuwese, W.I. Milne, T.H. Oosterkamp, J.T. Oostveen, A.G. Rinzler, T. van Rooij, K. Schoots, G. Schwind, K.B.K. Teo, P.C. Tiemeijer, and G.N.A. van Veen. This work was supported by FEI Company, the Dutch Ministry of Economic Affairs, the EPSRC, and the EC.

232

Niels de Jonge

REFERENCES Barth, J. E., and Kruit, P. (1996). Addition of different contributions to the charged particle probe size. Optik 101, 101–109. Colbert, D. T., Zhang, J., McClure, S. M., Nikolaev, P., Chen, Z., Hafner, J. H., Owens, D. W., Kotula, P. G., Carter, C. B., Weaver, J. H., Rinzler, A. G., and Smalley, R. (1994). Growth and sintering of fullerene nanotubes. Science 266, 1218–1222. Crespi, V. H., Chopra, N. G., Cohen, M. L., Zettl, A., and Louie, S. G. (1996). Anisotropic electron-beam damage and the collapse of carbon nanotubes. Phys. Rev. B 54, 5927–5931. Crewe, A. V., Wall, J., and Langmore, J. (1970). Visibility of single atoms. Science 168, 1338–1340. de Jonge, N., Lamy, Y., Schoots, K., and Oosterkamp, T. H. (2002). High brightness electron beam from a multi-walled carbon nanotube. Nature 420, 393–395. de Jonge, N., Lamy, Y., and Kaiser, M. (2003). Controlled mounting of individual multi-walled carbon nanotubes on support tips. Nano Lett. 3, 1621–1624. de Jonge, N. (2004). The brightness of carbon nanotube electron emitters. J. Appl. Phys. 95, 673–681. de Jonge, N., Allioux, M., Doytcheva, M., Kaiser, M., Teo, K. B. K., Lacerda, R. G., and Milne, W. I. (2004). Characterization of the ﬁeld emission properties of individual thin carbon nanotubes. Appl. Phys. Lett. 85, 1607–1609. de Jonge, N., and Bonard, J. M. (2004). Carbon nanotube electron sources and applications. Phil. Trans. R. Soc. London A 362, 2239–2266. de Jonge, N., Doytcheva, M., Allioux, M., Kaiser, M., Mentink, S. A. M., Teo, K. B. K., Lacerda, R. G., and Milne, W. I. (2005a). Cap closing of thin carbon nanotubes. Adv. Mater. 17, 451–455. de Jonge, N., Allioux, M., Oostveen, J. T., Teo, K. B. K., and Milne, W. I. (2005b). Low noise and stable emission from carbon nanotube electron sources. Appl. Phys. Lett. 87, 133118-1–3. de Jonge, N., Allioux, M., Oostveen, J. T., Teo, K. B. K., and Milne, W. I. (2005c). The optical performance of carbon nanotube electron sources. Phys. Rev. Lett. 94, 186807-1–4. Dean, K. A., and Chalamala, B. R. (2003). Experimental studies of the cap structures of singlewalled carbon nanotubes. J. Vac. Sci. Technol. B 21, 868–871. Doytcheva, M., Kaiser, M., and De Jonge, N. (2006). In situ transmission electron microscopy investigation of the structural changes in carbon nanotubes during electron emission at high currents. Nanotechnology 17, 3226–3233. Edgcombe, C. J., and de Jonge, N. (2007). Deduction of work function of carbon nanotube ﬁeld emitter by use of curved-surface theory. J. Phys. D 40, 4123–4128. Fowler, R. H., and Nordheim, L. (1928). Electron emission in intense electric ﬁelds. Proc. Roy. Soc. London A 119, 173–181. Gao, R., Pan, Z., and Wang, Z. L. (2001). Work function at the tips of multiwalled carbon nanotubes. Appl. Phys. Lett. 78, 1757–1759. Good, R. H., and Mueller, E. W. (1956). “Field Emission”. Handbuch der Physik, XXI. S. Fluegge. Springer Verlag, Berlin, pp. 176–231. Groening, O., Kuettel, O. M., Emmenegger, C., Groening, P., and Schlapbach, L. (2000). Field emission properties of carbon nanotubes. J. Vac. Sci. Technol. B 18, 665. Hainfeld, J. F. (1977). Understanding and using ﬁeld emission sources. Scanning Electron Microscopy 1, 591–604. Hata, K., Takakura, A., and Saito, Y. (2001). Field emission microscopy of adsorption and desorption of residual gas molecules on a carbon nanotube tip. Surface Science (Netherlands) 490, 296. Hawkes, P. W., and Kasper, E. (1996). “Principles of Electron Optics II: Applied Geometrical Optics. Academic Press, London.

Carbon Nanotube Electron Sources for Electron Microscopes

233

Hytch, M. J., Houdellier, F., Hue, F., and Snoeck, E. (2008). Nanoscale holographic interferometry for strain measurements in electronic devices. Nature 453, 1086–1089. Kaiser, M., Doytcheva, M., Verheijen, M. A., and de Jonge, N. (2006). In-situ transmission electron microscopy observations of individually selected freestanding carbon nanotubes during ﬁeld emission. Ultramicroscopy 106, 902–908. Kruit, P., Bezuijen, M., and Barth, J. E. (2006). Source brightness and useful beam current of carbon nanotubes and other very small emitters. J. Appl. Phys. 99, 24315-1–7. Kruit, P., Jansen, G. H., and Orloff, J. (1997). “Space Charge and Statistical Coulomb Effects: Handbook of Charged Particle Optics.” CRC Press, Boca Raton, FL. Lacerda, R. G., Teh, A. S., Yang, M. H., Teo, K. B. K., Rupesinghe, N. L., Dalal, S. H., Koziol, K. K. K., Roy, D., Amaratunga, G. A. J., Milne, W. I., Chowalla, M., Hasko, D. G., Wyczisk, F., and Legagneux, P. (2004). Growth of high-quality single-wall carbon nanotubes without amorphous carbon formation. App. Phys. Lett. 84, 269. Lupini, A., Rashkeev, S., Varela, M., Borisevich, A., Oxley, M. P., van Benthem, K., Peng, Y., de Jonge, N., Veith, G. M., Chisholm, M. F., and Pennycook, S. J. (2007). Scanning transmission electron microscopy. Nanocharacterization. E. J. Kirkland and Hutchison, J. L. Cambridge, Royal Society of Chemistry. Mann, M., Teo, K. B. K., and Milne, W. I. (2006). Direct growth of multi-walled carbon nanotubes on sharp tips for electron microscopy. NANO 1, 35–39. Nellist, P. D., Chisholm, M. F., Dellby, N., Krivanek, O. L., Murﬁtt, M. F., Szilagyi, Z. S., Lupini, A. R., Borisevich, A., Sides, W. H., and Pennycook, S. J. (2004). Direct sub-angstrom imaging of a crystal lattice. Science 305, 1741. Paulmier, T., Balat-Pichelin, M., Le Queau, D., Berjoan, R., and Robert, J. F. (2001). Physicochemical behavior of carbon materials under high temperature and ion radiation. App. Surface Sci. (Netherlands) 180, 227. Purcell, S. T., Vincent, P., Journet, C., and Binh, V. T. (2002). Hot nanotubes: stable heating of individual multiwall carbon nanotubes to 2000◦ K induced by the ﬁeld-emission current. Phys. Rev. Lett. 88, 105502-1–4. Rinzler, A. G., Hafner, J. K., Colbert, D. T., Smalley, R. E., Ed.by, B., Bethune, D. S., Chiang, L. Y., Ebbesen, T. W., Metzger, R. M., and Mintmire, J. W. (1995). Field emission and growth of fullerene nanotubes. Sci. Technol. Fullerene Mater. Symp. 61. Saini, R., Jandric, Z., Gammell, J., Mentink, S. A. M., and Tuggle, D. (2006). Manufacturable MEMS miniSEMs. Microelectron-Eng. (Netherlands) 83, 1376–1381. Saito, R., Dresselhaus, G., and Dresselhaus, M. S. (1998). Physical properties of carbon nanotubes. Imperial College Press, London. Spence, J. C. H., Qian, W., and Silverman, M. P. (1994). Electron source brightness and degeneracy from Fresnel fringes in ﬁeld emission point projection microscopy. J. Vac. Sci. Technol. A 12, 542. Swanson, L. W., Schwind, G. A., and Jon, O. (1997). “A Review of the ZrO/W Schottky Cathode: Handbook of Charged Particle Optics.” CRC Press, Boca Raton New York. Tuggle, D., Swanson, L. W., and Orloff, J. (1979). Application of a thermal ﬁeld emission source for high resolution, high current e-beam microprobes. J. Vac. Sci. Technol. 16, 1699. Wang, Z. L., Gao, R. P., Poncharal, P., and de Heer, W. A. (2002). In situ imaging of ﬁeld emission from individual carbon nanotubes and their structural damage. Appl. Phys. Lett. 80, 856–858. Zhao, G. P., Zhang, Q., Zhang, H., Yang, G., Zhou, O., Qin, L. C., and Tang, J. (2006). Field emission of electrons from a Cs -doped single carbon nanotube of known chiral indices. Appl. Phys. Lett. 89, 263113-1–3. Zolotorev, M., Commins, E. D., and Sannibale, F. (2007). Proposal for a quantum-degenerate electron source. Phys. Rev. Lett. 98, 184801-1–4.

CHAPTER

4 Localized Waves: A Review Erasmo Recami* and Michel Zamboni-Rached†

Contents

1 Localized Waves: A Scientific and Historical Introduction 1.1 Introduction and Preliminary Remarks 2 A More Detailed Introduction 2.1 The Localized Solutions 3 Complementary Material: A Historical Perspective (Theoretical and Experimental) 3.1 Introduction 3.2 Historical Recollections—Theory 3.3 A Glance at the Experimental State of the Art 4 Structure of Nondiffracting Waves and Some Interesting Applications 4.1 Foreword 4.2 Spectral Structure of Localized Waves and the Generalized Bidirectional Decomposition 4.3 Space-Time Focusing of X-Shaped Pulses 4.4 Chirped Optical X-Type Pulses in Material Media 5 “Frozen Waves” and Subluminal Wave Bullets 5.1 Modeling the Shape of Stationary Wave Fields: Frozen Waves 5.2 Subluminal Localized Waves (or Bullets) 5.3 A First Method for Constructing Physically Acceptable Subluminal Localized Pulses 5.4 A Second Method for Constructing Subluminal Localized Pulses 5.5 Stationary Solutions with Zero-Speed Envelopes 5.6 The Role of Special Relativity and Lorentz Transformations 5.7 Nonaxially Symmetric Solutions: Higher-Order Bessel Beams Acknowledgments References

236 236 243 245 256 256 257 265 277 277 279 291 294 301 301 317 320 327 330 336 340 344 344

* Faculty of Engineering, University of Bergamo, Bergamo, Italy; and INFN, Sezione di Milano, Milan, Italy † Center of Natural and Human Sciences, Federal University of ABC, Santo André, SP, Brazil Advances in Imaging and Electron Physics, Volume 156, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01404-3. Copyright © 2009 Elsevier Inc. All rights reserved.

235

236

Erasmo Recami and Michel Zamboni-Rached

1. LOCALIZED WAVES: A SCIENTIFIC AND HISTORICAL INTRODUCTION 1.1. Introduction and Preliminary Remarks Diffraction and dispersion have long been known as phenomena that limit the applications of (optical, for instance) beams or pulses. Diffraction is always present and affects any waves that propagate in two-dimensional (2D) or three-dimensional (3D) unbounded media, even when homogeneous. Pulses and beams are composed of waves traveling along different directions, which produce a gradual spatial broadening (Born and Wolf, 1998). This effect is a limiting factor whenever a pulse is needed that maintains its transverse localization—for example, in free space communications (Willebrand and Ghuman, 2001), image forming (Goodman, 1996), optical lithography (Ito and Okazaki, 2000; Okazaki, 1991), electromagnetic tweezers (Ashkin et al., 1986; Curtis, Koss, and Grier, 2002), and so on. Dispersion acts on pulses propagating in material media and causes mainly a temporal broadening. This effect is known to be due to the variation of the refraction index with the frequency, so that each spectral component of the pulse possesses a different phase velocity. This entails a gradual temporal widening, which constitutes a limiting factor when a pulse is needed that maintains its time width—for example, in communication systems (Agrawal, 1995). Therefore, it is important to develop techniques capable of reducing such phenomena. The localized waves (LWs), also known as nondiffracting waves, can indeed resist diffraction for a long distance in free space. Such solutions to the wave equations (and, in particular, to the Maxwell equations, under weak hypotheses) were theoretically predicted years ago (Barut, Maccarrone, and Recami, 1982; Bateman, 1915; Caldirola, Maccarrone, and Recami, 1980; Courant and Hilbert, 1966; Recami and Maccarrone, 1980, 1983; Maccarrone, Pavsic, and Recami, 1983; Stratton, 1941) (also compare with Recami, Zamboni-Rached, and Dartora (2004), as well as Section 3 below), were mathematically constructed in more recent times (Lu and Greenleaf, 1992a; Lu, Greenleaf, and Recami, 1996; Recami, 1998; Shaarawi and Besieris, 2000a; Ziolkowski, Besieris, and Shaarawi, 2000), and soon thereafter were experimentally produced (Lu and Greenleaf, 1992b; Mugnai, Ranfagni, and Ruggeri, 2000; Saari and Reivelt, 1997). Today localized waves are well established theoretically and experimentally and have innovative applications not only in vacuum but also in material (linear or nonlinear) media, and are able to resist also dispersion. Their potential applications are being intensively explored, always with surprising results, in ﬁelds such as acoustics, microwaves, and optics, with great promise also in mechanics, geophysics, and even gravitational

Localized Waves: A Review

237

waves and elementary particle physics. Of note, the so-called frozen waves presented in the ﬁfth section of this chapter are already being applied in high-resolution ultrasound scanning of moving organs in the human body (Lu, Zou, and Greenleaf, 1993, 1994). To restrict our discussion to electromagnetism, recall the present-day studies on electromagnetic tweezers (Arlt et al., 2001; Garcés-Chavez et al., 2002; MacDonald et al., 2002; McGloin, Garcés-Chavez, and Dholakia, 2003), optical (or acoustic) scalpels, optical guiding of atoms or (charged or neutral) corpuscles (Arlt, Hitomi, and Dholakia, 2000; Fan, Parra, and Milchberg, 2000; Rhodes et al., 2002), optical litography (Erdélyi et al., 1997; Garcés-Chavez et al., 2002), optical (or acoustic) images (Herman and Wiggins, 1991), communications in free space (Lu and Greenleaf, 1992a; Lu and Shiping, 1999; Ziolkowski, 1989, 1991), remote optical alignment (Salo et al., 1990; Salo, Friberg, and Salomaa, 2001; Vasara, Turunen, and Friberg, 1989), optical acceleration of charged particles, and so on. The following two subsections provide a brief introduction to the theory and applications of localized beams and localized pulses, respectively (Recami, Zamboni-Rached, and Hernández-Figueroa, 2008). Before proceeding, as in any review article (for obvious reasons of space), we had to select a few main topics, and such a choice can only be a personal one.

1.1.1. Localized (Nondiffracting) Beams The term beam refers to a monochromatic solution to the considered wave equation with a transverse localization of its ﬁeld. Herein we explicitly refer to the optical case, but our considerations hold true for any wave equation (vectorial, spinorial, scalar, and in particular, for the acoustic case). The most common type of optical beam is the Gaussian one, whose transverse behavior is described by a Gaussian function. However all common beams are affected by diffraction, which spoils the transverse shape of their ﬁeld, widening it gradually during propagation. As an example, the transverse width of a Gaussian beam doubles when it travels a distance √ zdif = 3πρ02 /λ0 , where ρ0 is the beam initial width and λ0 is its wavelength. A Gaussian beam with an initial transverse aperture of the order of its wavelength will already double its width after traveling just a few wavelengths. It was generally believed that the only wave devoid of diffraction was the plane wave, which does not incur any transverse changes. Some authors have shown that it is not the only one. For instance, in 1941 Stratton developed a monochromatic solution to the wave equation whose transverse shape was concentrated in the vicinity of its propagation axis and represented by a Bessel function. Such a solution, now called a Bessel beam, was not subject to diffraction because no change in its transverse shape

238

Erasmo Recami and Michel Zamboni-Rached

took place with time. Courant and Hilbert (1966) later demonstrated how a large class of equations (including wave equations) admit “non-distorted progressing waves” as solutions. By 1915, Bateman and subsequently researches such as Barut and Chandola (1993), showed the existence of soliton-like, wavelet-type solutions to the Maxwell equations. However, despite the literature reports, they did not garner the attention they deserved. In Stratton’s case, this can be partially justiﬁed since that (Bessel) beam was endowed with inﬁnite energy (as much as the plane waves or Gaussian beams, incidentally), and, moreover it was not square-integrable in the transverse direction. An interesting problem, therefore, is investigating what would happen to the ideal Bessel beam solution when truncated by a ﬁnite transverse aperture. It was not until 1987 that a heuristic answer came from the experiment by Durnin, Miceli, and Eberly when it was shown that a realistic Bessel beam, endowed with wavelength λ0 = 0.6328 μm and central spot1 ρ0 = 59 μm, passing through an aperture with radius R = 3.5 mm can travel ∼85 cm with its transverse intensity shape approximately unchanged (in the region ρ << R surrounding its central peak). In other words, it was shown that the transverse intensity peak, as well as the surrounding ﬁeld, do not change appreciably in shape along a large “depth of ﬁeld.” As a comparison, recall again that a Gaussian beam with the same wavelength and with the central “spot”2 ρ0 = 59 μm, when passing through an aperture with the same radius (R = 3.5 mm) doubles its transverse width after 3 cm, and after 6 cm its intensity is already diminished by a factor of 10. Therefore, in the considered case, a Bessel beam can travel approximately without deformation a distance 28 times larger than a gaussian beam. This remarkable property is due to the fact that when the transverse intensity ﬁelds (whose value decreases with increasing ρ), associated with the rings that constitute the (transverse) structure of the Bessel beam, diffract, they reconstruct the beam itself, all along a large ﬁeld depth. This property depends on the Bessel beam spectrum (wave number and frequency) (Durnin, 1987; Herman and Wiggins, 1991; Salo et al., 1990; Salo, Friberg, and Salomaa, 2001; Vasara, Turunen, and Friberg, 1989), as explained in detail in Zamboni-Rached, Recami, and Hernández-Figueroa (2002). Let us stress that, given a Bessel and a Gaussian beam—both with the same power ﬂux, the same spot ρ0 , and passing through apertures with the same radius R in the plane z = 0—the percentage of the total energy E contained inside the central peak region (0 ≤ ρ ≤ ρ0 ) is smaller

1 Let us deﬁne the central “spot” of a Bessel beam as the distance, along the transverse direction ρ, at which

the ﬁrst zero occurs of the Bessel function characterizing its transverse shape. 2 In the case of a Gaussian beam, let us deﬁne its central “spot” as the distance, along the transverse direction

ρ, at which its transverse intensity has decayed by the factor 1/e.

Localized Waves: A Review

239

for a Bessel than for a Gaussian beam. This differing energy distribution on the transverse plane is responsible for the reconstruction of the Besselbeam central peak even at large distances from the source (and even after an obstacle, provided that its size is smaller than the aperture (Bouchal, Wagner, and Chlup, 1998; Grunwald et al., 2003, 2004, 2005; ZamboniRached, 1998); this nice property is also possessed by the localized pulses as examined below (Zamboni-Rached, 1998)). Most experiments in this area have been performed rapidly and often with use of rather simple apparatus. For example, for the generation of a Bessel beam, Durnin, Miceli, and Eberly (1987a) had recourse to a laser source, an annular slit, and a lens (Figure 1). In a sense, such an apparatus produces what can be regarded as the cylindrically symmetric generalization of a couple of plane waves emitted at angles θ and −θ, with respect to (w.r.t.) the z-direction, respectively (in which case the plane wave intersection moves along z with the speed c/ cos θ). These nondiffracting beams can be generated also by a conic lens (axicon) (see Herman and Wiggins (1991)), or by other means such as holographic elements (MacDonald et al., 1983; Salo, Friberg, and Salomaa, 2001). We stress that many interesting applications of nondiffracting beams are currently being investigated in addition to those of Lu et al. in acoustics. In the optical sector, Bessel beams are used as optical tweezers to conﬁne or move around small particles. In such theoretical and application areas, noticeable contributions include those presented by ZamboniRached (2004, 2006); Zamboni-Rached, Recami, and Hernández-Figueroa (2005), wherein, by suitable superpositions of Bessel beams endowed with the same frequency but different longitudinal wave numbers, stationary envelopes have been mathematically constructed in closed form. These ﬁelds possess a high transverse localization and, more important,

Annular aperture

Lens

␭

R f

a ␦a

Bessel beam

␦a << ␭f R

FIGURE 1 The simple experimental setup used by Durnin, Miceli, and Eberly (1987a) to generate a Bessel beam.

240

Erasmo Recami and Michel Zamboni-Rached

a longitudinal intensity shape that can be freely chosen inside a predetermined space interval 0 ≤ z ≤ L. For instance, a high-intensity ﬁeld with a static envelope can be created within a tiny region, with negligible intensity elsewhere. Section 3 addresses such “frozen waves.”

1.1.2. Localized (Nondiffracting) Pulses As noted in the previous subsection, the existence of nondiffractive (or localized) pulses was predicted years ago: (see Barut, Maccarrone, and Recami, 1982; Bateman, 1915; Caldirola, Maccarrone, and Recami, 1980; Courant and Hilbert, 1966; Maccarrone, Pavsic, and Recami, 1983; Recami and Maccarrone, 1980, 1983; Recami, Zamboni-Rached, and Dartora, 2004, and more recent articles such as Barut and Bracken (1992); Barut and Grant (1990)). Modern studies of nondiffractive pulses followed a development rather independent of those on nondiffracting beams, even if both phenomena are part of the same sector of physics: that of localized waves. In 1983, Brittingham set forth a luminal (V = c) solution to the wave equation (more particularly, to the Maxwell equations) that travels rigidly, i.e., without diffraction. The solution proposed by Brittingham, however, had inﬁnite energy, and once again the issue arose of overcoming such a problem. As far as we know, a solution was ﬁrst obtained by Sezginer (1985), who showed how to construct ﬁnite-energy luminal pulses, which do not propagate without distortion for an inﬁnite distance: As expected, Sezginer’s pulses travel with constant speed and approximately without deforming for a certain (long) depth of ﬁeld—much longer, in this case, too, than ordinary pulses like the Gaussian ones. In a series of subsequent papers (Besieris, Shaarawi, and Ziolkowski, 1989; Donnelly and Ziolkowski, 1993; Shaarawi, Besieris, and Ziolkowski, 1990; Ziolkowski, 1989, 1991; Ziolkowski, Besieris, and Shaarawi, 1993), a simple theoretical method was developed, termed bidirectional decomposition, for constructing a new series of nondiffracting luminal pulses. Eventually, at the beginning of the 1990s, Lu and Greenleaf (1992a,b) constructed, both mathematically and experimentally, new solutions to the wave equation in free space: namely, an X-shaped localized pulse, with the form predicted by the so-called extended special relativity (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 1980; Maccarrone, Pavsic, and Recami, 1983; Recami, 1986; Recami and Maccarrone, 1980, 1983); for the connection between what Lu et al. called “X-waves” and “extended” relativity, see, for example, Recami, ZamboniRached, and Dartora (2004), while brief excerpts of the latter theory can be found in Garavaglia (1998); Lu, Greenleaf, and Recami (1996); Maccarrone and Recami (1980); Pavsic and Recami (1982); Recami (1985, 1987, 1998, 2001); Recami, Fontana, and Garavaglia (2000); Recami et al. (2003);

Localized Waves: A Review

241

see also Shaarawi and Besieris (2000a); Ziolkowski, Besieris, and Shaarawi (2000). Lu et al.’s solutions (which can be called the classic ones) were continuous superpositions of Bessel beams with the same phase velocity (i.e., with the same axicon angle α) (Friberg, Fagerholm, and Salomaa (1997); Lu and Greenleaf (1992a); Lu, Greenleaf, and Recami (1996); Recami (1986, 1998); Shaarawi and Besieris (2000a); Ziolkowski, Besieris, and Shaarawi (2000)), so that they could maintain their shape for long distances. Such X-shaped waves resulted in interesting and ﬂexible localized solutions and have since been studied in a number of papers, even if their velocity V is supersonic or superluminal (V > c). Actually, when the phase velocity does not depend on the frequency, it is known that such a phase velocity becomes the group velocity. Remembering how a superposition of Bessel beams is generated (for example, by a discrete or continuous set of annular slits or transducers), the energy forming the LWs, coming from those rings is transported at the ordinary speed c of the plane waves in the considered medium (here c, representing the velocity of the plane waves in the medium, is the speed of sound in the acoustic case, and the speed of light in the electromagnetic case, and so on). Nevertheless, the peak of the LWs is faster than c. (Let us note that, when using a lens after an aperture located at its back-focus, as in Figure 2, then a classic Xshaped pulse can be generated even by a single annular slit, or transducer, illuminated however by a light, or sound, pulse; but the previous considerations about the actual transportation speed of the “energy” forming the X-shaped wave remain unaffected. The experimental setup in Figure 2, with various annular slits, is actually needed only for generating (e.g., X-shaped) pulses more complex than the classic one, namely, depending

Annular slits

Lens

␭

R

Bessel beam superposition aN21 a2N

aN ␦a

f

␦a << ␭f R

FIGURE 2 One of the simplest experimental setups for generating various Bessel beam superpositions.

242

Erasmo Recami and Michel Zamboni-Rached

on the coordinates z and t not only through the quantity ζ ≡ z − Vt; see the following). It is indeed possible to generate (in addition to the “classic” X-wave produced by Lu et al. in 1992) inﬁnite sets of new X-shaped waves, with their energy increasingly concentrated in a spot corresponding to the vertex region (Zamboni-Rached, Recami, and Hernández-Figueroa, 2002). It may therefore appear rather intriguing that such a spot (even if no violations of special relativity [SR] are obviously implied: all the results come from Maxwell equations, or from the wave equations (Barbero, HernándezFigueroa, and Recami, 2000; Brodowsky, Heitmann, and Nimtz, 1996; Shaaraawi and Besieris, 2000)) travels superluminally when the waves are electromagnetic. For simplicity, we shall use the term superluminal for all X-shaped waves, even when the waves are acoustic. Figure 3, which refers to an X-wave possessing the velocity V > c, illustrates the fact that, if its vertex or central spot is located at P1 at time t1 , it will reach the position P2 at a time t + τ, where τ = |P2 − P1 |/V < |P2 − P1 |/c: All these points are discussed below. Soon after having mathematically and experimentally constructed their “classic” acoustic X-wave, Lu et al. started applying the waves to ultrasonic scanning, obtaining very high-quality images. Subsequently, in a 1996 e-print and report, Recami et al. (see Lu, Greenleaf, and Recami, 1996; Recami, 1998) published the analogous X-shaped solutions to the Maxwell equations by constructing scalar superluminal localized solutions for each component of the Hertz potential. That approach showed that the localized solutions to the scalar equation can also be used, under very weak conditions, to obtain localized solutions to Maxwell’s equations. (Actually

P1

R ␰5␶v R

P2 ␶c

FIGURE 3 This figure shows an X-shaped wave, that is, a localized superluminal pulse. It refers to an X-wave, possessing the velocity V > c, and illustrates the fact that, if its vertex or central spot is located at P1 at time t0 , it will reach the position P2 at a time t + τ, where τ = |P2 − P1 |/V < |P2 − P1 |/c: This is somewhat different from the illusory scissor effect, even if the feeding energy, coming from the regions R, has traveled with the ordinary speed c (which is the speed of light in the electromagnetic case, or the speed of sound in acoustics, and so on).

Localized Waves: A Review

243

Ziolkowski, Besieris, and Shaarawi (1993) had found something similar, which they called slingshot pulses, for the simple scalar case, but their solution went almost unnoticed). In 1997, Saari and Reivelt announced the laboratory production of an X-shaped wave in the optical realm, thus proving experimentally the existence of superluminal electromagnetic pulses. Three years later, in 2000, Mugnai, Ranfagni, and Ruggeri produced superluminal X-shaped waves in the microwave region (their paper aroused various criticisms, to which those authors responded).

2. A MORE DETAILED INTRODUCTION Let us refer to the differential equation known as the homogeneous wave equation: simple, but so important in acoustics, electromagnetism (microwaves, optics), geophysics, and even, as we said, gravitational waves and elementary particle physics:

∂2 ∂2 1 ∂2 ∂2 + + − ∂x2 ∂y2 ∂z2 c2 ∂t2

ψ(x, y, z; t) = 0.

(1)

Let us write it in the cylindrical coordinates (ρ, φ, z) and, for simplicity’s sake, conﬁne ourselves to axially symmetric solutions ψ(ρ, z; t). Then, Eq. (1) becomes

∂2 ∂2 1 ∂ 1 ∂2 + + − ρ ∂ρ ∂z2 ∂ρ2 c2 ∂t2

ψ(ρ, z; t) = 0.

(2)

In free space, the solution ψ(ρ, z; t) can be written in terms of a Bessel–Fourier transform w.r.t. the variable ρ, and two Fourier transforms w.r.t. variables z and t, as follows:

ψ(ρ, z, t) =

0

∞ ∞

∞

−∞ −∞

¯ ρ , kz , ω) dkρ dkz dω, kρ J0 (kρ ρ) eikz z e−iωt ψ(k (3)

¯ ρ , kz , ω) is the where J0 (.) is an ordinary zero-order Bessel function and ψ(k transform of ψ(ρ, z, t). Substituting Eq. (3) into Eq. (2), one obtains that the relation, among ω, kρ , and kz ,

ω2 = kρ2 + kz2 c2

(4)

244

Erasmo Recami and Michel Zamboni-Rached

must be satisﬁed. As a consequence, by using condition (4) in Eq. (3), any solution to the wave equation [Eq. (2)] can be written

ψ(ρ, z, t) =

ω/c ∞

0

−∞

kρ J0 (kρ ρ) e

i ω2 /c2 − kρ2 z −iωt

e

S(kρ , ω) dkρ dω, (5)

where S(kρ , ω) is the chosen spectral function, when kz > 0 (and we disregard evenescent waves). The general integral solution in Eq. (5) yields, for instance, the (nonlocalized) Gaussian beams and pulses, to which we shall refer to illustrate the differences of the localized waves w.r.t. them. The Gaussian Beam A very common (nonlocalized) beam is the Gaussian beam (Newell and Molone, 1992), corresponding to the spectrum

S(kρ , ω) = 2a2 e−a

2 k2 ρ

δ(ω − ω0 ).

(6)

In Eq. (6), a is a positive constant, which will be shown to depend on the transverse aperture of the initial pulse. Figure 4 illustrates the interpretation of the integral solution in Eq. (5), with spectral function (6), as a superposition of plane waves. Namely, from Figure 4 one can easily realize that this case corresponds to plane waves propagating in all directions (always with kz ≥ 0), the most intense ones being those directed along (positive) z. Note that, in the plane wave case, kz is the longitudinal component of the wave vector, k = kρ + kz , where kρ = kx + ky .

k␳ kz

ê␳ S(k␳, ␻0)k êz

FIGURE 4 Visual interpretation of the integral solution [Eq. (5)], with spectral function [Eq. (6)], in terms of a superposition of plane waves.

Localized Waves: A Review

245

On substituting Eq. (6) into Eq. (5) and adopting the paraxial approximation, one meets the Gaussian beam

ψgauss (ρ, z, t) =

−ρ2 2a2 exp 4(a2 + i z/2k0 )

2(a2 + i z/2k0 )

eik0 (z−ct) ,

(7)

where k0 = ω0 /c. We can verify that such a beam, which suffers transverse diffraction, doubles its initial width ρ0 = 2a after having traveled the √ distance zdif = 3 k0 ρ02 /2, called diffraction length. The more concentrated a Gaussian beam happens to be, the more rapidly it gets spoiled. The Gaussian Pulse The most common (nonlocalized) pulse is the Gaussian pulse, which is obtained from Eq. (5) by using the spectrum (Zamboni-Rached, Hernández-Figueroa, and Recami, 2004)

2ba2 2 2 2 2 S(kρ , ω) = √ e−a kρ e−b (ω−ω0 ) , π

(8)

where a and b are positive constants. Indeed, such a pulse is a superposition of Gaussian beams of different frequency. On substituting Eq. (8) into Eq. (5), and again adopting the paraxial approximation, one obtains the Gaussian pulse:

a2 exp ψ(ρ, z, t) =

−(z − ct)2 −ρ2 exp 4(a2 + iz/2k0 ) 4c2 b2 a2 + iz/2k0

,

(9)

endowed with speed c and temporal width t = 2b, and suffering a progressing enlargement of its transverse width, so that its initial value is √ already doubled at position zdif = 3 k0 ρ02 /2, with ρ0 = 2a.

2.1. The Localized Solutions We now proceed to the construction of the two most renowned localized waves: the Bessel beam and the ordinary X-shaped pulse.

246

Erasmo Recami and Michel Zamboni-Rached

First, it is interesting to observe that, when superposing (axially symmetric) solutions of the wave equation in vacuum, three spectral parameters, (ω, kρ , kz ), come into play, which must satisfy the constraint (4), deriving from the wave equation itself. Consequently, only two of them are independent, and we choose3 here ω and kρ . The possibility of choosing ω and kρ already was apparent in the spectral functions generating Gaussian beams and pulses, which consist of the product of two functions—one depending only on ω and the other on kρ . We will see that further particular relations between ω and kρ (or, analogously, between ω and kz ) can be enforced to produce interesting and unexpected results, such as the LWs.

2.1.1. The Bessel Beam Let us start by imposing a linear coupling between ω and kρ (it could be actually shown (Durnin, 1987) that it is the unique coupling leading to localized solutions). Namely, let us consider the spectral function

ω δ kρ − sin θ c S(kρ , ω) = δ(ω − ω0 ), kρ

(10)

which implies that kρ = (ω sin θ)/c, with 0 ≤ θ ≤ π/2: a relation that can be regarded as a space-time coupling. This linear constraint between ω and kρ , together with Eq. (4), yields kz = (ω cos θ)/c. This is an important fact because it has been shown elsewhere (Zamboni-Rached, 1999, 2004; Zamboni-Rached, Recami, and Hernández-Figueroa, 2002) that an ideal LW must contain a coupling of the type ω = Vkz + b, where V and b are arbitrary constants. The interpretation of the integral function (5), this time with spectrum (10), as a superposition of plane waves is visualized in Figure 5, which shows that an axially symmetric Bessel beam is nothing but the result of the superposition of plane waves whose wave vectors lie on the surface of a cone, with the propagation line as its symmetry axis and an opening angle equal to θ; such θ being called the axicon angle. By inserting Eq. (10) into Eq. (5), one gets the mathematical expression of the so-called Bessel beam:

ψ(ρ, z, t) = J0

ω

3 Elsewhere we chose ω and k . z

ω c 0 sin θ ρ exp i cos θ z − t . c c cos θ 0

(11)

Localized Waves: A Review

247

x k kx ␪

y ky

kz

z

FIGURE 5 The axially symmetric Bessel beam is created by the superposition of plane waves whose wave vectors lie on the surface of a cone with the propagation axis as its symmetry axis and angle equal to θ (“axicon angle”).

This beam possesses phase velocity vph = c/ cos θ, and ﬁeld transverse shape represented by a Bessel function J0 (.) so that its ﬁeld in concentrated in the surroundings of the propagation axis, z. Moreover, Eq. (11) tells us that the Bessel beam maintains its transverse shape (which is therefore invariant) while propagating, with central “spot” ρ = 2.405c/(ω sin θ). The ideal Bessel beam, however, is not a square-integrable function and therefore has an inﬁnite energy; that is, it cannot be experimentally produced. However, we can have recourse to truncated Bessel beams, generated by ﬁnite apertures. In this case, the (truncated) Bessel beams can still travel a long distance while maintaining their transfer shape, as well as their speed, approximately unchanged (Durnin, Miceli, and Eberly, 1987a,b; Overfelt and Kenney, 1991); that is, they still possess a large depth of ﬁeld. For instance, the ﬁeld depth of a Bessel beam generated by a circular ﬁnite aperture with radius R is given by

Zmax =

R , tan θ

(12)

where θ is the beam axicon angle. In the ﬁnite aperture case, the Bessel beam can no longer be represented by Eq. (11), and it must be calculated by the scalar diffraction theory: for example, by using Kirchhoff’s or Rayleigh–Sommerfeld’s diffraction integrals. But up to the distance Zmax , Eq. (11) can still be used to approximately describe the beam, at least in the vicinity of the axis ρ = 0, that is, for ρ << R. To realize the extent to which a truncated Bessel beam succeeds in resisting diffraction, let us also consider a Gaussian beam with the same frequency and central “spot,” and compare their ﬁeld depths. In particular, let us assume for both beams λ = 0.63 μm and initial central spot size ρ0 = 60 μm. The Bessel beam will possess axicon angle θ = arcsin[2.405c/(ωρ0 )] = 0.004 rad.

248

Erasmo Recami and Michel Zamboni-Rached

|CBessel|2

|CGauss|2 1 0.8 0.6 0.4 0.2 0 3

1.2 1 2

0.8

1 0 ␳(mm)

0.6 21

0.4 22

0.2

1.2 1 0.8 0.6 0.4 0.2

1.2 1

3

0.8

2 1

Z (m)

23 0

␳ (mm)

0

0.6 21

0.4 22

(a)

Z (m)

0.2

23

(b)

FIGURE 6 Comparison between a Gaussian (a) and a truncated Bessel beam (b). The Gaussian beam doubles its initial transverse width already after 3 cm, while after 6 cm its intensity decays of a factor 10. By contrast, the Bessel beam maintains its transverse shape up to the distance of 85 cm.

Figure 6 shows the behavior of the two beams for a circular aperture with radius 3.5 mm. We can verify how the Gaussian beam doubles its initial transverse width already after 3 cm, and after 6 cm its intensity has become an order of magnitude smaller. By contrast, the truncated Bessel beam maintains its transverse shape until the distance Zmax = R/ tan θ = 85 cm. Thereafter, the Bessel beam rapidly decays as a consequence of the sharp cut performed on its aperture (such cut being responsible also for the intensity oscillations suffered by the beam along its propagation axis, and for the fact that eventually the feeding waves, coming from the aperture, at a certain point become faint). The zero-order (axially symmetric) Bessel beam is nothing but one example of a localized beam. Further examples are the higher-order (not cylindrically symmetric) Bessel beams

ψ(ρ, φ, z; t) = Jν

ω

ω c 0 sin θ ρ exp (iνφ) exp i cos θ z − t , c c cos θ (13) 0

or the Mathieu beams (Dartora et al., 2003), and so on.

2.1.2. The Ordinary X-Shaped Pulse Following the same procedure adopted in the previous subsection, let us construct pulses by using spectral functions of the type

ω δ kρ − sin θ c S(kρ , ω) = F(ω), kρ

(14)

Localized Waves: A Review

249

where here the Dirac delta function furnishes the spectral space-time coupling kρ = (ω sin θ)/c. Function F(ω) is, of course, the frequency spectrum; it is left for the moment undetermined. On using Eq. (14) in Eq. (5), one obtains

ψ(ρ, z, t) =

∞

−∞

F(ω) J0

ω

ω c sin θ ρ exp cos θ z − t dω. c c cos θ (15)

It is easy to see that ψ will be a pulse of the type

ψ = ψ(ρ, z − Vt)

(16)

with a speed V = c/ cos θ independent of the frequency spectrum F(ω). Such solutions are known as X-shaped pulses, and are localized (nondiffractive) waves in the sense that they obviously do maintain their spatial shape during propagation (see, Lu and Greenleaf, 1992a; Lu, Greenleaf, and Recami, 1996; Recami, 1998; Shaarawi and Besieris, 2000a; Zamboni-Rached, Recami, and Hernández-Figueroa, 2002; Ziolkowski, Besieris, and Shaarawi, 2000, and references therein; as well as the following). At this point, some remarkable observations can be stressed: (i) When a pulse consists of a superposition of waves (in this case, Bessel beams) all endowed with the same phase velocity Vph (in this case, with the same axicon angle) independent of their frequency, then it is known that the phase velocity (in this case Vph = c/ cos θ) becomes the group velocity V (Esposito et al., 2003; Zamboni-Rached, Fontana, and Recami, 2003; Zamboni-Rached, Recami, and Fontana, 2001); that is, V = c/ cos θ > c. In this sense, the X-shaped waves are called Superluminal localized pulses. [For simplicity, the group velocity (Garavaglia, 1998; Olkhovsky, Recami, and Jakiel, 2004; Pavsic and Recami, 1982; Recami, Fontana, and Garavaglia, 2000; Zamboni-Rached, Recami, and Hernández-Figueroa, 2002) can be regarded as the peak velocity. Here, we add only the observations: (a) that the group velocity for a pulse, in general, is well deﬁned only when the pulse has a clear bump in space, but it can be calculated by the approximate, simple relation V dω/dk only when some extra conditions are satisﬁed (namely, when ω as a function of k is also clearly bumped); and (b) that the group velocity can a priori be evaluated through the aforementioned, customary derivation of ω with respect to the wave number for the inﬁnite total energy solutions; whereas, for the ﬁnite total energy superluminal solutions, the group velocity cannot be calculated through such

250

Erasmo Recami and Michel Zamboni-Rached

an elementary relation, since in those cases there does not exist a one-to-one function ω = ω(kz )]. (ii) Such pulses, even if their group velocity is superluminal, do not contradict standard physics, having been found in what precedes on the basis of the wave equations—in particular, of Maxwell equations (Lu, Greenleaf, and Recami, 1996; Recami, 1998; Shaarawi and Besieris, 2000a; Ziolkowski, 1989; Ziolkowski, Besieris, and Shaarawi, 2000)—only. Indeed, as we shall see better in Section 3, their existence can be understood within SR itself (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 1980; Garavaglia, 1998; Lu, Greenleaf, and Recami, 1996; Maccarrone, Pavsic, and Recami, 1983; Pavsic and Recami, 1982; Recami, 1998, 2001; Recami, Fontana, and Garavaglia, 2000; Recami and Maccarrone, 1980, 1983; Recami, Zamboni-Rached, and Dartora, 2004; Recami et al., 2003; Shaarawi and Besieris, 2000a; Ziolkowski, Besieris, and Shaarawi, 2000), on the basis of its ordinary postulates (Recami, 1993). Let us repeat: They are fed by waves originating at the aperture and carrying energy with the standard speed c of the medium (the light velocity in the electromagnetic case, and the sound velocity in the acoustic case). We can become convinced about the possibility of realizing superluminal Xshaped pulses by imagining the simple ideal case of a negligibly sized superluminal source S endowed with speed V > c in vacuum, and emitting electromagnetic waves W (each one traveling with the invariant speed c). The electromagnetic waves will be internally tangent to an enveloping cone C with S as its vertex, and as its axis the propagation line z of the source (Recami, 1986; Recami, Zamboni-Rached, and Dartora, 2004): This is completely analogous to what happens for an airplane that moves in air with constant supersonic speed. The waves W interfere mainly negatively inside the cone C and constructively on its surface. We can place a plane detector orthogonally to z and record magnitude and direction of the waves W that hit on it as (cylindrically symmetric) functions of position and of time. It sufﬁces, then, to replace the plane detector with a plane antenna that emits—instead of recording—exactly the same (axially symmetric) space-time pattern of waves W, for constructing a cone-shaped electromagnetic wave C that will propagate with the superluminal speed V (of course, without a source any longer at its vertex), even if each wave W travels with the invariant speed c. Again, this is exactly what would happen in the case of a supersonic airplane (in which case c is the sound speed in air; for simplicity, assume the observer to be at rest with respect to the air); for further details, see the quoted references. By suitable superpositions and interference of speed-c waves, one can obtain pulses increasingly more localized in the vertex region (Zamboni-Rached, Recami, and Hernández-Figueroa, 2002): that is, very localized ﬁeld “blobs” that

Localized Waves: A Review

251

z

␳

FIGURE 7 The truncated X-waves considered in this chapter, as predicted by SR (all wave equations have an intrinsic relativistic structure!), must have a leading cone in addition to the rear cone, with such a leading cone playing a role for peak stability. For example, when producing a finite conic wave truncated both in space and in time, the theory of SR suggests recourse, in the simplest case, to a dynamic antenna emitting a radiation cylindrically symmetric in space and symmetric in time, for a better approximation to what Courant and Hilbert (1966) called an “undistorted progressing wave.”

travel with superluminal group velocity. This apparently has nothing to do with the illusory “scissors effect,” since such blobs, along their ﬁeld depth, are a priori able to get two successive (weak) detectors, located at a distance L, to click after a time smaller than L/c. Incidentally, an analysis of the above-mentioned case (of a supersonic plane or a superluminal charge) led, as expected (Recami, 1986), to the simplest type of “X-shaped pulse” (Recami, Zamboni-Rached, and Dartora, 2004). It might be useful to recall that SR (even the wave equations have an internal relativistic structure!) implies considering also the forward cone (Figure 7). The truncated X-waves considered in this chapter, for instance, must have a leading cone in addition to the rear cone, with such a leading cone playing a role for the peak stability (Lu and Greenleaf, 1992a). For example, in the approximate case in which we produce a ﬁnite conic wave truncated both in space and in time, the theory of SR suggested the biconic shape (symmetrical in space with respect to the vertex S) as a better approximation to a rigidly traveling wave (so that SR suggests recourse to a dynamic antenna emitting a radiation cylindrically symmetric in space and symmetric in time, for a better approximation to an “undistorted progressing wave”). (iii) Any solutions that depend on z and on t only through the quantity z − Vt, like Eq. (15), will appear with a constant shape to an observer traveling along z with the same speed V. That is, such a solution will propagate rigidly with speed V. This further explains why our X-shaped pulses, after having been produced, travel almost rigidly at speed V (in this case, a faster-than-light group velocity), all along their depth of ﬁeld. For greater clarity, let us consider a generic function,

252

Erasmo Recami and Michel Zamboni-Rached

depending on z − Vt with V > c, and show, by explicit calculations involving the Maxwell equations only, that it obeys the scalar wave equation. Following Franco Selleri (2000), let us consider, for example, the wave function,

a (x, y, z, t) =

, [b − ic(z − Vt)]2 + (V 2 − c2 )(x2 + y2 )

(17)

with a and b nonzero constants, c the ordinary speed of light, and V > c (incidentally, this wave function is nothing but the classic X-shaped wave in cartesian coordinates). Let us naively verify that it is a solution to the wave equation

∇ 2 (x, y, z, t) −

1 ∂2 (x, y, z, t) = 0. c2 ∂2 t

(18)

On putting

R ≡

[b − ic(z − Vt)]2 + (V 2 − c2 )(x2 + y2 ) ,

(19)

one can write = a/R and evaluate the second derivatives

1 a 1 a 1 a

c2 ∂2 3c2 = − [b − ic(z − Vt)]2 ; ∂2 z R3 R5 2 V 2 − c2 ∂2 2 2 = − + 3 V − c ∂2 x R3 2 V 2 − c2 ∂2 2 2 = − + 3 V − c ∂2 y R3

x2 ; R5 y2 R5

c2 V 2 3c2 V 2 1 ∂2 = − [b − ic(z − Vt)]2 ; a ∂2 t R3 R5 wherefrom

2 [b − ic(z − Vt)]2 1 ∂2 V 2 − c2 1 ∂2 2 2 − = − + 3 V − c , a ∂2 z c2 ∂ 2 t R3 R5 and

2 x2 + y2 V 2 − c2 1 ∂2 ∂2 2 2 + = − 2 + 3 V − c . a ∂2 x ∂2 y R3 R5

Localized Waves: A Review

253

From the last two equations, remembering the previous deﬁnition, one ﬁnally gets

1 a

1 ∂2 ∂2 ∂2 ∂2 + + − ∂2 z ∂2 x ∂2 y c2 ∂ 2 t

= 0,

which is nothing but the (d’Alembert) wave equation [Eq. (18)], q.e.d. In conclusion, function is a solution of the wave equation even if it does obviously represent a pulse (Selleri says “a signal”!) propagating with superluminal speed. At this point, readers should be however informated that all the subluminal LWs, solutions of the ordinary homogeneous wave equation, have until now appeared to present singularities whenever they depend on z and t only via the quantity ζ ≡ z − Vt. This is still an open, interesting research topic, which is also related to analogous results met in gravitation physics. After the previous three important comments, let us return to our evaluations with regard to the X-type solutions to the wave equations. Let us now consider in Eq. (15), for instance, the particular frequency spectrum F(ω) given by

F(ω) = H(ω)

a a exp − ω , V V

(20)

where H(ω) is the Heaviside step function and a a positive constant. Then, Eq. (15) yields

a ψ(ρ, ζ) ≡ X = 2 , V 2 (a − iζ) + c2 − 1 ρ2

(21)

with ζ ≡ z − Vt. This solution [Eq. (21)] is the well-known ordinary or “classic” X-wave, which constitutes a simple example of X-shaped pulse (Lu and Greenleaf, 1992a; Lu, Greenleaf, and Recami, 1996; Recami, 1998; Shaarawi and Besieris, 2000a; Ziolkowski, Besieris, and Shaarawi, 2000). Notice that function (20) contains mainly low frequencies, so that the classic X-wave is suitable for low frequencies only. Figure 8 depicts (the real part of) an ordinary X-wave with V = 1.1 c and a = 3 m. Solutions (15), and in particular the pulse [Eq. (21)], have an inﬁnite ﬁeld depth and inﬁnite energy. Therefore, as was done in the Bessel beam case, one should pass to truncated pulses, originating from a ﬁnite aperture.

254

Erasmo Recami and Michel Zamboni-Rached

Re(␺)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 3000 2000 1000 ␳(m)

FIGURE 8 a=3m.

0 ⫺1000 ⫺2000 ⫺3000

1

⫺1.5

⫺1

⫺0.5

0

1.5

0.5 ⫻ 104

␰(m)

Plot of the real part of the ordinary X-wave, evaluated for V = 1.1 c with

Afterward, our truncated pulses will keep their spatial shape (and their speed) all along the depth of ﬁeld

Z=

R , tan θ

(22)

where, as before, R is the aperture radius and θ the axicon angle.

2.1.3. Further Observations Let us put forth some further observations. It is not strictly correct to call nondiffractive the LWs, since diffraction affects, more or less, all waves obeying Eq. (1). However, all LWs (both beams and pulses) possess the remarkable self-reconstruction property: That is, the LWs, when diffracting during propagation, immediately rebuild their shape (Bouchal, Wagner, and Chlup, 1998; Grunwald et al., 2003, 2004, 2005) (even after obstacles with size much larger than the characteristic wavelengths, provided it is smaller than the aperture size), due to their particular spectral structure [as shown in detail, in Wiley (2008)]. In particular, the ideal LWs (with inﬁnite energy and ﬁeld depth) are able to rebuild themselves for an inﬁnite time; whereas, the ﬁnite-energy (truncated) ones can do it, and thus resist the diffraction effects, only along a certain depth of ﬁeld.

255

Localized Waves: A Review

1

1

0.8

0.8

0.6

0.6 |⌿x|2

|⌿x|2

The interest in LWs (especially from the point of view of applications) lies in the circumstance that they are almost nondiffractive, rather than in their group velocity: From this point of view, superluminal, luminal, and subluminal localized solutions are equally interesting and suited to important applications. In reality, LWs are not restricted to the (X-shaped, superluminal) ones corresponding to the integral solution (15) to the wave equation; and, as already stated, three classes of localized pulses exist: the superluminal (with speed V > c), the luminal (V = c), and the subluminal (V < c) ones—all of them with, or without, axial symmetry, and corresponding in any case to a single, uniﬁed mathematical background. This issue will be discussed again in this review. Incidentally, we have addressed elsewhere topics such as (i) the construction of inﬁnite families of generalizations of the classic X-shaped wave (with energy increasingly concentrated around the vertex; compare with Figure 9 from Zamboni-Rached, Recami, and Hernández-Figueroa (2002)); (ii) the behavior of some ﬁnite total-energy superluminal localized solutions (SLS); (iii) the techniques for building new series of SLS to the Maxwell equations suitable for arbitrary frequencies and bandwidths; (iv) questions related to the case of dispersive (and even lossy) media; (v) the construction

0.4

0.4 0.2

0.2

0 0.2

0 0.2 0.1 0 ␳ (m)

20.1 20.2

20.6

(a)

20.4

20.2

0.2

0

0.4

0.6

0.6

0.1 0 20.1

␨(m)

␳(m)

20.2

20.6

20.4

20.2

0

0.2

0.4

␨(m)

(b)

FIGURE 9 Panel (a) represents (in arbitrary units) the square magnitude of the classic, X-shaped superluminal localized solution (SLS) to the wave equation, with V = 5c and a = 0.1 m. Families of infinite SLSs, however, exist, which generalize the classic X-shaped solution—for instance, a family of SLSs obtained by suitably differentiating the classic X-wave (Zamboni-Rached, Recami, and Hernández-Figueroa, 2002): Panel (b) depicts the first of these waves (corresponding to the first differentiation) with the same parameters. As noted, the successsive solutions in such a family are increasingly localized around their vertex. Quantity ρ is the distance (in meters) from the propagation axis z, while quantity ζ is the “V-cone” variable (still in meters) ζ ≡ z − Vt, with V ≥ c. Because all these solutions depend on z only via the variable ζ, they propagate “rigidly”; that is, without distortion (and they are called “localized,” or nondispersive, for this reason). Here we are assuming propagation in vacuum (or in a homogeneous medium).

256

Erasmo Recami and Michel Zamboni-Rached

of (inﬁnite- or ﬁnite-energy) superluminal LWs propagating down waveguides or coaxial cables; (vi) determining localized solutions to equations different from the wave equation, such as Schrödinger’s; and (vii) using the above techniques for constructing, in general relativity, new exact solutions for gravitational ways. We discuss some of these issues in the second section of this chapter. Let us add here that X-shaped waves also have been easily produced in nonlinear media (Conti et al., 2003). A more technical introduction to the subject of LWs (particularly w.r.t. the superluminal X-shaped ones) is found in the second section of this review, and in papers such as Recami et al. (2003). The Appendix presents a historical perspective.

3. COMPLEMENTARY MATERIAL: A HISTORICAL PERSPECTIVE (THEORETICAL AND EXPERIMENTAL) This mainly “historical” appendix, written as much as possible in a (partially) self-consistent form, ﬁrst refers, from the theoretical point of view, to the most intriguing localized solutions to the wave equation: the superluminal wave solutions (SLS), and in particular the X-shaped pulses. As a start, we recall their geometrical interpretation within SR. Afterward, to help resolving possible doubts, we present a bird’s-eye view of the various experimental sectors of physics in which superluminal motions seem to appear; in particular, in experiments with evanescent waves (and/or tunneling photons), and with the SLSs. In some parts of this appendix the propagation line is called x (and no longer z) without creating any interpretation problems.

3.1. Introduction The question of superluminal (V 2 > c2 ) objects or waves has a long history. In pre-relativistic times, various relevant papers include those by J.J. Thomson and by A. Sommerfeld. It is well known, however, that with SR the conviction spread that the speed c of light in vacuum was the upper limit of any possible speed. For instance, in 1917 Tolman was believed to have shown by his “paradox” that the existence of particles endowed with speeds larger than c would have allowed sending information into the past. The problem was tackled again only in the 1950s and ’60s, particularly after the papers (e.g., Bilaniuk, Deshpande, and Sudarshan, 1962) by George Sudarshan et al. and, later, by one of the present authors (Recami, 1978; Recami, and Mignani, 1974) as well as those by Corben and others. The ﬁrst experimental attempts were performed by T. Alväger et al.

Localized Waves: A Review

257

We wish to address the still unusual issue of the possible existence of superluminal wavelets, and objects—within standard physics and SR—, since at least four different experimental sectors of physics seem to support such a possibility (apparently conﬁrming some long-standing theoretical predictions (Barut, Maccarrone, and Recami, 1982; Bilaniuk, Deshpande, and Sudarshan, 1962; Maccarrone, Pavsic, and Recami, 1983; Recami, 1978, 1986)). The experimental review is necessarily short, but we provide enough bibliographical information, limited for brevity’s sake to only the last century—updated to the year 2000.

3.2. Historical Recollections—Theory A simple theoretical framework was long ago proposed (Bilaniuk, Deshpande, and Sudarshan, 1962; Recami, 1986; Recami, and Mignani, 1974), based merely on the space-time geometrical methods of SR, that appears to incorporate Superluminal waves and objects, and in a sense predicts (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 1980; Maccarrone, Pavsic, and Recami, 1983; Recami and Maccarrone, 1980, 1983), among others, the superluminal X-shaped waves, without violating the Relativity principles. A suitable choice of the postulates of SR (equivalent to the other, more common, choices) includes the following: (i) the standard Principle of Relativity, and (ii) space-time homogeneity and space isotropy. It follows that one and only one invariant speed exists, and experience shows that invariant speed to be the light speed, c, in vacuum—the essential role of c in SR being just due to its invariance, and not to the fact that it be a maximal, or minimal, speed. No subluminal or superluminal objects or pulses can be endowed with an invariant speed; therefore, their speed cannot play in SR the same essential role played the vacuum light-speed, c. Indeed, the speed c turns out to be also a limiting speed; but any limit possesses two sides and can be approached a priori both from below and from above (Figure 10). As Sudarshan stated, from the fact

|E |

m0 c 2 2c

c

v

FIGURE 10 Energy of a free object as a function of its speed. (Bilaniuk, Deshpande, and Sudarshan, 1962; Recami, 1986; Recami, and Mignani, 1974).

258

Erasmo Recami and Michel Zamboni-Rached

that no one could climb over the Himalayan ranges, people of India cannot conclude that there are no people north of the Himalayas. Indeed, speed-c photons exist, which are born, live, and die just “at the top of the mountain,” with no need to perform the impossible task of accelerating from rest to the speed of light. (Actually, the ordinary formulation of SR has been too restricted. For instance, even leaving superluminal speeds aside, it can easily be widened to include antimatter (Garavaglia, 1998; Maccarrone and Recami, 1980; Pavsic and Recami, 1982; Recami, 1985, 1986, 1987; Recami, Fontana, and Garavaglia, 2000)). An immediate consequence is that the quadratic form c2 dt2 − dx2 ≡ dxμ dxμ , called ds2 , with dx2 ≡ dx2 + dy2 + dz2 , is invariant, except for its sign. Quantity ds2 , let us recall, is the four-dimensional (4D) length-element square, along the space-time path of any object. In correspondence with the positive (negative) sign, one gets the subluminal (superluminal) Lorentz “transformations” (LTs). More speciﬁcally, the ordinary subluminal LTs are known to leave, for instance, the quadratic forms dxμ dxμ , dpμ dpμ , and dxμ dpμ exactly invariant, where pμ is the component of the energyimpulse four-vector, whereas the superluminal LTs, by contrast, should change (only) the sign of such quadratic forms. This is enough to deduce some important consequences, such as the one that a superluminal charge has to behave as a magnetic monopole, in the sense speciﬁed in Recami (1986) and the references therein. A more important consequence for us is that the simplest subluminal object—namely, a spherical particle at rest (which appears as ellipsoidal [Figure 11], due to Lorentz contraction, at subluminal speeds v), will appear (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 1980; Maccarrone, Pavsic, and Recami, 1983; Recami, 1986, 1998; Recami and Maccarrone, 1980, 1983) as occupying the cylindrically symmetrical region bounded by a two-sheeted rotation hyperboloid and an indeﬁnite double cone (Figure 11d) for superluminal speeds V. In the limiting case of a pointlike particle, one obtains only a double cone. Such a result is obtained by writing down the equation of the “world tube” of a subluminal particle and transforming it simply by changing the sign of the quadratic forms entering that equation. Thus, in 1980–1982, it was predicted (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 1980; Recami and Maccarrone, 1980, 1983) that the simplest superluminal object appears not as a particle, but as a ﬁeld or rather as a wave: namely, as an “X-shaped pulse,” the cone

2 semi-angle α being given (with c = 1) by cotg α = V − 1. Such X-shaped pulses will move rigidly with speed V along their motion direction. In fact, any “X-pulse” can be regarded at each instant of time as the (superluminal) Lorentz transform of a spherical object, which of course moves in vacuum—or in a homogeneous medium—without any deformation as

259

Localized Waves: A Review

␳

␳⬘

z

(a)

␳⬙

z⬙

z⬘

(b)

(c)

(d)

FIGURE 11 An intrinsically spherical (or pointlike, at the limit) object appears in vacuum as an ellipsoid contracted along the motion direction when endowed with a speed v < c. By contrast, if endowed with a speed V > c (even if the c-speed barrier can be crossed neither from the left nor from the right), it would appear (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 1980; Maccarrone, Pavsic, and Recami, 1983; Recami, 1986; Recami and Maccarrone, 1980, 1983) no longer as a particle, but rather as an “X-shaped” wave traveling rigidly---namely, as occupying the region delimited by a double cone and a two-sheeted hyperboloid---or as a double cone, at the limit, and moving without distortion in the vacuum, or in a homogeneous medium, with superluminal speed V (the square cotangent of the cone semi-angle being (V/c)2 − 1). For simplicity, a space axis is skipped. Source: Barut, Maccarrone, and Recami (1982); Caldirola, Maccarrone, and Recami (1980); Recami and Maccarrone (1980, 1983).

time elapses. The 3D picture of Figure 11d is shown in Figure 12, where its annular intersections with a transverse plane are shown (see Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 1980; Recami and Maccarrone, 1980, 1983). The X-shaped waves considered here are merely the simplest ones. If one starts not from an intrinsically spherical or pointlike object, but from a nonspherically symmetric particle, from a pulsating (contracting and dilating) sphere, or from a particle oscillating back and forth along the motion direction, then their superluminal Lorentz transforms would be increasingly more complicated. The above “X-waves”, however, are typical for a superluminal object—much as the spherical or pointlike shape is typical for a subluminal object. Incidentally, it has long been believed that superluminal objects would allow sending information into the past, but such problems with causality seem to be solvable within SR. Once SR is generalized to include superluminal objects or pulses, no signal traveling backward in time is apparently left. For a solution of those causal paradoxes, see Bilaniuk, Deshpande, and Sudarshan (1962); Maccarrone and Recami (1980); Pavsic and Recami (1982); Recami, Fontana, and Garavaglia (2000), and especially Recami (1985, 1987), and references therein. When addressing the problem within this elementary appendix of the production of an X-shaped pulse like the one depicted in Figure 12 (maybe truncated, in space and in time, by use of a ﬁnite antenna radiating for

260

Erasmo Recami and Michel Zamboni-Rached

V1

V2

c

Z

V

FIGURE 12 Here we show the intersections of the superluminal object T represented in Figure 11d with planes P orthogonal to its motion line (the z-axis). For simplicity, we assumed again the object is spherical in its rest frame, and the cone vertex C coincides with the origin O for t = 0. Such intersections evolve in time so that the same pattern appears on a second plane—shifted by x—after the time t = x/V. On each plane, as time elapses the intersection is therefore predicted by (extended) SR to be a circular ring which, for negative times, continues to shrink until it reduces to a circle and then to a point (for t = 0); afterward, such a point again becomes a circle and then a circular ring that broadens (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 1980; Maccarrone, Pavsic, and Recami, 1983; Recami, 1986, 1998; Recami and Maccarrone, 1980, 1983). (Notice that, if the object is not spherical when at rest [but, e.g., is ellipsoidal in its own rest frame], then the axis of T will no longer coincide with x, but its direction will depend on the speed V of the tachyon itself). For the case when the space extension of the superluminal object T is finite, see Recami and Maccarrone (1983). Source: This picture is from Barut, Maccarrone, and Recami (1982); Caldirola, Maccarrone, and Recami (1980); Recami (1986); Recami and Maccarrone (1980, 1983).

a ﬁnite time), all the considerations expounded under point (ii) of the subsection “The Ordinary X-shaped Pulse” become in order; here, we simply refer to them. Those considerations, together with the present ones (e.g., related to Figure 12), suggest the simplest antenna to consist of a series of concentric annular slits, or transducers (as in Figure 2) that suitably radiate, following speciﬁc time patterns (see, Zamboni-Rached, 2006, and references therein). Incidentally, the above procedure can lead to a very simple type of X-shaped wave, as investigated below.

Localized Waves: A Review

261

From the present point of view, it is rather interesting to note that, during the past 15 years, X-shaped waves have been actually found as solutions to the Maxwell and wave equations (we repeat that the form of any wave equation is intrinsically relativistic). To further clarify the connection existing between what is predicted by SR (see, Figures 11 and 12) and the localized X-waves mathematically, and experimentally, constructed in recent times, let us tackle in detail the problem of the (X-shaped) ﬁeld created by a superluminal electric charge, by following a paper (Recami, Zamboni-Rached, and Dartora, 2004) recently appeared in Physical Review E.

3.2.1. The Particular X-Shaped Field Associated With a Superluminal Charge It is now well known that Maxwell equations admit of wavelet-type solutions endowed with arbitrary group-velocities (0 < vg < ∞). We again conﬁne ourselves to the localized solutions, rigidly moving, and, more in particular, to the superluminal ones (SLSs), the most interesting of which are X-shaped. The SLSs have been produced in a number of experiments, always by suitable interference of ordinary-speed waves. In this subsection we show, by contrast, that even a superluminal charge creates an electromagnetic X-shaped wave, in agreement with what was predicted (Barut, Maccarrone, and Recami, 1982; Caldirola, Maccarrone, and Recami, 1980; Recami, 1986; Recami and Maccarrone, 1980) within SR. In fact, on the basis of Maxwell equations, the ﬁeld associated with a superluminal charge can be evaluated (at least, under the rough approximation of pointlikeness). As announced in what precedes, it constitutes a very simple example of true X-wave. Indeed, the theory of SR, when based on the ordinary postulates but not restricted to subluminal waves and objects (i.e., in its extended version), predicted the simplest X-shaped wave to be the one corresponding to the electromagnetic ﬁeld created by a superluminal charge (Recami, 1986a; Recami, Zamboni-Rached, and Dartora, 2004). It seems to be important evaluating such a ﬁeld, at least approximately, by following Recami, Zamboni-Rached, and Dartora (2004).

The toy model of a pointlike superluminal charge. Let us start by considering, formally, a pointlike superluminal charge, even if the hypothesis of pointlikeness (already unacceptable in the subluminal case) is totally inadequate in the superluminal case (Recami, 1986). Then, let us consider the ordinary vector-potential Aμ and a current density jμ ≡ (0, 0, jz ; jo ) ﬂowing in the z-direction (notice that the motion line is still the axis z). On assuming the ﬁelds to be generated by the sources only, one has that Aμ ≡ (0, 0, Az ; φ),

262

Erasmo Recami and Michel Zamboni-Rached

which, when adopting the Lorentz gauge, obeys the equation Aμ = jμ . We can write such a nonhomogeneous wave equation in the cylindrical coordinates (ρ, θ, z; t); for axial symmetry (which requires a priori that Aμ = Aμ (ρ, z; t)), when choosing the “V-cone variables” ζ ≡ z − Vt; η ≡ z + Vt, with V 2 > c2 , we arrive at the equation

∂ 1 ∂2 1 ∂2 ∂2 ∂ ρ + 2 2 + 2 2 −4 Aμ (ρ, ζ, η) = jμ (ρ, ζ, η), −ρ ∂ρ ∂ρ ∂ζ∂η γ ∂ζ γ ∂η (23)

where μ assumes the two values μ = 3, 0 only, so that Aμ ≡ (0, 0, Az ; φ) and γ 2 ≡ [V 2 − 1]−1 . (Notice that, whenever convenient, we set c = 1.) Let us now suppose Aμ to be actually independent of η, namely, Aμ = Aμ (ρ, ζ). According to Eq. (23), we shall have jμ = jμ (ρ, ζ) too; and therefore jz = Vj0 (from the continuity equation) and Az = Vφ/c (from the Lorentz gauge). Then, by calling ψ ≡ Az , we obtain two equations, which allow us to analyze the possibility and consequences of having a superluminal pointlike charge, e, traveling with constant speed V along the z-axis (ρ = 0) in the positive direction, in which case jz = e V δ(ρ)/ρ δ(ζ). Indeed, one of those two equations becomes the hyperbolic equation

1 ∂ δ(ρ) ∂ 1 ∂2 − ρ + 2 2 ψ = eV δ(ζ) ρ ∂ρ ∂ρ ρ γ ∂ζ

(24)

which can be solved (Recami, Zamboni-Rached, and Dartora, 2004) in a few steps. First, by applying (with respect to the variable ρ) the Fourier–Bessel (FB) transformation f (x) =

∞

0

f ()J0 (x) d, quantity

J0 (x) being the ordinary zero-order Bessel function. Second, by applying the ordinary Fourier transformation with respect to the variable ζ (going on, from ζ, to the variable ω). And, third, by ﬁnally performing the corresponding inverse Fourier and FB transformations. Afterward, it is sufﬁcient to have recourse to formulas (3.723.9) and (6.671.7) of Gradshteyn and Ryzhik (1965), still with ζ ≡ z − Vt , to be able to write the solution of Eq. (24) in the form

⎧ ψ(ρ, ζ) = 0 ⎪ ⎪ ⎨

for 0 < γ | ζ |< ρ

V ⎪ ⎪ ⎩ ψ(ρ, ζ) = e 2 ζ − ρ2 (V 2 − 1)

for 0 ≤ ρ < γ | ζ | .

(25)

263

Localized Waves: A Review

cAz

(8␲)1/2␥Ve

(10 mm21)

2.5

21

2 20.5

1.5 1 0

␨(mm)

0.5 0.5

0 1

0.8

0.6

0.4

0.2

0 ␳(mm)

20.2 20.4 20.6 20.8

21

1

FIGURE 13 Behavior of the field ψ ≡ Az generated by a charge supposed to be √ superluminal, as a function of ρ and ζ ≡ z − Vt, evaluated for γ = 1 (i.e., for V = c 2). We skipped the points at which ψ must diverge, namely, the vertex and the cone surface. Source: Recami, Zamboni-Rached, and Dartora (2004).

Figure 13 shows our √ solution Az ≡ ψ, as a function of ρ and ζ, evaluated for γ = 1 (i.e., for V = c 2). We skipped the points at which Az must diverge, namely, the vertex and the cone surface. For comparison, recall that the classic X-shaped solution (Lu and Greenleaf, 1992a) of the homogeneous wave equation—which is shown in Figures 8, 9, and 12—has the form (with a > 0):

V . X=

(a − iζ)2 + ρ2 (V 2 − 1)

(26)

The second equation in Eqs. (25) includes expression (26), given by the spectral parameter (Zamboni-Rached et al., 2003; Zamboni-Rached, Recami, and Hernández-Figueroa, 2002) a = 0, which indeed corresponds to the nonhomogeneous case [the nonnegligible fact that for a = 0 these equations differ for an imaginary unit (Recami, 1986; Recami and Mignani, 1976) is discussed elsewhere]. At this point, it is rather important to notice that such a solution, Eq. (25), does represent a wave existing only inside the (unlimited) double cone C generated by the rotation around the z-axis of the straight lines ρ = ±γζ. This too is in full agreement with the predictions of the extended

264

Erasmo Recami and Michel Zamboni-Rached

theory of SR. For the explicit evaluation of the electromagnetic ﬁelds generated by the superluminal charge (and of their boundary values and conditions) we conﬁne ourselves here to merely quoting Recami, Zamboni-Rached, and Dartora, 2004. Incidentally, the same results found by following the above procedure can be obtained by starting from the four-potential associated with a subluminal charge (e.g., an electric charge at rest), and then applying to it the suitable superluminal Lorentz “transformation”. Also of note, this double cone does not have much in common with the Cherenkov cone (Folman and Recami, 1995; Recami, 1986a; Zamboni-Rached and Recami, 2008b), and a superluminal charge traveling at constant speed, in vacuum, does not lose energy (Figure 14). Outside the cone C, that is, for 0 < γ | ζ |< ρ, we get (as expected) no ﬁeld, so that one meets a ﬁeld discontinuity when crossing the doublecone surface. Nevertheless, the boundary conditions imposed by Maxwell equations are satisﬁed by our solution in Eq. (25), since at each point of the cone surface the electric and the magnetic ﬁelds are both tangent to the cone; for a discussion of this point, see Recami, Zamboni-Rached, and Dartora (2004). ␳

Emission

Absorption

R

et

ar

de d

Z

Ad

va

nc

ed

FIGURE 14 The spherical equipotential surfaces of the electrostatic field created by a charge at rest are transformed into two-sheeted rotation hyperboloids, contained inside an unlimited double cone, when the charge travels at superluminal speed (compare, Recami (1986a); Recami, Zamboni-Rached, and Dartora (2004)). This figures shows that a superluminal charge traveling at constant speed in a homogeneous medium like a vacuum does not lose energy (Folman and Recami, 1995; Recami, 1986a). We mention, incidentally, that this double cone has nothing to do with the Cherenkov cone (Zamboni-Rached and Recami, 2008b). Source: Recami (1986), figure 27, page 80.

Localized Waves: A Review

265

Here we stress that, when V → ∞, and therefore γ → 0, the electric ﬁeld tends to vanish, while the magnetic ﬁeld tends to the value Hφ = −πe/ρ2 . This agrees once more with what is expected from extended SR, which predicted superluminal charges to behave (in a sense) as magnetic monopoles. In the present paper we can only mention such a circumstance and refer readers to Recami (1978, 1986); Recami, and Mignani (1974); Recami and Mignani (1976), and papers quoted therein.

3.3. A Glance at the Experimental State of the Art Extended relativity can also allow a better understanding of many aspects of ordinary physics (Recami, 1986), even if superluminal objects (tachyons) did not exist in our cosmos as asymptotically free objects. Anyway, at least three or four different experimental sectors of physics seem to suggest the possible existence of faster-than-light motions, or, at least, of superluminal group velocities. The following sets forth some information about the experimental results obtained in two of those different physics sectors, with only mere mention of the others.

3.3.1. Neutrinos A long series of experiments, started in 1971, seems to show that the square m0 2 of the mass m0 of muon neutrinos, and more recently also of electron neutrinos, is negative, which, if conﬁrmed, would mean that (when using a naive language, commonly adopted) such neutrinos possess an “imaginary mass” and are therefore tachyonic, or mainly tachyonic (Baldo Ceolin, 1993; Giani, 1999; Giannetto et al., 1986; Otten, 1995; Recami, 1986). (In extended SR, however, the dispersion relation for a free superluminal object does become ω2 − k2 = −2 , or E2 − p2 = −m2o , and there is no need, at all, therefore, of imaginary masses.)

3.3.2. Galactic Microquasars As to the apparent superluminal expansions observed in the core of quasars (Zensus and Pearson, 1987) and, recently, in the so-called galactic microquasars (Gisler, 1994; Mirabel and Rodriguez, 1994; Tingay et al., 1995), we do not address that problem, except by mentioning that for those astronomical observations there also exist orthodox interpretations, based on Cavaliere, Morrison, and Sartori (1971) and Rees (1966), that are still accepted by the majority of astrophysicists. For a theoretical discussion, see Recami et al. (1986). Here, let us only emphasize that simple geometrical considerations in Minkowski space show that a single superluminal source of light would appear (Recami, 1986; Recami et al., 1986) (i) initially, in the “optical boom” phase (analogous to the acoustic “boom”

266

Erasmo Recami and Michel Zamboni-Rached

produced by an airplane traveling with constant supersonic speed), as an intense source that suddenly comes into view; and which, afterward, (ii) seems to split into two objects receding from each other with speed V > 2c (all of this is similar to what has been actually observed, according to Gisler (1994); Mirabel and Rodriguez (1994); Tingay et al. (1995)).

3.3.3. Evanescent Waves and “Tunneling Photons” Within quantum mechanics (and precisely in the tunneling processes), it had been shown that the tunneling time—ﬁrst evaluated as a simple Wigner’s “phase time” and later calculated through the analysis of the wave packet behavior—does not depend (Hartman, 1962; MacColl, 1932; Milonni, 2002; Olkhovsky and Recami, 1992; Olkhovsky, Recami, and Jakiel, 2001; Olkhovsky et al., 1995) on the barrier width in the case of opaque barriers (“Hartman effect”). This implies superluminal and arbitrarily large group velocities V inside long enough barriers (Figure 15).

1014 3 4

1 2

10216

␶pen (s)

␶pen (s)

10215

10217

10218

10219

0

2

4

6

8

10

x (Å)

FIGURE 15 Behavior of the average “penetration time” (in seconds) spent by a tunneling wave packet as a function of the penetration depth (in angstroms) down a potential barrier (from Olkhovsky et al., 1995). According to the predictions of quantum mechanics, the wave packet speed inside the barrier increases in an unlimited way for opaque barriers, and the total tunneling time does not depend on the barrier width (Hartman, 1962; MacColl, 1932; Milonni, 2002; Olkhovsky and Recami, 1992; Olkhovsky, Recami, and Jakiel, 2001; Olkhovsky et al., 1995).

Localized Waves: A Review

267

Experiments that may verify this prediction by, say, electrons or neutrons, are difﬁcult and rare (Olkhovsky, Recami, and Jakiel, 2004; Olkhovsky, Recami, and Zaichenko, 2005). Luckily, however, the Schrödinger equation in the presence of a potential barrier is mathematically identical to the Helmholtz equation for an electromagnetic wave propagating, for instance, down a metallic waveguide (along the z-axis): as shown, for example in Chiao, Kwiat, and Steinberg (1991); Japha and Kurizki (1996); Kurizki, Kozhekin, and Kofman (1998); Kurizki et al. (1999); Martin and Landauer (1992); Ranfagni et al. (1991); and a barrier height U bigger than the electron energy E corresponds (for a given wave frequency) to a waveguide of transverse size lower than a cutoff value. A segment of “undersized” guide—to go on with our example—does therefore behave as a barrier for the wave (photonic barrier), as well as any other photonic band-gap ﬁlters. The wave assumes therein—like a particle inside a quantum barrier—an imaginary momentum or wave number and, as a consequence, results exponentially damped along x (Figure 16). It becomes an evanescent wave (reverting to normal propagation, even if with reduced amplitude, when the narrowing ends and the guide returns to its initial transverse size). Thus, a tunneling experiment can be simulated by having recourse to evanescent waves (for which the concept of group velocity can be properly extended; see Recami, Fontana, and Garavaglia (2000)). The fact that evanescent waves travel with superluminal speeds (Figure 17) has been veriﬁed in a series of famous experiments. Various

1.5 1 0.5 ⫺6

⫺4

⫺2

2

4

6

⫺0.5 ⫺1 ⫺1.5

FIGURE 16 The damping taking place inside a barrier (Garavaglia, 1998; Recami, Fontana, and Garavaglia, 2000). Such damping reduces the amplitude of the tunneling wave packet, imposing a practical limit on the adoptable barrier length.

268

Erasmo Recami and Michel Zamboni-Rached

(a) b2

b1

b1

(b) a

n nc 2 nc1

(c)

Length

a tc

2 1 0

50

100

a (mm)

FIGURE 17 Simulation of tunneling by experiments with evanescent classical waves (see text), which were predicted to be superluminal also on the basis of extended SR (Mugnai et al., 1995; Recami, 1986). The figure shows one of the measurement results by (Brodowsky, Heitmann, and Nimtz, 1996; Nimtz and Enders, 1992, 1993; Nimtz, Spieker, and Brodowsky, 1994; Nimtz and Heitmann, 1997)—the average beam speed while crossing the evanescent region ( = segment of undersized waveguide, or “barrier”) as a function of its length a. As theoretically predicted (Hartman, 1962; MacColl, 1932; Milonni, 2002; Mugnai et al., 1995; Olkhovsky and Recami, 1992; Olkhovsky, Recami, and Jakiel, 2001; Olkhovsky et al., 1995; Recami, 1986a), such an average speed exceeds c for long enough “barriers.” Further results appeared in Longhi et al. (2002), and are reported below; see Figures 20 and 21 in the following text.

experiments performed since 1992 by Nimtz et al. in Cologne (Brodowsky, Heitmann, and Nimtz, 1994, 1996; Nimtz and Enders, 1992, 1993; Nimtz, Spieker, and Brodowsky, 1994), by Chiao, Kwiat, and Steinberg at Berkeley (Steinberg, Kwiat, and Chiao, 1993), by Ranfagni and colleagues in Florence (Mugnai, Ranfagni, and Ruggeri, 2000), and by others in Vienna, Orsay, and Rennes (Balcou and Dutriaux, 1997; Laude and Tournois, 1999; Spielmann et al., 1994), veriﬁed that “tunneling photons” travel with superluminal group velocities (such experiments also raised a great deal of interest (Begley, 1995; Brown, 1995; Landauer, 1993), in the popular press, and were reported in Scientiﬁc American, Nature, New Scientist, and other publications). In addition, extended SR had predicted (Mugnai et al., 1995; Recami, 1986b) evanescent waves to be endowed with faster-than-c speeds; the whole matter appears to be therefore theoretically self-consistent. The debate in the current literature does not refer to the experimental results (which can be correctly reproduced even by numerical simulations (Barbero, Hernández-Figueroa, and Recami, 2000; Brodowsky, Heitmann,

269

Localized Waves: A Review

Amplitude (10⫺2 V/m)

1.4 1.2

9 8 (a) 7 6 5 0.00

0.8 0.6 0.4

0.08 Time (ns)

0.11

0.27

1.2400 (b) 1.2400

1.2400 99.8

0.2 0.0

0.05

1.2500

Amplitude (V/m)

Amplitude (V/m)

1.0

10

99.9

100.0

100.1

100.2

Time (ns)

0

100

200

300

400

500

600

Time (ns)

FIGURE 18 The delay of a wave packet crossing a barrier (see Figure 17) is due to the initial discontinuity. We then performed suitable numerical simulations (Barbero, Hernández-Figueroa, and Recami, 2000) by considering an (indefinite) undersized waveguide, and therefore eliminating any geometric discontinuity in its cross section. This figure shows the envelope of the initial signal. Inset (a) depicts in detail the initial part of this signal as a function of time, while inset (b) depicts the Gaussian pulse peak centered at t = 100 ns.

and Nimtz, 1996; Shaaraawi and Besieris, 2000) based on Maxwell equations only; compare Figures 18 and 19), but rather to the question of whether they allow, or do not allow, sending signals or information with superluminal speed (see Milonni, 2002; Nimtz and Haibel, 2002; Shaarawi and Besieris, 2000b; Ziolkowski, 2001). In the above-mentioned experiments one meets a substantial attenuation of the considered pulses (see Figure 16) during tunneling [or during propagation in an absorbing medium]. However, by using “gain doublets”, undistorted pulses have been observed, propagating with superluminal group- velocity with a small change in amplitude (see Wang, Kuzmich, and Dogariu, 2000). We emphasize that some of the most interesting experiments of this series seem to be the ones with two or more “barriers” (e.g., with two gratings in an optical ﬁber (Longhi et al., 2002), or with two segments of undersized waveguide separated by a piece of normal-sized waveguide (Enders and Nimtz, 1993; Nimtz, Enders, and Spieker, 1993, 1994): see Figure 20).

270

Amplitude (10⫺2 V/m)

Erasmo Recami and Michel Zamboni-Rached

10

9 8 (a) 7 6 5 0.0

6

Amplitude (10⫺2 V/m)

Amplitude (10⫺2 V/m)

8

10

4

0.1

0.2 Time (ns)

0.3

22.38 22.37 22.36 (b) 22.35 22.34 22.33 99.8

2

99.9

100.0

100.1

100.2

Time (ns)

0

0

100

200

300

400

500

600

Time (ns)

FIGURE 19 Envelope of the signal in Figure 18 after traveling a distance L = 32.96 mm through the undersized waveguide. Inset (a) shows the initial part (in time) of such arriving signal, while inset (b) shows the peak of the Gaussian pulse that had been initially modulated by centering it at t = 100 ns. Its propagation took zero time, so that the signal traveled with infinite speed. The numerical simulation has been based on Maxwell equations only. Proceeding from Figure 18 to Figure 19 verifies that the signal strongly lowered its amplitute. However, the width of each peak did not change (and this might have some relevance when thinking of a Morse alphabet “transmission”; see text).

FIGURE 20 Interesting experiments have been performed with two successive barriers—evanescence regions. For example, with two gratings in an optical fiber. This figure (Garavaglia, 1998) refers to the interesting experiment (Enders and Nimtz, 1993; Nimtz, Enders, and Spieker, 1993) performed with microwaves traveling along a metallic waveguide: the waveguide being endowed with two classical barriers—undersized guide segments. See text for details.

For suitable frequency bands—namely, for “tunneling” far from resonances—we found that the total crossing time does not depend on the length of the intermediate (normal) guide; that is, that the beam speed along it is inﬁnite (Aharonov, Erez, and Reznik, 2002; Enders and

Localized Waves: A Review

271

Nimtz, 1993; Esposito, 2003; Nimtz, Enders, and Spieker, 1993; Olkhovsky, Recami, and Jakiel, 2004; Olkhovsky, Recami, and Salesi, 2002; Recami, 2004). This agrees with the quantum mechanics prediction for nonresonant tunneling through two successive opaque barriers (Olkhovsky, Recami, and Salesi, 2002; Recami, 2004) (Figure 21). Such a prediction was veriﬁed ﬁrst theoretically by Olkhovsky, Recami, and Salesi (2002), and then, a second time, experimentally by taking advantage of the circumstance that evanescence regions can consist of a variety of photonic band-gap materials or gratings (from multilayer dielectric mirrors, or semiconductors, to photonic crystals). Indeed, the best experimental conﬁrmation has come by having recourse to two gratings in an optical ﬁber: Longhi et al. (2002); see Figures 22 and 23; particularly the rather peculiar (and quite interesting) results represented by the latter. We cannot omit a further topic—which, being delicate, probably should not appear in an overview like this—because it is currently raising much interest (Wang, Kuzmich, and Dogariu, 2000). Even if all the ordinary causal

V0

V0

1

I

II 0

2

III

IV

␣

V L 1␣

L

FIGURE 21 Scheme of the (nonresonant) tunneling process through two successive (opaque) quantum barriers. Far from resonances, the (total) phase time for tunneling through the two potential barriers does depend neither on the barrier widths nor on the distance between the barriers (“generalized Hartman effect”) (Olkhovsky, Recami, and Jakiel, 2004; Olkhovsky, Recami, and Salesi, 2002; Recami, 2001, 2004; Recami et al., 2003). See text for details.

Transmitted pulse

Incident pulse

n (z )5n0 [112V (z )cos(2␲ z /L)]

L

z50

L0

L

z

z 5 L 12L0

FIGURE 22 Realization of the quantum theoretical setup in Figure 21 using, as classical (photonic) barriers, two gratings in an optical fiber. The corresponding experiment has been performed by Longhi et al. (2002).

Erasmo Recami and Michel Zamboni-Rached

Tunneling time (ps)

272

300

200

100

0 10

20 30 40 Barrier separation (mm)

50

FIGURE 23 Off-resonance tunneling time versus barrier separation for the rectangular symmetric double-barrier Fiber Bragg grating (FBG) structure considered in Longhi et al. (2002) (see Figure 22). The solid line is the theoretical prediction based on group delay calculations; the dots are the experimental points as obtained by time-delay measurements (the dashed curve is the expected transit time from input to output planes for a pulse tuned far away from the stopband of the FBGs). The experimental results (Longhi et al., 2002), as well as the early ones in Enders and Nimtz (1993); Nimtz, Enders, and Spieker (1993), do confirm the theoretical prediction of a “generalized Hartman effect,” in particular, the independence of the total tunneling time from the distance between the two barriers.

paradoxes seem to be solvable (Recami, 1985, 1986, 1987), nevertheless one must consider (whenever it is met an object, O, traveling with superluminal speed) the possibility of dealing with negative contributions to the tunneling times (Olkhovsky, Recami, and Jakiel, 2004; Olkhovsky et al., 1995; Recami, 1986), which should not be regarded as unphysical. In fact, whenever an “object” (particle, electromagnetic pulse) O overcomes (Maccarrone and Recami, 1980; Recami, 1985, 1986, 1987) the inﬁnite speed with respect to a certain observer, it will afterward appear to the same observer as the “anti-object” O traveling in the opposite space direction (Bilaniuk, Deshpande, and Sudarshan, 1962; Maccarrone and Recami, 1980; Recami, 1985, 1986, 1987). For instance, when going from the lab to a frame F moving in the same direction as the particles or waves entering the barrier region, the object O penetrating through the ﬁnal part of the barrier (with almost inﬁnite speed, as in Figure 15) will appear in the frame F as an antiobject O crossing that portion of the barrier in the opposite space direction. In the new frame F , therefore, such anti-object O would yield a negative contribution to the tunneling time, which could even result as negative (for clariﬁcations, see the quoted references). We stress that even the appearance of such negative times had been predicted within SR itself (Olkhovsky et al., 1995), on the basis of its ordinary postulates, and recently has been conﬁrmed by quantum theoretical evaluations too (Olkhovsky, Recami, and Jakiel, 2004; Petrillo and Refaldi, 2000, 2003; Refaldi, 2000). (In the case of

Localized Waves: A Review

273

a nonpolarized beam, the wave anti-packet coincides with the initial wave packet; however, if a photon is endowed with helicity λ = +1, the antiphoton will bear the opposite helicity λ = −1.) From the theoretical point of view, besides the above-quoted papers (in particular Hartman, 1962; MacColl, 1932; Milonni, 2002; Olkhovsky and Recami, 1992; Olkhovsky, Recami, and Jakiel, 2001, 2004; Olkhovsky et al., 1995), see more specifically Bolda, Chiao, and Garrison, 1993; Chiao, Kozhekin, and Kurizki, 1996; Garret and McCumber, 1970. On the (very interesting!) experimental side, see the intriguing papers by Chu and Wong (1982); Macke et al. (1987); Mitchell and Chiao (1997); Nimtz (1999); Segard and Macke (1985); Wang, Kuzmich, and Dogariu (2000). Let us add here that, via quantum interference effects, it is possible to obtain dielectrics with refraction indices that rapidly vary as a function of frequency, also in three-level atomic systems, with almost complete absence of light absorption (i.e., with quantum-induced transparency) (Alzetta et al., 1976). The group velocity of a light pulse propagating in such a medium can decrease to very low values, either positive or negative, with no pulse distortion. Experiments performed both in atomic samples at room temperature and in Bose–Einstein condensates have shown the possibility of reducing the speed of light to a few meters per second. Similar, but negative group velocities, implying a propagation with superluminal speeds thousands of time higher than those previously mentioned, recently have been predicted, in the presence of such an “electromagnetically induced transparency,” for light moving in a rubidium condensate (Artoni et al., 2001). Finally, faster-than-c propagation of light pulses can be (and has been, in same cases) observed by taking advantage of the anomalous dispersion near an absorbing line, nonlinear and linear gain lines (as already seen), nondispersive dielectric media, or inverted twolevel media, as well as of some parametric processes in nonlinear optics (see Kurizki et al.’s works).

3.3.4. Superluminal Localized Solutions (SLS) to the Wave Equations The X-shaped waves. The fourth sector (to leave aside the others) is no less important. It came into fashion again when it was rediscovered in a series of remarkable works that any wave equation (to ﬁx the ideas, let us think of the electromagnetic case) also admits solutions as much subluminal as superluminal (besides the luminal ones, having speed c/n). Starting with the pioneering works of (Bateman, 1915), it slowly became known that all wave equations admit soliton-like (or rather wavelet-type) solutions with subluminal group velocities. Subsequently, superluminal solutions also started to be written (in one case (Barut and Chandola, 1993) simply by the mere application of a superluminal Lorentz “transformation” (Recami, 1986)).

274

Erasmo Recami and Michel Zamboni-Rached

A remarkable feature of some of these new solutions (which attracted much attention for their possible applications) is that they propagate as localized, nondispersive pulses because of their self-reconstruction property. It is easy to realize the practical importance, for instance, of a radio transmission carried out by localized beams (independently of their speed), but nondispersive wave packets can be of use even in theoretical physics for a reasonable representation of elementary particles, and so on. Incidentally, from the point of view of elementary particles, the fact that the wave equations possess pulse-type solutions that, in the subluminal case, are ball-like (Figure 24) can have a bearing on the corpuscle/wave duality problem met in quantum physics (besides agreeing with Figure 11, for example). Further comments on this point are found below. We emphasize once again that, within extended SR, since 1980 it has been shown that—while the simplest subluminal object conceivable is a small sphere, or a point in the limiting case—the simplest superluminal objects by contrast appear as (see Barut, Maccarrone, and Recami, 1982), and our Figures 11 and 12) an “X-shaped” wave, or a double cone, as their limit, which moreover travels without deforming (i.e., rigidly) in a homogeneous medium. It is not without meaning that the most interesting localized solutions to the wave equations happened to be the superluminal ones, and with a shape of that kind. Moreover, since from Maxwell equations under simple hypotheses one proceeds to the usual scalar wave equation for each electric or magnetic ﬁeld component, the same solutions were expected to exist also in the ﬁelds of acoustic waves, seismic waves, and gravitational waves. Indeed, this has been suggested in the literature for all such cases, and demonstrated in acoustics. Such pulses (as suitable superpositions of Bessel beams) were mathematically constructed for the

␳

␨

FIGURE 24 The wave equations possess pulse-type solutions that, in the subluminal case, are ball-like (in agreement with Figure 11). For comments, see the text.

Localized Waves: A Review

275

ﬁrst time by Lu et al. in acoustics and were then called X-waves or rather X-shaped waves. (One should not forget that, however, LWs can be constructed in exact form even for other equations, such as Schröedinger’s and Einstein’s). It is important that the X-shaped waves have been produced experimentally, with both acoustic and electromagnetic waves; that is, X-pulses were produced that, in their medium, travel undistorted with a speed faster than sound, in the ﬁrst case, and faster than light, in the second case. In acoustics, the ﬁrst experiment was performed by Lu et al. in 1992 at the Mayo Clinic (and their papers received the ﬁrst Institute of Electrical and Electronics Engineer 1992 award). In the electromagnetic case, certainly more intriguing, superluminal localized X-shaped solutions were ﬁrst mathematically constructed (Compare with Figure 25) in Lu, Greenleaf, and Recami (1996); Recami (1998), and later were experimentally produced by Saari et al. (Saari and Reivelt, 1997) in 1997 at Tartu by visible light (Figure 26), and more recently by Ranfagni et al. at Florence by microwaves (Mugnai, Ranfagni, and Ruggeri, 2000). In the theoretical sector the activity has been no less intense, in order to derive, for example, analogous new solutions with ﬁnite total energy or more suitable for high frequencies, on the one hand, and localized solutions superluminally propagating

2 mm

2 mm Max. 5 1.0 Min. 5 0.0

1.0 4m

4m

Max. 5 9.5e6 Min. 529.5e6

0.0

YXBB0)␪} (b) Re {(E

(a) Re {FXBB0}

2 mm

2 mm Max. 5 2.5e4 Min. 522.5e4

4m

4m

Y XBB0)p} (c) Re{(B

21.0

Max. 5 6.1 Min. 521.5

Y XBB0)z} (d) Re{(B

FIGURE 25 Real Part of the Hertz potential and of the field components of the localized electromagnetic (“classic,” axially symmetric) X-shaped wave predicted, and first mathematically constructed for the electromagnetic case in Lu, Greenleaf, and Recami (1996), and Recami (1998). For the meaning of the various panels, see the quoted references. The dimension of each panel is 4 m (in the radial direction) × 2 mm (in the propagation direction). The values shown on the top-right corner of each panel represent the maxima and the minima of the images before normalization for display (IS units).

276

Erasmo Recami and Michel Zamboni-Rached

Pos.1

Pos.2

Pos.3 z

V L1 L 2 PH M

L3

FIGURE 26 Scheme of the experiment by (Saari and Reivelt, 1997), who announced (Physical Review Letters, Nov. 24, 1997) the production in optics of the beams depicted in Figure 25. The present figure shows the experimental results. The X-shaped waves are superluminal; indeed, they, running after plane waves (the latter regularly traveling with speed c), do catch up with the considered plane waves. An analogous experiment was performed later with microwaves at Florence by (Mugnai, Ranfagni, and Ruggeri, 2000) (Physical Review Letters of May 22, 2000).

ⱍ␺3Dⱍ2 1

3 2

0.5 1

0 4

0 3.5

␩

⫺1

3 ␳

2.5

⫺2

2 1.5

⫺3

FIGURE 27 This figure depicts two elements of the trains of X-shaped pulses, mathematically constructed in Zamboni-Rached et al. (2002), which propagate down a coaxial guide (in the Transverse Magnetic case). This picture is taken from (Zamboni-Rached et al., 2002), but analogous X-pulses exist (with infinite or finite total energy) for propagation along a cylindrical, normal-sized metallic waveguide.

even along a normal waveguide (see Figure 27), on the other hand; and so on. Let us eventually recall the problem of producing an X-shaped superluminal wave like the one in Figure 12, but truncated in space and in

Localized Waves: A Review

277

time (by use of a ﬁnite antenna, radiating for a ﬁnite time). In such a situation, the wave is known to keep its localization and superluminality only until a certain depth of ﬁeld (i.e., as long as it is fed by the waves arriving (with speed c) from the antenna), decaying often abruptly afterward (Durnin, Miceli, and Eberly, 1987a; Zamboni-Rached, Recami, and Hernández-Figueroa, 2002). Various authors, taking account of the time needed to foster such superluminal waves, have concluded that these localized superluminal pulses are unable to transmit information faster than c. Many of these questions have been discussed in preceding text and references; for further details, see Lu, Greenleaf, and Recami, 1996; Recami, 1998; Shaarawi and Besieris, 2000a; Ziolkowski, Besieris, and Shaarawi, 2000. In any case, the existence of the X-shaped superluminal (or supersonic) pulses seems to constitute (e.g., together with the superluminality of evanescent waves), a conﬁrmation of extended SR: a theory (Recami, 1986) based on the ordinary postulates of SR and that consequently does not appear to violate any of the fundamental principles of physics. It is curious that one of the ﬁrst applications of such X-waves (that takes advantage of their propagation without deformation) has been accomplished in the ﬁeld of medicine—precisely, of ultrasound scanners (Lu, Zou, and Greenleaf, 1993, 1994), whereas the most important applications of the (subluminal!) Frozen Waves will very probably once again affect human health problems such as cancer. After the digression in this appendix, we pass to a second part of the work with a slightly more technical (Zamboni-Rached, Recami, and Hernández-Figueroa, 2008) review about the physical and mathematical characteristics of LWs and some interesting applications. In the third part we shall deal with the ones endowed with zero speed (i.e., with a static envelope) and, more in general, with the subluminal LWs.

4. STRUCTURE OF NONDIFFRACTING WAVES AND SOME INTERESTING APPLICATIONS 4.1. Foreword Since the early works (Brittingham, 1983; Durnin, Miceli, and Eberly, 1987a; Zamboni-Rached, Recami, and Hernández-Figueroa, 2008) on the so-called nondiffracting waves (or Localized Waves), many articles have been published on this important subject from both the theoretical and the experimental points of view. Initially, the theory was developed taking into account only free space; however, in recent years, it has been extended for more complex media exhibiting effects such as dispersion

278

Erasmo Recami and Michel Zamboni-Rached

(Lu and Greenleaf, 1992b; Sõnajalg and Saari, 1996; Zamboni-Rached et al., 2003), nonlinearity, Conti et al. (2003), anisotropy, Salo et al. (1999), and losses (Zamboni-Rached, 2006). Such extensions have been carried out in addition to the development of efﬁcient methods for obtaining nondiffracting beams and pulses in the subluminal, luminal, and superluminal regimes (Besieris, Shaarawi, and Ziolkowski, 1989; Longhi, 2004; Zamboni-Rached, 1999, 2004a,b, 2006; Zamboni-Rached, Recami, and Hernández-Figueroa, 2002, 2005). This section addresses some theoretical methods related to nondiffracting solutions of the linear wave equation in unbounded homogeneous media, as well as some interesting applications of such solutions. ZamboniRached, Recami, and Hernández-Figueroa (2008). The usual cylindrical coordinates (ρ, φ, z) are used herein. We already know that in these coordinates the linear wave equation is written as

∂ 1 ∂2 ∂2 1 ∂2 1 ∂ ρ + 2 2 + 2 − 2 2 = 0. ρ ∂ρ ∂ρ ρ ∂φ ∂z c ∂t

(27)

In Section 4.2 we analyze the general structure of LWs, develop the socalled Generalized Bidirectional Decomposition, and use it to obtain several luminal and superluminal nondiffracting wave solutions of Eq. (27). In Section 4.3 we develop a space-time focusing method by a continuous superposition of X-Shaped pulses of different velocities. Section 4.4 addresses the properties of chirped optical X-shaped pulses propagating in material media without boundaries. Subsequently, we show at the beginning of Section 5 how a suitable superposition of Bessel beams can be used to obtain stationary localized wave ﬁelds with high transverse localization, and whose longitudinal intensity pattern can assume any desired shape within a chosen interval 0 ≤ z ≤ L of the propagation axis. Because of space constraints, we necessarily omit many interesting results. Let us brieﬂy mention, for example, that rather simple analytic expressions, capable of describing the longitudinal (on-axis) evolution of axially symmetric nondiffracting pulses, have been recently completed by Zamboni-Rached (2006) even for pulses truncated by ﬁnite apertures. Excellent agreement has been found by comparing what is easily provided by such expressions, for several situations (involving subluminal, luminal, or superluminal localized pulses) with the results obtained by numerical evaluations of the Rayleigh–Sommerfeld diffraction integrals. Therefore, those new closed-form expressions dispense with the need for time-consuming numerical simulations (and provide an effective tool for determining the most important properties of the truncated localized pulses).

Localized Waves: A Review

279

4.2. Spectral Structure of Localized Waves and the Generalized Bidirectional Decomposition An effective approach to understand the concept of the (ideal) nondiffracting waves is furnishing a precise mathematical deﬁnition of these solutions, so to extract the necessary spectral structure from them. Intuitively, an ideal nondiffracting wave (beam or pulse) can be deﬁned as a wave capable of maintaining indeﬁnitely its spatial form (apart from local variations) while propagating. We can express such a characteristic by saying that a localized wave has to possess the property (Besieris, Shaarawi, and Ziolkowski, 1989; Zamboni-Rached, 1999).

z0 (ρ, φ, z, t) = ρ, φ, z + z0 , t + V

,

(28)

where z0 is a certain length and V is the pulse propagation speed that here can assume any value: 0 ≤ V ≤ ∞. In terms of a Fourier-Bessel (FB) expansion, we can write a function (ρ, φ, z, t) as

(ρ, φ, z, t) =

∞ n=−∞

×

∞ −∞

0

∞

dkρ

∞

−∞

dkz

dω kρ An (kρ , kz , ω)Jn (kρ ρ)e

ikz z −iωt inφ

e

e

.

(29)

On using the translation property of the Fourier transforms T[ f (x + a)] = exp(ika)T[ f (x)], we have that An (kρ , kz , ω) and exp[i(kz z0 − ωz0 / V)]An (kρ , kz , ω) are the FB transforms of the left-hand side and right-hand side functions in Eq. (28). From this same equation we can obtain (Besieris, Shaarawi, and Ziolkowski, 1989; Zamboni-Rached, 1999) the fundamental constraint linking the angular frequency ω and the longitudinal wave number kz :

ω = Vkz + 2mπ

V , z0

(30)

with m an integer. Obviously, this constraint can be satisﬁed by means of the spectral functions An (kρ , kz , ω). Now, let us explicitly mention that constraint (30) does not imply any breakdown of the wave equation. In fact, when inserting expression (29)

280

Erasmo Recami and Michel Zamboni-Rached

in the wave equation (27), we derive

ω2 = kz2 + kρ2 . c2

(31)

So, to obtain a solution of the wave equation by expression (29), the spectrum An (kρ , kz , ω) must possess the form

An (kρ , kz , ω) = An (kz , ω) δ

kρ2

−

ω2 − kz2 c2

,

(32)

where δ(.) is the Dirac delta function. With this, we can write a solution of the wave equation as

(ρ, φ, z, t) =

⎡

∞

⎣

∞

dω 0

n=−∞

ω/c −ω/c

dkz An (kz , ω)Jn

⎛

⎞ ⎤ 2 ω × ⎝ρ − kz2 ⎠ eikz z e−iωt einφ ⎦ c2

(33)

where we have considered positive angular frequencies only. Equation (33) is a superposition of Bessel beams, and it is understood that the integrations in the (ω, kz ) plane are conﬁned to the region 0 ≤ ω ≤ ∞ and −ω/c ≤ kz ≤ ω/c. Now, to obtain an ideal nondiffracting wave, the spectra An (kz , ω) must obey the fundamental constraint in Eq. (30), and so we write

An (kz , ω) =

∞

Snm (ω)δ [ω − (Vkz + bm )] ,

(34)

m=−∞

where bm are constants representing the terms 2mπV/z0 in Eq. (30), and Snm (ω) are arbitrary frequency spectra. By inserting Eq. (34) into Eq. (33), we get a general integral form of the ideal nondiffracting wave in Eq. (28):

(ρ, φ, z, t) =

∞

∞

n=−∞ m=−∞

ψnm (ρ, φ, z, t)

(35)

Localized Waves: A Review

281

with

ψnm (ρ, φ, z, t) = e−ibm z/V

(ωmax )m

(ωmin )m

dω Snm (ω)

⎛ ⎞ 2 ω 1 1 2b b × Jn ⎝ρ − 2 ω2 + 2 ω − 2 ⎠ ei V (z−Vt) einφ , 2 c V V V (36) where ωmin and ωmax depend on the values of V as speciﬁed below: • for subluminal (V < c) localized waves: bm > 0, (ωmin )m = cbm /(c + V) and (ωmax )m = cbm /(c − V); • for luminal (V = c) localized waves: bm > 0, (ωmin )m = bm /2 and (ωmax )m = ∞; • for superluminal (V > c) localized waves: bm ≥ 0, (ωmin )m = cbm /(c + V) and (ωmax )m = ∞. Or bm < 0, (ωmin )m = cbm /(c − V) and (ωmax )m = ∞. It is important to notice that each ψnm (ρ, φ, z, t) in the superposition in Eq. (35) is a truly nondiffracting wave (beam or pulse) and the superposition of them [Eq. (35)] is just the most general form to represent a nondiffracting wave deﬁned by Eq. (28). Due to this fact, the search for methods capable of providing analytic solutions for ψnm (ρ, φ, z, t), [Eq. (36)], becomes an important task. Recall that Eq. (36) is also a Bessel beam superposition, but with constraint (30) linking the angular frequencies and longitudinal wave numbers. Despite the fact that Eq. (36) represents ideal nondiffracting waves, it is difﬁcult to obtain closed analytic solutions from it. Because of this, we develop a method capable of overcoming such a difﬁculty, providing several interesting LW solutions (luminal and superluminal) of arbitrary frequencies, including some solutions endowed with ﬁnite energy.

4.2.1. The Generalized Bidirectional Decomposition For reasons that will be clear soon, instead of dealing with the integral expression (35), our starting point is the general expression in Eq. (33). Here, for simplicity, we restrict ourselves to axially symmetric solutions, adopting the spectral functions

An (kz , ω) = δn0 A(kz , ω), where δn0 is the Kronecker delta.

(37)

282

Erasmo Recami and Michel Zamboni-Rached

In this way, we get the following general solution (considering positive angular frequencies only), which describes axially symmetric waves:

(ρ, φ, z, t) =

∞

dω 0

ω/c −ω/c

⎛ dkz A(kz , ω)J0 ⎝ρ

⎞ ω2 c2

− kz2 ⎠ eikz z e−iωt . (38)

As we have seen, we can obtain ideal nondiffracting waves, given that the spectrum A(kz , ω) satisﬁes the linear relationship in Eq. (30). Therefore, it is natural to choose new spectral parameters, in place of (ω, kz ), that make it easier to implement the mentioned constraint (Besieris, Shaarawi, and Ziolkowski, 1989; Zamboni-Rached, 1999). With this in mind, let us choose the new spectral parameters (α, β)

α≡

1 (ω + Vkz ); 2V

β≡

1 (ω − Vkz ). 2V

(39)

Let us consider here only luminal (V = c) and superluminal (V > c) nondiffracting pulses. With the change of variables [Eq. (39)] in the integral solution [Eq. (38)], and considering (V ≥ c), the integration limits on α and β must satisfy the three inequalities below:

⎧ 0<α+β<∞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ α ≥ c −Vβ c+V ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ α ≥ c + V β. c−V

(40)

Let us suppose that both α and β are positive [α, β ≥ 0]. The ﬁrst inequality in Eq. (40) is then satisﬁed, whereas the coefﬁcients (c − V)/(c + V) and (c + V)/(c − V) entering relations Eq. (40) are both negative (since V ≥ c). As a consequence, the other two inequalities in Eq. (40) are automatically satisﬁed. In other words, the integration limits 0 ≤ α ≤ ∞ and 0 ≤ β ≤ ∞ are “contained” in the limits of (40) and are therefore acceptable. Indeed, they constitute a rather suitable choice for facilitating all the subsequent integrations. Therefore, instead of Eq. (38), we shall consider the (more easily integrable) Bessel beam superposition in the new variables [with V ≥ c]

Localized Waves: A Review

(ρ, ζ, η) =

∞

dα 0

0

∞

283

dβA(α, β) J0

⎛ ⎞ 2 2 V V × ⎝ρ − 1 (α2 + β2 ) + 2 + 1 αβ⎠ eiαζ e−iβη , c2 c2 (41) where we have deﬁned

ζ ≡ z − Vt;

η ≡ z + Vt.

(42)

This procedure is a generalization of the so-called “bidirectional decomposition” technique (Besieris, Shaarawi, and Ziolkowski, 1989), which was devised in the past for V = c. From the new spectral parameters deﬁned in transformation (39), it is easy to see that the constraint in Eq. (30), i.e., ω = Vkz + b, is implemented just by making

A(kz , ω) → A(α, β) = S(α)δ(β − β0 )

(43)

with β0 = b/2V. The delta function δ(β − β0 ) in the spectrum in Eq. (43) means that we are integrating Bessel beams along the continuous line ω = Vkz + 2Vβ0 and, in this way, the function S(α) will give the frequency dependence of the spectrum: S(α) → S(ω/V − β0 ). This method constitutes a simple, natural way for obtaining pulses with ﬁeld concentration on ρ = 0 and at ζ = 0 → z = Vt. It is important to emphasize (Zamboni-Rached, Recami, and Hernández-Figueroa, 2002) that, when β0 > 0 in Eq. (43), superposition in Eq. (41) gets contributions from both backward and forward traveling Bessel beams, corresponding to the frequency intervals Vβ0 ≤ ω < 2Vβ0 (where kz < 0) and 2Vβ0 ≤ ω ≤ ∞ (where kz ≥ 0), respectively. Nevertheless, we can obtain physical solutions when rendering the contribution of the backward-traveling components negligible by choosing suitable weight functions S(α). It is also noteworthy that we adopted the new spectral parameters α and β just to obtain (closed-form) analytic localized wave solutions. The spectral characteristics of these new solutions can be brought into evidence by using transformations in Eq. (39) and writing the corresponding spectrum in terms of the usual ω and kz spectral parameters. The following subsections consider some cases with β0 = 0 and β0 > 0.

284

Erasmo Recami and Michel Zamboni-Rached

Closed analytic expressions describing some ideal nondiffracting pulses. Let us ﬁrst consider, in Eq. (41), spectra of the type in Eq. (43) with β0 = 0: A(α, β) = aV δ(β)e−aVα

(44)

√ A(α, β) = aV δ(β)J0 (2d α)e−aVα

(45)

A(α, β) = δ(β)

sin(dα) −aVα e , α

(46)

a > 0 and d being constants. We can obtain from the above spectra the following superluminal LW solutions, respectively: • from spectrum (44), we can use the identity (6.611.1) in Gradshteyn and Ryzhik (1965), obtaining the well-known ordinary X-wave solution (also called X-shaped pulse)

aV (ρ, ζ) ≡ X = 2 ; V 2 (aV − iζ) + c2 − 1 ρ2

(47)

• by using spectrum (45) and the identity (6.6444) of Gradshteyn and Ryzhik (1965), one gets

⎞ 2 V (ρ, ζ) = X J0 ⎝ 2 − 1 (aV)−2 d2 X 2 ρ⎠exp −(aV − iζ) (aV)−2 d2 X 2 ; c ⎛

(48) • the superluminal nondiffracting pulse

(ρ, ζ) = sin

−1

d 2 aV

X −2

+ (d/aV)2

+ 2ρd(aV)−2

+

X −2 + (d/aV)2 − 2ρd(aV)−2

V 2 /c2 − 1

−1 ⎤ ⎦ V 2 /c2 − 1

(49)

is obtained from spectrum (46) by using identity (6.752.1) of Gradshteyn and Ryzhik (1965) for a > 0 and d > 0.

Localized Waves: A Review

285

From the previous discussion, we learn that any solutions obtained from spectra of the type in Eq. (43) with β0 = 0 are free of noncausal (backwardtraveling) components. In addition, when β0 = 0, the pulsed solutions depend on z and t through ζ = z − Vt only, and so propagate rigidly (i.e., without distortion). Such pulses can be transversally localized only if V > c, because if V = c the function must obey the Laplace equation on the transverse planes (Besieris, Shaarawi, and Ziolkowski, 1989; Zamboni-Rached, 1999). Many others superluminal LWs can be easily constructed (ZamboniRached, Recami, and Hernández-Figueroa, 2002) from the above solutions simply by taking the derivatives (of any order) with respect to ζ. It is also possible to show that the new solutions, obtained in this way, have their spectra shifted toward higher frequencies. Now, let us pass to consider, in Eq. (41), a spectrum of the type in Eq. (43) with β0 > 0:

A(α, β) = aVδ(β − β0 ) e−aVα

(50)

with a a positive constant. The presence of the delta function, with the constant β0 > 0, implies that we are integrating (summing) Bessel beams along the continuous line ω = Vkz + 2Vβ0 . Now, the function S(α) = aVexp(−aVω) entails that we are considering a frequency spectrum of the type S(ω) ∝ exp(−aω), and therefore with a bandwidth given by ω = 1/a. Since β0 > 0, the interval Vβ0 ≤ ω < 2Vβ0 (or, equivalently, in this case, 0 ≤ α < β0 ), corresponds to backward Bessel beams—negative values of kz . However, we can obtain physical solutions by making the contribution of this frequency interval negligible. In our case, this can be obtained by making aβ0 V << 1, so that the exponential decay of the spectrum S with respect to ω is very slow, and the contribution of the interval ω ≥ 2Vβ0 (where kz ≥ 0) largely overruns the Vβ0 ≤ ω < 2Vβ0 (where kz < 0) contribution. Incidentally, once we ensure the causal behavior of the pulse by making aVβ0 << 1 in Eq. (50), we have that α = 1/aV >> β0 , and one can therefore simplify the argument of the Bessel function, in the integrand of superposition in Eq. (41), by neglecting the term (V 2 /c2 − 1)β02 . With this, the superposition in Eq. (41), with the spectrum (50), can be written as

(ρ, ζ, η) ≈ aVe−iβ0 η

∞

0

× eiαζ e−aVα .

⎛ ⎞ 2 2 V V dαJ0 ⎝ρ − 1 α2 + 2 + 1 αβ0 ⎠ c2 c2 (51)

286

Erasmo Recami and Michel Zamboni-Rached

Now, we can use identity (6.616.1) of Gradshteyn and Ryzhik (1965) and obtain the new localized superluminal solution called (Zamboni-Rached, Recami, and Hernández-Figueroa, 2002) Superluminal Focus Wave Mode (SFWM):

β0 (V 2 + c2 ) −1 (aV − iζ) − a VX , X exp V 2 − c2

−iβ0 η

SFWM (ρ, ζ, η) = e

(52) where, as before, X is the ordinary X-pulse in Eq. (47). The center of the SFWM is located on ρ = 0 and ζ = 0 (i.e., at z = Vt). The intensity, ||2 , of this pulse propagates rigidly, being a function of ρ and ζ only. However, the complex function SFWM (i.e., its real and imaginary parts) propagate with local variations, recovering their entire 3-D form after each space and time interval z0 = π/β0 and t0 = π/β0 V. The SFWM solution above, for V −→ c+ , reduces to the well-known Focus Wave Mode (FWM) solution (Besieris, Shaarawi, and Ziolkowski, 1989), traveling with speed c:

e−iβ0 η β0 ρ 2 FWM (ρ, ζ, η) = ac exp − . ac − iζ ac − iζ

(53)

We also emphasize that, since β0 > 0, Eq. (50) corresponds to angular frequencies ω ≥ Vβ0 . Thus, our new solution also can be used to construct high-frequency pulses.

Finite-energy nondiffracting pulses. In this subsection, we show how to obtain ﬁnite-energy LW pulses that can propagate for long distances while maintaining their spatial resolution (i.e., that possess a large depth of ﬁeld). Ideal nondiffracting waves can be constructed by superposing Bessel beams [see Eq. (38) for cylindrical symmetry] with a spectrum A(ω, kz ) that satisﬁes a linear relationship between ω and kz . In the general bidirectional decomposition method, this can be obtained by using spectra of the type in Eq. (43) in superposition (41). Solutions of that type possess an inﬁnite depth of ﬁeld; however, they are endowed with inﬁnite energy. To overcome this problem, we can truncate an ideal nondiffracting wave by a ﬁnite aperture, and the resulting pulse will have ﬁnite energy and a ﬁnite ﬁeld depth. Even so, such ﬁeld depths may be very large compared with those of ordinary waves.

Localized Waves: A Review

287

The problem in the present case is that the resulting ﬁeld must be calculated by diffraction integrals (such as the well-known Rayleigh– Sommerfeld formula) and, in general, a closed analytic formula for the resulting pulse cannot be obtained. However, there is another way to construct localized pulses with ﬁnite energy (Zamboni-Rached, Recami, and Hernández-Figueroa, 2002)— namely, by using spectra A(ω, kz ) in Eq. (38), whose domains are not restricted exactly to the straight line ω = Vkz + b, but are deﬁned in the surroundings of that line, wherein the spectra should have their main values concentrated (in other words, any spectrum must be well localized in the vicinity of that line). Similarly, in terms of the generalized bidirectional decomposition given in Eq. (41), ﬁnite-energy nondiffracting wave pulses can be constructed by adopting spectral functions A(α, β) well localized in the vicinity of the line β = β0 , quantity β0 being a constant. To exemplify this method, let us consider the following spectrum:

A(α, β) =

⎧ ⎨ a q V e−aVα e−q(β−β0 ) ⎩

for β ≥ β0 for 0 ≤ β < β0

0

(54)

in superposition (41), quantities a and q being free positive constants and V the peak’s pulse velocity (here, V ≥ c). It is easy to see that the above spectrum is zero in the region above the β = β0 line, while it decays in the region below (as well as along) such a line. We can concentrate this spectrum on β = β0 by choosing values of q in such a way that qβ0 >> 1. The faster the spectrum decay takes place in the region below the β = β0 line, the larger is the ﬁeld depth of the corresponding pulse. Once we choose qβ0 >> 1 to obtain pulses with a large ﬁeld depth, we also can minimize the contribution of the noncausal (backward) components by choosing aVβ0 << 1, similar to the SFWM case. Again by analogy with the SFWM case, when we choose qβ0 >> 1 (i.e., a long ﬁeld depth) and aVβ0 << 1 (a minimal contribution of the backward components), we can simplify the argument of the Bessel function, in the integrand of superposition (41), by neglecting the term (V 2 /c2 − 1)β02 . With the help of the observations above, one can write the superposition (41), with spectrum (54), as

(ρ, ζ, η) ≈ a q V

∞

β0

∞

dβ 0

⎛ ⎞ 2 2 V V dα J0 ⎝ρ − 1 α2 + 2 2 + 1 αβ⎠ c2 c

× e−iβη eiαζ e−q(β−β0 ) e−aVα

(55)

288

Erasmo Recami and Michel Zamboni-Rached

and, using identity (6.616.1) given in Gradshteyn and Ryzhik (1965), we get

(ρ, ζ, η) ≈ q X

2 +c2 V dβe−q(β−β0 ) e−iβη exp β 2 2 aV −iζ−aVX −1 , V −c

∞

β0

(56) which can be viewed as a superposition of the SFWM pulses [see Eq. (52)]. The above integration can be performed easily and results (ZamboniRached, Recami, and Hernández-Figueroa, 2002) in the so-called Superluminal Modiﬁed Power Spectrum (SMPS) pulse:

SMPS (ρ, ζ, η) = q X

exp[(Y − iη)β0 ] , q − (Y − iη)

(57)

where X is the ordinary X-pulse [Eq. (47)] and Y is deﬁned by

Y ≡

V 2 + c2 −1 (aV − iζ) − aVX . V 2 − c2

(58)

The SMPS pulse is a superluminal LW, with ﬁeld concentration around ρ = 0 and ζ = 0 (i.e., z = Vt), and with ﬁnite total energy. We will show that the depth of ﬁeld, Z, of this pulse is given by ZSMPS = q/2. An interesting property of the SMPS pulse is related to its transverse width (the transverse spot size at the pulse center). It can be shown from Eq. (57) that, for the cases where aV << 1/β0 and qβ0 >> 1 (i.e., for the cases considered by us), the transverse spot size, ρ, of the pulse center (ζ = 0) is determined by the exponential function in Eq. (57) and is given by

ρ = c

V 2 − c2 aV + , β0 (V 2 + c2 ) 4β02 (V 2 + c2 )2

(59)

which clearly does not depend on z, and so remains constant during its propagation. In other words, even though the SMPS pulse incurs an intensity decrease during propagation, it preserves its transverse spot size. This interesting characteristic is not met in ordinary pulses (like the Gaussian ones) where the amplitude of the pulse decreases and the width increases by the same factor. Figure 28 shows the intensity of a SMPS pulse, with β0 = 33 m−1 , V = 1.01c, a = 10−12 s, and q = 105 m, at two different moments, for t = 0 and

289

Localized Waves: A Review

|CSMPS|2

6

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

4

1 y (mm)

2

0.8

0

22

0.6

24 26 26

0.4

24

22

0

2

4

6

|CSMPS|2

0.2

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

4

1

2

0.8

0

22

0.6

24 26 26

0.4

x (mm)

|CSMPS|2

6

y (mm)

|CSMPS|2

24

22

0

2

4

6

x (mm)

0.2

0 20

0 20 10 0

␳ (mm)

210

220 26

24

22

0

2

4

6

z (mm)

10 0

␳ (mm)

210

220 26 24

(a)

22

0

2

4

6

␨ (mm)

(b)

FIGURE 28 Representation of a Superluminal Modified Power Spectrum Pulse [Eq. (57)]. Its total energy is finite (even without any truncation), and so it is deformed while propagating since its amplitude decreases with time. Panel (a) represents, for t = 0, the pulse corresponding to β0 = 33m−1 , V = 1.01c, a = 10−12 s, and q = 105 m. Panel (b) shows the same pulse after traveling 50km.

after 50 km of propagation where the pulse becomes less intense (precisely, with half of its initial peak intensity). Despite the intensity decrease, the pulse maintains its transverse width, as seen in the 2D plots in Figure 28, which show the ﬁeld intensities in the transverse sections at z = 0 and z = q/2 = 50 km. Three other important well-known ﬁnite-energy nondiffracting solutions can be obtained directly from the SMPS pulse: • The ﬁrst one, obtained from Eq. (57) by making β0 = 0, is the so-called (Zamboni-Rached, Recami, and Hernández-Figueroa, 2002) superluminal splash pulse (SSP)

SSP (ρ, ζ, η) =

qX . q + iη − Y

(60)

• The other two pulses are luminal. By taking the limit V → c+ in the SMPS pulse of Eq. (57), we get the well-known (Besieris, Shaarawi, and Ziolkowski, 1989) luminal modiﬁed power spectrum (MPS) pulse

a q c e−iβ0 η −β0 ρ2 . MPS (ρ, ζ, η) = exp ac − iζ (q + iη)(ac − iζ) + ρ2

(61)

Finally, by taking the limit V → c+ and making β0 = 0 in the SMPS pulse [or, equivalently, by making β0 = 0 in the MPS of pulse in Eq. (61), or, instead, by taking the limit V → c+ in the SSP of Eq. (60)], we obtain the well-known (Besieris, Shaarawi, and Ziolkowski, 1989) luminal splash

290

Erasmo Recami and Michel Zamboni-Rached

pulse (SP) solution

SP (ρ, ζ, η) =

aqc . (q + iη)(ac − iζ) + ρ2

(62)

It is also interesting to notice that the X and SFWM pulses can be obtained from the SSP and SMPS pulses (respectively) by making q → ∞ in Eqs. (60) and (57). The solutions SSP and SMPS can be viewed as the ﬁnite-energy versions of the X and SFWM pulses, respectively.

Some characteristics of the SMPS pulse. Let us examine the on-axis (ρ = 0) behavior of the SMPS pulse. On ρ = 0, we have SMPS (ρ = 0, ζ, η) = aqVe−iβ0 z [(aV − iζ)(q + iη)]−1 .

(63)

From this expression, we can show that the longitudinal localization z, for t = 0, of the SMPS pulse square magnitude is

z = 2aV.

(64)

If we now deﬁne the ﬁeld depth Z as the distance over which the pulse’s peak intensity is at least 50% of its initial value,4 then we can obtain [from Eq. (63)] the depth of ﬁeld

ZSMPS =

q , 2

(65)

which depends only on q, as we expected since q regulates the concentration of the spectrum around the line ω = Vkz + 2Vβ0 . Now, let us examine the maximum amplitude M of the real part of Eq. (63), which for z = Vt writes (ζ = 0 and η = 2z):

MSMPS ≡ Re[SMPS (ρ = 0, z = Vt)] =

cos(2β0 z) − 2(z/q) sin(2β0 z) . 1 + 4(z/q)2 (66)

Initially, for z = 0, t = 0, one has M = 1 and can also infer that: (i) when z/q << 1, namely, when z << Z, Eq. (66) becomes

MSMPS ≈ cos(2β0 z)

for z << Z

(67)

4 We can expect that, while the pulse peak intensity is maintained, the pulse also keeps its spatial form.

Localized Waves: A Review

291

and the pulse’s peak actually oscillates harmonically with “wavelength” z0 = π/β0 and “period” t0 = π/Vβ0 , all along its ﬁeld depth; (ii) when z/q >> 1, namely, z >> Z, Eq. (66) becomes

MSMPS ≈ −

sin(2β0 z) 2z/q

for z >> Z.

(68)

Therefore, beyond its depth of ﬁeld, the pulse continues oscillating with the same z0 , but its maximum amplitude decays proportionally to z. The next two sections show applications of the LW pulses.

4.3. Space-Time Focusing of X-Shaped Pulses This section shows how any known superluminal solution can be used to obtain a large number of analytic expressions for space-time focused waves, endowed with a very strong intensity peak at the desired location. The method presented here is a natural extension of that developed by Shaarawi et al. (Shaarawi, Besieris, and Said, 2003), where the space-time focusing was achieved by superimposing a discrete number of ordinary X-waves, characterized by different values θ of the axicon angle. In this section, based on (Zamboni-Rached, Shaarawi, and Recami, 2004), we proceed to more efﬁcient superpositions for varying velocities V, related to θ through the known (Brittingham, 1983; Durnin, Miceli, and Eberly, 1987a; Recami, 1986) relation V = c/cos θ. This enhanced focusing scheme has the advantage of yielding analytic (closed-form) expressions for spatiotemporally focused pulses. We begin by considering an axially symmetric ideal nondiffracting superluminal wave pulse ψ(ρ, z − Vt) in a dispersionless medium, where V = c/cos θ > c is the pulse velocity and θ is the axicon angle. As shown previously, pulses like these can be obtained by a suitable frequency superposition of Bessel beams. Suppose that we have now N waves of the type ψn (ρ, z − Vn (t − tn )), with different velocities c < V1 < V2 < . . . < VN , and emitted at (different) times tn ; quantities tn being constants, while n = 1, 2, . . . , N. The center of each pulse is located at z = Vn (t − tn ). To obtain a highly focused wave, we need all the wave components ψn (ρ, z − Vn (t − tn )) to reach the given point, z = zf , at the same time t = tf . On choosing t1 = 0 for the slowest pulse ψ1 , it is easily seen that the peak of this pulse reaches the point z = zf at the time tf = zf /V1 . So we obtain that, for each ψn , the instant of emission tn must be

tn =

1 1 zf . − V1 Vn

(69)

292

Erasmo Recami and Michel Zamboni-Rached

With this in mind, we can construct other exact solutions to the wave equation, given by

(ρ, z, t) =

Vmax

Vmin

dV A(V)ψ ρ, z − V t −

1 Vmin

1 zf , (70) − V

where V is the velocity of the wave ψ(ρ, z − Vt) that enters the integrand of Eq. (70). While integrating, V is considered a continuous variable in the interval [Vmin , Vmax ]. In Eq. (70), function A(V) is the velocity distribution that speciﬁes the contribution of each wave component (with velocity V) to the integration. The resulting wave (ρ, z, t) can have a more or less strong amplitude peak at z = zf , at time tf = zf /Vmin , depending on A(V) and on the difference Vmax − Vmin . Notice that the resulting wave ﬁeld will propagate with a superluminal peak velocity, also depending on A(V). When the velocity-distribution function is well concentrated around a certain velocity value, the wave in Eq. (70) can be effected to increase its magnitude and spatial localization while propagating. Finally, the pulse peak acquires its maximum amplitude and localization at the chosen point z = zf , and at time t = zf /Vmin , as we know. Afterward, the wave suffers a progressing spreading, and a decreasing of its amplitude.

4.3.1. Focusing Effects by Using Ordinary X-Waves Here, we present a speciﬁc example by integrating Eq. (70) over the standard, classic (Lu and Greenleaf, 1992a) X-waves, X = aV[(aV − i(z − Vt))2 + (V 2 /c2 − 1)ρ2 ]−1/2 . When using this ordinary X-wave, the largest spectral amplitudes are obtained for low frequencies. For this reason, one may expect that the solutions considered below will be suitable mainly for low-frequency applications. Let us choose, then, the function ψ in the integrand of Eq. (70) to be ψ(ρ, z, t) ≡ X(ρ, z − V(t − (1/Vmin − 1/V)zf )), viz.:

ψ(ρ, z, t) ≡ X =

aV −i z − V t −

aV 1 Vmin

2 2 . − V1 zf + Vc2 − 1 ρ2 (71)

After some manipulations, one obtains the analytic integral solution

(ρ, z, t) =

Vmax

Vmin

aV A(V) dV

PV 2 + QV + R

(72)

Localized Waves: A Review

with

P≡

a+i t−

Q≡2 t−

zf Vmin

zf Vmin

2

+

ρ2 c2

− ai (z − zf )

293

(73)

! R ≡ −(z − zf )2 − ρ2 . In the following examples, we illustrate the behavior of some new spatiotemporally focused pulses, by taking into consideration a few different velocity distributions A(V). These new pulses are closed analytic exact solutions of the wave equation.

Example 1. Let us consider our integral solution in Eq. (72) with A(V) = 1 s/m. In this case, the contribution of the X-waves is the same for all velocities in the allowed range [Vmin , Vmax ]. By using identity 2.264.2 listed in (Gradshteyn and Ryzhik, 1965), we get the particular solution a 2 2 (ρ, z, t) = PVmax + QVmax + R − PVmin + QVmin + R P ⎞ ⎛ 2 P(PV + QV + R) + 2PV + Q 2 min min aQ ⎜ min ⎟ + 3/2 ln⎝

⎠, 2 2P 2 P(PVmax + QVmax + R) + 2PVmax + Q (74) where P, Q, and R are given in Eq. (73). A 3D plot of this function is shown in Figure 29; where we have chosen a = 10−12 s, Vmin = 1.001 c, Vmax = 1.005 c, and zf = 200 cm. This solution exhibits a rather evident space-time focusing. An initially spread-out pulse (shown for t = 0) becomes highly localized at t = tf = zf /Vmin = 6.66 ns, the pulse peak amplitude at zf being 40.82 times greater than the initial one. In addition, at the focusing time tf the ﬁeld is much more localized than at any other time. The velocity of this pulse is approximately V = 1.003 c.

Example 2. In this case, we choose A(V) = 1/V s/m, and, using the identity 2.261 in (Gradshteyn and Ryzhik, 1965), Eq. (72) yields ⎞

2 a + QVmax + R) + 2PVmax + Q ⎟ ⎜ 2 P(PVmax (ρ, z, t) = √ ln⎝ ⎠. 2 P 2 P(PVmin + QVmin + R) + 2PVmin + Q ⎛

(75)

294

Erasmo Recami and Michel Zamboni-Rached

t 52.22 ns

t 5 0 ns

0.0245

0.1

0.05

0 ␳ (m)

20.5

t 5 6.66 ns

0.7 0.6 0.65 z (m)

1.9

2.1

0 0.5

1.35 1.4 1.25 1.3 z (m)

t 511.1 ns

0.04 |C|2

0.05

␳ (m) 20.5

20.5

t 5 8.88 ns

0.5

2 z (m)

0 ␳ (m)

0.1

0

0 0.5

0.75

|C|2

|C|2

0 z (m)

0 0.5

|C|2

|C|2

|C|2

0 ␳ (m) 20.5 20.1

1

0 0.5

0.1

0.02

0.0123 0 0.5

t 54.44 ns

0.04

0.02

0 ␳ (m)

20.5

2.7 2.6 2.65 z (m)

2.75

0 0.5

0 ␳ (m)

20.5

3.35 3.4 3.25 3.3 z (m)

FIGURE 29 Space-time evolution of the superluminal pulse represented by Eq. (74); the chosen parameter values are a = 10−12 s, Vmin = 1.001 c, Vmax = 1.005 c, while the focusing point is at zf = 200 cm. This solution is associated with a rather good spatiotemporal focusing. The field amplitude at z = zf is 40.82 times larger than the initial one. The field amplitude is normalized at the space-time point ρ = 0, z = zf , t = tf .

Other exact closed-form solutions can be obtained Zamboni-Rached, Shaarawi, and Recami (2004) by considering, for instance, velocity distributions like A(V) = 1/V 2 and A(V) = 1/V 3 . Again, we can construct many other spatiotemporally focused pulses from the above solutions simply by taking their time derivatives (of any order). It also is possible to show (Zamboni-Rached, Shaarawi, and Recami, 2004) that the new solutions obtained in this way have their spectra shifted toward higher frequencies.

4.4. Chirped Optical X-Type Pulses in Material Media The theory of LWs was initially developed for free space (vacuum). In 1996, Sõnajalg et al. (Sõnajalg and Saari, 1996) showed that the LW theory can be extended to include (unbounded) dispersive media. This was obtained by making the axicon angle of the Bessel beams vary with the frequency (Lu and Greenleaf, 1992a; Sõnajalg and Saari, 1996; ZamboniRached et al., 2003) in such a way that a suitable frequency superposition of these beams compensates for the material dispersion. Soon after this idea was reported, many interesting nondiffracting/nondispersive pulses were obtained theoretically (Lu and Greenleaf, 1992a; Sõnajalg and Saari, 1996; Zamboni-Rached et al., 2003) and experimentally (Sõnajalg and Saari, 1996). Despite the remarkable importance of such an extended method—working well in theory—its experimental implementation is not so simple.5

5 We refer readers to references (Lu and Greenleaf, 1992a; Sõnajalg and Saari, 1996; Zamboni-Rached et al.,

2003) for a description, theoretical and experimental, of that extended method.

Localized Waves: A Review

295

In 2004 Zamboni-Rached et al. (Zamboni-Rached, Hernández-Figueroa, and Recami, 2004) developed a simpler way to obtain pulses capable of recovering their spatial shape, both transversally and longitudinally, after some propagation. It consisted of using chirped optical X-typed pulses, while keeping the axicon angle ﬁxed. Recall that, by contrast, chirped Gaussian pulses in unbounded material media may recover only their longitudinal shape, since they undergo progressive transverse spreading while propagating. The present section is devoted to this approach. She start with an axially symmetric Bessel beam in a material medium with refractive index n(ω):

ψ(ρ, z, t) = J0 (kρ ρ) exp(iβz) exp(−iωt),

(76)

which must obey the condition kρ2 = n2 (ω)ω2 /c2 − β2 , which connects the transverse and longitudinal wave numbers kρ and β, and the angular frequency ω. In addition, we impose that kρ2 ≥ 0 and ω/β ≥ 0 to avoid a nonphysical behavior of the Bessel function J0 (.) and to conﬁne ourselves to forward propagation only. Once the conditions above are satisﬁed, we have the liberty of writing the longitudinal wave number as β = (n(ω)ω cos θ)/c and, therefore, kρ = (n(ω)ω sin θ)/c, where (as in the free space case) θ is the axicon angle of the Bessel beam. Now we can obtain an X-shaped pulse by performing a frequency superposition of these Bessel beams, with β and kρ given by the previous relations:

(ρ, z, t) =

∞

−∞

S(ω) J0

n(ω)ω sin θ ρ exp[iβ(ω)z] exp(−iωt) dω, c (77)

where S(ω) is the frequency spectrum and the axicon angle is kept constant. The phase velocity of each Bessel beam in our superposition [Eq. (77)] is different and given by Vphase = c/(n(ω) cos θ). So, the pulse represented by Eq. (77) will suffer dispersion during its propagation. As we said the method developed by Sõnajalg et al. (Sõnajalg and Saari, 1996), and explored by others (Sõnajalg and Saari, 1996; Zamboni-Rached et al., 2003), to overcome this problem consisted of regarding the axicon angle θ as a function of the frequency in order to obtain a linear relationship between β and ω. Here, however, we choose to work with a ﬁxed axicon angle, and we need to ﬁnd another way to avoid dispersion and diffraction all along a certain propagation distance. To do so, we might choose a chirped Gaussian spectrum S(ω) in Eq. (77).

S(ω) = √

T0 exp[−q2 (ω − ω0 )2 ] 2π(1 + iC)

with

q2 =

T02 , 2(1 + iC) (78)

296

Erasmo Recami and Michel Zamboni-Rached

where ω0 is the central frequency of the spectrum, T0 is a constant related with the initial temporal width, and C is the chirp parameter (we chose as temporal width the half-width of the relevant Gaussian curve when its heigth equals 1/e times its full heigth). Unfortunately, there is no analytic solution to Eq. (77) with S(ω) given by Eq. (78), so that some approximations are to be made. Then, let us assume that the spectrum S(ω), surrounding of the carrier frequency ω0 , is narrow enough to guarantee that ω/ω0 << 1, to ensure that β(ω) can be approximated by the ﬁrst three terms of its Taylor expansion in the vicinity of ω0 ; that is, β(ω) ≈ β(ω0 ) + β (ω)|ω0 (ω − ω0 ) + (1/2)β (ω)|ω0 (ω − ω0 )2 ; when, after using β = n(ω)ω cos θ/c, it results that

cos θ ∂n ∂β = n(ω) + ω ; ∂ω c ∂ω

∂2 β ∂2 n cos θ ∂n 2 +ω 2 . = c ∂ω ∂ω2 ∂ω

(79)

As we know, β (ω) is related to the pulse group velocity by the relation Vg = 1/β (ω). Here we can see the difference between the group velocity of the X-type pulse (with a ﬁxed axicon angle) and that of a standard Gaussian pulse. Such a difference is due to the factor cos θ in Eq. (79). Because of it, the group velocity of our X-type pulse is always greater than that of the Gaussian. In other words, (Vg )X = (1/ cos θ)(Vg )gauss . We also know that the second derivative of β(ω) is related to the group velocity dispersion (GVD) β2 by β2 = β (ω). The GVD is responsible for the temporal (longitudinal) spreading of the pulse. Here the GVD of the X-type pulse is always smaller than that of the standard Gaussian pulses due to the factor cos θ in Eq. (79)—namely; (β2 )X = cos θ(β2 )gauss . On using the above results, we can write

n(ω)ω T0 exp[iβ(ω0 )z] exp(−iω0 t) ∞ sin θ ρ (ρ, z, t) = dω J0 √ c 2π(1 + iC) −∞ % $ % $ ! iβ2 (ω − ω0 ) z − Vg t exp (ω − ω0 )2 z − q2 . × exp i Vg 2 (80) The integral in Eq. (80) cannot be evaluated analytically, but for us it is sufﬁcient to obtain the pulse behavior. Let us analyze the pulse for ρ = 0. In this case, we get

(ρ = 0, z, t) =

−(z − Vg t)2 (1 + iC) T0 exp[iβ(ω0 )z] exp(−iω0 t) exp . 2Vg2 [T02 − iβ2 (1 + iC)z] T02 − iβ2 (1 + iC)z

(81)

Localized Waves: A Review

297

From Eq. (81) one can immediately see that the initial temporal width of the pulse intensity is T0 and that, after a propagation distance z, the time-width T1 becomes

⎡ 2 2 ⎤1/2 Cβ β T1 z z 2 2 ⎦ . = ⎣ 1+ + T0 T02 T02

(82)

Equation (82) describes the pulse spreading behavior. Such behavior depends on the sign (positive or negative) of the product β2 C, as is well known to happen for standard Gaussian pulses (Agrawal, 2006). In the case β2 C > 0, the pulse will monotonically become broader and broader with the distance z. On the other hand, if β2 C < 0, the pulse will, in a ﬁrst stage, narrow and then (during the rest of its propagation) it will spread. Thus, there will be a certain propagation distance AT after which the pulse will recover its initial temporal width (T1 = T0 ). From Eq. (82), we can ﬁnd such a distance ZT1=T0 (considering β2 C < 0) to be

ZT1 =T0 =

−2CT02 . β2 (C2 + 1)

(83)

The maximum distance at which our chirped pulse, with given T0 and β2 , may recover its initial temporal width can be easily evaluated from Eq. (83), and it is Ldisp = T02 /β2 . We call such a maximum value Ldisp the “dispersion length”: It is the maximum distance the X-type pulse may travel while recovering its initial longitudinal shape. Obviously, if we want the pulse to reassume its longitudinal shape at some desired distance z < Ldisp , we need only to suitably choose a new value for the chirp parameter. The property of recovering its own initial temporal (or longitudinal) width may be veriﬁed to exist also in the case of chirped standard Gaussian pulses. However, the latter suffer progressive transverse spreading, which is not reversible. The distance at√which a Gaussian pulse doubles its initial transverse width w0 is zdiff = 3πw02 /λ0 , where λ0 is the carrier wavelength. Thus, optical Gaussian pulses with great transverse localization are spoiled within a few centimeters or even less. We now show that it is possible to recover the transverse shape of the chirped X-type pulse intensity; actually, it is possible to recover its entire spatial shape after a distance ZT1 =T0 . To do so, let us return to the integral solution in Eq. (80) and perform the change of coordinates (z, t) → (z, tc = zc /Vg ), with

⎧ ⎨ z = zc + z zc , ⎩ t = tc ≡ Vg

(84)

298

Erasmo Recami and Michel Zamboni-Rached

where zc is the center of the pulse (z is the distance from such a point), and tc is the time at which the pulse center is located at zc . We compare our integral solution in Eq. (80), when zc = 0 (initial pulse), with that when zc = ZT1 =T0 = −2CT02 /(β2 (C2 + 1)). In this way, solution (80) can be written, when zc = 0, as

(ρ, zc = 0, z)

−T02 (ω − ω0 )2 T0 exp(iβ0 z) ∞ = √ dω J0 (kρ (ω)ρ) exp 2(1 + C2 ) 2π(1 + iC) −∞ ' & (ω − ω0 )z (ω − ω0 )2 β2 z (ω − ω0 )2 T02 C , (85) + + × exp i Vg 2 2(1 + C2 )

where we have taken the value q given by Eq. (78). To verify that the pulse intensity recovers its entire original form at zc = ZT1 =T0 = −2 CT02 / [β2 (C2 + 1)], we can analyze our integral solution at that point, obtaining

% czc T0 exp iβ0 zc − z − cos θ n(ω0 )Vg (ρ, zc = ZT1 =T0 , z) = √ 2π(1 + iC) ∞ −T02 (ω − ω0 )2 × dω J0 (kρ (ω)ρ) exp 2(1 + C2 ) −∞ & ' (ω − ω0 )2 T02 C (ω − ω0 )z (ω − ω0 )2 β2 z × exp −i , + + Vg 2 2(1 + C2 ) (86) $

where we put z = −z . In this way, one immediately sees that

|(ρ, zc = 0, z)|2 = |(ρ, zc = ZT1 =T0 , −z)|2 .

(87)

Therefore, from Eq. (87) it is clear that the intensity of a chirped optical X-type pulse is able to recover its original 3D shape, just with a longitudinal inversion at the pulse center. This method thus is a simple and effective procedure for compensating diffraction and dispersion in an unbounded material medium, and it is a method simpler than varying the axicon angle with the frequency. We note again that one can determine the distance z = ZT1 =T0 ≤ Ldisp at which the pulse resumes its spatial shape by choosing a suitable value of the chirp parameter.

Localized Waves: A Review

299

We have shown that the chirped X-type pulse recovers its 3D shape after some distance, and we also have obtained an analytic description of the pulse longitudinal behavior (for ρ = 0) during propagation, by means of Eq. (81). However, we have not yet obtained the same information about the pulse transverse behavior. We just learned that it will be recovered at z = ZT1 =T0 . To complete the picture, we should also determine the transverse behavior in the plane of the pulse center z = Vg t. We would then obtain quantitative information about the evolution of the pulse shape during its entire propagation. However, we do not expound all the relevant mathematical details here; let us state only that the transverse behavior of the pulse (in the plane z = zc = Vg t), during its entire propagation, can approximately be described by

(ρ, z = zc , t = zc /Vg )

− tan2 θ ρ2 exp 8 Vg2 (−iβ2 zc /2 + q2 ) T0 exp[iβ(ω0 )z] exp(−iω0 t) ≈

√ 2π(1 + iC) −iβ2 zc /2 + q2 tan2 θ ρ2 n(ω0 ) ω0 sin θ ρ I0 × (1/2)J0 c 8 Vg2 (−iβ2 zc /2 + q2 ) +2

∞ p 2 (p + 1/2)(p + 1) p=1

(2p + 1)

tan2 θ ρ2 × 8 Vg2 (−iβ2 zc /2 + q2 )

J2p

n(ω0 ) ω0 sin θ ρ I2p c

,

(88)

where Ip (.) is the modiﬁed Bessel function of the ﬁrst kind of order p, quantity (.) being the gamma function, and q being given by Eq. (78). [Readers can check ref. (Zamboni-Rached, Hernández-Figueroa, and Recami, 2004) for details on how Eq. (88) is obtained from Eq. (80).] At ﬁrst glance, this solution appears to be very complicated, but the series in its right-hand side gives a negligible contribution. This circumstance renders our solution in Eq. (88) of important practical interest, and we use later. [For additional information about the transverse pulse evolution (to be extracted from Eq. (88)), readers can consult again (Zamboni-Rached, Hernández-Figueroa, and Recami, 2004). The same paper, analyzes how the generation by a ﬁnite aperture affects the chirped X-type pulses.]

300

Erasmo Recami and Michel Zamboni-Rached

The valuable methods developed in (Zamboni-Rached, HernándezFigueroa, and Recami, 2004) and partially revisited in this section are of general interest, and work is in progress for applying them, e.g., also to the (different) case of the Schrödinger equation.

4.4.1. An Example: Chirped Optical X-Type Pulse in Bulk Fused Silica For a bulk fused silica, the refractive index n(ω) can be approximated by the Sellmeier equation Agrawal (2006)

n2 (ω) = 1 +

N

Bj ωj2

j=1

ωj2 − ω2

,

(89)

where ωj are the resonance frequencies, Bj the strength of the jth resonance, and N the total number of the material resonances that appear in the frequency range of interest. For our purposes, it is appropriate to choose N = 3, which yields, for bulk fused silica Agrawal (2006), the values B1 = 0.6961663, B2 = 0.4079426, B3 = 0.8974794, λ1 = 0.0684043 μm, λ2 = 0.1162414 μm, and λ3 = 9.896161 μm. Let us consider in this medium a chirped X-type pulse, with λ0 = 0.2 μm, T0 = 0.4 ps, C = −1, and with axicon angle θ = 0.00084 rad, that correspond to an initial central spot with ρ0 = 0.117 mm. By using Eqs. (81) and (88) we obtain the longitudinal and transverse pulse evolution, which are represented in Figure 30. From Figure 30a, we can observe that the pulse initially suffers a longitudinal narrowing with an increase of intensity until the position z = T02 /2β2 = 0.186 m. After that point, the pulse starts broadening and

1.4 1.4

1.2

1 2

0.8 |C|

2

|C|

C 5 21

1.2

C 5 21

1

0.6 0.4

0.8 0.6 0.4

0.2

0.2

0 4

0 5

2 0 (Z/Vg 2 t)/T0

22

24

0

(a)

0.2

0.4 Z (m)

0.6

0 ␳(m) 3 102 4

0.6 0.4 25

0.2 0

Z 5 Vg t (m)

(b)

FIGURE 30 (a) Longitudinal shape evolution of a chirped X-type pulse, propagating in fused silica with λ0 = 0.2 μm, T0 = 0.4 ps, C = −1, and axicon angle θ = 0.00084 rad, which correspond to an initial transverse width of ρ0 = 0.117 mm. (b) Transverse shape evolution for the same pulse.

Localized Waves: A Review

301

decreasing its intensity, while recovering its entire longitudinal shape (width and intensity) at the point z = T02 /β2 = 0.373 m, as predicted. At the same time, from Figure 30b, we notice that the pulse maintains its transverse width ρ = 2.4 c/(n(ω0 )ω0 sin θ) = 0.117 mm (because T0 ω0 >> 1) during its entire propagation. The same does not occur, however, with the pulse intensity: Initially, the pulse suffers an increase of intensity until position zc = T02 /2β2 = 0.186 m; after that point the intensity starts decreasing and the pulse recovers its entire transverse shape at point zc = T02 /β2 = 0.373 m, as expected. In the calculations we could skip the series in the right-hand side of Eq. (88), because, as already stated, it yields a negligible contribution. Summarizing, from Figure 30 we can see that the chirped X-type pulse totally recovers its longitudinal and transverse shape at position z = Ldisp = T02 /β2 = 0.373 m, as expected. Let us recall that a chirped Gaussian pulse may recover just its longitudinal width, but with an intensity decrease, at the position given by z = ZT1 =T0 = Ldisp = T02 /β2 . Its transverse width, on the other hand, suffers progressive and irreversible spreading. The next text section “completes” our review by investigating the (no less interesting) case of the subluminal localized solutions to the wave equations, which, among others, will allow us to set forth remarkable considerations about the role of (extended) SR. For instance, the various superluminal and subluminal LWs are expected to be transformed one into the other by suitable LTs. We start by studying, in terms of various different approaches, the peculiar topic of zero-speed waves—namely, the question of constructing localized ﬁelds with a static envelope, consisting, for example, in “light at rest” endowed with zero peak velocity. We called such solutions, Frozen Waves they can have many applications.

5. “FROZEN WAVES” AND SUBLUMINAL WAVE BULLETS 5.1. Modeling the Shape of Stationary Wave Fields: Frozen Waves We begin this section by studying the peculiar topic of zero-speed waves, namely, the question of constructing localized ﬁelds with a static envelope (e.g., consisting in “light at rest” endowed with null peak velocity). We called such solutions; “Frozen Waves” they permit a priori many applications. In this section we develop a simple ﬁrst method (Longhi, 2004; Salo et al., 1999; Zamboni-Rached, 2004b), based on Section 3, by using superpositions of forward-propagating and equal-frequency Bessel beams, that allow controlling the longitudinal beam intensity shape within a chosen interval 0 ≤ z ≤ L, where z is the propagation axis and L can be much

302

Erasmo Recami and Michel Zamboni-Rached

greater than the wavelength λ of the monochromatic light (or sound) being used. Inside such a space interval, we succeed in constructing a stationary envelope whose longitudinal intensity pattern can approximately assume any desired shape, including, for instance, one or more high-intensity peaks (with distances between them much larger than λ), and which in addition results to be naturally endowed with a good transverse localization. Since the intensity envelopes remain static (i.e., with velocity V = 0), we called such new solutions frozen waves (FW) (Longhi, 2004; Salo et al., 1999; Zamboni-Rached, 2004b) to the wave equations. Although we deal here with exact solutions of the scalar wave equation, vectorial solutions of the same kind for the electromagnetic ﬁeld can be determined. Indeed, solutions to Maxwell’s equations may be naturally inferred even from the scalar wave equation solutions (Agrawal, 2006; Bouchal and Olivik, 1995; Recami, 1998). We present ﬁrst the method referring to lossless media (Longhi, 2004; Zamboni-Rached, 2004b) while, in the second part of this section, we extend the method to absorbing media (Zamboni-Rached, 2006).

5.1.1. Stationary Wave Fields With Arbitrary Longitudinal Shape in Lossless Media, Obtained by Superposing Equal-Frequency Bessel Beams We start from the well-known axis-symmetric zero-order Bessel beam solution to the wave equation:

ψ(ρ, z, t) = J0 (kρ ρ)eiβz e−iωt

(90)

with

kρ2 =

ω2 − β2 , c2

(91)

where ω, kρ , and β are the angular frequency, the transverse wave number, and the longitudinal wave number, respectively. We also impose the conditions

ω/β > 0 and kρ2 ≥ 0

(92)

(which imply ω/β ≥ c) to ensure forward propagation only (with no evanescent waves), as well as a physical behavior of the Bessel function J0 . Now, let us make a superposition of 2N + 1 Bessel beams with the same frequency ω0 , but with different (and still unknown) longitudinal wave

Localized Waves: A Review

303

numbers βm : N

(ρ, z, t) = e−i ω0 t

Am J0 (kρ m ρ) ei βm z ,

(93)

m=−N

where the m represent integer numbers and the Am are constant coefﬁcients. For each m, the parameters ω0 , kρ m , and βm must satisfy Eq. (91), and, because of conditions (92), when considering ω0 > 0, we must have

0 ≤ βm ≤

ω0 . c

(94)

Let us now suppose that we wish |(ρ, z, t)|2 , given by Eq. (93), to assume on the axis ρ = 0 the pattern represented by a function |F(z)|2 , inside the chosen interval 0 ≤ z ≤ L. In this case, the function F(z) can be expanded, as usual, in a Fourier series as follows: ∞

F(z) =

2π

Bm ei L mz ,

m=−∞

where

Bm =

1 L

L

2π

F(z) e−i L mz d z.

0

More precisely, our goal now is ﬁnding the values of the longitudinal wave numbers βm and the coefﬁcients Am of Eq. (93) to reproduce approximately, within the said interval 0 ≤ z ≤ L (for ρ = 0), the predetermined longitudinal intensity pattern |F(z)|2 . Namely, we wish to have

( (2 ( N ( ( ( iβm z ( 2 ( A e m ( ( ≈ |F(z)| ( m=−N (

with 0 ≤ z ≤ L.

(95)

Looking at Eq. (95), one might be tempted to take βm = 2πm/L, thus obtaining a truncated Fourier series, expected to represent approximately the desired pattern F(z). Superpositions of Bessel beams with βm = 2πm/L have been used in some works to obtain a large set of transverse amplitude proﬁles (Bouchal, 2002). However, for our purposes, this choice is not appropriate, for two principal reasons. First, it yields negative values for βm (when m < 0), which implies backward-propagating components

304

Erasmo Recami and Michel Zamboni-Rached

(since ω0 > 0), and second, when L >> λ0 , which is our interest here, the main terms of the series correspond to very small values of βm , which results in a very short ﬁeld depth of the corresponding Bessel beams (when generated by ﬁnite apertures), preventing the creation of the desired envelopes far from the source. Therefore, we need a better choice for the values of βm , which permits forward-propagation components only, and a good depth of ﬁeld. This problem can be solved by putting

βm = Q +

2π m, L

(96)

where Q > 0 is a value to be chosen (as we shall see) according to the given experimental situation and the desired degree of transverse ﬁeld localization. Due to Eq. (94), one gets

0≤Q±

ω0 2π N≤ . L c

(97)

Inequality (97) can be used to determine the maximum value of m, which we call Nmax , once Q, L, and ω0 have been chosen. As a consequence, for getting a longitudinal intensity pattern approximately equal to the desired one, |F(z)|2 , in the interval 0 ≤ z ≤ L, Eq. (93) has to be rewritten as

(ρ = 0, z, t) = e−iω0 t ei Q z

N

2π

Am ei L mz ,

(98)

m=−N

with

1 Am = L

L

2π

F(z) e−i L mz d z.

(99)

0

Obviously, this yields only an approximation of the desired longitudinal pattern, because the trigonometric series in Eq. (98) is necessarily truncated (N ≤ Nmax ). Its total number of terms let us repeat is ﬁxed once the values of Q, L, and ω0 have been chosen. When ρ = 0, the wave ﬁeld (ρ, z, t) becomes

(ρ, z, t) = e−iω0 t ei Q z

N m=−N

2π

Am J0 (kρ m ρ) ei L mz ,

(100)

Localized Waves: A Review

305

with

kρ2 m

=

ω02

2π m − Q+ L

2 .

(101)

The coefﬁcients Am will yield the amplitudes and the relative phases of each Bessel beam in the superposition. Because we are adding zero-order Bessel functions, we can expect a high ﬁeld concentration around ρ = 0. Moreover, due to the known nondiffractive behavior of the Bessel beams, we expect that the resulting wave ﬁeld will preserve its transverse pattern in the entire interval 0 ≤ z ≤ L. The present methodology addresses the longitudinal intensity pattern control. Obviously, we cannot get a total 3D control because the ﬁeld must obey the wave equation. However, we can use two ways to exert some control over the transverse behavior. The ﬁrst is through the parameter Q of Eq. (96). We have some freedom in the choice of this parameter, and FWs representing the same longitudinal intensity pattern can possess different values of Q. The important point is that, in superposition (100), using a smaller value of Q makes the Bessel beams have a higher transverse concentration (because on decreasing the value of Q, one increases the value of the Bessel beams’ transverse wave numbers), and this is reﬂected in the resulting ﬁeld, which presents a narrower central transverse spot. The second way to control the transverse intensity pattern is by using higher-order Bessel beams (see Section 6.1.1). We now present a few examples of our methodology.

Example 1. Let us suppose that we want an optical waveﬁeld with λ0 = 0.632 μm (i.e., with ω0 = 2.98 × 1015 Hz) whose longitudinal pattern (along its z-axis) in the range 0 ≤ z ≤ L is given by the function

F(z) =

⎧ (z − l1 )(z − l2 ) ⎪ ⎪ −4 ⎪ ⎪ (l2 − l1 )2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎨ ⎪ ⎪ (z − l5 )(z − l6 ) ⎪ ⎪ −4 ⎪ ⎪ ⎪ (l6 − l5 )2 ⎪ ⎪ ⎪ ⎪ ⎩ 0

for l1 ≤ z ≤ l2 for l3 ≤ z ≤ l4 (102) for l5 ≤ z ≤ l6 elsewhere,

where l1 = L/5 − z12 and l2 = L/5 + z12 with z12 = L/50; while l3 = L/2 − z34 and l4 = L/2 + z34 with z34 = L/10; and, at last, l5 = 4L/5 − z56 and l6 = 4L/5 + z56 with z56 = L/50. In other words, the desired longitudinal shape, in the range 0 ≤ z ≤ L, is a parabolic function for

306

Erasmo Recami and Michel Zamboni-Rached

l1 ≤ z ≤ l2 , a unitary step function for l3 ≤ z ≤ l4 , and again a parabola in the interval l5 ≤ z ≤ l6 , being zero elsewhere (within the interval 0 ≤ z ≤ L, as stated). In this example, let us put L = 0.2 m. We can then easily calculate the coefﬁcients Am , which appear in superposition (100), by inserting Eq. (102) into Eq. (99). Let us choose, for instance, Q = 0.999 ω0 /c. This choice yields for m a maximum value Nmax = 316, as can be inferred from Eq. (97). We emphasize that one is not compelled to use just N = 316, but can adopt for N any values smaller than it—more generally, any value smaller than that calculated via inequality (97). When using the maximum value allowed for N, a better result is achieved. In the present case, let us adopt the value N = 30. Figure 31a compares the intensity of the desired longitudinal function F(z) with that of the FW, (ρ = 0, z, t), obtained from Eq. (98) by adopting the mentioned value N = 30. One can verify that good agreement between the desired longitudinal behavior and our approximate FW is already obtained for N = 30. The use of higher values for N can only improve the approximation. Figure 31b shows the 3D intensity of our FW, given by Eq. (100). This ﬁeld possesses the desired longitudinal pattern and has good transverse localization.

Example 2. (Controlling the transverse shape too). We wish to take advantage of this example to address an important issue. We can expect that, for a desired longitudinal pattern of the ﬁeld intensity, by choosing smaller values of the parameter Q one will get FWs with narrower transverse width [for the same number of terms in the series entering Eq. (100)], because

1 1 0.8

0.8 0.6

|C|2 0.6

|C|2

0.4 0.2

0.4

0.5

0.2

0 ␳ (mm)

0 0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0.1 20.5

0

0.15

0.2

0.05 z (m)

z (m)

(a)

(b)

FIGURE 31 (a) Comparison between the intensity of the desired longitudinal function F(z) and that of the FW, (ρ = 0, z, t), obtained from Eq. (98). The solid line represents the function F(z), and the dotted one the FW. (b) 3D plot of the field intensity of the FW chosen in this case.

Localized Waves: A Review

307

the Bessel beams in Eq. (100) will possess larger transverse wave numbers, and, consequently, higher transverse concentrations. We can verify this expectation by considering, for instance, inside the usual range 0 ≤ z ≤ L, the longitudinal pattern represented by the function

⎧ (z − l1 )(z − l2 ) ⎪ ⎨−4 (l2 − l1 )2 F(z) = ⎪ ⎩ 0

for l1 ≤ z ≤ l2 (103) elsewhere,

with l1 = L/2 − z and l2 = L/2 + z. Such a function has a parabolic √ shape, with its peak centered at L/2 and with longitudinal width 2z/ 2. By adopting λ0 = 0.632 μm (that is, ω0 = 2.98 × 1015 Hz), let us use superposition (100) with two different values of Q. We obtain two different FWs that, despite having the same longitudinal intensity pattern, possess different transverse localizations. Let us consider L = 0.06 m and z = L/100, and the two values Q = 0.999 ω0 /c and Q = 0.995 ω0 /c. In both cases, the coefﬁcients Am will be the same, calculated from Eq. (99) using this time the value N = 45 in superposition (100). The results are shown in Figure 32. Both FWs have the same longitudinal intensity pattern, but the one with the smaller Q has a narrower transverse width. In this way, we can exert some control on the transverse spot size through the parameter Q. Equation (100), which deﬁnes our FW, is actually a superposition of zero-order Bessel beams, and because of this the resulting ﬁeld is expected to possess a transverse localization around ρ = 0. Each Bessel beam in superposition (100) is associated with a central spot with transverse size, or width, ρm ≈ 2.4/kρ m . On the basis of the expected convergence of the series (100), we can estimate the width of the transverse

|C|2

0.8

0.8

0.6

0.6

0.4

|C|2

0.4 0.2

0.2

50 ␳ (␮m)

60 0

250

40 20 0

(a)

z (mm)

50 ␳ (␮m)

60 0

250

0

40 20 z (mm)

(b)

FIGURE 32 (a) The FW with Q = 0.999 ω0 /c and N = 45, approximately reproducing the chosen longitudinal pattern represented by Eq. (103). (b) A different FW, now with Q = 0.995 ω0 /c (but still with N = 45) forwarding the same longitudinal pattern. In this case (with a lower value for Q) a higher transverse localization is obtained.

308

Erasmo Recami and Michel Zamboni-Rached

spot of the resulting beam as follows:

ρ ≈

2.4 kρ, m=0

2.4 = , ω02 /c2 − Q2

(104)

which is the same value as that for the transverse spot of the Bessel beam with m = 0 in superposition (100). Relation (104) can be useful: Once we have chosen the desired longitudinal intensity pattern, we can even choose the size of the transverse spot, and use relation (104) to evaluate the corresponding needed value of parameter Q. For a more detailed analysis concerning the spatial resolution and residual intensity of FWs, see ref. Zamboni-Rached, Recami, and Hernández-Figueroa (2005). The FWs, corresponding to zero group velocity, are a particular case of the subluminal LWs. As in the superluminal case, the (more orthodox, in a sense) subluminal LWs can be obtained by suitable superpositions of Bessel beams. Until now they have been largely neglected, however, because of the mathematical difﬁculties in deriving analytic expressions for them, difﬁculties associated with the fact that the superposition integral runs now over a ﬁnite interval. In Zamboni-Rached and Recami (2008a) we have shown, by contrast, that one can arrive at exact (analytic) solutions in the case of general subluminal LWs—both in the case of integration over the Bessel beams’ angular frequency ω and in the case of integration over their longitudinal wave number kz . We return to this point in the following text.

Increasing control on the transverse shape by using higher-order Bessel beams. Here, we argue that it is possible to increase even more control on the transverse shape by using higher-order Bessel beams in the fundamental superposition (100). This new approach can be understood and accepted on the basis of simple and intuitive arguments (not presented here but found in Zamboni-Rached, Recami, and Hernández-Figueroa (2005)). A brief description of that approach follows. The basic idea is obtaining the desired longitudinal intensity pattern not along the axis ρ = 0, but on a cylindrical surface corresponding to ρ = ρ > 0. To do that, we ﬁrst proceed as before. Once we have chosen the desired longitudinal intensity pattern F(z), within the interval 0 ≤ z ≤ L, we calculate the coefﬁcients A m as before; that is, )L Am = (1/L) 0 F(z) exp(−i2πmz/L) dz, and kρ m = ω02 − (Q + 2πm/L)2 . Afterward, we simply replace the zero-order Bessel beams J0 (kρ m ρ), in superposition (100), with higher-order Bessel beams, Jμ (kρ m ρ), to get (ρ, z, t) = e

−iω0 t i Q z

e

N m=−N

2π

Am Jμ (kρ m ρ) ei L mz .

(105)

Localized Waves: A Review

309

From this result, and on the basis of intuitive arguments, ZamboniRached, Recami, and Hernández-Figueroa (2005), we can expect that the desired longitudinal intensity pattern, initially constructed for ρ = 0, will approximately shift to ρ = ρ , where ρ represents the position of the ﬁrst maximum of the Bessel function (the ﬁrst positive root of the equation d Jμ (kρ, m=0 ρ)/dρ)|ρ = 0). By using such a procedure, one can obtain interesting stationary conﬁgurations of ﬁeld intensity, as “donuts,” cylindrical surfaces, and much more. In the following example, we show how to obtain, for example, a cylindrical surface of stationary light. To obtain it, within the interval 0 ≤ z ≤ L, let us ﬁrst select the longitudinal intensity pattern given by Eq. (103), with l1 = L/2 − z and l2 = L/2 + z, and with z = L/300. Moreover, let us choose L = 0.05 m, Q = 0.998 ω0 /c, and use N = 150. Then, after calculating the coefﬁcients Am by Eq. (99), we use superposition (105), choosing, in this case, μ = 4. According to the previous discussion, one can expect the desired longitudinal intensity pattern to appear shifted to ρ ≈ 5.318/kρ, m=0 = 8.47 μm, where 5.318 is the value of kρ, m=0 ρ for which theBessel function J4 (kρ, m=0 ρ) assumes its maximum

value, with kρ, m=0 = ω02 − Q2 . Figure 33 shows the resulting intensity ﬁeld. Figure 33a shows the transverse section of the resulting beam for z = L/2. The transverse peak intensity is located at ρ = 7.75 μm, with a 8.5% difference with respect to the predicted value of 8.47 μm. Figure 33b shows the orthogonal projection of the resulting ﬁeld, which corresponds to nothing but a cylindrical surface of stationary light (or other ﬁelds). We can see that the desired longitudinal intensity pattern has been approximately obtained, shifted, as desired, from ρ = 0 to ρ = 7.75 μm; and the resulting ﬁeld resembles a cylindrical surface of stationary light |C|2/ |C|2max

1

1

50

20

0.9

40

0.9

15

0.8

30

0.8

10

0.7

20

0.7

5

0.6

10

0.6

0

0.5

25

␳ (mm)

x (mm)

|C|2/ |C|2max 25

0

0.5

0.4

210

0.4

210

0.3

220

0.3

215

0.2

230

0.2

220

0.1

240

0.1

0

250 24.75 24.8 24.85 24.9 24.95 25 25.05 25.1 25.15 25.2 25.25 z (mm)

0

225 225 220 215 210

25

0 5 y (mm)

(a)

10

15

20

25

(b)

FIGURE 33 (a) Transverse section at z = L/2 of the considered higher-order FW. (b) Orthogonal projection of the 3D intensity pattern of the same higher-order FW.

310

Erasmo Recami and Michel Zamboni-Rached

with radius 7.75 μm and length 238 μm. Donut-like conﬁgurations of light (or sound) are also possible.

5.1.2. Stationary Wave Fields With Arbitrary Longitudinal Shape in Absorbing Media: An Extension of the Method When propagating in a nonabsorbing medium, the so-called nondiffracting waves maintain their spatial shape for long distances. However, the situation is not the same with absorbing media. In such cases, both the ordinary and the nondiffracting beams (and pulses) suffer the same effect: an exponential attenuation along the propagation axis. We present an extension Zamboni-Rached (2006a) of the method given above with the aim of showing that, through suitable superpositions of equal-frequency Bessel beams, it is possible to obtain even in absorbing media nondiffracting beams, whose longitudinal intensity pattern can assume any desired shape within a chosen interval 0 ≤ z ≤ L of the propagation axis z. As a particular example, we are going to obtain new nondiffracting beams capable of resisting the loss effects, maintaining amplitude and spot size of their central core for long distances. It is important to stress that the energy absorption by the medium continues as normal, but the new beams have an initial transverse ﬁeld distribution, so to reconstruct (notwithstanding the presence of absorption) their central cores for distances considerably longer than the penetration depths of ordinary (nondiffracting or diffracting) beams. In this sense, the new method can be regarded as extending, for absorbing media, the selfreconstruction properties (Grunwald et al., 2004) that usual LWs are known to possess in lossless media. In the same way as for lossless media, we construct a Bessel beam with angular frequency ω and axicon angle θ in the absorbing materials by superposing plane waves, with the same angular frequency ω, and whose wave vectors lie on the surface of a cone with vertex angle θ. The refractive index of the medium can be written as n(ω) = nR (ω) + inI (ω), quantity nR being the real part of the complex refraction index and nI the imaginary one, responsible for the absorbtion effects. For a plane wave, the penetration depth δ for the frequency ω is given by δ = 1/α = c/2ωnI , where α is the absorption coefﬁcient. Therefore, a zero-order Bessel beam in dissipative media can be written as ψ = J0 (kρ ρ)exp(iβz)exp(−iωt) with β = n(ω) ω cos θ/c = nR ω cos θ/c + inI ω cos θ/c ≡ βR + iβI ; kρ = nR ω sin θ/c + inI ω sin θ/c ≡ kρR + ikρI , and so kρ2 = n2 ω2 /c2 − β2 . Thus, the result is ψ = J0 ((kρR + ikρI )ρ)exp(iβR z)exp(−iωt)exp(−βI z), where βR , kρR are the real parts of the longitudinal and transverse wave numbers, and βI , kρI are the imaginary ones, while the absorption coefﬁcient of a Bessel beam with axicon angle θ is given by αθ = 2βI = 2nI ω cos θ/c, its penetration depth being δθ = 1/αθ = c/2ωnI cos θ.

Localized Waves: A Review

311

Because kρ is complex, the amplitude of the Bessel function J0 (kρ ρ) starts decreasing from ρ = 0 until the transverse distance ρ = 1/2kρI , and afterward it starts growing exponentially. This behavior is not physically acceptable, but it occurs only because an ideal Bessel beam needs an inﬁnite aperture to be generated. However, in any real situation, when a Bessel beam is generated by ﬁnite apertures, that exponential growth in the transverse direction, starting after ρ = 1/2kρI , will not occur indeﬁnitely, stopping at a given value of ρ. Moreover, we emphasize that, when generated by a ﬁnite aperture of radius R, the truncated Bessel beam (Zamboni-Rached, Recami, and Hernández-Figueroa, 2005) possesses a depth of ﬁeld Z = R/ tan θ and can be approximately described by the solution in the previous paragraph, for ρ < R and z < Z. Experimentally, to guarantee that the mentioned exponential growth in the transverse direction does not even start, so as to meet only a decreasing transverse intensity, the radius R of the aperture used to generate the Bessel beam should be R ≤ 1/2kρI . However, as noted by Durnin, Miceli, and Eberly (1987a) the same aperture also must satisfy the relation R ≥ 2π/kρR . From these two conditions, one can infer that, in an absorbing medium, a Bessel beam with only a decreasing transverse intensity can be generated only when the absorption coefﬁcient is α < 2/λ; that is, if the penetration depth is δ > λ/2. The present method does refer to these cases—it is always possible to choose a suitable ﬁnite aperture size in such a way that the truncated versions of all solutions, including the general one given by Eq. (111), do not develop any unphysical behavior. We now outline the method Zamboni-Rached and Recami (2008a). Let us consider an absorbing medium with the complex refraction index n(ω) = nR (ω) + inI (ω), and the following superposition of 2N + 1 Bessel beams with the same frequency ω:

(ρ, z, t) =

N

+ * Am J0 (kρRm + ikρIm )ρ ei βRm z e−iωt e−βIm z ,

(106)

m=−N

where the m are integer numbers, the Am are constant coefﬁcients (yet unknown), quantities βRm and kρRm (βIm and kρIm ) are the real (the imaginary) parts of the complex longitudinal and transverse wave numbers of the mth Bessel beam in superposition (106); the following relations being satisﬁed

kρ2m = n2

ω2 2 − βm c2

βRm nR = , βIm nI where βm = βRm + iβIm , kρm = kρRm + ikρIm , with kρRm /kρIm = nR /nI .

(107) (108)

312

Erasmo Recami and Michel Zamboni-Rached

Our goal is to ﬁnd the values of the longitudinal wave numbers βm and the coefﬁcients Am to reproduce approximately, inside the interval 0 ≤ z ≤ L (on the axis ρ = 0), a freely chosen longitudinal intensity pattern that we call |F(z)|2 . The problem for the particular case of lossless media Longhi (2004); Zamboni-Rached (2004b)—when nI = 0 → βIm = 0—was solved in the previous subsection. For those cases, it was shown that )L the choice β = Q + 2πm/L, with Am = 0 F(z)exp(−i2πmz/L)/L dz, can be used to provide approximately the desired longitudinal intensity pattern |F(z)|2 in the interval 0 ≤ z ≤ L, and, at the same time, to regulate the spot size of the resulting beam by means of the parameter Q. Such parameter, incidentally, also can be used to obtain large ﬁeld depths and moreover to inforce the linear polarization approximation to the electric ﬁeld for the Transverse Electric (TE) electromagnetic wave (see details in Longhi (2004); Zamboni-Rached (2004b)). However, when dealing with absorbing media, the procedure described in the last paragraph does not work, due to the presence of the functions exp(−βIm z) in the superposition (106), because in this case that series does not became a Fourier series when ρ = 0. On attempting to overcome this limitation, let us write the real part of the longitudinal wave number, in superposition (106), as

βRm = Q +

2πm L

(109)

with

0≤Q+

2πm ω ≤ nR . L c

(110)

where inequality (110) guarantees forward propagation only, with no evanescent waves. In this way, the superposition (106) can be written

(ρ, z, t) = e−i ω t ei Qz

N

* + 2πm Am J0 (kρRm + ikρIm )ρ ei L z e−βIm z , (111)

m=−N

where, by using Eq. (108), we have βIm = (Q + 2πm/L)nI /nR , and kρm = kρRm + ikρIm is given by Eq. (107). Obviously, the discrete superposition (111) could be written as a continuous one (i.e., as an integral over βRm ) by taking L → ∞, but we prefer the discrete sum due to the difﬁculty of obtaining closed-form solutions to the integral form. Now, let us examine the imaginary part of the longitudinal wave numbers. The minimum and maximum values among the βIm are (βI )min = (Q − 2πN/L)nI /nR and (βI )max = (Q + 2πN/L)nI /nR , the central one being

Localized Waves: A Review

313

given by βI ≡ (βI )m=0 = QnI /nR . With this in mind, let us evaluate the ratio = [(βI )max − (βI )min ]/βI = 4πN/LQ. Thus, when << 1, there are no considerable differences among the various βIm , because βIm ≈ βI holds for all m. In the same way, there are no considerable differences among the exponential attenuation factors, since exp(−βIm z) ≈ exp(−βI z). So, when ρ = 0 the series in the right hand side, of Eq. (111) can be approximately considered a truncated Fourier series multiplied by the function exp(−βI z), and, therefore, superposition (111) can be used to reproduce approximately the desired longitudinal intensity pattern |F(z)|2 (on ρ = 0), within 0 ≤ z ≤ L, when the coefﬁcients Am are given by

Am =

1 L

L

F(z) eβI z e−i

2πm L z

dz,

(112)

0

the presence of the factor exp(βI z) in the integrand being necessary to compensate for the factors exp(−βIm z) in superposition (111). Since we are adding zero-order Bessel functions, we can expect a good ﬁeld concentration around ρ = 0. In short, we have shown in this section how to obtain, in an absorbing medium, a stationary wave ﬁeld with a good transverse concentration, and whose longitudinal intensity pattern (on ρ = 0) can approximately assume any desired shape |F(z)|2 within the predetermined interval 0 ≤ z ≤ L. This method is a generalization of a previous one, Longhi (2004); Zamboni-Rached (2004b), and consists of the superposition in Eq. (111) of Bessel beams whose longitudinal wave numbers are individuated by the real and imaginary parts given in Eqs. (109) and (108), respectively, while their complex transverse wave numbers are given by Eq. (107). Finally, the coefﬁcients of the superposition are given by Eq. (112). The method is justiﬁed when 4πN/LQ << 1; fortunately, this condition is satisﬁed in many situations. With regard to the generation of these new beams, once we have an apparatus capable of generating a single Bessel beam, we can simply use an array of such apparatuses to generate a sum of Bessel beams, with the appropriate longitudinal wave numbers and amplitudes/phases (as speciﬁed by our method), thus producing the desired ﬁnal beam. For instance, we can use, Longhi (2004); Zamboni-Rached (2004b), a laser illuminating an array of concentric annular apertures (located at the focus of a convergent lens), with the appropriate radii and transfer functions to yield both the required longitudinal wave numbers (once a value for Q has been chosen) and the coefﬁcients An of the fundamental superposition (111).

Examples For generality’s sake, let us consider a hypothetical medium in which a typical XeCl excimer laser (λ = 308nm → ω = 6.12 × 1015 Hz) has a

314

Erasmo Recami and Michel Zamboni-Rached

penetration depth of 5 cm—that is, an absorption coefﬁcient α = 20m−1 , and therefore nI = 0.49 × 10−6 . In addition, let us suppose that the real part of the refraction index for this wavelength is nR = 1.5 and therefore n = nR + inI = 1.5 + i 0.49 × 10−6 . Note that the value of the real part of the refractive index is not as important for us because we are dealing with monochromatic wave ﬁelds. A Bessel beam with ω = 6.12 × 1015 Hz and with an axicon angle θ = 0.0141 rad (thus, with a transverse spot of radius 8.4 μm), when generated by an aperture, say, of radius R = 3.5 mm, can travel in vacuum a distance equal to Z = R/ tan θ = 25 cm while resisting the diffraction effects. However, in the material medium considered here, the penetration depth of this Bessel beam would be only zp = 5 cm. We now consider two interesting applications of this method.

Example 1. Almost undistorted beams in absorbing media. We can use the extended method to obtain, in the same medium and for the same wavelength, an almost undistorted beam capable of preserving its spot size and the intensity of its central core for a distance many times larger than the typical penetration depth of an ordinary beam (nondiffracting or not). To this purpose, let us suppose that, in the considered material medium, we want a beam (with ω = 6.12 × 1015 Hz) that maintains amplitude and spot size of its central core for a distance of 25 cm—a distance ﬁve times greater than the penetration depth of an ordinary beam with the same frequency. We can model this beam by choosing the desired longitudinal intensity pattern |F(z)|2 (on ρ = 0), within 0 ≤ z ≤ L, to be given by the function $ 1 F(z) = 0

for 0 ≤ z ≤ Z elsewhere,

(113)

and by putting Z = 25 cm, with, for example, L = 33 cm. Now, we can use the Bessel beam superposition (111) to reproduce approximately the selected intensity pattern. Let us choose Q = 0.9999ω/c for the βRm in Eq. (109), and N = 20 (notice that, according to inequality (110), N could assume a maximum value of 158.) After having chosen the values of Q, L, and N, the values of the complex longitudinal and transverse wave numbers of the Bessel beams happen to be deﬁned by relations (109), (108), and (107). Eventually, we have recourse to Eq. (112) and determine the coefﬁcients Am of the fundamental superposition (111), that deﬁnes the resulting stationary wave ﬁeld. (Note that the condition 4πN/LQ << 1 is perfectly satisﬁed in this case.) Figure 34a shows the 3D ﬁeld intensity of the resulting beam. The ﬁeld possesses a good transverse localization (with a spot size smaller than 10 μm) and is capable of maintaining spot size and intensity of its central

Localized Waves: A Review

315

10log10[|C|2/|Cmax|2] [dB] 0 4000 1.2

210

3000

220

1 2000

230

0.8 1000 ␳ (mm)

0.6 2

|C|

0.4

240

0 250

21000

0.2

260

22000

4000

␳ (mm)

270

23000

2000 0 22000 24000

0

50

100

150

200

250

300

280

24000

290

z (mm)

0

(a)

50

100

150 200 z (mm)

250

300

(b)

FIGURE 34 (a) Three-dimensional field intensity of the resulting beam. (b) The resulting beam, in an orthogonal projection and in logaritmic scale.

core up to the desired distance (a better result could be reached by using a higher value of N). It is interesting to note that at that distance (25 cm), an ordinary beam would have its initial ﬁeld intensity attenuated 148 times. As noted, the energy absorption by the medium continues as normal; the difference is that these new beams have an initial transverse ﬁeld distribution sophisticated enough to be able to reconstruct (even in the presence of absorption) their central cores up to a certain distance. For a better visualization of this ﬁeld intensity distribution and of the energy ﬂux, Figure 34b shows the resulting beam, in an orthogonal projection and in logarithmic scale. It appears clear that the energy comes from the lateral regions in order to reconstruct the central core of the beam. On the plane z = 0, within the region ρ ≤ R = 3.5 mm, there is an uncommon ﬁeld intensity distribution; it is very dispersed instead of concentrated. This uncommon initial ﬁeld intensity distribution is responsible for the construction of the central core of the resulting beam and for its reconstruction all along the distance z = 25 cm. Due to absorption, the beam (total) energy, ﬂowing through different z planes, is not constant, but the energy ﬂowing in the beam spot area, and the beam spot size itself, are conserved up to the distance (in this case) z = 25 cm.

Example 2. Beams in absorbing media with a growing longitudinal field intensity. We consider again the previous hypothetical medium, in which an ordinary Bessel beam with θ = 0.0141 rad and ω = 6.12 × 1015 Hz would have a penetration depth of 5 cm. We now want to construct a beam that, instead of possessing a constant core intensity up to the position z = 25 cm, presents, on the contrary, a (moderate) exponential growth of its intensity, up to that distance (z = 25 cm).

316

Erasmo Recami and Michel Zamboni-Rached

Let us assume we wish to get the following longitudinal intensity pattern |F(z)|2 , in the interval 0 < z < L:

& F(z) =

exp(z/Z)

for 0 ≤ z ≤ Z

0

elsewhere,

(114)

with Z = 25 cm and L = 33 cm. Using again Q = 0.9999 ω/c, and N = 20, we can proceed as in the ﬁrst example, calculating the complex longitudinal and transverse wave numbers of the Bessel beams, and ﬁnally the coefﬁcients Am of the fundamental superposition (111). In Figure 35 we can see the 3D ﬁeld intensity of the resulting beam. The ﬁeld presents the desired longitudinal intensity pattern with a good transverse localization (a spot size smaller than 10 μm). Obviously, the amount of energy necessary to construct these new beams is greater than that necessary to generate an ordinary beam in a nonabsorbing medium. It is also clear that there is a limitation on the depth of ﬁeld of these new beams. In the ﬁrst example, for distances larger than 10 times the penetration depth of an ordinary beam, the ﬁeld intensity in the lateral regions would be higher than that at the core, and the ﬁeld would lose

8 7 6 5 4 |C|2 3 2 1 4000 2000 0 ␳(mm) 22000 24000

200

0

50

100

250

300

150 z (mm)

FIGURE 35 Three-dimensional field intensity of the resulting beam in an absorbing medium with a growing longitudinal field intensity.

Localized Waves: A Review

317

the usual characteristics of a beam (transverse ﬁeld concentration), not to mention the greater energy demand.

5.2. Subluminal Localized Waves (or Bullets) In this subsection, we face the more general problem of obtaining, in a simple way, localized (nondiffractive) subluminal pulses as exact analytic solutions to the wave equations, Zamboni-Rached and Recami (2008a). These new ideal subluminal solutions, which propagate without distortion in any homogeneous linear media, will be obtained here for arbitrarily chosen frequencies and bandwidths, avoiding in particular any recourse to the noncausal (backward-moving) components that so frequently plague the previously known LWs. The new solutions are suitable superpositions of—zero-order, in general—Bessel beams, which can be performed either by integrating with respect to the angular frequency ω, or by integrating with respect to the longitudinal wave number kz ; both methods are expounded in this review. The ﬁrst appears to be powerful enough, but we also present the second method since it allows dealing once more (from a different starting point) also with the limiting case of zero-speed solutions (and furnishes a new way, in terms of continuous spectra, for obtaining our FWs, so promising also from the point of view of applications). Moreover, we also pay attention to the known role of SR, and to the fact that the LWs are expected to be transformed one into the other by suitable LTs. Finally, we brieﬂy treat the case of non–axially symmetric solutions in terms of higher-order Bessel beams. The analogous pulses with intrinsic ﬁnite energy, or merely truncated, are considered elsewhere. We devote our attention especially to electromagnetism and optics—but we reiterate that results of the same kind are valid whenever an essential role is played by a wave equation (as in acoustics, seismology, geophysics, elementary particle physics [as we veriﬁed even in the slightly different case of the Schrödinger equation], and gravitation [for which we recently got stimulating new results], and so on).

5.2.1. Foreword For self-consistency, let us repeat here the following considerations. For more than 10 years, the so-called (nondiffracting) LWs, which are new solutions to the wave equations (scalar, vectorial, spinorial etc.), have been in fashion, both in theory and in experiment. In particular, rather well known are those with luminal or superluminal peak velocity, Hernández-Figueroa, Zamboni-Rached, and Recami (2008)—like the socalled X-shaped waves (see Lu and Greenleaf (1992a); Zamboni-Rached, Recami, and Hernández-Figueroa (2002) and references therein; for a

318

Erasmo Recami and Michel Zamboni-Rached

review, see Recami et al. (2003)), which are supersonic in acoustics, Lu and Greenleaf (1992b), and superluminal in electromagnetism (see Recami (1998) and references therein). Since the work of Bateman (1915) and later Courant and Hilbert (1966), it has been known, e.g., that luminal LWs exist, which are solutions to the wave equations. More recently, some attention also has been paid to the subluminal LWs. Let us recall that all LWs propagate without distortion—and in a self-reconstructive way—in a homogeneous linear medium (apart from local variations): in the sense that their square magnitude maintains its shape during propagation, while local variations are shown only by its real, or imaginary, part. As in the superluminal case, subluminal LWs can be obtained by suitable superpositions of Bessel beams, Zamboni-Rached and Recami (2008a). Until recently, as we know, they have been almost neglected because of the mathematical difﬁculties in getting analytic expressions for them, difﬁculties associated with the fact that the superposition integral runs over a ﬁnite interval. Here we readdress the question of such subluminal LWs, showing, by contrast, that exact (analytic) solutions can indeed be determined—both in the case of integration over the Bessel beams’ angular frequency ω and in the case of integration over their longitudinal wave number kz . As already claimed, this work is devoted to the exact, analytic solutions: that is to ideal solutions. The corresponding pulses with ﬁnite energy, or truncated, are presented elsewhere. In the past, not enough attention was paid to Brittingham’s 1983 paper, Brittingham (1983), wherein he showed the possibility of obtaining pulsetype solutions to the Maxwell equations, which propagate in free space as a new kind of speed-c “solitons.” That lack of attention was partially due to the fact that Brittingham had been unable to either obtain correct ﬁnite-energy expressions for such “wavelets” and to make suggestions about their practical production. Two years later, however, Sezginer (1985) was able to obtain quasi-nondiffracting luminal pulses endowed with a ﬁnite energy. Finite-energy pulses no longer travel undistorted, as we know, for an inﬁnite distance, but they can nevertheless propagate without deformation for a long ﬁeld depth, much larger than the one achieved by ordinary pulses like the Gaussian ones (see Arlt et al. (2001); GarcésChavez et al. (2002); Lu, Zou, and Greenleaf (1993, 1994); MacDonald et al. (2002); McGloin, Garcés-Chavez, and Dholakia (2003), and references therein). Only after 1985 was the general theory of LWs extensively developed, Arlt, Hitomi, and Dholakia (2000); Ashkin et al. (1986); Erdélyi et al. (1997); Fan, Parra, and Milchberg (2000); Goodman (1996); Herman and Wiggins (1991); Rhodes et al. (2002); Willebrand and Ghuman (2001); Ziolkowski

Localized Waves: A Review

319

(1989, 1991), both in the case of beams and in the case of pulses. For reviews, see Besieris et al. (1998); Recami (2001); Recami et al. (2003); Recami, Zamboni-Rached, and Hernández-Figueroa (2008) and citations therein. For the propagation of LWs in bounded regions (like wave guides), see Barbero, Hernández-Figueroa, and Recami (2000); Zamboni-Rached, Fontana, and Recami (2003); Zamboni-Rached, Recami, and Fontana (2001); Zamboni-Rached et al. (2002), and references therein. For the focusing of LWs, see the fourth section of this review (as well as Shaarawi, Besieris, and Said (2003); Zamboni-Rached, Shaarawi, and Recami (2004) and citations therein). With regard to the construction of general LWs propagating in dispersive media, see references Conti et al. (2003); HernándezFigueroa, Zamboni-Rached, and Recami (2008); Longhi (2003); Porras, Valiulis, and Di Trapani (2003); Porras et al. (2003); Salo et al. (1999); Sõnajalg and Saari (1996); Zamboni-Rached, Hernández-Figueroa, and Recami (2004); Zamboni-Rached et al. (2003), and for lossy media compare with ZamboniRached (2006a) and references therein. Finally, for ﬁnite-energy, or truncated, solutions, see references Besieris et al. (1998); Zamboni-Rached (2004, 2006); Zamboni-Rached, Fontana, and Recami (2003); ZamboniRached, Recami, and Hernández-Figueroa (2002); Ziolkowski, Besieris, and Shaarawi (1993), and work in progress. By LWs have been experimentally produced (Lu and Greenleaf (1992b); Mugnai, Ranfagni, and Ruggeri (2000); Saari and Reivelt (1997); Valtua, Reivelt, and Saari (2007)), and are being applied in ﬁelds ranging from ultrasound scanning (Lu and Greenleaf (1995); Lu, Zou, and Greenleaf (1993, 1994)) to optics (e.g., for the production, of a new type of tweezers Zamboni-Rached et al. (2005)). We now know that nondiffracting pulses can travel with an arbitrary peak speed v, that is, with 0 < v < ∞, whereas Brittingham and Sezginer had conﬁned themselves to the luminal case (v = c) only. Superluminal and luminal LWs have been, and continue to be, intensively studied, but the subluminal ones have been neglected. The few papers dealing with them until now have been restricted to the paraxial (Newell and Molone, 1992) approximation (Besieris and Shaarawi, 2004) or to numerical simulations (Salo and Salomaa, 2001), because of the mathematical difﬁculty in obtaining exact analytic expressions for subluminal pulses. Indeed, only one analytic solution was known, Besieris and Shaarawi (2002, 2004); Donnelly and Ziolkowski (1993); Lu and Greenleaf (1995); Mackinnon (1978); Rodrigues, Vaz, and Recami (1994), biased by the physically unconvenient facts that (1) its frequency spectrum is very large, (2) it does not possess a well-deﬁned central frequency, and (3) that backward-traveling components (ordinarily called “non causal” because they should be entering the antenna or generator) were needed to construct it. The next text subsections show that subluminal localized exact

320

Erasmo Recami and Michel Zamboni-Rached

solutions can be constructed with any spectra, in any frequency bands, and for any bandwidths—without using (Recami, Zamboni-Rached, and Hernández-Figueroa (2008); Zamboni-Rached, Recami, and HernándezFigueroa (2002)) any backward-traveling components.

5.3. A First Method for Constructing Physically Acceptable Subluminal Localized Pulses Axially symmetric solutions to the scalar wave equation are known to be superpositions of zero-order Bessel beams over the angular frequency ω and the longitudinal wave number kz : That is, in cylindrical coordinates,

(ρ, z, t) =

∞

dω 0

ω/c −ω/c

⎛

⎞ 2 ω dkz S(ω, kz )J0 ⎝ρ − kz2 ⎠ eikz z e−iωt , (115) c2

where kρ2 ≡ ω2 /c2 − kz2 is the transverse wave number. Quantity kρ2 must be positive since evanescent waves cannot come into play. The condition characterizing a nondiffracting wave is the existence (Besieris et al. (1998); Zamboni-Rached and Hernàndez-Figueroa (2000)) of a linear relation between longitudinal wave number kz and frequency ω for all the Bessel beams entering superposition (113); that is, the chosen spectrum must entail, Zamboni-Rached, Recami, and Hernández-Figueroa (2002), for each Bessel beam a linear relationship of the type:6

ω = v kz + b,

(116)

with b ≥ 0. Requirement (116) also can be regarded as a speciﬁc space-time coupling, implied by the chosen spectrum S. Equation (116) must be obeyed by the spectra of any one of the three possible types (subluminal, luminal, or superluminal) of nondiffracting pulses. With the choice in Eq. (116) the pulse regains its initial shape after the space interval z1 = 2πv/b. But the more general case also can be considered, Zamboni-Rached, Recami, and Hernández-Figueroa (2002, 2008)) when b assumes any values bm = m b (with m an integer), and the periodicity space interval becomes zm = z1 /m . We are referring to the real (or imaginary) part of the pulse, since its modulus is known to be endowed with rigid motion.

6 More generally, as shown in Zamboni-Rached, Recami, and Hernández-Figueroa (2002), the chosen spec-

trum has to call into play, in the plane ω, kz , if not exactly the line (116), at least a region in the proximity of a straight line of that a type. It is interesting that in the latter case one obtains solutions endowed with ﬁnite energy, but possessing a ﬁnite “depth of ﬁeld,” that is, nondiffracting only for a certain ﬁnite distance.

Localized Waves: A Review

321

In the subluminal case, of interest here, the only exact solution known until recently, just represented by Eq. (124) below, was the one found by Mackinnon (1978). Indeed, by taking into explicit account that the transverse wave number kρ of each Bessel beam entering Eq. (115) must be real, it can be easily shown (as ﬁrst noticed by Salo and Salomaa (2001) for the analogous acoustic solutions) that in the subluminal case b cannot vanish, but must be larger than zero: b > 0. Then, on using conditions (116) and b > 0, the subluminal localized pulses can be expressed as integrals over the frequency only:

z (ρ, z, t) = exp −ib v

where now

kρ =

1 v

ω+ ω−

ζ dω S(ω) J0 (ρkρ ) exp iω , v

2bω − b2 − (1 − v2 /c2 )ω2

(117)

(118)

with

and with

ζ ≡ z − vt

(119)

⎧ b ⎪ ⎪ ⎨ ω− = 1 + v/c ⎪ b ⎪ ⎩ ω+ = . 1 − v/c

(120)

As anticipated, the Bessel beam superposition in the subluminal case proves to be an integration over a ﬁnite interval of ω, which clearly shows that the backward-traveling (noncausal) components correspond to the interval ω− < ω < b. It could be noticed that Eq. (117) does not represent the most general exact solution, which on the contrary is a sum (ZamboniRached, Recami, and Hernández-Figueroa (2008)) of such solutions for the various possible values of b mentioned above—that is, for the values bm = m b with spatial periodicity zm = z1 /m . However, we can conﬁne ourselves to solution (117) without any real loss of generality, since the actual problem is evaluating in analytic form the integral entering Eq. (117). For mathematical and physical details, see Zamboni-Rached, Recami, and Hernández-Figueroa (2008). Now, if one adopts the change of variable

ω ≡

b v (1 + s), c 1 − v2 /c2

(121)

322

Erasmo Recami and Michel Zamboni-Rached

Eq. (117) becomes, Salo and Salomaa (2001),

1 b v b b exp −i z exp i ζ (ρ, z, t) = v 1 − v2 /c2 c 1 − v2 /c2 v 1

ρ 1 b b 2 ds S(s) J0 1 − s exp i ζs . (122) ×

c 1 − v2 /c2 c 1 − v2 /c2 −1 In the following we adhere to some symbols standard in SR (since the topic of subluminal, luminal, and superluminal LWs is strictly connected (Recami (1998); Recami, Zamboni-Rached, and Dartora (2004); Recami et al. (2003)) with the principles and structure of SR [compare Barut, Maccarrone, and Recami (1982); Recami (1986) and references therein], as we shall mention also in the concluding remarks); namely:

β≡

v ; c

1 γ ≡

. 1 − β2

(123)

As stated, Eq. (122) has until now yielded one analytic solution for S(s) = constant only (for instance, S(s) = 1), which means S(ω) = const.; in this case, one gets the Mackinnon solution, Donnelly and Ziolkowski (1993); Lu and Greenleaf (1995); Mackinnon (1978); Zamboni-Rached (2006),

b b (ρ, ζ, η) = 2 v γ 2 exp i βγ 2 η c c + b2 2 * 2 × sinc γ ρ + γ 2 ζ2 , 2 c

(124)

which, however, because of its previously mentioned drawbacks, is endowed with little physical and practical interest. In Eq. (124) the sinc function has the ordinary deﬁnition

sinc x ≡ (sin x)/x, and

η ≡ z − Vt,

with V ≡

c2 , v

(125)

where V and v are related by the de Broglie relation. (Note that in Eq. (124), and in the following ones, is eventually a function [besides of ρ] of z, t via ζ and η, both functions of z and t.)

Localized Waves: A Review

323

However, we can use a simple method to construct new subluminal pulses corresponding to any spectrum, and devoid of backward-moving (i.e., “entering”) components, by taking advantage of the fact that in Eq. (122) the integration interval is ﬁnite; that is, by transforming it into a good, instead of a harm. Let us ﬁrst observe that Eq. (122) does not admit only the already known analytic solution corresponding to S(s) = constant, and more in general to S(ω) = constant, but it also yields an exact, analytic solution for any exponential spectra of the type

i2nπω , S(ω) = exp

(126)

with n any integer number, which means for any spectra of the type S(s) = exp [inπ/β] exp [inπs], as can be easily seen by checking the product of the various exponentials entering the integrand. In Eq. (126) we have set ≡ ω+ − ω− . The solution in this more general case is written as:

b 2 (ρ, ζ, η) = 2bβ γ exp i β γ η c 2 b2 2 2 b 2 π sinc γ ζ + nπ . γ ρ + × exp in β c c2 2

(127)

Notice that in Eq. (127) quantity η also is deﬁned as in Eq. (125), where V and v obey the de Broglie relation vV = c2 , where the subluminal quantity v is the velocity of the pulse envelope, and V plays the role (in the envelope’s interior) of a superluminal phase velocity. The next step, as anticipated, consists of taking advantage of the ﬁniteness of the integration limits for expanding any arbitrary spectra S(ω) in a Fourier series in the interval ω− ≤ ω ≤ ω+ : ∞

S(ω) =

n=−∞

2π An exp +in ω ,

(128)

where (we went back, now, from the s to the ω variable),

1 An =

ω+

ω−

2π dω S(ω) exp −in ω

quantity being deﬁned as above.

(129)

324

Erasmo Recami and Michel Zamboni-Rached

On recounting the special solution in Eq. (127), we can infer from expansion (126) that, for any arbitrary spectral function S(ω), a rather general axially symmetric analytic solution for the subluminal case can be worked out as

b 2 (ρ, ζ, η) = 2bβ γ exp i β γ η c 2 ∞ π b 2 b2 2 2 sinc γ An exp in γ ρ + ζ + nπ , × β c c2 n=−∞ 2

(130) in which the coefﬁcients An are still given by Eq. (129). We repeat that our solution is expressed in terms of the particular Eq. (127), which is a Mackinnon-type solution. The present approach presents many advantages. We can easily choose spectra localized within the preﬁxed frequency interval (optical waves, microwaves, and so on) and endowed with the desired bandwidth. Moreover, as we have seen, spectra can now be chosen such that they have zero value in the region ω− ≤ ω ≤ b, which is responsible for the backward-traveling components of the subluminal pulse. We stress that, even when the adopted spectrum S(ω) does not possess a known Fourier series [so that the coefﬁcients An cannot be exactly evaluated via Eq. (129)], one can calculate approximately such coefﬁcients without any problem, since our general solutions in Eq. (130) will still be exact solutions. Some examples are presented below.

5.3.1. Examples In general, optical pulses generated in the laboratory possess a spectrum centered at some frequency value, ω0 , called the carrier frequency. For instance, the pulses can be ultrashort, when ω/ω0 ≥ 1; or quasimonochromatic, when ω/ω0 << 1, where ω is the spectrum bandwidth. These types of spectra can be mathematically represented by a Gaussian function or functions with similar behavior.

Examples 1 and 2. Let us ﬁrst consider a Gaussian spectrum a S(ω) = √ exp [−a2 (ω − ω0 )2 ], π

(131)

whose values are negligible outside the frequency interval ω− < ω < ω+ over which the Bessel beams superposition in Eq. (117) is made, it being ω− = b/(1 + β) and ω+ = b/(1 − β). Of course, the relation in Eq. (116) must

Localized Waves: A Review

325

still be satisﬁed, with b > 0, to obtain an ideal subluminal localized solution. Notice that, with the spectrum in Eq. (131), the bandwidth (actually, the full-width half-maximum (FWHM)) results in ω = 2/a. We emphasize that, once v and b have been ﬁxed, the values of a and ω0 can afterward be selected in order to eliminate the backward-traveling components that correspond to ω < b. The Fourier expansion in Eq. (128), which yields, with the above spectral function (131), the coefﬁcients

2π −n2 π2 1 , exp −in ω0 exp An W a2 W 2

(132)

constitutes an excellent representation of the Gaussian spectrum [Eq. (131)] in the interval ω− < ω < ω+ (provided that, as we requested, the Gaussian spectrum does get negligible values outside the frequency interval ω− < ω < ω+ ). In other words, on choosing a pulse velocity v < c and a value for the parameter b, a subluminal pulse with the above frequency spectrum [Eq. (131)] can be written as Eq. (129), with the coefﬁcients An given by Eq. (132)—the evaluation of such coefﬁcients An being rather simple. Even if the values of An are obtained via a (rather good, by the way) approximation, our basis is the exact solution Eq. (130). For instance, exact solutions representing subluminal pulses for optical frequencies can be obtained. Let us determine the subluminal pulse with velocity v = 0.99 c, carrier angular frequency ω0 = 2.4 × 1015 Hz (that is, λ0 = 0.785 μm), and bandwidth (FWHM) ω = ω0 /20 = 1.2 × 1014 Hz, which is an optical pulse of 24 fs (which is the FWHM of the pulse intensity). For a complete pulse characterization, the value of the frequency b must be chosen; let it be b = 3 × 1013 Hz; as a consequence ω− = 1.507 × 1013 Hz and ω+ = 3 × 1015 Hz. (This is exactly a case in which the considered pulse is not plagued by the presence of backward-traveling components, since the chosen spectrum possesses totally negligible values for ω < b.) The construction of the resulting pulse is already satisfactory when considering about 51 terms (−25 ≤ n ≤ 25) in the series entering Eq. (130). Figures 36 show the pulse evaluated by summing the 51 terms. Figure 36a depicts the orthogonal projection of the pulse intensity, and Figure 36b shows the 3D intensity pattern of the real part of the pulse, which reveals the carrier wave oscillations. The ball-like shape7 for the ﬁeld intensity typically should be associated with all the subluminal LWs, whereas the typical superluminal

7 It can be noted that each term of the series in Eq. (130) corresponds to an ellipsoid or, more speciﬁcally,

to a spheroid, for each velocity v.

326

Erasmo Recami and Michel Zamboni-Rached

FIGURE 36 (a) The intensity orthogonal projection for the pulse corresponding to Eqs. (131) and (132) in the case of an optical frequency (see text). (b) The 3D intensity pattern of the real part of the same pulse, which reveals the carrier wave oscillations.

ones are known to be X-shaped (Lu and Greenleaf (1992a); Recami (1998); Recami, Zamboni-Rached, and Dartora (2004)), as predicted by SR in its non-restricted version (see Barut, Maccarrone, and Recami (1982); Recami (1986, 1998); Recami et al. (2003) and references therein). For instance, a second spectrum S(ω) would be the “inverted parabola,” centered at the frequency ω0 ; that is,

S(ω) =

⎧ −4 [ω−(ω −ω/2)][ω−(ω +ω/2)] 0 0 ⎨ for ω0 − ω/2 ≤ ω ≤ ω0 + ω/2 ω2 ⎩

0

otherwise,

(133) where ω, the distance between the two zeros of the parabola, can be regarded as the spectrum bandwidth. One can expand S(ω), given by Eq. (133), in the Fourier series (128), for ω− ≤ ω ≤ ω+ , with coefﬁcients An that—even if straightforwardly calculable—are complicated, so we omit reporting them here. Here we mention only that spectrum (133) may be easily used to determine, for instance, an ultrashort (femtosecond) optical nondiffracting pulse, with satisfactory results even when considering very few terms in expansion (128).

Example 3. As a third interesting example, we consider the simple case when (within the integration limits ω− , ω+ ) the complex exponential spectrum in Eq. (124) is replaced by the real function (still linear in ω) S(ω) =

a exp [a(ω − ω+ )], 1 − exp [−a(ω+ − ω− )]

(134)

Localized Waves: A Review

327

with a a positive number (for a = 0, one reverts to the Mackinnon case). The spectrum in Eq. (134) is exponentially concentrated in the proximity of ω+ , where it reaches its maximum value and becomes increasingly concentrated (on the left of ω+ ) as the arbitrarily chosen value of a increases, its frequency bandwidth being ω = 1/a. Recall that, on their turn, quantities ω+ and ω− depend on the pulse velocity v and on the arbitrary parameter b. By performing the integration as in the case of Eq. (126), instead of the solution in Eq. (127), in the present case one eventually reaches the solution

(ρ, ζ, η) =

2abβγ 2 exp [abγ 2 ] exp [−aω+ ] 1 − exp [−a(ω+ − ω− )]

b b 2 2 −2 2 2 × exp i β γ η sinc γ γ ρ − (av + iζ) . (135) c c After Mckinnon’s, Eq. (135) appears to be the simplest closed-form solution, since both do not require any recourse to series expansions. In a sense, our solution in Eq. (135) may be regarded as the subluminal analog of the (superluminal) X-wave solution; the difference being that the standard X-shaped solution has a spectrum starting with 0, where it assumes its maximum value, while in the present case the spectrum starts at ω− and increases afterward, until ω+ . More important is that the Gaussian spectrum has a priori two advantages with respect to Eq. (134): It may be more easily centered around any value ω0 of ω, and, when increasing its concentration in the surrounding of ω0 , the spot transverse width does not increase indeﬁnitely, but tends to the spot width of a Bessel beam with ω = ω0 and kz = (ω0 − b)/V, at variance with what occurs with Eq. (134). Regardless, Eq. (135) is noteworthy since it is the simplest solution. Figure 37 shows the intensity of the real part of the subluminal pulse corresponding to this spectrum, with v = 0.99 c, with b = 3 × 1013 Hz (which results in ω− = 1.5 × 1013 Hz and ω+ = 3 × 1015 Hz), and with ω/ω+ = 1/100 (i.e., a = 100). This is an optical pulse of 0.2 ps.

5.4. A Second Method for Constructing Subluminal Localized Pulses The previous method appears to be very efﬁcient for determining analytic subluminal LWs, but it loses its validity in the limiting case v → 0, since for v = 0 it is ω− ≡ ω+ and the integral in Eq. (117) degenerates, furnishing a null value. By contrast, we also are interested in the v = 0 case, since it corresponds to some of the most interesting, and potentially useful, LWs—namely, the stationary solutions to the wave equations endowed with a static envelope, which we have called frozen waves.

(Re C)2

328

Erasmo Recami and Michel Zamboni-Rached

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.2 0.1

0.1 0.05

0 ␳ (mm)

0

20.1 20.2 20.1

20.05

␰ (mm)

FIGURE 37 The intensity of the real part of the subluminal pulse corresponding to spectrum (134), with v = 0.99 c, with b = 3 × 1013 Hz (which results in ω− = 1.5 × 1013 Hz and ω− = 3 × 1015 Hz), and with ω/ω+ = 1/100 (i.e., a = 100).

Before proceeding, recall that the theory of FWs was initially developed in Zamboni-Rached, Recami, and Hernández-Figueroa (2005) by having recourse to discrete superpositions to bypass the need for numerical simulations. In the case of continuous superpositions, some numerical simulations were performed by Dartora et al. (2006). However, the method presented in this section does allow determining analytic exact solutions (with no further need for numerical simulations) even for FWs consisting of continuous superpositions. We will show that the present method works regardless of the chosen ﬁeld intensity shape and in regions with size of the order of the wavelength. It is possible to reach such results by starting from Eq. (115), with the constraint in Eq. (114), but going on (this time) to integrals over kz , instead of over ω. It is enough to write Eq. (116) in the form

kz =

1 (ω − b) v

(116’)

for expressing the exact solutions in Eq. (115) as

(ρ, z, t) = exp [−ibt]

kz max kz min

dkz S(kz ) J0 (ρkρ ) exp [iζkz ],

(136)

Localized Waves: A Review

with

⎧ −b 1 ⎪ ⎪ ⎨ kz min = c 1+β ⎪ b 1 ⎪ ⎩ kz max = c 1−β

329

(135)

and with

kρ2 = −

kz2 b b2 βk + 2 + , z c γ2 c2

(138)

where quantity ζ is still deﬁned according to Eq. (119), always with v < c. The unique exact solution previously known, Mackinnon (1978), may be rewritten in the form of Eq. (136) with S(kz ) = constant. Then, on following the same procedure in our ﬁrst method (previous Section 5.3), we can obtain the new exact solutions corresponding to

i2nπkz , S(kz ) = exp K

(139)

where

K ≡ kz max − kz min, by performing the change of variable [analogous, in its ﬁnality, to the one in Eq. (121)]

kz ≡

b 2 γ (s + β). c

(140)

At the end, the exact subluminal solution corresponding to spectrum in Eq. (139) results as

b 2 b 2 (ρ, ζ, η) = 2 γ exp i β γ η c c 2 b 2 b2 2 2 γ γ ρ + ζ + nπ , × exp [inπβ] sinc c c2

(141)

We again observe that any spectra S(kz ) can be expanded, in the interval kz min < kz < kz max, in the Fourier series:

S(kz ) =

∞ n=−∞

2π An exp +in kz , K

(142)

330

Erasmo Recami and Michel Zamboni-Rached

with coefﬁcients given now by

An =

1 K

kz max

kz min

2π dkz S(kz ) exp −in kz , K

(143)

quantity K having been deﬁned previously. At the end of the process, the general exact solution representing a subluminal LW, for any spectra S(kz ), can be eventually written as

(ρ, ζ, η) = 2

×

b 2 b γ exp [i β γ 2 η c c ∞

An exp [inπβ] sinc

n=−∞

b2 2 2 γ ρ + c2

b 2 γ ζ + nπ c

2 , (144)

whose coefﬁcients are expressed in Eq. (143), and where quantity η is deﬁned as above, in Eq. (125). Interesting examples could easily be worked out, as at the end of the previous Section 5.3.

5.5. Stationary Solutions with Zero-Speed Envelopes Here we refer to the (second) method, expounded in the previous Section 5.4. Our solution in Eq. (144), for the case of envelopes at rest, that is, in the case v = 0 (which implies ζ = z), becomes ∞ b (ρ, z, t) = 2 exp [−ibt] An sinc c n=−∞

b2 2 ρ + c2

b z + nπ c

2 , (145)

with coefﬁcients An given by Eq. (143) with v = 0, so that its integration limits simplify into −b/c and b/c, respectively. Thus,

An =

c 2b

cπ dkz S(kz ) exp [−in kz . b −b/c b/c

(143’)

Equation (145) is a new exact solution, corresponding to stationary beams with a static intensity envelope. Observe, however, that even in this case there is energy propagation, as it can be easily veriﬁed from the power ﬂux Ss = −∇R ∂R /∂t (scalar case) or from the Poynting

Localized Waves: A Review

331

vector Sv = (E ∧ H) (vectorial case: the condition being that R be a single component, Az , of the vector potential A) (Recami, 1998). We have indicated by R the real part of . For v = 0, Eq. (116) becomes

ω = b ≡ ω0 , so that the particular subluminal waves endowed with null velocity are actually monochromatic beams. It may be stressed that this method does yield exact solutions, without any need for the paraxial approximation, which is so often used when looking for expressions representing beams like the Gaussian ones. We recall that, when having recourse to the paraxial approximation, the obtained beam expressions are valid only when the envelope sizes (e.g., the beam spot) vary in space much more slowly than the beam wavelength. For instance, the usual expression for a Gaussian beam, Newell and Molone (1992), holds only when the beam spot ρ is much larger than λ0 ≡ ω0 /(2πc) = b/(2πc): so that those beams cannot be very much localized. By contrast, our method overcomes such problems, since we have seen that it yields exact expressions for (well-localized) beams with sizes of the order of their wavelength. Notice, moreover, that the already known exact solutions—for instance, the Bessel beams—are nothing but particular cases of our solution in Eq. (145). Example. On choosing (with 0 ≤ q− < q+ ≤ 1) the spectral double-step function

S(kz ) =

⎧ ⎪ ⎨ ⎪ ⎩

c ω0 (q+ − q− )

for q− ω0 /c ≤ kz ≤ q+ ω0 /c

0

(146) elsewhere,

the coefﬁcients of Eq. (145) become

An =

ic e−iq+ πn − e−iq− πn . 2πnω0 (q+ − q− )

(147)

The double-step spectrum in Eq. (146) corresponds, with regard to the longitudinal wave number, to the mean value k z = ω0 (q+ + q− )/2c and to the width kz = ω0 (q+ − q− )/c. From such relations, it follows that kz /k z = 2(q+ − q− )/(q+ + q− ). For values of q− and q+ that do not satisfy the inequality kz /k z << 1, the resulting solution will be a nonparaxial beam.

332

Erasmo Recami and Michel Zamboni-Rached

FIGURE 38 (a) Orthogonal projection of the 3D intensity pattern of the beam (a null-speed subluminal wave) corresponding to spectrum (146); (b) 3D plot of the field intensity. The beam considered in this example is highly nonparaxial.

Figures 38 shows the exact solution corresponding to ω0 = 1.88 × 1015 Hz (i.e., λ0 = 1 μm), and to q− = 0.3 and to q+ = 0.9. It results as a beam with a spot-size diameter of 0.6 μm, and, moreover, with a rather good longitudinal localization. In the case of Eqs. (144) and (145), about 21 terms (−10 ≤ n ≤ 10) in the sum entering Eq. (143) are already satisfactory for a good evaluation of the series. The beam considered in this example is highly nonparaxial (with kz /k z = 1 ), and therefore could not have been obtained by ordinary Gaussian beam solutions (which are valid only in the paraxial regime).8

5.5.1. A New Approach to Frozen Waves A noticeable property of our present method is that it allows a spatial modeling of even monochromatic ﬁelds (that correspond to envelopes at rest; so that, in electromagnetic cases, one can speak, for example, of the modeling of “light ﬁelds at rest”). Recall that such a modeling (rather interesting, especially for applications, Zamboni-Rached et al. (2005)) was already performed in ZamboniRached (2004, 2006); Zamboni-Rached, Recami, and Hernández-Figueroa (2005) and has been exploited at the beginning of the previous section [compare with Eqs. (99) and (100)], in terms of discrete superpositions of Bessel beams. However, the method presented in the previous subsections allows use of continuous superpositions to obtain a predetermined longitudinal (onaxis) intensity pattern, inside a desired space interval 0 < z < L. In fact, the

8 Here we consider only scalar wave ﬁelds. In the case of nonparaxial optical beams, the vector character

of the ﬁeld has to be taken into account.

Localized Waves: A Review

333

continuous superposition, analogous to Eq. (100), now can be written as

(ρ, z, t) = e

−iω0 t

ω0 /c −ω0 /c

dkz S(kz ) J0 (ρkρ ) eizkz ,

(148)

which is nothing but the previous Eq. (136) with v = 0 (and therefore ζ = z). In other words, Eq. (149) does represent a null-speed subluminal wave. To clarify, recall that the FWs were expressed in the past as discrete superposition because it was not then known how to analytically treat a continuous superposition like that in Eq. (148). However, the previous approach to FWs can now be extended to the case of integrals, without numerical simulations, but in terms of analytic solutions. Indeed, the exact solution of Eq. (148) is given by Eq. (145), with coefﬁcients in Eq. (143’); and one can choose the spectral function S(kz ) in such a way that assumes the on-axis prechosen static intensity pattern |F(z)|2 . Namely, the equation to be satisﬁed by S(kz ), to such an aim, is derived by associating Eq. (148) with the requirement |(ρ = 0, z, t)|2 = |F(z)|2 , which entails the integral relation

ω0 /c −ω0 /c

dkz S(kz ) eizkz = F(z).

(149)

Equation (149) would be trivially solvable in the case of an integration between −∞ and +∞, since it would merely be a Fourier transformation, but obviously this is not the case, because its integration limits are ﬁnite. Actually, there are functions F(z) for which Eq. (149) is not solvable at all, in the sense that no spectra S(kz ) exist obeying the last equation. For instance, if we consider the Fourier expansion

F(z) =

∞

−∞

dkz , S(kz ) eizkz ,

when , S(kz ) does assume nonnegligible values outside the interval −ω0 /c < kz < ω0 /c, then in Eq. (149) no S(kz ) can forward that particular F(z) as a result. However, some procedures can be devised, such that one can nevertheless ﬁnd a function S(kz ) that approximately (but satisfactorily) complies with Eq. (149). The ﬁrst procedure consists of writing S(kz ) in the form

1 S(kz ) = K

∞ n=−∞

2nπ F K

e−i2nπkz /K ,

(150)

334

Erasmo Recami and Michel Zamboni-Rached

where, as before, K = 2ω0 /c. Then, Eq. (150) can easily be veriﬁed as guaranteeing that the integral in Eq. (149) yields the values of the desired F(z) at the discrete points z = 2nπ/K. Indeed, the Fourier expansion (150) is already of the same type as Eq. (142), so that in this case the coefﬁcients An of our solution in Eq. (145), appearing in Eq. (143’), do simply become

2nπ 1 . An = F − K K

(151)

This is a powerful way to obtain a desired longitudinal (on-axis) intensity pattern, especially for tiny spatial regions, because it is not necessary to solve any integral to determine the coefﬁcients An , which by contrast are given directly by Eq. (151). Figures 39 depicts some interesting applications of this method. A few desired longitudinal intensity patterns |F(z)|2 have been chosen, and the corresponding FWs are calculated by using Eq. (145) with the coefﬁcients An given in Eq. (151). The desired patterns are enforced to exist within very small spatial intervals only, to show the capability of the method to model (Zamboni-Rached and Recami, 2008a) the ﬁeld intensity shape under such strict requirements. In the four examples below, we considered a wavelength λ = 0.6 μm, which corresponds to ω0 = b = 3.14 × 1015 Hz. The ﬁrst longitudinal (on-axis) pattern considered by us is that given by

& F(z) =

ea(z−Z) 0

for 0 ≤ z ≤ Z elsewhere,

that is, a pattern with an exponential increase, starting from z = 0 until z = Z. The chosen values of a and Z are Z = 10 μm and a = 3/Z. The intensity of the corresponding FW is shown in Figure 39a. The second longitudinal pattern (on-axis) taken into consideration is the Gaussian one, given by

& F(z) =

z 2

e−q( Z ) 0

for − Z ≤ z ≤ Z elsewhere,

with q = 2 and Z = 1.6 μm. The intensity of the corresponding FW is shown in Figure 39b.

Localized Waves: A Review

335

FIGURE 39 Frozen Waves with the on-axis longitudinal field pattern chosen as (a) exponential, (b) Gaussian, (c) super-Gaussian, and (d) zero-order Bessel function.

In the third example, the desired longitudinal pattern is supposed to be a super-Gaussian:

& F(z) =

z 2m

e−q( Z ) 0

for − Z ≤ z ≤ Z elsewhere,

where m controls the edge sharpness. Here we have chosen q = 2, m = 4, and Z = 2 μm. The intensity of the FW obtained in this case is shown in Figure 39c. Finally, in the fourth example, we chose the longitudinal pattern as being the zero-order Bessel function

& F(z) =

J0 (q z) 0

for − Z ≤ z ≤ Z elsewhere,

336

Erasmo Recami and Michel Zamboni-Rached

with q = 1.6 × 106 m−1 and Z = 15 μm. The intensity of the corresponding FW is shown in Figure 39d. Any static envelopes of this type can be easily transformed into propagating pulses by the mere application of LTs. Another procedures exists for evaluating S(kz ), based on the assumption that

S(kz ) , S(kz ),

(152)

which constitutes a good approximation whenever , S(kz ) assumes negligible values outside the interval [−ω0 /c, ω0 /c]. In such a case, there is recourse to the method associated with Eq. (142), and , S(kz ) itself can be expanded in a Fourier series, eventually yielding the relevant coefﬁcients An by Eq. (143). Let us recall that it is still K ≡ kz max − kz min = 2ω0 /c. It is worthwhile to call attention to the circumstance that, when constructing FWs in terms of a sum of discrete superpositions of Bessel beams (as done in our Section 5.1 and in the references Zamboni-Rached (2004, 2006); Zamboni-Rached, Recami, and Hernández-Figueroa (2005); Zamboni-Rached et al. (2005)), it was easy to obtain extended envelopes like, for example, “cigars”—where “easy” means by using only a few terms of the sum. By contrast, when we construct FWs as continuous superpositions, then it is easy to get highly localized (concentrated) envelopes. Moreover, the method presented in this subsection furnishes FWs that are no longer periodic along the z-axis (a situation that, with our old method Zamboni-Rached (2004, 2006); Zamboni-Rached, Recami, and HernándezFigueroa (2005), was obtainable only when the periodicity interval tended to inﬁnity).

5.6. The Role of Special Relativity and Lorentz Transformations On one hand, strict connections exist between the principles and structure of SR and, on the other hand, the entire subject of subluminal, luminal, and superluminal localized waves, in the sense that it was expected for years that a priori they are transformable one into the other via suitable LTs (compare references Barut, Maccarrone, and Recami (1982); Mignani and Recami (1973); Recami (1986), in addition to ours in progress).9 We ﬁrst conﬁne ourselves to the cases faced in this text section. Our subluminal localized pulses, which may be called “wave bullets,” behave

9 We call attention to a paper by Saari and Reivelt (2004) only recently noticed by us, wherein the relativistic

connections among the LWs were also investigated in terms of suitable LTs. We are happy to quote such a reference here because it appears inspired by a philosophy, which—going back in part to papers like Barut, Maccarrone, and Recami (1982); Recami (1986)—has been constantly shared by our old, and recent, papers. Let us mention also a further interesting article by Besieris et al. (1998), wherein ordinary LTs were already used, correctly, in the context of subluminal LWs (although the superluminal LTs used in that paper seem, however, to be partially defective).

Localized Waves: A Review

337

as particles. Indeed, our subluminal pulses (as well as the luminal and superluminal [X-shaped] ones, that have been amply investigated in the past literature) do exist as solutions of any wave equations, ranging from electromagnetism and acoustics or geophysics, to elementary particle physics (and even, as we discovered recently, to gravitation physics). From the kinematical point of view, the velocity composition relativistic law holds also for them. The same is true, more generally, for any LWs (pulses or beams). For simplicity we start by considering, in an initial reference frame O, just a (ν-order) Bessel beam:

(ρ, φ, z, t) = Jν (ρkρ ) eiνφ eizkz e−iωt .

(153)

In a second reference frame O , moving with respect to O with speed u—along the positive z-axis and in the positive direction, for simplicity—it will be observed (Saari and Reivelt, 2004) the new Bessel beam

(ρ , φ , z , t ) = Jν (ρ k ρ ) eiνφ eiz k z e−iω t ,

(154)

obtained by applying the appropriate LT (a Lorentz “boost”) with γ =

[ 1 − u2 /c2 ]−1 :

k ρ = kρ ; k z = γ(kz − uω/c2 ); ω = γ(ω − ukz );

(155)

for example, this can be easily seen by putting

ρ = ρ ; z = γ(z + ut ); t = γ(t + uz /c2 )

(156)

directly into Eq. (154). We now pass to subluminal pulses. We can investigate the action of a LT, by expressing them either via the ﬁrst method (Section 4.3) or via the second one (Section 4.4). Let us consider for instance, in the frame O, a v-speed (subluminal) pulse, given by Eq. (155). We proceed to a second observer O moving with the same speed v with respect to frame O, and, still for the sake of simplicity, passing through the origin O of the initial frame at time t = 0. The new observer O will see the pulse (Saari and Reivelt, 2004)

(ρ , z , t ) = e

−it ω0

ω+

ω−

dω S(ω) J0 (ρ k ρ ) eiz k z ,

(157)

338

Erasmo Recami and Michel Zamboni-Rached

with

k z = γ −1 ω/v − γb/v; ω = γb; k ρ = ω 0 /c2 − k z , 2

(158)

as determined from the LTs in Eq. (155) or in Eq. (156), with u = v [and γ given by Eq. (123)]. Notice that k z is a function of ω, as expressed by the ﬁrst of the three relations in Eq. (158), and that here ω is a constant. If we explicitly insert into Eq. (157) the relation ω = γ(vk z + γb), which is nothing but rewriting the ﬁrst relation of Eq. (158), then Eq. (157) becomes

(ρ , z , t ) = γv e−it ω0

ω0 /c

−ω0 /c

dk z S(k z ) J0 (ρ k ρ ) eiz k z ,

(159)

where S is expressed in terms of the previous function S(ω), entering Eq. (157), as follows:

S(k z ) = S(γvk z + γ 2 b).

(160)

Equation (159) describes monochromatic beams with axial symmetry (and does coincide with what derived in our second method, in Section 13, when posing v = 0). The remarkable conclusion is that a subluminal pulse, given by our Eq. (117), which appears as a v-speed pulse in a frame O, will appear (Saari and Reivelt, 2004) in another frame O (traveling with respect to observer O with the same speed v in the same direction z) just as the monochromatic beam in Eq. (159) endowed with angular frequency ω0 = γb, whatever be the pulse spectral function in the initial frame O: even if the kind of monochromatic beam to which we arrive does of course depend10 on the chosen S(ω). The opposite is also true, in general. Let us set forth explicitly an observation that until now has been noticed only by Zamboni-Rached and Recami (2008a). Namely, we mention that, when starting not from Eq. (117) but from the most general solutions which, as already seen, are sums of solutions (117) over the various values bm of b, then a LT will lead to a sum of monochromatic beams—actually, of harmonics (rather than to a single monochromatic beam). In particular, if the goal is to obtain a sum of harmonic beams, a LT must be applied to more general subluminal pulses.

10 One gets in particular a Bessel-type beam when S is a Dirac delta function: S(ω) = δ(ω − ω ). Moreover, 0

let us notice that, on applying a LT to a Bessel beam, one obtains another Bessel beam, with a different axicon angle.

Localized Waves: A Review

339

In additions, the various superluminal localized pulses are transformed (Saari and Reivelt, 2004) one into the other by the mere application of ordinary LTs, while it may be expected that the subluminal and the superluminal LWs are to be linked (apart from some known technical difﬁculties that require particular caution) by the superluminal Lorentz “transformations” expounded long ago (see Barut, Maccarrone, and Recami, 1982; Mignani and Recami, 1973; Recami, 1986; Recami and Rodrigues, 1985 and references therein.)11 Recall that in the years 1980–1982, the theory of SR, in its nonrestricted version, predicted that, while the simplest subluminal object is obviously a sphere (or, in the limit, a space point), the simplest superluminal object is, on the contrary, an X-shaped pulse (or, in the limit, a double cone) (see Figure 11,12 from Barut, Maccarrone, and Recami, 1982; Recami, 1986). The circumstance that the localized solutions to the wave equations follow the same behavior is rather interesting, and is expected to be useful for example, in the case of elementary particles and quantum physics, for a deeper comprehension of de Broglie’s and Schrödinger’s wave mechanics. With regard to the fact that the simplest subluminal LWs, solutions to the wave equation, are “ball-like,” let us present in Figure 40, in ordinary 3D space, the general shape of Mackinnon’s solutions, as expressed by Eq. (124) for v << c: In such ﬁgures we graphically depict the ﬁeld isointensity surfaces, which results (as expected) to be just spherical in the considered case. We have also seen, among the others, that, even if our ﬁrst method (Section 5.3) cannot directly yield zero-speed envelopes, such envelopes “at rest,” Eq. (145), however can be obtained by applying a v-speed LT to Eq. (130). In this way, one starts from many frequencies [Eq. (130)] and ends with one frequency only [Eq. (145)], since b gets transformed into the frequency of the monochromatic beam.

11 The topic of superluminal LTs is a delicate one, to the extent that the majority of the recent attempts to

readdress this question and its applications seem to be defective (sometimes they do not even maintain the necessary covariance of the wave equation itself). 12 Recall, more speciﬁcally, that Figure 11 depicts the following. Let us start from an object that is intrinsically spherical (i.e., that is a sphere in its rest frame [Panel (a)]. Then, after a generic subluminal LT along x (i.e., under a subluminal x-boost), it is predicted by SR to appear as ellipsoidal due to Lorentz contraction [panel (b)]. After a superluminal x-boost (Mignani and Recami, 1973; Recami, 1986; Recami and Rodrigues, 1985) (namely, when this object moves (Recami, Zamboni-Rached, and Dartora, 2004) with superluminal speed V ), it is predicted by SR—in its nonrestricted version, “extended relatively” (ER)—to appear (Barut, Maccarrone, and Recami, 1982) as in panel [d], i.e., as occupying the cylindrically symmetric region bounded by a two-sheeted rotation hyperboloid and an indeﬁnite double cone. The whole structure, according to ER, is expected to move rigidly and, of course, with the speed V , the cotangent square of the cone semi-angle being (V/c)2 − 1. Panel [c] refers to the limiting case when the boost speed tends to c, either from the left or from the right [for simplicity, a space axis is skipped]). It is remarkable that the shape of the localized (subluminal and superluminal) pulses, solutions to the wave equations, appears to follow the same behavior; this can also play a role for better comprehension of de Broglie and Schrödinger wave mechanics. See also Figure 40.

340

Erasmo Recami and Michel Zamboni-Rached

FIGURE 40 In Figure 39 we saw how SR, in its nonrestricted version (ER), predicted (Barut, Maccarrone, and Recami, 1982; Recami, 1986) that, while the simplest subluminal object is obviously a sphere (or, in the limit, a space point), the simplest superluminal object is, on the contrary, an X-shaped pulse (or, in the limit, a double cone). The circumstance that the localized solutions to the wave equations do follow the same pattern is rather interesting and is expected to be useful—for example, in the case of elementary particles and quantum physics—for a deeper comprehension of de Broglie and Schrödinger wave mechanics. With regard to the fact that the simplest subluminal LWs, solutions to the wave equations, are “ball-like,” let us depict in these figures, in ordinary 3D space, the general shape of Mackinnon solutions as expressed by Eq. (124), numerically evaluated for v << c. Panels (a) and (b) graphically represent the field iso-intensity surfaces, which in the considered case are (as expected) just spherical.

5.7. Nonaxially Symmetric Solutions: Higher-Order Bessel Beams To this point, we have directed attention to exact solutions representing axially symmetric (subluminal) pulses only: that is, to pulses obtained by suitable superpositions of zero-order Bessel beams. However, it is interesting to look for analytic solutions representing nonaxially symmetric subluminal pulses, which can be constructed in terms of superpositions of ν-order Bessel beams, with ν a positive integer (ν > 0). This can be attempted both in the case of Section 4.3 (ﬁrst method) and in the case of Section 4.4 (second method). For brevity’s sake, let us consider only the ﬁrst method (Section 4.3). One is immediately confronted with the difﬁculty that no exact solution is known for the integral in Eq. (122) when J0 (.) is replaced with Jν (.). This difﬁculty can be overcome by following a simple method that allows obtaining “higher-order” subluminal waves in terms of the axially symmetric ones. Indeed, it is well known that, if (x, y, z, t) is an exact solution to the ordinary wave equation, then ∂/∂x and ∂/∂y are also

Localized Waves: A Review

341

exact solutions.13 By contrast, when working with cylindrical coordinates, if (ρ, φ, z, t) is a solution to the wave equation, quantities ∂/∂ρ and ∂/∂φ generally are not solutions. Nevertheless, it is not difﬁcult to reach the noticeable conclusion that, once (ρ, φ, z, t) is a solution, then

(ρ, φ, z, t) = e

iφ

i ∂ ∂ + ∂ρ ρ ∂φ

(161)

also is an exact solution. For instance, for an axially symmetric solution of the type = J0 (kρ ρ) exp[ikz ] exp[−iωt], Eq. (161) yields = −kρ J1 (kρ ρ) exp[iφ] exp[ikz ] exp[−iωt], which is actually one more analytic solution. In other words, it is enough to start for simplicity from a zero-order Bessel beam, and to apply Eq. (161), successively, ν times, to get as a new solution = (−kρ )ν Jν (kρ ρ) exp[iνφ] exp[ikz ] exp[−iωt], which is a ν-order Bessel beam. In this mannar, when applying ν times Eq. (161) to the (axially symmetric) subluminal solution (ρ, z, t) in Eqs. (128–130) [obtained from Eq. (117) with spectral function S(ω)], we get the subluminal nonaxially symmetric pulses ν (ρ, φ, z, t) as new analytic solutions, consisting, as expected, of superpositions of ν-order Bessel beams:

n (ρ, φ, z, t) =

ω+

ω−

dω S (ω) Jν (kρ ρ) eiνφ eikz z e−iωt ,

(162)

where kρ (ω) is given by Eq. (118), and quantities S (ω) = (−kρ (ω))ν S(ω) are the spectra of the new pulses. If S(ω) is centered at a certain carrier frequency (it is a Gaussian spectrum, for instance), then S (ω) also will approximately be the same type. If we wish the new solution ν (ρ, φ, z, t) to possess a predeﬁned spectrum S (ω) = F(ω), we can ﬁrst take Eq. (117) and put S(ω) = F(ω)/(−kρ (ω))ν in its solution [Eq. (130)] and afterward apply to it, ν times, the operator U ≡ exp[iφ] [∂/∂ρ + (i/ρ)∂/∂φ)]. As a result, we obtain the desired pulse, ν (ρ, φ, z, t), endowed with S (ω) = F(ω).

Example. On starting from the subluminal axially symmetric pulse (ρ, z, t), given by Eq. (130) with the Gaussian spectrum in Eq. (131), we can get the subluminal, nonaxially symmetric, exact solution 1 (ρ, φ, z, t) by simply calculating 1 (ρ, φ, z, t) =

∂ iφ e , ∂ρ

13 Let us mention that even ∂n /∂zn and ∂n /∂tn will be exact solutions.

(163)

342

Erasmo Recami and Michel Zamboni-Rached

which actually yields the “ﬁrst-order” pulse 1 (ρ, φ, z, t), which can be more compactly written in the form:

∞ b b π 2 2 ψ1n (164) 1 (ρ, φ, η, ζ) = 2 v γ exp i β γ η An exp in c c β n=−∞ with

ψ1n (ρ, φ, η, ζ) ≡ where

b2 2 γ ρ Z−3 [Z cos Z − sin Z] eiφ , c2

Z≡

b2 2 2 γ ρ + c2

b 2 γ ζ + nπ c

(165)

2 .

(166)

This exact solution corresponds to the superposition in Eq. (162), with S (ω) = kρ (ω)S(ω), quantity S(ω) being given by Eq. (131). It is represented in Figure 41. The pulse intensity has a “donut-like” shape.

5.7.1. Concluding Remarks for Section 5 In this section we started by developing, by suitable superpositions of equal-frequency Bessel beams, a ﬁrst theoretical and experimental methodology to obtain localized stationary wave ﬁelds, with high transverse localization, whose longitudinal intensity pattern can approximately assume any desired shape within a chosen interval 0 ≤ z ≤ L of the propagation axis z. Their intensity envelope remains static (i.e., with velocity v = 0), so we named these new solutions to the wave equations (and, in particular, to the Maxwell equations) “Frozen Waves” (FW). Inside the envelope of a FW only the carrier wave propagates. In addition, the longitudinal shape, within the interval 0 ≤ z ≤ L, can be chosen in such a way that no nonnegligible ﬁeld exists outside the predetermined region (consisting, e.g., of one or more high-intensity peaks). Such solutions are noticeable also for the different and interesting applications they can have, especially in electromagnetism and acoustics, such as optical tweezers, atom guides, optical or acoustic bistouries, various important medical apparatus (mainly for destroying cancerous cells), and so on. We later addressed the more general subject of the subluminal LWs and showed that, as in the well-known superluminal case (HernándezFigueroa, Zamboni-Rached, and Recami, 2008), the subluminal solutions can be obtained by superposing Bessel beams (Zamboni-Rached and Recami, 2008a). Such solutions were barely considered in the past because the superposition integral must run in this case over a ﬁnite interval (which

Localized Waves: A Review

343

|c(␳,␾,␩,␰)|2

20

0.9

15

0.8 10 0.7

␳ (L.m)

5

0.6

0

0.5

25

0.4 0.3

210

0.2 215 220 230

0.1 220

210

0 0 ␰(L.m)

10

20

30

FIGURE 41 Orthogonal projection of the field intensity corresponding to the higher-order subluminal pulse represented by the exact solution Eq. (163), quantity being given by Eq. (128) with the Gaussian spectrum in Eq. (131). The pulse intensity here happens to have a “donut-like” shape.

makes it mathematically difﬁcult to determine analytic expressions for them). However, we have shown how it is possible to obtain, in a simple way, nondiffracting subluminal pulses as exact analytic solutions to the wave equations: for arbitrarily chosen frequencies and bandwidths, and avoiding any recourse to the backward-traveling components. Until recently, only one closed-form subluminal LW solution, ψcf , to the wave equations was known, Mackinnon (1978). It was obtained by choosing in the relevant integration a constant weight function S(ω), whereas all other solutions had previously got only by numerical simulations. On the contrary, a subluminal LW can be obtained in closed form by adopting, for instance, any spectra S(ω) that are expansions in terms of ψcf . In fact, the initial disadvantage—dealing with a limited bandwidth—may be turned into an advantage because, in the case of “truncated” integrals, the spectrum S(ω) can be expanded in a Fourier series. More generally, it has been shown how exact solutions can be determined both by integration over the Bessel beams’ angular frequency ω and by integration over their longitudinal wave number kz . Both methods have been expounded above.

344

Erasmo Recami and Michel Zamboni-Rached

The ﬁrst method appears to be comprehensive enough; we studied the second method as well, however, because it provides a new way, in terms of continuous spectra, to tackle the limiting case of zero-speed solutions (i.e., for obtaining the FWs). We also brieﬂy treated the case of nonaxially symmetric solutions, that is, of higher-order Bessel beams. Finally, some attention was directed to the role of SR and to the fact that the LWs are expected to be transformable one into the other by suitable LTs. Moreover, our results seem to show that in the subluminal case the simplest LW solutions are (for v << c) ball-like, as expected since 1982 Barut, Maccarrone, and Recami (1982) on the mere basis of SR, Recami (1986). (Indeed, we recall that in the years 1980–82 it had already been predicted that, if the simplest subluminal object is a sphere (or, in the limit, a space point), then the simplest superluminal object is an X-shaped pulse (or, in the limit, a double cone); and vice versa). It is rather interesting that the same pattern appears to be followed by the localized solutions of the wave equations). For the subluminal case, see Figure 40. The subluminal localized pulses, endowed with a ﬁnite energy, or merely truncated, are the subject of another paper.

ACKNOWLEDGMENTS Work partially supported by FAPESP and CNPq (Brazil), and by MIUR and INFN (Italy). We are grateful to Peter Hawkes for his kind invitation to contribute this paper to the AIEP series, as well as S. Narayana and other AIEP staff for their extreme dedication, and to H. E. Hernández-Figueroa for his continuous help and collaboration over the years. For useful discussions we also wish to thank, among others, I. M. Besieris, R. Bonifacio, R. Chiao, C. Conti, A. Friberg, G. Degli Antoni, F. Fontana, M. Ibison, G. Kurizki, I. Licata, J-y. Lu, A. Loredo, M. Mattiuzzi, P. Milonni, P. Nelli, P. Saari, A. M. Shaarawi, E. C. G. Sudarshan, M. Tygel, and R. Ziolkowski. Thanks are ﬁnally due to M. Tenório de Vasconselos for her willingness and for patient cooperation.

REFERENCES Agrawal, G. P. (1995). “Nonlinear Fiber Optics,” 2nd edition. Academic Press, New York. Agrawal, G. (2006). “Nonlinear Fiber Optics,” 4th edition. Academic Press, New York. Aharonov, Y., Erez, N., and Reznik, B. (2002). Superoscillations and tunneling times, Phys. Rev. A 65, no. 052124[5 pages]. Alzetta, G., Gozzini, A., Moi, L., and Orriols, G. (1976). An experimental method for the observation of the RF transitions and laser beat resonances in oriented Na vapor. Nuovo Cimento B 36, 5–20. Arlt, J., Hitomi, T., and Dholakia, K. (2000). Atom guiding along Laguerre-Gaussian and Bessel light beams. Appl. Phys. B 71, 549–554. Arlt, J., Garcés-Chavez, V., Sibbett, W., and Dholakia, K. (2001). Optical micromanipulation using a Bessel light beam. Opt. Comm. 197, 239–245. Artoni, M., La Rocca, G. C., Cataliotti F. S., and Bassani, F. (2001). Highly anomalous group velocity of light in ultracold Rubidium gases. Phys. Rev. A 63, no. 023805.

Localized Waves: A Review

345

Ashkin, A., Dziedzic, J. M., Bjorkholm, J. E., and Chu, S. (1986). Observation of a single-beam gradient force optical trap for dielectric particles. Opt. Lett. 11, 288–290. Balcou, Ph., and Dutriaux, L. (1997). Dual optical tunneling times in frustrated total internal reﬂection, Phys. Rev. Lett. 78, 851–854. Baldo Ceolin, M. (1993). “Review of neutrino physics.” Invited lecture at the XXIII International Symposium on Multiparticle Dynamics (Aspen, CO; September 1993). Barbero, A. P. L., Hernández-Figueroa, H. E., and Recami, E. (2000). On the propagation speed of evanescent modes. Phys. Rev. E 62, 8628–8635. Barut, A. O., and Bracken, A. J. (1992). Particle-like conﬁgurations of the electromagnetic ﬁeld: an extension of de Broglie’s ideas. Found. Phys. 22, 1267–1285. Barut, A. O., and Chandola, H. C. (1993). Localized tachyonic wavelet solutions of the wave equation. Phys. Lett. A 180, 5–8. Barut, A. O., and Grant, A. (1990). Quantum particle-like conﬁgurations of the electromagnetic ﬁeld. Found. Phys. Lett. 3, 303–310. Barut, A. O., Maccarrone, G. D., and Recami, E. (1982). On the shape of tachyons. Nuovo Cimento. A, 71, 509–533. Bateman, H. (1915). “Electrical and Optical Wave Motion.” Cambridge University Press, Cambridge. Begley S. (1995). Faster than what? Newsweek, 26, p. 44. Besieris, I. M., Shaarawi, A. M., and Ziolkowski, R. W. (1989). A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation. J. Math. Phys. 30, 1254–1269. Besieris, I., Abdel-Rahman, M., Shaarawi, A., and Chatzipetros, A. (1998). Two fundamental representations of localized pulse solutions to the scalar wave equation. Prog. Electromagn. Res. 19, 1–48. Besieris, I., and Shaarawi, A. (2002). Three classes of Courant-Hilbert’s progressing solutions to the scalar wave equation. J. Electr. Waves Appl. 16(8), 1047–1060. Besieris, I., and Shaarawi, A. (2004). Paraxial localized waves in free space. Opt. Express 12, 3848–3864. Bilaniuk, O. M. P., Deshpande, V. K., and Sudarshan, E. C. G. (1962). “Meta” relativity. Am. J. Phys. 30, 718–723. Bolda, E. L., Chiao, R. Y., and Garrison, J. C. (1993). Two theorems for the group velocity in dispersive media. Phys. Rev. A 48, 3890–3894; Bouchal, Z., and Olivik, M. (1995). Non-diffractive vector Bessel beams. J. Mod. Opt. 42, 1555–1566. Bouchal, Z., Wagner, J., and Chlup, M. (1998). Self-reconstruction of a distorted nondiffracting beam. Opt. Comm. 151, 207–211. Bouchal, Z. (2002). Controlled spatial shaping of nondiffracting patterns and arrays. Opt. Lett. 27, 1376–1378. Born, M., and Wolf, E. (1998). “Principles of Optics—Electromagnetic Theory of Propagation, Interference and Diffraction of Light,” 6th edition. Cambridge University Press; Cambridge. Brittingham, J. N. (1983). Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode. J. Appl. Phys. 54, 1179–1189. Brodowsky, H. M., Heitmann, W., and Nimtz, G. (1994). Photonic tunneling times. J. Phys. I 4, 565–570. Brodowsky, H. M., Heitmann W., and Nimtz, G. (1996). Comparison of experimental microwave tunneling data with calculations based on Maxwell’s equations. Phys. Lett. A 222, 125–129. Brown, J. (1995). Faster than the speed of light. New Scientist, 146, pp. 26–30. See also R. Y. Chiao, P. G. Kwiat, and A. M. Steinberg: Faster than light?, Scientiﬁc American, Aug. 1993, already quoted].

346

Erasmo Recami and Michel Zamboni-Rached

Caldirola, P., Maccarrone, G. D., and Recami, E. (1980). Second contribution on solving the ‘Imaginary Quantities’ problem in Superluminal Lorentz Transformations. Lett. Nuovo Cim. 29, 241–250. Cavaliere, A., Morrison, P., and Sartori, L. (1971). Rapidly changing radio images. Science 173, 525–528. Chiao, R. Y., Kwiat, P. G., and Steinberg, A. M. (1991). Analogies between electron and photon tunneling. Phys. B 175, 257–262. Chiao, R. Y., Kwiat, P. G., and Steinberg, A. M. (1993) Faster than light?, Scientiﬁc American 269(2), 52–60. Chiao, R. Y., Kozhekin, A. E., and Kurizki, G. (1996). Tachyon-like excitations in inverted two-level media. Phys. Rev. Lett. 77, 1254–1257. Chiao, R. Y., and Steinberg, A. M. (1997). Tunneling times and superluminality, in “Progress in Optics”, vol. 37 (E. Wolf, ed.), pp. 347–406. Elsevier, Amsterdam. Chu, S., and Wong, W. (1982). Linear pulse propagation in an absorbing medium. Phys. Rev. Lett. 48, 738–741. Conti, C., Trillo, S., Di Trapani, P., Valiulis, G., Piskarskas, A., Jedrkiewicz, O., and Trull, J. (2003). Nonlinear electromagnetic X-waves. Phys. Rev. Lett. 90, no. 170406. Courant, R., and Hilbert, D. (1966). “Methods of Mathematical Physics,” vol. 2, p. 760. J. Wiley, New York. Curtis, J. E., Koss, B. A., and Grier, D. G. (2002). Dynamic holographic optical tweezers. Opt. Comm. 207, 169–175. Dartora, C. A., Zamboni-Rached, M., Nobrega, K. Z., Recami, E., and Hernández-Figueroa, H. E. (2003). General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams. Opt. Comm. 222, 75–80. Dartora, C. A., Nóbrega, K. Z., Dartora, A., Viana, G. A., and Filho, H. T. S. (2006). A general theory for the frozen waves and their realizations through ﬁnite apertures. Opt. Commun. 265, 481–487; Study of frozen waves’ theory through continuous superpositions of Bessel beams. Opt. Laser Technol. 39, 1370–1373. Donnelly, R., and Ziolkowski, R. W. (1993). Designing localized waves. Proc. Roy. Soc. Lond. A, 440, 541–565. Durnin, J. (1987). Exact solutions for nondiffracting beams: I. The scalar theory. J. Opt. Soc. Am. A4, 651–654. Durnin, J., Miceli, J. J., and Eberly, J. H. (1987a). Diffraction-free beams. Phys. Rev. Lett. 58, 1499–1501. Durnin, J., Miceli, J. J., and Eberly, J. H. (1987b). Comparison of Bessel and Gaussian beams. Opt. Lett. 13, 79–80. Enders, A., and Nimtz, G. (1993). Photonic tunneling experiments, Phys. Rev. B 47, 9605–9609. Erdélyi, M., Horvath, Z. L., Szabo, G., Bor, Xs., Tittell, F. K., Cavallaro, J. R., and Smayling, M. C. (1997). Generation of diffraction-free beams for applications in optical microlithography. J. Vac. Sci. Technol. B 15, 287–292. Esposito, S. (1997). Classical solutions of Maxwell’s equations with group-velocity different from c, and the photon tunneling effect. Phys. Lett. A 225, 203–209. Esposito, S. (2003). Multibarrier tunneling, Phys. Rev. E 67, no. 016609[6 pages]. Esposito, S., Majorana, E. Jr., van der Merwe, A., and Recami, E., eds. (2003). ”Ettore Majorana—Notes on Theoretical Physics.” Kluwer, New York. Fan, J., Parra, E., and Milchberg, H. M. (2000). Resonant self-trapping and absorption of intense Bessel beams. Phys. Rev. Lett. 84, 3085–3088. Folman, R., and Recami, E., (1995). On the phenomenology of tachyon radiation. Found. Phys. Lett. 8, 127–134. Friberg, A. T., Fagerholm, J., and Salomaa, M. M. (1997). Space-frequency analysis of nondiffracting pulses. Opt. Comm. 136, 207–212.

Localized Waves: A Review

347

Garavaglia, R. (1998). Thesis work (Dip. Sc. Informazione, Università statale di Milano; Milan, G. Degli Antoni and E. Recami supervisors). Garcés-Chavez, V., McGloin, D., Melville, H., Sibbett, W., and Dholakia, K. (2002). Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam. Nature, 419, 145–147. Garrett, C. G. B., and McCumber, D. E. (1970). Propagation of a Gaussian light pulse through an anomalous dispersion medium. Phys. Rev. A 1, 305–313. Giani, S. (1999). Experimental evidence of superluminal velocities in astrophysics and proposed experiments. In “Space Technology and Applications International Forum 1999” (M. S. El-Genk, ed.), pp. 881–888. American Institute of Physics, Melville, NY. Giannetto, E., Maccarrone, G. D., Mignani, R., and Recami, E. (1986). Are muon neutrinos faster-than-light particles? Phys. Lett. B.178, 115–120. Gisler, G. (1994). Astrophysics: a galactic speed record. Nature 371, 18. Goodman, J. W. (1996). “Introduction to Fourier Optics,” 2nd edition. McGraw-Hill, New York. Gradshteyn, I. S., and Ryzhik, I. M. (1965). “Integrals, Series and Products,” 4th edition. Academic Press, New York. Grunwald, R., Kebbel, V., Griebner, U., Neumann, U., Kummrow, A., Rini, M., Nibbering, A. T. J., Piché, M., Rousseau, G., and Fortin, M. (2003). Generation and characterization of spatially and temporally localized few-cycle optical wavepackets, Phys. Rev. A 67, 063820. Grunwald, R., Griebner, U., Neumann, U., and Kebbel, V. (2004). Self-reconstruction of ultrashort-pulse Bessel-like X-waves. CLEO/QELS 2004, San Francisco, May 16–21, 2004, Conference Digest (CD-ROM), no. CMQ7. Grunwald, R., Neumann, U., Langer, S., Stibenz, G., Steinmeyer, G., Kebbel, V., Piché, M., and Néron, J.-L. (2005). Ultrabroadband spectral transfer in extended focal zones: truncated few-cycle Bessel-Gauss beams. Conference Digest (CD-ROM) (paper CTuAA6) CLEO 2005, Baltimore, May 23–27. Hartman, T. E. (1962). Tunneling of a wave packet. J. Appl. Phys. 33, 3427–3433. Herman, R. M., and Wiggins, T. A. (1991). Production and uses of diffractionless beams. J. Opt. Soc. Am. A8, 932–942. Hernández-Figueroa, H. E., Zamboni-Rached, M., and Recami, E., eds. (2008) “Localized Waves” (387 pages). J. Wiley, New York. Ito, T., and Okazaki, S. (2000). Pushing the limits of lithography. Nature, 406, 1027–1031. Japha, Y., and Kurizki, G. (1996). Superluminal delays of coherent pulses in nondissipative media: A universal mechanism. Phys. Rev. A 53, 586–590. Kurizki, G., Kozhekin, A. E., and Kofman, A. G. (1998). Tachyons and information transfer in quantized parametric ampliﬁers. Europhys. Lett. 42, 499–504. Kurizki, G., Kozhekin, A. E., Kofman, A. G., and Blaauboer, M. (1999). Optical tachyons in parametric ampliﬁers: How fast can quantum information travel? Opt. Spektrosc. 87, 505–512. Kukhlevsky, S. V., and Mechler, M. (2004). Diffraction-free subwavelength-beam optics at nanometer scale. Opt. Comm. 231, 35–43. Landauer, R. (1993). Light faster than light? Nature, 365. Laude, V., and Tournois, P. (1999). Superluminal asymptotic tunneling times through onedimensional photonic bandgaps in quarter-wave-stack dielectric mirrors, J. Opt. Soc. Am. B16, 194–198. Longhi, S., Laporta, P., Belmonte, M., and Recami, E. (2002). Measurement of superluminal optical tunneling times in double-barrier photonic bandgaps. Phys. Rev. E, 65, 046610. Longhi, S. (2003). Spatial-temporal Gauss-Laguerre waves in dispersive media. Phys. Rev. E 68, no. 066612 (6 pages). Longhi, S. (2004). Localized subluminal envelope pulses in dispersive media. Opt. Lett. 29, 147–149.

348

Erasmo Recami and Michel Zamboni-Rached

Lu, J.-y., and Greenleaf, J. F. (1992a). Nondiffracting X-waves—exact solutions to free-space wave equation and their ﬁnite aperture realizations. IEEE Trans. Ultrason. Ferr. 39, 19–31. Lu, J.-y., and Greenleaf, J. F. (1992b). Experimental veriﬁcation of nondiffracting X-waves. IEEE Trans. Ultrason. Ferr. 39, 441–446. Lu, J.-y., and Greenleaf, J. F. (1995). Comparison of side lobes of limited diffraction beams and localized waves. In “Acoustic Imaging,” vol.21, (J. P. Jones, ed.), pp. 145–152. Plenum; New York. Lu, J.-y., Greenleaf, J. F., and Recami, E. (October 1996). Limited diffraction solutions to Maxwell (and Schroedinger) equations [Los Alamos National Lab (LANL) Archives e-print physics/9610012], report INFN/FM–96/01 (INFN; Frascati). Lu, J.-y., and Shiping, H. (1999). Optical X wave communications. Opt. Comm. 161, 187–192. Lu, J.-y., Zou, H.-H., and Greenleaf, J. F. (1993). Producing deep depth of ﬁeld and depth independent resolution in NDE with limited diffraction beams. Ultrasonic Imaging 15, 134–149. Lu, J.-y., Zou, H.-H., and Greenleaf, J. F. (1994). Biomedical ultrasound beam forming. Ultrasound Med. Biol. 20, 403–428. Maccarrone, G. D., and Recami, E. (1980). Two-body interactions through tachyon exchange, Nuovo Cimento A 57, 85–101. Maccarrone, G. D., Pavsic, M., and Recami, E. (1983). Formal and physical properties of the Generalized (sub- and Super-luminal) Lorentz Transformations. Nuovo Cim. B 73, 91–111. MacColl, L. A. (1932). Note on the transmission and reﬂection of wave packets by potential barriers. Phys. Rev. 40, 621–626. MacDonald, R. P., Chrostowski, J., Boothroyd, S. A., and Syrett, B. A. (1983). Holographic formation of a Diode Laser non-diffracting beam. Appl. Optics 32, 6470–6474. MacDonald, M. P., Paterson, L., Volke-Supelveda, K., Arlt, J., Sibbett, W., and Dholakia, K. (2002). Creation and manipulation of three-dimensional optically trapped structures. Science 296, 1101–1103. Macke, B., Queva, J. L., Rohart, F., and Segard, B. (1987). Linear pulse propagation in a resonant medium: The adiabatic limit. J. de Physique 48, 797–808. Mackinnon, L. (1978). A nondispersive de Broglie wave packet. Found. Phys. 8, 157–158. Martin, Th., and Landauer, R. (1992). Time delay of evanescent electromagnetic waves and the analogy to particle tunneling. Phys. Rev. A 45, 774–776. McGloin, D., Garcés-Chavez, V., and Dholakia, K. (2003). Interfering Bessel beams for optical micromanipulation. Opt. Lett. 28, 657–659. Mignani, R., and Recami, E. (1973). Generalized Lorentz transformations and superluminal objects in four dimensions. Nuovo Cimento A 14, 169–189; A16, E206. Mignani, R., and Recami, E. (1974). The possibility of superluminal sources and their Doppler effect. Gen. Relat Grav. 5, 615–620. Milonni, P. W. (2002). Controlling the speed of light pulses. J. Phys. B 35, pp. R31–R56. Mirabel, I. F., and Rodriguez, L. F. (1994). A superluminal source in the galaxy. Nature 371, 46–48, with an editorial comment: Mitchell, M. W., and Chiao, R. Y., (1997). Negative group delay and ’fronts’ in a causal system: An experiment with very low frequency bandpass ampliﬁers. Phys. Lett. A 230, 133–138. Mugnai, D., Ranfagni, A., Ruggeri, R., Agresti, A., and Recami, E. (1995). Phys. Lett. A 209, 227–234. Mugnai, D., Ranfagni, A., and Ruggeri, R. (2000). Observation of superluminal behaviors in wave propagation. Phys. Rev. Lett. 84, 4830–4833. Newell, A. C., and Molone, J. V. (1992). “Nonlinear Optics.” Addison & Wesley; Redwood City, CA. Nimtz, G., and Enders, A. (1992). On superluminal barrier traversal. J. Phys. I 2, 1693–1698. Nimtz, G., and Enders, A. (1993). Zero-time tunneling of evanscent mode packets. J. Phys. I., 3, 1089–1092.

Localized Waves: A Review

349

Nimtz, G., and Enders, A. (1993). Evanescent-mode propagation and quantum tunneling. Phys. Rev. E. 48, 632–634. Nimtz, G., Enders, A., and Spieker, H. (1993). Photonic tunneling experiments: Superluminal tunneling, in “Wave and Particle in Light and Matter” (A.van der Merwe and A. Garuccio, eds.), Plenum Press, New York, 1993. Nimtz, G., Enders, A., and Spieker, H. (1994). Photonic tunneling experiments, J. de Physique-I (France) 4, 565–570. Nimtz, G., Spieker, H., and Brodowsky, H. M. (1994). Tunneling with dissipation. J. de PhysiqueI. 4, 1379–1382. Nimtz, G., and Heitmann, W. (1997). Superluminal photonic tunneling and quantum electronics. Prog. Quant. Electron, 21, 81–108. Nimtz, G. (1999). Evanescent modes are not necessarily Einstein causal. Eur. Phys. J. B 7, 523–525. Nimtz, G., and Haibel, A. (2002). Basic os superluminal signals, Ann. der Physik (Leipzig) 11, 163–171. Okazaki, S. (1991). Resolution limits of optical lithography. J. Vac. Sci. Technol. B 9, 2829–2833. Overfelt, P. L., and Kenney, C. S., (1991). Comparison of the propagation characteristics of Bessel, Bessel-Gauss, and gaussian beams diffracted by a circular aperture. J. Opt. Soc. Am. A8, 732–745. Olkhovsky, V. S., and Recami, E. (1992). Recent developments in the time analysis of tunneling processes. Phys. Rep. 214, 339–357. Olkhovsky, V. S., Recami E., and Jakiel, J. (2001). Uniﬁed time analysis of photon and nonrelativistic particle tunneling (report INFN/FM-01/01) Infn; Frascati, Apr.2001), e-print quant-ph/0102007. Olkhovsky, V. S., Recami, E., and Jakiel, J. (2004). Uniﬁed time analysis of photon and nonrelativistic particle tunneling. Phys. Rep. 398, 133–178, 152. Olkhovsky, V. S., Recami, E., Raciti, F., and Zaichenko, A. K. (1995). More about the tunneling times, the dwell time and the Hartman effect. J. de Phys. I, 5, 1351–1365, 1361. Olkhovsky, V. S., Recami, E., and Salesi, G. (2002). Tunneling through two successive barriers and the Hartman (Superluminal) effect, Europhys. Lett. 57, 879–884; Olkhovsky, V. S., Recami E., and Zaichenko, A. K. (2005). Resonant and non-resonant tunneling through a double barrier. Europhys. Lett. 70, 712–718. Olkhovsky, V. S., Recami, E., Raciti, F., and Zaichenko, A. K. (1995) More about tunneling times, the dwell time and the Hartman effect. J. de Physique-I 5, 1361–1365. Otten, E. W. (1995). Squeezing the neutrino mass with new instruments. Nucl. Phys. News 5(1), 11–15. Pavsic M., and Recami, E. (1982). Charge conjugation and internal space-time symmetries. Lett. Nuovo Cimento 34, 357–362. Petrillo, V., and Refaldi, L. (2000). A time asymptotic expression for the wave function emerging from a quantum-mechanical barrier. Opt. Comm. 186, 35–41. Petrillo, V., and Refaldi, L. (2003). Relativistic analysis of a wave packet interacting with a quantum-mechanical barrier. Phys. Rev. A 67, no. 012110[7 pages]. Porras, M. A., Borghi, R., and Santarsiero, M. (2003). Suppression of dispersion broadening of light pulses with Bessel-Gauss beams. Opt. Comm. 206, 235–241. Porras, M. A., Valiulis, G., and Di Trapani, P. (2003). Uniﬁed description of Bessel X waves with cone dispersion and tilted pulses. Phys. Rev. E 68, no. 016613. Porras, M. A., Trillo, S., Conti, C., and Di Trapani, P. (2003). Paraxial envelope X waves. Opt. Lett. 28, 1090–1092. Ranfagni, A., Mugnai, D., Fabeni P., and Pazzi, G. P. (1991). Delay-time measurements in narrowed wave-guides as a test of tunneling. Appl. Phys. Lett. 58, 774–776. Ranfagni, A., Fabeni, P., Pazzi, G. P., and Mugnai, D. (1993). Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time, Phys. Rev. E, 48, 1453–1460.

350

Erasmo Recami and Michel Zamboni-Rached

Recami, E., ed. (1978). “Tachyons, Monopoles, and Related Topics” (295 pages). NorthHolland, Amsterdam. Recami, E. (1985). The Tolman-Regge antitelephone paradox: its solution by tachyon mechanics. Lett. Nuovo Cimento 44, 587–593; Recami, E. (1986). Classical tachyons and possible applications. Riv. Nuovo Cim. 9(6), 1–178. Recami, E. (1986a). Classical tachyons and possible application (see pp. 80–81 and Fig. 27). Riv. Nuovo Cim. 9(6). Recami, E. (1986b). Classical tachyons and possible applications. Riv. Nuovo Cim. 9(6), p. 158 and pp. 116–117. See also Mugnai et al. (1995). Superluminal processes and signal velocity in tunneling simulation. Phys. Lett. A 209, 227–234. Recami, E. (1987). Tachyon mechanics and causality; a systematic thorough analysis of the tachyon causal paradoxes. Found. Phys. 17, 239–296. Recami, E. (1998). On localized X-shaped Superluminal solutions to Maxwell equations. Physica A, 252, 586–610. Recami, E. (2001). Superluminal motions? A bird’s-eye view of the experimental status-ofthe-art. Found. Phys. 31, 1119–1135. Recami, E. (2004). Superluminal tunneling through successive barriers: does quantum mechanics predict inﬁnite group velocities?, J. Mod. Optics 51, 913–923. Recami, E., Fontana, F., and Garavaglia, R. (2000). Superluminal motions and special relativity: a discussion of recent experiments. Int. J. Mod. Phys. A, 15, 2793–2812, 2807. Recami, E., and Maccarrone, G. D. (1980). Solving the ‘Imaginary Quantities’ problem in Superluminal Lorentz Transformations. Lett. Nuovo Cim. 28, 151–157. Recami, E., and Maccarrone, G. D. (1983). Are classical tachyons slower-than-light quantum particles? Lett. Nuovo Cim. 37, 345–352. Recami, E., and Mignani, R. (1974). Classical theory of tachyons (special relativity extended to superluminal frames and objects). Rivista Nuovo Cim. 4, 209–290, E398. Recami, E., and Mignani, R. (1976). Magnetic monopoles and tachyons in special relativity. Phys. Lett. B 62, 41–43. Recami, E., Zamboni-Rached, M., and Dartora, C. A. (2004). The X-shaped, localized ﬁeld generated by a superluminal electric charge. Phys. Rev. E, 69, no. 027602 (4 pages). Recami, E., Zamboni-Rached, M., and Hernández-Figueroa, H. E. (2008). Localized waves: a historical and scientiﬁc introduction. In “Localized Waves” (H. E. Hernández-Figueroa, M. Zamboni-Rached and E. Recami, eds.), pp. 1–42. J. Wiley, New York. Recami, E., Castellino, A., Maccarrone G. D., and Rodonò, M. (1986). Considerations about the apparent superluminal expansions observed in astrophysics. Nuovo Cimento B 93, 119–144. Recami, E., Zamboni-Rached, M., Nobrega, K. Z., Dartora, C. A., and Hernández-Figueroa, H. E. (2003). On the localized superluminal solutions to the Maxwell equations. IEEE J. Sel. Top. Quant. Electron. 9(1), 59–73. Recami, E., and Rodrigues, W. A. (1985). A model theory for tachyons in two dimensions. In “Gravitational Radiation and Relativity” (vol. 3 of the Proceedings of the Sir Arthur Eddington Centenary Symposium; J. Weber and T. M. Karade, eds.), pp. 151–203. World Scientiﬁc, Singapore. Recami, E., Zamboni-Rached, M., and Hernández-Figueroa, H. E., eds. (2008). Localized waves: a historical and scientiﬁc introduction (pp. 1–42). In “Localized Waves.” J. Wiley, New York. Recami, E., Zamboni-Rached, M., and Dartora, C. A. (2004). The X-shaped, localized ﬁeld generated by a Superluminal electric charge. Phys. Rev. E 69, no. 027602 (4 pages). Rees, M. J. (1966). The appearance of relativistically expanding radio sources. Nature 211, 468–470. Refaldi, L. (2000). Analytic and numerical analysis of tunneling through a potential barrier. MSc. thesis, Physics Department, Milan University.

Localized Waves: A Review

351

Rhodes, D. P., Lancaster, G. P. T., Livesey, J., McGloin, D., Arlt, J., and Dholakia, K. (2002). Guiding a cold atomic beam along a co-propagating and oblique hollow light guide. Opt. Comm. 214, 247–254. Rodrigues, W. A., Vaz, J., and Recami, E. (1994). Free Maxwell equations, Dirac equation, and non-dispersive de Broglie wave packets. In “Courants, Amers, Écueils en Microphysique” (G. Lochak and P. Lochak, eds.), pp. 379–392. Foundation Louis de Broglie, Paris. Saari, P., and Reivelt, K. (1997). Evidence of X-shaped propagation-invariant localized light waves. Phys. Rev. Lett. 79, 4135–4138. Saari, P., and Reivelt, K. (2004). Generation and classiﬁcation of localized waves by Lorentz transformations in Fourier space. Phys. Rev. E 69, no. 036612. Salo, J., Fagerholm, J., Friberg, A. T., and Salomaa, M. M. (1990). Nondiffracting bulk-acoustic X-waves in crystals. Phys. Rev. Lett. 83, 1171–1174. Salo, J., Fagerholm, J., Friberg, A. T., and Salomaa, M. M. (1999). Nondiffracting bulk-acoustic X waves in crystals. Phys. Rev. Lett. 83, 1171–1174. Salo, J., Friberg, A. T., and Salomaa, M. M. (2001). Orthogonal X-waves. J. Phys. A: Math. Gen. 34, 9319–9327. Salo, J., and Salomaa, M. M. (2001). Subsonic nondiffracting waves. Acoust. Res. Lett. Online 2(1), 31–36. See also the large lists of references contained in the book Localized waves, edited by H. E. Hernandez-Figueroa, M. Zamboni-Rached and Erasmo Recami (J. Wiley; New York, Jan. 2008). Segard, B., and Macke, B. (1985). Observation of negative velocity pulse propagation. Phys. Lett. A 109, 213–216. Selleri, F. (2000). Lecture presented at the 2nd Cracow-Clausthal workshop on “Extensions of Quantum Theory,” Krakow (unpublished); and in Selleri, F. (2003). “Lezioni di relatività da Einstein all’etere di Lorentz.” Progedit pub.; Bari. Sezginer, A. (1985). A general formulation of focus wave modes. J. Appl. Phys. 57, 678–683. Shaarawi, A. M., Besieris, I. M., and Ziolkowski, R. W. (1990). Anovel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations. J. Math. Phys. 31, 2511–2519. Shaarawi, A. M., and Besieris, I. M. (2000). Superluminal tunneling of an electromagnetic X wave through a planar slab. Phys. Rev. E 62, 7415–7421. Shaarawi, A. M., and Besieris, I. M. (2000a). On the Superluminal propagation of X-shaped localized waves J. Phys. A: Math. Gen 33, 7227–7254. Shaarawi, A. M., and Besieris, I. M. (2000b). Relativistic causality and superluminal signalling using X-shaped localized waves. J. Phys. A 33, 7255–7263. Shaarawi, A. M., Besieris, I. M., and Said, T. M. (2003). Temporal focusing by use of composite X-waves. J. Opt. Soc. Am. A 20, 1658–1665. Sõnajalg, H., and Saari, P. (1996). Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators. Opt. Lett. 21, 1162–1164. Sõnajalg, H., Rätsep, M., and Saari, P. (1997). Demonstration of the Besswel-X pulse propagation with strong lateral and longitudinal localization in a dispersive medium, Opt. Lett. 22, 310–312. Spielmann, Ch., Szipocs, R., Stingl, A., and Krausz, F. (1994). “Tunneling of optical pulses through Photonic Band Gaps”, Phys. Rev. Lett. 73, 2308–2311. Steinberg, A. M., Kwiat, P. G., and Chiao, R. Y. (1993). Measurement of the single-photon tunneling time, Phys. Rev. Lett. 71, 708–711. Stratton, J. A. (1941). “Electromagnetic Theory” (p. 356). McGraw-Hill, New York. Tingay S. J., Jauncey, D. L., Preston, R. A., Reynolds, J. E., Meier, D. L., Murphy, D, W., Tzioumis, A. K., Mckay, D. J., Kesteven, M. J., Lovell, J. E. J., Campbell-Wilson, D., Ellingsen, S. P., Gough, R., Hunstead, R. W., Jonos, D. L., McCulloch, P. M., Migenes, V., Quick, J., Sinclair,

352

Erasmo Recami and Michel Zamboni-Rached

M. W., and Smits, D. (1995). Relativistic motion in a nearby bright X-ray source. Nature 374, 141–143. Valtua, H., Reivelt, K., and Saari, P. (2007). Methods for generating wideband localized waves of Superluminal group-velocity. Opt. Comm. 278, (2007) 1–7. Vasara, A., Turunen, J., and Friberg, A. T. (1989). Realization of general nondiffracting beams with computer-generated holograms. J. Opt. Soc. Am. A 6, 1748–1754. Wang, L. J., Kuzmich, A., and Dogariu, A. (2000). Gain-assisted superluminal light propagation. Nature, 406, 277–279. Willebrand, H. A., and Ghuman, B. S. (2001). Fiber optics without ﬁber. IEEE Spectrum 38(8), 40–45. Zamboni-Rached, M. (1999). Localized solutions: structure and applications. M.Sc. thesis, Universidade Estadual de Campinas. Zamboni-Rached, M. (1998). Personal communications. Zamboni-Rached, M. (2004). “Localized waves in diffractive/dispersive media.” PhD thesis, 2004. Zamboni-Rached, M. (2004a). Localized waves in diffractive/dispersive media. PhD Thesis, Universidade Estadual de Campinas, DMO/FEEC. (http://libdigi.unicamp.br/ document/?code=vtls000337794). Zamboni-Rached, M. (2004b). Stationary optical wave ﬁelds with arbitrary longitudinal shape by superposing equal frequency Bessel beams: frozen waves. Opt. Express 12, 4001–4006. Zamboni-Rached, M. (2006). Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by ﬁnite apertures. J. Opt. Soc. Am. A23, 2166–2176. Zamboni-Rached, M. (2006a). Diffraction-attenuation resistant beams in absorbing media. Opt. Express 14, 1804–1809. Zamboni-Rached, M., and Hernàndez-Figueroa, H. E. (2000). A rigorous analysis of localized wave propagation in optical ﬁbers. Opt. Comm. 191, 49–54. Zamboni-Rached, M., Hernández-Figueroa, H. E., and Recami, E. (2004). Chirped optical X-shaped pulses in material media. J. Opt. Soc. Am. A 21, 2455–2463. Zamboni-Rached, M., Fontana, F., and Recami, E. (2003). Superluminal localized solutions to Maxwell equations propagating along a waveguide: the ﬁnite-energy case. Phys. Rev.E 67, no. 036620 (7 pages). Zamboni-Rached, M., Nobrega, K. Z., Recami, E., and Hernández-Figueroa, H. E. (2002). Superluminal X-shaped beams propagating without distortion along a coaxial guide. Phys. Rev. E 66, no. 036620 (10 pages). Zamboni-Rached, M., Nobrega, K. Z., Hernández-Figueroa, H. E., and Recami, E. (2003). Localized Superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth. Opt. Comm. 226, 15–23. Zamboni-Rached, M., and Recami, E. (2008a). Subluminal wave bullets: exact subluminal localized solutions to the wave equations. Phys. Rev. A77, no. 033824. Zamboni-Rached, M., and Recami, E. (2008b). Cherenkov radiation has nothing to do with X-shaped Localized Waves (Comments on “Cherenkov-Vavilov formulation of X-waves”), e-print arXiv:0807.4301v2 [physics.Optics] 19 Aug 2008. Zamboni-Rached, M., Recami, E., and Fontana, F. (2001). Localized superluminal solutions to Maxwell equations propagating along a normal-sized waveguide. Phys. Rev. E 64, no. 066603 (6 pages). Zamboni-Rached, M., Recami, E., and Hernández-Figueroa, H. E. (2002). New localized Superluminal solutions to the wave equations with ﬁnite total energies and arbitrary frequencies. Eur. Phys. J. D 21, 217–228. Zamboni-Rached, M., Recami, E., and Hernández-Figueroa, H. E. (2005). Theory of frozen waves: modeling the shape of stationary wave ﬁelds. J. Opt. Soc. Am. A, 22, 2465–2475. Zamboni-Rached, M., Recami, E., Hernández-Figueroa, H. E., Dartora, C. A., and Nóbrega, K. Z. (2005). Method and apparatus for producing stationary intense waveﬁelds

Localized Waves: A Review

353

of arbitrary shape. Application/patent nos. 05743093.6-1240/EP2005052352, European Patent Ofﬁce. Zamboni-Rached, M., Recami, E., and Hernández-Figueroa, H. E. (2008). Structure of the nondiffracting waves, and some interesting applications In “Localized Waves” (M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa ed.) (pp. 43–78). J. Wiley, New York. Zamboni-Rached, M., Shaarawi, A., and Recami, E. (2004). Focused X-shaped pulses. J. Opt. Soc. Am. A 21, 1564–1574. Zensus, J. A., and Pearson, T. J., eds. (1987). “Superluminal Radio Sources.” Cambridge University Press, Cambridge, UK. Ziolkowski, R. W. (1989). Localized transmission of electromagnetic energy. Phys. Rev. A 39, 2005–2033. Ziolkowski, R. W. (1991). Localized wave physics and engineering. Phys. Rev. A 44, 3960–3984. Ziolkowski, R. W., Besieris, I. M., and Shaarawi, A. M. (1993). Aperture realizations of exact solutions to homogeneous wave equations. J. Opt. Soc. Am. A 10, 75–87. Ziolkowski, R. W., Besieris, I. M., and Shaarawi, A. M. (2000). Aperture realizations of exact solutions to homogeneous-wave equations. J. Opt. Soc. Am. A10, 75–87. Ziolkowski, R. W. (2001). Superluminal transmission of information through an electromagnetic metamaterial, Phys. Rev. E 63, no. 046604.

CONTENTS OF VOLUMES 151–155

VOLUME 151* C. Bontus and T. Köhler, Reconstruction algorithms for computed tomography L. Busin, N. Vandenbroucke, and L. Macaire, Color spaces and image segmentation G. R. Easley and F. Colonna, Generalized discrete Radon transforms and applications to image processing T. Radlicka, Lie agebraic methods in charged particle optics ˇ V. Randle, Recent developments in electron backscatter diffraction

VOLUME 152 N. S. T. Hirata, Stack filters: From definition to design algorithms S. A. Khan, The Foldy–Wouthuysen transformation technique in optics S. Morfu, P. Marquié, B. Nofiélé, and D. Ginhac, Nonlinear systems for image processing T. Nitta, Complex-valued neural network and complex-valued backpropagation learning algorithm J. Bobin, J.-L. Starck, Y. Moudden, and M. J. Fadili, Blind source separation: The sparsity revolution R. L. Withers, “Disorder”: Structured diffuse scattering and local crystal chemistry

VOLUME 153 Aberration-corrected Electron Microscopy H. Rose, History of direct aberration correction M. Haider, H. Müller, and S. Uhlemann, Present and future hexapole aberration correctors for high-resolution electron microscopy O. L. Krivanek, N. Dellby, R. J. Kyse, M. F. Murfitt, C. S. Own, and Z. S. Szilagyi, Advances in aberration-corrected scanning transmission electron microscopy and electron energy-loss spectroscopy

*Lists of the contents of volumes 100–149 are to be found in volume 150; the entire series can be searched on ScienceDirect.com

355

356

Contents of Volumes 151–155

P. E. Batson, First results using the Nion third-order scanning transmission electron microscope corrector A. L. Bleloch, Scanning transmission electron microscopy and electron energy loss spectroscopy: Mapping materials atom by atom F. Houdellier, M. Hÿtch, F. Hüe, and E. Snoeck, Aberration correction with the SACTEM-Toulouse: From imaging to diffraction B. Kabius and H. Rose, Novel aberration correction concepts A. I. Kirkland, P. D Nellist, L.-Y. Chang, and S. J. Haigh, Aberration-corrected imaging in conventional transmission electron microscopy and scanning transmission electron microscopy S. J. Pennycook, M. F. Chisholm, A. R. Lupini, M. Varela, K. van Benthem, A. Y. Borisevich, M. P. Oxley, W. Luo, and S. T. Pantelides, Materials applications of aberration-corrected scanning transmission electron microscopy N. Tanaka, Spherical aberration-corrected transmission electron microscopy for nanomaterials K. Urban, L. Houben, C.-L. Jia, M. Lentzen, S.-B. Mi, A. Thust, and K. Tillmann, Atomic-resolution aberration-corrected transmission electron microscopy Y. Zhu and J. Wall, Aberration-corrected electron microscopes at Brookhaven National Laboratory

VOLUME 154 H. F. Harmuth and B. Meffert, Dirac’s difference equation and the physics of finite differences

VOLUME 155 D. Greenfield and M. Monastyrskiy, Computational charged particle optics

INDEX

A Absorbing media with growing longitudinal ﬁeld intensity, 315 nondiffracting beams in, 310 stationary wave ﬁelds in, 310 undistorted beams in, 314 Azimuth angle, 41–44, 47–48

B BDA. See Beam-deﬁning aperture Beam-deﬁning aperture (BDA), 225–226 Bessel beams, 237–238, 246–248, 295, 302 comparison between Gaussian and, 248 experimental setup for generating, 239, 241 higher-order, 308, 340–342, 344 superposition of, 241, 281 zero-order, 302, 305, 308, 340 Bidirectional decomposition, 240 generalized, 278, 281–291 Bidirectional reﬂectance distribution function (BRDF), 9–11 Binocular stereo, 17–18. See also Photometric stereo BRDF. See Bidirectional reﬂectance distribution function Brightness measurements, 219–221

C Camera calibration techniques, 39 Carbon nanotube electron sources for electron microscopes, 222–229 FEM of apex of, 207–208 ﬁgure of merit, 221–222 under realistic conditions, 224 lifetime testing, 224–225 mounting procedure of, 206 noise level, 222–223 in SEM, 228–229 source module, 225–228 speciﬁcations of, 229–230 TEM investigations of, 208–213 Carbon nanotube emitter, 222 Carbon nanotubes advantage of, 208 emission instability, causes of, 216–217

emission stability measurements, 217–219 mounting, 205–208 cap closing, 206–207 cleaning, 206 procedure, 205–206 Carrier frequency, 296, 324, 341 Cast-shadow, 6 Cauchy–Kowalewskaia (CK) extension procedure, 64 Cauchy–Riemann operator, 62–63, 83 Cauchy transforms, 84 CFEG. See Cold ﬁeld-emission gun Chromaticity, 28 Classical Fourier transform and Clifford–Fourier transform, 140–143 operator exponential representation of, 113–114 Classical fractional Fourier transform, 87–90 Classical multidimensional Hermite ﬁlters, 151–153 Classical multidimensional Hermite transform, 148–151 Classical one-dimensional Gabor ﬁlters, 159–161 Clifford algebra, 57, 59–62, 96 Clifford analysis, 57. See also Clifford–Fourier transform central notion in, 62 generalization, 73 toolkit, 58–86 Clifford differential operators, 64 with polynomial coefﬁcients, CHM decomposition of, 107–113 Clifford ﬁlters, 143–166 Clifford–Fourier kernel, 58, 97, 132, 134 Clifford–Fourier theory, 58 Clifford–Fourier transform, 58, 95–96 of box function, 142–143 change of scale property, 118 and classical Fourier transform, 140–143 of 2D and 3D signals, 96 deﬁnition, 115–118 differentiation rule, 119 of Ebling and Scheuermann, 141 integral kernel of, 122–127 linearity property, 118

357

358

Index

Clifford–Fourier transform (continued) mixed product rule, 120 multiplication rule, 119 operational calculus, 118–122 properties, 118–120 two dimensional case. See Two dimensional Clifford–Fourier transform Clifford–Gabor ﬁlters, two-dimensional. See Two dimensional Clifford–Gabor ﬁlters Clifford–Heaviside functions, 85 Clifford–Hermite (CH)-anti-monogenic operators, 103, 106 Clifford–Hermite CWT, 156–158 Clifford–Hermite ﬁlters, 145–158 Clifford–Hermite monogenic decomposition, 106 of Clifford differential operators, 107–113 Clifford–Hermite monogenic operators, 98–100 deﬁnition and properties, 102–107 operators O1 and O2 , 100–102 Clifford–Hermite polynomials, 73, 151–153, 195 deﬁnition, 74 generalized, 73 differential equations, 77–79 Mehler formula for, 93–95 square integrable functions, 79 Clifford–Hermite transform, 154–156 Clifford numbers, 122–123 Clifford polynomial operators, 63, 98–99 Clifford vector, 178 Cold ﬁeld-emission gun (CFEG), 221, 224 Cold ﬁeld-emitter, 204, 216 Color bands, 26 Color photometric stereo, 26 Conﬂuent hypergeometric function, 75, 177 Continuous wavelet transform (CWT), 156 Clifford–Hermite, 156–158 Convolution theorems, 96, 129 CWT. See Continuous wavelet transform Cylindrical Fourier spectrum, 175, 187 calculations of, 195 of Gaussian, 176, 181, 186 with Clifford vector, 178 Cylindrical Fourier transform, 167 deﬁnition, 167 kernel of, 169 properties, 170–175

D de Broglie relation, 323 Dichromatic reﬂectance model, 15–17 Differential equations, 77–79 Diffraction length, 245 Dirac operator, 57–58, 62, 69, 97, 115

E Eigenfunctions of Fourier transform, 80–82 Electron emission process, 213–216 Electron microscopes for carbon nanotube electron sources, 222–229 Electron optical performance, ﬁgure of merit of, 216–222 Emission instability, causes of, 216–217 Emission stability measurements, 217–219 Error analysis in image acquisition stage, 39 photometric stereo, 38–45 Euler operator, 63–64 Evanescent waves, 266–273

F FEM of carbon nanotube electron sources, 207–208 Filters Clifford, 143–166 Clifford–Hermite, 145–158 Gabor. See Gabor ﬁlters Hermite. See Hermite ﬁlters linear shift-invariant, 159 two dimensional Clifford–Gabor, 162–166 Finite-energy luminal pulses, 240 Finite-energy nondiffracting pulses, 286–290 Focus wave mode (FWM) solution, 286 Forward Hermite transform, 147 Fourier analysis, 56 Fourier–Bessel (FB) transformation, 262, 279 Fourier image, 56 Fourier invariant operators, 116 Fourier kernel, 68, 96–97, 142 Fourier transform, 56. See also Clifford–Fourier transform; Fractional Fourier transform classical, 113–114, 140–143 cylindrical, 167, 169–175 eigenfunctions of, 80–82

Index

fractional, 86 Gaussian function, 68 inverse, 56 quaternionic, 95–96 and spherical harmonics, 68–73 Fowler–Nordheim model, 213–214 Fractional Fourier transform (FrFT), 86 calculation, 89 classical, 87–90 integral representation, 87 multidimensional, 87, 90 Frequency contents, 56 FrFT. See Fractional Fourier transform Frozen waves, 301–302 approach to, 332 Full-width half-maximum (FWHM), 325 Funk–Hecke theorem, 69, 192 application of, 70

G Gabor ﬁlters, 144 of Ebling and Scheuermann, 161–162 one-dimensional classical, 159–161 complex, 159–160 real, 160–161 quaternionic, 161 two-dimensional, types, 161–162 Gabor model, 144 Galactic microquasars, 265–266 Gamma function, 71 Gaussian beams, 237, 244–245 comparison between truncated Bessel and, 248 Gaussian derivative theory, 144 Gaussian function, 68 Gaussian pulses, 245, 296–297 Gaussian spectrum, 324–325 Generalized bidirectional decomposition, 278, 281–291 Generalized Clifford–Hermite CWT, 157 Generalized Clifford–Hermite transform, 154–156 Group velocity dispersion (GVD), 296

H Hardy spaces, 83 Hartman effect, 266 generalized, 271–272 Hermitean conjugation, 60 Hermite ﬁlters, 144 classical multidimensional, 151–153 Hermite model, 144

359

Hermite polynomials, 73, 88 Hermite transform, 145–151 classical multidimensional, 148–151 forward, 147 inverse, 147 one-dimensional, 145–148 Higher-order Bessel beams, 308, 340–342, 344 Hilbert–Clifford module, 67 Hilbert transform, 84

I Image acquisition stage, error analysis in, 39 Image formation, 6–9 Image irradiance, 8. See also Surface radiance equation for Lambertian surface, 34, 36–37 Imaging geometry, 3 Integral kernel of Clifford–Fourier transform, 122–127 Inverse Fourier transform, 56 Inverse Hermite transform, 147 IR albedo, 32 Iso-brightness contours, 19–20 Iso-intensity contours, 46

K Kernel Clifford–Fourier, 58, 97, 132, 134 of cylindrical Fourier transform, 169 Fourier, 68, 96–97, 142 of two dimensional Clifford–Fourier transform, 126–127 Kummer function, 75, 178

L Laguerre polynomials, 75, 187 Lambertian reﬂectance model, 12–14 Lambertian surface image irradiance equation for, 34 reﬂectance map for, 19 Laplace operator, 57, 62, 97 Legendre polynomials, 69–70, 182 Light calibration methods, 40 Linear shift-invariant (LSI) ﬁlters, 159 Localized beams, 237–240 in absorbing media, 310 Localized pulses, 240–243 ﬁnite-energy, 286–290 Localized solutions, 245–256 Localized superluminal pulse, 242

360

Index

Localized waves (LWs), 236–243 luminal, 318–319 propagation of, 319 spectral structure of, 279–281 subluminal, 301, 308, 317 superluminal, 301, 318 theory of, 294 Lorentz transformations, 336 LWs. See Localized waves

M Masking, deﬁnition of, 6 Material media Bessel beam in, 295 optical X-type pulses in, 294–300 Mehler’s formula, 86, 93–95 Metal cold-ﬁeld emitters, 223 Modiﬁed power spectrum (MPS) pulse, luminal, 289 Monochromatic beam, 338 Monogenic functions, 57, 62 Monogenic operators, 98–100. See also Clifford–Hermite monogenic operators MPS pulse. See Modiﬁed power spectrum pulse Multidimensional FrFT, 87, 90 deﬁnition, 90 operator exponential form, 90

N Neutrinos, 265 Noise level of carbon nanotube electron sources, 222–223 Nonaxially symmetric solutions, 340–342, 344 Noncommutative algebra, 57 Nondiffracting beams. See Localized beams Nondiffracting pulses. See Localized pulses Nondiffracting waves. See Localized waves

O Occlusion. See Masking One-dimensional complex Gabor ﬁlters, 159–160 One-dimensional Hermite transform, 145–148 One-dimensional real Gabor ﬁlters, 160–161 Optical X-type pulses, 294 in bulk fused silica, 300–301

GVD of, 296 in material media, 294–300 Optimal illumination conﬁguration, for photometric stereo, 46–51 Ordinary X-wave, focusing effects by, 292

P Parabivectors, 127, 131, 133 Parseval formula, 56, 68 Parseval’s theorem, 146 Phong reﬂectance model, 14–15 ambient component, 14 diffuse component, 14 specular component, 14 Photometric stereo, 20–25 algorithm, sensitivity of, 42 color, 26 dynamic and motion, 30–33 error analysis, 38–45 of extended light source, 28–30 with highlights and shadows, 26–28 narrow infrared (IR), 32 optimal illumination conﬁguration for, 46–51 perspective, 33–38

Q Quaternionic Fourier transform, 95–96

R Radiometry, 6–9 Reﬂectance map, 46–47 for Lambertian surface, 19 Reﬂectance model dichromatic, 15–17 Lambertian, 12–14 Phong, 14–15 Rodrigues formula, 73

S Schottky emitter, 204, 223–226, 228 Self-shadows, 6 SEM, prototype carbon electron source in, 228–229 SFS. See Shape from shading SFWM. See Superluminal focus wave mode Shape from shading (SFS) from single image, 18–20 Slingshot pulses, 243 SLS. See Superluminal localized solutions

Index

SMPS. See Superluminal modiﬁed power spectrum Space-time focused waves, 291 Special relativity (SR), 257–261, 336–340 Spectral multiplexing, 31–32 Spherical harmonics, 66, 72 Fourier transform and, 68–73 Spherical monogenics, 65 Square integrable functions, orthonormal basis for, 79 SR. See Special relativity Stationary wave ﬁelds, 301 in absorbing media, 310 frozen waves, 301–302 in lossless media, 302–310 Subluminal localized waves, 317 Bessel beam superposition in, 321 constructing physically acceptable, 320–330 Subluminal Lorentz transformations, 258 Subluminal wave bullets, 301 Superluminal charge toy model of pointlike, 261–265 X-shaped ﬁeld with, 261–265 Superluminal focus wave mode (SFWM), 286, 290 Superluminal localized pulses, 249 Superluminal localized solutions (SLS) ﬁnite total-energy, 255 to wave equations, 273–277 Superluminal modiﬁed power spectrum (SMPS), 288–289 characteristics of, 290 Surface albedo, 11, 23 Surface gradients, 20, 23, 42 Surface normals noise ratio for, 50–51 sensitivity analysis of, 40, 49 variance of noise, 49 Surface orientation, 24, 42 Surface radiance, 7–8, 13 Surface recovering methods, 17 Szegö projections, 85

T TEM of cap closing, 209–210 of carbon nanotube electron sources, 208–213 at high currents, 210–211

361

of individual free-standing carbon nanotubes, 209 in situ specimen holder, 208–209 Temporal multiplexing, 31 Tomasi–Kanade factorization algorithm, 33 Transverse Electric (TE) electromagnetic wave, 312 Tunneling, 271 simulation of, 268 Tunneling photons, 266–273 Two dimensional Clifford–Fourier transform, 122, 162 kernel of, 126–127 as linear operator, 127–140 properties, 139 Two dimensional Clifford–Gabor ﬁlters, 162–166 deﬁnition, 162–163 localization in spatial and frequency domain, 164–166 relationship with Gabor ﬁlters, 163–164 Two dimensional Gabor ﬁlters, types of, 161–162

U Undistorted progressing wave, 251

W Wave bullets, 317, 336 Wave equations, SLS to, 273–277 Wavelet, 156 Weight function, 146

X X-shaped pulses, 259 focusing effects by, 292 in material media, 294–300 ordinary, 248–254, 260 space-time focusing of, 291–292 X-shaped waves, 273–277, 317

Z Zenith angles, 42–44, 47–48, 51 Zero-order Bessel functions, 302, 305, 308, 340 Zero-speed envelopes, stationary solutions with, 330–336 Zonal monogenics, 152

Advances in Imaging and Electron Physics, Volume 156

Advances in Imaging and Electron Physics, Volume 148 (Advances in Imaging and Electron Physics) (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 127 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 132 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 143 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 120 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 121 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 111 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 102 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 113 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 144 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 128 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 125 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics (Volume 112) (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 113 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 150 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, volume 136 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 111 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 101 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 135 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 130 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 99 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 141 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 146 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 123: Advances in Electron Microscopy and Diffraction (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 134 (Advances in Imaging & Electron Physics)

Advances in Imaging & Electron Physics - Volume 100 Cumulative Index (Advances in Imaging and Electron Physics)

Advances in Imaging & Electron Physics, Volume 122 (Advances in Imaging and Electron Physics)

Advances in Electronics and Electron Physics, Volume 71 (Advances in Imaging and Electron Physics)

Advances in Electronics and Electron Physics, Volume 80 (Advances in Imaging and Electron Physics)

Advances in Electronics and Electron Physics, Volume 65 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 156

Advances in Imaging and Electron Physics, Volume 148 (Advances in Imaging and Electron Physics) (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 127 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 132 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 143 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 120 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 121 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 111 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 102 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 113 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 144 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 128 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 125 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics (Volume 112) (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 113 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 150 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, volume 136 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 111 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 101 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 135 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 130 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 99 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 141 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 146 (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 123: Advances in Electron Microscopy and Diffraction (Advances in Imaging and Electron Physics)

Advances in Imaging and Electron Physics, Volume 134 (Advances in Imaging & Electron Physics)

Advances in Imaging & Electron Physics - Volume 100 Cumulative Index (Advances in Imaging and Electron Physics)

Advances in Imaging & Electron Physics, Volume 122 (Advances in Imaging and Electron Physics)

Advances in Electronics and Electron Physics, Volume 71 (Advances in Imaging and Electron Physics)

Advances in Electronics and Electron Physics, Volume 80 (Advances in Imaging and Electron Physics)

Advances in Electronics and Electron Physics, Volume 65 (Advances in Imaging and Electron Physics)

Recommend Documents