Progress in Optics, Volume 40

EDITORIAL ADVISORY BOARD G. S. AGARWAL,

Ahmedabad, India

T. ASAKURA,

Sapporo, Japan

M.V BERRY,

Bristol England

C. COHEN-TANNOUDJI,

Paris, France

V. L. GINZBURG,

Moscow, Russia

E GORI,

Rome, Italy

A. KUJAWSKI,

Warsaw, Poland

J. PE~NA,

Olomouc, Czech Republic

R. M. SILLITTO,

Edinburgh, Scotland

H. WALTHER,

Garching, Germany

PREFACE With the publication of the fortieth volume of Progress in Optics, a significant milestone has been reached. The first volume was published in 1961, a year after the invention of the laser, an event which triggered a wealth of new and exciting developments. Many of them have been reported in the 228 review articles published in this series since its inception. The present volume contains six review articles on a variety of subjects of current research interests. The first, by T.R. Wolifiski, is concerned with polarimetric optical fibers and sensors. These devices have created a novel generation of powerful sensory-oriented techniques. The article reviews the main efforts and achievements in this field within the last two decades. It discusses the physical origin of polarization phenomena in birefringent fibers, both at the fundamental and the applied levels, and various deformation effects due to pressure, strain, twist and temperature on propagation of the lowest-order mode in fibers. The second article, by J. Tanida and Y. Ichioka, presents a review of recent researches on digital optical computing. After introducing the basic concepts needed for understanding the developments in this field, some feasibility experiments as well as software studies are discussed. The article by V. Pefinov~ and A. Luk~ which follows, deals largely with photodetection from the standpoint of the theory of open systems, bordering on novel techniques for testing irreversibility via quantum trajectories. Both destructive and non-destructive models of the process of photodetection are discussed. The fourth article, by Z. Zalevsky, D. Mendlovic and A.W. Lohmann, presents an account of modern theories of resolution in optical systems, based on the concepts of communication theory. The next article, by J. Turunen, M. Kuittinen and E Wyrowski, is concerned with the design of microstructured optical elements by the use of electromagnetic diffraction theory. Such an approach is required when the paraxial approximation is inadequate to describe their performance, or when it becomes necessary to take into account the state of polarization of the light. Diffractive elements based on linear or modulated gratings which operate in zero-order, first-order and multiorder modes are discussed.

vi

PREFACE

The concluding article by Z. Ficek and H.S. Freedhoff deals with the theory underlying the interaction of an atom with an intense polychromatic driving field, with particular reference to certain experiments. Several different systems which have been studied to date are discussed, including subharmonic resonances in the absorption spectrum of a strong probe, the fluorescence, near-resonance absorption and the Autler-Townes absorption by the entangled driven systems. In publishing this fortieth volume it is appropriate to acknowledge the substantial help which I have received over the years. There are too many persons to acknowledge individually. Three of them, however, deserve special mention: Mr. Jeroen Soutberg, director of ISYS Prepress Services in the Netherlands, is largely responsible for the production of these volumes. He must be credited for consistently maintaining the highest possible standards. I wish to thank Dr. M. Suhail Zubairy, one of my former students and now Professor at a University in Islamabad, Pakistan for preparing, for many years, the subject indexes for these volumes. I also wish to express my appreciation to Dr. Joost Kircz, a former publisher of Elsevier, who provided much help and advice with the publication of earlier volumes in this series. Finally, I wish to thank members of the Editorial Advisory Board of Progress in Optics for their part in having made this series such a successful enterprise. Emil Wolf

Department of Physics and Astronomy University of Rochester Rochester, New York 14627, USA October 1999

E. WOLF, PROGRESS 1N OPTICS XL 9 2000 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED

POLARIMETRIC OPTICAL FIBERS AND SENSORS

BY

TOMASZ R. W O L I l q S K I

Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa, Poland phone: (+48 22) 660-8212,-7262, fax: (+48 22) 628-2171; e-mail: [email protected], http://www.if.pw.edu.pl./-wolinski

CONTENTS

PAGE w 1.

INTRODUCTION

w 2.

POLARIZATION

w 3.

BIREFRINGENCE

w 4.

DEFORMATION FIBERS

. . . . . . . . . . . . . . . . . . . PHENOMENA

IN O P T I C A L F I B E R S

1N O P T I C A L F I B E R S

.

. . . . . . . . .

3 .

.

4 22

E F F E C T S IN H I G H L Y B I R E F R I N G E N T

. . . . . . . . . . . . . . . . . . . . . . .

28

w 5.

APPLICATION OF POLARIZATION EFFECTS . . . . . . .

52

w 6.

CONCLUSIONS

69

AND FUTURE PERSPECTIVES

ACKNOWLEDGEMENTS REFERENCES

. . . . . .

. . . . . . . . . . . . . . . . . .

70

. . . . . . . . . . . . . . . . . . . . . . .

70

w 1. Introduction Over the last two decades significant progress has occurred in optical fiber technologies from the use of intensity (amplitude) modulation to that of modulation of the optical polarization of the electromagnetic wave propagating along a fiber. At the same time new possibilities have opened up for both optical communication and also optical fiber sensors and systems. The key to successful construction of these new sensing devices and coherent communication systems is in high-performance polarimetric optical fibers and sensors. They are mostly based on highly birefringent (HB), polarization-maintaining (PM) fibers, which have aroused great interest from both theoretical and practical points of view. Although polarization effects in optical fibers initially played a minor role in the development of lightwave systems, their importance is still growing, due to an enormous increase in optical path lengths that can be achieved with singlemode fibers and also to an increase in bit rates in digital systems, as reviewed by Poole and Nagel [ 1997]. These two events recently precipitated a rediscovery of polarization phenomena in lightwave systems. Before 1980 it was impossible to exploit the polarization modulation in a fiber for sensing applications, since the conventional single-mode fibers manufactured for telecommunication use do not hold the optical wave amplitude in a particular polarization state. The appearance of HB fibers created a new generation of fiber-optic sensors known as polarimetric fiber sensors, which use polarization (phase) modulation within these fibers or at their output due to various external perturbations describing the physical environment. The aim of this chapter is to review the foremost achievements and efforts in research activities related to the development of a new generation of polarimetric optical fibers and sensors at both fundamental and applied levels during the past twenty years. The review underlines the physical origin of the perturbations (e.g., those induced by pressure, strain, bend, twist, temperature) on the lowestorder mode propagation in HB polarization-maintaining fibers together with their impact on applications in optical fiber sensors and systems. Several papers and chapters in textbooks have been published on polarizationmaintaining fibers and polarization effects in fibers, for example, by Kaminow [ 1981 ], Payne, Barlow and Ramskov-Hansen [ 1982], Rashleigh [ 1983a], Noda,

4

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 2

Okamoto and Sasaki [1986], Tsao [1992], and Huard [1997]. However, they have not addressed all aspects of the rapidly growing part of fiber optics that is analyzed in this review.

w 2. Polarization Phenomena in Optical Fibers The phenomenon of polarization was discovered by Huyghens in 1690 by passing light through two calcite crystals, and only in 1808 Malus passed partially reflected light through a calcite crystal and found that it was polarized (Born and Wolf [1993]). In analogy to magnetic bodies, Malus called oriented light "polarized light" (Kliger, Lewis and Randall [1990]). Optical fibers exhibit particular polarization properties, (e.g., Kaminow and Ramaswamy [1979], Cancellieri, Fantini and Tilio [1985], Tsao [1986], Shafir, Hardy and Tur [1987]). Contrary to ordinary plane waves in bulk media, of which the amplitudes are constant in the wave plane, guided electromagnetic fields in optical-fiber waveguides are called inhomogeneous plane waves, since their amplitudes are no longer stable within the plane wave (Huard [ 1997]) and the fields are generally characterized by non-transverse components. Two approaches are generally used in the description of polarization phenomena in optical fibers (Tsao [1992]). The first approach treats an optical fiber as an optical waveguide, in which light as a kind of electromagnetic wave of optical frequencies can be guided in the form of waveguide modes. This approach identifies basic polarization eigenmodes of a fiber and relates them to the polarization state of the guided light. Changes in output polarization are described in terms of polarization-mode coupling due to birefringence changes acting as perturbations along the fiber. The polarization coupling, except for some simple cases, is described by the coupled mode theory (Rashleigh [ 1983a]). The second approach treats an optical fiber like any other optical device that transmits light, and the fiber can be divided into separate sections that behave like polarization state shifters. Here, polarization evolution in a fiber can be described by one of the three general formalisms: by the Jones vectors and matrices formalism, by the Stokes vectors and Mueller matrices formalism, or by the Poincar6 sphere representation. Since optical fibers allow large propagation distances, even extremely small birefringence effects can accumulate along the fiber, and their random distribution over such distances generally make it difficult to determine the polarization properties of guided light; this applies both to the state and the degree of polarization.

I, w 2]

POLARIZATION PHENOMENA IN OPTICAL FIBERS

5

2.1. M O D E S OF O P T I C A L F I B E R

An optical fiber consists of a core of dielectric material with refractive index nco and a cladding of another dielectric material with a refractive index ncl less than nco. The exact description of the modes propagating in a fiber is complicated, since they are six hybrid-field components of great mathematical complexity. Detailed analyses of guided-wave modes in cylindrical optical fibers have been conducted in many review papers and textbooks by, among others, Snitzer [1961], Marcuse [1974], Clarricoats [1976], Kaminow and Ramaswamy [1979], Snyder and Love [1983], Cheo [1985], Cancellieri, Fantini and Tilio [1985], Tsao [1986], Yeh [1987], Shafir, Hardy and Tur [1987, 1988], and Huard [1997]. The modes with a strong electric E- field compared with the magnetic Hz field along the direction of propagation (z-axis) are designated as EH modes. Similarly, those with a stronger ~ field are called HE modes. These modes are hybrid since they consist of all six field components (3 electric and 3 magnetic) and possess no circular symmetry. The propagating modes are discrete and require identification by two indexes (/,p): HE0, and EHI,, both of which are integer indexes. The first, 1=0,1,2,3,..., separates the variables in the scalar wave equation, whereas the second, p = 1,2,3,..., indicates the pth roots of the Bessel function of the first kind Jl and the modified Bessel function KI. F o r / = 0 , the hybrid modes are analogous to the transverse-electric (TE) and the transverse-magnetic (TM) modes of planar waveguides, and two linearly polarized sets of modes exist that are circularly symmetric with vanishing either the E or H longitudinal field components" TE0t~ (E_- = 0) and TM0p (/-/: = 0). The lowest-order transverse modes TE01 and TM01 have cutoff frequencies V = Vc = 2.405, where V is the normalized frequency, defined as 2 = v/u 2 + w 2 V = -2~J a V n/ c o2 - ncl

(2.1)

with a the core radius, X the free space wavelength, n~o (nd) the refractive index of the fiber core (cladding), V/ nc2o- nd2 = NA the numerical aperture of the fiber used in optics to express the ability of the system to gather light, and u and w are parameters defined as follows: u=a

ncok0 -

,

w=a

-n:lk~,

where/3 is the propagation constant and k0 = 2x/2.

(2.2)

6

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 2

The lowest-order mode of cylindrical waveguide is the HEll mode, which has zero cutoff frequency. This is the fundamental mode of an optical fiber and is also the only mode propagating in the region of frequencies 0 < V < 2.405. Hence, in this region a fiber is considered to be single-mode. The field distribution E(r, t) corresponding to the HEll mode, has three non-zero components Ex, E y and Ez (in Cartesian coordinates), among which either Ex or E y dominates. Even a single-mode fiber is not truly single-mode however, since the electric field of the HE l~ mode has two polarizations orthogonal to each other that constitute two polarization modes of a single-spatial-mode fiber. A significant simplification in the description of these modes is based on the fact that most fibers for practical applications use core materials in which the refractive index is only slightly higher than that of the surrounding cladding; that is, nco - ncl << 1.

(2.3)

This assumption leads to the so-called "weakly guiding approximation", in which instead of a six-component field only four field components need to be considered (Marcuse [1974]). For a weakly guiding fiber (nco.~ncl), approximate mode solutions are defined as linearly polarized LPtp modes of different azimuthal (l) and radial (p) mode numbers. The idea of LP modes was originally introduced by Gloge [ 1971 ], who showed that for the lower-order modes, the combination modes have the electric field configuration resembling a linearly polarized pattern (see figs. 1 and 2 below). 2.2. POLARIZATIONOF THE MODES A significant difference occurs between polarization properties of the monochromatic plane wave in an infinite dielectric medium and an electromagnetic wave confined in an optical fiber waveguide. In the first case, the state of polarization is defined by temporal evolution of the electric vector E, which has the same direction and amplitude in the plane wave, whereas in the second case, polarization of the propagating wave can be described in terms of polarization properties of the waveguide modes. However, finding a proper definition of polarization states of the guided modes creates a serious problem, because hybrid modes possess an axial component (along the direction of propagation) and that direction of the field is not the same in every point of the wave plane. The weakly guiding approximation (2.3), assuming a small core-to-cladding index difference (of about 10-3 , which is the case of real single-mode fibers in the

I, w 2]

POLARIZATION PHENOMENA IN OPTICAL FIBERS

7

LPol (HEl 1) Fig. 1. Electric and magnetic field distributions (Ex, Ev, Hx, Hv), for the LP01 (HEll) mode.

spectrum ranging from visible to the near infrared), removes this difficulty by neglecting longitudinal field components. Thus, it appears that the propagating field is almost transverse, composed of quasi-transverse, linearly polarized TEM modes. In this sense, linear polarization states of the guided light can be directly attributed to those of LP modes with arbitrary basic linear polarization directions due to the cylindrical symmetry of an isotropic fiber. The problem of polarization of the fiber-optic modes was analyzed by Varnham, Payne and Love [1984], Zheng, Henry and Snyder [1988], and Eflimov and Bock [1998]. The LPlp modes count the group of modes appearing together as a single mode, and they are an example of pseudomodes with the property of changing their cross-sectional intensity and polarization pattern as the mode propagates. However, the LPlp modes are superpositions of the true, generally hybrid, waveguide modes, namely HEl+l,p and EHl_l.p modes. The EH/_l,p modes of l = 1 lose their hybrid nature and, consequently, become transverse magnetic TM0p and transverse electric TE0p waveguide modes. In an isotropic fiber, modes of zero azimuthal order (/=0) are twofold degenerate: two polarization modes, LP~p and LPop, are possible, whereas modes of non-zero azimuthal order (l > 0) are fourfold degenerate: twofold orientational degeneracy (even and odd) and twofold polarization degeneracy (x and y). In this case four polarization modes, x, e, v, e x, o, ,. o, can be guided along the fiber. namely LPtp LP~p , LPlp and L p ip In the isotropic case single-mode fibers (normalized frequency parameter V ~<2.405) with perfect circular cores support two degenerated orthogonal polarization modes HE{1 and HE~1 of the propagation mode HEll with the same spatial intensity distribution, which is exactly the linearly polarized LP01 mode in the weakly guiding approximation. The fundamental or lowest-order HEll mode, named after Gloge [1971] the LPoI mode, exhibits an amplitude of a revolutionary symmetry and may be linearly polarized either along the x or the y axis (fig. 1). Its two degenerated polarization modes LP~1 and LP0'1 have the same propagation constant fl on a perfect cylindrically symmetrical fiber. These

8

[I, w 2

POLARIMETRIC OPTICAL FIBERS AND SENSORS

TMol

LP~~ " TMo~ - HE21

HE21

HE21

LPI~"TEo~+ HE2~

LPI~>" TEo~ - HE2~

TE01

LP,~-" 9 TMo~+ HE2~

Fig. 2. Electric field vector of the waveguide modes, TE01, TM01, HE21 (two polarizations), and their four independent linear contributions to the LPIl spatial mode. polarization modes constitute a basis of two orthogonal and normalized states of linear polarizations and the electric field of these modes are given by Jo (ur/a) Eo--l~t-yZ~, ~ , ~0

E~,y =

0 ~< r < a ,

~b4t~

Ko (ur/a) E0 K0(u) '

(2.4) r>a.

The normalization constant amplitude E0 can be determined from the power relation (Yeh [ 1987]):

wJo+( zo)u +( zo)

E o - V Jl(U)

;ra2ncl

V Kl(w)

.Tra2ncl '

where zo = ~u/ko is the plane-wave impedance in a vacuum, Jl and Kl (1= 0, 1) are the Bessel functions of the first kind and the modified Bessel functions, respectively, and ~0 is the angular frequency corresponding to the free space wavelength ]l. The next four higher-order modes TE01, TM01, l_.llTeven =-~21 and H E ~ d (with 2.405 < V < 3.832), have slightly different propagation velocities and almost the same cross-sectional optical intensity distributions. A new method for measuring cutoff frequencies of TE01, TM01, and HE21 modes was proposed by Kato and Miyauchi [ 1985]. In the weakly guiding approximation these four second-order modes become fourfold degenerate and are denoted as LP~ modes. The field distributions of four independent linear combinations of the waveguide modes, TM01-HE21, TE01-HE21, TE0~ +HE21 and TM0~ +HE21, shown in fig. 2, constitute the

I, w 2]

POLARIZATIONPHENOMENAIN OPTICAL FIBERS

9

Table 1 Polarization modes of an isotropic weakly guiding two-mode fiber

Fundamental LPol mode:

Normalized frequency:

V~< Vc=2.405

Waveguide modes:

HE i"l' HE;'I

Linearly polarized modes:

LPoVl, LPo 1

Propagation constants:

ridx = riO' (twofold degeneracy)

Second-order LPll model:

Normalized frequency:

2.405 < V < 3.832

Waveguide modes:

TM01 -HE21, TE01-HE21, TE01 +HE21, TM01 +HE21

Linearly polarized modes:

LP~,

Propagation constants:

fi~x = fi~y= fi~x= fi~-" (fourfold degeneracy) ~ even = )odd cut "~cut

Cutoff wavelengths:

~1'

LPll,

OX

LPll,

01'

LP(l

linearly polarized second-order (LPll) modes as a single linear electric field vector. Fibers operating in this regime (2.405 < V < 3.832) are two-mode (or bimodal) fibers. In fact, the two-mode fiber supports six modes: two polarizations of the fundamental LP0~ mode and two polarizations of each of two lobe orientations (even and odd) of the second-order LPI1 mode" LP~l and LP~'1 (table 1). Consequently, the LP02 mode is the sum of TE02, TM02, and the HE22 modes, and the LP21 mode is the sum of HE31 and EHI1 modes, and so on. 2.3. OPTICAL FIBERS SENSITIVE TO POLARIZATION EFFECTS

An ideal isotropic fiber propagates any state of polarization launched into the fiber unchanged. However, the realization of the perfectly isotropic singlemode fiber demands huge manufacturing requirements with respect to the ideal circularity of the core and lack of mechanical stress. Since in the ideal cylindrical fiber the fundamental LP01 mode contains two degenerated orthogonally polarized modes they are propagating at the same phase velocity. In real single-mode fibers that possess non-zero internal birefringence, both orthogonally polarized modes have randomly different phase velocities, causing fluctuations of the polarization state of the light guided in the fiber. The absolute magnitude of birefringence in such fibers is typically of the order

10

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 2

l[3y- fix] ,~ 1 m -1 (Ulrich [1994]). This means that after propagation through a length of about 1 m the polarization will be modified in an unpredictable way and consequently these fibers cannot preserve any state of polarization launched into the fiber. Over the past twenty years numerous authors have developed and analyzed different types of optical fibers sensitive to polarization effects (e.g., Ramaswamy, French and Standley [1978], Okamoto, Edahiro and Shibata [1982], Simpson, Stolen, Sears, Pleibel, Macchesney and Howard [1983], Snyder and Riihl [1983], Burns, Moeller and Chen [1983], Okamoto, Varnham and Payne [1983], Noda, Shibata, Edahiro and Sasaki [1983], Okamoto [1984], Snyder and Riihl [1984], Marrone, Rashleigh and Btaszczyk [1984], Chen [1987], Alphones and Sanyal [ 1987], Hayata and Koshiba [ 1988]).

2.3.1. Low-birefringence polarization-maintaining fibers The concept of polarization preservation in single-mode fibers may be realized either by manufacturing fibers with negligible birefringence or by enhancing internal birefringence to very high values. In the first case we obtain lowbirefringence (LB) fibers fabricated by special techniques so as to exhibit particularly low birefringence of the order lily- fi~] ~< 10-2m -1. The LB polarization-maintaining fibers (PMFs) can preserve any and all polarization states (Schneider, Harms, Papp and Aulich [1978], Eickhoff and Brinkmeyer [1984]), whichever is injected, but they require the high stability of the environment since the polarization state is extremely sensitive to external perturbation effects.

2.3.2. Highly birefringent polarization-maintaining fibers In highly birefringent (HB) polarization-maintaining fibers, the difference between the phase velocities for the two orthogonally polarized modes is sufficiently high to avoid coupling between these two modes. Fibers of these class have a built-in, well-defined, high internal birefringence obtained by designing a core and/or cladding with non-circular (mostly elliptical) geometry or by using anisotropic stress applying parts built into the cross-section of the fiber. Various types of HB polarization-maintaining fibers are presented in fig. 3. These include (a) elliptical-core, (b) stress-induced elliptical internal cladding, (c) bow-tie, or (d) PANDA fibers. The magnitude of the internal birefringence is characterized by the beat length of the two polarization modes, 2Jr

Z~- IG-/~xl'

(2.6)

I, w2]

POLARIZATIONPHENOMENAIN OPTICALFIBERS

11

Fig. 3. Various types of HB polarization-maintaning fibers: (a) elliptical core; (b) elliptical internal cladding; (c) bow-tie; (d) PANDA.

Fig. 4. Definition of the beat length of the fundamental mode LP01 in a single-mode HB PM fiber.

and is responsible for phase difference changes along the longitudinal axis z of the HB fiber. The spatial period of these changes reflects the changes in the polarization states along the fiber (fig. 4). Since linearly birefringent (anisotropic) optical fibers have a pair of preferred orthogonal axes of symmetry (birefringence axes), two orthogonal quasilinear polarized field components HE{1 and HE~ 1 of the fundamental mode HEll (LP01), which propagate for all values of frequency (wavelength), have electric fields that are polarized along one of these two birefringence axes. Hence, light polarized in a plane parallel to either axis will propagate without any change in its polarization but with different velocities. However, injection of any other input polarization excites both field components HEi"1 and HEiVl, and since these two orthogonal mode components are characterized by different propagation constants fix and [3y (degeneracy of the fundamental mode is lifted), they run into and out of phase at a rate determined by the birefringence of the HB fiber. At the same time they produce a periodic variation in the transmitted polarization state from linear through elliptic to circular and back again (see fig. 4). It is apparent that transmission properties of such fibers, when propagating only the fundamental modes, are similar to those of anisotropic crystals, in that the fiber has a pair of optical axes (Snyder and Love [1983]).

12

[I, w 2

P O L A R I M E T R I C O P T I C A L FIBERS A N D S E N S O R S

Table 2 Degeneracy lifting of polarization modes in HB two-mode fiber fundamental LP01 m o d e

second-order LP l I mode

two-mode HB fiber (LP 01 + LP~I )

Operating wavelength:

aeven ~' > "~cut

'~ < aodd "~cut < ~even "~cut

~odd "~cut < ~" < aeven "~cut

Normalized frequency:

V~< Vc = 2 . 4 0 5

2.405 < V < 3 . 8 3 2

Polarization modes:

LPbVl, LP~' 1

LP~~, L P Iev I , LPIIox , LPI"ov!

ev LPbVI , LP~)'I, LP~'~, LP(1

Propagation constants:

flo'c ~ fi0v

fi~x ~ file.'" ~ fi~x ~ fl~y

riOv ~ riO'v fi~x = fi(, file.`.- fi~'

Polarization birefringences:

Aft0 = r0 r -fi~'

Afi~ = fl~x_ file.,. Afi~ = fi~x _ fi/y

Aft0 = fi0~ - f i 0 v Aft I = fiiv _ fiiv

Aft) = f i ~ - fi~x Aft I, = file``- ill"'

Aft x = fi6~- _ fii~-

Modal birefringences:

-

Cutoff wavelengths:

2.4. P O L A R I Z A T I O N BIREFRINGENT

Aft,, = rid'- fi( Aft0 - Afil = Afix - A[3y

Relation 9~even/aodd "cut -'~cut = (3e + 1)/(3 + e)

EVOLUTION

IN T H E L O W E S T - O R D E R

(e-core ellipticity)

MODES

OF A

FIBER

Two important effects are a direct consequence of birefringence properties of the fiber in the polarization evolution in the lowest-order modes of the HB fiber. The first effect is lifting the degeneracy of the modes, which means that the different polarization modes will have different propagation constants and the greater the birefringence the greater the difference. The second effect is that the even and the odd LPll modes - in the case of a two-mode fiber- will have different cutoff wavelengths that can reduce several propagating modes in the HB fiber (table 2). For wavelengths slightly shorter than a critical value (cutoff wavelength), the next higher-order mode with greater propagation velocity compared with that of the fundamental mode is guided (fig. 5). The relevant feature of HB twomode fibers is that only two second-order modes (LP~I) propagate instead of four. This means that over a large region of the optical spectrum, the two-mode HB fiber guides only four polarization modes: two orthogonal linearly polarized 1' ex fundamental LPdl and LP61 eigenmodes and the even second-order LPll and LPle~ spatial modes, the propagation constants of which are denoted by fi6", rio,

I, {} 2]

POLARIZATION PHENOMENA IN OPTICAL FIBERS

13

Fig. 5. Spectral distribution of the lowest-order linearly polarized modes in a two-mode fiber; ~.o and ~e stand for cutoff wavelengths for odd and even LPll modes

Fig. 6. Mode pattern orientation of the even LPll mode in an elliptical-core and a bow-tie fiber.

/31x and fil instead of six as in the case of isotropic fibers with perfect circular cores. This effect is clearly observed in elliptical-core (e-core) and bow-tie HB fibers in which the orientation of the even LP~ mode is shown in fig. 6. The schematic diagram of the propagation constants and the corresponding mode patterns is shown in fig. 7. The separation between the cutoff wavelengths for the LP~l and the LP~'1 modes will increase with increasing birefringence (table 2). To describe quantitatively polarization transformation due to birefringence changes (intrinsic and induced) in HB fiber, both the Jones matrix formalism and the formalism using Stokes vectors and Mueller matrices can be applied. The Jones formalism is limited to the strictly monochromatic light sources when the light propagating in the fiber is completely polarized, whereas in the quasimonochromatic case the evolution of the state and the degree of polarization along a birefringent fiber is described by the Mueller-Stokes matrix formalism. For visualization and graphic representation the Poincar6 sphere is convenient. Many authors have addressed the question of the polarization evolution in birefringent fibers (e.g., Stolen, Ramaswamy, Kaiser and Pleibel [1978], Ulrich [ 1979], Wagner, Stolen and Pleibel [ 1981 ], Sakai, Machida and Kimura [ 1982], Crosignani and Di Porto [1982], Rashleigh and Marrone [1982], Love, Hussey, Snyder and Sammut [1982], Varnham, Payne and Love [1984], Sakai [1984],

14

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 2

Fig. 7. Propagation constants in a single- and two-mode HB fiber.

Takada, Okamoto, Sasaki and Noda [1986], Zheng, Henry and Snyder [1988], Jaroszewicz [ 1994], Menyuk and Wai [ 1994], Eftimov and Bock [ 1998]).

2.4.1. The Jones formalism Polarization properties of a birefringent fiber can be adequately described in the monochromatic case by a 2 x 2 unitary complex matrix, the Jones matrix (Jones [1941]): E ~

--

T(X, ~). E

in,

(2.7)

where ~, is the wavelength of propagating light, and M is the propagation matrix depending on the physical environment represented by vector V and usually expressed as a product of three terms (Culshaw and Dakin [1989]) M = Te io J,

(2.8)

where T is the scalar transmittance, 0 denotes the mean phase retardance, and J is the birefringence (Jones) matrix of the fiber. The matrix becomes the identity matrix I in the case of an isotropic fiber with a perfect cylindrical symmetry.

I, w 2]

POLARIZATION PHENOMENA IN OPTICAL FIBERS

15

For a linearly birefringent fiber, J = Jg =

e iAO/2 0 ] 0 e -iAr '

(2.9)

where Aq~ denotes linear relative phase retardance between the eigenmodes and the fiber behaves like a simple linear retarder. For a circularly birefringent fiber, B=Bc=[C~ ~] sin6O cos6q~ '

(2.10)

where 26r denotes circular relative phase retardance between the eigenmodes and the fiber behaves like a simple circular retarder. The general Jones matrix for any optical fiber with evenly distributed retardations, including linear retardance A0, circular retardance 26q~ and axial rotation, was analyzed and described by Ysao [1992]: Q p,

,

(2.11)

where P=cos~-j(Ar

sin

& ,

Q=(r+6O)

Z ~ ,

sin

/

~=~Aq~ 2+(r+6q~) 2.

(2.12) If the fundamental mode LP01 of the linearly birefringent, weakly-guiding twomode fiber is labeled with an index 0, the LPll mode is labeled with an index 1, and the electric field excitation coefficient is denoted by It, the resultant electric field in the monochromatic case can be expressed as coherent superposition of the electric fields E0 and E1 of each of the modes: (2.13)

E ( r , q~,z) = ltoEo(r,z) + ltlEl (r, r

where E o ( r , z ) : Ax(z)fo(r)ex + A.,.(z)fo(r)e ,,

(2.14)

E l ( r ' O ' z ) : Bx(z)fle(r'O)ev + Bv(z)fle(r' O)e"

(2.15)

+ Cx(Z)fl~

O)e, ~ + C.,.(z)fl~

O)e,.,

where/to and/t, are the mode excitation coefficients (It2 + lt~ = 1), j6(r), f~e (?. , q~) andfl~ r, 0) are the corresponding modal spatial distributions, and e~ and ey are

16

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 2

Fig. 8. Propagation constants for the lower-order waveguide and LP modes of an HB fiber (after Eftimov [1995]).

unit vectors along the birefringence axes. As the modes propagate along the fiber (z-axis), the amplitudes of polarization modes evolve. The general case for all LP01 and LPll (even and odd) modes was analyzed by Eftimov [1995]. Figure 8 shows propagation constants for the lower-order waveguide and LP modes of a linearly birefringent HB fiber, where A / ~ = firM- fiNE, AfiE = firE- fiHE, and /3 = (]~M + 2fiHE + riTE)/4 = (fie +/30)/2, and Aft0 = fi6" - rio" The stability of the second-order mode is set by the stability parameters introduced by Snyder and Love [ 1983], defined as

fi e _

~o

= ~ . AM(E) 2AfiM(E)

(2.16)

Isotropic fibers are characterized by degenerated values of second-order birefringencies (fie = rio), hence the stability parameters turn zero and the second-order modes are unstable, changing their polarization and orientation along the fiber. In HB anisotropic fibers fie_/3o >> A/~, AfiE, which means that stability parameters tend to infinity (AM,AE~ZX~), stabilizing the LPll modes. Limiting the analysis to the practical case when only fundamental and second-

I, w 2]

POLARIZATION PHENOMENA

IN

OPTICAL FIBERS

17

order LP~I modes are excited, polarization evolution along the fiber can be described by the coupled-mode equations

A,(z) A,(z) Bx(z)

d By(z)

(2.17)

B,(~)

[Ji "

The transformation of the amplitudes A,-(z), A ,.(z) and B,-(z), B,,(z) along the fiber is given by the Jones matrices jr and jill as: Ax(z)

l

A~(O)

AI~(O)

'

B,,<:)

B].(O)

,

(2.18)

with J(i)=EeiAB'z/20 e_iAB, 0 z/2 leiB'z ,

(219)

fii = (fii x + fiv)/2, and Afii = fii" - fi/ for i - 0, 1. Based on the coupled-mode equations (Efiimov and Bock [1993]), this evolution can be expressed in terms of the Jones matrices j(0) and J(~) of the fundamental and the second-order mode. Hence, the resultant electric field at the output of the fiber can be represented as a superposition of the fields parallel to the principal axes of the fiber E(r, O, z) = Ex(r, O, z) e~ + E ,.(r, (/), z) e ,.,

(2.20)

where E~(r, O,z) = ltoAl(z)fo(r) + ItlBl(z)fl+(r, 0), Ey(r, O,z) = ltoAz(z)fo(r) + ltlB2(z)fle(r, 0).

(2.21)

Equation (2.20) contains information about the mode pattern distribution of each polarization mode of the HB two-mode fiber, its attenuation, excitation efficiency, and amplitude and phase evolution, which enables the description of polarizationsensitive intermodal interference. From the viewpoint of Zheng, Henry and Snyder [1988], the polarization (direction of the field lines) of the fundamental fiber mode has been represented

18

POLARIMETRICOPTICAL FIBERS AND SENSORS

--

m

[I, w 2

+

+

Fig. 9. Field pattern of circular fiber resolved into two plane-polarized components perpendicular to each other: (top) incorrect pattern (Varnham, Payne and Love [1984]); (bottom) corrected pattern (Zheng, Henry and Snyder [1988]).

incorrectly at least over the last 70 years (fig. 9); this also has implications for the polarization analysis of birefringent fibers. Generally, in most polarization analyses the modes in the HB fiber are commonly considered as LP modes, but the higher the birefringence of the fiber, the more the mode departs from the LP model. Varnham, Payne and Love [1984] and Zheng, Henry and Snyder [1988] showed that the fundamental hybrid modes are composed of a dominant linearly polarized component and a corrective orthogonally polarized component. Eftimov and Bock [1998] recently analyzed the polarization behavior of hybrid modes in HB fibers, introducing the spatial polarization Jones matrix for the hybrid modes. It has been observed that the hybrid nature of the mode reduces polarization holding and is also responsible for the parasitic polarimetric response and its wavelength dependence of the HB fiber.

2.4.2. The Mueller-Stokes formalism For not strictly monochromatic sources (generally for optical fiber sensors driven by laser diodes or light-emitting diodes), the Jones vector and Jones matrix formalism are no more valid (Franceschetti and Smith [1981]). For practical broadband sources a quasi-monochromatic model is more appropriate. According to Born and Wolf [ 1993], light is said to be quasi-monochromatic if the wavelength range A2 is small compared with the mean wavelength: A~,

<< 1 or

A~o fo

<< 1.

(2.22)

I, w 2]

POLARIZATION PHENOMENA 1N OPTICAL FIBERS

19

The Stokes vector and the Mueller matrix formalism are necessary to describe polarization phenomena in such a case. Contrary to polarization properties of a plane wave for which four Stokes parameters are only z dependent, Eftimov and Bock [1993, 1997, 1998], Eftimov [ 1995], Bock, Wolifiski and Eftimov [1996] proposed and explored the establishment of a relation between the resultant polarization and the interference pattern within the weakly-guiding birefringent fiber derivation of the spatial Stokes parameters. The spatial Stokes parameters are the Stokes parameters that are not only depend on z but also on r and q~ (or x, y), and are defined as

So(r, d/),z) =

IEx(r, q~, z)[ 2 + IE,,(r, q~,z)]2 ,

Sl(r, ~0,z) = IEx(r, q~,z)l2 - Ev(r, O,z) 2, S2(r, r

= 2Re[Ex(r, O,z)E,.(r, O,z)] ,

S3(r, r

= 2Im [E* (r, r

(2.23)

E,.(r, (),z)] .

By knowing the input Stokes vector of the wave exciting the LP01 and LPll modes in the HB two-mode fiber, the spatial Stokes parameters can fully describe the state of polarization of the interference pattern at any point (r, q~,z) in the cross-section of the fiber. With the help of eq. (2.20), Eftimov and Bock described the resultant output distribution in the quasi-monochromatic case by

I(r, 99,z) = r/0f02(r)+ r/1A2(r, q~) + 2r/01jC0A(r, qg) ElF,/,,- cos(A~,z) + IF,, I,:,, cos(Ag, z)]

Afix = N - fil~,

A/3~ = g ' -

(2.24)

fii'-

The first two terms of the above expression are the contributions of each mode, whereas the third is the interferometric term. The factors !,-,- and I>), are given as

Ixx = Ax(O) B~(O) = ~(s8 + s?),

Iv. - A, (O) By(O) = 1(S~ - S~),

(2.25)

where S~ and S~' are the first two parameters of the input Stokes vector S O = {S ~ S ~ S ~ S o }. The factors IF,-] and IF, I are the correlation functions between the LP~Cl, LP~I and the LP0~, LF~'1 modes, which describe the fading of the intermodal interference in the quasi-monochromatic case.

20

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 2

The state of polarization of the modes can be expressed by the output Stokes vectors 'S Oand "S O of each mode and which are obtained from the input Stokes vector through matrix relations

'S(z) = 'M(z) S ~

"S(z) = "M(z) S ~

(2.26)

where 1 0 0 01 1 0 1 0 0 0 0 17o,,]sin(Afl~ -17o,, sin(Aft~ 1z) (2.27) 0 0 170,11sin(Afi~ are the Mueller transformation matrices of each spatial mode. Thus, 'M(z) describes the polarization transformation between the Lp~cl and LPol modes, and "M(z) stands for the polarization transformation between the even LP{1 and LPYl modes. Both matrices are of the same type and are distinguished by the propagation constant differences Aft~ for the LP01 mode and Aft ~ for the LPll mode (see fig. 7, above). In the quasi-monochromatic case the propagation-constant differences Aft0, Afll, Afix and Afly a r e developed into a series, and are written as

'M(z), "M(z) :

i

Ir0,,Icos(A/38,1z)

d(Aflk) 6rk - - - , (k = 0, 1,x,y), (2.28) dw where 6rk are the corresponding dispersions among which 6r0 and 6rl are the polarization dispersions of the LP01 and LPll modes and cause their depolarization through the mutual correlation functions ]7'0] and [71] appearing in the Mueller matrices, whereas 6rx and 6r,. are the intermodal polarization dispersions causing the fading of intermodal interference between the LP~I, LPi~I and the LPol , LPll polarization modes, which is accounted for by the correlation functions ]Ix] and ]F,,]. The correlation functions are defined as the Fourier transform of the form factor g(~o): Aflk = Afiz~+6rk(co- o~0),

I~,0,,I, Ir~,~,l

-

g(co) exp[j6rk(o) - o0)] do,

(k = 0, 1,x,y).

For a Lorentzian source (as a laser diode is often assumed to be), (3o92 g(o)) = 6o92 + (~o - (-o0)2' ]~'~1 = exp (-6rk 6e). z).

(2.29)

(2.30)

2.4.3. The PoincarO representation For visualization and graphic representation of polarization and birefringence effects, it is helpful to use a sphere of unit radius, called the Poincar6 sphere

I, w2]

POLARIZATIONPHENOMENAIN OPTICALFIBERS

21

lcp

V

H

rcp

P

lcp

rcp

P

rcp Fig. 10. Representation of polarization by points on the Poincar6 sphere: H,V: horizontal, vertical polarizations; Q,P: +45~ linear polarizations; lcp,rcp: left, right circular polarizations; •B: birefringence vector.

(fig. 10a). Any point on the sphere represents a state of polarization with "latitude" 2to, where co - + arctan(b/a) is the ellipticity of the polarization ellipse (a and b are the lengths of the major and minor axes, respectively), and "longitude" 2a, where a is the azimuth of the polarization ellipse. The different states of polarization are represented by points on the Poincar6 sphere. The equator of the sphere is the locus of all linearly polarized states (zero ellipticity: to=0), which can have different azimuths. The northern hemisphere represents all left-handed elliptically polarized states and the southern hemisphere represents all right-handed elliptically polarized states. Two poles of the sphere represent the two orthogonal states of circular polarization: north being left-handed (lcp) and south being right-handed (rcp). Polarization evolution in birefringent fiber represented by the Poincar6 sphere representation was used by many authors, including Simon and Ulrich [1977], Ulrich and Simon [ 1979], Eickhoff, Yen and Ulrich [ 1981 ], Monerie [ 1981 ], Rashleigh [1983a], Tsao [1992], Ulrich [1994], and Huard [1997]. This approach is simple, and relies on the rotation of all input points on the sphere to their corresponding output points. This rotation, which is characterized by the rotation

22

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 3

vector g2B, can be performed by rotating about the direction of J2B, an input state through the angle given by the magnitude I~BI. The rotation vector 12B characterizes the birefringent properties of the fiber and is also known as the birefringence vector. Linearly birefringent fibers are characterized by the rotation vector I2B in the equatorial plane (fig. 10), fibers with circular birefringence possess the rotation vector QB along their polar axis (fig. 10c), and any other direction of the rotation vector g2B is connected with elliptical birefringence. It should be noted that points on the Poincar6 sphere along the directions -+-g2Brepresent two eigenstates of polarization of the fiber; in particular, both poles are eigenstates of circular polarization (fig. 10c), and points along the direction +QB in the equatorial plane (fig. 10b) are two linearly polarized eigenstates of the linearly birefringent fiber.

w 3. Birefringence in Optical Fibers An ideal isotropic fiber has no birefringence. It propagates any state of polarization launched into the fiber unchanged. Real fibers possess some amount of anisotropy because of an accidental loss of circular symmetry. This loss is due to either a non-circular geometry of the fiber or a non-symmetrical stress field in the fiber cross-section. When birefringence is introduced into an isotropic fiber, the circular symmetry of the ideal fiber is broken, thus producing the anisotropic refractive index distribution into the core region. The asymmetry results from either intrinsic birefringence including a geometrical deformation of the core and stresses induced during the manufacturing process or from material anisotropy due to induced (extrinsic) elastic birefringence. 3.1. INTRINSIC B I R E F R I N G E N C E

Intrinsic birefringence is introduced in the manufacturing process. It is a permanent feature of the fiber that was analyzed by numerous authors (e.g., Imoto, Yoshizawa, Sakai and Tsuchiya [1980], Sakai and Kimura [1981 ], Sakai and Kimura [1982], Namihira, Ejiri and Mochizuki [1982], Varnham, Payne, Barlow and Birch [1983], Chu and Sammut [1984], Eickhoff and Brinkmeyer [1984], Chen [1987], Chiang [ 1987], Wu [ 1992], Krasifiski, Majewski and Hinata [ 1993]). It comprises any effect that causes a deviation from the perfect rotational symmetry of the ideal fiber. A non-circular (elliptical) core gives rise

I, w 3]

BIREFRINGENCE IN OPTICAL FIBERS

23

to geometrical (shape) birefringence, whereas a non-symmetrical stress field in the fiber cross-section creates stress birefringence. Geometrical birefringence has been discussed by many authors (e.g., Varshney and Kumar [1984], Chiang [1987]), whereas stress-induced birefringence was addressed by Okamoto, Hosaka and Edahiro [1981], Okamoto, Edahiro and Shibata [1982], Rashleigh and Marrone [1983b], Stolen [1983], Barlow and Payne [1983], Chu and Sammut [1984], Hayata, Koshiba and Suzuki [1986], Vassallo [1987], Chiang [1992]). In the case of the elliptical-core fiber of which the slow axis of the birefringence is aligned with the minor axis of the elliptical core, the birefringence for fibers with small-core ellipticity, ( a / b - 1) << 1 and in the vicinity of the cutoff frequency can be approximated, according to Payne, Barlow and RamskovHansen [1982] and Rashleigh [1983a], by

2~(a )

A/3~0.2-~--

~-1

(An) 2,

(3.1)

where 2a and 2b are the major and minor core diameters and An is the refractive index difference between the core and cladding regions. For more elliptical cores the birefringence is independent of the core ellipticity, 2yg

Aft ~ 0.25--X- (An) 2 .

(3.2)

Stress birefringence is induced by the photoelastic effect during the fiber manufacturing process (Kaminow and Ramaswamy [1979]). The main idea is to use materials exhibiting slightly different thermal expansions coefficients, which induce permanent stresses on the fiber core (Eickhoff [1982]) Aft -

C

2zr

1-Vp ~

AaAT~

A -B

A+B'

(3.3)

where C = 0.5n 3 ( p l , - p l 2 ) ( 1

+ ~))

(3.4)

is the strain-optical coefficient, 2A and 2B are the major and minor axes of the elliptical cladding, no is the mean refractive index of the fiber, pll and pl2 are the components of the strain-optical tensor, vn is Poisson's ratio, Aa is the difference between the expansion coefficients of the cladding and of the outer medium, and AT is the difference between the softening temperature of the cladding and

24

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 3

the ambient temperature. Asymmetrical lateral stresses may also be induced by surrounding the circular core by stress applying parts, that is, zones with bow-tie shapes (bow-tie HB fibers) or two circles (PANDA HB fibers). 3.2. INDUCED (EXTRINSIC) BIREFRINGENCE

Birefringence can also be created whenever a fiber undergoes elastic stresses resulting from external perturbations (Ulrich and Rashleigh [1982]) such as hydrostatic pressure, longitudinal strain, squeezing, twisting, bending, and other factors acting on the fiber from outside. The perturbation induced in the permittivity tensor through the photoelastic effect lifts the degeneracy of the linearly polarized modes and induces extrinsic birefringence. Several important perturbations and the resulting induced birefringences include internal stress anisotropy, external lateral force, bending, transverse electric field, elastic twist, axial magnetic field, axial strain, temperature, and hydrostatic pressure, which were summarized by Rashleigh [1983a] and Ulrich [1994], and also addressed by Sakai and Kimura [1981 ], Mochizuki, Namihira and Ejiri [1982], Rashleigh and Marrone [1983a], Varnham, Payne, Birch and Tarbox [1983], Imai and Ohtsuka [1987], Wu and Yip [1987], Garth [1988], Vengsarkar, Fogg, Murphy and Claus [1991], Ruffin and Sung [1993], Bock, Wolifiski and Eflimov [1996], Fontaine, Wu, Tzolov, Bock and Urbaficzyk [1996] and Storoy and Johannessen [ 1997]. It should be noted that, contrary to the long-standing assumption that birefringent fiber has uniform elastic properties, the birefringence of HB polarization-maintaining fibers depends primarily on external deformations induced by hydrostatic pressure and axial strain (Bock, Domafiski and Wolifiski [1990]). This will be discussed in detail in the following sections. 3.3. NONLINEAR BIREFRINGENCE

Nonlinear bireffingent properties and polarization changes in fibers are discussed by Botineau and Stolen [1982], Caglioti, Trillo and Wabnitz [1987], Kimura and Nakazawa [ 1987], Agrawal [ 1995], and Garth and Pask [ 1992]. Nonlinear or selfinduced birefringence relies on the nonlinear coupling between the orthogonally polarized components of an optical wave, which changes the refractive index by different amounts of Anx and An,, due to nonlinear contributions Anx = n2 [IEx[ 2 + 5

]

(3.5)

I, w 3]

BIREFRINGENCE IN OPTICAL FIBERS

Any

=

n2 [[Ev 2

2 [g,~12]

25

(3.6)

-k- -5

where n2 is the nonlinear-index coefficient given by the formula

rt2 :

3 - - R e ( . (Z x3x)x x ) x 5n

(3.7)

-

and defined by one component of the fourth-rank nonlinear susceptibility tensor

xxx x In the case of silica fibers, where the dominant nonlinear contribution is of electronic origin, the nonlinear-index coefficient has a value 1.2 • 10- 2 2 m2/V or 3.2 x 10-22 m 2/W. As the wave propagates along the fiber, it acquires an intensity-dependent nonlinear phase given by 2Jr nl_ 2Jr --ff-Lgnx = - ~ - L

r

r -, LE,-[- + 3 2

E,12]

(3.8)

. = T L A n y = T L n 2 [E,.] + ~ E,-2].

(3.9)

2~r

2~

n2

2

The first term in the brackets is responsible for self-phase modulation (SPM), whereas the second term results from the phase modulation of one polarization (wave) by the copropagating orthogonal polarization (wave) and is responsible for so-called cross-phase modulation (XPM). The XPM-induced nonlinear coupling between the field components E, and E ~ creates nonlinear birefringence, which changes the state of polarization (SOP) if the input light is elliptically polarized. The nonlinear coupling between the two orthogonally polarized components of the optical wave is referred to as nonlinear or selfinduced birefringence, and it has many device applications.

3.3.1. Optical Kerr effect The optical Kerr effect involves transmission changes of a weak probe beam due to nonlinear birefringence induced by a strong pump beam. In all-fiber realization both pump and probe beams are linearly polarized at 45 ~ to each other at the fiber input9 A crossed polarizer blocks the fiber output in the absence of the

26

POLARIMETRICOPTICALFIBERS AND SENSORS

[I, w 3

pump beam. The presence of the pump modifies the phase difference between orthogonal components of the probe at the output of a fiber of length L: Aq) = A0 e +Aq) nl =--~L2Jr [Ant + n K Epumpl 2]

(3.10)

where An e - n x - n y is the linear birefringence of the fiber, nn is the Kerr coefficient, and Epump is the pump intensity. The probe transitivity Tp is given by Tp = sin 2(AO/2).

(3.11)

3.3.2. Pulse shaping

Nonlinear birefringence induced by a pulse can be used to change its own shape. Here, the signal itself produces the nonlinear birefringence and modifies its own state of polarization (SOP). If an input beam is polarized in such a way that it excites both orthogonal polarizations, the field components Ex and E y change the refractive indices nx and n y by the amount Anx (3.5) and Any (3.6). Hence, the resulting phase shift at the fiber output is given by A0nl - 2JrLn2 [

~, 3

2 ] levi-[Eyl 2 ,

(3 12)

and the transmitted power depends on input polarization angle and input power. Generally speaking, an accurate description of the polarization effects in HB fibers requires simultaneous consideration of both intrinsic linear birefringence and induced nonlinear birefringence effects. One of the most spectacular effects is so-called polarization instability, which manifests itself as large changes in the output SOP when the input power or input SOP is changed slightly. When the input beam is polarized close to the slow axis, the nonlinear birefringence adds to the intrinsic birefringence and the fiber is more birefringent. When the input beam is polarized close to the fast axis, however, the nonlinear birefringence decreases the intrinsic birefringence. When the input power is close to the critical power defined by

Pcr- 3 A/3,

(3.13)

2 ), where A/3 is the intrinsic (linear) birefringence and ), is a nonlinear parameter, the effective beat length becomes infinite (intrinsic birefringence vanishes). A further

I, w 3]

BIREFRINGENCE IN OPTICAL FIBERS

27

increase in the input power makes the fiber birefringent again, but the roles of the slow and fast axes are reversed. Hence, any slight changes in the input power close to the critical power cause large changes in the output SOP. The polarization instability first observed by Trillo, Wabnitz, Stolen, Assanto, Seaton and Stegemann [ 1986] proves that both birefringence axes (slow and fast) of an HB PM fiber are not entirely equivalent. This effect can also lead to chaos in the output SOP if the intrinsic birefringence is modulated along the fiber length (Kimura and Nakazawa [ 1987]). Birefringence can also significantly affect soliton propagation in optical fibers (Essiambre and Agrawal [ 1997]). In HB fibers the beat length (inverse of intrinsic birefringence) is much lower than the nonlinear length (LB << LNL), and solitons remain stable whether launched close to the slow or fast axis. By contrast, in weakly birefringent fibers (LNL << LB), solitons remain stable along the slow axis but become unstable along the fast axis. Under certain conditions the two orthogonally polarized solitons can trap one another and move at a common group velocity despite the polarization dispersion. This phenomenon, called soliton trapping, was initially observed by Islam, Poole and Gordon [1989], and can be applied in soliton-dragging logic gates as proposed by Sauer, Islam and Dijali [1993]. Other applications of the nonlinear birefringence than nonlinear switching include high-resolution distributed fiber sensing (Zhao and Bourkhoff [1993]) and passive mode-locking of fiber lasers. 3.4. BASIC M E A S U R E M E N T S : B I R E F R I N G E N C E A N D P O L A R I Z A T I O N M O D E DISPERSION

Modal birefringence Aft and polarization mode dispersion (PMD) A r/L are the most important parameters characterizing HB fibers. Both parameters are interrelated according to Ar L

d(Afi) dtO

(

1 AnHB+ tO c dtO

(3.14) '

where A r/L is usually expressed in units of picoseconds per kilometer of fiber length, AnHB is the differential effective index of refraction for the slow and fast polarization modes, and oa = 2:rc/)~ is the angular frequency of light. For HB fibers with stress-induced birefringence AnHB is nearly wavelength independent (Bock and Urbaficzyk [1993]), and the chromatic dispersion of the modal birefringence is negligible. Hence, for this type of fiber the measurements of birefringence and PMD are equivalent. This situation totally changes for

28

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 4

elliptical-core HB fibers with geometrical birefringence AnHB, where both measurements are required to characterize the fiber. To date, many methods have been reported for measuring the modal birefringence of HB fibers. These include Rayleigh scattering (Eickhoff and Krumpholz [1976]), magneto-optical and elasto-optical modulation (Calvani, Caponi, Cisternino and Coppa [1987]), wavelength sweeping (Kikuchi and Okoshi [1983]), optical time-domain reflectometry (OTDR) twist-induced variations of birefringence (Huang and Lin [ 1985]), and white-light interferometry (Bock and Urbaficzyk [1993]). Birefringence measurements under stress were performed by Wolifiski [1993] and Wolifiski and Bock [1995] and are discussed later. Numerous methods were also developed to measure PMD (e.g., Mochizuki, Namihira and Wakabayashi [ 1981 ], Barlow, Ramskov-Hansen and Payne [ 1981 ], Burns and Moeller [1983], Shibata, Tsubokawa, Ohashi, Kitayama and Seikai [1986], Okamoto and Hosaka [1987], Th6venaz, de Coulon and Von der Weid [1987], Tsubokawa, Shibata, Higashi and Seikai [1987], Bock and Urbaflczyk [1993], Menyuk and Wai [1994], Karlsson [1998]. Other methods of birefringence and PMD measurements were reported by Papp and Harms [1975], Smith [1979], Yen and Ulrich [1981], Eickhoff, Yen and Ulrich [ 1981 ], Kim and Choi [ 1981 ], Rogers [ 1981 ], Ross [ 1982], Okamoto, Edahiro and Shibata [1982], Rashleigh [1983b], Eickhoff and Brinkmeyer [1984], Nakazawa, Shibata, Tokuda and Negishi [1984], Shibata and Tokuda [1984], Ohtsuka, Ando, Imai and Imai [1987], Farhadiroushan and Youngquist [1990].

w 4. Deformation Effects in Highly Birefringent Fibers 4.1. SYMMETRICALDEFORMATIONEFFECTS The modal behavior of the lowest-order mode HB fibers under various external deformations is of special interest for sensors and device applications (Rashleigh [1983b], Huang, Blake and Kim [1990], Wolifiski [1994]). Several physical quantities can be measured on the basis of two-mode HB fibers, including hydrostatic pressure, strain, vibration, temperature, acoustic wave, and other factors. A symmetrical deformation effect influences propagation constant/~i in every mode due to the changes in fiber length (L) and the refractive indices of the core and cladding. This leads to changes in the phase q)i - AfiiL along the fiber: 6~i = 6(Afii) L + Afii6L,

(4.1)

I, w 4]

Fig.

I(z

DEFORMATION EFFECTS IN HIGHLY BIREFRINGENT FIBERS

29

11. Deformation effect in an HB fiber modulates light intensity after the analyzer: = L) = 89[ 1 + cos 2 a 9 cos 20 + [)'i sin a 9 sin 20 9 cos A q~i ], where A q~i signifies the differential phase o f the light exiting the HB fiber and 7i is a mutual correlation function.

where i=0, 1,x,y, and 6q)~- and 6q~, are responsible for changes in the intensity distribution of light exciting the fiber under deformation, and 6
6ci)i = [ O(A[3i) . + A[3i OL ]

2:r - e

(4.2)

If two external perturbations denoted by ~, and ~ are simultaneously acting on the fiber, the resultant phase shift will be

~i

2x "-- ~ ~

2~8~.

-t- Ti,~

=

Ai,ff6~ + A,.~6~

(4.3)

with

OL 2zr _ 0(A/3i). Ai,r - Ti,~ -~_ L + A[3i--~,

2:/" _ 0(A/3;) L + A[3i OL Ai.:. - Ti.:: 0-----~_ -~.

(4.4)

The quantities Ti,z (i = 0, 1,x,y and Z - ~,~) have the dimension of the measurands ~ and ~, whereas Ai,/ have the inverse dimension. These are experimentally measurable parameters and they determine the sensitivity of the sensor to a given external perturbation. In the case of an elliptical-core HB fiber (with ellipticity e - a/b), Huang, Blake and Kim [ 1990] developed analytical expressions for deformation-induced

30

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 4

birefringence changes 6(Afii) and differential phase shifts 6(A~i) in terms of only Afii and its wavelength dispersion (5(Aft/) _ [ O(nl - n2) 1 (3~ [ 0~ nl - n2

10V V 0~

Afi~

;~ o v O(AI3,) --, v o~ o~

(4.5)

where IOV lob 1 ( Onl On2) V O ~ - b O ~ + n2 - n 2 n , --ff-~ - n 2 - ~ ,

(4.6)

V = L~bv/n ~ - n 2 is the normalized frequency, /1. is the optical wavelength, b is the semiminor axis of the elliptical core, n l and n2 are refractive indices of the core and cladding, respectively, and ~ signifies a deformation. Hence, the differential phase shift 6(A q~i) is given by the formula

6(Aq)i) _ { [ g3~

10(nl-n2) n, - n2

O~

I OV 1

V -~

0l}

l+ --~

~ OV O(Afii) l.

A[~i

V O~

O~

(4.7) Both formulas hold true for cases in which the core ellipticity (e= a/b) remains constant under the deformation. It means that the deformation should be isotropic over the fiber cross-section, and the fiber should have homogeneous elastic and thermal characteristics so that no anisotropic transverse strains and refractive index changes will be induced. The effects of some isotropic radially symmetrical perturbations including deformations induced by hydrostatic pressure, axial strain, and then temperature on mode propagation in the weakly-guiding two-mode HB fiber are reviewed in following sections. Then, as an example of asymmetrical perturbation, a twisting effect in the HB fiber is discussed and, in particular, its impact when combined with isotropic deformation as hydrostatic pressure or axial strain acting on the HB fiber. 4.2. HYDROSTATIC P R E S S U R E

Contrary to the long-standing assumption that the birefringent fiber has uniform elastic properties, the birefringence of HB PM fibers depends primarily on external hydrostatic pressure, as shown experimentally for both bow-tie and elliptical-core HB fibers by Bock and Domafiski [1989] and Bock, Domafiski and Wolifiski [1990]. An interesting application of HB single-mode fibers for measuring high hydrostatic pressure up to 200 MPa was reported by Bock,

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Wolifiski and Barwicz [ 1990]; it is based on the direct effect of hydrostatic stress on the polarization birefringence of the fundamental LP01 mode in these fibers. Theoretical studies as well as birefringence measurements based on the Rayleigh scattering carried out by Bock, Domafiski and Wolifiski [1990] indicated that in the case of HB single-mode bow-tie fibers, hydrostatic pressure up to 100 MPa decreased beat length with a mean coefficient of 1/LBoldLB/dp] - 0.15%/MPa. A similar decrease of beat length was observed when uniaxial longitudinal strain was applied to the bow-tie fiber. Further experimental research on the effect of hydrostatic pressure on mode propagation was conducted for a HB two-mode bow-tie fiber by Bock and Wolifiski [1990a]. They found that hydrostatic pressure produced a large phase shift between both orthogonal polarizations in fundamental and second-order modes and also between the two lowest-order spatial modes in the HB bow-tie fiber. It appeared that pressure phase sensitivity between two spatial modes, ~ 1.3 rad/MPa.m, was about one order lower than for polarization modes (table 3). Further investigations on pressure effects in HB PM fibers were carried out by Xie, Dabkiewicz and Ulrich [ 1986], Bertholds and Diindliker [1986], Jansen and Dabkiewicz [1987], Chiang [1990], Passy, Gama, Gisin and von der Weid [1992], Bock and Eftimov [1994], Bock and Urbaficzyk [1996], and Clowes, Syngellakis and Zerras [ 1998], including also liquid crystal HB fibers (Wolifiski [1999]). In a two-mode HB fiber (i = 0, 1,x,y), the variation of the phase difference under hydrostatic pressure (6~ = 6p) may be written as 6(AOi) _ Afii OL 0 (Aft/) 6/) ~+L~p

(4.8)

In terms of the birefringences Aft;, one can define four beat lengths LB, i as 237

LB, i - Afii

(i = O, 1,x,y).

(4.9)

In the case of the LP0~ mode, the beat length (LB, 0) has a simple interpretation as the periodicity of the radial scattered intensity. For incident polarization with 0 = 45 ~ according to birefringence axes at z---0, the polarization becomes circular for Aq) = 3:/2, linear with 0 = - 4 5 ~ for A0 = 3:, circular for Aq) = 33:/2, and linear again with 0 = 45 ~ for Aq) = 23:. The length LB corresponding to the reproduction of the original linear polarization is called the "beat length." Therefore, the beat length can be observed directly by means of the dipole (Rayleigh) scattering from the fiber. Since the radiation pattern of a dipole has a null along

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POLARIMETRIC OPTICAL FIBERS AND SENSORS

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Fig. 12. The method of beat length determination under hydrostatic pressure in a HB fiber based on direct observation of dipole (Rayleigh) scattering from the fiber at ~ = 633 nm.

the dipole axis and a maximum normal to the axis, a fiber viewed along the direction of the incident polarization exhibits a series of dark and bright bands with period 45 ~ It is thus possible to determine Aft from the observed beat length. The birefringence beat-length parameter was first determined by Bock, Domafiski and Wolifiski [1990] for an HB single-mode fiber subjected to hydrostatic pressure conditions. The experimental procedure (fig. 12) involved the application of the Rayleigh scattering method for direct measurement of beat length (LB,0) variations in the LP01 mode under hydrostatic pressure in the range up to 100 MPa. By pressure variations the fiber image created by the Rayleigh (dipole) scattering from the fiber was displaced: dark and bright fringes moved only while the pressure was changed. The direction of the movement unmistakably defined the direction of the beat-length changes under pressure. The results indicated that in HB single-mode bow-tie fibers, beat length LB, 0 in the LP01 mode decreased with pressure characterized by the mean coefficient of 1/ZB,oldLB/dpl-- 0.15%/MPa, and consequently, polarization birefringence Aft0 increased. The same experiment was also repeated for elliptical-core HB fibers, confirming the direction of birefringence changes under hydrostatic pressure. Based on the results, a more developed semiphenomenological theory

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Fig. 13. Far-fieldradiation pattems for three values of hydrostatic pressure from the HB-800 bow-tie fiber as a result of superposition of LP01 and LP~I spatial modes (there is a phase shift of :r between the site patterns) (Bock and Wolifiski [1990a]).

Fig. 14. Far-fieldradiation pattems for three values of hydrostatic pressure from the HB-800 bow-tie fiber for three types of launching conditions: (a) polarization coupling in the fundamentalLP01 mode, (b) polarization coupling in the second-orderLP~ mode, and (c) intermodal coupling between both spatial modes (LP01-LP~I) (Bock and Wolifiski [1990a]). was proposed by Bock, Domafiski and Woliflski [1990] that accounted for the observed direction of birefringence changes under hydrostatic pressure. Hydrostatic pressure effects on mode propagation in HB two-mode bowtie fibers were investigated by Bock and Wolifiski [1990a] (figs. 13 and 14). Here, four modes: two orthogonal linearly polarized eigenmodes for each of the fundamental (LP01) and the second-order (LP~!) spatial modes are guided, and four possible types of interferences (polarization and intermodal) were observed. Hydrostatic pressure produced a large shift between both orthogonal polarizations in fundamental and second-order modes and also between the

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Table 3 Hydrostatic pressure effects in HB two-mode fibers

Beat length under hydrostatic pressure Without pressure

L ( p = O ) : L ~B,i n

Under pressure

L ( p > O) = LB, i(n + k)

Beat length under pressure

LB, i(p) = L 0B , i[1 + p L 0 , i/(TpiL)] -1 ,

Pressure-induced change in beat length

ALB, i(P) = -LOB, i[ 1 + Tp,iL/(pL 0, i)]-1

Birefringence under hydrostatic pressure Afli(p ) = Aft? + p2Jr/( Tp,iL )

Phase changes under hydrostatic pressure 6(Ar _ 2:r OL ~ 2Jr Op rp, i + A[ji Op - Tp, i Experimental data, HB bow-tie fiber (L = 195 mm, p = 100 MPa) Tp, o = Tp, l ~ 3 . 7 M P a , Tp, x = Tp,y = 2 4 . 0 M P a A/3o ( p ) / A f l ~

Afix(p)/Afl~

= 0) ~ A f l l ( p ) / A f l ~

= 0) : 16.0%

= O) ~ A f l y ( p ) / A f l ~

= 0 ) = 0.1%

i = 0, 1:

d(AOi)/Lp ~ 9.0 rad/MPa.m

i = x,y:

d(AOi)/L p "~ 1.3 rad/MPa.m

two lowest-order spatial modes in the HB bow-tie fiber. The pressure-induced effects were expressed in terms of birefringence changes and pressure phase sensitivity, both described by the experimental parameter Tp (i = 0, 1,x,y). The results indicated that pressure phase sensitivity between two spatial modes is approximately one order of magnitude lower than for polarization modes. Table 3 summarizes all the results of hydrostatic pressure effects on mode propagation obtained for HB single-mode and two-mode bow-tie fibers by Bock, Domafiski and Wolifiski [1990] and Bock and Wolifiski [1990a], where L is the length of the fiber, n is a number and k has to be an integer, p is pressure, and Tp (i = 0, 1,x,y) is an experimental parameter corresponding to the amount of pressure required to induce a 2:r phase shift of the polarized light observed at the output. 4.3. A X I A L S T R A I N

The influence of strain on polarization and birefringence properties of HB fibers

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was analyzed by Varnham, Barlow, Payne and Okamoto [1983], Blake, Huang and Kim [1987], Blake, Kim and Shaw [1986], Blake, Huang, Kim and Shaw [1987], Zhang and Lit [1992], Farahi, Webb, Jones and Jackson [1990], and Bock, Wolifiski and Efiimov [1996].

4.3.1. Differential phase evolution and birefringence changes The effect of longitudinal stress on the mode coupling is to modulate the relative phase retardation between the two orthogonal polarizations (in LP01 and LPll modes) and between the spatial LP01 and LP~I modes. When a light beam propagates along a fiber of length L, the phase difference due to polarization or modal birefringence is given by

cI)i : AfiiL,

i = O, 1,x,y.

(4.10)

Similarly to eq. (4.2), the variation of Aq~/under longitudinal axial strain (e) can be expressed as

6(A~-) _ A[3iOL O(Afli) 6e ~ + L 0--------7-'

(4.11)

where modal or polarization birefringence Afii (i = 0, 1,x,y) depends on the value of deformation (longitudinal axial strain), as demonstrated by Bock, Domafiski and Wolifiski [1990]. They demonstrated that the birefringence sensitivity to strain is then expressed in terms of the parameter Ti, e. It also appears that some types of birefringences may increase with strain (as in the case of hydrostatic pressure), but some birefringences may also decrease with strain depending on the HB fiber used. The general formula describing the birefringence sensitivity to strain is then expressed in terms of the parameter T/, as

2zr Afli(e) = A/3i~ + sgn d(A/3;) e - de L Ti.~.'

(4.12)

where A/3~ signifies unperturbed modal (polarization) birefringence of a fiber, and the function sgn[d(Afli)/de] has two values: +1 o r - 1 depending on the sign of changes in the relative modal (polarization) birefringence with strain. Ti, e is the amount of strain (e) required to induce a 2zr phase shift of a polarized light observed at the output, and L is the total optical path of the fiber.

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From eq. (4.2) the phase changes 6q~i due to the strain e = 6L/L are

~)~)i

=

2r [ O(Afii) OL ] A i, e6 = ~ e = [ 06 L + A [3i - ~ 6.

(4.13)

The second term on the right-hand side of this expression is negligible with respect to the first, so that

6q~i ~- O(Afii) L e - O(Afii) 6L. Oe Oe

(4.14)

As seen from this expression, the phase changes of the polarimetric and polarization sensitive interferometric responses are proportional to the absolute elongation 6L and are independent of the length L of the sensing region. This means that the parameters Ai, 6l and Ti,,~t are constant and length independent. For an e-core fiber Huang, Blake and Kim [1990] found that 6(A05i) = [0.953Afii + 0.605~ O(Afii)o~ 6L,

(4.15)

where ~, is the wavelength, whereas for a bow-tie fiber the phase shift for the polarimetric response of the fundamental mode was reported by Varnham, Barlow, Payne and Okamoto [1983] as follows: 6(A05o )

-

v2 - Vl Afio 6L, a 2 - a l T - Ts

(4.16)

where T is the ambient temperature, Ts is the softening temperature, and a l, Vl and a2, v2 are the expansion coefficients and the Poisson's ratios of the cladding and stress-producing regions, respectively. Under the influence of a longitudinal axial strain, eq. (4.14) can be approximated with the use of formulas (4.12) in terms of the experimental parameter T/, e: 6(A cI)i) _ Afii OL d(Afii) 2~ _~ sgn d(Afli) 2~ 6e O-e + sgn d - - - ~ ' T i , e d--------~ " T/,E

(4.17)

4.3.2. Optical fibers under strain 4.3.2.1. Birefringence changes under strain. For i - 0 (LP01 mode), eq. (4.12) was experimentally verified for different lengths L of the fiber (Bock and

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Table 4 Axial strain effects in HB bow-tie and elliptical-core fibers

Beat length LB, i(e ) = L B,i 0 [1 +sgn{d(Afii)/de} eL~ i/(Ti cL)] -1

Birefringence A[~i(6 ) = A[~0 + sgn{d(Afii)/de } 9 e . 2Jr/( Ti. eL ) HB-800 bow-tie fiber:

sgn{d(A/3i)/de} = + 1

Andrew elliptical-core fiber:

sgn { d(A/31 )/d e } = + 1

sgn { d( A[3o )/ d e } = -1

Wolifiski [ 1990b]). It appeared that in the case of York HB bow-tie fibers operating in a single-mode regime, uniaxial longitudinal stress increased birefringence (sgn[d(Afi)/rmde] = +1) with a mean coefficient of 1/A~[d(A[3)/dc] = 1.8%/mE. Similar behavior was observed for PANDA fibers. This effect is caused by the difference between the compressibility coefficients for the stressapplying regions and those for the surrounding cladding of an HB circularcore fiber. However, for Andrew elliptical-core fibers operated at the fundamental mode LP01, uniaxial longitudinal strain reduces the polarization birefringence of the LP01 mode (sgn(6Afii/6e)=-1) with a mean coefficient of 1/Afi0[d(Afi)/de] - 0.15%/me, which is considerably less than the previously cited value for the HB bow-tie fibers, as was also confirmed by Huang, Blake and Kim [1990]. This feature of Andrew fibers, observed due to the Rayleigh measurement of polarization birefringence under strain (Bock, Domafiski and Wolifiski [1990]) is of particular interest, since hydrostatic pressure applied to the same fiber increased its polarization birefringence. Table 4 summarizes axial strain effects in HB bow-tie and elliptical-core fibers.

4.3.2.2. Strain sensitivity of HBfibers. The following three classes of HB optical fibers have been examined by Bock, Wolifiski and Eftimov [1996]: (1) York bow-tie fibers: HB-600 (single-mode regime) and HB-800 (two-mode regime); (2) Andrew elliptical-core HB fibers: E-type fiber having a core size of 1 • ~tm and a cutoff wavelength about 600nm, and D-type fiber having a core size of 1.25• used in the fundamental LP01 or second-order LPll operating regime; and (3) PANDA fibers: SM63 (single-mode regime) and SM85 (twomode regime).

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Table 5 Sensitivities to axial strain deformations of different HB fibers operating in fundamental (LP01) and second-order (LP11) polarization-mode regimes, and intermodal regime (LP01-LP1 l), at 633 nm (Te is the period of the output signal)

T,,,L (~tm)= TaeL (/~strain.m)

Type of fiber LP01

LPli

LP01 ~ LP 11 X

Andrew

York (bow-tie)

PANDA

Y

E-type

1100 a

154

112

D-type

900

420

114

88

HB 600

64

HB 800

53

72

373

204

SM 63

50

SM 85

38

45

e

o

e

o

43

40

425

950

a The results were obtained for 845 nm.

10 8 o 41mm

6 "~

,

4

0 0

1000

2000

3000

4000

7mm

' 5000

Strain [microstrain] Fig. 15. Typical response to any symmetrical deformation of two different lengths of HB fiber. The

data provided are for strain sensor based on HB-600 bow-tie (Fibercore) fiber.

Table 5 contains numerical values for the measured period of an output signal T~ multiplied by the length of the sensing element for different HB fibers operating in the fundamental (SM, LP01) and second-order (TM, LPll) polarization-mode regimes, and in the two-mode regime (LP01-LPll) with two polarizations (X, Y) measured at a wavelength of 633 nm. For an SM85 PANDA fiber both pairs of second-order spatial modes (even, e and odd, o) are presented at the wavelength of 633 nm. A general feature of an HB fiber is that the output signal of the polarimetric strain sensor based on HB fibers is a periodic function of longitudinal strain, as demonstrated by Bock, Wolifiski and Eftimov [1996] and shown in fig. 15.

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This response of a polarimetric sensor is typical of any symmetrical perturbation acting on the sensing fiber. Since e = 6L/L, T~. ~ = T~.,~L/L. Since Ti, ,~L = const., it follows that L. T/, E = const. 9Jl,

(4.18)

and this formula was initially established for bow-tie fibers (Bock and Woliflski [1990b]), where L is the length of the sensing part of an HB fiber. Hence, the period of this function Ti, e can be conveniently adjusted by choosing an appropriate length of the sensing fiber. The dimension of T;,~. is ~tstrain.m or also ~tm, and consequently, Ti,~L is dimensionless. For a wavelength of 633 nm, the value of the product L. T;,~ = T~,6L was the lowest for an SM85 PANDA fiber, 38 ~tstrain.m, and achieved the highest level, 900~tstrain.m, for the D type Andrew fiber, both in the fundamental mode of operation. This means that the sensitivity of the Andrew fiber to strain is about an order of magnitude lower than the sensitivity of bow-tie and PANDA fibers. Similarly, for an e-type fiber this product was found to be 1100 ~tstrain.m for the fundamental mode at 845 nm. However, in the prospective strain gauges operating in the two-mode regime, LP01 - LPll intermodal beating in both polarizations X and Y, the sensitivity of the Andrew fiber to strain is generally higher than that of HB circular-core fibers, except for X polarization of intermodal beating in an SM85 PANDA fiber. All the results summarizing the values of the product (4.18) for different fibers operating in various modes are presented in table 5. The foregoing three general classes of HB fibers offer a variety of possibilities to construct fiber-optic strain gauges precisely adjusted to particular needs and applications. 4.4. N O N S Y M M E T R I C D E F O R M A T I O N EFFECT: TWIST

This section provides an analysis of twisting effects in HB fibers, mostly based on the results obtained by Ulrich and Johnson [1979], Ulrich and Simon [1979], Fujii and Sano [ 1980], Monerie and Jeunhomme [1980], Smith [1980], Monerie and Lamouler [1981], Okoshi, Ryu and Emura [ 1981], Barlow and Payne [ 1981 ], Barlow, Ramskov-Hansen and Payne [ 1981 ], Barthelemy and Arnaud [1982], Sakai, Matsuura and Kimura [1983], Itoh, Saitoh and Ohtsuka [1987], Xiaopeng, Hao and Jingren [1991 ], and Brown and Bak [1995]. We also discuss simultaneous twist and hydrostatic pressure as well as uniaxial longitudinal strain effects in twisted highly birefringent fibers from the standpoint of the Marcuse mode-coupling theory. The problem is analyzed in terms of local normal modes

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POLARIMETRIC OPTICAL FIBERS AND SENSORS

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of the ideal fiber and in the limit of weak twist, where large linear birefringence dominates over twist effect, and therefore twist coupling between local modes is not effective. For simplicity we consider in this section an HB fiber that can support fundamental polarization eigenstates (i = 1) and whose polarization directions coincide with the principal birefringence axes. Hence, a linearly polarized light input along either axis will maintain its polarization state along the length of the fiber. When the HB fiber is twisted, however, the output polarization state is changed due to both a geometrical effect and an elasto-optic effect.

4.4.1. Theory of coupled local modes 4.4.1.1. Expansion in terms of local normal modes. The guided (and also radiation) modes propagating in optical fibers form a complete and orthogonal set. It is possible to express the general field solution of Maxwell's equation for an arbitrary refractive index distribution n = n(x,y,z) in terms of modes of the ideal fiber. Modes are characterized by the fact that their z dependence can be expressed as exp(-ifiz). The ideal fiber is defined by a refractive index distribution no = no(x, y) that is independent of z. The ideal modes satisfy Maxwell's equation with n replaced by no. Another approach consists of expressing an arbitrary field in the optical fiber in terms of modes belonging to a fictitious waveguide that coincides in width locally at the point z, at which the field expansion is being considered. Instead of no = no(x,y), we now use no = n(x,y,z), where z appears as a parameter in the solutions. This approach is called expansion in terms of local normal modes. Mathematically, local normal mode field solutions are identical in form to the ideal normal mode expressions. However, all the fiber dimensions appearing in these field expressions are functions of z. The local normal modes are orthogonal among each other at each cross-section z along the waveguide, even though they depend on z as a parameter. The local modes form a complete set of modes, but they are not themselves the solutions of Maxwell's equations. The polarization evolution of the fundamental and second-order modes in a twisted HB two-mode fiber can be obtained by using the Marcuse local coupledmode theory (Marcuse [1974]). We can define the local coordinates xt,y ~, which follow the principal axes of the twisted HB fiber. We assume that no coupling occurs between the fundamental and second-order modes caused by the twisting, so that these two spatial modes can be treated independently. Two fundamental polarization modes LP6~I and LP0'1, propagating in the +z direction, couple to each other when the fiber is twisted along its axis.

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Assuming that the electric (and magnetic) field in the imperfect (twisted) fiber is represented as superposition of the modes of the ideal fiber, we can express the general transverse electric field in the imperfect fiber in any given plane z = const. as

Et(x,y,z) = Z

Am(z)Etm(X'Y)'

(4.19)

m

(4.19) where Am(z) = am(z)exp{ifi,,,(z)dz} is the mode amplitude (m = x,y), and tim(Z) is the propagation constant of the local mode, which according to the local mode theory, depends on z. In a twisted HB weakly guiding fiber operating in a single-mode regime, the refractive index profile rotates along the fiber at the same rate as its principal (birefringence) axes. Hence, transverse electrical fields of the two local polarizations of the fundamental mode LPi~1 and LP01 propagating along the twisted fiber follow the direction of the principal axes which rotate along the fiber length. It is evident that these two polarization modes are characterized by propagation constants fix and/3,,, which are independent of z. In this way the only difference between the local modes and the modes of a translationally invariant fiber consists of rotation of electrical field vectors that follow, as a result of the direction of the principle (birefringence) axes of the HB fiber.

4.4.1.2.

Continuum condition. The continuum condition requires that the

twisting effect should be negligible at beat length between the two polarization modes; that is, T(z)

I& -

2~

(4.20)

where r(z) = (1/n2)6n2/gbz is the local twist period (rate) and ] & - fiy]/2sr denotes the beat length reciprocal. In the weakly guiding approximation a twisted HB single-mode fiber can be analyzed in terms of local coupled-mode theory. This formalism is based on the assumption that any deformation of a cylindrical symmetrical fiber will be changing slowly along the fiber. The theory distinguishes the two general cases of weak-mode and strong-mode coupling conditions.

4.4.1.3.

Weak mode coupling. For a sufficiently small value of the twist rate r(z), a weak twist-induced mode coupling between polarization modes LP~ 1 and LPol occurs. Marcuse [ 1974] showed that weak coupling conditions are fulfilled

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POLARIMETRIC OPTICAL FIBERS AND SENSORS

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if the coupling coefficient N 12 between propagating forward LPr~I and LPffl "local normal modes" is much smaller than the average deviation in the propagation constant. The coupling coefficient is given by

_ N12 = N2*1

-

-

oae0 4(fi~-fl~)

/~/~ ~

0n 2 .~ --~Tze~e,.dxdy

(4.21)

with /3x,/3y propagation constants, ex, e,, normalized field distributions of the polarization modes LPr~I,LP0'l, and 6n2/6z denoting the change in the square of the index distribution along the z-axis, and the deviation in average propagation constant is A]~ -- I z

[~i(z) ZF ~-(z)]

dz,

(4.22)

which in the case of a twisted HB single-mode fiber yields Aft = ]fix- [3y]. Hence, the cross-coupling length LN- 1/Nij will be much longer than the local beat length LB - 2at/Aft. This is known as the condition of slow parameter variation, and it means that any irregularities or deformations cause changes in fiber parameters on the length that are much longer than the beat length parameter LB. The coupling coefficient N~2, calculated by Snyder and Love [1983] from equation (4.21), yields N12(z)-r(z). The condition of slow parameter variation then leads to the relation

Ir(z)l << I/3x-gl,

(4.23)

(4.23) where the linear birefringence Aft = ]fix- fiyl is the dominant effect. This condition is equivalent to the assumption that the modal fields of the uniform fiber are slightly affected by the twist and it means that the input linear polarization is practically unaffected by the twist and rotates as if it were rigidly attached to the fiber. The weak mode-coupling limit is realized by the HB polarization-maintaining fiber, the operation of which is based on a particularly large birefringence and which is the subject of our interest.

4.4.1.4. Strong mode coupling. In the case of strong coupling imposed by a strong deformation effect (high twist rate), a significant coupling occurs between local normal modes, and radiation modes and the solution of coupled-mode equations is much more complicated (Snyder and Love [1983]). However, for resonance conditions (energy transfer between two local modes on half beat length due to the at/2 twist)

12rl - I/~x -/3yl,

(4.24)

coupling to the radiation modes can be omitted. The power transmitted by local modes can be calculated from the relation Pi = IE/[2, (i = x,y). Assuming initial

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conditions in the form Ex(O) = 1 and E,.(0) = 0, power distribution along the fiber yields Px = 1 -

~

sin 2 89

and

P,, =

~

sin 2 89

(4.25)

giving for the resonance condition 50% power transfer.

4.4.2. Mode coupling in a twisted HB fiber A straight HB fiber supports linear polarization eigenstates, the polarization directions of which coincide with the principal birefringence axes. Hence, a linearly polarized light input along either axis will maintain its polarization state along the length of the fiber. When the HB fiber is twisted, however, the output polarization state is changed due to both geometrical and elasto-optic effects.

4.4.2.1.

Geometrical effect. Let us consider after Monerie and Jeunhomme [ 1980] the coupling between the linearly polarized local polarization eigenmodes LP~I and LP~l of an HB single-mode fiber, which arises from variations in the orientation of the local birefringence axes along the fiber. As was mentioned previously, this variation of their orientation may be due to a physical twist of the HB fiber, comprising both a geometrical effect and an elasto-optic effect. The idea of the geometrical effect is schematically presented in fig. 16, where x,y represent local (rotating) axes of the fiber, which for the input conditions (z = 0) are identical to the fixed laboratory coordinates. Assuming a uniform twist rate, r [rad/m] defined as r = dO/dz, where 0 is rotational angle, we arrive at Ex(z) = [Ex(0) cos t~ + Ey(O) sin O] exp(-ifi.,z), Ey(z) = [-Ex(0) sin 0 + E:,(0) cos O] exp(-ifiyz).

(4.26)

y(O) \y(z)

E

.,.(:.)

x(O) Fig. 16. Mode coupling in twisted fiber due to geometrical effect: (x,y) represent rotating axes of the fiber; r is twist rate.

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POLARIMETRIC OPTICAL FIBERS AND SENSORS

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Series expansion of the cos, sin and exp functions and reducing to first order yields coupled-mode equations

dz Ey

-r-ifiv

(4.27)

Ev

Hence, the pure geometrical effect leads to mode coupling between the polarization eigenmodes LPorl and LP0 1 of an HB single-mode fiber described by a real mode-coupling coefficient equal to the twist rate r.

4.4.2.2. Elasto-optic effect. The elasto-optic effect was calculated on the basis of perturbation theory developed by Ulrich and Simon [1979]. The transverse electric field in a fiber in any given plane z = const, is

Et(x'Y'Z) = Z

am exp(ifimz)Etm(X,y),

(4.28)

m

where Etm(x,y) is the transverse electric field of the normal mode m of the perfect fiber, tim is its propagation constant, and am is the mode amplitude, which in the perfect fiber will be independent of the longitudinal position z. In any imperfect fiber the amplitude will be z dependent, which leads to coupled-mode equations for the amplitudes am(z):

dam(z) dz

- iZ

tCm.an(z)exp[i(fin - tim) Z],

(4.29)

m

where tCmn describes the coupling coefficients that depend on the nature of a specific fiber deformation. In the case of a single-mode weakly guiding fiber, the perturbation theory gives the following expressions for the coupling coefficients: I (ml)n + 1* m(2) n

tCm, = ~

,

(4.30)

Q

where I,~2,) = - i k V . (~E,,) dx dy,

Q = 4r

/

j2r dr,

k is the flee-space propagation constant of light, no is the refractive index of the medium, fi is the propagation constant of two polarization normal modes LPdl and LPol, J(r) is the radial field distribution function, and ~ is the perturbation

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DEFORMATION EFFECTS IN HIGHLY BIREFRINGENT FIBERS

45

of the dielectric constant of the medium, which causes the mode coupling (in general, 5 will be a second-rank tensor). In the coupled-wave equations the diagonal coefficients toil and tCz2 describe a detuning of the two previously degenerated modes, whereas the off-diagonal coefficients tr and tr describe the mutual coupling of the modes (interchange of power). For the lossless HB fiber we have K'12=tr Assuming that the elastic properties and elasto-optic tensor are uniform throughout the fiber, we can calculate coupling coefficients resulting from twist. When an HB fiber is twisted at the rate r, the only nonvanishing components of the strain tensor ers are e4 = rx and e5 = - r y . These components change the impermeability tensor and, hence, they cause changes in dielectric permeability 5. In the amorphous material of the optical fiber, two elements of 5 become nonzero, namely, e4 =-p44n4rx, 55 =p44n4ry, where p44 = 51 ( p ~ - p 1 2 ) = - 0 . 7 5 is an element of the elasto-optic tensor. After integration it can be shown that for weakly guiding fibers of arbitrary index profile tell = K'22 = 0 and K'12 = -tr - -in~v44r/2. Since the coupling coefficients tqz, tr are imaginary, they cause circular birefringence (optical activity). This results from the coupling between a longitudinal (E;~,) and a transverse field component (Ef,) which are :r/2 out of phase. The optical activity defined as a =-2itr is proportional to the twist rate, a = gr,

(4.31)

where g = -n2p44 ~ 0.16 (for silica fibers) is a photoelastic coefficient. It is evident that the elasto-optic effect introduces a coupling coefficient equal to -gr/2. Thus, the total influence of both geometrical and elasto-optic twist effects will result in a real mode-coupling coefficient proportional to the twist rate and equal to (1 - g/2) r.

4.4.2.3.

Coupled-mode equations. Assuming the z dependence of the two orthogonal quasilinear polarized field components HElix and HEll, of the propagation mode HElm (or LP01) in the form El(Z) = Ei exp(i/3/z) (i = x,y), the coupling between the modes can be described by the following coupled-mode equations:

d

dz Ey(z)

i IN,, N,2

N21 N22

E,(z) ]' E,(:)

(4.32)

where i = x/Z--1 and Nij (i,j = 1,2) are coupling coefficients with N21 =N~2. The coupled-mode equations can be solved practically only when all the

46

POLARIMETRICOPTICAL FIBERSAND SENSORS

[I, w 4

birefringences under consideration are constant along the z-axis. This depends on the reference axes chosen and also on the nature of the base vectors. In the base represented by the linearly polarized eigenstates of the non-disturbed HB fiber (HElix, HElly) and under the influence of both geometrical and elasto-optic twist effects, the coefficients defined in local coordinates are expressed as N i l =/~x,

N22 = fly,

N12 = i ( 2 - g)r/2,

(4.33)

leading to the following form of the coupled-mode equations:

dEx,(Z) (2 - g ) r - ifixE~,(z) + ~ E , . , ( z ) dz 2 dE y, (Z) _ ifivEr,(Z ) _ (2 - g ) r E ~ , ( z ) dz 2 "

(4.34)

Assuming a uniformly anisotropic fiber structure, hence z-independent coupling coefficients, one can define a difference in propagation constants fii (i - x,y) of the eigenpolarization modes HElix and HEllv as the total birefringence

(4.35)

6 3 = v/(N11 - N22) 2 + 4lN12l

or after appropriate substitution

(4.36)

6fi = v/(Afi) 2 + [(2 - g)r] 2.

The coupled-mode equations (4.34) can be easily solved by the Laplace transform method, giving the two polarization modes at the output end of the fiber at a distance z

6fiz -i~-~E~-(0) Aft sin --~ 6fiz , Ex,(Z) = e -i/2q~x+l~-)z E~(0)cos --~sin 6fiz - ~ + (2 -6~)rE,.(O) -,. [E, 6fizAfi 6fiz(2-g)r Ey,(Z) = e -~2(l~-'+/~.)~ ,(0)cos --~ + i~--fiE,.(0) sin 2 6 ~

E~(0)sin

6_~_]

'

(4.37) where Ex(O) and E :,(0) signify initial conditions and are the electric fields in the two polarizations at the input end of the fiber (z = 0).

I, w 4]

DEFORMATION EFFECTS IN HIGHLY BIREFRINGENT FIBERS

47

4.5. TWIST UNDER PRESSURE/STRAIN Under the influence of external deformation, such as hydrostatic pressure p or axial strain e, the polarization birefringence of a non-twisted HB bow-tie fiber changes according to the following formula, similar to eq. (4.12):

Aft i(x)=A[3i,o+sgn d(

')

.x.

2~ LTi..,. '

(4.38)

where x (x =p or x = e) defines the specific deformation. For a pressure-induced deformation, the function sgn(6Afii/6p) is always positive (yields +1). However, for strain-induced deformation, sgn(6Afi;/6e) has two values, + 1 or -1, depending on the sign of changes in the relative modal (polarization) birefringence with strain. When a light beam propagates along a fiber of length L, the phase difference due to polarization is given by A 0 = 6fi(x, r)L, where 6fi(x, r) is defined by eq. (4.36). Wolifiski [ 1993] and Wolifiski and Bock [1993, 1995] considered deformation effects induced by simultaneous hydrostatic pressure (axial strain) and twisting on mode propagation in an HB single-mode fiber. These effects cause modulation of the differential phase according to d(A0) dx

0r

2 + [(2- g)r] 2 V/ 0x L+ (Aft) 2+[(2

g) r] 20L Ox'

(4.39)

which can be considerably simplified to d(A0)

Aft(x) 2:r

6Nx, r) rx'

(4.40)

where phase sensitivity to x (pressure or strain) is defined by the relative birefringence (the birefringence under x deformation normalized by the total birefringence that comprises both effects) multiplied by the constant 2;r/T~.

4.5.1. Birefringence measurement under stress On the basis of the solutions (4.37) of coupled-mode equations describing the electric fields Ex,(Z) and E,.,(z) in local (rotating) coordinates, we can now calculate light intensity at the output end of the twisted HB fiber, at a distance z = L. We assume that the linearly polarized light is injected at an angle a with respect to principal birefringence axes, E,(z - O) = Eo(z)cos a,

48

POLARIMETRICOPTICALFIBERSAND SENSORS

[I, w 4

Ey(z = O) = Eo(z)sin a, and that the analyzer at the output of the fiber makes an angle y with the local fiber axes. Then, the output electric field will be given by Eout = Ex,(L) cos y + Ey,(L) sin y. Taking into account the relations (4.37), the light intensity I = IEout]2 received by the detector yields

I = I0

[

613L (2 - g) r sin -6~Z T s i n ( a - y) 12 cos ~ - c o s ( a - y) + 6/3

+

sin - - ~ cos(a + y)

]2}

(4.41)

.

This formula will be greatly simplified if linearly polarized light is injected at a = 4 5 ~ to the axes and the output intensity is observed at V = 45 ~ to the fiber local axes:

I =/o COS2

{L~I ~

A/~) + sgn

(O(Afl))2Jr Ox

XLT-Tx + [ ( 2 - g) 1"12 } ,

(4.42)

where Io=E 2 is the input light intensity, x stands either for pressure (p) or for strain (e), and sgn(rAfi/rx) = +1 when birefringence rises under the influence of the stress x. The output response is periodic with the fiber length, and can be affected by external parameters such as pressure (or strain) and twist. According to eq. (4.42), the total output intensity is expressed as a function of two perturbation parameters, pressure p (strain e) and twist r, which can be simultaneously applied to a fiber. This opens the new possibility of measuring the intrinsic birefringence of an HB fiber under hydrostatic pressure (longitudinal strain) by twisting the fiber. Here, to measure birefringence under pressure we have applied a twist method introduced by Huang and Lin [ 1985], which was especially formulated for fibers with very short beat lengths (high birefringence). Based on a three-section fiber model, this method uses a derived birefringence formula expressed in terms of the position of the first maximum and its adjacent minimum on the curve of the measured polarization ellipticity in relation to the twist rate of the fiber. The experimental setup to measure intrinsic birefringence in HB fibers by the twist method is shown in fig. 17. For measurement in a high-pressure environment, the sample was placed inside a specially designed, high-pressure test chamber described elsewhere (Bock, Woliflski and Barwicz [1990]). To determine the birefringence of the HB bow-tie fiber under high hydrostatic pressure, we first

I, w 4]

DEFORMATION EFFECTS IN HIGHD" BIREFRINGENT FIBERS

49

Fig. 17. The experimental setup to measure the intrinsic birefringence in HB fibers by the twist method: P, polarizer; L, lens; C, quarter-wave plate; A, analyzer; D, detector; HE high-pressure chamber; DWT, deadweight tester; IN-E input fiber; OUT-F, output fiber; LOCK-IN, lock-in amplifier; DVM, digital multivoltmeter. found a test curve sin20 (which is the measured polarization ellipticity of the fiber) versus twist; then, from this curve we found the first maximum (b) and the adjacent minimum (a) after symmetrization sweeping of the curve. According to Huang and Lin [1985] (their formula 18), the beat length can be expressed as LB =

L V/[(2 - g)2(a2 - b2)] 2 - (2 - g)2 b'

(4.43)

where L is the twisted fiber length, g is an elasto-optic coefficient equal to ~0.16, and a and b are pressure dependent; that is, a = a(p) and b = b(p) are the adjacent extremes, expressed in the number of turns. Figure 18 shows the test curves sin20 (polarization ellipticity) versus twist obtained for two selected values of pressure at ambient temperature (T = 20~ by twisting the HB bow-tie fiber. The test curve without external pressure (0MPa) is symmetrical about the point of zero twist rate, with negligible polarization ellipticity (sin20 is close to 0) corresponding to linear output polarization. Applying hydrostatic pressure to the fiber without twisting produces a variation in the transmitted polarization state of light and, hence, causes a change in the polarization ellipticity of the HB fiber (sin 2 0 is above 0). However, the simultaneous influence of twist and hydrostatic pressure effects transforms the initial curve (0 MPa) to a new position defined by the current value of the pressure exerted on the fiber. This displacement of the zero pressure test curve is responsible for pressure-induced changes in birefringence of the HB fiber. Hence, based on the measured test curves and on eq. (4.43), one can calculate beat-length (birefringence) changes with pressure. The results are summarized in

50

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 4

1.0

0.8

0.6

A

.M

0.4

I OMVa/

1

/

94.5 MPa

/ 0.2

0.0

-25 -20 -15 -10

-5 0 5 Twist, turns/m

10

15

2t

"~z5

Fig. 18. Polarization ellipticity versus twist as obtained for 94.5MPa by twisting at ambient temperature a York HB 600 fiber 781 mm in length. The figure also shows a test curve obtained without the influence of external hydrostatic pressure.

1.3

E

E

1050

beat,ength/

1.2~

,000

,,~JJ

950

0

t,,, 1.1

900 J

.9

850

0

.....

birefri ngence] . . . . .

20

= L.

tn 1.0 p~

9 r

i

40

,

,

,

,

.I,,. m

800

.

60

80

100

Pressure, MPa Fig. 19. Beat-length and birefringence changes with hydrostatic pressure for the HB 600 single-mode bow-tie fiber as measured by the twist method.

fig. 19. Figure 20 shows results obtained for elliptical-core HB fibers subjected to the simultaneous action of strain and twist effects. The polarizer (P) and analyzer (A) were kept at a 45 ~ angle to the fiber local axis, so that the light

I, w 4]

DEFORMATION EFFECTS IN HIGHLY BIREFRINGENTFIBERS

51

1.2 •

1.0

1000 ~e 2000 ge 3000 pe

0.8

9 v,,,~

0.6

o "~

0.4 0.2

|

0.0 1'.0

'

1'.5

21.0

'

21.5

i

3.0

|

Twist rate, t u r n s / m Fig. 20. Responses to twist under strain of an elliptical-core HB fiber (UMCS, Poland).

intensity coming out of the fiber was strain and twist dependent. The curve without strain (0 ~te) is cosZ-like, as predicted by theory. By applying strain to the fiber, one can observe a shift of the twist characteristics, since the linear birefringence is strain dependent. For a given value of twist rate, there is a fight shift of the characteristics with increasing value of the strain. This means that for the elliptical-core fibers operating at the LP0~ fundamental mode, uniaxial longitudinal strain reduces the polarization birefringence and sgn(6Afl/6e) = -1. This behavior of elliptical-core fibers was independently confirmed by Bock, Domafiski and Woliflski [1990] resulting from the Rayleigh scattering method of birefringence measurement. This is of particular interest, since hydrostatic pressure applied to the same fiber increased its polarization birefringence with the mean coefficient of 1/Afid(Afi)/dp = 17%/MPa. In an HB bow-tie fiber, since intrinsic birefringence Aft is of the order of 1000 turns/m and the twist rate r exerted on an HB fiber does not usually exceed the order of 10 turns/m, twisting does not significantly change the total birefringence (4.36) of the HB fiber. It only slightly disturbs pressure- (strain-) induced phase modulation (4.40), described by the experimental parameter rip. For r = 20 turns/m and without external hydrostatic pressure ( p = 0 MPa), the relative birefringence Afl(p = O)/6fi(p = 0, r = 20) is 99.90%. An analogy between twisted HB fibers and (anisotropic) chiral liquid crystals subjected to the influence of hydrostatic pressure was presented by Wolifiski [1997].

52

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 5

w 5. Application of Polarization Effects 5.1. POLARIMETRIC FIBER-OPTIC SENSORS AND SYSTEMS

The linearly polarization eigenmodes of an HB fiber are associated with phase retardation q~f and q~s, where the subscripts f and s stand for fast and slow azimuths, respectively. The total relative phase retardation between the two perpendicularly polarized eigenmodes propagating in an HB fiber of length L can be expressed as Aq~ = Of - q~s = (2:r/~,) AnL,

(5.1)

where ~, is the wavelength of the light, and An = nf - ns is the difference between the effective indices of the polarization modes. This phase retardation can be easily changed by external factors (pressure, temperature, different stresses, etc.), and it creates the background of polarimetric fiber optic sensors. The first study on polarimetric temperature sensing was done by Eickhoff [1981], on polarimetric stress sensing by Carrara, Kim and Shaw [1986], on strain sensing by Rashleigh [ 1983a], and on polarimetric high pressure sensing by Xie, Dabkiewicz, Ulrich and Okamoto [1986]. To date numerous papers and chapters in textbooks have been published on polarimetric optical fiber sensors of different physical parameters as well as of various fiber-optic configurations (e.g., by Sheem and Giallorenzi [1979], Layton and Bucaro [1979] (acoustical signal detection), Rashleigh and Ulrich [1979], (magneto-optic current sensing), Varnham, Payne, Barlow and Tarbox [1984], Chardon and Huard [1986] (temperature sensor), Tatam, Pannell, Jones and Jackson [1987], Noda, Okamoto and Yokohama [1987] (devices with PANDA fibers), Kim, Blake, Huang and Shaw [1987] (elliptical-core twomode fiber devices), Kersey, Davis and Marrone [ 1989] (differential polarimetric sensor system), Chua and Chen [1989] (stress sensor), Gusmeroli, Vavassori and Martinelli [1989] (structural monitoring), Murphy, Miller, Vengsarkar and Claus [1990] (elliptical-core two-mode fiber sensor), Picherit and Mineau [1990] (force and temperature sensor), Wang, Wang, Murphy and Claus [1991] (hightemperature measurement), Wong [1992], Lo, Sirkis and Richtie [1995] (passive demodulation technique for polarimetric sensors)). Two subcases are of great importance in the polarimetric fiber-sensing applications: the single-mode polarimetric and the two-mode intermodal interferometric sensors.

I, w 5]

APPLICATION OF POLARIZATION EFFECTS

53

5.1.1. Single-mode regime of operation. the truly polarimetric sensor In the single-mode polarimetric sensor, only one spatial mode LP01 (~70= 1, r/l=0) or LPll (170=0, r/1 = 1) is excited at a fiber's input. If the fiber is bimodal at the operating wavelength, the single-mode regime can be achieved by adjusting the launching conditions. However, the practically popular solution is to use a single-mode optical fiber in which only the fundamental LP01 mode is propagating. No intermodal interference is observed in this case because the second mode is absent. If quasi-monochromatic light linearly polarized at an angle q~ with respect to the fiber's x-axis is launched into the fiber and an analyzer turned to an angle a is placed at the output of the fiber, the optical intensity detected will be I = ~1 [1 + c o s 2 a c o s 2 c # + Iyol sin 2a sin 2cg cos q~o] ,

(5.2)

with q)0 = Afl~L the phase. The same dependence is valid for the LPi"1 to LP(I polarimetric interference, but Aft ~ should be substituted instead of Aft~ When external perturbations are introduced, they lead to changes in the phase q~o = Afi~ of the fundamental LP01 mode (or correspondingly q~l = Afi~ for the LPll mode). Consequently, it will lead to a cosine variation of the observed intensity I measured after the analyzer, which is, in fact, the polarization interference. The setup is then a polarimetric sensor. The interfering beams in this case are the LP~I and LP0 1 polarization modes. An input polarizer (if the light is not linearly polarized) acts as a splitter, and the analyzer acts as a recombiner. The visibility of the observed interference pattern, defined as V -- /max -- Imin

(5.3)

/max + I m i n '

is obtained to be sin 2a sin 2q~ V = [Yl 1 + cos 2a cos 2q~"

(5.4)

If we define kl = sin 2 c49and k2 = sin 2 a as the power coupling coefficients of the splitter and recombiner, the expression for visibility becomes V = [YI 4v/klk2(1 - kl)(1 - k2) 1 + (1 - 2 k 2 ) ( 1

(5.5)

- 2k~)

This expression is analogous to the well-known formulas for the visibility in classical two-beam interferometry. The maximum visibility is obtained if

54

POLARIMETRICOPTICALFIBERS AND SENSORS

[I, w 5

kl = k2 = 0.5 or equally for q~ = a = 45 ~ This is when both LP~Cl and LPffl polarization modes are equally excited by launching -+-45~ linearly polarized light; that is, S ~ = { 1,0, + l , 0}. In this case the observed intensity and the visibility become 1 = 31 [1 + ]~t0[ COS ( A / ~ L ) ]

with

V--]Y0[.

(5.6)

As is evident, if a monochromatic source is used, visibility will become unity, since I 'ol = 1. The same effect can be achieved if right/left circularly polarized light (i.e., S ~ - { 1,0, 0, + 1}) is launched at the input.

5.1.2. Two-mode regime of operation." the intermodal interference sensor In the two-mode regime of operation, the interference between either X- or Ypolarized LP01 and LPll spatial modes is observed. By letting q~x = Aft~ and 9 y = Afi~ the intensity observed at the output of the fiber excited with X- or Y-polarized quasi-monochromatic light is written as

Ix, y(r, 99,z) = r/ofo2(r)+ r/lfl2(r, qg)+ 2r/Olj~(r)J](r, 99)IFx,y[ cos ~x,y.

(5.7)

Depending on the detection setup, the three important subcases that can readily be outlined include visibility at a point (pixel), visibility when half the pattern is detected, and visibility when an offset single-mode fiber is used. In all three cases the visibility is presented in the form

V(z, n~)= fr~,y(Z)[ v0,

(5.8)

where V0 is a factor depending on the relative excitation coefficient of the second mode and on the detection scheme (table 6). In all of these cases maximum visibility is obtained if the source is monochromatic and both modes are equally excited; that is, rl0 = ?~1 --0.5. To increase the measurement range with the highest possible sensitivity, a multisensor system can be designed. The system was initially proposed for pressure measurements by Bock, Wolifiski and Barwicz [1990] and then for polarimetric strain measurements by Bock, Wolifiski and Eftimov [1996]. The primary idea of such a 3-sensor system is based on the use of steep monotonic segments of the characteristics of two reference transducers (made of identical long parts of the sensing fiber), correlated with readings of the third much less sensitive (short fiber) transducer that has monotonic characteristics in the required range of strain. This idea was seen in fig. 15, showing idealized

I, w 5]

APPLICATION OF POLARIZATION EFFECTS

55

static characteristics of two strain transducers for a constant temperature. The transducer with monotonic characteristics enables us to estimate the quantity to be measured (measurand) without ambiguity but with little accuracy. This rough estimate is successively used first to identify the appropriate reference transducer and second to determine a final, much more accurate estimate of the measurand. Another interesting approach to process the output signal of fiber-optic strain sensors based on polarimetric effects involves the application of neural networks, especially in the case of few-mode devices, where the strain-modulated, far-field pattern emitted from the output of the sensor may be complex. In many applications in civil engineering, such as mining, large dams, construction projects, and tunneling, it is important to evaluate the response of a structure for safety purposes, including structural integrity and damage assessment in foundations, pillars, or anchors, and total ground pressure and load evaluation. For static or quasistatic conditions this evaluation is usually done by

56

POLARIMETRIC OPTICAL FIBERS AND SENSORS

LD-830

HB-800 IN

X

HB-800

45 ~

X 45 ~

HB-800

[I, w 5

.....

OUT

(a) FP ~) LD-790 ~ ' ~

MPC

A LB-800

SM-800 (b)

Fig. 21. Two configurations of all-fiber polarimetric sensors: (a) a laser diode LD-830 pigtailed to an HB 800 PMF; (b) a laser diode LD-790 pigtailed to a single-mode SM-800 fiber. X is a fusion splice; FP, fiber polarizer; M P C , polarization controller, IN (OUT), input (output) fiber; A, analyzer; and PD, photodetector. Configuration (a) requires two 45 ~ splices.

monitoring real stress (or pressure) distribution in concrete, rock, or earth. Many specific requirements relative to sensing pressure in these environments can be best met by using emerging fiber-optic sensing technology. Bock, Voet, Beaulieu, Wolifiski and Chen [1992] proposed replacing a widely used, often difficult and cumbersome technique of hydraulic evaluation of stress in concrete materials with a fiber-optic measurement device, consisting of an HB fiber. Selected applications of polarimetric optical fiber sensors in engineering mechanics were discussed by Bock, Urbaficzyk and Voet [1995]. In addition, HB polarization-maintaining fibers had a highly significant impact on the fiber-optic gyroscopes, in that they allowed the achievement of good longterm stability, which is essential for inertial navigation applications (Jaroszewicz, Ostrzy2ek and Szustakowski [ 1993], Burns [ 1994]). 5.1.3. Polarimetric all-fiber sensor

Practical implementations of the optical fiber pressure and strain sensors are envisaged as all-fiber polarimetric sensors (Bock, Wolifiski and Domafiski [1992]), in which bulk polarization-controlling components are replaced by all-fiber polarization controllers and all-fiber polarizers. The all-fiber polarimetric sensors can be assembled by using the two configurations (Domafiski, Woliflski and Bock [1995]) shown in fig. 21. The first configuration (a) involves a semiconductor laser diode (LD) pigtailed to the same HB fiber used as a sensing element. To allow the incoming light to excite both fundamental polarization eigenmodes equally in the sensing element, the laser pigtail has to be fusion-spliced to it at

I, w 5]

APPLICATIONOF POLARIZATIONEFFECTS

57

Fig. 22. Optical instrumentation and processing system for the polarimetric FOSGM: IF (OF), input (output) HB fiber; P, polarizer; k/4, quarter-wave plate; SBC, Soleil-Babinet compensator; WP, Wollaston prism; DI, D2, detectors; C, computer; VM, voltmeter; PL, plotter. 45 o relative to the birefringence axes of both HB fibers. High accuracy of the azimuthal alignment is required in this case to avoid instabilities and diminution of the sensor sensitivity. A second configuration (b), based on a semiconductor LD pigtailed to a single-mode fiber, involves the use of a fiber polarizer (FP) and polarization controller to rotate the linearly polarized input light signal appropriately relative to the eigenaxes of the sensing element. Different configurations of the output fibers have been used to deliver the modulated output signal from the sensing fiber to the detectors. These include either HB or LB single-mode fibers, each of which exhibits particular advantages and disadvantages. 5.1.4. Polarimetric strain-gauge manometer Bock, Wolifiski and Wigniewski (Bock, Wigniewski and Wolifiski [1992], Bock, Wolifiski and Wigniewski [1993]) developed an idea of indirect pressure measurement, where a polarimetric strain sensor can be configured to measure high hydrostatic pressure in process control, in cases where the direct introduction of the sensing element inside the pressure region is not possible (e.g., high temperature). The transducer, which takes the form of a fiber-optic strain gauge manometer (FOSGM), uses an active element configured in the form of an infinite-like cylinder with free ends and a fiber-optic strain sensor epoxied to the outer wall of the cylinder. The deformation of this cylinder depends exclusively on the value of the internal pressure delivered from outside the transducer, with the longitudinal strain el registered by the fiber-optic sensor being directly related to the pressure p through the following expression:

el-

vpD 2dE'

(5.9)

where E is the Young modulus, v is the Poisson ratio, d is the thickness of the cylinder, and D is its inside diameter. Figure 22 shows an optical instrumentation system designed to deliver a controlled polarization light signal and to detect the pressure-modulated output signal of the sensing element. The use of the

58

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 5

difference-over-sum processing, ( ~ l - 12)/(I1 +/2) allows significant reduction of the system fluctuations. The total operation range of the FOSGM (100 MPa, corresponding to 100~te in the strain scale) can be easily adjusted with the help of a Soleil-Babinet compensator to a quasilinear steep region of a sinelike characteristics. No high pressure lead-through is needed in this case, which obviously simplifies the transducer construction and avoids some uncertainties and false readings usually associated with such a lead-through. The fiber-optic strain gauge manometer demonstrates significantly increased sensitivity (at least 0.1 MPa -1) over similar devices based on electrical strain gauges. 5.2. THE INFLUENCEOF TEMPERATURE The temperature field acting on an HB fiber causes a change in the dimensions of the fiber and in the refractive indices, which was discussed by many authors (e.g., Eickhoff [1981], Mochizuki, Namihira and Ejiri [1982], Rashleigh and Marrone [ 1983a], Chardon and Huard [ 1986], Picherit and Mineau [ 1990], Wang, Wang, Murphy and Claus [1991 ], Zhang and Lit [1992], Ruffin and Sung [1993], Zhang and Lit [ 1993]). Assuming that the fiber has homogenous thermal characteristics, the core and cladding will have the same thermal coefficient a and the same thermo-optic coefficient ~. For the HB elliptical-core fiber, the general formula (4.7) describing the differential phase change under the influence of temperature will be considerably simplified. Substituting in eq. (4.7) 6~ = 6T (which is positive on heating) and, hence, 8 b / S T = ab, 6 L / S T = aL, 6 n i / 6 T = ~ni (i = 1,2), we obtain 8(Aq,i) 6T

~, O(A[3i)L(ot + ~),

(5.10)

0~

and also a change in birefringence induced by the deformation effect of temperature in every mode (i = 0, 1,x,y): ~)(A[~i) -

6T

aAfii

" O(Aj~i) L ( a + ~).

(5 11)

- z--~

From these formulas it follows that the thermal effects in HB fibers depend only on the wavelength dispersion of birefringence Afii. Typical values of the thermal expansion coefficient and thermo-optic coefficient for fused silica yield a ~ 5 • 10-7/K, ~ ~ 10-5/K, leading to the data for the HB elliptical-core Polaroid fiber, d(Afi0)/dT ~ -0.16 rad/m.K, d(Afll ) / d T -- -0.18 rad/m.K. The effect of temperature on bireffingence is similar to the effect induced by pressure or strain, and can be described in terms of an experimental

I, w5]

APPLICATIONOF POLARIZATIONEFFECTS

59

Table 7 Temperature effects in HB bow-tie and elliptical-core fibers

Beat length

LB, i(T) = LO B,i [1- T. L~ B,i/(TiTL)]-1 Birefringence

Afli(T) = Aft/0- T. 2Jr/(TiTL) Data at ~ =633 nm

HB-600 bow-tie fiber:

d(Afio )/d T ~ -8. 0 rad/mK

Andrew ET elliptical-core fiber:

d(Afi0)/dT ~ -1.05 rad/mK

Polaroid elliptical-core fiber:

d(A/J0)/dT ~-0.16 rad/mK; d(Afil )/dT ~ -0.18 rad/mK

parameter TT, which is equal to the amount of temperature required to induce a 2:r phase shift at the fiber output. From the Rayleigh scattering measurements performed by Bock, Domafiski and Wolifiski [1991] it appeared that the beat length (birefringence) of the HB fibers increases (decreases) with temperature, as was predicted by the theory presented above. Table 7 summarizes the influence of temperature in both elliptical- (Andrew ET, Polaroid) and circular-core (bow-tie) HB fibers. The negative sign in the formula describing birefringence dependence in both types of HB fibers indicates a relieving of the thermal stresses with heating. In an analogy to formula (4.18), the following equation can be deduced: L . TT = const.. ~,.

(5.12)

The experimental results show that any polarimetric or intermodal interferometric strain or a pressure sensor is inevitably temperature dependent, and thus ambient thermal fluctuations cause instabilities in the output reading. Therefore, the temperature compensation becomes of prime importance for the proper work of any polarimetric fiber-optic sensor. Different approaches for temperature stabilization can be used, depending on whether the sensor is single-mode polarimetric or two-mode interferometric. 5.2.1. Methods o f temperature compensation 5.2.1.1. Temperature compensation in a polarimetric sensor. To reduce the disturbing influence of temperature on the characteristics on an HB fiber-

60

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 5

Fig. 23. Temperature-compensatedpolarimetric fiber-optic sensor: (a) in transmission configuration, after Dakin and Wade [1984]; (b) in reflective configuration, after Bock and Wolifiski [1990b]. based, single-mode polarimetric sensor, Dakin and Wade [1984] proposed a compensation scheme in which two pieces of the same fiber with identical lengths are spliced with birefringence axes rotated at 90 ~ to each other. This is a polarization mode interchanger, in which the fast mode in the first section is coupled to the slow mode of the second section. An improved version of this temperature compensation scheme was proposed by Bock and Wolifiski [1989, 1990b] and Bock and Urbaficzyk [1996]. The scheme is based on the polarization-rotated rotation introduced by Enokihara, Izutsu and Sueta [1987] as well as by Marrone and Villarruel [1987], and Marrone, Villarruel, Frigo and Dandridge [1987]. This is pictured in fig. 23, showing a fiber-optic strain sensor in a compensation-reflective configuration operating in the fundamental mode regime. A linearly polarized light beam is launched into one of two principal axes of a single-mode HB input/output lead fiber, spliced at 45 ~ with a compensation section (L~) of an HB fiber and spliced again at 90 ~ with a sensing section (L2) of the same HB fiber. The output Stokes vector in the improved reflective configuration is

S = M(L1)M90oM(L2) IM(L2)M90oM(L1)S ~

(5.13)

where M90o is the Mueller matrix of the 90 ~ rotator, 1 0

0 0

0-1 0 0 M9~176 0 0 -1 0 0 0 0 1

'

(5 14)

and I is the unit matrix of the reflector, which is assumed to be ideal and polarization insensitive, whereas the matrices M(Ll) and M(L2) are those of the appropriate fiber sections in the monochromatic case (see w2.3.2).

I, w 5]

APPLICATION OF POLARIZATION EFFECTS

61

After multiplication and subsequent integration over the source's spectrum and assuming the 45 ~ input excitation and the 45~ analyzer, the light intensity is found to be I = ~1 { 1 + Iy(r, - r2)l cos [2 (Afi, Ll

-

(5.15)

A~L2)]}

with rl = 26~o6rl and r2 = 26~o6r2. This result proves that if both sections are identical (A/31 = A/32 and L1 = L2) and are subjected to the same perturbations, no changes will be observed at the output. In addition, the coherence function in the ideal case will be unit, and a lower coherence diode can be used. Since for a reflective configuration the effective length of an HB sensor is twice its actual length so that L = 2L; (i = 1,2), the total relative phase retardation between two perpendicularly polarized eigenmodes propagating in an HB fiber is given by (5.16)

6 ~ = 6q)(Ll) - 6q~(L2), where 2:r 6q~(Li) = -~Ani2Li

4Jr

=--s

(5.17)

is the wavelength used, and An; = Afii/k is the difference in effective refractive index between the polarization modes in the ith sections (i = 1,2). Temperatureinduced phase retardation can be expressed by d(rq0_4sr

dr

~ Z

d_~

i

4st

(AniLi)= ~ - Z

( d(Ani) + AnidLi ) i

Li dT

-d--T '

(5.18)

where A n i = n x i - nyi, and nxi and n ,.; are the refractive indices for two orthogonal linear polarization modes of the ith section. Since L~ = L2 := L and both parts are rotated about their longitudinal axes one fiber relative to the other by 90 ~, we obtain An2 = -Anl (Y~'~Ani = 0), and finally d(rq0 4srd( dT - 2. dT

) LZAn~

=0

(5.19)

i = const. From eq. (5.19) it follows that the strain sensor will be immune to the ambient temperature or any other physical parameter acting on both its parts. However, if only one part of the sensor (L l) is exposed to the influence of a physical or 6q~(T)

62

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 5

parameter, the sensor will measure this parameter, for example, pressure or strain. In this case, assuming a monochromatic source, eq. (5.14) is written in the form 1 I = ~

1 + cos 2

2Jr

4~ + T1, e(Ll)

6e+~6T-4

T1, r(L1)

~

T2, r(L2)

' (5.20)

and if L1 = L2, then T1, 7-(Ll)= T2,T(L2) and I=~

1(

l+cos

E( 2

q~0_q~o+ T1, ~(L1)

and the sensor will respond only to one measured physical parameter (e.g., pressure, strain). However, if the lengths are slightly unequal or they are equal but the ambient temperature around both of them is not the same, the temperature compensation will not be complete. The temperature should also be the same around both parts. Another effect that is still present, even for a perfectly manufactured sensor, that is, one with exactly equal sensing and compensating parts, a residual temperature response related to the measurand-temperature cross-sensitivity effect. In the case of the polarimetric strain sensor, Bock, Urbaficzyk, Buczyfiski and Domafiski [ 1994] showed that the residual temperature response is proportional to the temperature change and to the ratio of the strain-temperature crosssensitivity (second-order effect) to the first-order strain sensitivity of the HB fiber. It appeared that the cross-sensitivity determines the physical limit for the temperature stability of the perfectly compensated polarimetric strain gauge. For example, in the temperature range At = +50~ the best achieved stability was --1.5% of the gauge's full range, which is more than enough to satisfy industrial requirements for the temperature stability. Bock, Urbaficzyk and Barwicz [ 1996]) recently obtained experimental data on temperature drift of the compensated polarimetric strain gauge based on the Panda HB fiber, which is within about 1% range of the full scale (FS) in the temperature range At = 32~ The reflection configuration of the sensor-incorporated, temperature-drift compensation is particularly well suited for multiplexing of polarimetric sensor arrays. An example of this type of multiplexing system using polarizationpreserving couplers (PPCs) is shown in fig. 24 (Bock, Wolifiski and Barwicz [1990]). The total number of multiplexed sensors in the array will depend ultimately on the power budget of the system, with the quality of splices being one of the most important factors.

I, w 5]

APPLICATIONOF POLARIZATIONEFFECTS

63

k/2 plate Polarizer ~

i Laser ~ . ~

~

,

$45 ~

HB Fiber

$90

~ Mirror

XCPXCp /

~

HB Fiber

S~45~ S 90~ Mirror )"( CP )"( CP ~ 45~ S 90~ Mirror CP ~ CP ~

[ ~ ~ ~ ~ ~

HB Fiber

/

k*

Analyzers Fig. 24. Multiplexing system for polarimetric sensors in reflection configuration: PPC, polarizationpreserving coupler, C(S)P, compensating (sensing) part; S, splicing at 45~ (90~ Di, detector; Pi, processing (i = 1,2, 3 .... ).

5.2.1.2. Temperature compensation in a two-mode interferometric sensor. The technique of splicing two identical fibers at 90 ~ used to suppress ambient temperature fluctuations is difficult, if not impossible, to use in two-mode interferometric arrangements, because several factors, such as modal excitation efficiency, and an additional amplitude and polarization modulation in the second segment occur. Instead of compensation, a polarization-sensitive bimodal sensor allows the simultaneous measurement of both strain (pressure) and temperature. In its essence this is a two-parameter sensing device, and in its simplest form two wavelengths are used. At the higher wavelength (850 nm) the sensor is single mode, whereas at the lower (633 nm) it is two mode. The sensitivity to strain (pressure) and to temperature of the polarimetric and interferometric responses is different, and this is the basis of the simultaneous two-parameter sensing. We found that it is not necessary to use two wavelengths in order to obtain two different signals, however, since intermodal interference in an HB two mode fiber is polarization sensitive. This means that if the fiber is excited with a 45 ~ linearly polarized wave, a Wollaston prism will detect the two responses proportional to COS2(~~ + ~)~x) and to cos2(q~~ + 6q~.), where

[

6~ Ay, t Ay, e

6t 6e

I"

(5.22)

64

[I, w 5

POLARIMETRIC OPTICAL FIBERS AND SENSORS

Fig. 25. Experimental setup for simultaneous pressure and temperature measurement employing LP01-LPll fiber-optic polarization-sensitive intermodal interferometer (after Bock and Eftimov [1994]).

Since the parameters Ai, z (i = x , y and Z = t , e ) can be determined experimentally for a given fiber, as a result the system can be solved with respect to 6t and 6e using the well-known Kramer's rule. Hence, to detect both strain (pressure) and temperature changes 6e and 6t simultaneously, two requirements are posed: (1) The determinant of eq. (5.22) must be D ~ 0, which means A~t ' ;~

AX

Ay, t

Ay, e

,E

or equally

., T v. , t ;~ TvF Tx, t Tx, e

(5.23)

(2) The X and Y sensitivities must be different, that is, Ax t

'

A y, t

~1,

Ax E

'

A y,e

;~1,

or equally

Tv t

'

Tx,,t

~1,

Tv E

~'

~1.

(5.24)

Tx, e

As was proved by Bock, Wolifiski and Eflimov [ 1996], these two requirements are best met for a bow-tie fiber HB-800 compared with the commonly used e-core fiber. Experimental evidence for simultaneous pressure and temperature measurement employing an LP01-LP~l fiber-optic polarization-sensitive intermodal interferometer was demonstrated by Bock and Eflimov [ 1994]. Figure 25 shows the experimental setup for such a measurement. 5.3. POLARIMETRIC FIBER-OPTIC SENSORS WITH PARTIALLY POLARIZED LASER SOURCES

The degree of polarization (DOP) of light propagating in an HB fiber diminishes along the propagating distance. This effect is of particular interest in optoelectronics, since the quasi-monochromatic semiconductor sources are not perfectly coherent. Therefore, measurement of the DOP of the light propagating along the fiber can be directly applied to determine the coherence characteristics of

I, w5]

APPLICATIONOFPOLARIZATIONEFFECTS

65

laser diodes. The issue also concerns applications in polarimetric sensing with HB fibers. Most currently available polarimetric fiber-optic sensors have been constructed and tested in research laboratories, where traditional (e.g., He-Ne) laser sources that are characterized by good coherence properties and a high value of the DOP are in common use. However, practical field implementations of polarimetric fiber-optic sensors require the use of cost-effective laser diodes not necessarily single-mode operated and generally characterized by poor coherence properties and a low value of the DOE Thus, a question arises concerning how the use of low-cost laser-diode sources will influence the performance of polarimetric fiber-optic sensors. Domafiski, Wolifiski, Karpierz and Kujawski [1995] analyzed the problem of fading degree of polarization in polarimetric fiber-optic sensors in view of the optimizing performance of real polarimetric sensors with HB fibers. They concluded that since the maximum modulation depth of the output signal from the polarimetric sensor is equal to the DOP of the semiconductor laser source, the totally polarized part of the light entering the sensing fiber should be linearly polarized. In this case the disturbing effect of the imperfectly polarized light source will be minimized. 5.4. POLARIMETRICSMART STRUCTURES At present much effort is being expended to apply polarimetric fiber-optic sensors to measure strain in aircraft or concrete structures, using the concept of so-called smart skins and structures. In recent years several researchers have published initial results of their investigations of polarimetric sensors that are structurally embedded in composite materials. These include, for example, polafimetfic strain sensors (Chardon and Huard [1986], Spajer, Carquille and Maillotte [ 1986]), polafimetric fiber-optic strain rosette (Measures, Hogg, Turner, Valis and Giliberto [1988]), coherence multiplexed polarimetric sensors for aerospace applications (Taylor and Ranshaw [1992]), HB fibers embedded in concrete structures (Calero, Wu, Pope, Chuang and Murtha [1994]), and structurally embedded polafimetric sensors (Lo, Sirkis and Richtie [1995]). Smart structures and smart skin are structural components, particularly prepared for advanced aircraft and space vehicles, with networks of fiber-optic sensors directly embedded within their composite material matrices. They are valued by the aerospace industry for their light weight and high strength. The composite materials are made with epoxies or polyimides. Structurally integrated polafimetfic optical fiber sensors have emerged as an important part of the sensors used for smart structure applications. A more ambitious and complex

66

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 5

Fig. 26. HB fiber embedded in a cylindrical epoxy structure. The lead-in and lead-out HB 600 fibers are spliced at different angles.

use of smart structures involves linking fiber-optic sensors with a real-time computer-control system on advanced aircraft. In this model, fiber-optic sensors are embedded in a panel to be integrated with the wing. The sensors monitor environmental effects, such as strain and bending, around and within the panel. In smart structures, the fiber-optic sensors become a part of the wing itself and are, of course, not affected by electrical disturbances. In response to the computer output, a fiber-optic link could drive remote actuators. A complete smart structure not only would detect problems, but would also respond to them instantaneously. Wolifiski, Konopka and Domafiski [1998] recently demonstrated a model of a smart structure composed of the HB bow-tie fiber embedded in an epoxy cylinder. The HB fiber-based structure was subjected to selected deformation effects mostly induced by hydrostatic pressure (up to 300 MPa) and temperature, whereas polarization properties of the transmitted optical signal were investigated. The test samples were composed of the separated HB fibers, as well as the same HB fibers embedded in epoxy structures (fig. 26). It appeared that the presence of the epoxy coating modified the output characteristics of the HB fiber (mostly due to elastic properties of the structure) as well as the level of the induced birefringence of the HB fiber, and hence influenced polarization properties of the light propagating in the fiber. The characteristic period Tp does not depend on the angle between the birefringence axes of the sensing structures and lead-in fibers, whereas the modulation depth varies significantly with this angle (fig. 27). The hydrostatic pressure characteristics of smart structures have a fundamental period Tp-100 MPa, which is twice as large as the period of the separated HB-600 bow-tie fiber (Tp ~ 50 MPa) (fig. 28). Hence, the presence of the epoxy

I, w 5]

APPLICATION OF POLARIZATION EFFECTS

9 45[deg] 9 30[deg] *_ lO[deg____]]

1.0"

67

//p

0.8

=

0.6-

olalll r~

0.4-

@ 0.2

0.0

I

9

I

0

9

I

20

'

I

40

'

I

60

'

I

80

'

I

lO0

120

Pressure, MPa Fig. 27. Optical responses of smart structures ( 12 mm in length) under hydrostatic pressure at various input polarization angles.

9 HB-600 fiber, L= 10 mm 9 H B - 6 0 0 s t r ~ r e , D=7mm, L=10 mm

9

250 -

9

/ 200

> 150 e~O 9r~ "

9~

100

50

I

0

9

I

20

'

I

40

'

I

60

'

I

80

'

I

100

'

I

120

'

I

140

'

I

'

160

Pressure, MPa Fig. 28. Comparison of the pressure characteristics of separated HB fiber and HB fiber-based polarimetric smart structure.

structure significantly modifies the output characteristics of the HB fiber. As a consequence, the induced birefringence of the embedded HB fiber changes less markedly than for the separated HB fiber, probably because pressure-induced

68

POLARIMETRIC OPTICAL FIBERS AND SENSORS

[I, w 5

stresses in the structure are separated between both the epoxy coating and the embedded HB fiber. 5.5. DYNAMICPOLARIMETRICOPTICAL FIBER SENSING In dynamic stress/strain measurements, the influence of temperature may be neglected due to its long period changes compared with high-speed mechanical changes. An example of a dynamic fiber-optic polafimetric sensor is shown in fig. 29. The sensor was proposed by Domafiski, Karpierz, Sierakowski, Switto and Wolifiski [1997] as part of an early-detection system for wheel-flat of a moving train. A laser diode pigtailed to a single-mode fiber and operating at 780nm wavelength was used as a light source. The fiber was wound onto a piezoceramic cylinder, which generated polarization modulation of the light passing inside the single-mode fiber. An additional fiber-optic phase shifter controlled the polarization coupling into the measuring HB fiber. The running wheel causes stress, which is transformed into strain on the bottom part of the rail and consequently into strain on the measuring fiber. The fiber-optic polarizer then converted the polarization modulation into intensity modulation. The idea of wheel-flat test systems is based on the application of an all-fiber polarimetfic strain sensor with the measuring HB fiber glued between pillars of a rail.

Fig. 29. Configurationof a polarimetric optical fiber sensor for dynamic strain measurement.

I, w6]

CONCLUSIONSANDFUTUREPERSPECTIVES

69

In field tests, a multipoint semidistributed polarimetric sensor with compensated birefringence was used. The multipoint sensor with the HB fibers measuring a few meters long was spliced 90 ~ in the middle of the sensing fiber to compensate birefringence in both parts of the HB fiber and then to resist deterioration of the DOP along the sensing fiber.

w 6. Conclusions and Future Perspectives Polarimetric optical fibers and sensors have emerged as a result of investigations of polarization fluctuation in long-distance optical fiber communication systems. They have created a new generation of powerful, mostly sensing-oriented techniques, in which polarization of the guided optical field is the important issue. The rapid development of optical fiber technology over the last two decades has led to extensive research activities and progress in the field of polarimetric optical fibers and sensors. The use of highly birefringent optical has been stimulated by a significant decrease in the costs of manufacturing polarization-sensitive fibers and polarization-preserving optical fiber elements as well as semiconductor light sources. Current research trends in the field include multiplexed and distributed sensors (Udd [1991 ], Belgnaoui, Picherit and Turpin [1994], Bock and Karpierz [1999]), few-modes polarimetric fibers for multiparameter sensing (Bock and Eftimov [ 1994], Wolifiski and Muszkowski [ 1995], Eftimov and Bock [ 1998]), combined dynamic and static polarimetric sensing (Charasse, Turpin and le Pesant [1991 ]), as well as the development of new types of polarization-sensitive fibers configured in polarimetric structures suitable for environmental monitoring (Woliliski, Konopka and Domafiski [1998]), for modern industrial civil engineering (Calero, Wu, Pope, Chuang and Murtha [1994]), and for machinery. These new structures also include side-hole, elliptical-core fibers (Bock, Urbaflczyk, Wdjcik and Beaulieu [1995], Fontaine, Wu, Tzolov, Bock and Urbaflczyk [1996]), and liquid-crystal-core fibers (Yuan, Li and Palffy-Muhoray [1991], Chen and Chen [1995], Wolifiski, Szymafiska, Nasflowski, Konopka, Karpierz, Kujawski and Dgbrowski [1998]) which are extremely sensitive to external parameters (Wolifiski, Nasitowski, Szymafiska, Konopka, Karpierz, Domafiski and Bock [1997], Woliflski [1999]). Very recently (Wolifiski, Szymaflska, Nasitowski, Nowinowski-Kruszelnicki and Dobrowski [1999]) it also became apparent that the elliptical-core liquid-crystal fiber could act as a single-polarization, fewmode fiber in which only one polarization is guided. More basic problems arise during distributed polarimetric sensing. The early work by Kurosawa and Hattori [1987] showed that high birefringence of fibers

70

POLARIMETRICOPTICAt, FIBERSAND SENSORS

[I

used in distributed polarimetric sensors did not permit finding a location of small external disturbance, such as local fluctuation of hydrostatic pressure, strain, and stress, since a total birefringence compensation in this location was required. New research indicates that white-light interferometry (Urbaficzyk, Kurzynowski, Wo~niak and Bock [ 1997]) with different fiber lengths and special kinds of splices in between might find significant use. One of the most encouraging potential applications of polarimetric optical fibers and sensors is to embed them directly inside various ceramic and composite materials, and to measure strain distribution in different structures (e.g., aircraft, bridges, highways, concrete structures), using the concept of socalled smart skins and structures. The identification of polarization phenomena existing in optical fibers opens up new perspectives on basic physical effects that occur when light is confined to optical fiber waveguides. This simultaneously creates new opportunities for applications in modern optical fiber-sensing technology that holds still greater potential for optical fiber telecommunications.

Acknowledgements I am greatly indebted to two colleagues from the Faculty of Physics at Warsaw University of Technology: Professor Adam Kujawski for his encouragement to write this review, and Dr. Andrzej Domafiski, for his inspiration to conduct research in fiber optics and his continued valuable collaboration since the late 1980s. The technical assistance of Agnieszka Szymafiska, a doctoral student, during the preparation of the chapter is gratefully acknowledged. I apologize to all distinguished individuals whose papers may have been omitted in this review due to space limitations. This research was partially supported by the Warsaw University of Technology, Warsaw, Poland.

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Blake, J.N., S.-Y. Huang, B.Y. Kim and H.J. Shaw, 1987, Opt. Lett. 12, 732. Blake, J.N., B.Y. Kim and H.J. Shaw, 1986, Opt. kett. 11, 177. Bock, W.J., and A.W. Domafiski, 1989, J. Lightwave Technol. LT-7, 1279. Bock, W.J., A.W. Domafiski and T.R. Woliflski, 1990, Appl. Opt. 29, 3484. Bock, W.J., A.W. Domafiski and T.R. Woliflski, 1991, Proc. SPIE 1267, 50. Bock, W.J., and T.A. Eftimov, 1994, IEEE Trans. Instrum. Meas. 43, 337. Bock, W.J., and M.A. Karpierz, 1999, in: Wiley Encyclopedia of Electrical and Electronics Engineering, ed. J.G. Webster (Wiley, New York) Vol. 7, p. 376. Bock, W.J., and W. Urbaficzyk, 1993, Appl. Opt. 32, 5841. Book, W.J., and W. Urbaficzyk, 1996, Appl. Opt. 35, 6267. Bock, W.J., W. Urbaficzyk and A. Barwicz, 1996, IEEE Trans. Instrum. Meas. 45, 556. Bock, W.J., W Urbaficzyk, R. Buczyflski and A.W. Domaflski, 1994, Appl. Opt. 33, 6078. Bock, W.J., W Urbaficzyk and M.R.H. Voet, 1995, in: Applications of Photonics Technology, ed. G.A. Lampropoulos (Plenum, New York) p. 311. Bock, W.J., W. Urbaficzyk, J. W6jcik and M. Beaulieu, 1995, IEEE Trans. Instrum. Meas. 44, 694. Bock, W.J., M.R.H. Voet, M. Beaulieu, T.R. Wolifiski and J. Chen, 1992, IEEE Trans. Instrum. Meas. 41, 1045. Bock, W.J., R. Wi~niewski and T.R. Woliflski, 1992, IEEE Trans. Instrum. Meas. 41, 72. Bock, W.J., and T.R. Woliflski, 1989, in: Springer Proceedings in Physics, Vol. 44, eds H.J. Arditty, J.P. Dakin and R.Th. Kersten (Springer, Berlin) pp. 464-469. Bock, W.J., and T.R. Wolifiski, 1990a, Opt. Lett. 15, 1434. Bock, W.J., and T.R. Wolifiski, 1990b, Proc. SPIE 1370, 189. Bock, W.J., T.R. Wolifiski and A. Barwicz, 1990, IEEE Trans. Instrum. Meas. 39, 715. Bock, W.J., T.R. Wolifiski and A.W. Domafiski, 1992, Proc. SPIE 1511, 250. Bock, W.J., T.R. Wolifiski and T.A. Eflimov, 1996, Pure Appl. Opt. 5, 125. Bock, W.J., T.R. Wolifiski and R. Wi~niewski, 1993, US Patent No. 5,187,983 (US Govt Printing Office, Washington, DC). Bom, M., and E. Wolf, 1993, Principles of Optics, 6th Ed. (Pergamon Press, Oxford). Botineau, J., and R.H. Stolen, 1982, J. Opt. Soc. Am. 72, 1592. Brown, C.S., and A.E. Bak, 1995, Opt. Eng. 34, 1625. Bums, WK., 1994, Polarization control with PM fibers, in: Optical Fiber Rotation Sensing, ed. WK. Bums (Academic Press, San Diego, CA) ch. 6, pp. 175-210. Bums, W.K., and R.P. Moeller, 1983, Opt. Lett. 8, 195. Bums, W.K., R.P. Moeller and C.-L. Chen, 1983, J. Lightwave Technol. LT-1, 44. Caglioti, E., S. Trillo and S. Wabnitz, 1987, Opt. Lett. 12, 1044. Calero, J., S.-P. Wu, C. Pope, S.L. Chuang and J.P. Murtha, 1994, J. Lightwave Technol. LT-12, 1081. Calvani, R., R. Caponi, E Cistemino and G. Coppa, 1987, J. Lightwave Technol. LT-5, 1176. Cancellieri, G., P. Fantini and M. Tilio, 1985, J. Opt. Soc. Am. A 2, 1885. Carrara, S.L.A., B.Y. Kim and H.J. Shaw, 1986, Opt. Lett. 11,470. Charasse, M.N., M. Turpin and J.P. le Pesant, 1991, Opt. Lett. 16, 1043. Chardon, D., and S.J. Huard, 1986, J. Lightwave Technol. LT-4, 720. Chen, C.-L., 1987, J. Lightwave Technol. LT-5, 53. Chen, T.-J., and S.-H. Chen, 1995, J. Lightwave Technol. LT-13, 1698. Cheo, P.K., 1985, Fiber Optics: Devices and Systems (Prentice-Hall, Englewood Cliffs, NJ). Chiang, K.S., 1987, J. Lightwave Technol. LT-5, 737. Chiang, K.S., 1990, J. Lightwave Technol. LT-5, 737. Chiang, K.S., 1992, J. Lightwave Technol. LT-10, 1850. Chu, P.L., and R.A. Sammut, 1984, J. Lightwave Technol. LT-2, 650.

72

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[I

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E. WOLE PROGRESS IN OPTICS XL 9 2000 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED

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JUN TANIDA AND YOSHIKI ICHIOKA Department of Material and Life Sciences, Graduate School of Engineering, Osaka University, Suita, Osaka 565-0871, Japan

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PAGE w 1.

INTRODUCTION

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METHODS

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DEMONSTRATION

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CONCLUSION

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w 1. Introduction Optical computing is a broad term indicating concepts and technologies of information processing based on optical applications. Because of large flexibilities in the representation of information, way of processing, and architectures of processing systems, many interesting methods and technologies have been presented in this research area. Digital optical computing is a sub-area of optical computing in which a digital computing scheme is utilized as the foundation. In the digital optical computing scheme, information is represented by discrete signals and is processed in the same way as used in digital electronics. As a result, digital optical computing is suitable for cooperating with the current electronic technologies. Whereas the final goal of the optical computing technique is still vague, digital optical computing has a relatively clear target because it is based on the practical evolution of electronic technologies. As a result, digital optical computing can be viewed as one of the most promising application fields in optical technologies. In this article, various interesting ideas and technologies are reviewed to clarify the whole image of digital optical computing and to identify the promising applications. In w2, basic concepts involved in digital optical computing are explained. In w3, elemental components required for implementing a digital optical computing system are described, and in w4, methods for constructing a desired circuit are presented. In w5, experimental systems which are constructed as demonstrators of the digital optical computing scheme are mentioned. In w6, studies related to software development for digital optical computing are explained. In w7, future directions of the digital optical computing technology are discussed.

w 2. Basic Concepts The basic concepts of digital optical computing are logical operation, the logic gate array, free-space optical interconnection, the methodology for logic construction, smart pixels, the optical computing system, and software for problem solving. 79

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2.1. LOGICAL OPERATION

The fundamental basis of digital optical computing is the same as the digital computing scheme widely used in current computer science. In the digital computing scheme, information is represented by a set of binary bits, i.e., O's and l's, and is processed by logical operations to convert it into another one. The general form of the operation is y = f(x),

(1)

where x and y are sets of binary data, X -" ( X 0 , x l , . . .

,xm),

(2)

y = (yO,yl,...

,y,,).

(3)

Since any complicated operation can be decomposed into a combination of simple logical operations, if a basic set of logical operations is developed, any computing system can be constructed based on the digital computing scheme. This is an important point for constructing a digital optical computing system. 2.2. LOGIC GATE ARRAY

One of the useful features of optics is the capability of processing and transferring data in parallel format. Conventional optical elements, e.g., a lens and a mirror, can transfer information as an image. Assuming such a parallel image communication, an array of logic gates placed on a plane substrate is a reasonable form of a logic element for the digital optical computing scheme. As shown in fig. 1, a logic gate array consists of multiple logic gates functioning in parallel.

Fig. 1. Concept of logic gate array.

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There are two forms of parallel processing to identify the operation. One is a single instruction stream multiple data stream (called SIMD), and the other is a multiple instruction stream multiple data stream (called MIMD). Referring to fig. 1, the case in which the function of all logic gates is identical is SIMD and the case in which the function is different is MIMD. An MIMD system can provide flexible computing capability, but is difficult to control. An SIMD system is easy to control, but processing flexibility is inferior. Due to ease of implementation, most of logic gate arrays developed for optical computing are assumed to execute the SIMD type of parallel processing. 2.3. FREE-SPACEOPTICAL 1NTERCONNECTION Once logic gate arrays are implemented, an individual logic gate on the gate array must be connected to the others to form a logical circuit for a desired operation. In digital optical computing, free-space optical interconnection is considered as a fundamental technique for this purpose. As shown in fig. 2, appropriate optical elements, e.g., a lens, a mirror, and a holographic optical element, are used to configure interconnection between the individual logic gates. As a rule of thumb, information capacity handled by an optical system is estimated by the space bandwidth product of the system. When the sizes and the maximum spatial frequency of the transmitted image are x, y, fx and f., the space-bandwidth product is calculated as SBWP =

32XyfxL.

(4)

Since the maximum spatial frequency is limited by diffraction of the optical system, the information capacity of the system can be estimated easily. For example, an imaging system with a typical camera lens has the value of more than one million. As a result, a huge amount of information can be transferred by simple imaging.

Fig. 2. Free-space optical interconnection.

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Although such an interconnection line is established in three dimensional space, the individual lines do not interfere with each other, so that the free-space interconnection scheme provides a large degree of freedom in connectivity. In addition, using specific optical elements capable of changing their characteristics, such as a spatial light modulator, a reconfigurable interconnection network is also implemented.

2.4. METHODOLOGYFOR LOGIC CONSTRUCTION To implement a desired operation by a combination of logical gates, specific procedures are required in the design process. Whereas many methodologies exist in the field of information science, those are not necessarily effective for the digital optical computing scheme. Since huge numbers of logic gates on a plane substrate are connected with more relaxed constrainsts than in electric interconnection, a different type of logic construction can be developed to utilize the potential capability of the optical digital computing scheme. The physical characteristics of logic gate arrays and free-space optical interconnection should be taken account of in the methodology. As described later, various physical phenomena are utilized in logic gates and optical interconnection. For effective use of these technological components, consideration of their operational characteristics is indispensable.

2.5. SMARTPIXELS A practical way of implementing functions in digital optical computing is the positive use of a new category of optoelectronic devices called smart pixels. The smart pixel is a device composed of optical signal emitters or modulators, optical signal receivers, and electronic signal processing circuits. Such a configuration is effective to utilize the characteristics of both optics and electronics. Optics serves in interconnection while electronics are dedicated to processing tasks. In addition, we can effectively design and optimize functional blocks for processing, detection, and modulation. Since state-of-art techniques and environments can be used for the development, high performance systems are expected to be realized. Note that this form of system construction can go along with the evolution of electronics; therefore, we can effectively utilize the fruitful results of current VLSI (very large scale integration) technologies.

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2.6. OPTICALCOMPUTINGSYSTEM One of the most sophisticated systems based on the digital scheme is the computer. Highly organized digital circuits provide extremely large capability and flexibility in computing. With the same methodology used in the current computer, a computing system can be constructed within the digital optical computing scheme. To demonstrate the capabilities of the digital optical computing scheme, various types of optical computing systems have been designed and developed experimentally. The architectures aim toward both general purpose and special purpose computing. The experimental systems are expected to provide useful information on practical system construction and the requirement for logic gate arrays and optical interconnection. 2.7. SOFTWAREFOR PROBLEM SOLVING Based on the methodologies for logic construction, a wide range of problems can be solved within the framework of optical digital computing. Such efforts are considered as software development specified for optical digital computing systems. The targets are image processing, numerical processing, emulation of parallel processes, logical processing, etc. Although the methodologies for logic construction are not necessarily the same as in current computer science, the accumulated resources, e.g., data representation schemes and algorithms, can be utilized to develop sophisticated methods for various problems.

w 3. Logic Gates To develop a logic gate array, various methods have been considered. They are categorized into optical logic devices, coded pattern processing, and other procedural techniques. In addition, data representation is relevant to each implementation, so that we also focus on the format for data representation. 3.1. OPTICALLOGIC DEVICES The most straightforward way to get functionality of logic gates is to find appropriate physical phenomena for logic operations and to apply it to device embodiment. To achieve a logic operation, a kind of nonlinear response is required. As a result, a variety of nonlinear optical phenomena have been studied and adopted for optical logic gate devices.

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In the early days, intrinsically nonlinear materials were investigated and various optical logic devices were developed. For example, the change of the refractive index was used in a Fabry-Perot etalon device to achieve optical bistability (Gibbs, McCall and Venkatesan [1980]). Although the principle is simple and interesting, the slow time response and high operation power are disadvantages of this approach. An effective way to get large nonlinearity for optical signals is to use a hybrid effect of the optical light wave and other physical properties; e.g., electric field, magnetic field, elastic field, and so on. Although Fabry-Perot types of bistable optical devices were fabricated in the early stage (Smith, Turner and Maloney [ 1978]), simpler device structures have been developed successfully. In general, optical devices capable of controlling optical signal distribution in a twodimensional format are called spatial light modulators. Spatial light modulators are not necessarily designed for logic operations, but their nonlinear response and capability of parallel operation are ideal for parallel optical logic gates. PROM (Feinleib and Oliver [ 1972]) based on the electro-optical effect, magnetooptical spatial light modulators (Farhat and Shae [1989]), liquid crystal light valves (Fatehi, Wasmundt and Collines [ 1981], Mukohzaka, Yoshida, Toyoda, Kobayashi and Hara [1994]), and deformable mirror devices (Pape and Hornbeck [ 1983]) are typical examples of spatial light modulators. In terms of speed and driving power, spatial light modulators fabricated of semiconductor materials are the most promising. The multiple quantum well structure of semiconductor materials shows interesting characteristics for optical signals. With the help of self-feedback electric field, high speed and power effective optical logic devices have been developed. SEED families (Miller, Chemla, Damen, Gossard, Wiegmann, Wood and Burrus [1984], Lentine, Hinton, Miller, Henry, Cunningham and Chirovsky [1988], Lentine, Tooley, Walker, McCormick, Morrison, Chirovsky, Focht, Freund, Guth, Leibenguth, Smith, D'Asaro and Miller [1992]), EARS (Amano, Matsuo and Kurokawa [1991]), and an optical thyristor device (Tasiro, Ogura, Sugimoto, Hamao and Kasahara [1990]) are mentioned as typical examples. The data formats of the optical logic devices presented here are rather simple. Each bit of information is expressed by optical intensity or the state of polarization of the light signal for most logic devices. Two different intensity levels or two distinguishable states of polarization, e.g., horizontal and vertical linear polarization, are assigned to a logical one and a logical zero. As a special case, the intensity ratio of a couple of light signals is used for S-SEED to increase operational tolerance (Lentine, Hinton, Miller, Henry, Cunningham and Chirovsky [ 1988]).

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Fig. 3. Schematic diagram of coded pattern processing.

3.2. CODED PATTERN PROCESSING

The effective use of optical linear processing is another way for logic operation. Although logical operations cannot be implemented only by linear processing, a spatial coding technique enables us to achieve them. Figure 3 shows a schematic diagram of the method. Information to be processed is converted into a spatial pattern which is then processed by optical linear methods. The processed pattern is retrieved to the same form as the original information, which provides the result of the logic operation. Optical shadow casting (Tanida and Ichioka [1983]) and spatial filtering logic (Bartelt, Lohmann and Sicre [1984]) are good examples of this technique. Figure 4 shows a conceptual diagram of optical shadow casting for a logic gate SLOAUNRCE~

INPUT PLANE

CODED INPUT B

SCREENs ~ "" ~Jn

DECODING MASK CODED INPUT A

Fig. 4. Optical shadow casting for logic gate array.

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Fig. 5. Conceptual diagram of symbolic substitution. array. Arbitrary logical operation between two binary images is accomplished by this technique. Each couple of pixels at the same location on the input images is converted into any one of four different spatial patterns according to the coding rule shown by the table in the figure. The coded image composed of the spatial pattems is illuminated by an array of point light sources and their shadowgrams are projected onto the screen. Then, the optical signals through the decoding mask provide a result of a logical operation. In this technique, a uniform operation is executed for all data set on the input images in parallel. The contents of the operation are specified by the pattern of the on-state sources. The optical processing used here is considered as a specific version of a cross-correlation called discrete digital correlation. Several extended versions of optical shadow casting have been developed with the introduction of a space-variant decoding mask or polarization (Yatagai [1986], Li, Eichmann and Alfano [ 1986], Karim, Awwal and Cherri [ 1987]). For the case of spatial filtering logic (Bartelt, Lohmann and Sicre [1984]), information is converted into a spatial pattern within a pre-determined set of spatial frequencies. For example, grating patterns with different spatial frequencies or random dot patterns with different grain sizes are used for the coding. Since signals with the same spatial frequency are converged into the same position on the spectrum plane, signals with a specific condition can be selected by the spatial filtering technique. Therefore, this processing is implemented by a typical coherent optical system (Goodman [1996]). As a more interesting technique for logic operation, symbolic substitution has been presented (Brenner, Huang and Streibl [1986]). Figure 5 shows the conceptual diagram of symbolic substitution. Although this technique is quite intuitive, symbolic substitution is recognized as a two-dimensional extension of Boolean logic. A logic operation takes a set of bit information as the input and puts out another set of bits according to a pre-determined transition rule. In contrast, symbolic substitution converts a spatial pattern of bit information into

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another one. The concrete procedure of symbolic substitution is to find specific spatial patterns and to substitute them with other ones according to a substitution rule describing the processing contents. As shown in fig. 5, with specific spatial patterns and substitution rules, desired logic operations can be implemented with this technique. For the coded pattern processing, various kinds of spatial patterns of optical signals, e.g., light intensity or polarization states, play an important role in data representation. Although those techniques require more space on the image plane than direct data representation, flexibility in processing is obtained with the penalty. 3.3. OPTICAL PROCEDURAL TECHNIQUES

To utilize the multi-state nature and parallelism involved in optical signals, various interesting techniques have been presented. These are residue arithmetic (Huang, Tsunoda, Goodman and Ishihara [1979]), multi-level logic (Abraham [1986]), and table lookup processing (Guest and Gaylord [1980]). The first two techniques are useful to suppress carry generation during arithmetic operations, and the last one provides a flexible method for logic generation with a simple optical setup. Residue arithmetic (Huang, Tsunoda, Goodman and Ishihara [1979]) is a number representing system using a set of residue numbers of different divisors. In this system, a non-negative integer x is represented by a set of integers, ( r l , r 2 , . . , rk). ri is the residue of x/yi, and the y;'s are arbitrary prime numbers which are different from each other. With this number representation, any x satisfying the inequality k

O <~x <~H y i - 1

(5)

i=l

can be expressed explicitly. The most salient feature of the residue arithmetic is that operations for individual residues are independent and can be evaluated in parallel. As a result, the parallelism of optics is utilized fully. Various optical implementations of residue arithmetic are considered by Huang, Tsunoda, Goodman and Ishihara [1979]. Multi-level logic (Abraham [1986]) is a logic system in which numbers other than binary are used. Although several variations of multi-level logic exist, an interesting technique for digital optical computing is modified signed digit

88

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DIGITAL OPTICAL COMPUTING

(a)

(b)

Fig. 6. Schematic diagram of table lookup processing: (a) recording truth-table hologram and (b) example of addition (Guest and Gaylord [1980]). number representation (Avizienis [1961]) and its derivatives. In this number representation system, an integer x is represented by

x =Z

ai2i'

(6)

i

where ai is any one o f - 1 , 0, or 1. Applying the modified signed digit number representation to arithmetic operations, we can reduce carry propagation and increase processing efficiency. Table lookup processing (Guest and Gaylord [1980]) is a simple approach to extract parallelism of optics. As shown in fig. 6, the desired response of a logic operation is described in the form of a table in which all combinations of input bits corresponding to logical one (zero) are stored. If input bits are matched to any one stored in the table, a logical one (zero) is returned. Arbitrary logic functions can be implemented by configuration of the table. Since the matching process is achieved in parallel, this technique can effectively utilize the parallelism of optics. The table lookup processing is also known as content addressable memory.

w 4. Methods for Logic Construction We next turn our attention to the topic of how to construct a desired operation using the logic gates described in w3. A straightforward method is connecting

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the gates by optical fibers and/or optical wave guides. However, light wave propagation in free space is preferable in optical computing because of the excellent features afforded by this approach. Since no solid line is required for free-space optical interconnection, large flexibility can be obtained. In this section, we overview several interesting methods, including holographic interconnection, a multistage interconnection network, and a reconfigurable function module. 4.1. HOLOGRAPHIC INTERCONNECTION

An elegant implementation of a logic circuit by a logic gate array and freespace interconnection was demonstrated by a group at the University of Southern California (Jenkins, Sawchuk, Strand, Forchheimer and Softer [1984]). The idea is based on repeated use of a logic gate array with a feedback optical system inserting a holographic optical element as an interconnection configurator. As shown in figure 7, an output signal from each logic gate on the logic gate array device is transferred to the input ports of the other logic gates on the same device via a small facet of the holographic optical element array. Each facet holds interconnection patterns between the logic gates. It can be said that the holographic optical element array stores information on virtual wires in freespace. Because the traveling time of the light wave on the feedback path and the response time of the logic gate device function as delay elements, arbitrary logic circuits, including sequential ones, can be implemented. In this scheme, the density of the logic circuits is limited by that of the holographic optical element. To overcome this problem, three types of interconnection methods,

SYSTEM ~ OUTPUTS.<

~ 9 .

I NTE RCON N EC T IONS [ .... ; ~ .__

GATE

I,--, I GATE

INPUTS I -V,U I OUTPUTS _

.

' '~MB

SYSTEM INPUT S

J

INITORIAL ' OPERATIONS

Fig. 7. Functional diagram of optical sequential logic (Jenkins, Chavel, Forchheimer, Sawchuk and Strand [ 1984]).

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i.e., space-invariant, space-variant, and hybrid interconnection, are considered (Jenkins, Chavel, Forchheimer, Sawchuk and Strand [1984]). 4.2. MULTISTAGEINTERCONNECTIONNETWORK As a constraint of currently available optical logic devices, the small number of fan-out due to the low contrast ratio of the device should be considered. For a SEED device, only two or three fan-outs are possible. To solve the problem, a multistage interconnection network was proposed (Murdocca [1990]). As shown in fig. 8, multiple stages of logic gates and quasi-regular interconnection

Fig. 8. Multistage interconnection network (Murdocca [ 1990]).

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networks are used to construct a logic circuit. The fan-in and fan-out numbers of all logic gates are two, which satisfies the requirement imposed by low-contrast devices. For the multistage interconnection network, there exist several options. Those are perfect shuffle (Stone [1971]), banyan (Jahns [1990]), and crossover (Jahns and Murdocca [1988]), all of which are equivalent in the multistage interconnection network. Based on this methodology, various logic circuits have been constructed, including an exchange network (Murdocca and Cloonan [ 1989]), a sorting network (Desmulliez, Tooley, Dines, Grant, Goodwill, Baillie, Wherrett, Foulk, Ashcroft and Black [1995]), a memory circuit (Murdocca and Sugla [ 1989]), and so on. A general procedure for an arbitrary logic circuit is also presented using the concept of programmable logic arrays (Murdocca, Huang, Jahns and Streibl [ 1988]). 4.3. RECONFIGURABLEFUNCTIONALMODULE One of the interesting features of digital optical computing is reconfigurability in logic circuits using free-space interconnection. Using this feature, a reconfigurable functional module is considered instead of simple logic circuits. Figure 9 compares the forms of circuit embodiment. As seen from the figure, a reconfigurable functional module is an effective means of versatile logic construction. Optical array logic (Tanida and Ichioka [1988]) is a good example of an implementation technique for the reconfigurable functional module. Optical array logic is an extension of the optical shadow-casting logic (Tanida and Ichioka [1983]), whose processing procedure is generalized for arbitrary logic operation. Since logic configuration of the optical shadow casting technique is achieved by switching of the source array, the feature is utilized for the reconfigurable

Fig. 9. Forms of circuit embodiment: (a) simple logic circuit and (b) reconfigurable functional module.

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Fig. 10. Processing procedure of optical array logic.

ab

cd

ef

l

(a) g

i

j

k

o

m

ZDCABZZDADAZ

Do

(c)

'

:

"

i

j

"

k

f

m

"

!

!

Fig. 11. Concept of computational Origami: (a) an original circuit, (b) a regularized circuit, and (c) a folded circuit (Huang [1989]).

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functional module. As shown in fig. 10, any logic function, or even complicated processing, can be implemented by an optical array logic system. The pattern named the operation kernel is equivalent to the pattern of the on-state sources in the optical shadow casting technique. Changing the pattern of the operation kernel is nothing but changing the function of the function module. As a result, a reconfigurable functional module can be realized. An interesting idea called computational Origami (Huang [1989]) is presented to construct an arbitrary logic circuit using reconfigurable functional modules. Computational Origami provides a procedure for converting a function of a parallel processing array into a control sequence of functions for a reconfigurable functional element. Origami is a Japanese traditional art for making a threedimensional object by folding a sheet of paper. The name of computational Origami seems to be after the capability of changing dimensions by a folding process. As shown in fig. 11, a logical circuit mapped on an array of logic gates is converted into a control sequence for a reconfigurable functional element. Computational Origami is useful for mapping functions onto a limited size of processing elements. It also provides a method to expand a sequential process onto a parallel processing element.

w 5. Demonstration Systems Using the devices and the circuit construction techniques presented in the previous sections, we can execute various kinds of processing on the basis of digital optical computing. However, in practice, there are many problems encountered in the construction of an optical computing system. To clarify the issues and to show the capabilities of digital optical computing, several demonstration systems have been presented.

5.1. TSE COMPUTER The TSE computer (Schaefer and Strong [1977]) is a pioneer system that provides a concrete concept of digital optical computing. The term "TSE" means a Chinese character for "letter", which is used to suggest a system capable of processing two-dimensional patterns in parallel instead of a sequential stream of bit data. In the TSE computer, logic gate arrays are connected by bundles of optical fibers to construct a logic circuit.

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5.2. FREE-SPACE FLIP-FLOP AND DOCIP

Based on the free-space optical interconnection with holographic optical elements in w several experimental systems were constructed (Jenkins, Sawchuk, Strand, Forchheimer and Softer [1984], Huang, Sawchuk, Jenkins, Chavel, Wang and Weber [ 1993]). As shown in fig. 12, a liquid-crystal light valve (LCLV) is used for a logic gate array and a computer generated hologram serves as an interconnection configurator. A master-slave flip-flop was demonstrated, which showed stable operation of the system.

M2 ,.

- ~,~---Motor control

~.+//,i/1" .g" A,, I

L5 .z/l//',4~S'-" L2 /l

I I

/{," M!,~, i I. ]{'

,~"i'~"~ P2

LP

PC

. Mask Hologram

/2~

t_ /

n

n

P3

Ar Laser

BS1

L4

L

LED Array !

Fig. 12. Experimental setup of optical sequential logic (Huang, Sawchuk, Jenkins, Chavel, Wang and Weber [ 1993]).

As an extension of the demonstrator, an image processing system was designed and its elemental component was constructed. The system is called the digital optical cellular image processor (DOCIP) (Huang, Sawchuk, Jenkins, Chavel, Wang and Weber [ 1993]) and is specialized for general image processing. Logic circuits of a processing element dedicated for processing on a pixel of the image were implemented. Since the DOCIP is a SIMD type of parallel processing system, demonstration of one processing element is sufficient to describe the whole system.

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5.3. BELL LABORATORIESSYSTEMS A group at AT&T Bell Laboratories (currently Lucent Technology) has contributed actively to research on digital optical computing systems. They have achieved a wide spectrum of accomplishments, ranging from device development to opto-mechanical assembling. Their target was set to a high performance exchange system in optical communication, and several experimental systems have been constructed (Hinton, Cloonan, McCormick, Lentine and Tooley [1994]).

Fig. 13. Schematic diagram of smart pixels. Recently, the Bell Laboratories group (Lentine, Reiley, Novotny, Morrison, Sasian, Beckman, Buchholz, Hinterlong, Cloonan, Richards and McCormick [1997]) focused on a system based on the smart pixel technology. Figure 13 shows a schematic diagram of a smart pixel system. In the system, most of the processing is executed by electronic circuits, and optics is used just for data communication. Such a system has advantages in design, development, and performance improvement. Especially for the case of optical communication, information itself is an optical signal, so that such an optical exchange system is effective because there is no requirement for undesirable optical-to-electronic and electronic-to-optical signal conversions. 5.4. O-CLIP The optical-cellular logic image processor (O-CLIP) (Walker, Craig, McKnight, Redmond, Snowdon, Buller, Restall, Wilson, Wakelin, McArdle, Meredith, Miller, MacKinnon, Taghizadeh, Smith and Wherrett [1991 ]) was developed as an optical version of the cellular logic image processor (CLIP). The O-CLIP achieves general image processing based on cellular logic (Preston and Duff [1984]). Figure 14 shows the optical setup of the O-CLIP. A holographic optical element provides a space-invariant interconnection pattern; namely, all data from a logic gate arrays are transferred with the same interconnection

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Fig. 14. Optical setup of O-CLIP (Walker, Craig, McKnight, Redmond, Snowdon, Buller, Restall, Wilson, Wakelin, McArdle, Meredith, Miller, MacKinnon, Taghizadeh, Smith and Wherrett [1991]). pattern, which provides advantages in fabrication and packaging density. Systems with 16 x 16 pixels by ZnSe nonlinear interference filter devices (Walker, Craig, McKnight, Redmond, Snowdon, Buller, Restall, Wilson, Wakelin, McArdle, Meredith, Miller, MacKinnon, Taghizadeh, Smith and Wherrett [ 1991 ]) and with 16x8 pixels by S-SEED's (Wakelin, Tooley and Smith [1993]) were presented. 5.5. OPALS Based on optical array logic, an optical parallel array logic system (OPALS) has been presented by the authors' group as a general-purpose parallel computing system (Tanida and Ichioka [ 1986]). The OPALS has a structure in which the output of an optical array logic processor is fed back as one of the inputs of the processor to effectively utilize hardware. As experimental systems, an optoelectronic hybrid version called H-OPALS (Tanida, Miyazaki and Ichioka [1992]) and a pure optical version called P-OPALS (Tanida, Konishi and Ichioka [1994]) were constructed. The H-OPALS processes 16 x 16 pixels in parallel at 20000 frames per second, whereas the P-OPALS does 100x300 pixels at 3 frames per second. Figure 15 depicts an encoded image used in the P-OPALS. In this system, encoding is achieved by optically using a birefringent crystal.

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Fig. 15. Image data used in P-OPALS (Tanida, Konishi and Ichioka [1994]).

5.6. SPE-4K

An optoelectronic digital computing system called SPE-4k has been developed by a group at the university of Tokyo (Ishikawa [1994]). Figure 16 shows a schematic diagram of the system. The SPE-4k is composed of an array of microprocessors with optical input/output ports for communication. This system is designed to minimize the number of electronic logic gates per processing element for compact integration. A processing element for a pixel can be I processingelement photo detector -*

0

0

*~- LD "

2

1

SPE

pattern ~

output irkerconnection

ree~nligurable hologram Fig. 16. Schematic diagram of SPE-4k (Ishikawa [1994]).

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constructed with 337 gates. Various applications, including real-time image processing and dynamic solution of partial differential equations have been demonstrated on the system. A compact optical system using selfoc rod lenses has also been built (McArdle, Naruse, Okuto and Ishikawa [1998]).

5.7. SMARTPIXEL SYSTEMS Among the most promising architectures for digital optical computing are those based on the smart pixel technologies. In addition to the Bell Laboratories' system, various demonstration systems have been constructed using this strategy. The reasons why so many research efforts are proceeding are the maturity of optoelectronic devices, the presence of concrete procedures for system construction based on the LSI (large-scale integration) design methodology, and coincidence between the demand from the users and the features provided by this architecture. With a procedure similar to that used in designing LSI devices, one can design smart pixels and construct various types of optoelectronic processing systems. One sophisticated example of systems based on smart pixel technology is an optical sorting system presented by a group at Heriot-Watt University

Input SLM

Memory array

iiii

Output SLM

iii

~ .....

# fo: ..." : .. f.:" f 9

...' f. :

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.

.... .........Sorting node array

9 9

/ ....: .,

~ f:

...

........

".................... 7".... ..." of~

..

/"

f. .:"

' 2-D folded ...................... ...... perfect shuffle

Fig. 17. Schematic diagram of optical sorting module (Desmulliez, Tooley,Dines, Grant, Goodwill, Baillie, Wherrett, Foulk, Ashcroft and Black [1995]).

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(Desmulliez, Tooley, Dines, Grant, Goodwill, Baillie, Wherrett, Foulk, Ashcroft and Black [ 1995]). Figure 17 shows a conceptual diagram of the system. Inside the optical feedback loop, smart pixel devices functioning as sorting nodes and a memory array are connected by free-space optical interconnection with a perfect shuffle interconnection pattern. The sorting node and the memory array are fabricated by CMOS integration contacted on an InGaAs/GaAs SEED for optical signal modulation and detection. The flip chip bonding technique (Goodwin, Caswell, Parsons, Bennion and Stewart [1989]) is used to precisely assemble those devices. Since perfect shuffle provides a non-local interconnection pattern with two fan-outs, the system satisfies the requirement imposed by the low contrast characteristics of semiconductor based devices such as a SEED device.

PBS+QWP

Macro relay lenses

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Y Fig. 18. Schematic diagram of optical backplane system (Liu, Robertson, Boisset, Ayliffe, Iyer and Plant [1998]). Another practical example of the smart pixel system is the backplane interconnection bus developed by a group at McGill University (Liu, Robertson, Boisset, Ayliffe, Iyer and Plant [1998]). In the system, four-stage unidirectional ring free-space optical interconnection is implemented. Figure 18 shows a schematic diagram of the overall optical system. Modularization is employed positively for effective system design and fabrication. The system is fitted into a 6U commercial VME electronic backplane chassis. CMOS-SEED devices are

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used in the system. The remarkable points of the system are that detailed system design has been made and that a good mechanical tolerance was obtained on the system. This example shows the importance of overall system design to realization of a practical system. 5.8. OAL-NC

In a common sense, processing and interconnection are different tasks and they are implemented by different techniques. As the smart pixel systems become dominant in digital optical computing, the main function of optical technology seems to be limited to signal communication. However, the optical technology can provide much more than signal transfer. Based on this idea, a computational architecture called optical array logic network computing (OAL-NC) is presented below (Tanida and Ichioka [1998]).

Fig. 19. Conceptual diagram of optical array logic network computing.

The OAL-NC is a general-purpose computing architecture based on optically linked electronic multiprocessors. Figure 19 shows the conceptual diagram of the OAL-NC. The system is composed of two layers: an electronic multiprocessor layer and an optical network processor layer based on optical array logic. The key point of the OAL-NC is that each processing element on the electronic multiprocessor layer passes optical signals to the optical network processor layer with the spatially encoded form of the optical array logic scheme. As a result, the signals coming from the electronic processing elements can be processed directly in the optical network processor layer according to the processing scheme of optical array logic. The optical network processor executes not only data transfer but also data processing; e.g., data broadcasting, conditional data transfer, data matching, and so on. As a result, effective processing performance is expected. From the point of view of computational architecture, the OAL-NC is a kind of heterogeneous computing system. Cooperation

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of different characteristic processors provide large flexibility and processing power.

w 6. Software Research

A computer cannot perform any task without a program code describing how to manipulate data using the hardware. The same situation exists for the digital optical computing scheme. The digital optical computing scheme is considered as an extension of conventional logic system, but specific consideration is required to utilize its features. In this section, interesting research related to software for digital optical computing are described. 6.1. PARALLEL LOGIC SYSTEMS FOR PROGRAMMING

To utilize the parallel nature of optical technology, several logic systems have been proposed. As examples of them, cellular logic, symbolic substitution, optical array logic, binary logic algebra, and image logic algebra are explained.

6.1.1. Cellular logic Cellular logic (Preston and Duff [1984]) is a two-dimensional extension of the combinatorial logic system. Originally, cellular logic was invented for electronic computing, but its suitability for parallel processing is valuable for the optical computing scheme. The cellular logic transforms one data array into another according to the values of the neighboring data. As seen from fig. 20, each pixel signal on the input image is spread onto the neighboring pixels. This process is expressed as

~(x,y) =

SL(X ,y ) k L ( x - x , y - y )cbc' d3/, f

! !

/

(7)

/

where SL(X,y) and kL(x,y) are the input data and operation kernel. The output value of the processing, rL(x,y), is determined by the thresholding function to the convolved signals ~(x,y);

rE(x,y) =

1 if ~(x, y) > Z, 0 otherwise.

(8)

By configuring the kernel function and the thresholding value Z, a variety of processing can be implemented; e.g., image enhancement, noise reduction, and simulation of parallel processes.

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Fig. 20. Processingprocedure of cellular logic.

6.1.2. Symbolic substitution As mentioned in w3.2, symbolic substitution (Brenner, Huang and Streibl [ 1986]) is a powerful technique for digital optical computing. Not only logic operations but also higher levels of processing can be achieved by the symbolic substitution scheme. The procedure is as follows: (1) Define code patterns to represent the information to be processed; (2) Under this definition, find substitution rules to accomplish the desired processing; (3) With the substitution rules, execute the procedure of symbolic substitution; and (4) Retrieve the processed patterns into the initial form of information. Symbolic substitution has large flexibility in data representation and processing algorithms, which shows the potential capability of the scheme. However, no systematic study for describing arbitrary processing has been done for symbolic substitution.

6.1.3. Optical array logic Optical array logic (Tanida and Ichioka [1988]) described in w is also a logic system suitable for the digital optical computing scheme. Complicated processing can be described and executed by the optical array logic scheme. Compared with symbolic substitution, optical array logic uses a conservative way of processing description; viz., all processing in optical array logic is described by logical operations for pixels within a neighborhood area on the two input

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SOFTWARE RESEARCH

103

p r o g r a m addition; var i, N; image imagea=../image/add.attr; image imageb=../image/add.data; kernel add; N:7;

/* N bit

add:

I 1. I I_0. I *

+I 1.1 I_o. I

*/

*

.0 .i .i .0

+1_1. I I 0.1 * t _ . l l [ .11@(o,i); for i:l to N do i m a g e b : exec( end; imout

add.result

imagea,

imageb,

add);

imageb;

end addi t ion; Fig. 21. Program example used in optical array logic.

images. A general form of the operation executed by optical array logic is represented as follows: K

L

L

k=l

m=-L

n=-L

where a i + m , j + n and b i + m , j + n are input images and c;.j is the output image, f ( . ) indicates any one of two-variable binary logic functions, and L and K are the size of the neighborhood area and the number of product terms, respectively. Arbitrary logical operations can be executed by configuring f,,.,,: k's for individual m, n and k. Unfortunately, it is difficult for us to capture the meaning of this general form, so that a specialized notation language is prepared. Using the language, processing in optical array logic can be written as the program format shown in fig. 21.

6.1.4. Binary image algebra Binary image algebra (Huang, Jenkins and Sawchuk [1988]) is an algebraic system specialized for compact description of operations on binary image data.

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Binary image algebra is constructed by an image space and a family of operations including three fundamental operations and five elementary images; viz.,

BIA - (P(W); -, U, |

(10)

where P(W) is an image space which is the power set of a predefined universal image W. The fundamental operations are complement, -, union, U, and dilation, | Each elementary image consists of just one pixel located at either the origin or one of its neighboring positions. The binary image algebra is applied to design effective logical circuits for the DOCIP (Huang, Sawchuk, Jenkins, Chavel, Wang and Weber [1993]).

6.1.5. Image logic algebra Image logic algebra (Fukui and Kitayama [1992]) is designed as a genetic language for parallel image processing. In image logic algebra, binary image processing is treated as a sequence of logical template matching. Each logical template matching is described by a kernel pattern used in a discrete correlation for the target image. A neighborhood configuration pattern is introduced for compact description of processing. For example, a transformation from A[M; N] to B[M; N] is expressed as follows:

B[M;N] = A[M;N] |

Ilrll ,,,,

(11)

where [[r[[(k,t) is the neighborhood configuration pattern. Image logic algebra comprises three operations of images, six transformations of images and three operations of the neighborhood configuration patterns. As shown in fig. 22, image casting, multiple imaging, and test transformation are unique

Fig. 22. Transformations defined in image logic algebra: (a) image casting and (b) multiple imaging (Fukui and Kitayama [1992]).

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transformations. In addition, extended erosion and dilation are defined with the help of the neighborhood configuration pattern. 6.2. APPLICATIONSOF PARALLELLOGIC SYSTEMS Various kinds of problems have been solved by the parallel logic systems explained in the previous section. Applications of the parallel logic systems cover a wide field, including image processing, numerical processing, parallel process emulation and non-numerical processing.

6.2.1. Image processing Image processing is an essential application of the digital optical computing system. Since an image is a primitive data format of the digital optical computing scheme, it can be processed effectively, and a variety of studies has been made of the subject. Due to differences in processing methods, image data are often categorized into two major classes: binary and multi-valued images. The former is composed of binary pixels, which can be processed by simple logic operations. The latter consists of an array of multi-level pixels, which requires a complicated procedure for data representation and processing. As the data format of the multivalued image, a bit expansion and a bit slice format are considered, as shown in fig. 23. Once the target image is represented by a binary form, it can be processed by logical operations in the digital optical computing scheme. For simple image processing, heuristic approaches are effective, as shown by Brenner, Huang and Streibl [1986] and Tanida and Ichioka [1988]. For flexible processing, techniques based on digital filtering are effective. Goodman and

Fig. 23. Data format for multi-valued image: (a) bit expansion and (b) bit slice.

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Rhodes [1988] present a procedure using the symbolic substitution, and Tanida and Ichioka [ 1988] show a counterpart by the optical array logic scheme. Another method is the use of morphological operations (Serra [1982]). Huang, Jenkins and Sawchuk [1988] and Fukui and Kitayama [1992] show how to implement image processing based on morphological operations within the frameworks of binary image algebra and image logic algebra, respectively.

6.2.2. Numerical processing The parallelism provided by digital optical computing is attractive for scientific computing. In scientific computing, large-size matrices often appear and their efficient calculation holds the key for high-performance computing. Therefore, various methods for matrix computation have been studied actively. The four elementary operations, i.e., addition, subtraction, multiplication, and division, are implemented by many researchers. For the case of symbolic substitution, addition was demonstrated by Brenner, Huang and Streibl [1986] and multiplication and division are described by Hwang and Louri [ 1989]. For optical array logic, addition, subtraction, and multiplication are discussed by Tanida, Fukui and Ichioka [ 1988]. Also, multiplication by image logic algebra is discussed by Fukui and Kitayama [ 1992]. Although most parallel logic systems treat data in the SIMD manner, numerical processing usually requires minute data manipulation. To overcome the problem, various sophisticated techniques have been developed. Figure 24 shows a processing procedure of addition by optical array logic (Tanida, Fukui and Ichioka [1988]). In addition to an image containing the data, an auxiliary image is prepared for storing data attributes. Use of the attribute image is a key point of the implementation. Whereas the data image holds both addends and augends, the attribute image keeps the information of location of pairs of an addend and an augend. Addition is achieved by repeated application of the following operations calculating sum and carry of each bit position: sum / = 2iy i + xiy; i, carry/+ 1

_

.

xiyi,

(12)

(13)

where x ~and y/are the ith bit position of the addend and the augend, respectively. Because the addend and the augend are placed on the data image, the actual operation for the images becomes Ci, j = ~ l i , j a i _ l , j b i , j l ) i _ l , j + ~ l i , j a i _ l , j l ) i , j b i _ l , j + a i , j a i + l , j b i , j + l b i + l , j + l ,

(14)

where a, b, and c are pixels in the attribute, data, and output images, respectively. The suffix indicates the address of the pixel on the image. Note

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Fig. 24. Processing procedure of addition by optical array logic (Tanida, Fukui and Ichioka [1988]). that a product of a's, e.g., ~li,jai_l,j, becomes 1 only if the condition is satisfied. Therefore, such a product can be used to identify the location where x and y are placed on the image. Although the operation in eq. (14) is more complicated than those in eqs. (12) and (13), we do not take care of the position and the length of each couple of the addend and augend. This feature should be emphasized as an important feature of the implementation.

6.2.3. Emulation of parallel processes Because of its inherent parallel nature, the digital optical computing scheme seems to be suitable for emulation of various parallel processes. Actually, by describing a primitive process or behavior of an individual element by logic operations, we can effectively emulate a complicated phenomenon consisting of large number of elements. Although the main purpose of the emulation is to analyze a complicated phenomenon, it can be extended as a computing technique. Based

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Transition rule

~~

(a)

(b)

l'l I'I [o1,Iol

(c)

1-1

l'~.Jl

(d)

Fig. 25. Virtual logic circuit emulated by symbolic substitution (Murdocca [1987]).

on the idea, an interesting technique was proposed with symbolic substitution (Murdocca [1987]). As shown in fig. 25, a logic circuit is constructed using spatial patterns on an image. Then an information token is placed on the image as a spatial pattern. Applying a simple substitution rule appearing in the figure, one can emulate information propagation along with the logic circuit. Since all processes are driven in parallel, large processing capability can be expected. The Turing machine is emulated by two parallel logic systems to prove their computational capability (Brenner, Huang and Streibl [1986], Tanida, Nakagawa, Yagyu, Fukui and Ichioka [1990]). Although the Turing machine itself is not an optimized system, the same technique used in the Turing machine can be used to emulate various types of virtual processing systems. Reconfiguring a processing system for a given problem, we can virtually construct an effective processing system for the problem. As an example, matrix processing was performed by emulation of a systolic processor (Fey and Brenner [1990], Fukui, Tanida and Ichioka [1990]). Emulation of a data flow machine by optical array logic was reported (Iwata, Tanida and Ichioka [1993b]). 6. 2.4. Non-numerical processing The large parallelism provided by digital optical computing seems to be ideal for tasks handling a large amount of information. Processing of huge amounts of data is one of the important applications of digital optical computing. As an example, processing for database management is considered as a promising application of digital optical computing.

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Fig. 26. Processing example of database managementby optical array logic: (a) relational database image, (b) selection rule, and (c) resultant image.

An example of optical implementation of a relational database is reported by Iwata, Tanida and Ichioka [1993a]. Figure 26 shows a processing example of database management by optical array logic. Mapping data onto an image plane as a database, one can manage the database by logical operations on the image. As an extended application of database management, inference operations are demonstrated by optical array logic (Iwata, Tanida and Ichioka [1992]). Using a specially designed database in which relations between objects are stored, objects satisfying a given condition are extracted. An expert system was constructed with the same technique.

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w 7. Further Directions Without a doubt, today's electronic computing systems are highly developed and their performance is increasing steadily. It seems to be difficult for new technologies to take the place of the steady reign of electronics. Instead, a 'newcomer' should cooperate with electronics to perform specialized functions. This is true for digital optical computing. A reasonable strategy for optics is to implement specialized functions supplementing weak points of electronics and to enlarge its application field more and more. Reflecting this situation, the research field of 'optical computing' is gradually shifted to that of 'optics in computing'. A promising architecture of the digital optical computing system is that based on smart pixel technologies. In this architecture, the state-of-art technologies for device design and fabrication can be utilized effectively. This form of application is nothing but an extension of optical interconnection technology; viz., free-space optical interconnection is employed effectively to connect elemental processors comprising a high-performance parallel information processing system. In addition, such a smart pixel system has an advantage in compactness. With appropriate packaging methods, a compact and reliable information processing system could be constructed. In terms of applications suitable for digital optical computing, a signal exchange system for optical communication and a processing system for multimedia applications are promising. The importance of an information highway like the Internet is beyond discussion. On the network various forms of information including sounds, images, and movies are transferred frequently. Because of the large amount of information and the tremendous number of potential users, the communication system is required to handle effectively extremely large quantities of information. For the demand, development of a new parallel communication network system, which has an ultra-high bandwidth capable of transmitting high-resolution images at high speed would be expected. For the new network system, the digital optical computing system would play an important role. In addition, development of a practical system will be an additional impetus to promote digital optical computing.

w 8. Conclusion Various interesting ideas and technologies related to digital optical computing have been reviewed to clarify the concept of the scheme. From basic concepts to practical applications, some of the results accumulated in this field have

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been presented in overview form. The important features of the digital optical computing scheme are parallelism and familiarity with current electronics. Due to these features, the digital optical computing scheme is a promising candidate for high-performance information processing. As a result, digital optical computing would be expected to be one of the most important and most fundamental techniques in the information processing area in the near future.

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E. WOLE PROGRESS IN OPTICS XL 9 2000 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED

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CONTINUOUS M E A S U R E M E N T S IN QUANTUM OPTICS

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V P E ~ X I O V . , i, AND A. L U K S

Laboratory of Quantum Optics, Faculty of Natural Sciences, Palaclo) University, Tr Svobody 26, 771 46 Olomouc, Czech Republic

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CONTENTS

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INTRODUCTION

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THE M A N D E L AND SRINIVAS-DAVIES APPROACHES TO

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M O D E L I N G PHOTODETECTION . . . . . . . . . . . . w 3.

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REVERSIBILITY AND IRREVERSIBILITY OF PHOTOCOUNTING . . . . . . . . . . . . . . . . . .

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MODELS OF CONTINUOUS M E A S U R E M E N T

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PROBABILITY-DENSITY FUNCTIONAL FOR QUANTUM

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PHOTODETECTION PROCESSES . . . . . . . . . . . .

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QUASICONTINUOUS SCHEMES OF P H O T O D E T E C T I O N .

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SPECIAL STATES OF AN OPTICAL MODE BY MEANS

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OF CONTINUOUS AND NEARLY C O N T I N U O U S M E A S U R E M E N T S ON THIS MODE . . . . . . . . . . .

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PRODUCTION OF CORRELATED PHOTONS

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CONDITIONAL GENERATION OF SPECIAL STATES USING

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IDEAL STATE R E D U C T I O N OF ENTANGLED FIELDS.

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w 10. SPECIAL STATES OF ONE OR TWO OPTICAL MODES BY MEANS OF CONTINUOUS M E A S U R E M E N T ON A DIFFERENT MODE . . . . . . . . . . . . . . . .

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w 11. EIGENVALUE PROBLEMS FOR INTELLIGENT STATES GENERATED 1N IDEAL AND C O N T I N U O U S M E A S U R E M E N T S . . . . . 116

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ACKNOWLEDGEMENTS

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Appendix A. SUPEROPERATOR-VALUED M E A S U R E AND RELATED CONCEPTS. .

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Appendix B. ITO'S CALCULUS WITHOUT ITO'S DIFFERENTIAL

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w I. Introduction The description of the photodetection process is an indispensable part of quantum optical courses, and yet many papers have been published contributing to this topic. The researchers have taken into account more and more items of detail, such as polarization, special characteristics, and inefficient detection. Both thermal and laser radiations were studied using the photodetection equation, first in the framework of the semiclassical approach to the photodetection (Mandel [1958, 1959]), and then respecting the quantum optical approach to the photon-number measurement (Mandel [1963a], Kelley and Kleiner [1964], Glauber [1965], Mollow [1968]). In accordance with this, the formally perfect model of the photon absorption by a detector (Srinivas and Davies [1981]) was not first accepted and its use in quantum optics was debated for some time (Mandel [ 1981 ], Srinivas and Davies [ 1982]). The Srinivas-Davies model can be generalized to involve a variable absorption rate which, in a particular case of its time dependence, gives a constant intensity of the photocount process. It could be interesting with respect to photon bunching and antibunching, which were first defined for a stationary field (Mandel and Wolf [1995], Torgerson and Mandel [ 1997]). The relationships of these concepts to the subPoissonian behavior have been treated (Singh [1983], Jakeman [1986], Zhou and Mandel [ 1990]), and they have been generalized to nonstationary processes (Singh [1983], Dung, Shumovsky and Bogolubov [1992]), whilst Srinivas and Davies [1981] considered these concepts only for their nonstationary process. Believing that the quantum theory is not complete without the projection postulate, we will mention the attempts to avoid this postulate (Omn+s [ 1994], Haus and K~irtner [ 1996]) as well as a scheme of successive measurements which is based on this postulate (Wigner [1963]). On the other hand, it is important that the master equation in the Lindblad form (Lindblad [ 1976]) describing the photon absorption by the detector suggests its unraveling and also the unraveling of its solutions; i.e., their expression as expectation values of the time-dependent statistical operators respecting the detection process. The master equation relates to the evolution equation for an unnormalized statistical operator between registrations and a transformation of the unnormalized statistical operator due 118

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to a registration [Ueda, Imoto and Ogawa [1990a], Carmichael [1993a], Knight and Garraway [1994], Ban [1997a]). A general description is applied both to the case of an ordinary photon counter which does not have a logically reversible dynamics, and to the case of a quantum counter (Mandel [1966]), with a logically reversible dynamics (Ueda and Kitagawa [1992], Ueda, Imoto and Nagaoka [ 1996]). Concentrated on atoms, some papers have been concerned with the reversibility of quantum jumps (Mabuchi and Zoller [1996], Pellizzari, Beth, Grassl and Mfiller-Quade [1996], Ekert and Macchiavello [1997], Mensky [ 1996a], Nielsen and Caves [ 1997], Bennett, Brassard, Cr6peau, Jozsa, Peres and Wootters [ 1993]). The time evolutions, quantum jumps, the quantum trajectory idea, and the stochastic wave-function evolution concept have been applied in numerical simulations and formalized in the quantum stochastic calculus together with the Gaussian (or diffusive) continuous measurement, stochastic equations and Monte Carlo simulation schemes. Although based in part on formal measurement arguments, the literature also proves that the quantum trajectories arise naturally (cf. Carmichael [ 1996]). We will develop an independent stochastic formulation of the detection theory, although a standard approach exists (Parthasarathy [1992]). We will base the considerations on the damped quantum harmonic oscillator (Lax [ 1966], Louisell [ 1973], Gardiner [ 1991 ]). First we will expound the Heisenberg-Langevin approach to the description of the damped harmonic oscillator and more general open systems interacting with a thermal reservoir. Besides the It6 integral (It6 [ 1944]), we will introduce the stochastic Stratonovich integral (Stratonovich [1963, 1964]). We will then introduce the quantum stochastic Schr6dinger equation for a system interacting with a reservoir. We will outline a derivation of a stochastic nonlinear Schr6dinger equation for the state conditioned on the outcomes of photon-number measurements (Carmichael [ 1993a], Wiseman and Milburn [ 1993a], Barchielli and Belavkin [ 1991 ]). We will mention a nonlinear Schr6dinger equation for a state conditioned on the results of measurement of continuous observables (Gisin [1984], Gisin and Percival [ 1992, 1993a,b]). Even the continuous quantum nondemolition measurement of photon number can be treated as a particular case of the back-action of the continuous measurement and the photocount registration on the radiation field (Ueda, Imoto, Nagaoka and Ogawa [ 1992]). A quantum theory of feedback has been established (Wiseman and Milburn [1993b, 1994], Wiseman [ 1994a,b,c]). The developed formalism can also be applied in the case of the microscopic models of photonnumber measurements (Imoto, Ueda and Ogawa [ 1990], Ueda, Imoto, Nagaoka and Ogawa [ 1992]).

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The probability-density functional for a quantum photodetection process has been invented to provide complete information on the photocount process dependent on the initial statistical operator (Ueda [1989a,b, 1990]). We will focus our attention on the generating functionals, which in comparison with the probability-density functional, present fewer difficulties and describe well the random-point process. An analogy to continuous measurement is also useful in the description of an apparatus consisting of many lossless beam splitters and photodetectors, which can be called a quasicontinuous photodetection scheme with relationship to a destructive continuous photon-number measurement (Ban [1994]). There exists an analogous device with the parametric amplifiers equivalent to the quantum counter, and one with four-wave mixers equivalent to the quantum nondemolition continuous measurement. The continuous destructive photon-number measurement can be used for generating a Schr6dinger-cat state from an ordinary squeezed state (Ogawa, Ueda and Imoto [ 1991 a]). By means of continuous photodetection, one produces novel states of a one-mode optical field conditioned on specific situations during the photodetection. Ideal and continuous measurements lead to a peculiar behavior when applied to one of two entangled fields which can be produced either on a beam splitter or in a parametric down-conversion process. Cascaded down-conversion processes can produce entanglement of three and four modes. In addition to these theoretical proposals, encouraged by experiments which were positive in clear violation of the Bell inequality (Walls and Milburn [ 1994], Pefina, Hradil and Jur6o [1994]), there are some experiments which were implemented in the continuous-wave regime. Interest in the generation of the sub-Poissonian light dates back to Yuen [1986a], concentrating on the use of the parametric amplifier and the photonnumber measurement on one of the correlated modes. One can also consider the complex amplitude measurement or quadrature measurement on one of these fields (Watanabe and Yamamoto [1988]). A general analysis also applies to a down-conversion stimulated by a two-mode input coherent state (Agarwal [1990]). Many of the quantum statistics of the special states due to the photon-number measurement have been calculated (Luke, Pefinovfi and K~epelka [ 1994]). For the beam splitter also, the photon-number measurement, along with the complex amplitude and quadrature measurements, can be analyzed (Ban [1996a]). This single device has also been proposed as a means of generating the Schr6dinger-cat states (Dakna, Anhut, Opatrn~,, Kn611 and Welsch [1997], Dakna, Kn611 and Welsch [1998]). In addition to these suggestions, we will pay attention to the states arising in the measurement of three and four correlated

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modes after down-conversions with complete or partial alignment of the idler modes (Luis and Pefina [ 1996]). A somewhat more difficult analysis is needed in this connection for the description of the continuous destructive photon-number measurement carried out simultaneously with the nondegenerate parametric amplification (Holmes, Milburn and Walls [1989]). We will concentrate rather on a situation where the counting is done after the interaction which produced the correlated state. In the case of measurement on one of two entangled modes, the evolution of the moments of the measured and unmeasured fields depends on the variances and covariances of the amplitude of these fields (Ueda, Imoto and Ogawa [1990b]). The evolution of many conditional statistics of the unmeasured mode has been obtained in the case of destructive continuous photon-number measurement on the idler mode of a two-mode squeezed state (Pefinovfi, Luk~ and K~epelka [1996a,b]). The conditional statistics have also been obtained in the case of the nondemolition continuous photon-number measurement (Pefinovfi, Luk~ and K~epelka [ 1996b]). The continuous quadrature measurement has also been analyzed (Breslin and Milburn [1997]) and the scheme of the quantum nondemolition continuous-wave detection of the quadrature has been implemented in experiments (Schiller, Bruckmeier, Schalke, Schneider and Mlynek [1996], Bruckmeier, Schneider, Schiller and Mlynek [1997], Bruckmeier, Hansen and Schiller [1997]). Luke, Pefinovfi and K~epelka [1994] and Luis and Pefina [1996] have formulated eigenvalue problems and analyzed the ideal reduction of the correlated state of light fields. A generalization of this theory to the case of the destructive continuous state reduction has been accomplished (Pefinovfi, Luk~ and K~epelka [ 1996a]).

w 2. The Mandel and Srinivas-Davies Approaches to Modeling Photodetection With the discovery of the photoelectric effect, photometry has gained an important tool. Whereas purely classical apparatuses present no problems, photoelectric detectors should be described in a more sophisticated way. 2.1. S E M I C L A S S I C A L APPROACH TO P H O T O D E T E C T I O N

The semiclassical theory of photoelectric detection of light describes the situation where the light intensity is converted into irregular releases of

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photoelectrons. In the simple case of a quasimonochromatic electromagnetic wave, the energy E of the wave, considered within the finite interval from t to t + T, is converted into several photoelectrons in dependence on the detector efficiency r/, and it should be regarded as consisting of m photons of energy hco, where h is Planck's constant divided by 2st, and underlining indicates that a variable is stochastic. Since in this situation the Poisson process is appropriate and the equation E = (hmm) may express that the wave and particle aspects are not separated, we arrive at the probability p'(m, t, T) for m photons to occur within the interval from t to t + T,

I (E)

p'(m, t, r ) = ~

~-~

m

(

E)

exp -~-~

.

(2.1)

The photocount distribution takes into account the efficiency r/, and it reads

p(n, t, T) = ~1 W" exp(-W),

(2.2)

with the parameter W = rlE/(hoo). Since the field need not be quasimonochromatic, we write

f

t+T

W = ~1

I(t') dt',

(2.3)

dt

where I(t) is the 'light intensity'. Strictly speaking, this quantity, having the dimension of time -1 , cannot be just the light intensity in the nonquasimonochromatic case. The Poisson process has the property that the numbers of counts in the nonoverlapping intervals are distributed according to the respective lengths and are statistically independent. Let us consider the numbers of counts n l , . . . , ~ and the intervals [tl, tl + T i ) , . . . , [tj, tj + Tj). The statistical independence implies a joint photocount distribution of the form d

p(nl,. . . ,nj, tl, T1, . . . , tj, Tj) = H p ( n j , tj, Tj),

(2.4)

j=l

where the photocount distributions are

p(nj, tj, Tj) = 1 Wj~jexp(-Wj), nj!

j = 1 "'

J,

(2.5)

with

Wj = rl

f

d t;

ts+~

I(t') dt'.

(2.6)

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When the optical field fluctuates, the light intensity is a random process/(t), and the relation (2.6) takes the form t/+r/ (2.7) W_j = 7/ /(t') dt'. at/ The relation (2.4) becomes

/

p ( n l , . . . , r t j , tl, T1,. . . , tj, T j ) =

rl--~.

(2.8)

"

j=l

When the intervals [tj, tj + T/), j - 1,... ,J, do not overlap, we consider the joint probability density P ( W 1 , . . . , W j , tl, T l , . . . , tj, T j ) and we can write the averaging operation as an integral: p ( n l , . . . , n j , tl, T1, . . . , tj, T j )

-fo ~

9

~0"~

,,

ii

j:l

1 Wj+ exp(- W/)P(W1

~

, ' ' ' 9

nj!

W/t,,T1, ,

" ' ' ,

tiT/) ,

"

fi

d~.

j:l

(2.9) The case J = 1 is known as the Mandel photodetection formula (Mandel [1958, 1959]). Using the notation of the left-hand side of relation (2.4), we can introduce other notation due to Ueda [1988] (cf., a notation on the probability distributions of triggered counting (Ueda [1989b]); e.g., (l(h), n, l(t2)) =p(1,n, 1 , t l , A t l , t l + A t l , t 2 - tl - A t l , t 2 , A t 2 ) , (2.10) ( e ( t l ) , n , l(t2))=p(n, 1,tl + Atl, t2 - tl -Atl,t2, At2). We can obtain the following expansions (Ueda [1988]): lim

(l(h),n, l(t2)) _

At, -+ O, 6t2 ~ 0 lim

A h At2

,~ m 0 ill

~

=

0k

( e ( t l ) ' n ' l(t2)) = - ~

At2

At2 ----+0

=

0

0

Otl Ot2 p ( k ' tl , t2 - tl ),

~2P(k,

(2.11) t ~ , h_-

t~).

k=0

We mention the rest of the notation due to Ueda [1988]: P~,.cl(tl, t2), P c e l ( t l , t2), P s l l ( h , t 2 ) , P c l l ( t l , t 2 ) . Particularly, to quote only the simplest formulas, P s e l ( t l , t2) -"

lim

(e(tl),O,l(t2))

At2 --+ 0 O(3

Pcel (tl, t2) = Z

At2

( e ( t l ) , n , 1(t2))

lim At2 --' 0

(2.12)

At2

n=0

Applications to the fundamental statistics, that is, the Poisson and BoseEinstein statistics, have been provided. For the Poisson statistics, the probability

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distribution p(n, T) of n photoelectrons being registered during time T is given by (2.13)

p(n, T) = ~ ( I T ) " exp(-IT),

where I is the average number of photoelectrons per unit time. We can obtain that Psll(T)

= Psel(T) =

I exp(-IT),

Pc1 I ( T ) = P c e l ( T ) = L

(2.14)

The facts that Psll(T) and Poll(T) are equal to Psel(T) and Pcel(T), respectively, indicate that no two photoelectrons in the Poisson process are correlated. On the other hand, for the Bose-Einstein statistics two photoelectrons are correlated or bunched if their time interval is comparable to or shorter than the coherence time rc of light. The photocount distribution p ( n , T ) for the Bose-Einstein statistics is given by (Mandel [1958, 1959, 1963b]): p(n, T) =

(IT)" (1 + IT) n+l '

(2.15)

where T is required to be much shorter than re. We can obtain that (Glauber [1967]): Pse~(T) =

I (1 + IT) 2'

P~e~(T) = I.

(2.16)

For short time intervals such that IT < 1, Psll(T)-

21 > Psel(T), (1 + IT) 3

(2.17)

which shows that photons which obey the Bose-Einstein statistics tend to be registered in bunches. P~ll(T) is twice as large as P~eI(T): (2.18)

Pcll(T) = 2I = 2Peel(T).

Two probability distributions under the same initial condition, but for different time intervals, have been compared (Mehta and Wolf [1964a,b]), P~ll(r) = 2P~ll(r'),

r << re,

r ' > > rc,

(2.19)

as an expression of photon bunching in a Gaussian field. The relation (2.18) connects two probability distributions for the same time interval under different initial conditions.

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Ueda, Kuwata, Nagasawa, Urakami, Takiguchi and Tsuchiya [1988] measured the statistics Ps(r) - Psll(r), Pc(r) =-- P~ll(r), r ~< 190ps, and p ( n , r), where r = 300 ps. Their analysis took into account the detector dead time rd - 9.7 ps (cf. Cantor and Teich [1975]). Transforming the photoelectron events onto the photocurrent using a familiar formula (Saleh and Teich [1991]), the Schottky-Campbell formula for the variance or/.z of the photocurrent, (2.20)

~r2 = 2 e l o B h ,

where e is the elementary charge, I0 is the mean photocurrent, and Bh is the detection bandwidth, can be improved (Winzer [1997]). 2.2. QUANTUM OPTICAL APPROACH TO PHOTODETECTION

The quantum theory of photoelectric light detection attempts to express the light intensity in terms of the negative and positive frequency parts ~il-)(t) and ii(+)(t), respectively, of an appropriate component ~i(t) of the vector potential, ~i(t) = A(+)(t) + ~i(-)(t), (Pef-ina [1991]): I(t) = ~i(-)(t) ~i~+)(t).

(2.21)

The relation (2.7) must be replaced by ~. = 7/ ftj. tj+~.I(t') dt'.

(2.22)

Not only cannot }(t) be exactly the light intensity, but also ~it-)(t) and ~it+)(t) cannot exactly be the parts of the vector potential component. Rather, one recalls the Mandel photodetection operator (Mandel [ 1964]). When ~ii-)(t') and ~it+)(t'') commute whenever t t ~ l it, Di(+)(t),~i(-)(t')] =

t')i,

6(t-

(2.23)

we can use the results of the nonrelativistic theory of a scalar boson field and we replace the relation (2.9) by p(nl,.

. . , n j , tl, T1,. . . , t j , T j )

_-3/0"~... f0 ~

__1 j = l nj!

; exp(-Wj)P v(WI,..

., W j t l , T I , . ,

.

., t j T j ) ,

dWj,

j=l

(2.24) where P a r ( W 1 , . . . , W j , tl, T 1 , . . . , tj, T j ) is the joint quasidistribution of (r/times) the light intensity integrals. Great attention has been paid to the case J = 1,

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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and this photodetection equation has been derived by Mandel [1963a], Kelley and Kleiner [1964], and Glauber [1965]. The relation (2.24) is seldom used for J >> 1, because one assumes that Tl,..., Tj a r e small in this case. Particularly, t j , 0) = 0,

p ( 1 , . . . , 1, t l , 0 , . . .

OT1

(2.25)

OTjp(l'''''l'q'Tl'''''tJ'Ts) 9

"

= (r/S](q)" " "](tJ))" Ti

"

.

.

.

.

.

Tj

= 0

(2.26)

The relation (2.26) represents the probability density for the joint J-fold photoelectric detection probability. Considering for simplicity a free single-mode electromagnetic field, we may find functions W(A;)(a, tj, Tj.) which correspond to the operators (2.22) and express the joint quasidistribution PAr(W1,

. . . , Wj,

tl,

T1,

. . . , tj,

Tj)

d

= f + (a)ri

Tj-))d2a,

(2.27)

j=l

where q~A;(a) is the quasidistribution of the complex amplitude related to the normal ordering of field operators, and 6(W) is the Dirac delta function. Then eq. (2.24) can be written as p(nl,.

. . ,

n j, tl,

= /rI j=l

In particular,

T1,.

. . ,

tj, Tj)

l[w(H)(a'tJ ' Ts)]n' exp[-W(~C)(a' tJ TJ)]~A/'(a) d2cy"

(2.28)

nj!

1/ [W(~C)(ot,t,T)]" exp[-W(A;)(a,t,T)]~A;(ot)d2a.

p(n,t,T) = ~

(2.29)

On substituting into eq. (2.26) according to eq. (2.28) and using the property W(A;)(a, tj, Tj)I r =0 = 0,

j=l,...,J,

(2.30)

which corresponds to the similar property of operators (2.22) and implies eq. (2.25), we obtain that
= IT/

f

. .I(a, tj) Ar(a)d2a,

(2.31)

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SOME APPROACHES TO MODELING PHOTODETECTION

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where

I(a, tj) =

~

a, 6,

j = 1,... ,J.

(2.32)

~=0 Special attention has been paid (Mandel [1981 ]) to a paper by Srinivas and Davies [1981], who arrived at an analogue of the photocount formula (2.28) in the case of a concrete model, with

W(sZr)(a, tj, T/) = l a l 2 { e x p ( - R t / ) - e x p [ - R ( t / +

~.)]},

j = 1,... ,J;

(2.33)

particularly,

W~ar)(a, t, T) = lal2 {exp(-Rt) - exp[-R(t + T)I},

(2.34)

where R is a parameter with the dimension of frequency that characterizes the coupling between the field and the detector. For definiteness, the above assumption of nonoverlapping intervals can be formulated as

tl < tl + T1 ~ t2 < . . . ~ tj < tj + Tj.

(2.35)

We recall here the observation of (Mollow [1968]) that in this situation the past techniques were able to derive only

WtMH)(a, 6, Tj.) = laIZRT/,

j = 1,... ,J;

(2.36)

particularly, WtMH)(a, t, T) = laI2RT.

(2.37)

Let us emphasize that the discussion proceeded with J - 1 and t - 0. The form (2.28) with (2.33) is applicable to a closed system; e.g., one in which the electromagnetic field and the detector are contained within a resonant cavity (Mandel [ 1981 ]). The field then interacts continuously with the detector and the light intensity decays to zero in time as a result of measurement. According to this statement, the system is closed, although it is treated by open systems theory. Besides the generalization to the multimode case, the formally identical generalization to the case of continuous mode expansion is important, serving for the description of stationary fields. The objections had been associated with the belief that the form (2.28) with (2.36), provided incorrect probabilities for R(T1 + . . . + Tj) > 1. The discussion makes it clear, although not very explicitly

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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(cf. also Mandel and Wolf [ 1995]), that the detection begins at t = 0 and ends at t = R - 1 , and all other conclusions suit this assumption. The quantum theory of open systems is quite general (Davies [1976]) and it can also be applied in the case of continuous mode expansion, as mentioned in the response by Srinivas and Davies [ 1982]. Continuous modes can be assumed, as in the paper by Davies [1977], and have been investigated by Chmara [1987]. The review paper by Srinivas [ 1996] states, besides these facts, that the quantum theory of open systems can be applied in monitoring atoms which led subsequently to generalizations of this framework. Intuitively, the detector counts photons in the output mode of a quantum system, which motivates to codify clearly the relation between input, internal motion, and output in a class of well-defined systems (Gardiner and Collett [ 1985]). The Srinivas-Davies model presents a good approximation of some microscopic models of continuous photodetection, as we will see below. This can be held as a solution of the question of arbitrariness of the Srinivas-Davies model. Also, the placement of the detector inside the cavity can be amended as a location outside the cavity with an appropriate analysis (Huttner and BenAryeh [1988a]). 2.3. G E N E R A L I Z E D SRINIVAS-DAVIES M O D E L

It is quite clear that the decreasing light intensity in the simplest Srinivas-Davies model is attributed to the attenuation of the radiation mode at a constant decay rate. The time-independent light intensity in the simplest Mandel description can be connected, as an apparent paradox, with an attenuation of the mode at an ever increasing decay rate. As the paper by Srinivas and Davies [1981] does not comprise the appropriate, rather trivial generalization, we outline it in what follows. Firstly, the master equation for the statistical operator/5 - / 5 ( t ) describing the damped quantum harmonic oscillator can be generalized to involve a variable damping rate,

Ot

- Latt(t)/~,

(2.38)

with Latt(t) a superoperator such that

Latt(t)/~- IR(t)(2~/~t

_ ~t~/~_/~t~),

(2.39)

where R(t) > 0 expresses the damping rate, and h and h t are the photon annihilation and creation operators, respectively. To be more explicit with respect

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to this superoperator, we introduce the annihilation superoperators h+, h_ with the property

a+/, = aft,

ati, = at/,,

a_h = ffa*,

at_/, = ffa,

(2.40)

so that

Latt(t) =

1R(t)(2a+a_-

(2.41)

ata+ - a*_a_).

The evolution equation (2.38), with an initial condition

,b(t)[,= o =/5(0),

(2.42)

has the solution r

Latt(t') dt'

= exp

h(O).

(2.43)

More generally, /5(t + T) = G[,, ,+r) /5(t),

(2.44)

with the affine (completely positive and trace preserving) superoperator G[,,,+r) =

exp

Latt(t') dt' .

bit

(2.45)

The generalization made has the consequence that the language of semigroups cannot be used (Davies [1976]); instead, one has: G{t2, tz+T2) G[t,,t,+rl)

=

G{tl,t2+l))

if

tl + Tl = t2.

(2.46)

There exists a resolution of the affine superoperator in terms of superoperators N[t, t+r)(M), oo

b:,,,+~,- Z ht,,,+~(M),

(2.47)

M=0

where ~r[t,t+r)(M) = ~

(1-A[t,,+r,)

S[,.,+r, + _,

(2.48)

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

with M the number of counts, the attenuation coefficient

IS

A[t,t+r) = exp -

R(t') dt'

]

,

(2.49)

and

~[t,t+T)

-- ~[[t,t+T)(O)

--

1

R(t')dt'(ht+h+ + hi_h_) .

exp ,]t

(2.50)

The superoperator S[t,t+r) describes the change of the system conditioned on the event that within the interval [t, t + T) no counts have been observed. The superoperators N[t,t+T)(M) can be used to determine the probability

p ( M , t, T) = Tr{~r

h(t)}.

(2.51)

The superoperators (2.47) represent a superoperator-valued measure. As an easy generalization of this kind (cf. Srinivas and Davies [1981]), we present the probability

p(M1, . . . , M j , h, T1, . . . , tg , Tj ) (2.52)

= Tr{~l[tj,tj+vj)(Mj)~7[tj_,+rj_,,t.,)...

FC[t,,t,+r,)(gl)~[o,t,)h(O)}.

Upon formal manipulation, we can arrive at the formulas (2.29) and (2.28), with

W ( N ) ( a , tj, Tj) = [a 2A[0,tj) (1 - A[tj, t,+rj)),

j = 1 ,... , J

(2.53)

The relation (2.32) yields

I ( a , tj.) = [a[2R(tj)A[o, tj),

j = 1,... , J ,

(2.54)

and obviously

W(-~)(a,

tj, ~) = f"+~ I ( a , t') dt',

j = 1,...,J.

(2.55)

Quite surprisingly, letting tmax denote the time instant after which I ( a , t) = 0, we can obtain I(1,t) -

1

tmax

,

t < tma•

(2.56)

by merely setting

R(t)-

1 - - , tmax - t

t < tma•

(2.57)

Let us note that this rederivation of the so-called Mandel's formula proceeds in the ungrateful way of generalizing the Srinivas-Davies formalism.

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The resolution (2.47) of the affine superoperator can be refined by taking into account the times r l , . . . , rM at which one count is registered, and by resolving the superoperators [t,t+T)(M) =

at

,.it

N[t,t+T)(M, r l , . . . , r~t-1, r t t ) d r l . . , drM-1 dTm

...

(2.58) in terms of superoperators, N[t,t+T)(M,

T1,...,

TM-1, TM) =

S[r,~.t+T)Jr,~S[r,~_,.

r,~)'''

JnS[t.

(2.59)

n),

with r l , . . . , rM meaning the times at which one count is registered, and J r = R(r)h+ h_.

(2.60)

In the generalized Srinivas-Davies model, it holds that: 2

N[t,t+T)(M, TI,..., TM-1, TM) = P[t.~+ri(rl,..., t~t-l, rMIM)N[t.t+rt(M), (2.61) where I ]--[,x.t /(l.r,) M. l ~i-- 1 ll,>:,(l, t. r) P[t,t+r)(rl,..., TM-1, rMIM)= fort ~< ri < . . . < r.lt-1 < Lw < t + T, 0 otherwise. (2.62) This probability density can be interpreted as that of ordered M initially independent and identically distributed random variables. Each of them would have the probability density I(1, r)/W('~')(1, t, T). Since W('~'t(1,0, ~ ) = 1, the light intensity I(1, r) is a probability density and the renormalization takes into account the restriction to the interval [t, t + T). Considering both sides of the relations (2.58) and (2.61) right multiplied by /3(0) and taking into account the trace, we see that p(M,t,T)=

..

,It

dt

P[t,t+ll(M,

r l , . . . , r~t-l, r M ) d r l . . , drM_l drM

"

(2.63) and P[t,t+T)(M, r l , . . . , TM-1, TM)= P[t,,+ri(rl,... , r.~l_l, r;~ilM)p(M,t, T),

(2.64) where the probability density is ,I

(2.65) In a slightly different context, the importance of a formula such as the relation (2.62) has been emphasized (Shapiro [1998]).

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2.4. BUNCHING AND ANTIBUNCHING

The bunching of counts is usually considered in terms of the coincidence probability in the dependence on some interval To between the counts (Mandel and Wolf [ 1995]). This theory is connected to stationary fields more than it can be expected, but the definitions are also relevant to more general fields, in which we are interested. Of course, the stationary fields occur regularly (Torgerson and Mandel [1997]). We shall generalize the well-known result (Srinivas and Davies [1981]) for the conditional probability c(to + T0[to) that, given that a count occurs at to, another (not necessarily next) count is registered around to + To, expressed by:

c(to + T0]to) =

Tr{),o+roGfto,,o+r,,,)to~fO, to,[~(O)} .

(2.66)

Tr {Jto ~[o, to,/3(0)) Introducing the quasidistribution q~x(a, t),

1 to) a5X q~Ar(a, to) = A[o,

v/A[a 0, to) )

(2.67)

'

we obtain that

f ]Of4q~Af(a, t0) d2a c(to + To Ito) = R(to + To)A[ro, to+ro)f Iot 2@A/.(a, t0) d2(zEvoking the fourth-order degree of eq. (2.68) in the form

coherence

(2.68)

y~'2)(tl,t2), we can rewrite

c(to + To It0) = R(to + To)A[to, ro+ro)(h)(to) y~'z)(to, to),

(2.69)

where

(h(h- ])... [ h - ( k -

1)]])(t) = f ]al2k~N(a,t)d2a, A/-(2'2)(t,

t) = (h(h - i))(t) [ (~/) (t)]2

"

k ~> 1,

(2.70) (2.71)

This conditional probability is used for the comparison of detection of an arbitrary state with the coherent state of the same mean photon number (h)(to). Obviously, this amounts to the use of the reduced fourth factorial moment

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SOME APPROACHES TO MODELING PHOTODETECTION

133

(to, to)- 1 for the comparison with zero. Since any coherent state has the

Poisson statistics of photons, a more usual means of comparison is the Fano factor,

d(t) = ((Ah)2)(t),

Ah = h - ( h )

<~)(t)

i.

(2.72)

The sub-Poissonian behavior is obtained for d(to) < 1, whereas a kind of antibunching is encountered for ~.(2.2) . (to, t o ) - 1 < 0. These two conditions are equivalent. The established definition of antibunching is based rather on the analysis of the behavior of the coincidence probability in dependence on To (Mandel and Wolf [1995]). In our application, it provides only bunching independent of the Fano factor, which reminds one of the considerations involved in the papers of (Singh [1983], Zhou and Mandel [1990], Dung, Shumovsky and Bogolubov [1992], and Aliskenderov, Dung and Kn611 [1993]. It seems that the analysis of the coincidence probability for To ---+ 0+ as described by Jakeman [1986] does not apply here. Moreover, using assumption (2.57), we obtain a uniform coincidence probability, tmax--t0

.(2.2 {h)(to) ~,~(to to)

for

To <~ tmax- to, (2.73)

c(to + To It0) 0

for

T0~>tmax-t0.

The coincidence probability is rivaled by the conditional probability ~(t0 + T0lt0) that given that a count occurs at to, the next count occurs around to + To, which is expressed as

O(to + Tolto) =

Tr { ff ,o+foS[,,,.,,,+f,,t J,,, +[O. ,,,,P(O) }

(2.74)

Tr {J,,, ~[0. ,,,,/5(0) } The variant of the relation (2.68) reads as O(to + T01t0) = R(to + To)Apo.,,,+r,, 1

x

f la[ 4 exp[-[al2(1 - AE,,,.,,,+T,,t)] q~\r(a, t0) d2a {h)(to)

(2.75)

Since the vacuum state implies infinite waiting time, we have that

fo

~ O.(to + To[to)dTo = 1 - p ' ( O , to),

(2.76)

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CONTINUOUS MEASUREMENTSIN QUANTUMOPTICS

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where

p'(O, to) = (Ol~'(to)lO),

(2.77)

p'(to) =

(2.78)

with

Jt~176

= h/)(to) ht Tr {)to,b(to) } (h)(to)

Using again assumption (2.57), we obtain a To-dependent variant of eq. (2.73):

~(to + To ]to) =

f l a l 4 e x p _l al2r~ ci)H(a, to)d2a ( t.... -t0) for (tmax - to)(h)(to) 0

for

To<~tmax-to,

To/> tmax- to.

(2.79) In this simplified situation, we approach the derivative with respect to To, To ~> 0, at To = 0: ~~(to

[(n)(t~ y~/3I(to to), (tmax - t0)2

+rolto)

OTo

To= 0

(2.80)

with ffN.(3,3)tt (n(n -- i)(h - 2i))(t) ~,~, t) = [(~}(t)] 3 ,

(2.81)

as a particular instance of the sixth-order degree of coherence },~;'3)(tl, t2). We cannot establish a connection with the sub-Poissonian behavior (cf. Jakeman [ 1986]), because

o. 0 > ~ 0 CFock(tO +

[

Tolto) To=0

o

> ~Cc~

+

Tolto)

(2.82) 7o=O

for the Fock state In) and the coherent state [a), with ]a[ 2 = n ~> 1, but it still results in bunching. Whereas the decrease of the waiting time probability density is faster for the coherent state than that for the equivalent Fock state, so that the detection process appropriate to the Fock state is antibunched in the sense of this comparison, the analysis for small To performed just as in the case of coincidence probability here also indicates bunching.

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w 3. Reversibility and Irreversibility of Photocounting The status of irreversibility was doubted at the beginning of the statistical mechanics era. A maw-particle system which is closed or isolated from the external world has a reversible evolution in classical physics. On the other hand, the maw-particle system should have an irreversible evolution of gas or a solid. This apparent paradox has been cleared up long ago. The considerations of quantum statistical physics are analogous. A simple quantum system, either mechanical or optical, which is closed or isolated, has a genuinely reversible evolution. Open quantum mechanical systems frequently exhibit irreversible behavior as a consequence of interaction with their environment. Such systems are described by master equations, which do not comprise external world variables. According to this general view, master equations can be derived by reduction of a Hamiltonian dynamics describing the interaction between the system and the environment. The reduced description would be of a less fundamental nature. The common knowledge that quantum mechanics cannot be formulated as a Markov process has been discussed recently (Gillespie [1996a,b, 1997], Hardy, Home, Squires and Whitaker [1997]). A genuine irreversibility enters the physical conditions via von Neumann's projection postulate. In the Copenhagen interpretation, an external observer 'perceives' only classical eigenvalues of a quantum observable and he or she 'causes' a collapse of the quantum state onto the appropriate eigenspace of the observable. When so markedly formulated, this interpretation can be doubted, and indeed the fundamentals of quantum theory comprise alternatives (Omn~s [1994], Haus and K~irtner [1996], Hay and Peres [1998]). In this work, we adhere to the Copenhagen interpretation. Von Neumann pointed out the two fundamentally different dynamics for describing the change of a quantum state in quantum theory: The umtary continuous evolution of a closed system and the instantaneous but unpredictable projection of its state due to a measurement causing a reduction of information on the history.

3.1. S C H E M E OF S U C C E S S I V E M E A S U R E M E N T S

Basic principles of nonrelativistic quantum mechanics are formulated for state vectors and discrete observables (Ozawa [1997]). Generalizations to statistical operators and continuous observables are possible, but the projection postulate 'generalized' to continuous observables can no longer be consistent with the statistical formula.

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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The time evolution of the system is given by the unitary transformation

,~p(t+ T)) = exp(-~ TH) ,qJ(t)),

(3.1)

where h is the reduced Planck constant and/2/is the Hamiltonian of the system, as long as it is time independent. We let ~i denote the observable under consideration and assume that it has a discrete spectral decomposition, L

= Z a~A (ak),

(3.2)

k=l

where L is either an integer or infinity, EA(ak) are projection operators and ak, k = 1 , . . . , L , are eigenvalues of A. The probability distribution of the outcome A of the measurement of the observable A in the state ]~), (~ ~) = 1, is given by the statistical formula ^

Prob{A = a} =

(%1%)= (N %),

(3.3)

where I~pa) = L'J(a) I~p),

(3.4)

with a an eigenvalue of ~i. The function L'J( ) can be continued as a projection-valued measure; i.e., it can be defined for all parts of the set { a l , . . . ,dE} if L < oc and of {al,a2,...} if L = oc, and for all Borel subsets I of the real line R. Particularly, ~'A(R) i. Such a measure also exists for a continuous observable. One need not speak of measurement of an observable and see it as an operator, but one can apply the concept of a probability operator measure (Helstrom [1976]). The projection postulate is formulated as follows (cf. Ozawa [1997]): The state change caused by the measurement of an observable ~i with the outcome a is given by the relation: I~Pla) =

Ira> v/Prob{A_ = a}

(3.5)

Note that (~plal~Pla) = 1 and the statistical formula can be used again after the system evolves in time. For a continuous observable/} we may define the operator-valued density P#(b), /2"b(b) =

lim L'b([b'b+ Ab)) Ab --, 0+ Ab '

(3.6)

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REVERSIBILITY AND IRREVERSIBILITY OF PHOTOCOUNTING

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and we obtain the probability density pk(b) = <~'l tP(b)),

(3.7)

where I ~(b)) = P~(b)IVJ).

(3.8)

Trying to define the 'state change' by the relation I~(b)) ~ , I tPlb) = v/P k(b)

(3.9)

we obtain (~lbl %h) = oc, and see that we cannot use the statistical formula any more. The dominance of the Markovian property in the quantum measurement theory can be related to the scheme of successive measurements (Wigner [1963], Ozawa [1997]). We let ~ij, j = 1,... ,J, denote observables and assume that they have discrete spectral decompositions, ^

(3.10) k=l

where Lj each is either an integer or infinity, E"4'(ai) are projection operators and a#, k = 1 , . . . , Lj, are eigenvalues of~ii. If the observables ii/in a system initially in the state ]~p(O)) are measured at the times 0 ~< tl < ... < tj, then the joint probability distribution of the respective outcomes Aj is given by Prob{A__l(tl) = a l , . . . ,A__j(tj)= a j } = (~,,, .....,~(tj +0)l~p, , .....aj(tj +0)), (3.11) with Ilpa, .....aj(tj + O))

= E'dJ(aj)exp

--h(tj-

tj_,)H

...

'(a~)exp -ht,/2/

I~p(0)),

(3.12) where tj + 0 indicates that the quantity dependent on it is the limit for t slightly greater than tj. More generally,

[i

[lPa, .....aj(t)) = exp - - h ( t - t j ) H

l

tp,, .....,1(tj +0)),

(3.13)

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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for t E (tj, tj+l], where tj+l is the time of the next measurement or for t > tj if there is no further measurement. We associate with this measurement a time series with the Markovian property ((l~(t)), t ~ [O, tl]),(lW__(tl)),Al),(]~p(t)),t C [tl + 0, t2]), (]lP(t2)),A2), 9 .., (IE(tJ)),~),

I~_(tj + 0))),

(3.14) where the notation t E [t/+ 0, tj+l] means that ]~p(tj-)) is replaced by ]~(tj. + 0)), j = 1 , . . . , J - 1. We speak of time series because we disregard the intervals [0, tl ], (tl, t2],..., (tj_l, tj], where unitary evolution proceeds. On the condition that the event of which the probability is given by eq. (3.11) has occurred,

I__W(t))--IhOla,.....aj(t)),

(3.15)

where I~pla,.....aj(t)) =

]~Pa,.....,,j(t)) . V/(~Pa, .....~(t)l~Po, .....~(t))

(3.16)

When the eigenvalues aj can be summed, as for example when the operators Aj are number operators, we may consider a time series: ((]~(t)), t c [0, tl]), (]~(tl)),A_l), (A_l, ]~(t)), t ~ [tl + 0, t2]), (A__l, ]~(t2)),A_2), J-1

...,

J

A_j, j=l

+ 0)>)) j=l

(3.17) This time series also has the Markovian property. 3.2. UNRAVELING THE MASTER EQUATION

In what follows, we will generalize the Srinivas-Davies model in another direction than in w2.3. Again assuming that R is time independent, we study a composite system of a single mode and a detector described by a master equation:

Ot

-L/~,

(3.18)

with an initial condition /5(t)lt= o =/5(0),

(3.19)

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REVERSIBILITY AND IRREVERSIBILITYOF PHOTOCOUNTING

139

r

where L is a superoperator having the property

Li, =

i

R(2Of~Ot_O, Of~_DO, O),

bl + 7

(3.20)

w i t h / 2 / a Hamiltonian which does not include the interaction with the detector and with 0 the operator which affects the state of the mode whenever a count is detected. The generalization to more modes and even to a composite system comprised of an electromagnetic field and an atomic system will be obvious. In the following, we assume [ / = 6 because essentially everything is clear from the case of a free field, the free-field Hamiltonian not being explicit due to the use of the interaction representation. Since a continuous measurement is under consideration, there is a back-action of the detector also between the successive counts when no other counts are detected (Srinivas and Davies [1981]). The operation of the detector can be described by the related random process M(t), oo

M(t) = Z O(t- r__j),

(3.21)

j=l

where 0 is the Heaviside unit-step function. Formally, we may treat the detector as another degree of freedom with a Hilbert space Hdet enlarging the Hilbert space H, 7-{ ~ ~-l~m = ']"{@ '}-{det, where the subscript m stands for 'modified'. Macroscopically, the detector does not admit the superposition principle (the Schr6dinger-cat paradox excluded), and we work only with the mixtures of the number states described by ]M)det det (M]. Nevertheless, it is useful to define the shift operators, 0(2)

elx'p(iq)det)

= i @ Z [M)det det(M + 1 , M=O

eIx"p(--iq)det) = [e~p(i~det)] t ,

(3.22)

but to rely only on their pairwise use, as in the characteristic relation e~p(--i~det)lM)det det(Mle'xp(iqgdet) = M + 1)det det(M + 11.

(3.23)

Let us mention the single-mode scalar product such that: det (Mlm,Mt)rn =

~MM'In).

(3.24)

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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Now we pass from/5(0 to the statistical operator/Sm(t) and introduce the master equation 0,, 1 1 ^ (t)~tm~m] -~pm(t) = R [~)mPm(t)()tm- -~()tm ()m[gm(t)_ SPm

,

(3.25)

where (3.26)

Om = Oetx-"p(-i~det),

and the initial condition is /~m(t) t=0 = r

10)det det(0 I.

|

(3.27)

The perturbative solution to the master equation (3.25) with the initial condition (3.27) for t = T is of the form O(3 / S m ( T ) -- Z ~/[0, T ) ( M ) / ) ( 0 ) ( ~ M=0

(3.28)

IM)det d e t ( M 9

^

Here the superoperator hi0, r)(M) is defined by the property fi[0, v)(M)/5(0)=

JO JO

•

hi0, r)(M, r l , . . . , rg-1, rg)

...

(3.29)

... drg_~ drg,

where u[o, v)(M, r l , . . . , rM),b(O)= h[o, r)(M, r l , . . . ,

rM)~(O)h~o,r)(M, r l , . . . ,

rM), (3.30)

with b/[0, T ) ( M , T I , . . . ,

TM)= R M/2 exp [ - 8 9

• Oexp[- 89

rM)Ot0]

rg_,)OtO] ...Oexp(- 89 (3.31)

To get closer to the relation (2.59), we may write ^

^ r)(M, fi[o,

T1,. 9 9 , TM) =

~ ~ ...),~ S[r~,,r)JS[r,,_,,r,,)

[o, r,),

(3.32)

where the superoperator Sir,,_., r,,), having the property S[t,t+r) = S[o, r),

(3.33)

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REVERSIBILITY AND IRREVERSIBILITY OF PHOTOCOUNTING

141

is such that

S[o,r)p=exp(-1RTOtD)pexp(- 89

(3.34)

"k

and the superoperator J is such that

)~ = RO~O t.

(3.35)

^

We introduce h[t,t+r)(M, q , . . . , r~t) generally. Substituting {,bm(T)}

/ 3 ( T ) = Trdet

-

u[0.rt(M, rl - t , . . . , rM - t) more

(3.36)

into the left-hand sides of eqs. (3.18) and (3.19), using the achieved property (3.25) and the cyclic property of Yrdet, we derive that eq. (3.36) is the solution of eqs. (3.18) and (3.19) for t - T. Substituting eq. (3.28) into eq. (3.36), we arrive at the form /5(T) = ~[o, r)/5(0),

(3.37)

where (cf. eq. 2.47) (x)

u[0, r)(M),

~[0, r) = Z

(3.38)

M=0

and either a coarser expansion,

/,(r)

p(M, O,

=

(3.39)

M=0

or a finer one,

~(T) = Z

..

P[o,r)(M,r , , . . . , r u)/SL.,1.r~.....~,,(T)dri ... drM_, drM,

M=0

(3.40) where p(M, 0, T) is the probability of M counts being registered during the interval [0, T),

p(M, 0, T) = Tr ( ,ha4(T) },

(3.41)

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CONTINUOUS MEASUREMENTSIN QUANTUMOPTICS

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with ^

(3.42)

/SM(T) = u[o, r)(M)/5(0),

and P[o, r)(M, rl,..., rM) is the probability density of M counts being registered at the times rl,..., rM in the interval [0, T), (3.43)

P[o, r ) ( M , T1,... , TM) = Tr {/3M, r, ..... r,t ( T ) } ,

with ^ RM,

1"1 . . . . . TM

(3.44)

(T) = u[o, r)(M, r , , . . . , rM)/5(0).

The normalized statistical operator filM(T) describes the resulting single-mode state on the coarser condition, 1

'OIM(T) =

p(M, O,T) DM(T)'

(3.45)

and ,blM,r,.....r,(T) describes the resulting state on the finer condition, 1

PlM,rl

...., r M ( T )

"-"

(3.46)

P ,. P[o, r)(M, rl,..., rM) -M'r'

Taking into account the decomposition (3.28), we obtain coupled master ^ equations for unnormalized statistical operators ~M(T)- u[0, r)(M)~(0), 0 ^ ~-~pM(T) =

R

[

(9/3M-l(T)O* ~l b t O p M ( y ) _ -

1^ ~pM(T) ~)t O]

(3.47)

Tracing over the mode under study in eq. (3.47), using eq. (3.41) and the fact that Tr{O t

O~M(T)} = Tr{~IM(T)Ot O} p(M, 0, T) = Tr{~M(T)O t 0},

Tr{0/SM_I(T) 0t } = Tr{/)IM_I (T) 0* O}

p(M-1,0,

(3.48) (3.49)

T),

we arrive at the rate equations:

oOp(M, O,T) = -R [Tr {/51M(T) 0 t ~)} p(M, O,T)

0}

0,

(3.50)

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By speaking of unraveling the master equation (3.25), or more commonly, of unraveling the master equation (3.18) with (3.20), we understand introducing a random process ~m(t) such that pm(t) = E(~m(t)),

(3.51)

where E means the expectation value (ensemble average) of the quantities whose stochastic character must be taken into account. We choose a random process with an almost obvious Markovian property, and such that /~m(t) = ~(t) Q M(t))det det(M(t)l.

(3.52)

NOW, the process has simply two components, /5(t) and M_M_(t). The initial condition of the process (3.52) can be chosen deterministic, /gm(t)] t-- 0 = ,bm (0).

(3.53)

This random process is defined explicitly with respect to the component M(t), but it also implies a definition of/~(t). Let us use the transition probabilities within the time interval [t, t + At) together with their typical asymptotic behavior for small At: Prob ( M ( t + At) = MI/5(0 ) = ,b(0), M(t) = M,_r 1 = r , , . . . ,_Lv = r~t) Tr{C3M,~, ..... r,,(t + At)} Tr{/SM,r, ..... rM(t)}

= Tr{h[t,t+at)(O)~lM,r , ..... r,,(t)}

(3.54)

= 1 -RTr{/51M,r , .....rM(t) O* 0 } At + ol(At) Prob(M___(t + A t ) = M[~_(t)=/5!.,,.r, .....r,, (t), M ( ' ) = M ) , ProblM(t + At) = M + 1 [/5_(0) = ,b(0), M(t) = M, _rI = r , , . . . , _r;,.t = r @ m

f,+at Tr{/SM+l,r, l

"'" ' TM +

1

(t + At)} dr,,+,

Tr {,bM,r, .....r,, (t)} = Tr

{it+At ,,

h[t,t+at)(1, rM+l)dr~t+l/51.~t.~ 1..... ~,,(t)

}

,It

= RTr{/)IM,r , .....r,,(t) O* O}At + o2(At) Prob ( M ( , + A t ) = M + ll/5_(t) = r

.....r,, (t), M ( ' ) = M ) ,

(3.55)

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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

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Prob = o3(At), (3.56) where oj(At), j = 1,2, 3, have the property that o j ( A t ) / A t ~ 0 for At ~ 0+. From this it follows that we define a random process with the Markovian property with respect to both of the components. To complete the definition of the Markovian process, we must define /5(t + At) on the condition that ,b(t) =/51M,~, .....r~ (t) and M ( t + At) - M(t) = 0" ^

/5(t + At) = -

h[t't+At)(O)plM'r' .....r,,(t)

,

(3.57)

Tr{u[t, t+At)(O) [~lM,r, .....r,, (t)}

and we must define /5(t + At) on the condition that /5(t) = /51M,r, .....r~,(t), M ( t + A t ) - M(t) - 1, and that _r = r, where _r is a continuous-time random variable such that M(__r + O) - M ( _ r - O) - 1" ^

/5(t + At) =

--

h[t, t+At,(1, r)/SiM,r , .....r,, (t)

(3.58)

Tr{u[t,t+At)(1, r),biM.r , .....r,,(t)}'

the fight-hand side being independent of r. We need not define /5(t + At) on the condition that M ( t + A t ) - M(t) ~ 0, 1, because this event has a negligible probability. On the condition that M ( t + T) = M and still M(rj + 0 ) M ( r j - 0 ) = 1, j = 1,... ,M, we have the deterministic value /5(t + T) =/51M,r , .....~, (t + T).

(3.59)

For the operator 0 chosen appropriately, e.g., ( ) = h (Ueda, Imoto and Ogawa [1990a]) and 0 - h t (Ueda and Kitagawa [1992]), it holds that (3.60)

/'lM.~. .....~.(T) =/,IM(T);

i.e., the result does not depend on the times at which the counts have been registered. In this case, the 'rate equation' (3.50) is a genuine one for the sodefined Markov process. The superoperator (3.32) simplifies to ^

^

h[o, r)(M, r l , . . . , rM) = P[0, r ) ( r l , . . . , rM [M) hE0' r)(M),

(3.61)

with an appropriate conditional probability density P[0, r ) ( r l , . . . , rM M); cf. eq. (2.61). The nonunitary operator (3.31) has a similar form" h[o, r)(M, r l , . . . , rM) = v/P[o, T ) ( r l , . . . , rM IM) h~o,v)(M),

(3.62)

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where the nonunitary operator h[o. T)(M) is defined implicitly by this relation, and u[0, T)(M) /5 = fit0, T)(M) ~h~o' rl (M).

(3.63)

In the situation where the initial statistical operator/5(0) corresponds to a pure state I~p(0)), the random process/5(t) remains pure for all times. This means that there exists a random process I~p(t)) such that

iS(t) = I,p(t))(,p(t)l.

(3.64)

Indeed, because ,blM,r, .....r~,(t)simplifies as

I~,,,,

.....r,,(t))(~Pl.,,.,, .....r,,(t), the relation (3.57)

h[,,,+A,~(0)

Iq,(t+At))=

tpl.~,t.r , ..... r,,(t))

,

(3.65)

V/(IPlM,r, .....,,,(t)lh* [,.,+~,t(0) "I,.,+~,t(0) ~PI.~.,, ..... ,,,(t)) the relation (3.58) simplifies as

~[,.,+A,,(1, r)I~'t~l.,, ..... ,,,(t))

I q,(t + At)) = V/

{VJiM,,, ..... ,,,(t) I~ [,.,+Ati(1, t r) u[,.,A,)(1, T) q~l.~1.,,.....,,,(t)) (3.66)

and

I,p(t + At)) ~

Ol~,l~,,,,, _

V/(IPlM.r,

,,,(t)> .....

..... r,,

(3.67)

(t)10* 01,j,~,,.,, .....,,, (t))

In the foregoing exposition, we have touched on the essential concepts from work by Ueda, Imoto and Ogawa [1990a], Carmichael [1993a] and Knight and Garraway [ 1994], deviating slightly from their notation. It has been confirmed for - ~ that the total entropy of the cavity photons and the photodetector does not decrease, and the entropy change and information gain in the photon-counting measurements have been investigated in detail (Ban [1997a]). The master equation (3.18) with (3.20) is frequently referred to as the master equation in the Lindblad form (Lindblad [1976]), which is however much more general, since it contains several Lindblad operators O/. Holland [1998] has considered unraveling statistical operator evolution in the frequency domain and continuously sweeping between use of the time and frequency domains. He has presented a master equation in the Lindblad form, but he let the number of the

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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Lindblad operators depend on the size of a cascaded array of filters. Jack, Collett and Walls [ 1999a] have presented a general form for non-Markovian quantum trajectories, focusing on the trajectories that simulate real-time spectral detection of the light from a localized system. 3.3. L O G I C A L R E V E R S I B I L I T Y

IN Q U A N T U M

MEASUREMENT

We will call a quantum measurement process logically reversible if there exists an operator b(0,rj(M, rM,..., r~) such that b(0,rl(M, rM,..., rl)u[0, r)(M, r l , . . . , rM)= i.

(3.68)

This new operator is the left inverse of the operator ~[0, r)(M, r l , . . . , I'M), which is nonunitary. The concept of logical reversibility was introduced by Ueda and Kitagawa [1992] and analyzed by Ueda, Imoto and Nagaoka [1996], who have also given a general condition for physical reversibility. If the left inverse exists, we can arrive at the initial value of the quantum trajectory from its actual value, ]~p(0)) =

-, ~ T r { (E(t) I_bt__bE(t)) }

(3.69)

where b_V_- b(o,rl(M(t), _rM~t),..., __r1).

(3.70)

The choice O - h corresponds to the usual photon counter (Srinivas and Davies [1981 ]), and O - h t to the quantum counter (Mandel [1966]). The case of the photon counter has been treated in w2, and formulas (2.29) and (2.34) read, for t = 0, [ W-(Ar)t~, ,0, T)]Mexp[-Wr176

p(pc)(M, O, T) = ~

0, T)] q)Ar(a) d2a, (3.71)

where ~(Ac)(Ot pc ~. , 0, T) =

lalZ[1 -

exp(-RT)].

(3.72)

For the photon counter, the conditional probability density (2.62) and the implicit one in eq. (3.61) reads: MW l 9(pc) [ ~ [ 0 , T) ~ . r l ' " 9 9 , "/'M IM )

exp -R

1 - exp(-RT)

" =

rj j= 1

for

0

0~<

T1 <''"

otherwise.

<

rM


(3.73)

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147

Unfortunately, the left inverse does not exist, and we encounter a case of logical irreversibility. For the quantum counter, the conditional probability density implicit in eq. (3.6 1) is of the form

p(qc) [0, T){,TI'" " " '

" exn~PTarM [M)

exp R

~. (3.74)

j-- 1

=

for t ~< rt < . . . < r~i < T, 0

otherwise.

In this case, we can modify eq. (3.62) for the left inverse:

b(o,vl(M, rM,..., r l ) =

v/P[o, r ) ( r l , . . . , rx~t[M)

blo. rl(M),

(3.75)

where the nonunitary operator b(o,rl(M) is again defined implicitly. Since

~(qc) [exp(RT)- 1] M/2 1RThht) (hi) ~t [~ r)(M) = x/-M! exp (-~

(3.76)

we obtain that (0,T](M)

[exp(RT)- 1]g/2

h + ~h

exP( 89

).

(3.77)

The operator b(0,Tl(M, r g , . . . , rl) can be obtained from eq. (3.75). A variant of the relation (3.69) reads 8(qc)

[~(0) =

,,(qc)~

v(~ ~/Tr {r

f~IM(T) v(~ (M)

(3.78)

(r)~(qc)t ,,(qc) } u(0,r] (M) V(o,r](M)

The counterpart of the photodetection formulas (3.71), with (3.72), is:

1/ [W(A)ta,

p(qc)(M,O,T) = ~ .

'.

(]SA(a) d2a,

(3.79) where

W('a)r qc ',

,

0, T ) = [a[2[exp(RT) - 1]

(3.80)

Whereas in the case of the formulas (3.71), with (3.72), we may recover (r/= 1), the photon-number distribution in the limit RT ~ ~ , the connection of

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the photocount-number distribution (3.79), with (3.80), to a photon-number distribution will always be indirect. 3.4. REVERSIBILITYOF QUANTUMJUMPS The open systems approach can be applied to monitoring atoms. In that case, the statistical operator describes mainly the atomic system, and the operator O corresponds to the back-action of the detector on the atom; i.e., to a quantum jump. Mabuchi and Zoller [1996] described a scheme for inversion of quantum jumps in systems which decay by emission of quanta into a continuously monitored reservoir. They noted that jump operators are not invertible on the entire system Hilbert space. They required that a jump operator be extended unitarily to the whole Hilbert space, which can be generated by an appropriate feedback Hamiltonian. For instance, in the field of cavity quantum electrodynamics, an operator,4 has the property,4(c0 ]2, 0) + c, 10, 2)) = c011,0) + c, 10, 1). They proposed how to achieve the photon doubling and stabilization of the subspace spanned by the Fock states 12, 0) and I0, 2). In the context of quantum computation, their scheme provides a means for dissipation-free storage of quantum bits. Stabilization of quantum states against decoherence and decay has been connected to existing error correction schemes with respect to the theory of quantum noise and the open systems approach (Pellizzari, Beth, Grassl and Mfiller-Quade [1996]). Ekert and Macchiavello [1997] discussed conditions under which it is possible to restore an unknown state of a composite quantum system after performing measurements on some, but not on all, of its components. They assumed that the protection against disturbances and the restoration are based on quantum-error-correcting schemes. Mensky [1996a] has noted on the report of Mabuchi and Zoller [ 1996] that an invertible quantum jump is not a measurement or, equivalently, is a degenerate form of a measurement giving no information. This restriction must be taken into account as one of the difficulties in creating quantum computers. Nielsen and Caves [1997] generalized the scheme of Mabuchi and Zoller [1996], and provided an application to the teleportation scheme described by Bennett, Brassard, Cr6peau, Jozsa, Peres and Wootters [1993]. Probabilistically reversible measurements can be considered and may serve as probabilistic quantum error corrections (Koashi and Ueda [ 1999]).

w 4. Models of Continuous Measurement

Much work on the continuous measurement employs the quantum trajectory idea. It originated in the quantum jump work of Cohen-Tannoudji and Dalibard

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149

[1986] and Zoller, Marte and Walls [1987]. It was established by Carmichael, Singh, Vyas and Rice [1989], who also made the explicit connection between the mathematics of Srinivas and Davies [1981] and Mandel's physics (Mandel [1981]). Time evolutions according to Ueda, Imoto and Ogawa [1990a] and Ueda, Imoto, Nagaoka and Ogawa [1992] are close to quantum trajectories in a sense. The stochastic wave-function evolution was studied by Dalibard, Castin and Molmer [ 1992], Molmer, Castin and Dalibard [1993] and Molmer and Castin [ 1996]. Numerical simulations are combined with more formal parts by Gardiner, Parkins and Zoller [1992]. The use of methods of quantum stochastic calculus is typical of Barchielli (Barchielli [1990], Barchielli and Belavkin [1991], Barchielli and Paganoni [1996]). Barchielli considered not only direct detection of photon number, but also homodyne detection of a quadrature and heterodyne detection of two quadratures, and the Gaussian (or diffusive) continuous measurement (Barchielli, Lanz and Prosperi [1983], Barchielli and Lupieri [1985a,b], Barchielli [1986]). Stochastic equations which are needed for the description can be found in publications by Carmichael [1993a] and Wiseman and Milburn [1993a]. These equations can also be arrived at on the basis of formal measurement theory arguments (Gisin [1984], Gisin and Cibils [1992], Gisin and Percival [1992]). Setting aside the complexity of computations, the interest in simulating measurement records naturally leads to connections with quantum measurement theory. Several methods for estimating the number of trajectories required to recover the statistical operator and ensemble operator averages have been developed for the case of the quantum kicked rotor (Nielsen [1996]). The researchers, namely, Gardiner [1991], Gardiner, Parkins and Zoller [1992], Molmer, Castin and Dalibard [ 1993], Carmichael [1993a], Wiseman and Milburn [1993a], and Gisin and Percival [1992], have shown how trajectories arise naturally for a wide class of physical systems. Furthermore, the trajectories obtained by these researchers are often simulated easily on a computer. A detailed study of the Monte Carlo simulation schemes has been published by Steinbach, Garraway and Knight [1994]. 4.1. STOCHASTICREPRESENTATIONOF DETECTIONTHEORY The exposition of the detection process in w2 and w3 was somewhat strange, because the system consisting of the system of interest and the detector was described less with respect to the measured system than with respect to the detector. The system of interest obeys a master equation (in general, an operation-valued measure), and the detection process is described statistically

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by means of a probability-operator measure, which is not projection-valued. The simplest master equation for nonreferring process, i.e., one in which the readout (detection outcome) can be neglected, is familiar in open systems theory, but it is often derived by the elimination of the heat bath or an atomic system (Louisell [ 1973]). On the other hand, any probability-operator measure can be made to be projection-valued using an appropriate extended Hilbert space, as stated in the Naimark theorem (Helstrom [ 1976]). Fortunately, the concept of quantum noise (Gardiner [1991 ]) can serve both purposes: to derive the master equation (and the operation-valued measure; equivalently, the superoperator-valued measure) and to derive the probability-operator measure under consideration. Let us start with the measured system connected to a detector modeled as a quantum noise. We will work with the Hilbert space 7-/e = 7-/| 7Y[o,~), where is the single-mode separable Hilbert space and 7-/[0,~) is the 'continuous tensor product'; i.e., the symmetrical Fock space of the quantum noise. We choose the interaction Hamiltonian /2/in t --

h[G*(t,t)h(t)+ H . c . ] ,

(4.1)

where H.c. means the Hermitian conjugate term to the previous one and t~(s, t) for s ~> 0 is the quantum noise obeying the commutation relations

[(~(s, t), ~t (s', t)] = R 6 ( s

- s')

ie,

[G(s, it), G(s', it)] = 0e.

(4.2)

We assume that the quantum process, (~(s,t), s /> 0, is in the ground state initially; i.e.,



= o,

<8t(~, o)OC, o)) = o, (8(s, o)8*(s', o)) = R 6 ( s (&(~, 0)8(~', 0)) = <8*(s,

(4.3) s'), O)8*(s', 0)) = O.

The Heisenberg equation for the annihilation operator of the field reads

d ~(t)

dt

i

= - ~ [a(t),/2/1 - i(~(t, t),

(4.4)

while those for the quantum noise read d

^

dtG(s, t) = - i R b ( s - Oh(t),

s >~ O.

(4.5)

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151

We make use of the solution

CJ(s, t) =

{ (~(s, 0) G(s, O) - l iRh(s) G(s, O) - iRh(s)

for for for

t < s, t - s, t > s.

(4.6)

Substituting from eq. (4.6) into eq. (4.4), we obtain the usual description of the damped harmonic oscillator using the quantum Langevin equation:

d h(t) = - ~i [h(t),/2/]- 89 dt

iG(t, 0),

(4.7)

into which the Langevin force, L(t) = -i(~(t, 0),

(4.8)

can be introduced. This example would mean a restriction to the usual photon counter (the Srinivas-Davies model), but it is simple enough to begin with. Nevertheless, we will understand the master equation (3.18), with (3.20), which is known to be related to the Hamiltonians /7/ and E/int --

(4.9)

h[t~t(t, t) 0 + H.c.].

The Heisenberg equation for the annihilation operator of the measured mode is deceptively simple: d^ i -~a(t) = - - h [ h ( t ) , H ] - i { G ( t , t ) [ h ,

()t]}s-i{Gt(t,t)[h,

~)]}s,

(4.10)

but it is still more complicated than the relation (4.4). Here the fluctuating quantity [h, b t] multiplies the quantum noise G(t,t), and the subscript S is used to indicate that this peculiar product is defined according to Stratonovich (Stratonovich [1963, 1964]). The same is true of the rest of the expression. Actually, a 'classical' stochastic Stratonovich integral has been defined (Stratonovich [ 1963, 1964]), but we introduce

T

{ G ( t ' , t ' ) C ( t ' ) } s dt ' = lim

J- 1 Z T/_I

A --. 0j__~

G(t', t') dt' C(rj) + C(~+l) 2 (4.11)

where 0 = r0 < T1... < Tj

= T,

A = max ,(rj+l- rj),

(4.12)

C'(t) = [h(t), ot(t)].

(4.13)

O <~j
The limit in eq. (4.11) is understood as a limit in the mean square as far as it is applied in the original classical form. It can also be endowed with meaning

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in the quantum stochastic calculus (Parthasarathy [1992]). Such multiplication is known to occur in quantum optics already in the simple case of the damped quantum harmonic oscillator (cf. Lax [1966]), when the evolution of the number operator in the Heisenberg picture is formulated, although it is not usual to even speak of the Stratonovich stochastic integral. Substituting G(t,t) (cf. eq. 4.6) according to the formula G(t, t) = G(t, 0) - I iRO(t)

(4.14)

into eq. (4.10), we obtain the usual quantum stochastic differential equation: d^

i

1

-~a(t) = - ] [fi(t),/2/]- 2R{O[~, o r ] _ Ot[fi, ~]}

(4.15)

+ {L(t)[h, 0t]}s - {Lt(t)[h, O]}s. Using the theorem (see Appendix A) stating that the Stratonovich product can be decomposed into the It6 product and an increment reflecting the dynamics of the quantum process C, we have:

{G(t, t)[~, ot]}s : {G(t, t)[~, 0t]}l- 2'iR[[~,0 t ], O1, {Gt(t, t)[a, O]}s : {Gt(t, t)[a, 0 1 } , - 2'iR[[z,* , O] , O t 1.

(4.16)

In analogy to the classical stochastic integral (It6 [ 1944]),

ZT

J- 1 Zr~+ l G(t',t')dt' (7(r~), {G(t" t') C(t')}I dt' = a-,olim /~o--

(4.17)

where eqs. (4.12) and (4.13) hold. Substituting eq. (4.16) into eq. (4.10), we arrive at

d h(t)= - ~i [h(t), [ / ] - ~R{[[h, 1 Ot ], Ol + [[h, O], Ot 1}

dt

(4.18)

- i{G(t, t)[h, 0 t ] } i - i{Gt(t, t)[h, 0]}I. Eliminating the dynamics of the quantum noise in the same manner as from the original Heisenberg equation (4.10), we obtain the It6 quantum stochastic differential equation (Lukg and Pefinovfi [1987]),

d h(t)= - ~i [h(t), [/] - 89 dt

OtlO - or[h, Ol)

(4.19)

+ {L(t)[h, 0 t ] } ~ - {Lt(t)[h, O]}~. Although the above considerations may be relevant to the nonreferring (nonselective) measurement, the nonreferring and referring (selective) measurements,

I I I , w 4]

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153

in general, require the Schr6dinger picture. Using the Stratonovich rule, we write the Schr6dinger equation as d i^ dt I~P~(t)) - -~HIW~(t)> + {[L(t)O t - Lt(t)0]l q~e(t)) }s

(4.20)

or

d

i^

dtlWe(t)) = - ~HIW~(t))- 89

(4.21)

+ {[L(t)b t - Lt(t)O]l V:~(t))},. Description (4.21) is referred to as one using an 'effective' Hamiltonian, which is not, however, Hermitian. The statistical operator obeys the quantum stochastic differential equations d i dt/)e(t) = _ ~ [/2/,/~e(t)] + {[L(t)O t - Lt(t)O]/)e(t)}s -

{~e(t)[L(t)O

t -

(4.22)

Lt(t)0]}s,

and i

~f'e(t) = -- ~ [/?/,/Se(t)] + 2R[2O/5~(t)O t - Ot O/5~(t)-/Se(t)OtO] + {[L(t)Ot

- Lt(t)O]k~(t)}~

- { k~(t)[L(t)O

t -

Lt(t)O]},.

(4.23) Tracing over the quantum-noise Hilbert space, we arrive at the usual master equation (3.18), with (3.20), for ,b(t) = Tr[0, ~){,be(t)}.

(4.24)

The derivation of the master equation (4.23) from the Schr6dinger equation (4.21) must use the It6 table (Hudson and Parthasarathy [1984]). Since the quantum noise always comes in the ground state, the relation (4.21) can be simplified as d IlPe(t)) = - ~iH^l W e ( t ) ) - 89 dt

V:~(t))- {Lt(t)0 V:~(t))}, .

(4.25)

The evolution can be approximated as IWe(t + At)) ~

Iq'e(t))-

i

gAt[/Iv:~(t)) j,f

~RAtOtOh~p~(t))

-

l ,j!

(4.26)

t+'~t

Lt(r) dr 0[ V:~(t)) "

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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

[III, w4

We may imagine that at the time t + At the operator A tA is measured, where A =

i

t+At

L(r') dr',

(4.27)

,Jt

having the property [A,A t] =

RAtie.

(4.28)

Let us observe that

A*A = RAti | i[0,,)| ~

nln)tt,,+At)t,,t+At)(nl | itt+At,~),

(4.29)

n=l

where [n)[t,t+At) are number states in the subspace 94(0) '"[t,t+At) of ~-[[t,t+At) related to the wave function 1 / x / ~ . Let us consider an approximation to the eigenvalue problem for the operator A tA. One eigenvalue vanishes and the other is RAt, and we denote the corresponding eigenvectors by ]O)[t,t+At) and ]l)[t,t+At), respectively. The annihilation and creation operators, CX3

A = v/RAt i @ i[0,,)| Z v/n + l ln)t,,,+A,)t,,,+A,}(n + 1[ | ][t+A,,oc), (4.30) n=0

and OG

At =

x/RAt i | i[0,t)| Z v/n + 1 n + 1)[,,,+At)[t,t+at)
(4.31) act on the vector

Iw~(t)) = IW(t)) Q Ipast)[o,,)Q Io)t,,,+A,, o Io)t,+~,,~),

(4.32)

where [past) [0, t) comprises all the history of quantum noise we will introduce inductively. We consider the projection operator [-l[t,t+At)(O) onto the ground state [O)[t,t+At) and the projection operator [-l[t,t+At)(1) onto the excited state. Multiplying both sides of the relation (4.26) by the two projection operators, we obtain the nonunitary evolutions i

[-ltt, t+At)(O)lWe(t+ At)) ~ IWe(t)) - ~At/2/[ ~Pe(t)) -- 2gAtOtOl~Pe(t)), (4.33) and

fl[t,t+at)(1)llPe(t + At)) ,~ -iv/RAt bll)[t,t+At)[t,t+at)(O[lPe(t)),

(4.34)

occurring with probabilities given according to the relation (3.11). We may assume for simplicity that t is an integral multiple of At. Provided that the

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MODELS OF C O N T I N U O U S MEASUREMENT

155

measurement is performed for such t starting from the time origin, we may evoke the scheme of successive measurements and apply the relation (3.11) to the vectors ..... , J - , , n ( t + A t + O ) ) =

where n 1 , . . . ,

n j - l , with J -

f/[0,At)(nl),...

,Fl[t-At,t)(nj-1).

fI[,,,+a,)(n)l~P,,, ..... ,,,,(t+At)),

n=0,1,

(4.35)

1 = t/At, correspond to the projection operators It is advantageous, with respect to the limiting procedure, to divide these probabilities by (At) tt, where M is a total number of registered photocounts. The limiting procedure then yields unnormalized probability densities for the times of registration. Taking into account the random process M ( t ) , we can derive the stochastic nonlinear Schr6dinger equation: d

dt I__W(t)) = - h/t ___W(t))- 89

- (q~(t) Ot01~(t))][ W(t))

{ [

J }

(4.36)

d 0 - ] ~p(t)) d~[M(t)] V/(V~(t)Ot 0 ~#(t)) i

The form of this equation is analogous to eq. (4.21). Such equations have been derived many times in the literature (cf. Carmichael [1993a], Wiseman and Milburn [ 1993a], Barchielli and Belavkin [ 1991 ]). Acting on both sides of the relation (4.26) by an appropriate pair of projection operators, we can obtain the nonunitary evolutions (4.33), (4.34), and the respective probabilities with the replacements b ~ 0~ = b + ~ i e, _.

lihR (~, 0

(4.37)

~t)

(4.38)

and with new projection operators. We can use the projection operators

f-l[t,t+At)(q-)

(4.39)

=

/e/[t,t+A,)(--) = I--)[t,t+At)[t,t+At)(--

(4.40)

,

where

(1 - 89

+ R~~

~. 1)t,.,+A,>,

(4.4])

156

CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS ]-) [t, t+At) ~ Rx/-R-~_~* O)[t,t+At) + (1 - 89

[III, w4

(4.42)

2) 1)[t,t+At),

and we obtain that the relations (4.33) and (4.34) reappear:

(4.43)

/~/tt,,+A,)(+)lq'e(t + At)) ~ Iq'~(t))- ~At/c/zl~P~(t))-' ~RatO~O~l~Pe(t)), ^*

Htt, t+At)(-) lPe(t + At)>

~ - v / R A t ()~

(4.44)

-)t,,,+A,~I,.,+A,,(+I We(t)).

This possibility can also be derived by the standard unraveling and the observar

tion that the superoperator L in eq. (3.20) is invariant under the transformation consisting of the replacements (4.37) and (4.38) (Wiseman [1994a]). If again 0 = h, we may solve for the unitary evolution, which is not combined with the measurement. It is the description of the damped harmonic oscillator. We may obtain closed solutions for some initial states; namely, a coherent state, a number state, a squeezed coherent state, and a thermal state. For example, for the coherent state ]~(0)) at the beginning, we obtain the solution to eq. (4.20) or (4.21), (4.45)

I*Pe(t)) = I~:(t)> o Ih01(s, t))[o, ~,,

where ~(t) = ~(0)exp(- 89 is the complex amplitude of the coherent state [.~(t)), ~Pl(S, t) is a one-photon unnormalized wave function,

1/01 (S, t) = {

-v/R~(s) for s < t, 0 for s ~> t,

(4.46)

and ]~Pl(s, t))[0,~) is the coherent state of the quantum noise,

I~p~(s,t))[o,~ ) = exp -

x

{

0)[0, oc)+

I~p~(s,t)l 2 ds

jo, jo ""

)

lpl(sl,t)"" lPl(S,,,t)ln; s l , . . . , s , , ) [ 0 , ~ ) d s l - " d s n

n=l

)

(4.47) Here

In;s~,..., s,)[o, ~> [o,~(n;s~,

...,

are number states with the property I

s, ln"s,,

I

I

, 9 9 9 , Sn,)[O, ~c ) = (~n,,'(~(S1--S1)""

"~(Sn--Stn)

9

(4.48)

9

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MODELSOFCONTINUOUSMEASUREMENT

157

In general, for any initial state we obtain

IWe(t)) = ~~0=

""

]~Pn,s, ...... ~ . , ( t ) ) l n ; s l , . . . ,

s,,)[o. ~:t dSl.., ds,,,

(4.49)

where I~P~,s,..... s,,(t)) = (-1) ~ [ o , , ) ( n , s l , . . . , s,,)l~P(O)).

(4.50)

Introducing the continuous measurement of the quantum noise, we encounter a generalization of the situation described at the beginning of w3. Unfortunately, we can see that the number states are not normalized according to eq. (4.48), which complicates the description. Reducing the appropriate superoperatorvalued and operator-valued measures, we obtain the Srinivas-Davies model. The nonreferring process is described by the master equation; i.e., the process with introduced continuous measurement is after reduction described by the same equation as the damped harmonic oscillator without introducing measurement, but with the reduction. The replacements (4.37) and (4.38) enable one to model the homodyne detection, although in this case another limit is taken, also leading from Poissonian to the Gaussian noise. Direct detection, which is essentially continuous-time photocounting, is considered to be one of three well-known paradigms for high-sensitivity photodetection. The other two are homodyne detection and heterodyne detection. The theoretical analysis neglects polarization and special characteristics and frequently focuses on unity quantum efficiency direct detection. In a development of the work of Shapiro, Yuen and Machado Mata [1979] and Yuen and Shapiro [1980], an orthogonality relation similar to eq. (4.48) has been treated by Shapiro [1998]. Steimle and Alber [1996] refer to literature concerned with the quantum measurement process from a general viewpoint. They point out the substantial progress achieved in the theoretical description of individual photoelectric detection processes, in particular, those which involve homodyning or heterodyning with a classical intense source. In this context, the time evolution of a continuously observed photon source has been described by stochastic nonlinear Schr6dinger equations (Wiseman and Milburn [1993a,c]). Subsequently, various classes of stochastic nonlinear Schr6dinger equations have been proposed as general dynamical approaches to the process of state reduction and as means of describing arbitrary quantum measurement processes which do not necessarily involve photodetection processes. In one of these approaches, namely, the quantum state diffusion model (Gisin [ 1984], Gisin and Percival [ 1992, 1993a,b]), the completion of an individual quantum measurement process requires a

158

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characteristic time Tm. In this model, the time evolution of a state of an individual quantum system is described by a stochastic differential equation, d i^ dt I~(t)) = -

-hH]lp(t)) J

+

{Z

d },

J9

I

(4.51) where Lj are Lindblad operators and W4 are complex Wiener processes. Steimle and Alber [1996] developed a quantum-mechanical description of the StemGerlach effect, which has been designed as a method to measure the spin state of an electron confined in a Penning trap (Dehmelt [1986a,b]). Brun, Gisin, O'Mahony and Rigo [1997] have shown how the use of quantum trajectories, both continuous and discontinuous, illuminates properties of systems on the mesoscopic scale where neither a purely quantum nor a purely classical description is practical. They presented quantum jump trajectories which approach a diffusive limit similar to quantum state diffusion. Going to the classical limit, they found the rise of classical orbits for both regular and chaotic systems. A nonlinear stochastic equation for pure states describing non-Markovian diffusion of quantum trajectories has been presented (Didsi, Gisin and Strunz [ 1998]). It is not connected very closely to the measurement theory, but it can be used in measurement-like situations. Results of an example numerical simulation of trajectories for atoms radiating into a reservoir with a non-flat density of states have been presented and compared with those of an extended system approach (Jack, Collett and Walls [ 1999b]). 4.2. D E S T R U C T I V E A N D N O N D E M O L I T I O N M E A S U R E M E N T S

A very detailed analysis of the Srinivas-Davies model was carried out by Ueda, Imoto and Ogawa [1990a]. For an initial number state, the referring process is described by the unitary evolution, the state I~p(t)) always being a number state. At times r~, r2,..., the photon number decreases by one. For an initial coherent state, the referring measurement is the same as the nonreferring one; i.e., the coherent state remains coherent for all times. The initially squeezed state can be chosen as a sub-Poissonian (amplitude-squeezed) state, and its trajectory during the referring measurement always comprises sub-Poissonian states and is appropriately closer to that of the number state. The trajectory

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of/5(0 for an initial thermal state always consists of super-Poissonian states. Its behavior contrasts with that of the number state, because the mean photon number increases at the times r~, T2,..., whereas in the meantime it decreases faster than the appropriate nonreferring process. The model of continuous quantum nondemolition measurement of photon number is obtained from the scheme in w3.2 by setting 0 = h. This view is inherent in the paper by Ueda, Imoto, Nagaoka and Ogawa [1992]. For the achievements related to the general concept, see Walls and Milburn [1994]. For an initial number state, the state Iq~(t)) is always the initial number state in the referring process. For an initial coherent state, the trajectory of ]q~(t)) consists of ever more sub-Poissonian states, and it converges to a number state. At the times rl, r 2 , . . . , the mean photon number increases. The same can be said about an initial squeezed state and a thermal state, although the thermal state is superPoissonian for an initial period. Almost surely, m

lim ~-fT_h//(T) = n 2 T

----+

cx2

--

(4.52)

~

where n is an integer-valued random variable having the distribution p,(0) = p,,(0) = (nl/5(0)In),

(4.53)

which describes the measurement of the number operator. In other words, with the probability 1,/5(T) converges to a number state. Breslin, Milburn and Wiseman [1995] defined an ideal optimal quantum measurement as that measurement for which the average algorithmic information in the measurement record is minimized. To illustrate the discussion, they considered the quantum dynamics of a parametrically kicked nonlinear optical oscillator. The information in the optimal measurement record increases similarly as the von Neumann entropy. Breuer and Petruccione [1996] continued the work related to a microscopic system-meter-reservoir model of quantum measurement (Walls, Collett and Milburn [ 1985]). Following Wiseman [ 1993], they decomposed the variance of any Hermitian operator A in the stochastic vector liP(t)) as ^

var(A) = varl (A)+ var2(A),

(4.54)

where var(A) = Tr{ ,b(t) A 2 } - [Tr{ t5(t) A}] 2,

/,(t) = E(l~(t))
~il ~,(t))]2},

(4.55)

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where var(x) = E(x_2) - E2(x_),

(4.56)

with x any random variable. They performed a stochastic simulation to estimate these variances for A a constant of motion, whose variance remained constant throughout the simulation, whereas varl(.4) decreased to zero and var2(.4) increased monotonically from zero until its magnitude was equal to var(A). Ban [1998] has investigated in detail the information-theoretical properties of a sequence of quantum nondemolition measurements of a discrete observable, particularly of the photon number. In general, the choice of O as any self-adjoint operator leads to a continuous projection measurement or a nondemolition measurement. Here one has to say 'self-adjoint' instead of a more usual Hermitian operator, because the Hermitian operators with unnormalizable 'eigenvectors' do not satisfy the nondemolition condition (Belavkin [1990]). Therefore, no quantum nondemolition scheme has been introduced for measuring the phase and reducing the field to an unnormalizable phase state. The approach adopted by Belavkin and Bendjaballah [1994] is rather related to the quantum phase observable introduced by Pegg and Barnett [ 1989a]. A quantum nondemolition scheme has been presented for measuring the phase quadrature of a coherent field using an optical parametric amplifier (Smith, Collett and Walls [ 1993]). To evaluate the scheme, they used the symmetrized form of the quantum nondemolition criteria introduced by Holland, Collett, Walls and Levenson [1990]. Karlsson and Bj6rk [1995] analyzed schemes for linear measurements of the photon number of a light beam. They classified quantum nondemolition measurements as indirect measurements and found direct measurements to be quantum destructive measurements. They investigated and compared the effectiveness of such schemes in single and repeated measurements. They discussed and extended the criteria based on linear mean square estimation theory used, e.g., by Holland, Collett, Walls and Levenson [ 1990]. Bruckmeier, Schneider, Schiller and Mlynek [1997] performed improved quantum nondemolition measurements with a system based on a proposal by Smith, Collett and Walls [ 1993, 1994]. Bruckmeier, Hansen and Schiller [1997] reported on repeated quantum nondemolition measurements of continuous optical waves. Sinatra, Roch, Vigneron, Grelu, Poizat, Wang and Grangier [1998] analyzed a quantum nondemolition experiment using cold atoms in a magnetooptical trap as a nonlinear medium. Concepts such as monitoring an observable A and a quantum nondemolition observable are discussed by Mensky [1996b]. The term 'selective' corresponds to 'referring' used in the present chapter. Mensky [ 1996b] formulated in different

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forms and analyzed the action uncertainty principle for continuous quantum measurements investigated by a method based on restricted path integrals. This approach has been developed by Mensky [1993]. Audretsch and Mensky [1997] modeled a continuous measurement of energy for a two-level atomic system. A parameter they called the 'level resolution' time specifies a sharp (perfect) measurement, the Zeno regime of measurement, and an unsharp (fuzzy) measurement, the Rabi regime of measurement. They introduced fuzziness with the help of restricted path integrals and discussed both two-level and multilevel systems. In quantum optics, the multilevel free system is, in this case, realized as a mode of the electromagnetic field. Camacho and Camacho-Galvfin [1998] calculated the propagators of a particle caught in a Paul trap associated with a continuous quantum measurement of the x and z coordinates. They chose a Gaussian weight functional which renders a Gaussian path integral. In contrast, continuous projection measurements have been studied using linear quantum trajectories (Jacobs and Knight [1998]). Peculiar behavior is connected to the choice of 0 as a unitary operator. Milburn [1991] has proposed an elegant model of intrinsic decoherence [cf. eq. (3.18), where/2/= 0 and 0 = Within the framework of this model, Bu2ek and Kon6pka [1998] have studied the dynamics of open systems with specific system Hamiltonians. Doherty, Tan, Parkins and Walls [1999] have established that for a simple class of systems quantum-trajectory theories allow the determination of a unique state after a continuous measurement from the measurement record alone. The Gardiner-Collett input-output theory of quantum noise (Gardiner and Collett [1985]) has been adapted for the description of unidirectional coupling (Gardiner [1993]) and the quantum trajectory theory has been formulated for cascaded open systems, those comprising unidirectional couplings (Carmichael [1993b]). The techniques of feedback have been realized in optical systems to control laser intensity or phase stability. Wiseman and Milburn [1993c] studied three schemes for measuring the position-like quadrature of a cavity field (i.e., a mode). The first is a quantum nondemolition measurement of this quadrature via a coupling to that of another, heavily damped mode. They analyzed simple homodyne measurements and balanced homodyne measurements. In modeling these three schemes of quadrature measurements, they used extensively the concept of quantum trajectories due to Carmichael (Carmichael and Tian [1990]). Wiseman and Milburn [1993b] presented a quantum theory of feedback in which the homodyne photocurrent controls the dynamics of the source cavity,

exp(-iHsys/hR)].

d dt/5(t) =

1 ~2 t~(t)+k(h~(t)+~_(t)hi)+ ~--~K ~_(t)+{ v/-~(t)(H + ~-'K)/5(t)}, (4.57)

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where L is the superoperator of the dynamics, /~/ is the superoperator of the homodyne detection, K is the superoperator corregponding to the control, r/ is the efficiency of the detection system, and ~(t) is real delta-correlated noise. Wiseman and Milburn [1994] presented a quantum theory of optical cavity feedback mediated by homodyne detection, with an arbitrary time delay. As an application, they studied a sub-Poissonian pumped laser. They showed that extracavity measurement cannot produce nonclassical light unless the cavity dynamics can do so without feedback. To the contrary, an intracavity quantum nondemolition quadrature measurement allows arbitrary squeezing to be produced by controlling the driving field. Wiseman [1994a,b,c] has attempted a general theory of quantum limited feedback for continuously monitored systems. He compared two approaches, one based on quantum measurement theory and one on Hamiltonian systembath interactions. In practice, the all-optical schemes are still complicated and difficult to achieve, whereas electro-optic feedback has been in wide use. The theory can be equally widely applied as more devices are becoming quantum limited and controlling noise becomes important. Giovannetti, Tombesi and Vitali [ 1999] have solved exactly the non-Markovian dynamics of a cavity mode in the presence of a feedback loop based on homodyne measurement with a non-unit detection efficiency in the case of a nonzero feedback delay time. Wiseman and Vaccaro [1998] have focused on the ensemble that results from monitoring the environment, and have defined the survival time as the time at which such an ensemble becomes obsolete. They have shown that, for the class of continuous Markovian unravelings, an analytical solution is possible for the parametric optical oscillator. Some techniques are related to quantum nondemolition measurement (Yamamoto, Imoto and Machida [1986]). An approach to treating a particular feedback quantum nondemolition system based on the Kerr effect and external homodyne detection is based on the use of nonexclusive probability densities (eq. 5.12 below) (Wiseman [1994a,b,c], Wiseman and Milburn [1994], Jann and Ben-Aryeh [ 1997]). 4.3. MICROSCOPIC MODELS AND OBSERVATION OF ATOMS

Microscopic models of continuous detection use atoms to measure a radiation field. As a consequence, they can only be approximated by the continuous detection concept. The following derivation can be completed with a limit procedure resembling that for the quantum Zeno effect (Misra and Sudarshan [1977]). While the quantum Zeno effect is caused by always 'stating' at the

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same atom (the 'watch-dog' effect), characteristic of the following models is a succession of atoms (an atomic beam). To be consistent with the previous exposition, we use the relation (3.12), but generalized to the form i ^ ] ]~Pa, ..... aj(tj + 0)) = Ed"(aj)exp - ~ ( t j - tj_l)H) 'nt)

(4.58) 9-./~A'(al)ex p

" (int) tlH l

-

lp(0)),

where/2/j!int)__/_~(int) (ti_l,t/],J = 1,... ,J,

are the interaction Hamiltonians in the time interval (tj-1, tj], to = 0. The proposed microscopic models consist of two-level atoms, and describe the interaction by the Hamiltonian /:/s! mr) =

hg(O&,j + O+&,s),

(4.59)

where 0 = h (Imoto, Ueda and Ogawa [ 1990]) and 0 = h (Ueda, Imoto, Nagaoka and Ogawa [1992]), &,j is the level-lowering operator for the jth JaynesCummings two-level atom, and 6+../= (&.i)t. In more detail, &,s = Ig)jj(el,

0+,s =

le)sx{gl,

(4.60)

J = 1,...,J,

where Ig)j and le}; are vectors for the ground and the excited state of the atom, respectively. Associating numerical values to these states, we have: EJJ(0) = ]g)jy(g,

L'A'(1) = e}i/{e,

j = 1,... , J

(4.61)

Up to the second order of perturbation, I*Pa, ..... a.,(ts)) ~ I~Pa, ..... .,,(tS-~ + 0))

-ig(tj - tj_ 1 ) [ 0 & , / - q -

,

0 -~(~)-./] I ~a, ..... a/_, ([/-1 -t- 0))

[

]

2 [g(tj - ti-l)]200tk.~, (i) + 0 +OL":4'(0) ~p,,,..... ,,,_, (t/-I + 0)). (4.62) We suppose that the atoms begin to interact in the ground state. So it holds effectively that I~pa, ..... aj_l (tj)) e~ I~Ja, ..... al_l (t/_ l -~- 0)) --

ig(tj - t/-~ )O&./] ~Jal ...... I/ 1 (t/-I -~- 0)) 1 ~.[g(t/-t/-~)] 2 b*

01~iJ,,

,

.....

., ~ ,

(t/--I + 0)). .

(4.63)

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After their interaction, the observable ~ij is measured,

I a, ..... a,,,o(tj "]-0)) ~ ]~.Jal .....

al

l(tj-- 1 71-0))

~[g(t/-tj-1)]2

(4.64)

. + 0)), O+ 0 ~p.,..... ,,,_, (tJ-~

where 0 _= aj means that the atom has been detected in the ground state, and IW~, ..... aj-,,~(ty + o)) ~ -ig(tJ - tJ-~ )O&.ylWa, ..... ,, , (t/-J +0)),

(4.65)

where 1 = aj means that the atom has been detected in the excited state. Comparing with eqs. (4.33) and (4.34), we can see that the proposed atomic models are similar to the approximations outlined in eqs. (4.33) and (4.34). Fukuo, Ogawa and Nakamura [1998] have investigated the time evolution of the continuously measured Jaynes-Cummings model without the rotating-wave approximation as one of the candidates that has the capacity of showing temporal chaos within a quantum theory. In the continuous quantum nondemolition measurement process, results of a measurement come into existence in an indeterministic way. Thus the quantum chaos in the continuous quantum nondemolition measurement inevitably involves stochasticity. A further application of continuous measurement can be realized in a twomode radiation field and three-level atoms. For the purposes of description, we must modify the Hamiltonian (4.59) as done by Agarwal, Graf, Orszag, Scully and Walther [1994]:

/2/j!int) -- h [ga e)gj.(g[ + gb e)j.j(g'] + H.c.] ,

(4.66)

where le)j is a vector for the excited state and Ig)~, Ig')y are vectors for two ground states (the A system). Quantum nondemolition measurement as a modification of the SrinivasDavies (destructive) continuous measurement may correspond to decay of offdiagonal Fock-basis matrix elements of the unconditional statistical operator, but unraveling of such evolution means reduction of the field to a Fock state. Quantum nondemolition schemes in which the detector is a probe field coupled quadratically to an electromagnetic-field mode in a transparent solid medium have been proposed (Braginsky and Khalili [1980a,b], Milburn and Walls [1983], Imoto, Haus and Yamamoto [ 1985]) and demonstrated. Brune, Haroche, Lefevre, Raimond and Zagury [1990] described the use of three Rydberg levels o f f , e, and i of an atomic vapor. The detuning between the cavity mode and the e ~ i transition was large enough to cause only a phase shift. First they considered

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the dependence of the transition probability from .f to e on L (mod (Or2:r~,0 ), -- (t)e/ where L is the distance between Ramsey fields, v0 is the mean velocity of the atomic beam, or is the angular frequency of the Ramsey fields and coc/ is the angular frequency of the transition f ~ e. The field must be in the same initial state before each atom interacts with it. They then analyzed another experiment where the field was not sensibly relaxing between atoms (very high Q cavity). Repeated measurements were performed on the field. A quantum nondemolition method to measure the number of photons stored in a high-Q cavity has been described in detail (Brune, Haroche, Raimond, Davidovich and Zagury [1992]). It is based on the detection of the dispersive phase shift produced by the field on nonresonant atoms. This shift can be measured by atomic interferometry. The information acquired modifies the field step by step, until it eventually collapses into a Fock state. The field phase simultaneously undergoes a diffusive process as a result of the reaction of the measurement on the variable conjugate with the photon number. Comparing with Brune, Haroche, Lefevre, Raimond and Zagury [1990] and Brune, Haroche, Raimond, Davidovich and Zagury [1992], Onofrio and Viola [1998] have developed a model for the description of the decay of coherence in continuous quantum nondemolition measurements involving photons and atoms. They considered not only monitoring the photon field using nonresonant atoms as the probe system, but also monitoring the atomic level via an electromagnetic field. Agarwal, Graf, Orszag, Scully and Walther [1994] generalized the theory of continuous measurement so as to be applicable to a microscopic model. After eliminating the atomic system, we obtain a usual situation, except that instead 1 (~ + ~)e-iq). of 0 = h one has 0 = ~_ Cirac, Gardiner, Naraschewski and Zoller [1996] described the evolution of two Bose condensates in an interference experiment with the help of a master equation of equal damping constants. They have equivalently reformulated the description so that it suggests a 0-parametrized system of measurements (0 the relative phase) instead of a measurement on each mode. Dalibard [1996] presented a dramatic effect in the simulation of counting by two detectors, namely, an interference effect was observed for number-state fields. Such an interference effect can occur even when the statistical operator treatment predicts no interference (Molmer [1997a,b]). This work has a close connection to a number of recent approaches to interference phenomena in degenerate Bose gases. Englert, Gantsog, Schenzle, Wagner and Walther [1996] reconsidered the oneatom-maser setup of Wagner, Brecha, Schenzle and Walther [ 1992a,b, 1993] and proposed the appropriate phase-sensitive measurements.

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In order to better understand photon statistics and photon counting processes for single atoms, in particular a A system, the quantum jump approach was developed in Germany (Hegerfeldt [1993]). In the derivation of the quantum jump approach, one assumes photon measurements in rapid succession at times At apart. Between photon detections, the time development of an atom is described by a 'conditional' Hamiltonian, the condition being that no photons are found. Once a photon is detected, the atomic state must be reset. A simplified expression for the reset state has been connected to a reset operator (Hegerfeldt and Sondermann [1996]). A probe pulse can couple the ground state of a two-level atom to a third, auxiliary, level of the atom, and it has been investigated (Beige, Hegerfeldt and Sondermann [1996]) to what extent this procedure results in an effective measurement close to an ideal (projectionpostulate) measurement. Beige and Hegerfeldt [ 1998] have investigated transition from antibunching to bunching for two dipole-interacting atoms and touched on the quantum-jump approach. Plenio, Knight and Thompson [ 1996] have pointed out that the experiment of Itano, Heinzen, Bollinger and Wineland [1990] to verify the quantum Zeno effect was a success due to a quadratic short-time dependence of the probability to find the system in its initial state. Although in this experiment the inhibition of a coherent transition was demonstrated, a suppression of incoherent exponential decay of the population of an atomic level (Misra and Sudarshan [1977]) was regarded as not accessible to experimental verification. It is experimentally feasible according to the proposal by Plenio, Knight and Thompson [1996]. Belavkin and Melsheimer [1996] have given an explicit stochastic Hamiltonian model of discontinuous unitary evolution for quantum jumps as in a system of atoms in quantum optics, or in a system of identical particles which interact with bubbles in a bubble chamber. They pointed out that the quantum measurement theory based on the ordinary von Neumann reduction postulate does not even apply to instantaneous observations with continuous spectra, which motivated von Neumann to perhaps the first model of instantaneous unsharp measurement. Herkommer, Carmichael and Schleich [1996] used the method of quantum trajectories to study the time evolution of a two-level atom interacting with an optical field in a cavity while a continuous homodyne measurement is made on the field transmitted through the cavity mirrors. Mabuchi and Wiseman [ 1998] have investigated the strongly driven Jaynes-Cummings system with dissipation whose classical dynamics suggests that two orthogonal quantum states are mixed in the steady-state q~A quasidistribution. They have studied the retroactive quantum jumps occurring in the conditional evolution when the cavity output is continuously observed via homodyne detection. Wiseman and Toombes [1999] have commented on the

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theory of quantum jumps proposed by Teich and Mahler [1992] who claimed that the quantum jumps in their theory corresponded to photon detection. Nevertheless, the Teich-Mahler trajectories are different from the quantum trajectories from direct photodetection. Casagrande, Lulli and Ulzega [1999a] have presented a quantum-trajectory treatment of dynamics of a high-Q-cavity mode interacting with a low-density atomic beam in the strong coupling regime. They have considered up to three-atom collective effects for mean number of atoms in the cavity both corresponding to the micromaser or microlaser and approaching a mesoscopic regime. By applying a quantum-trajectory approach to a master equation for the statistical operator of a cavity mode, Casagrande, Lulli and Ulzega [ 1999b] have described the dynamics of a micromaser or microlaser including the effects of cavity temperature, two-atom events, atomic velocity spread, and arbitrary level of excitation of the atomic beam. The general theory of path integral propagators for the solution of linear quantum-state diffusion stochastic Schr6dinger equations describing open quantum systems has been developed by Strunz [1996a]. Although the theoretical description of open quantum systems with the help of the path integral goes back to Feynman and Vernon [1963] and Feynman and Hibbs [1965], the approach of Strunz [1996a] is new as it is based on stochastic Schr6dinger equations. They have been introduced in both linear (e.g., Belavkin and Staszewski [1992], Barchielli and Belavkin [ 1991 ], Goetsch and Graham [ 1994], Goetsch, Graham and Haake [1995], Wiseman [1995], Goetsch, Tombesi and Vitali [1996], Strunz [1996a,b]) and nonlinear (e.g., Belavkin [1988], Gisin [1989], Belavkin and Staszewski [ 1989], Wiseman and Milburn [ 1994], Carmichael [ 1994]) versions, the nonlinear one being used primarily for numerical purposes (for a review, see Jacobs and Knight [1998]). Di6si [1994] and Strunz [1996b] have expressed the general Feynman-Vernon path integral propagator for open quantum systems as the classical ensemble average over a stochastic pure state propagator. However, they have had to consider non-Markovian dynamics of quantum state diffusion. Whereas Di6si [1994] constructed a real Gaussian process, Strunz [1996b] demonstrated the relevance of complex Gaussian stochastic processes. Carmichael [1997] commented on Einstein's stochastic process governing spontaneous emission, stimulated emission, and absorption, which has been widely applied in quantum optics. He analyzed amplification without population inversion in a resonant V-type atomic medium, which 'cannot occur' according to Einstein theory, and used the theory of quantum trajectories. The concept of photodetection as a continuous quantum measurement (introduced by Srinivas and Davies) can be extended to the detection of resonance fluorescence (Jones and Lee [1998]).

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Kilin, Maevskaya, Nizovtsev, Shatokhin, Berman, yon Borczyskowski, Wrachtrup and Fleury [ 1998] studied the stochastic dynamics of a laser-driven four-level system serving as a model in single-molecule spectroscopy. At low temperature (_~ 2 K), a chromophore molecule can be modeled by such a system, each state being the lowest energetic state for a corresponding local well of asymmetric double-well potential for the ground and excited electronic states. The theory of continuous quantum measurement has been used to describe the complex stochastic dynamics experienced by the molecule due to tunneling jumps. Casagrande, Garavaglia and Lulli [1998] have used the quantum trajectory method to investigate the microscopic dynamics and to describe the effects produced on trapping states by temperature, atomic velocity spread, presence of atom pairs, and partial atomic excitation. Vaccaro and Richards [ 1998] studied a microscopic model of quantum counter (O - h t) and its generalizations. They addressed the evolution of an optical linear amplifier conditioned on the state-selective detection of the atoms. They have been interested in the use of the stochastic Schr6dinger equation to represent different unravelings of the master equation. By applying a classical pulse to each atom before the detection process, they generate different stochastic Schr6dinger equations, including that describing the quantum counter. In a suitable limit, they have obtained a formal consistence up to the interchange of h t and h with the result for a measurement scheme based on homodyne detection (Carmichael [1993a]), where 0 = h. Assuming that the phase of the classical pulses varies rapidly, they have been able to replace a stochastic Schr6dinger equation by the quantum-state diffusion equation introduced by Gisin and Percival [1992, 1993a,b]. Vitali, Tombesi and Milburn [1998] have shown how an initially prepared quantum state of a radiation mode in a cavity can be preserved for a long time using a feedback scheme based on the injection of appropriately prepared atoms. Di6si and Halliwell [1998] have used continuous quantum measurement theory to construct a phenomenological description of the interaction of a quasiclassical variable X with an operator ~. They have borne in mind quantum field theory in curved space-time and the semiclassical Einstein equations, which could be valid in a very limited set of circumstances. They have taken into account that the mean field equations would not produce intuitively sensible results in the key case of superposition states. They considered a classical particle with position X and a harmonic oscillator with the position operator ~. Trying to describe the coupling of classical and quantum variables, they suggested thinking of the quasiclassical particle as in some sense 'measuring' the position operator of the quantum particle. They derived that one standard Gaussian white noise

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enters both the stochastic nonlinear Schr6dinger equation describing the quantum subsystem, and the stochastic ordinary differential equation for the quasiclassical variable X.

w 5. Probability-density Functional for Quantum Photodetection Processes The probability-density functional has been invented to give complete information on the statistical properties of any random-point process (Ueda [1989a]). Any point process can be considered to be a Poisson process whose parameter is time dependent and random. Frequently, this parameter is called a probability-density function. In the process of photodetection, it is proportional to the intensity of light. The probability-density functional is used to obtain the characteristics of the random-point process, which are considered to be expectation values of other functionals of the probability-density function. It is assumed that a generalized integral can be taken of the product of the probabilitydensity functional and another functional of the probability-density function. The probability-density functional is obtained by the inverse generalized Fourier transformation of a generating functional (or characteristic functional). From the mathematical point of view, the generating functional presents a few difficulties; therefore we will focus on such functionals, and we will treat the probabilitydensity functional only marginally. 5.1. GENERATING FUNCTIONALS FOR RANDOM-POINT PROCESS

We will start with the characteristic functional of a random-point process,

CPP[iu(t)]

=

E

exp i Z

u(r/)

,

(5.1)

j=0

u(t)

where is an arbitrary auxiliary function, M___(~c) is the total number of registered photoelectrons in the interval (0, ~c), and r/, j = 1,... ,M_M_(oc), are times at which the registrations were made. This functional can be expressed using the generating functional of the probability-density function, CPP[iu(t)]

= c(~l)[eiu(t) - 1],

(5.2)

where

C~')[iu(t)] = (exp[i f~ u(t) rlI(t)dt] ) .

(5.3)

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Denoting by n the number of registered photoelectrons in the interval [t, t + T), n = M ( t + T ) - M ( t ) , and by W the integrated intensity in this interval (cf. eq. 2.7), we will introduce a notation for concepts treated by Pef-ina [ 1991]"

(5.4) (5.5)

C(")(is) = (exp(isn)), c ( W ) ( i s ) = (exp(isW)).

Using DeFinetti's symbol (t ~< t' < t + T) equal to unity in the interval [t, t + T) and zero elsewhere, we may observe that C(n)(is) = CPP[is(t <~ t' < t + T)],

(5.6)

c ( W ) ( i s ) = C("I)[is(t <, t' < t + T)].

(5.7)

In this situation, the relation (5.2) becomes the familiar relation C(n)(is) = C(W)(e is - 1).

(5.8)

The usual application of rules concerning the relation (5.6) leads to the photocount probability distribution

1 d"

p ( n , t, T ) -

_ _

n! dos)"

cr

,

(5.9)

is = 0

1 d"

(5.1o)

--c(W)(is)

n! dos)"

is = -1

The moments of the fluctuating parameter W are known to be

(__Wk(t, T))

-

1

dk __CtW)(is)

(5.11)

k! d(is) k

is = 0

A generalization of the relation (5.11) is a formulation of the joint probability distribution of multicoincidence, the so-called nonexclusive probability densities,

~,(tl,... ,to,)=

lim

p(1,..., 1,tl,Atl,... ,tj,Atj).

{At; ~ 0+}

Atl 999Atj

(5.12)

Since /~(tl, . . . , tg) = r/g ( / ( t l ) . . . I ( t j ) ) ,

(5.13)

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such a generalization reads as 1 6 J C(O1)[iu(t)] ij 6 u ( t l ) . . . 6 u ( t y )

/~(tl,... , t j ) -

(5.14) {iu(t)=0}

where ~u--N~,), etc., means functional differentiation (Volterra [1959]). Alternatively, the functional C("I)[iu(t)] can be constructed from these probabilities:

i f0 /0

C("I)[iu(t)] = Z

"'"

J

)~(tl,. . . , t j ) u ( t l ) .

. . u(tj)dtl

. . . dtj.

(5.15)

Exploiting the notation of the left-hand side of eq. (2.9), we can introduce the probability distribution of the forward-recurrence times (the so-called exclusive probability densities): P[t,t+r)(M, r l , . . . , rM-1, rM)

=

1

lim {Ar~~0+} Arl ... ArM

x p(O, 1 , 0 , . . . , O, 1, O, t, rl - t, rl, Arl, Tl + Arl, r2 - rl - A r l , . . . , rM-1 + ArM-l, rM -- rM-I -- AL~t-1, r~1,Ar~t, r~.t + A L ~ t , t + T -

TM - ARM).

(5.16) The numerator of the fraction in eq. (5.16) is the probability of no counts in the interval [t, rl), one count in the interval [r~, r~ + A r~), no counts in the interval [rl + A rl, r2),..., no counts in [r~l_l + A r~,~4_l, rM), one count in [rM, rM + Arm), and no counts in [r~t + A r~, t + T). The counterparts of relations (5.13) and (5.14) read as

P[t,t+r)(M, r l , . . . ,

rM-1, TM)

/ = r/M / ( r l ) ' " / ( r M - 1 ) / ( r ~ ) e x p

[

~.t+r -r/

-)

(5.17)

I(t')dt'

and P[t,t+r)(M, r l , . . . , rM-1, rM) _

1

6 MC("I)[iu(t')]

[

(5.18)

iM 6U(rl)? ~~6--~M) I {iu(t')=-(t ~
(5.19)

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the generating functional of probability distribution of forward-recurrence times

G(2)[u(t)] = G(l)[u(t) + i],

(5.20)

and he has also considered the generating functional of the probability distribution for the number of counts GI3)[u(t)] with the property

1 &G(3)[u(t')] p(n, t, T) - inn! 6[u(t + T)]"

(5.21) {mr)=0}

In several solvable quantum-optical models, the above functionals can be expressed as was done by Ueda [1989a, 1990] for 0 = h and R = constant. In the generalized Srinivas-Davies model, we have: J

(5.22)

X(t,,... ,tj) = ~ H I ( l , t/) Tr{ /5(0) h t J d }, j=l

where I(a, tj) is defined in eq. (2.54) with (2.49). Substituting eq. (5.22) into eq. (5.15), we obtain

C("t)[iu(t)] = Tr {~(O) { exp

ih*h~~~ I(1,t') u(t')dt'l }x } ,

(5.23)

where {}H is a normal ordering symbol that indicates rearranging the operator in the normal order without the use of the commutation relation. Assuming the statistical operator in the Glauber-Sudarshan diagonal form,

/~(o) = f 4.A:(.. o)pa>
(5.24)

where q~x(a, 0) is the appropriate quasidistribution, the relation (5.23) becomes

C(OO[iu(t)] = =

I /

[ /0 [/0

q)x(a, 0) exp i ale t~ q~ar(a,O)exp it/

I(1, t') u(t') dt' d 2a

I(a,t')u(t')dt'

1

d2a.

(5.25) (5.26)

The form of this functional suggests that the appropriate density function has a very simple property which can be expressed, provided q~N(a, 0) 'exists', as follows. The probability densities at distinct times are not only correlated, but they are connected formally. All the values at t > 0 can be determined from

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173

I(a, 0).

Ueda [1989b, 1990] assumes 'existence' of the functional T'[r/I(t)] with the property <.M[r/I(t)])

= / .M[rlI(t)] 7)[r/I(t)] 6[tfl(t)],

(5.27)

where .M[r/I(t)] is the functional to be averaged. From what we have said, it follows that

<M[.i(t)]> = fI cI)~:(a,O)A4[rlI(a, t)] d e a,

(5.28)

J

while we refer to the original papers for the functional 7)[r/I(t)].

w 6. Quasicontinuous Schemes of Photodetection Ban [1994] proposed a quasicontinuous measurement with lossless beam splitters and photodetectors. Futhermore, he considered a measurement by means of a parametric amplifier and a photodetector and a measurement in terms of a four-wave mixer and a photodetector. He discussed the relationships to the continuous destructive quantum measurement of photon number, the continuous measurement of photon number with a quantum counter, and the continuous quantum nondemolition measurement of photon number. 6.1. MEASUREMENT WITH BEAM SPLITTERS

In the Heisenberg picture, the beam splitter is characterized by the relation a(1) = ta(O) + r'b(O),

b(1) = rb(O)+ t'b(O),

(6.1)

where h(0), at(0) and b(0), bt(0) are the annihilation and creation operators of the input signal and reference modes, and a(1), at (1) and b(1), Dt (1) are those of the output modes. The coefficients t, r, r', t ' have the properties Itl 2 + Ir[ 2 = 1,

Ir'l 2 + It'] 2 = 1,

tr'*

+ rt'*

= O.

(6.2)

Here we assume that t ' = t > 0, since phase shifts are not important for our purpose. The transmittance 7- and the reflectance 7-s of the beam splitter are given by = itl2,

j~

=

ir 2.

(6.3)

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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In the following, we assume that the input state of the reference mode is the vacuum state. In the Schr6dinger picture, the output state/3(1) of the beam splitter is given by /3(1) = V'[p(a)(0) @ [O)b b(Ol]V t,

(6.4)

~"

(6.5)

= exp(~]+ - ~*J_),

where ,b(~)(O) is the statistical operator of the signal mode input and [O)bb (0 is the statistical operator of the reference mode input. In eq. (6.5), r

- - ]-~ arccos t

(6.6)

and J + and J0 are the generators of the SU(2) Lie algebra given by J+ =

;Ttb,

hb* ,

J_ =

Jo = :'(a*a -

(6.7)

b'b),

which satisfythe commutation relations [J+,J_] = 2Jo and [fro,J+] = +J• Here we use the simplified notation h - h(0), h t - ht(0), and t) - b(0), ~)t - Dr(0). The relation (6.4) reflects the well-known fact that a lossless beam splitter is characterized by the SU(2) Lie algebra (Campos, Saleh and Teich [1989]). If the photodetector for the output of the reference mode measures photons, we obtain that (6.8)

,bm(1 + 0) =/5(~)(1 + 0) | ]m)h b(ml, where

p(ma)(1+0)-

s189189 m!

(6.9)

Now we introduce the superoperators k+, k0 as ~+ll~/= ~t)l~/~,

~ _ ~ / = ~/t~ t,

~0)1~/=~ t ~ / + ll~/~t,

(6.10)

for an arbitrary operator l~/ (Pefinovfi, Luk~ and K~epelka [1996b]). The superoperators (6.10) obey the commutation relations [k_,k+] [~0,~+] = +2k+. Then r

=

k0 and

+ 0)is expressed as follows"

/5~)(I+0)= u(T, m)/5(m(0),

(6.11)

where u(T, m) = (1 - T)m T89 m! The normalized output state of the signal mode is given by

~)(a) O) Im (1 + =

u(T' m) p(a)(o)

Tra{U(T, m) f)(a)(O)}

(6.12)

(6.13)

where Tra is the trace operator over the Hilbert space of the signal mode.

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175

It is interesting to remark that the output state (6.9) (or 6.11) is equal to the state of the light beam, after a photon counter registers M - m photons, in the continuous measurement of photon number described by the quantum Markov process. When we do not refer to the result of the measurement on the output of the reference mode, the output state of the signal mode becomes 0(3

/5(a)(1 + O) = Z/5(ma)(o)'

(6.14)

m----O

= T 89(i'~ i )-Z75(")(0),

(6.15)

where Tra/~(a)(1 + 0 ) = Tr,/5(")(0)= 1 is satisfied. It is easily seen that this state is merely attenuated (Pe~inovfi, Lukg and K~epelka [1996b]). Using the statistical operator (6.13) of the signal mode output, we can calculate the mean (hth)lm, the second-order moment of the photon number ((hth)2)l,,,, and the Fano factor decreased by unity (Mandel [1979]):

fm

((hth)2>lm-
(hth)~"'

(6.16)

'

Making use of eq. (6.16), we obtain the following results for special input states: (i) The number eigenstate. When the input state of the signal mode is the eigenstate of photon number, such that/5("i(0) = ]n0>(n0l, with no > m, we obtain (hth>l m = l0 - m = n 0 - m,

((hth)2>l m = (no - m) e,

(6.17)

fm = --1. These are trivial results, indicating that no input photons are divided into m reference output photons and ( n o - m) signal output photons by the beam splitter. (ii) The binomial state (Stoler, Saleh and Teich [1985]) is given by N

IN; b) = Z

V/p(nlN)In>'

(6. l S)

n=0

where

p(nlN ) =

N!

n!(N-n)!

p"(1 _p)X-,,

(6.19)

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

IN; b)(N; b

Thus, for ,bta)(0)= (hth)l m =

(

m)

1- ~

[III, w 6

(N > m), we get (Ban [1994])

Up

(hth)l o = ( N - m) _---L-~p, 1

((h*h)2)l m = (N - m) 1 - ~ p

(N - m - 1) 1 - ~ p +1

(6.20)

~rp fro--

1--~p"

We see sub-Poissonian photon statistics from f,,, < O. (iii) The coherent state. When the input state is a coherent state, such that /5(~ = la)(a[ and n0 = ]a 2, we obtain

(hth)l., = (~t~)to = T-~o, (6.21)

((hth)2)lm = 7-~o(7"ffo + 1), fm = 0.

(iv) The thermal state. When the input signal mode is in the thermal state, oo

,o(a)(o) = Z p ( n )

n)(nl,

(6.22)

n=0

where --//

p(n) =

no

(1 + n0)n+l

(6.23) '

with if0 the mean photon number, we get the results (hth)l, , = ( m + l ) ( h t h ) l 0 = (m + 1)T~o

((hth)2)lm =

1 + 7~-no

(m + 1)Tff0 l+Rff0 '

(m + 2)'T~o

1

1 + 7~-no + lj ,

(6.24)

T-~o fm = 1 + ~-~0 We have super-Poissonian photon statistics (f,,, > 0). We find bunching correlation of photon numbers in the thermal state, since the mean photon number in the output state of the signal mode increases if the photodetector for the output of the reference mode registers the photons, (htgt)l m >1 (hth)l o. It is a possible definition of this property. We find

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177

antibunching correlation of photon numbers in the binomial and number states, since the mean photon number in the output state decreases if the photodetector registers the photons, (gttgt)l,, , <~ {hth)lo . When the input state is the coherent state, the mean photon number is independent of how many photons the photodetector of the reference output registers. 6.2. ANALOGY TO CONTINUOUS MEASUREMENT

Let us now discuss the similarity between the continuous measurement of photon number described by the quantum Markov process and a series of J lossless beam splitters such that the output signal of the jth beam splitter is used as the input signal of the (j + 1)th beam splitter. We assume that the input state of the reference mode of each beam splitter is vacuum, and that the output of the reference mode is measured by the photodetector subject to the projection postulate. If the jth photodetector measures mi photons, the unnormalized output state of the signal mode of the last (Jth) beam splitter is given by:

p(ma]..... mj(J +O) = u('-l-j,mj)

...

~(T~,m,

)h<,,,(o),

(6.25)

where u ( 7 - , m ) is defined by eq. (6.12), ~ is the transmittance of the jth beam splitter, and/5(")(0) is the input state of the signal mode of the first beam splitter. The probability that such a detector measures m/photons (j - 1,..., J) is given by p(ml

'

"

"

"

'

m j ) = Tra{ t3ta) I"1111,

....

m,/

(J + 0)}

(6.26)

"

Now we assume that each of the J beam splitters has a sufficiently low reflectance (7~k << 1) and that the probability of the photodetector for the output state of the reference mode registering more than one photon is negligible. Under this assumption, we can approximate u/(~, m/) as fii(Tjj, 0 ) = S[0,At,),

fti(~, 1) ~ JAr~,

z)/(T/, mi > 1) ~ 0,

(6.27)

where RAtj is defined by Tjj = exp(-RAt/) and the superoperators S[0. r ) a n d ) are defined by S[o, r) = exp -89

- ~) ,

J - Rk_.

(6.28)

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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

[III, w 6

Then the probability that given M photodetectors, for example, the /lth, /2th, ...,/Mth detectors, measure one photon and the other J - M photodetectors do not register photons is obtained from

Pexample(ml,... , m j )

~

PLo.r)(M, r l , . . . , rM)Arl...ARM,

(6.29)

with

P[o,T)(M, Tl,..., TM)=Tra {S[r,,,T))~[r,t_,,r,,))"" ")~[0, r,)/3(a)(o)} =R M exp -R

rk

k=l • Tra {,bta)(o)h *M exp(-era*a)aM}, (6.30) where we set rk = ~j~l Atj, k = 1,..., M. Let us compare this result with eqs. (3.43), (3.44), and (3.32). The normalized output state of the signal mode is given by ~,

'kr

"k

"k'2

S[r,,.r) JS[r,,_,.r,,)J"" JS[o, r, )/3(a)(0) Tra { ~[r,.r)JS[ r,,_,.r,,, ) " " )~[0, r, ) /5(")(0) } exp

(6.31)

(-89

~lMp(a)(O)atMexp (-89 Tra {/5(a)(O)htM exp(-Rrhth) hM}

which is equal to that obtained in the continuous measurement of photon number. If none of the results exhibited by the n photodetectors are referred, the output state of the signal mode becomes r

= exp [-89

- i - 2~_)1/5/"'(0).

(6.32)

Therefore, it is found that the measurement in terms of a series of lossless beam splitters with very low reflectances and of photodetectors subject to the projection postulate can lead to results equivalent to those obtained by the continuous measurement of photon number described by the quantum Markov process. Let us remark that the input-output relations (6.1) for an optical beam splitter can be generalized to allow for a linear absorption by the medium forming the mirror (Barnett, Gilson, Huttner and Imoto [1996], Barnett, Jeffers, Gatti and Loudon [ 1998]).

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179

6.3. MEASUREMENT BY MEANS OF PARAMETRIC AMPLIFIERS AND FOUR-WAVE MIXERS

So far, we have been expounding the quasicontinuous measurement of photon number with beam splitters. Now we consider measurements by means of other optical devices. We first consider the measurement using the nondegenerate parametric amplifier and the photodetector. The Hamiltonian of the nondegenerate parametric amplifier with the classical pump field is given by: /2/_ h (O)a~t~ + (ohbtb + g~tbte-i("' + g*~bei("') ,

(6.33)

where h, h t and b, ~t are the annihilation and creation operators for the signal and reference (idler) modes and ge -i''~ stands for the classical pump field, including coupling constant (Mollow and Glauber [1967]). If we assume ~o = o9, + ~oh, we obtain the time-translation generator of states in the interaction representation: /-~/int = h(gh t ~t + g, hb).

(6.34)

The output state t5(1) of the nondegenerate parametric amplifier is given by /5(1) = ~'r[~)(a)(0)@ 10>b b(0 ] ~ ' t

(6.35)

= exp(~/s - ~*k_),

(6.36)

where ,b(a)(0) is the statistical operator of the signal mode input and ]0)h b (01 is the statistical operator of the reference mode input. In eq. (6.36), ~ = -igt, where t is the transit time of the photon passing through the parametric amplifier, and K+ and K0 are the generators of the SU(1,1) Lie algebra given by /~+ = ~tbt,

k?_ = ~b,

Kf, = ~(h~h+bbt),

(6.37)

which satisfy the commutation relations [K+, k ] -- -2K0 and [K0, K+ ] = -+-K• Thus, if the photodetector for the output of the reference mode measures m photons, we obtain that /gm(1 + 0 ) =/5},")(1 + 0 ) |

Im)h t,<ml,

(6.38)

with h(ma)(1 + 0) = ~(T, m) h(")(0),

(6.39)

where

u(T,m) = (7-- 1)m 7-_89

(6.40) m! with 7-= cosh 2 I~l- When we do not refer to the state of the photodetector, the output state of the signal mode becomes /3(a)(1 + 0 ) =

T~+-89

which is equivalent to the output state of a linear amplifier.

(6.41)

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

[III, w 6

When we consider a measurement in terms of a series of nondegenerate parametric amplifiers with weak parametric coupling constants and photodetectors, we may use the relations (6.25) and (6.26), but we approximate hj(Tj, my) as ^

?~j(~, o) = Sto,~,,~,

uj(Tjj, l) ~ )Atj,

/~j(Tjj, mi > 1) ~ o,

(6.42)

where RAtj is defined by ~ = exp(RAtj ) and the superoperators ~[o, r) and ) are defined by

~[0, r)= exp -1RT(~o + ~) ,

J = Rk+.

(6.43)

Using the same discussion as in the derivation of eqs. (6.29) and (6.31), we can obtain the probability distribution P[0, r)(M, r~,..., rM) that the /lth, /2th, ..., lMth photodetectors measure one photon (the other ( J - M) photodetectors do not register photons) and the output state IJjM,r~ ~(") .....r~l(T) of the field. Here we set rk = }--~J~2-1Atj-, k = 1,... ,M. By replacing ]r with ~+ in eqs. (6.30) and (6.31), we have P[o. r)(M, r , , . . . , rM) = Tra { ~[~,,,r))~[r,,_, .~,,, ) " " )S[o, ~,, r = R M exp R Z

}

rr~ {~"~(0)a Mexp(-eraa*)a*~ },

rk

k=l

(6.44) and the normalized output state of the signal mode is given by /~(,,) IM.r,.....rM(T) =

S[r,,,r) JS[

J

JS[o. )/5(0)(0)

r,,_,.r,,) "'" r, Tra { ~[r,,,r))~[ r,,_, ,r,,)J"" )~[0. r,)/}(a)(0) }

(6.45)

exp (-89 h'M/5(")(0) h M exp (-89 Tra{/5(")(0) h M exp(-RThht ) him } It is seen that eqs. (6.44) and (6.45) are equivalent to eqs. (3.43) and (3.46) [(3.44), (3.62), (3.74), (3.76)], respectively. Therefore, the measurement using parametric amplifiers and photodetectors subject to the projection postulate yields results equivalent to those obtained in the continuous measurement of photon number with a quantum counter described by the quantum Markov process. Finally, we consider the measurement with a four-wave mixer, instead of a beam splitter, and a photodetector with which the measurement obeys the

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181

projection postulate. We assume here that one mode of the four-wave mixer is highly excited and treated classically, and that the Hamiltonian is given by

= h [maata + ~mgtg + ata(g*be i''" +

gbte-i"")],

(6.46)

where h, h t and b, ~t are the annihilation and creation operators for the signal and reference (idler) modes and ge -i"" stands for the classical pump field, including the coupling constant. If we assume co/, - o) and g to be real, we obtain the time-translation generator of states in the interaction representation /2/int =

hgata(g + bt).

(6.47)

When the input state of the reference mode is a vacuum state, the output/3(1) of the four-wave mixer is given by P(1) = P~(a)(0) | 10>bh
(6.48)

f/

(6.49)

= exp[_igthth(b + bt)],

where t is the transit time of the photon passing through the four-wave mixer. When the photodetector for the output of the reference mode measures m photons, the unnormalized output state of the signal mode becomes /}~)(1 + 0) = b<m/5(a)(1)lm>t,

_ (gt)2m h),,, e x p [- ~ (gt)2 m! (ht (hth)

1

2 r

(6.50)

x exp[-~(htgt) 2] (hth) ''', where [m)b is a number eigenstate of the reference mode. It is found that the state /5~m~)(1+ 0) is identical to that obtained by the continuous quantum nondemolition measurement of photon number described by the quantum Markov process, provided we set (gt) 2 = RT. The probability p(m, O, T) of the photodetector measuring m photons is given by

p(m, O, T) - (gt)2mTra{D(")(O)expI-(gt)2(hth)21 (hth) 2''' }

(6.51)

m!

which is equal to that obtained in the continuous measurement. Therefore, it is found that the measurement using the four-wave mixer and the photodetector

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[III, w7

gives the same results as those obtained in the continuous quantum nondemolition measurement described by the quantum Markov process.

w 7. Special States of an Optical Mode by Means of Continuous and Nearly Continuous Measurements on this Mode

In the nonreferring (nonselective) measurement, the photon counter and the quantum counter act on the light field as an attenuator and an amplifier, respectively. Any compensation for attenuation by an equal amplification (amplification after attenuation) introduces noise or causes thermalization. The amount of noise introduced is less when the order of attenuation and amplification is reversed (attenuation after amplification), 1 - A < 1 _ 1, where A is the attenuation. Vourdas [1988] studied number eigenstates with thermal noise, and has concluded that number eigenstates can play a positive role in optical communications. In practical experiments, the quantum efficiency of photodetectors is much lower than 100%. Ogawa, Ueda and Imoto [199 la], however, have confined themselves to the case of perfect quantum efficiency (100%) in order to propose the generation of the Schr6dinger-cat state by continuous photodetection. A few models of continuous photodetection have the remarkable property that the conditional statistical operator/31M(T ) is pure, kiM(T) - I PIM(T))< IM(T)I, some ]WIM(T)). The Sfinivas-Davies model of the photon counter has this property. Ogawa, Ueda and Imoto [1991a] selected an amplitude-squeezed state as the initial state. They found that the quasidistributions of the state [~PlM(T)), M = 1,2, have two distinct peaks indicating the Schr6dinger-cat state. The amplitude-squeezed state Ifl, r) = S(r)[3) is chosen as an initial state, /3(T)lr= o - Ifi, r)(fi, r I, where S(r) is the squeezing operator with a squeeze parameter r; i.e., 5'(r) = exp [~l r(h2 - ht2)].

(7.1)

They confined themselves to the case where fi is real and the squeeze parameter is positive, r ~> 0. The quasidistribution related to the antinormal ordering reads 4~A(a, T)lr=o

=

exp(-[a]2) { [~ exp Re(a 2) - 2fi Re a :r~t g

(~t - v) fi2 g

]} ' (7.2)

where kt = cosh r,

v = sinh r.

(7.3)

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183

S P E C I A L S T A T E S OF AN O P T I C A L M O D E

They introduced the normalization constants

ZM(T)=2MOM[411 /exp ( -

(1

+y)~tv

(7.4)

'

particularly 1

Zo(T) =

V/1 _y2 exp (1

(7.5)

+y)Itv '

where

1/

y = ~ exp(-RT).

(7.6)

While no photons are detected (M = 0; no-count process), the q~A representation evolves as exp(-la[ 2) 4'A(a, rio) = JrZo(r)

x exp {- [Y~exp(-RT) Re( a 2) 2fi ~

exp(- 51RT) Re a]

}

(7.7) However, as soon as one photon is detected (M = 1), the q~A representation changes abruptly into v z0(r) q~a(a, T[1) = 2 - ~

12Zl(T)

• {Iexp(-~1RT)R e a - f iv

a.

+ [exp(- 89

a] 2 q~A(a, T 0).

(7.8) The subsequent one-count event further emphasizes the two-peaked character of the q~A representation, which is given by q'A(a, TI2) Zo(r) [ v2 /32 = 4Z2(T) / 1 + 1/314~ exp(-2RT) + 4--1t: exp(-RT)

(

+ 2 fi_2_2 /32 /*v ~ - 1 + 4/3 exp(-1RT)

la

) ( +2

E 1

fi2 ~-1

Itv

~exp(-RT)Re(a 2) v exp(-RT)

~t

a2

Re a

}

~A(a, T 0).

(7.9) If the quantum counter is used for measuring an initial vacuum state, conditional states are reminiscent of photon-added thermal states. Although

184

CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

[III, w 7

such states need exhibit neither squeezing nor sub-Poissonian statistics, their nonclassical character can be tested using a quantitative criterion developed by Agarwal and Tara [1992]. Agarwal, Puri and Singh [1997] studied the properties of wave-packet states with vortex structure. The quantum system can be a two-dimensional harmonic oscillator or a two-mode radiation field. They demonstrated the generation of such states by the interaction of A systems with squeezed radiation in a twomode cavity. The state of each mode is mixed, even though Agarwal, Puri and Singh [ 1997] define the state of the two-mode field as a pure state. With 0 = 1 (h + e-iqb), we have an essentially one-mode measurement, which is interesting since it influences two modes. Agarwal, Graf, Orszag, Scully and Walther [1994] examined the cases of an initial two-mode coherent state la)al/3)b or the product of a coherent state with a number state ]a)a]N)b. They paid attention to the mode b and assumed ]al 2 = ]/3]2 or ]a] 2 = N initially. They found that as the counting interval increases, bimodal quasidistributions evolve for the mode b starting from an initial number state. The phase distribution (Pegg and Barnett [1988, 1989a,b]) first narrows and then bifurcates. The state of the separate mode described by the statistical operator /~(b) IM = Tra {/51M} (7.1 0) is mixed, and a numerical calculation has shown that /~(b)

1

IM = 5(IA+)(A+[ +

A_)(A_ ),

(7.11)

where IA+) are states which have bimodal (/)A quasidistribution. Ban [ 1995] investigated photon statistics and quadrature squeezing in the even and odd coherent states (Klauder and Skagerstam [ 1985]) and the Yurke-Stoler state (Yurke and Stoler [ 1986]) under the influence of a continuous measurement of photon number. He found that the photon statistics of the cavity mode oscillate between sub-Poissonian and super-Poissonian distributions, and that the quadrature fluctuation oscillates between squeezing and nonsqueezing at each time when one photon of a cavity mode is registered by a counter. He has also treated a 'phase cat' (Schleich, Pernigo and Le Kien [1991]): [~p) = - ~1 ([ae i~c/2) + [ae-'q~/2 )),

(7.12)

where a is a positive real parameter and the normalization constant Z becomes

{1 + exp

cos, o 2 sin

},

to show the statistical oscillation in the continuous measurement of photon number. These oscillatory behaviors have been related to the oscillation in the

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185

photon-number distribution, (cf. Ogawa, Ueda and Imoto [1991 b]). A possibility of taking into account an imperfection of the photon counting is provided by the hybrid model of photon-counting measurement proposed by Lee [1994]. In this model, there are some photons taken away from the cavity without being registered by the counter. When we use the hybrid model, the results are to be modified. Cirac, Gardiner, Naraschewski and Zoller [1996] used continuous measurement theory to describe the evolution of two Bose condensates in an interference experiment. They have shown how the system evolves into a state with a fixed relative phase. The quasicontinuous schemes of photodetection should be completed by a connection to the generation of special states. Ban [1997b] derived the equivalence relation between a lossless beam splitter and a nondegenerate parametric amplifier with respect to the conditional output state. He found that, passing from the beam splitter to the nondegenerate parametric amplifier, the roles of the input state and the postmeasurement state of the reference mode are exchanged. In more detail, we assume that the input state is I~p)~,~,(~p[ and the vacuum state is found in the output port of the reference mode. The equivalent input state of the reference mode of the nondegenerate parametric amplifier is the vacuum state, and the equivalent postmeasurement state of this mode is with oo

I~p*)b = Z(h(nl~P)b)* [n)h.

(7.14)

n=O

Photon-counting measurement, heterodyne detection, and homodyne detection are considered as the conditional quantum measurements. Although a mismatch factor e -i'~-~2f in the effective interaction Hamiltonian (6.34), with g replaced by ge -i'~21, leads to the inhibition of emission, it cancels with its complex conjugate in the limit of continuous measurement with O(t) = e-i'~th t. It indicates an enhancement of the emission in the scheme analyzed by Luis and Sfinchez-Soto [1998a]. This scheme comprises an instantaneous, yet destructive, measurement.

w 8. Production of Correlated Photons We proceed with correlated photons in nonlinear optical processes. The correlated photons are produced in a variety of optical devices such as parametric

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amplifiers, four-wave mixers, and down-converters. The photons produced in a nonlinear optical process such as four-wave mixing or parametric downconversion are known to have unusual correlation properties which result in many nonclassical aspects of the radiation field. Agarwal [ 1990] noted that the amount of correlation depends on the nature of the nonlinear process, the strength of the pumping field, the losses in the medium, and the transmission from the mirror if nonlinear processes in cavities are considered. The correlated photons have been used in a number of studies on nonclassical aspects of light, including questions like the Bell inequalities. Correlation between the signal and readout modes is present in the optical back-action evasion (La Porta, Slusher and Yurke [1989], Song, Caves and Yurke [1990]) using a single-pass parametric amplifier (parametric down-conversion process). Smith, Collett and Walls [1993] have considered quantum nondemolition measurements with a degenerate parametric amplifier driven by a classical pump field, which is able to provide good quantum nondemolition correlations. A beam splitter is one of the key optical devices in quantum optical experiments and quantum communication systems, which is mathematically equivalent to the Mach-Zehnder interferometer and linear directional coupler. The input-output relationships of these devices are characterized conveniently in the language of the SU(2) Lie group when dissipation can be ignored (Yurke, McCall and Klauder [1986], Prasad, Scully and Martienssen [1987], Ou, Hong and Mandel [1987], Fearn and Loudon [1987], Huttner and Ben-Aryeh [1988b], Campos, Saleh and Teich [1989], Janszky, Sibilia and Bertolotti [1991], Lai, Bu~ek and Knight [1991], Leonhardt [1993], Luis and Sfinchez-Soto [1995]). In optical signal detections such as the homodyne and heterodyne detections, a beam splitter is used to mix a signal mode with a local oscillator mode. It is also indispensable for interferometric experiments. In some cases the signal mode is divided into two parts by the beam splitter, and the two output modes of the beam splitter are correlated to each other (Ban [ 1996a]). Some authors speak of an 'anticorrelation' since an increase of the photon number in one output mode is equivalent to a decrease of the photon number in the other. Ban [1996b] has explained the model of a degenerate four-wave mixer. The photon number of the signal mode is conserved in this model, and only the phase of this mode changes. The rate of phase shift is given by the positionlike quadrature of the reference mode. The momentum-like quadrature of the reference mode shifts and the rate of quantum shift is given by the photon number of the signal mode. The initial state of the reference mode is vacuum so that the change of the phase of the signal mode is slow at the beginning. The process produces correlation between the output modes.

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187

Joobeur, Saleh and Teich [1994] developed a theory for the second- and fourthorder spatiotemporal coherence properties of spontaneous parametrically downconverted light, assuming a crystal of finite length and a pump of finite spectral width. Joobeur, Saleh, Larchuk and Teich [1996] extended this theory to include the effect of pump beam waist (or equivalently, of the pump transverse width). Kono, Koashi, Hirano and Matsuoka [1996] performed correlation measurements of the outputs of a nondegenerate optical parametric amplifier with a weak coherent input field. By varying the relative phase between the input and pump fields, they observed positive and negative correlations between the signal and idler photons. The cross-correlation function between the signal and idler photons is g~2) = (as,

ailb~bsb~bilas, ai>

(8.~)


ai = ]al exp(iri),

g = -ir exp(i2r/)

(8.2)

and adopting the model Hamiltonian in eq. (6.34), we obtain in the Heisenberg picture that 4]al 2 sinh(2r)[sinh(2r) - cosh(2r) cos Z] + sinh2(2r) g~2)= 1+ 4{l~176 c~ X] + sinh2 r} 2 ,

(8.3)

where the relative phase between the pump and the signal-idler product of complex amplitudes is 2' = rs + r i - 2r/.

(8.4)

The result of Kono, Koashi, Hirano and Matsuoka [1996] agrees with those of Caves, Zhu, Milburn and Schleich [1991] and Selvadoray, Kumar and Simon [1994], where joint photon-number distributions have been treated. Cascaded down-conversion processes are of great interest. Luis and Pefina [1996] have started with the investigation of entanglement of three modes due to two parametric down-conversion crystals with aligned idler beams. Considering partial correlation of the idler beams, they addressed the role of entanglement of four modes, l~ehfi6ek and Pefina [1996] studied a correlation between two

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signal output photon numbers, as well as a correlation between two signal output quantum phases. Schiller, Bruckmeier, Schalke, Schneider and Mlynek [1996] realized the scheme of Smith, Collett and Walls [1993] for continuous-wave quantum nondemolition detection of optical amplitude quadratures. A central property of the optical parametric amplifier is the correlation of the output waves. They determined the correlation of the photocurrent fluctuations experimentally. Since the input wave is verified to not carry significant excess noise, the measured optical correlation is essentially a quantum correlation. The two output waves are entangled. The output waves represent individually squeezed twin beams. Casado, Marshall and Santos [ 1997] developed the theory of parametric downconversion to the point of calculating the autocorrelations of the signal and idler beams, and Casado, Fem~indez-Rueda, Marshall, Risco-Delgado and Santos [ 1997] have calculated the auto-correlation and cross-correlations of the signal and idler beams. Torgerson and Mandel [1997] have shown that the downconverted light exhibits photon antibunching. Levenson, Bencheikh, Lovering, Vidakovic and Simonneau [ 1997] have shown that quantum optics enables noiseless amplification and distribution of optical signals via optical parametric amplification. They focused on the traveling-wave configuration (i.e., without a cavity). They demonstrated that two outputs of the amplifier (signal and idler) are correlated far better than the shot noise. They used twinned amplified outputs as an amplifying quantum optical tap for the distribution of information encoded in an optical beam. Presenting an idea to synthesize Fock states for pulsed light beams, Steuernagel [ 1997] used and studied an array of beam splitters. Vedral, Plenio, Rippin and Knight [1997] presented conditions which every measure of entanglement must satisfy, and have shown that there is a whole class of 'good' entanglement measures which generalize the von Neumann mutual information. The central idea of the construction is that we calculate the distance from a given state to the manifold of all possible disentangled states. An extreme approach to quantifying entanglement would present a reconstruction of the state of a two-mode quantum field. The measurement schemes, however, rely on some simplifying assumptions, which by now limit this application; see, e.g., the paper by Walser [1997]. Bruckmeier, Schneider, Schiller and Mlynek [ 1997] have shown that improved quantum nondemolition measurements are realized by injecting 10o.34 times amplitude-squeezed light into the meter input port of the degenerate optical parametric amplifier. This achieves increased squeezing and correlation of the output signal and meter waves.

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189

Bruckmeier, Hansen and Schiller [1997] have demonstrated three-beam entanglement of individually squeezed output beams. They achieved it by combining a degenerate optical parametric amplifier and a squeezed-light beam splitter. Breitenbach, Illuminati, Schiller and Mlynek [1998] obtained a 'spectrum' of squeezed quantum states of the light field emitted by a continuouswave degenerate optical parametric amplifier. The squeezing may be associated with a particular frequency, although it results from a coupling of a pair of frequencies on either side of resonance (Gardiner [1991]).

w 9. Conditional Generation of Special States Using Ideal State Reduction of Entangled Fields The prospect of signal-to-noise improvement in many optical systems led to suggestions for generating sub-Poissonian light via measurement feedback (Imoto, Haus and Yamamoto [1985], Saleh and Teich [1985]), Jakeman and Walker [1985], Yamamoto, Imoto and Machida [1986], Hong and Mandel [1986]). Even if the quantum detection efficiency could never be brought close to unity, one could still realize the full potential of a number state by optical preamplification via a 'noiseless photon amplifier' (Yuen [1986b]). Yuen [ 1986a] considered the parametric interaction described by the Hamiltonian f/int =

+

^ ^t a aia ),

:~^

(9.1)

where hs, hi, and ap are the annihilation operators for the signal, idler, and pump modes, respectively. From eq. (9.1), the quantum Manley-Rowe relation

,~s(t)- ,~(o) = hi(t)- ~(o)

(9.2)

follows easily. It is clear from eq. (9.2) or, immediately, from eq. (9.1), that each time the pump photon creates an idler photon, it must also create a signal photon. On assuming a spontaneous process, it is obvious that after one counts n photons on the idler mode, the corresponding signal mode is in the state In). One may thus suggest the generation of In) on the signal mode by stopping the signal after n counts on the idler. Yuen [1986a] proposed the use of a parametric amplifier or oscillator, and corresponding four-wave mixer configurations. However, the fields need to be coupled out of the cavity through a mirror with reflectance 7~ > 0. This imposes a limitation on the achievable Fano factor similar to that due to the

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imperfect photodetection. He characterized, presenting the mean and variance of the number sum, the states (the joint state) of M independent signal modes conditioned on a total number of counts over M independent idler modes. He assumed that pump quantization and depletion can be neglected and that all of the down-converted modes have the same real coupling constant, g = tC(hp). Yuen [ 1986a] presented the proposal in more detail. An optical shutter is used to turn off the signal output after photodetection on the idler. Since the response time of the optical-shutter should be taken into account, he has contemplated to feed back the idler counts for them to vary the pump intensity according to a suitable control law. A great advantage of near-number-state beams compared with squeezed light may be that no phase sensitivity is required in the system. While one added noise photon destroys squeezing completely, it would not affect such a beam above the tolerance level. This is another key point (Yuen [1986a]). Yuen provides more detail on a demonstration experiment and the Fano factor for the only feedback on the shutter. 9.1. CORRELATEDPHOTONS One may expect that near-number state beams and low-loss fibers contribute an application in communication and data processing systems. Furthermore, nonclassical states lead to significant improvement in precision interferometry. According to Bondurant and Shapiro [1984], a correlated photon-number state for the input modes can be generated from two independent number-state beams via spatial interference. Nonlinear optical processes such as four-wave mixing and parametric down-conversion can possess interesting correlation properties. In several papers (Watanabe and Yamamoto [1988], Agarwal [1990], Luke, Pe~inov/l and I~epelka [ 1994]), the correlated photons are studied as correlated or entangled modes. It is well known that the state reduction carried out on the second mode has no effect on the first mode when the two modes are disentangled. Generally, after measurement on the second mode, the state of the first mode is changed. In the case of the parametric oscillator (down-conversion), the measurement of the photon number on the idler mode has been studied (Agarwal [1990]) and the statistics of the result in the signal mode has been described by the quasidistribution related to the antinormal ordering of the signal mode operators. Agarwal [1990] considered a two-mode radiation field consisting of modes 1 and 2. The applicability of the projection postulate for a description of the measurement in the situation under consideration does not depend on whether one measures 'on both modes' or perhaps on mode 2 only. It still holds that if

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191

the observable A has a continuous spectrum, we have no projection postulate for the measurement of A (Ozawa [ 1997]); cf. w3. In spite of this, a generalization of the statistical formula can be used, which has a form reminiscent of the projection postulate. To illustrate such a generalization, we will use the example of quadrature measurement on mode 2, A = ~l(h2 + a^t2). We will begin with the case of photon-number measurement on mode 2, A = h2. The number state in the product Hilbert space will be denoted as ]ml,m2) = ]ml)l | [m2)2, and the statistical operator/5 describing the state of the optical system under investigation can be expanded as oo

P= Z

oc

Z

2(n2~lm2)2|

'

(9.3)

m 2 = 0 n2 = 0

where ]m2)2, 1n2)2 are the number states of the second mode on which the measurement is carried out. The single-mode scalar product (projection) has the property

2(n2lml,m2) = 6n2,m2lml),

(9.4)

where Iml) is the number state of the signal mode. As a consequence of the measurement of the operator h2, the state/5 is reduced to a state oo

/5' = Z

2(M ~IM)2 | M)2 x(M

(9.5)

X(1) Vl M | p(M) ]M)2 2(MI,

(9.6)

M=O o~ ~- Z M=O

where the reduced state, on the result of the measurement being known, reads

IM

p(Z)(M ) 2(M/5 M)2,

(9.7)

with P(2)(M) = 2 ( M I T r l { f)} m ) 2 .

(9.8)

Thus, the statistical mixture (9.6) is the reduced state on the result of the measurement not being known.

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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In the case of quadrature measurement (that is, of homodyne detection, - ~(h21 -k- h~)), we have no projection postulate, and analogues of relations (9.5) and (9.6) do not exist. We still realize a reduced state, f)(1)hom

]Re a = x

_

1

p(2)(x ) 2(Re a = x[/51Re a = x)2,

(9.9)

where P(2)(x) = 2(Rea = x Trl {f~}]Re a = x)2.

(9.10)

In the case of amplitude measurement (i.e., of double homodyne or heterodyne detection), the reduced state reads ~(1)het =

I~

1 jrq,(2)(~) 2(~1F31~)2,

(9.1 l)

where 1

~(2)(~) = ~ 2(~[Tr, {t3}[~52.

(9.12)

A generalization exists for a quantum measurement on mode 2 resulting in the random variable A when the reduced state is of the form ^(z)

1

Pin = Prob(A E g2)Tr2{/5/I2(g2)}'

(9.13)

where Prob(A E g2)= Tr {/5/I2(g2)},

(9.14)

with [/2(g2) an arbitrary mode-2 probability operator-valued measure of a measurable subset g2 of the real axis. Particularly, we obtain the relations (9.7) and (9.8) using [/2(M) = ]M)2 2(M I

(9.15)

and, accepting operator-valued densities, we get the relations (9.9), (9.10) and (9.11), (9.12) using /jh~ f/het(o0

=

IRe a = x)2 2 (Re a = x], 1

= ~ [0~)2 2 (a].

(9.16) (9.17)

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193

Agarwal [ 1990] considered a two-mode field characterized by a q~A quasidistribution related to the normal ordering of the statistical operator/5 of the form

q'A(a,,

1

=

( a , , a li (*lla,, a2),

1

{

(9.18)

2

~- 2

~2 x / ~ exp [ - Z EiA ai - gi j= 1

(9.19)

- [F,2A (a~ - ~ ) ( a ~ - ~ ) + c.c.] },

where LA = (81.482.,4 -1C1212) 2, E~A E2A -

B2A

B~A

F12A -

'

(9.20)

'

C12 v/LA "

The parameters ~., ByA, C12 are related to the mean values, ~=(hj),

j = 1,2,

BjA = Bja; +1 = (hJ hj ) + 1 -

~j :,

j-

1,2,

(9.21)

C12 = (all~2) -- ~1 ~:2.

The Wigner function cI)8(al, a2) of such a field is given by eq. (9.19); however, the parameters Bj8 should be substituted, which have a slightly different definition: Bjs = (h~hj)+ 51

]~jl 2 , J = 1,2.

(9.22)

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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(a) For the amplitude measurement, the operator-valued density (9.17) has the representation 6 c ( a 2 - a2), where 6c(a) = 6(Rea)6(Im a), and the unnormalized operator Tr2{/5//2het(~)} has the q~.a quasidistribution / ~5C((Y2-- ~ 2 ) ~A(al, a2) d2a2 = q~A(al, ~2)-

(9.23)

The joint probability density of the two quadratures is /. q~(2)(~2) =

/ OA(Ol'a2)d2~

(9.24)

-

The normalized state is described by the q~A quasidistribution q~A(a, a2)

het

q~A (ala2)=

(9.25)

(j5(2)(~2)

It can be seen that q~)(a2)

=

exp(

1

~B2A

~2 -- ~212) ,

(9.26)

92,4

and 1

qbhet(a .A 1~2)

exp

YgB1.A

(

--

lal-~12 ~I.A

)

(9.27) '

where -

-B1A -

XB 2/A~

'

-

-~1 = ~l-4-

C12

B2A ( a 2 -

~2)*

(9.28)

(b) For the photon-number measurement, the projector (9.15) has the representation x

M!

exp(la2l 2)

0 TM

(9.29)

Oot~tOa~M6c(a2),

and the unnormalized operator Tr2{/3/I2(M)} has the q~.a quasidistribution

f

,7"g

0 TM

M.I exp(la212) 0a~t Oot~M6C(a2) qDA(al, 0~2)d20~2 Jr 02M - M! 0a~t0a~ M [exp(]a212)q~A(al' a2)]

(9.30) a~ = 0

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195

The probabilities are given similarly, P(2)(M) = ~r

f

0TM M [exp(]a2]2)q:'A(al, a2)] OaMOaf

d2oq 02 = 0

_

)]

2

0 TM

,7"g

(t2 =0

(9.31)

where C)((22)

f J ~A(al, a2) d 2at.

(9.32)

The normalized state is described by the q).a quasidistribution a TM

2)

~. oCo,U [exp(la2l q~A(al, a2)] q'.a(a~ IM) = p(2)(M )

] a2 = 0

(9.33)

According to Mollow and Glauber [ 1967], p(2)(M) = B~4Ac _~2]2 Iz B ~ exp (___)LOM ( - 1_~2 ) B2A B2A/'B2A

(9.34)

where the Laguerre polynomials M (-zy L~(z) = F ( M + v + 1) j~o= j ! ( M - j)! F(j + v + 1)'

whereas, at least for ~ 3"g

(9.35)

> B l.a,

0 TM

M! OaYOafM [exp( a2l 2) (~)A((Yl, (22) ]

( ) ( /~2A

where

(o

,

q~)(al) - ~B1A exp -

Bi.a

ct~ = 0

/~2A/'/~2A

~

C12

B1A

-- ~1 -

'

' (9.37)

B1A ~2 = ~2 + ~ ( 0 ~ 1

)

(9.36)

)*

=

C12 B1A

(al

+

-~,

)* '

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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

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with

( c;~ )-' ~* =-~1 + k B--~A ~,

(9.38)

so that

(B2HB2A)Mexp (

1

~BIA--

B2AB2.A/"

-]a 1)__ y 2

BIA

zO(

'-~2'2 1 [~2A/'B2A

L~ ( BE.A/B' EAI-~2]2 '

(9.39) where

C12 Y -- ~1 -- -'~-----~ . /~2A--

(9.40)

(c) For the quadrature measurement, the operator-valued density (9.16) has the representation 6(Re a 2 - x), but the Wigner function is used. The unnormalized operator Tr2{/3[/2h~ has the Wigner function 6(Re 02 - x) ~s(Otl, a2) d2 a2 =

cI)s(O~l,x+iy)dy.

(9.41)

~x5

The probability density of the position-like quadrature is p(2)(x ) =

l~S(al,X + iy) dy d20cl

0<3 =

o~

(9.42)

q~)(x + iy)dy,

where

q)~)(a2) = / qlSS(al, Or2)d2al .

(9.43)

The normalized state is described by the Wigner function

~(1)hom

_ L~:~ CI)s(OgI,X + iy) dy

(a, Ix) -

p~2)(x )

(9.44)

It can be seen that p(2)(x ) =

1

((x-Re~2) 2)

V/YgB2S exp -

B2s

(9.45)

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CONDITIONAL GENERATION OF SPECIAL STATES

197

and qg~)h~

1

1 X)=

YCC K l s

..-

x exp - ~ l s

_

-

(al--~l

])

+C.C.

,

(9.46) where --2

K l s = B1s - C1 ]2,

-

B1s = B l s -C1-

ICl2l 2 ~ , 2B2s

--2 Cl2

(9.47)

2B28'

C12 (x - Re _~2). ~1 = ~1 -t- 2 B l s

--

Lukg, Pefinovfi and K~epelka [ 1994] have illustrated the role of the correlation (entanglement), which leads to a transfer of properties of the state resulting in the idler mode to the signal mode. Actually, their attention to the near-number states was attracted by the work of Hradil [ 1991 ], in which a connection to the numberquadrature minimum uncertainty states was emphasized. A thorough study of the statistical properties of these states has been provided (Pefinovfi, Lukg and K~epelka [ 1994]). The contour diagrams of the q~A quasidistributions are shown to have crescent shapes. These minimum uncertainty states after displacement and rotation in the signal mode can be described equivalently as eigenstates of an operator, depending on the number operator. In w 11 we will derive that the joint definition of the initial coherent state by two eigenvalue equations for the signal and idler modes can be reduced to the signal-mode eigenvalue problem. The simple definition of number-quadrature minimum uncertainty states and the easy generalization enable us to express not only the ~ t quasidistribution, but also the Wigner function and principal quadrature distributions. The increasing correlation between the signal and idler modes causes the crescent shape of the contours of the q~A quasidistribution and a trend to the near-number state. Let us consider two-mode parametric down-conversion without losses described by the interaction Hamiltonian

I2/int = h ( g a^t^t l a 2 + g * Cll a2),

(9.48)

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CONTINUOUSMEASUREMENTSIN QUANTUMOPT|CS

[III, w 9

where hi, a2 (tl~, a[) are the annihilation (creation) operators of a photon in the signal and idler mode, respectively. The solution of the Heisenberg equations of motion of the field operators can be written ig

h,(t) = cosh( glt)h,(O)- ~ sinh(lglt)h~(O), (9.49) a2(t) = - Tig ~ sinh(g]t)h~(O) + cosh(lg t) h2(0). For an initial coherent state 1~1(0), ~2(0)), with the amplitudes ~l(O), ~2(0), the expectation values evolve as follows: ig (&~) -- ~ (t) = cosh(lglt) _~1(0) -- ~-~ sinh(lglt) ff~(0), (9.50)

ig (h2) - ~e(t) = -]~-~ sinh(]glt) if?(0) + cosh([glt) ~2(0), and the second-order moments are of the form (AhjAhJ) -- BjA(t) = cosh2( g t),

j = 1,2,

((Ahj) 2) -- Cj(t)= 0, j = 1,2, (9.51)

(Ah,Ah~) - B,2(t)= 0, ig

(AhlAh2) -- C12(t) = - ~ - ~ sinh( g t)cosh(Ig]t ). The dynamics of this optical system evolving from the coherent state is described by the antinormal characteristic function

CA(fll,fi2, t) = exp

[-BjA(t) Ifljl2 + (9.52)

+ (-B12(t) fi~fl2 + C~2(/) fil]~ + c.c.)

+

(g,

- c.c.)

j= 1

Performing the Fourier transformation

,/ CA(fil,[32, t)exp

(aj~.* - aft.)

q~.a(al, a2,t) - ~ -

j--1

}

d2fil d2/32, (9.53)

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199

we obtain the quasidistribution q)A(0q, 0t2, t) = ~1 (al , a2 ]~5(N) (t)la~,

a2),

(9.54)

related to the normal ordering of the statistical operator/5(t), of the form 1 qlSA(al, a2, t) = N2 X / ~

exp

{

- Z

Ej(t)1o(.i - ~.(t) 2

j=l

- [F~z(t) (a~ - ~ (t)) (a] - ~ (t)) + c.c.] }, )

(9.55) where

L(t) = cosh4(]g]t), Ej(t)= l, j - 1 , 2 ,

(9.56)

ig Flz(t) = ~ tanh(g]t).

Complete formulas for the function (/).a(al, a2, t) were given by Pefinovfi and Pefina [1981]. The expectation values and the antinormally ordered moments are given by an integral, <~k~tl p,mp, tn '*1'~1 '*2"2 )(t)=

/ d

0t~~1.,1 (x2 .m_,n (*2 qbA(al, (Y2,t) d 2 0tl d20t2.

(9.57)

In the operator notation, the produced state is related to a state vector ]~p(t)), /5(t) = [~p(t)) (~p(t)],

(9.58)

where ]~p(t)) =/)1 ( ~ l ( t ) , - ~ ( t ) ) b 2 (~2(t),-~(t))]lpA(t)),

(9.59)

with the displacement operators

b~ (+
~*<,)~,),

j = 1,2,

(9.60)

and

i~+> - 1 ~ exp(-~,2+ ~ ) Io,o>

<961)

The centered state vector ]~PA(t)) is a two-mode squeezed vacuum resulting from the spontaneous down-conversion described by the condition ~1(0) = ~2(0) = 0.

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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Properties of entangled states can be derived from their Schmidt decompositions (Schmidt [1907], Riesz and Sz6kefalvi-Nagy [1955]). The states (9.59) are appropriate for the study of entanglement, because the pertinent Schmidt decompositions can be easily expressed in terms of the displaced number states. An entanglement leads to an increase of marginal entropies of the separate modes. These entropies are the same in both modes, and can be used as a measure of the degree of the strength of quantum correlation between the modes (Barnett and Phoenix [1989], Bu~ek and Drobn~, [1993]). The von Neumann entropies (Luk~ and Pefinovfi [1989]) for eq. (9.59) are expressed as

Sj = BjA(t) ln[BjA(t)] - BjA;(t) ln[BjA;(t)] = 2 cosh2(lglt) ln[cosh(lglt)] - 2 sinh2(lg t)ln[sinh(lglt )]

(9.62)

= ln[lGz(t) ] - c o s h ( Z l g l t ) ln[IFl2(t ) ], with the second-order central moment

BjAc(t) = B j A ( t ) - 1;

(9.63)

the subscript .A/"indicates the normal ordering of the field operators. They diverge to infinity as t ~ ~ . The appropriate information measure is I12 = Sl + $2 = 2S1 = 2S2.

(9.64)

For a normalized measure of quantum correlation, the functional introduced by Araki and Lieb [ 1970] can serve; i.e., o<3

'-'l~'c~ = '-'2~c~ = 1 - Z

c"14 '

(9.65)

n=0

where cn occur in the Schmidt decomposition, Ic,,] being the so-called singular values. Here the squared singular values ]c,

12 =

~n

(n + ly+l ,

n = B~A;(t) = B2A;(t).

(9.66)

With the aid of eqs. (9.51) and (9.59), we express eq. (9.65) as follows: 2~ 1 ,vcorr _ - 1~'J 2~ + 1 cosh(21g}t )'

j = 1 2. '

(9.67)

A linearization of the information measure reads as corr = 2s~orr __ 2s~orr 12

(9.68)

The entanglement of the two modes can also be measured by the group correlation coefficient Pal,a2 of the column vectors (al, a~) r, (a2, a~) r, where

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CONDITIONALGENERATIONOF SPECIALSTATES

201

the expectation values of the symmetrically ordered field operators are used (Pefinovfi, Lukg, I~epelka, Sibilia and Bertolotti [ 1991 ]), p2

: 1'~=

Here I

IWl IVllllV221 "

(9.69)

I denotes the determinant, and

V JJ =

(Bjs cj ) Cj* B j s

'

(9.70)

j:l,2,

Bls C1 B12 C12 W :

C~' B1S C~2 B~2 B~2 C12 B 2 s C2

(9.71) "

C~2 B12 C~ B2S In eqs. (9.69)-(9.71), the time dependence is omitted for the sake of simplicity. The coefficients B j s read 8is - &s(t)= &A(t)

1

(9.72)

2.

Substituting from eq. (9.51), we obtain /921 ,a2 __ pa,,a~ 2 (t) = 1 -

(9.73)

cosh4(2lglt)"

This means that the quantum correlation increases from zero at t = 0 to unity in the large time limit, and this maximum can be attained on the pump not being depleted. lcorr, and p~, ,a2 depend only on Remarkably, the measures of correlation I12, ~12 the centered state vector [~PA(t)). In other words, these measures do not depend on the expectation values ~l(t), ~2(t) for the process under study. Of course, a similar independence of the initial conditions emerges in the limit t ---, oc, as seen from the asymptotic formula

t ~ ~ x2 X / ~

exp

-

lajl 2 _

ig a~(a~ + c.c.

j= 1

(9.74) The derivation of this formula is based on the assumption that the intensity of the is bounded. A similar asymptotic approximation can also

field

lajl2

202

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

be obtained on the condition that the photon number n~ + n2 cannot exceed 2s, and we have

1

1 ig~'6,,,:

(n,,nelW(t)) t__+% ~

-Igl,/

'

'

nl, n2 ~< s.

(9.75)

On a renormalization, we can consider the asymptotic equality [lP(t))t --+o~ ~ [~PA(t)) t +,.~_ ~ s(t) 1+ l

_ -~1 ig

In, n),

(9.76)

where s(t) is any sequence of positive integers diverging to infinity as t ---+ oo. Expressing ~.a(al, a2,t) in terms of the basis --

~A(a2;

In2)(m2[) =

1

1

~ 2{02[n2)2 2(m2[a2)2 -

1

(~n2 .*n2

Jr v/m2 !n2!

"2 ,

(9.77)

m2, n2 = 0 , . . . , oo, and respecting the fact that

p(M,t) =

(sinh2(Iglt))M ( I-~2(t)12 (cosh2(lglt))M+ 1 exp -cosh2(lglt ) (

• L~

l~2(t) 2 ) cosh2(Iglt) sinh2(glt) '

(9.78)

we arrive at the quasidistribution conditioned on the result of measurement,

q~A(Otl, tlM, t ) -

1 1 1 (tanh2(lg t))g la 1 + ~,(t)12M p(M, t) Jr cosh2(lglt) M! • exp { - ] a l + y* (012 + [al + y* (t)]*[y(t) + -y* (t)]

+ [al + ~*(t)][y(t) + ~*(t)]*

(9.79)

-IY(O + -Y*(t)]2 cosh2( glt)} 1 1 [LO(_ly(t) +-~*(t)[2)]-I ]al + y*(t)l TM arM! • e x p [ - l a l - y(t)]2] , where (cf. eqs. 9.40 and 9.38):

ig tanh( g]t)~(t), Y(O = ~l(t) + -~] (9.80) y*(t) = -~l(t)

_

ig __

-

-

1

Igl tanh(lglt)

~,2 ( t ) .

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203

Noticing the inverse relation to eq. (9.50), we obtain simply that 1

y - y(t)=

~l(0), cosh(Iglt) ig 1 y* -= -~*(t)--~(0). Ig] sinh(]g]t)

(9.81)

Adapting ourselves to the single-mode situation, we omit the subscript 1, a=_al. Independently of the time evolution, the formula (9.79) for the quasidistribution q~A can be recovered in the form

9 A(a) = r

[LO( - Y + y, [2)]-' [a + y* 2M exp (- a - 1'[2) .

(9.82)

We introduce the phase-dependent Hermitian quadrature operators Q(r) = exp(-ir) h + exp(ir) h t, /~(r) = -i [exp(-ir) h - exp(ir) ht],

(9.83)

and put y+y* exp(ir) =

ly+?*l

for

y + ?* ~ 0.

(9.84)

For y + ~* = 0, the study applies to the displaced number state. The general quasidistribution (9.82) can be reduced to any of the following three normal (canonical) forms (a)

1 [L0(--4/']2)]ll ~Ao(a)= ~M! la + r/olTM exp ( - [ a - r / o 2 ) ,

(9.85)

= @A(eira + ~'), with (b)

1 y* r/o : ~[)'+ I,

@Ao(a)=

~,

1 = ~(Y--Y*);

1 ~M!

(9.86)

'

10t[TM exp ( - [ a - y o [ 2 ) ,

(9.87)

- ~A(eira - ~*), with (c) with

Yo = [Y+Y*[;

[

1 LO(_~ ) qSAo(a)= ~M! = q~a(ei~a + y), Yo = lY +7.1.

]'

(9.88) la + -~olTM exp

(-lal 2) '

(9.89) (9.90)

A study of the above specializations of parameters is very useful for gaining insight into the shape of the quasidistribution ~A(a) in the general case. Indeed,

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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we derive for M > 0 that the three-dimensional graph of the quasidistribution has the typical forms, which are, due to the contour diagrams, called crescents. Let us suppose that 70, Y0 are real numbers such that -7o < Yo. It can be easily shown that the quasidistribution (9.82) with 7, 7 substituted by Yo, 70, respectively, has a local maximum at a = x~, a saddle point at a - x2, and a local minimum at a =-Y0. Here,

x l,2 -

2

2

+ M.

(9.91)

It can also be proved that qlSA0(Xl) > qDA0(X2) ,

(9.92)

where (9.93)

q ' A o ( a ) = q ' A ( a ) l ; , ~ :,o ,:~ :~,,,

or the value at the local maximum is greater than that at the saddle point. It holds that (9.94)

X2 <--'Y0 < Y 0 - ~0 < ]tO < Xl.

2

Moreover, q{a0(a*) = q{a0(a); i.e., the graph of the quasidistribution is symmetric with respect to the plane perpendicular to the imaginary axis at the origin. For all that, we can see that the graph has the shape of an open kettle (cup), with the opening at the saddle point x2, the bottom at the point -70 and with the edge at the maximum xl. Of course, the favorable case of the open kettle can be compared with cases of near-coherent states, where the probability of measurement of values corresponding to the minimum and the saddle point is negligible, and accordingly these notions are practically undefined. An easy generalization provides the same general picture of the quasidistribution 4'A(a) with the minimum at a = - ~ * , the maximum at a = al, and the saddle point at a = a2, with a l , 2 = Y + [y+_~,[

_ l iv+

I+

4

+M

.

(9.95)

In this general case, the symmetry relates to the plane passing through the points -~*, y. Again, the notions of the minimum and the saddle point should not be defined when connected with improbable quadrature measurement.

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205

Although the Wigner function cannot be a proper probability function, since it takes negative values for all non-Gaussian pure states, it has been pointed out recently (Leonhardt and Paul [1993]) that the smoothed Wigner function can actually be measured in quantum optics. We use the fact that the symetrically (Weyl) ordered operator y1 In)(m] after the substitution h ~ a, h t --+ a* maps into the function (-1)" ~,,'.,',.~,,-,,.m-,,IA

-

V ~~J

"~"

~-,lal

2) 2 exp(_2l a 2) for m ~> n,

cbs(a, ln) (ml) =

[q~s(a, In)(nl)]*

for m < n. (9.96)

In case (c), the eigenstate is given by the finite expansion M

Po

=

M

Z

(9.97)

Z c,c,*, n)(m],

n=0 m=0

where c, - c,,(-~o,M) are expressed as follows:

Cn --

I

1 O" q72 VLM(-'~

M v (7o)u-'' (M-n)!

0

f o r n ~<

M,

(9.98)

for n > M.

We obtain the Wigner function M

9 so(a) = Z

M

Z c,c~,-~s(a, In)(ml).

(9.99)

n=0 m=0

The general form of the Wigner function is

9 s(ot) = qbso(e-ir(a- y)) M

=Z

M

Z

exp(inr)c,[exp(imr) c,,,]* ~s(a - y, n)(ml).

(9.100)

n=0 m=0

Here exp(ir) is given in eq. (9.84). It is well known that the distributions of the generalized position and momentum operators are given as marginal integrals of the Wigner function (Beck, Smithey and Raymer [1993]), and the distributions of quadrature operators Q(0),/3(0) are linked to them by a simple rescaling. Nevertheless, Luke, Pefinovfi and I~epelka [1994] derived the formulas for the distributions

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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of these quadratures using the technique to be described in w11, with the choice r = 0, r/ > 0 (which corresponds to )' and 7 real and such that-7 < Y). The distribution of quadrature operator Q(0) is

~Q(Q)= (QIPIQ)=I(Q W) I2 , where I~p)= Y'~'~n~oCnln).Here, IQ) have the properties

(9.101)

Q(O)IQ) = QIQ),

(9.102)

(Q'IQ)=6(Q'-Q).

(9.103)

and

Using the formula

(Q]M)=HM(---~2) [2MMI!x/~exp (-Q-~2) ] ~/2

(9.104)

with the Hermite polynomials [~]

gM(x) = ~

M~

k ! ( M - 2k)! (--llk(2x)M-2k'

k=O

(9.105)

M forM even, M5---2for M odd, we derive the quadrature representation (Q]~p): (Q]V) = exp(-r/2) [L~

•

-1/2 exp[r/(Q - 2~)]

~

x/2 J [2MM 1,-x( Q / ~-e2x~p)(2 .

2

)] 1/2 (9.106)

and q~Q(Q) = exp(-2r/2) [L~ •

x/~] -1 exp[2r/(Q - 2~)]

(Q-2-~)21 [HM(Q-2-~2 ~

..)] ,2

(9.107)

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207

with -

l yl z- lYl z

Ix z + 2 y ~ + 171 z

rt = 21y + Y*I'

rl =

2ly +-Y*I

(9.108)

'

the parameter ~ is real and rl is a complex number generally. Similarly, the distribution of the quadrature operator/5(0) is 05~(P) :

(PIPlP):

(9.109)

I(PIW)I z ,

where [P) are the quadrature eigenstates, P(O)IP) =

PIP),

(9.11 O)

normalized as (P'IP) = 6(P'-

n).

(9.111)

With the aid of the formula p2

(P[M)=(--i,MHM(~2) ( 2MMIv/~_~ exp (--~-) ]

1/2 ,

(9.112,

we get the quadrature representation (PIW)" (P ~p) = exp(-r/2)[L~

-1/2 exp(-i~P)

x (--i)MHM( P +i2r/X/~) [2MM}x/~-~ exp( (P +i2r/)22

)] 1/2

p

(9.113) and q~(P) = [L~

exp (-~-~2) (9.114)

• HM

X/~

Let us note that the distribution (9.114) does not depend on the parameter r/(r/ real), which can be easily related to the same property of the integral expression q~(P) = ~1 / ~

qss ( 2x )+ i

dx.

(9.115)

O(3

A shifting property of the parameter ~ is obvious from eq. (9.107). Purely imaginary displacement from eq. (9.114) has its counterpart in eq. (9.107) as

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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

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an exponential factor exp[2r/(Q- 2~)] amplifying on the right (r/ > 0) and attenuating on the left. In general, the formulas (9.107) and (9.114) can be applied to quadrature components Q(r),/5(r) ((9.83)), which are, as can be seen easily, the principal quadrature operators (LukL Pe?inovfi and Pe?ina [1988]). The condition r = 0 can be reached by using a suitable phase shifter in the local oscillator beam. Nevertheless, r/is complex generally, and we thus use the formula (9.114), with the modification r / + Re r/and with a displacement by 2Im r/. Let us note that the quadrature variances used to measure the amount of squeezing are invariant in displacement. The squeezing of vacuum fluctuations occurs in the principal component with the variance ((AQ(r)) 2) for suitable values of the parameters 7, ~ and each M > 0. No squeezing is possible in the quadrature ~b(r) (Pe?inovfi, Luk~ and K~epelka [1994]). We will investigate the unitary dynamical time evolution followed by the measurement process on the idler mode. For the description of this evolution, three quantities are used and are important; namely, --9*(t), al(t), a2(t), given by the relations (9.81) and (9.95). For chosen initial complex amplitudes and the outcome M of the photon-number measurement, we see the motion of these points in fig. 1. The maximum al(t) moves from ~1(0) over an arc to the point

(9.116)

The saddle point a2(t) moves from infinity (and from low probability) to a2(oo) = - a l (cx~).

(9.117)

The minimum o f - 7 * (t) moves from infinity to the origin over a ray. At an equal time the three points are collinear. Straight-line segments are plotted to express the time dependence of connected points. The line segment linking the points al (t), a2(t) may play the role of the diameter of a three-dimensional plot of the q).a quasidistribution. In the unitary dynamics, the time evolution is reflected in the increasing correlation between the signal and idler modes. Whenever the measurement process closes the evolution, the amount of correlation causes leakage of the properties of the measurement outcome (the resulting Fock state of the idler mode) to the signal mode. However, we assume the Fock state ]M) to be the outcome, so that for suitable times its phase uncertainty manifests itself in

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209

10-

Ins

8 (o) -2-4

I

i

I

'

I

~

~

'

I

-4

Re Fig. 1. The evolution o f the quantities - y * ( t ) , al(t) and a2!t) for t E [10 -2, 10] and the sweep o f the crescent axes for times 9 = exp{ln 10 -2 + ( l n 1 0 - I n 1 0 - 2 ) ~ } , j = 13 . . . . . 20, for ~1(0) = 6, qJ1 = 0, [_~2(0) = 6, g[ = 1, lp = 0, and M = 10.

the signal mode, and the parametric down-conversion can evolve to very strong correlations. Thus, for very large times the signal mode possesses an identical copy of the number state IM). To this extent the photon number of the signal mode is modulated for the purposes of quantum communication. On the contrary, for t = 0, when the unitary dynamics is not present, the measurement process does not affect the coherent state of the signal mode. For suitable times crescent states result and the straight lines represent their axes. In figs. 2-4 we present the quasidistributions q~.a(a, t) describing the reduced state. Figure 2 demonstrates a more-or-less coherent state for t = 0.001. At time t-- 3.5, a weak correlation causes an annulus to arise. An increasing correlation in fig. 3 makes the annulus more pronounced, with a well-defined saddle in the superstructure graph for t = 4. Figure 4 (t = 10) corresponds to a Fock state with undefined saddle point. For t = 0.001, it holds t h a t - ~ * ( t ) ,~ 6000i, a l ( t ) ~ 6.0, and a2(t) ~ 6000i, and the points al(t), a2(t) fall outside the figure, although already for t = 3.5 they are contained in the figure. The appropriate Wigner functions easily demonstrate the time reversal. For t = 10, we almost obtain the familiar picture of the number state with two

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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

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Fig. 3. As fig. 2, but for t = 4.

local maxima presented in fig. 5, one corresponding to the flat m a x i m u m of q ~ t ( a , t ) in fig. 4 and the other situated at the origin. In fig. 6 (t = 4), the local m a x i m u m at the origin becomes weaker and the m a x i m u m preserved in the quasidistribution q~A(a,t) increases. The oscillations connected with the

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CONDITIONALGENERATIONOF SPECIALSTATES

Fig. 5. The quasidistribution ~s(a,t) for I_~r 6, ~1 and t = 10.

= 0,

211

1_~2(0)1= 6, Igl = 1, ~p - 0, M = 10,

occurrence of the negative quasiprobabilities survive persistently, and only for t = 0.001 does the Gaussian form prevail over nonclassical properties (fig. 7). In the following, we will demonstrate the distributions of quadrature operators by applying eqs. (9.107) and (9.114). With the special choice o f initial phases of qg~ = 0, ~p = - ~1 Jr, we achieve that the principal quadratures coincide with the basic ones for all interaction times. Similarly, as in the previous considerations,

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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Fig. 7. As fig. 5, but for t = 0.001. we obtain the result that the variance of the position-like quadrature can be squeezed, whereas that of the momentum-like quadrature cannot. In fig. 8, the squeeze of the position-like quadrature variance manifests itself with the crest followed by branching into oscillations of the near-number states. The Gaussian shape applies for near-coherent states. The oscillations in the Q-quadrature distribution correspond to the bimodality in the P-quadrature distribution and they replace the squeezing when the promontories in the q~s quasidistribution begin to manifest themselves and, eventually, the kettle

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213

Fig. 9. Another view of the Q-quadrature distribution from fig. 8. closes. This bifurcation is analogous to the situation with number and quantum phase in the ordinary squeezed state (Schleich and Wheeler [1987]). From another viewpoint, we can distinguish in fig. 9 the effect of a symmetry related to the quasidistribution q~A(a, t) of crescent contour lines. Here we do not emphasize the marginal property of the Wigner function, because for moderate times the oscillations are not present, although the crest has disappeared and the peak has fallen. Of course, the study of squeezing based on using all of the

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CONTINUOUSMEASUREMENTS|N QUANTUMOPTICS

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Fig. 10. The P-quadrature distribution for ~l(0)[ = 20, q~l = 0, ~2(0) = 6, g = 1, ~p= -~, and M=10. quadrature distribution is not so abstract and simple as a plot of the variance only. The momentum-like quadrature squeezing is not present, as can be seen in fig. 10, where the Gaussian shape evolves into the Hermitian oscillations without going through a stage of crest formation. Moreover, the distribution is symmetric regardless of special initial conditions. The quasidistributions of the complex amplitudes provide not only distributions of the quadratures, but also those of the optical phase. Along with such phase distributions, the canonical phase distribution ranks, which is derived more straightforwardly from the number-state representation (Pefinovfi, Luk~ and ~(1) Pefina [1998]). The reduced statistical operator PlM = ]~PlM)(~PlM , with the vector [~plM) --[lPlM(t)), has the number-state representation

C.IM -- C.IM(t)= (nl~PlM(t)).

(9.118)

The optical phase is multivalued, in principle, which is reflected in the 2:r periodicity of the following 'raw' phase distribution:

1 oc Praw(q~'tlM't) = ~

Z

[2 exp(-inq~)c"lM(t)

"

(9.119)

n=0

One of the phase characteristics is the time-dependent preferred phase ~,

- ~ - -~(t[M,t)- arg [(etx"p(iq~))(t)],

(9.120)

where argz = Im(lnz) and e~p(iq~) is an exponential phase operator of Susskind and Glogower (Susskind and Glogower [1964], Lukg and Pe~inovfi [ 1991 ], Luke,

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CONDITIONALGENERATIONOFSPECIALSTATES

215

Pefinovfi and Kfepelka [1992]). The expectation value ( ) has been computed according to the formula o~

(etx"p(iq~)) = Z

C~'IM(t) cn+'IM(t)"

(9.121)

n=O

This 'raw' preferred phase is multivalued, in principle, as an example shows, in which the comb of the peaks in the raw phase distribution determines a multivalued preferred phase. The peaks have a period of 2r they do not provide a single value. Therefore, we confine the preferred phase ~ to the interval [-r r Having a single-valued preferred phase ~ - ~_~ at our disposal, we generate nonperiodic phase distributions: P(99, tiM, t) = ~ Praw(qg' tiM, t) for ~ C [~ - Jr, ~ + Jr), otherwise. / 0

(9 122) "

Because the preferred phase is time dependent, we are led to the continuous multivaluedness of functions. The graph of a continuous multivalued function consists of many continuous component curves preserving a vertical spacing. Here the graph preserves 2Jr vertical spacing. Of course, the restriction to the interval [-Jr, zr) should be required only for t = 0, and everything else follows from the continuity principle. We name the function defined by this connected component of the graph ~cont- The phase distribution (9.122) has not been produced only formally, because this distribution is possessed by a physically important random phase-valued variable confined to the interval [ ~ - Jz, 99 + ~). The phase distribution (9.122) may be interpreted as^ the distribution of eigenvalues of an orthogonally generated phase operator ~ _ ~ (Lukg and Pefinovfi [1994]). But the preferred phase can be subtracted from these eigenvalues and the appropriate distribution becomes

P(cp,-~, tiM, t) = P(q) + -~, t M, t).

(9.123)

Quantum-mechanically, the phase distribution (9.123) should be understood as that of eigenvalues of the rotated phase operator ~_~(~), ~-:r(~) = exp(i~h) ~_~ exp(-i~h) = q)-r - ~ i ;

(9.124)

the second relation was established by Luk~ and Pefinovfi [ 1993] when studying s-phase formalisms.

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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

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Fig. 11. Dependence of the preferred phase on the measured photon number for ~i (0) = 6, ~2(0) = 6, Igl- 1, and arg g = 0.

The dynamics of the preferred phase is obvious from fig. 11, which comprises the arguments of mean values of the exponential phase operator. For t = O, the preferred phase independent of M corresponds only to that of the signal mode. It emerges that, for the mode entanglement increasing with t, the dependence on the measured photon number manifests the initial phase of the idler mode, which is not measured, but enters the result via the optical parametric process. Particularly for very large t, when the determination of the preferred phase is mostly the matter of an exact computation, it is demonstrated that in the large-M limit the preferred phase tends to arg(-~* (e~)). In fig. 12, the conditional phase distribution P(qJ, t[M, t) is plotted, which is located at q0 = 0 (mod 2:0 for t = 0 and which for increasing t moves at first very quickly towards negative values of q~. (The t discretization manifests itself in the cock crest.) The diffusion of the phase also continues successively, which seems to moderate the decrease of the preferred phase, so that it does not reach the value arg (~*(~)). The phase distribution flattens ultimately. The transformation, which makes the preferred phase zero, leads to the centered distributions P(cp,-~, t]M, t) plotted in fig. 13. This picture can be compared with fig. 12. The similarity is enhanced by the phase diffusion, which dominates over

III, w 9]

CONDITIONALGENERATIONOF SPECIALSTATES

217

Fig. 13. Evolution of the phase distribution P(q~, ~,t M,t), q~ E [-:r, Jr), related to the rotated phase operator ~9-~_~(~) for _~1(0) = 6, ~2(0) = 6, g = 1, arg g - 0, and M = 10.

the shift o f the p r e f e r r e d phase. T h e effect o f the shift o f the p h a s e distribution is e l i m i n a t e d a n d the p h a s e diffusion m a y be m o r e p e r s p i c u o u s .

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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

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9.2. ENTANGLEMENTVIA OTHER TECHNIQUES Ban [ 1997c] has shown that a nondegenerate parametric amplifier can produce a single-mode squeezed state with any squeeze parameter and coherent amplitude. He assumed that a homodyne detection as a conditional measurement on the idler mode and controlled beam splitting for the signal mode are combined. He obtained the statistical operator of the output state when the idler mode is in the vacuum state at the initial time. The unnormalized quantum state of the signal mode after the measurement on the idler mode, when the initial state is pure, is given as I~P(q)) = fih~

(9.125)

where ]~p) is the initial state vector of the signal mode and t~h~

~1 r

exp( ~/-T

q2~exp

,, -~

~^t2 - ~-~ a

/2~

^t

V --T--q a

1 2 -~

.

(9.126) If in addition the initial state vector of the signal mode is a vacuum, I~p)s = [0)s, we obtain by comparison of the scalar product (a]~p(q)) with the well-known formula for (alfi) (Yuen [1976]) that fi = v / 2 - R r q / v / r + R and [~p(q)) is an unnormalized squeezed state,

]~P(q))= ~/Jr(T+74) exp

- @ ~ ( T + 7 4 ) exp

I 2(7-+7"4)q2 '

~, 2 ' 0

2(T+7"4)q2 b

b

T+7-4 q

v/T+7.4 q

~ 2'0

10)

]0),

(9.127) where 0 = ln(T + 7"4), and S (r, rl) is the single-mode squeezing operator, 1 (h2e-i2~ - ht2e i2~)], S(r, r/)= exp [~r

(9.128)

where r is a nonnegative squeeze parameter and r/is the phase of squeezing. Next the single-mode displacement operator has been used, /3(/3) = exp(/3h* -/3" h),

(9.129)

where/3 is a complex displacement. The squeeze parameter of this state can obviously be controlled, but its coherent amplitude comprises a random factor q.

I I I , w 9]

CONDITIONAL GENERATION OF SPECIAL STATES

219

Since even this random factor is known after the homodyne measurement, we can set the coherent amplitude to any value. To achieve this, the controlled displacement of the complex amplitude of the signal mode is performed. Ban [ 1996a] assumed that an observation is carried out at one of the two output ports of a lossless beam splitter, and that the input state of the reference mode is the vacuum state. He investigated photon statistics of a conditional output state at the unobserved output port for input states of two kinds; viz., for the Fock state input and the thermal state input. He considered photon counting, heterodyne detection, and homodyne detection as the means of observations at the output port. The unnormalized output state at the unobserved output port when the input state is pure is given as

lip(m) ) = USU(2)(m)] ,,pc IP), I~,(a)>

= "het

(9.130)

USU(2)(a)] IP), I~P(q)) = Usu(2)(q)l*P), ^hom

where I~p) is the input state of the signal mode, and ~Pu(2)(m)

1

-~m'. ,,het

1

,,hom

1

Usu(2)(a ) = ~

Usu(2)(q) = ~

7-4 ~ h,,T~

(9.131)

~

(~_~) exp -

(q2)

exp --~-

exp

exp

(

-~--~a +

7 -89

qa 7-~

(9.132)

(9.133)

Pegg, Phillips and Barnett [1998] consider even a generalized measurement. The authors are concerned with the operator ]event)b b(event l, which is not a projector, since levent)b is an unnormalized vector, [event)b =c (CIR* ]1)b [0)~ 1

- v~(r~lO)~ + iy~[1)b),

(9.134)

where k = exp [88

+ b*~)]

(9.135)

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

[III, w 9

is the unitary operator describing the operation of a second beam splitter (in terms of the annihilation operators/~ and ~ for modes b and c), and O(3

[C)e - ~-~ ?',,[n)c.

(9.136)

n=0

Their first beam splitter is described similarly as given in relation (9.13 5) (but in terms of the annihilation operators h and/~ for modes a and b). They assumed that a single-photon field is incident on the input port a and have shown that, after the detection of the event [event)b, the output a is in the state K'(Y010)a+ Y~ll)a), where K" is a normalization constant. In other words, the field in output mode a is the same as that obtained by truncating the state [C) after its first two photon-number components and then renormalizing the state. In their analysis, the authors also take account of detector inefficiency, r/< 1. Dakna, Anhut, Opatm~,, Kn611 and Welsch [1997] proposed a simple beamsplitter scheme for generating a Schr6dinger-cat-like state of a single-mode optical field. They assumed that a squeezed vacuum is injected in one of the input channels (the second input channel being unused) and photons are counted in one of the output channels. They have shown that the conditional states in the other output channel exhibit properties of superpositions of two coherent states with opposite phases. Further, they discussed the effect of realistic photocounting on the states. Dakna, Kn611 and Welsch [1998] have shown that conditional output measurement on a beam splitter may be used to generate photon-added states for a large class of single-mode quantum states, such as thermal states, coherent states, squeezed states, and displaced photon-number states. They assumed that the beam splitter combines a mode prepared in such a state and a mode prepared in a photon-number state, and that no photons are detected in one of the output channels. Photon-added states can be highly nonclassical ones. Photonadded coherent states are non-Gaussian squeezed states. Photon-added squeezed vacuum states exhibit all of the typical properties of Schr6dinger-cat-like states. Photon adding to a squeezed vacuum can therefore be regarded as a method for producing Schr6dinger cats. Dakna, Clausen, Kn611 and Welsch [ 1998] have compared single-detector photocounting with N-fold photon chopping. The photon-added states were introduced by Agarwal and Tara [ 1991 ]. They studied the mathematical and physical properties of such states and discussed how such states can be produced in practice. They considered the passage of initially excited atoms through a cavity, and modeled atoms as two-level atoms. The interaction Hamiltonian has the form (cf. eq. 4.58) /t~/int

=

h ( g h ?r+ + g'hi(r_).

(9.137)

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CONDITIONALGENERATIONOF SPECIALSTATES

221

The initial state of the atom-field system is [a)]e), where [a) is the coherent state of the field and [e) is the excited state; the combined system is (approximately) equivalent to a beam splitter with a low reflectivity and the input state [a)[ 1). If the atom is detected to be in the ground state [g), then the state of the field is reduced to a photon-added coherent state. Back-action evasion is achieved by a quantum noise evader (Yurke [1985]). The optical back-action-evading apparatus can be described by an effective interaction Hamiltonian /-)in, = hKDsglR,

(9.138)

where K is an effective coupling constant, the subscripts S and R stand for signal and readout, respectively, and ^

glj = --~ Qj = v/-2Re(hj), j = S,R, = 1 _ ~ , . = x/2Im(hj),

j = S,R.

(9.139) (9.140)

The coupling between the photon number and a quadrature has been described in eqs. (6.46)-(6.49), but here we expound that between two quadratures. In the Heisenberg picture, this coupling is described by the following transformation of the signal and readout modes (Song, Caves and Yurke [1990]):

(qs) OR

=(1 r) (qs) out

01

OR

(9.141) in'

(/Ss) =( 1 0) (/Ss) /3R out -r 1 /3R

(9.142)

in'

where r = Kt = 2 sinh r, and r is the squeeze parameter. Let the input state be described by a Wigner function of the form q~,~(qs,Ps, qR,PR) = q~s(qS,PS)

exp(-q~ _p2).

(9.143)

In the Schr6dinger picture the 'back-action evasion' is described by the relation for the output Wigner function q~,~Ut(qs,Ps, qR,PR) = qs~n(qs -- rqR,PS, qR, rps +PR),

(9.144)

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

[III, w 9

or, according to eq. (9.143): z

1

(•)souttqs,Ps, qR,PR) = q~s(qs -- rqR,PS)~

exp[-q 2 - ( - r p s +pR)2].

(9.145) The momentum-like quadrature operator fiR is measured. When it has a realization PR, we encounter the signal mode in an unnormalized state: pout

--

s (qS,PS,PR) =

f

OG

pout z

-

S tqS,PS, qR,PR)dqR

OG

-~r.out z

(9.146) 1

- 'vS tqs,PSl~R)--~_ exp[--qSR + rps)2] 9 V'Jr Here the conditional state is described by the Wigner function

q~,~Ut(qs,Ps]fiR) =

(y2)

~ CI)s(qs --Y, Ps) V/~ r exp --~Z

dy.

(9.147)

The readout quadrature f i R can, in principle, measure the quadrature operator/3s with an arbitrary large signal-to-noise ratio since r can be large. The back-action noise is 'evaded' and appears in the position-like quadrature component of the transmitted signal (see the convolution in the relation (9.147)). Song, Caves and Yurke [ 1990] described a method for generating superpositions of classically distinguishable quantum states using the optical back-actionevading apparatus shown by La Porta, Slusher and Yurke [ 1989]. They assumed that two modes of the electromagnetic field, the signal and readout modes, are correlated through a back-action-evading device consisting of a nondegenerate parametric amplifier and polarization rotators. They took advantage of the correlation between signal and readout to generate a Schr6dinger cat on the signal mode. They assumed that both modes are injected in the vacuum state and the number of photons in the readout is measured at the output. To separate the superposed states more distinctly, they proposed to process the signal mode through a degenerate parametric amplifier. Yurke, Schleich and Walls [ 1990] proposed to inject a squeezed vacuum at the signal frequency instead of amplifying the signal after processing by the backaction-evading apparatus and the measurement. Ban [1996b] proposed a scheme to generate the Fock state via a degenerate four-wave mixing and partial measurement on the reference mode whose initial state he assumed to be the vacuum. As an example of the initial state of the signal mode, he considered the Fock state, coherent state, and superposition of

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CONDITIONAL GENERATION OF SPECIAL STATES

223

two Fock states. He considered the direct, heterodyne, and homodyne detection measurements on the reference mode as conditioning for the signal mode. The unnormalized output state of the signal mode after the measurement on the reference mode, when the initial state is pure, is given as:

I~P(r, m)) = ~PC(r, re)lip(O) ), I*P(r, a)) = cthet(1", a)l,p(0)), Iq,(r, q)) = ~h~ q)l~P(O)),

(9.148)

where I~p(O)) is the input state of the signal mode, and ~pc(T,m ) -- ( - i~v Y ) m exp(- 89 ~/het(g, a )

= - ~1

~h~

= ~

1

m,

(9.149)

, ^2), exp(- 89lal 2) exp (_ix/~ra,h - ~grn

(

)

e x p ( - l q 2) exp i 2v/~grrqh ,

(9.150) (9.151)

where gr = (/./1-)2, and r is the interaction time of the four-wave mixing. In the cases of photon counting and heterodyne detection, the conditional signal state of the degenerate four-wave mixer reduces to a Fock state in the limit gr --+ cx). The same is valid for the homodyne detection, with the exception of the case treated by Ban [1996b]. The phase of the local oscillator has been chosen such that we cannot obtain any information of the signal mode. If we get some information of the signal mode, the scheme generates a Fock state; otherwise, the homodyne detection of the reference mode does not induce the reduction of the conditional signal state to a Fock state. Luis and Pefina [ 1996] studied the states generated from the vacuum in two parametric down-conversion crystals with aligned idler beams when the idler beams are completely or partially connected. First they assumed that the idler modes are perfectly superimposed and aligned. The parametric interaction in the two nonlinear media is described by the effective interaction Hamiltonians/~/lint and/-~/2int, /2/jint =

h(gj~l~a[+gf&jai),

j = 1,2,

(9.152)

where hsj and ai are the annihilation operators for the corresponding signal and idler beams and gj are parameters depending on the pump and the

224

CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

[III,w 9

nonlinear characteristics of the media. Luis and Pefina introduced the unitary transformations (9.153) i^

Tj-= exp ( - ~/-/jint Tj) , j = 1,2,

(9.154)

where rj. are the corresponding interaction times. In the Heisenberg picture, the operator N = hsl + hs2 - h i , with hsj = h~hsj, hi = h~hi, is conserved. It can be derived that

/Wout--/Qin,

(9.155)

where the subscripts in and out mean that the operator N is calculated using input and output photon-number operators, respectively. The operator N behaves similarly to the operator 7",

~/'out-- ~'in,

(9.156)

the operator ]~[in commutes with 7" and/Vout- 7't]Qinff'- The study is realized in the Schrfdinger picture. The relation between the input lip) and output ]~p) field states of the whole device is provided by the unitary operator T, I~) = 7"I~P)9

(9.157)

Luis and Pefina [1996] first assumed that a photon-number measurement is performed on the idler beam. They derived that the conditional states are the SU(2) coherent states, while the photon-number statistics of the idler beam are Bose-Einstein with the mean ~i,

ni = ~12~/~- 1,

/tj = cosh([gjl~-),

j - 1,2.

(9.158)

They further considered a measurement on the second signal beam. The conditional states are the SU(1,1) coherent states. The photon-number statistics of the second signal beam are Bose-Einstein with the mean ~s2,

-ns2 =/t~lv2] 2,

gJ sinh(gjlrj), vj =-i[--~j[

j = 1 2.

(9.159)

They commented on a result of the same quality in the case of the first signal beam. Here, nsl = / t ~ - 1 = [Vl [2.

III, w 9]

CONDITIONAL GENERATION OF SPECIAL STATES

225

To show that other special states can be produced, they considered partial coupling of the idler modes. One way to accomplish this is by inserting a beam splitter in the idler beam between the two crystals. The beam splitter couples the output idler mode of the first crystal with an 'escape' mode they described by the annihilation operator h0. The beam splitter was described by real reflection, r, and transmission, t, coefficients with a Jr phase change in one of the reflections. Its action on the field state is given by the unitary operator I"BS : exp[y(a~ai- a~a0)],

(9.160)

where ~' = arctan(-~). The relation between the states entering, I~p), and leaving, I~p), the whole device is now given by the unitary operator 7" = 1"2TBsl"l. An example with decoupled idler modes indicates that the conditional states are products of number states in the limit. First they studied the case corresponding to a measurement of the photon number on the h0 and ai modes. They did not present the joint photon-number distribution of the two modes (it is more complicated than that of the four modes), but they described the states as solutions to eigenvalue problems. This will be expounded in w 11. They obtained 'photon-added' SU(2) coherent states (with respect to the first signal mode, the photons are subtracted in the second one). They demonstrated a connection to the ix-J,, intelligent states. They also considered measurements on the other two output modes. First, they again treated the two output signal beams, but one obtains similar special states as above. A measurement on the h0 and as2 modes leads to 'photon-added' SU(1,1) coherent states. As many photons are added as escape. The authors related these states to the Ky-Kz intelligent states. A measurement on the hsl and ai modes leads to SU(1,1) special states spanned by a finite number of photon-number states. Here also, a relation to the K,,-k- or the k~-kv intelligent states can be found. In the Heisenberg picture, the operator N' = hsl + hs2 - h0 - hi, where ho = h~ho, is conserved; i.e., ^I __ j~qT./ Nout m"

(9.161)

From such conservation laws, one can simply discern which case leads to the SU(2) Lie group and which one leads to the SU(1,1) Lie group. De Martini [ 1998] dealt with a situation which resembles somewhat the case of complete alignment and detection of one photon on the first signal beam. The second crystal is, due to polarization, described as two equal and independent

226

CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

[III, w 10

optical parametric amplifiers. The author proposed a novel optical device, the quantum-injected, entangled optical parametric amplifier, which produces fourmode SU(1,1) Schr6dinger-cat states. Steuernagel [1997] proposed a quantum optical selection scheme to generate Fock states 12q) on pulsed light beam. His idea uses an array of (2 q - 1) beam splitters with 2 q input ports which are fed with single-photon Fock states. As a source, spontaneous down-conversion crystals are suggested, since this allows one to apply a time gating technique to all of the detectors involved. The detectors survey (2 q - 1) output ports; the free port is the one to produce the sought Fock state. It has been shown that the use of balanced beam splitters and symmetrical input states should give best results. Luis and Sfinchez-Soto [1998b] analyzed a conditional-generation scheme consisting of a parametric down-converter and a generalized measurement on some of its output modes. They found that this scheme generates approximately the state associated with the generalized measurement. In more detail, the scheme generates the desired state up to the action of a nonunitary operator and the normalization. This enables one to associate the desired state with a suitable generalized measurement in many situations.

w 10. Special States of One or Two Optical Modes by Means of Continuous Measurement on a Different Mode The destructive and nondemolition versions of the process of continuous measurement have the property in common that they converge to the ideal photon number and even to the collapsed wave function in the long-time limit when some assumptions are observed. As a consequence, the continuous measurement on one of two or three quantum-correlated modes presents a natural generalization of the conditional generation of special states using the ideal measurement. 10.1. CONTINUOUS STATE REDUCTION Yuen [1986a] proposed generation of high-intensity photon-number eigenstate fields from parametric processes with measurement feedback. According to that paper, strong correlation and ideal photodetection can entail that, after one counts n photons on the idler mode, the corresponding signal mode is just in state In) (see w9). One may thus consider the generation of the number state In) on the signal mode by stopping the signal after n counts on the idler. In addition to a 'continuous generation' of a near-number state, which we do not analyze in detail

III, w 10]

SPECIAL STATES

227

in this chapter, Yuen [ 1986a] obviously assumed a continuous measurement of the photon number, even though this is not the same as the one we have focused on here. Let us remember that the theory in w3 has been restricted to the single-mode fields only for the sake of concreteness, but it is valid in general for an arbitrary number of modes provided that one defines on which mode the measurement is being carried out. Holmes, Milburn and Walls [1989] assumed the measurement was on the signal, not on the idler mode. Nevertheless, in their notation, the measurement was still carried out on the mode b. Their model includes the interaction in the photodetection process; i.e., they performed the unraveling of the master equation (3.18), (3.20), where H is the interaction Hamiltonian of a nondegenerate parametric amplification. They introduced coupled master equations of the kind of eq. (3.47). They solved them in the representation by q~A quasidistributions (Q functions) for initial coherent states. They determined the conditional state of both modes and the conditional state of separate a (idler) mode. For simplicity, they assumed the initial vacuum state in a sequel. Similarly as in a somewhat different case (Agarwal [1990]; w9 here), one can obtain only an amplified number state on this condition on the idler mode. They compared the uncompleted measurement using a perfect detector with the completed measurement limited by a low quantum efficiency. As is usual in the analysis of optical parametric amplifiers including losses on the idler mode (Pefina [ 1991 ], p. 146), they discerned two cases dependent on the constants of the model, namely, whether the amplification dominates the damping or the damping dominates the amplification. Obviously, the latter case would be closer to a situation in which the counting is done, after the interaction which produces the correlated state is turned off (Walls and Milburn [1994]). They focused on this case, although they could only approximate it when the losses on the idler mode did not occur. Ueda, Imoto and Ogawa [1990b] developed a general theory that describes continuous state reduction of an arbitrary two-mode state by continuous photodetection on one of the modes. It is not very difficult to generalize again, and we present such an attempt here. We assume that the operator 0 acts only on the idler mode. The average photon number of the idler field for a postmeasurement state just after the one-count event is obtained from the relation (3.58) as lira (hz)(s)= s---~ t +

(Oth20)(t) (0*0)(t)

=

(10.1)

(Oth2b)(t)- (~2)(t)(OtO)(t) (n2)(t) + (t)

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CONTINUOUSMEASUREMENTSINQUANTUMOPTICS

[III, w 10

Keeping in mind the special ordering of the operators 0 t, n2, 0, we can write eq. (10.1) in the form lim (a2)(s) = (n2)(t) + s --+ t +

[ (Art2A(b i O)) (/)]ord (t)

(10.2)

where the subscript ord indicates the considered ordering. Particularly, for 0 = h2 (Ueda, Imoto and Ogawa [ 1990b]), the ordering ord is the normal ordering of the annihilation and creation operators h2, h~ and the second term on the right-hand side in eq. (10.2) is the Fano factor decreased by unity. In a similar way, it can be shown that the kth moment of the photon number of the idler field for a postmeasurement state has the form lim (hw

[(Anw

= (hzk)(t)+

S ----+ t +

tO)) (t)lord (0t0>(t)

(10.3)

where the subscript ord means a special ordering of the operators 0 t, n2, O. The time development of the average photon number of the idler field in a no-count process satisfies the following differential equation: d dt (h2) = -R cov(h2, 0 t O), (10.4) where we have introduced a 'covariance' (Merzbacher [1970])

cov(&, bt O) = ' [(Ah2A(O*O)) + (A(0*O)A&)].

(10.5)

The average photon number of the signal mode just after the one-count event is given by

lim (h,)(s) =

s-~,+

(bt;~,O)(t)

(b*b)(t)

= (~)(t) +

cov(a,, OtO)(t)

(10.6)

(OtO)(t)

Here there are no ordering problems in the definition of the covariance. The kth moment of the postmeasurement signal photon number is given by lim (h~)(s)= (h~)(t)+

s~,+

coy(a{ O*O)(t) ' (OtO)(t)

(10.7)

For the average photon number of the signal mode in a no-count process, we have d (hi) = - R c o v ( ~ , O* b). (10.8) dt The conditioning on the number M of photocount registrations in the interval [t,t + T) is simply written (or reasonably complicated) only if

III, w 10]

SPECIAL STATES

229

the postmeasurement state is not dependent on the times of registration rl, ~'2,..., rM, U[0, T)(M, Tl, 1"2,..., TM) = v/P(T1, T 2 , . . . , TM IM)h[o, r)(M).

(10.9)

In analogy to eq. (10.7),

(h*,)wM(T) =
coy (h'l,

(0)

(lo.lo)

(h]o' r)(M)h[o, r)(M)) (0)"

Ueda, Imoto and Ogawa [1990b] discussed the nonunitary time evolution of the quantum-mechanically correlated photon fields generated by parametric frequency down-conversion from the vacuum state. In a somewhat changed notation, we can write the total statistical operator conditioned on the number M of photocount registrations as DIM(t, T) = [lplM(t , T))
(10.11)

where [~lg(t, T)) = [A(t, T)] (g+')/2 exp [r/(t, T)K+] [M, 0),

(10.12)

A(t, T ) - 1 -]F12(t)] 2 exp(-RT), r/(t, T) = -F,2(t) exp (-89

(10.13)

with

The reduced statistical operator of the signal field is given by

IM =

(10.14) Similarly, the reduced statistical operator of the idler field is given by x(2) (t, T) = [A(t, T)] g+l Z /JIM

k +k M [1 -A(t, Tllk ]k)2 2(k].

(10.15)

k=0

Thus we find that the postmeasurement state is a two-mode SU(1,1) (generalized) coherent state. The expansion (10.14) is an amplified number state IM)~I (M],

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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

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whereas the expansion (10.15) is a version of this state for the idler mode shifted exactly by the number of detected photons M. 10.2. DESTRUCTIVE CONTINUOUS PHOTON-NUMBER MEASUREMENT ON THE IDLER MODE

Pefinovfi, Luk~ and K_fepelka [1996a,b] applied the theory of continuous measurement of the photon number to a two-mode light field, and assumed that the second mode is being measured. They supposed an entanglement of the two-mode state. We arrive at the description of the destructive continuous measurement of photon number on the idler mode on substituting O = he in eq. (3.20). We find that the formula (3.63) can be rewritten as U2dem(M,

T)~(t) =

/~2dem(M, T) p(l) U2dem ^t (M, T),

(10.16)

where 1 U2dem(M, T) = x/--~.. [ 1 -

exp(-RT)] g/2 exp (-89RTh2) h~.

(10.17)

The formula (3.45) becomes

~ IM

^

(dern),~.

it, T) =

Pdem(M,t,T)

U2dem(M, T)/3(/),

(10.18)

where the appropriate form of eq. (3.41) reads

Pdem(M, t, T) = Tr2 {u2dem(M, T) Tr, { f~(t)} } (10.19)

= Tr2{Tr,{f~(t)}H2dem(M,T)}, with [-/2dem(M, T) = ^U2dem(M, t T)/~2dem(M, T).

(10.20)

The positive operators [/2dem(M, T) resolve the identity, O(3

i2 = Z [/2dem(M, T), M=O

as usual in the situation of measurement.

(10.21)

III, w 10]

SPECIAL STATES

231

Switching to the quasidistributions, we see that we can write the quasidistribution cI)a(al, a2]M, t, T) and the photocount distribution p(M,t, T) in the forms /)A(dem)

(al, a2[M, t, T) =

1

1

- - [ 1 - exp(-RT)] M exp(-la212) Pdem(M, t, T) M! 0 0

•189

+a2~2)]

02M • OaMOa~M [exp(la212)q)a(al,az,

t)], (10.22)

and

Pdem(M,t, T) =

1

~.v[1 •

•

/

exp(-RT)] M

[-89 (a$-~2o + a2-~20) oayoa~ M f cI,.a(al,a2,t)d2ald2a2. 0 TM

exp(-la212)exp

(10.23) Here and in what follows, we use the property

exp

E-89 (a~-~2 + a20-~2 ~

V(al, a2, t) = V(al, exp

(-2RT)

a2, t),

(10.24)

where V is an analytic function in aj, a~, j = 1,2, and the explicit dependence on the conjugates is omitted. Using the reordering rule for the quantum characteristic function, (exp [is ( h i - h2)])= (exp [ ( 1 - e -i') la, Iz + ( 1 - e i~) laz 2]) A ,

(10.25)

we may verify that the quasidistribution (10.22) leads to the property \(dem)(t, T) = exp(isM), (exp [is(hi - n2)]/Im

(10.26)

when the measurement starts from the two-mode squeezed vacuum. The quantum Manley-Rowe relation hi - h 2 = const, can be generalized to the process of destructive measurement only in the case of photon-twin states (Ueda, Imoto and Ogawa [ 1990b]).

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

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Substituting from eq. (9.19) into eq. (10.22) and using the definition (9.33), we obtain

,t~l[M,t, T) = [A(t, T ) ] M+I [LOg(-A(t, T)[yl(t, T) + ~'*(t)[2)] -l :rM!

~A(ldem)l -

• la~ + ?*(t)l

where

A(t, T)

[-A(t, T ) [ a l - yl(t, T)I2], (10.27) was introduced in eq. (10.13), and [1 - exp(-RT)]

)'l (t, T) = ~l (t) + ?(t) = -~f(t)

A(t,T) -

1

Fl2(t)

TM exp

F12(t) ~] (t), (10.28)

~2(t).

Here the normalization of the quasidistribution has been obtained using the closed formula { I-~2(t,T)[ 2 } Pdem(M, t, T) = [B2B(~IT)-I]M (t, T) exp B2A(t, T) ( [~-2(t'T)12 ) • L~ -B2A(t, T)[B2A(t, T)- 1] '

(10.29)

where ~2(t, T) = V/1 - exp(-RT) ~2(t),

B2A(t, T) = [1 - exp(-RT)] B2A(t).

(~0.30)

Using the resolution of the positive operator/~/2dem(M, T ) and T r l {,b(t)} into the idler-mode photon-number projectors, we obtain from eq. (10.19) that the photocount distribution is simply related to a photon-number distribution of the second mode: 0(2

Pdem(M, t, T) = Z

Wdem(M[n2,1 - exp(-RT))p~2)(n2, t),

(10.31)

n2 = M

where

Wdem(Mln2, 1 - exp(-RT)) -

n2!

M!(n2 -M)! • [ 1 - exp(-RT)] M[exp(-RT)] "2-g.

(10.32) This relation is quite general, and it holds independent of not only the input of the down-conversion, but also of the chosen process itself. Mathematically,

III, w 10]

SPECIAL STATES

233

the coefficient Wdem(M[n2,1 - e x p ( - R T ) ) in eq. (10.31) represents a conditional distribution of the readout given a photon number n2, and the formula (10.32) indicates that, in the destructive continuous measurement, this conditional distribution is binomial. The photon-number distribution of the idler mode just after the downconversion has a quantal character,

t.:

exp{

Rn2+l

),0 -

1] " (10.33) Let us note that for t - . ~ the resulting squeezed vacuum ceases to be physical and with respect to the idler mode it is characterized by the limit property ! independent of n2, n 2,

"2.4 (t)

BzA(t)

n~

BzA(t)[BzA(t)-

p(2)(n~ t) lim = 1. t ~ ~ p(2)(n2, t)

(10.34)

Introducing the inverse probability,

Wdem(rt2, t M, exp(-RT)) =

Wdem(M[n2, 1 - exp(-RT))p(2)(n2, t) Pdem(M,t,T)

(10.35)

we derive that the postmeasurement state, IJlM=(ldem)(t, T)

,, (dem)

= Tr2 {PlM

(t, T)}

(10.36)

has the expansion (Pefinovfi, Luk~ and K~epelka [1996a]) CXD ~(ldem)l.

IalM tt, T ) = Z W d e m ( n 2 , t[M, exp(-RT))P(l,i](t).

(10.37)

II 2 = M

In other words, the state after the destructive continuous photon-number measurement can be expanded in terms of the states after an ideal photon-number measurement. In the strong correlation limit t ---. co, we observe that X( 1dem) t .

lim PlM

t ----+O~

it, T) = [1 - exp(-RT)] O<3 •

~

Wde m ( M I n 2 ,

1

-exp(-RT))In2)l , I n 2 ] .

n2 =M

(10.38)

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CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

[ I I I , w 10

Here, we use the fact that t lim ~ / - , ix(1) n 2 ( t ) - In2), ,(n21,

(10.39)

and observe that the negative binomial distribution of the states after the ideal photon-number measurement results in (cf. eq. 10.34) lim Wdem(n2,tiM, exp(-RT)) = [1 -exp(-RT)]Wdem(M n2, 1 - exp(-RT)). t ---~cxD (10.40) The relation (10.37) holds independent of which process is used to generate quantum correlated signal and idler (or reference) modes of the light field. For instance, a degenerate four-wave mixing can be considered, and a quite similar relation derived (Ban [1996b]). To this end, we modify the relation (10.31) to the form OG

Pdem(m, T, T) = Z

Wdem(mlm'

1-exp(-RT))p(b)(m' r),

(10.41)

m =m

where the probability

(10.42)

p(b)(m, r ) = (~p(r, m) lp(r, m)), with [~p(r, m)) given in eq. (9.148). Modifying the relation (10.35) to Wdem(m, r M, exp(-RT))=

Wdem(M[m, 1 - exp(-RT))p(b)(m, r) Pdem(M, r, T)

(10.43)

we see that O<3

~(adem)l... k'lM t t ' T ) = Z

Wdem(m,r]M,

exp(-RT))]tPlm(r))(~Plm(r)[,

(10.44)

m=M

where

I~P(r, m))

(10.45)

I~Plm(r)) = V/(lp(r,m)l~P(r,m)) In the case of the destructive continuous photon-number measurement, we aim at the optimum approximation of the ideal photon-number measurement conditioned on Mid. From eq. (10.37) it is obvious that the nonideal photonnumber measurement should be conditioned on a small value mde m with the

III, w 10]

SPECIAL STATES

235

property that, e.g., the peak of the distribution Wdem(n2,tlMdem,exp(-RT)) is located at the photocount number Mid. To obtain a tractable expression for Mdem - - M ( M i d , t, T), we impose the limit t ~ oc, and derive that Mde m is characterized by the property Mde m ~ (Mid q- 1) [ 1 - e x p ( - R T ) ] ,

(10.46)

Mdem > Mid [1 - e x p ( - R T ) ] ,

but for small RT, these original inequalities for the peak Mia may not be obeyed by any Mdem. Performing an analogous consideration for the Bernoulli distribution, we obtain the inequalities Mdem ~ (Mid nt-

1 ) [ 1 - exp(-RT)]

(10.47)

< Mde m q- 1,

or the formula Mdem = [(Mid q- 1)(1 - e-Rr)],

(10.48)

where [ and ] denote the integer part. Applying the formula

Pdem(nIM, t, T) = ~JF

02n

-dic(Otl) exP(la~ Iz) rr~(ldem)z-~'A ~'Ul IM, t, T ) -O0~(nOo(~

d2Otl,

(10.49) where be(el) is the Dirac delta function of a complex variable [cf. w9.1(a)], we arrive at the conditional probability of the signal-mode photon number given a readout of the detector: n~

Pdem(n M,t,T)= -~. [LOM(-A(t,T)lyl(t,T)+ ~'*(t)lZ)] -' [A(t,T)] M+' x exp [-A(t,

T)I y, (t, T)I ~] k=0

[1 A(t, T)] n-k (n - k)!

i~,(t) 2(M-k)

x lLf-k(lA(t, T) y,(t, T) ~,(t))l z (10.50) The appropriate dependence of the conditional photon-number distribution on the rescaled detection time for the readout Md -- Mdem, Mid = 10, can be observed in fig. 14. For RT <~ 2.3, the dislocations exhibit the control of the photocount number Md preventing the distribution from the drift to higher values of n. Nevertheless, for very small RT, this control fails because the distribution is too flat. Numerical results have demonstrated that this control

236

CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

[III, w 10

Fig. 14. The dependence of the conditional photon-number distribution given a readout of the detector on the rescaled detection time RT. The destructive continuous photon-number measurement of the ideally entangled state is assumed and the readout in the reserved time is M d = [ 1 1 ( 1 - e-eT)].

Fig. 15. The dependence of the conditional mean photon number given a readout of the detector on the rescaled detection time RT and on the ideal readout M - Mid. The destructive continuous photon-number measurement of the ideally entangled state is assumed and the readout in the reserved time is Md = Mdem.

111, w 101

SPECIALSTATES

237

proceeds unindicated by the Fano factor. In fig. 15 the control is marked by some steps for fixed M - Mid and decreasing RT, and these steps stabilize the mean photon number. 10.3. NONDEMOLITION CONTINUOUS PHOTON-NUMBER MEASUREMENT ON THE IDLER MODE

To obtain a description of the quantum nondemolition continuous measurement of the photon number on the idler mode, we assume O - n2 in eq. (3.20). The formula (3.63) can be rewritten as U2nondem(M, T)/3(t) = t22nondem(M,T)/3(t) U2nonde "* m(M, T),

(10.51)

where

1 (RT) M/2 exp (-~I RTh 2) h~.

b/2nondem(M, T) =

(10.52)

The formula (3.45) becomes t,t 1 /•(nondem) IM ~, T) = Pnondem(M, t, T) U2nondem(M, T)/5(t),

(10.53)

where the appropriate form of eq. (3.41) reads

Pnondem(M,t, T) = Tr2 {U2nondern(M, T)Trl {/5(t)} } (10.54) = Tr2 {Trl { P} f/2nondem(M, T) } ,

with

JP/2nondem(M,T) =

^t U2nonde m(M, T)/A2nondem(M, T).

(10.55)

The identity resolution into the positive operators f/2nondem(M, T) reads O<3

(10.56)

i2 = Z [/2nondem(M, T). M=0

Using the resolution of the positive operator [/2nondem(M, T) into the idlermode photon-number projectors, we obtain the distribution of the number of counts being registered in the interval [t, t + T): oo

Pnondem(M, t, T) = M.v Z H-~ = O

exp(-RTn~) (RTn~)Mp(2)(n2, t).

(10.57)

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CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

[III, w 10

Denoting a conditional distribution of the readout by

1 (RTn~)M exp(-RTn~), Wnondem(MlRTn~)= -~.

(10.58)

we can rewrite eq. (10.57) as oo

Wnondem(M RTn~)p(2)(n2,t).

Pnondem(M, t, T) = Z

(10.59)

n2 = 0

The distribution (10.33) is transformed using the conditional distribution (10.58), which is Poisson. Introducing the inverse probability, Wnondem(n2, tlM, RT) = W n ~

t),

(10.60)

Pnondem(M, t, T) we find that the postmeasurement state

/~(lnondem)i. IM

it, T) = Tr2

{ ~(nondem

PlM

)(t, T)

}

(10.61)

has the property (Pefinovfi, Lukg and K~epelka [1996b]) oo

PlM I't'T)=Z ~..(1nondem)i.

Wnondem(n2,/M,

RT)p([1)(I)9 n2

(10.62)

n2 = 0

Again, the state after the nondemolition continuous measurement of photon number can be expanded in terms of the states after an ideal photon-number measurement. In the strong correlation limit t ---, oc, we obtain that

lnondem,,

lim /JIM

1-1

Wnondem(M[RTn~)

i,t, T) =

t ----, o o

"2=~

(10.63)

0(5

X Z

Wn~

(MlRrnl) n2),, (n21.

n2 =0

Here, we used the asymptotic property (10.39), and we see that a distribution of the states after the ideal measurement centered at the value v/M/RT arises: lim Wnondem(n2,t t ----+ ~

M, RT) = Wnondem(n2,ocIM, RT)

=IntO wnOndem(M[RTtli2),=Wn~ =

I~~2 ,_- 0 ~ 1. . (RTn'2:) M exp (-RTn; 2)

~1 (RTn~)M exp (-RTn~). (10.64)

III, w 10]

SPECIAL STATES

239

Expressing explicitly the dependence on the method and on the detection time, M _= Mnonaem(T), and considering for a while that Mid denotes the outcome of the ideal photon-number measurement, we recognize the relation (10.64) as the exact formulation of the fundamental physical result (Ueda, Imoto, Nagaoka and Ogawa [ 1992]) M_nonde m ( T)

,~ RTM~ d .

(10.65)

On the contrary, assuming that T goes to infinity, and adopting the asymptotics

M '~ RTMi2d,

(10.66)

with a number Mid, we obtain the situation of generating crescent states similar to that of ideal photon-number measurement with M fixed and T ~ ~ . Formally, to obtain a definition of M for all T, we forget about the interaction time t and put Mnondem =

[RTM~d ],

(10.67)

where [ and ] denote again the integer part. There is no closed formula for Pnondem(nlM, t, T). In figs. 16 and 17, the effect of control of readout M on the conditional photonnumber distribution is shown. This control does not affect the graph with visible discontinuities almost anywhere because they should occur with the step of the rescaled detection time RT, A(RT) = (mid) -2 = 0.01. This is as fine as the plot. We see that in fig. 16 the control prevents the photon number from being attenuated for long RT. Nevertheless, the increase in variance connected with RT decreasing cannot be controlled except for very small Mid, as is obvious from fig. 17. The result of control of the mean photon number is very good even when the rule (10.67) is not perfect in that it yields Mnd ~ Mnondem = 0 for RTM~ < 1. Some exceptional discontinuities in fig. 17 may be situated on the curve RTMi2d - 1. Schiller, Bruckmeier, Schalke, Schneider and Mlynek [1996] implemented a scheme for continuous-wave quantum nondemolition detection of opticalamplitude quadratures, and Bruckmeier, Schneider, Schiller and Mlynek [1997] improved it (see w8). The device has the particular property of permitting quantum state preparation for both output beams: Measuring one beam will lead to a state preparation on the other beam. Bruckmeier, Hansen and Schiller [ 1997] demonstrated two-step quantum state preparation and three-beam entanglement.

240

CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

[III, w 10

Fig. 16. The dependence of the conditional photon-number distribution given a readout of the detector on the rescaled detection time RT. The nondemolition continuous photon-number measurement of the ideally entangled state is assumed and the readout in the reserved time is Mnd = [ 100RT].

Fig. 17. The dependence of the conditional Fano factor given a readout of the detector on the rescaled detection time R T and the readout M -= Mid. The nondemolition continuous photonnumber measurement of the ideally entangled state is assumed and the readout in the reserved time is Mnd ----Mnondem.

111, w 11]

EIGENVALUE PROBLEMS FOR INTELLIGENT STATES

241

Breslin and Milburn [1997] paid attention to the continuous spectra of the observables in the Einstein-Podolsky-Rosen (EPR) paper, and referred to the experiment performed by Ou, Pereira, Kimble and Peng [1992], who followed the original EPR proposal and suggestions by Reid [ 1989] and Drummond and Reid [1990]. Essentially, they described a homodyne quadrature measurement on one of two modes. Their paper does not only differ from that of Holmes, Milburn and Walls [1989] by the measured observable, but also by the same damping rate from the cavity for both modes, and by the description using a stochastic differential equation. w 11. Eigenvalue Problems for Intelligent States Generated in Ideal and Continuous Measurements

Eigenvalue problems which are considered to be useful mathematical definitions of special states, namely the minimum-uncertainty or intelligent states, can be in a closer connection to the experiment, derived by means of a unitary evolution and an ideal state reduction of correlated light fields. Generalizations to the case of destructive continuous state reduction are presented. 11.1. I D E A L S T A T E R E D U C T I O N

Luke, Pefinovfi and K~epelka [ 1994] derived eigenvalue problems for intelligent states generated using ideal state reduction. They connected the time evolution of the parametric down-conversion and the event of the measurement with the eigenvalue problem for the reduced state. Let us start with the fact that the initial coherent state ]~p(0)) = [{~.(0)}) has the property hj(O)l~p(o)) = ~.(O)[W(O)), j = 1,2.

(11.1)

Inserting the identity operator ] = ~ t U, where U = exp{-~/2/t}, we obtain that

(Jhj(O)O t O'l~(O)) = ~j(O)Olq,(O)) ,

(11.2)

hj(-t) ~p(t)) -- ~(O)[~(t)),

(11.3)

or

where I~P(t)) = U]Ip(0)) is the state I~P(0)) evolved in the Schr6dinger picture and hj(-t), j = 1,2, result from eq. (9.49) in the Heisenberg picture with time reversal. Substituting from eq. (9.49) and using the expansions OC

hi(0) -- h(0) |

Z n~ = -1

Ina + 1)2 2(n2 + l[,

(1 1.4)

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C O N T I N U O U S M E A S U R E M E N T S IN Q U A N T U M OPTICS

[III, w 11

where h(0) is a single-mode annihilation operator, oo

~:(o) = i, o ~

~nz + l lnz>z ~_
(11.5)

n2 = 0

and oc

i=i,o

~

(11.6)

Inz+l>zz
n2 = - 1

where 11 is a single-mode identity operator, the first-mode one, we get the following recursive relations" [a(O)- y(t)i] 2(n2l~P(t)) =--Fl2(t)v/l~

2
v ~ 2(n2lW(t)) =

-F,2(t)

[hi(O)+ y(t)i] 2
The use of the expansions (11.4) and (11.5) is connected with the modification of the formula (9.3) for the pure states, oo

l~> I Z

z o

Inz)z.

(11.8)

n2 = 0

From eqs. (11.7), we arrive easily at a 'solved' eigenvalue problem: [ht(o) + y(t)l] [h(O)- y(t)i] 2(MI~P(t)) = M 2(Ml~P(t)),

(11.9)

or

h(O) + y(t)h(O)- y(t)hi(O)- y(t)

y(t)i] I~lM(t)) = Ml*PIM(t)),

(11.10)

where the normalized state 1

1

l~PiM(t))= v/p(M) 2(MllP(t))2 = V/p(M) 2(MI~P(t)).

(I I.I I)

Let us observe that the eigenstates (11.11) are described by the quasidistribution (9.79). Not taking the time evolution into account, we simplify the decoupled equations (11.10) to the form (11.12)

III, w 11]

EIGENVALUEPROBLEMSFORINTELLIGENTSTATES

243

where

A=M+yy.

(11.13)

The problem (11.12) has the solution

Ito> = .A/b(~,, Y)IM),

(11.14)

where the nonunitary (or, exceptionally, unitary) displacement operator D(y, y) = exp(yh t + yh),

(11.15)

and the constant N" can be determined in the form

H : exp{-89

z + Re(yy)]}

[L~

+ -y* 12)] ll/2

(11.16)

For y + ~* = O, the study applies to the displaced number state, which is beyond our interest. The eigenvalue problem (11.12) is then reformulated as

[h-~0(r) + ir/P(r)] I~> : XI~>,

(11.17)

with ~ and r/given in eq. (9.108). The general eigenvalue problems (11.12) or (11.17) can be reduced to any of the following three normal (canonical) forms:

(a) where

[h + ir/o/3(0)] I~Po)= Al~o), I~Po) = 0tl~p),

~r - 0(~', r ) = b(~')exp(irh),

(11.18) (11.19)

with r/o and ~' given in eq. (9.86);

(b) where

(h-),oht)l~po>= ~1~Oo>,

(11.20)

[~Po) = Ertl~P),

(11.21)

~r -- ~(_~*, r) = b(-~*)exp(irh),

with Yo given in eq. (9.88); and

(c) where

(h + ~oa)[tOo) = XltOo), ]~Po) = ~r*l~p),

~ - ~r(y, r ) = D(y)exp(irh),

(11.22) (11.23)

with Yo given in eq. (9.90). In all the cases, the unitary displacement operator D(fl) is defined in eq. (9.129). It can be demonstrated that by using an

244

CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

[III,w 11

appropriate rotation followed by a suitable displacement, any normal state can be transformed into any solution of the more general eigenvalue problem. Indeed, the normal forms rest on a displacement and a rotation of the points -~*, ), situating them on the real axis as the points -r/0, r/0 (case a), 0, )'0 (case b), and -Y0, 0 (case c). For small ),, ~, the near-number state is understood, which occurs for t going to infinity in the evolution. A consideration of normal eigenvalue problems is not very convincing with respect to the quasidistributions qsa, since we obtain only special forms of the formula (9.79). Nevertheless, it is important that problem a) (eq. 11.18) is the definition of the number-quadrature minimum uncertainty state (Pefinovdt, Luk~ and I~epelka [1994]). These states have been studied in connection with the quantum phase problem, when the problematic quantum phase operator has been replaced by the quadrature operator. For further detail, see Pe[inov~, Luk~ and Pe~ina [ 1998]. The representation of the state I~p) in the number state basis for case (a) is m~ u

exp(-~)

n-M n-M

v/G(_4,~,) r/(j

-..

LM (-r/02) for n ~> M,

c. = (n[ ~p) =

(11.24) n~ exp(-'~

v'G(-4,~)

r/y-nLM-n(--r/2)for n ~< M.

Applying the formula (11.14) to the definition
(11.25)

we arrive at [L~


' E~,--0 (~,)(-r/0) k-"' E',--0 (',)(-r/0)'-" amn , k ~< l, k >~ 1,

,,

(11.26) where

am,=

I(m+M)! r xn-mrn-m ~" ,4~,,2x for m ~< n, M ! ~/" rIO1 l~m+M ~--"e l]O]

(11.27) (n+M)! t o n ~m-nrm-n i A~2~ M! ~./-ulO) ~n+M~,--'.ttlo)

for m ~> n.

Particularly, 1

(h)

-_

r/o LO(_4r/2

(h2) = L~

)

~0(_l)l_..~,,r. = ,. ~M(--4r/2),

(11.28)

L (2n)(-1)2-"2"L'/w(-4r/2)"

(11.29)

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245

The coherent state lr/0) solves the problem for M = 0. In case (b) (eq. 11.20), the role of the quadrature is taken over by the photon creation operator, and the number state representation of the appropriate eigenstate is (Yamamoto, Machida, Imoto, Kitagawa and BjBrk [1987])

( )

n<M,

,/2

Cn =

exp -~-

n!

(11.30)

1 y(~-M

v/LOM(_yg) (n--M)!

,

n >~ M.

The expectation values of the antinormally ordered field operators read 1

=

(k+M)! . l - k r l - k

Lo(_ro~) M~ Yo ~k+M(-Y2),

k ~< 1,

,,

k > t.

(11.31)

For M = 0, the coherent state ]Y0) results. In case (c) (eq. 11.22), the eigenvalue problem is related to the number operator and the photon annihilation operator, which results in the finite number state representation (9.98). The moments are expressed as follows:

k

(~k~t/)=

Z~

1

2)

l

m~~ (km) (--frO )k-m ~ =

(In)(--ffo)l-namn,

(11.32)

B=0

where (m+M)!q-in-mln-m M!

IO

L'm+M

(__y2)

for m ~< n, (11.33)

amn = (n+M)! [q-i ~ m - n l m - n M! ~.1"01 L'n+M

(__y2) for m ~> n.

For M = 0, the vacuum state results for all values of the parameter Y0. Luis and Pefina [1996] presented a number of eigenvalue problems, and outlined ingenious derivations. Since the states are to be generated from the vacuum, the roles of relations (11.1) are played by

Tasl~'tl~> = (lUlasl q- tv1V;as2 - tVl~2t~ [ - Z~vla;)[~> = 0 ,

(11.34)

Tas2ftl~> = (/t2hs2- v2h[)lip) = 0,

(11.35)

7'aiT't[-~>

= (t2~l~2ai-k- r[21a 0 - Vla~l- t/A 1v2a~2 ) 11/9>= 0,

(11.36)

taofrtl~>

= ( - ~ 2 a i + tao + ~2a~:)I~> - o,

(11.37)

when one considers partial coupling of the idler modes. They started by first studying the case corresponding to a measurement of the photon number on the

246

CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

[III, w 11

'escape' and idler modes by means of photodetectors placed at the output of the beam splitter and at the output idler beam. If the results of such a measurement are no and ni, respectively, the state of the signal modes ]~Pln~,n0) is given by the projection of ]~p) on ni)iln0)0 (and a normalization). Equations (11.36) and (11.37) can be solved for hi]~P> and h0]~P). Equations (11.34) and (11.35) are solved for h~ ]~) and h~l~), but solutions are presented somewhat implicitly: =

-

,

(11.38)

^tl~)

rvl

(as, - ~asz)lq,> = tt-Tao

,

where ~ = tV1/(V2gl) , while (explicit solutions) it2

(11.39)

r v1

aol'lp> = --~1 h~l]~).

The recursive relations (11.7) can be avoided here. On multiplying the first and second equations in (11.39) by hi and h~, respectively, and on appropriate substitutions, we obtain that

~ i ] ~ ) = ( ~ , + ~2)~s21~), tl~ao]~> -- a ~ l ( a s l -

(11.40)

~:as2)l~>-

We can project these two equations over photon-number states of the idler and 'escape' modes, respectively. We obtain easily that

dz -

Nl~Plni,no) = (no + ni) q~ln~,,o)' ~J+)l~P,n,,no) - ~(no - ni)l-~ln,,no>,

(11.41)

1

where /~r = ~:ll~S 1 -t- l~2as2 ,

) z = 1 ( a : l a s , - t~2tffs2 ) ,

) + = asll~s2 9

(11.42)

Taking into account the commutation relations

Dz,J+] = J + ,

[/V,J+] : O,

(11.43)

we have ,Jz - ~,J+ = exp(~J+) ,Jz exp(-~J +),

(11.44)

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247

which gives the solution as = jVexp(

j+)l

o)s.

ni- k

k=0

where N" is a normalization constant, ni ni-k

i

N'=

~ k=0

no + k k i_~lzk

(11.46)

By means of an SU(2) transformation (i.e., the action of a beam splitter), an SU(2) intelligent state is obtained from I~Pln~,,,0) (the state 11.45). Luis and Pefina [1996] discussed the kinds of states which arise when the photon-number measurement is performed on other pairs of modes. They assumed that the two detectors were placed at the output signal modes. When the outcomes of such photon-number measurement are n l and n2 for the first and second signal modes, respectively, the state of the idler and 'escape' modes [~Pln,,~2) is given by the solution of the equations

_

~

(11.47)

d z + ~ff-)IlPlnl,n z) = ~ ( r t l - rt2)[~ln~,n2),

where ~ - t/(/~2 r), and

= a[Zti +a~Zto, Jz

= 1 (a~a0- a[ai) ,

J+ = a~ai.

(11.48)

The solution of eqs. (11.47) reads 1~1,,,,:) = N" exp(~9-)ln2)ilnl )i,

"(c )(;,)

= ./V. Z k=0

\

n2 + k n2

~k n2 + k)iIn, - k)o,

(11.49)

where N" is a normalization constant, 1

N =

_n2 \

(11.50) n2

Note that the notation is light, and its meaning is somewhat different from that in relation (11.41). To the contrary, the properties of the states I~pl,,,,2) are the same as those of I~pln,,n0).

248

CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

[III, w 11

If the measurement is performed on the modes as2 and h0, giving n2 and no, respectively, the state of the idler and first signal modes ]lpln2,n0) is to be found. Equations (11.36) and (11.37)can be solved for h0]~) and h~21-~). Equations (11.34) and (11.35) are solved for hs2]-~), h~]-~), but solutions h~2 ~), h~ ]-~) are presented somewhat implicitly:

~2~ ~/,

(11.51) as21~> = V ~01~,>-

-~

I~>,

where r/= tVl/(ttltt2). On multiplying the third and fourth equations in (11.51) by h~2 and h~, respectively, and on appropriate substitutions, we obtain that h~ho]~p) = a~,(as, - r/a[)l~ ),

(11.52)

aZ=as21~>= a~(a~- ~as,)l~>. ^t can project these two equations over photon-number states of the second and 'escape' signal modes, respectively. We obtain easily that

We

Rl~,:,no>

=

(no - n: - 1)l~ln2,.,,>, 1

(k:-

(11.53)

~k+)l~Pln:,n,,> = ~(no + n:

+

1)l~ln:..,,),

where

"

_

_ ,

'(a~ asl-k-a[a,-k-i)

K+ = asia ^t "ti 9

(11 .54)

From the commutation relations [k~,K+] = K+,

[K,K+] = 0,

(11.55)

we have K~ - r/K+ = exp(r/K+) K~ exp(-r/K+),

(11.56)

which leads to the solution ]~lnz,no> = .A/"exp(r/K+)lno)sl ]n2>i, ~ ~(n2+k )( )no+k = ./V'Z ~kln0 + k>sl n2 + k>i, k=o \ n2 k

(11.57,

III, w11]

EIGENVALUEPROBLEMSFORINTELLIGENTSTATES

249

where the normalization constant is given by

.IV.= [ L (n2ff-k/(no k k=0 \ n2 +

k)

[?][2k]

-89

(11.58)

By means of an SU(1,1) transformation, an SU(1,1) intelligent state is obtained from Ilpl,2,n0) (the state 11.57). Finally, when we measure the photon number on the first signal and idler modes, with outcomes n l and n2, respectively, we search for the state of the second signal and 'escape' modes. Equations (11.36) and (11.37) can be solved for hi[~) and h~l [~). Equations (11.34) and (11.35) are solved for hs, I~P), h~ [~), but solutions h~, ~), hi [~p) are presented somewhat implicitly:

=

' --

~s211~) ---- "V2 ~.l-Ilp)

,

'

//2

Iv,)

=

(11.59)

--~1("as2 --t--

,

V2

where r/= t:/(rv2). On multiplying the third and fourth equations in (11.59) by ^ t and h~ respectively, and on appropriate substitutions, we obtain that asl (11.60) We can project these two equations over photon-number states of the first signal and idler modes, respectively. We obtain that ^

m

Xl~lnl,n,>

(Kz

-

(nl

-

ni -

1)l~lnl,,,.>,

(11.61)

+ r/K-)l~l,2,,0 ) = ~(n0 1 + n2 + 1)IIPL,,1,,,,),

where

R : a~ao- a~2as2- i,

k , -,-.=.

l (a~ao-+-a~2~/s2q-i)

~

K- = aohs2. (11.62)

250

CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

[III, w 11

Again, the notation does not mean the same as that in relation (1 1.53). Taking into account suitable commutation relations, we arrive at the solution [~ln,,n,) = A/" exp(rlk_)lni>s2ln~ )0,

(()(:1) ni

= "A/" Z

ni - k

k=0

n ~ l n i - k>s21n, - k)0,

1163.

where the normalization constant is given by

.1.64. After distinct SU(1,1) transformations for [rll > 1 and I~1 < 1, SU(1,1) intelligent states are obtained from I~l,l,~i) (the state 11.63). The case corresponding to a measurement of the photon number on the first signal and 'escape' modes yields (only) an SU(1,1) coherent state,

~22as2ai

I~pl,,,,no) =.A/'exp =.A/"'~.

m + nl - no

~0=

]O)s2ln~-n0)i "v2

]m)s2]m + nl - n0)i,

(11.65)

m

where the normalization constant is given by 1

.A/. = [~-~ (m+nl-no)

v2 2ml ~

m

m=0

~22

(1)n,-n0+l = ~22 "

(1 1.66)

When we measure the photon number on the second signal and idler modes, the state of the first signal and escape modes is an SU(1,1) coherent state, -

I~]n2,ni)

frvl

"--

^,

.A/'exp ~ - ~ a s , h ~/ ]ni- n2)i]O)0 N

I

*

(11.67)

= .A/" Z(x:)~l ni

-

n n2

[n)sl In -- ni-+- n2)0,

n2

where the normalization constant is given by

.A/" =

m + n i - n2 m

r 'v1

-~l

=

1-

r gl

71

(11.68)

m=O Luis and Pefina [1996] also considered input coherent states, mainly on the assumption of the complete coupling of the idler modes.

III, w 11]

EIGENVALUE PROBLEMS FOR INTELLIGENT STATES

251

11.2. C O N T I N U O U S STATE R E D U C T I O N

Agarwal and Tara [ 1992] investigated a 'necessary condition' of the positiveness or a classical character of the P function (@• quasidistribution). Its violation is sufficient for the nonclassical character of this function. Such a condition can be applied to the states under study. They provided the example of the photonadded thermal state and that of the superposition of two coherent states similar to an amplitude cat (Schaufler, Freyberger and Schleich [1994]). Pef-inovfi, Lukg and I~epelka [1996a] derived a generalization of the problem (11.12). They observed that it is equivalent to an eigenvalue-eigenoperator problem: (11.69) We will derive a generalization of the problem (11.69) using the property of rescaling in the @.a quasidistribution. Starting from a modification of the problem (11.69),

where

~' = M + A y ~ , ,

(11.71)

introducing the A-dependent statistical operator (11.72) where the superoperator S means amplification by the factor ~A' and using the rescaling property in the form (11.73) we obtain the problem +

-

/5(A, y, ~,)= ~'/5(A, y, ~,).

(11.74)

Here the superoperators h+, ht+ are given in eq. (2.40) and

h+ = ht+h+.

(11.75)

252

CONTINUOUS MEASUREMENTSIN QUANTUMOPTICS

[III, w 11

Since ,~-1

Sh+S

= Aa*+a*_ + a*+(a+ - a*),

~-l 1 sa+s = v~a*_ + ~ ( a + ,,

(11.76)

a*),

9,-1

~a*+s = ~a*+, where the superoperators h_, ht_ are given in eq. (2.40), we arrive at the eigenvalue-eigenoperator problem Aa*~a + a*[a,/,] +Ap~a + ~,[h,[~]-Ayh*[~ = 2,'/5,

/5 -/5(A, y, p). (11.77) Let us remark that the amplification superoperator can be expressed as 1

) 2a+ta-t-ala-t-a+a+t

= v/Aexp [(1-A)ht+ht] (x/A) a+ta+(v/A) a-a~ CO

= x / ~ Z (1-A)Jht+j j!

(11.78)

(V/--~)ht+h+~ltJ (V/~)a-ht-

j=0

From the relation min(n, m)

PnmIM(A , Y, }')= Z

/'+) "-" '+) (A)P,,-j,m-jlM(1 ' x/-Ay, x/A~) , VP,,In-jta)Pmlm-/

j=0

(11.79) where the (displaced) negative binomial distribution pnlk(A) =

(1 -

,

n=k,k+l,k+2,...,

(11.8o)

and PnmlM(1, x/My, v/-A~) -- CnlM(1, x/Ay, v/A~)C,*,IM (1, X/~y, V ~ ' ) , (11.81) with C,IM(1, Y, Y)--

C ~.. n! [t~ • exp(-5 lyl

~. 12)] -l

L~-n(-y~),

(11.82)

III, w 11]

EIGENVALUE PROBLEMS FOR INTELLIGENT STATES

253

we derive the formula

[Lo (-AIy + P*lZ)] -' PnmlM($, 7, ~)- v~!m! M!

A M+I exp(-AlYl2)

min(n,m)

x

Z k=0

(1 - A) k ~Ml.+k rMl.+k ( - A yr/) k! ~.-k

(11.83)

x (~,)M-m+k ~,,,-krM-m+k( - A y* ~*).

Applying displacement and rotation operators to the density operator r a situation can be achieved with real parameters 70, ~'0 such that-~'0 < 70. Substituting particular values of 7, )' into the eq. (11.83), we obtain formulas pertinent to the following three cases: (i) For 7' = ~*, denoting )70 = 7' = ~'*, the matrix elements can be written as

P, mIM (At = ~

1

x/m!n!A M+I

exp(-A ]r/012 ) L~ ]r/0 [2)

min(m, n)

•

Z

(1 - A ) k rl~t_,,,+k rM_m+k(_AlrlOi2 ) k!

k=0

(11.84)

~,,-k

x ()7~) M-"+k "-'.-k'M-"+k(--A]~o]2)"

For A = 1 the appropriate form of the coefficients (11.82) generalizes the formula (11.24), removing the constraint r/0 > 0. (ii) For ~, = 0, denoting 70 = Y, the matrix elements simplify dramatically; i.e., 1

P,,mIM (A) = --~.. x/~m!n!A M+'

•

min(m-M,n-M)~ Z...,

k=0

exp(-A ]y0 ]2) L~ ]y0]2)

(1 -

A)k(Ay~)"'-M-k(Ayo) "-M-l<

k! (m - M - k)! (n - M - k)!

(11.85) The corresponding coefficients (11.82) for A = 1 generalize the canonical case (b) characterized by )'0 > 0 (eq. 11.30) (Lukg, Pe~inovfi and K~epelka [1994]), and which for complex 7'0 was studied by Agarwal and Tara [1991 ]. For A ~< 1 and M - 0, we also obtain a generalization of the well-known formula for the superposition of coherent and chaotic fields (Pefina [ 1991 ]).

254

CONTINUOUS MEASUREMENTSIN QUANTUMOPTICS

[III, w 11

(iii) For ), = 0, denoting 70 = ~', the matrix elements read

PnmlM (,4) = M! x/~?n? A m+l

L~

1

min(m, n)

•

(1

Z

k -- max(0, m - M, n - M)

--A)k~'~)M-m+kyg-n+k

k! (m - k)V ( M - m + k)! ( n - k)V ( M - n + k)! "

(11.86) The appropriate coefficients (11.82) under the assumption A = 1 generalize those for the canonical case (c) (eq. 9.98) in the paper by Lukg, Pefinovfi and K~epelka [ 1994]. The output state of the down-conversion process has the property (h~ + yi)(h, - yi)/5(t) = n2/3(t).

(11.87)

On multiplying eq. (11.87) by U2dem(M, T) given in eq. (10.17) from the left and by its Hermitian conjugate U2dem(M ^t , T) from the fight, reordering the operators, and taking the trace over the idler mode, we obtain the following property: ~_.(ldem)i. (a~ q" Yi 1)(a 1 - Yi 1)~)M tt,

T)=Mp~dem)(t,T)+Tr2

{ h2PM ~"(dem)(t, T)}

(11.88) where = U2dem(M, / M3 ( d T)e m ) ( t , T)/5(t), ,, (ldem)

PM

{~(dem)l.

(t, T) = Tr2 k'M

it, T)

}

~(ldem)E t

= Poem(M, t, T) ~'IM

(11.89)

~., T).

Dividing both sides of eq. (11.88) by the probability Pdem(M, t, T), we arrive at

(~l~+-yll)(~ll-yll)f)(]lMdem)(t,T)=''^(ldem)(l,T)+Tr2{ "~'x(dem)l" MPlM ,t2P[M ~t, T) } . (11.90) From the explicit expression for the q~,a quasidistribution, the following property can be derived (Pefinovfi, Luk~ and K~epelka [1996a]):

A(h~ + ~i 1)/5(')h, + (h~ + ~i 1)[hl,,b (')] - A y l h ~ (1) = Jl',b(1) = (M + AylY)/3 (1),

(11.91)

where f ) ( 1 ) _ ~(ldem)l.

r, iM

~, T).

(11.92)

III, w 11]

EIGENVALUE PROBLEMS FOR INTELLIGENT STATES

255

Using the relation 1

]/1 -- ~ ~t +

1 -A_, A y '

(11.93)

we rewrite eq. (11.91) in the form (a~ -{- y i 1)(al -- ]/i 1)1~
-A)(h~ +~i,)/5<')(hl

+ ? * i 1).

(11.94) Comparing formulas (11.88) and (11.94), we arrive at the connecting relation Tr2

~A
(11.95)

We have shown that eq. (11.91) generalizes the eigenvalue problem for the ideal postmeasurement ket (cf. eq. 11.9): (&~ + ? i l ) ( & l - yil)l~P (1)) :

(11.96)

this problem can be reformulated as + yi,)o

- y i l ) p <') =

(11.97) (11.98)

On the replacement M H ~, the relation (11.94) can be approached as the usual eigenvalue problem in analogy to eq. (11.96). According to eq. (11.14), the solution of eq. (11.96) has the form 1r 1)) -I~pllM))

o< DI(y,~)IM)I,

(11.99)

where b l ( ) t , - y ) --

exp(yh~+ -~al).

(11.100)

Similarly, we assume that the operator t~ (1) in eqs. (11.97) and (11.98) has the form p<,) : N : b l ( y ,

y)E1)b

(y, ?),

(11.101)

where /5~ ) is another desired operator. The normalization constant N "2 will be determined later as soon as it is required that the operators /5(1), /5~ )

256

CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

[III, w 11

be normalized. Multiplying eq. (11.97) by the operators [~)l(y,y)] -l and [ ~ ( y , y-)]-l from the left and from the fight, respectively, we get hlf3~) = ~43~).

(11.102)

Equation (11.102) facilitates observation of not only the equivalence of eq. (11.96) and eqs. (11.97) and (11.98), but also an interesting analogy. Similarly, we obtain from eq. (11.94) the equation hlp~ ) - (1 - A ) ^alp t ~_(1)^ A al = ~,/~1)

(11.103)

This equation can be approached as an eigenvalue problem; e.g., using the thermofield notation. Again, the unambiguity is kept at the level of eq. (11.96) using the symmetry property (11.98). Such an analysis provides a unique solution in the form ,,tm /5(1) = A/'Z[A(t' T)]M+lb~(Y(t)' Y(t)) al (11.104) • [1 - A(t, T)] '~'h~4b~ (y(t), ~(t)), where

A/. 2 = exp{-Re[y(t) + Y*(t)]y(t)-Aly~(t, T)I: + Aly(t)l:}

L~

T)lyl(t, T)

+

-~*(t)] 2)

(11.105)

In the ideal case, when the produced state is pure, it is interesting to see its broad-sense crescent shape. This can be realized using three canonical forms of the eigenvalue problem (11.96) (Luke, Pefinovfi and K~epelka [1994]). Of these, the case when y = if* = ~70 is most interesting, because it is related to a characteristic of the particular crescent state as an intelligent state with respect to the number and quadrature operators. Recalling the generalized classification presented by Pefinovfi, Luk~ and I~epelka [1996a], we have presented three particular forms of eq. (11.83), and now we add those of a generalized eigenvalue problem:

(h~ +yil)(hl-yil)~(1)-[1-A(t,T)](h~

+yi~)/5(l~(hl +y*i~) = ~l/5(~) (11.106) Case (i), a generalization of case (a) from the paper of Luke, Pefinovfi and I~epelka [ 1994], is encountered for Y1 = Y* = r/o,

(11.107)

and leads to the eigenvalue problem [hi + il ~0 [/3(arg ~]0)] [) (1) - ( 1 - A ) ( h ~ + rl~ ] l)/3(1)(al- r](~i l) -- ,~t~)(1) (11.108) The appropriate eigenstates are a generalization of the intelligent states with respect to the number and quadrature operators. The generalization is seen from

III,w 12]

CONCLUSION

257

the second term in the left-hand side of eq. (11.108), and can be understood as a thermalization of the pure state. Case (ii) is very close to case (b) described by Luke, Pefinov~ and K~epelka [ 1994], because it is subject to the same condition ~ - 0. By )' = Y0,

(11.109)

a parameter Y0 is introduced, and the eigenvalue problem (11.106) becomes ( h , - yoh{)/5(1)- ( I - A ) h{/5(l)hl = ~/5(')

(11.110)

Case (iii) is more complicated, and corresponds to case (c) described by Luke, PeHnovfi and I~epelka [1994]. It is characterized by a generalized condition )'1 = 0. By m

m

Y= Yo,

(11.111)

another parameter is introduced. The eigenvalue problem (11.106) can be rewritten as (hi +Yohl)/5(1)- (1 -A)(h~ +yoi,)/5(1)hl = ~/5(1)

(11.112)

The nonclassical character of all these special states of the signal mode can also be tested using a criterion developed by Agarwal and Tara [1992].

w 12. Conclusion The measurement of quantum objects can be described simply only if the meter is, at least in part, described classically. A collection of unnormalized statistical operators are appropriate for a joint statistical description of a quantum system and classical variables. In fact, these statistical operators must be 'jointly normalized'. Their traces will provide a marginal distribution of the classical variables and the sum or integral of all these operators will give a marginal statistical description of the quantum system. Similarly, as with a joint distribution of bare classical variables, a statistical description conditioned on values of some classical variables can be derived, even in general. Taking all of them into account, one obtains a normalized conditional statistical operator. An instantaneous measurement is assumed by the von Neumann postulate of a joint statistical description of the postmeasurement quantum system and the outcome

258

CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

[III, App. A

of the measurement. The postulated measurement is connected simply with a measured operator, and it is ideal just in this sense. Great attention has been paid to measurement on one component of an entangled (a quantum correlated) two-component quantum system. This may be due to the fact t h a t - independent of the realization of the ideal measurement on the considered component, which may partly defy the von Neumann postulate - one obtains states very close to the postulated ones on the other component of this system. Correlated photons at the output from a down-converter are to help produce states close to number states on the signal mode, even if the realization of the ideal photon-number measurement on the idler mode fulfills the von Neumann postulate only in part. The von Neumann postulate itself indicates a Markovian property of quantum measurement. The distribution of the measurement outcome depends on the premeasurement state of the quantum system, but it is not affected further by any older state of the quantum system or by any measurement outcome. The theory of continuous measurement supplements features of the quantum measurement, which are at least mediate consequences of the von Neumann postulate. Moreover, these features are also at least mediate consequences of the interaction between the quantum system and a meter, so that the theory of continuous measurement has enriched the description of the interaction between radiation and an apparatus by more detail. This makes it possible to investigate properties of states conditioned on specific situations during the photodetection, such as the number of photons measured within a certain time interval. The destructive and nondemolition versions of the process of continuous photonnumber measurement enable one to measure an ideal photon number in the long-time limit. In the continuous quantum nondemolition measurement also, the wave function converges to the collapsed one. As a consequence, the continuous measurement on one of two or three quantum-correlated modes presents a natural generalization of the conditional generation of special states using the ideal measurement.

Acknowledgements The authors would like to thank Ing. J. I~epelka and Dr. Jan Pefina, Jr. for careful preparation of figures. They acknowledge Prof. J. Pe[ina, DSc, for reading the manuscript.

Appendix A. Superoperator-valued Measure and Related Concepts We identify the space g2 of all possible outcomes of the experiment with the set of nondecreasing step functions, which are defined on [0, c~), are left-

III, App. A]

SUPEROPERATOR-VALUED MEASURE

259

continuous, vanish at the time origin, assume only integer values, and assume them 'successively'. We will call these functions trajectories. We define a random process M ( t ) as the identical mapping on Q; i.e., the map that sends any trajectory onto itself. We consider an interval [0, T). On a random event, i.e., on a subset,

( M ( T ) >~j ) = { M E E2;M(T) >~j } ,

j = 1,2, 3 , . . . ,

(A.1)

we define a random variable, i.e., a functional, rj - r_j[M(t)] as the time t that M ( t ) = j - 1, but M ( t + 0) = j . We will let M stand for the random variable (the functional) M(T). In what follows, let m be a natural number, k l , . . . , km be non-negative integers such that kl ~< ... ~< km, and let t l , . . . , tm be real numbers from [0, T) such that t~ < ... < tm. We define a a-field .Tr as the smallest a-field containing the random events (subsets of s (

Am, tl,kl .....

tin,kin : ~ M C s

M(T) t> k.,,

_rl[M] < " "

< r_k,[M] < tl <, _rk,+l[M] < ' ' '

< _rk2[M] < t2 ~<... < tm <~ _rk,,,+l[M] < . . .

< __rM(r)[M]}.

(A.2) For kl < 1 or k2 < kl + 1 o r . . . or M ( T ) < km + 1, the respective chain inequality is absent. We will associate with any random event A from 9t-r a sequence of Borel sets B1 C R, B2 C I1~2,... , where B1 = T I ( A f-'l (g__m-- 1)), B2 = ('El, ~ 2 ) ( A f-'l ( M = 2)),

(A.3)

9, o

where (__.T1,__.T2), ... are random vectors, i.e., vector-valued functionals, and ___T1, (__r1,_r2), ..., although modified to assign to set arguments set values, are not distinguished by any change in notation. It is also convenient to consider a one-element set R ~ and to define B0 = R ~ if (M = O) C A, B0 = 0 (the empty set) if A N (M = O) = O. Now we define the superoperator-valued measure,

II(A) = H(Bo) + H(B1) + H(B2) + . . . ,

(A.4)

260

CONTINUOUS MEASUREMENTS IN QUANTUM OPTICS

[III, App. A

where /~/(Bo) = { 0u[~r)(0) forf~BOBo==R~ /~/(B1) = f u[o, r)(1, rl) drl,

(A.5)

Bl

h(B2) =

ff ~[o,r)(2, rl, r2) drl dr2, B2

9

.

.

^

with h[o,r)(M, rl,..., rM) defined in eq. (3.32). The referring and nonreferring measurements are treated as particular cases of a general concept similarly as a one-element event (subset) and the certain event s are only particular random events. The convex and semigroup structure of the set of operations was studied by Kraus [ 1971]. The probability of the measurement having the outcome in A is given as (A.6)

Prob(A) = Tr {/3(0)/)(A) }, where the probability operator measure [/(A) has the property Tr{/3/I(A))=Tr(/~/(A)/3),

any /5.

(A.7)

In more detail, [I(A) =//(Bo) + f/(B1) +//(B2) + ' " ,

(A.8)

where [/(Bo) =

{U]o, r)(0)fi[o, r)(0) for Bo = •o, 6 for Bo = O,

//(B1) = f h~o,r)(1, rl)h[o, r)(1, rl) drl, B1

[/(B2) =

ff h~o,r)(2, rl, r2) h[o,r)(2, rl, r2) drl dr2, B2

(A.9)

III, App. B]

ITO'S CALCULUSWITHOUTITO'S DIFFERENTIAL

261

with u[0, r)(M, r l , . . . , rM) defined in eq. (3.31). The definition of an instrument was given by Davies and Lewis [ 1970]. After the random event A, the measured system is in the conditional state

~A(T)

(A.10)

/,).(r) = Tr{ff.(r)}' where ff A ( r ) =

n(A) :,(O).

(A.11)

The probability operator/'/(f2) = i and the superoperator H ( I 2 ) = G[0, r).

Appendix B. It6's Calculus Without It6's Differential

In the quantum stochastic calculus, an operator .s is localized in (s, t) if it depends only on system operators and on the fields L(r) and Lt(r) only for times r between s and t. An adapted process is a time-parametrized family of operators localized in (0, t) for every t; for more precise definitions, see Hudson and Parthasarathy [ 1984]. Using the finite difference notation

&:i(t) = .i(t + a t ) - . i ( t ) ,

(B.1)

a t > o,

we obtain the tautology A[/l~/(t)N(t)] = [A/~(t)] N(t) +/l~/(t)[A/V(t)] + A/~(t) AdV(t),

(B.2)

where ~/(t) and N(t) are adapted operator-valued processes. Dividing both sides of eq. (B.2) by At and passing to a limit, we obtain the Leibniz formula with the It6 correction:

+ lim At ~ o+

(B.3)

AM(t)AN(t). At

Using the Stratonovich product where necessary, we persevere on the Leibniz formula d d--~[~Vl(t) N(t)] = ( [d ]f/l(t)lN(t)} s

+

}s

262

CONTINUOUSMEASUREMENTSIN QUANTUMOPTICS

[III

where d

^

1

AM(t) AIV(t) At

{ /~/(t) [~t N(t) ]}

s

lim AI~(t)AIV(t) = { ~l(t) [d~-~N(t) } + 21 At---,0+ At

, (B.5) (B.6)

I

The indicated limits do not vanish, as for instance

lim 1 f,+A, L(r) dr atf,+A, Lt(r') dr'

At---~0+ At at

= Ri

'

(B.7)

although lim h1 ft+a, Lt(r,) dr,

At ~ 0+ At

1 atlim --. o+ A-t

t

[ftt+At

it+AtL(r) dr = 0, Jt

L(r) dr j 2 = Ate0+ lim At1

[/

t+At Lt(r)dr ]2 = ~'

(B.8)

(B.9)

where L(t) is quantum noise according to eq. (4.8), with the properties given in eq. (4.3).

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E. WOLF, PROGRESS IN OPTICS XL 9 2000 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED

IV

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

BY

ZEEV Z A L E V S K Y , DAVID M E N D L O V I C

Tel-Aviv University, Faculty of Engineering, 69978 Tel-Aviv, Israel

AND

ADOLF W. L O H M A N N

University of Erlangen, Lab. far Nachrichtentechnik, Cauer Str. 7, 91058 Erlangen, Germany

271

CONTENTS

PAGE w 1.

INTRODUCTION

. . . . . . . . . . . . . . . . . . .

w 2.

T H E S P A C E - B A N D W I D T H P R O D U C T (SW) AS A T O O L F O R SUPER RESOLUTION STUDIES

273

. . . . . . . . . . . .

275

w 3.

S U P E R R E S O L U T I O N AS S W A D A P T A T I O N

w 4.

S U P E R R E S O L U T I O N B A S E D ON T E M P O R A L C O N S T R A I N T S 293

w 5.

THE GENERALIZED WIGNER FUNCTION FOR THE

. . . . . .

A N A L Y S I S OF S U P E R - R E S O L U T I O N S Y S T E M S

282

. . . . .

w 6.

S U P E R R E S O L U T I O N F O R O B J E C T S W I T H F I N I T E SIZE

w 7.

WAVELENGTH-MULTIPLEXING SUPER RESOLUTION

w 8.

CONCLUSIONS . . . . . . . . . . . . . . . . . . .

312 .

320

. .

329 338

L I S T OF A B B R E V I A T I O N S A N D S Y M B O L S . . . . . . . . . .

339

ACKNOWLEDGEMENTS

. . . . . . . . . . . . . . . . .

339

. . . . . . . . . . . . . . . . . . . . .

340

REFERENCES

272

w 1. Introduction

Speaking in general terms, 'resolution' means that information about small details, 6x, of an object is available as output of an optical system. 'Being available' usually means that the object is displayed as an image. But in a broader sense, 'resolution' means the possibility to somehow infer object information, such as a small size 6x, from the observed output data. In this study the final output will be an image, but on its way from input to output, the information about the object may be totally unrecognizable due to some coding schemes. The common goal of all projects described in this chapter is to achieve 'super resolution'. The attribute 'super' refers to the capability of obtaining more information about the object than could be expected when considering only the lenses and the apertures of a standard optical instrument, such as a microscope or a telescope. Additional components, such as moving gratings, spectral prisms, polarization components, etc., are used to improve the performance of the system; i.e., to create 'super resolution' (SR). The field of super resolution can be divided into 'classical SR' and 'modern SR'. The latter is also called 'near-field microscopy' (a recent paper on that subject by Blattner, Herzig and Dandliker [ 1998] may serve as an entrance into the literature on near-field microscopy). Another categorization considers the various causes, which limit the resolution. Foremost in this review are the limits caused by diffraction. We refer to this category as 'diffraction resolution' (DR). Another category, called 'geometrical resolution' (GR), is concerned with the discrete structure of certain detectors, such as CCD arrays. The third category, called 'noise equivalent resolution' (NER), is the most fundamental limitation of all information gathering systems, and deals with the ability of each detector cell to distinguish a signal out of a noise. However, in most of the existing image forming systems, it is the DR that one wants to improve. This has been accomplished with some success, as the following sections will show. 273

274

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 1

1.1. DIFFRACTIONRESOLUTION (DR) According to Abbe (Lummer and Reiche [1910]), the diffraction limit of spatial resolution, in the camera plane, may be expressed as 6XD~F~ 1.22~.F#,

(1)

where/l is the wavelength and F# is the F number of the imaging system. This spatial resolution is related to the size of the aperture, since

f

F#- D'

(2)

where D is the diameter of the imaging lens and f is its focal length. Toraldo Di Francia [1955] said that if one only wants to know the lateral distance Ax between two stars, there is no diffraction limit on the accuracy of Ax. It could very well be Ax < 6x = 2f/D. However, if Ax is smaller than 6x, the recorded image is very similar to the single star case (Ax = 0). So, the signal-to-noise ratio will set the limit. What this case teaches us is that 'image formation' (here seeing two separate bright points) is only a very luxurious case of 'information gathering'. The fewer questions we ask, the more accurate will be the answer, if the system is properly tailored to these particular questions. Many techniques have been suggested to improve the diffraction limit. As will be further specified, all of those techniques were based on an a priori knowledge about the object. This knowledge was used to synthesize in effect an extra large aperture of the imaging system. One can characterize the a priori information types according to the following groups: object shape, temporally restricted object, wavelength-restricted object, one-dimensional object, and polarizationrestricted object. 1.2. GEOMETRICALRESOLUTION (GR) This second type of resolution is related to the finite size of the detector pixels. Assuming that this size is Ax and the focal length of the lens is f , then if such a system is aimed at a scene located a distance R from the camera, the spatial resolution in the scene's plane is Ax

8x = -~R.

(3)

For instance, assuming that Ax = 30 gm, R = 10 km a n d f = 300 mm, one obtains 8x = 1 m. According to the Johnson Criterion (Waldman and Wootton [1993]),

IV, w 2]

THE SPACE-BANDWIDTH PRODUCT (SW) AS A TOOL FOR SUPER RESOLUTION STUDIES

275

for a probability of 50% one needs 1.5 pixels in order to detect an object, 6 pixels to recognize it, and 12 pixels to identify it. Thus, having an object with a size of AL = 3 m, the above mentioned distances will be

Rdet

-

Rrec

-

Riden

-

ALf 3m .0.3m = 20 km, 1.5Ax 1.5 930 gm ALf 3m.0.3m = 5 km, 6Ax 6 . 3 0 gm ALl 3m .0.3m = 2.5 km. 12Ax 1.5 930 ~tm

(4) (5)

(6)

The algorithm optimal for achieving this type of super resolution is coined a sub pixeling algorithm and it is related to the Gabor transform (Gabor [1946]). Briefly, the procedure for obtaining the improvement is to record N images. Between two recordings the camera is shifted by a sub pixel distance of Ax/N. Then the images are properly merged, a Fourier transform is performed, the result is divided by G ( - v ) (which is a Fourier transform of the pixel's shape), and eventually an inverse Fourier transform is calculated. 1.3. N O I S E

EQUIVALENT

RESOLUTION

(NER)

This third type of resolution limit is related to the noise developed in each one of the detector's cells. As had been mentioned before, noise may be the limiting factor, for example if one wants to measure the distance of two stars (Toraldo Di Francia [1955, 1969]). The cause of noise might be stray light, or temperature dependent detector noise, or quantum noise if the light level is very low, or even quantization noise which is caused due to the fact that the camera has a finite number of sampling bits. We assume in our study that these types of resolution impairments are negligible compared with the influence of diffraction upon the resolution. As an overall rule, an averaging operation, whether it is temporal or spatial, is helpful for improving the NER.

w 2. The Space-Bandwidth Product (SW) As a Tool for Super Resolution Studies As previously mentioned, the improvement of resolution requires a priori knowledge about the signal. The term 'signal' will be used instead of the narrower

276

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 2

term 'object'. One can characterize the a priori information types according to the following classes: object shape (Toraldo Di Francia [1955, 1969]), temporally restricted signal (Francon [1952], Lukosz [1966]), wavelength-restricted signal (Kartashev [1960]), one-dimensional signal (Bartelt and Lohmann [1982]) and polarization-restricted signal (Gartner and Lohmann [1963]). Mendlovic and Lohmann [1997] described super resolution as a rearrangement of the signal's degrees of freedom and the transmitting system's degrees of freedom such that these two sets of freedom are well matched. A trivial example may serve to illustrate the concept. Suppose that a set of objects is 1 mm wide, and that the finest details are 1 ~tm wide. Suppose also that a detector (e.g., photographic film) can resolve details of 100 ~tm in size. However, the size of the detector is 100 mm. In this case, an ordinary magnification is enough to adapt the shape of the signal SW to the shape of the detector SW. Or, for instance, let us assume that the spatial aperture of a system is small and that some of the signal's information is lost due to this fact. If it is also known a priori that the signal's information is the same for all wavelengths, one may convert part of the spatial information into wavelength information, in a way that the aperture of the system is expanded synthetically. Based on the distinction between the signal information and system's capabilities, Lohmann, Dorsch, Mendlovic, Zalevsky and Ferreira [1996] and Mendlovic and Lohmann [1997] proposed a way to adapt the signal to the system. The Wigner chart is often a useful conceptual tool. A brief introduction can be found as an appendix of an earlier article in Progress in Optics (Lohmann, Mendlovic and Zalevsky [1998]). Super resolution apparently had an early phase in the 1960s, and a recent phase in the 1990s. In between these two phases the field was almost dormant, apart from a few isolated attempts (Cox and Sheppard [1986]). 2.1. THE W I G N E R D I S T R I B U T I O N F U N C T I O N

The Wigner distribution function (WDF), introduced by Wigner [1932] in the context of quantum mechanics, is a space-frequency representation of a signal which was applied to optical signals by Bastiaans [1979a,b]. A recent review was presented by Dragoman [ 1997]. The Wigner chart is a wave-optical generalization of the 'Delano diagram' (ray optics Y~" diagram). Several wellknown optical transformations can be performed in the Wigner domain simply by changing coordinates. For example, the Fourier transform (FOU) is represented by a 90 ~ rotation of the Wigner distribution function (WDF). A Fresnel transform or free space propagation (FSP) corresponds to an x-shearing of the WDE Passage through a lens (LENS) means v-shearing. Therefore the FSP and the

IV, w 2] THE SPACE-BANDWIDTHPRODUCT (SW)AS A TOOL FOR SUPER RESOLUTION STUDIES

277

(a)

(f) l MAG

I"

FT

FRT

FSP

~

,=x

4. 1/

(e)

V

.,'-

/%

(d)

V

,,

,x

(d)

1"

~x

Fig. 1. Basic transformation applied over the WDE (a) Input, (b) lens, (c) free space, (d) Fourier, (e) fractional Fourier, (f) magnification.

LENS operation are Fourier conjugates. A fractional Fourier transform (FRT) corresponds to a rotation of the WDF by an arbitrary angle (Lohmann [ 1993]). These basic transforms are illustrated in fig. 1. The mathematical definition of the WDF is

W(x, v) = f cx:~u(x + X-~t )u *(x - X-~t ) exp(-i2a'vx') dx'.

(7)

oo

Apparently the Wigner chart simultaneously presents spatial and spectral information. It doubles the number of dimensions; thus, a 1D object has a 2D Wigner chart. The WDF is closely related to the intensity lu(x)] 2 and to the power spectrum Ifi(v) l2 as

W(x, v)dv = lu(x)l 2

f

~ m(x, v)dx = I~(v)l 2,

(8)

oc

where (x)

u(x) e x p ( - 2 m vx) dx.

fi(v) =

(9)

O(3

The double integral yields the total energy

f o ~ / ~ 1 7 W(x, 6 v) dx dv = Etot" oo

oo

(10)

278

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 2

The way back from the WDF to the signal domain is possible by

f_

~ W ('-~x, v) exp(2:rivx) d v = u(x)u* (0)

(11)

2.2. THE S P A C E - B A N D W I D T H PRODUCT OF SIGNALS: SWI

2.2.1. The SWI-number The letter 'I' in SWI refers to the second letter of 'signal', in contrast to the 'Y' in SWY, which alludes to the space-bandwidth product of systems. Suppose that a signal is bounded in the space domain and also in the frequency domain:

Ix[

u(x) ~ 0

only within

< Ax/2,

(12)

fi(v) ;~ 0

only within Iv[ < Av/2.

(13)

From eq. (13) it follows that u(x) may be represented as a sampling series: u(x)= Z

u(m6x) sinc ( X - 6xm6X

"

(14)

The sampling step, 6x, which is the inverse of the bandwidth, determines the finest detail within u(x): 6x = 1/Av.

(15)

Strictly speaking, only one of the two bounds (12) and (13) can be valid rigorously. However, if the product of the two bounds AxAv is large, say 100, both equations may be valid in good approximation (Marks [ 1993]). In that case, the series (14) contains only a finite number N of samples: Iml ~ N/2;

Ax N - 6x - AxAv.

(16)

The number N counts the degrees of freedom of the signal u(x), as expressed in a physicist's language (VanderLugt [1992]). In communications theory, AxAv is called the 'space-bandwidth product'. We will refer to this product as the SWI-number. Such a number makes sense not only in the context of a particular signal but also for a set of signals with common bounds (eqs. 12 and 13).

IV, w 2] THE SPACE-BANDWIDTH PRODUCT (SW) AS A TOOL FOR SUPER RESOLUTION STUDIES

279

One brief comment about the case where N is not large: a more careful definition of signal size Ax and bandwidth A v is required; for example, as

x21u(x)l 2 dx f_~ lu(x ) 2 dx '

(17)

f _ ~ ]~(V)]2 dv

(18)

The product AxAv has a lower bound, which is actually reached if u(x) is a Gaussian function. This is the essence of the uncertainty principle (Marks [1993]).

2.2.2. The SWI area In w2.1 on the Wigner distribution function WDF, we illustrated how the shape of W(x, v) is modified if the signal u(x) itself experiences a process like free space propagation (FSP), transmission through a lens, Fraunhofer-Fourier diffraction (FOU), self imaging, magnification, or fractional Fourier transformation (FRT). The size of the area remains the same, but the shape may change drastically (FSP, lens, magnification). A rectangle, which encloses the WDE and whose boundaries are parallel to the two axes (x, v), may now have a larger area than the WDF area (FSP, lens, FRT). For instance, a detector whose SWY is rectangular, has to be wider in x if the signal has traveled through free space, as shown in fig. l c. In the case of magnification, the shape has been changed, but not the size, because the boundaries remain parallel to the axes (x, v). These considerations lead us to define the space-bandwidth product SWI of a set of signals u(x) as the area occupied by the set of associated WDFs (Lohmann [1967]). Notice that this (x, v) definition of the SWI has in common with the SWI-number definition that the amount of SWI area remains unchanged if the SWI number also remains the same. However, the shape of SWI(x, v) may change even if the SWI number remains constant. This tells us that the SWI shape is more informative than the SWI number. This fact will turn out to be crucial for the various super-resolution methods, where the shape of the SWI is manipulated, but not the SWI number. 2.3. T H E S P A C E - B A N D W I D T H P R O D U C T OF A S Y S T E M - S W Y

2.3.1. The SWY number Suppose a photographic camera has an image size of 25 mm (for simplicity, in one dimension only), and the resolution may be 25/tm. Such a system can handle

280

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 2

= 25 mm/25 ~tm - 1000 pixels. This number 1000 is in this case the SWY number. This concept can be extended to higher dimensions: imagine a TV set which is switched on for one hour (At = 3600 s). The frame rate may be 25 per second (6t = ~ s). 400 lines with 500 pixels per line mean a SWY number of N~v = 400 x 500 = 2 9 105. Together with the temporal 'SWY number' Nt = 3 6 0 0 / ~ = 9 9 106, the overall SWY number is N,~,,Nt = Ntotal = 1.8 * 1012. This number may be called 'space-time-bandwidth product', to be consistent. Another generalization from spectroscopy: the usable wavelength range may be A~. = 100nm and the spectral resolution 6/l = 0.01 nm. This leads to NA = 104. Ax/6x

2.3.2.

The S W Y area

The three examples mentioned in w2.3.1 have in common that they correspond to rectangular domains in phase space, with the boundaries parallel to the axes. There exists a good reason for these rectangular SWY areas, which will now be illustrated by a counter example. Suppose a cheap lens is used as a wide-angle photographic objective. The image resolution will decrease from the center of the image field towards the edges. Expressed in another way: the local bandwidth capability will vary as A v ( x ) = A - B x 2,

Ix] ~< Ax/2.

(19)

The problem with such a system is that two identical object details, one located near the center (x = 0), the other close to the edge (x = Ax/2), will yield different images. Or, the other way around, if two image details look alike, the two objects may actually be quite different. Hence, an unbiased image interpretation is not possible. 2.4. THE SW AS A 2D F U N C T I O N

A certain signal can only pass through a certain system if the following relation holds: SWI c SWY.

(20)

Note that this is a graphical relation. That is, the SWI area must be included in the SWY area. In other words, not only the pure number but also the shapes of SWI and of SWY are important. Otherwise some information of the signal is

IV, w 2] THE SPACE-BANDWIDTH PRODUCT (SW) AS A TOOL FOR SUPER RESOLUTION STUDIES

281

lost in passing through the system. The SW of the system may cut the SW of the signal. Again, this statement can be visualized easily in the Wigner domain. For a signal having a rectangular shape, the SWY plot of the system could be the area of the conventional space-bandwidth product SW. In conclusion, no information is lost if a necessary and also a sufficient condition are satisfied. It is necessary that the SWI and SWY numbers satisfy SWI(#) ~< SWY(#).

(21)

A sufficient condition is that the SWI and SWY plots obey SWI(plot) c SWY(plot).

(22)

The definition of the SW can be generalized in two ways. First, so far we dealt with a Wigner chart of a single signal. Now, let us present the ensemble average of Wigner charts due to a set of signals that may enter the optical system (Mendlovic and Lohmann [1997]). So far the SW has been a pure number. Now, as a second generalization, SW(x, v) is a binary function of two variables, for a 1D object, with the following definition:

SWB(x, v ) =

1 (W(x, v)) > WTR, 0

otherwise.

(23)

The symbol (.) means an ensemble average operation, and WTR is a certain threshold value. Now, the SW is a binary function that is suitable for the estimation of Ax and A v. To be more precise, one should consider that the total area of W(x, v) is related to the total energy (eq. 10). For the following discussion it is advantageous to keep this property also for the SW chart definition. Thus, we define

SW(x, v)= sT swB(x, v),

(24)

when ST is selected in such a way that

(25) As a result, we get:

sT = f f swB(x, v) W(x, v)dx dv f f SWB(X, v)dx d v

(26)

2.4.1. The SW in a hyper space So far, our SWB(x, v) has been binary. In other words, the average properties of a set of signals, or the dynamic range capabilities of the system (which includes

282

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 3

a detector) are the same, wherever the SW is non-zero. In other words, the set of signals, or the capabilities of the system, are space invariant. That may not always be the case. For example, the signals (images) may be brighter than average around the center, or scattered light may affect predominantly the corners of the image plane. Having signals with wider dynamic range and a detector that can sense it, influences the number of degrees of freedom and the shape of the SW function. For example, consider having a priori information that a certain point source is a binary point source that can be located at 100 possible locations. This means that the input signal has 100 degrees of freedom. We may assume that this point source is imaged by a CCD camera. For a binary detector (pixel of the CCD) one needs 100 detector cells in order to know the exact location of the point source. Now assume that the detector has an infinite dynamic range. Then, based on the a priori information about the image (a point source), by only one pixel one may find the exact location of the point source since the intensity readout of the detector is proportional to the position of the point source. Thus, the dynamic range also affects the number of degrees of freedom. Therefore, instead of binarizing the Wigner chart, one should leave it as it is and define an SW hyper space (having a non-binary value per each x, v): SW(x, v ) =

(W(x, 0

v))

(W(x, v)) > WVR otherwise.

(27)

Now, the volume of the shape defined by SW is the number of degrees of freedom. The WTR is a suitable threshold value.

w 3. Super Resolution as SW Adaptation 3.1. LOSSLESS TRANSMISSION THROUGH A SYSTEM

For some optical system given by its SWY(x, v) and a given input signal described by its SWI(x, v), a necessary condition for transmitting the whole signal without losing information is SWI(x, v) C SWY(x, v).

(28)

Note that in this context C is a graphical relation comparing two shapes. As a result of this graphical relation we obtain a numerical condition: (Volume{SWI} -)

Nsigna I ~ Nsystem

(-- Volume{SWY}).

(29)

If the last two conditions are not fulfilled, some information of the input signal may be lost while passing through the system.

IV, w 3]

SUPER RESOLUTION AS SW ADAPTATION

283

In many cases the condition (29) about the SW numbers N may be satisfied, whereas the two SW shapes do not obey eq. (28), or SWI(x, v) r SWY(x, v),

Nsigna 1 ~ Nsystem.

(30)

That is the situation where super-resolution methods can be useful. 3.2. SUPER-RESOLUTION STRATEGY

We assume that the number of degrees of freedom of the system is larger (or at least not smaller) than the degrees of freedom of the input signal. Thus, from the information capacity point of view, the system should be able to handle the signal. However, let us assume that the SWI shape is not included in the SWY shape. For such a case we propose the SW-adaptation strategy, which adapts SWI to be included in SWY (Mendlovic and Lohmann [1997]). The adaptation of the SWI shape can be accomplished by using one of the basic optical processes: 9 x-sheafing of the SWI using free space propagation; v-sheafing using a lens; 9 rotation caused by the FRT; 9 x shift, or v shift due to a prism or grating; 9 changing of the aspect ratio (x scaled by a, v by 1/a) by means of image magnification; 9 any combination of these processes. Figure 1 portrayed those shape distortions. Based on this list of processes, the possibilities of obtaining the SW adaptation vary widely as the following examples show. SW adaptation may be accomplished by several of those processes in cascade, as illustrated in fig. 2. We start with a given SWI and SWY. We notice that SWI and SWY satisfy the condition of eq. (29) since they have the same area but different locations and orientations. First, we should adapt SWI to be enclosed by SWY. Now the adapted signal can be transmitted by the system. Since we transformed the original signal, in some cases there is a need for performing an inverse adaptation process after passing through the system (using the same list of processes mentioned above, but in opposite sequence). The final result is the output. In the example of fig. 2, the adaptation process contains three steps. First, a prism shifts the signal along the v direction. Then a FOU rotates the SW chart by 90 ~ Another prism then shifts the SWI chart to be included in the SWY chart. This example is trivial, of course, but it serves to illustrate our concept in general. 9

284

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OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

I swi(x, v)

Ifneeded Inverse

SW ~ adaptation

...

. pu[

V

SWY

SWl I X

2rism y

;X

ET.

;x

Prism r

Fig. 2. Schematic illustration of the SW adaptation process.

3.3. HYPER SPACEADAPTATION Having information with wide dynamic range and a detector with low dynamic range means that the system will not be able to reconstruct the complete information of the signal. It will show perhaps all of the frequencies existing in the signal, but not in their correct intensities. The tool used for the dynamic range adaptation is a grating, since in the Wigner domain a multiplication of a signal by a grating causes its Wigner chart to be replicated while the height of each replica is decreased. Thus, a grating conserves the volume (the energy) but it enlarges the area of the function occupied in the (x, v) plane. In other words, the SW hyper space adaptation process actually consists of two stages. In the first stage the dynamic range is adapted using a grating. Then, a 2D (area) SW adaptation process is performed as illustrated in fig. 2 in order to feed the frequency distribution in the (x, v) plane of the signal to the acceptance SW area of the system. The Fourier coefficients of the grating regulate the magnitudes of the v-shifted replicas of the original W(x, v). The geometrical super resolution in terms of SW adaptation is illustrated schematically by fig. 3. This is the case where the spatial resolution of the viewed background is much finer than the spatial resolution to be viewed by the sensing device. Figure 3a is the SW function of the signal (SWI), and fig. 3b is the accepted SW of the system (SWY). In order to increase spatial resolution, the following SW adaptation process is applied. First, some of the dynamic range degrees of freedom are converted to spatial degrees of freedom, for example, by a grating. The SWI after this stage is shown in fig. 3c. Then, based on time

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285

Fig. 4. SW hyper space adaptation process illustrating sub-pixeling operation. (a) SWI. (b) SWY. (c) SWY after dividing each pixel into three regions.

multiplexing (using the temporal degrees of freedom), each time slot, a part of the SWI shown in fig. 3c, is transferred. Figure 4 demonstrates another example where SW hyper space adaptation

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OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 3

is performed. In this case the adaptation was performed with the help of the detector which is a part of the system. The adaptation was done by dividing each detector's pixel into three regions. This operation allowed the detector to sense spatial frequencies which are three times higher, but it decreased its dynamic range. This operation adapted the SWY to the SWI charts and allowed a full transfer of the signal's information. 3.4. G E N E R A L I Z A T I O N S

So far, the SW chart is a function of the spatial parameters x and v. This can be generalized by taking into account all other parameters: temporal information, wavelength, polarization, etc. This leads to the following SW chart: S W ( x , Vx, y, Vy, t, ~,, POL, . . .).

(31)

This generalized definition is useful for the application of the SW-adaptation process not only with spatial parameters but also with all other proposals for achieving super resolution. Now the adaptation process is done on the M-dimensional SW function where M is the number of parameters involved; for example, M = 7 in eq. (31). For example, if it is known a priori that the signal is temporally constant, then the SW space can be divided into many time slots. Each time slot handles different spatial windows. As a result, the total spatial resolution is increased. This approach is equivalent to the 'time multiplexing' approach for super resolution that was suggested by Francon [1952] and by Lukosz [1966]. 3.5. SURVEY OF T H E E X P L O I T E D SIGNAL C O N S T R A I N T S

The classification and the demonstration of the SW-adaptation process has been discussed by Mendlovic, Lohmann and Zalevsky [1997]. Below is a brief summary. Francon [1952] as well as Lukosz [1966] previously considered this issue. 3.5.1. Restricted object shape

The first family of examples is concerned with the spatial information of the object. One example is the trade-off between the finest detail of the object and its extent Ax. For instance, assume an object with finest detail of 6x that should be captured by a CCD camera whose pixel size is 6XCCD = M 6 x , where M is

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SUPER RESOLUTION AS SW ADAPTATION

287

Fig. 6. Adaptation using a lens.

a magnification factor. We assume that the signal and the camera (system) have the same number of degrees of freedom (number of pixels). Figure 5 illustrates the example. The ratio A x / 6 x = Ns is not changed by the magnification process. In that sense the process of magnification is a simple case of SW adaptation. Another example of this type of adaptation is a human eye looking at a distant bird. If the bird contains too fine details, once again an adaptation can be executed by a magnification device. Here it is common to use a telescope (Kepler or Galilei). Figure 5 illustrates this type of adaptation. In both examples the object had to be magnified in order to adapt the image resolution to the resolution capability of the detector. The price to be paid is a smaller object field. A third example is connected with coupling an optical signal into a GRIN fiber. Conceptually, the acceptance shape of the fiber in the Wigner plane is a distorted rectangle, sheared along the frequency axis (fig. 6). This phenomenon can be explained in the following manner: Input locations which are close to the upper outer part of the fiber can contribute only negative ray directions (spatial frequencies) which are inserted into the fiber. Locations which are close to the lower entrance of the fiber can contribute only positive ray directions. Input points which are located in the center of the entrance plane (the core) contribute

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OPTICAL SYSTEMS WITH IMPROVED RESOLVINGPOWER

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Fig. 8. Adaptation using a fractional Fourier transform.

a certain range of positive and negative directions. On the other hand, a shearing of the Wigner chart of the input signal along the frequency direction can be done using a lens. The focal power of the lens determines the amount of the shearing and should be matched to the GRIN acceptance Wigner shape (fig. 6). The fourth example concerning adaptation of restricted object shapes is related to the fact that the SW of many objects have a higher bandwidth around the center and lower bandwidth toward the edges. For example, a portrait photo with a neutral background belongs to that category. On the other hand, a common SW shape of a system is rectangular. Figure 7 illustrates this phenomenon. In order to transmit all of the signal's information through the system, a relatively big rectangular system SW should be used, which is expensive. Using the adaptation process illustrated in fig. 8, one can reduce the requirements and the price of the system. The input signal is minified so that its Wigner shape will be a rotated square. Then this square should be rotated by 45 ~ using the fractional Fourier transform

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289

Fig. 9. Adaptation using the moir6 effect.

(Lohmann [1993]) which can be easily implemented optically. Eventually, another magnification is performed for full adaptation to the SW shape of the system. These three adaptation steps are implemented by simple optical elements such as lenses and free space propagation distances. Thus, the fact that three adaptation steps are used barely affects the total price of the system. If needed, after transmission, inverse steps of this process might be performed in order to return to the original representation of the signal. Note that the final remark is relevant to most of the SW-adaptation examples. The fifth example is related to the moir6 effect, which was essential for a project called 'spatial pulse modulation' (Lohmann and Werlich [1971]). A spatial high-frequency signal was down-modulated by superposing it with a high-frequency grating. Then, a low-frequency filter eliminated unwanted terms, which were caused by unsuitable Fourier components of the demodulating grating (only two unwanted terms are shown in fig. 9). The output (far fight) is adapted to the low-frequency capabilities of the detector.

3.5.2. Temporally restricted signals A second type of a priori information is related to the time coordinate. In many cases, it is known a priori that the signal changes only slowly as a function of time, or not at all. This allows one to achieve super resolution using time multiplexing. We denote this action as 'temporal adaptation' of the generalized SW function. A different part of the SW chart is transmitted in each time slot. We must assume that the signal is constant over the scan duration. Figure 10 illustrates the first method for performing such a temporal adaptation that is based on synchronized moving pinholes. This method was introduced by Francon [ 1952]. Notice that the quality of the lens influences the light efficiency, but not the resolution. The resolution depends only on the size of the scanning

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OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

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Fig. 11. Time multiplexing by using a pair of scanning gratings.

pinhole. Francon's idea is the root of today's scanning confocal microscopes (Osterberg and Smith [ 1964], Lewis [ 1994]). The second approach was introduced by Lukosz and Marchand [ 1963] and by Lukosz [ 1967]. The method is similar to that of Francon, but instead of scanning pinholes, scanning gratings were used. The system is illustrated in fig. 11. In this suggested system the light efficiency is better and the integration time is shorter. The temporal adaptation of the generalized SW is done as follows: Each diffraction order carries a different section of the spatial frequency spectrum of the object. Each section has its own temporal carrier frequency. The analysis of the encoded information is done with the second synchronously moving grating. A detector with temporal integration is needed at the output. The spatial resolution of the transmitting system is low. Hence, most of the spatial information of the object must be converted into temporal information. w4 contains the details of temporal super resolution.

3.5.3. Waoelength-restricted signals Another type of signal adaptation is connected with wavelength multiplexing (codification) (Kartashev [1960], Bartelt [ 1979a,b]). If the object is color neutral

IV, w3]

White

SUPER RESOLUTION AS SW ADAPTATION

1 Prism

Io(x)

-

Spectrum

291

transparency

Io(a2)

Fig. 12. ~, multiplexing adaptation.

(black-gray-white), one may use the wavelength ~, as a parameter which replaces the x coordinate. An optional system along this direction is shown in fig. 12. Using dispersive prisms, the temporal spectrum of the white light is spread spatially over the spatial input information Io(x). Each slot of the spatial information is encoded with a different wavelength and then sent into a fiber for transmission. The ~, multiplexed signal is transmitted through the fiber and reconstructed (decoded) again in the output using the same prism type: /o(X)

,/o(aA) =r TRANSMISSION :=~/B(a,~)

,/B(X).

(32)

3.5.4. One-dimensional signals A one-dimensional signal with high resolution may be represented as a twodimensional signal with less resolution along each direction (Lohmann and Werlich [1971 ], Bartelt and Lohmann [ 1982]). We start with an SW function that contains two spatial axes, of which one is unused (a 6 function). After adaptation, the SW function contains information in both spatial axes. Then, one performs the transmission through the system, and if needed, there should be an inverse adaptation step that returns the information into a single spatial axis. An example of such an implementation is a moir6 pattern of a one-dimensional highresolution object and a slightly rotated grating which produces a low-resolution two-dimensional raster representation (Grimm and Lohmann [ 1966]).

3.5.5. Polarization-restricted signals The last type of a priori information is connected with the polarization state of the signal. We assume the object to be unpolarized. But the optical system is able to transmit two sets of data in two orthogonal states of information. Here

292

OPTICAL SYSTEMSWITH IMPROVEDRESOLVINGPOWER

%(v)

[IV, w 3

P(v)

<

AVo

(a)

(b)

U L ( V - AVp / 2)P(v)

[,

,'1

AVp = Av o / 2

UR(V + AVp / 2)P(v)

V

I., i-~

AVp = Av 0 / 2 (c)

(d)

~

J. 1

V

AVp = Av 0 / 2

Fig. 13. Super resolution based on polarization. (a) Spectrum of the input. (b) Pupil transfer function. (c) Left part of the input spectrum shifted to the transfer band. (d) Right part of the input spectrum.

I

~-- Output

Input J Prism Fig. 14. System that performs polarization codification.

the resolution adaptation may be called 'polarization multiplexing.' A signal with high spatial resolution may be separated into two signals with lower spatial resolution. Each one of those two signals is transmitted using different polarization (Gartner and Lohmann [1963]). Notice that the two frequency sidebands are combined coherently. Figure 13 illustrates graphically the Fourier domain of the polarization-restricted signal before and after the encoding. Two polarizers at 45 ~ orientation must be inserted before the first Wollaston prism and behind the second Wollaston prism. The system that implements the polarization codification is illustrated in fig. 14. In this figure, the first Wollaston prism causes the different polarizations to be shifted to different spatial locations in the Fourier domain. Thus, through the aperture of the Fourier plane, half of the spectral information is passed with

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SUPER RESOLUTION BASED ON TEMPORAL CONSTRAINTS

293

polarization POLL and the other half is passed with polarization POLR (see fig. 13). Altogether, no spectral information is lost despite the fact that only half of the spectrum is passed in each one of the polarizations individually.

w 4. Super Resolution Based on Temporal Constraints The essence of the previous section on 'SW adaptation' was that optical signals are often less general than what the transmitting optical system would allow. The system is over-specified, which is a waste. 'SW adaptation' is a general concept for exploiting the unused capacities of the system. One dramatic waste has to do with the temporal aspect of optical signals and systems. Expressed in simple terms, a lens performs equally well for timeconstant objects as for objects which vary at a rate of 100GHz. For human consumption, a lens with 100 Hz temporal bandwidth would be sufficient. The human detectors (i.e, our eyes), cannot resolve higher frequencies, which was apparently good enough for mankind's evolution. Technical detectors may be faster, but they are still very slow compared with a lens (chromatic aberrations dictate the tolerance for the temporal modulation of optical signals). Based on these comments it is very promising to attempt to achieve super resolution for objects which do not vary in time, or at least vary only very slowly. Francon [ 1952] proposed a super resolution system for such objects. His proposal is conceptually very simple, but not easy to implement, and furthermore is poor in terms of light efficiency. Francons's idea and Lukosz's improvement thereof [1966, 1967] have been sketched in w3, figs. 10 and 11. We will now describe Lukosz's method and its generalizations. The original Lukosz method achieved one-dimensional super resolution for spatially incoherent objects. The same setup also works for coherent objects, but the theory is more involved. The extension to two-dimensional super resolution is not trivial. The one-dimensional time is supposed to carry two-dimensional (x,y) information through the optical system. Nonetheless, it can be done, and it has been done. When Lukosz performed his experiments about 30 years ago, he had at his disposal only Ronchi rulings as gratings. Today, with Dammann gratings, all of those super resolution experiments give better results. The super-resolved high spatial frequencies may now be transmitted with the same transfer factor as the lower frequencies. This is because the Fourier coefficients of Dammann gratings can be synthesized almost arbitrarily. This recent design freedom of Dammann type gratings is exploited in w4.3 on 'high-frequency enhancement'.

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OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

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As we have seen, Francon's super-resolution idea can be viewed as the nucleus of the temporal super-resolution methods described so far in w4. But there are also other methods which can be presented as Francon derivatives. Historically, those methods did not evolve as a consequence of Francon's paper, which is apparently almost unknown. At any rate, we will mention in w4.4 on 'related topics' some systems which can be considered as temporal super-resolution experiments, at least in hindsight.

4.1. 1D TIME MULTIPLEXING SUPER RESOLUTION

4.1.1. System analysis." incoherent The analyzed system is shown in fig. 11. It differs from an ordinary imageforming system by two gratings, G l and G2, in the object plane and image plane, respectively. Their periods D = l/v0 are alike, assuming unit magnification. The gratings move synchronously with velocity v. 1 GP(X)

~

Ii II (a)

D

(b)

D

X

1

X

Fig. 15. Dammann gratings: (a) binary phase gratings; (b) binary amplitude gratings.

Lukosz [1966, 1967] used Ronchi gratings, since at that time more suitable gratings, the 'Dammann gratings', had not yet been developed (Dammann and Gortler [ 1971], Dammann and Klotz [ 1977]). The Dammann gratings are either binary phase gratings (with a Jr phase jump) or binary amplitude gratings as seen in fig. 15.

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BASED ON TEMPORAL

CONSTRAINTS

295

(Ap)2

TT T

I T T

II,

v

(a) -)

(AQ)-

t

t

t

t

t

t v

(b) Fig. 16. Fourier coefficients of Dammann gratings: (a) binary phase gratings; (b) binary amplitude gratings. Those gratings are related as 1 GO(X ) = -~(Gp(x) + 1).

Gp(x) = 2GQ(x) - l,

(33)

Their Fourier coefficients correspond as

Ap(m)

= f2AQ(m)

/

2Ao(0 ) - 1

if if

m*0, m=0.

(34)

The special feature of Dammann gratings is that their coefficients are almost constant, up to a limit. More specifically:

[Ap(m)]2 =

1 0

if Iml ~< ( N - 1)/2, otherwise.

(35)

If such gratings are used as diffraction objects, one would observe N equally bright diffraction orders. A Dammann grating with N > 200 has been realized. In our case, N = 7 is reasonable. It is already valuable since N is the gain in resolution. For amplitude gratings with the same x-structure, the Fourier coefficients are

[Ao(m)] 2 =

1 ?-~ 1

if

m~0,

if

m=0.

(36)

Figures 16a and b show the two sets of gratings orders, respectively, for N = 7. The Dammann gratings are the essential components of the super-resolution system seen in fig. 17.

296

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OPTICAL SYSTEMS WITH IMPROVEDRESOLVINGPOWER

Fig. 17. Optical system for obtaining an increased effective aperture by using moving gratings. Now let us focus the analysis on the system proposed by Lukosz [ 1967]. G(x) is the moving mask. The movement is performed with a velocity o along the x direction. The input object is assumed to be illuminated by temporally coherent (monochromatic) light. The complex amplitude at z = 0 is denoted by Uo(x). We will start with the analysis of the spatially incoherent illumination case, which is mathematically easier. Then we will continue with the spatially coherent case. If one denotes by Uo(x) the input amplitude and by U(x) the output amplitude, one obtains in the static case

g(x) = G2(x)

Uo(x') G1(x') h(-x - x') dx',

(37)

where G1 and G2 are the input and output gratings, respectively, and h is the amplitude point spread function of the imaging system. Note that the output is expressed according to the same axis as the input. The output intensity is

I(x) = Ig(x)l 2= IGz(x)l z go(xl) go*(x2) GI(Xl) G~(x2)h(-x -xl)h*(-x-x2)dXl dx2.

x (3O

(X)

When moving the input and the output gratings with a velocity v, one obtains

I(x,t) = ]U(x,t)l 2 = [G2(x + ot)l 2 X

g0(x1 ) g g (x2) GI (Xl -- u t ) Go

G~ (X2 -- u t ) h(-x

- x1 )

h* (-x

- x2) dXl dx2.

(3o

(39)

For incoherent illumination, the ensemble average of the intensity is E{Uo(Xl) Uo(x2)*} =

IUo(xl)I26(x,--X2),

(40)

which yields

I(x,t) = IGe(x+vt)

igo(x,)12lG~(x,_vt)

2

CX2

2

h(_x_x,)]2 dx'.

(41)

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297

Decomposing the intensity transmissions of the gratings into Fourier series yields [Gl(x)l 2= ZAmexp(2:rimxvo),

]G2(x)[2= ~-~B, exp(2:rinxvo),

m

n

(42) where v0 is the basic frequency of the gratings. By substituting those expressions into eq. (41), one obtains

I(x, t) = Z

Z B,Am exp(2:rinxv0)

n

m

x

(43)

Io(x') exp(2~imx' Vo)lh(-x - x')l 2 dx' O0

x exp[2~ivovt(n- m)], where Io(x)= [e0(x)[ 2. The time duration for moving one grating period is r -

1

Vov

.

(44)

The detector should perform temporal integration over one period r or over many periods r. Due to the integration operation, the output of such a detector obeys

if

exp[2zrivoot(n - m)] dt = { 1

. . . . "t"

-r/2

0

(n - m) = 0,

(45)

( n - m) ~ 0.

Thus, the expression for the output intensity becomes

I(x) = Z B,A_n exp(2srinxvo) tl

(46)

x

Io(x') exp(-2~inx' Vo)[h(-x - x')[ 2 dx'. CX3

Performing a Fourier transform of the output intensity yields

I(v) = Z B,A_, tl

Io(x') exp(-Z:rinx' Vo)[h(-x o(3

oc

-

x')l 2 dx t

(47)

x exp[-2~ix(v - nvo)] dx. Since a Fourier transform of a correlation is the multiplication of the Fourier transforms of the correlated functions, one may rewrite the last expression as

](v) = ]o(V) Z B,A_,So(v- nvo), n

(48)

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OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

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where So is the optical transfer function (OTF) of the imaging system. Assuming that the aperture (the coherent transfer function, CTF) of the system is denoted by Po(v), then:

" So(V) =

f

CX3

'v) Po* ( v ' - ' v -~ ) Po (v' + -~

dr' ,

(49)

O(3

So(O) = trian(v/Vo)

if

Po(v) = rect(v/Vo).

By trian(v/vo), we mean a triangle of width 2v0 and unit height. Observing again eq. (48) yields an extended synthetic aperture existing for the input intensity. The corresponding total OTF is shown in fig. 18. This is the essence of the

,~ OTF(v)

V 6Av Fig. 18. Total OTF due to super resolution with amplitude gratings. super-resolution effect. The common fundamental spatial frequency of the two gratings v0 should match the bandwidth A v of the pupil function, A v = v0. If the two gratings are alike, the coefficient in eq. (48) will be A,A_, = IAnl2. Both gratings consist of binary transmission elements (zero or one) for the intensities. The object was assumed to be spatially incoherent. Hence we need to use GQ(X), the amplitude grating (fig. 15). The Fourier power coefficients IA,]2 have been shown graphically for N = 7 in fig. 16b. The OTF plateau reaches as far as predicted, but the plateau has a low level. A totally flat OTF is possible, if a low-resolution image (image transmitted through the original aperture of the optical system) of proper strength is subtracted digitally from the super-resolved image.

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4.1.2. System analysis." coherent Turning now to the coherent illumination case, we note the complex amplitude just after the first grating:

Uo(x) G1 (x + ot) = Uo(x) ~

Am exp[2Jrimvo(x + ot)] = Jo(x, t),

(50)

m

where v0 is the fundamental spatial frequency which should match the bandwidth A v of the pupil function, A v = v0. Equation (50) is actually a decomposition of the grating function G~(x) into a Fourier series. Using a spatial Fourier transform operation, one gets the spatial spectrum of Jo(x, t):

)o( V, t) =

f

cx~

Jo(x, t) exp(-2Jri vx) OG

= ZA,,,

Uo(x)exp{2~i[x(mvo- v)+ mvootl}

(51)

OC m

= Z A m ~ r O ( V - mvo)exp(2Jrimvoot). m

The pupil function at z - 2f is v

P0(V) = rect (~---~) .

(52)

Note that for the coherent input illumination case, the pupil function is the coherent transfer function (CTF). At plane z = 4f, just before the second moving grating, the arriving complex amplitude is

J(x,t) =

Po(V)fo(V,t)exp(2Jrivx)dv.

(53)

O(3

The output intensity just before being multiplied by the second grating can be expressed as

I(x, t) = J(x, t)J(x, t)* = Z~-'~A,,A~ n

]

Po(v)Uo(v-n'vo)exp[2Jri(xv+n'vovt)]dv

II t

OG

x

Pg(v)~(v-nvo)exp[-2Jri(xv+nvovt)]dv (N2~

.

(54)

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OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

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This instant intensity is to be multiplied by the intensity transmittance of the second grating G2 I(x, t) Gz(x - t)t) = Io(x, t),

(55)

G2(x) = Z Bm exp(2srimvox),

(56)

m

which yields Io(x,t) = Z

Po(V,)['~)(v2)(-/o(V, - n'vo) Uo*(V2- nvo)

A17,A~,Bm

nt ,n,m

cx~

oc

• {exp[2sri(x(vl + mvo - v2) + ( n ' - n - m)voot)]} dvl dr2. (57) One may notice that the temporal frequencies of Io(x, t) are ( n ' - n - re)you. It is obvious that the basic spatial frequency Vo relates to the grating period d as Vo - 1/d. Thus, the time duration for moving by one grating period is

1 r

d

-

-

(58)

v0u o" The intensity distribution of eq. (57) is recorded by the detector as shown in fig. 17. The detector should perform temporal integration over one period r or over many periods r. Due to the integration operation, the output of such a detector should follow 1 [ r/2 {1 ( n ' - n - m ) = O , . . . . exp{2sri[vout(n'-n-m)]} dt = (59) T d-r/2 0 ( n ' - n - m) ~ O.

By substituting/U 1 - I(x)

=

1 It/2

n'

/50(/tl + n' Vo)P~ (/t2 + nvo)

A17,A17Bm * f__~f~

Z for all

and/t2 = v2- nvo, the detected image is

Io(x,t)dt

7 d-r/2

=

V 1 --n'vo

-

n -

m = O

cxz

(60,

~c

x Uo(ktl) ~rff(/t2) exp{Zsri[x(/t, -/t2)]} d~, d/t2. This equation, after separating the two integrals, may be rewritten as I(x) = ~

A17'Po(~l + ntvo) ~ro(~l) exp(2alix~l) d~l

ZB17,_17 17

n t

0<3

1

A,,[90(lt2 + nvo) ~ro(/t2) exp(-2srix/t2) d/t2

X

]*

~X3

(61)

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301

If Bn'-n is constant for all nl, n with An,A~, ~ 0, we obtain 2

I(x) =

Z

A'fls~

+ nv0) Lr0(g) exp(2Jrixg) dg

,

(62)

n

since then the two integrals are separated terms. This intensity is apparently the modulus square of the image amplitude from a coherent image-forming system with effective coherent transfer function

P(~) = Z An[~~ + nvo).

(63)

n

Note that the indices n',n of An,A;, were confined to n and [n'[ not to exceed ( N - 1)/2 (N is the number of times the bandwidth of the pupil function P0(v) should be enlarged). Hence the Fourier coefficients Bm of the second grating ought to be non-zero and constant within the wider range Ira[ < N - 1. We summarize our demands on the two gratings: G1 should have a complex amplitude transmission such that [An[2 is constant within [n] ~< ( N - 1)/2 and zero otherwise, and the intensity transmission of the second grating ought to have homogeneous Fourier coefficients Bm up to [m] = N - 1. The demand on the second grating G2 causes a problem because G2 is an intensity transmission. That fact limits the freedom of finding a grating with the desired coefficients. One solution is to design the synthetic aperture to be wider than the object's maximal spatial frequency, which for itself is wider than the width of the system's aperture. Another solution is an amplitude grating with slits as narrow as 1/N of the grating period, and otherwise opaque. Hence, the light efficiency is very poor. A lenticular grating (an array of very small cylinder lenses) may replace the narrow-slit grating. The lenticular grating should be located on focal length before the image plane. Its light efficiency is obviously much better. The cause of our problem is obviously the second grating, which acts as an intensity transmittance immediately before the intensity of the light is recorded by the detector. It is possible, however, to let the detector record the complex amplitude of the image, if holography is employed. In other words, a tilted coherent reference wave must be added. This concept is shown in black box fashion in fig. 19 (overleaf). The recording step from I,(x) to u(x) can also be performed digitally. Notice that now both gratings can be phase gratings and no conditions o n Bm are required to obtain n' ..~ n.

4.1.3. Experimental verification So far, we have shown theoretically how to achieve an effective transfer function that is N times wider than the actual pupil function of the system. In this

302

OPTICAL SYSTEMSWITH IMPROVEDRESOLVINGPOWER Gl(x - vt)

u0,,

~)---~

~0(v)

[IV, w 4

UREF(X) = exp(2~iVRx)

Fourier ] - - ~ '

ou.er , - ~ .

~___~____~

I-- I

G~(x + vt) UREF(X)

Psynthelic Fourier

J

" [Fourier

I

~--

Fig. 19. The concept of super resolution for coherent illumination via holographic recording. section we present a laboratory experiment that demonstrates the operation of the proposed system (Mendlovic, Lohmann, Konforti, Kiryuschev and Zalevsky [1997]). In order to get automatically synchronized motion of the two gratings, we used a folded setup (in this case Gl = G2), as shown in fig. 20. First, a folded 4f imaging system was aligned. The system provides a magnification factor of -1. Then, another relay optics was used to transmit the image into a CCD camera. This camera also acted as the time integrator with a typical r of 20-40 ms. After aligning this system, a slit was added just in front of the folding mirror. This slit mimics the limited pupil of the system. The input pattern used for the demonstration was a Ronchi pattern of 12 lines per mm. The focal length f of the lens was 600 mm. The slit was adjusted to be 5 times smaller than the minimal size needed for transmitting the Ronchi image. An angular grating ('Roseta pattern') with a groove shape of a Dammann grating (N = 5) was used instead of a regular Cartesian grating. The use of a Roseta pattern was necessary for getting easily the moving effect. It also allowed us to Rotating Dammann Detector grating Roseta (CCD camera) / t-

Input object

f

Mirror

f

Fig. 20. Optical setup used for experimental demonstration.

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SUPER RESOLUTION BASED ON TEMPORAL CONSTRAINTS

303

Fig. 21. The Roseta mask is an angular grating with a Dammann groove profile.

adjust the most appropriate Dammann grating basic spatial frequency. The design of the Dammann grating and its alignment followed the condition A v = v0; i.e., the basic Dammann grating spatial frequency was 24 lines per cm. Figure 21 shows this mask. To get a uniform extrapolation of the aperture, we used a phase Dammann grating. The Roseta mask rotated with a speed of about 4 cycles per second. The image of the Ronchi object apeared on the CCD camera display as soon as we started the rotation of the Roseta mask. The illuminating source was a H e - N e laser. Figure 22 depicts the experimental results obtained by this system. It presents

Fig. 22. Output captured by the CCD camera with (a) clear aperture, (b) closed aperture without rotating Roseta mask, and (c) after adding the Roseta mask.

304

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 4

the imaging with a clear aperture and open slit (fig. 22a), the imaging after closing the slit as described above (fig. 22b) and the captured image after adding the rotating Roseta mask (fig. 22c). One can easily notice that the rotating Roseta mask improved dramatically the imaging ability of the system. The shape of the slit was rectangular. The fact that the grating lines are not strictly parallel does not cause any problem. The important feature is that the temporal up-modulation at the object plane is perfectly compensated by the temporal down-modulation in the image plane, due to the two gratings, which move synchronously.

4.2. 2D TIME MULTIPLEXINGSUPER RESOLUTIONIN x AND IN y 4.2.1. System analysis

Figure 17 (above) presented the setup for achieving the time multiplexing super resolution effect. Note that the synchronization of two moving gratings presented in fig. 17 is a tough problem. This problem was solved by folding the setup of fig. 17 and having both gratings on the same substrate, which guarantees synchronization independent of the actual velocity of the gratings. This folded setup, used in the acquisition of experimental results, is depicted in fig. 20. A fundamental mathematical analysis of the system of fig. 17 for one-dimensional objects was presented by Mendlovic, Lohmann, Konforti, Kiryuschev and Zalevsky [1997]. An extension of such an analysis for two dimensions is straightforward. Hence, it will be presented only briefly here. Contrary to the system described by Mendlovic, Lohmann, Konforti, Kiryuschev and Zalevsky [1997], here the modulation at the input plane is now performed by a two-dimensional Dammann grating Gl(x,y). A simple set of identical orthogonal gratings is not suitable, as can be concluded from

G,(x + vt,y) = Z Z A11Am exp[2zrivo(mx + ny)+ 2;rivomvt], tl

(64)

m

where v0 is the basic spatial frequency of G1 and v is its velocity in the x direction. Apparently, none of the spatial y-frequencies nvo is modulated temporally. Hence, the partial signals, modulated by carrier frequencies v), = nvo, are irreversibly mixed up. Rotation or shearing can solve this problem. For instance, rotating the grating G~ by an angle or, as seen in fig. 23a, yields Gl (x, y) --+ Gl (x cos ot - y sin ot, y cos ot +xsin a).

(65)

IV, w 4]

SUPER RESOLUTION BASED ON TEMPORAL CONSTRAINTS

305

y

x

(a)

(b)

Fig. 23. Schematic illustration of a sheared 2D grating: (a) Cartesian; (b) polar/spiral.

Moving the grating in the x direction, one obtains

G1 [(x + vt) cos a - y sin a , y cos a + (x + vt) sin a]

=Z

Z AnAm exp{2:rivo[m(x + ot)cos a - my sin a]

n

(66)

m

+ 2:rivo[ny cos a + n(x + ot)sin a ] } , which equals

Z n

Z AnAm exp{2Jriv0 [x(m cos a + n sin a) m

(67)

+ y(-m sin or + n cos a) + ot(m cos a + n sin a)] }. Note that the x-carrier frequency, vx(m, n) = vo(m cos a + n sin a), and the t-carrier frequency, /tin, = v0u(m cos a + n sin a), are directly proportional. By choosing a suitable angle, one can assume that no two temporal carrier frequencies ~mn are alike. This is demonstrated in fig. 24a for the case of Ira[ ~< 2, [n[ ~< 2. A suitable rotation angle for this case is 1 tan a - 5"

(68)

The main point in 2D time multiplexing super resolution is that no pairs

[vx(m,n), vv(m,n)] and [vx(m',n'), v,.(m',n')] should have the same temporal carrier frequency Itm,, in order to be distinct when decoded at the output plane. Since the x-components of the spatial carrier frequency are always proportional to the temporal cartier frequency with the same index pair, this requirement can be deduced from fig. 24 by down-projecting the spatial frequency vectors onto the vx-axis.

306

OPTICAL SYSTEMS WITH IMPROVEDRESOLVINGPOWER

VY AV o AVe, ' ~ 1 7 6 1 7 6 :

9

9 9

9

9

m,

9

9

9

9

9

9

9

9

9 9

Av Ao-,~--), ~

9w

t d

9

9

99 9

Ave9 o

9 9

99

Vx

[IV, w 4

Vy 9

9

9

9

9

9

9

9

9

9

9 9

9 9

9

9

9

9

9

9

9

9

Vx

9

(a)

(b)

Fig. 24. Schematic illustration of the spectral replica: (a) Cartesian; (b) polar/spiral.

Similar claims may be made for the Spiral Roseta grating illustrated in fig. 23b. This grating is rotated around its center, while the input illumination illuminates a region in its perimeter. Due to the comparatively small size of this region and its distance from the center of the Spiral Roseta grating, the grating there may be assumed to be a regular Cartesian grating with the spectral distribution seen in fig. 24b. The radial and the angular directions of the Spiral Roseta grating may be approximated as x- and y-axes, respectively, for the Cartesian grating. Due to the spiral structure, both angular and radial movement are created. Thus, the Cartesian grating is shifted by different velocities (which are once again chosen and designed carefully) in its x- and y-axes. Due to the time multiplexing, a synthetic aperture similar to the one described in w4.1 is generated:

P(Vx, vv)=

Z Z A,,,mPo(vx- nvo",~,- mVo'), m

(69)

11

where Po(vx, vv) is the original aperture of the system. Note that Vo~ and Vo~ are the basic spatial frequencies of Gn(x,y). For symmetric apertures, one should choose Vox = v o' = Vo.

4.2.2. Experimental verification Thus far, we have explained how to achieve an effective coherent transfer function that is N times wider than that obtained by the actual pupil function of the system. Now we present a two-dimensional super-resolution experimental result obtained by time multiplexing (Mendlovic, Kiryuschev, Zalevsky, Lohmalm and Farkas [1997]). As previously mentioned, in order to get automatically synchronized motion of the two gratings, we used the folded setup seen in fig. 20. First, a folded

IV, w 4]

SUPER RESOLUTION BASED ON TEMPORAL CONSTRAINTS

307

Fig. 25. The two objects used for the experiment. This figure presents the output plane with clear aperture. (a) 20 ~ tilted Ronchi grating pattern, 0.8 mm period. (b) 2D board of squares with period size 0.4 mm.

4f imaging system was aligned. The system provides a magnification o f - 1 . Then, another relay optics was used to input the image into a CCD camera. This camera also acted as the time integrator, with the integrating time approximately 20-40ms. Note that this integration time is needed for the decoding procedure. After aligning this system, a slit was added just in front of the folding mirror. This slit mimics the limited pupil of the system. The focal length f of the lens was 1500 mm. The slit was adjusted to be 5 times smaller than the minimal size needed for transmitting the input image. In the next step, a Roseta pattern that included a Dammann grating with N - 5 was produced. The Dammann grating has a spatial frequency of 1 line per mm. To get uniform extension of the aperture, it was a phase Dammann grating. The Roseta pattern was placed as shown in fig. 20 and rotated with a speed of about 4 revolutions per second. The input pattern used was a Ronchi grating pattern with a period of 0.8 mm which was placed at about 20 ~ (in order to examine the super resolution effect in both axes) and a two-dimensional board of squares with period size 0.4 mm. Figures 25a and 25b depict the two input patterns used. These patterns are seen in the output plane with a clear aperture. Figures 26a and 26b illustrate the output corresponding to figs. 25a and 25b, respectively, when the aperture was added but the Roseta pattern was not yet rotated. Then, we rotated the Roseta and a super resolution of the input patterns appeared in the CCD camera. The results thus obtained are illustrated in figs. 27a and 27b, and correspond to figs. 25a and 25b, respectively. One may see a good similarity between the input patterns and the patterns obtained with the super resolution effect.

308

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 4

Fig. 26. Output plane with aperture and without the time multiplexing super resolution. (a) 20~ tilted Ronchi grating pattern, 0.8 mm period. (b) 2D board of squares with period size 0.4 mm.

Fig. 27. Obtained output results with aperture and with time multiplexing super resolution. (a) 20~tilted Ronchi grating pattern, 0.8 mm period. (b) 2D board of squares with period size 0.4 mm.

4.3. HIGH-FREQUENCY ENHANCEMENT BY MEANS OF A SUPER-RESOLUTION OPTICAL SYSTEM FOR TEMPORALLY RESTRICTED OBJECTS As previously seen, the time multiplexing super-resolution approach divides the object's spatial spectrum into adjacent spatial frequency slots, each slot with a width of the cut-off frequency of the optical system. Since frequency enhancement is a case of spectrum distortion, the meaning of giving each decoding/encoding slot a different 'strength' is spatial filtering. In this section we propose to synthesize a power spectrum distortion by using modified Dammann gratings (Dammann and Klotz [ 1977]). The major advantage of this means of spatial filtering is the ability to enhance high spatial frequencies (i.e., frequencies which are higher than the transmission bandwidth of the system) with no filter in the Fourier domain. Every optical system suffers from low-pass filtering due to the size of its apertures and the limitations of the Fourier domain. Using this approach, an efficient enhancement of the high frequencies may be done in order to compensate this phenomenon. In addition, the spatial

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SUPER RESOLUTION BASED ON TEMPORAL CONSTRAINTS

309

frequency enhancement may be used for optical signal processing and patternrecognition applications. As was proven by Mendlovic, Lohmann, Konforti, Kiryuschev and Zalevsky [1997], the output intensity distribution for coherent illumination is described by I(x,y) =

(70) x Ui(Vx, Vv) exp[2zri(xvx +yv),)ldvxdvv where An,m a r e the Fourier series coefficients of the grating attached to the input object, P0 is the aperture in the Fourier plane, and Ui is the spatial spectrum of the input object. In order to achieve such a series, the Dammann gratings should be modified in order to provide a non-uniform impulse response. This type of grating is coined by us a 'generalized Dammann grating' (GDG). As one may see from eq. (70), the spectral information of the object is passed through a synthetic aperture (or a filter) of [9(Vx, Vy) = Z

An,m[90(vx - nvo, v,. - mvo),

(71)

n,m

which is wider than P0 itself, and thus more spatial frequencies of the input object may be transmitted through the optical system without being distorted. This is the effect of the time multiplexing super resolution. In our experiment (Mendlovic, Farkas, Zalevsky and Lohmann [1998]), we choose the Fourier series coefficients An,m in such a way that the synthetic aperture (the filter)/5 is a high-pass filter. Thus a high-pass filtering enhancement is achieved for a wide band of frequencies, without actually using a filter transparency or using lenses with huge apertures. The iterative equations used in order to find the transition points in the GDG are M

4 Z(-1)m+lxm + (-1) M+I if n = 0, An=

(72)

m=l

2 M ~-~ Z ( - - 1 ) m+l sin(ZJrnZm)

if n r 0,

m=l

where An are the coefficients and Zm are the transition points of the Dammann grating as seen in fig. 28. These equations are solved in an iterative manner.

310

OPTICAL SYSTEMS WITH IMPROVED RESOLVINGPOWER

-0.5

- Z2

-Z1

[IV, w 4

f

Z1

Z2

0.5

Fig. 28. Transition points of the Dammann grating.

I -

2V 0

I -'., -

V0

I V0

2v0

Fig. 29. Schematic illustration of the synthetic aperture. The thick arrows are the spectrum of the grating. The thin arrows are the spectrum of the input object.

Notice that, in contrast to the conventional Dammann case, our coefficients A, are planned to be unequal. In the GDG case, the Fourier coefficients are intended to represent a certain filtering function, such as a high-pass filter in our case (fig. 29). The coefficients An are samples of a high pass. Note that in the spatially coherent case, the GDG may be designed as a phase-only grating which has no DC term (A0). This design leads to high light efficiency. However, when spatially incoherent illumination is involved, an amplitude-only Dammann grating is required. This leads to an inherent distortion of the DC term and reduces the light efficiency. The Fourier coefficients of the GDG grating that was used in the experiment fulfilled A2 > A1. A0 > A1 since an amplitude GDG was used (see fig. 28). The coefficients of An for n > 2 are not of interest since in the experiment we used an input object whose spectrum was negligible for the spatial frequencies corresponding to n > 2. The A0 coefficient is the DC term of the grating. This term may be reduced if a phase grating is used (for the case of coherent light illumination). The experimental verification was similar to the previous cases of temporal

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SUPER RESOLUTION BASED ON TEMPORAL CONSTRAINTS

311

Fig. 30. Experimental results: (a) input grating; (b) scan of its spectrum; (c) output after high-pass enhancement. (d) scan of the output spectrum.

super resolution. The results are shown in fig. 30. The third order in fig. 30d is now stronger than the first order, which is what we wanted to achieve. 4.4. RELATEDTOPICS Super resolution has also been achieved with microwaves. A detailed review is beyond the scope of this article. However, we ought to mention the antenna arrays used in radio astronomy. In synthetic aperture radar, an antenna array is put together by sequentially replacing a single antenna at different locations. The impact of such array concepts upon optics has been reviewed by Goodman [1970]. Michelson's famous double slit telescope may be considered as the first optical array antenna. It yielded the autocorrelation of astronomical objects with a resolution beyond what the turbulent atmosphere allows. Labeyrie [ 1976] improved Michelson's concept considerably, and Weigelt [1991 ] improved it even further. He obtained direct images with a resolution more than 50 times better than what is achieved with an ordinary telescope of 6 m aperture. Confocal microscopy is another technique which achieved super resolution, not only laterally, but also longitudinally (Gu and Sheppard [1995]).

312

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 5

w 5. The Generalized Wigner Function for the Analysis of Super-Resolution Systems The aim of this section is to show a different point of view for examining superresolving systems. This alternate outlook arises by observing the different stages of the super-resolving system in the Wigner domain. Our presentation is not an exact quantitative one, but more qualitative and intuitive. 5.1. MOTIVATION The Wigner function provides a representation of optical wavefields which is intuitively appealing and has specific mathematical properties worthy of application to optical systems whose phase-space description is relevant. In this section we study the transformations undergone by a signal in the process of time multiplexing for the purpose of super resolution (Mendlovic, Lohmann, Konforti, Kiryuschev and Zalevsky [ 1997]) while using the generalized Wigner-function representation in three coordinates: position, momentum (space frequency) and wavelength, proposed by Wolf [1996]. Dependence on the wavelength has not been used formerly because the commonly known Wigner quasi-probability distribution function formalism (Wigner [1932], Lee [1995]) is borrowed from quantum mechanics. Note that there is a more generalized version of the Wigner representation that also includes the temporal coordinate (Bastiaans [1979a,b], Mendlovic and Zalevsky [1997]). However, for the purposes of this section the 3-coordinate representation is sufficient (Wolf, Mendlovic and Zalevsky [ 1998]). The system we analyze in this section is the time multiplexing super-resolution setup seen in fig. 17 (w4.1), which is capable of sending a one-dimensional object signal through a pupil, by segmenting it into parts separated by small differences in wavelength, and reconstituting the signal thereafter. An illustration of such a multiplexer consists of the following modular steps: (A) The object (input) signal u(x) is modulated by a moving grating G(x). (B) A Fourier transform takes place. (C) The signal passes the pupil. (B) A Fourier transform takes place. (A) The counter-moving grating (7 reconstitutes the signal as fi(x). 5.2. QUASI-MONOCHROMATICAND POLYCHROMATICWIGNERFUNCTION In the model of monochromatic paraxial optics, given a signal u(x,&) of wavelength ~, its Wigner function is a bi-linear functional of u, and a function of

IV, w 5]

GENERALIZED WIGNER FUNCTION FOR ANALYSIS OF SUPER-RESOLUTION SYSTEMS

313

position x and its spatial frequency v, defined by Wigner [ 1932] and introduced into information optics by Bastiaans [ 1979a,b], W ( x , v; X ) =

f

~ u (x + ~x ~ ' ,,~ )

u (x-

I , X) * exp(-2Jrix'v) dx'. 5x,

(73)

(ND

The polychromatic paraxial model (Wolf [1996]) allows for 0 < X c ~ , and results in the same formula. Since the wavelength shifts, AX, needed for multiplexing are small, the general formalism is dispensable, and we may allow X to be simply the third dimension orthogonal to the x - v phase-space plane. The change of scale due to AX is assumed to be negligible. The Wigner function has built into it the important property of covariance under inhomogeneous linear transformations, i.e., under translations of position and momentum, and under general linear maps of phase space produced by free propagation, thin lens, and compositions thereof, such as rotations corresponding to fractional Fourier transforms (Lohmann [ 1993]). Under these, geometric and wave optics remain in one-to-one correspondence (Castanos, Lopez-Moreno and Wolf [1986]). Changes in wavelength due to relative cross motion between object and screen are not included in the theory, and here we introduce them 'by hand' as paraxial, classical Doppler transformations depending on the small parameter )' = v/c << 1, where u is the relative velocity and c is the velocity of light. Then, for motion across the optical axis, v > v + y/~, ~ > ~(1 + y ~ v ) , W(x, v; ~) > W(x, v + y/~; ~(1 + y~v)),

(74)

where we disregard terms of order ),2. As shown in fig. 31, a monochromatic

Fig. 31. Transformation of the Wigner function under relative motion between signal and grating by y = v/c (the value of )' << 1 is grossly exaggerated). It comprises shift in v by 1'/)~ and projection in A by A/t =/12vT. A single level curve is plotted.

314

[IV, w 5

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

signal represented by a Wigner function in (x, v; ~) thus shifts in direction v and slants the phase plane in wavelength.

5.3. W I G N E R F U N C T I O N OF A SIGNAL S U P E R I M P O S E D BY A GRATING

Assume that an input signal u(x, ~) is transmitted through a grating G of period L, whose Fourier series expansion is

G(x) =

-~1

x/~lfL/2G(x) exp(-2~imx/L) dx.

Am -

Z Am exp(2~imx/L),

,I-L~2

m

(75) Then, the Wigner function of the signal after it has passed through the grating is

WG(X, V; ~) = fR dx' u (x - -~x ' ', ~) - ~1 Z Am exp(2crim(x - lx')/L) m

[

1

x u (x + 89 ~) - ~ Z Am, exp(2s~im'(x + 89 m t

1

exp(-2~ix' v)

LZA:~A~, d~'u(x- l'~x,x)*

-

m,m

•

[ (m'm (m'+m)) t

2;ri

=

L

x-

v-

2~

x'

u

,

Z AmA"-m exp(Z~i(n - 2m)x/L) W x, v - ~--L; m

=Z

(

")

rv2(~)ve x , , , - S Z ; x

,

H

(76) where we have replaced the summation index n = m + m ~. We see thus that the effect of a grating is to produce multiple copies of the original Wigner function spaced apart in spatial frequency v by 1/2L. In the last expression, the coefficient W~(x) represents the intensity of each copy, and is directly related to the shape of the grating. To give an example, consider a cosinusoidal grating,

a(x)-~

1(

2~x)

l+cos--if-

,

(77)

IV, w 5] GENERALIZED WIGNER FUNCTION FOR ANALYSIS OF SUPER-RESOLUTION SYSTEMS

315

Fig. 32. The five replicas of a Wigner function produced by a cosinusoidal grating G(x) = 1 (1 + cos[2Jrx/L])of period L in x. The replicas stand apart in the v axis. Above are the maxima of the coefficients WG(x)(they sum to unity); below are their r.m.s values (WG(x,t)). whose Fourier coefficients (eq. 75) are A - I = ~ 1 x/L,

A0 = ~1 x/L,

A I=~ 1 v~,

(78)

and all others are zero. Then, the summation in eq. (76) includes five cross terms for n = - 2 , - 1 , 0 , 1,2, and the grating will give five replicas of the Wigner function:

Wc(x, v; z) = ~6 w 1

0 1)

+ ~ cos +

, v-

Z ;~

( l ) "~

~2:rXw x,vL

~'

(1~ + g c1o s 4:rX ) w(x, v; ~) L

+ ~1 c o s2~x -~W

(l)

(x

, v + ~ 1; X

+~6 w x,v+Z;x

(79)

)

.

Note that the two extreme terms are true replicas, the middle term (n = 0) is positive, but presents oscillations in x of period L/2, while the n = + 1 terms oscillate in x with period L (see fig. 32).

316

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 5

Next consider a moving grating G ( x - ot), whose Fourier coefficients (eq. 75) will be Am(t)= Am exp(-2Jrimut/L). The Wigner function of the moving grating (eq. 76) will follow eq. (74) for all replicas, and the coefficient in brackets in eq. (76) will have the further time dependence

1

[

-ot

(80)

W~ (x, t) = -~ Z AmAn_m exp 2Jri(n - Zm) x L m

In the example of eq. (79), the oscillation can be pictured in fig. 32. Optical sensors which integrate over time will not see those terms whose coefficients oscillate. The time r.m.s, average of the coefficients in eq. (80) is

(W~(x, t)) = Z1 Z AmAn-'6D(n - 2m) = l lAn/212

(81)

m

for n even, and zero for n odd [6D(n- 2m) = 1 if n = 2m and 0 otherwise]. Therefore, in fig. 32, the n = +1 replica will vanish, while the central one, n = 0, will reduce to its constant term. The time-averaged Wigner function of eq. (76) will similarly halve the number of terms to n even, becoming

<Wc(x,v;A))=Z~-~lAk w x,v+-~-Z;A

1+

v-~

2)'

(82)

k

for integer k, where we have also replaced the effect of the relative motion on the phase space and wavelength coordinates. This is the effect of a moving grating on the Wigner function of a signal. In the actual multiplexer, moreover, we should be aware that AlL is a very big quantity compared to the v extent of the signal, so instead of having clearly separated replicas, the single original Wigner function W(x, v; 2) will unfold into superposed copies of itself, separated by wavelength A2

~.

-

k'-'---

Lc

for

k=0,+l,+2,...,

(83)

as suggested in fig. 33. The time averaging contained in eq. (82) refers to the Wigner function of the beam in the middle section of the multiplexer of fig. 17, just after the moving grating. 5.4. FOURIER T R A N S F O R M I N G AND PASSING T H R O U G H A SLIT

Once the beam leaves the moving grating, having unfolded into several superposed copies distinguished by small shifts in color, it is ready to undergo

IV, w 5] GENERALIZEDWIGNER FUNCTION FOR ANALYSISOF SUPER-RESOLUTIONSYSTEMS

317

Fig. 33. Multiple copies of an originally monochromatic Wigner function separated by wavelength.

passage through a constricting neck in phase space. If the neck is a pupil, it will restrict the horizontal spread of fig. 33. Since a Fourier transform produces a 90 ~ counter-clockwise rotation of the Wigner distribution in the x, v plane (Lohmann [1993]), the middle section of fig. 17 is equivalent to applying a vertical restriction spread slit Rw(V). We remind the reader that the passage of a signal u(x, ~) through a rectangular slit function of width w,

Rw(x) =

1 0

for - ~ wl < x < ~w, 1 otherwise,

(84)

, Rw(x)u(x, ~). has the effect of multiplying the signal by this function, u(x, )0 From its defining equation (73), it is easy to see that the support of the Wigner 1 function Wslit(x , V; ~) is then also restricted to - s 1w < x < ~w. In fig. 34 we show a simple signal (a Gaussian), a rectangular slit function, the signal after passing through the slit, and their corresponding Wigner functions. At this point, we wish to indicate that the discussion above is related to the following Wigner theorems: 9 If u(x) = UA(X)" UB(X), then W(x, v) = f W A ( X , V t) WB(X, V -- v')dv', which means that an x-multiplication in signal space means a 'blur' only in the v-coordinate of the Wigner chart.

318

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 5

o

.,..~

~0

o

.,..~

o

~z

o

. ,...~

~0 r~

,4

IV, w 5] GENERALIZEDWIGNER FUNCTIONFOR ANALYSISOF SUPER-RESOLUTIONSYSTEMS

319

Fig. 35. Multiple copies of the original Wigner function at the plane behind the slit. In this way each copy carries a different portion of the signal and is capable of passing the information through the spatial neck of the pupil.

9 If fi(v) = UA(~t) UB(V), then W(x, v) = f WA(X' , V) WB(X- X t, V)dx t, since fi is the Fourier transform of u:

(85) The multiplexed signal, after Fourier transformation by a lens (90 ~ counterclockwise rotation of the Wigner chart, Lohmann [ 1993]) and passage through the slit, will have the Wigner function shown in fig. 35. If the width of the slit, w, is such that it corresponds to the separation in v of the multiplexed copies, as shown in the figure, namely A v = w / ~ f = l/L, then no part of the Wigner function will be lost. If the pupil is wider than this quantity, there will be redundancy in the information. Note that f is the focal length (fig. 17). Following the neck, the transformation is undone by a further Fourier transform and a grating moving in the opposite direction. When the Lukosz setup is modified by reflecting the beam closely after the pupil (Mendlovic, Lohmann, Konforti, Kiryuschev and Zalevsky [1997]), the same optical elements act in reverse order, the original rotating Roseta grating is traversed by the light beam rotating in the opposite direction, and a CCD camera does the rest by timeintegrating the obtained output.

320

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 6

w 6. Super Resolution for Objects With Finite Size In this approach, super resolution is achieved by placing several gratings at specific locations within the imaging setup (Zalevsky, Mendlovic and Lohmann [ 1998]). In contrast to w4, the gratings are stationary. Hence the experiments are now simpler. As we will see below, the improvement of the spatial resolution comes at the expense of decreasing the field of view of the imaging setup. 6.1. THE SETUP AND THE LAYOUT OF THE THEORY

Figure 36 shows the optical setup. Notice that the object is much smaller than the gratings. The setup contains three gratings placed between the input and the output planes. The locations of the gratings are chosen such that super resolution will be achieved in the output plane. The grating periods must be chosen properly. In order to avoid any distortion of spatial frequency spectrum of the input, the gratings ought to be Dammann gratings. The Fourier coefficients of a Dammann grating are constant for indices Inl < I ( N - 1) and zero otherwise. We will now describe simultaneously the details of the setup and the sequence of the mathematical steps of the theoretical analysis. Figure 37 illustrates the 'computational path' of the theory. The various optical and mathematical steps are: 9 uo(x, O) ~ uo(x, zo); free space propagation. ~ u o ( x , z o) ~ uo(x,z-~); passing through grating Gl. 9 uo(x,z-~) ~ uo(x, 0); virtual back propagation. 9 uo(x, 0) ~ uo(#, 2 f - ) = uo (#//lf); optical Fourier transformation. 9 uo(#, 2f-) ~ u0(#, 2f-) rect(#/A#) - uo(#, 2f+); passing through an aperture. 9 uo(#, 2f +) ~ uo(x, 4f); optical Fourier transform. 9 uo(x, 4f) ~ uo[x, (4f - Zl)-]; virtual back propagation. 9 uo[x, ( 4 f - z l ) - ] ~ u0[x, (4f-zl)+]; passing through grating G2. ~ u0[x, (4f - Zl) +] --. u0(x, 4f); virtual forward propagation. 9 uo(x, 4f) ~ uo[x, (4f - z2)-]; virtual back propagation. ~ uo[x, ( 4 f - z2)-] ~ uo[x, ( 4 f - z2)+]; passing through grating G3. ~ uo[x, ( 4 f - z2)+] ---. uo(x, 4f); free space propagation. Figure 38 covers the front part of the setup. The upper two rays of this figure show how the finite aperture restricts the admissible deflection angles in the object domain. The lowest ray would be stopped by the aperture, unless the grating reduces the angular deviation from the axis. The grating performs a 'modulation' or 'encoding' in order to adapt the object signal to the channel

IV, w 6]

SUPER RESOLUTION FOR OBJECTS WITH FINITE SIZE

Fig. 38. Front part of super resolution setup.

321

OPTICAL SYSTEMS WITH IMPROVEDRESOLVINGPOWER

322

[IV, w 6

capacity, expressed in communication terminology. Later on, we will discuss the 'decoding' or the demodulating scheme. Note that we sacrifice the size of the object. When tracing the deflected ray in fig. 38 back to the object plane, we realize an ambiguity. The ray could have arrived at the aperture plane either from the object center, with deflection by the grating, or from an off-center object point, but then without deflection by the grating. This ambiguity problem becomes even more severe if the grating deflects not only into one diffraction angle but into many of them, even two-dimensionally. The ambiguity can be removed completely, as we will see later. The price to be paid is a decrease of the permissible object field, but that price is reasonable because the number of usable pixels remains unchanged. In other words, field size divided by pixel size is invariant. However, we do lose some light to redundant ghost images, as we will see later. 6.2. THEORYOF THE COHERENT CASE The input field distribution may be described as a superposition of spatial frequencies:

uo(x, z = O) =

oo

f

rio(V) exp(2Jrixv) d v.

(86)

oo

After a free space propagation of z0, we get

uo(x, zo)=

fcx~ fi0(v)exp [2~i (xv+-~-x/'l Z0 - ~2V2. )] dr. oo

(87)

Passing through the first grating yields

uo(x, z-~) = Z Am exp(2Jrixmvo )uo(x, zo ) m

f ~ AmfiO(v)exp[2~i (x(v+ m

=Z

v 0 ) + ~zo v/1 -

~2 v2)] dr,

oo

m

(88)

where v0 is the basic period of the first grating. We now go back to the input plane (in order to simplify the analysis):

uo(x,z = O) = ~ m

/cx~ c~

ZO

Am~o(v)exp[2~i (x(v+mvo)+ -~1)] dr,

(89)

IV, w 6]

SUPER RESOLUTIONFOR OBJECTSWITH FINITESIZE

323

where = V/1 -/~2V2 - V/1 - ~2( g q- mVo)2

r

/~2m 21/2

(90)

+ ~,2m Vo V.

From there we proceed to the pupil plane, z = 2f, and by assigning ~ to be Zf (v +mvo), we obtain u0(/t,z = 2f+) =

ZAmblO(/~~-mvo )

.z0 exp[2orl~+2]

(91)

.

m

(")

Next we multiply by the pupil function rect ~ aperture in the Fourier plane, and

~2 =

7 1 -(- / ~ 2 --~-mVo It_:, 1 2-7 ( [1-~, 1 g )2

while A/t is the size of the

~

f

2

9

(92) Notice that the V/1- (...)2 are always a cosine, describing the z-component of a wave vector. The so-called paraxial approximation corresponds to c o s a = V / 1 - s i n 2 a ~ 1 - 2 1 sin2a

(93)

Continuing further on fig. 37, we propagate the complex amplitude to the output plane without including as yet the effect of the second and third gratings:

uo(x,z

4f)

=

= ~ Am m

fi0

- mY0 exp 2ari ( ~

z0 + ~r

/t ] rect (~--~) d/t.

c~

(94)

Back propagating a distance of zl yields

uo[x,z =

=

(4f - zl)-]

Am m

uo --s - mvo o(3

X exp[2JlEi ( X ~

where ~3 =

( ()2 1--/~ 2

-t- Zo -~--02- Zl-~--~3)

rect()

(95)

/t

~2

2f2"

(96)

324

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 6

Multiplying the complex amplitude by the second grating y~,,B,, exp(2:rinvl~) (vl is the basic period of the second grating) yields uo[x,z = ( 4 f - z,) +] = Z Z AmBn m

rect

?to -~ - m v o

?/

• exp [2:ri (x~ + nVllA + z0 --~)2 -- zl -~q~3 )1 dlt. (97) After additional free space propagation to the output plane, one obtains uo(x,z = 4f)= Z Z A,,B,, m

n

/~ ( , , )

?~o --~ - mvo rect

oc

( [x(~-~ + nvl )zo

• exp 2n'i

+ --~02-

(.)

z, Zl]) --~03 + --~)4

dl t,

(98) where q~4=

1 - X 2 TT+nVl

(99)

.

Another backwards free space propagation brings us to the location of the third grating: uo[x,z = (4f - z2)-] =

AmB.

m

n

?to It _ mvo rect

{ [~(~ )zo

• exp 2~i

It + nvl

Zl Zl

+ - ~ 0 2 - - ~ 0 3 "q- - ~ 0 4 -

~1) 04

dlt.

(100) Multiplying the complex amplitude by the third grating ~-'~lCl exp(2Jrilv2#) (with basic period v2) yields u o [ x , z = ( 4 f - z 2 ) +]

=Z m

Z

Z n

AmB, CI l

{ [~(~~

• exp 2Jri

?~o -~ - m v o oo

-~ + nvl + lv2

rect

)zo z, z, ~1) + -~02 -

-Z03 + -ZO4 --

04

dl~,

(101)

IV, w 6]

SUPER RESOLUTION FOR OBJECTS WITH FINITE SIZE

325

and finally in the output plane,

uo(x,z

= 4f)

= ZZZAmBnC, m n l

{

exp 2:ri

x

rect

?to - ~ - f - m v o oc It + nvl

)z0 + --s

Zl + z,- - s

- --s

--s

+ --s

d/t, (102)

where r =

1 -- R 2

~i + nVl + lv2

(103)

.

Changing the variables v = f f - m vo results in fvc Uo(X,Z = 4 f ) = Z

Z

m

Z

f/

~o(v) rect

AraB, C, l

(v+mVo) Ag/Xf

~

(104)

x exp{2Jri [x(v + mvo + nvl + lv2) + Ct]} dr, where ~t =

zo~ m2 v 2 - zl~mnvovl

V[zoJ~mvo - z l ~ n v 1 - z2)~lv2] + --2-

(105) zl J~n2 v21 - z2/lmlvo v2

- -T

z2 ~ 12 v 2 - z2)~nlVl v2. -5-

This equation is based on the paraxial approximation of eq. (93). The superresolution effect is expressed as a synthetic wide-band aperture through which the spectral information of the input is passed. In order to obtain the superresolution effect in our case, we need three conditions to be fulfilled: m vo + n Vl + I v2 = O, zomvo-zlnvl-z21v2

(106) = 0,

z~ m2 V2 - Z l /~mn vo Vl

2

(107)

z1 ~/,/2

-5-

v2

z2 ~ 12 - z z ) ~ m l Vo v2 - --2v~ - z z ) ~ n l

(108) Vl v2 = N ,

where N is an integer number. The first condition is needed so that all the spectral slots passed through the different parts of the synthetic aperture will be added

326

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 6

with the same linear phase. The second condition is needed so that all the spectral slots passed through the different parts of the synthetic aperture will appear at the same lateral location of the output plane. The third condition is needed so that all the spectral slots passed through the different parts of the synthetic aperture will be added with the same constant phase factor. Note that the third condition is akin to the Talbot effect condition. In order to avoid distortions of the synthetic aperture, one needs AmB,,CI = const, for all m, n, l that are relevant for the image region in the output plane. From these three conditions it is easily understood why three gratings are needed. If, for instance, only two gratings are used (indicated here as v2 = 0) and assuming that v0 = v~, one obtains m = -n,

z0 = -z].

(109)

Thus, the two gratings are in imaging position to one another. Hence, those two gratings cannot both be between object and image plane. Having the first grating already before the object is inconvenient because the object domain may not be accessible for additional hardware. Having the second grating behind the primary image plane is even worse, since one needs an imaging lens so that the virtual image obtained in z = 4f will be imaged to the observation plane. This lens must be big enough so that no spatial information will be lost. Thus, indeed the 4f setup itself is a super-resolving setup. However, in order to view the superresolution effect, one needs very big lenses whose sizes are larger than the size of the aperture in the Fourier plane by as many times as the increment of the synthetic aperture. If three gratings are used, zo,zl and z2 may be all positive and smaller than 4f. For instance, one may choose Vo = v],

v2 = 2Vo,

z2 = zo/2,

zo = zl,

(110)

and the solution is then

n = 3m,

l = -2m,

N

zo -

3JtvS"-~m2

M

3JtvS"-~'

(111)

where M is an integer number. The expression for the output amplitude for this case is

uo(x,z = 4f) = ~ AmB3mC-2m m

f~

uo(v) rect

(v+mvo) Alt/~,f

exp[2Jrixv] dr.

vc

(112)

IV, w 6]

SUPER RESOLUTION FOR OBJECTS WITH FINITE SIZE

327

~CTF(v)

'< v0 -" z~B/2f

v

Fig. 39. Synthetic aperture. For resolution improvement we also need v0 -

Art

V

(113)

Hence, the grating frequency matches the size of the basic aperture. For this case the synthetic aperture is as seen in fig. 39. 6.3. GHOST IMAGES The three-fold sum of the result (104) contains some undesirable terms which do not satisfy the important second condition (107). The undesirable terms fall into two groups, which we call 'ghost images' and 'spurious images', respectively. Mathematically, let us separate all the other terms for which the conditions of eqs. (106)-(108) are not fulfilled into two groups: terms for which the condition z o m v o - z l n vl - z 2 1 v 2 = 0 is fulfilled and terms for which it is not fulfilled. All the summation terms that do fulfill condition z o m v o - z l n v l z21v2 - 0 will be shifted to a location of Xm,n,l =

~ ( z o m Vo - Z l n vl - z21 v2)

(114)

in the output plane, since the expression exp[2:riXm,,,lv] appears in the inverse Fourier integral of eq. (104). Thus, the resolution improvement requires a reduction of the relevant output field of view, since in order to view the image it must be restricted to a size of min{xm,,, t} in the output plane and thus also in the input plane (the magnification is one). The images that will appear outside of our output field of view are coined by us as 'ghost images'. Regarding the terms that do fulfill condition z o m v o - z l n v l -z21v2 = 0 but do not fulfill part of the other conditions of eqs. (106)-(108): Those terms would appear inside our observation window in the output plane and they would damage the resolved output image. Thus, we will require the Fourier coefficients A m B n C t of those m, n, l terms to be zero. For instance, in the case demonstrated

328

OPTICAL SYSTEMS WITH IMPROVEDRESOLVINGPOWER

[IV, w 6

by eq. (111) if the resolution improvement is a factor three, then one needs A-l,O,1 :g: 0, B-3,0,3 :g: 0 and C2,0,2 :g: 0. 6.4. THEORY OF THE INCOHERENT CASE

We now develop the theory of the incoherent case by using w6.2, which was devoted to the coherent case. We assume the same setup as before, with three gratings. Suppose our object is a single point at x = x'. According to eq. (104), the associated image amplitude will be

P~(x;x') = E E E AmB,,Ci m

n

l

rect

Ala/~.f

exp[2~rixv]dv.

(115)

vc

The associated image intensity is

F(x; x')

= IP~(x; x')l 2.

(116)

If the source location is within the legal region of x ~ < Ax/2, and if we ignore the ghost images outside of that region, the 'point spread function' F assumes a form suitable for convolution: F(x; x') = F(x- x'). Now we let a whole ensemble of mutually incoherent point sources start at z = 0:

f

Io(x) = J Io(x') 6(x - x') dx',

(117)

where Io(x) is the intensity of the input object. The resulting output image intensity will be cx)

IB(X) =

f

Io(x')F(x-x')dx'.

(118)

oo

The associated super resolution OTF is /.

~'(v) = J F(x) exp[-2Jri vx] dx.

(119)

We may also compute it in the spirit of Duffieux as the autocorrelation of the super-resolution transfer function of the coherent case: Y(v)= f

1 * (v-~ I v t )dr', fis(V+~V')fi~

where Ps is the CTF (a Fourier transformed point spread function Ps).

(120)

IV, w7]

WAVELENGTH-MULTIPLEXING SUPER RESOLUTION

329

If, for instance, the optical setup was suitable to generate an expanded square box coherent transfer function, the OTF would be a triangle: 'V

Ps(V) = rect (AvsR)

(121)

F(v) = trian AvsR

(122)

where A vsR is the expanded synthetic aperture achieved due to the super resolution effect.

w 7. Wavelength-Multiplexing Super Resolution In this section, we deal with wavelength-multiplexing super resolution. The fundamentals of this approach have been constructed by Kartashev [ 1960]. Later on, Armitage, Lohmann and Paris [1965] proposed an optical implementation for one-dimensional wavelength multiplexing, for encoding objects and for correlation. In their setup, a white light point source and a dispersive prism were used for encoding the input object. The light was then collected and inserted into the fiber. The decoding setup was similar. An extension to two dimensions and a white light correlator for two 2D spatial distributions were suggested. However, the 2D feature has been achieved using a moving slit (time multiplexing). Optical laboratory demonstrations of these proposals were performed by Bartelt, who used the different wavelengths of the white light source for the transmission of signals (Bartelt [1979a]), image correlation (Bartelt [1979b]), coordinate transformation (Bartelt [1981]) and for height contouring (Bartelt [1984]). The disadvantage in Bartelt's approach is that his setups contained moving elements (for time multiplexing in the second axis). Later, Paek, Zah, Cheung and Curtis [1992] suggested a wavelength real-time multiplexing approach for transmitting 2D information through a single-mode fiber. However, in their approach the encoding of the input pattern was done with an exotic element coined 2D MC SELDA. Friesem, Levy and Silberberg [1983] have suggested various optical setups for parallel transmission of images through single optical fibers. They suggested three main approaches for transmission of 2D images: angular-time multiplexing, wavelength-time multiplexing and angular-wavelength multiplexing. The disadvantage of the first two is that they contain moving elements, and the third approach suffers from low light efficiency due to the use of a slit. In addition, the transmission fiber is a multimode fiber and not a single-mode fiber as in our suggestion.

330

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IM w 7

In the approach suggested here, the white light source illuminates a diffractive optical element (DOE) in order to obtain the color spread. Every 2D spatial location is encoded with a specific pre-determined wavelength. Then, the wavelength-multiplexing approach is implemented for encoding 2D images, transmitting them via a single-mode fiber and then decoding them by a system which is similar to the encoding system. The final system is static (i.e., without moving components) and flexible to take into account the exact spectrum of the light source. A terminological note about notation (taken from the discussion part of Armitage, Lohmann and Paris [ 1965]): "If a given lens with additional devices has a larger passband for spatial frequencies than this lens would have when used alone, then we talk about super resolution". In this section, a fiber with a single-mode spatial bandwidth but a sufficient wavelength bandwidth is used. Thus, we coin our system a 'wavelength-multiplexing system'. 7.1. THE SUGGESTED SYSTEM

The suggested setup for the transmission of 2D images is illustrated in figs. 40 and 41. The system consists of two parts. The first part performs the encoding of the spatial information contained in the input pattern into wavelength information (planes 0-3). Then the encoded distribution is collected by a lens into a single spatial mode for transmission (plane 4). The second part of the system performs the decoding of the information received of the fiber (planes 5, 6). At first, a white light point source is collimated into a plane wave using a lens. The collimated light reaches the first DOE that consists of grating stripes (as seen in fig. 42). The number of stripes is determined by the spatial information along the vertical axis of the input pattem. For instance, if the filter consists of only one stripe, a 1D super-resolution effect may be obtained. Each stripe of the DOE creates a 1D rainbow spread over the input plane (see fig. 42). Each particular range of wavelengths illuminates the corresponding stripe of the input object. The spatial frequencies of the gratings in each stripe are determined in such a way that a particular range of wavelengths will illuminate particular stripes of the input object. Thus, every 2D spatial coordinate of the input object is encoded by a different wavelength. The period of the grating in each stripe must fulfill two conditions: first, that even the highest period rate is small enough to ensure a full spatial separation between the spectrum of the different diffraction orders, and second, as previously mentioned, the wavelength distribution coming from each stripe is shifted enough so that inside the aperture of the input pattern each pixel is illuminated by a different wavelength. If, for instance, the experimental

IV, w7]

WAVELENGTH-MULTIPLEXING SUPERRESOLUTION DOE1 Y

fY,

fx = fy, +fy 2 fY2

Input plane

fY2 fx = fYl + fY2

~

l'y~

331

DOE2

Single m o d e fiber

J_J --fy,"-I-- fY2 ~

~ fY2

plane 1

"-I-plane 2

~'2

Y

fY2

"7

--fy, - [ plane 3

plane 4

(a)

I

Ifi

li

-I-

plane 0 plane I

fy~

4 -f~,:-

J-J

fy~ q~fy -I

plane 2

plane 3

I

I plane 4

(b)

in, D U D ~176

I-_i

--fy, "-I-~ fY2 ~

I@

plane I

Du" D ~ fY2

_l_ J-J "]-fY2 ..]--fy, r]~ plane 2

fY]

J-J "]--fy,-I plane 3

I plane 4

(c) Fig. 40. Encoding part of suggested optical setup: (a) encoding part; (b) x - z projection of encoding part; (c) y - z projection of encoding part. demonstration is designed for 4 • pixels inputs, the DOE may contain only two stripes. The illustration of the color distribution over the input pattern, for this case, is seen in fig. 43. If the above requirements are formulated mathematically for the two-stripe type DOE, the conditions to be fulfilled are /lo _ / l o A1 A2 2/10~ ~ 89> A1

' Jl0 + 2A/l A2

(123) (124)

where Ai denotes the period of the ith grating's stripe in the DOE and i = 1 designates the highest period rate, 2o is the central wavelength of the white light

332

OPTICAL

SYSTEMS

WITH

IMPROVED

RESOLVING

POWER

[IV, w 7

DOE3 Y

f

f\ = f~, + fy~ -

Single mode fiber

t:,, plane 4

-+- f,~

Output plane

1~,

.,

fy~ "-I

plane 5

plane 6

(a) x

Tn

f-I

Iv

u

f-i v U

plane 5

i

i plane 6

(b) y

TAI

I v t

I~

f f:, J plane 5

fl h

v

u f,_~

(c)

v

:'

I

i;

'

f:_~ "-I

plane 6

Fig. 41. Decoding part of suggested optical setup: (a) decoding part; (b) x - z projection of decoding part; (c)y-z projection of decoding part. Note: three anamorphic systems perform imaging in the y-direction, and spreading or collecting in the x-direction.

point source, and A/I is the spectral bandwidth of the source. For a lens with focal length f , the horizontal dimension of the input pattern Lx should be

(

A0

~,0 )

A2 Ai f=Lx.

(125)

Equation (123) ensures the proper displacement between the wavelength distribution caused by each stripe so that different pixels of the input object are illuminated by different wavelengths. Equation (124) ensures that a sufficient separation exists between the first and second diffraction orders. These considerations are illustrated graphically in fig. 43. In plane 2, the spectrum is spread. It illuminates the input pattern placed in the first diffraction order created by the first DOE. According to our design of the DOE, each spatial pixel of the input is illuminated by a different wavelength. Thus, a spatial-wavelength transformation is achieved. Note that the Fourier

IV, w 7]

WAVELENGTH-MULTIPLEXING SUPER RESOLUTION

333

IIIIIIIII

I Il[llllllllllJl IIIllllllllllll Illl IIIIIlllllllllllllllllll]llllllll lllllllllllllllllllllllllllllllllllllllllllllll[ll IY i

~x y

Fig. 42. DOE grating structure.

First diffraction order

Zero

order

e

460 nm I !

460 nm

-

-

I 760 nm

Region of interest

Fig. 43. Illustration of color distribution over the input pattern.

transform is performed only along the wavelength spread direction (horizontal direction), while imaging is performed in the vertical direction (see figs. 40 and 41, above). In plane 3, an additional Fourier transform is performed. There, all of the spatial information of the input plane is encoded by wavelengths. Thus, in plane 3 the light distribution obtained is a summation of plane waves, while each plane wave has a different wavelength and its amplitude is proportional to the spatial transmission of the specific pixel of the input pattern that is encoded with that specific wavelength. The Fourier transform, performed in order to propagate to plane 3, is once again anamorphic: a Fourier transform in the horizontal direction and imaging vertically. Note that plane waves of different colors are tilted since they start from different locations at plane 2. In order to correct the direction of propagation, an additional DOE is used. This DOE is identical to the first one. The direction correction is critical in order to be able to collect the light efficiently and to enter it into the fiber. This second DOE is also placed in the first diffraction order of the first DOE. In plane 4, the plane wave is focused into a point in order to be transmitted via a single-mode fiber. The second part of the system is the decoding setup. It consists of a lens that enlarges the fiber point source data back to its original locations. Then, the plane wave is multiplied by a DOE which is the same grating as the two before. An anamorphic Fourier transform (Fourier in horizontal axis and imaging in the vertical axis) is

334

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 7

performed while the beam propagates from plane 5 to plane 6. In this plane a full restoration of the input pattern is obtained. No spatial information was lost, despite the fact that the image was transmitted through a single-mode optical fiber. 7.2. T H E O R E T I C A L ANALYSIS OF T H E S Y S T E M

In this section, a 1D analysis of the suggested system is performed. An extension to 2D is straightforward. The spectral intensity distribution of the white light point source is denoted by S(~). In our analysis, we deal at first with complex amplitudes, such as ~ e x p [ i t p ( ~ ) ] . The phase tp(~) is neither supposed to be known, nor needed further on. Eventually, when computing the output intensity, we apply a modulus square operation: S(~) = ] X / ~

exp[i ~(/~)]]2;

(126)

we will ignore this S(~) during first few steps of our analysis. Only at the end, when we perform the transition from complex amplitude, will S(~,) be reintroduced. Since the white light source is a point source, a collimated plane wave is obtained in plane 1. Thus, the field distribution after the first diffractive optical element (DOE) is ul(xl, ~,) = exp

~xl

,

(127)

where A is the period of the DOE grating. After performing the additional Fourier transform in order to propagate the light to plane 2, the field distribution is U2(X2, X) = ~) X2 -- - - ~

(128)

,

where f is the focal length of the lens. When multiplied by the input pattern u0(x2), the result is U2(X2,/~) = Uo(X2) 6

2 -- - ~ -

(129)

9

By passing to plane 3, another spatial Fourier transform is performed: U3(X3'/~) =

cx~

U2(X2,X) exp \

~

dx2 (130)

= exp ( - 2 ; ix3 ) u0 ( ~ - ) .

IV,w7]

WAVELENGTH-MULTIPLEXING SUPERRESOLUTION

335

When this distribution is multiplied by the second DOE, which is the decoding grating, the result is

u3+(x3"~')=u3(x3"~')exp( 2~ix3 A ) = u0 (f~A) .

(131)

Thus, according to eq. (131), u3+ is a plane wave with no spatial information. All of the spatial information of uo(x) has been encoded as wavelength information. Thus, it may be reduced into a point source in order to allow a transmission throughout a single-mode fiber. This light collection is obtained in plane 4:

U4(X4,1~)=~)(X4) 9uo(f~A).

(132)

Now, u4 is ready to be transmitted via a single-mode device such as a fiber. After the collimating lens, the light distribution of plane 5 again becomes us(xs, ~) = u0 ( ~ - ) .

(133)

It is then multiplied by the correcting grating: us(xs, ~)= exp ( 2rA

) u0 ( ~ - )

.

(134)

Performing the Fourier transform in order to propagate the beam to plane 6 yields

) U6(X6,/~)= foc~ us(x5,~,) exp ( --2Y/Tix5x6 Xf ( ~ ) ~( X6 f/~ .

_

.

.

.

dx5

u0,._.

(135)

Note that this Dirac delta is a physicist's delta; hence, its square is again such a delta. The chromatic intensity distribution is

[U6(X6,/~)[2=]U0(~) ]26(X6 _ fX ~-) .

(136)

Finally, we compute the panchromatic intensity distribution, with the source spectrum as weighting function:

(xDS(I~) u 0

6 x 6 - --~

dR = S(x6A/f)Iuo(x6 ) 2.

(137)

The size Ax0 of the object should be chosen such that S(~) is practically constant. Otherwise the edges of Io(x) - ]u0(x)]2 would not be recovered at the output.

336

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

Note that eq. (137) implies that the image due to S(xA/f).

Io(x) is

[IV, w 7

colored in rainbow fashion

7.3. EXPERIMENTALRESULTS The optical setup illustrated in figs. 40 and 41 has been implemented (Mendlovic, Garcia, Zalevsky, Marom, Mas, Ferreira and Lohmann [1997]). Three DOE elements with a single stripe were produced by a Scitex plotter and then reduced into sizes of 10 mm • 10 mm using a high-resolution reduction lens. The grating was designed to work with a broad-band light point source: a halogen lamp with a significant spectral range of 520-720 nm (~,0 = 620 nm, A2, - 200nm). Most of the spectrum of such a lamp is concentrated around the red and the infrared part of the spectrum. As seen in fig. 44a, the grating was fully reconstructed. Note that the different spatial coordinates of the input grating image were obtained in different wavelengths. In fig. 44b, one may see the information reconstructed only by the green range of the spectrum; in fig. 44c the information is carried by the yellow range, in fig. 44d by the orange range of information, and in fig. 44e by the red range. The camera that captured the experimental results was a CCD camera in which the infrared filter was removed.

Fig. 44. (a) Reconstruction in the output plane. (b) Information carried in the green spectral range. (c) Information carried in the yellow spectral range. (d) Information carried in the orange spectral range. (e) Information carried in the red spectral range.

IV, w 7]

WAVELENGTH-MULTIPLEXING

SUPERRESOLUTION

337

Fig. 45. (a) Input pattern. (b) Information carried by 490nm wavelength. (c) Information carried by 600nm wavelength. (d) Information carried by 640nm wavelength. (e) Information carried by 710nm wavelength. (f) Information carried in the red spectral range. (g) Information carried in the green spectral range. (h) Overall reconstruction in the output plane.

The system needed for 2D objects is almost exactly the same. The only difference is in the structure of the DOEs, which for the used input pattern has to contain two grating stripes. Since the system has a large number of elements, the DOE must be efficient. Moreover, all three DOEs must be identical. The DOE in this case contained two stripes with periods of A1 = 10 Bm and A2 = 14 ~tm so that the two rainbows created by each stripe caused the color distribution seen in fig. 43. The input object contained two stripes of gratings with periods of 1 and 2 mm (see fig. 45a). As previously explained, each spatial position of the output is

338

OPTICAL SYSTEMS WITH IMPROVED RESOLVING POWER

[IV, w 8

obtained in a different wavelength. Figures 45b through 45e illustrate this effect. They were obtained by placing narrow spectral filters of 490, 600, 640, and 710nm, respectively, in the output plane. Figures 45f and 45g were obtained by placing a broad red and a broad green spectral filter, respectively, in the output plane. The output obtained after multiplexing the information, transmitting it through a single-mode fiber, and demultiplexing it, is seen in fig. 45h. One may see the reconstruction of the input object.

w 8. Conclusions There are two types of super resolution: the 'classical' and the 'modem'. Classical super resolution deals with propagating plane waves, while the modem version relies on evanescent waves. The modem version is known as near-field microscopy. The classical super resolution may be divided into three subfields: geometrical super resolution, which is related to the spatial structure of the detector; noise-equivalent resolution, which is related to the internal noise developed in each cell of the detector; and diffraction super resolution, which is related to the resolution limitation due to the finite wavelength. This chapter reviews predominantly the recent developments in the field of diffraction-limited classical super resolution. The main novelties exhibited by the chapter are not only the new experimental tools and approaches, but also a new and general approach of adaptation between the signal's and the system's degrees of freedom and their space-bandwidth charts. The adaptation process serves to find the tools which exploit the a priori information. This is the nucleus of the entire article. The types of adaptation are one-dimensional and two-dimensional time multiplexing, adaptation based on a restriction of the object's field of view, and wavelength multiplexing. Some other topics related to diffraction-limited far-field super resolution were beyond the scope of this chapter. Among those approaches, we may mention triple correlation and the Synthetic Aperture Radar (SAR). Yet more examples are mentioned in w4.4. We hope that our article will stimulate the generation of new opportunities and approaches in the field of super resolution. For example, some of the methods described here, may perhaps be combined with near-field microscopy. Finally, we apologize to all those whose publications may not have been mentioned properly in this article with a finite length.

IV, w . A C K N O W L E D G E M E N T S ]

!!

339

List of Abbreviations and Symbols CCD

Charge-coupled device

CTF

Coherent transfer function

DOE

Diffractive optical elements

DR

Diffraction resolution

FOU

Fourier transformation

FRT

Fractional Fourier transformation

FSP

Free space propagation

GR

Geometrical resolution

NER

Noise equivalent resolution

OTF

Optical transfer function

SR

Super resolution

SW

Space-bandwidth product, in general

SWI

Space-bandwidth product of a signal

SWY

Space-bandwidth product of a system

WDF

Wigner distribution function

D

Diameter of the aperture

F#

F-number (stop number)

f

Focal length

Y

Spatial frequency

Av

Bandwidth

R

Distance

x,y

Cartesian coordinates in a plane

6x Ax

Size of a resolution element

u(x) ~(v)

Size of a CCD detector element, or size of a signal Complex amplitude of the object, or signal Fourier transform of u(x); spatial frequency spectrum

Acknowledgements The authors would like to acknowledge constructive interactions with R.G. Dorsch, D. Farkas, C. Ferreira, J. Garcia, I. Kiryuschev, N. Konforti, E. Marom,

340

OPTICALSYSTEMSWITH IMPROVEDRESOLVINGPOWER

[IV

D. Mas and K.B. Wolf. A. Shemer and G. Shabtay are acknowledged for improving the final quality of the manuscript. The authors realize the significance of primer publications, especially those of W. Lukosz. A.W. Lohmann also acknowledges support by DFG. D. Mendlovic and Z. Zalevsky acknowledge the support of the Israeli Ministry of Science.

References Armitage, J.D., A.W. Lohmann and P.D. Paris, 1965, Jpn. J. Appl. Phys. 4, 273. Bartelt, H.O., 1979a, Opt. Commun. 28, 45. Bartelt, H.O., 1979b, Opt. Commun. 29, 37. Bartelt, H.O., 1981, Opt. Commun. 38, 239. Bartelt, H.O., 1984, Opt. Commun. 49, 17. Bartelt, H.O., and A.W. Lohmann, 1982, Opt. Commun. 42, 87. Bastiaans, M.J., 1979a, J. Opt. Soc. Am. A 69, 1710. Bastiaans, M.J., 1979b, Opt. Acta 26, 1333. Blattner, R., H.P. Herzig and R. Dandliker, 1998, Opt. Commun. 155, 245. Castanos, O., E. Lopez-Moreno and K.B. Wolf, 1986, Canonical transforms for paraxial wave optics, in: Lie Methods in Optics, eds J. Sanchez-Mondragon and K.B. Wolf, Vol. 250 of Lecture Notes in Physics (Springer, Heidelberg) pp. 159-182. Cox, I.J., and C.J.R. Sheppard, 1986, J. Opt. Soc. Am. A 3, 1152. Dammann, H., and K. Gortler, 197 l, Opt. Commun. 3, 312. Dammann, H., and E. Klotz, 1977, Opt. Acta 24, 505. Dragoman, D., 1997, in: Progress in Optics, Vol. XXXVII, ed. E. Wolf (North-Holland, Amsterdam) ch. I. Francon, M., 1952, Nuovo Cim. Suppl. 9, 283. Friesem, A.A., U. Levy and Y. Silberberg, 1983, Proc. IEEE 71,208. Gabor, D., 1946, J. Inst. Electr. Eng. 93, 429. Gartner, W., and A.W. Lohmann, 1963, Z. Phys. 174, 18. Goodman, J.W., 1970, in: Progress in Optics, Vol. VIII, ed. E. Wolf (North-Holland, Amsterdam) ch. I. Grimm, M.A., and A.W. Lohmann, 1966, J. Opt. Soc. Am. A 56, 1151. Gu, M., and C.J.R. Sheppard, 1995, Opt. Commun. 114, 45. Kartashev, AT, 1960, Opt. Spectry. 9, 204. Labeyrie, A., 1976, in: Progress in Optics, Vol. XIV, ed. E. Wolf (North-Holland, Amsterdam) ch. II. Lee, H.W., 1995, Phys. Rep. 259, 147. Lewis, A., 1994, Diffraction unlimited optics, in: Current Trends in Optics, ed. J.C. Dainty (Academic Press, New York) pp. 233-254. Lohmann, A.W., 1967, Research Paper RJ-438 (IBM San Jose Research Laboratory, San Jose, CA) p. 1. Lohmann, A.W., 1993, J. Opt. Soc. Am. A 10, 2181. Lohmann, A.W., R.G. Dorsch, D. Mendlovic, Z. Zalevsky and C. Ferreira, 1996, J. Opt. Soc. Am. A 13, 470. Lohmann, A.W., D. Mendlovic and Z. Zalevsky, 1998, in: Progress in Optics, Vol. XXXVIII, ed. E. Wolf (North-Holland, Amsterdam) p. 263.

IV]

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341

Lohmann, A.W., and H. Werlich, 1971, Appl. Opt. 10, 2743. Lukosz, W, 1966, J. Opt. Soc. Am. 56, 1463. Lukosz, W., 1967, J. Opt. Soc. Am. A 57, 932. Lukosz, W., and M. Marchand, 1963, Opt. Acta 10, 241. Lummer, O., and E Reiche, 1910, Die Lehre vonder Bildentstehung im Mikroskop von E. Abbe (Vieweg, Braunschweig). Marks, R.J., 1993, Advanced Topics in Shannon Sampling and in Interpolation Theory (Springer, New York). Mendlovic, D., D. Farkas, Z. Zalevsky and A.W. Lohmann, 1998, Opt. Let. 23, 801. Mendlovic, D., J. Garcia, Z. Zalevsky, E. Marom, D. Mas, C. Ferreira and A.W. Lohmann, 1997, Appl. Opt. 36 8474. Mendlovic, D., I. Kiryuschev, Z. Zalevsky, A.W. Lohmann and D. Farkas, 1997, Appl. Opt. 36, 6687. Mendlovic, D., and A.W. Lohmann, 1997, J. Opt. Soc. Am. A 14, 558. Mendlovic, D., A.W. Lohmann, N. Konforti, I. Kiryuschev and Z. Zalevsky, 1997, Appl. Opt. 36, 2353. Mendlovic, D., A.W. Lohmann and Z. Zalevsky, 1997, J. Opt. Soc. Am. 14, 563. Mendlovic, D., and Z. Zalevsky, 1997, Optik 107, 49. Osterberg, H., and L.W Smith, 1964, J. Opt. Soc. Am. 54, 1073. Paek, E.G., C.E. Zah, K.W Cheung and L. Curtis, 1992, Opt. Lett. 17, 613. Toraldo Di Francia, G., 1955, J. Opt. Soc. Am. 45, 497. Toraldo Di Francia, G., 1969, J. Opt. Soc. Am. 59, 799. VanderLugt, A., 1992, Optical Signal Processing, Wiley Series in Pure and Applied Optics (Wiley, New York). Waldman, G., and J. Wootton, 1993, Electro-optical Systems Performance Modeling (Artech House, Boston) p. 190. Weigelt, G., 1991, Progress in Optics, Vol. XXIX, ed. E. Wolf (North-Holland, Amsterdam) ch. IV. Wigner, E., 1932, Phys. Rev. 40, 749. Wolf, K.B., 1996, Opt. Commun. 132, 343. Wolf, K.B., D. Mendlovic and Z. Zalevsky, 1998, Appl. Opt. 37, 4374. Zalevsky, Z., D. Mendlovic and A.W. Lohmann, 1998, Super resolution optical systems using fixed gratings, Opt. Commun. 163, 79.

E. WOLE PROGRESS IN OPTICS XL 9 2000 ELSEVIER SCIENCE B.V. ALL RIGHTS RESERVED

V

DIFFRACTIVE OPTICS: ELECTROMAGNETIC APPROACH

BY

JARI

TURUNEN, MARKKU

KUITTINEN

University of Joensuu, Department of Physics, PO. Box 111, FIN-80101 Joensuu, Finland

AND

FRANK W Y R O W S K I

Friedrich Schiller University of Jena, Institute of Applied Physics, Max-Wien-Platz 1, D-07743 Jena, Germany

343

CONTENTS

PAGE w 1.

INTRODUCTION

w 2.

SOME RESONANCE-DOMAIN EFFECTS . . . . . . . . .

346

w 3.

E L E C T R O M A G N E T I C T H E O R Y OF G R A T I N G S . . . . . .

349

w 4.

L I N E A R G R A T I N G S O P E R A T I N G IN F I R S T - O R D E R M O D E .

359

w 5.

L I N E A R G R A T I N G S O P E R A T I N G IN Z E R O T H - O R D E R M O D E

364

w 6.

M O D U L A T E D G R A T I N G S O P E R A T I N G IN F I R S T - O R D E R MODE .

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M O D U L A T E D G R A T I N G S O P E R A T I N G IN Z E R O T H - O R D E R MODE .

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G R A T I N G S O P E R A T I N G IN M U L T I - O R D E R M O D E

....

377

w 9.

CONCLUSIONS . . . . . . . . . . . . . . . . . . .

383

REFERENCES

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w I. Introduction The need to apply rigorous electromagnetic theory in the analysis of spectroscopic diffraction gratings was understood in the 1 9 5 0 s - the calculation of diffraction efficiency curves and the explanation of effects such as Wood anomalies (Wood [1902]) required exact solutions of Maxwell's equations. The need to use the electromagnetic approach arises because ruling engines and optical interferometry permit the fabrication of highly dispersive gratings with periods in the optical wavelength range, where polarization-dependent resonance and multiple-scattering effects can not be ignored. The electromagnetic theory of gratings is now well established, and a wide selection of numerical methods exist, which permit the analysis of most linear gratings. Many of these methods are described by Petit [1980]. Microstructures that are much more complicated than linear gratings are common in diffractive optics. There are several reasons to apply rigorous diffraction theory in wave-optical analysis and design of such microstructures. The rigorous approach must typically be employed in the non-paraxial domain of diffractive optics; i.e., if wavelength scale features exist in the profile or if the angle of incidence is large. Polarization-dependent wave transformations are often needed, which can be realized with wavelength-scale structured isotropic materials if the rigorous electromagnetic design approach is employed. Furthermore, use of wavelength-scale features can provide advantages such as increased wavefront conversion efficiency, even if the optical function is polarization independent and could be satisfied with larger features. Despite these reasons, rigorous electromagnetic theory became a standard tool in diffractive optics only in the 1990s. One reason is the emergence of microlithographic technology for fabrication of finely structured surface profiles. Another reason is the computational complexity of rigorous analysis of diffraction from complicated surface profiles, which is a challenge even with modem computers. This review is devoted to that domain of diffractive optics in which rigorous electromagnetic theory is necessary or useful. Some examples of such situations are discussed in w2. In w3, we describe one approach to rigorous analysis of gratings and discuss several others. Sections 4 and 5 deal with linear gratings, which employ the first and zeroth diffraction orders, respectively. Diffractive 345

346

DIFFRACTIVE OPTICS: ELECTROMAGNETIC APPROACH

[V, w 2

elements obtained by modulating these gratings are discussed in w6 and w 7. In w 8, we consider linear gratings which employ more than one diffraction order, and some final remarks are given in w9. Certain restrictions are made on the scope of this review. We consider only surface-relief type diffractive elements made of isotropic materials. Fabrication issues are only touched upon, because they have been treated in recent books edited by Herzig [1997] and Turunen and Wyrowski [ 1997].

w 2. Some Resonance-Domain Effects We begin with some striking effects which occur when wavelength-scale features appear in the profile of a diffraction grating. These effects cannot be anticipated if the electromagnetic nature of light is i g n o r e d - they illustrate the usefulness of rigorous electromagnetic theory as a design tool in diffractive optics. 2.1. TOTALABSORPTION OF LIGHT BY A GRATING One of the most striking phenomena in the interaction of light with gratings is the total absorption of light by a metallic grating, first predicted by Maystre and Petit [ 1976] and observed by Hutley and Maystre [ 1976]. Let us assume, in the geometry of fig. 1, that 2, = 633 nm, 0 = 11 o, h = 75 nm, d = 508 nm, c = d/2, and h - 3.54 + i4.36 (Cr). Now the grating period is so small that, according to the grating equation, no orders other than the specularly reflected zeroth order can propagate. If the incident field is polarized linearly with the electric field vector in the plane perpendicular to the grooves (TM polarization), the diffraction

Fig. 1. Incidence of a plane wave of wavelength ~, at an angle 0 on a binary reflection grating with groove depth h, groove width c, and period d, in a metal substrate with complex refractive index h = n+itr

V, w 2]

SOMERESONANCE-DOMAINEFFECTS

efficiency of the zeroth reflected order is r/0 ~

347

10-5; i.e., TM-polarized light

is absorbed almost totally by the grating. If, on the other hand, the electric field vector is parallel to the grating grooves (TE polarization), the efficiency of the reflected zeroth order is r/0 = 59.0%. For comparison, the reflectance o f a plane interface between air and Cr at 0 = 11 ~ is 64.8% for TE polarization and 63.7% for TM polarization. Obviously the grating could be used as a polarizer, which reflects TE-polarized and absorbs TM-polarized monochromatic light.

2.2. P H A S E R E T A R D A T I O N B Y A R E F L E C T I O N G R A T I N G

Let us assume, in fig. 1, 2. = 633 nm, 0 = 0, h = 87 nm, c = 200 nm, d = 400 nm, and h = 1.37 + i7.62 (A1). Again only the zeroth order can propagate: r/0 = 89.2% in TE polarization and r/0 = 71.4% in TM polarization. More interestingly, the phase delay in TM polarization is Jr/2 radians larger than in TE polarization, and thus the grating acts as a reflective quarter-wave plate. Because of different TE and TM reflectances, a circularly polarized output wave is obtained from a linearly polarized input wave if the grating grooves are oriented at an angle q~ = 51.3 ~ with respect to the incident electric field vector (at r = 45 ~ an elliptically polarized wave with eccentricity e = 0.895 is obtained). 2.3. BRAGG EFFECT IN A BINARY GRATING Consider next fig. 2, with nr = n l = 1.46, nt = n2 = 1, and Bragg incidence 0 = ~ , where sin 0B = ~/(2nrd). Let us choose d = 2. = 633 nm, c = 373 nm, and h = 1310 nm. N o w the (minus) first and zeroth transmitted and reflected orders can propagate. The diffraction efficiencies in TE polarization are r/_l = 97.8% and r/0 = 0.1% for transmitted orders, and r/_l = 0.9% and r/0 = 1.2% for reflected orders. In TM polarization, r/_l = 97.7% and r/0 = 0.4% for transmitted orders,

nr

T,

h.

i-~

nl

nt

I

c

~i

n2

n2 l'q '

d

Fig. 2. Incidence of a plane wave upon a lamellar dielectric grating.

348

DIFFRACTIVE

OPTICS:

ELECTROMAGNETIC

APPROACH

[g, w 2

and r/_l = 1.4% and ~0 = 0.5% for reflected orders. Thus the binary surfacerelief grating is an efficient beam deflector, much like a thick volume grating (Solymar and Cooke [ 1981 ]). 2.4. GUIDED-MODE RESONANCE EFFECT Extreme anomalous effects, known as resonance anomalies, can occur in the interaction o f light with a dielectric grating if d ~ ~, and the average refractive index of the modulated region is higher than the refractive indices o f the surrounding media. For example, 100% reflectance from a dielectric grating can be obtained over a narrow wavelength range, as (apparently) first predicted by Vincent and Nevi+re [ 1979]. This is illustrated in fig. 3, where the parameters in the geometry o f fig. 2 are nr = 1, nl = 1.9, n2 = n t = 1.46, h =/~, d = 0.6225~,, and c = 0.50~. At 0 = 0, we have 100% reflectance in TE polarization. When 0 (or ~,) is detuned the reflectance is reduced sharply: the grating is a narrow-band reflector. is I l 0.8

a

a

o

l

= 0.6 r o

0.4

0.2

0

-5

0

5

0 [deg] Fig. 3. Reflectance of a binary waveguide grating as a function of the angle of incidence. Several resonance anomalies with 100% reflectance are observed. Solid line: TE polarization. Dotted line: TM polarization.

2.5. BINARY BEAM DEFLECTOR FOR NORMAL INCIDENCE The inversion symmetry ~-m Tim o f the diffraction efficiencies o f any binary grating profile with even symmetry at 0 = 0 can be broken by the introduction -

o f wavelength-scale features. Assume, in fig. 4, nr = 1.46 and h = 0.12 + i3.29 (Au at X = 6 3 3 n m ) , x l = 6 8 n m , x2 = 2 7 0 n m , x3 = 5 1 7 n m , d = 6 1 3 n m , and h = 61 nm. The efficiencies of the reflected orders in TM polarization are

V, w3]

ELECTROMAGNETICTHEORYOF GRATINGS

349

Fig. 4. Binary reflection grating with two pulses per period under TM-polarized normally incident plane-wave illumination. 77-1 70.2%, r/0 = 2.4%, and ?]1 5.4% (22.0% is absorbed). Beam deflectors of this type were demonstrated by Saarinen, Noponen, Turunen, Suhara and Nishihara [ 1995]. -

=

w 3. Electromagnetic Theory of Gratings In this section some mathematical tools are presented, which permit one to analyze structures of the type considered in w2. We concentrate on linear gratings because a large majority of elements that can be designed using the electromagnetic approach is periodic, at least locally. 3.1. GRATING-DIFFRACTIONGEOMETRY Let us divide the space into two half-spaces with uniform refractive index, z < 0 and z > h, separated by an inhomogeneous slab in 0 < z < h. Region z < 0, which contains the incident field Ei(r) and the reflected field Er(r), has a real refractive index nr. The index distribution in 0 < z < h, where the field is denoted by Eg(r), may be complex: h(r) = n ( r ) + ire(r). Finally, the refractive index in region z > h, which contains the transmitted field Et(r), is real (nt) for transmission-type elements and complex (ht = nt + iK't) for reflectiontype elements (other possible geometries exist, such as a reflective dielectric multilayer-coated surface profile). Our task is to solve the unknown fields Er(r), Eg(r), and Et(r) from known Ei(r) and the refractive-index distribution. Maxwell's equations reduce to two

350

DIFFRACTIVE OPTICS: ELECTROMAGNETIC APPROACH

[V, w 3

x

n=nt

n =n r

n=n(x)

_ ~ r m Rm

O,n

z

0

h

Fig. 5. Diffraction of a plane wave by a grating located in the region 0 < z < h between two uniform media in half-spaces z < 0 and z > h.

independent sets in the y-invariant geometry of fig. 5. One set describes TE polarization, in which E(x,z) points in the y direction and all other nonvanishing electromagnetic field components (H~ and/-/~) can be expressed in terms of Ey. The other set describes TM polarization, in which E(x,z) lies in the xz plane and the non-vanishing components Ex and E~ can be expressed in terms of Hr. Consequently, the full solution of the electromagnetic diffraction problem can be obtained by solving two independent scalar differential equations instead of the vectorial Maxwell equations. 3.2. RAYLEIGH EXPANSIONS AND GRATING EQUATIONS

Let us assume a periodic two-dimensional index distribution h(x + d, z) = h(x, z) for all z, noting that this condition applies both inside the grating and (trivially) in the uniform regions around it. Then every scalar component U(x,z) of the twodimensional electromagnetic field satisfies the Floquet-Bloch pseudoperiodicity condition, U(x + d,z) = U(x,z) exp(iad),

(3.1)

where a = k n r sin 0, k = 2:r/~., and 0 is the angle of incidence. This condition discretizes the angular spectrum representations of all electromagnetic field components in the homogeneous regions z < 0 and z > h. So-called Rayleigh expansions, Ur(x,z < O) = Z m=--oo

Rm

exp[i(amx- rmZ)],

(3.2)

V, w 3]

ELECTROMAGNETICTHEORYOFGRATINGS

351

O0

Ut(x,z > h) = Z

Tm exp{i[amx + tm(Z-- h)]},

(3.3)

m=--oo

are obtained, where

am = a + 2arm~d,

(3.4)

= / [ ( k n r ) 2 - ~ ] 1/2 rm

if laml <

knr,

i [Ot2m- (knr)2] I/2 if laml >

knr,

(3.5)

and, if we assume for simplicity that region z > h is dielectric,

tm

=

[(knt)2-a2] I/2

if [aml ~knt,

i [ a 2 _ (knt)2]l/2

if lam[ > knt.

(3.6)

If the region z > h is metallic, the square root is taken such that the sum of the real and imaginary parts of tm is positive. Expansions (3.2) and (3.3) contain several types of plane waves with complex amplitudes Rm and Tm. All waves with real-valued rm and tm are homogeneous backward- and forward-propagating plane waves, respectively. Waves with imaginary-valued rm or tm are evanescent, with the surfaces of constant amplitude and phase perpendicular to each other. If region z > h is metallic, all plane waves in it are inhomogeneous: the surfaces of constant amplitude and phase are neither parallel nor perpendicular. Inhomogeneous waves (including evanescent waves) decay exponentially with the distance from the reference plane z - 0 or z - h. For homogeneous waves the conventional plane-wave interpretation gives am = knr sin 0 m in region z < 0 and am = knt sin Om in region z > h, where Om are the propagation directions in the appropriate half-spaces. Equation (3.4) gives the grating equation m~,

nr sin Om = nr sin 0 + d

(3.7)

for reflected diffraction orders, and m~

nt sin Om = nr sin 0 + --d-

(3.8)

for transmitted orders. The law of reflection and Snell's law follow from eqs. (3.7) and (3.8) when m = 0.

352

DIFFRACTIVEOPTICS: ELECTROMAGNETICAPPROACH

[V, w 3

3.3. DIFFRACTION EFFICIENCIES

The diffraction efficiency r/m of order m is the fraction of incident flux directed in that order. Its measure is the z-component of the time-averaged Poynting vector, calculated for the mth plane wave in the appropriate Rayleigh expansion. For reflected propagating orders we obtain COS 0m Om -- COS 19

iRm[2

(3.9)

while for transmitted orders, ~m-

Om

nt c o s

nr cos 0

ITml

2

(3 10)

in TE polarization, and ~m -

nr COS Om nt c o s 0

ITml

2

(3 11)

in TM polarization. Evanescent waves do not carry energy in the z direction; i.e., r/m = 0. Hence, for a dielectric grating, energy conservation requires that the sum of the efficiencies of the reflected and transmitted propagating orders is one. For metallic gratings the sum of the efficiencies of reflected orders is less than one and the difference is a measure of absorption. 3.4. EXACT EIGENMODE MODEL FOR BINARY GRATINGS

Let us assume a dielectric lamellar grating profile of fill factor f = c/d in region 0 < z
n(x,z) = [ n l

I

n2

if0~<x
(3.12)

if c <~ x < d.

It follows from Maxwell's equations that every scalar component Ug(x,z) of the electromagnetic field satisfies the Helmholtz equation

V2Ug(x,z)+(kn)2Ug(x,z) = 0

(3.13)

in regions with uniform refractive index n - nl and n = n2. Separation of variables Ug(x,z) = X ( x ) Z ( z ) in 0 < z < h leads to ordinary differential equations: d2 d x 2 X ( x ) + (k2n 2 _/~2)

X(x) -- 0,

(3.14)

V, w3]

ELECTROMAGNETICTHEORYOFGRATINGS

d2 dz2Z(x) + fl2Z(x) = 0,

353 (3.15)

where [32 is the separation constant. The solutions of eqs. (3.14) and (3.15) may be expressed as X ( x ) = {Alexp(iylx)+Blexp(-iylx) if0 <x
(3 16)

Z(z) = a exp (ifiz) + b exp [-ifi (z - h)],

(3.17)

and

where yl2 = k2n~- [32, y2 = k2n 2 _ [32, and A,, A2, BI, Be, a and b are free parameters. In TE polarization the boundary conditions require the continuity of X(x) and dX(x)/dx across the boundary x = c. In TM polarization, X(x) and n-2dX(x)/dx are continuous. By applying pseudoperiodicity to connect the fields at x = 0 and x = d, we obtain a transcendental eigenvalue equation (see Yariv and Yeh [ 1977] and Yeh, Yariv and Hong [ 1977]):

cos(Tlc)cos[~,2(d-c)]--~ 1 ( R ~Y1 + ~1~1 ) sin(~/lc)sin[~,2(d-c)] = c o s ( a d ) , (3.18) with R = 1 in TE polarization and R = (n2/nl) 2 in TM polarization. This equation has an infinite number of solutions for/32. Each solution represents a mode of the periodic waveguide defined by eq. (3.12), with a propagation constant/3 and a transverse field distribution given by eq. (3.16). 1 Equation (3.18) has at least one positive zero solution for/32. Each positive zero corresponds to a homogeneous eigenmode with a propagation constant [3m and effective refractive index

Nm =fin&

(3.19)

The solutions with negative values of/32 correspond to inhomogeneous eigenmodes, which decay exponentially in the z-direction and are, in a sense, analogous to evanescent waves in homogeneous media. The superposition CX3

Ug(x,z) - Z Xp(x) { ap exp (i[3pZ) + bp exp [-i[3p (z - h ) ] ) ,

(3.20)

p=l

with free ap and bp, is the desired field representation in 0 < z < h.

1 Parameters A2, B1, and B 2 c a n be expressed in terms of Al, which can be fixed by orthonormalization of the modes.

354

DIFFRACTIVE OPTICS: ELECTROMAGNETICAPPROACH

[V, w 3

It remains to match eq. (3.20) with the Rayleigh expansions (3.2) and (3.3) at z = 0 and z = h. In TM polarization, the method of moments leads (see Botten, Craig, McPhedran, Adams and Andrewartha [ 198 l a] and Sheng, Stepleman and Sanda [1982]) to a set of linear equations: oo

E

[(gmp~m 4- (~mp)am + (f~mp-gmp~m) exp(iymh) bm] = 2Ip(a),

(3.21)

[((~mp-Lmp~m) exp(iymh)am + (LmpYm+ 6mp)bm] = O,

(3.22)

m=l oo

E m=l

where

Kmp = n r2 E

r q- IIm(Otq)Ip*(%),

(3.23)

tqllm(O~q)I; (O~q)'

(3.24)

q = -o(:)

oo

Lmp=n~ E q = -oo

and the integrals

1 ~d n-2(x) Xp(x) exp (-iOtqX) dx Ip(Olq) = -~

(3.25)

can be evaluated analytically (in the TE case, slightly simpler expressions are obtained). The coefficients ap and bp are determined from eqs. (3.21) and (3.22), and the complex amplitudes are obtained from o<3

=

2-, -nrrp E ym[am-bn, exp(iyn, h)]Irn(Otp),

(3.26)

m=0 oo

Tp

-1 = n2t tp E Ym [am exp (iymh) - bm] Im(ap).

(3.27)

m=O

Optimum truncation of the infinite sets is important: a sufficient number of modes and Rayleigh orders must be retained to achieve adequate convergence, but an excessive number increases the computational cost. We have restricted our attention to binary dielectric gratings. Metallic (binary) gratings are discussed by Botten, Craig, McPhedran, Adams and Andrewartha [ 198 l b], multilevel profiles by Li [ 1993a], gratings with an arbitrary number of

V, w3]

ELECTROMAGNETIC

THEORYOFGRATINGS

355

grooves within the period by Miller, Turunen, Noponen, Vasara and Taghizadeh [ 1994], and the case of conical incidence (the wave vector of the incident wave is not in the x z plane) by Li [1993b]. 3.5. OTHERMETHODSOF GRATINGANALYSIS The exact eigenmode method presented above is perhaps the most natural approach for binary gratings. Unfortunately, for metallic gratings, finding the roots of eq. (3.18) in the complex plane is challenging (see Botten, Craig and McPhedran [1981] and Tayeb and Petit [1984]). A different implementation of the modal method is the Fourier-expansion approach developed by Burckhardt [1966], Kaspar [1973], Peng, Tamir and Bertoni [1975], Knop [1978a], and Nyyssonen and Kirk [1988] (see Turunen [1997] for a review). The widely used approach known as rigorous coupled-wave analysis, introduced by Moharam and Gaylord [1981] for volume gratings and applied to surface-relief gratings by Moharam and Gaylord [1982] (see also Gaylord and Moharam [1985]), is equivalent with the Fourier-expansion modal method. The differential method described by Nevi~re [ 1980] also employs similar principles. Fourier-expansion methods are directly applicable to almost arbitrary grating structures, whether they are of volume or surface-relief type. One simply divides the grating profile into a sufficiently large number of slices in the z direction, solves independently the eigenvalue problem in each slice, and matches all solutions using the electromagnetic boundary conditions. Stable algorithms for this purpose have been presented by Moharam, Grann, Pommet and Gaylord [1995] and Li [1996a]. The convergence problems for metallic gratings in TM polarization, described by Li and Haggans [ 1993], have been solved by a reformulation of the eigenvalue problem presented independently by Lalanne and Morris [1996] and Granet and Guizal [1996] (partial success was achieved by Turunen [1996]). The improved convergence was explained by Li [1996b] (see also Lalanne [1997]). Three-dimensionally modulated gratings can be modeled by extensions described by Vincent [1978], Moharam and Gaylord [1983], Br/iuer and Bryngdahl [1993], Noponen and Turunen [1994b], and Li [1997]. With today's workstations, ~20x 20-50 x 50 orders can be retained in the analysis of three-dimensionally modulated gratings (see Lalanne and Lemercier-Lalanne [1996] for efficient treatment of symmetrical profiles). The maximum size of the grating period for which reliable results can be obtained is ~10~,x 10~. for metallic and ~20A x 20A for dielectric gratings. Another important class of grating-analysis techniques is formed by integral methods (Petit [1980]). Kleemann, Mitreiter and Wyrowski [1996] presented a

356

DIFFRACTIVE OPTICS: ELECTROMAGNETICAPPROACH

[V, w 3

generalization that permits the analysis of profiles which contain non-functional (such as vertical or undercut) surface profiles. Other approaches include coordinate-transformation methods such as the C method (see Chandezon, Maystre and Raoult [1980], Li and Chandezon [1996]), the volume integral method of Greffet [1989], as well as finite-element and boundary-element methods described by Delort and Maystre [1993], Lichtenberg and Gallagher [1994], Mirotznik, Prather and Mait [1996], Hirayama, Glytsis and Gaylord [ 1997], and Prather, Mirotznik and Mait [ 1997]. No comprehensive comparisons between different analysis methods exist, and such a comparison is outside the scope of this review. However, we recommend Fourier-expansion modal methods because of their straightforward implementation, universal validity, and the ease of checking convergence by increasing the number of retained Rayleigh orders. However, other approaches have superior features in special circumstances, such as in the case of multilayer coated surface relief gratings.

3.6. MODELS FOR NON-PERIODIC GEOMETRIES

The assumption of pseudoperiodicity often simplifies considerably the solution of the diffraction problem, but space-domain methods such as integral, finiteelement, and boundary-element methods do not require this assumption. Hence they are directly applicable to the analysis of non-periodic elements but are restricted, due to computational constraints, to structures which are not orders of magnitude larger than ~. Thus non-periodic elements with macroscopic dimensions cannot be analyzed in a fully rigorous manner. Frequently, however, the wave transformation is encoded into variations of the local period of a carrier grating (local-linear-grating approach). Here the profile shape of the cartier grating determines the local diffraction efficiency, which must be predicted by a rigorous electromagnetic approach if the cartier period is in the wavelength scale. Local determination of the complex amplitudes of the scalar components of the electromagnetic field permits one to construct the field behind the element. This field can be propagated into other planes using efficient numerical techniques reviewed by Stamnes [1986]. Another useful technique is the field stitching method introduced by Layet and Taghizadeh [ 1996]. Here the aperture is divided into partly overlapping sections which are sufficiently small to be analyzed by rigorous theory. The overlap is chosen such that the fields constructed independently in adjacent sections can be matched. Once the total field is constructed, it can be propagated to another plane.

V, w3]

ELECTROMAGNETICTHEORYOFGRATINGS

357

3.7. THIN-ELEMENTAPPROXIMATION When the transverse scale of a surface profile expands but the depth scale is kept constant, the results given by rigorous theory approach stable asymptotic values (an important special case is a linear grating in the limit d / 2 ~ ~ ) . The local slope of the surface profile becomes so small asymptotically that one may ignore refraction and imagine that light rays propagate through the structure along straight lines, experiencing only a phase shift that depends on the profile shape (and a reduction of amplitude if the element has absorption). Similarly, for reflection-type elements, light rays travel along straight lines to the bottom of the profile and back, again experiencing a phase shift proportional to the local profile depth. In both cases the field at the exit plane of the element can be constructed easily and propagated to another plane using Fresnel and Fraunhofer formulas (with paraxial illumination, the transmitted and reflected fields are paraxial and the state of polarization of the incident field is preserved). The assumption of small local profile slope is not valid at vertical step boundaries, which always act as scattering centers but are globally insignificant in the asymptotic limit defined above, which we refer to as the thin-element approximation. It is straightforward to develop more elaborate models than the thin-element approximation by taking into account factors such as the change of propagation direction of light rays (treated locally as plane waves) upon refraction or reflection at a locally planar interface, polarization effects that result from Fresnel coefficients also in case of non-paraxial incidence, and even multiple scattering of rays within the structure. However, little attention has been paid so far to such extensions (Swanson [1991 ], Rossi, Bona and Kunz [1995], Noponen, Turunen and Vasara [ 1993]). 3.8. EFFECTIVEMEDIA The results given by rigorous grating theory approach stable asymptotic values also in the limit d/~, --~ O. In this limit only the zeroth reflected and transmitted orders can propagate, and light can no longer resolve the structure, which may be treated as a slab of effective material with two refractive indices NTE and NTM, which are in general different. Consequently, a subwavelength-period grating can modify the reflectance and transmittance of an interface, as well as the state of polarization of light. For a binary grating the effective refractive index No of the dominant eigenmode, defined by eq. (3.19), can be evaluated from eq. (3.18) for both states of polarization. The indices obtained, NTE and NTM, depend on d/A. The lowest-

358

DIFFRACTIVEOPTICS:ELECTROMAGNETICAPPROACH

[V, w 3

order approximations in the limit d//~ ~ 0 are obtained by Taylor expansions of the sine and cosine terms in eq. (3.18). The results,

JUTE -- [fn 2 + (1 - f ) n ~ ] 1/2,

(3.28)

NTM = [ f n l 2 + (1 - f ) n 2 2 ] -'/2 ,

(3.29)

and

can also be obtained by elementary considerations presented by Born and Wolf [1980]. Equations (3.28) and (3.29) are valid for binary profiles only. For an arbitrary refractive-index profile n ( x , z ) we have:

NTE(Z) =

[/o l

1/2

n 2 (x, z) dx

1

(3.30)

and

NTM(Z) =

~1 f0 d n -2(x, z) dx ]-1/2 ,

(3.31)

with eqs. (3.28) and (3.29) as special cases. The difference between NTE and NTM in gratings made of isotropic media is sometimes called form birefringence: NTE and NTM represent the ordinary and extraordinary refractive indices, respectively. Closed-form higher-order approximations for the effective indices have been derived by Rytov [ 1956], Bell, Derrick and McPhedran [1982], and Lalanne and Lemercier-Lalanne [1996]. The case of conical incidence (wave vector not in the xz plane) has been discussed by Haggans, Li and Kostuk [1993]. Two homogeneous modes propagate in any two-dimensionally periodic structure with a subwavelength-scale period, but they are degenerate at normal incidence if the structure is symmetric. In this case the polarization dependence disappears (see Grann, Moharam and Pommet [1994], Kikuta, Yoshida and lwata [ 1995], and Kikuta, Ohira, Kubo and Iwata [ 1998]). It appears difficult to derive unambiguous lowest-order expressions such as eqs. (3.30) and (3.31) for two-dimensionally periodic gratings. Proposals have been presented by Coriell and Jackson [1968], Br/iuer and Bryngdahl [1994], Kuittinen, Turunen and Vahimaa [1998], and Lalanne and Lemercier-Lalanne [ 1996]; the latter is quite accurate but requires numerical computations.

V, w 4]

LINEAR

GRATINGS

OPERATING

IN F I R S T - O R D E R

(a)

(b) 08

0.8 .,.=,

/".

0.6

~:~

' i"

'"

~

0.2 0 0

,

=~ o 6

9... "..

04

359

MODE

2 ................. .-r..':'.~.-'..-'..':'..-:"-"

,

,

2

4

6

8

0.4

~""

"'-

~ -. ...........................

"-" ''-"

10

~

~

;

6

8

~o

d/~

Fig. 6. Diffraction efficiency curves for orders m = 0 and m = + 1 of binary gratings with A p = ~,. (a) TE polarization. (b) TM polarization. Solid lines: f = 89 Dashed lines: f = 1. Dotted lines: f - 3.

w 4. Linear Gratings Operating in First-Order Mode We now proceed to the analysis of linear diffraction gratings, which produce at least two transmitted or reflected orders, the zeroth and (minus) first orders. Diffraction efficiencies of these orders, r/0 and r/_l, are plotted against d/)~, the normalized grating period 2, and the results are compared with those given by the thin-element approximation. Some design issues, i.e., finding a profile that maximizes r/_l in TE mode, TM mode, or simultaneously in both modes, are also considered. 4.1. BINARY GRATINGS

Assume, in fig. 5 and eq. (3.12), that ni = nl = 1.46, no = n2 = 1, and 0 = 0. In view of the thin-element approximation, r/0 = T [/2 + (1 ~]m - -

_f)2 + 2f(1 - f ) cos (kAp)],

4T f ZsincZ(mf) sin 2 (kAp/2) ,

(4.1) (4.2)

(m r 0), where sinc(x) = sin(:rx)/(:rx), Ap = ( n l - n2)h, and T is the Fresnel transmission coefficient (T ~ 0.965 for n = 1.46). Choosing h = /l/0.46 and f = 1/2, we have r/0 = 0 and r/_l = 4 / y t "2 for all d/,~. With f = 1/4 or f = 3/4, we obtain r/0 = 1/4 and r/_l = 2 / ~ 2. The results given by rigorous theory are shown in fig. 6, where zeroth-order and first-order efficiency curves for a g i v e n f

In spectroscopy (Hutley [1982]), it is natural to plot ~0 and r/_! against k/d, but the convention adopted here is more appropriate with general diffractive elements, which may be considered locally as gratings and are illuminated by monochromatic light. 2

360

DIFFRACTIVE OPTICS: ELECTROMAGNETICAPPROACH

[V, w 4

share the line style but are distinguishable because the latter approach zero when d/A ~ 1. These curves approach the thin-element results when d >> ~, but the convergence is rather slow, in particular i f f = 1/4 or f - 3/4. A comprehensive analysis of the limits of the thin-element approximation is outside the scope of this review because of the large number of parameters to be considered: d, c, h, n, 0 in both TE and TM polarization (see Pommet, Moharam and Grann [1994]). We saw in w2.3 that binary dielectric transmission gratings can be efficient beam deflectors at Bragg incidence if the period is chosen such that only orders m = 0 and m = -1 can propagate. Early discussions of this concept are due to Loewen, Nevibre and Maystre [ 1979] and Moharam and Gaylord [1982]; further details can be found in papers by Gupta and Peng [ 1993], Noponen and Turunen [1994a], and Gerritsen and Jepsen [1998]. The Bragg-type 3 solution presented in w2.3 is nearly polarization-independent. However, its aspect ratio (ratio of groove depth and minimum feature size) a ~ 5.0 is rather high. The aspect ratio can be reduced if the grating is designed for either TE or TM polarization. If f = 1/2 and h = 1.6342, (a ,~ 3.27), we obtain r/-i = 97.7% in TE polarization. Similarly, i f f = 1/2 and h = 2.1542, (a ,~ 4.3), we have r/_l = 97.9% in TM polarization. Enger and Case [1983] obtained experimentally r/_l > 85% in SiO2 using interference lithography and reactive ion etching. More recently, Glaser, Schr6ter, P6hlmann, Fuchs and Bartelt [1998] obtained r/_l - 96%. Nguyen, Shore, Bryan, Britten, Boyd and Perry [1997] measured r/_l = 94% for a grating with d = 350nm and demonstrated that corrugated SiO2 surfaces are as resistant to laser-induced damage as flat SiOe surfaces, which confirms the applicability of transmission-mode diffractive optics in high-power visible and UV laser technology. The Bragg-angle selectivity as well as the dependence of r/_l o n f and h/2, are analyzed in fig. 7 for the polarization-insensitive solution presented in w2.3. The angular selectivity is much weaker than in conventional volume gratings because of smaller relief depth and higher index modulation (small value of the KleinCook parameter). Reasonable fabrication errors in the fill factor f are tolerated, especially in TM polarization, and the design is not exceedingly sensitive to depth errors. Consider a metallic binary reflection grating with real refractive indices nr = n l = n and complex n2 = nt = h in fig. 2. The thin-element approximation still gives eqs. (4.1) and (4.2), but now with A p - 2nh. Some limits of

3 Note that these gratings are not Bragg gratings in the classical sense: the Klein-Cook parameter Q = 2;r(Md)(h/d), which should satisfy Q >> 1, is only ~ 10.

V, w 4]

LINEAR GRATINGS OPERATING IN FIRST-ORDER MODE

(a)

(b)

1[~,~~'~ 0.8 J

1 /

(c)

~

1

0.8

06~fi~

0.8

~ 06

0.4

~

0.2 0

~ 06

0.4

~

0.2 5

10

15

20

361

25

30

0 0.3

35

o.,

0.2 0.4

0.5

0.6

0 [deg]

0.7

0.8

0.9

0 1.2

1.5

2

f

2.5

3

h/~,

Fig. 7. Dielectric binary Bragg gratings. Dependence of diffraction efficiency on (a) angle of incidence, (b) fill factor, and (c) groove depth. Solid lines: TE polarization. Dashed lines: TM polarization.

Table 1 Efficiencies r/_ 1 of binary reflection gratings made of real metals but optimized by assuming perfect conductivity (Bragg incidence) Metal

A = 488nm

A = 633 nm

A = 1064nm

TE

TM

TE

TM

TE

TM

Ag

0.874

0.634

0.919

0.184

0.972

0.956

A1

0.892

0.736

0.877

0.805

0.936

0.914

Au

0.228

0.137

0.940

0.695

0.983

0.966

Cr

0.596

0.423

0.535

0.423

0.519

0.434

Cu

0.484

0.300

0.899

0.600

0.962

0.923

Mo

0.473

0.367

0.436

0.365

0.600

0.424

Ti

0.352

0.239

0.419

0.294

0.434

0.342

this approximation have been evaluated for perfectly conducting gratings by Gremaux and Gallagher [1993]. Hessel, Schmoys and Tseng [1975], Jull, Heath and Ebbeson [1977], and Cheo, Schmoys and Hessel [1977] have shown that r/_l = 100% can be achieved for many combinations of d/~, f and h, and simultaneously for TE and TM polarization. If only one polarization state is of interest, one possible solution in TM mode is d = ~, n = 1, f = 0.5 and h = 0.234~.. In TE mode, deeper and wider grooves are needed; for example, d = ~, n = 1, f = 0.75, and h = 0.407~. Little attention has been paid to finitely conducting gratings even though it is known that the assumption of perfect conductivity fails at near-infrared and visible regions. As illustrated in table 1, where the two structures presented above are considered, the efficiency depends strongly on the choice of metal

362

DIFFRACTIVE OPTICS: ELECTROMAGNETICAPPROACH

(a)

(c)

(b)

1

1

0.8

0.8

1 0.8

~ ""

i"

r

~

[V, w 4

0.6

0.6

0.4

0.4

0.2

0.2

0 10

20

30

40

50

0 [deg]

0 0.8

~ ~

0.6 0,4 0.2 0

0.9

1

f/,/i~

11

2

0.8

0.9

1

1.1

1.2

h/h o

Fig. 8. Binary AI Bragg gratings at A = 633 nm. Dependence of r/_ l on (a) angle of incidence, (b) normalized fill factor and (c) normalized groove depth. Solid lines: TE polarization. Dashed lines: TM polarization.

and the operating wavelength. At ~. - 488 nm the efficiency is poor for all metals, especially in TM polarization. At 2. = 633 nm, A1 is reasonably good also for TM polarization, and at 2. = 1064 nm several metals give acceptable values of r/_l. The efficiencies can be improved by readjusting f and h: for A1 gratings we obtain 17-1 = 0.898, 0.882, and 0.940 for 2 = 488 nm, 633 nm, and 1064 nm, respectively, in TE polarization. In TM polarization the corresponding efficiencies are 0.865, 0.854, and 0.926; i.e., the improvement is substantial in visible light. Figure 8 illustrates more closely the properties of the reflection-mode A1 gratings at 2. = 633 nm. Here j~ and h0 represent the (different) ideal values of fill factor and groove depth for each state of polarization. The near absence of Bragg selectivity, which is a characteristic of reflection-mode solutions in TM polarization, is evident from fig. 8a. In TE mode the solution is nearly independent o f f over a wide range. Yokomori [ 1984] showed that many profile shapes in addition to the binary profile can give high efficiencies at Bragg incidence if only two orders can propagate, and Miller, de Beaucoudrey, Chavel, Turunen and Cambril [ 1997] demonstrated slanted binary gratings, which display the Bragg effect at normal incidence and give high first-order efficiencies. 4.2. MULTILEVEL GRATINGS

High-efficiency deflection of an on-axis light beam is accomplished traditionally using triangular (blazed) grating profiles or, for convenience of fabrication by certain lithographic methods, their Q-level staircase approximations. Considering transmission gratings with step height ~.(Q- 1)/Q(n- 1), the thin-element approximation gives r/_l = TsincZ(1/Q),

(4.3)

V, w 4]

LINEAR GRATINGS OPERATING IN FIRST-ORDER MODE (a)

(b)

0.8 =

(D . ,...,

363

0.8

~=

0.6

o.6

04

0.4

0.2

0.2 0 2

4

6

8

0

1

2

4

d/A,

6

8

10

d/A,

Fig. 9. Efficiencies of Q-level stair-step SiO2 gratings. (a) TE polarization. (b) TM polarization. Solid lines: Q = 16. Dashed lines: Q = 8. Dotted lines: Q = 4.

(a)

(b) ,

0.8

1

0.8

0.6

.

.

.

.

.

.

.

.

.

.

.

.

.

0.6

-

t~

~ o.4

0.4

0.2

0.2 '

2

4

6

dA

8

10

0

1

2

4

6

8

10

d/k

Fig. 10. Same as fig. 9, but for a silicon grating.

where T is the Fresnel transmission coefficient. In fig. 9 we assume normal incidence from substrate (n = 1.46) to air: the results of eq. (9) significantly exceed the rigorous results when d is comparable to ~. A sharp drop occurs when d ~ 2~, followed by a peak at d ~ 1.5+l; an explanation in terms of multiple scattering was offered by Noponen, Turunen and Vasara [1993] (the peak disappears in the case of normal incidence from air to substrate). Some further comparisons with approximate methods were presented by Pommet, Moharam and Grann [1994]. Gratings made of high-index semiconductor materials for infrared applications suffer a far less dramatic reduction of efficiency than SiO2 gratings when the period is reduced, because the profile is considerably shallower. Figure 10 shows the efficiency curves for ideal multilevel Si gratings for +l = 1550nm (n = 3.474). In TM polarization, r/_l actually reaches its maximum at the 2Z Wood anomaly. Because of high n, the asymptotic values of ~7-1 in the limit d/~ ~ ~ are relatively low, but use of antireflection coatings significantly

364

DIFFRACTIVE OPTICS: ELECTROMAGNETIC APPROACH (a)

[V, w 5

(b)

.

0.8

0.8

0.6

~ 0.6

0.4

0.4

0.2

0.2 0 d/~

10 d/X

Fig. 11. Efficiencies of four-level A1 gratings. (a) TE polarization. (b) TM polarization. Solid lines: A = 1064nm. Dashed lines: ~. = 633 nm. Dotted lines: A = 488nm.

improves the efficiencies (see Pawlowski, Engel, Ferstl, Ffirst and Kuhlow [1994]). For example, a single-layer ~,/4 coating (TazOs, n = 2.054, thickness 0.122~,) improves the efficiency of a four-level grating with d = 5Z from 51.9% to 75.7% in TE polarization, and from 57.2% to 79.2% in TM polarization. Additional results for coated Q-level gratings were presented by Kleemann and Gfither [ 1998]. For metallic reflection gratings the groove depth is also rather shallow, ~,(Q- 1)/2Q, and the reduction of the efficiency is consequently modest, as shown by Shiono, Kitagawa, Setsune and Mitsuyu [1989] in TE polarization. Figure 11 illustrates the efficiency curves of four-level metallic gratings for three different wavelengths in both TE and TM polarization. In TE polarization, the efficiency is nearly constant down to d ~ 3~ and then drops rapidly. In TM polarization, strong Wood anomalies occur at integer values of d/Z, but efficiencies substantially higher than the thin-element value are observed for small d/~,. Shiono and Ogawa [ 1991 ] have used depth compensation to improve the efficiency in case of off-axis incidence.

w 5. Linear Gratings Operating in Zeroth-Order Mode According to the grating equations (3.7) and (3.8), only the zeroth transmitted and reflected orders can propagate if the period d is sufficiently smaller than Z. One might think that little can be done with gratings in this domain, but in fact such subwavelength-period gratings can introduce substantial wavelengthdependent effects in the division of power between the two zeroth orders, much like thin-film stacks (c.f. w2.4). Additionally, the transmittance and reflectance properties of subwavelength-period corrugated surfaces may depend strongly on

V, w 5]

LINEAR GRATINGS OPERATING IN ZEROTH-ORDER MODE

(a)

(b)

9. TM 0.8

1

"". . . . . . . . . . . . . . . . "'.

0.8

0.6 ;.7..1

365

............ '..

~ o.6

0.4

~

0.2

~

0

~

0.4 0.2 0

0

0.5

0

d/Z,

0.5

1

d/'X

Fig. 12. Efficiencies of the zeroth transmitted orders of metal stripe gratings made of (a) A1 and (b) Au. Solid lines: h = 50nm. Dotted lines: h = 200 nm. polarization, facilitating the construction of polarization components as pointed out in w2.2. 5.1. M E T A L L I C S U B W A V E L E N G T H T R A N S M I S S I O N GRATINGS

Perhaps the best known subwavelength-period grating is the conducting wire-grid polarizer, which transmits TM-polarized light but reflects TE-polarized light. These components are employed extensively at long wavelengths, such as in radio engineering. Subwavelength-period wire grids are difficult to fabricate for visible light, but the same effect is achieved with an array of metal stripes (thickness h, width d/2, rectangular cross section) on a dielectric substrate. These structures can be made using, e.g., electron-beam or phase-mask lithography followed by lift-off or electroplating. Figure 12 illustrates rigorously calculated TE and TM mode transmission curves for A1 and Au metal-stripe gratings for two values of h. When d = 200 nm, good polarizers are obtained when d ~ 0.5~. or smaller. This is indeed the conventional domain of operation of metal-stripe polarizers. With smaller values of h, such as 50nm, the TE transmission increases in the range d << ~. and the polarization ratio becomes unacceptable. However, an anomalous reversal of transmittance is observed (in particular for Au), where the TM-mode efficiency reduces close to zero but the TE efficiency is ~ 50%. This effect has been verified experimentally by Honkanen, Kettunen, Kuittinen, Lautanen, Turunen, Schnabel and Wyrowski [1999]. More detailed theoretical discussions of metal-stripe polarizers have been presented by Lochbihler and Depine [1993], Lochbihler, Polnau and Predehl [1995], and Lochbihler [1996]. Tamada, Doumuki, Yamaguchi and Matsumoto [1997] have presented experimental results for Al-stripe polarizers at ~. = 780nm, ob-

366

DIFFRACTIVE OPTICS: ELECTROMAGNETIC APPROACH (a)

(b)

0.8

0.8

-~ 0.6

0.6

~ o.4

0.4

~ o.2 0 200

[V, w 5

~ -

'

500

1000

~, [nm]

0.2 0

1500

2000

200

500

1000

1500

2000

~. [nm]

Fig. 13. Transmittance of a free-standing inductive A1 grid with square aperture, period 4 0 0 n m • 400nm, and fill factor 1/2. (a) Thickness h = 100rim. (b) h = 300nm. Solid lines: combined efficiency of all transmitted orders. Dotted lines: ~0- (Courtesy of V. Kettunen.)

taining TM/TE transmission ratios in excess of 1000 with d = 390 nm. Schnabel, Kley and Wyrowski [ 1999] considered the choice of material and period suitable for lithographic techniques and provided experimental demonstrations of Crstripe polarizers with periods close to the upper limit. Gruntman [1995] has presented experimental results for free-standing grids (no substrate). Another interesting device, which is in standard use in, e.g., radio telescopes and microwave ovens, is the inductive grid. This is a regular array of apertures in a conducting screen of thickness h; see McPhedran and Maystre [1977] for a detailed rigorous analysis under the assumption of perfect conductivity. The grid reflects long wavelengths very effectively, but produces several transmitted orders in the wavelength range A < d. Kettunen, Kuittinen, Turunen and Vahimaa [ 1998] analyzed the effect of finite conductivity in the performance of inductive grids, demonstrating that the assumption of perfect conductivity fails in the visible region. Figure 13 illustrates the transmittance of an A1 grid with square apertures, which fill 50% of the grid period (the zeroth-order transmittance is independent of the state of polarization because of the inversion symmetry of the structure). The goal to attenuate infrared (IR) radiation while transmitting visible and UV regions of the spectrum, preferably in the zeroth order, is clearly achieved. Use of thicker grids or smaller apertures improves the IR rejection, but reduces the transmission of visible light. Yet another potentially interesting component is the capacitive grid, which is the complement of the inductive grid; i.e., an array of metal islands on a substrate. According to Ulrich [ 1967a,b], a perfectly conducting capacitive grid features increasing transmittance when ,~/d is increased. Figure 14 illustrates the spectral transmittance of a structure complementary to that considered in fig. 13,

V, w 5]

LINEAR GRATINGSOPERATINGIN ZEROTH-ORDERMODE (a)

(b)

0.8

0.8

~ 0.6

0.6

~ 0.4

~ 0.4

~ 0.2

0.2

.,..~

0 200 500

1000

367

1500

2000

[nm]

0 200 500

1000

1500

2000

~ [nm]

Fig. 14. Same as fig. 13, but for the complementary (unsupported) capacitive grid. (Courtesy of V. Kettunen.)

with the metal islands unsupported in air 4. A thin grid (100 nm) behaves in the expected manner, but a thick grid (300 nm) has a transmittance maximum, which is not anticipated in view of the perfect-conductor model. 5.2. GRATING ANTIREFLECTION SURFACES

The concept of effective media discussed in w3.8, together with basic knowledge of thin-film theory, enables one to design subwavelength-period dielectric grating antireflective (AR) surfaces: a uniform layer of refractive index N = ~ and thickness h = A/4N between media with refractive indices n~ and n2 yields zero reflectance at normal incidence. Thus a binary dielectric subwavelengthperiod grating acts as an AR layer if its fill factor and groove depth are chosen appropriately. A corrugated AR surface etched in, e.g., SiO2 has an important advantage over a single-layer thin film: the fill factor f can be chosen to provide zero reflectance, while solid materials with sufficiently low refractive indices for this purpose do not exist 5. Obviously the single-material corrugated AR layer also has a high laser-induced-damage threshold, which is important in many highpower laser applications. However, a linear grating cannot provide polarizationindependent AR response because NTE ;~ NTM. The problem can be solved using, e.g., two-dimensional arrays of pillars or holes. Improved spectral performance of AR surfaces is obtained with subwavelength-period triangular or pyramidtype gratings. Again the analogy with thin-film technology is obvious: a grating

4 The presence of a substrate does not change the results significantly. 5 Porous sol-gel coatings satisfy this requirement, but it appears appropriate to classify them under effective media as well.

368

DIFFRACTIVE OPTICS: ELECTROMAGNETIC APPROACH

[V, w 5

with a depth-dependent fill factor f(z) is analogous to a film with a depthdependent refractive index N(z). If N(z) varies smoothly from nl to n2, a wideband AR effect is achieved. The literature on grating AR surfaces is so extensive that it cannot be reviewed in detail here. Early experiments were performed by Clapham and Hutley [1973] and Enger and Case [1983]. Early rigorous modeling of binary gratings may be found in the paper by Gaylord, Baird and Moharam [1986], and multilevel elements were considered by Ono, Kimura, Ohta and Nishida [ 1987]. Pyramid structures were discussed by Southwell [1991 ] and optimized by Grann, Moharam and Pommet [1995]. Comparisons between thin-film and grating AR layers were made by Glytsis and Gaylord [1992]. High-index IR materials were studied by Motamedi, Southwell and Gunning [ 1992], Raguin and Morris [1993], Brundrett, Glytsis and Gaylord [1994], and Lalanne and Morris [ 1997]. Applications to solar cells were discussed by Heine and Morf [ 1995] and Heine, Morf and Gale [ 1996]. 5.3. GRATING RETARDERS

The form birefringence of subwavelength-period gratings permits the construction of elements capable of polarization conversion using isotropic optical materials. Enger and Case [1983], Flanders [1983], and Cescato, Gluch and Streibl [ 1990] were among the first to perform experiments on grating retarders, relying on the formula a r = kh (NTM - NTE).

(5.1)

For example, a quarter-wave plate is obtained by choosing h and f such that A0 = :r/4. This yields deep grating profiles if n ~ 1.5, but shallower profiles are obtained with high-index materials. Kikuta, Ohira and Iwata [ 1997] presented a remarkable design of an achromatic/l/4 retarder that makes use of the dispersion of form birefringence in the region d ~/l. Haggans, Li, Fujita and Kostuk [1993] described a number of polarization compensators, wave plates, and polarization rotators that employ the zeroth reflected order, considering both metallic reflection gratings and dielectric gratings with incidence from the substrate above the critical angle. They show, e.g, that any elliptically polarized light can be converted to linearly polarized light with 100% efficiency by internal reflection from a grating of appropriate depth. Liu and Azzam [1996] designed dielectric binary gratings on a flat metallic substrate for /l = 10.6llm, while Kettunen and Wyrowski [1998]

V, w 5]

LINEARGRATINGSOPERATINGIN ZEROTH-ORDERMODE

369

Table 2 Transmission-mode and reflection-mode quarter-wave and half-wave retarders A0

0

nr

nt a

n1

n2 a

Jr/4

0

1.46

1.00

1.46

1.00

0.349 0.685

1 . 2 1 9 0 . 9 7 0 0.979

Jr/2

0

1.46

1.00

1.46

1.00

0.346 0.685

2.769 0.977

Jr/4

0

1.97

1.00

1.97

1.00

0.572 0.400 0.452

Jr/2

0

1.97

1.00

1.97

1.00

0.572 0.505

1 . 0 4 0 0 . 9 2 6 0.926

Jr/4

0

1.00

A1

1.46

1.00

0.527 0.918

0.362

0 . 9 1 0 0.905

Jr/2

0

1.00

A1

1.46

1.00

0.487 0.985

0.438

0.883

c/)t

d//~

h/)t

riTE

rITM

0.989

0 . 9 6 2 0.962

0.883

Jr/4

0

1.00

A1

1.00

AI

0.387

0.867 0.379

0.877

0.877

Jr/2

0

1.00

A1

1.00

AI

0.499

0.960 0.330

0.838

0.838

Jr/4

45 ~

1.46

1.00

1.46

1.00

0.385 0.391

0.453

1.000

1.000

Jr/2

45 ~

1.46

1.00

1.46

1.00

0.520 0.400 0.916

1.000

1.000

Jr/4

45 ~

1.46

A1

1.46

A1

0.520

0.401

0.284 0.859

0.767

Jr/2

45 ~

1.46

AI

1.46

A1

0.637

0.388

0.251

0 . 8 4 4 0.768

a A1 indicates h = 1.37 + i7.62.

presented a reflection-mode design that c o m b i n e s a grating with a multilayer dielectric stack. Applications o f polarization conversion elements in highpower laser t e c h n o l o g y were pointed out by Okorkov, Panchenko, Russkikh, Semigonov, Sokolov and Yakunin [1994], who also provided experimental results at ~, - 10.6/tm. E x p e r i m e n t a l results on polarization conversion with reflectiontype sinusoidal surface relief gratings were provided by Watts and Sambles [1997]. Richter, Sun, X u and F a i n m a n [ 1995] provided comparisons o f approximate and rigorous m o d e l s for retarders. The use o f the rigorous approach to design retarders is quite straightforward; some binary designs for quarter-wave and halfwave plates are collected in table 2 (the notation is that o f fig. 2). Here dielectric transmission gratings, dielectric reflection gratings backed by a reflecting A1 layer, and metallic surface-relief gratings are considered for A = 633 nm. Note that in some cases rtrE ~ rlTM, which implies that the output b e a m is slightly elliptically polarized. Brundrett, Glytsis and Gaylord [1996] provided designs with multilevel structures.

370

DIFFRACTIVE OPTICS: ELECTROMAGNETICAPPROACH

[V, w 5

5.4. GUIDED-MODE RESONANCE FILTERS

The guided-mode resonance effect, introduced in w2.4, is explained by the excitation of a guided mode in the leaky waveguide formed by the grating when its average refractive index exceeds the refractive indices of the surrounding media. The excited waveguide mode is leaky and it produces a specularly reflected wave at the exact resonance. However, in off-resonance conditions the specular reflection decreases rapidly. The actual form of the grating profile is not important for the effect to occur: the principle was demonstrated with a coated sinusoidal grating by Mashev and Popov [1985]. A theoretical analysis predicting 100% efficiency for an infinite plane wave was given by Popov, Mashev and Maystre [1986], but the use of a finite beam was shown by Avrutsky and Sychugov [1989] to reduce the efficiency, because only one of the plane-wave components of such a beam satisfies precisely the resonance condition. Saarinen, Noponen and Turunen [1995] considered the effect of finite grating size in the reflection efficiency, which is also reduced by grating irregularities and even slight absorption in the waveguide. Binary volume-type modulation was considered by Magnusson and Wang [1992], who also showed (Wang and Magnusson [1994, 1995]) that the addition of uniform layers helps to provide symmetric line shapes and low reflectance around the passband. Tibuleac and Magnusson [1997] presented an extensive comparison of these gratings with traditional multilayer film stacks. Wang and Magnusson [1993] proposed a number of potential applications of these filters as, e.g., polarizing resonator mirrors and notch filters, which remove a single wavelength from a collimated light beam. Sharon, Rosenblatt, Friesem, Weber, Engel and Steingrueber [ 1996] proposed active light modulation by control of one of the refractive indices in the filter structure. In recent experiments, Peng and Morris [1996] demonstrated polarization-independent operation with approximately 60% peak reflectance using arrays of circular holes in a waveguide. Norton, Morris and Erdogan [1998] provided experimental results and discussed various factors that may reduce the peak reflection efficiency. Magnusson, Shin and Liu [1998] designed and demonstrated highefficiency (experimentally 94%) filters in TM mode at the Brewster angle, where the reflectance without the grating is zero. Liu, Tibuleac, Shin, Young and Magnusson [1998] demonstrated 98% efficiency at normal incidence for a filter with 2.2nm linewidth. These experiments prove that guided-mode resonance filters can become useful optical components. Lemarchand, Sentenac and Giovannini [ 1998] introduced doubly periodic structures, which increase the angular tolerance of the filter without affecting its spectral selectivity.

V, w 6]

MODULATED GRATINGS OPERATING IN FIRST-ORDER MODE

371

w 6. Modulated Gratings Operating in First-Order Mode Modulation of parameters which define the profile shape of a linear grating, such as the period, fill factor, and aspect ratio, is an essential design tool in diffractive optics. In particular, the idea of employing the first order of a carrier grating is fundamental in classical optical holography and in the synthetic approach put forward by Lohmann in the 1960s. Such spatial modulation of a grating structure is related closely to analog cartier-coding of information in radio transmission. 6.1. HIGH-CARRIER-FREQUENCYDIFFRACTIVE ELEMENTS Often the phase O(x,y) of the field to be generated by the diffractive element is known analytically, or at least in a discrete form. If a linear carrier phase Oc(x) = 2:rx/dc is added, the fringe center lines of the resulting element are obtained from O(x, y) + 2 ~

X

= 2~q,

(6.1)

where q is an integer. When d ~ ~, and O(x,y) represents the phase of a paraxial field, the element appears like a linear grating with slightly perturbed fringes 6 the phase information is encoded into these perturbations and the local profile shape of the grating determines the local first-order diffraction efficiency r/-1 (x,y). Thus any amplitude information a(x,y), which may be present in the field to be generated, can be encoded by controlling the local efficiency according to the formula _

r/-l(X,y) = r/max

a(x,y) maxlx,y) ~ A a(x, y)

(6.2)

where A denotes the aperture of the element and /]max denotes the maximum attainable local diffraction efficiency. Obviously one would prefer to have ?']max = 100% 7. AS discussed in w167 2.3 and 4.1, r/max = 100% can be approached using a binary cartier grating in the Bragg configuration (see Turunen, Blair, Miller, Taghizadeh and Noponen

6 No significant polarization conversion occurs when the element may be considered locally as a y-directional grating. 7 Note that r/max = 100% does not guarantee 100% conversion efficiency, which is still restricted by the upper bound introduced by Wyrowski [1991] in the paraxial domain.

372

DIFFRACTIVE OPTICS: ELECTROMAGNETICAPPROACH

[V, w 6

[1993], Noponen and Turunen [1994a], Tervonen, Turunen and Pekola [1994], Ehbets, Herzig, Kuittinen, Clube and Darbellay [1995], and Blair, Taghizadeh, Parkes and Wilkinson [1995]). In the case of phase-only wave transformations, the local fill factorf is kept constant across A. Control of amplitude information by modulation of r/_l can be achieved by modulating either f or the groove depth h, but care must be taken to avoid unintentional phase modulation (Noponen and Turunen [1996]). If f is modulated and h is kept constant, the groove shape must be kept symmetric with respect to the fringe center line, but if h is modulated a lateral shift is needed to compensate the local first-order phase shift. Despite the off-axis configuration, the high-carrier-frequency technique offers some attractive features. A continuous phase profile (p(x,y), which in a conventional axial configuration would be realized with a continuous surface profile, can be generated with a binary profile. Amplitude information can be encoded, without employing absorption, by control of the local diffraction efficiency of the cartier grating. This is also a feature of paraxial-domain carrier techniques, which however require more sophisticated profile shapes for high r/m~x. Additionally, in the high-frequency-cartier technique, the stray light appears in the zeroth order of the carrier grating, which is well separated from the signal, while it is divided among numerous cartier orders in paraxial schemes. The field to be encoded in the element frequently is available as a set of values defined in a discrete set of P x R points arranged in a rectangular grid in the xy plane. Now the direct use of eq. (6.1) is not possible. One may, however, transform the discrete x-directional data into a piecewise-continuous profile individually in R stripes before adding the cartier frequency. This leads to a structure consisting of rectangles, which can be easily fabricated by lithographic techniques (Tervonen, Turunen and Pekola [1994]). 6.2. HIGH-NUMERICAL-APERTURE DIFFRACTIVE LENSES

Diffractive lenses (Nishihara and Suhara [ 1987]) are among the most important components in diffractive optics. Regular arrays of cylindrical diffractive lenses can be analyzed by rigorous electromagnetic theory of gratings if the size of an individual lens, i.e., the grating period, is at most ~ 1002. (see Noponen, Turunen and Vasara [1993] and Schmitz and Bryngdahl [1997]). Finite-element methods (Lichtenberg and Gallagher [1994]), boundary element methods (Hirayama, Glytsis, Gaylord and Wilson [1996] and Glytsis, Harrigan, Hirayama and Gaylord [1998]), hybrid boundary element-finite element methods (Mirotznik, Prather and Mait [ 1996]), and boundary-integral methods (Prather, Mirotznik and

V, {} 6]

MODULATED GRATINGS OPERATING IN FIRST-ORDER MODE

373

Mait [1997]) have been applied to the analysis of individual cylindrical lenses. Layet and Taghizadeh [1997] and Schmitz and Bryngdahl [1998] have applied the field-stitching method to analyze lenses with larger aperture. Circular diffractive lenses with macroscopic dimensions cannot be analyzed rigorously. However, they can be considered locally as gratings with a slowly varying period d(r): for a focusing lens of focal length f ,

d(r) -- k. [1 +

(f/r) 2] 1/2

(6.3)

Therefore, large diffractive lenses can be modeled rather accurately by applying the thin-element approximation in the central region, where the local period is large, and using rigorous diffraction theory elsewhere to predict the local diffraction efficiency. The global conversion efficiency is determined by integrating the product of the local efficiency (taking into account the local state of polarization) and the incident intensity over the lens aperture, and dividing the result by the integrated intensity of the incident field. The field in the focal region can be calculated from the electromagnetic field components constructed locally just behind the lens using appropriate coordinate transformations and wavepropagation methods as described by Vahimaa, Kettunen, Kuittinen, Turunen and Friberg [1997], Sheng, Feng and Larochelle [1997], Sergienko, Stamnes, Kettunen, Kuittinen, Turunen, Vahimaa and Friberg [ 1999], and Kleemann and Gfither [1998]. Similar techniques may also be applied to the analysis of other locally linear diffractive elements, including off-axis lenses based on binary surface-relief-type cartier gratings discussed in w and diffractive lenses designed to modulate the shape of the focal spot into, e.g., a flat-top profile. Considering fig. 9 and eq. (6.3), one can appreciate that the local efficiency is low over a substantial fraction of the aperture of a high-numerical-aperture diffractive lens. Noponen, Turunen and Vasara [1992, 1993] overcame this problem by optimizing the local grating profile: the three independent depthlevel transition points of a four-level grating were readjusted to optimize r/_l, and the valley seen in fig. 9 around d ~ 2~ was avoided. Kleemann and Gfither [1998] optimized the profile height and the thickness of the antireflection layer of a triangular-profile grating with the same goal. Local optimization of the grating-profile shape, taking into account fabrication constraints, is a subject that requires a considerable amount of further study. It is important to make sure that the first-order phase of the optimized grating is independent of the local period- otherwise aberrations are introduced. Fortunately, they can be eliminated by an appropriate lateral cyclic shift of the local grating profile at the end of efficiency optimization (note, however, that exact simultaneous phase correction

374

DIFFRACTIVE OPTICS: ELECTROMAGNETICAPPROACH

IV, w 7

in TE and TM polarization is not possible for small grating periods). Schmitz and Bryngdahl [1997] demonstrated that high-numerical-aperture diffractive lenses can also be designed by optimization of the local phase of the transmitted field. High-quality diffractive lenses with multilevel profiles have been demonstrated by Urquhart, Stein and Lee [1993], Mikolas, Bojko, Craighead, Haas, Honey and Bare [1994], Zarschizky, Stemmer, Mayerhofer, Lefranc and Gramann [1994], and Finlan, Flood and Bojko [1995] using electron beam lithography, but according to Kuittinen and Turunen [1997], the fabrication of such profiles by conventional photolithography appears exceedingly difficult because of mask alignment errors. Kunz and Rossi [1993] and Hessler, Rossi, Kunz and Gale [ 1998] have employed direct-write laser beam lithography to produce so-called harmonic, or modulo-2JrQ profiles 8. The greatest problem in the fabrication of these profiles is the accurate generation of the vertical side walls.

w 7. Modulated Gratings Operating in Zeroth-Order Mode Spatial variations of the parameters of subwavelength-period gratings can produce interesting effects. For example, the effective refractive index of a material can be modulated, which permits the simulation of continuous surface profiles. On the other hand, modulation of the groove direction can produce spatially variable polarization components. 7.1. SPATIALLY VARIABLE EFFECTIVE MEDIA

It is clear from eqs. (3.28) and (3.29) that modulation of the local fill factor f of a binary subwavelength-period grating alters the effective refractive index and therefore the phase shift experienced by a wave passing through a slab of thickness h. The implications of this fact were realized independently by Stork, Streibl, Haidner and Kipfer [ 1991 ] and Farn [ 1992]: continuous surface profiles can be simulated with binary-structured effective media. Early theoretical and experimental investigations of this scheme were performed by Haidner, Kipfer and Streibl [ 1992], Haidner, Kipfer, Stork and Streibl [ 1992], Haidner, Sheridan, Schwider and Streibl [1993], Haidner, Kipfer, Sheridan, Schwider, Streibl, Collischon, Hutfless and M~irz [1993], Haidner, Sheridan and Streibl [1993],

8 In this manner one can realize achromatic lenses with a single material, as discovered independently by Faklis and Morris [ 1995] and Sweeney and Sommargren [1995].

V, w 7]

MODULATED GRATINGS OPERATING IN ZEROTH-ORDER MODE

375

Collischon, Haidner, Kipfer, Lang, Sheridan, Schwider, Streibl and Lindolf [ 1994], and Kipfer, Collischon, Haidner, Sheridan, Schwider, Streibl and Lindolf [ 1994] for both dielectric and metallic surface profiles. Equations (3.28) and (3.29) are accurate in the design of spatially variable effective media only in the limit d << ~., but because of fabrication constraints one would prefer cartier gratings with periods slightly below the onset of diffraction orders m - + 1 in the substrate. In this domain the dependence of ~70 on f must be determined by rigorous theory: see Grann, Moharam and Pommet [1994], Kikuta, Yoshida and Iwata [1995] and Richter, Sun, Xu and Fainman [1995] for comparisons of rigorous and approximate methods for two- and three-dimensionally modulated profiles. Because of form birefringence, threedimensional modulation is necessary for polarization-independent operation. Chen and Craighead [1995] and Miller, de Beaucoudrey, Chavel, Cambril and Launois [1996] provided experimental results for beam deflectors and 1 ~ 3 beam splitters. The aspect ratios required in fabrication of subwavelength-period effective media can be reduced by use of high-index dielectric materials because then a given phase modulation is obtained with a thinner layer (Astilean, Lalanne, Chavel, Cambril and Launois [1998]). The conversion efficiency obtained with the simple construction principle presented above reduces when the transverse scale of the diffractive element is reduced or, in other words, when the signal wave becomes non-paraxial the situation is analogous to Q-level surface-relief profiles considered in fig. 9 and w6.2. Consequently, Zhou and Drabik [ 1995], Warren, Smith, Vawter and Wendt [ 1995], Kuittinen, Turunen and Vahimaa [ 1997], Kuittinen, Turunen and Vahimaa [1998] and Lalanne, Astilean, Chavel, Cambril and Launois [1998] optimized subwavelength-period grating parameters to increase the efficiency in this domain, considering both two- and three-dimensionally modulated structures and presenting experimental results. 7.2. S U B W A V E L E N G T H - S T R U C T U R E D DIFFRACTIVE LENSES

Diffractive lenses are among the obvious applications of effective media realized with subwavelength-period gratings. Chen and Craighead [ 1996] have demonstrated a diffractive f : 20 lens with 53% efficiency using a binary threedimensionally structured subwavelength-period profile. Prather, Mait, Mirotznik and Collins [1998] used the boundary integral method to analyze and design a subwavelength-structured diffractive lens and a focusing 1 ~ 2 beam splitter. Mait, Prather and Mirotznik [1998] considered the limitations imposed by fabrication constraints in the diffraction efficiency.

376

DIFFRACTIVE OPTICS: ELECTROMAGNETICAPPROACH

[V, w 7

7.3. POLARIZATION MULTIPLEXING WITH EFFECTIVE MEDIA

Use of birefringent materials permits independent generation of two different waves from a single diffractive element illuminated by TE and TM polarized light (see Xu, Ford and Fainman [1995]). Richter, Sun, Xu and Fainman [1995] and Schmitz, Br/iuer and Bryngdahl [1995b] considered the design of such polarization-multiplexed elements on the basis of isotropic materials, employing form birefringence. To understand the principle of this polarization multiplexing scheme, consider a sandwich structure with fixed total thickness h: (1) a slab of dielectric material of refractive index n and thickness t, (2) a subwavelength-period binary grating of thickness H and fill factor f , with effective refractive indices NTE and NTM, and (3) a slab of air of thickness h - H - t > 0. In view of the effective-medium theory the phase delay experienced by a plane wave in transmission through the slab is q~rE = k [ ( n -

1)t +(NTE- 1)H + h]

in TE mode, and analogously for q~rM in TM mode. Since NTE and NTM depend on f , one can obtain any desired combination of q~rE and t/)rM in a 2:r range by choosing appropriate values for t and f provided that H is sufficiently large to allow modulation of q~rE- q~rM over a 2:r range when f is varied over the range [0, 1]. Thus, by modulating t and f , one can store two independent phase profiles in a single diffractive surface. The main problem of this construction principle is that it leads to rather thick surface profiles with high aspect ratios unless high-index materials are used. The principle is also applicable in the reflection mode, in which considerably shallower profiles result. In constructing reflection-mode polarization-multiplexed surfaces it is useful to keep in mind that TE-polarized light is, to a first approximation, reflected from the top surface of a subwavelength-period grating, while TM-polarized light sees the grating as a slab of effective dielectric medium and reflects from the bottom of the slab (see, e.g., Kuittinen, Turunen and Vahimaa [1997]). 7.4. SPATIALLY VARIABLE POLARIZATION COMPONENTS

Spatial modulation of the properties of subwavelength-period gratings, such as profile depth, fill factor, and most notably the fringe orientation, provides the greatest fundamental advantages of diffractive polarization components over conventional polarization components based on anisotropic materials.

V, w 8]

GRATINGS OPERATING IN MULTI-ORDER MODE

377

Perhaps the most obvious example of spatially variable polarization components is spatial modulation of the fringe direction of a wire-grid (or metalstripe) polarizer. As discussed in w5.1, such an element transmits only the field component perpendicular to the grooves. Hence spatially variable grating polarizers can produce locally linearly polarized fields with rather arbitrary intensity distributions. Kley, Schnabel, Htibner and Zeitner [1996], Kley, Schnabel and Zeitner [1997], and Schnabel and Kley [1997] have used this approach to produce a circular wire-grid polarizer. Because of the continuous variation of the local polarization direction, the circular grating can be used as a high-resolution polarimeter: in combination with a circular first-order grating, the state of polarization can be measured simultaneously for different wavelengths. The use of pixel-structured metal-stripe gratings for polarization multiplexing has been demonstrated by Zeitner, Schnabel, Kley and Wyrowski [ 1997, 1999]. Diffractive half-wave plates also offer interesting prospects for the generation of spatially variable polarization components. Recall that if a linearly polarized wave is incident upon a half-wave plate with its electric vector at an angle a with respect to the optical axis of the crystal, the output wave will be linearly polarized with its optical axis at an angle - a with respect to the optical axis. In the case of grating half-wave retarders, the optical axis is perpendicular to the grating grooves. Therefore a subwavelength-period, space-variant, half-wave grating retarder can transform a globally linearly polarized incident field into a locally linearly polarized output field with a spatially variable direction of the electric field vector. This scheme was proposed by Davidson, Friesem and Hasman [1992], but their experimental demonstration was not convincing because of inadequate fabrication accuracy. Thus further experiments are desirable.

w 8. Gratings Operating in Multi-Order Mode In this section we consider gratings with optical functions requiring at least two diffraction orders. The simplest example is a beam splitter, which divides an incident TE (or TM) polarized plane wave into two equal-intensity plane waves which propagate in different directions. More complicated tasks include polarization-selective beam splitting and multiple beam splitting. Attempts are finally made to determine the limits of the thin-element approximation for arbitrary diffractive elements.

378

8.1. T W O - B E A M

DIFFRACTIVE OPTICS: ELECTROMAGNETIC APPROACH

[V, w 8

SPLITTERS

If the field incident upon a linear grating is either TE or TM polarized, each diffraction order will be similarly polarized. Binary dielectric and metallic gratings act naturally as such polarization-preserving beam splitters at normal incidence because of their symmetrical diffraction patterns: one needs only to choose f and h such that the zeroth order is minimized. Similarly, orders m = 0 and m = -1 of a binary grating can be employed at Bragg incidence: in fact, the grooves are shallower and fabrication is therefore easier than in the case of deflection gratings with maximum r/_l. If, for example, n~ = r t l = 1.46, n2 = nt = 1, d - ~,, f = 0.62, and h = 1.10~. in the geometry of fig. 2, we have ~10- ~ - 1 -- 48.7% in TE polarization and ~70= r/_l = 49.2% in TM polarization. Conversion efficiencies well above the paraxial upper bound (Wyrowski [ 1991 ]) r/u = 8 / : r 2 are obtained routinely with beam splitters in the non-paraxial domain: see Vasara, Taghizadeh, Turunen, Westerholm, Noponen, Ichikawa, Miller, Jaakkola and Kuisma [1992], Noponen, Vasara, Turunen, Miller and Taghizadeh [1992], Walker, Jahns, Li, Mansfield, Mulgrew, Tennant, Roberts, West and Ailawadi [ 1993], and Kipfer, Collischon, Haidner, Sheridan, Schwider, Streibl and Lindolf [1994] for a number of designs in different geometries. Noponen, Turunen and Wyrowski [ 1995] demonstrated that r/u can be exceeded also in the paraxial domain if subwavelength features are introduced in the profile. One of the most interesting applications of polarization-preserving twobeam splitters is their use as phase masks in ultraviolet exposure of volume Bragg gratings in optical fibers as described by Hill, Malo, Bilodeau, Johnson and Albert [1993] and Hegedus [1997]. A cube beam splitter that separates the two states of linear polarization which may be present in the incident beam is a standard component in optical engineering. Similarly operating, compact elements may be realized with gratings by directing the TE- and TM-polarized field components in different diffraction orders. Apparently the first polarization beam splitter of this type was proposed by Roumiguieres [1976] and demonstrated by Knop [1978b] (see also Noponen, Vasara, Turunen, Miller and Taghizadeh [ 1992] and Lima, Soares, Cescato and Gobbi [1997]). Bragg incidence upon a binary metallic grating was used, with the fill factor and groove depth optimized to provide a high diffraction efficiency for order m = 0 in TM mode and order m = -1 in TE mode. Roumiguieres [ 1976] assumed a perfectly conducting metal, but considering A1 as the grating material for ~. - 633 nm, we obtain the efficiencies 77-1 = 88-00//0 and r/0 = 0.0% in TE mode, and r/_l = 0.1% and r/0 - 81.6% in TM mode, with 0 = 30 ~ d - ~,, c = 0.17d, and h = 0.392A. Gratings with high r/0 in TE mode

V, w 8]

GRATINGS OPERATING IN MULTI-ORDER MODE

379

and high r/_~ in TM mode can also be designed, but tend to be less efficient. The use of exact Bragg incidence is not necessary: reflecting polarizing beam splitters may also be designed to operate at off-Bragg and conical-incidence conditions. Corresponding transmission mode polarizing beam splitters have been designed by Noponen, Vasara, Turunen, Miller and Taghizadeh [1992], Habraken, Michaux, Renotte and Lion [ 1995] and Schmitz, Br~iuer and Bryngdahl [ 1995a]. Lalanne, Hazart, Chavel, Cambril and Launois [1999] provide experimental results using high-index material to reduce the aspect ratio. Glytsis and Gaylord [1992] consider a reflection/transmission polarizing beam splitter that employs the two zeroth orders, while Tyan, Salvekar, Chou, Cheng, Scherer, Sun, Xu and Fainman [ 1997] combine a form-birefringent subwavelength grating with a multilayer film stack. More complicated optical functions are also possible. A grating that splits TE-polarized light into two equal-intensity beams propagating in directions -t-45 ~ (0+l = 47.5%), but transmits TM polarization directly (70 = 97.9%), is obtained if c = 0.2712 and h = 0.7552. A grating that splits TM polarization in two beams (r/il = 49.4%) but transmits TE polarization (r/0 = 92.1%) is obtained with c = 0.569/l, h = 1.5842. 8.2. ARRAY ILLUMINATORS

Beam splitters which generate more than two diffraction orders with equal efficiency are known as array illuminators. In the paraxial domain a large variety of such elements with binary, multilevel, and continuous surface profiles has been designed: see Aagedal, Wyrowski and Schmid [1997] and the references cited therein. We consider the limits of the thin-element approximation in the analysis and design of these elements, as well as their rigorous design techniques. It is obvious that the thin-element approximation fails when the period of an array illuminator is reduced. Some initial results were given by Johnson and Kathman [ 1991 ]. Vasara, Noponen, Turunen, Miller and Taghizadeh [ 1991 ] analyzed binary gratings with a symmetric set of orders - N ~< m ~< N to generate an odd number M = 2N + 1 of equal-efficiency beams, and gratings that produce an even number M = 2N of beams by utilization of only the odd-numbered orders in the r a n g e - ( 2 N - 1) ~< m ~< 2 N - 1. They found that, when the period is reduced, the designs which employ only the odd-numbered orders (evennumbered array) perform significantly better than the designs which employ all orders (odd-numbered array): the uniformity error of odd-numbered array illuminators increases smoothly when d/2 is reduced, but for even-numbered

380

DIFFRACTIVE OPTICS: ELECTROMAGNETICAPPROACH (a)

0.4 0.3

0.3

0.2

0.2

0.1

0.1 0

10

20 Omax

(b)

0.4

15 ",...9 !1 / - A ~ / " '1111.3) . ~ . . . 5

[g, w 8

30

0

8 "12

..... ?,2

'~i~64 --.. -

0

20

40

60

Oma,,

Fig. 15. Uniformity error as a function of maximum beam deflection angle for binary array illuminators. (a) Odd-numbered array in M central orders. (b) Even-numbered array in M odd orders.

array illuminators the error remains small until a sharp turn upwards takes place in the region where the smallest transverse feature size is reduced b e l o w - ~. This unexpected behavior is illustrated in fig. 15; the uniformity error is defined as

E =

7]max- r]min

,

(8.1)

/']max 4- /']min

are the highest and the lowest diffraction efficiency within the intended array. The results are presented in fig. 15 by plotting E as a function of 0max, the diffraction angle of the marginal order in the array, to obtain an idea of the limit of the paraxial domain and the validity of the thin-element approximation 9. The exact shape of the curve depends on both M and the particular thinelement solution found by optimization, which was performed using the iterative Fourier-transform algorithm (Bryngdahl and Wyrowski [1990]). Nevertheless, some general conclusions can be drawn. If we choose E = 10% as a criterion for acceptable uniformity error, we see that the limit of the thin-element approximation is reached at 0max "~ 5-10 ~ for odd-numbered arrays and at 0max "~ 20-30 ~ for even-numbered arrays. No physical explanation of this significant difference is known. Multilevel array illuminators provide substantially higher conversion efficiencies than their binary counterparts, in particular if the array is non-inversionsymmetric. Such multilevel array illuminators have been analyzed rigorously w h e r e ?]max and/Tmin

In the case of odd-numbered arrays, r/0 is excluded in determination of E because it is the main source of error. However, 7/0 can be made equal to the average efficiency by adjusting h.

9

V, w 8]

GRATINGS OPERATING IN MULTI-ORDERMODE 0.8

381

15

iiiiiiiiii~ 0.6

0.4

15.=0

0.2 .

o

0

.

.

.

.

.

5

10

15

20

25

30

On-lax Fig. 16. Uniformity error as a function of maximum beam deflection angle for eight-level array illuminators.

by Miller, Taghizadeh, Turunen, Ross, Noponen and Vasara [1993] and Sidick, Knoesen and Mait [1993]. Some results are plotted in fig. 16 as a function of 0max. The limit of the thin-element approximation (with the criterion E = 10%) appears to be l0 0ma• ~ 2-3 ~ for both M = 15 and M = 16. The curve labeled as 15_0 shows the results for M = 15 without the zeroth order. The general behavior is similar to that observed with odd-numbered binary array illuminators" E grows smoothly when 0ma• is increased. Similar behavior is obtained with more than four quantization steps, and with continuous surface profiles. Array illuminators realized by the high-cartier-frequency technique discussed in w6.1 and the subwavelength pulse-width-modulation technique discussed in w7.1 have been evaluated by Noponen and Turunen [1994a]. Similar trends are observed as in figs. 15 and 16. Rigorous analysis of the limits of the thin-element approximation is not possible for three-dimensionally modulated elements which generate two-dimensional arrays. However, one may expect the values of 0ma• to be similar as in the case of one-dimensional arrays. The uniform array of equal-intensity beams is just one example of possible signals which may be generated with diffractive elements. Nevertheless, one is tempted to draw general conclusions concerning the limits of the thin-element approximation in diffractive optics from the results presented in figs. 15 and 16. Doing this, one could say that the thin-element approximation is reasonably accurate for binary diffractive elements if the angular spectrum of the signal wave is located within 5-10 ~ from the axis; for multilevel elements the maximum

l0 The calculations could not be extended to smaller values of 0max because of computer memory limitations.

382

DIFFRACTIVE OPTICS: ELECTROMAGNETIC APPROACH

[V, w 8

spread is 2-3 ~. It must be stressed, however, that these general results have not been properly verified either numerically or experimentally. In the transition region, where the thin-element approximation begins to fail, one may attempt to improve the uniformity by adjusting the parameters that define the profile and evaluating the result by rigorous theory after each modification (the process is, however, computationally time-consuming). An obvious strategy is to start from a large value of d/)~, for which the array is still reasonably uniform according to rigorous analysis. The period is then decreased in small steps, starting the optimization of the transition points from the previous solution. Of course, such a procedure may stagnate at some stage. This is particularly likely if some symmetries which simplify the design in the paraxial region are lost and the number of design parameters is thereby inadequate in the non-paraxial domain. Such stagnation is typical with binary elements with inversion symmetry /7_m = ~m in the paraxial domain. The problem can be solved only by adding new features in the structure, as illustrated by Br/iuer and Bryngdahl [ 1995b]. On the other hand, non-symmetric arrays can be generated in the non-paraxial domain with binary gratings (Br~uer and Bryngdahl [ 1995a]). Deep in the resonance domain, where the number of signal diffraction orders is comparable to the total number of propagating diffraction orders in the appropriate half-space, direct optimization of the parameters which define the interface is feasible, at least if M is small. This kind of beam splitter is useful in the construction of compact optical systems: one example, described by Wyrowski and Zuidema [1994], is the coupling of light from a high-power Nd:YAG laser into a fiber bundle in an industrial environment. Another example is the splitting and distribution of light in planar-integrated, or substrate-mode optical systems (Jahns [1994]), in which the deflection angles must be highly non-paraxial (Walker, Jahns, Li, Mansfield, Mulgrew, Tennant, Roberts, West and Ailawadi [1993]). Table 3 contains some examples of beam splitters optimized to generate M diffraction orders with equal efficiency over a large angular range (Vasara, Taghizadeh, Turunen, Westerholm, Noponen, Ichikawa, Miller, Jaakkola and Kuisma [1992]). Normal incidence from nr = 1.46 to nt = 1 is assumed. Outcoupling from a planar-integrated system can also be combined with array illumination. Assuming nr = nl = 1.46, n2 = nt = 1, and 0 = 45~ we obtain three equal-efficiency orders m - -1, m = -2, and m = -3 with JTtot = 78.7% if d - 1.82/~, h = 1.20~,, Xl = 0.21~,, x2 = 0.32~,, x3 = 0.464~, x4 = 0.84~. In this geometry the propagation angle of order m = - 2 is 0-2 = 0 and the other two orders make angles +45 ~ with respect to the surface normal. Noponen and Turunen [1994b] presented a number of high-

V, w 9]

CONCLUSIONS

383

Table 3 Multiple-beam-splitter solutions in TE polarization

M

d/~,

3

1.5

0

~tot

E

xl/d

x2/d

x3/d

x4/d

0

96.7

0.0

0.0532

0.8965

-

-

x5/d

3

1.9

0

92.7

0.0

0.0507

0.9146

-

-

4

2.0

9.59

90.7

0.0

0.0788

0.4342

0.7389

-

4

2.4

7.98

83.0

0.0

0.1623

0.3035

0.4189

0.9649

5

2.5

0

95.7

0.0

0.1616

0.5594

0.6022

-

5

2.9

0

90.6

0.0

0.0598

0.2385

0.2636

0.4682

0.7015

6

3.0

6.38

89.7

0.0

0.0266

0.2789

0.4680

0.6339

0.8108

6

3.4

5.63

87.7

0.1

0.1140

0.6474

0.7587

0.8151

0.8380

7

3.5

0

95.6

0.4

0.0949

0.6366

0.6968

0.7094

0.7644

7

3.9

0

94.2

0.1

0.0586

0.3677

0.4862

0.5724

0.6909

efficiency designs with two-dimensionally periodic gratings, which generate twodimensional arrays of highly non-paraxial beams. In the designs considered above, all propagating diffraction orders have been employed as signal orders. This represents the largest possible angular spread of the array. If the angular range of the signal wave is reduced, the efficiencies attainable with binary gratings are, in general, also reduced. However, as shown by Noponen, Turunen and Wyrowski [1995], high conversion efficiencies from one to M beams can be obtained even in paraxial conditions with subwavelengthstructured profiles - the values obtained exceed clearly the upper bounds of Wyrowski [ 1991 ] because of redistribution of energy in propagation through the modulated structure 11. Unfortunately, because of small features and high aspect ratios, the designed profiles are difficult to fabricate.

w 9. Conclusions It is noteworthy that a large majority of the literature cited in this review has been published in the 1990s. This indicates the recent emergence and rapidly growing

11 Note that the upper bounds cannot be exceeded using the pulse-position and pulse-width coding schemes of w167 6.1 and 7.1. They can only be exceeded with the help of volume or multiple scattering effects.

384

DIFFRACTIVEOPTICS: ELECTROMAGNETICAPPROACH

[V

importance of rigorous electromagnetic diffraction theory as a modeling and design tool in diffractive optics. Together with improved fabrication technology, the use of rigorous theory has pushed diffractive optics beyond the limitations of the paraxial approximation, and therefore significantly broadened its applicability in optical engineering.

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[V

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E. WOLE PROGRESS IN OPTICS XL 9 2000 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED

VI

S P E C T R O S C O P Y IN POLYCHROMATIC FIELDS

BY

ZBIGNIEW F I C E K

Department of Physics and Centre for Laser Science, The University of Queensland, Brisbane, Australia 4072

AND

HELEN S. FREEDHOFF Department of Physics and Astronomy, York University, Toronto, Ontario, Canada M3J 1P3

389

CONTENTS

PAGE w 1.

INTRODUCTION

. . . . . . . . . . . . . . . . . . .

391

w 2.

SUBHARMONIC

RESONANCES . . . . . . . . . . . . .

394

w 3.

THEORETICAL

w 4.

BICHROMATIC FIELDS

w 5.

POLYCHROMATIC

METHODS

. . . . . . . . . . . . . .

398

. . . . . . . . . . . . . . .

409

. . . . . . . . . . .

431

. . . . . . . . . . . . . . . . . . .

438

REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

439

ACKNOWLEDGMENTS

F I E L D S , p ~> 3

390

w 1. Introduction The interaction of a single two-level atom with an intense electromagnetic field has proved a fertile testing ground for the fields of quantum optics and laser spectroscopy. In the earliest studies, the atom was driven by an intense radiofrequency field, and the absorption and dispersion measured of a weak probe field resonant with a transition from one of the driven levels to a third atomic level (Autler and Townes [ 1955]). The subject gained greatly in interest, however, when laser intensities and stability grew high enough that it became possible to pursue these studies in the optical frequency domain. Mollow [ 1969] calculated the fluorescence spectrum of a two-level atom driven by a nearly resonant (monochromatic) laser, whose Rabi frequency 2s was greater than the atomic damping constant F; the spectrum was observed subsequently by various groups (Schuda, Stroud and Hercher [1974], Wu, Grove and Ezekiel [1975], Hartig, Rasmussen, Schieder and Walther [1976]). It consists of a sharp coherent part at the laser frequency tOL together with a three-peaked incoherent part with lines at WL and tOE + 292, which has come to be known as the 'Mollow triplet'. In analogy with the familar splitting of atomic spectral lines in a dc electric field, this splitting was called the dynamic or ac Stark effect. The absorption and dispersion spectra of a weak probe beam monitoring the strongly driven atom were also calculated (Mollow [1972]) and observed (Wu, Ezekiel, Ducloy and Mollow [ 1977]). The Mollow absorption spectrum contains features at the same frequencies and with the same linewidths as their counterparts in the fluorescence spectrum, but with widely differing intensities. When the atom is driven by a resonant laser, the central component of the spectrum disappears and the probe absorption profile exhibits dispersion-like features at the Rabi sidebands O)L + 2g2. For an off-resonant driving field, the dominant features of the spectrum are emissive and absorptive components at the Rabi sidebands, indicating that in one sideband stimulated emission outweighs absorption and the probe beam is amplified at the expense of the driving field. In the Autler-Townes spectrum, a doublet is observed, similarly demonstrating the Rabi splitting of the atomic levels. The doublet is symmetric for a resonant driving field and becomes asymmetric when the field is detuned from atomic resonance. The Mollow fluorescence triplet is shown in fig. 1, the near-resonance 391

392

SPECTROSCOPY IN POLYCHROMATICFIELDS

[VI, w 1

3'

-20

0

(~

-

20

~OIr

Fig. 1. Fluorescence spectrum of a two-level atom driven by a strong monochromatic field of Rabi frequency 2s = 20F, where F is the spontaneous emission rate.

II A v

3:

l/ tI I i

-20

i

i

0

i

i

20

(~ -(~L)Ir Fig. 2. Near-resonance absorption spectrum of a weak probe beam monitoring a strongly driven two-level atom with 292 = 20F, and detunings A = 0 (solid line), A = F (dashed line).

(Mollow) absorption spectrum in fig. 2, and the Autler-Townes absorption spectrum in fig. 3. In the near-resonance absorption spectrum, the structure above zero corresponds to amplification of the probe, and that below zero to

VI, w 1]

INTRODUCTION

393

V

!

!

9

-20

0

20

(~ - o d ) / r Fig. 3. Autler-Townes absorption spectrum of a weak laser beam probing a driven two-level transition to a third level for 2g2 = 20F and spontaneous emission rate of the third level Fd = 1 F.

its absorption. These constitute the standards with which the spectra presented for polychromatic driving should be compared. Throughout the 1970s, the monochromatically driven, two-level atom was studied in all its aspects by many authors. For a review of the literature, see for example Knight and Milonni [ 1980], or Mollow [ 1981 ]. The study of atoms in intense fields also led directly to the discovery of such dramatic phenomena as photon antibunching (Carmichael and Walls [1976], Kimble and Mandel [1976], Kimble, Dagenais and Mandel [1977]) and quantum jumps (Cook [1990]), and to the cooling of atoms and ions and atom optics (Walls and Milbum [1994], Mandel and Wolf [ 1995]). However, in this article we will confine ourselves to the spectroscopic aspects of intense driving fields, and describe the extension of the ac Stark effect to polychromatic driving. The physical situation involving polychromatic driving is much more complex, and the fluorescence and absorption spectra display significantly different properties than those observed with a monochromatic field. Much work has been focused on the appearance of the 'subharmonic resonances', displayed by the absorption spectrum of a strong probe beam monitoring the strongly driven atom. The fluorescence and absorption spectra of a polychromatically driven atom depend in an intricate way on the number, Rabi frequencies and detunings of the driving field components, and are rich in detail. In this chapter, we review the work performed to date on spectroscopy with intense polychromatic fields, and

394

[VI,

SPECTROSCOPY IN POLYCHROMATIC FIELDS

w2

present a unified physical picture of the many observed phenomena. Section 2 is devoted to the subharmonic resonances and their applications. Section 3 gives a brief overview of the main theoretical methods used. In w4, we review results on the fluorescence, absorption and Autler-Townes spectra with bichromatic driving fields, and in w5 we present results on more complicated field configurations involving trichromatic and multiple driving fields. w 2. S u b h a r m o n i c

Resonances

In 1970, the group of Bucci and Santucci studied the NMR absorption spectrum of a nearly-resonant probe field by a two-level molecule (chloroform) driven near resonance by an intense monochromatic field (Bucci, Cavaliere and Santucci [1970], Bucci, Cavaliere, Santucci and Serra [1970], Bucci, Martinelli and Santucci [1970], Bucci and Santucci [1970]). This time, however, the probe field was itself strong. They observed a series of very narrow resonances in the absorption spectrum, which they attributed to multiple quantum transitions. However, there does not appear to have arisen much related activity at the time in the use of intense probe fields; perhaps this was because the subject of intense driving fields itself did not receive a great deal of attention until the prediction by Mollow [1969] of the fluorescence triplet of an atom driven by an intense field in the optical regime. The interest in strong probes began in earnest in 1979, with the work of BonchBruevich, Vartanyan and Chigir [1979]. They used a probe field of (resonant)

u

~

"~

..~ ~

I -25

4----- 11

9 I 0 Probe Detuning (kHz)

I 25

Fig. 4. Experimental result for the dispersive response of a two-level system driven at resonance by a strong field of Rabi frequency 2Q = 21 kHz, and probed by a strong field corresponding to a = 1.43 (from Manson, Wei and Martin [1996]).

VI, w 2]

SUBHARMONIC RESONANCES

395

? o x ur)

(~ ...,,

:3 o }:

,.....

I .

l

i

l

l

l

,

,

|

.

,

.

,

,

(b)

v

1:3

"X -

0

-

o

,,

10

20

(,~

- <~o)/r

Fig. 5. (a) Near-resonance absorption and (b) dispersion spectra of a two-level atom driven on resonance by a strong field with 292 -- 20F and probed by an another weaker field with 2f22 - 12F (solid line) and 2922 = 1.2F (dashed line).

Rabi frequency 2Y22 = a2Y2 such that Y22 > F, and observed a series of narrow resonances in the probe spectrum. For a < 1, these appeared near the frequencies oy2 = COo + 2_~; i.e., at detunings from atomic resonance by a subharmonic of the strong field Rabi frequency 2Y2. Hence, they were termed 'subharmonic resonances'. It was apparent immediately that they correspond to n-photon absorption by the driven atom between its energy levels (Toptygina and Fradkin [1982], Tsukada, Fujiwara, Tokuda, Kanamoto and Nakayama [1989], Kumar, Pons and Eberly [ 1991 ]). Figure 4, from a more recent investigation (Manson, Wei and Martin [ 1996]), shows the dispersive response of a two-level atom driven at resonance by a strong 'pump' field of Rabi frequency 2~2 = 21 ld-Iz, and 'probed' by a strong field corresponding to a = 1.43. Because of the excellent experimental resolution, up to 11 subharmonic resonances are clearly resolved. In figs. 5a and 5b, we

396

SPECTROSCOPY IN POLYCHROMATICFIELDS

[VI,

w2

plot theoretical profiles of the absorption W(oJ) and dispersion D(~o) of a probe beam having a = 0.6, together with the corresponding curves for a weak probe having a = 0.06. In this system, the strong driving field is exactly at atomic resonance. The spectra consist of a series of features near the subharmonic resonances, being dispersive in shape in the case of W(oJ) and absorptive in the case of D(oJ) (Manson, Wei and Martin [1996]). Note that the resonances are shifted from the exact subharmonic detunings of 2Y2/n. These shifts will be explained in w4.3.1. Maximum absorption from the driving fields corresponds to maximum probability Pe for the atom to be pumped into its excited state. In some experiments, maximum absorption is therefore monitored by observing the integrated intensity of fluorescence by the atom as the frequency of the probe field is scanned. In this way, Thomann [1980] observed the subharmonic resonances for sodium atoms driven by an intense amplitude modulated (am) field. Similarly, Chakmakjian, Koch and Stroud [1988] and Papademetriou, Chakmakjian and Stroud [1992] observed the resonances for sodium atoms driven by a 100% am field, resolving up to six resonances, with maxima appearing at the zeros of the Bessel function J0 (-492/6). This behavior will be explained in w4.1. In the latest experiment, Lounis, Jelezko and Orrit [ 1997] observed the subharmonic resonances for a driven single (impurity) molecule in a solid, the first such observation in the optical regime in condensed matter. 2.1. EXPERIMENTAL STUDIES

The experiments in this field can be divided into two main groups: those in the radiofrequency regime and those in the optical regime. In the radiofrequency experiments, the radiative broadening is negligible; the result in general is in spectra having very narrow linewidths and high resolution. The early experiments were performed on a single-proton-spin system in chloroform (Bucci and Santucci [1970]) and on a hyperfine nuclear transition in cadmium atoms in a vapor (Bonch-Bruevich, Vartanyan and Chigir [ 1979]); the more recent ones involve a nitrogen-vacancy color center in diamond (this system is described in detail by Wei and Manson [1994]). They are unable, however, to detect fluorescence by the atom. In the optical experiments, both pump and probe fields are provided by lasers, and the signal is monitored by measuring the integrated intensity of fluorescence by the atom. The experiments by Thomann [ 1980] and the Rochester experiments (Chakmakjian, Koch and Stroud [1988], Papademetriou, Chakmakjian and Stroud [1992], Papademetriou, Van Leeuwen and Stroud [1996]) involved an

vI, w2]

SUBHARMONICRESONANCES

397

atomic beam of sodium, and the Oregon experiments (Zhu, Wu, Lezama, Gauthier and Mossberg [1990], Yu, Bochinski, Kordich, Mossberg and Ficek [1997]) involved a beam of barium atoms. Although their resolution is in general not as good as those performed at radiofrequencies, the advantage of these experiments lies precisely in the fact that one is able to detect fluorescence, allowing for measurements to be made of quantities such as radiative widths and photon correlations, and thus contributing to a detailed understanding of quantum optics. Most recently, experiments have been performed on single (impurity) molecules in a solid matrix (Jelezko, Lounis and Orrit [ 1997], Lounis, Jelezko and Orrit [1997], Brunel, Lounis, Tamarat and Orrit [ 1998]) in which the effect is observed of both intense optical and radio fields on the two-level system. 2.2. APPLICATIONSOF SUBHARMONICSPECTROSCOPY 2.2.1. Two-photon laser In a search for a two-photon laser operating in the optical regime, Zhu, Wu, Morin and Mossberg [1990] observed a feature associated with continuouswave two-photon gain in the strong-probe absorption spectrum of barium atoms. Subsequently, Gauthier, Wu, Morin and Mossberg [1992] used barium atoms driven by a strong laser field and coupled to an optical cavity mode tuned to the n = 2 subharmonic resonance to demonstrate two-photon lasing at that frequency, m 0 - g2. In both cases, the gain medium was provided by the atom plus driving field. The gain is most easily pictured in terms of a transition between the dressed states I(N + 1)-) and I ( N - 1 ) + ) of the dressed atom, pictured in fig. 15a (below). The driving field is tuned to the red of the atomic transition frequency, producing a population inversion between the states (CohenTannoudji and Reynaud [1977]). 2.2.2. Lasing without population inversion The observed two-photon lasing is an example of lasing without population inversion. No inversion is created between the bare atomic states of the system studied by Gauthier, Wu, Morin and Mossberg [1992], because (S~) is always negative; however, the lasing occurs between dressed states between which a population inversion does exist. This type of inversion is often called a 'hidden' inversion (Mossberg [ 1995], Gawlik [ 1993], Lu and Berman [ 1991 ]). An example in which gain is possible without any population inversion at

398

SPECTROSCOPYIN POLYCHROMATICFIELDS

[VI, w3

all is a two-level atom driven by a resonant laser field. In this case, the amplification observed at one side of the dispersive features at the Rabi sidebands of the absorption spectrum (Mollow [ 1972]) occurs without population inversion between either the bare or dressed states. This type of gain can be interpreted as arising from coherences between the dressed states (Lu and Berman [ 1991 ], Szymanowski, Keitel, Dalton and Knight [1995]) or understood as quantum interference between various scattering channels of the driving and probe fields (Grynberg and Cohen-Tannoudji [1993]). Using a bichromatic field the gain without any population inversion can be enhanced significantly (Zhou and Swain [1997], Szymanowski, Wicht and Danzmann [1997]).

2.2.3. Laser cooling If an atom moves with velocity v in a standing-wave laser field of wave vector kL and frequency WE, the field at its position is equivalent to the sum of two traveling waves, of frequencies OJL + kL" v: a bichromatic field. Thus the calculation of its fluorescence intensity and spectrum is basically the same as that for nearly symmetric excitation, described in w4.2. The force versus velocity curve for the system displays many sharp resonances, referred to as dopplerons by Kyrol~i and Stenholm [ 1977], and observed experimentally for the first time by Bigelow and Prentiss [1990] and Tollett, Chen, Story, Ritchie, Bradley and Hulet [1990]. Another technique of laser cooling related to bichromatic driving is the 'sideband cooling' of an ion oscillating harmonically at frequency g2 in a trap (Wineland and Dehmelt [1975]). In the Lamb-Dicke limit, this form of cooling is most efficient when the laser is tuned to the first sideband below the atomic resonance, COL= 0~0 -- g2 (Cook [1990], Cirac, Blatt, Zoller and Phillips [1992]).

w 3. Theoretical Methods The subharmonic resonances observed in the near-resonance absorption and dispersion profiles of a strong field probing the monochromatically driven, two-level atom sparked renewed interest in these systems. However, a 'strong probe' is a contradiction in terms: It is an intense field which itself alters the characteristics of the system it is supposed to be probing. We will show that the system is both in principle and in practice more profitably regarded from the point of view that both 'pump' and 'probe' fields 'dress' the atom, and probe the resulting bichromatically dressed atom with a third weak field: the vacuum, in the case of fluorescence, or a weak applied field nearly resonant with either

VI, w 3]

THEORETICAL METHODS

399

the driven transition (Mollow absorption) or a transition from one of the driven levels to a third atomic level (Autler-Townes absorption). In this section, we will describe two principal theoretical methods used to calculate the response of an atom to a polychromatic field. One is the Floquet approach, in which the driving fields are treated classically and the atomic evolution is obtained from the master equation and described in terms of harmonics of the atomic variables. The other is the dressed-atom approach, in which the atom and driving fields are considered to form a single, entangled quantum system, and the master equation is solved in the basis provided by the dressed states of that system. The advantage of the Floquet method is that it can be used for all values of the system parameters; it does not, however, provide physical insight into the dynamics of the system. By contrast, the dressed-atom method is limited in the range of parameters for which it can be used; its predictions, however, are then in complete agreement with those of the Floquet method. Most importantly, the dressed-atom method provides an understanding of the physics of the system. 3.1. FLOQUET METHOD

3.1.1. M a s t e r equation

The dynamics of a system of atoms or molecules coupled to an external field are conveniently studied in terms of the reduced density operator p of the system. The time evolution of the density operator is described by the Markovian master equation (Lehmberg [ 1970], Louisell [ 1973], Agarwal [ 1974]): 0/9 Ot

i [Ho + Hint, p] +12p, h

(1)

where H0 is the atomic Hamiltonian, Hint is the interaction between the system and the external fields, and s is the Liouvillean operator representing the damping of the system by spontaneous emission. We consider a two-level atom, with excited state le), ground state Ig), transition frequency tOo and transition dipole moment/~. The Hamiltonian H0 can be written as Ho = htooS z,

(2)

and the operator s as s

= ~IF ( 2 S - p S + - p S + S - _ S + S - p ) ,

(3)

SPECTROSCOPY IN POLYCHROMATIC FIELDS

400

[VI, w 3

where F is the spontaneous emission rate, S: = ~1 ( ] e ) ( e [ - g)(g[) is the inversion operator, and S + = ]e) (g] and S- = [g) (el are the atomic raising and lowering operators, respectively. The interaction Hamiltonian Hint, in the electric-dipole and rotating-wave (RWA) approximations, is given by Hint = - ~ 1.

E(+) (t) s - - h.c.,

(4)

where E<+)(t) is the positive-frequency component of the electric field. For a classical polychromatic field composed of p discrete frequency components, E<+)(t) can be written as P

e<+(t) - Z eJ +'ei+',

(5)

j=l

where E) +) is the amplitude of the jth component and ~. is its frequency. The interaction Hamiltonian is explicitly time dependent, although for a monochromatic driving field, in a frame rotating at frequency tO1, Hint can be written as Hint =

- ~ p1

9 E(I+)S-

(t)

h.c.,

(6)

where S-(t) = S-ei<~

(7)

the Hamiltonian (6) is then independent of time. If the driving field is composed of more than one frequency component, however, the time dependence of Hint is quite complicated and in general involves p different parameters. This renders the problem difficult to solve, except in those special cases in which the time dependence can be expressed in terms of a single parameter. We introduce the average frequency tos of the driving field components and the detuning 6/= ~.-tOm of the jth component from tOs. An example of such a special case occurs when 6/= n6, (n = 0, + 1, +2,...). With a single parameter, the problem becomes one involving one-dimensional recurrence relations (Risken [1984]), whose solution we will describe in the following sections.

3.1.2. Optical Bloch equations The master equation (1) with the Hamiltonian (6) leads to a closed set of three equations of motion for the expectation values of the atomic operators. This set

VI, w 3]

THEORETICAL METHODS

401

of equations, known as optical Bloch equations, can be written in matrix form as

d

-~X(t) = U(t)X(t) - 1FI,

(8)

where X ( t ) = col({S-(t)), (S+(t)), {S:(t)}), U(t) is the 3 •

/ ,

0

- ( ~F +

U(t) =

.C2jei"~Jt

- (31F - iA)

0 _~--";__

g * e-i/); t

P

matrix

2 ~-~'P~j=l ~

* e - i 0 /9t

/

,

(9)

ei~ t

I is a column vector with the components I~ = 12 = 0 and 13 = 1, A = coo - COs, and 2~.-

l~ . Ej

(10)

h

is the Rabi frequency associated with component j of the driving field. The optical Bloch equations (8) are first-order differential equations with timedependent coefficients. In principle, the equations can be solved numerically by direct integration. This method, however, will not be used as it does not provide physical understanding of the problem. Instead, we shall use other methods, which involve both analytical and numerical analysis.

3.1.3. Recurrence relation Since the coefficients of the system of equations (8) are periodic in time with periodicity 6j, we can study their dynamics by applying the Floquet method (Stenholm [1984], Chu [1985]). We decompose the Bloch vector components X;(t) into amplitudes which oscillate at frequencies 6j and their harmonics, by means of a p-dimensional Fourier series: o~

Xi(t) = Z II = - o o

oo

~ 12 = - O c

oo

"'" ~

Xi(i''i2..... 1;,)r

(11)

1t, = - ~ c

where Xi<;'';2..... 6,) are slowly varying amplitudes. Substituting eq. (11) into eq. (8), we find that the slowly varying amplitudes satisfy a p-dimensional recurrence relation: d X(O(t)= _ 1F6,c, o&: dt ~ , ,0" p

- A 6 X(t)(t) - Z j=l

&, ol ,,

(12)

P

Bt; X~t- l)' (t) - ~ j= 1

Dt'Xtt + 1)'(t)'

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SPECTROSCOPY IN POLYCHROMATICFIELDS

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where (g) = (gl, g 2 , . . . , gp), (g + 1)j = (g,, g2, . . . , gj + 1 , . . . , gp) and

( 89 + i A + i y'~jgj6j) A6 =

B6 =

0

0 ('F - iA+i2jgS6j) \~

0

(00 0 0

0 0

0 0

0

,

D,,j=

(000) 0 0-2E2j0

(13) .

(14)

We present the solution of this recurrence relation for the case of a bichromatic driving field, p = 2, n = + 1. Equation (12) is then reduced to the one-dimensional recurrence relation

d X(O(t) = _89F6~,ol - A~X(C)(t) - BoX ~-l)(t) - D c X I~+l)(t) dt

(15)

where Ae, Be and De are the matrices

(2

At =

0

(~1 F - i A + i g 6 )

0

Be=

(o o

0 0 -2ag2 a ~ ~2 0

0 ,

Dl=

0 (F+igb)

(o o

0 0 -292 E2 aE2 0

(16)

,

(17)

with g21 - g2, a = g22/g2 and 6 = ~1 ((_D1 --602). One method of solving the recurrence relations is to use continued fractions. However, we choose instead to solve in terms of the eigenvalues and eigenvectors of the infinite-dimensional (Floquet) matrix, which we construct by arranging the amplitudes X(e)(t) in the order

X(1)(t)

Y(t) =

X(~

(18)

X(-1)(t)

Equation (15) can then be written as the matrix differential equation d

- - Y(t) = K_Y(t) + P, dt

(19)

where K__is an infinite-dimensional tridiagonal (Floquet) matrix composed of the 3 >3 matrices Ae, Br and Dr, and P is an infinite-dimensional vector with the non-zero c o m p o n e n t - ~ 1F6~,0I.

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The matrix equation (19) is a simple differential equation with timeindependent coefficients, and is solved by direct integration. For an arbitrary initial time to, the integration of eq. (19) leads to the following formal solution for Y(t): Y(t) = Y(to)e ~ t - (1 - eKt) K-1p.

(20)

In order to proceed further, we must truncate the dimension of the vector Y(t). The validity of the truncation is ensured by requiring that the solution (20) does not change as the dimension of Y(t) increases or decreases by one. Because the determinant of the finite-dimensional (truncated) matrix K is different from zero, there exists a complex invertible matrix T which diagonalizes K, and/~ = T-1K T is the diagonal matrix of complex eigenvalues. By introducing L = T-IY and R = T-1p, we can rewrite eq. (20) as L(t) = L(to)e )~t- ( 1 - e '~t) ~,-IR,

(21)

or, in component form, q

Zi(t) = Zi(to)e )~it- Z

(/~-l)ij ( 1 - e zjt) Rj,

(22)

j=l

where q is the dimension of the truncated matrix. To obtain solutions for X/.~0(t), we determine the eigenvalues ~.; and eigenvectors Li(t) by a numerical diagonalization of the matrix K. The steady-state values of the components X/It)(t) can be found from eq. (22) by taking t ~ ~ , or more directly by setting the left-hand side of eq. (19) equal to zero. Thus, q Y/(oo) = - Z (K-1)/j j=l

PJ"

(23)

In subsequent sections, we will use the solution (23) to calculate the stationary fluorescence intensity, Is = [89 +X3~~

,

(24)

and absorption and dispersion coefficients of the probe fields, W(~o) = Z'(O)) = -Im [X~-')(~)],

(25)

D(o)) = Z " ( o ) ) = - R e [X~-')(oo)],

(26)

where 2'~(0)) and 2'"(09) are the real and imaginary parts, respectively, of the field susceptibility. We will use eq. (22), together with the quantum-regression

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SPECTROSCOPY IN POLYCHROMATIC FIELDS

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theorem (Lax [1968]), to calculate the incoherent and coherent parts of the fluorescence spectrum:

F(m) = ReX(~

(27)

oo

=

6 (m- ms + g6),

(28)

~ ---- - - ( X )

where v = ( m - ms)/F and X( 0 = limt ~ ~

X~)(t).

3.2. D R E S S E D - A T O M M E T H O D

In this section, we present the dressed-atom method of calculating the fluorescence and absorption spectra of a two-level atom driven by a polychromatic field. The method was first introduced by Cohen-Tannoudji and Reynaud [1977] for a monochromatic driving field. In this picture, every field mode whose Rabi frequency 292 is sufficiently large that g2 > F is considered to 'dress' the atom, and to form along with it a single, entangled quantum system. This reflects the fact that photons are exchanged between the atom and driving field modes via absorption and stimulated-emission processes many times between successive spontaneous emissions by the atom into the vacuum field.

3.2.1. Dressed states The Hamiltonian of the entangled system is written as H = HA +HE + W = H 0 + W,

(29)

where HA is the Hamiltonian of the atom, given by eq. (2), P

HL=hZoi(a~ai+ 89

(30)

i=l

is the Hamiltonian of the driving field, P

W=hZgi(a~S-+S+ai )

(31)

i=1

is the interaction (in the RWA) between the atom and the driving field, and the coupling constant between the atom and the ith field mode.

gi is

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THEORETICALMETHODS

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The 'undressed' states of the system are the eigenstates of H0. They can be written as direct products of the form la) | ]Nl) | | ]Np), where ]a) is an atomic state (a = e, g), and Ni is the number of photons in driving mode i. We diagonalize the total Hamiltonian (29) in the basis formed by these undressed states. The interaction W has non-vanishing matrix elements only between those undressed states between which the atom has a non-vanishing dipole moment and the number of photons in one of the field modes changes by one; the matrix elements are thus proportional to v/Ni + 1 or x/~/. However, for strong driving fields (Ni) >> 1, we can approximate

gi v/Ni + 1 ~ gi ~

~ . . . . 2-Qi,

(32)

where 2g2i is the (on resonance) Rabi frequency of mode i. The eigenstates of H, denoted by In, N), are the dressed states of the system, and must be calculated individually for every system considered. In general, they are grouped into manifolds, which are labelled s where N is the total number of excitations of the states in oe(N). The dressed-atom method is most useful when the energy differences between the dressed states In, N) and Im, N) are large compared with the damping rate F: ](_Dnm] > 1". It is then possible to make the secular approximation, in which we neglect coupling between diagonal and off-diagonal elements of the density matrix. The calculations performed within this approximation are valid only at the lowest order in F~ [(_Dnm].

3.2.2. Fluorescence Next, the dressed atom is allowed to interact with the vacuum field. This results in spontaneous emission by the dressed atom down its energy manifold l a d d e r - equivalent to fluorescence in the bare-atom picture. The temporal and spectral properties of this emission/fluorescence are a signature of the entangled system, its energy levels and their populations. Additionally, the system may be probed by measuring the absorption or dispersion of a weak applied field, nearly-resonant either with the driven transition (Mollow absorption) or with a transition of the atom between one of the driven levels and a third level ]d) (Autler-Townes absorption). The fluorescence and Mollow absorption processes involve transitions between dressed states of the system, and depend on nonvanishing matrix elements of the atomic dipole moment operator between states in which the numbers of driving field photons remain constant. If the atom has no permanent dipole moment (see w4.3.3), these occur only between neighboring manifolds, and we denote them by/t,,, = l/t[ (n,N IS+[ m , N - 1).

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SPECTROSCOPYIN POLYCHROMATICFIELDS

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The probability of a transition between the dressed states [n,N) and m, N - 1) is Fnm = F [/tnml2, and the total spontaneous emission decay rate from In, N) into the manifold below is given by

Fn=ZFnm.

(33)

m

3.2.3. Populations The populations of the dressed states P n , N = (n,N [p[ n,N) are the diagonal elements of the density matrix of the system, and are found by projecting the master equation (1) onto In, N) on both the left and the fight. In the secular approximation, the reduced populations P, = ~-~X P n , N (Cohen-Tannoudji, Dupont-Roc and Grynberg [ 1992]) satisfy the following set of coupled equations of motion: d -~Pn(t) :-FnPn(t) + ~ F,mPm(t). (34) m

The first term on the right-hand side of eq. (34) is due to transitions out of In, N) into the manifolds below, and the second term to transitions into In, N) from the states Im, N + 1) of the manifolds above. The intensity of every spectral line which corresponds to a transition originating in In, N), whether in fluorescence or in absorption, is proportional to Pn. The populations P~ can be used as well to calculate Pc, the population of the excited state le) of the strongly driven atom. The total intensity of fluorescence by the atom is then FPe.

3.2.4. Coherences The projection of the master equation (1) onto ]re,N - 1 ) on the right and (n,N on the left gives the equations of motion for the coherences flnm,N -- (n,N [p[ re, N - 1 ) between the dressed states of the neighboring manifolds. If there is only one transition which has (Bohr) frequency r the reduced coherence Pnm = ~--~X Pnm,U obeys the equation -~ Dnm

[iOOnm+ F~"m)]Dnm,

(35)

where F (nm) is the coherence damping rate, F(nm) _. -21(F n q_ Fm ) q_Dnm,

(36)

and D n m takes into account the 'transfer of coherence' between manifolds (Cohen-Tannoudji, Dupont-Roc and Grynberg [ 1992]). The transition frequen-

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THEORETICALMETHODS

407

cies O)~m give the frequencies of the lines in both the fluorescence and nearresonance absorption spectra. If more than one transition has the same Bohr frequency, however, the evolution of their corresponding coherences may be coupled. A simple general expression cannot then be given for the evolution; each case must be studied individually.

3.2.5. Fluorescence spectrum The steady-state fluorescence spectrum is given by the real part of the Fourier transform of the two-time correlation function of the dipole moment operators, F(co) = Re f o o d r e i'~ lim (/t + (t + r)/t-(t)) t --+ OO

(37) d r e i~~ lim Ztl,,m(a~m(t+r)ll-(t)), = Re f0 ~176 t ----+ OO nm

where (Ynm = ] n , N ) ( m , N - 1 [ . From the quantum regression theorem (Lax [1968]), it is well known that for r > 0 the two-time average (a.m (t + r ) g - ( t ) ) satisfies the same equation of motion as the one-time average (a.m(t)) with the initial condition

(38)

(Onm(t)ll-(t)) : FnmPn,

where rnm is the probability of the transition from n to m and Pn is the reduced steady-state population of level n. The one-time average (a,m(t)) satisfies the same equation of motion as the coherence pmn(t). When the spectral lines do not overlap, the fluorescence line corresponding to the transition n =~ m is simply (39)

Fnm((-1)) = F n m P n L n m ( ~ ) ,

where 1

r (nm)

L.m(OO) -

(40) \

/

The line is centered at the Bohr frequency O~,m, has a width determined by the damping rate of the coherence F~~"'), and an intensity (in the steady state)

408

SPECTROSCOPY IN POLYCHROMATIC FIELDS

[VI, w 3

proportional to the product of the steady-state population of state n and the transition rate I " n m from n to m. The integrated fluorescence intensity Is = f F(~o)d~o is proportional to the population Pe of the excited state le) of the bare atom, found using the dressed populations it',.

3.2.6. Near-resonance absorption spectrum The absorption spectrum of a weak probe beam monitoring the driven system is given by the real part of the Fourier transform of the two-time commutator ([W(t),W(t + r)]). The term (W(t)W(t')) is associated with absorption, and the term (/t+(t~)/t-(t)) with stimulated emission of the probe beam: the net absorption between the levels is equal to the difference between absorption and stimulated emission by the atom. Using the same procedure as for the fluorescence spectrum, we find that the absorption line corresponding to a transition from n to m is (41)

J/Vnm(O))- - I-'nm(Pm - Pn) Lnm(fO),

where Lnm(O)) is given by eq. (40). The components of the absorption spectrum have the same positions and widths as their counterparts in the fluorescence spectrum, but widely different intensities. The net absorption at any frequency is proportional to the transition rate F,,m and the difference between the populations of the lower and upper levels in the transition. If the population difference is positive, the probe is absorbed by the system, whereas it is amplified if the difference is negative. The central component of the absorption spectrum is not given correctly by the dressed-atom method within the secular approximation. This component is reproduced correctly only in higher order (Grynberg and Cohen-Tannoudji [1993]). We can also calculate the refractive index of the probe beam, which is proportional to the imaginary part of the two-time commutator. The index is given by a sum of dispersive profiles, where the profile corresponding to the transition n :=~ m is D,m(~O) = 1 F~m (Pm -- P,)

6O- 09,,,,

2"

(42)

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409

3.2.7. Autler-Townes absorption spectrum The structure and the population distribution of the dressed states can also be studied by monitoring the system with a weak probe beam coupled to a third atomic level Id), connected to either Ig) or ]e), which has a transition frequency tod different from tOo (Autler and Townes [1955]). In the transition from Ig, N) to Id, N), for example, the number of drivingfield photons remains constant. The atomic feature is split by the presence of the driving fields into components, whose intensity is the product of Pn and a weight factor I(g,N n,N)l 2. Thus the number of components is equal to the number of dressed states 'contaminated' by Ig, N). The shift of each component is determined by the shift from NhtOo of the energy of In, N), and the width by 1_ 2 (Fd + Fn), where Fd is the natural width of level Id)

w 4. Bichromatic Fields The subharmonic resonances appear in the absorption and dispersion spectra of an intense field used to probe a monochromatically strongly driven atom; i.e., the so-called 'strong probe'. However, a 'strong probe' is a contradiction in terms. In reality, the system consists of an atom that is being driven by two strong fields, and which should be probed by a (third) weak field. If this weak field is simply the vacuum, we observe fluorescence by the system. Therefore, in this case one could expect significantly different spectral properties of the fluorescence from those observed with monochromatic driving. Zhu, Lezama, Wu and Mossberg [1989] measured the fluorescence spectrum of a two-level atom driven by a bichromatic field consisting of two components of equal amplitude displaced symmetrically by frequency 6 about the atomic resonance. The observed spectrum was composed of a central component at the atomic resonance frequency together with a series of sidebands separated equally by 6. The most striking features of the spectrum were that the positions of the sidebands were independent of the Rabi frequency, which, however, determined their number and relative intensities (in a complicated manner). This was in complete contrast with the situation for the Mollow triplet observed for monochromatic driving (Schuda, Stroud and Hercher [1974], Wu, Grove and Ezekiel [1975], Hartig, Rasmussen, Schieder and Walther [1976]), in which the displacements of the sidebands depended (linearly) on the Rabi frequency, while their number (2) and intensities did not. These observations were explained by a number of theoretical analyses of the fluorescence spectrum (Kryuchkov

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SPECTROSCOPYINPOLYCHROMATICFIELDS

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[ 1985], Tewari and Kumari [ 1990], Freedhoff and Chen [ 1990], Agarwal, Zhu, Gauthier and Mossberg [1991], Agarwal and Zhu [1992], Ficek and Freedhoff [ 1993]). The specific structure of the dressed states was also confirmed in another experiment by Papademetriou, Van Leeuwen and Stroud [ 1996], who observed the Autler-Townes spectrum of a bichromatically driven, two-level transition in a three-level cascade atom. There have now been several experimental investigations as well which examined the response of a two-level atom to a bichromatic field composed of one strong and one weaker component. This special case of bichromatic excitation has led to the discovery of many interesting effects, such as phasedependent atomic dynamics (Wu, Gauthier and Mossberg [1994a, 1994b]), the double Stark effect (Yu, Bochinski, Kordich, Mossberg and Ficek [1997]), and the multi-photon ac Stark effect (Greentree, Wei, Holmstrom, Martin, Manson, Catchpole and Savage [ 1999]). If the time evolution of the fluorescence intensity is observed (Wu, Gauthier and Mossberg [1994a, 1994b]), it exhibits a modulation, and the atomic response depends on the phase even if the interaction begins with the atoms in their ground states (Chien, Wahiddin and Ficek [1998]). In the double Stark effect, the components of the Mollow triplet are each split into a triplet, and additional triplets appear at harmonics of the Rabi frequency (Ficek and Freedhoff [1996]). The multiphoton ac Stark effect appears whenever the frequency of the weaker component is near the multiphoton resonances induced by the strong component, and is manifested by the appearance of additional triplets at subharmonics of the Rabi frequency. A theoretical interpretation of these results has been given in terms of Floquet states and with a doubly-dressed atom model that provides a simple physical explanation of the observed features (Rudolph, Ficek and Freedhoff [1998], Rudolph, Freedhoff and Ficek [1998a,b]). 4.1. SYMMETRICEXCITATION We consider first a symmetric excitation in which the components of the bichromatic field have equal Rabi frequencies and are displaced equally about the atomic resonance. We calculate the dressed states of the entangled atom-field system, and use these to explain the observed spectra. 4.1.1. Dressed states

The Hamiltonian of the non-interacting atom and bichromatic field is given by Ho = h~ooS z + h (090 + 6) at+a+ + h (O9o - 6) at_a_,

(43)

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BICHROMATICFIELDS

411

and has the eigenvalue equations Ho IZn, 2 N ) = [EzN + 2nh6] IZn, 2 N ) , Ho IZn - 1 , 2 N ) - [E2N + (2n - 1) h6] IZn - 1,2N),

(44)

where a• ( a ~ ] are the annihilation (creation)operators for the driving modes \

/

of frequencies (-Oo + 6, E 2 N = 2Nhooo, and Ig, N + n, N - n) = 12n,2N) and ]e, N + n - 1, N - n) - 12n - 1,2N) are the 'undressed' states of the noninteracting system corresponding to energy manifold E(2N). When we include the interaction between the atom and the bichromatic field, the states recombine to form a new ladder of states (dressed states) (Freedhoff and Chen [1990]): 2N, ~ , m

=

J,-m - - - g

In, ZN),

(45)

n=--OO

corresponding to energies E~N) = E2N + mh6,

m = 0, 4-1, + 2 , . . . .

(46)

In a similar way, the undressed states ]g,N + n + 1 , N - n) and le, N + n , N - n) of manifold E (2N + 1) recombine to form the eigenstates of that manifold. Each manifold contains an infinite number of states. Neighboring manifolds are separated by frequency co0, while neighboring states within each manifold are separated by 6. We now introduce the interaction between the atom and the vacuum field, leading to a spontaneous emission cascade by the dressed system down its energy manifold ladder. It is straightforward to show that there are two distinct transition rates between manifolds of even and odd numbers of excitations. The transition rate from [2N + 1,-~, m) to IZN, ~, ~ m' ) is Fmm' = -~F 6ram' + (-1)mJm'-m

-~

(47)

,

Q m t ) is while the rate from IZN, -8,m) to 12N - 1, ~, I'm,,, : ~F

6mm' -- (-1)'J~,_,,,

-~

(48)

The total transition rates out of 12N + 1,-~,m) and 12N, ~, ~2 m) are then given by J0 - ~

(49)

where (+) and (-) stand for the rates from the dressed states ]2N + 1 -g,m) e and 12N, ~e , m ), respectively. ,

412

SPECTROSCOPYIN POLYCHROMATICFIELDS

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d

(5

(5 . i

o4

d 1--

d

0

0

1o

20

3o

2fl/r

(b) O

,

,

i

i

1 /0

I

i

i

0

,

.

20

,

,

.

l

30

n/r Fig. 6. Stationary total fluorescence intensity for a symmetric bichromatic field plotted (a) as a function of 2#2 and constant 6 = 5F, (b) as a function of 6 and constant 2g2 = 20F. The p o p u l a t i o n s o f the dressed states are then found f r o m the rate equations. F r o m them, the steady-state p o p u l a t i o n s o f the atomic levels are f o u n d to be

ee,g((X)) =

1(

~

4Jo(-4#2/6))

1 :t: 3 + Jo ( - 8 . Q / O )

"

(50)

The steady-state total intensity o f fluorescence by the b i c h r o m a t i c a l l y driven a t o m is Is =

FPe(oO). This

expression was also derived by K r y u c h k o v [1985],

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BICHROMATICFIELDS

413

using a method based on quasi-energy states, and is plotted in fig. 6a for constant 6 = 5F as a function of 2s and in fig. 6b for constant 2f2 = 20F as a function of 6. These figures display the 'subharmonic-resonance' behavior observed experimentally by Chakmakjian, Koch and Stroud [ 1988], with maxima appearing in Is or, equivalently, minima in the absorption by the 'strong probe', at the zeros of the function J0(-4f2/6). 4.1.2. F l u o r e s c e n c e

spectrum

We obtain an analytical expression for the fluorescence spectrum by studying the evolution of the coherences between dressed states in neighboring manifolds. There are four principal (reduced) coherences, O0s and o0, both of which correspond to polarization of the system at frequency COo, and azk and azk-l, corresponding to polarizations at even and odd sideband frequencies, respectively. These can be shown to satisfy the equations of motion

-

d ~ I F +) ~o~ = - (~i0)o dt -

d

+

d dtO2k =~do 2 k - ~ td

= -

[ [

OOs, (3 _ jo ( _ 8 ~ )

i (0)o + 2 k 6 ) + l F i (~o + (2k -

(( 8J))l ((82))1 3 - Jo - - -

1) 6) + 1F

(51)

o2k,

3+Jo

-~

O2k-1.

At frequency 0)0 there are two lines, with widths ~l F a n d G = ~1 F [ 3 - J o ( - ~ )] 9 The even sidebands are located at frequencies 0)o + 2 k 6 and also have widths Fe. The odd sidebands are located at 0)o + ( 2 k - 1)6, and their widths are Fo = ~1 F [ 3 + J o ( - 8 ~-2- ) ] . The full spectrum is given by the expression Fe [Q + J0 (-4.(2/6)] J0 (-4f2/6) F { F 1 + QJ0 (-4s + F(0)) = ~-~ ~ (0)_ 0)0)2 + (F/2)z 4 (~0- ~00)2 + r 2

/;

O(9

+ Z k=-oG

J2k (-4f2/6)

(o) - ~o - 2k 6) 2 + F 2

OO

+ Z k=-o~

J2k-I(-4.(2/6)

(0)-0)0-(2k-1)6)2+Fo 2

' (52)

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SPECTROSCOPY IN POLYCHROMATICFIELDS

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(*) 3 8~

v I~

O

-10

0

10

0

10

(b)

3 v i,

tl~ O

6

-10

(:~

-

%)/r

Fig. 7. Fluorescence spectrum for a symmetric bichromatic field with constant 6 = 2 . 5 F and two different Rabi frequencies: (a) 2s = 10F; (b) 2s = 6/'. The parameter values correspond to the experimentally observed spectra, presented in figs. 2(a) and 2(b) of Zhu, Wu, Lezama, Gauthier and Mossberg [ 1990].

where 4J0 (-4g2/6) O = 3 + J0 ( - 8 Q / O ) "

(53)

This expression is plotted in fig. 7 for 6 = 2 . 5F and two different Rabi frequencies: 2g2 = 10/" (fig. 7a) and 2g2 = 6/" (fig. 7b). The theoretical curves are in excellent agreement with the experimental observations, presented in figs. 2a and 2b of Zhu, Wu, Lezama, Gauthier and Mossberg [1990]. The intensities of the sidebands at +k6 are proportional to J~ (-492/6), so the number of sidebands observed increases with (2/6. The widths of the odd sidebands in the experimental spectra are somewhat less than those in their theoretical counterparts. This narrowing results from the contribution to the odd sidebands of the coherent scattering (Agarwal, Zhu, Gauthier and Mossberg [ 1991 ]), which is not included in the theoretical expression (52).

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415

Other techniques have also been used to study the interaction of a twolevel atom with a symmetric bichromatic field. Wilkens and Rza~ewski [1989] proposed a purely numerical technique based on the It6 prescription of the integration of the time-dependent differential equations. Tewari and Kumari [1990] and Kryuchkov [1985] have derived analytical formulae for the spectrum using an iterative procedure on the damping parameter and the separated lines approximation, respectively. This latter technique has also been used to calculate photon statistics and squeezing under bichromatic excitation (Kryuchkyan [1991 ], Kryuchkyan and Kheruntsyan [1992], Kryuchkyan, Jakob and Sargsian [1998]), and to explain the experiments on the two-photon decay of Rydberg atoms in a bichromatically driven cavity (Lange and Walther [1994], Lange, Walther and Agarwal [ 1994]).

4.1.3. Near-resonance absorption spectrum A nearly-resonant probe field can also be used to monitor the energy structure and population distribution of the system. We calculate the spectrum with the Floquet method, however, because the dressed-atom approach will reproduce the structure at the central frequency only in higher order (Grynberg and CohenTannoudji [1993]). In fig. 8, we plot the absorption spectrum for a symmetric bichromatic field with 292 = 14F and 6 = 5F. The spectrum shows a series of dispersive features located at n6, (n = -t-1, + 2 , . . . ) with no structure at the central frequency too. As in the fluorescence spectrum, the separations of the sidebands are independent of the Rabi frequency of the driving field, but their number increases with g2. However, depending on g2 the spectrum can also exhibit a large absorption peak at the central frequency. This is shown in fig. 9, where we plot the absorption spectrum for 6 - 5F and 292 = 15F. The dispersive structure of the sidebands and their constant separation can be explained quantitatively by the dressed-atom model. To explain the oscillation of the central peak amplitude with the Rabi frequency, we refer to the optical Bloch equations (8) and the oscillatory properties of the steady-state fluorescence intensity, shown in fig. 6a. For A = 0 and a = 1, we can write the absorption spectrum as 1

W(to) = ~F

(i s _

1

/-'

2) !F2 + ( t o - too) 2 + 2Re Ul~

=

-i(,o)-~oo)/r'

(54)

4

where Is is the steady-state fluorescence intensity and U~~ is the Laplace transform of the zeroth-order harmonic component of U(t)= ~1 [(5'-(t)) + (S+(t))].

416

SPECTROSCOPY IN POLYCHROMATIC FIELDS

[VI, w 4

? o x If)

I

v o 3=

? o x ut) I

I

I

I

I

o

-20

I

2o

(o - % ) / r Fig. 8. Near-resonance absorption spectrum for a symmetric bichromatic field with 2f2 = 14F and di = 5F.

v

d

-

20

.

.

0

.

.

20

(~ -~o)/r Fig. 9. Near-resonance absorption spectrum for a symmetric bichromatic field with 2~2 = 15F and 6 = 5F.

We see from eq. (54) that the central component of the absorption spectrum is distinct from the remaining features, depending only on Is. The oscillations with ~2 of its amplitude reflect the oscillations of the fluorescence intensity, shown in fig. 6a. Whenever Is ~ ~, 1 there is no central line in the absorption spectrum. [There is also a contribution (dispersive-type) to the central component

VI, w 4]

BICHROMATIC FIELDS

417

from U(~ its amplitude, however, is very small and it does not affect the spectrum.] 4.1.4. Autler-Townes absorption spectrum

In fig. 10, we plot the spectrum of the Autler-Townes absorption to a third atomic level Id) when a symmetric bichromatic field drives the Ig) --* Ie) transition. The spectrum shows a series of peaks separated by 6, revealing the constant b-dependent separation of the dressed states.

3'

O_4o

.

. - 2 0.

.

!

0

20

4O

Fig. 10. Autler-Townes absorption spectrum for a symmetric bichromatic field with 2s 6=5FandF d= IF.

= 20F,

This multi-peaked Autler-Townes spectrum was observed experimentally by Papademetriou, Van Leeuwen and Stroud [1996] in atomic sodium. In the experiment the absorption was measured on the 3P3/2 --* 4D5/2 transition of a three-level cascade atom in which the 3S~/2 ---* 4P1/2 transition was driven by a 100% amplitude-modulated field. 4.2. NEARLY SYMMETRIC EXCITATION

4.2.1. Fluorescence spectrum

When the average frequency of the field components is detuned from the atomic resonance (,4 ~ O) and/or the Rabi frequencies of the two fields are unbalanced (a ~ 1), symmetry is destroyed and the positions of the sidebands in the

418

SPECTROSCOPY IN POLYCHROMATICFIELDS

[VI, w 4

tN

ci

ci

ci

g ci

0

-20

0

20

(,~ - ,~,)/r Fig. 11. Fluorescence spectrum for a detuned bichromatic field with 2s = 8F, a = 1, A = 3F and 6 = 5F.

V

!

-20

0

20

(~ - ~,0)/r Fig. 12. Fluorescence spectrum for an asymmetric bichromatic field with 2g2 = 8F, A = 0, 6 = 5F and a = 0.75. f l u o r e s c e n c e s p e c t r u m d e p e n d o n the R a b i f r e q u e n c i e s a n d d e t u n i n g s . In fig. 11, w e p l o t the s p e c t r u m for 292 = 8 F , a = 1, 6 = 5 F a n d A = 3 F ; i.e., for e q u a l R a b i f r e q u e n c i e s b u t a s y m m e t r i c d e t u n i n g . T h e s p e c t r u m s h o w s an i n t e r e s t i n g d e v i a t i o n f r o m that for A = 0: It c o n t a i n s m o r e p e a k s , w h i c h arise f r o m the splitting o f the c e n t r a l line a n d the even s i d e b a n d s into doublets. T h e p o s i t i o n s

VI, w 4]

BICHROMATIC FIELDS

419

of the odd sidebands are unaffected by A. A similar modification to the spectrum appears for A = 0 and a , 1. This is shown in fig. 12, where we plot the spectrum for A = 0, 6 = 5F, 2g2 = 8F and a = 0.75. As in the case of A , 0, the central line and the even sidebands split into doublets, whereas the odd sidebands remain unchanged. The splitting of the spectral lines into doublets might suggest that non-zero detunings or unequal Rabi frequencies have the effect of splitting some of the dressed states into doublets. This, however, is not the case. The splitting in fact arises from shifts of the dressed-state energies. We show this by considering the asymmetry to be a perturbation on the symmetric (a - 1, A = 0) Hamiltonian, and using first-order perturbation theory to find the corrected energies of the system (Ficek and Freedhoff [1993]). Because the eigenstates of the even manifold ,5' (2N) are different from those of the odd ,5 ( 2 N - 1), the shifts of their dressed states occur in different directions. For example, for a = 1 and A ;~ 0, the energies are E~N) = E2N

+

1

m6 + -~A [1 + (-1)mJ0 (-4g2/6)],

E(m) 1 2N-1 = E z N - 1 + mt~ + ~A[1

(55)

- (-1)mJ0 (-492/6)].

The central line and the even sidebands result from 2m =~ 2m' and from 2m + 1 =~ 2m' + 1 transitions within adjacent manifolds. Because the shifts of these states occur in opposite directions, the transition frequencies are to = too + 2 (m - m') 6 + AJ0 (-4g2/6),

(56)

and the lines split into doublets. The odd sidebands, however, are due to transitions 2m =~ 2m' + 1 and 2m + 1 =~ 2m' within neighboring manifolds. These states are shifted in the same direction, so the transition frequencies are unshifted. Similarly, the perturbed energies for a , 1, A - - 0 are given by E~N)

= E2N +

mb + (-1)rag2 ( a - 1) J1 (4Y2/b),

(m) = E2N-1 + 2N-1

mb-(-1)mg2(a

(57)

- 1) J1 (4g2/6)

As a result, the central line and even sidebands are split by an amount 4g2 ( a - 1) J1 (4g2/6), while the odd sidebands are unaffected. The total fluorescence intensity and the fluorescence spectrum have been calculated by Agarwal and Zhu [ 1992] for both symmetric and nearly symmetric bichromatic driving for a range of experimental parameters using the Floquet method.

420

SPECTROSCOPYIN POLYCHROMATICFIELDS

[VI, w4

d

I

-20

I

,,

i

0

20

(o -~,)/r Fig. 13. Near-resonance absorption spectrum for a bichromatic field with 292 = 8F, a = 1, 6 = 5F, Fd = l F, and A = 5F.

4.2.2. Near-resonance absorption spectrum The dispersive shape of the sidebands of the absorption spectrum and the oscillatory properties of the central line are characteristic of symmetric bichromatic driving. As with the fluorescence spectrum, the absorption spectrum changes drastically when A ~ 0 or a ~ 1. This is shown in fig. 13, where we plot the absorption spectrum for a = 1,292 = 8F, 6 = 5F and A = 2F. In this case, the central component and the even sidebands of the absorption spectrum are composed of absorption-emission doublets, whereas the odd sidebands remain dispersive for all values of A and a. With a nonsymmetric driving field there are more regions of frequency where the probe field can be amplified instead of being absorbed by the atom. The properties of the absorption spectrum are explained qualitatively by the dressed-atom model. The splitting of the spectral lines is related directly to the shifted dressed-atom frequencies. The absorptive and emissive behavior is related to an unequal population of the dressed states. As in the case of a monochromatic driving field detuned from atomic resonance, the dressed states are populated unequally, giving a net absorption or amplification of the probe field on transitions between states of unequal population.

4.2.3. Autler-Townes absorption spectrum Figure 14 shows the Autler-Townes spectrum for a driving field with a - 1, A - 5F. In this case, the spectrum is asymmetric and composed of pairs of

VI, w 4]

BICHROMATIC FIELDS

421

<

o

o

-20

0

(,~

-

20

%)/r

Fig. 14. Autler-Townes absorption spectrum for a detuned bichromatic field with 2 Q = 20F, a = 1, b = 5F, F d = + and A = 5F.

peaks whose positions depend on the detuning and the Rabi frequency, which again can be interpreted in terms of the dressed states of w4.2. The Autler-Townes spectrum for nearly symmetric excitation was observed experimentally by Papademetriou, Van Leeuwen and Stroud [ 1996], and found to be in agreement with theoretical predictions (Ficek and Freedhoff [ 1993], Van Leeuwen, Papademetriou and Stroud [1996]). 4.3. DOUBLY DRIVEN ATOM

The multi-peaked spectra of w4.2 are the signature of a bichromatic driving field whose components have equal or almost equal Rabi frequencies and are symmetrically or almost symmetrically detuned about the atomic resonance. Another group of systems which has been studied extensively is highly nonsymmetric" The atom is driven firstly on resonance by an intense field of Rabi frequency 2C2. The resulting (singly) dressed atom is then driven near one of its resonances by a second, usually weaker, intense field of frequency 092 and (on-resonance) Rabi frequency 2~c22 = a292. These systems can be regarded as doubly driven atoms, and three such examples will be discussed in this section; viz.,

2.

2g2 0)2 ~ co0 + - - , n 0)2 ~ o90,

3.

o92 ~ 2C2.

1.

n = 4-1,+2,...,

422

SPECTROSCOPY IN POLYCHROMATIC FIELDS

[VI, w 4

4.3.1. The multiphoton ac Stark effect In this system, the second driving field is tuned to one of the subharmonic resonances of the singly dressed atom, discussed in w2:o92 = ~o0 + 2s where n is an integer. The undressed states of the non-interacting system of singly dressed atom plus field mode 2 are then degenerate in pairs: I(N + n - m)) @ M - n + m) - [a~,)) and

I ( N - m ) - ) @ IM + m) -

IbW>

(58)

both correspond to energy (N + M) hog0 + h__~[2 (M + m) - n]. This degeneracy is due to an n-photon coupling between singly dressed states, as indicated for n = 2 by arrows in fig. 15a. In eq. (58), M + m represents the number of photons in field mode 2; IN+) are the singly dressed eigenstates

1

IN-+-) = ~ (Ig, N) + le, N v'2

1)).

(59)

The product states (58) group into manifolds g (N + M), each containing an infinite number of degenerate doublets (fig. 15b). When the interaction is included between the atom and field mode 2, the degeneracy is split (fig. 15c). For n = 1, the doublet states are coupled directly via a 1-photon interaction, resulting (at resonance and to lowest order in a) in the doubly dressed states

I(N+M),m-+-) = - ~1

~ (]a~l)) • b(m)>),

(60)

1 with energies (N + M)ho90 + hi2 ( 2 M - 1 + 2m) + ~f22 (Ficek and Freedhoff [1996]). For n ~> 2, the doublet states are coupled only by n-photon operators and the calculation of their splittings and the doubly dressed eigenstates is a lengthy exercise in high-order degenerate perturbation theory (Rudolph, Ficek and Freedhoff [ 1998], Rudolph, Freedhoff and Ficek [ 1998a]) resulting for both in series expansions in a. With these dressed-state energies and eigenstates, the fluorescence, absorption, and Autler-Townes spectra may be calculated in the standard manner. All spectra consist of a series of multiplet features: triplets in the case of fluorescence and nearly-resonant absorption, and doublets in the case of Autler-Townes absorption. The most intense of these features are centered at the frequencies of their monochromatic-driving counterparts. In addition, there are subsidiary features centered at the harmonics m2s of the strong-field Rabi frequency, and

VI, w 4]

BICHROMATIC FIELDS

423

9

9

E(N+I)

-~---

03o

IN+> + i

"QI1 Iboz

2hi

lat]>

[

,(N+M)O+> I(N+M)O->

E(N)

i i

IN->

o

99

o" 9

9

9

9

(a)

(b)

E(N-1)

(c)

Fig. 15. Energy levels of a doubly driven atom: (a) singly dressed states driven by a strong field of frequency to2 = too + g2; (b) energy levels of the non-interacting singly dressed atom plus the second mode; (c) doubly dressed states of the interacting singly dressed atom and the second mode.

for n ~> 2, at the subharmonic frequencies m 2•17 as well. The higher a, the greater the number and the intensity of the subsidiary features, whose splittings and spectral distributions display an intricate dependence on n and on a. As examples, we present the fluorescence spectrum for n - 1, the dispersive profile of the probe for n - 2, and the Autler-Townes absorption spectrum for n - 1, 2 and 3. The correlation between successive photons scattered by the atom into individual lines of its fluorescence spectrum has also been studied theoretically, by Ben-Aryeh, Freedhoff and Rudolph [ 1999].

4.3.1.1. Fluorescence spectrum. In fig. 16, we plot the theoretical spectrum corresponding to parameters 292 = 220 MHz and several values of 2s along with the fluorescence spectrum observed experimentally (Yu, Bochinski, Kordich, Mossberg and Ficek [1997]) for the same parameters. In this case, to

424

SPECTROSCOPYIN POLYCHROMATICFIELDS (a) ~ 2 = 0 M H z

E

[VI, w4

(c) D- 2 = 100 M H z

ii

c~ (b) D- 2 = 60 M H z

(d) D- 2 = 140 M H z

g

-I ~

ii

u_

-440-220

o

220 4 4 0

V c -- V a

-440-220

o

220 440

(MHz)

Fig. 16. Experimentallyobserved spectra (lines i) togetherwith theoretical spectra (lines ii) of doubly driven Ba atoms with a strong resonant field of Rabi frequency 2(2 = 220 MHz and a weaker field of several Rabi frequencies 2f22 and detuned from the resonance by 292 (from Yu, Bochinski, Kordich, Mossberg and Ficek [1997]). obtain agreement it was necessary to include as well in the theoretical curve both the elastic scattering and the effect of the different isotopes of the Ba atoms.

4.3.1.2. Near-resonance index of refraction. The modifications of the absorption spectrum, especially the vanishing absorption at the central frequency, may prove useful in the preparation of optical materials having a large index of refraction accompanied by vanishing absorption. In fig. 17 we plot the dispersive profile of the probe field for n = 2. At the lower frequency sideband (where the probe is amplified strongly), the refractive index varies rapidly with frequency and vanishes at the point of maximum amplification. The situation is different at the central frequency (where the absorption vanishes). Here, the refractive index is different from zero with a strong negative dispersion. Moreover, near the central frequency both the absorption and dispersion change slowly with frequency. This property makes the system a convenient candidate for practical applications, since it does not require a precise matching of the probe field frequency to the point of vanishing absorption. The central structure is remarkably stable against variation in frequency.

VI, w 4]

BICHROMATIC FIELDS

425

q 0

o

d

v

,

-50

,

,

.

,

,

,

0

.

.

.

,

,

50

(= -=0)/r Fig. 17. Dispersive profile of the probe beam monitoring a doubly driven atom with 2g2 = 50F, n = 2 and a = 0.4.

4.3.1.3. Autler-Townes absorption spectrum. In fig. 18 (overleaf), we plot the theoretical Autler-Townes absorption spectra for n = 1, 2 and 3, 2g2 = 40F, and a = 0.45. Autler-Townes absorption and dispersion have been measured by Greentree, Wei, Holmstrom, Martin, Manson, Catchpole and Savage [1999] in a nitrogen-vacancy center in diamond. The resolution of these experiments is so high that the splitting of the doublets is measured as a function of a and of the detuning of the weaker field from exact (subharmonic) resonance. It is found that the minimum splitting occurs for o92 shifted slightly from ~o0 + ~-, a feature we discuss in the next section. 4.3.1.4. Shifts of the subharmonic resonances. The minima in the splittings of the Autler-Townes doublet features appear as well, of course, in the corresponding features of the fluorescence and near-resonance absorption spectra, and can be explained fully by the dressed-atom picture in terms of dynamic Stark shifts (Rudolph, Freedhoff and Ficek [1998b]). Furthermore, at those frequencies O,)(mni)n = (D O q- 2~2 + An , the central component of the n fluorescence spectrum F(to0) vanishes identically, a fact which allows one to pinpoint sharply the positions of tOmi ~')n 9 Lastly, those same frequencies t~mi "(")n are the frequencies which correspond to the shifts of the subharmonic resonances shown in figs. 4 and 5. All these phenomena can be unified and explained in the dressed-atom picture.

426

SPECTROSCOPY IN POLYCHROMATIC FIELDS

[VI, w 4

n--1

!

9

,

I

....

,

,

,

n-2

:3 vt.~

Jt~

A * A

n---3

-40

-20

0

20

40

(~ - ~n)/r Fig. 18. Autler-Townes absorption spectra for 2~2 = 4 0 F , a = 0.45, F d = 89F and different n.

The shifts of the subharmonic resonances were first predicted theoretically by Berman and Ziegler [1977] and Ruyten [1989, 1992a,b] in a numerical calculation which involved the solution of the optical Bloch equations for the system. Because the shifts appear in the same positions as do the Bloch-Siegert shifts in the Bloch equations for monochromatic driving without the RWA (Allen and Eberly [ 1975]), Ruyten termed them the 'generalized Bloch-Siegert shifts'. For bichromatic driving, however, they occur within the RWA, and are much

VI, w4]

BICHROMATICFIELDS

427

larger than the original Bloch-Siegert shifts (Bloch and Siegert [1940]), which makes it possible to experimentally observe them. The physical origin of the shifts can be explained with the energy-level diagram of fig. 15b (above). Before calculating the splitting of the doublet states [a~m ")) and ]b~ )) using (higher-order degenerate)perturbation theory, we first take into account the shifts of the two levels caused by all doublets within ,5 (N + M) having m' ;~ m. These (second-order) shifts are of order of magnitude g22/g2, represent a dynamic Stark shift of the levels of doublet m due to the second field mode, and shift the levels in opposite directions. The result is a shift of their anticrossing point, given (to lowest order in a) by An =

!a2g2 8 ,a 2 2(~,2_1)(2

for n = 1 ' for n/> 2.

(61)

At the shifted frequencies tOmi n'(n) = (DO + -n--2f2+ An, the splittings are easily shown to be a minimum, and equal to the matrix element of the effective n-photon coupling interaction between are then simply given by

IN + M,m+)=

~

1

t,) ) [am

and ,Ib~,~')) - Furthermore, the dressed states

(62)

([a~m"') • Ibm,7')),

causing a vanishing of the fluorescence at ~o0 due to destructive interference between the transition amplitudes from la~,~')) and ]b~,~')). Finally, these shifted I

l

i

t

energies determine the true subharmonic resonances of the doubly driven system, at the shifted points of anticrossing.

4.3.2. Twofields with equalfrequencies The phenomena described in the previous sections arise from the excitation of a two-level atom by two fields having different frequencies. In this section, we consider driving the atom with two fields having equal frequencies. For a fixed relative phase between the fields, this is equivalent to interaction with a monochromatic field whose electric field is the vector sum of those of the two fields. However, for an arbitrary relative phase between the fields the problem is more complicated and, in fact, is similar to the phase diffusion effect (Zoller [ 1979], Agarwal and Lakshmi [ 1987], Lobodzinski [ 1997]). A complementary view of this phase diffusion is obtained by studying this system as an example of 'double dressing'. The two-level atom is driven by two

428

SPECTROSCOPYIN POLYCHROMATICFIELDS

[VI, w4

field modes which have equal frequencies but are otherwise distinguishable (e.g., by different k,), with 292 > 2922. For simplicity, we assume that the frequencies are equal to the atomic transition frequency o~0. The interaction of the atom with the strong field results in the singly dressed states ]N-+-) of eq. (59). The Hamiltonian H0 of the noninteracting dressed atom + weaker field satisfies the eigenvalue equation (Freedhoff and Ficek [ 1997]) So I(N - n) • n) = h [No)o :t: .Q] I(N - n) +, n),

(63)

where I ( N - n ) + , n ) = ] ( N - n ) + ) | In), and n is the number of photons in the weaker field. We see from eq. (63) that the eigenstates form an infinite ladder of doublets, each composed of an (effectively) infinite number of degenerate states. When we include the interaction between the singly dressed atom and the weaker field, the degeneracy is lifted. The resulting doubly dressed states are found by diagonalizing the matrix of the interaction W in the basis of the undressed states (assuming negligible interaction between + a n d - manifolds, true for g2 > g22). The matrix elements of W are W(~ ) = ((N - n) :i:, n IWI (N - m) ::k, m)

(

~hg2 v / n + 1 6 . + l , m + ~ 6 . _ l , m

)9

(64)

This matrix has the same form as that which represents the position operator in the basis of the energy eigenstates of the one-dimensional harmonic oscillator (Cohen-Tannoudji, Dupont-Roc and LaloE [1977]). The eigenvalues of manifold g (N) are -cx~ < A < cx~,

(+) A = h [N~0 • (g2 + 89

(65)

and the corresponding eigenvectors are

IN • ~.) = n~O

-q- - ~

I( N - n ) + , n ) ,

(66)

where q~,(x) =

(VS- 2"n! )-,J2 H,(x) e-~, x-,

(67)

and H,(x) is the Hermite polynomial of order n. The energy levels of the doubly dressed atom thus consist of an infinite ladder of doublet continua.

VI, w4]

BICHROMATICFIELDS

429

Transition rates between the continua in neighboring manifolds are given by

G+--FI(N+~,IS+I(N-1)•

2 =a16(/~,

- ~,) ,

(68) Is+I( N - 1) m A') 2 =a16( ,~, q- ,~,,) 9

GT--FI(N+'a.

Thus transitions between pairs of +(-) doublets can occur only between dressed states corresponding to the same eigenvalue ~' = ~,, resulting in the central component of the fluorescence spectrum being unchanged, while transitions corresponding to the sidebands of the Mollow triplet (4- =~ :F) occur to 2,'=-~,, producing a broad continuum9 The total populations of the + and - continua are equal for resonant driving9 Within each continuum, however, the population distribution depends on ~,: If there are M photons in field mode 2, the populations P (~,) are given by

P(~) = [~M (]~/V/2) [2, where

OM(~,/V/2) is

given by eq. (67). The population

distribution has maxima for ~,/x/~ in the vicinity of the classical turning points of the harmonic oscillator eigenfunction CM ( V A// 2 )

1 ; i.e., for AM ~ +2 C M + ~,

or for energies 1 I~Mlhg2 ~ h g 2 v ~ = 89 For I~1 < I~MI the populations P (zl) are smaller, but nonzero, while for I~1 > I'~MI ,P (~) goes rapidly to zero. Following the standard dressed-atom procedure, we find that the incoherent part of the fluorescence spectrum is given by iF

F

s~o~)- ~

~o-~o0/+ (89 2 1

+ 8

/+

( )[

dA q~M

~

+

A

~

2

3_4F

(to - co0 - 2s _ ~,g2)2 + (3 F) 2

4 (co- ~Oo + 2 s

Xg2)2 + ( 3/-') 2

l}

(69) This spectrum is plotted in fig. 19 for 2f2 = 20F and 2f22 = 7F. The graph shows a central component which is the same as that in the Mollow triplet, together with sidebands consisting of a convolution of Lorentzian functions, centered at r 4- 2g2- ~g2 and having width 3F/4, multiplied by a weight factor 9]q~M --(~/X/2)]2. Therefore, the overall width of the sidebands is determined by I

I

the population distribution P(~,) 9 Because q~M ( ~ / V ~ ) ] 2 has maxima near the \

/

i

classical turning points, the sidebands display peaks near to0 4- 2~2 4- s weaker continua centered at COo 4- 2s of width ~ 2922.

with

430

SPECTROSCOPYIN POLYCHROMATICFIELDS ~,,,

,

,

,

,

,

[VI, w4 ,,,

r

c~

c~

c~

0

-20

0

20

(~, - ,%)/r

Fig. 19. Fluorescence spectrum of a two-level atom driven by two fields of equal frequencies, 292 = 20F and 2922= 7F. The Autler-Townes spectrum can also be calculated. It is found theoretically that each line of that spectrum is replaced by a continuum of width Y22. These continua were observed by Wei [1994] and by Greentree, Wei, Holmstrom, Martin, Manson, Catchpole and Savage [1999]. They studied the response of a two-level atom which is driven by two lasers with the same frequency, but with significantly different intensities and without phase locking. The response was monitored by recording the (Autler-Townes) absorption spectrum corresponding to a transition to a third atomic level.

4.3.3. Driving at the Rabi frequency If the atom has a permanent dipole moment, transitions can occur at the Rabi frequency between the states iN+) of the same energy manifold (Freedhoff and Smithers [1975], Freedhoff [1978], Dalton and Gagen [1985]). These transitions have been observed by measuring the absorption of a (weak) probe field tuned near the Rabi frequency 292 of the strong resonant field (Barrett, Woodard and Lafyatis [1992], Holmstrom, Wei, Windsor, Manson, Martin and Glasbeek [1997]). When a strong field is applied at the Rabi frequency, the dressed atom is dressed again. The doubly dressed states of manifold E (N) are IM (N) ~) = ~

1

( N+, M - 1) • i N - , M ) ) ,

(70)

VI, w5]

POLYCHROMATIC FIELDS, p/> 3

431

corresponding to energies E M ( N ) • -- h [Nw0 + (2M - 1) g2 + G]. Here we have [N+,M) - I N + ) | IM), where IN+) are the singly dressed states (59) and M is the number of photons in the field at Rabi frequency 2s 2G is the 'Rabi of the Rabi frequency' g2 v/M, where the atomic dipole moment element in g2 is "~1 (,Ue e -- ~gg). Thus each manifold contains an infinite number of doublets, with interdoublet splitting 2g2, and intradoublet splitting 2G, very similar to the energy level spectrum in w4.3.1. Subharmonic resonances in the absorption of a 'strong probe' tuned near the Rabi frequency 2g2 of the resonant driving field have been observed by Brunel, Lounis, Tamarat and Orrit [1998], who monitored the integrated intensity of fluorescence by their driven molecule as a function of the probe frequency. The near-resonance absorption spectrum of this doubly driven system has been measured by Windsor, Wei, Holmstrom, Martin and Manson [1998], and interpreted in terms of the doubly dressed states (70).

w 5. Polychromatic Fields, p t> 3 5.1. AMPLITUDE-MODULATEDDRIVING FIELDS The earliest theoretical studies of an atom driven by a polychromatic field involved semi-classical amplitude-modulated (am) fields of the form (Armstrong and Feneuille [1975], Feneuille, Schweighofer and Oliver [1976], Thomann [1976, 1980], Blind, Fontana and Thomann [1980], Agarwal and Nayak [1984, 1986], Friedmann and Wilson-Gordon [ 1987], Ruyten [ 1990], Smirnov [ 1994])

E(+)(t) - E 0~+)e --i~OLt ( l + a c o s b t ) ,

(71)

where a is the modulation amplitude and 6 the modulation frequency. We can rewrite the field (71) in the form

E~+)(t) = E~+) [e-i~~

1 (e-i(wL-~)t+ + ~a

e-i(~oL+0)t)] ,

(72)

showing that the am field is in general equivalent to a trichromatic field. For a >> 1, the carrier frequency is effectively suppressed, so that a 100% modulated field corresponds to bichromatic driving with a = 1, considered in w4. In this section, we consider a to be finite.

5.1.1. Floquet treatment We consider a trichromatic driving field (p = 3) with 61 = 0, 02 = - 6 3 = 6, Rabi frequencies of the detuned fields 2s - 2s and Rabi frequency of the resonant

432

SPECTROSCOPYIN POLYCHROMATICFIELDS

[VI, w5

central component 2s = 2g2. For this situation, the optical Bloch equations (8) reduce to the (Laplace transform) recurrence relation

AeX
(73)

where

(z + ~1F + ig6) Ae =

Be=

0 g2

0

(z + ~IF + ig6) Q

0 0 -2ag2 ) 0 0 -2ag2 , ag2 ag2 0

-292 (z + F + if6)

)

(74)

(75)

and X~t(O)

xe(~ = l_

/

+

(76)

Relation (73) is then rewritten in the form of a Floquet matrix and the resulting equation solved by matrix inversion. Calculation of all spectra then follows exactly as in w3.

5.1.2. Dressed-atom treatment A. Weak modulation. We treat the system in this case in a manner similar to that of w4.3.3, as a 'double dressing' in which the atom is first dressed by the strong resonant field, and the resulting singly dressed atom is then dressed again by the weaker bichromatic sidebands. The undressed states of the noninteracting singly dressed atom and driving sideband fields are denoted by

In, N-t-) =_ ](N - 2M) + , M + n,M) ,

(77)

with corresponding energies En, N + -- h

[Ncoo + g2 + (2M + n) 6],

(78)

so that manifold g (N) consists of two submanifolds g (N+), separated by 292, each containing an (effectively) infinite number of states. For g22, g23 < g2, the submanifolds are not coupled by the sideband fields.

VI, w 5]

POLYCHROMATIC FIELDS,p/> 3

433

The total Hamiltonian can then be diagonalized within each submanifold in a manner similar to that for a = 1 bichromatic driving (w 4.1.1). The eigenstates are (Rudolph [1998])

N+,--~-,m

J,-m

=

-

-

-

In, N+),

(79)

?1=--0C

F(m) and correspond to energies ~N+ = h (N to0 + g2 + m6). B. Strong modulation. Even though the interaction between the (bare) atom and the field sidebands is stronger than its interaction with the weak central component, it is profitable nonetheless to introduce as a basis for manifold ,5' (N + 2M) of this system the states

12n, ( N

+ 2M) +) = IN•

[2n + 1, (N + 2M) •

+ n,M - n),

-- I(N -

1) + , M + n + 1,M - n) ;

(80)

these states correspond to energies h [(N + 2M) ~o0 + 2n6] and h [(N + 2M) ~o0 + (2n + 1) 6], respectively. In ]N+,M + n , m - n), e.g., IN-+-) are the eigenstates of eq. (59), and M -+-n are the numbers of photons in the sidebands at frequency ~o0 + 6. We have not yet included the interaction of the atom with any of the driving field modes. Because the interaction with the sidebands is stronger, we first dress the system with these, and then with the weaker central component. Guided by the system in w4.1.1 and using the summation properties of the Bessel functions, we can show that manifold E (N + 2M) again separates into two separate + manifolds, with eigenstates

N +2M,-~,m+

=

Jn-m

T-~

]n,(N + 2 M ) + ) ,

(81)

B=--O0

and energies h [(N + 2M) ~o0 + m6 + g2]. Comparing equations (81) and (79), we see that for both cases considered, weak or strong modulation, the eigenstates and energies of the system are the same. In fact, this result can be traced to the symmetry of the system, which produces the Bessel function eigenstates, whose summation properties in turn produce the non-mixing of the + and - manifolds. For the carrier frequency exactly on atomic resonance, this will occur for all modulation amplitudes a. With the energies and eigenstates, we can now proceed in the standard manner to calculate Is and the spectra of the system. The intensities of the lines in

434

SPECTROSCOPY IN POLYCHROMATIC FIELDS

[VI, w 5

(a) o ~ o :3 v Is. ~1" 0 0

20

-20

(b) o ~ o

o

i

-20

0

20

(~ - l o ) / r Fig. 20. Fluorescence spectrum for an amplitude modulated field with: (a) weak modulation, 6 = 3F, 2~2 -- 4 0 F , and a = 0.1, (b) strong modulation, 292 = 1.5F, 6 = 10F, and a = 8.

the fluorescence and near-resonance absorption spectra are proportional to the transition rates between the dressed states in neighboring manifolds. It is easy to show that the transition rates F,,+,m,• are zero for m * m'. Thus, the central components of these spectra are unchanged from their counterparts in the corresponding spectra for monochromatic driving. The sidebands of the spectra are however split into multiplets, as illustrated for the fluorescence spectrum F(oJ), plotted in fig. 20a for a = 0.1 and in fig. 20b for a = 8. The spectra are very different in appearance, but in fact represent the same expression. The Autler-Townes absorption spectrum is given by the expression

A(~o)= Fa ~ j2

4

- ~ k=-cx:)

x{

r ( ~ 0 ) - ~od - Q -

k6) 2 +/-2

( 0 9 ) - O)d + ~ -

} k 6 ) 2 +/-'2

(82)

'

VI, w 5]

POLYCHROMATIC FIELDS,p > 3

435

A v


o -40

I

I

I

I

-20

I

I

0

I

20

40

((~ - ~ ) / r Fig. 21. Autler-Townes absorption spectrum for an amplitude-modulated field with 2g2 = 20F, a=l,b=5FandFd= 89

"-'d,d-

9

*

9

.

9

i

i

*

10

9

I

20

*

9

9

9

30

~/r Fig. 22. Stationary fluorescence intensity as a function of 6 for an amplitude modulated field with 292 = 20F and a - ~.

where F =

1

(/"d -t-/-'k)-

The multiplets comprising A(o)) are shown in fig. 21 for

a=l.

In fig. 22 we plot the integrated fluorescence intensity Is and in fig. 23 the absorption W(to) of one of the sidebands as a function of 6 for 2Q = 20F

436

SPECTROSCOPY IN POLYCHROMATICFIELDS

[VI, w 5

vo 3:

9

9

I

9

I

l

I

I

"

10

9

20

(~

-

lo)/r

Fig. 23. Near-resonance absorption spectrum for one of the sidebands of an amplitude modulated field with 212 = 20F and a = 0.4.

and a = g. 1 The intensity Is shows sharp minima, and the absorption W(o9) shows distinct dispersive features at exactly the positions of the subharmonic resonances. Note that in a number of respects this is in sharp contrast with the situation for bichromatic driving: For symmetric bichromatic driving, maxima appear in Is and minima in W(o)) at frequencies determined by the zeros of the Bessel function J0 (-492/6). For a resonant strong driving field and a weaker field having a < 1, dispersive features appear in W(o)) of the weaker field near 2g2/n, but dynamic Stark-shifted away from them. The occurrence of the minima in Is in fig. 22 at exactly the subharmonic resonances is explained by expressions obtained by Blind, Fontana and Thomann [1980], using the optical Bloch equations. The stationary fluorescence intensity is given by 3

Is = 1 - 4

FZj2(_4922/6 )

n=-c~Z ec

(2f2 + ntS)2 +

(~3/_,)2"

(83)

This expression shows clear minima for 2Y2 + n6 = 0; i.e., at the subharmonic resonances. 5.2. DRIVING AT SUCCESSIVE RABI FREQUENCIES

Windsor, Wei, Holmstrom, Martin and Manson [ 1998] have measured the near-

VI, w 5]

POLYCHROMATIC

'

I

'

'

'

I

'

'

'

1

'

FIELDS,

'

'

I

'

p/> 3

'

'

I

437

'

'

'

I

'

'

'

Rabi of the Rab~

r ..O x__

t.__O .t-, C~ O

<

i

!

i

2

i

I

4

6 8 10 Probe frequency (MHz)

12

14

Fig. 24. Experimentally observed absorption spectra of successively driven Rabi transitions (from Windsor, Wei, Holmstrom, Martin and Manson [ 1998]). resonance absorption spectra of a two-level atom driven by the following three fields: 1. 2. 3.

O) 1 --

O) 0

o92 ~ 2Y21 ~03 ~ 2Y22

of Rabi frequency 25"21, of Rabi frequency 2g22, of Rabi frequency 2g23.

The observed absorption spectra are shown in fig. 24 and clearly demonstrate the production of triply dressed states. 5.3. MULTIPLEFIELDS,p > 3 In a similar way, the study of the response of atoms to polychromatic fields has been extended to increasingly higher p. Wilkens and Rza2ewski [1989], Zinin and Sushilov [1995], Yoon, Chung and Lee [1999] and Yoon, Pulkin, Park, Chung and Lee [ 1999] have analyzed theoretically complex excitations involving four and even five driving fields. The fluorescence and absorption spectra

438

SPECTROSCOPY IN POLYCHROMATIC F [ I ~ , ~ 9"-"

I

.

.

.

.

.

.

.

.

.

I

.

.

.

.

.

.

t-

.

.

IZ

=

. ACKNOWLEDGMENTS

.

I

4 = 1 5 0 kHz - - - _

.

C

._o .,.., 0 t,",l

<

-1

'

-0.7

0

I

1

I

'

-0.6

-0.5

'

-0.4

-0.3

I

0.3

0.4

Ap/X1 Fig.

25.

four

strong

Experimental fields

094 = w 0 - 2s

'

I

0.5

'

0.6

0.7

Ap/X1

Autler-Townes

of frequencies

'

~ol

absorption =

~o0, o)2

spectrum =

of a two-level

w0 - 2Y21, 0)3

=

system

driven

by

~o0 - 2Y21 - 2g22,

- 2K22 - 2g23 ( f r o m G r e e n t r e e , Wei a n d M a n s o n [1999]).

show increasingly more complicated structures. For example, the fluorescence spectrum under tetrachromatic excitation exhibits many irregular structures, and for some parameters the structure disappears. Greentree, Wei and Manson [ 1999] have measured the Autler-Townes spectra of a two-level atom driven by a succession of four driving fields of frequencies ~oi and Rabi frequencies 2Y2i such that g2i < g2i-1, i = 1 , . . . , 4. The first field was resonant with the atomic transition (o91 = w0), and each successive field wi was resonant with a transition between a pair of dressed states created by the atom and the i - 1 stronger fields. Each field thus induced a splitting in the energy levels of the system, redressing the states. The measured Autler-Townes spectrum for p = 4 is shown in fig. 25.

Acknowledgments This work was supported by the Natural Science and Engineering Research Council of Canada and by the Australian Research Council. We acknowledge gratefully the many contributions to the research reviewed in this paper by T.G. Rudolph, A. Greentree, N.B. Manson, T.W. Mossberg, C. Wei, A. Windsor and C.C. Yu.

VI]

REFERENCES

439

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AUTHOR

INDEX

FOR VOLUME

XL

Ban, M. 119, 120, 145, 160, 173, 176, 184186, 218, 219, 222, 223, 234 Barchielli, A. 119, 149, 155, 167 Bare, H.E 374 Barlow, A.J. 3, 22, 23, 28, 35, 36, 39, 52 Barnett, S.M. 160, 178, 184, 200, 219 Barrett, T.E. 430 Bartelt, H.O. 85, 86, 276, 290, 291,329, 360 Barthelemy, A. 39 Barwicz, A. 30, 48, 54, 62 Bastiaans, M.J. 276, 312, 313 Beaulieu, M. 56, 69 Beck, M. 205 Beckman, M.G. 95 Beige, A. 166 Belavkin, V.P. 119, 149, 155, 160, 166, 167 Belgnaoui, Y. 69 Bell, J.M. 358 Ben-Aryeh, Y. 128, 162, 186, 423 Bencheikh, K. 188 Bendjaballah, C. 160 Bennett, C.H. 119, 148 Bennion, I. 99 Berman, P.R. 168, 397, 398, 426 Bertholds, A. 31 Bertolotti, M. 186, 201 Bertoni, H.L. 355 Beth, Th. 119, 148 Bigelow, N.P. 398 Bilodeau, E 378 Birch, R.D. 22, 24 Bj6rk, G. 160, 245 Black, P. 91, 98, 99 Blair, P. 371,372 Blake, J.N. 28, 29, 35-37, 52 Btaszczyk, P.E. 10 Blatt, R. 398 Blattner, R. 273 Blind, B. 431,436

A Aagedal, H. 379 Abraham, G. 87 Adams, J.L. 354 Agarwal, G.S. 120, 164, 165, 184, 186, 190, 193,220, 227, 251,253, 257, 399, 410, 414, 415, 419, 427, 431 Agrawal, G.P. 24, 27 Ailawadi, N.K. 378, 382 Alber, G. 157, 158 Albert, J. 378 Alfano, R.R. 86 Aliskenderov, E.I. 133 Allen, L. 426 Alphones, A. 10 Amano, C. 84 Ando, T. 28 Andrewartha, J.R. 354 Anhut, T. 120, 220 Araki, M. 200 Armitage, J.D. 329, 330 Armstrong, L. 431 Arnaud, J. 39 Ashcrofl, S. 91, 98, 99 Assanto, G. 27 Astilean, S. 375 Audretsch, J. 161 Aulich, H. 10 Autler, S.H. 391,409 Avizienis, A. 88 Avrutsky, I.A. 370 Awwal, A.A.S. 86 Ayliffe, M.H. 99 Azzam, R.M.A. 368 B Baillie, D. 91, 98, 99 Baird, W.E. 368 Bak, A.E. 39 443

444

AUTHOR INDEX FOR VOLUME XL

Bloch, E 427 Bochinski, J.R. 397, 410, 423, 424 Bock, W.J. 7, 14, 17-19, 24, 27, 28, 30-39, 47, 48, 51, 54, 56, 57, 59, 60, 62, 64, 69, 70 Bogolubov Jr, N.N. 118, 133 Boisset, G.C. 99 Bojko, R. 374 Bojko, R.J. 374 Bollinger, J.J. 166 Bona, G.L. 357 Bonch-Bruevich, A.M. 394, 396 Bondurant, R.S. 190 Born, M. 4, 18,358 Botineau, J. 24 Botten, L.C. 354, 355 Bourkhoff, E. 27 Boyd, R.D. 360 Bradley, C.C. 398 Braginsky, V.B. 164 Brassard, G. 119, 148 Br~iuer, R. 355, 358, 376, 379, 382 Brecha, R.J. 165 Breitenbach, G. 189 Brenner, K.-H. 86, 102, 105, 106, 108 Breslin, J.K. 121, 159, 241 Breuer, H.-P. 159 Brinkmeyer, E. 10, 22, 28 Britten, J.A. 360 Brown, C.S. 39 Bruckmeier, R. 121, 160, 188, 189, 239 Brun, T.A. 158 Brundrett, D.L. 368, 369 Brune, M. 164, 165 Brtmel, Ch. 397, 431 Bryan, S.J. 360 Bryngdahl, O. 355, 358, 372-374, 376, 379, 380, 382 Bucaro, J.A. 52 Bucci, P. 394, 396 Buchholz, D.B. 95 Buczyfiski, R. 62 Buller, G.S. 95, 96 Burckhardt, C.B. 355 Burns, W.K. 10, 28, 56 Burrus, C.A. 84 Bu~ek, V. 161, 186, 200 C Caglioti, E. 24 Calero, J. 65, 69

Calvani, R. 28 Camacho, A. 161 Camacho-Galvfin, A. 161 Cambril, E. 362, 375, 379 Campos, R.A. 174, 186 Cancellieri, G. 4, 5 Cantor, BT 125 Caponi, R. 28 Carmichael, H.J. 119, 145, 149, 155, 161, 166-168, 393 Carquille, B. 65 Carrara, S.L.A. 52 Casado, A. 188 Casagrande, E 167, 168 Case, S.K. 360, 368 Castanos, O. 313 Castin, Y. 149 Caswell, L.T.D. 99 Catchpole, K.R. 410, 425, 430 Cavaliere, P. 394 Caves, C.M. 119, 148, 186, 187, 221,222 Cescato, L. 378 Cescato, L.H. 368 Chakmakjian, S. 396, 413 Chandezon, J. 356 Charasse, M.N. 69 Chardon, D. 52, 58, 65 Chavel, P. 89, 90, 94, 104, 362, 375, 379 Chemla, D.S. 84 Chen, C.-L. 10, 22, 52 Chen, ET. 375 Chen, J. 56, 398 Chen, S.-H. 69 Chen, T.-J. 69 Chen, Z. 410, 411 Cheng, C.-C. 379 Cheo, L.S. 361 Cheo, P.K. 5 Cherri, A.K. 86 Cheung, K.W. 329 Chiang, K.S. 22, 23, 31 Chien, S.E 410 Chigir, N.A. 394, 396 Chirovsky, L.M.E 84 Chmara, W. 128 Choi, S.S. 28 Chou, H.-P. 379 Chu, P.L. 22, 23 Chu, ST 401 Chua, T.H. 52

AUTHOR INDEX FOR VOLUME XL Chuang, S.L. 65, 69 Chung, M.S. 437 Cibils, M.B. 149 Cirac, J.I. 165, 185, 398 Cisternino, E 28 Clapham, P.B. 368 Clarricoats, P.J.B. 5 Claus, R.O. 24, 52, 58 Clausen, J. 220 Cloonan, T.J. 91, 95 Clowes, I.R. 31 Clube, ES.M. 372 Cohen-Tannoudji, C. 148, 397, 398, 404, 406, 408, 415, 428 Collett, M.J. 128, 146, 158-161, 186, 188 Collines Jr, S.A. 84 Collins, J.P. 375 Collischon, M. 374, 375, 378 Cook, R.J. 393, 398 Cooke, D.J. 348 Coppa, G. 28 Coriell, S.R. 358 Cox, I.J. 276 Craig, M.S. 354, 355 Craig, R.G.A. 95, 96 Craighead, H.G. 374, 375 Cr~peau, C. 119, 148 Crosignani, B. 13 Culshaw, B. 14 Cunningham, J.E. 84 Curtis, L. 329

445

Davies, E.B. 118, 127-130, 132, 139, 146, 149, 261 Davis, M.A. 52 de Beaucoudrey, N. 362, 375 de Coulon, V. 28 De Martini, E 225 Dehmelt, H. 158, 398 Delort, T. 356 Depine, R. 365 Derrick, G.H. 358 Desmulliez, M.P.Y. 91, 98, 99 Di Porto, P. 13 Dijali, S.P. 27 Dines, J.A.B. 91, 98, 99 Di6si, L. 158, 167, 168 Doherty, A.C. 161 Domafiski, A.W. 24, 30-35, 37, 51, 56, 59, 62, 65, 66, 68, 69 Dorsch, R.G. 276 Doumuki, T. 365 Drabik, T.J. 375 Dragoman, D. 276 Drobn~,, G. 200 Drummond, P.D. 241 Ducloy, M. 391 Duff, M.J.B. 95, 101 Dung, H.T. 118, 133 Dupont-Roc, J. 406, 428

E D

Dabkiewicz, Ph. 31, 52 Dobrowski, R. 69 Dagenais, M. 393 Dakin, J.P. 14, 60 Dakna, M. 120, 220 Dalibard, J. 148, 149, 165 Dalton, B.J. 398, 430 Damen, T.C. 84 Dammann, H. 294, 308 Dandliker, R. 273 D~indliker, R. 31 Dandridge, A. 60 Danzmann, K. 398 Darbellay, Y. 372 D'Asaro, L.A. 84 Davidovich, L. 165 Davidson, N. 377

Ebbeson, G.R. 361 Eberly, J.H. 395, 426 Edahiro, T. 10, 23, 28 Eftimov, T.A. 7, 14, 16-19, 24, 31, 35, 37, 38, 54, 64, 69 Ehbets, P. 372 Eichmann, G. 86 Eickhoff, W. 10, 21-23, 28, 52, 58 Ejiri, Y. 22, 24, 58 Ekert, A. 119, 148 Emura, K. 39 Engel, H. 364, 370 Enger, R.C. 360, 368 Englert, B.-G. 165 Enokihara, A. 60 Erdogan, T. 370 Essiambre, R.-J. 27 Ezekiel, S. 391,409

446

AUTHOR INDEX FOR VOLUME XL

F

Fainman, Y. 369, 375, 376, 379 Faklis, D. 374 Fantini, P. 4, 5 Farahi, E 35 Farhadiroushan, M. 28 Farhat, N.Y. 84 Farkas, D. 306, 309 Farn, M.W. 374 Fatehi, M.T. 84 Fearn, H. 186 Feinleib, J. 84 Feneuille, S. 431 Feng, D. 373 Fernfindez-Rueda, A. 188 Ferreira, C. 276, 336 Ferstl, M. 364 Fey, D. 108 Feynman, R.P. 167 Ficek, Z. 397, 410, 419, 421-425, 428 Finlan, J.M. 374 Flanders, D.C. 368 Fleury, L. 168 Flood, K.M. 374 Focht, M.W. 84 Fogg, B.R. 24 Fontaine, M. 24, 69 Fontana, P.R. 431,436 Forchheimer, R. 89, 90, 94 Ford, J.E. 376 Foulk, P.W. 91, 98, 99 Fradkin, E.E. 395 Franceschetti, G. 18 Francon, M. 276, 286, 289, 293 Freedhoff, H.S. 410, 411,419, 421-423, 425, 428, 430 French, W.G. 10 Freund, J.M. 84 Freyberger, M. 251 Friberg, A.T. 373 Friedmann, H. 431 Friesem, A.A. 329, 370, 377 Frigo, N.J. 60 Fuchs, H.-J. 360 Fujii, Y. 39 Fujita, T. 368 Fujiwara, K. 395 Fukui, M. 104, 106-108 Fukuo, T. 164 Fiirst, W. 364

G Gabor, D. 275 Gagen, M. 430 Gale, M.T. 368, 374 Gallagher, N.C. 356, 361,372 Gama, A.L. 31 Gantsog, T. 165 Garavaglia, M. 168 Garcia, J. 336 Gardiner, C.W. 119, 128, 149, 150, 161, 165, 185, 189 Garraway, B.M. 119, 145, 149 Garth, S.J. 24 Gartner, W. 276, 292 Gatti, A. 178 Gauthier, D.J. 397, 410, 414 Gawlik, W. 397 Gaylord, T.K. 87, 88, 355, 356, 360, 368, 369, 372, 379 Gerritsen, H.J. 360 Giallorenzi, T.G. 52 Gibbs, H.M. 84 Giliberto, M.J. 65 Gillespie, D.T. 135 Gilson, C.R. 178 Giovannetti, V. 162 Giovannini, H. 370 Gisin, N. 31, 119, 149, 157, 158, 167, 168 Glasbeek, M. 430 Glaser, T. 360 Glauber, R.J. 118, 124, 126, 179, 195 Gloge, D. 6, 7 Glogower, J. 214 Gluch, E. 368 Glytsis, E.N. 356, 368, 369, 372, 379 Gobbi, A.L. 378 Goetsch, P. 167 Goodman, J.W. 86, 87, 311 Goodman, S.D. 105 Goodwill, D.J. 91, 98, 99 Goodwin, M.J. 99 Gordon, J.P. 27 Gortler, K. 294 Gossard, A.C. 84 Graf, M. 164, 165, 184 Graham, R. 167 Gramann, W. 374 Granet, G. 355 Grangier, P. 160 Grann, E.B. 355, 358, 360, 363, 368, 375

AUTHOR INDEX FOR VOLUME XL Grant, N.L. 91, 98, 99 Grassl, M. 119, 148 Greentree, A.D. 410, 425, 430, 438 Greffet, J.J. 356 Grelu, Ph. 160 Gremaux, D.A. 361 Grimm, M.A. 291 Grove, R.E. 391,409 Gruntman, M. 366 Grynberg, G. 398, 406, 408, 415 Gu, M. 311 Guest, C.C. 87, 88 Guizal, B. 355 Gunning, W.J. 368 Gupta, M.C. 360 Gusmeroli, V 52 Guth, G.D. 84 Gtither, R. 364, 373 H Haake, E 167 Haas, E 374 Habraken, S. 379 Haggans, C.W. 355, 358, 368 Haidner, H. 374, 375, 378 Halliwell, J.J. 168 Hamao, N. 84 Hansen, H. 121, 160, 189, 239 Hao, H. 39 Hara, T. 84 Hardy, A. 4, 5 Hardy, L. 135 Harms, H. 10, 28 Haroche, S. 164, 165 Harrigan, M.E. 372 Hartig, W. 391,409 Hasman, E. 377 Hattori, S. 69 Haus, H.A. 118, 135, 164, 189 Hay, O. 135 Hayata, K. 10, 23 Hazart, J. 379 Heath, J.W. 361 Hegedus, Z.S. 378 Hegerfeldt, G.C. 166 Heine, C. 368 Heinzen, D.J. 166 Helstrom, C.W. 136, 150 Henry, J.E. 84 Henry, W.M. 7, 14, 17, 18

447

Hercher, M. 391,409 Herkommer, A.M. 166 Herzig, H.P. 273, 346, 372 Hessel, A. 361 Hessler, T. 374 Hibbs, A.R. 167 Higashi, T. 28 Hill, K.O. 378 Hinata, T. 22 Hinterlong, S.J. 95 Hinton, H.S. 84, 95 Hirano, T. 187 Hirayama, K. 356, 372 Hogg, D. 65 Holland, M. 145 Holland, M.J. 160 Holmes, C.A. 121,227, 241 Holmstrom, S.A. 410, 425, 430, 431, 436, 437 Home, D. 135 Honey, D.A. 374 Hong, C.K. 186, 189 Hong, C.-S. 353 Honkanen, M. 365 Hombeck, L.J. 84 Hosaka, T. 23, 28 Howard, R.E. 10 Hradil, Z. 120, 197 Huang, A. 86, 87, 91-93, 102, 105, 106, 108 Huang, K.-S. 94, 103, 104, 106 Huang, S. 28, 48, 49 Huang, S.-Y. 28, 29, 35-37, 52 Huard, S. 4, 5,21 Huard, S.J. 52, 58, 65 Hiibner, H. 377 Hudson, R.L. 153, 261 Hulet, R.G. 398 Hussey, C.D. 13 Hutfless, J. 374 Hutley, M.C. 346, 359, 368 Huttner, B. 128, 178, 186 Hwang, K. 106

I

Ichikawa, H. 378, 382 Ichioka, Y. 85, 91, 96, 97, 100, 102, 105-109 Illuminati, E 189 Imai, M. 28 Imai, Y. 24, 28

448

AUTHOR INDEX FOR VOLUME XL

Imoto, N. 22, 119-121, 144-146, 149, 158, 159, 162-164, 178, 182, 185, 189, 227-229, 231,239, 245 Ishihara, S. 87 Ishikawa, M. 97, 98 Islam, M.N. 27 Itano, W.M. 166 It6, K. 119, 152 Itoh, K. 39 Iwata, K. 358, 368, 375 Iwata, M. 108, 109 Iyer, R. 99 Izutsu, M. 60 J

Jaakkola, T. 378, 382 Jack, M.W. 146, 158 Jackson, D.A. 35, 52 Jackson, J.L. 358 Jacobs, K. 161, 167 Jahns, J. 91,378, 382 Jakeman, E. 118, 133, 134, 189 Jakob, M. 415 Jann, A. 162 Jansen, K. 31 Janszky, J. 186 Jaroszewicz, L.R. 14, 56 Jeffers, J. 178 Jelezko, E 396, 397 Jenkins, B.K. 89, 90, 94, 103, 104, 106 Jepsen, M.L. 360 Jeunhomme, L. 39, 43 Jingren, Q. 39 Johannessen, K. 24 Johnson, D.C. 378 Johnson, E.G. 379 Johnson, M. 39 Jones, G.N. 167 Jones, J.D.C. 35, 52 Jones, R.C. 14 Joobeur, A. 187 Jozsa, R. 119, 148 Jull, E.V. 361 Jur6o, B. 120 K

Kaiser, P. 13 Kaminow, I.P. 3-5, 23 Kanamoto, K. 395 Karim, M.A. 86

Karlsson, A. 160 Karlsson, M. 28 Karpierz, M.A. 65, 68, 69 Kartashev, A.I. 276, 290, 329 K~irtner, EX. 118, 135 Kasahara, K. 84 Kaspar, EG. 355 Kathman, A.D. 379 Kato, Y. 8 Keitel, C.H. 398 Kelley, P.L. 118, 126 Kersey, A.D. 52 Kettunen, V. 365, 366, 368, 373 Khalili, El. 164 Kheruntsyan, K.V. 415 Kikuchi, K. 28 Kikuta, H. 358, 368, 375 Kilin, S.Ya. 168 Kim, B.Y. 28, 29, 35-37, 52 Kimble, H.J. 241,393 Kimura, T. 13, 22, 24, 39 Kimura, Y. 24, 27, 368 Kipfer, P. 374, 375, 378 Kirk, C.P. 355 Kiryuschev, I. 302, 304, 306, 309, 312, 319 Kitagawa, M. 119, 144, 146, 245, 364 Kitayama, K. 28, 104, 106 Klauder, J.R. 184, 186 Kleemann, B.H. 355, 364, 373 Kleiner, W.H. 118, 126 Kley, E.-B. 366, 377 Kliger, D.S. 4 Klotz, E. 294, 308 Knight, P.L. 119, 145, 149, 161, 166, 167, 186, 188, 393, 398 Knoesen, A. 381 Kn611, L. 120, 133, 220 Knop, K. 355, 378 Koashi, M. 148, 187 Kobayashi, Y. 84 Koch, K. 396, 413 Konforti, N. 302, 304, 309, 312, 319 Konishi, T. 96, 97 Kono, K. 187 Kon6pka, M. 161 Konopka, W. 66, 69 Kordich, T.M.V. 397, 410, 423,424 Koshiba, M. 10, 23 Kostuk, R.K. 358, 368 Krasifiski, Z. 22

AUTHOR INDEX FOR VOLUME XL Kraus, K. 260 K~epelka, J. 120, 121, 174, 175, 190, 197, 201,205, 208, 214, 230, 233,238, 241,244, 251,253,254, 256, 257 Krumpholz, O. 28 Kryuchkov, G.Yu. 409, 412, 415 Kryuchkyan, G.Yu. 415 Kubo, H. 358 Kuhlow, B. 364 Kuisma, S. 378, 382 Kuittinen, M. 358, 365, 366, 372-376 Kujawski, A. 65, 69 Kumar, A. 23 Kumar, M.S. 187, 395 Kumari, M.K. 410, 415 Kunz, R.E. 357, 374 Kurokawa, T. 84 Kurosawa, K. 69 Kurzynowski, P. 70 Kuwata, M. 125 Kyrolfi, E. 398 L La Porta, A. 186, 222 Labeyrie, A. 311 Lafyatis, G.P. 430 Lai, W.K. 186 Lakshmi, P.A. 427 Lalanne, Ph. 355, 358, 368, 375, 379 LaloE, E 428 Lamouler, P. 39 Lang, A. 375 Lange, W. 415 Lanz, L. 149 Larchuk, T.S. 187 Larochelle, S. 373 Launois, H. 375, 379 Lautanen, J. 365 Lax, M. 119, 152, 404, 407 Layet, B. 356, 373 Layton, M.R. 52 Le Kien, E 184 le Pesant, J.P. 69 Lee, C.T. 167, 185 Lee, H.W. 312, 437 Lee, S.H. 374 Lefevre, V. 164, 165 Lefranc, G. 374 Lehmberg, R.H. 399 Leibenguth, R.E. 84

449

Lemarchand, E 370 Lemercier-Lalanne, D. 355, 358 Lentine, A.L. 84, 95 Leonhardt, U. 186 Leonhardt, V. 205 Levenson, J.A. 188 Levenson, M.D. 160 Levy, U. 329 Lewis, A. 290 Lewis, J. 4 Lewis, J.T. 261 Lezama, A. 397, 409, 414 Li, L. 69, 354-356, 358, 368, 378, 382 Li, Y. 86 Lichtenberg, B. 356, 372 Lieb, E. 200 Lima, C.R.A. 378 Lin, Z. 28, 48, 49 Lindblad, G. 118, 145 Lindolf, J. 375, 378 Lion, Y. 379 Lit, J.W.Y. 35, 58 Liu, S. 368 Liu, Y. 99 Liu, Z.S. 370 Lo, Y.L. 52, 65 Lobodzinski, B. 427 Lochbihler, H. 365 Loewen, E.G. 360 Lohmann, A.W. 85, 86, 276, 277, 279, 281, 283, 286, 289, 291,292, 302, 304, 306, 309, 312, 313, 317, 319, 320, 329, 330, 336 Lopez-Moreno, E. 313 Loudon, R. 178, 186 Louisell, W.H. 119, 150, 399 Lounis, B. 396, 397, 431 Louri, A. 106 Love, J.D. 5, 7, 11, 13, 16, 18, 42 Lovering, D.J. 188 Lu, N. 397, 398 Luis, A. 121, 185-187, 223, 224, 226, 245, 247, 250 Lukosz, W. 276, 286, 290, 293, 294, 296 Luke, A. 120, 121, 152, 174, 175, 190, 197, 200, 201,205, 208, 214, 215, 230, 233, 238, 241,244, 251,253,254, 256, 257 Lulli, A. 167, 168 Lummer, O. 274 Lupieri, G. 149

450

AUTHOR INDEX FOR VOLUMEXL

M

Mabuchi, H. 119, 148, 166 Macchesney, J.B. 10 Macchiavello, C. 119, 148 Machado Mata, J.A. 157 Machida, S. 13, 162, 189, 245 MacKinnon, G. 95, 96 Maevskaya, T.M. 168 Magnusson, R. 370 Mahler, G. 167 Maillotte, H. 65 Mait, J.N. 356, 372, 375, 381 Majewski, A. 22 Malo, B. 378 Maloney, P.J. 84 Mandel, L. 118, 119, 123-128, 132, 133, 146, 149, 175, 186, 188, 189, 393 Mansfield, W.M. 378, 382 Manson, N.B. 394-396, 410, 425, 430, 431, 436-438 Marchand, M. 290 Marcuse, D. 5, 6, 40, 41 Marks, R.J. 278, 279 Marom, E. 336 Marrone, M.J. 10, 13, 23, 24, 52, 58, 60 Marshall, T.W. 188 Marte, M. 149 Martienssen, W. 186 Martin, J.ED. 394-396, 410, 425, 430, 431, 436, 437 Martinelli, M. 52, 394 M/~rz, M. 374 Mas, D. 336 Mashev, I. 370 Mashev, L. 370 Matsumoto, S. 365 Matsuo, S. 84 Matsuoka, M. 187 Matsuura, M. 39 Mayerhofer, E 374 Maystre, D. 346, 356, 360, 366, 370 McArdle, N. 95, 96, 98 McCall, S.L. 84, 186 McCormick, EB. 95 McCormick Jr, EB. 84, 95 McKnight, D.J. 95, 96 McPhedran, R.C. 354, 355, 358, 366 Measures, R.M. 65 Mehta, C.L. 124 Melsheimer, O. 166

Mendlovic, D. 276, 281,283, 286, 302, 304, 306, 309, 312, 319, 320, 336 Mensky, M.B. 119, 148, 160, 161 Menyuk, C.R. 14, 28 Meredith, P. 95, 96 Merzbacher, E. 228 Michaux, O. 379 Mikolas, D. 374 Milburn, G.J. 119-121, 149, 155, 157, 159, 161, 162, 164, 167, 168, 187, 227, 241,393 Miller, D.A.B. 84 Miller, J.M. 95, 96, 355, 362, 371,375, 378, 379, 381,382 Miller, M.S. 52 Milonni, P.W. 393 Mineau, J.L. 52, 58 Mirotznik, M.S. 356, 372, 375 Misra, B. 162, 166 Mitreiter, A. 355 Mitsuyu, T. 364 Miyauchi, M. 8 Miyazaki, D. 96 Mlynek, J. 121, 160, 188, 189, 239 Mochizuki, K. 22, 24, 28, 58 Moeller, R.P. 10, 28 Moharam, M.G. 355, 358, 360, 363, 368, 375 Mollow, B.R. 118, 127, 179, 195, 391, 393, 394, 398 Molmer, K. 149, 165 Monerie, M. 21, 39, 43 Morf, R.H. 368 Morin, E.S. 397 Morris, G.M. 355, 368, 370, 374 Morrison, R.L. 84, 95 Mossberg, T.W. 397, 409, 410, 414, 423,424 Motamedi, M.E. 368 Mukohzaka, N. 84 Mulgrew, P. 378, 382 Mtiller-Quade, J. 119, 148 Murdocca, M.J. 90, 91, 108 Murphy, K.A. 24, 52, 58 Murtha, J.P. 65, 69 Muszkowski, M. 69 N Nagaoka, H. 119, 146, 149, 159, 163, 239 Nagasawa, N. 125 Nagel, J. 3 Nakagawa, J. 108 Nakamura, K. 164

AUTHOR INDEX FOR VOLUME XL Nakayama, T. 395 Nakazawa, M. 24, 27, 28 Namihira, Y. 22, 24, 28, 58 Naraschewski, M. 165, 185 Naruse, M. 98 Nasitowski, T. 69 Nayak, N. 431 Negishi, Y. 28 Nevi6re, M. 348, 355, 360 Nguyen, H.T. 360 Nielsen, M.A. 119, 148, 149 Nishida, N. 368 Nishihara, H. 349, 372 Nizovtsev, A.P. 168 Noda, J. 3, 10, 14, 52 Noponen, E. 349, 355, 357, 360, 363, 370373, 378, 379, 381-383 Norton, S.M. 370 Novotny, R.A. 95 Nowinowski-Kruszelnicki, E. 69 Nyyssonen, D. 355

O Ogawa, H. 364 Ogawa, T. 119-121, 144, 145, 149, 158, 159, 163, 164, 182, 185,227-229, 231,239 Ogura, I. 84 Ohashi, M. 28 Ohira, Y. 358, 368 Ohta, Y. 368 Ohtsuka, Y. 24, 28, 39 Okamoto, K. 3, 10, 14, 23, 28, 35, 36, 52 Okorkov, V.N. 369 Okoshi, T. 28, 39 Okuto, A. 98 Oliver, D.S. 84 Oliver, G. 431 O'Mahony, P.E 158 Omn~s,R. 118, 135 Ono, Y. 368 Onofrio, R. 165 Opatrn~,, T. 120, 220 Orrit, M. 396, 397, 431 Orszag, M. 164, 165, 184 Osterberg, H. 290 Ostrzy~ek, A. 56 Ou, Z. 186 Ou, Z.Y. 241 Ozawa, M. 135-137, 191

451

P

Paek, E.G. 329 Paganoni, A.M. 149 Palffy-Muhoray, P 69 Panchenko, V.Ya. 369 Pannell, C.N. 52 Papademetriou, S. 396, 410, 417, 421 Pape, D.R. 84 Papp, A. 10,28 Paris, P.D. 329, 330 Park, J.R. 437 Parkes, W. 372 Parkins, A.S. 161 Parkins, H.S. 149 Parsons, A.D. 99 Parthasarathy, K.R. 119, 152, 153, 261 Pask, C. 24 Passy, R. 31 Paul, H. 205 Pawlowski, E. 364 Payne, D.N. 3, 7, 10, 13, 18, 22-24, 28, 35, 36, 39, 52 Pegg, D.T. 160, 184, 219 Pekola, J. 372 Pellizzari, T. 119, 148 Peng, K.C. 241 Peng, S. 370 Peng, S.T. 355, 360 Percival, I.C. 119, 149, 157, 168 Pereira, S.E 241 Peres, A. 119, 135, 148 Pefina, J. 120, 121, 125, 170, 187, 199, 208, 214, 223,224, 227, 244, 245, 247, 250, 253 Pef'inovfi, V. 120, 121, 152, 174, 175, 190, 197, 199-201,205, 208, 214, 215, 230, 233, 238, 241,244, 251,253, 254, 256, 257 Pemigo, M. 184 Perry, M.D. 360 Petit, R. 345, 346, 355 Petruccione, E 159 Phillips, L.S. 219 Phillips, W.D. 398 Phoenix, S.J.D. 200 Picherit, E 52, 58, 69 Plant, D.V. 99 Pleibel, W. 10, 13 Plenio, M.B. 166, 188 P6hlmann, R. 360 Poizat, J.-Ph. 160 Polnau, E. 365

452

AUTHOR INDEX FOR VOLUME XL

Pommet, D.A. 355, 358, 360, 363, 368, 375 Pons, M.L. 395 Poole, C.D. 3, 27 Pope, C. 65, 69 Popov, E. 370 Prasad, S. 186 Prather, D.W. 356, 372, 375 Predehl, P. 365 Prentiss, M.G. 398 Preston Jr, K. 95, 101 Prosperi, G.M. 149 Pulkin, S.A. 437 Puff, R.R. 184

Rosenblatt, D. 370 Ross, J.N. 28 Ross, N. 381 Rossi, M. 357, 374 Roumiguieres, J.L. 378 Rudolph, T.G. 410, 422, 423,425, 433 Ruff-m, P.B. 24, 58 Riihl, E 10 Russkikh, B.V. 369 Ruyten, W.M. 426, 431 Rytov, S.M. 358 Ryu, S. 39 Rza• K. 415, 437

R

S

Raguin, D.H. 368 Raimond, J.M. 164, 165 Ramaswamy, V. 4, 5, 10, 13, 23 Ramskov-Hansen, J.J. 3, 23, 28, 39 Randall, C.E. 4 Ranshaw, M.J. 65 Raoult, G. 356 Rashleigh, S.C. 3, 4, 10, 13, 21, 23, 24, 28, 52, 58 Rasmussen, W. 391,409 Raymer, M.G. 205 Redmond, I.R. 95, 96 l~ehfi~ek, J. 187 Reiche, E 274 Reid, M.D. 241 Reiley, D.J. 95 Renotte, Y. 379 Restall, E.J. 95, 96 Reynaud, S. 397, 404 Rhodes, W.T. 105 Rice, P.R. 149 Richards, D. 168 Richards, G.W. 95 Richter, I. 369, 375, 376 Richtie, K.T. 52, 65 Riesz, E 200 Rigo, M. 158 Rippin, M.A. 188 Risco-Delgado, R. 188 Risken, H. 400 Ritchie, N.W.M. 398 Roberts, C.D. 378, 382 Robertson, B. 99 Roch, J.E 160 Rogers, A.J. 28

Saarinen, J. 349, 370 Saitoh, T. 39 Sakai, J. 13 Sakai, J.-I. 13, 22, 24, 39 Saleh, B.E.A. 125, 174, 175, 186, 187, 189 Salvekar, A.A. 379 Sambles, J.R. 369 Sammut, R.A. 13, 22, 23 Sfinchez-Soto, L.L. 185, 186, 226 Sanda, P.N. 354 Sano, K. 39 Santos, E. 188 Santucci, S. 394, 396 Sanyal, G.S. 10 Sargsian, A.S. 415 Sasaki, Y. 3, 10, 14 Sasian, J.M. 95 Sauer, J.R. 27 Savage, C. 410, 425, 430 Sawchuk, A.A. 89, 90, 94, 103, 104, 106 Schaefer, D.H. 93 Schalke, M. 121, 188, 239 Schaufler, S. 251 Schenzle, A. 165 Scherer, A. 379 Schieder, R. 391,409 Schiller, S. 121, 160, 188, 189, 239 Schleich, W. 184, 187, 213, 222 Schleich, W.P. 166, 251 Schmid, M. 379 Schmidt, E. 200 Schmitz, M. 372-374, 376, 379 Schmoys, J. 361 Schnabel, B. 365, 366, 377 Schneider, H. 10

AUTHOR INDEX FOR VOLUME XL Schneider, K. 121, 160, 188, 239 Schr6ter, S. 360 Schuda, E 391,409 Schweighofer, M.G. 431 Schwider, J. 374, 375, 378 Scully, M.O. 164, 165, 184, 186 Sears, EM. 10 Seaton, C.T. 27 Seikai, S. 28 Selvadoray, M. 187 Semigonov, V.N. 369 Sentenac, A. 370 Sergienko, N. 373 Serra, A.M. 394 Serra, J. 106 Setsune, K. 364 Shae, Z.Y. 84 Shafir, E. 4, 5 Shapiro, J.H. 131, 157, 190 Sharon, A. 370 Shatokhin, V.N. 168 Shaw, H.J. 35, 52 Sheem, S.K. 52 Sheng, P. 354, 373 Sheppard, C.J.R. 276, 311 Sheridan, J.T. 374, 375, 378 Shibata, N. 10, 23, 28 Shibata, X. 28 Shin, D. 370 Shiono, T. 364 Shore, B.W. 360 Shumovsky, A.S. 118, 133 Sibilia, C. 186, 201 Sicre, E.E. 85, 86 Sidick, E. 381 Siegert, A. 427 Sierakowski, M. 68 Silberberg, Y. 329 Simon, A. 21, 39, 44 Simon, R. 187 Simonneau, C. 188 Simpson, J.R. 10 Sinatra, A. 160 Singh, R.P. 184 Singh, S. 118, 133, 149 Sirkis, J.S. 52, 65 Skagerstam, B. 184 Slusher, R.E. 186, 222 Smirnov, M.Z. 431 Smith, A.M. 28, 39

453

Smith, C.P. 18 Smith, G.J.EE. 84 Smith, G.R. 96 Smith, L.W. 290 Smith, P. 160, 186, 188 Smith, P.W. 84 Smith, R.E. 375 Smith, S.D. 95, 96 Smithers, M.E. 430 Smithey, D.T. 205 Snitzer, E. 5 Snowdon, J.E 95, 96 Snyder, A.W. 5, 7, 10, 11, 13, 14, 16-18, 42 Soares, L.L. 378 Softer, B. 89, 94 Sokolov, VT 369 Solymar, L. 348 Sommargren, G.E. 374 Sondermann, D.G. 166 Song, S. 186, 221,222 Southwell, W.H. 368 Spajer, M. 65 Squires, E.J. 135 Srinivas, M.D. 118, 127, 128, 130, 132, 139, 146, 149 Stamnes, J.J. 356, 373 Standley, R.D. 10 Staszewski, P. 167 Stegemann, G. 27 Steimle, T. 157, 158 Stein, R.S. 374 Steinbach, J. 149 Steingrueber, R. 370 Stemmer, A. 374 Stenholm, S. 398, 401 Stepleman, R.S. 354 Steuemagel, O. 188, 226 Stewart, W.J. 99 Stolen, R.H. 10, 13, 23, 24, 27 Stoler, D. 175, 184 Stone, H.S. 91 Stork, W. 374 Storoy, H. 24 Story, J.G. 398 Strand, T.C. 89, 90, 94 Stratonovich, R.L. 119, 151 Streibl, N. 86, 91, 102, 105, 106, 108, 368, 374, 375, 378 Strong III, J.E 93

454

AUTHOR INDEX FOR VOLUME XL

Stroud Jr, C.R. 391,396, 409, 410, 413, 417, 421 Strunz, W.T. 158, 167 Sudarshan, E.C.G. 162, 166 Sueta, T. 60 Sugimoto, M. 84 Sugla, B. 91 Suhara, T. 349, 372 Sun, P.-C. 369, 375, 376, 379 Sung, C.C. 24, 58 Sushilov, N.V 437 Susskind, L. 214 Suzuki, M. 23 Swain, S. 398 Swanson, G.J. 357 Sweeney, D.W. 374 Swit|o, M. 68 Sychugov, VA. 370 Syngellakis, S. 31 Sz6kefalvi-Nagy, B. 200 Szustakowski, M. 56 Szymanowski, C. 398 Szymafiska, A. 69 T Taghizadeh, M.R. 95, 96, 355, 356, 371-373, 378, 379, 381,382 Takada, K. 14 Takiguchi, Y. 125 Tamada, H. 365 Tamarat, Ph. 397, 431 Tamir, T. 355 Tan, S.M. 161 Tanida, J. 85, 91, 96, 97, 100, 102, 105-109 Tara, K. 184, 220, 251,253,257 Tarbox, E.J. 24, 52 Tasiro, T. 84 Tatam, R.P. 52 Tayeb, G. 355 Taylor, R.M. 65 Teich, M.C. 125, 174, 175, 186, 187, 189 Teich, W.G. 167 Tennant, D.M. 378, 382 Tervonen, E. 372 Tewari, S.P. 410, 415 Th6venaz, L. 28 Thomann, P. 396, 431,436 Thompson, R.C. 166 Tian, L. 161 Tibuleac, S. 370

Tilio, M. 4, 5 Tokuda, M. 28 Tokuda, Y. 395 Tollett, J.J. 398 Tombesi, P. 162, 167, 168 Tooley, EA.P. 84, 91, 95, 96, 98, 99 Toombes, G.E. 166 Toptygina, G.I. 395 Toraldo Di Francia, G. 274-276 Torgerson, J.R. 118, 132, 188 Townes, C.H. 391,409 Toyoda, H. 84 Trillo, S. 24, 27 Tsao, C. 4, 15, 21 Tsao, C.Y.H. 4, 5 Tseng, D.Y. 361 Tsubokawa, M. 28 Tsuchiya, H. 22 Tsuchiya, Y. 125 Tsukada, N. 395 Tsunoda, Y. 87 Tur, M. 4,5 Turner, E.H. 84 Turner, R.D. 65 Turpin, M. 69 Turunen, J. 346, 349, 355,357, 358, 360, 362, 363, 365, 366, 370-376, 378, 379, 381-383 Tyan, R.-C. 379 Tzolov, VP. 24, 69 U Udd, E. 69 Ueda, M. 119-121, 123, 125, 144-146, 148, 149, 158, 159, 163, 169, 171-173, 182, 185, 227-229, 231,239 Ulrich, R. 10, 13, 21, 24, 28, 31, 39, 44, 52, 366 Ulzega, S. 167 Urakami, T. 125 Urbaficzyk, W. 24, 27, 28, 31, 56, 60, 62, 69, 7O Urquhart, K.S. 374 V Vaccaro, J.A. 162, 168 Vahimaa, P. 358, 366, 373, 375, 376 Valis, T. 65 Van Leeuwen, M.E 396, 410, 417, 421 VanderLugt, A. 278

AUTHOR INDEX FOR VOLUME XL Varnham, M.P. 7, 10, 13, 18, 22, 24, 35, 36, 52 Varshney, R.K. 23 Vartanyan, T.A. 394, 396 Vasara, A. 355, 357, 363, 372, 373, 378, 379, 381,382 Vassallo, C. 23 Vavassori, P. 52 Vawter, G.A. 375 Vedral, V. 188 Vengsarkar, A.M. 24, 52 Venkatesan, T.N.C. 84 Vernon, EL. 167 Vidakovic, P. 188 Vigneron, K. 160 Villarruel, C.A. 60 Vincent, P. 348, 355 Viola, L. 165 Vitali, D. 162, 167, 168 Voet, M.R.H. 56 Volterra, V. 171 von Borczyskowski, C. 168 vonder Weid, J.P. 28, 31 Vourdas, A. 182 Vyas, R. 149 W

Wabnitz, S. 24, 27 Wade, C.A. 60 Wagner, C. 165 Wagner, R.E. 13 Wahiddin, M.R.B. 410 Wai, P.K.A. 14, 28 Wakabayashi, H. 28 Wakelin, S. 95, 96 Waldman, G. 274 Walker, A.C. 95, 96 Walker, J.G. 189 Walker, S.J. 378, 382 Walker, S.L. 84 Walls, D.E 120, 121, 146, 149, 158-161, 164, 186, 188, 222, 227, 241,393 Walser, R. 188 Walther, H. 164, 165, 184, 391,409, 415 Wang, A. 52, 58 Wang, G.Z. 52, 58 Wang, J.-M. 94, 104 Wang, K. 160 Wang, S.S. 370 Warren, M.E. 375

455

Wasmundt, K.C. 84 Watanabe, K. 120, 190 Watts, R.A. 369 Webb, D.J. 35 Weber, A.G. 94, 104 Weber, H.G. 370 Wei, C. 394-396, 410, 425, 430, 431,436438 Weigelt, G. 311 Welsch, D.-G. 120, 220 Wendt, J.R. 375 Werlich, H. 289, 291 West, L.C. 378, 382 Westerholm, J. 378, 382 Wheeler, J.A. 213 Wherrett, B.S. 91, 95, 96, 98, 99 Whitaker, M.A.G. 135 Wicht, A. 398 Wiegmann, W. 84 Wigner, E. 276, 312, 313 Wigner, E.P. 118, 137 Wilkens, M. 415,437 Wilkinson, C.D.W 372 Wilson, D.W. 372 Wilson, R.A. 95, 96 Wilson-Gordon, A.D. 431 Windsor, A.S.M. 430, 431,436, 437 Wineland, D.J. 166, 398 Winzer, P.J. 125 Wiseman, H.M. 119, 149, 155-157, 159, 161, 162, 166, 167 Wi~niewski, R. 57 W6jcik, J. 69 Wolf, E. 4, 18, 118, 124, 128, 132, 133, 358, 393 Wolf, K.B. 312, 313 Wolifiski, T.R. 19, 24, 28, 30-39, 47, 48, 51, 54, 56, 57, 59, 60, 62, 64-66, 68, 69 Wong, D. 52 Wood, R.W. 345 Wood, T.H. 84 Woodard, N.G. 430 Wootters, WK. 119, 148 Wootton, J. 274 Wo2niak, W.A. 70 Wrachtrup, J. 168 Wu, B. 24, 69 Wu, C. 24 Wu, EY. 391,409 Wu, Q. 397, 409, 410, 414

456

AUTHOR INDEX FOR VOLUME XL

Wu, R.-B. 22 Wu, S.-P. 65, 69 Wyrowski, E 346, 355, 365, 366, 368, 371, 377-380, 382, 383 X Xiaopeng, D. 39 Xie, H.M. 31, 52 Xu, E 369, 375, 376, 379 Y Yagyu, E. 108 Yakunin, V.P. 369 Yamaguchi, T. 365 Yamamoto, Y. 120, 162, 164, 189, 190, 245 Yariv, A. 353 Yatagai, T. 86 Yeh, C. 5,8 Yeh, P. 353 Yen, Y. 21, 28 Yip, G.L. 24 Yokohama, I. 52 Yokomori, K. 362 Yoon, T.H. 437 Yoshida, H. 358, 375 Yoshida, N. 84 Yoshizawa, N. 22

Young, P.P. 370 Youngquist, R.C. 28 Yu, C.C. 397, 410, 423,424 Yuan, H.J. 69 Yuen, H.P. 120, 157, 189, 190, 218, 226, 227 Yurke, B. 184, 186, 221,222 Z Zagury, N. 164, 165 Zah, C.E. 329 Zalevsky, Z. 276, 286, 302, 304, 306, 309, 312, 319, 320, 336 Zarschizky, H. 374 Zeitner, U.-D. 377 Zerras, M.N. 31 Zhang, E 35, 58 Zhao, W. 27 Zheng, X.-H. 7, 14, 17, 18 Zhou, P. 398 Zhou, X.T. 118, 133 Zhou, Z. 375 Zhu, C. 187 Zhu, Y. 397, 409, 410, 414, 419 Ziegler, J. 426 Zinin, Yu.A. 437 Zoller, P. 119, 148, 149, 165, 185, 398, 427 Zuidema, R. 382

CUMULATIVE I N D E X - VOLUMES I-XL

ABELI~S,E, Methods for Determining Optical Parameters of Thin Films ABELLA,I.D., Echoes at Optical Frequencies ABITBOL, C.I., s e e Clair, J.J. ABRAHAM, N.B., P. MANDEL, L.M. NARDUCCI, Dynamical Instabilities and Pulsations in Lasers AGARWAL,G.S., Master Equation Methods in Quantum Optics AGRANOVICH,V.M., V.L. GINZBURG,Crystal Optics with Spatial Dispersion A6RAWAL,G.P., Single-Longitudinal-Mode Semiconductor Lasers A6RAWAL,G.P, s e e Essiambre, R.-J. ALLEN, L., D.G.C. JONES,Mode Locking in Gas Lasers ALLEN, L., M.J. PADGETT,M. BABIKER,The Orbital Angular Momentum of Light AMMANN,E.O., Synthesis of Optical Birefringent Networks APRESYAN,L.A., s e e Kravtsov, Yu.A. ARIMONDO,E., Coherent Population Trapping in Laser Spectroscopy ARMSTRONG,J.A., A.W. SMITH,Experimental Studies of Intensity Fluctuations in Lasers ARNAUD,J.A., Hamiltonian Theory of Beam Mode Propagation ASAKURA,T., s e e Okamoto, T. ASAKURA,T., s e e Peiponen, K.-E. ASATRYAN,A.A., s e e Kravtsov, Yu.A. BABIKER,M., s e e Allen, L. BALTES, H.R, On the Validity of Kirchhoff's Law of Heat Radiation for a Body in a Nonequilibrium Environment BARABANENKOV,Yu.N., Yu.A. KRAVTSOV,V.D. OZRIN, A.I. SAICHEV, Enhanced Backscattering in Optics BARAKAT,R., The Intensity Distribution and Total Illumination of Aberration-Free Diffraction Images BARRETT, H.H., The Radon Transform and its Applications BASHKIN, S., Beam-Foil Spectroscopy BASSETT, I.M., W.T. WELFORD, R. WINSTON, Nonimaging Optics for Flux Concentration BECK~IANN,P., Scattering of Light by Rough Surfaces BERAN, M.J., J. Oz-VO6T, Imaging through Turbulence in the Atmosphere BERNARD,J., s e e Orrit, M. BERRY,M.V., C. UPSTILL,Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns BERTERO, M., C. DE MOL, Super-Resolution by Data Inversion BERTOLOTTI,M., s e e Mihalache, D. 471

II, 249 VII, 139 XVI, 71 XXV, XI, IX, XXVI, XXXVII, IX, XXXIX, IX, XXXVI, XXXV,

1 1 235 163 185 179 291 123 179 257

VI, 211 XI, 247 XXXIV, 183 XXXVII, 57 XXXIX, 1 XXXIX, 291 XIII,

1

XXIX,

65

I, 67 XXI, 217 XII, 287 XXVII, VI, XXXIII, XXXV,

161 53 319 61

XVIII, 257 XXXVI, 129 XXVII, 227

472

CUMULATIVEINDEX- VOLUMESI-XL

BERTOLOTTI,M., s e e Chumash, V. BEVERLYIII, R.E., Light Emission From High-Current Surface-Spark Discharges BIALYNICKI-BIRULA,I., Photon Wave Function BJ6RK, G., s e e Yamamoto, Y. BLOOM, A.L., Gas Lasers and their Application to Precise Length Measurements BOUMAN,M.A., W.A. VAN DE GRIND,P. ZUIDEMA,Quantum Fluctuations in Vision BOUSQUET,E, s e e Rouard, E BROWN, G.S., s e e DeSanto, J A. BROWN, R., s e e Orrit, M. BRUNNER, W., H. PAUL, Theory of Optical Parametric Amplification and Oscillation BRYNGDAHL,O., Applications of Shearing Interferometry BRYNGDAHL,O., Evanescent Waves in Optical Imaging BRYNGDAHL, O., E WYROWSKI, Digital Holography - Computer-Generated Holograms BRYNGDAHL,O., T. SCHEERMESSER,E WYROWSKI,Digital Halftoning: Synthesis of Binary Images BURGH, J.M., The Metrological Applications of Diffraction Gratings BUTTERWECK,H.J., Principles of Optical Data-Processing BU2EK, V., EL. KNIGHT,Quantum Interference, Superposition States of Light, and Nonclassical Effects

XXXVI, XVI, XXXVI, XXVIII, IX, XXII, IV, XXIII, XXXV,

XXXIV,

1

CAGNAC, B., s e e Giacobino, E. CASASENT,D., D. PSALTIS,Deformation Invariant, Space-Variant Optical Pattern Recognition CEGLIO, N.M., D.W. SWEENEY, Zone Plate Coded Imaging: Theory and Applications CHARNOTSKII,M.I., J. GOZANI,V.I. TATARSKn,V.U. ZAVOROYNV,Wave Propagation Theories in Random Media Based on the Path-Integral Approach CHIAO, R.Y., A.M. STEINBERG,Tunneling Times and Superluminality CHmSTENSEN, J.L., s e e Rosenblum, W.M. CHRISTOV,I.P., Generation and Propagation of Ultrashort Optical Pulses CHUMASH, V., I. COJOCARU,E. FAZIO, E MICHELOTTI,M. BERTOLOTTI,Nonlinear Propagation of Strong Laser Pulses in Chalcogenide Glass Films CLAIR, J.J., C.I. ABITBOL,Recent Advances in Phase Profiles Generation CLARRICOATS,P.J.B., Optical Fibre Waveguides- A Review COHEN-TANNOUDJI,C., A. KASTLER,Optical Pumping COJOCARU,I., s e e Chumash, V. COLE, T.W., Quasi-Optical Techniques of Radio Astronomy COLOMBEAU,B., s e e Froehly, C. COOK, R.J., Quantum Jumps COURTI~S,G., P. CRUVELLIER,M. DETAILLE,M. SA'I'SSE,Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects CREATH,K., Phase-Measurement Interferometry Techniques CREWE, A.V., Production of Electron Probes Using a Field Emission Source CRUVELLIER,P., s e e Court~s, G. CUMMINS, H.Z., H.L. SWINNEY,Light Beating Spectroscopy

XVII,

85

DAINTY, J.C., The Statistics of Speckle Patterns DXNDLIKER,R., Heterodyne Holographic Interferometry

1 357 245 87 1 77 145 1 61

XV, 1 IV, 37 XI, 167 XXVIII,

1

XXXIII, 389 II, 73 XIX, 211

XVI, 289 XXI, 287 XXXII, XXXVII, XIII, XXIX,

203 345 69 199

XXXVI, XVI, XIV, V, XXXVI, XV, XX, XXVIII,

1 71 327 1 1 187 63 361

XX, 1 XXVI, 349 XI, 223 XX, 1 VIII, 133 XIV, XVII,

1 1

CUMULATIVEINDEX- VOLUMESI-XL DATTOLI, G., L. GIANNESSI,A. RENIERI, A. TORRE, Theory of Compton Free Electron Lasers DE MOL, C., s e e Bertero, M. DE STERKE,C.M., J.E. SIPE, Gap Solitons DECKER JR, J.A., s e e Harwit, M. DELANO, E., R.J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters DEMARIA, A.J., Picosecond Laser Pulses DESANTO, J.A., G.S. BROWN,Analytical Techniques for Multiple Scattering from Rough Surfaces DETAILLE,M., s e e Courtbs, G. DEXTER, D.L., s e e Smith, D.u DRAGOMAN,D., The Wigner Distribution Function in Optics and Optoelectronics DREXHAGE,K.H., Interaction of Light with Monomolecular Dye Layers DUGUAY,M.A., The Ultrafast Optical Kerr Shutter DUTTA, N.K., J.R. SIMPSON,Optical Amplifiers DUTTA GUPTA, S., Nonlinear Optics of Stratified Media EBERLY, J.H., Interaction of Very Intense Light with Free Electrons ENGLUND,J.C., R.R. SNAPP,W.C. SCHIEVE,Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity ENNOS, A.E., Speckle Interferometry ESSIAMBRE,R.-J., G.P. AGRAWAL,Soliton Communication Systems FABELINSKII,I.L., Spectra of Molecular Scattering of Light FABRE, C . , s e e Reynaud, S. FANTE, R.L., Wave Propagation in Random Media: A Systems Approach FAZIO, E., s e e Chumash, V. FICEK, Z. AND H.S. FREEDHOFF,Spectroscopy in Polychromatic Fields FIORENTINI,A., Dynamic Characteristics of Visual Processes FLYTZANIS, C., E HACHE, M.C. KLEIN, D. RaCARD, PH. ROUSSIGNOL,Nonlinear Optics in Composite Materials. 1. Semiconductor and Metal Crystallites in Dielectrics FOCKE, J., Higher Order Aberration Theory FORBES, G.W., s e e Kravtsov, Yu.A. FRANqON, M., S. MALLICK, Measurement of the Second Order Degree of Coherence FREEDHOFF,H.S., s e e Ficek, Z. FREILIKHER,V.D., S.A. GREDESKUL,Localization of Waves in Media with OneDimensional Disorder FRIEDEN,B.R., Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions FROEHLY,C., B. COLOMBEAU,M. V.~/IPOUILLE,Shaping and Analysis of Picosecond Light Pulses FRY, G.A., The Optical Performance of the Human Eye GABOR, D., Light and Information GAMO, H., Matrix Treatment of Partial Coherence GANOJBAmtCHE,A.H., G.H. WEISS, Random Walk and Diffusion-Like Models of Photon Migration in Turbid Media GANTSOG,Ts., s e e Tana~, R. GHATAK, A., K. THYAGARAJAN,Graded Index Optical Waveguides: A Review

473

XXXI, XXXVI, XXXIII, XII, VII, IX,

321 129 203 101 67 31

XXIII, XX, X, XXXVII, XII, XIV, XXXI, XXXVIII,

1 1 165 1 163 161 189 1

VII, 359 XXI, 355 XVI, 233 XXXVII, 185 XXXVII, XXX, XXII, XXXVI, XL, I,

95 1 341 1 389 253

XXIX, 321 IV, 1 XXXIX, 1 VI, 71 XL, 389 XXX, 137 IX, 311 XX, 63 VIII, 51 I, 109 III, 187 XXXIV, 333 XXXV, 355 XVIII, 1

474

CUMULATIVEINDEX-VOLUMES I-XL

GHATAK,A.K., s e e Sodha, M.S. GIACOBINO,E., B. CAGNAC,Doppler-Free Multiphoton Spectroscopy GIACOBINO,E., s e e Reynaud, S. GIANNESSI,L., s e e Dattoli, G. G1NZBURG,V.L., s e e Agranovich, V.M. GINZBtrRG, V.L., Radiation by Uniformly Moving Sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena GIOVANELLI,R.G., Diffusion Through Non-Uniform Media GLASER,I., Information Processing with Spatially Incoherent Light GNIADEK,K., J. PETYKIEWICZ,Applications of Optical Methods in the Diffraction Theory of Elastic Waves GOODMAN,J.W., Synthetic-Aperture Optics GOZANI, J., s e e Charnotskii, M.I. GRAHAM,R., The Phase Transition Concept and Coherence in Atomic Emission GREDESKUL,S.A., s e e Freilikher, V.D. HACHE, E, s e e Flytzanis, C. HALL, D.G., Optical Waveguide Diffraction Gratings: Coupling between Guided Modes HARIHARAN,P., Colour Holography HARIHARAN,P., Interferometry with Lasers HARIHARAN,P., B.C. SANDERS,Quantum Phenomena in Optical Interferometry HARWIT, M., J.A. DECKERJR, Modulation Techniques in Spectrometry HASEGAWA,A . , s e e Kodama, Y. HEIDMANN,A., s e e Reynaud, S. HELLO, P., Optical Aspects of Interferometric Gravitational-Wave Detectors HELSTROM,C.W., Quantum Detection Theory HERRIOT, D.R., Some Applications of Lasers to Interferometry HORNER, J.L., s e e Javidi, B. HUANG, T.S., Bandwidth Compression of Optical Images ICHIOKA,Y., s e e Tanida, J. IMOTO,N., s e e Yamamoto, Y. ITOH, K., Interferometric Multispectral Imaging JACOBSSON,R., Light Reflection from Films of Continuously Varying Refractive Index JACQUINOT,P., B. ROIZEN-DOSSIER,Apodisation JAHNS, J., Free-space Optical Digital Computing and Interconnection JAMROZ,W., B.E STOICHEFF,Generation of Tunable Coherent Vacuum-Ultraviolet Radiation JAVIDI, B., J.L. HORNER, Pattern Recognition with Nonlinear Techniques in the Fourier Domain JONES, D.G.C., s e e Allen, L. KASTLER, A., s e e Cohen-Tannoudji, C. KELLER, O., Local Fields in Linear and Nonlinear Optics of Mesoscopic Systems KHOO, I.C., Nonlinear Optics of Liquid Crystals KIELICH, S., Multi-Photon Scattering Molecular Spectroscopy KINOSlTA, K., Surface Deterioration of Optical Glasses

XIII, XVII, XXX, XXXI, IX,

169

85 1 321 235

XXXII, 267 II, 109 XXIV, 389 IX, VIII, XXXII, XII, XXX,

281 1 203 233 137

XXIX, 321 XXIX, XX, XXIV, XXXVI, XII, XXX, XXX, XXXVIII, X, VI, XXXVIII, X,

1 263 103 49 101 205 1 85 289 171 343 1

XL, 77 XXVIII, 87 XXXV, 145

V, 247 III, 29 XXXVIII, 419 XX, 325 XXXVIII, 343 IX, 179 V, XXXVII, XXVI, XX, IV,

1 257 105 155 85

CUMULATIVEINDEX- VOLUMESI-XL

KITAGAWA,M., s e e Yamamoto, Y. KLEIN, M.C., s e e Flytzanis, C. KLYATSrdN, V.I., The Imbedding Method in Statistical Boundary-Value Wave Problems KNIGHT,P.L., s e e Bu2ek, V. KODAMA,Y., A. HASEGAWA,Theoretical Foundation of Optical-Soliton Concept in Fibers KOPPELMAN, G., Multiple-Beam Interference and Natural Modes in Open Resonators KOTTLER,E, The Elements of Radiative Transfer KOTTLER,E, Diffraction at a Black Screen, Part I: Kirchhoff's Theory KOTTLER,E, Diffraction at a Black Screen, Part II: Electromagnetic Theory KRAVTSOV,Yu.A., Rays and Caustics as Physical Objects KRAVTSOV,Yu.A., s e e Barabanenkov, Yu.N. KRAVTSOV,YtJ.A., L.A. APRESVAN,Radiative Transfer: New Aspects of the Old Theory KRAVTSOV, Yu.A., G.W. FORBES, A.A. ASATRYAN,Theory and Applications of Complex Rays KUBOTA, H., Interference Color KUITTINEN,M., s e e Turunen, J. LABEYRIE,A., High-Resolution Techniques in Optical Astronomy LEAN, E.G., Interaction of Light and Acoustic Surface Waves LEE, W.-H., Computer-Generated Holograms: Techniques and Applications LEITH, E.N., J. UPATNIEKS,Recent Advances in Holography LETOKHOV,V.S., Laser Selective Photophysics and Photochemistry LEUCHS, G., s e e Sizmann, A. LEVI, L., Vision in Communication LIPSON, H., C.A. TAYLOR,X-Ray Crystal-Structure Determination as a Branch of Physical Optics LOHMANN, A.W, D. MENDLOVIC, Z. ZALEVSKY, Fractional Transformations in Optics LOHMANN,A.W., s e e Zalevsky, Z. LOUNIS, B., s e e Orrit, M. LUGIATO,L.A., Theory of Optical Bistability L t m ~ , A . , s e e Pefinovfi, V. LtJKg, A . , s e e Pefinovfi, V.

1VIACHIDA,S.,

s e e Yamamoto, Y. MAINFRAY, G., C. MANtJS, Nonlinear Processes in Atoms and in Weakly Relativistic Plasmas MALACARA,D., Optical and Electronic Processing of Medical Images MALACARA,D., s e e Vlad, V.I. MALLICK,S., s e e Frangon, M. MANDEL, L., Fluctuations of Light Beams MANDEL, L., The Case For and Against Semiclassical Radiation Theory MANDEL, P., s e e Abraham, N.B. MANUS, C., s e e Mainfray, G. MARCHAND,E.W, Gradient Index Lenses MARTIN, P.J., R.P. NETTERfiELD,Optical Films Produced by Ion-Based Techniques

475 XXVIII, 87 XXIX, 321 XXXIII, XXXIV,

1 1

XXX, 205 VII, III, IV, VI, XXVI, XXIX,

1 1 281 331 227 65

XXXVI, 179 XXXIX, 1 1,211 XL, 343 XIV, XI, XVI, VI, XVI, XXXIX, VIII,

47 123 119 1 1 373 343

V, 287 XXXVIII, XL, XXXV, XXI, XXXIII, XL,

263 271 61 69 129 115

XXVIII, 87 XXXII, XXII, XXXIII, VI, II, XIII, XXV, XXXII, XI, XXIII,

313 1 261 71 181 27 1 313 305 113

476

CUMULATIVEINDEX- VOLUMESI-XL

1VIASALOV, A.V., Spectral and Temporal Fluctuations of Broad-Band Laser Radiation MAYSTRE,D., Rigorous Vector Theories of Diffraction Gratings MEESSEN, A., s e e Rouard, P. MEHTA, C.L., Theory of Photoelectron Counting MENDLOVIC,D., s e e Lohmann, A.W. MENDLOVIC,O., s e e Zalevsky, Z. MEYSTRE, P., Cavity Quantum Optics and the Quantum Measurement Process. MICHELOTrI,E, s e e Chumash, V. MIHALACHE,O., M. BERTOLOTTI,C. SIBILIA,Nonlinear Wave Propagation in Planar Structures MIKAELIAN,A.L., M.L. TER-MI~ELIAN,Quasi-Classical Theory of Laser Radiation MI~ELIAN, A.L., Self-Focusing Media with Variable Index of Refraction MILLS, D.L., K.R. StmaaswaMv, Surface and Size Effects on the Light Scattering Spectra of Solids MILON)qI,P.W., B. StYNOAr~M,Atoms in Strong Fields: Photoionization and Chaos MIRANOWICZ,A., s e e Tana~, R. MIYAMOTO,K., Wave Optics and Geometrical Optics in Optical Design MOLLOW,B.R., Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence MtmATA, K., Instruments for the Measuring of Optical Transfer Functions MUSSET, A., A. TnELEN, Multilayer Antireflection Coatings NAKWASKI, W., M. OSI~SKI, Thermal Properties of Vertical-Cavity SurfaceEmitting Semiconductor Lasers NAROUCCI,L.M., s e e Abraham, N.B. NAVR~TIL,K., s e e Ohlidal, I. NE~ERfiELD, R.P., s e e Martin, P.J. Nlsnina~, H., T. StmA~, Micro Fresnel Lenses OHLiDAL, I., K. NAVRATIL,M. OHLiDAL, Scattering of Light from Multilayer Systems with Rough Boundaries OHLiDAL,M., s e e Ohlidal, I. OHTSU, M., T. TAKO,Coherence in Semiconductor Lasers OKAMOTO,T., T. ASAKURA,The Statistics of Dynamic Speckles OKOSHI, T., Projection-Type Holography OOUE, S., The Photographic Image OPATRNY,T., s e e Welsch, D.-G. ORRIT, M., J. BERNARD,R. BROWN, B. LOUNIS, Optical Spectroscopy of Single Molecules in Solids OSI~SKI, M., s e e Nakwaski, W. OSTROVSKAYA,G.V., Yu.I. OSTROVSKV,Holographic Methods of Plasma Diagnostics OSTROVSKY,YU.I., s e e Ostrovskaya, G.V. OSTROVSKY, YU.I., V.P. SHCHEPINOV, Correlation Holographic and Speckle Interferometry OUGHSTUN,K.E., Unstable Resonator Modes Oz-VOGT, J., s e e Beran, M.J. OZRIN, V.D., s e e Barabanenkov, Yu.N. PADGETT,M.J., s e e Allen, L. PAL, B.P., Guided-Wave Optics on Silicon: Physics, Technology and Status

XXII, XXI, XV, VIII, XXXVIII, XL, XXX, XXXVI,

145 1 77 373 263 271 261 1

XXVII, 227 VII, 231 XVII, 279 XIX, 45 XXXI, 1 XXXV, 355 I, 31 XIX, 1 V, 199 VIII, 201 XXXVIII, 165 XXV, 1 XXXIV, 249 XXIII, 113 XXIV, 1 XXXIV, XXXIV, XXV, XXXIV, XV, VII, XXXIX,

249 249 191 183 139 299 63

XXXV, 61 XXXVIII, 165 XXII, 197 XXII, 197 XXX, XXIV, XXXIII, XXIX,

87 165 319 65

XXXIX, 291 XXXII, 1

CUMULATIVEINDEX- VOLUMESI-XL

477

PAOLETTI, D., G. SCHIRRIPA SPAGNOLO, Interferometric Methods for Artwork Diagnostics PATORSKI,K., The Self-Imaging Phenomenon and Its Applications PAUL, H., s e e Brunner, W. PEGIS, R.J., The Modern Development of Hamiltonian Optics PEGIS, R.J., s e e Delano, E. PEIPONEN,K.-E., E.M. VARTIAINEN,T. ASAKURA,Dispersion Relations and Phase Retrieval in Optical Spectroscopy PE~NA, J., Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media PE0ffNOVA,V., A. LUKe, Quantum Statistics of Dissipative Nonlinear Oscillators PEI~aNOVX,V., A. LUKe, Continuous Measurements in Quantum Optics PERSHAN, ES., Non-Linear Optics PETYKIEWICZ,J., s e e Gniadek, K. PICHT, J., The Wave of a Moving Classical Electron PoPov, E., Light Diffraction by Relief Gratings: A Macroscopic and Microscopic View PORTER, R.P., Generalized Holography with Application to Inverse Scattering and Inverse Source Problems PRESNYAKOV, L.P., Wave Propagation in Inhomogeneous Media: Phase-Shift Approach PSALTIS,D., s e e Casasent, D. PSALTIS, D., Y. QIAO, Adaptive Multilayer Optical Networks QIAO, Y.,

see

Psaltis, D.

RAYMER,M.G., I.A. WALMSLEY,The Quantum Coherence Properties of Stimulated Raman Scattering RENIEPd, A., s e e Dattoli, G. REWAUD, S., A. HEIDMANN,E. GIACOBrNO,C. FABRE, Quantum Fluctuations in Optical Systems R I c A l ~ , D., s e e Flytzanis, C. RJSEBER6, L.A., M.J. WEBER,Relaxation Phenomena in Rare-Earth Luminescence RJSrd~N, H., Statistical Properties of Laser Light RODDIER, E, The Effects of Atmospheric Turbulence in Optical Astronomy ROIZEN-DOSSIER,B., s e e Jacquinot, P. RONCHI, L., s e e Wang Shaomin ROSANOV,N.N., Transverse Patterns in Wide-Aperture Nonlinear Optical Systems ROSENBLUM, W.M., J.L. CHRISTENSEN, Objective and Subjective Spherical Aberration Measurements of the Human Eye ROTrmERG, L., Dephasing-Induced Coherent Phenomena ROUARD, P., P. BOUSQUET,Optical Constants of Thin Films ROUARD, P., A. MEESSEN, Optical Properties of Thin Metal Films ROUSSIGNOL,PH., s e e Flytzanis, C. RtmINOWICZ,A., The Miyamoto-Wolf Diffraction Wave RUDOLPH,D., s e e Schmahl, G.

SAICHEV,A.I., s e e Barabanenkov, Yu.N. SAISSE,M., s e e Court6s, G. SAITO, S.,

see

S AKAI, H . , s e e

Yamamoto, Y. Vanasse, G.A.

XXXV, 197 XXVII, 1 XV, 1 I, 1 VII, 67 XXXVII,

57

XVIII, XXXIII, XL, V, IX, V,

127 129 115 83 281 351

XXXI, 139 XXVII, 315 XXXIV, 159 XVI, 289 XXXI, 227 XXXI, 227

XXVIII, 181 XXXI, 321 XXX, XXIX, XIV, VIII, XIX, III, XXV, XXXV,

1 321 89 239 281 29 279 1

XIII, XXIV, IV, XV, XXIX, IV, XIV,

69 39 145 77 321 199 195

XXIX, 65 XX, 1 XXVIII, 87 VI, 259

478

CUMULATIVEINDEX- VOLUMESI-XL

SALEH, B.E.A., s e e Teich, M.C. SANDERS,B.C., s e e Hariharan, P. SCHEERMESSER~T., s e e Bryngdahl, O. SCHIEVE, W.C., s e e Englund, J.C. SCHIRRIPASPAGNOLO,G., s e e Paoletti, D. SCHMAHL,G., D. RUDOLPH,Holographic Diffraction Gratings SCHUBERT, M., B. WILHELMI, The Mutual Dependence Between Coherence Properties of Light and Nonlinear Optical Processes SCHULZ, G., J. SCHWIDER,Interferometric Testing of Smooth Surfaces SCHULZ, G., Aspheric Surfaces SCHWlDER,J., s e e Schulz, G. SCHWIDER,J., Advanced Evaluation Techniques in Interferometry SCULLV,M.O., K.G. WHITNEY,Tools of Theoretical Quantum Optics SENITZKV, I.R., Semiclassical Radiation Theory Within a Quantum-Mechanical Framework SHARMA, S.K., D.J. SOMERFORD,Scattering of Light in the Eikonal Approximation SHCHEPINOV,V.P., s e e Ostrovsky, Yu.I. SIBILIA,C., s e e Mihalache, D. SIMPSON,J.R., s e e Dutta, N.K. SIPE, J.E., s e e Van Kranendonk, J. SIPE, J.E., s e e De Sterke, C.M. SITTIG, E.K., Elastooptic Light Modulation and Deflection SIZMANN,A., G. LEUCHS,The Optical Kerr Effect and Quantum Optics in Fibers SLUSHER,R.E., Self-Induced Transparency SMITH, A.W., s e e Armstrong, J.A. SMITH, D.Y., D.L. DEXTER, Optical Absorption Strength of Defects in Insulators SMITH, R.W., The Use of Image Tubes as Shutters SNAPP, R.R., s e e Englund, J.C. SODHA, M.S., A.K. GHATAK,V.K. TRIPATHI, Self-Focusing of Laser Beams in Plasmas and Semiconductors SOMERFORD,D.J., s e e Sharma, S.K. SOROKO, L.M., Axicons and Meso-Optical Imaging Devices SPREEUW,R.J.C., J.P. WOERDMAN,Optical Atoms STEEL, W.H., Two-Beam Interferometry STEINBERG,A.M., s e e Chiao, R.Y. STOICHEFF,B.P., s e e Jamroz, W. STROHBEHN,J.W., Optical Propagation Through the Turbulent Atmosphere STROKE, G.W., Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy SUBBASWAMV,K.R., s e e Mills, D.L. SUHARA,T., s e e Nishihara, H. StrNDARAM,B., s e e Milonni, P.W. SVELTO, O., Self-Focusing, Self-Trapping, and Self-Phase Modulation of Laser Beams SWEENEu D.W., s e e Ceglio, N.M. SWINNEV,H.L., s e e Cummins, H.Z. TAKO, T., s e e Ohtsu, M. TANAKA,K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets TANAS,R., A. MIRANOWICZ,TS. GANTSOG,Quantum Phase Properties of Nonlinear Optical Phenomena

XXVI, XXXVI, XXXIII, XXI, XXXV, XIV,

1 49 389 355 197 195

XVII, XIII, XXV, XIII, XXVIII, X,

163 93 349 93 271 89

XVI, XXXIX, XXX, XXVII, XXXI, XV, XXXIII, X, XXXIX, XII, VI, X, X, XXI,

413 213 87 227 189 245 203 229 373 53 211 165 45 355

XIII, XXXIX, XXVII, XXXI, V, XXXVII, XX, IX,

169 213 109 263 145 345 325 73

II, 1 XIX, 45 XXIV, 1 XXXI, 1 XII, 1 XXI, 287 VIII, 133 XXV, 191 XXIII, 63 XXXV, 355

479

CUMULATIVEINDEX- VOLUMESI-XL TANGO,W.J., R.Q. TWISS,Michelson Stellar Interferometry TANIDA,J., Y. ICHIOKA,Digital Optical Computing TATARSKII,V.I., V.U. ZAVOROTNYI,Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium TATARSKII,V.I., s e e Charnotskii, M.I. TAYLOR,C.A., s e e Lipson, H. TEICH, M.C., B.E.A. SALEH,Photon Bunching and Antibunching TER-MIKAELIAN,M.L., s e e Mikaelian, A.L. THELEN, A., s e e Musset, A. THOMPSON,B.J., Image Formation with Partially Coherent Light THYAGARAJAN,K., s e e Ghatak, A. TONOMURA,A., Electron Holography TORRE, A., s e e Dattoli, G. TRIPATHI,V.K., s e e Sodha, M.S. TSUJIUCHI, J., Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering TURUNEN, J., M. KUITTINEN, E WYROWSKI,Diffractive Optics: Electromagnetic Approach TwIss, R.Q., s e e Tango, W.J.

XVII, 239 XL, 77 XVIII, XXXII, V, XXVI, VII, VIII, VII, XVIII, XXIII, XXXI, XIII,

204 203 287 1 231 201 169 1 183 321 169

II, 131 XL, 343 XVII, 239

UPATNIEKS,J., s e e Leith, E.N. UPSTILL,C., s e e Berry, M.V. USHIODA,S., Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids

VI, 1 XVIII, 257

VAMPOUILLE,M., s e e Froehly, C. VAN DE GRIND,W.A., s e e Bouman, M.A. VAN HEEL, A.C.S., Modern Alignment Devices VAN KRANENDONK,J., J.E. SIPE, Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media VANASSE,G.A., H. SAKAI,Fourier Spectroscopy VARTIAINEN,E.M., s e e Peiponen, K.-E. VERNIER,P.J., Photoemission VLAD, VT, D. MALACARA,Direct Spatial Reconstruction of Optical Phase from Phase-Modulated Images VOGEL,W., s e e Welsch, D.-G.

XX, 63 XXII, 77 I, 289

WALMSLEY,I.A., s e e Raymer, M.G. WANGSHAOMIN,L. RONCHI,Principles and Design of Optical Arrays WEBER, M.J., s e e Riseberg, L.A. ~TEIGELT,G., Triple-Correlation Imaging in Optical Astronomy WEISS, G.H., s e e Gandjbakhche, A.H. WELFORD,W.T., Aberration Theory of Gratings and Grating Mountings WELFORD,W.T., Aplanatism and Isoplanatism WELFORD,W.T., s e e Bassett, I.M. WELSCH,D.-G., W. VOGEL,T. OPATRN~',Homodyne Detection and Quantum-State Reconstruction WHITNEY,K.G., s e e Scully, M.O. WILHELMI,B., s e e Schubert, M. WINSTON,R., s e e Bassett, I.M. WOERDMAN,J.P., s e e Spreeuw, R.J.C.

XIX, 139

XV, VI, XXXVII, XIV,

245 259 57 245

XXXIII, 261 XXXIX, 63 XXVIII, XXV, XIV, XXIX, XXXIV, IV, XIII, XXVII,

181 279 89 293 333 241 267 161

XXXIX, X, XVII, XXVII, XXXI,

63 89 163 161 263

480

CUMULATIVEINDEX- VOLUMESI-XL

WOL1-NSKI,T.R., Polarimetric Optical Fibers and Sensors WOLTER,H., On Basic Analogies and Principal Differences between Optical and Electronic Information WYNNE, C.G., Field Correctors for Astronomical Telescopes WVROWSKI,E, s e e Bryngdahl, O. WYROWSKI,E, s e e Bryngdahl, O. WYROWSKI,E, s e e Turunen, J. YAMAGUCHI,I., Fringe Formations in Deformation and Vibration Measurements using Laser Light YAMAJI, K., Design of Zoom Lenses YAMAMOTO,T., Coherence Theory of Source-Size Compensation in Interference Microscopy YAMAMOTO,Y., S. MACHIDA,S. SAITO,N. IMOTO, T. YANAGAWA,M. KITAGAWA, G. BJ6RK, Quantum Mechanical Limit in Optical Precision Measurement and Communication YANAGAWA,T., s e e Yamamoto, Y. YAROSLAVSKY,L.P., The Theory of Optimal Methods for Localization of Objects in Pictures YOSHINAGA,H., Recent Developments in Far Infrared Spectroscopic Techniques Yu, ET.S., Principles of Optical Processing with Partially Coherent Light Yu, ET.S., Optical Neural Networks: Architecture, Design and Models ZALEVSKY,Z., s e e Lohmann, A.W. ZaLEVSKY, Z., D. MENDLOVIC,A.W. LOHMANN,Optical Systems with Improved Resolving Power ZAVOROrm', VU., s e e Charnotskii, M.I. ZAVOROTSYI,V.U., s e e Tatarskii, VI. ZUIDEMA,P., s e e Bouman, M.A.

XL,

1

I, X, XXVIII, XXXIII, XL,

155 137 1 389 343

XXII, 271 VI, 105 VIII, 295

XXVIII, 87 XXVIII, 87 XXXII, XI, XXIII, XXXII,

145 77 221 61

XXXVIII, 263 XL, XXXII, XVIII, XXII,

271 203 204 77

SUBJECT

INDEX

FOR

A ac Stark effect 422 acoustical signal detection 52 amplitude-squeezed light 188 Autler-Townes absorption 399, 405, 417, 420, 422, 425 spectrum 391-394, 417, 421,423, 426, 430, 438

VOLUME

XL

E

elasto-optic effect 44, 45 electro-optical effect 84 F Fabry-Perot etalon device 84 Fano factor 190, 228 fiber, highly birefringent 3, 10-13, 28-34, 38-41, 43, 48-51, 62, 63, 66-69 -, low-birefringent 10 -, numerical aperture of 5 -, polarization effects in 3 , maintaining 3, 10 , phenomena in optical 4 , preserving 69 , sensitive 69 -, single-mode 3 -, two-mode 9 optic sensor 66 flip-flop 94 Floquet method 399, 401, 415 Floquet-Bloch condition 350 fluorescence spectrum 413, 415, 417-4 19, 423,430 four-wave mixer 179-181, 186, 189, 222 fractional Fourier transform 277

B

back-action evasion 222, 223 Bell inequality 120, 186 binomial state 175 birefringence, geometrical 22 -, induced 24 -, intrinsic 22 -, nonlinear 24 -, stress 23, 27 Bloch-Siegert shift 426, 427 Borel subset 136 Bose-Einstein statistics 123, 124 C cellular logic 101 coherent state 176, 224 - transfer function 298 continuous measurement 120, 167, 226 , models of 148-169 state reduction 226 Copenhagen interpretation 135

G Gabor transform 275 geometrical resolution 273, 274 grating, binary 359 -, Bragg effect in a binary grating 347 -, electromagnetic theory of 349-358 -, modulated 371 -, multilevel 362 -, phase retardation by a reflection grating 347 -, total absorption of light by 346 - antireflection surfaces 367

D

Dammann grating 293-295, 302-304, 307, 309, 310, 320 DeFinetti's symbol 170 Delano diagram 276 diffraction efficiency 347, 352, 375 - resolution 273 dressed-atom method 404 states 410, 412

457

458

SUBJECT INDEX FOR VOLUME XL

grating (cont'd) equations 350 - retarders 368 -

I-I HB fiber, see fiber, highly birefringent Heisenberg equations of motion 198 Helmholtz equation 352 holographic interconnection 89 homodyne measurement 218

I

image logic algebra 104 - processing 105 input-output relation 178 It6 calculus 261 - integral 119 - product 152

J

Jaynes-Cummings model Johnson Criterion 274 Jones formalism 13, 14 -matrix 15, 18 - vector 18

164

K

Kramers' rule

64

L Lamb-Dicke limit 398 laser cooling 398 lasing without inversion 397 Lie algebra 174, 179 Lindblad form 118 - operators 158 liquid crystal light valve 84 logic gate array 79, 80, 85, 86 gates 83-88

M

magneto-optic current sensing 52 Mandel photodetection formula 123, 130 Manley-Rowe relation 189, 231 Markov process 135, 144 , quantum 177, 182 Maxwell's equations 40, 352 MIMD 81

modulation, cross-phase 25 Mollow absorption spectrum 391,399, 405 - triplet 391, 410 Monte Carlo simulation 119 Mueller matrices 4, 13, 19 Mueller-Stokes formalism 13, 18-20 N near-field microscopy 273 negative binomial distribution 252 noise equivalent resolution 275 nonstationary process 118 number eigenstate 175

O open system, quantum theory of 125 optical array logic 91, 102 - - - network computing 100 - bistability 84 - Bloch equations 400, 426 - cellular logic image processor 95 computing 79, 83 - -, digital 79, 80, 93, 110, 111 - fiber, birefringence in 22-28 - - , modes of 5 sensor 3 - Kerr effect 25 - logic devices 83 -parallel array logic system 96 shadow casting 85, 86 - transfer function 298 P

P function 251 parametric amplifier 120, 160, 179, 227 - down-conversion process 120, 197 paraxial approximation 325 path integral propagator 167 Paul trap 161 Penning trap 158 phase distribution 214 - operator 214 photocount-number distribution 148 photodetection, quantum optical approach to 125 -, quasicontinuous schemes of 173-182 -, semiclassical approach to 118, 121 - p r o c e s s 118, 120, 169 photoelastic effect 23

SUBJECT INDEX FOR VOLUME XL photon antibunching 118, 393 bunching 118 - number distribution 147 - measurement 118, 120, 208 - - - , destructive continuous 230, 234 - - - , nondemolition continuous 121,237 Poincar6 sphere representation 4, 13, 20, 21 Poisson process 122 23, 57 statistics 123 polarimetric all-fiber sensor 56 - fiber-optic sensor 52 strain-gauge manometer 57 polarization mode dispersion 27 - preserving coupler 62 pulse shaping 26 pupil function 299 -

-

- r a t i o

-

-

Q quantum characteristic function 231 -jump 119,393 , reversibility of 148 - kicked rotor 149 nondemolition measurement 119, 120, 161, 188 , continuous 159 - trajectory 146, 148, 158 -

R

Rabi frequency 401,410, 415, 417, 419, 421 - sidebands 398 Ramsey field 165 Rayleigh expansion 350 - scattering 32, 59 residue arithmetic 87 resonance fluorescence 167 Ronchi pattern 302 - ruling 293 Roseta grating 306 - mask 303 - pattern 307

459

SIMD 81 smart pixels 79, 82, 98 Soleil-Babinet compensator 58 soliton trapping 27 space-bandwidth product 81,275 - - - of signals 278 .... system 279 spatial filtering logic 85 light modulator 84 squeezed state 120, 182 - vacuum, two-mode 199 squeezing of vacuum fluctuations 208 Srinivas-Davies model 118, 128, 138, 151, 157, 158 , generalized 128-131 Stokes parameters 19 - v e c t o r s 4, 13, 19 Stratonovich integral 119, 151 - product 152 sub-Poissonian light 120 - photon statistics 176, 184 - - pumped laser 162 state 184 super-Poissonian photon statistics 176, 184 symbolic substitution 102 synthetic aperture radar 338 -

-

T TSE computer 93 Turing machine 108 two-photon laser 397

W wavelength-multiplexing super resolution 329 Wiener process 158 Wigner chart 277, 281,282, 284, 319 - function 196, 197, 2 0 5 , 2 2 1 , 2 7 6 , 312-317, 319 - theorems 317 Wollaston prism 63,292

S

Schmidt decomposition 200 Schrrdinger-cat paradox 139 - - state 120, 182, 226 equation, nonlinear 169 , stochastic 119, 168 - picture 224

Y Young modulus

57

-

Z Zeno effect, quantum

162, 166

CONTENTS OF PREVIOUS VOLUMES

VOLUME I (1961) I II III

The Modem Development of Hamiltonian Optics, R.J. I~GIs Wave Optics and Geometrical Optics in Optical Design, K. MIYAMOTO The Intensity Distribution and Total Illumination of Aberration-Free Diffraction Images, R. BA~KAT IV Light and Information, D. GABOR V On Basic Analogies and Principal Differences between Optical and Electronic Information, H. WOLTER VI Interference Color, H. KUBOTA VII Dynamic Characteristics of Visual Processes, A. FIORENTINI VIII Modem Alignment Devices, A.C.S. VAN HEEL

1- 29 31- 66 67-108 109-153 155-210 211-251 253-288 289-329

VOLUME II (1963) I

Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy, G.W. STROKE II The Metrological Applications of Diffraction Gratings, J.M. BURCH III Diffusion Through Non-Uniform Media, R.G. GIOVANELLI IV Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering, J. TSUJIUCHI V Fluctuations of Light Beams, L. MANDEL VI Methods for Determining Optical Parameters of Thin Films, E ABELI~S

1- 72 73-108 109-129 131-180 181-248 249-288

VOLUME III (1964) I II III

The Elements of Radiative Transfer, E KOYrLER Apodisation, P. JACQUINOT,B. ROIZEN-DosSIER Matrix Treatment of Partial Coherence, H. GAMO

1- 28 29-186 187-332

VOLUME IV (1965) I II III IV V VI VII

Higher Order Aberration Theory, J. FocKE Applications of Shearing Interferometry, O. BRVNGDAHL Surface Deterioration of Optical Glasses, K. KrYOSITA Optical Constants of Thin Films, R ROUARD,R BOUSQUET The Miyamoto-Wolf Diffraction Wave, A. RtmIyOWICZ Aberration Theory of Gratings and Grating Mountings, W.T. WELFORD Diffraction at a Black Screen, Part I: Kirchhoff's Theory, E KOYrLER 461

1- 36 37- 83 85-143 145-197 199-240 241-280 281-314

462

CONTENTS OF PREVIOUS VOLUMES VOLUME V (1966)

I Optical Pumping, C. COHEN-TANNOUDJI,A. KASTLER II Non-Linear Optics, P.S. PERSHAN III Two-Beam Interferometry, W.H. STEEL IV Instruments for the Measuring of Optical Transfer Functions, K. MURATA V Light Reflection from Films of Continuously Varying Refractive Index, R. JACOBSSON VI X-Ray Crystal-Structure Determination as a Branch of Physical Optics, H. LIPSON, C.A. TAYLOR VII The Wave of a Moving Classical Electron, J. PICHT

1-- 81 83--144 145--197 199--245 247--286 287--350 351-370

VOLUME VI (1967) Recent Advances in Holography, E.N. LEITH,J. UPATNIEKS Scattering of Light by Rough Surfaces, P. BECKMANN Measurement of the Second Order Degree of Coherence, M. FRAN~ON, S. MALLICK Design of Zoom Lenses, K. YAMAJI Some Applications of Lasers to Interferometry, D.R. HERRIOT Experimental Studies of Intensity Fluctuations in Lasers, J.A. ARMSTRONG, A.W. SMITH VII Fourier Spectroscopy, G.A. VANASSE,H. SAKAI VIII Diffraction at a Black Screen, Part II: Electromagnetic Theory, E KOTTLER

I II III IV V VI

1- 52 53- 69 71-104 105-170 171-209 211-257 259-330 331-377

VOLUME VII (1969) I

II III IV V VI VII

Multiple-Beam Interference and Natural Modes in Open Resonators, G. KOPPELMAN Methods of Synthesis for Dielectric Multilayer Filters, E. DELANO, R.J. PEOIS Echoes at Optical Frequencies, I.D. ABELLA Image Formation with Partially Coherent Light, B.J. THOMPSON Quasi-Classical Theory of Laser Radiation, A.L. MIKAELIAN,M.L. TER-MIKAELIAN The Photographic Image, S. OOUE Interaction of Very Intense Light with Free Electrons, J.H. EBERLY

1- 66 67-137 139-168 169-230 231-297 299-358 359-415

VOLUME VIII (1970) Synthetic-Aperture Optics, J.W. GOODMAN The Optical Performance of the Human Eye, G.A. FRY Light Beating Spectroscopy, H.Z. CUMMINS, H.L. SWINNEY Multilayer Antireflection Coatings, A. MUSSET, A. THELEN Statistical Properties of Laser Light, H. RiSKEN Coherence Theory of Source-Size Compensation in Interference Microscopy, T. YAMAMOTO VII Vision in Communication, L. LEVI VIII Theory of Photoelectron Counting, C.L. MEHTA

I

II III IV V VI

1-- 50 51--131 133--200 201--237 239--294 295--341 343--372 373-440

VOLUME IX (1971 ) I 11 III IV

Gas Lasers and their Application to Precise Length Measurements, A.L. BLOOM Picosecond Laser Pulses, A.J. DEMARIA Optical Propagation Through the Turbulent Atmosphere, J.W. STROHBEHN Synthesis of Optical Birefringent Networks, E.O. AMMANN

1- 30 31- 71 73-122 123-177

CONTENTS OF PREVIOUS VOLUMES V Mode Locking in Gas Lasers, L. ALLEN,D.G.C. JONES VI Crystal Optics with Spatial Dispersion, V.M. AGRANOVICH,V.L. GINZBURG VII Applications of Optical Methods in the Diffraction Theory of Elastic Waves, K. GNIADEK,J. PETYKIEWICZ VIII Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions, B.R. FRIEDEN

463 179-234 235-280 281-310 311-407

VOLUME X (1972) I II III IV V VI VII

Bandwidth Compression of Optical Images, T.S. HUANG The Use of Image Tubes as Shutters, R.W. SMITH Tools of Theoretical Quantum Optics, M.O. SCULLY,K.G. WHITNEY Field Correctors for Astronomical Telescopes, C.G. WYNNE Optical Absorption Strength of Defects in Insulators, D.Y. SMITH, D.L. DEXTER Elastooptic Light Modulation and Deflection, E.K. SITTIG Quantum Detection Theory, C.W. HELSTROM

1- 44 45- 87 89-135 137-164 165-228 229-288 289-369

VOLUME XI (1973) I II III IV V VI VII

Master Equation Methods in Quantum Optics, G.S. AGARWAL Recent Developments in Far Infrared Spectroscopic Techniques, H. YOSHINAGA Interaction of Light and Acoustic Surface Waves, E.G. LEAN Evanescent Waves in Optical Imaging, O. BRYNGDAHL Production of Electron Probes Using a Field Emission Source, A.V CREWE Hamiltonian Theory of Beam Mode Propagation, J.A. ARNAUD Gradient Index Lenses, E.W. MARCHAND

1- 76 77-122 123-166 167-221 223-246 247-304 305-337

VOLUME XlI (1974) I

II III IV V VI

Self-Focusing, Self-Trapping, and Self-Phase Modulation of Laser Beams, O. SVELTO Self-Induced Transparency, R.E. SLUSHER Modulation Techniques in Spectrometry, M. HARWIT, J.A. DECKERJR Interaction of Light with Monomolecular Dye Layers, K.H. DREXHAGE The Phase Transition Concept and Coherence in Atomic Emission, R. GRAHAM Beam-Foil Spectroscopy, S. BASHKIN

1- 51 53-100 101-162 163-232 233-286 287-344

VOLUME XIII (1976) I

On the Validity of Kirchhoff's Law of Heat Radiation for a Body in a Nonequilibrium 1- 25 Environment, H.P. BALTES 27- 68 II The Case For and Against Semiclassical Radiation Theory, L. MANDEL III Objective and Subjective Spherical Aberration Measurements of the Human Eye, 69- 91 WM. ROSENBLUM,J.L. CHRJSTENSEN 93-167 IV Interferometric Testing of Smooth Surfaces, G. SCHULZ,J. SCHWIDER V Self-Focusing of Laser Beams in Plasmas and Semiconductors, M.S. SODHA, 169-265 A.K. GHATAK,V.K. TRIPATHI 267-292 VI Aplanatism and Isoplanatism, W.T. WELFORD

464

CONTENTS OF PREVIOUS VOLUMES

VOLUME XIV (1976) I II III IV V VI VII

The Statistics of Speckle Patterns, J.C. DAINTY High-Resolution Techniques in Optical Astronomy, A. LABEYRIE Relaxation Phenomena in Rare-Earth Luminescence, L.A. RISEBERG,M.J. WEBER The Ultrafast Optical Kerr Shutter, M.A. DUGUAY Holographic Diffraction Gratings, G. SCHMAHL,D. RUDOLPH Photoemission, P.J. VERNIER Optical Fibre Waveguides- A Review, P.J.B. CLARRICOATS

1- 46 47- 87 89-159 161-193 195-244 245-325 327-402

VOLUME XV (1977) 1- 75 I Theory of Optical Parametric Amplification and Oscillation, W. BRUNNER,H. PAUL 77-137 II Optical Properties of Thin Metal Films, P. ROUARD,A. MEESSEN 139-185 III Projection-Type Holography, T. OKOSHI 187-244 IV Quasi-Optical Techniques of Radio Astronomy, T.W COLE V Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media, J. VAN 245-350 KRANENDONK,J.E. SIPE

VOLUME XVI (1978) I Laser Selective Photophysics and Photochemistry, V.S. LETOKHOV II Recent Advances in Phase Profiles Generation, J.J. CLAIR,C.I. ABITBOL III Computer-Generated Holograms: Techniques and Applications, W.-H. LEE IV Speckle Interferometry, A.E. ENNOS V Deformation Invariant, Space-Variant Optical Pattem Recognition, D. CASASENT, D. PSALTIS VI Light Emission From High-Current Surface-Spark Discharges, R.E. BEVERLYIII VII Semiclassical Radiation Theory Within a Quantum-Mechanical Framework, I.R. SENITZKY

1- 69 71-117 119-232 233-288 289-356 357-411 413-448

VOLUME XVII (1980) I II III

Heterodyne Holographic Interferometry, R. DANDLIKER Doppler-Free Multiphoton Spectroscopy, E. GIACOBINO,B. CAGNAC The Mutual Dependence Between Coherence Properties of Light and Nonlinear Optical Processes, M. SCHUBERT,B. WILHELMI IV Michelson Stellar Interferometry, W.J. TANGO,R.Q. TwIss V Self-Focusing Media with Variable Index of Refraction, A.L. MIKAELIAN

1- 84 85-161 163-238 239-277 279-345

VOLUME XVIII (1980) I II

Graded Index Optical Waveguides: A Review, A. GHATAK,K. THYAGARAJAN Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media, J. PEI~INA III Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium, V.I. TATARSKII,V.U. ZAVOROTNYI IV Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns, M.V. BERRY, C. UPSTILL

1-126 127-203 204-256 257-346

CONTENTS OF PREVIOUSVOLUMES

465

VOLUME XIX (1981) I

Theory of Intensity Dependent Resonance Light Scattering and Resonance 1- 43 Fluorescence, B.R. MOLLOW II Surface and Size Effects on the Light Scattering Spectra of Solids, D.L. MILLS, 45-137 K.R. StmBASW~X4Y III Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids, 139-210 S. USHIODA 211-280 IV Principles of Optical Data-Processing, H.J. BUTrERWECK 281-376 V The Effects of Atmospheric Turbulence in Optical Astronomy, E RODDIER

VOLUME XX (1983) I II III IV V

Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronom1-61 ical Objects, G. COURT]~S,P. CRUVELLIER,M. DETAILLE,M. SA'I'SSE Shaping and Analysis of Picosecond Light Pulses, C. FROEHLY, B. COLOMBEAU, 63-153 M. VAMPOUILLE 155-261 Multi-Photon Scattering Molecular Spectroscopy, S. KIELICH 263-324 Colour Holography, P. H A ~ ~ Generation of Tunable Coherent Vacuum-Ultraviolet Radiation, W. JAMROZ, 325-380 B.P. STOICHEW

VOLUME XXI (1984) I II III IV V

Rigorous Vector Theories of Diffraction Gratings, D. MAYSTRE 1-- 67 Theory of Optical Bistability, L.A. LUGIATO 69--216 The Radon Transform and its Applications, H.H. BARRETT 217--286 Zone Plate Coded Imaging: Theory and Applications, N.M. CEOLIO,D.W SWEENEY 287--354 Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity, J.C. ENGLUND,R.R. SNAPP, WC. SCHIEVE 355-428

VOLUME XXII (1985) I Optical and Electronic Processing of Medical Images, D. MALACARA II Quantum Fluctuations in Vision, M.A. BOUMAN,W.A. VAN DE GRIND, E ZUIDEMA III Spectral and Temporal Fluctuations of Broad-Band Laser Radiation, A.V. MASALOV IV Holographic Methods of Plasma Diagnostics, G.V. OSTROVSKAVA,YU.I. OSTROVSKY V Fringe Formations in Deformation and Vibration Measurements using Laser Light, I. YAMAGUCHI VI Wave Propagation in Random Media: A Systems Approach, R.L. FANTE

1-- 76 77--144 145--196 197--270 271--340 341--398

VOLUME XXIII (1986) I

Analytical Techniques for Multiple Scattering from Rough Surfaces, J.A. DESANTO, 1- 62 G.S. BROWN 63-111 II Paraxial Theory in Optical Design in Terms of Gaussian Brackets, K. TANAKA 113-182 III Optical Films Produced by Ion-Based Techniques, P.J. MARTIN,R.P. NETTERfiELD 183-220 IV Electron Holography, A. TONO~mA 221-275 V Principles of Optical Processing with Partially Coherent Light, ET.S. Yu

466

CONTENTS OF PREVIOUS VOLUMES VOLUME XXIV (1987)

I II III IV V

Micro Fresnel Lenses, H. NISHIHARA,T. SUHARA Dephasing-Induced Coherent Phenomena, L. ROTHBERG Interferometry with Lasers, P. HARIHARAN Unstable Resonator Modes, K.E. OUGHSTtm Information Processing with Spatially Incoherent Light, I. GLASER

I

Dynamical Instabilities and Pulsations in Lasers, N.B. ABRAHAM, P. MANDEL, L.M. NARDUCCI 1-190 Coherence in Semiconductor Lasers, M. OHTSU, T. TAKO 191-278 Principles and Design of Optical Arrays, WANG SHAOMIN,L. RONCHI 279-348 Aspheric Surfaces, G. SCrlULZ 349-4 15

1- 37 39-101 103-164 165-387 389-509

VOLUME XXV (1988)

II III IV

VOLUME XXVI (1988) I II III IV V

Photon Bunching and Antibunching, M.C. TEICH, B.E.A. SALEH Nonlinear Optics of Liquid Crystals, I.C. Krtoo Single-Longitudinal-Mode Semiconductor Lasers, G.P. AGRAWAL Rays and Caustics as Physical Objects, Yu.A. KRAVTSOV Phase-Measurement Interferometry Techniques, K. CREATH

1-104 105-161 163-225 227-348 349-393

VOLUME XXVII (1989) The Self-Imaging Phenomenon and Its Applications, K. PATORSKI Axicons and Meso-Optical Imaging Devices, L.M. SOROKO Nonimaging Optics for Flux Concentration, I.M. BASSETT, W.T. WELFORD, R. WINSTON IV Nonlinear Wave Propagation in Planar Structures, D. MIHALACHE,M. BERTOLOTTI, C. SIBILIA V Generalized Holography with Application to Inverse Scattering and Inverse Source Problems, R.P. PORTER I

II III

1-108 109-160

161-226 227-313 315-397

VOLUME XXVIII (1990) I II

III IV V

Digital Holography- Computer-Generated Holograms, O. BRWGDAHL,E WYROWSKI 1- 86 Quantum Mechanical Limit in Optical Precision Measurement and Communication, Y. YAMAMOTO, S. MACHIDA, S. SAITO, N. IMOTO, T. YANAGAWA,M. KITAGAWA, G. BJ6RK 87-179 The Quantum Coherence Properties of Stimulated Raman Scattering, M.G. RAYMER, I.A. WALMSLEY 181-270 Advanced Evaluation Techniques in Interferometry, J. SCHWIDER 271-359 Quantum Jumps, R.J. COOK 361-416

CONTENTS OF PREVIOUS VOLUMES

467

VOLUME XXIX (1991) I

Optical Waveguide Diffraction Gratings: Coupling between Guided Modes, 1- 63 D.G. HALL II Enhanced Backscattering in Optics, Yu.N. BARABANENKOV, Yu.A. KRAVTSOV, 65-197 V.D. OZRrN, A.I. SAIClqEV 199-291 III Generation and Propagation of Ultrashort Optical Pulses, I.P. CHRISTOV 293-319 IV Triple-Correlation Imaging in Optical Astronomy, G. WEIGELT V Nonlinear Optics in Composite Materials. 1. Semiconductor and Metal Crystallites 321-411 in Dielectrics, C. FLYTZANIS,F. HACHE, M.C. KLEIN, D. RICARD,PH. ROUSSIGNOL

VOLUME XXX (1992) I

Quantum Fluctuations in Optical Systems, S. REVNAUO,A. HEIDMANN,E. GIACOBINO, 1- 85 C. FABRE II Correlation Holographic and Speckle Interferometry, Yu.I. OSTROVSKV,V.P. SHCHEP87-135 INOV III Localization of Waves in Media with One-Dimensional Disorder, V.D. FREILIKHER, 137-203 S.A. GREDESKUL IV Theoretical Foundation of Optical-Soliton Concept in Fibers, Y. KODAMA, 205-259 A. HASEGAWA 261-355 V Cavity Quantum Optics and the Quantum Measurement Process., P. MEYSTRE

VOLUME XXXI (1993) I II

Atoms in Strong Fields: Photoionization and Chaos, P.W MILONNI, B. SUNDARAM Light Diffraction by Relief Gratings: A Macroscopic and Microscopic View, E. PoPov III Optical Amplifiers, N.K. DUTTA, J.R. SIMPSON IV Adaptive Multilayer Optical Networks, D. PSALTIS,Y. QIAO V Optical Atoms, R.J.C. SPREEUW, J.P. WOERDMAN VI Theory of Compton Free Electron Lasers, G. DATTOLI, L. GIANNESSI,A. RENIERI, A. TORRE

1-137 139-187 189-226 227-261 263-319 321-412

VOLUME XXXII (1993) Guided-Wave Optics on Silicon: Physics, Technology and Status, B.P. PAL Optical Neural Networks: Architecture, Design and Models, ET.S. Yu The Theory of Optimal Methods for Localization of Objects in Pictures, L.P. YAROSLAVSKY IV Wave Propagation Theories in Random Media Based on the Path-Integral Approach, M.I. CHARNOTSKII,J. GOZANI, V.I. TATARSKII,V.U. ZAVOROTNY V Radiation by Uniformly Moving Sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, V.L. GINZBURG VI Nonlinear Processes in Atoms and in Weakly Relativistic Plasmas, G. MAINFRAY, C. MANUS I

II III

1- 59 61-144 145-201 203-266 267-312 313-361

468

CONTENTS OF PREVIOUSVOLUMES VOLUME XXXIII (1994)

I

The Imbedding Method in Statistical Boundary-Value Wave Problems, V.I. KLYATSKIN II Quantum Statistics of Dissipative Nonlinear Oscillators, V. l~0ayovX, A. Ltmg III Gap Solitons, C.M. DE STEVa~, J.E. SIPE IV Direct Spatial Reconstruction of Optical Phase from Phase-Modulated Images, V.I. VLAD,D. MALACARA V Imaging through Turbulence in the Atmosphere, M.J. BERAN, J. Oz-VOGT VI Digital Halftoning: Synthesis of Binary Images, O. BRYNGDAHL,T. SCHEERMESSER, E WYROWSm

1-127 129-202 203-260 261-317 319-388 389-463

VOLUME XXXIV (1995) I

Quantum Interference, Superposition States of Light, and Nonclassical Effects, V. BU2EK, P.L. KNIGHT II Wave Propagation in Inhomogeneous Media: Phase-Shift Approach, L.P. Pva~SNYAKOV III The Statistics of Dynamic Speckles, T. Og~vlOTO, T. ASAKta~ IV Scattering of Light from Multilayer Systems with Rough Boundaries, I. OHLiD~, K. NAVRATIL,M. OHLiDAL V Random Walk and Diffusion-Like Models of Photon Migration in Turbid Media, A.H. GANDJBAKHCHE,G.H. WEISS

1-158 159-181 183-248 249-331 333-402

VOLUME XXXV (1996) I II

Transverse Patterns in Wide-Aperture Nonlinear Optical Systems, N.N. ROSANOV Optical Spectroscopy of Single Molecules in Solids, M. ORRIT, J. BERNARD, R. BROWN,B. LOUNIS III Interferometric Multispectral Imaging, K. ITOH IV Interferometric Methods for Artwork Diagnostics, D. PAOLE~I, G. SCHIRRIPA SPAGNOLO V Coherent Population Trapping in Laser Spectroscopy, E. A~MONDO VI Quantum Phase Properties of Nonlinear Optical Phenomena, R. TAYAg, A. MIRANOWlCZ, TS. GANTSOG

1- 60 61-144 145-196 197-255 257-354 355-446

VOLUME XXXVI (1996) I

Nonlinear Propagation of Strong Laser Pulses in Chalcogenide Glass Films, V. CHUMASH,I. COJOCARU,E. FAZIO,E MICHELO'Iq'I,M. BERTOLOTrI II Quantum Phenomena in Optical Interferometry, P. H ~ ~ , B.C. SANDERS III Super-Resolution by Data Inversion, M. BERTERO,C. DE MOL IV Radiative Transfer: New Aspects of the Old Theory, Yu.A. I~VTSOV, L.A. APRESYAY V Photon Wave Function, I. BIALYNICKI-BIRULA

1-- 47 49--128 129-178 179--244 245-294

VOLUME XXXVII (1997) I II III

The Wigner Distribution Function in Optics and Optoelectronics, D. DRAGOMAN Dispersion Relations and Phase Retrieval in Optical Spectroscopy, K.-E. PEIPONEN, E.M. VARTIAINEN,T. ASAKURA Spectra of Molecular Scattering of Light, I.L. FABELINSKII

1- 56 57- 94 95-184

CONTENTS OF PREVIOUS VOLUMES

IV Soliton Communication Systems, R.-J. ESSIAMBRE,G.E AGRAWAL V Local Fields in Linear and Nonlinear Optics of Mesoscopic Systems, O. KELLER VI Tunneling Times and Superluminality, R.Y. CHIAO, A.M. STEINBERG

469 185--256 257--343 345-405

VOLUME XXXVIII (1998) I Nonlinear Optics of Stratified Media, S. DUTTA GUPTA II Optical Aspects of Interferometric Gravitational-Wave Detectors, P. HELLO III Thermal Properties of Vertical-Cavity Surface-Emitting Semiconductor Lasers, W. NAKWASKI,M. OSINSKI IV Fractional Transformations in Optics, A.W. LOHMANN,D. MENDLOVIC,Z. ZALEVSKV V Pattern Recognition with Nonlinear Techniques in the Fourier Domain, B. JAVIDI, J.L. HORNER VI Free-space Optical Digital Computing and Interconnection, J. JAHNS

1-- 84 85--164 165-262 263--342 343-418 419--513

VOLUME XXXIX (1999) I

Theory and Applications of Complex Rays, Yu.A. KRAVTSOV, G.W. FORBES, 1- 62 A.A. ASATRYAN II Homodyne Detection and Quantum-State Reconstruction, D.-G. WELSCH,W. VOGEL, 63-211 T. OPATRNY III Scattering of Light in the Eikonal Approximation, S.K. SHARMA,D.J. SOMERFORD 213-290 291-372 IV The Orbital Angular Momentum of Light, L. ALLEN, M.J. PADGETT,M. BABIKER 373--469 V The Optical Kerr Effect and Quantum Optics in Fibers, A. SIZMANN,G. LEUCHS