Linear Ray and Wave Optics in Phase Space Bridging Ray and Wave Optics via the Wigner PhaseSpace Picture
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Linear Ray and Wave Optics in Phase Space Bridging Ray and Wave Optics via the Wigner PhaseSpace Picture
For the cover design: 9 Ren6 Magritte, La corde sensible, BY SIAE 2005
Linear Ray and Wave Optics in Phase Space B r i d g i n g Ray and W a v e O p t i c s via the W i g n e r PhaseSpace Picture
Amalia Torre
ENEAUTS Tecnologie Fisiche A vanzate Frasca ti (Rome), Italy
G
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To my m o t h e r whose caress still lingers on my cheek. To my father whose smile still shines into my eyes. To my country whose colours feed my avid love of life.
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Preface Amo i colori, tempi di un anelito inquieto, irresolvibile, vitale spiegazione umilissima e sovrana dei cosmici " p e r c h , " del mio respiro. . . .
A. Merini, Colori (from A. Merini, Fiore di poesia, Einaudi, Torino, 1998)
Ray, wave and quantum concepts are central to diverse and seemingly incompatible models of light. Each model particularizes a specific "manifestation" of light, and then corresponds to adequate physical assumptions and formal approxilnations, whose domain of applicability is well established. Accordingly each model comprises its own set of geometrical and dynamical postulates with the pertinent mathematical means. Geometrical optics models the light field as an aggregate of incoherent light rays, naively perceived as the trace of the motion of the "luminous corpuscles", which, emitted by the source, move through spa.ce in obedience to the usual laws of mechanics. It treats light rays as lines in 3space dimensions and is accordingly concerned with the dynamical laws establishing how the rays bend when propagating in inhomogeneous media described by the refractive index function. Geometrical optics is not suited to explain interference, diffraction and quantum coherence effects, but, in contrast, it provides a particularly convenient mea,ns for the design of optical systems, which is based on the purely geometrical rules of ray tracing. Geometrical optics has developed its own mathematical framework, which can remarkably be brought into correspondence with that of the Hamiltonian mechanics of pointparticles, with "time" corresponding to the arclength along the ray path and the mechanical "potential" to the refractive index of the optical medium. Wave optics accounts for the wave characteristics of light. Originating directly from the classical electromagnetic theory, it shares with this theory the same system of theoretical principles and methods, which can notably be put in correspondence with those proper to relativistic quantum mechanics. Then, the geometry of light rays is replaced by the geometry of "luminous" waves, whose propagation is geometrically pictured as the transfer of the interference shaped vibrations from one portion of the medium to the contiguous one. Wave optics treats the light waves as complex functions of position in 3space dimensions and is accordingly concerned with the dynamical laws establishing how the wave function changes as the optical wave propagates through inhomogeneous media.
viii
Linear Ray and Wave Optics in Phase Space
Q u a n t u m optics recovers the grainy view of geometrical optics, picturing the light ray as a stream of particlelike entities, the photons. Whereas geometrical optics deals with the influence at a macroscopic level of the medium on the trajectory of tile photon streams, q u a n t u m optics is typically concerned with the wavelike question relevant to the coherence properties of tim t)hoton 1)earns and to the relevance of those properties on the interactioll ()f ligllt with llla,ttcr, whi(:h (:an (:orrcspondingly t)(; tr('~a,t('d qllantllln m('~r Cohere;lit an(1 stpu;r state.s of light arc th(; lmilding (:on(:('pts of (tlm,llt11~ll ()t)ti(:s. Wigner optics 1)ri(tges t)etween ray an(1 wa.ve ()I)tics. It ofli;rs th('~ optical t)ha,se sl)a,(:('~ a,s th(; amt)i(;nc('~ a,n(1 tlm Wign(;r flliwtion )ascd t(;(:lllli(tll(; a.s the nla, thelllati(:al lna.('llilmry to a,('(:t)llllll()(la,t(~ )(~tw(~ell tim two ()t)t)()sit,e extr(mms ()f ligtlt r(~I)res('~llta,tioll: th(~ l()(~a.liz(:(l ray ()f g('~()Ill(~tri(:a,1 ot)ti(:s a.ll(1 tll(~ lnll()ca,liz(~(t wa,ve flln('ti()n ()f wave ()l)ti(;s. N()ta.)ly (tllald, Ulll ()t)tics till(Is a, t:()llV(~llient f()rllnlla,tion ill tl~e ])l'()])(~.r I)hase st)a,ce with tlm (:()~se(tll(mt g(~()~ml,ri(:al vi('.w ()f ('.()ll(~I'(~llt an(l s(t~mez(~d states a,s circles a,nd ellil)s(~s. Tim Wig~mr fll~(',l,ion ~(:l,l~()(ls ('a,n suita,)ly lm applied t,() (t~m,~tl~ ()l)t,i(:s as w(;ll t()(:~a.t)l(; (@~('t,iv(~ al~a.lyti('a,1 ~n('m~s f()l" ('a.l('.~fla,ting ext)e(:l, atiol~ va,lu(~s a,n(1 tra.~sil,i(n~ l)r()l)a,)ilities f()r t,lm a,f()r(nn(~ld,i(nm(1 states. Tim l)UI'I)()se ()f t,lm 1)()()k ix t.() i~t,l'()(lu(:(~ tim r(m.(l(~r t() t,l~(~ ()t)t,i('al I)ha.sesl)a,('e a,n(1 t() tim al)l)r()aclms 1,()()l)ti(:s 1)a.se(1 ()~ 1,1~(~Wig~mr (lisl,ril)~l,i()n fltn(:l,i(n~, tt~a.t ha.v(~ 1)(:(,l~ (hw(4ot)e(1 ()v(:r tlm t)ast 25 years or st) i~ s(w(~ral s('ient, ific titles. These yiel(l tlm f()n~m.1 (:()ll,(~Xt, where (:()~(:(~t)ts m~(t lnetl~t)(ls ()f 1)ott~ ray an(t wave ot)ti(:s (:()alesc(~ into a ~ i f v i n g formalis~n. In this reslmct, elnt)hasis is givei~ to the Li(~ alge)ra r(~t)r(~seId,ation of ()i)ti(:a,1 systems m~(l a,(:(:()rdingly to the Lie algebra view of light I)r()t)agation thr()~gh oI)tical s y s t ( ~ s . The book is nm(le as selfc()~ta.in(~(1 as t)ossi)le. Chapter 1 1)rcs(mts tile Halniltonian equation,s of motions, wl~i(:h are basic to the (tevelot)~mld, of t)oth the tra,nsfer matrix fonna,lism, a,i~t)roi)riate to t)araxial ray ot)ti(:s (Chapters 2 and 3), and the trai~sfer operator formalism, suited to para,xial wave optics (Chapters 4 a.nd 5). The re,la.tio~ of both formalisms to the Lie algebra inethods is gently (tisplayed. Chapter 6 introduces tile Wigner distribution flmction, elucidating its origin taken in q u a n t u m mechanics and illustrating its properties. A host of diverse optical signals are considered and the relevant Wigner distribution functions arc analytically evaluated and graphically shown to help the intuitive perception of the siinultaneous account of the signal in the space and spatial frequency domains, conveyed by tile Wigner distribution function. Chapter 7 frames tile Wigner distribution function within the broad reahn of tile phasespace signal representations, and illustrates the procedure, and the relevant optical architectures, for displaying the Wigner distribution function of a given signal. In Chapter 8 the laws for the transfer of the Wigner distribution function through linear optical systems are derived. Attention is drawn to the
Preface
ix
relation between the Wigner distribution function and the fractional Fourier transform, which is a revealing and effective tool for the spacefrequency representation of signals (optical or not). Chapter 9 is concerned with the moments of the Wigner distribution function and their propagation laws. The Wigner representation is presented on the fascinating borderline between quantum mechanics and signal theory. Chapters are made as selfconsistent as possible. Indeed, the Introduction to each chapter is conceived as a summary of the basic results of previous chapters, which are central to those that are going to be presented. A basic role is assigned to the diagrams, which illustrate the syllabus of each chapter, and the figures, which confer physical reality to conceptual architectures. A wide bibliography is given in relation to topics both carefully investigated and briefly mentioned. Throughout the book the calculations are kept at an accessible level; most ina,thematical steps are justified. Difficulties Inight be encountered in connection with the algebra of operators, which do not obey the familiar rules of the algebra, of scalars. Careful and illustrative comments on the peculiar behavior of operators are provided in w 1.4.1 in order to help the readers who are not acquainted with the operator algebra. It is my hope to give the flavour of the fascinating feature of optics that enables a visible account of abstra,ct inathenmtical entities, like, for instance, symplectic ma.trices and inetaplectic operators, represented through integral transforms. Symplectic matrices and integral transforms, which essentially provide the formal structures for the considerations developed in Chapters 1 to 5, are intimately related, being indeed different representations of the same Sp(2, N) ~ Mp(2, IR) group element. Firstly recognized within a purely quantum mechanical context, this relation has been applied in optics in connection with the fractional Fourier transform. The link between ray matrices a,nd transfer operators from the alternative viewpoint of linear ca,nonical transformations and relevant representations, is elucidated in w5.6. This is an example of those parallel paths, that, explicitly illustrated or implicitly suggested in the text or in the problems, are intended to improve the feeling for the specific topic under consideration and to gain some insight and intuition for unforeseen correspondences and analogies between totally different physical problems. I am pleased to express my deep gratitude to Professor W.A.B. Evans, whose stimulating discussions, critical comments and technical suggestions have been precious to the completion of tile book. I am greatly indebted to Dr. A. De Angelis for his enlightening suggestions, and to Professor A. Reale and Professor A. Scafati for their helpful comments. It is dutiful of me to thank Dr. G. Dattoli, who introduced me to the Lie algebra theory during the
x
Linear Ray and Wave Optics in Phase Space
stage of our collaboration on the quantum picture of the Free Electron Laser dynamics. I am grateful to Dr. S. Bollanti, Dr. F. Flora and Dr. L. Mezi for their useful comments, and to Mrs. G. Gili, Mr. S. Lupini, Mrs. G. Martoriati, Mrs. M.T. Paolini, Mrs. L. Santonato, Dr. S. Palmerio, Dr. B. Robouch, Dr. N. Sa(:(:hetti and Dr. V. Violante for their c o n s t a n t and invah> able sympathy. I t h a n k our librarians, Mrs. C. De Palo and Mrs. M. Liberati, wtlo at certain t)erio(ls have t)atiently a(:(:epte(t the roh; of "nly" librarians. It is a t)lea.sllre to t,tm,~lk the ()t)ti(:al Society of A~imri(:a for kin(tly givi~lg me the t)ennissi(m t() r(:I)r()(t11(:('. Ilm.l,('.rial fl'()~li At)plied ()t)ti(:s and ()l)ti('s Letters, and Einmuli for t)ernfit, ting m e t() ret)r()(hl(:e the lines fr(),n Merilfi's t)()enl, which ()t)en(',(1 this Prcfa(:(;, lily literal tra,nslati()n ()f whi(:h ll()w (:h)s(;s it (t)elow). I (;xt)ress xlly at)t)I'(;('iatioIl t() l)I'()fess()r A. L()lnlm,llll f()," lfis t)r(nlll)t ml(t kill(t ,'cst)(nlse to my re(llu;st ()f ret)r()(llu'ing nmteria,1 fl'()m t)at)ers )y tlinls(;lf a.n(t his ('()w()rk('rs, whi('tl at)I)(;ar(;(1 ill ()I)ti('s C()~lmnl~fi(:ati()Ii. I a lll I)h;as(:(l l,()(;xI)i'css lily gratitu(tc t() l,ll('. F()ll(tal, i(nl Magritt('. for alh)wiIlg l,ll(', r(~I)r()(lll(:l,i()ll ()f the (w()(:atively e~(~l,i(n~a,1 Magritt(; I)ai~ti~g La (:ordu. s('.nsibl(' f(n" 1,1~; (:(~ver. I ain also i~(l(;)t(;(t t,() l,l~e Els(;vier l)r()(l~(:ti(m teal~ f()I" exl)ertly i~l)le~e~d, ing n~y i(h;as in relati(n~ I,,) l,h(; (:()v(:t'. A j()yflfl "Tl~m~k y(n~, s()relli~a" is (tirecte(t t() l)r. 1". M~u'('i, f(n" (;~th~siasti(:ally list(u~il~g 1,() l,l~e (h's(:ril)l,i(n~ ()f my "(:(n~(:et)t~ml (:asl,h's". A 1)()w is for 1,t~(; fl'i(;n(ls wh(~ sire,r(; ~ y t)assi(n~ f(n" the th(;a.l,r('~, fin" f()rgivil~g n~y at)scn(:es fl'()~ the t)relil~s, l)eil~g f()l'getfl~lly e~rat)ture(t i~ "1~i(; a.(l()rat(: f()n~n~linc". It is with intense elll()ti()iI tlm,t lily t h a n k s g()es als() t() zia Ai(ta a~(t Maria, wh() (:()lfl(t not see the (:()l~l)l('~l,i()li ()f the book.
I love colours, times of a yearning restless, irresolvable, vital, very lmmble and supreme explanation of the cosmic "why" of my breath. . . .
Colours
Contents
I. Hamiltonian Picture of Light Optics. FirstOrder Ray Optics 1.1 Introduction 1.2 Hamiltonian picture of lightray propagation 1.3 Halniltonian picture of lightray propagation: formal settings 1.4 Hamilton's equations for the lightray 1.5 Lie transformations in the optical phase space 1.6 Linear ray optics and quadratic Hamiltonian functions 1.7 Planar model of firstorder optical systems 1.8 ABCD matrix and focal, principal and nodal planes 1.9 Summary Problems References
1 3 9 19 24 30 36 44 53 53 55
2. FirstOrder Optical Systems: The RayTransfer Matrix 2.1 Introduction 2.2 Rayensemble description of light propagation 2.3 Quadratic monomials and symplectic matrices 2.4 Quadratic monomials and firstorder optical systems 2.5 Quadratic nlonomials in phase space 2.6 Summary Problems References
59 62 88 93 99 105 106 107
3. The Group of 1D FirstOrder Optical Systems 3.1 3.2 3.3 3.4
Introduction Ray matrix of composite optical systems The subgroup of free propagation and thin lens matrices Optical matrices factorized in terms of freemedium sections and thin lenses 3.5 WeiNorman representation of optical elements: LST synthesis 3.6 Rotations and squeezes in the phase plane 3.7 Iwasawa representation of optical elements: LSF ~ synthesis 3.8 Canonical and noncanonical representations of symplectic matrices 3.9 Integrating the equation for the ray transfer matrix 3.10 Summary Problems References
111 113 115 120 131 134 151 153 156 162 162 164
4. WaveOptical Picture of FirstOrder Optical Systems 4.1 Introduction 4.2 Essentials of the scalar wave model of light. The paraxial wave equation in a quadratic medium
167 169
xii
Linear Ray and Wave Optics in Phase Space
4.3 Ray and wave optics 4.4 From the rayoptical matrix to the waveoptical opera,tor 4.5 Eigenfunctions of ~ and ~: pointlike and spatial ha,rmolfiC waveforms 4.6 Spatial Fourier ret)resentation of optical wave, fields 4.7 Summary Prol)lems I/ef(;ren(:es
5. 1D FirstOrder Optical Systems: The HuygensFresnel Integral 5.1 IId,r()(lu(',ti()ll 5.2 Qlm,(lrati(: Ha lililtr a ll(1 lllr Li(', alg(',l)ra 5.3 Wav(~()I)ti(:al I I'a,IlSf(w r(4ati()xls fi)r a Il A B ( : I ) syst,tull 5.4 Tim ()I)ti(',al F()lll'ier t l'allS[t)rlll 5.5 R(~(:overing t ll(' ray()l)ti('al (h's('riI)ti()II 5.6 Waveot)ti(',al t)r()l)agatt)rs as llllitary ret)resr162 ()f linear ca Il()llical t,l'allSfl)rlllal,i()llS 5.7 Sunlmary Problems Referen(;es 6. The Wigner Distribution Function: Analytical Evaluation 6.1 Introducti(m 6.2 The optical Wigll~'~r (tisl,ril)111i()ll flnl(:tion: ])asi(: (:()ll(:~l)lS 6.3 The WigIler (listril)llti()ll f'llll(:t,i()ll: t)a,si(: t)roI)(;rti(;s 6.4 The Wigner r fllll(:t,i()ll ()f light sig2mls: flll'l]l(',r r 6.5 Summary Prot)leInS 1{eferences 7. The Wigner Distribution Function: Optical Production 7.1 Introducti(m 7.2 The slidingwimh)w Fourier transform 7.3 The Wigner distribution flm(:tion and the general (:lass of spacefrequency signal representations 7.4 The ambiguity flm(:tion 7.5 Understanding tile Wigner and ambiguity functions from the the viewpoint of the mutual intensity function 7 . 6 0 p t i c a J production of the Wigner distribution function: general considerations 7.7 Wigner processor for 1D real signals: basic configurations 7.8 Wigner processor for 1D complex signals: basic configurations 7.9 The smoothed Wigner distribution function and the crossambiguity function: optical production 7.10 Summary Problems References
174 186 194 198 214 215 216
221 224 234 242 257 261 266 267 268
271 277 282
303 333 333 335
341 343 354 358 369 379 384 394 398 400 400 403
Contents
8. 1D FirstOrder Optical Systems: Transfer Laws for the Wigner Distribution Function 8.1 Introduction 8.2 From the wave fimction to the phasespace representation 8.3 Firstorder optical systems: propaga, tion law for the Wigner distribution function 8.4 The Wigner distribution function and the optical Fourier transform: linking Fourier optics to Wigner optics 8.5 Transport equation for the Wigner distribution function 8.6 Summary Problems References 9. 1D FirstOrder Optical Systems: Moments of the Wigner Distribution Function 9.1 Introduction 9.2 Basic notions on moments 9.3 Preliminaries to the calculation of the moments of the Wigner distribution function 9.4 Wigner distribution function: local and global moments 9.5 Gaussian Wigner distribution functions: the variance matrix and its evolution 9.6 Propagation laws for the moments of the Wigner distribution function in firstorder optical systems 9.7 Higherorder moments of the Wigner distribution function 9.8 Summary Problems References
xiii
409 411 424 438 451 456 457 458
463 466 472 477 492 499 512 514 515 516
A. Lie algebras and Lie groups: basic notions
519
Index
523
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1 Hamiltonian Picture of Light Optics. FirstOrder Ray Optics
1.1
Introduction
The phase space representation of light optics naturally arises from the Hamiltonian formulation of geometrical optics. Geometrical optics gives a, simple model for light behaviour, in which the wave character of light is ignored. It is valid whenever light waves propagate through or around objects which are very large compa, red to the wavelength of the light and when we do not examine too closely what is happening in the proximity of shadows or foci. Accordingly, it does not account for diffraction, interference or polarization effects. Geometrical optics employs the concept of light ray [1], which we may give the naive view as a,n infinitesimally thin beam of light. Several formal definitions of light ray have been elaborated within both the corpuscular and wave theory to accommodate geometrical abstraction and physical observability. All definitions work well in certain situations, but in others are confronted with intrinsically physical difficulties. Thus, for instance, the corpuscular view of rays as trajectories of "luminous" corpuscles confronts with the problem that the energy density may become infinite. Likewise the wavelike view of rays as orthogonal trajectories to the phase fronts of the light wave confronts with the difficulty of individualizing a defined wave front in the twowave overlap distribution. Indeed, the ray must be thought of as a convenient and successful model which supports our perception, and hence facilitates the formal description, of a wide class of light phenomena. Geometrical optics establishes the geometrical rules governing the propagation of light rays through optical systems. The analogy of geometrical optics of light rays to Hamiltonian mechanics of material particles is well established and effectively exploited. The Hamiltonian formalism was originally developed by Hamilton for optics in his 1828 paper Theory of Systems of Rays and in subsequent papers and brief notes, published during the years from 1830 to 1837 [2.1]. In his papers, Hamilton
2
Linear Ray and Wave Optics in Phase Space
formulates the problem of s t u d y i n g the g e o m e t r y of light rays as they pass t h r o u g h optical systems in terms of welldefined relations between the local coordinates of the rays entering and emerging from the system, specified with respect to the optical axis and prot)erly chosen planes across tile axis. He shows that, if the ray coordina.tes a,rc suitably defined, the i n p u t  o u t p u t rcla; tions configure a,s symplectic tra, nsfornmtions, generated by a, function of the ray varia,t)lcs, the charact('~ri,~tic function, whose flm(:tiona.1 form is determined s()lely )y tlle ()t)ti('al I)rot)erties ()f t,tw, syste~ll. Im.t(;r, tlmlfilt(m realiz(~(1 tha.t the Sa,lll(; nlet, h()(t (:olfl(t t)(~ a,I)t)lie(1 lmt:hange(1 t() nl(;('lm,lfi(:a,1 systems, rct)la(:ing the ot)ti(:al axis t)y the ti~llC a,xis, the light rays 1)y tim t)arti(:h; traj(;(:tories and the ray(:()()r(linatcs t)y l,ll(; lll(;(:lm,ni(:a,1 I)lm,s(;sI)a(:(; va,ria,lfics [2.2]o T h e t)hase st)a(:(; rct)r('~s(;ld, a.ti()ll is a, fa.llfilia,r nlct, ll()(t wil, llill tll(; Hallfilt(mia,n fornmlation ()f ('lassi(',a,1 nwclm~fi(:s, wlfi(:ll dcscril)(;s t,tlc (lylm,llfics ()f a. Ill(;(:lm,ni(:a.1 syst,(;nl witll 'll~ (l(~.grt',cs ()f fl'(',c(l()lll ill t(;rlllS ()f '/l~ g('.l~('.ralizc(t ill(l(;I)entttmt ('()()r(tinat(',s (q~, q~, ..., (1,,,) aal(l t,l~c sa~(', ~ f i ) ( ' . r ()f ('aI~()~i(:a,lly ('()~.j~ga,t(', va,rial)lcs (p, ,p~, ...,p,,,) [3]. TI~(: l~(;(:l~a,l~i(:al l)lm,s(; st)a(:c is t,l~(; Cart(;sim~ st)a.(:c of tl~(;se 2,ttt, (:()()l'(li~m.t(,.s. F()r ('Xa,liq)lc., ill(; st, at(; ()f a fl'(',e t)a,rti(:h' at a, (:crta.in t,in~c is r(~.t)r('.s(;nt('.(t in tim. i)r()t)cr 6 D t)hasc st)a,(:(; 1)y a 't'(:pr(:,s('.'nlativ(: point, st)e(:ific(t l)y t,h('. Ca rt('.sia~ ('()()r(li~mt(;s q = (q.,,, %, q:) m~(l tl~(', r(;l(;vant m()~(',l~l,a. p = (p:,,, p,/, p:). ~l'l~(' ~()l,i(n~ ()f tl~(', t)a.rt,i(:l(; i~ r(;al st)a.(:e (:()rl'(;st)()n(is t() a, t,ra,j(~.(:t()ry in 1)lm,se sI)a(:(;. Then, l,l~(~,st, a.t,('. ()f an (ms(',~fl)l('. ()f i(l(;~d,i(:al an(t n()nint, era(:ti~g t)a,rti(:l(',s a,l, a givc~ t,ilil(', (:()rI'(',sI)()ll(ts t() a, s(;l, ()f I)()iI~t,s i~ t,l~(; 6D t)has(' st)a('(:. Tl~(' (l()~min ()('('~q)i(;(l t)y this set ()f t)()i~t,s ~()v(:s t.lm)~gh i)has(; sI)a,(:e a.s th(', I)a.rti(:l(',s ~()v('. ii~ i'(',al sl)a,(:c, tt()w(wer, a,s t,l~(', t()ta,l ~n~fi)cr ()f t)a,rti(:le.s r(',l~m,i~ (:()nsta,nt, s() will the t()tal mmfi)(;r ()f I)has(; sI)a,(:(; I)()ints. Evi(lently a, rca,1 (t(;~sity can t)(; ass()(:ia.t(;(1 with the r(;t)I'(',S(;llta,I,iv(; l)()ilits in phase st)a,(:c, a,n(t (:()rre, sI)()~(tingly a. (tistrit)~d,i()I~ flln(:ti(m ()f (l(;~sity p(q, p, t) (:a.n be (lefin(.~(1 so tha,t p(q, p, t)dV st)e('ifies the m~nfl)cr of rct)r('.S(;ld;ativ(; t)()ints in the element of v()luIne dV in ttw, vicinity ()f the t)()int (q, p). Li()uville's theorem sta,tes the. i~w~,rian('e of the (lensity of ret)resentativ(', t)()i~d,s a,l()~g I,lm trajectory of any t)()int, and a(:('or(tingly ()f th('. volmne of the I)hasc spa,('.e domain, even though its sha.pc may (:l~a,ng(; (:onsi(t(;rably (luring the motion. Likewise the geometri(:ot)tica,1 t)ha,se st)ace is the 4D C a r t e s i a n space of the ray position and m o m e n t u m coordinates (qz, qy, pz, py). However, tile phase st)aces of classical mechanics and geometrical optics are globally different. The t)a,rticle m o m e n t u m of classical mechanics is not restricted in value, whilst the ray m o m e n t u m of geometrical optics is confined within a circle determined by the local refra,ctive index t h r o u g h the inherent form of the optical Hamiltonian. In the lin(;ar a p p r o x i m a t i o n the ray m o m e n t u m is assumed to range w(;ll below its na,tural limit, which then is ignored. T h u s the geometricoptical phase space of linear optics comes to be similar to the mechanical phase space. Section 1.2 reviews the H a m i l t o n i a n formulation of geometrical optics and
Hamiltonian Picture of Light Optics. FirstOrder Ray Optics Lagrangian picture
Hamilton's principle
Fermat's principle
Solves Lagrange's equations for the
3
Particle trajectory in real space
Legendre transformation Hamiltonian picture
Solves Hamilton's equations for the
Particle trajectory in phase space
Lagrangian picture
Solves Lagrange's equations for the9 9
Ray trajectory in real space
9
.
Legendre transformation Hamiltonian picture
Solves Hamilum's equations for th~
Ray trajectory in phase space
F I G U R E 1.1. T h e F e r m a t e x t r e m a l p r i n c i p l e b a s e d f o r m u l a t i o n of g e o m e t r i c a l optics m i r r o r s t h a t of classical m e c h a n i c s , b a s e d on t h e H a m i l t o n m i n i m a l principle.
introduces the related concept of geometricoptical phase space. Section 1.3 emphasizes the symplectic nature of ray propagation, and details the suited mathematical settings (Poisson brackets and Lie operators) to approach the integration of Hamilton's equations for the light ray. In Sect. 1.4 the raytransfer operator is introduced and the relevant Lietransformation based formalism is describe(t. Illustrative examples of phasest)ace transformations are given in Sect. 1.5. Sections 1.6 and 1.7 illustrate the linear approximation to lightray propagation, which naturally yields the raytransfer matrix formalism. Finally, Sect. 1.8 clarifies the link between the raymatrix approach and the cardinal point (and planes) method.
1.2
Hamiltonian picture of lightray propagation
We will give a brief account of the Hamiltonian formulation of geometrical optics in order to fix the notations we adopt and to trace the conceptual path towards the I)hase space representation and the inherent geometry. Hamiltonian optics develops from Fermat's principle of extremal optical path, which is the optical analog of Hamilton's principle of least action (Fig. 1.1). From Hamilton's principle one can derive both the Lagrangian and Hamiltonian mechanics, related through the Legendre transformation [3]. Likewise from Fermat's principle one can develop the Lagrangian as well as the Hamiltonian formulation of optics [4]. The former yields the equations for the ray variables in real space, while the latter the equations for the ray variables in phase space. We will cursorily illustrate the basic steps leading to both
Linear Ray and Wave Optics in Phase Space
4
x
\
s
r (s) = ( x , y , : )
n (x, y,:) Z
FIGURE 1.2. Ge()Iimtri('al ()I)tics (h;scril)cs t,tle Ine(li~In by tt,e refl'm'tive i,l(lex flu,ction 'n(x, y, z) m~(l t,l,e ligld, rays 1)y l,lie 3vc(:t()r ()f f,l,,(:ti(),~s r(s) = (:r(.~), y(.s), z(.s)) ()f the arc le~gt~ .s Ineas~m;(l alo~g l,~e ray I)al,]~. t)i(:t, lu'es, a,(t(h'essilig t,li(' I'(~a,(h;r t() [4] fl)r a lii()I'e (t(~taih;(t t,r(;atili(',lit. As a , m t l m d s(:(~iiari() f()r ilitr()(lli(:illg F(;nlmt's t)rill(:it)h; [4, 5], w(; (:()llsi(ter a,n inh()nl()g(m(;()lts 111(;(lilnll, ()(:(:lH)yillg a, (:(',rl,a,ill regi()n ill 1,11(; 3 D Sl)a(:e , w h e r e w(; sllt)l)()s(; a (~,a.rl,(;sia,ll sysI,(;lli ()f (:()()r(lilml,(;s (x, y, z) )(; a ssiglle(l. 'I'11(; ()t)ti(:a,1 i)i'()t)ert, i(;s ()f tim lll(',tilull a,r(; tyI)i('a,lly (t(;s(:i'il)e(1 )y t,ll(' r(',fl'a,(:tiv(; in(tex 'n,(:r, y, z), giv('~Ii as a. s(~ala,r fllll(:t,i()II ()f sire(:(; (''). A light ray is I)r()t)aga,ting in l,h(; in(;(lilnn a,h)ng s()~ll(', I,raj(;(:t()ry. R,(;gar(h',l a,s a, line in l,ll(; 3 D st)a,(:(;, the ray (:a,,, a,(:(:oi'(tiIlgly t)e (h;s('rit)(;(1 l)y t,tl(; l)()sit, i()11 ve(;t~()r r ( s )  (.r(.s),y(,s),z(s)) for t)oint~s ()n 1,11(; ray, w i t h th(; (:()()r(lina,t,;s t)eing flm(:t,i()ns ()f l,ll(; a,I'(: length ,ll(;a,slu'e(t al()llg 1,11(; ray t)a,tll witll rest)e(:t, t() a (:hosen t)()ild, (Fig. 1.2). Fernm, t,'s t)I'ill(:it)h; (:()1111)ill(;s 1,11(;ge()nlet~ri(:a,1 an(1 t)hysi(:al a,sl)(;(:ts ()f the ray t)rot)a, gatioll t,tlr()llgh the, ('()n(:(',i)t, ()f optical path. W e recall t h a t give,l tw()l)()ild, s P, a,n(t P~ an(t a (:llrve C ('(),ln(;(:tillg t h e m , the geometrical t)a,t,tl lengttl 12 (C) from P~ t,() P2 along C is (letille(1 as the l e n g t h of C a,n(t hen(:(; is f()nna,lly given by t h e line integrM P2
11 (C) 
d,s,
(1.2.1)
1
t)erformed a l o n g C f r o m P1 tO P2; s d e n o t e s tile arc l e n g t h m e a s u r e d along t h e p a t h a n d ds = v / a x 2+ay2+dz 2 is the infinitesimal arc length. C o r r e s p o n d i n g l y , t h e optical p a t h l e n g t h s (C) a l o n g t h e ray t r a j e c t o r y C We will consider only linear spatially nondispersive isotropic media, whose refractive index fllnction is accordingly dependent on position and independent of direction. Hence we will distinguish only between homogeneous and inhomogeneous media, according to whether the scalar index function is uniform or changes from point to point within the medium. a
Hamiltonian Picture of Light Optics. FirstOrder Ray Optics
5
from P1 to P2 is defined as the line integral along C of the refractive index"
s (C) 
n(x, y, z)ds.
(1.2.2)
1
If the medium is homogeneous, so that n(x, y, z)  no, the optical path length is the geometrical path length multiplied by the refractive index: s  n 0 s Slightly correcting tile originM formulation, as given by Fermat in his (Evreus (1891) [5.1]" "Je reconnois premi~rement...la vdritd de ce principe, que la nature agit toujours par les voies les plus courtes," Fermat's principle states that, among all the possible paths C connecting the points P1 and P2, the light ray would follow the path C whose optical path length /2 (C) is an extremum. Therefore it may be a minimum, which is the most frequent case, a maximum, or it may be stationary with respect to the optical path lengths of other paths closely adjacent to C. In mathematical terms, the actual ray path C is identified as the extremal of the variational problem ~;z; (C)  ~
n(x, y, z ) d s  O,
(1.2.3)
1
where the ~ variation is intended for small deviations with respect to C of the integration path between the two fixed endpoints P1 and P2. The formal correspondence of Fermat's principle (1.2.3) with Hamilton's principle becomes apparent, once changing the integration variable from s to one of the Cartesian coor(tinates, say z. Then, evi(lencillg the (tifferential element dz in the line element ds, we may cast Fermat's principle (1.2.3) exactly in the same form as Hamilton's principle, i.e.,
(~
{~(X, y, Z)v/l+x'2+y'2}dz=O,
(1.2.4)
1
where the integrand function is naturally identified as the optical Lagrangian
L ( x , y , x ' y' z ) 
n(x y z)v/l+x'2+y '2
(1.2.5)
with
x~
dx dz '
y~
dy d~"
(1.2.6)
The Lagrangian system of equations plainly follow as
d (OL_=__:.) OL dz
~"
d OL dz (~yu
O~
)
OL
Oy
O, (1.2.7)
o,
6
Linear Ray and Wave Optics in Phase Space
and, on account of (1.2.5) and (1.2.6), can be rearranged into a, single vectorial differential equation of second order, known as the ray equation, d (ndr ds (~s)  Vn,
(1.2.8)
relating the ray tra,jcctory t() the ot)tica,1 t)rot)erties ()f the ~ne(tilml [6]. Indeed, _d~ is the tang(ult t(~ the ra,y t)a.tl~, a,n(1 V n  (~~ o,,,,o:,j o,,,,o~) h~t(;restingly, Eq. (1.2.8) shows a noti(:oat)l('~ resellfl)lall(:e t() the e(tlm.ti(m ()f (:lassi(:al rclativisti(', nle(:ha,Ili(:s F  ~dt ~ tlmt I'llh',s the (lynanfi(:s ()f a I)(filltt)a,rti('le, th(; f()rc(; F (tetenllilfi~lg the rate of va,riati()ll i1~ t i ~ e ()f the t)a.rticlc lll()llle, lltlllll p. W i t h time ret)la.(',e(t l)y t,l~e ar(: ltu~gtl~ s a.~(t the ~tm(:lm,ifi(:al t)()to, ntia,1 })y tlw, ro,fra(:tive index '~,, tl~e gra(lie~t V'n ~imy l)c ild,(;rt)r(;t,e(1 as t,l~(; "f()r(:e", a,('ti~g ahmg the gra.(ti(nd, ()f the t)(~t(ud, ial 'n(ir, y, z), a,~(t ' n dr ~ a,s the l~(n~O,l~t~ll~ (ff tim tm,I'ti(:h'Y, whose rate ()f va,ria,ti(~l 1)ci~lg (teter~lfi~m(t t)y tlie fi~r(:c V~, [7]. Tlie t,ralisiti(ni if(nlx tlie Im,gra,~lgia,ii t() the IIa,xiiilt(niia,li l)i(:tt~re is then t)urmm(t ~n(t(;r Legend(Ire tra,~sf()r~m,ti()n, whM~ takes .r' a,~l(t y' t() the ray 'm,omr p:~: m~(l p:~ t,lir()~gti d,N
~
OL P:"
~
"
.
OL
0.," '
P:'/
(1.2.9)
Oy"
an(1 a,('(:or(lingly l,tl('~ Im,grmlgiml fllll('ti()xl t() the (q)ti(:a,1 tIa,lllill,()Iliml a,s ~
9
(1.2.10)
By (1.2.5) the gelleraliz(;(l nlOIllcnta p:,. and p , ext)lit:itly write a.s (til;
.r/.
PX II(;U, ill, Z) x/l_F:~2+y/2 y' p:,j  '.(:r, y, z) v h + x ' ~ + . r
~

I t  

l t , ~
its dy ds'
(1.2.11)
in accordance with tile nl(;(:hani(:al analogy suggeste(t above,. As we see, px and py equal the local value ()f the refractive index times the direction cosines (d.~ d;s, ~ds) of the light ray relative to x and y axis, respectively. Hence, they are also termed optical direction cosines of the ray. We express the function (1.2.10) in terms of its proper variables to obtain the Hamiltonian of the ray in the form y,


y,


(1.2.12)
which is just the negative of the optical direction cosine of the ray along the z direction.
Hamiltonian Picture of Light Optics. FirstOrderRay Optics
7
Finally, in analogy with mechanics we may write Hamilton's equations for the light ray as dqx OH dqv OH dz Opx ' dz Opy _
~
_
.
_
_
dpx
OH
dpy
OH
dz
Oqx '
dz
Oqy '
(1.2.13)
thus completing tile Hamiltonian formulation of geometrical optics. We changed x + qx and y ~ qy for purpose of future use. Evidently, within the mechanical analogy the light propagation can be assimilated to a dynamical problem with two degrees of freedom, the length coordinate z being given the distinguished role as the independent (evolution) variable. We emphasize that the z direction can be arbitrarily fixed, and hence conveniently chosen to coincide with some privileged direction of the system. Ordinary optical systems are commonly designed to have a plane of symmetry, where an ideal ray can be identified, which represents tile path of a ray through tile system corresponding to adequately assigned input conditions. This ideal ray is understood as the optical axiso We restrict our attention to centered optical systems, where all of the optical components are aligned with their optical axes lying on the same stra,ight line, which then may be designa,ted as the optical axis for the overall system; the z axis may be chosen a,s this common axis. The optica,1 axis of a system of coaxial lenses, for instance, is just tile COllltilOil axis of tile lenses and can conveniently be taken as z axis. Throughout the book we refer to the optical axis as the reference z axis. Equations (1.2.13) provide tile formal answer to the basic problem of geolnetrical optics of determining the final variables of the ray after passing through an optical system, once assigned the ray variables before propagation and specified the optical properties of the system. In fact, given the va,lues of the position and direction coordinates of the ray at some zi, i.e., the initial values for the equations of motion (1.2.13), it is possible in principle to solve (1.2.13) for the ray position and direction coordinates at any other z. Following again the suggestion of Hamiltonian mechanics, we can construct the geometricaloptical phase space as the Cartesian space of the four rayvariables (qx,qy,px,py). Notably, according to (1.2.11), the refractive index n represents the normalization factor to turn the geometrical variables dqx/ds and dqv/ds into the phasespace variables px and pv. The corresponding step in mechanics is to replace velocity with momentum, the mass being the proper normalization factor. In the geometrical optical phase space the light ray is individualized by the trajectory of the point whose zevolving coordinates (q~, qy,Pz,Pv) from the specified initial values are determined by the equations of motion. A collection of rays distributed over a range of possible positions and directions
8
Linear Ray and Wave Optics in Phase Space
fills a, certain region in phase space, with which we can associate a density d i s t r i b u t i o n function p(q.~,qy,px,py, z). T h e change in the ray bundle as it p r o p a g a t e s t h r o u g h optical m e d i a reflects into the m o t i o n of the representative points as t h e y move t h r o u g h phase space in a,ccord with the equations of m o t i o n (1.2.13). While the exact phasespace m o t i o n of each representative t)oint (i.e., ca,oh ray in the t)lm(th~) is uni(tucly dctcrmin(',t by the initia,1 con(tit, i()llS, it, is r a t h e r illlt)i'a,(:t,i(:a,)le to (:a,l(:lfla,t,e a,n exa,(:t s()lllt, i()~l f()r t,tle whole 1)un(llc of rays. It is thcref()rc convenient to 1)rovide a, sta,tisti(:a,1 des(',rit)tion of the bchavioln" of tllc a.sscllfl)ly of ra,ys, rt'ga,I'dCd a,s a,ll (uls(Ulfl)le ()f i(lentica,1 syst, cnls (tiffcI'i~lg ()ver a, rmlge ()f initial (:()II(tit,i()IlS, anti a(:(:()r(tingly t() follow the cvolld, ion ()f t,h(~ (tensity (listril)llti(m flm(:ti()n p. (~c(mlctric ()I)tics is t)ri~lmrily ('o~lcer~l(;(l with the t)ehavi(nlr ()f a, sillglc ray when it, t)a,sscs ttlr()llgtl ot)ti(:a,l nmdia,. We will firstly ta,t'kle tim g(ulera,l t)rol)lcm ()f the cvolld, i()ll lln(tcr the Itallfiltoniml (1.2.12), intro(tlu:illg tlle 1)asi(: tools ()f tim Ha,miltollLie f()nlm.lislll. Tlmll we will f()t:l~s r att(ud,i()~ ()~ li~ma,r rayot)tics a.nd hen(:e on the rela,tcd ray~na, trix forma,lisn~. We will ilh~stratc the a,)ility ()f the ~m,trix n~ct,l~()(t a,s a, trm:ii~g t()()l ft)r )()tl~ a, si~gl(' I)a,raxia.1 ray a,n(t a,n (u~stu~fl)le ()f t)a,raxia,1 rays, tln~s tm.ralleling tlm sil~glt~Im,rtit'le an(t l)arti(:lt~(;nstu~flfle l)i(:t~res ()f (:la.ssit:a.1 sta,tistit:al met:tm,nit:s. I~ later (~lmt)ters, we will illustrat(~ t,l,e s~fl)sta,**tia]ly (liffeI'(u,t t()()ls of linear wavc()t)ti(:s, (lcalil,g with light (tistri)~,ti()**s i**st('a(1 ()f )~m(ll(~s ()f rays. Th(~rc we will i**tr()(l~,(:(~ tim 'waveoptical pha.s'(', ,s'pa('(:, wl~()s(', <x)()r(tina.t(~s arc (:~lst()nm.rily set a,s I,l~(; st)a,tial (:()()r(tina,tc a,n(t the Sl)a,tia,1 frc(t~u~**(:y. T h e g('.onmtl'i(: a.n(1 wavc()l)ti(:a,1 t)hasc st)a,ccs are ba,si(:a, lly (liffcr(ud,, si**(:c a, "t)()i,d," in t)ha,sc sI)a,'e, i(l(ud, itic(t )y a, (t('finit, e set, of all t)lmsest)a(:(', (:()()I'(linat(~.s, has no t)t,ysi(:al **ma,**i**gi** wave()I)tics, due to tim ~n~(:(~rta,i,d,y r(fa,ti()n ()f F()~,rier analysis. Tim wav(~ ()t)tical phase space is in a, sense " g r a n u l a r " , there being a m i n i m u m t)()ssi})lc t)hase st)ace volume asso(:ia,t(xt with a, s()~,r('c ()r a,n ()t)tica,1 system in ('()nscqueIme of the wave t)r()t)erties of light. In this Cha,t)ter we (tea,1 with tim ge(),netrica,1optical phase spa,ce. Finally we evidence a basic feature of the optical phase space, whict, globally differentiates from the mechanical pha, se space. In fact, the functional form (1.2.12) of the ot)tical H a m i l t o n i a n liinits the optical m o m e n t a accor(ting to p~ + P~ _< n ~ . Thus, the r a y  m o m e n t u m coordinates px and py arc confined within a circle, whereas the rayposition coordinates q, and % are b o t h allowed to range over all the real line. As a consequence, the geometricaloptical phase space configures as the product of a circle for the R2plane: C~ x R ~, in contrast with the mechanical phase space being the well known ~ 4 __ i~2 X I~ 2. As we will see, firstorder optics ignores the above limit on the ray inomenta; it assumes t h a t b o t h the ray coordinates and m o m e n t a r e m a i n small t h r o u g h the p r o p a g a t i o n , so that p , and p~ are well inside their natural range, whose limits can therefore be ignored.
Hamiltonian Picture of Light Optics. FirstOrder Ray Optics
9
o'! ~
!
~.~ o~*~
9"" gli i! . , . " ' 9 Input plane i ."" Ty; !! s 0~
.s
.s"
.~"
[Io
i. I
~....~ Output planei .
Yo
i
s . S"
;
,'"'"
i
i
z.
i
i
i
.1o
i
(qx~176
i i
,.'" o,~s~
I
i
9 ,,. i . . , , ' ' ~
i
~"
s" s"
i.' .." ."'~qx, ' q Yt )
l
i!
~ .S~
F I G U R E 1.3. An o p t i c a l s y s t e m a n d t h e relevant i n p u t a n d o u t p u t planes, to which t h e c o o r d i n a t e s of the i n c o m i n g a n d o u t g o i n g rays are referred.
1.3
Hamiltonian picture of lightray propagation" formal
settings
We consider a light ray propagating along an optica.1 system, which we suppose to be completely characterized by the refractive index function, given at each point before and after the system and in the system itself. The optical zaxis is commonly oriented as the direction of light propaga, tion, assumed to occur from left to right. The light rays therefore enter the system at the left and emerge from it at the right. It is natural to describe the evolution of the ray along the axis by determining a,t any a,xia,1 position the intersection point of the ray with a plane orthogonal to the axis, a,nd the optical direction cosines of the ray at that point. Thus, we fix two planes, IIi and IIo, across the optical axis, located at positions zi and Zo, respectively at the left and the right of the system (Fig. 1.3). We will refer to Hi as the in p u t plane and to IIo as the output plane. Also, we will equip each plane with a coordinate system. In the following, unless otherwise specified, rectangular coordinate systems will be taken in both the reference planes. The orientation and the position of the axes can be chosen arbitrarily; it is a bare matter of convenience to choose tile origins lying on tile optical zaxis and the coordinate axes parallel, and equally scaled, from one to the other plane. The incoming ray is unambiguously identified by the coordinates (q,~, qw) of the intersection point with the input plane IIi, and by the optical direction cosines (Px~, Py~) at that point. Explicitly, we can write Pxi  ni sin ~x~ , py~  ni sin ~y~,
(1.3.1)
Linear Ray and Wave Optics in Phase 5pace
10
qx
q"l
ray s e c t i o n
"~ i
"'"'""
.
(a)
sect i o n
~..,o
..".... ,..".....
"'"'"
(b)
F I G U R E 1.4. T h e aIlglcs (a) r~. al~(l (b) rh,, clltc.rillg tile (lcfillitioxl ()f tile ()ptical l n o m e n t a p,. an(l p.~,, a r c t h e ('ox**plc**m,d,s to 2 of tim m,glcs ()f tim r a y to tim ;r a,,(l !/;txis, r e s p e c t i v e l y .
whcr(, 'n,~ ~_ 'n,(q.,:~, q:,~,, z i) is tl~(: l()(:al va,l~t(; ()f tl~(; rcfl'a(:tiv(; i~r at the intcrsc(:tion I)oint m~(I ((Lr,, (~'U,) axe l,l~(; a.~gh;s, tl~(; i~(:()~i~g ray (lcfil~(;s with the assign(;(l t)la,n(;s yiz a l~(l .riz (Fig. 1.4). Sinfilarly, th(; ()~tl,g()i~g ray is st)c(:ificd )y the coor(li~m,tt;s (q.r,,, q.q,,) of l,~(; i~d,crsccl, i{)~ with I,l~(; t)~l,l)~t, t)la~(; IIo anti l)y I,l~(; ()I)l,i(:al (lirt;('l,i{)~ ('.{)si~cs (p.,., p~/,,) a,t l,l~is I)lmm: P.r,,   1}o S i l l ~kr,,
,
py,,
  lt o S i l l ( ~ y , ,
(1.3.2)
whcl'C '~.o ~ 'n(q,:,,, qv,,, z,,) a.~r (,~:,,,,, ~:,/,) arc tl~(; a.I~gl(;s ()f tll(: ra,y I'clativc to the t)lancs yoz a ll Po is symplectic.
Hamiltonian Picture of Light Optics. FirstOrder Ray Optics
1.3.1
11
Ray propagation as a symplectic transformation
We clarify the formal a,nd physical significance of the ray p r o p a g a t i o n as a symplectic t r a n s f o r m a t i o n [3, 9]. To this end we firstly note that, like the mechanical phase space, the optical phase space naturally configures as a 4D linear vector space, the 4vectors u being formed by the zdepending raycoordinates (qx, qy,Px,Py). Similarly it can conveniently be equipped with the euclidean metric defined as usual by the scalar product of vectors" 4
ujvj  u T  v ,
(u, v)  ~
(1.3.4)
j=l
where u n is the transpose vector and, for notational convenience, the vector c o m p o n e n t s are indexed by the integers from 1 to 4. Conforming with the c o m m o n practice, by "vectors" we mean column vectors; then, row vectors are identified by transposition, i.e., u  (qx, %,p~,py)7 _ (Ul, u2, u3, u4) r . Exploiting the vector notation, we can write H a m i l t o n ' s equations (1.2.13) in one form duj 4 OH dz = ~ ,ljl Oul' j  1, .., 4, (1.3.5) l=1
with the ai(t as well of tile suitably (tefined coefficients Jjl, which a,re allowed to be only 1,  1 or 0 according to their own indices"
,]jl

~il, j + 2 ,
j 
,]jl
1, 2;

~l,j2,
j 
3, 4.
(1.3.6)
Evidently, Jjl's ca,n conveniently be a,rra,nged into the 4 x 4 a,n t i s y m m e t r i c m a t r i x J as
J
(0I) I
0
'
(1.3.7)
where I denotes the 2 x 2 unit m a t r i x as 0 tile 2 x 2 null m a t r i x (b) . The m a t r i x J is known a,s the symplcctic unit matrix. It has the nice properties j1 _ jr_
_j,
j2 _  I ,
detJ 1,
(1.3.8)
j T denoting the transpose matrix. We suppose now to take the vector u to the vector v  (Vl, v2, Va, v4) under the t r a n s f o r m a t i o n Vj  Vj(Ul,...
, U4) ,
j

1,
.., 4 ,
(1.3.9)
b I t is a c o m m o n c o n v e n t i o n t o r e p r e s e n t 2 m • 2 m m a t r i c e s as p a r t i t i o n e d i n t o f o u r m • m s u b m a t r i c e s [8].
12
Linear Ray and Wave Optics in Phase Space
identified by the associated J a c o b i a n m a t r i x S, whose entries Sii = O v i / O u j (tet)en(t in general on uj, j = 1, .., 4. It is ca.sy to prove t h a t v obeys equa.tions dvj = dz
S:)~,,I~,~;Sha O H
,
j  1, .., 4,
(1.3.10)
Ovh
sig~ls ()v(w l"(:I)(;a,l,(;(1 i~l(ti(:(:s t)(:i~g (mfittc(t. W(; r(;(p~ir(; that th(: tra~lsf()rllmti()~l (1.3.9) h;av(;s tll(: f()n~l ()f Hanfill,()n's (;(t~ati()ns inva,rimd,, i.(:., tlmt v ()t)(:ys th(; san~(: (:(t~m,ti()IlS (1.3.5) a,s u, na,m(;ly tll( ~, s ~ n ~ m t i ( ) ~
d~,j _ .lj O H dz i)~,t '
j  1,.., 4.
(1.3.11)
Witll tlfis r(;(t~u:st al)t)li(;(1 t()(1.3.1()), w(: (;~(1 ~1t) witll tll(: ~mtrix r(:la,ti(nl: S J S m  J.
(1.3.12)
It r(~t)r(~s(~llts tll(' ('()ll(liti(nl f()r S t.() 1)(~ a syllll)l('('ti(: llmtrix a.t rely l)()illt in I)llas(: sI)a(:(; [9]. As ll()t(:(l, ill fa(:t, S lllay (l(:t)(;ll(l ill g(',21(;ral ()zi 1,11(; v(:(:t()r u. Ill ('()ld,ra,s,, tll(: I)l'()(lll('t S J S T r(:slllts lilt()tll(; lllatI'iX J, ill(l(:l)(:ll(l(:~d, ()f U. ';vi(le.~d,ly tll(: llllit llmtrix I m~(l ill(: llmtrix J a r(: sylllt)l(:(:ti(' as w(:ll. W(: r(:(:a,ll tlla.t ill g(;l~(:ral ll(nlsi~lgllla.r s(t~m,I'(: ~m.tri(:(:s fl)r~ a. gr()~ 1) ~nl(l(;r tl(: r()w(:()l~lnlll ~mtrix ~11~ltit)li(:a, ti()~l, tl~(: (:xist(:~(:(: ()f (,11(: i(l(:~til,y a n(l the ~nli(t~(: i~v(;rs(: as w(:ll as tl~(: ass()('iativity 1)(:i~g a, I)lah~ (:(n~s(:(l~uul('(: ()f th(; ~m,trix ~n~ltit)li(:al,i()~ l)r()t)(:rti(:s. I~ t)a,rti(:~la.r, l,l~(: ~la.tri(:(:s ()l)(;yi~g (1.3.12) (1() f()rnl a. gr()~q)(1~(; t() tl~(: (:l()m~r('. I)r()t)crty ()f tl~(', sy~q)le,(:ti(: (:(n~(li,i()ll, whi(:h ix I)rcs(;rv(:(l )y ~ml,rix ~n~ltiI)li('al,i()~; i~ flu:t, if S l a~(t S~ ())(;y (1.3.12), s() will th(; I)r()(t~mt S~ S~. W(: i~vit(: tll(: rca,(lcr t() v(:rify that, if S is sy~q)h;('tic, the ii~v(:rsc S  1 exists a,n(t is syn~t)h;(:tic a,s well a,s the tra,nst)()s(: S T. Ttl(: group of the 4 x 4 sy~I)l(:(:ti(: r(',al ~mt, i'i('(;s ix (t(;~()t(;(t a.s S p ( 4 , R). Since (let J = 1 m~(t (tot S = (t('.t S T, t)y (1.3.12) it follows t h a t (let S = +1.
(1.3.13)
It can be shown that actually foI" a, symplectic m a t r i x it is always (let S = +1 [.().2]. Remarkably, in tile 2 x 2 case tile necessary and sufficient condition for a m a t r i x to be symplcctic is t h a t it has d e t e r m i n a n t +1. Consequent to the symplectic condition (1.3.12), we also find t h a t the 16 entries of the m a t r i x S are not all independent; only 10 can be arbitrarily chosen, the others being fixed by (1.3.12). This reflects the general p r o p e r t y of the 2m x 2 m symplectic matrices, for which only m ( 2 m + 1) of the 4 m 2 elements can arbitrarily be assigned ( P r o b l e m 1).
Hamiltonian Picture of Light Optics. FirstOrder Ray Optics
13
Interestingly the Jacobian matrix S relates the infinitesimal vectors dv and du (see also Problem 2). Indeed, being 4
dvi  ~
OVi
~
4
duj  E Sij duj,
j=l
(1.3.14)
j=l
we can concisely write dv  S du.
(1.3.15)
It is therefore a direct consequence of the symplectic condition (1.3.12) that the quantity ( d U l , J d u 2), for any choice of du 1 and du2, remains unchanged under the symplectic transformation (1.3.15), which takes u to v and accordingly dul to d v 1 and du 2 to dv 2. Explicitly, we can easily prove that
(1.3.16)
( d v 1 , Jdv2) = ( d U l , Jdu2).
In particular, if the new and old vectors, v and u, arc linked by a linear transformation as vi  ~ = 1 aijuj, i  1, .., 4, the entries of S are just the uindependent coefficients of the transformation, i.e., S/j = aij. Hence, relations (1.3.15) and (1.3.16) hold true for the finite vectors as well. Explicitly, if
v=Su,
(1.3.17)
with S symplcctic and uindependent, then (V 1 , J v 2 ) = ( U l ,
(1.3.18)
Ju2).
It is evident that if the transformation which takes the vector u to v is zdependent: Vj   V j ( U l , ..., U4; Z), j = 1, .., 4, (1.3.19) the simple derivation we have sketched for the symplectic condition (1.3.12) no longer holds. But, indeed, this is just the case of our main interest. In fact, as wc said earlier, the propagation of light rays through optical systems is formally described by the solution to Hamilton's equations (1.3.5), which, for assigned input conditions, yield the ray variables, and so the relevant phase space vector u(z), at each z along the optical axis through the axial interval we are interested in, say from zi to Zo. Therefore, what we must verify is that the ray propagation from zi to Zo, we may picture as a transformation of the initial rayvector u(zi) to the final rayvector U(Zo):
u(zg)
, U(Zo) = u(z~ + Az),
Az =
Zo 
ruled by Hamilton's equations (1.3.5), be symplectic.
z~,
(1.3.20)
14
Linear Ray and Wave Optics in Phase 5pace
To prove this, it is enough to demonstrate that any infinitesimal transformation ahmg the path (1.3.20) is symplectic, i.e., the relevant Ja.cobian matrix obeys relation (1.3.12). The continuous evolution of the ray vector u(z) along the finite space interval Az can be viewed as a succession of infinitesima,1 symt)lecti(: transformations, which, in virtue of the pro(hlct t)reserva.tion t)r()t)crty menti(nic(t a,t)ovc, result into a. single finite symt)lc(:ti(: transformatioil. Theli, let ~lS (:o~si(l('x a,n iI~finit(;si~m,1 tra,nsf()r~m,ti()~ a,long the (;v()h~tion t)a,tl~ (1.3.20) l,aki~g th(; i~fitial ray v(;(:t()r u(z,~) at zs t()tl~(; ray vc(:t()r u(zi + d z ) at the illfinit(;si~mlly (:l()sc axia,1 t)()siti(m zi + dz. F()r (:()nv(;nien(:(;, w(: set w
u( i) vu(zi+dz)w+
~du I:, dz
(1.3.21)
wh(;rc, ()f (:()In's(;, ()nly tirst()r(lcr tcrnls in the (;l('~nlclltal (list)la.(:('.nl(;nt dz have })(;(;n r('.taill('~(l. In a(:(:()r(l witll Ha,lllilt(m's (;(tlmti()llS (1.3.5), ()ll('~ ll~LS duj dz
,ljl
0ti
(1.3.22) Zi
wlficll, (hi(:(; ills('rt,(',(t illt,() tll(' s(;(:()~l(l ()f (1.3.2(}), yi('l(ls OH
dz.
(1.3.23)
~. "7
Tll(,~ ,Ja,(:()])ia,]l llmtrix (:(rams t() t)(;
Ot,j Ou~j 0 2H Sjl ..... O'u'! ..... O'u'! _of ,]jl,: OulOUl,: where the se(:ond or(tot derivatives of tile Hamiltonian flmction, H[~. 
(1.3.24)
o,,,ouk z~
which fornl tim s()(:a,lh;(1 Hcssia,n matrix H "  (H[~)~k, a,rc cva,lua,tcd a,t u(zi) and zi. Dlu; to the asslHncd interchangeability of the derivative order with respect to the vector components in the Hessiai~ entries: Hi'~  H ~ , relation (1.3.24) (:an finally be cast in the matrix form S

I + JH"(w,zi)dz.
(1.3.25)
In order to assess the symplecticity of S we must take the product S J S qwhich, consistently to first order, is easily proved to be: S J S T  (I + J H " ( w , z i ) d z ) J ( I H"(w,zi)Jdz) = J + J H " ( w , z i ) J d z  J H " ( w , z i ) J d z  J.
(1.3.26)
Hamiltonian Picture of Light Optics. FirstOrder Ray Optics
15
The symplecticity of the infinitesimal transformation mapping the initial ray vector u(z~) to the infinitesima,lly displaced vector u(z~ + dz) along the finite evolution path (1.3.20) governed by Hamilton's equation (1.3.5) is then demonstrated. Accordingly, we may think of the ray propagation as a continuous sequence of symplectic transformations, under which the ray vectors in phase space arc taken one to the other, the axial coordinate z being the stepping parameter. We may say that optical systems operate symplectic transformations on the ray variables q and p [10]. It is worth emphasizing that the foregoing discussion, we have developed with specific reference to the light ray propagation, frames within the general issue of Hamiltonian mechanics [3], stating that any canonical transformation (i.e., such to map phasespace vectors obeying equations of motion of Hamiltonian form into phasespace vectors obeying as well equations of motion of Hamiltonian form) is symplcctic, whether it depends or not on the evolution variable inherent in the dynamical problem under examination. Readers interested in a rigorous treatment of symplectic groups are directed to [9]. Additional comments are given in Chapters 2 and 3. The Appendix provides an elementary exposition of the basic notions pertaining to Lie algebras and Lie groups.
1.3.2
Poisson brackets and Lie operators
Central to our approach to the geometrical optics problem of deterinining the transformation (1.3.3) of the lightray coordinates, generated by tile Hamiltonian (1.2.10) through Eqs. (1.2.13), are the strictly related concepts of Poisson bracket and Lie operator. Poisson brackets and Lie operators represent the formal tools to introduce the Hamiltonian equations of motion into the appropriate Lie algebraic ambience, where their intrinsic symmetry can more clearly be evidenced and effectively exploited. The definitions and main properties of both the Poisson brackets and Lie operators are recalled below. The discussion is developed at a general level. Thus, (q, p) denotes a set of canonically conjugate variables for some HaIniltonian system and z the indepeildent variable, independently of their specific meaning in ray optics.
Poisson brackets Let f and 9 be any two continuously differentiable functions of the canonical variables (q,p). The Poisson bracket of f and g, denoted by {f, g}, is still function of (q,p), formed according to [3, 10.2, 11]
{f' g} =~~ OL
Of Og Oq~ Op~
Of Og "~ Op,~ Oq~ J '
(1.3.27)
16
Linear Ray and Wave Optics in Phase 5pace
where tile sum runs over all tile pairs of conjugate variables (q~,p~) for the problem at hand. Since both the functions f and g may depend on the independent variable z, the Poisson bracket {f, g} may depend on z as well. Clearly the Poisson bra('ket ot)era,tion creates a binary relation: (f, 9) {f, g} of every pair of dyim,mi(:al varia.bles to one dynamical variable, identified thro~@l the rule (1.3.27). In parti(:~flar, the relatioi~s
{ q,,, p/~ } = (5~,~;~,
(1.3.28)
(:hara(:terize the (:m~()ni(:a.lly (:(m.j~ga.te wtriat)les (q,~, p,,), as the f()ll()wing { q"' "f } 
i) f
i~'p,~ '
{P"' "f } 
O.f
Oq,, '
(1.3.29)
esta.t)lish tll(; rlll(~s I,() tin'ill l,lle P()iss(m t)ra(:kets ()f aaly tm.ir ()f (:(nljllga.tc varia.1)h's (q,,,p,,) witl~ a fi~t:l,i(n~ .f ()f (q,p). In t)a,rtit:~llar, if .f is ta.ke~ as the tIa,llfilto~ia,~ fl~xmti(m H, tl~c l)(fisso~ t)ra,(:kets (1.3.29) rcI)r()(l~(:e tt~e right lm,~(t si(les of Ha,~ilt,()~'s ('.(l~m,t,i()~s, wl~i('t~ a,(:(:()I'(li~gly write also a.s
dq,,
dz = { q'' ' I I } ,
dp,, dz = { p ' ' ' H } '
(1.3.30)
f~)r ('very t)air ()f (:()Iljllgat(' varial)l(;s (q,,, p,,). Evi(h;ntly the P()iss(m t)ra.(:k(;t ()f .f and th(, v(,(:torva.llw(l fll~l(:t,it)n g (.q, (q,p), ..,g .... (q,p))Y is tile ve(:t()r fllll(:(,i()II { f , g } = ({f,.q, }, .., {f,.q,,, })T The Poiss(m )raI,i~' flWl~mlis~, {les(:ril){'~l ix~ t,hc Im'viCnts se(:ti~ms, alh~ws ~s to i(lc~tify a tm,tl~ t{~ ild,(:gra.t,(, I I m ~ i l t ~ ' s ~;~t~m.ti{n~s fin" tl~{', light ray t{~ the corrcst)ol~(li~g syl~l)le(:l,i(: l,ral~sf~)r~m,t, iol~ i~ tl~(; gt'~()~cl, ri(:a,l()t)l, ica,1 t)]m,sc st)a,ce: II
Ell
Poisson bracket operation
...... Exponentiation
~'(z''zi)Lll .
(1.5.1)
T h e r~'s~ll, il~g t)lm.s{: st)a,c~, l,rm~sf{)r~na.l,i()~: A
1[][(Zi)
by,, ( . . . . i)L//'
U(Zo)
( '(:z')'~ll U(Z.i),
(1.5.2)
l"Cfl(~,(:,S [,llO, ew~llll,i(~ll ~f tilt; ray a l()llg l,ll(', ~l)l,i(:al zaxis. It, is (;vi~l~;ld, l,lm,t, as excllq)lili(',~l ill (1.5.1) fi)r ~t)l,i(:a,1 Ita,l~dlt,(~nialls, we (:an g~',ll~',ral,(; 1)lms~'Sl)aC~' l,ra,]lsf()rxl~ml,i()]ls, lmvillg ~r ll~t, a (tir(~,('t, lillk with the Zev~)lllt,i~nl ~)f tll~' ligllt, ray. We llmy ~:~)l~si~l~'x" m~y ~n~t,ianu)usly ~liff~'rcntial)Ic, I)tias~;slm.~:~' varia,l~l~; .[(q,p) a,~t t,lxe~ f~n'~ t,l~e ~:~n'rest)~n~ling Lie l,rm>for~na,t,ion, wl~s(', "wdi(lity" as a I)has(',spa,(:(; l,ra,nsfi)r~m,l,i~)~ is (;sl,at)lisll(;(l a(:(:or(ling 1,() wl~(;l,l~er il, ('~wre('l, ly ~m,1)s l,l~(; g(;(~el,ri(>()I)l,it'a,1 l)t~a,s(;st)a,('(; i~d,(~ itself. We recall i~ flu:t, tim,l, tim ()l)tica.1 I)tm,se st)a,ce lm.s t,l~(', l)(W~flia.r 672 • IR2 st, ru('talre, in C()lll,rasl, wil, l~ t,l~: IR4 str~u:t~rc ~)f l,l~e n~e(:ha,ni~:a,1 l)hasc st)a('e. Wc will giw; ,SOl~e cxm~t)lcs of Lic tra,x~sf(wl~m.t,i~n> in the ot)t, ical t)hasc st)a(:(',, h~ l)a,rl,i(:~lar, i~ w 1.,5.1 an(t 1.5.2 w(; will (:tmsidcr a,s I)lm,scst)a.ce fim(:ti~n~s .f(q,p) lixma.r a n~t q u a d r a t i c t)olynomiMs in q and p. In tha.t case, we cm~ ~;a.sily ~))tail~ tl~c explicit expressions of the tra,nsformed plmsc space veer(w, si~l)ly ~six~g l,l~e ext)onentia,1 series exI)a,nsio~ (1.4.5). T h e n in w 1.5.3 we will ])rielty ~les(:ril)e the factorizationba,se(t mcth{)d widely used to deal with the Lie transfor~nations generated by higherorder polynomials. To simplify ~otatioI~s we will consider the single pair of conjugate variables (q, p), and hence we will a,ccount for the proper I  n , n] x IR s t r u c t u r e of the relevant 2 D phasespa.ce. We will denote by ( the real p a r a m e t e r of .... the expo
nentiation from the Lie o p e r a t o r L, to the Lie t r a n s f o r m a t i o n eCLf. We will not detail the physical m e a n i n g of ~, whict~ depends, of course, on t h a t of the varia,ble f(q, p) in order t h a t the exponent, ~ k, be dimensionless. Thus, if f ( q , p) can be interpreted as an optical Hamiltonian, ~ may acquire the meaning a,s the evolution variable, and accordingly the relevant Lie transformation that Ks t,he raytransfer operator.
Hamiltonian Picture of Light Optics. FirstOrder Ray Optics
25
qT ("
;
t,,,, I .,"i p.
:: ................. "
r Z
(a)
(b)
F I G U R E 1.5. As the tip of the r a y  m o m e n t u m vector n = (p, pz) is confined on the circle of radius n, (a) the " m o m e n t u m translation" is a rotation of the vector, with the tip slipping along the circle, which a m o u n t s to (b) a variation of the direction of the ray with respect to the qz reference axes.
1.5.1
L i n e a r p o l y n o m i a l s in q a n d p
The momentum variable p generates a shift of the position coordinate of the phasespace vector, without affecting the relative momentuni coordinate. In fact, being (1.5.3)
kp = Oq '
the pair (q,p) is turned into
( ) q( 0) or virtual ( f < 0), the sign of f allows a characterization of the system as a converging or diverging device. When f > 0, rays diverging from F are imaged into parallel rays and correspondingly parallel rays are imaged into rays converging to F'. Conversely, when f < 0, the system turns rays converging to F into parallel rays and parallel rays into rays diverging from F'. Also, on account of (1.8.12) and (1.8.13), the lowerleft entry C of the ray inatrix silnply relates to f as 1
C 
P.
(1.8.17)
f Remarkably, if the locations of the focal and principal points (or planes) relative to the assigned input and output planes are known, the behaviour of the system with respect, to these planes is completely determined, being the relevant ray matrix known by (1.8.8), (1.8.10)and (1.8.11). Thus, tile system can effectively be modeled by its two focal and principal planes. In tile next section, we discuss a characterization of tile optical systems, for which the focal or principa.1 planes are a,ssumed as reference planes. Let us establish now the condition under which an incident ray with slope ~9i  p i / rt01 at Hi emerges from Ho with tile same slope ~9o  P o / no 2  tgi. According to the expression (1.8.3) for the raymomentum magnification Adp, it is evident that the input ray coordinates must satisfy the relation .A4p  D + C qi Pi
n~ .
(1.8.18)
~Ol
Thereby, any incident ray, whose real (virtual) part originates at (is directed toward) the point N, located in the object space at
SN 
~'~01 ( ~'~'02  D)
C

fl (f2
__ D)
fl
S01
(1.8.10)
'
is transformed into a ray with the same slope" tgo  ~9i, whose real (virtual) part is directed toward (originates at) the point N', located at
SNt in the image space.
•02 ( •01 C Tb02
A)

L(7
flA) J2
,
(1.8.20)
Linear Ray and Wave Optics in Phase Space
50
T h e points N and N' are called nodal points and accordingly the nodal planes II N and II~, are tile planes t h r o u g h N an(t N' n o r m a l to tile axis. It is clear t h a t N and N' are conjugate points. If the m e d i u m on b o t h sides of the s y s t e m is tile same, the no(tal t)oints (:oinci(te with the t)rincit)al t)oints: N  H an(t N'  H'. In general, wh(m 'n ol r 'n02,, the two set of t)()ints are set)arat(;, a,lth(n@l th(: (tista,n(:(; t)etw(;(m the no(ta.1 t)oints is exa,ctly ('qual to l,l~(; (tista.n(:(; l)ctw(;('n the t)ri~:(:it)a,1 t)()i~ts: NN' = Ittt'. In a,(t(titi()n, tl~(', fl~rther r(;la.ti(n~s: F N = F'tt' a,~(1 F ' N ' = FIt t)('tw('~(;~ tl:(: f()(:al a,~(t ~()(tal I)()i~:ts hol(t. ()tl~(;r (:a,r(li~ml I)()i~l,s a,r(', (l(:ti~(',(l wl~i(:l~ a,r(; ()f s(n~(; int(;r(;sl,, as th(', 'negative ~:a'~'di'n,al a.n(1 nodal point,s, r(,st)(;(:tiv(;ly (:ha.ra,(:t(:riz(;(1 t)y a, ~l~il, ~(;gal,iv(; lat(:ral m~(1 m~g~fla,r ~mg~fifi(:a,l,i()x~ (s(;(; t)r()t)l(;l~ 11). Finally, t() fln'th(;r (',vi(h;l~(',e tl~(; a,na,h)g~V 1)(:l,w(;(:n the (:hara,(:l,(;riza.ti()n ()f th(; ilnagi:~g t)r()l)(;rti(;s ()f m~ ()t)l,i(:al sysl,(;n~ t)y tl~(', (:a.r(li~ml t)()ild,S (a.l~(l t)la,:~(;s) a.~i(t tlm.t t)y tl~(; ra.y l~mtrix, w(; (l(;(l~u:(; fl'()~ (1.7.1(i) l,l~(; relati()l~
[
'll,() l
11) ] + [ ,
'll,() 2
IA ]
2
~,
_ (7,
(1.8.21)
It(), 2
wl~(;r(; a,gai~ s~ m~(l s~ give l,lm axial ('()()r(ti~mt('s ()f th('~ t)()il:ts P~ m~(t P2 a,t whi(:l~ tl~(; i~('i(l('~l, m~(! r(;fi'a,'l,('(l rays i~l,('rs('('l, l,l~(; ()l)ti('al axis r(;sl)(,('l,ively in tl~(' ()l).i(;(:t m~(l i~a.g(; Sl)a.(:(;. ()~ a.('(:(n~t ()f (l.8.11) an(l (1.8.15), tl~(; al)()v(; relal,i()x~ (:m~ I)(, (:ast i~ 1,11(; fi)r~ ()f l,t~(; well k~()w~ thin lens fo'~"m,'u,la +


:~, ;v~ .f' with 741a,n(1 7~ (t(;ll()tillg tll(; axial (:()(n'(lilm,tt:s ()f tll(', t)()illts P l a,:l(t I),,. r(:fi;:'en(:ed t,) tile I)rinr l)lmms II~ a ll(t II,,,. By a, filrtlmr ~Im.lfitnflati()ll, (1.8.21) (:a,xl t)(; giv(;ll the Nc'wtonian form (1.8.23) or nlore c(nl(:is(~ly ,~, ~2  f, L ,
(1.8.24)
where the axia,1 c o o r d i n a t e s '~1 and '~2 of the points P1 and P2 are referenced to the focal planes II~, and IIv, , respectively. 1.8.4
Special Optical Systems
T h e ray m a t r i x takes specific forms w h e n the focal or the principal planes are t a k e n as input or o u t p u t planes [16, 17]. In fact, w h e n tile principal planes IIH, IIii , are t a k e n as input a n d o u t p u t planes tile front and p r i m a r y focal points as well as the back a n d s e c o n d a r y focal points coincide: s~ = fl and s~, = f2T h e ray m a t r i x from the p r i m a r y to the s e c o n d a r y principal plane is therefore
Hamiltonian Picture of Light Optics. FirstOrder Ray Optics Hi
l[o
I
Ilo
[]i
Optica[ System
qi I
51
Optica[ System
B=O
D=0
qi I r
(b)
(a) VIo
/@p
Hi
1lo
Optical System
Optica[ System
A=0
C=0 r
FIGURE
1.10. O p t i c a l s y s t e m s h a v i n g one v a n i s h i n g e n t r y in their ray t r a n s f e r m a t r i x . (a) s y s t e m s ( D  0), (c) focusing s y s t e m s (A  0)
imaging s y s t e m s (B  0), (b) collimating a n d (d) telescopic s y s t e m s ( C  0).
M. ,.,
(1 0) C
1
'
(1.8.25)
where by (1.8.17) C =  1 / f . The form of MH_+H, evidences the property of the principal planes to be conjugate, points of II H being imaged into points of IIH,. In particular, (1.8.25) produces a unit positive image magnification for any input ray. In fact, the principal planes are also defined as conjugate planes for which Adq = 1, and accordingly called unit planes. Matrix (1.8.25) is a particular form of the most general optical matrix Mimagi"g  
(A 0) C
1/A
'
(1.8.26)
which, having B = 0, relates conjugate planes. In fact, as qo = Aqi, the position of the image ray depends only on the position of the object ray; thus, all the rays through the point at qi on IIi are brought to the point at qo on IIo (Fig. 1.10.a)). Evidently, AJq = A. Optical systems described by matrices having B = 0 are called imaging systems (or, magnifiers). Special systems are also obtained when the focal planes are taken as reference planes. In particular, the ray matrix for the propagation from the front
52
Linear Ray and Wave Optics in Phase Space
foca,1 pla,ne to a, given o u t p u t plane comes to have D = 0, n a m e l y Mc~

C
0
(1.8.27)
'
T h e sign of C =  1 / f establishes if the object a,nd image distributions are real ( f > 0) or virt~m,1 (.[ < 0). ()ptica,1 systems, (tescrit)e(t by the ra~ m a t r i x (1.8.27), into,go a, t)()int int() a tnn~(tlc ()f tm,rallcl ra.ys (Fig. 1.1().1))) sim'e Po (Tqi f()r all Pi. 'I'tl(;y are r(;f(,rr(,(l 1,() as (:olli'mo, ti~t,fl syste~lls. C(nlvt'~rst;ly, (,lit' t)r()l)a.ga,t,i(nl fl'(nll a, givtul illt)ll(, t)l~nl(; 1,()(,ll('~ lm,(:k f()(:a.1 t)la,nc is (:ha,ra,(:tcrizc(t l)y A = (), a,ll(t ht;n(:(; t)y th(; ray l~m,trix Mr'"""~"~ =
C
D
"
(1.8.28)
Syst(:l~s (1.8.28) i~mg(: a t)~(ll(, ()f I)arall(:l rays i~t()~,. I)()i~t (Fig. 1.1().(:)) as qo =  P i / ( 7 f()r all qi. Tll(:y at(: (:alh:(l .fiw'tt.si,~,yl sysl,(:l~S. As w(: will s(:(: la,t(:i', tt~(: (:()1~(li(,i()l~ A = () is 1,1~(: r(:(l~isil,(: f()r (,1~(: ()l)ti(:a.1 sysl,(u~ (,() l)(:rf()n~ tt~(: i'mlw.'~fc zi. For ea,(:h z > z,i w(; (:all (',wdlla,t(; 1,11('~(l('~llsity p(P; Z) at P, lll()villg witll it, a h)llt4 its own t)ath. W('~ till(t tlmt it (l()(;s 11()1,(:lla.~lg(;, i.e., p(P; z) = p(I'; z~) V z > zi. T h e nl()ti()ll ()f tll('~ ra,y('~lls('nfl)l(,~ r('l)l'(;s('~ld,ativ('~ 1)()ild,s ill t)lms(' sl)a(:(; is 1,11(~11lik(; ttm,t ()f a,I1 ill('(nllI)r(;ssit)l('~ tllli(l ill l'cal Slm(:e. Ill ()tiler w()r(ls, p(q, p; z) t)clmves like a "t)r()t)(;rty" ()f th(; t)()illl, in t)llas(; sI)a,(:(;, ml(t a,(:(:()I'(tiligly it, is tra,nst)oI'tt;d t)y the t)()ilit, a,l(nig its ()w~l traje(:t()ry wll(;~l it, is r(;t)r(;se~lta,tiv(; ()f a ligtit ray. Tl~c i~va,riml(:(' of 1,11(;1)llascst)a,('.(; (lc~lsity p(q, p; z) inlt)lics 1,11(;i~ma,l'ian(:(; of the t)ha,s(;st)a(:(; v()l~tnle 12 ()f 1,11(;regi()ll (m(:lt)sillg tim rayl)~ultlle r(,.t)i'escnta,tiv(; t)()ints. In fi~.(:t, s~t)t)()st; t() s(;l('(:t at a (:(;rtah~ z = zi a ('(;rt, ai~ ~nmfl)(;r ()f ret)rescntative I)oix~ts aroml(t a (:hose~ t)()int,, througtl wlli(:ll tllc (l(;~lsity takes ()I1 t h e COllStallt va,l~lc Pi. Let Ni l)(; the volcanic cn(:l()sing the (:h()s(;n t)oints. T h e tota,1 Inlnfl)(;r ()f t)oi~lts in Ni evi(lt;~ltly is Ni = piN,:.
(2.2.14)
Following the points as t h e y move t h r o u g h t)ha,scspacc, we see tha,t the boundaries of the enclosing volmne change with z > zi, but by definition the total n u m b e r of points contained inside it does not va,ry: N(z) = Ni. Also, the density of points keeps on the value p ( z ) = Pi by virtue of (2.2.10). Thus, for any z > zi we can write =
which by comparison with (2.2.14) immediately leads to V(z) = V i .
(2.2.16)
The volume enclosing a fixed n u m b e r of representative points in phase space remains constant in z, although, due to the motion of the enclosed points,
1D FirstOrder Optical Systems: The RayTransfer Matrix
69
its shape may sensibly change. So, the total phasespace vohlme 12 of tile beam remains constant in z, regardless of the motion of each ray and the corresponding motion of the representative points through phase space: 12o V(z) = lPi (Fig. 2.3). This result is rather intuitive if we think of a flowing "fluid". If tile total quantity of fluid and the density of the fluid do not change through the motion, the volume occupied by the fluid remains constant as well, even though it may deform significantly during the fluid's flow. The invariance of the phase space volume applies obviously also to the volume elements dip 's: d12o = d~)i along any ray path. In w 2.4 we will elaborate a geometrical interpretation of the volume element dV and its invariance along the ray path in connection with the conceptual definition of the intuitive perception of the elementary bundle of rays. Finally, we note that the ray transformation from the phasespace point P~(q~,p~) to the point Po(qo, po) is accompanied by a transformation of the phasespace volume elements from dl)i to dl2o (b). We known that if S denotes the Jacobian matrix associated with the transformation P i + P o (see w 1.3.1), the relevant volumes dl)i and dl)o are linked by the determinant of S as dl2o det S dlPi. The invariance of the volume elements under ray propagation implies therefore that det S = 1, thus solving between the two possibilities offered by Eq. (1.3.13) for the determinant of the symplectic matrix associated with the ray transfer. Viceversa, the unimodularity of the symplectic Jacobian matrix associated with the ray transformation implies the invariance of the phase space volume. This offers a clear geometrical interpretatioil of the syInplectic nature of ray propagation, signifying that elementary volumes along the ray path in phasespace remain constant. Evidently, areas play the role as volumes within a 1D picture, we are mainly concerned with (see w 1.7.4). 2.2.2
Basics of Lie transformations
As noted in w 1.3.2, given a dynamical variable f(q, p) we can generate the Lie operator k s by Poisson bracket operation" kf  {, f},
(2.2.17)
and the Lie transformation TI by computing the power series" t
cx~ ~j Aj
Ts (~)  ~
~. Ls
_
_
e~f
~
~ e R,
(2.2.18)
j=O
In the present discussion the concept of volume in phase space has been left on the same intuitive footing as t h a t in physical space. Actually, phase space is a m a t h e m a t i c a l space in which indeed the infinitesimal volume is defined in t e r m s of the linear a n t i s y m m e t r i c nondegenerate 2nform, and finite volume are accordingly obtained from it by integration over the domain of interest. For an accurate analysis the reader is directed to [3.3].
Linear Ray and Wave Optics in Phase Space
70
A
where ~ is a real p a r a m e t e r and the j  t h power of Lf is obviously Aj L/ 
{{{..{,f}..},
f}, f}.
(2.2.19)
j times
As evidenced )y the cxponc~I~tia,1 nota,tion on tile rightha,nd side of (2.2.18), the power series is n o t h i n g trot tile cxt)()~lcnl, ia,1 flm('.tion of tile ot)crator ( k f . Evi(h;,~tly, 'Y/(())is t,h('~ i(]cld, il,y ()t)('~ra,t()l': "Y/(())  I [6]. Lie tra,nsf()rmatio~ls ha,v(; ninny i~lI,(;r(;sting a,n(] ~s(;fl~l t)r()l)crl, i(;s, whi(:h dire(:t, ly follow K()~ t,t~(; t)rot)eI'ties of t,l~c Li(; ()t)cra, t,ors a,~(l t11(; cxt)()~m1~tiation. We shall ll()I, give a. (tct, aih,(t a.t:(:()ll~d, ()f a,ll tim t)r()t)(;rt, i(;s ()f 1,1~(;Li('~ tra,lisforlna,tio~s, for wlfi(:t~ tim r('a(h,r is (lirc(:t(;(l t,() [(;], t)~I, slm.ll ( ' ( ) ~ ( , ~ t , ill Im,rti(:ula,r ()~l I,h()s(; l,hal, arc ~s(;fifl l,() ga.iIl insight int() t,tl(; })(;tmvi()~r ()f l,h('~ l)lms(;st)a,ce (hqlsity fl(q,p; z) tl~r()~gl~ E(t. (2.2.13). Wc ('l~t)lm,sizc tim, I, I,l~(; ~ml, l~('~m,tica.1 setti~g }1()I'(~ is ~()l, t)r()I)erly (:()~t)h;t(;; (t~mst, i()l~s r(;h;va,~d,, f()r i~sl, m~(:('~, t() tile a lm,lyti(:ity ()r t,lm ra,(Ii~s ()f (:()llV(;l'~(;ll(:O ()f l,ll(; t)()wcr s(;I'io,s (;Xl)m~si()I~S ()r of ()t)era.l,()r i(h;ntiti('s, we will ]~(' (:()l~(x,rll(;(l wit, l~, will n()t 1)(; aI)t)roa.(:tm(t. 77w. trl'od'm~t tnY;.se'l"tTatimt, tn'ope'l'ty We t)r()w; tile pt'od'tt,:t p'l','.sr
tio'u p'rope'l'ty, wllic:ll t,lu'()ltgll l,ll(; r(;la,ti()ll
~: zi, evolving into p(q, p; z) = gi(q  q(z))rS(p  p ( z ) ) .
(2.2.70)
Here, q(z) and p(z) are the ray variables at z propagated from the input data (q~, p~) through the transfer operator 9)l (z, z~) or, in the linear model, through the transfer matrix M(z, z~) (see also Problem 4). Accordingly, throughout the remaining of the chapter we will work within the singleray picture of linear geometrical optics. In due account of the established significance of the raytransfer matrix, in the next sections we will proceed deriving the ray matrices for assigned zindependent model Hamiltonians. Their correspondence to recognizable optical systems will then be investigated. Before closing this section we wish to illustrate the relation of the phase space density p(q,p; z) to those measurable quantities that the traditional radiometry uses to characterize the light distribution and its propagation in connection with the radiative properties of the source. A
80
2.2.4
Linear Ray and Wave Optics in Phase Space Phase space density and radiance
As a n n o u n c e d , in this pa, ragra, ph we elucidate tile connecting link between the ba,sic q u a n t i t y of the phase spa,(:(; represent~tion of gcometrica,1 optics, i.e., tile t)hase st)a,('e density p(q,p; z), a,n(t the basic q u a n t i t y of radiometry, i.e., tim radian(:e (~). T h e (tisr is strictly aimc(t a,t giving a, feeling for the t)hysi(:a,1 con(:rcteness of the t)ha,sc spa(:c density. Accordingly we merely derive the f()rnm,1 relati()~l of p t() tim ra,(timlcc fll~l(:t,ir a(l(lressing tim tea(let to [9] for a fllller a(:(:()llld, ()f ])()tllt, tl(; ra(tio~lletri(" lll()(l(;1 a,ll(l tll(; relatively re('.cntly inv(;stiga,te(t lillk wit, tl tilt, statisti(:al wave; I,tl(;(~ry. W(; ('lllI)tlasiz(; tim.t, a,s g(;()Ill(;tri(:al ()t)ti(:s, ra(tit)lllt;tI'y a n(t the r(;la.tc(t theory ()f ra.(lia.tive (ulergy l,ra,llSl)t)rt are )a,se(l ()ll 1,11(' (:(nlt:el)t ()f t,lm light ray, lm(t(;rst()()(t a,s tll(' gt;(nll('tri(:al ,l'aj(;(ti,()ry a l(nlg wlli(:ll t,ll(' ()l)ti('al t;llt;I'gy, ra(tia,te(t )y tile s(nlr(:e, I)rt)lm,gates [l()]. Tlnls, ar t() I,lm ra(li()metl'i(: nl()(tel, tlle rar (;lmrgy is l()('alize(I )()t,l~ i~ sI)a('(; a~(l i~ (lir(;('ti()~. W(; will s('(; tlmt the rar is (h;fil~(;(l as a fim('ti(m ()f )()t,l~ l)()sil,i()l~ a,l~(l (lire(:ti()~. P()sit, i()n a,i~(t r162 a,r(; tl~(; ln~il(li~g Wtl'ial)lt;s ()f tl~(, g(;()~('tri(:al()I)tir I)lm,s(' sI)a,(:e, whi(:h tht;n s(;(;~s i,()()f[t;r 1,1~(;lmt~ral fi'an~(; ft)r tim ra.(litn~ml,ri(' th(:()ry a.s well. I{a,r t~as I)(;liev(;(l I,()t)r()t)(;l'ly (h;st'ril)(; l,l~e I)el~avi()r ()f ra, (). Also, it, vanish(:s a.(, l)()ints q i~ (,h(: s()~n'c(: l)la,n(: l y i n g ()~It, si(l(: (,I~(: s()~r(:(: a,r(:a, S, i.e., /3(q, v )  ()
if q ~ S,
(2.2.73)
f()r a,ll tll(: t)()ssit)l(: (lir(:('tillr(:(: surfa,(:(: S. Evi(t(:n(,ly, (,t1(: tota.1 t)()w(:r ra,(tiate(l )y t h e s o u r c e is
p
co.. 0 s
v).S,
74)
e Oil account of the spectral content of the emitted radiation, it is custoinary to consider the contributions to the radiated power for each frequency component separately. The symbol dP~ [Wrim1] is typically used with the meaning as the limiting vahle of the ratio AP/ACe for freqlxency interval Ace being made infinitesimally small; A P is clearly intended as tile amount of radiant power contained within Aw around the specific frequency co. Needless to say, dP~ obeys the law (2.2.71) in terms of the spectral radiance B ~ ( q , p ) [W. m 2 sr1 . riml]. We will ignore here these frequencyrelated aspects of the radiometric model; accordingly we will not consider the possible frequencydependence of the refractive index.
1D FirstOrder Optical Systems: The RayTransfer Matrix
83
the integrations extending over the source area S and over the 27rsolid angle formed by all the directions v pointing into the halfspace z > 0. The radiance as a phasespace f u n c t i o n
Being a function of both position in space and direction from that position, the brightness comes naturally to be a kind of phase space function. Evidently the pair (q, v) identifies a ray emerging from the source point Ps (q) into the v direction (Fig. 2.4.b)), the ray m o m e n t u m p being obtained just scaling the unit vector v by the refractive index value n o of the medium. Thus the radiance can as well be understood as a flmction of the proper rayvariables (q,p): B(q, v) ~ , B ( q , p ) . Resorting to the phasespace picture of light distributions in terms of the density of rays in phase space, it is evident that the radiating surface S corresponds to a phasespace density function, we may denote as Ps (q, P) Likewise the sample ray through the source point Ps(q) extending in the v direction corresponds to the point (q,n0v) in phase space. Then, the rays emitted by the surface element dS a,t Ps within the solid angle df~ around v individualize the element dl; of volume in phase space, which encloses the sample ray. According to the geometricoptical picture of uncorrelated rays, giving independent energy contributions, we can say that ea,ch ray radiating from the surfa, ce c~I'ries a certa,in amount of power independently of the other emitted rays. Evidently, dP is the power carried by the phasespace volume dI;, and so it is also obtain~tble as dP = (5Pps (q, p)dl2,
(2.2.75)
Here (SP roughly denotes the average power carried by each ra,y radia,ted by the surface S. The product ~PPs(q,P) acquires the meaning as the optical power density in phase space of the light field across the ra,dia,ting surface. It is worth noting that the linear addition of power represented by the integral (2.2.74) for the total power is consistent with the view of incoherent energy contributions emerging from all elements of the source into a.ll directions, thus implying that there are no interference effects present. We consider in some detail the phasespace volume d12 = dqxdqydpxdpy, which we may rewrite as d)2 = d S d p x d p y , (2.2.76) in terms of the area element dS = dqxdqy enclosing the chosen point Ps (q.~, %) in the source plane. Recalling from Eq. (1.3.1) the expressions of the optical momenta pz, py: Px = no sin ctx,
dpy = n o sin C~y = n o cos c~x sin ~)y,
(2.2.77)
Linear Ray and Wave Optics in Phase Space
84
qx
Z
FI(;[JI/E 2.5. (h~(n,~ctry of tl}c miglcs rclcva~t to Eqs. (2.2.77). wher(~ 1,1}(~. g(~.()l~el, ry ()f l,ll(~ i~v()lve(l a.l~gl(~s is (:la,rifi('~(l ill Fig. 2.5, tim (lilli;renrials (tp.,, a,l~(l dp,.q wri(,(~ as
dp.r Ac(:()r(li~gly,

'It() ('()S

py
(t.rd(t.,:,

' I t o (:(),'; ( l / : r
(:()S
l)y S~l(:('(:ssiv(: sl,(;l)S w(: (:~l.ll (:lal)()ral,(:
d F  I~)dS ('()s (~.,: (:()s,)u
.,,
f()r d F t,11(; (;xl)r(:ssi()]~
')
.
'l)yd't):q.
.
'2
(2.2.79) a.ft(:r I'(:a,lizi]lg l, lm.t (:()s~.,, (:()s'0 u  (:()s 0 ml(t t i m (:h:lll(:]d,a.l s()li(l migl(: a,r()lm(t
(tt~
I,h(: v  ( l i r ( : ( ' l , i ( n l
is
E(t]m.ti()ll
(2.2.79)
= (:()s
(~:,:d(~:,,(t'Oy.
(:stal)lish(:s
a.ll illl,(:r(:sl, i n g l i n k
t)(;l,w(;(:n I,ll(; I)ms(;st)a,(:(:
(t~ il,ll(t S ( ) l l l O g(~()lll(q,ri(',a,1 (tlla.lll, il,i(~,s ill t)hysi(:a.1 Sl)a(:e, lik(' ~I,I'(~IIN ~1,I1( s()li(t a,ngl(~s. Inscrti~g (2.2.79) i~lt() (2.2.75) a,n(t (:()]~t)a,ring with (2.2.71), w(~ a,r(; (;md)hxl to relat(~ t,tle ra,(tianc(~ B(q, p) t() tile ()I)ti(:al t)()w(~r (t(nlsity i1~ l)lmsc space; t)reciscly, we (:an write v()llllll( ~,
/3(q, p) 
'n~bPps
(q, p).
(2.2.s0)
The dei)en(tent:e, on z of t)oth B~ and Ps above ca,n ta,citly be asslmled beca,use, as noted, the radia, ting s~lrfa('c may be a,rbitrarily located in space. E q u a t i o n (2.2.80) provides the key to turn from the real spa,('e representation to the pha,se space picture. A mca,sura,ble qua,ntity associated with the optical field in real space, i.e., the distribution of radiant energy at a given plane z, relates to the corresponding density of rays in phase space, thus reflecting as well the intuitive view t h a t the ray density in phase space reproduces the spatial and angula,r energy distribution of the optical beam. In this connection, the concept of emittance domain in phase space arises quite naturally as the region t h r o u g h the representative phasespace volume of the ray
1D FirstOrder Optical Systems: The Ray Transfer Matrix
8,5
bundle, which encloses most of the beam energy, say 86% or 99% according to tile adopted criterion. By Liouville's theorein, this region, or more specifically its boundary, can loosely be taken as a representative region, and accordingly as a representative contour, to provide through its shape evolution a pictorial view in phase space of the propagation of the ray bundle in real space. Remarkably, by (2.2.80) tile invariance of the phase space density Ps along the ray trajectory implies the invariane of the quantity ~ along the ray path ( /
in a homogeneous lossless and gainless medium (see also Problem 7); this is a well established result of traditional radiometry [9.1][9.3]. As already noted, relation (2.2.80) applies to the classical context of geometrical optics and spatially incoherent sources. Under these conditions, in fact, we can loosely define an energy density distribution in phase space, and interpret it in terms of the radiance of the classical theory of radiometry. In the modern theory various expressions for the radiance have been elaborated, which account for coherent and partially coherent sources, being linearly expressed in terms of various secondorder correlation functions of the optical distribution across the source. However, they cannot be interpreted the same as the radiance of the classical theory, i.e., as the radiated power distribution in position and direction. Likewise, waveoptical analogs of the phase space density have been defined in order to combine in a single view, comprising both position and direction, tile two waveoptical description of light signals, separately pertaining to the spatial and spatial frequency (i.e., angular) domains. As an example of such phasespace (i.e., spacefrequency) distributions, in Chapter 6 we will consider the Wigner distribution function. We will see that it cannot strictly be interpreted as an energy density distribution in phase space, even though a direct relation with a generalized radiance function can be established, thus resembling the classical relation (2.2.80). For completeness' sake, we report as well the definitions of other two measurable quantities, which are obtained separately integrating the brightness 13 over the source area S or over the 27rsolid angle formed by all the possible raydirections p. Thus, we obtain the radiant intensity J ( p ) [ W  s r 1] in a given direction by the integral J ( p )  cos 0 fs B(q, p)dS,
(2.2.81)
which specifies the angular distribution of the power radiated by the source. Correspondingly the irradiance g(q) [ W  m  2 ] , specifying the radiated power by the unit area of the source at any given point q, results from the integral g(q)  f J2
B(q, p) cos Odft.
(2.2.82)
71"
As the radiance, the radiant intensity and the irradiance are typically referred to a specific interval of frequency. Also, as the radiance, they were
Linear Ray and Wave Optics in Phase Space
86
origina,lly defined for r a t h e r incoherent thermal sources and later generalized to sources of any sta,te of coherence. T h e generalization preserves the forreal expressions (2.2.81) and (2.2.82) tn~t in terms of the generalized radiance flm(:tion. Thus, as the ra.diance, the generalized irra,(tia, n(:e does not t)reserve the I)hysi(:a,1 meaning a,s the (:orrest)on(tiI~g (p~a,ntity of tra,ditiona,1 ra(tiometry. In contrast, the generalized radiant intensity retains its t)hysi(:al n~ca,ning a,s r(;t)r(;s(;~t,i~g t,t~(; a.x~glflar (tistri)~ti()~ r tl~c ra,(tia,tr162 t)()wcr [:).:), :).~0, .~.4].
Ele'me'ntar;q beam of '~'ays Before; r
tllis t)a,ra,gl'a,t)ll wc 1)ricfty r162
()ll tile ra,(ti(mlctric r
of
elcm, e,'ntar!l beam of light. ExI)l(fitiIlg S~Xll~; ge~)xlletri(:a,1 a,rglnllcnts a,ll~l rela,ted inwl,riml(:(:s, it is l)r t,~ giw~ a,ll "~)I)~ra,tiw~" (lefilfiti~nl ~f tl~e at)stra,(:t ('(m('(;I)t ()f ray, (:(n~r i~g t,lw, ~m,tlw,~m,tical i(lca, ()f t,t~('~ ray as a tl~i~ t)cncil r rar witt~ va.~dshing s~li~t a,ngh; m~l ('r(~sssc(:ti~m,1 area., a.nr lm~(:c not (:at)a.1)h; ()f (:a,rryi~g a ti~it,(; a,~(ntl~t ()f c~crgy, a,n(l 1,1~(; l)l~ysi('a.1 ~(;('~(1 f()r an "clarity" t'a,l)a/)h;, ill C()lltI'ast, ()f ca,rryi~g a. fi~fite a,ll(1 n~c.a.s~md)lc,, CVel~ tl~ough i~di~fit(;si~m.lly s~all, a~l~(nl~d, ()f c~mrgy. Rest)rti~g t() t,l~(; t)a.sir g(',()Ill(',l,I'y ()f Fig. 2.4.1)), w(: r(;(:()l~si(h:r t,l~(: (I)r fictit, i(n~s) ('l('l~('l~t,a.l s~rfi~.('t' dS~ ('cl~t,('~r(;(i ar(nn~r l)()i~t, P ~ m~(t tl~('~ light ray ('~liicrgilig fl'()lii l ) iill,() l,ll(; (lil'(;(:t,i()li v (Fig. 2.6) I~ct (iS 2 l)(; ,lit' (I)()ssi)ly fi(:titi(nls) ch;nl('~ltal sln'fi~,('c, tha,t, r a,I'()lnl(l l)()i~lt P2 at (lista,n(:r d fl'()In I ) al()~lg v, r162 a,ll tllr r a y s ()rigi~m,t,ixig fl'()~li t)()i~lts ()f dS l The clt)a.~a.ti()ll, wliose ~;fI'(;(:t in the t)hase l)lml~' is l)i(:t, lu'er as a shorn" ~)f the initial ~list,z'it)ld, i()ll a,l~)llg t,ll(; qaxis, as will sI)e('ifir 1)(: (;vi(h;ll('e(t ill !i 2.5. Ill g(;llcz'a,1, wc will s('~(; t,lla.t, (]allssia,n (listril)llt,i()]ls l'eilla,ill (]allssiaxi lnl(h'~r l)rr t]n'()llgli a,lly [irst()r(tcr ot)tical syst,(:lli, i.(;., syst,(;~is t,]la.t at(' (h'sli ()f the b e a m at the waist" A l  7rw~, and a(x:ordingly a.s soli(t angle t~ ttm t s u b t e n d e d at the waist by the (:ross section of the b e a m at some z far fr()m the waistf~  7rw2(z)/z 2  A2/Trw~o. T h e I)roduct A~ t~ is then the zindet)cndent value
A1~'~1
2.3
 ,~2.
(2.2.89)
Quadratic monomials and symplectic matrices
As briefly recalled in the i n t r o d u c t o r y notes, the optical Hamiltonian, suited to the 1D picture of geometrical linear optics, takes the q u a d r a t i c form (w 1.7)
H(q, p) _ 2__~0 2 p12 + _~q2,
(2.3.1)
1D FirstOrder Optical Systems: The Ray Transfer Matr&
Quadratic monomiats:
A
Matrix representatives:
N
p2/2, q2/2, qp/2
K_, K + , K3 ~ ,V~(2, P, )
> 0). Scllellmt,ic of tim plm.se l)lmm trmlsforIimt,i()lls of (a) tlm reI)rt,'scld,ative point of a sillgle ray, ml(l of tim rr seld, ative (listributio~s of a I)~(lle of rays witl~ (b) angular extent only, (c) Sl)atial ext,o~t o~fly a.~M (d) m~g~flar m~(l Sl)atial cxt,c~ts. tri)lltcd over r l) 0. T h e r e p r e s e n t a t i v e point Pi = (qi,Pi) of the incoming ray slips along the relevant h y p e r b o l a b r a n c h to tile r e p r e s e n t a t i v e point Po  (mqi, ~m ) of the o u t g o i n g ray.
1D FirstOrderOptical Systems: The RayTransfer Matrix
. !~ "~~
103
~"x input
Pi
"\
.,"
.s'po
~9
. ~..
".
~..
~"'~",~ I
q
q
output
q p = q, P,
(a)
FIGURE 2.12. The scaling matrix in the phase plane. The transformations experienced by (a) a single ray and (b) a spatially and angularly extended bundle of rays. In Fig. 2.12.b) we show the transformation acted by (2.4.13) on the phase plane rectangular region, representing a bundle of rays uniformly distributed over position and direction through the respective ranges Aqi and Api.The spatial and angular extents of the beam are changed to Aqo = m A q i and Apo = A p i / m . So, it is possible to reduce the spatial extent of the bundle but only producing a spread in the directions of the rays in the bundle. The behaviour of the system (2.4.13) gives signs, in geometrical optics, of the "uncertainty" principle. Mutuing the wording from the q u a n t u m mechanical context, we may say that the pure magnifier (2.4.13) produces a squeeze (g) in the phase plane contracting the raybundle representative area, in one direction and enlarging it in the orthogonal direction. Finally, we consider the effects of the raytransformations (2.4.4), (2.4.8) and (2.4.13) on the phaseplane Gaussian distributions of the type (2.2.84). Specifically, we show the deformations experienced by the relative contours of constant density, which are clearly exemplified by the ellipses q2
p2 +
= r
(2.5.16)
for assigned values of the positive number c. The value of c determines the fraction of the total energy enclosed within the relevant elliptical contour; thus, for instance, the contour corresponding to c = 2 comprises about 86.5% of the optical energy of the beam. Thus, in Figs. 2.13.b) and 2.13.c) we m a y respectively see the q and pshears of the phaseplane contours (being circular 9 The term squeeze is largely used in quantum mechanics to indicate that some of the quantum fluctuations are squeezed out of one observable and into the second noncommuting observable.
104
Linear Ray and Wave Optics in Phase Space
FIGURE 2.13. ~iYmlsforlnatiolls of the plla.scl)lalle contollrs of tile (]mlssia, l raydensity 2~,,,~,, (' ' ' ( a , ) ' I ' t ~ e circ~lar cont(),~rs of the initial ~listrib~tio~ are ( b ) qsl~eared by the propagation thrcJtlgh a ho~nogeIleolls medilml, ( c ) psheared by t.lm refraction at a twometlia iIlterface, an~l (c) sqlleezetl ill ttle p tlirection by a scaling trm~sfi)r~n.
a,t, the int)111, plane) r162 t() t,lu', t)rot)a, ga,ti()n thr o.
(3.3.17)
The Group of the 1D FirstOrder Optical Systems
119
.... < ........................................................ < ............................................!
__2,!
d>0
no
,
~o
!
i
!
1.
d 1
]
n
,
0
]
i
9
I 13o
i.f !A ill
.f
.1~ A!
A ii
If!
!
I
i
d 1). Tim ot)ti(:al (:()~lfigurati()Ils t)i(:tllrc(l ill Figs. 3.9 ml(t 3.10 nmy tlmll 1)e lnl(lcrst()()(t as tll(~ ()t)ti(:al mlal()gs ()f I)llysi(:a,1 I)r()('esses, (les(:ril)e(1 t)y 2 x 2 sylllI)l(~(:t,i(: t)ln'ely (tiag()na,1 llm.tri(:es with ll(~ga,tive all(l I)()sitiv(~ (~t,ries, r(~sI)(~(:tiv(~ly. Also, as a, t)()sitiv(; ~a,g~ifi(w ('i.1,111)(~,a,rra,~g'(~(t })y (:a,s(:a~li~g tw() n(;gat,iv(~ l~mgnifying systems, f(n~r I)()sitive F(n~rier traaM'()rnfing (:()nfig~mtti()ns (:a,n )e (:()n(:a,t(;na,t(;(t to sy~thesize the ()I)ti(:a,1 mm, l()g ()f m~y I)l~ysi(:a,l I)r()('ess (tes(:ri)e(l t)y a, 2 x 2 synq)le('ti(: (tiag(nm,1 I)()sitive ~na,trix. Finally, we n()te tha,t the (:()nfig~mtti()n (3.4.16) (:a,n als() t)e (mla,rge(t to (omprisc a,n a,(t(titi()~m,1 thin le~s thus lca,(li~g t() tim thr(~(~leI~S sct~t) (3.4.22)
D(.A4)  L ( . f , ~ ) T ( d , ~ ) L ( f ) T ( d I )L(.fl ).
We invite the reader to find out the explicit expressions linking the involved t)arameters d I , d 2, fl, f2 and f. Also, wc suggest to prove the rcalizability of the (:onfigura,tioIl with fl  di, f2  d2, 7I _ ~1 + ~ 1 a,n(t accordingly A/I
_12
which is interpreted as the sequence of two Fourier tubes.
The subgroup o f pure m a g n i f i e r s
Tile ideal magnifiers form a proper subgroup of the set of 1D linear optical systems. This straightforwardly follows from the properties of the unimodular diagonal matrices, we have denoted by D ( 3 / ) . In fact, as already shown, the p r o d u c t of two matrices in the set { D ( 3 d ) , Ad real} belongs to the set as weli, a,nd hence just rewriting (3.4.18) we have D(.A41)D(.M2)  D(.A/[2)D(.M1)  D(Ad),
~4

.A/~IM
2 .
(3.4.23)
The Group of the 1D FirstOrder Optical Systems
129
The inverse of a pure magnifier is the magnifier with reciprocal magnification: [D(.s
1  D ( . / ~  I ) ,
(3.4.24)
and the unit matrix I can be interpreted as the optical matrix of the magnifier with A/l = 1: D(1) = I. (3.4.25) We note that, according to (3.3.7) and (3.3.8), the unit matrix can be optically synthesized by cascading two, positive and "negative", free propagation sections or two thin lenses having opposite focal lenghts. Due to (3.4.25), the identity matrix can also be implemented by cascading two positive or two negative pure magnifiers with A42 = M1I ' and hence it can be realized as two, positive and negative, Fourier tubes nested one into the other or as four positive Fourier transforming systems concatenated one to the other with the relative focal distances being related by flf3 = f~ f4.
Pure magnifiers without inversion Positive ideal magnifiers constitute a proper subgroup of the group of unimodular purely diagonal matrices {D(AJ), ,k4 real}, because cascading two positive magnifiers produces a positive magnifier. In w2.4.3 we have elaborated for the pure magnifiers without inversion the exponential representation, rewritten in (3.1.2), involving the traceless matrix K3, representative of the mixed quadratic monomial lpq. As we will see later, under specific restrictions the scale matrix S(m) plays a crucial role as a building "block" for the representation of optical matrices in a factored product form involving as well the T and L matrices. Therefore, we further coinInent here on the system S(m), suggesting in particular some possible optical realizations of it. We may rephrase relatively to the scaling matrix S(m)
_
m1 ) ~
?It _
r
(3.4.26)
the formal considerations developed in w 3.3 in respect to the free propagation section and thin lens inatrices, T(d) and L ( f ) , thus completing the view of the three one parameter subgroups of @(2, IR), generated by the algebra basis {K_, K+, K3} by exponentiation through a real parameter. Evidently, Slike matrices constitute a proper subgroup of Sp(2, IR); more precisely, the set {S(m), m > 0} is included into the subgroup {D(A/I), A/I real} of unimodular diagonal matrices D(A/I). Then, as particular cases of (3.4.23) and (3.4.24), we can state the closure property S(T~I)S(~2) = 8(~2)8(~1)
= 8(77~),
m  m l m 2 > 0
(3.4.27)
130
Linear Ray and Wave Optics in Phase Space
for ma,trices (3.4.26), and hence the existence and unicity of the inverse being
[S(I~,.)] 1  S(lrt 1).
(3.4.28)
In a,ddition, the previously developed considerations concerning the optical realiza,tion of puremagnification devices with ,A/I > 0 apt)ly to the system (3.4.26) as well. Thus, tim symplectic diagonal ma,tric(',s, generated by the a,lg('])ra, nmt, rix Ka, (:an ()t)ti(:a,lly t)e realiz(;(t )y (:()n(:a,t(',~m,ting tw() systems like (3.4.16)()r two Fouri(',r tra,~sformers (3.4.13), the relewml, fo(:a,1 l(;ngt, hs )eing related t() th(; nm.gnifi('a,ti()~ '~, a,s
.f~ = ,,,,,.f,.
(3.4.29)
An (;xa,~q)le is giv(;~ i~ Fig. 3.10. Also, we ('m~ (:as(:a,(l(; tw() ~mgat,iv(; ~m,gnifiers t() sy~tl~(',siz(', tim ~m,t,rix (3.4.26), tim resull, i~g (:(mfig~n'a,ti()~ nmy t)e regar(te(t a.s ()t)ti(:a.1 a,I~al()gs ()f tl~(', ~a.trix ~,,sKa, tim, i~v()lv(;(t f()(:al l(;~gt,l~s fl, f2, .f:~, .f~ 1)(;ing s~ita)ly (:l~()s('a~ t() satisfy tim r(;la,ti(m
f.e.f4  (,,~/2.f, f,a. 3.4.4
(3.4.30)
Sympler:tic mat's'ices as optical mat's'ices
At)t)r{)t)ria.te (:()ll.jllr {)f T, L a.ll{t Dlike ilm.l,rir {:a,Ii 1)e a.rrmlged to be eqlliva,lent t() ally giv(',ll sylllI)lectic llmtrix, (;x(:llutiIlg tim a.ld,iclia,g()lm,1 nm,t,ri(:es, for whi(:h, a,s (lis(:llsse(1 ill .~ 3.4.2, the 2.f sysl, ellls a.ll(t t,lm F(nn'ier tut)es t)rovide the slfit, a,t)le ()l)l,i(:al rea,liza,ti(ms. In tiffs (:()llll('~(:l,i()ll, il, is w()rt,h emphasizing once a,ga.iil tim (litf'erence, relatively t{) tim gr{nlI) I)r()perty, of t,he matrices T ' s a.(:(:or(ting to whether they are rega,r{te(l as In~r('ly ~m,thematical entities or interI)reted a.s fi)rnm.1 l,ools for n~o{telling ot)tir syst, e~ns. Ma,thematica,lly, the 2 x 2 ~nin~o{lular real upper triangular ma,trir (3.3.1) do form a, subgroup of tl~c synq)lcctic grout) Sp(2, I~). In co~trast, {)ptically ttmy represent ra,yt)roI)aga,ti(m througl~ I)ortions of a, l~(m~ogeimous nm(timn, a,~(t hence do not form a. subgroup because the inverse of a, fr(;epropagation section cannot be synthesized a,s a freet)ropagation section as well; it should demand in fact for sections of "negatiw~" length and thereby is not physically realizable. In the following we rega,rd T(d) just as the upper triangula,r matrix ot)tically synthcsizabh; by portions of a homogeneous medium or by more complex arrangements of lenses separated by uniform medium sections according to whether the parameter d takes on a positive or negative value. It is easily proved t h a t every symplectic matrix with non vanishing upperleft entry can be realized as the ray matrix of an optical system, composed by the sequence of a free propagation, a pure magnifier and a thin lens, namely
(Ac#0 DB)  L ( f ) D ( . M ) T ( d )
(3.4.31)
The Group of the 1D FirstOrder Optical Systems
131
with the parameters d, 3/[ and f being related to the entries of the matrix as
d
_B A ' MA
'
f=
A C"
(3.4.32)
Needless to say, the optical arrangement on the right of (3.4.31) is basically composed by thin lenses and free propagation sections. Interestingly, since D represents an optical system with vanishing effective length and focal power, the presence of the matrices T and L in the synthesis (3.4.31) directly relates to the entries B and C, respectively. Thus, if B = 0 or C = 0, correspondingly T or L disappears from (3.4.31). If A = 0, and hence D ~ 0, the reader ma,y easily verify that the reverse ordering TDL may be effectively applied:
,
1.
(3.4.33)
However, following the suggestion put forward in w 3.4.2 we can verify the feasibility of the optical equivalence
(~ D C1 )  F (  ~
)T(~),
(3.4.34)
which unlike (3.4.33) coInprises tile case D = 0 as well. Oil account of tile optical realization of F ( f ) by a 2 f system, it is evident that the product in (3.4.34) describes tile propagation from an input plane, placed at distance d from the primary focal plane of the lens, to the secondary focal plane. The presence of T ( d ) reveals in fact that system (3.4.34) performs the imperfect Fourier transform of tile input signal, and so ma.rks tile difference with respect to a perfect Fourier transforming system, corresponding to d = 0, i.e. D = 0. Equation (3.4.31) and all others, obtainable changing the order of the optical components in the product configuration on the right, represent the conclusive step of the investigation presented in this section. We may accordingly state that every 2 x 2 symplectic matrix can arise as an optical matrix; in other words, every Sp(2, R) system can be realized a,s an optical system composed by a suitably arranged and characterized set of thin lenses separated by (positive) free propagation sections. This establishes the homomorphism between the symplectic group Sp(2, R) and the group of the 1D linea,r optical systems.
3.5
WeiNorman representation of optical elements: LST synthesis
In the previous section it has been shown that every real symplectic matrix with non vanishing diagonal is expressible in a product form involving the matrices T, L and D. In general, the matrix T is to be considered from a purely
132
Linear Ray and Wave Optics in Phase 5pace
mathematical viewpoint as a real unimodular u p p e r  t r i a n g u l a r matrix, admitting the exponential reprcsenta, tion in t e r m s of the algebra, m a t r i x K , and hen(:e describing from the optical viewpoint a single freepropagation section or a suitable a,rrangement of freeine(tilm~ sections and thin lenses. Here we will see that, resorting to the well known W e i  N o r m a n t h e o r e m of Lie group theory [4.1], sy~t)le(:ti(: nmtrices having a positive A or D entry a,d~fit a,n ordered t)ro(tu(:t forn~ rel)rese~tati()~ in t e n ~ s of T, L a,n(t Slike ~lm,tI'i('(;s; Im,~ll(;ly i~l I,('r~ls ()f a l~t)t)(;r trimitL~la,r ~lm,trix witli ~mit (tia,g(mal entl'i(;s, a, ~lninl()(l~fla,r t)()sitiv('(h'finitc (lia,g()nal matrix, ml(l a h)wer triangular IIm,ti'ix with uliit (lia,g()lial (;~ltI'ies. In fa,('t, tll('~ ()t)ti(:a,1 a,rI'a,llgelllellt (~ B)
_
L(7))S(.,,,.)T(d )
(3.5.1)
(h;~xm,~(ls for tll(; I)ara,~(;t(;rs P , 't~, m~(1 d 1)(; rela,te(l t() tl~(; ~m,tl'ix eh;l~ents t)y 7'
_c. ,
',,,
A ,
,t
B_.4'
(;3.5.2)
.j~st a,s (3.4.32) witl~ M rel)la(:(;(l t)y 'ttt. As 't~t > (), i(h;~d,ity (3.5.1) is a,(l~fissit)h; ()nly f()r llmtri('.es witll t)()sitive (liagoilal (;IlI,ry A, wllih; D is ill general a,lh~w('~(l t() rmlg~'~ fr(~lll C~ t,(~ OO. Tll(; rea,(l(;r iimy w;rify tlmt th(' fa,('.t()riza,t,i(~ils wllere t,tl('~ limt, rix L is (~ll tll(; l(;ft ()f tll('~ xlmtrix T (i.e., t,llc I~ST, SLT a,n(1 LTS (:(mfigllra,ti(ms) are sllital)le t,() syntllesize sYnll)h;(:ti(: xlm,tri(:(;s having A > () a,~(t  ~ < I ) < c~. I~ (:~)ld,ra,st, t,tle fa,(:t()riza,ti~)~ls wll('xo, tlm nm,trix L is (m th(; right (~f T (wtli('ll lea,(l t,(~ t tlo TSL, STI~ a~l(t TI~S ('(~Ilfigl~rati(~s) (:a,n a,(:(:()~nd, for D > () a,x~(l A a,rlfitra,rily ra,~gi~g t,l~r(~gl~ t,l~(; real line. T h e eq~fiva,h;~t:e (;xt)I't,,ss(:(t i~l (3.5.1) a,s w(;ll a,s in all tl~e ()tl~er fa,(:t()I'e(t forms, obta,ina,)h; chm~gi~g tl~e ()rtter r the (:omt)onent ~m,t,ri(:es in the t)r()(l~lct on the right, sta,tes the I)ossil)ility of representing every Sp(2, I~) element with t)ositive ut)t)erh;fl, or lowerright entry a,s the or(lere(1 1)ro(tuct of elements (trawn from the on(~t)a,ra,~neter subgrout)s generated )y the corresponding a,lgel)ra, ma,tri('es K , K+, and K 3. Thus, on ac('ou~t of the ext)onential ret)resentatioI~s of the ma,tri(:es involved in the t)ro(hlct, we (:a,n write
with 7~, s and d defined t)y (3.5.2); namely, 7~ cA, s   2 1 n A , d  ~ . Accordingly, every 2 x 2 symplectic matrix with positive upperleft or lowerright entry can be interpreted as an optical matrix arising from the appropriate conjuctions of (positive or negative) freepropagations, thin lenses and recipro(:al s(:aling systems. W i t h i n tile (:ontext of 1D linear optics tile decoInposition (3.5.3) acquires an utmost relevance when a p p r o a c h i n g p r o p a g a t i o n problems. In that case the initial values of the ray matrix entries are set in order to reproduce the identity matrix. The continuity of the process assures that in a
The Group of the 1D FirstOrder Optical Systems
133
Hi i
r
no I
i
i i
!
I
i
I
0
ADB('=I
i
i
Q.
I I
i
0
i
""
d freepropagation
m
section
positive magnifier
f
thin lens
(positive or negative) .m
>
O JE
ADBC=I lowertriangular
El. :~
unitdiagonal matrix
LO
unitdiagonal matrix
(5'p(2,R) parabolic subgroup)
O~
uppertriangular
(,,;p(2,R)hyperbolic subgroup) unitdiagonal matrix (5;p(2,R) parabolic subgroup)
y,  ~9
( (,
q
$ A~.~'I t~ t"
........ /  q
q
] qshear
overall scaling
pshear
F I G U R E 3.11. W e i  N o r m a n synthesis of an optical m a t r i x .
neighbourhood of the initial position, i.e. in a neighbourhood of the identity, the entries A and D remain greater than zero. The representation (3.5.3) is therefore "locally" allowed in that neighbourhood, where we can write M(z, zi)  c p(z)K+ cs(z)Kac d(z)K ,
M(zi, zi)  I,
(3.5.4)
thus decomposing the overall problem of determining the transfer matrix M(z, zi) into the "partial" problems of determining the parameters 7)(z), s(z) and d(z). Notably the factors in (3.5.4) directly relate to the "dynamics" associated respectively with the quadratic monomials l q2 , lqp and ~l p 2 , which can easily be investigated and possibly expressed through definite functional forms of the relevant characteristic parameters P(z), s(z) and d(z) (see w167 1.5.2 and 2.4). As an example, when considering the propagation in a parabolic index profile medium, we will see that the differential equations for the matrix ele
Linear Ray and Wave Optics in Phase Space
134
ments A, B, C, D turn into differential equations for the parameters 7), s, d of the decomposition (3.5.4), which, offering an alternative parametrization of the problem, may provide a deeper feeling for the behaviour of the system. Figure 3.11 sumnm,rizes the vari(ms i n t e r p r e t a t i o n s of the LST de(:omposition (3.5.1) within the grmlt) theorcti(:a,1 context as well as within the optica,1 context in terms of b o t h optical system a,rrangeme~lts a,Ii(t t)haset)la,Im transf()rnm,ti()n sequen(:es. M()reover, as in tile lilmar a t)I)r()xinmtion tim ray tA,ra,nsfer m a t r i x M is the Iim,i,rix r('4)n'st'lltativ('~ ()f t,tl('~ ray transfer ()t)cra,t()r 9)l, relat, i(m (3.5.4) a.lso sta,tes t]lat t]le ray tI'a,llS[(W ()i)(;ra,tt)r ()f a,]ly ()]It tirstt)I'(h'x 1D ()t)ti(:al systenl is fa,(:tt)rizal)h'~ in a ll('~iglfl)t)llrh()()(t ()f the (;ntra,ll('c t)la,nc illl,() tlw, t)r()(tu(:t of tile Lie transfornmti(ms g('al('xa.te(1 )y tlm lll()n(nllials ~2 ml(l a,s ~ ( z , z~)  (,P(~)K+ r
9~l(Z~,Zi)

I,
(3.5.5)
where K , K~ ml(l K:~ are tile Lic ()l)cra, t()rs ass()(:ia, te(t witll tlm (tll()tc(l mono,,rials (see .~ 1.5.2 an(l 2.3) [4]. Rt'~stri(:tly to tl~(: li~mar al)t)r()xi~m, tiol~ the ~m.trix an(l ot)eI'a.t()r i(lentities, (:].5.4) /I,11( (:].5.5), tn'(wi(h: tlm a,~swer t,(, l,l~(; I,,(,],1(',~ (,f z,w(h,ri,~g, whi(.h lma,v()i(lal)ly arises wl~(;~ al)l)I'(m(:l~i~g tim i~t, egrati,m ()f ttm~ilt,()~'s e(l~ations in tl~e (:a,se of a, z(h;t)el~(li~g IIa~ilt,()~im~, (t~(; t,() 1,1~(' i~ gel~(;ra.1 ~()~(:()~mn~tatiw'~ nat,~tre ()f ~m,tri('es a,n(t ()I)erat()rs. In w 1.5.3 w(; n~(;~iti()I~('(t tha,t the fa(~l,()rizati()~i ()f 9/1 ca~ 1)(, (~()l~t,i~nt(;(t t,() i~(:l~(h, higher or(h'r t(,rn~s, a(:(',()unti~g fl)r t,lic a.1)crra,ti()~s. Ea~'l~ ,~f 1,1~(' a,(htit, i,)~m,1 fi~:t()rs tak(;s tl~(; fi)n~ of a, Lie tra,nsfi)n~m,tio~, a,ltl~()~lgl~ it, ('a,~()t t)e giwu~ a 2 x 2 ~m,trix ret)rcse~ta,tion.
3.6
Rotations and squeezes in the phase plane
In w 1.5.2, when discussing the t)hasest)ace d y n a m i c s g e n e r a t e d by the Lie tI'ansfornmtions ass()('iate(t with (tynami(:al flm(:tions qua(h'ati(: in tile t)osition an(t m ( ) m e n t u m ray varia)les, we have (:onsi(tere(t the t)olynomials 1(p2 + q2) q2). We hav(~ sh()we(t t h a t they give rise to (tefinitely different and ff1 (p2 motions in the phase plane, the corresponding trajectories lying respectively along a circle and a branch of hyt)erbola. Here we reconsider in detail the above bynomials in order to identify the optical systems they may describe. 
3.6.1
Attractive oscillator, phaseplane rotation and fractional Fourier transform
We denote as Ha.o. the quadratic homogeneous polynomial Hao 1 .
.
2 + ~_~12q2 ,
"Js
(3.6.1)
The Group of the 1D FirstOrder Optical Systems
135
which typically models the attractive oscillatorlike dynamics. The space coordinate q has been explicitly scaled by a,n a p p r o p r i a t e length factor f s ' whose optical interpretation is clarified below. The corresponding Lie o p e r a t o r ka.o. is A
~0 L~.o.  p Oq
q 0
f 2 Op'
(3.6.2)
t..
~..
which we rewrite in terms of the operators K_ and K+, defined by (2.3.5), in the convenient form 1" __ 1 _ 1 K+) (3 .6 .3 ) L~.o.  fs ( L K + ~ 9
Accordingly tile m a t r i x representative Ha.o. linearly relates to the sp(2,]R) basis matrices K _ and K+ as
.ao. . ( 1/fs2 ~
1
__
~s_sUl(fs),
(3.6.4)
where we have denoted by K~ (fs) the algebra element K 1 (fs)  l ( f s K
+ ~1K + ) 

 1 (  1 /0f ~
f s. ) 0
(3.6.5) "
Tile convenience of such a, seemingly odd definition will become a p p a r e n t later. In order to identify the optical system described by the harmonic oscillatorlike Hamiltonian (3.6.1), we exponentiate the traceless m a t r i x (3.6.4) t h r o u g h the axial distance A z  z  zi. As a result, we obtain the "transfer matrix" for the ray propagation from zi to z in the form (a)
F~(fs) . .cAzHa . . . _c2~Kl(fs) ~  (  ~ c~162 1 sin r
.fssinr cos r
'
(3.6.6)
having interpreted the dimensionless a r g u m e n t ZXz of the circular functions as tile angle 6, c o m m o n l y measured in unity of ~ t h r o u g h the t)araIneter (~: r
~
Z
fs
Zi
7r
 c~(z)~.
(3.6.7)
Note that, as fs has the dimension of length, the offdiagonal entries in (3.6.6) have the dimensions of length and (length) 1 respectively, as it should be for an optical m a t r i x in the raycoordinate and m o m e n t u m representation. a As in the case of the matrices K_, K+ and K a (see w 2.4), we may compute the exponential function of the matrix K 1(fs) (as well of K2(fs ) below) by simply applying the exponential power series, since powers of g 1(fs) (and g 2(fs)) are easily computed. The interested reader may consult [14] in ch. 2 for a general account of the methods of computing exponential functions of matrices.
136
Linear Ray and Wave Optics in Phase 5pace
   "  ~ P (z)
FIGURE 3.12. I)hasct)lane (ly~mmic generate(l by the attractive oscillator Hmniltonian (3.6.1). The ray represe~tative poi~t slips alo~g the ellipse (leter~ni~e(1 by tim i~fitial data and the frostily parm~mter fs.
The tra~sfl)r~m.ti(n~ a('t(;(l 1)y tim ~xm.trix F " ( f s ) ()~ th(; t)lm.s(,'l)la,n(; wtria)h;s (q,p) fr()~ l,lm i~itia.l (la.ta (q,s,Pi) is (l(;s(:ril)e(l 1)y
q(z) ..... qi (:()s q5 + Pi.fs sil~ (/), p( z ) 
1
 ~ qi si~ 4) + pi (:()s 05,
whicll is ()f (:(Jllrs(; ill a(:(:(n'r wittl rcslllt (1.5.23). The (:y(:li(: lm.t,llre (ff tim l)llas(; l)la,n(; lll()ti(m (3.6.8) is al)l)ar(;llt. As tile ray I)r()t)agal,(;s (,llr()llgll (,ll(; "()l)(,i(:al" sys(,(;Ill (3.6.6), (,ll(; r(;I)r(;s(;~d,ativ(; I)()i~t P in tim qp t)lan(; l~()v(;s al()l~g (,l~e (;llit)s(;
q2 2 e   ~ + Pi'
q(fz2)2, s + P ( z ) 2  ~'
(3.6.9)
The consta,~t e is (t(;t(;rn~i~m(t t)y the t)aran~etcr fs a,n(l the initial Wd~leS; r specifies fi)r ea,(:tl va,hu', ()f tllc axial t)arameter z tllc a,xlgle of tlm rotation of the (:orrcst)on(li~g representative t)oint P(q(z),p(z)) with rest)cot to the initial t)()int P~(q~,p~) (Fig. 3.12). The rotation is ('h)(:kwise or (:o~nter(:lockwisc accor(tiIlg to the sigll of fsEvi(tcntly the nmtrix F'~(fs) relates to a rotation by r ct~ along a I l in general cllipti(:al contour in the phaseplane. In fact, as will be clarified later, it can t)asically be interpreted as a phaseplane rotator. In particular, with fs = 1, F'~(1) takes the appearance of a pure rotation matrix by the angle r  sin r K
cos r
where K 1  Z 1 (1) = =2+K+ , the unit a r g u m e n t being omitted. Accordingly the representative point Inoves along a circle centered around the origin with radius v(,  v/q] + p~.
The Group of the 1D FirstOrder Optical Systems
Interestingly, with r form matrix (3.4.13),
137
~ (i.e. c ~  1) we recover the perfect Fourier trans
F(fs) 
0 1/f
fS )  (~TrKl(f S ) S
0
(3.6 11) '
"
(the unit order being omitted as well), which then is seen to relate to the phase plane rotation by rr/2. The matrix F~(fs) is reported in the literature as the fractional Fourier transfoTwz matrix of order c~ and family parameter fs" The term "fractional" evokes the charming vision of the "extension to continuum" of a process that occurs through finitesize steps. It is in fact in conformity with this "idea of fragmentation" that tile fractiona,1 Fourier trailsform has been introduced within different contexts of both mathematics and physics. Condon, for instance, investigated about a continuous group of functional transformations isomorphic with the group of rotations of a plane about a fixed point by multiples of an arbitrary angle, thus generalizing the property of the Fourier transform group, which in fact corresponds to rotations by multiples of tile right a,ngle [5.1]. Likewise, Namias ela,bora,ted tile functional form of Fourierlike operators, a,dlnitting, as the Fourier tra,nsform, the HerlniteGauss eigenfunctions, but with eigenvalues evenly separa,ted by a fraction of the imaginary unit [5.2]. The definition of the Fourier transform of fractional order within the optical context has been inspired by the performance of an optical system perceived as obtainable by fragmenting the Fourier transform configuration into a larger and larger number of shorter and shorter freespace sections interleaved with weaker and weaker lenses [5.a], or equivalently as capable of extending to arbitrary angles the property of the Fourier tra,nsforming system to rotate by a right angle the Wigner chart in the inherent Wigner phase plane [5.4]. The rotation by a right angle in the optical phase plane is the underlying canonical transformation acted by the Fourier operation on the conjugate variables q and p. Letting the rotation angle vary continuously leads to the fractional Fourier operation. Thus, the finitestep transformations marked by multiples of the right angle, corresponding to Fourier transform systems possibly concatenated one to the other, become specific events within the smooth evolution of the process governed by the continuously varying angle qS(z) over the 2re range, and so by the continuously varying axial parameter z. Such a phase plane rotation by a continuously varying angle may be achieved by sectioning the basic Fourier transforming configurations into an increasing number of similar configurations appropriately designed, as we will see later. The relation, and the relevant visualization in the optical phase plane, between the Fourier transform and the Wigner distribution function is clarified in w 8.4.1.
Linear Ray and Wave Optics in Phase Space
138
After its i n t r o d u c t i o n into the field of optics, the (:oncept of fractional Fourier t r a n s f o r m and the relevant forma, lism gave rise to a huge variety of applications, investigations and new formulations in an increasingly enriched ()t)tics scenario. In the forthcoming paragrat)hs we will simply cla,rify the con(:cI)t of fractional Fourier t r a n s f o r m at a very basic l(;v(;1, a,(t(lr(;ssing the rea,(lcr to the related lit,era, t~re for a wider mid (teet)er t,rcatnie~lt [5].
FT"actional t:owrit'.'r t'ra'n.@~Twt matT"let's as a sy'mplt'ctic subgT"oup Rcnmrkat)ly, fin" a fixc(l wdlu; ()f th(' fanlily Im,ra.nlcter fs, tllc ma.tri(:cs F ~ span a t)rot)cr Ab(;liall slfl)grr162 the symplc(:tic gr()llt) Sp(2, R) with rcst)c('t to the order (~, wtfi('tl Ilmy ra.llge ill t)riIl(:it)lc tllr(nlgll all tile r(;a,1 lilm. In fact, with (~  0 w(: (fi)ta,in the i(lcntity nm,trix; lla,llmly, F~
I.
(3.6.12)
Also, l)y th(: a,(l(li(,i()ll f()nlnllas ()f ill(: (:ir(:lfla.r flult:i,i()llS, w(: lmv(: F'~' ( f s ) F ' ~ ( f s )

, + ~ . T h e ad(litivi(,y t)r()l)('rty (3.(i. 13) with r(;sI)('~(:t (,() tim ()r(ler (~ lea(Is t()i(tcntify tt~('~ i~v('xsc t() F " (fs) a,s (,Ira ~m,t,rix (:()I'r(;st)()~(li~g t() (,lm ()I)l)()si(,('~ vahm ()f (~, [F" (.fs)]'

F')(fs ) 
((:()s 0 or b < 0. E i t h e r cases, the a,symptotes are the straight lines p = +q. T h e d y n a m i c s of the ray vector in the phase plane under the action of B(~) is simila, r to t h a t under S ( m ) , which, a,s noted in w 2.5, identifies the h y p e r b o l a qp = qiPi. It is evident, in fact, t h a t the h y p e r b o l a s qp = qiPi a,nd q2 _ p2 q~ _ p/2 can be t u r n e d one into the other by a p r o p e r r o t a t i o n by 4" Then, as phaseplane r o t a t i o n s are a c t u a t e d by fractional Fourier transfornfing systems, we m a y guess for B(T) a t h r e e  s t e p realization in t e r m s of pure r o t a t i o n s and squeezes as B(T)  F 1/2 S ( c ~ ) F 1/2. (3.6.45) Notably, (3.6.45) a,llows us to identify a,n optical realization of the symplcctic m a t r i x B(T), on account of the optical realizations of the fl'a(:tional Fourier t r a n s f o r m i n g and pure magnifying s y s t e m s discussed in w 3.6.1 and w 3.4.3. We invite the reader to prove t h a t a. s y m m e t r i c singlelens realization of m a t r i x (3.6.45) is actuabh', with d  t a n h ( ~ ) a n d f   s i n h  l ( T ) when 7 > 0, whilst a more complex configuration d e m a n d i n g for more t h a n two lenses is needed to i m p l e m e n t negative values of T. In order to clearly show t h a t the m a t r i x B(T) o p e r a t e s in the phase plane as a scale t r a n s f o r m a t i o n in a similar fashion as S ( m ) , but w i t h respect to a reference direction Inaking an angle 0  ~ w i t h the qaxis, we consider again the Gaussian d i s t r i b u t i o n (3.6.30) ~nd the relevant r e p r e s e n t a t i v e c = 2 phaseplane ellipse. For illustrative purposes, we set Crq = crp = or, and accordingly consider the circle C of radius r = 2or centered a r o u n d the origin, q2 + p 2 _
r2,
(3.6.46)
s A matrix A is positive definite if the scalar product (u, Au) is positive for any vector u in the linear space acted by A. It is evident that in the case of matrix (3.6.43) we have (p2 + q2) cosh ~ + 2pq sinh T > 0 on account of the inequalities [sinhr[ < cosh r and 2pq 0, f()r inst, a.ii(:e, t,he xlm,jor axis (:(nlies t,o lie a,l()ng l,ll(: (lirect, ion m a k i n g a, { angle wil, h the qa,xis (see t,he la,st, grat)h in Fig. 3.17), st)ecifically being a = cTr an(t b = ('.Tr. T h e init, ial t)ha,seplane area, is then stretched up to the factor m  c r in the dire(:tion at { to the qaxis and orthogonally squeezed down t)y the same factor. T h e system B(T) is indeed referred to as h y p e r b o l i c ezpa'r~,der ()r 'l'eductor a,(:t:()r(ting to w h e t h e r r > 0 or r < 0. Figure 3.17 shows the r o t a t i o n  s c a l i n g  r o t a t i o n sequence, whose overall effect is the e x p e c t e d squeeze of the phase t)lane area in the direction at ~ to tile qaxis. As in the graphs of Fig. 3.16, the scaled variables qOq and ~cr have been used. It is possible to p r o d u c e squeeze in any direction 0, com[~ining scaling and r o t a t i o n m a t r i c e s t h r o u g h a similarity t r a n s f o r m a t i o n analogous to (3.6.45) with F 1/2 replaced by F ~. In fact, the reader can easily verify t h a t the m a t r i x
The Group of the 1D FirstOrder Optical Systems
B(r, 0)  F ~ S(cr
149
~  ( cosh r + cos(20)sinh 7sin(20) sinh r ) sin(20) sinh r cosh r  cos(20)sinh r / ' k
(3.6.49)
produces a scale transformation with respect to the direction making the angle 0  c, 2 with the qaxis. Like (3.6.43), the matrix B(r, 0) is symmetric and positive definite. Also, it is characterized by two parameters specifying the "entity" and the "direction" of the squeeze. In accord with the previously adopted symbology, we identify the specific determinations of matrix (3.6.49) corresponding to 0  0 and 0  { as S(m) and B ( r ) respectively; namely, B(r, 0)  S(m), m  e ~ and B(r, {)  B ( r ) . Interestingly, although the matrices S and F ~ separately form proper subgroups of the group Sp(2, R), the matrices B ( r , 0) do not span a subgroup. In fact, successive squeeze transformations along different directions do not compose into one scale transformation, but into a squeeze preceded or followed by a rotation (see Problem 10). It is worth evidencing the formal similarity between the squeeze matrix B ( r ) and the Lorentz transformation matrix in the Cartesian system of z and ct, positive and negative values of r respectively corresponding to boosts and antiboosts. Likewise, the pure magnifier matrix S(m) has the form of the Lorentz transforInation matrix in the Dirac's lightcone coordinate system, which is rotated in the Euclidean plane (z, ct) by ~. A(:(:()rdingly, we may note that as successive ilOIl collinear squeezes do not combine into one squeeze, Lorentz boosts in different directions do not multiply into one Lorentz boost, but into a Lorentz transformation preceded or followed by the Wigner rotation. The Lie algebra sp(2, R) is in fact isomorphic to the underlying algebra so(2, 1) of the group S0(2, 1) of the Lorentz transformations in three (two space and one time) dimensions. This isomorphisIn, a.nd in particular the analogy in the behaviour of optical scale transformations and Lorentz boosts, has suggested optical experiments to test the abstract properties of the Lorentz group [6]. Finally, we briefly comment on the factored representation of the squeeze matrix (3.6.36), which is naturally suggested by (3.6.45) in the form B(T, fs)  F  1 / 2 ( f s ) S ( e r ) F 1 / 2 ( f s )  S ( f : / 2 ) B ( T ) S ( f ( 1 / 2 ) ,
(3.6.50)
paralleling expression (3.6.28) for the fractional Fourier transform matrix F ~ ( f s ) . The parameter fs retains its role as the focal length of the Fourier transform unit associated with the fractional Fourier transforming systems with c~  t1, entering (3.6.50). The transformation acted by B(r, fs) produces therefore the further scalings by f s 1/2 and f l / 2 with respect to that acted by B(~).
Linear Ray and Wave Optics in Phase Space
150 3.6.3
Generator matrices of pure rotations and squeezes
We frame the matrices S 1 and K,~ within the proper algebraic context. Defined a,ccording to relati(ms (3.6.5) and (3.6.35) for a unit value of the length t)a.rameter .fs, the ma.tri(:es K1 a.nd K~ ha,ve been re('ognized as the generators of pure r()ta,tions a,nd S(l~W.ezes in the t)ha,se plane, and a.sso(:ia.ted with optical systems (tist)la,ying r(~st)(;('tiv(@ a (tefinit( ~,f()(:~si~g a,n(t (tefi)(:l~sing })ehaviour. I~(~t l~s rewrite t,l~e l)erti~(nd, (;xl)ressi()~s, i.e. K,  ~~ ( K
+K, ) ,
K~  ~(U
K,),
(3.6.51)
witl~ tl~e (;xI)li('it ~mtrix f
i
i
:
n
.o 4~
I
I
ADBC= I
i
i
i
i
,
0
i
m
fractional Fourier transformer
positive magnifier
thin lens
m 9
r lowertriangular unitdiagonal matrix
O_ "~ 0 L
unitdiagonal matrix (5;p(2,P,) h y p e r b o l i c s u b g r o u p )
antisymmetric matrix (5'/,(2,R) elliptic s u b g r o u p )
(,~;p(2,R) p a r a b o l i c s u b g r o u p )
o~
._o > r
o/..o.)
0, exhibits a maximum on the optical axis and decreases radially from it. The general solution of Eq. (3.9.11) writes in terms of the harmonic functions as y(z)
:
C,
C 1 COS ~ 31 C2 s i n
(3.9.12)
where c 1 and c 2 denote two arbitrary constants, determined by the initial values, and ( is the dimensionless axial coordinate ~ ( z )  ~nZ(z%
(3.9.13)
 zi).
Determining c I and c 2 according to the initial values appropriate to the entries A, B, C or D, we obtain A  cos ~,
~
__
sin ~,
1
(3.9.14) C =  v / n 0 n 2 sin ~,
D = cos ~.
It is apparent that the field ray corresponds to the cosine solution of Eq. (3.9.11) as the axial ray to the sine solution. The optical ma,trix M(z, zi) writes therefore
(
Mfo r ( z, zi ) 
cos~
~
1
 v/no n e sin ~
siIl~ )
cos ;
(3.9.15) '
and the ray variables change with the a,xial coordinate according to q(z) 
qi c o s ~ +
P~
sin~, (3.9.16)
p(z) =  qi v/n0 n2 sin ~ + Pi cos ~. The ray oscillates back and forth ~cross the axis with the spatial period Z
27r~,
(3.9.17)
known as the p i t c h (Fig. 3.19). The maximum transverse excursion of the ray to
qm x
+
m
xim.m
ray
forms with the axis along its path is / ) m a x ~ P m a x / n o = V / n 2 / n o q m a x . If the width of the material is greater than 2qmax, the ray remains confined within the medium, which then acts as a light guide. The matrix (3.9.18) has the same form as the fractional Fourier transform matrix (3.6.6). In fact, we can write Mfoc(Z, zi )  F~(Z)(fs) ,
(3.9.18)
Linear Ray and Wave Optics in Phase Space
160
fli
flo
i (qi,Pi
~
i~ \
i
/ \
(qo,Po)
_
r
FI(~,URE 3.19. Ray t r a j e c t o r y i~ a foc~mi~g ~ n e ( l i ~ .
where the stm~(la,r(t f()ca,1 hu~gtl~ .fs is i(tentifier a,s fs ~
a,n(t tim or(ter ~ ra,I~ges r a,(:(:()r(ting t() relati(n~ (3.6.7), wlfir (:(z)

(3.9.19)
1
witl~ t,l~e axial gives tl~en ~
zz
(:r
te z
as exI)e(:te(t
i
As alrea,tly ll()l,e(l, ill fat:t, a, (tlm,tlra,tic gra,tletti11(lex fi)t:llsi~lg llle(lilllll t)rovitles the ()t)tit'al sysl,eIll I,() iIllI)lement the fl'at:titmal Fr tra,nsf()rlning ot)era,tion with a. (:(mtimit)llsly ranging ortter. T h e r ()f the ra,y I)a.th a,(:ross the ot)tit'al a,xis (:orI'eSl)()ii(l to the (:()IltilnlOllS rota,titnl of tlw, ret)resent,a,tive I)oint ill l,lle l)tm,se, l)lmle, wtli('ll dest:rit)es tim ellit)se (3.6.9) (letermine(t t)y the mc'(ti11111 Imra,meters 'n,0 a,n(t 'n2, a,n(t the initial (ta,ta,. As the ()rr ~ is r lillke(t to the t)rr wt,ria,t)l(; z, tim system nmu a,('t a,s a, F()lu'ier tra, nsf()rmillg s y s t e m a,t the t)r()t)er (tist, a,ll(:e L F fl't)m the input screen. In fa(:t, if L v is sllch t h a t a ( L v + z~)  1, a,n(t hen(:e Lv
Z 
2 ~/~,,, V "
(3.9.20)
4'
the p r o p a g a t i o n t h r o u g h th(; m e d i u m over the distance L v from the input pla, ne results in the o r d i n a r y Fourier t r a n s f o r m a t i o n with foca,1 lenth fs This establishes the possibility of optically realizing a Fourier t r a n s f o r m a t i o n as well by a piece of a quadra,ti(: m e d i u m of proper length L~. At distances c~Lv from z i , tile s y s t e m behaves as a Fourier transformer of fractional order c~. T h e n we can imagine the g r a d e d  i n d e x m e d i u m as formed by a c o n t i n u u m of planes from zi along the optical axis; on each plane at z we z Lv  z ~ of the object can observe the fractional Fourier t r a n s f o r m of order (~ _
at zi. At planes regularly placed from the i n p u t plane at distances LF, 2L~,
The Group of the 1D FirstOrder Optical Systems lI 4(h
]7411 [1
rl4hll
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4h~
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i 1}
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~x/7770n 2
Z
[F(.l.)l,lh,
:
[V(./)]
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i
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,
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e
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F(./)
[F(.f)]4h,'~
:
[F(.f)]
,
e 4
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FIGURE 3.20. A quadratic focusing medium is like a sequence of Fourier transformers. 3Lr, and so on, we may detect the successive powers of the Fourier transform of the object; specifically, at distance j L ~ we observe the j  t h power of the Fourier transform of the object (Fig. 3.20). For j = 4h, we recover the identity matrix and hence the input signal, according to the periodicity of the Fourier transform operation (a). The overall medium can be understood as a sequence of identical Fourier transforming configurations, L~ fixing the zsize of the Fourier transform units, by which the medium may be thought as composed. D e f o c u s i n g quadratic m e d i u m
For a, defocusing index profile: n 2 < 0, which exhibits a, minitnum on the a,xis a,nd increa,ses fa,r from it, Eq. (3.9.11) is solved by hyperbolic sine a,nd cosine functions. Hence the optical matrix writes as ( M d e r ( Z ' Z i ) 
cosh ~ V/rt0•2 sinh{
1 % sinh~) v,,_,~o
,
~ ( z )  i _2a. (z  zi)
cosh{
%
(3.9.21) which has evidently the same form as the squeeze matrix (3.6.36), being indeed Mdef(Z , Zi)


B(~, fs),
.fs 
1 v/_% ~2 9
(3.922)
As the height of the ray with respect to the axis increases while propagating from the input plane, the representative point in the phase plane moves along the branch of the hyperbola (3.6.39). The propagation along a defocusing quadratic medium physically realizes the threestep phaseplane transformation (3.6.50) involving squeezes and scaled rotations. a More precisely, as it will be shown in w 5.4, the Fourier transformed signal is affected by a phase factor, which obviously cannot be accounted for within the rayoptics description of the performance of the optical system.
Linear Ray and Wave Optics in Phase Space
162
B e i n g (:oshsc > 0, t h e W e i  N o r m a n d e c o m p o s i t i o n c a n safely b e a p p l i e d t h r o u g h t h e w h o l e { (i.e., z) d o m a i n . T h i s is left as a n e x e r c i s e to t h e r e a d e r .
3.10
Summary
It ha,s )cell t)r()vc(1 tha, t e v e r y first()r(ter ()t)ti(:al systcxll (:a,ll t)c l u l ( l c r s t o o d a.s a s('xtll(',~l(:(', ()f tllill l(',~lscs s(',l)a,ra,t(;(l l)y ff(;('~I)r()t)aAati()ll s('~(:ti(ms, visllaliza.t)lc a S('qll('~ll(:('~ ()f v(;rti(:a,1 a ll(l ll()l'iz()Id, a.1 stl(;a.rs in tilt; t)llas(; i)lmle view. Tile ()t)ti(:al syst,enls, ~(',ll(;ra,te(l l)y t,]l(', at,tra,(:tiv(; a,]l(l l'(;1)lllsiv(', ()s(:illa,tof like ttmlfilt()llia,llS 1721)2+ ~q21' a,ll(l :jp I 2 _ }1q2 , lm.vc l)(',(~,ll (',Xa,lllillc(1. q'lw, f()rnlcr is rc(:()glfiZ(',(t as I)erf()nldllg s(:al(',(t r()t,a.ti()lls ill l,ll(', I)lm.s(; l)la,~(;, wl~(;r(',a,s t h e l a t t e r a,(:ts as a s(p~(',czing trm~sf()rnm,ti(m. T w o i~t,cI'eStil~g rc, a liza.ti()~s (~f ()I)ti(:al ~m,ti'i(:e,s l~a.w', 1)e,(;~ (lcs(:ril)e,(l i~ (:or
,'(,st)o,,(l(;1,(:(, wit,l, t,l,c tw,)il.lg,~,},I'a. ],a.sis { K
, K,,
K:,} a.,,(l { K
, K , , K:,}.
( ) h e realizati()~l, wlii(:ti al)l)li(;s l()(:a,lly Ii(',a,r tll(; (',ld,ra~l(:(; s(:re(;]l, syld, ltcsises t,l~e ()l)ti('a.l ~ m t r i x as a s(;(ll~(;~(:(; ()f a. (1)()sitiv(; ()r ~l(;gativ(;) fl'(;(~l)r()l)a,ga,ti()n s(;(:ti()n, a. I)()sitiv(;(l(;linit(; l)llr(; ~m,guili(;r m~(l a, t h i ~ l(;~s, a,u(l a(x:()r(li~g;ly is vislm,liz(;(l i~ t,l~e ()I)ti(:al 1)lm,s(; I)la.~(; a,s a l~()riz()~d,a,1 sl~(',ar f()ll()wc(l I)y a s(:a,ling; a,n(1 a, v(',l'l,i(:al sl~(m,l'. TI~(; ()t,l~(',r r(;a.lizati()~ all()ws ~s t,() (l(',s(:ril)(', t,l~(; syste~n a.s a, s(',(t~(;l~(:(: ()f a fl'a,(:ti()~a,1 F()~rier I,l'a,]~sf()rl~i]~g~ sysl,(;l~, a, I)()sitiv('(l(;fi~fitc t)urc lna,g;l~ilier a,l~(t a, t,l~i~ lel~S, m~(t h(',ll(:e is t)i(:t~rc(I i~ tl~c t)ha,s(', l)la,n(; a,s a, r()t,a,ti()~ f()ll()w(;(l )y a, s ( : a l i ~ m~(1 l,t~(',~ t)y a v('rti('a.1 st~(',a,r. In t,ll(', ~(;xt, (;lmt)t,(',r, tl~(', ligl~t, t)r()t)a,ga,ti()l~ will 1)(', r(;(:()~si(tcrc(l witlfin t h e fram(',w()rk ()f wav(; ()t)ti(:s. W e will i~v(~,st,ig;a,t,(~ t,h(', ra,ywa,ve (:(mne(:ti()]~, w()rking ()~d, t h e ()I)(;rat()r r(;t)r(',s(;~ta,ti()l~ ()f tt~(', litmar 1D ()t)ti(:al systcn~s tl~r()ugh a s()rt ()f (t~m,~xtiza,ti()~ t)r()('(',(il~re, sit~filar t() tl~a,t a,t)t)lic(i t,() (:la,ssi(:a,1 ~x~(',(:ha,ni(:s t() yicl(t (l~m~d,~m~ m(',(:tm,~i('s. R a y vc(:t()rs will t)e rct)la(:(',(t )y w a v e fim(:tions a,n(t t,h(', ~mi~()(lula,r nmtri(:(;s ()f t,h(', symI)lc(:ti(: g;r()ut) ~p(2, ~) will t)c rcpla, ced t)y t h e mfita,ry ()t)(',rators ()f t,l~c I~C,tapl(',(:ti(: grout) AJp(2, ]~).
Problems 1. Co~si(ler a thick lens, co~sisfing of two refracting surfaces with curvature radius/~t mid /~2 separ~rte(t by a distance d on the optical axis. Deilote by 'n(,~ and 'rt(,~ the refractive
indices in the object aim image space, and by n the refractive index of the lens material. Take the reference planes imincdiately at the left and at the right of the lens (Fig. 3.3.a)). (a) Find tt~c positions of the principal, focal aim nodal points of a the lens. (b) Express the ttfick lens matrix (3.2.4) in terms of the coordinates of the principal points s H and SH,. (C) Write down the explicit expression of the reduced focal length of a thin lens in terms of the relevant parmncters, i.e., n(,1, n,,.e, n, R~ and R~. 2. Enlarge configuration (3.3.13) by (a) an additional freepropagation section placed on the right or the left, and (b) two additional freepropagation sections placed on both sides. Write down in both cases the relations for the relevant parameters and individualize some configuration in accord with the obtained relations. 3. Specialize the parameters of the LTL configurations, which may synthesize symplectic matrices having (a) D = 0, (b) A = 0, and (c) A = D = 0. Specify in every case the
The Group of the 1D FirstOrder Optical Systems
163
restrictions on the non null matrix elements. 4. Specify the parameters 7), m and d for the "dual" decomposition: (A ~) _ T ( d ) S ( m ) L ( 7 ) ) ,
D > 0,
which reproduces what in the group theory is called the antinormal ordering, hi turn, the LST realization (3.5.1) reproduces the so called normal ordering. 5. Prove relation (3.6.15) for c ~  1~.n ~ m  1 , 2, 9.(Hint: Rewrite (3.6.15) as [F~(fs)] 1/~ F ( f s) and use Sylvester's theorem according to which
(A)ml(Asin(mO)sin(m1)O BD
  sin(~nO)
B sin(rn, O) Dsin(rn~)sin(m1)O
Csin(rnO)
A+D
) ,
COS(~  ~.)
6. Decompose the fractional Fourier transforming matrix F ~ ( f s ) into the form S1Fc~S for negative values of fs. 7. Write down the relation between the matrix entries (A, B, C, D) and the the parameters (7), rn, r of the K:M.hf decomposition: (A B) _ F ~ S ( m ) L ( 7 ) ) . 8. Consider two symplectic matrices M 1 and M2, whose relative N'.AK; decompositions are given with the appropriate parameters being 7)1, m l , (751 and 7)2, rn2, r Prove t h a t the parameters 7), m, r of tile LSF ~ synthesis of the product matrix M 2 M 1  L ( 7 ) ) S ( m ) F ~ are
mmm~,
where
r 1 6 2 +r
PP~ +~,
~~2 __ , H 2 COS 2 (~)2 t (?Tt.12 ~ 1H.127) 2) SiIl 21~)2   77L127)1 S i I l ( 2 0 2 ) ,
7) __ 2_._~_~ .1 
[2rn~7), cos(24)2)+ (m~  m72  rn~P~2) sin(202)], t a n 4) 2
tan r  ,,~ ( l _ p I tan 4)2)"
(Hint" Write the Iwasawa decomposition of tile internal sequence F~2L(7), )S(rn, )  L(7)) S ( ' ~ ) F ~ in tile product M 2 M 1 ; then combine tile resulting sequential rotations, lenses and magnifiers). 9. Identify the parameters (7), m, 4)) of the A/'MK~ decomposition of the squeeze matrix (3.6.49)" B ( r , 0)  L(7))S(rn)F ~, in terms of the squeeze parameters r, 0. 10. Prove that two squeezes into different directions do not multiply into a single squeeze. (Hint: Consider as an example the product B ( % , 0)B(%,01); identify the parameters of the relevant N'AK; decomposition; then, verify t h a t they do not reproduce the results of Problem 9, thus proving that B ( % , 0 ) B ( % , 01) = B ( r , 0)N ;~. It is also possible to express (r, 0, ~p /32 ) in terms of (%, %, 01 ).) 11. Verify t h a t the matrices  cos 4)  sin 4)
sin 4) ~ cos 4)]
and
(  cosh r sinh T sinh r   c o s h z]
,
are not of exponential type. (Hint: Write down the above matrices in the neighbourhood of the identity, i.e., respectively as I+(~Q 1tO((~ 2) and I + r Q 2 +O(r2). Verify t h a t Q1 and Q2 do not belong to sp(2, R).) 12. Infer the equations for the parameters 7), rn and 4) relevant to the Iwasawa decomposition of the optical matrix M ( z , zi) of the parabolic index medium.
164
Linear Ray and Wave Optics in Phase Space
References [1]
For the ray m a t r i x apt)roach to linear ot)tics see [12] of oh. 2. For tl~cr a n d at)t)licative (:onsiderations on the symt)lectic Lie algebra and gr()~ll) , see [9] alitt [1()] ii~ t']~. 1. Nit)re re('ent ilWestigatit)I~S, ii~ relatioil a,s well to get)~etrit:al ()t)tit:s, t:a~ 1)r finn~(1 i~ [11] ()f t:t~. 2. T h e ~()~ eXl)()~le~t, ial ~mt~u'e ()f
the syn~t)lc(:lir gr()~t) S p ( 2 , IR) is ana lyse(l in [2.1] R. Si~o,~ m~(l N. Ni~k~(la, "Tl~e l,wo
>
ion,
(4.6.17)
t~
The relations above are used in w 4.6.6 to identify the momentum representation of the operators ~ and ~, and accordingly the frequency representation of the position and frequency operators ~, ~ (there introduced).
202
Linear Ray and Wave Optics in Phase Space
Ailticipa,ting the results of later sections, we mention that tile convolution (4.6.11) and the product (4.6.13) m a y be regarded as the tra, nsformation laws for light signa,ls t)ropaga.ting respectively through freespa,(:e se(:tions a,nd thin lenses. One of the functions in (4.6.11) and (4.6.13) (:an t)(: interpreted as the int)ut wavefonn a,nd the other as the impulse rest)oxlse of tlm freesI)acc section or (:orrest)oll(lingly tlle t)ha.semo(tulation fllll(:tion of the h:lls. A(:cor(tingly, tile l)i'()(lli(:(, (4.6.12)a.1~(l tl~(: (:()I~v()h~ti()n (4.6.14) r(:l)resel~t th(: (,ra,nsfer la,ws f()I" the a l)ove (tll()(,(:(t syst(ullS ill tll(: sI)atia.lfre(llU:llt:y (t()lim,i11. As a (:()llvt)llltivclike r(:la,tioIl ill the qdollm.i1~ (,~n~s i~t,() a, ~n~ltiI)li('ativ(~lik(: r(:lati(n~ i~ tl~(: F()~rier (:onj~ga,t(: (t()~mil~, i.(:., (,l~(: h'(h)l~m,i~, sil~ila,I'ly a l~nfl(,iI)li(:a,tiv(:like r(:lati()n in the q(t(n~m,i~ t,~n~s i~t() a (',()l~v()lutiv(:lik(: r(:lati(n~ i~ t,l~(: h;(l()~m,i~. C()nvol~lti(n~ m~(I ~nll(,il)li(:a.(,i()~ a.r(: (:()~ti,~ga.(,(: ()I)(:ra.ti()~s wi(,]~ r(:sl)(:(:(, (,() F()~ri(:r trm~sf()r~m.(,i()~. Fr(:(: l)r()l)agati()~ m~(l l(:~s (,ra~sf(:r (m~(l, Iii()r(: i~ general, sl)a(:einvaria.~t m~(l sl)r(:a(lh'ss sys(,(n~s) at(' .i~s( (,I~(' ('(n~('r(:t(: r(:a.liza(,i()n ()f s~(:tl a, ~lm,tll(:~la,t,i(:al r(:la(,i()~l. Ill tiffs (:()llll(:(:(,i()ll, WO. ll()tO. (,ha,I, (h:l(,a a,~(l Imn~(n~i(" fln~('t,i()~s ar(: (:()~tj~ga,te witl~ resl)(:(:t (,() l'(n~ri(,r (,ra~isf()r~im(,i()~ a.s l)r()v(:(l ])y (,If(: r(:lat,i()~is _
q)

.1

1
.1",
(5(h ' t  ~)  if4,["
,ie,'(q'q)dH,, 

.1

")dq,
(4.6.18)
,,:),t,,',
(:xl)r(:ssing (,ll(: fl)r~m,l li~k l)(:(,w(:(:~l (,l~(:s(: (,w()i(h:a,liz(:(l ()l)ti(:a,l (list,~Ii'l)ml(:es, wll()s(: )ella vi()~H's i~l t,ll(: q m~(l h" (l(nlmi~is at(: (:()~l)h:~ll(:~It, ary t()()~l(: a,~()tl~er. ll,(:lati()l~S (,I.6.18) fi~rl~isll (,I~(: ~ImI,ll(:llm,(,i(:a,l gr()lll~(I f()r (,If(: l)(:rf()nl~a,~l(:(: ()f (,l~e ()l)(,i(:a.l F()~ri(:r (,i'a.llSf()rlll(:r, wl~i(:l~ i~mg(:s a ln~(ll(: ()f l)arall(:l rays i~l(,() a I)()i~(, (i.e., sl)atia, l ]la,i'lll()lli(:s ill(,() bfinl(:ti()~s) ill (,l~(: F()IIri(:r l)lalm a~(l (:()xlversely a l)oint i~l (,I)(' i~l)~t l)la~(: i~(() a l)~l(ll(: ()f l)aralh;l rays (i.('., bfi~l('(,i()~is into spatial harl~()~fi(:s). 4.6.2
Inputoutput 7"elations in the spatial .frequency representation: the. wave,~'trre.ad function
The integral tra, nsf~)rm (4.3.13) is the ma,thematica,1 model for the light wave propa, gation through oI)tical systems, exenlt)lified in ternls of a. relation between the opti('a] (tisturt)a,ll(:es entering to and emerging from the system. In it both the input a.nd olltt)llt signals are (h:scribed in the space domain and hence are ret)i'esentc(t t)y th(: (:oinplex amt)litudes ~o:,(q, z;) an(t Po (q, Zo) a,t the a,ssigned int)ut and o u t t m t reference planes. Equiva,lent inputoutI)ut relations can be elabora,ted, in which the input and o u t p u t signals are (tescribe(t in (tiffereIlt domains. One can relate, for instance, the frequency spectra of the input and o u t p u t signals @; (t~,, z ) and @o(~c, z o), or the field distribution at one plane and the Fourier transform at the other plane, namely p,:(q, z,:) and @o(~,Zo), or inversely ~5~(~, zg) and Po(q, Zo). In any case
WaveOptical Picture of FirstOrder Optical Systems
203
the transfer relation takes the form of ail integral transform similar to (4.3.13), with the inherent spread functions being specific of the involved domains. Evidently, the existence of four equivalent forms of inputoutput relations with the respective spread functions is a direct consequence of the existence of four pairs of inputoutput domains in which the signals can be represented, i.e., spacespace, frequencyfrequency, spacefrequency and frequencyspace. This can be brought into relation with the existence of exactly four Hamilton's cha,ra,cteristics, i.e., the point characteristic, the angle characteristic and the two mixed characteristics. In fact, a definite relation can be shown to hold between tile four system functions of wave optics and tile corresponding Hamilton's characteristics of geometrical optics. As it is beyond the scope of this book to present a, more deta,iled ana, lysis of the subject, the interested reader is referred to [18]. Here we deduce the relation between the Fourier transforms of the object and image distributions ~5~ and @o Fourier transforming both sides of (4.3.13) with respect to qo and then expressing the input amplitude p~ (q, zi) in terms of the relevant frequency spectrum @~(~,z~) by (4.6.1), we end up with the integral relation
@o(h;o,Zo)  f dh;~h(h;o,h;~;Zo,Z~)@~(~;~,z~),
(4.6.19)
formally identical to (4.3.13). The weighting function h is explicitly given by
h(~;o, ~;~,9Zo, z~) _ ~1 / / d q ~ d q o (~i(~o qo~q~) g(qo, q~; Zo, z~),
(4.6.20)
with the reciprocal expression of g versus h,
g(qo,qg;Zo,Z~) __ ~I / / d l ~ i d t i , ~ ci(~o qo~iqi)
h(/~ ~ , h;i;
Zo , zi).
(4.6.21)
It is evident that h and g are related by a Fourier transformlike relation; however, due to the minus sign in the exponent (~o qo  t~q~), they do not constitute a Fourierintegral pair (~). For ease of writing, as in the case of g(qo, q~) we simply write h(rco,rC~) instead of h(~o, ~ ; Zo, z~). Interestingly, if the input to the system is a spatial harmonic ~ (q~, z~) eit~,i qi
, the system responds producing the output
~o(qo Z o ) 
1 f dqig(qo qi)c i~q~
c The Fourier transform of a 2D function /~'~ ( ~ 1 , t%)  ~1
(4.6.22)
f(ql, q2) is defined by the 2D integral
f f d%dq~ e   i ( ~ 1
ql t ~2 q2 )
f(%, q2).
W h e n the inner product (n~ql + ~2q2) in the exponent is replaced by the skewproduct (~1 q~  ~2q2), the resulting transform is also reported as syrnplectic Fourier transform.
204
Linear Ray and Wave Optics in Phase Space
which first by (4.6.21) a,nd t h e n by (4.6.18) is seen to be ~o(qo, Zo)  ~ .1
/
dh; o h(h;o, t;,i )(,it% qo . .
(4.6.23)
T h e flmction h yiehls therefore tlm st)e(:tra,1 ret)rcscIlta.tiol~ of tllc systenl rcst)()Ilse to a st)atia,1 lmn~l()ni(: wav(;fornl, whit:h is r(;t)I'Cs(mt(;(1 1)y a &fim(:ti(m in h',st)a.(:('~" ~ (h,,, z, )  b(h:h;,), th;n(:,;, h ix r(;t)ort,;(l as ,,he wa't,,'.,,q,'t','.ad ((,r 'wa'~,~: 'r(:spo'n.s~:) .flt'n(:lio'n (~f tll(; syst(',lll. TtI(' il~t)~t()~tl)~t r('lati(n~ (,1.6.19) lm.s ttl(; sm~l(' sigIliti(:ml('(' as (4.3.13), si~l(:(' ttl(; r('.Sl)()~ls(' ()f tll(, sysl,(;lll ill the h;&st, il~nlli b ( h ;  h',), (:()rr(;sl)()~l(li~lg l,() fl'(;(t~l(;ll(:y ('~ig('llWa,W;S ill l,ll('~ q(l()l~m,in. As a (:()Ii(:l~lsi(~l ~f ,llis a.('('()~l, a)()~l, l,lw, wav(', st)r('a.(t flnl(:l,i~)~l, W(; say tlmt t~(;iIlg h a.Ii(t g li~lk('(l ])y (4.6.2{)) ml(l (4.6.21), all rela.ti()IlS f i w g (:()rr(;sl)(m(t t() a,Im,l()g()~s r('lati()~s fin" h, a,~l(l vi(:(', versa.. Ill 1)arti('~flar, relal, i(~l (4.3.21) (;xt)r(;ssing 1,11(; h)ssl(:ssl~:ss ()f ,l~(; ~)l)l,i(:al sysl,(;l~l is I11irr()r('~(l ill,~
/&.
~
o
*
t
o
*
t
ti,
) i
"
Tll(' r(;a(l('r ('m~ ~:asily (l~;(l~u'~' t,l~' ('~n~lI~(~sit.i(n~ r~l(; fiw h, wlli('ll Imrall('ls, ()f ('()~rs(;, r('la,,i~)~ (.1.3.18) fl)r g. 4.6.3
S"pac(:inva'ria'nt a,n d sprcadh~,s',s ,s'ystcm,s
Spa, cci'n.'~,aria,'nt syst~'lIls, wlli('ll i,l.r(, ('llara('t~'riz~'(l t~y l ll(' r(,lllarkalfi~, l)I'()l)(;l'l,y l,lmt tll(; il~m,g~; (listl'it)lll,i()ll (I~J~;S ll()l, (1('I)('11(1 ()ll t,tl~; St)e(:iti(' h)(:a.ti()tl ()f tim ()t)je(:t in the illl),lt, t)lml~; [17]. 'l'lnls, wu'ia,ti()llS in t,lxe lo(:a,ti~)~l ()f the illI)11t, (listri])lltion ()w'r l,lm ()l)j(;(:t l)lml(' (1() n()t a,lf'e('t 1,11(; st, rlu:t~r~: ()f t,lm ()~t,t)~t, (listriln~ti(m, t)~l, ()I~ly its l()('ati(nl ill l,ll~; iina,g(, I)la.ne, t)(;ing ill(h:e(l sllift(;(t t)y the sa,n~c il,lll()llIll, ~1,"; 1,1~(' ()])j~',(:, (lis,ril)~ti(m ('t). Shifting the il~l)~, ()v('r ,h~' ()t)j(;(:t I)la,ne as gp~(q~  Q , z~) , t,l~(; (n~tt)~lt is (:orrest)(m(tingly sllift('(l ,hr(~gll 1,t1(' ilna.g('~ l)lml(; ill,() r ( q o  Q , z ) fl~r a.ny shift a,n~ount Q. Ttlis il~lI)lies ttla,t ttl(' st)r(;a.(t finmti(nl (lo(;s not (1(;I)(;11(1 s(;t)aratcly on its a,rg~m~t;nl, s t)~t, ()lily t,hr()~gh th(; (liffer(;n('(; qo  q~" g ( q . , q~ )  ~(qo  q~ ).
(4.G.25)
T h e iIlt)utoutt)llt relation for stfiftinvariant s y s t e m s writes t h e n in the form ~o(qo, Zo)  f d q ~ ( q o  q,~)~(q~, z~)  ~ 9 ~,~, J
(4.6.26)
d In practice, optical systems are seldom spaceinvariant over all tile object plane, but it is usually possible to divide the input plane into small regions through which the system is approximately invariant. Thus, within the linear approximation of our concern, the condition for spaceinvariance may be checked over a small region of tile object plane near tile axis.
WaveOptical Picture of FirstOrder Optical Systems
205
thus showing that the image distribution is the convolution of the object distribution with the pointspread function of the system. Taking advantage of the convolution property of the Fourier transforin, the above relation takes the simple multiplicative form
JPo(t~, Zo)  b(t~)~, (t~, z,). in the t~space. The function b(n) is, apart from a ~ 1 transform of the pointspread function g(q) according to
(4.6.27) factor, the Fourier
I~(~)  / d q c i~q g(q)  v"2~(t~).
(4.6.28)
It is reported as the optical transfer function. The "local" character of the inputoutput relation (4.6.27) is evident" no mixing of the spatial frequencies is produced by the system, each spectral component of the input being transmitted into the output signal though roodulated by the transfer function b In particular, spatial tmrmonics are simply transferred to the output plane. Naturally, by an appropriate design of the optical system, i.e., of the relative bfunction, one may remove or preferentially pass certain desired spatial frequency component of the signal; in other terms, one may operate a spatial frequency filtering of the input signal. Equation (4.6.27) relates the Fourier transform of the output signal @o to the Fourier transform of the input ~?~ through a multiplication by the system function b. This suggests an alternative procedure to that indicated by the spacedomain convolution (4.6.26) to find the system output, requiring the often simpler sequence with a detour through the frequency domain:
spaceinvariant systems: g(qo, q~)  g(qo  q ~ ) qspace
~space
~
Fourier transformation
convolution by ~
multiplication by [} 0r
1 ~o
'~
1 inverse Fourier transformation
O'q2 O'~
d
qp*~qFdq
_
12
1 2>"
(4.6.42)
The conclusive inequa, lity above follows from the estimation of the intermedia,te quantity through the steps: (4.6.43)
tile la,st equa,lity resulting from a,n integration by parts of the two integrals embraced by the modulus symbol under the assumption that the ('onlt)lex amplitude •(q) goes to zero a,t infinity more ra.pidly than v~" 1 Rela,tion (4.6.34) with definitions (4.6.35) for the signal variances in the space a,nd frequency doma,ins is strongly reminiscent of Heisenberg's uncerta,inty relation of qua,ntum mechanics. Of course, the analogy is formal only, although the use of the same word "uncertainty" may be misleading. In this respect, we further remark tha,t (4.6.34) expresses a, definite rela, tion between the energy density of the signal in space and in spatial frequency, respectively, represented by I~12 and I@ 12. Such a relation takes the ultimate form (4.6.34) as a consequence of the mathematical definitions (4.6.35) being adopted as mea,sures of the spa, ce and frequency extents of the signa,1, a,nd above all a,s consequence of the Fourier relation holding between p and @. The plain f We recall that the Schwarz inequality states that b
b
f(x)g* (x)dx a
()I)l,i(:s (l(;scI'iI)t,i()ll ()f ligtd, t)r()l)a,gat, ion ('v('n wll(:ll a ray (l(;llsil,y (lisl,ril)l~l,i(n~ ix ilw()lv(;(t, sin(:(; 1,1~('~ray (l(;ilsil,y (:m~ )(; ev()lv(;(l l,l~r(n~gl~ I,lw tra,j(;('l,()ri('s ()f 1,1~(' si~gl(; rays i~ 1,1~(; I)~(ll('. Evi(l(;~d, ly a, wavc()t)l,i(:al t)lm,s(;Sl)a,('(; r(;t)rcsent, a t,i()l~ ()f ligld, t)r()l)a,ga.l,i(n~ ix still ~()i, f(,asil)h; il~ l,('~n~s ()f 1,1~(;l~il,l~('rt,()('la,1)()ra,l,e(l (t('s(:ril)l,i()~s ()f ()I)l,i(:al wa,v(; fi(;hts as l,lwy i~v()lv(; S(:l)aral,(:ly 1,1~('~sl)a.l,ial wa.v(:fl)n~, sig~iti('~(l l)y w(q), m~(l I,l~(; fr(;(pu;~(:y st)(;('.l,r~t~, sigl~ifi(;(l I)y @(h'). Tll(;r(;fl)r(;, a .i()i~ll, r('t)r(:s(;~d,a,l,i(n~ ()f ()t)l,i(:a.1 wav(; ti(;hls l,()g(,l,lll,r i~ Slm(:(' a,ll(l fl'(;(l~t(;~(:y ~l~sl, 1)(; ('la,l)()l'al,('~(l, a,ll(l a,(> (:()r(li~@y tl~(' rt'l('wug t,rm~sfl,r laws i(l('~d, ifi('(l, i~ ()l'(l('r t,()('~al)h' a I)l~a,s('st)a(:(' (l(;s(:rit)ti(n~ ()f ()l)l,i(:al sig~m,ls m~(l l,l~('ir l)rt)l)aga,l,i(m t,l~l'()~gl~ ()l)l,i(:al sysl,(;lllS. H()wt'vt;r, w(' l~my g~lt;ss tim.l, (;v('~ wl~(;~ ,1~(' (l(;sir(;(l jtfild, Sl)a(:t;fl'(;tl~U;nt:y rcl)rest;~d, at, i()~ ()f oI)ti(:a,l wa.v('~t'(n'~s will 1)(' (;la.)oI'a.t,(:(l, "I)(fi~d,s" i~ wave()I)l,i('al I)l~as(; Sl)a('(; will still l'(;l~m.ill t)t~ysit'ally ~(;a~i~glt'ss. Si~(',(: l,l~t, sillllllt, a~(;(n~s a.n(l l)r('(:is(' sI)('(:itit:ity ()f m~y sig~m,l i~ sI)a,('(; a,ll(l fl'l;tllll;llt:y ix i~l~(;r('~d, ly li~fit('(t t)y tl~(' ~ ( : ( ' r t a i ~ t y r('lati(~, w(; ~my ('Xl)('ct tl~at ally jt)illt, s i ~ a l rt't)r(;s(~'~d,a,t,i(n~ will retle(:I, s~u'l~ a. lil~fita.ti(n~, lla,lllely, Sll(:ll il.ll ilm('.(:(;ssil)ility ()f "t)()i~t,s" i~ l,l~(; wav(;()l)ti(:a,1 l)l~a,s(; st)a('(;. Tln~s, whilst a. 1)]~as('~ sire.(:(; (list, titration lik(' a ( q  qo)(~(p p,,) ix l(,git, i~m,t('ly (:()n(:('iva,)l(' i~ g('()~('t, ri(: ()t)t,i(:s, ret)res(;~d,i~g a, si~gle ray at, % i~ tt~(; (lir(;(',ti()~ P0, tlw, a,~a,l()g()~s I)lm.sest)a,(:e dist, ri)~t,i(m (5(qq0)(~(h;h',,) ix n()t ('(n~(:eiva,)le in wave ()t)t,i(:s. Ii~ this respect, the m i n i ~ n ~ llllcert~aillt, y t)r()(t~ct, wa,v('forlllS, i.e., t;h(; Gaussia,~ wave packets like (4.6.49), m a y be int, e~(led as the signals closest to the ra.yconcept of geometri(: ()t)t,i(:s, having the l~m,ximmn a,llowed space and K e q u e n c y localization. Finally, w(; note t h a t t,h(; g e o m e t r y of a space is of course d e t e r m i n e d by the invarian('e laws u n d e r the inherent t r a n s f o r m a t i o n s of t h a t space inl;o itself. T h e geon~etI'icoptical p h a s e space is indeed symplectic since areas are preserved t)y the p r o p a g a t i n g ray t,ransforma,t, ions in accord with the m~(terlying H a m i l t o n i a n dynamics. Likewise the g e o m e t r y of the waveoptical phase space will be d e t e r m i n e d by the invariance laws of the individualized spacefrequency d e s c r i p t o r w h e n evolved by propagatioi~ t h r o u g h optical systems. Specifically it has been proved t h a t in the W i g n e r r e p r e s e n t a t i o n the wave optical phase
213
WaveOptical Picture of FirstOrder Optical Systems
space (also called Mock space) is an affine space [21]. As a conclusion, it is interesting to reInark that, oil account of the definite relation between the optical m o m e n t u m p and the spatial frequency ~, the matrix formalism of ray optics may also be considered to yield relations between the spatial coordinates and the spatial frequencies of the light waveform from the input to the output plane. In fact, taking account of (4.5.14), the inputoutput relations of ray optics, involving the ray position and slope (q,p) and the A B C D matrix, can be turned into analogous relations, involving the space and frequency variables q and t~ and a slightly different transfer matrix, where 1 the offdiagonal elements are scaled by ~0 and lco with respect to those in the rayoptics matrix. In fact, it is easy to verify the correspondence: canonically conjugate variables (q,p)
Fourier conjugate variables (q,t~)
(qo)
Of course, the matrix relation on the left is visualizable in phase space identifying the meaningful trajectory of the ray under paraxial propagation. In contrast, the matrix relation on the right has only a mathematical valence linking the conjugate variables q, n from the input to the output plane. As earlier said, it may acquire a physical content only if it is brought into relation with the transfer laws for some signal representative in the waveoptical phase space. Indeed, we will see in w 5.5 that the centroids ~ and g of any waveform p(q) and its spectrum @(t~) evolve under paraxial propagation through the matrix relation on the left of (4.6.52), i.e., just as the raycoordinates of geometrical optics. Also, such a relation comes to be crucially functional to the paraxial propagation law for the Wigner distribution function, whose arguments, in fact, transform just through that relation (see w 8.3). 4.6.6
Coordinate and frequency operators in the q and ec representations
Expressions (4.4.1) and (4.4.2) define the operators ~ and ~ in the coordinate representation, i.e., through their action on the spatial waveform p(q); namely, 0 ~(q) P~(q)   k o Oq "
qF(q)  qF(q)'
(4.6.53)
The corresponding expressions in the m o m e n t u m representation identify the operators ~ and ~ through their action on the m o m e n t u m wave function ~5(p). We recall that if the operator A defines a mapping of s into itself, every operator 6 is changed to O '  ~,OA 1, thus bearing the same relation to the transformed wavefunctions Ap as the original operator 0 to the original wavefunctions ~, i.e., A
if
p
6,
~
then
A
A (~s
5'=ASAI ' A~.
(4.6.54)
Linear Ray and Wave Optics in Phase Space
214
Then, confirming rela,tions (4.6.17), one can give the ot)erators ~, ~ in the preI)resentation a,s *"
q 
i
Oq
"
% Op'
P 
(4.6.55)
p'
It is (:onvenient in the i)reseId, (:o~text t() intr()(hwe the st)a.tia,1 fre(l~mncy ()t)eI'a,t()r ~,, just ti'a,nsla, ting into ot)era,t()r f()l'ni the s(:a.la,r relation h:  / % p , i.e.,
W(', will I,~()stly (l(;al i,~ tlm f()ll()wi,~g witl~ tl~e I,air (~,~) i,~stea.(l ()f (~,~). Evi(l(;ntly, l,l~(; ()I)(;ra.l,()r ~ (:a.~ 1)(; i(l(;~tili(;(! as 1,1~(;(lifli;r('~d, ia.ti(n~ ()l)(;l'a,t()r
i) Oq
a   i   .
(4.6.57)
when a(:tillg (hi ~(q), a,n(t as l,h(; llnfltiI)li(:ati()ll ()l)(;ra,t()r
awhen acting ()n ~(h'). Likewise, t,ll('~(:()()r(lillal,(; ()t)erat()r ~ ill tll(; h'r(;l)r(;s('~nta.ti()n writes a,s i)
io7,
ml(t ll(;ll(:(; l,lle (:()l~lllnll,a,l,i()zl l)ra.(:k(;t l)(:l,w(:(:zl ~ ml(t a, (:(nll(;S l,() !)(:
a ]  ii. The Hermitia.n ()t)era,tors {~, k:,,i}, wil,ll ill,: reh:w,,,lt Lie In',,(l,u:ts [~. ~ ]  i]', A
A
[~, I]  (), [~,, I]  (), SI)a,ll tl,(; Ileise, fl)(,rgW(,yl a,lg(;l)ra is, tl,(; sa,,m way as the ot)erator set {~, ~ , I } , lm.vi,,g il,(le,xl l,l,e sa.,,,e (:(,,,m,,,tal,(,rs (4.4.5).
4.7
Summary
We ha,ve presented a fornmliza,tion of wa,ve ()t)tics, simila,r to that of quantum mechanics 5, la Schr6dingerDirac, namely in terms of wave fllnctions ~(q) t,elonging to s and unitary tra,nsforma,tions M acting on s As g)(q) is the mathema, tical representation of light (tistributions, which are subject to an evolution under propagation through optical systeins, M is the Ina,thelnatical representa,tion of the "agents" of this evol~ltion, i.e., the optical systems. The representation of optical systems in terms of unitary operators M has been obtained by means of a quantizationlike procedure applied to ray optics, by turning variables into appropriately defined operators. As a first step, the ray varia,bles q and p have been replaced by the position and momentum A
A
WaveOptical Picture of FirstOrder Optical Systems
2 15
i o, whose Lie bracket [~, ~ ]  ~i mirrors the Poisson operators ~ and b k00q bracket of the classical variables {q, p}  1. The complete sets of eigensta,tes of ~ and b have been seen to be basic to two complementary representations of optical wavefields, carrying information about the relevant energy distributions in space and spatial frequency (i.e., direction). Owing to the Fourier relation linking these two signal representations, one can establish an inequality for the product of the space and frequency variances of the signal, only formally similar to Heisenberg's inequality. Problems 1. Consider the set of uptosecond degree polynomials in q and p: ~ 0) or (b) diverging from the virtual secondary focal point F' (if f < 0).
Transfer relation in the qrepresentation: lens linespread function gf As in the spacecoordinate representation the operator ~ has the effect of multiplying the wave function by the coordinate value q, the thin lens operator k(f) acts as a phaseonly transformer, affecting the incident wave simply by tile quadraticphase factor ci% q2/2f. The complex amplitudes p~ (q) and Po (q) of the optical waves incident on and emerging from the lens arc then linked by
_i #'Oq2 (q)

(q),
tile reference planes being a~ uslml locate(] immediately in front of and t)ehind tile lens. Unlike a section of homogeneous inediuin, a thin lens a.cts as a modulator with the phaseonly modulating function ci% q2/2f. Since, as previously noted, the chirp factor in (5.2.22) may be interpreted as the paraxial approximation to the circular profile of a spherical wave, the above relation means that a thin lens transforms a normally incident unit amplitude t)lane wave (p~ (q)  1) into a spherical wave centered about a point on tile lens axis. In particular, as illustrated in Fig. 5.3, if the lens is convergent (f > 0), the emerging wave is convergent toward the real secondary focal point F'; correspondingly, if the lens is divergent ( f < 0), the emerging wave is divergent about the virtual secondary focal point F' (w 1.8.3). Accordingly a thin lens maps linear profiles into circular profiles (and hence planar wave fronts into spherical wave fronts, within a proper 2D picture of light propagation) in the proximity of the optical axis. Under non paraxial conditions the emerging wave fronts exhibit deviations from perfect sphericity, as an effect of the aberrations. The inputoutput relation (5.2.22) can easily be cast in the integral form
Po (q)  f dq' gf (q, q')F~ (q'), J
(5.2.23)
232
Linear Ray and Wave Optics in Phase Space
ret)ro(tucing the genera,1 form (15.1.8). The imt)ulse response g~(q, q') m'ites a,s .._~q~ g~ (q, q')  k ( f ) ~ ( q  q')  c~*~/ b(q  q'). T h e a,t)()ve rela.tion (:an a.ls() t)e rega,r(h;(t a,s tim eig(;x~va.l~('~ e(t~a.ti()~ fi)r the thin lens ()t)(;rat()r k ( f ) , with the b flm(:t,i(n~s t)eing tt~(; (;ig(;~fim('ti()~s ()f ~. The (h's:(mq)()siti(m ()f tlx(; int)~t sit4na,1 i~ tel'IllS ()f ~ fllll(:t,i()llS l)r()vi(h's tl~eref()r(' 1,1~(; syste~n. I~ (,l~is r('sI)c(:(,, w(; r('(:all thai,, a(:(:()r(li~t4 t() 1,1~('~nkv ()I)tit:s I)i(:t,~r('~, a (,lfin lens (l()(;s 11(), afl'e(:t 1,1~(;ray I)()sit, i(n~, l)(,ix~g q,, = q,. C~l('arly tll(', i~l)~l, ()~lt,l)~ll, relati()x~ (5.2.22) (l()(;s 1~()1, a.(',(:()llld, ft)r l,l~(; axial t)hase slfifl, a.s t,l~(; wa.ve t)r()I)aga.tes l,l~r()~tZ]~ l,l~(; lens, (t~u' f()r i~si,a,~(:(' t() tl~e fi~i(,(:, a.Ii,l~()~gl~ ~(:gligil)h:, (,l~i('kn(:ss ()f (,l~(: h:ns. TI~(:~, if ~(:(:(h:(l, (,h(: axial I>has(: (May ~'it'()"'~'~ I~n~s(, l)(: a (), wlli(:l~ a,1]()ws ~s t,() lil~k t,ll(; s(:a,h; t,ra,llsf()r~m,ti()ll s()h;ly t,() K:~, w(; ()l)t,a,in f()r t,l~(' i~l~('i'(;~t, wa.w;()l)ti('al ()l)Cra,t()r t,l~(', ()i'(tcr(:(l I)r()(l~(',t f()r~ [~..~]I~.7] ~
M(M)

()~""~ K+ (' ;2~,.,,~"(A)K:'~ '~;/'''~K , I.t
(5.3.17)
wt~i(:tl ('~vi(lellt, ly tI'a,llsla,t(;s ill t,('rl~lS (~f lll(~ta,1)l('~('ti(' (>l)(~I'a,t()l'S 1,11(' W~',iN()rnm,n (l(~(:o~lI)()sit,i(n~ ()f si~lll)l(',(:t,i(: llla, l,l'it:cs ill ,tWillS ()f a, ~1"(~(~ t)r()l)a.ga.l,i(nl, a. t)ositivc s(:a,lil~g mitt a, lens 1,1"l'l,llSftw, slll).j(~('l, l,()th(~ (:(m(liti()l~ t h a t A > () (!i 3.5)"
M

C B L(~)S(A)T(~).
(5.3.18)
It has t)c(u~ n o t e d i~l w 3.5 tlmt ((;(:()~q)()siti()n (5.3.18) is a,lh)w('~( i~l a neighborhoo(t of the i n p u t t)lan(; in any t)r()t)agation problem. This m e a n s t h a t any prot)a,gation t)roblem can 1)(; ~m,thcnm,ti(:a,lly t h o u g h t of as a, suita)ly parameterizc( sequence of a Fr(,s~l(,1 tra.nsf()nn, a scale t r a n s f o r m and a ('hirp modulation, a,nd optically a,s a, s u i t a b l y t)aramcterized cascade of a, freemedium section, a positive scaler and a thin lens. T h e e x p o n e n t i a l opera.tor tra,ns('ription of the transfer scheme (5.3.16)in
the frequency domain needs, of course, to represent the basis operators K , K+, K 3 in the Fourier space. Using indeed the appropriate expressions given in (5.3.13), the reader may easily obtain the dual form of (5.3.17) and verify the transfer relation in the ~domain having the form (5.3.9) with (5.3.10). A
A
1D FirstOrder Optical Systems: The HuygensFresnel Integral
5.3.4
241
Imaging systems ( B O)
The case B = 0 deserves some comments since expression (5.3.5) of the line spead function seems to be meaningful only for B ~: 0. We recall from w 1.8.4 that the vanishing Belement in the optical matrix characterizes imaging systems, with the matrix entry A specifying the scaling factor by which the object is expanded or contra,cted in the image plane. A thin lens is the basic example of an imaging system, with unit magnification. In terms of wave optics, imaging means tha,t the amplitude distribution of the input signal is reproduced on the output plane, apart from possible scaling, folding about the origin and phase shift effects. The image is perfect when the corresponding amplitude distribution has not phase factors other than those present in the amplitude distribution of the input signal. A thin lens always produces an imperfect image, since, according to (5.2.22), the image amplitude has a quadratic phase term that is not present in the object amplitude. It is convenient to recast the spread function (5.3.5) in the form g(q, q')  i 2~iB ko eis2, (q'~ )2ci s2A ,i_2,
(5.3.19)
which, on a,ccount of the limit relation (4.5.6), immediately yields
V)
/2" "_~k~ 2

v/
c '
(5.3.20)
Imaging systems are therefore ('hara(:terized )y intmtoutput relations like 
2.
:,(
) 
(.
 o),
(5.3.21)
the back foca,1 length f2 A being introduce( in the last expression. c Equation (5.3.21) genera.lizes the imaging relation (5.2.22) of a, thin lens to arbitrary imaging systems. The input signal is scaled into the output signal by the matrix entry A; the factor ~/~ preserves norma,lization. As for a, thin lens, the image is not perfect due to the presence of the quadratic phase term eik~ It represents (in the parabolic approximation) a circular wa,ve converging towa, rd (if f2 > 0) or diverging from (if f2 < 0) the secondary focal point F' (Fig. 5.5), whose wavefront profile So with curva,ture radius 7~ = f~ is then ta, ngent to the output plane IIo, where the emerging signal is observed. The qua,dra.tic pha,se term in (5.3.19) is compensated if the output signaJ is observed on So, which thereby represents the profile where a perfect image ma,y be detected. If C  0 (i.e., 7~ : f2 ~ cxD), the pha, se factor disappears and the perfect image is observed on the straight profile of the endplane [i o. In that case, (5.3.21) yields the transfer relation for the pure magnifier, /
:o
(q)

/5.3.22)
242
Linear Ray and Wave Optics in Phase Space Hi
[]o
i
!~
So
(' ol !I
..t
i
:,
,
./: FIGURE 5.5. Ilnagixlg syst, czl~ (// .... ()). Tlu:
pcTfi:cl image ()f tim i~l>~(, sigzml lll~ty t)e observed
oi~ tim ('irc~lar I)r()tih' $,,. wlfi(:l~ Ira,s, ()f (:()~rs(:, tl~(: sm~(' f()nl~ as (5.2.33), In~(, ~ a y a.('(:(n~t as w(',ll fi)r a, f()l(lil~g ()f (,li(: s i g l m l a l)()~(, 1,I~(: ()rigil~.
5.4
The optical Fourier transform
11, Ires l)(:(:ll (:a,l'li(:r n:llm,rk(;(l l,lla,(, (,ll(: l:(nu'i(;r (,ra,llsl'~)rlll ()f ~1,11()l)(,i('al siglm,1 is ll()t, a,ll al)st, ra,(:t (:~d,ity. Ill fa.(:l,, it, is ())s(:rwd)l(:, f()r illsl,ml(:(:, ()ll (,If(: ])a(:k f()(:a,l l)Im)(: ()f a, (,llill l(:llS. 'l'll(: l"()llri(:r (,ra.llsf()nll is (,If(: llla,(,ll(:~m,(,i(:al (,()()I (,() (,urn fr()I~ (,I~(: Sl)a,(:(:(l()l~mi~ r(:l)r(:s(:~l,a(,i()~ (,() (,l~(: Sl)a,l,ialfr(:(1)U:l~(:y (l()~m,ii~ r(:l)r(:s(:~(,a,(,i0 )
FC~ ~:
)"
2 .//~
1
(fs) (a(L)=TrVn0n2"g, f S  v / n o n
2 )"
(5.4.8) In particular, with c~ = 1, the resulting optical systems perform the ordinary Fourier transform of the object. Specifically, it amounts to choosing all the lengths involved in (5.4.7) equal to fs, and correspondingly the length of the quadratic medium in (5.4.8) as (or, as quartic multiples of) LF  ~ ~ . Finally, we recall from w 3.6.1 the single exponential representation of the 1 matrix (8.4.6). It writes in terms of the matrix K 1(fs)  ~(fs K  + ~1K + ) !2 ( 0.  1 / f s /s)which is nothing but one half the Fourier transform matrix with 0.. ' focal length fs Explicitly, one finds F~ (fs)  ~2r
(fs).
(5.4.9)
All the above reviewed results can be read from the waveoptical perspective just replacing the inherent scalar functions of q and p by operator fimctions of ~ and ~, and accordingly the inherent matrices by integral transforms. We can therefore say that the qua,ntum harmonic oscilla,torlike Hamiltonian "" 1"2 + ~ $ 2 ~ 2 , H~.o.~p
(5.4.10)
genera, tes the optical propagator ~c~ i2k~ Is  C

+ , A ~SK+].
(5.4.11)
As F~(fs) ' F~fs is intended to connect two planes separated by a finite or infinitesima,1 distance according to whether the order ct is given a, definite specification within the proper interval ct E (2,2] or is allowed to range continuously in conformity with the propagation variable z. Of course, IF;s]  1 
F~ fs ' The formal inputtooutput relation
~o(q)
F;s  [Ffs] ~
(5.4.12)
F~is [p~(q)]
(5.4.13)
for the fractional Fourier transforming systems (5.4.7) finds its own integral representation through rule (5.3.5), which indeed yields )90 (q) 
i
ko
2TOilssin r
sin r [q' 2 cos r + q2 cos r  2qq'] / c i 2fs ko
~i ( q t ) d q t,
~  c[2"
(5.4.14)
Linear Ray and Wave Optics in Phase Space
246
Of (:ourse, the above equa,tion Inky describe as well the paraxia,1 propagation of a, light disturbance through a, quadratic index medium, if a a,nd fs are given a,s in (5.4.8), an(t so directly linked to the t)ropaga,tion length z  zi and the n,e(tiunl t)a,ra.meters ~t(), ~t~. Therefore, it t)rovides the ext)li(:it s()hlti(),, to the t)a,ra)oli(" wa,ve e(l~mti()~ f()r a (tua,(h'a,ti(" f()('~lsing nle(ti~u1~, wh()se ray ()t)ti(:s a,ccount ha,s t)ee~ (letaile(1 i~l ~ 3.9. Equa,ti~)n (5.4.14) is .jllst tlw, wa.w',ot)tical I,ra,l~s(:rit)l,i()l~ ()f I,l~(', ra,y t,ra,~sfiw rela,t,ir (3.9.1(i). It rel)r(;sents a.1~exa,nq)le of l,]w, i~d,eI'l)la:v l)(;l,w(',(',ll i~m.l,rix a.]l(l ()l)(Wa,l,t)r ]~l(;I,]~()(ls l,]~a,l, (:a,]~ (',[t'(',(:l,ivt',ly 1)een ('xt)l()it(',(l t() s()lv(; (;v()l~lti()ua,ry (liI[er(mtia,1 (Xt~la.l,i()llS. Tl~e, a:(l()~m.il~ r(;t)r(',s(;~l, al,i()11 ()f t,l~e i~l)~ll,()~lt,t)]ll, r(~la,l,i()]l (5.4.1d) is easily ()l)ta.ine(l l,l~I'()]ltKl~ l,l~(; r~h~ (5.3.1()), t)y ~()l,i~t4 tlm.t tl~e (l,lal ~m.trix F'~(.fs) writes j,lst a,s (5.4.(i) witl~ fa,l,fily l)ara,ll,t,t.(;r f~s' 1,m,,t;ly, (.fs)(.f~)Th(~ si~gh;(;xI)()~,;~tial f()rlll (5./1 11) ()f t,l~(; ()I)(wa,t,()r F '~ is t)ara,lh;h;(l t)y the 9 t~ liutll,iext)()~('~fl,ial f(n'l~s, (lir(,('l,ly ()])tail~(;(l fl'()]l~ (5.,1.7) })y a, l~(;r(' r(;l)la(:ement ()f ma,l, rir l)y ()l)('ra,l,()rs. I~1 fad',t, l,l~e wa,v(',()t)ti(:al l,ra.]~s(',ril)ti()~s ()f (5.4.7) yiel(1 1,11(;f()ll(~wi~g (,> chirp modulation,
,~(,(q)
(5.4.15) Tlle rea,~ter is ilwit~'(1 t~) writ~,, (t()wll the, t)111k ~)I)ti~:s r~'alizati~)ll ~)f tlw, (hla,1 ,)t,era.tor Ffs (,:,)rr,',st,,)l,,li,~g t , ) t h e ,hml l,la,trix F " ( f ~ ) ) , wl~,)s,; tyI)e I m~d type II i1111)]elll(;11ta,ti()]IS 1,1]nl o11t to l)e .illst the (llla]s resl)ective]y ~)f tile type II and t,yI)t', I rea,liza,tit)lls t)f F '*fs (and so F '* (.fs))" We (:()11(:111(t(', t,llis se(:l,i(nl resorting to the fre(tlwafl, ly outlill('(l a,~m,logy of the ray a,n(1 wa,veot)ti(:s spaccpropaga,tion un(ler the (tuadrati(: Hamiltonia,ns (5.4.5)a,~l(t (5.4.10) with the t i m c  e w ) h l t i o n of a, mechanica,1 (respectively, cla, ssica,1 aal(t (t~m~fl,~lm) lm,r~noific oscilla,tor. In the, light of this analogy, realizations (5.4.7) and (5.4.8)(:a,n t)e interpreted as opti(:al a,nalogs of the harmonic oscilla,tor (tyna,mi(:s, whi(:t~ is indeed optically repro(hl(:e(t at "sampled times" a,nd at "continuous times". The ana, logy with the qua,nt~m~ harmonic oscillator dynamics is further evidenced by the eigenvalue equa, tion for the fractionM Fourier tra, nsform opera,tor F~ , s
A
Fa
(q) 
(q)
(5.4.16)
1D FirstOrder Optical Systems: The HuygensFresnel Integral
247
Physically, it identifies light distributions Fn(q), whose transverse profile is not altered (apart from the complex factor I~, I nl 2  1) when propagating through a fractional Fourier transforming system. Since F~ fs can equally be interpreted as the timeevolution operator of the quantum harmonic oscillator described by the Hamiltonian (5.4.10), it is natural to look for ~ ' s in the form of the oscillator energy eigenstates, and hence satisfying the equation Ha.o. Fn(q)  En~n(q).
(5.4.17)
Resorting to the well known oscillator energy eigenstates, we then write (q) 
1
1/2
(5.4.18)
where Hn denotes the Hermite polynomial of degree n. Also, on account of tile already mentioned correspondences: oscillator
mass
+
oscillator frequency Planck's constant
1
+ &1 + ~o
(5.4.19)
tile parameter w ill (5.4.18) and tile "energy" eigenvaluc cc,~ specialize as w
@So
'
g~
1 ~ (n + ~).
kof s
(5.4.20)
A
Then, applying F~ to ~n we obtain the explicit expression of the propagation ' fs factor A,,~; namely,
F~spn(q) ci~2("+ 89
(i)~(n+ 89
(5.4.21)
The fractional Fourier transform operator F~ fs can equally be identified through the eigenvMue equation (5.4.21) being required to be solved by the HermiteGauss wave functions (5.4.18) with eigenvalues An  (  i ) ~(n+l). In particular, with c~ = 1 one recovers the wellknown property of the HermiteGauss functions being eigenfunctions of the ordinary Fourier transform operator: t"
1
Ffs %On(q)  (   i ) n +  ~ n ( q ) .
(5.4.22)
Relation (5.4.22) for n = 0 is in accord with (4.6.51) on account of (5.4.4). It is worth noting that in the original formulation by Namias [7] an eigenvalue equation similar to (5.4.21) (see Eq. (5.4.42) below) was proposed as defining equation of the fractional Fourier transform operator, which is then
248
Linear Ray and Wave Optics in Phase Space
conceive(t to have the same eigenfllnctions of the o r d i n a r y Fourier t r a n s f o r m but with eigenva,lues sepa,rated by a, fra,ction of the, inmgina, ry unit. E q u a t i o n (5.4.21) merely ret)roduces the time evohltion of the, os(:illator energy eigenstate, s, whi('tl in fa,(:t ()(:curs t h r o u g h the, t)lmse fa,(:tor (~is 1 (i)'~("'~+~). It howev('~r states s()nlethillg new, sllggesting t() i~ltert)rel, tim timeevohlti()ll ()t)era,tor ()f tlle (tlla.lltllnl ()s(:illa,tor a,s a, fra,(:ti()lml F(mrier t,ra,l~sform ()t)era,t()r. Tl~is is a rat, l~(;r g('al(;l'a,1 r(;s~flt rega,r(ling il~(h:e(l ally (l~a.(lrati(: tta,nfilt,()liiali a.ll(1 ll(;ll('('~ rely l)a.raxia,1 wav('~ l)l'()I)aga,ti()li, sill(:(;, as w(: will s(:('~ ill t,he ~(;xl, 1)aragra,I)h, a,lly AH(7I)ild,(;gra,1, i.('~, m~y l)araxial ()t)ti(:al l)r()l)~,g~tl,()r, is r('la,t,(:(l t() t,l~(; fl'a(:ti()~al F()~ri(;r 1,ra,llSf()rlll. A(l(liti()l~ally, sil~(:(', t,lm, ql)r()tih; ()f tl~(; (;ig(',~u~()(l(:s ()f a (l~m,(lI'a,l,i(" f()('~sing nn',(ti~n~ ~()(l(:l .i~sl, as (5.4.18), E(t. (5.4.21) (',a,l~ a.s well 1)(: i~m:rt)ret(;(t ('n+ 7 ) eigel~l~()(l(;s, t,l~(; (:()rresl)(nnli~g t)r()l)aga,ti()~ (:()~st, a,~ts t)(;i~g s() s  ~ . ) ~ . Tl~is, ()f (:(n~rs(;, (:(n~tir~s tin: (;arli(;r ~eld, i()~(;(l view ()f t,l~(; fl'a.(:t,i()lml F()~riex l,rmisf()r~ ()l)('rat()r as tin; ()l)ti('al l)r()l)agat,()r il~ a (l~a.(lI'a,ti(' f()(:~si~g ~(',(li~l~. lh;lati(n~ (5.4.21) ('(n~l(l (:(t~ally I)(; ()l)tail~(:(l fl'()~ (5.4.22) ()ll H(:(:()llllt ()f l,l~e s(',(:()l~(l ()f (5.4.12)', l,lm (lis('~ssi(n~ a,l)()ve el~q)l~asizes l,lw r()le ()f F '~Is a,s a,n l
A
5.4.3
AHCI)
i'ntq]'ml and fl'actional t, bv,'~'icr t'ra'n,@~'lv.,. Opt'tarot
('quivah':nt of the: Jwa,sau,a matri:r ,s'ynth.c,s'i,s A(:(:()r(li~lg t() tin; Iwasawa (l(;(:(nlll)()sit,i()ll ()f syllll)l(x:t,i(', llla,l,l'i(:(;S (!i 2{.7), every tirst()r(ler ()l)ti('a.1 sysl,(;lll (:ml 1)(~ sylgll(;size(l 1)y a S(:(llWll('(: ()f a i)lmset)la,ne r()tat()r, a I)()sitive s('ah;r a ll(l a t,llill l('~ls, as (;xI)li('itly ('xl)r('ss(:(l )y
M = L(7))S('~,~)F '' (.f.~), the real I)a,ra,Ineters 'I1~,, 7) ml(t 4) )eiIlg B2 tan r ~tt~'2  A2 + 7~s'
m "f s '
T) 
~ m2
( A C + D~.~s, ) "
(5.4.24)
Here an in genera,1 non 11nit vahm of fs has been (:onsidered with respect to the parameterizatioIl re, p o r t e d in Eq.(3.7.3). Thus, simply tra,nslating i~ opera, tor form the above ma,trix p r o d u c t , we see t h a t cve, ry optical t)r()t)agator ca,n be represented a,s the ordered product of a lens ot)erator, a scale o p e r a t o r and a fractional Fourier t r a n s f o r m operator: A
A
A
A
M(M)  L('P)S(m)F~fs'
(5.4.25)
with relations (5.4.24) for the involved parameters. The exponential representations of the three components of the product above in terms of the algebra basis {K_, K+, Ka} are respectively given by (5.2.21), (5.2.27) and (5.4.11).
1D FirstOrder Optical Systems: The HuygensFresnel Integral A
A
249 A
Clearly. while F~fs generates the diffraction integral (5.4.14), S(m) and L(7)) manifest on the incoming signal Pi(q), respectively, through the scaling operation (5.2.33) and the chirpmodulation (5.2.22). Any ABCD system can therefore be represented by the transfer relation 2
o(q) 
1
[Fs
ikoT)~2 ~"c~
m
(5.4.26)
which expresses the amplitude distribution ~Po(q) over the output plane as the scaled fractional Fourier transform of the amplitude distribution pi(q) over the input plane with a further quadratic phase term, and hence as the imperfect scaled fractionM Fourier transform of the input (~). The above offers an equivalent representation of the ABCD integral, which then ca,n alterimtively be characterized by the scale factor m, the curvature parameter 7) and the angle r and hence the order ~ of the fractional Fourier transform. Evidently, 7) specifies the curvature radius T~   ~ of the circular profile on which the perfect (scaled) fractional Fourier transform can be observed. When 7)  0, the quadratic phase factor disappears and the perfect (scaled) fractional Fourier transform is observed on a planar profile ( T ~  c~). The pure fra,ctional Fourier transform corresponds to m  1 and 7)  0 . The ABCD integra,1 represents the explicit fim('tional solution to any paraxial propagation problem ruled by a polynomial Hamiltonian quadratic in q and p. Ae('.ordingly, the LST and LSF (~ decompositions (5.3.17) and (5.4.25) yield a valuable solving tool permitting, in the same way as their matrix counterparts (5.3.18) and (5.4.23) do for the relevant matrix equation (4.3.4), to decompose the overall problem into three subprobleIns, whose dynamics might be more easily investigated. This may simplify the solving procedure and/or suggest interesting and enlightening analogies. It is therefore evident the physical and mathematical relevance of decomposition rules in relation with evolution problems ruled by secondorder differential equations. In particular, relation (5.4.26) will be used in w 5.4.5 to relate the freemedium propagator to the quadratic medium propagator, i.e. the Fresnel transform to the fractional Fourier transform.
5.4.4
Fresncl and Fourier transform
It is interesting to investigate the link between the Fourier transform of a wave field and the relevant farfield pattern. To this end, we transform back to the qdomain the frequency spectrum @o(~, Zo) of the freepropagated signal c One m a y talk of imperfect fractional Fourier t r a n s f o r m in the same sense as one talks of imperfect Fourier transform. An imperfect fractional Fourier tr a n s f o r m displays a residual quadratic phase factor in the o u t p u t variable with respect to the basic performance (5.4.14).
Linear Ray and Wave
250
Optics
in Phase Space
amplitu(te ~o(q, Zo) over the distance z o  %. T h e n , using (5.2.18), we o b t a i n
1 / ,/5;
/ cigq~ ~o(q, Zo)_ ,~if;. l dh; J 92o(fl;,Zo)
(/h;
citW i(z~ c
2"()to
~i(fl', z i )
'"
(5.4.27) W r i t i n g z o  %5 f a,n(1 r(;lmn(tling the a,rgmllent ()f the (;xt)()ll(;ld, ia,1 flm(:ti()n, 1,11(; a l)()ve i n t e g r a l ('a,n t)e r(;(:a.st in tlx; form ~,,(q,z
,
+(')
/'/ d,~'  , i ~""~"' ~' (,_,,,~k(),., ' )
 ~l , ' i ~ , F

wlfi(',h, ()11 a.(:(:()lnfl, ()f tim r(,lal,i()11 (4.5.(i), in tll(: lilllii, t'~ OC a ll(l fillit,(; extent
()f the, signa,1, tllrlls i11t() 2,
f ,oo
V
if
~ i
(
f
, z,) 9
(5.4.29)
TII(; a,l)()v(; r(;lati()ll sillll)lY says tlmt 1,11(; farIi(:l(l l)att(:r11, i.(;., tll(: fi'(;ely t)r()t)a,ga,t(;(t fiel(t (listril)llt, i()ll at larg(; (listml('es fl'()lll tim illl)l~t, 1)la,n(; (f* oc), is (;(t~iwd(;Ilt, a I)a.r(, fl'()~ a. (l~a(lra,ti(: l)l~as('~ fa,('t()r, t() tim sI)al, ial F()~n'ier I,ra.nsf()n~ ()f tl~(~ i~lln~t sig~ml (tl~(', ~,~arfi(',l(l (listritn~ti(n~), (wal~m.t('(l at fi'('.(l~l('a~(:ies "o%q h', f . A(:(:()r(lil~gly, tl~('~ ()l)s(~rv(;(1 sigxm.1 a(:r()ss tl~(; 1)a(:k fi)(:al l)lm~(; ()f a. tllin l('~s (:an t)e ~(l('rsl()()(! as l,l~(' far ti(;l(l I)al,l,('r~ ()f 1,1~(' i~l),~l, sig~al, if fr(:(;ly t)r()t)a,gat(;(l ()vex la,rg(; (listm~(:(;s. R,(;la,t,i()~l ( 5 . 4 . 2 9 ) (',xl)r(;sses 1,1~(; w(;llk~x)w~ r(;s~ll, ()f (lilh'a.(:l,i()~ tl~(:()ry a(:(:or(ling to whi(:ll fi)r large (lisl,m~(:(;s a,ll(1 fillit,(~ ('.xl,(;lll, ()f l,l~('~sig;~m,1, t,l~(; Fr(;snel (tiffra(:tion t)ass('~s i~d,() tile l"ra,~u~h()fi;r (lifli'a.(:ti()l~ [(i]. W(~ will see i~1 tlx; next subse(:tion tha.t 1,11(; "gal)" ])(;l,w(;e~l t,h(; Fres~M a.ll(l Frm~l~l~()fi;r trmlsf()r~ns is filled t)y a ('ont, ilnn~x~ ()f fl'm:ti()~m.1 F()~u'i(;r traa~sfi)r~s, tln~s r(~l)r()(l~(:i~lg f()r a, h o m o g e n e o u s n l e , ( l i ~ 1,11(; v i e w ()f t)r()t)a.ga.ti()~ tl~l'(),~gl~ fl'a.(:ti()~m.1 trmlsform t)la,nes s~ggest(;(t f()r a, (l~m(lra,ti(: f()(:~lsing me(li~n~l.
5.4.5
Fvcsncl and fi'actional kbuvicv transform
Interestingly, tlle ext)ressi()II (5.2.9) of the Fremml illtegral (:()llll(',(:tillg two pla: na, r surfa(:es can t)e a(tai)te( t() the ext)ressi(m (5.4.14) ()f the fra(:ti(mal Fourier tra,nsform, thus pa, r a l M i n g the relation of the Fralmh()fer integral to tlle sta, nd a r d Fourier tra,nsfonn, previously discussed. In fact, it is ea.sy to see t h a t the Fresnel t r a n s f o r m ('a,n )e i n t e r p r e t e d a,s a ~lm,gnifie( fra(:tional Fourier t r a n s f o r m with a, residlml t)hase(:urvature. Ext)li(:itly, by (5.4.26) one m a y (ecomt)ose the f r e e  m e d i u m t)r()t)a,ga, t()r T(d) into T(d)  L(P)S(m)F~fs '
(5.4.30)
w i t h the parameters m, 7) , (b specifically given as m2
_
1.5
d2 fs2 '
7)
_ ~2fs2 d ,
tanr
~d.
(5.4.31)
1D FirstOrder Optical Systems: The HuygensFresnel Integral Wavefront profiles
q
t~...,..~._
I
~
i
: w z)/ I
9
I
251
~
1
',. I
I
.
,
..,
l
i
Z
FIGURE 5.6. Schematic of a Gaussian beam, showing the parabolic profiles of the wave fronts and the hyperbolic profiles of the beam width. Here, the scale p a r a m e t e r can be conveniently set to fs = k0T h e Fresnel diffraction p a t t e r n can therefore be i n t e r p r e t e d as the scaled fractional Fourier t r a n s f o r m of the signal diffracting from the e n t r a n c e plane Hi. T h e t r a n s f o r m is observed on the spherical surface Eo with c u r v a t u r e radius n o  d(1 + ~ ) ,
(5.4.32)
a,t dista,nce d from Hi. Tile order of the t r a n s f o r m increases monotonically with the (tistan(:e d. As d ~ oc, tile angle 4) apt)roaches ~ and hence tile fractional t r a n s f o r m turns into the ordina,ry Fourier transform. Correspondingly, since d and ~ o ~ d, one recovers the farfield diffraction p a t t e r n , which according to (5.4.29) is the Fourier tra.nsform of the incoming signal. A unifying m a t h e m a t i c a l description of the diffraction in a free m e d i u m is therefore provided by the fractional Fourier t r a n s f o r m formalism, which embraces the ordinary Fourier formulation as limiting case. Notably, tile a m p l i t u d e distributions on the t)lane IIi and on the spherical surface Eo of curva,ture radius ~ o are directly connected by a fractionM Fourier t r a n s f o r m a t i o n . T h e expression (5.4.32) of T~o closely resembles t h a t for the c u r v a t u r e radius R of the wavefront at d of a Gaussian b e a m having its wa,ist at z2
the plane IIi, i.e. R ( d )  d ( l +  ~ ) ([8], see also w 9.5.1). T h e scale p a r a m e t e r fs takes t h e n the m e a n i n g as the Rayleigh length z R of the beam: z R _ ~lkw2o , Wo denoting the waist radius (a). Besides, the magnification factor m has the form of the normalized b e a m width ~ at distance d from the waist, which, according z2 to the law ~ 2 ( z )  ~2(z) 1 + ~ , yields rn 2 ~s in (5.4.31) with z  d and 2 w0
R
zR = fs. Finally, the order c~ of the fractional t r a n s f o r m is p r o p o r t i o n a l to the d We recall that when dealing with the propagation laws of Gaussian beam parameters, the z coordinate is commonly measured with respect to the beam waist.
Linear Ray and Wave Optics in Phase 5pace
252
G o u y I)hase ( of the bea,m, which writing
((z)  ta,nl(~z ),
(5.4.33)
turns int() the a,ngh~ O ()f tl~e tra,nsfl)i'u~ by t)roperly setting z = d a,~(t z u = fsTlu~s, we (:a.n say t h a t t,l~e t)r()t)aga,ti(m ()f a, Gm~ssia,n t)ea.m i~ fl'ec st)a,(:e lm,t~u'a,lly ()(:(:~lrs tl~r()~gl~ s(:al('~(t fra(:ti()na,1 F()~rier tra.~sfl)r~mti()~s, wl~()se or(t(:r is (lir('('tly li~k(,(l t() l,l~(, (,~()~y I)l~a,s(; sl~ift wl~(~r(,as (,1~(' s('a,l(' fa('~.()r a('('()~u~ts fi)r tt~(~ variati()~ ()f tl~(~ 1)(,a~a wi(ltl~ witl~ l)r()t)agati()~l. 5.4.6
7'he A H C I ) integral as a scaled fi'actional b'ov,'l'ic'l' t'l'a'nsform of a scaled i'npu, t
Un~h,r (h;tilfite, (:(m(lit, i()lls, the ra.y lim.trix M ca,ll 1)(' (le(:()llq)()se(l as M
(5.4.34)
S( ,,,.2)F'~(.f.~)S( ,,,, ).
(:()llv(~yillg l,l~(~ vi('w ()f t,ll(', ()l)ti(:al syl,(~lll (les(:ril)(',(l I)y M a,~ l)(',rfi)rllfillg tim fi'a,(:ti()lml F()ln'i(',r trmlsfi)rlll ()f tim i111)llt, sigxml witll tw() S(:a.l(~ fa(~t()rs, ()ira fi)r S()lvi~g (5.4.34) fl)r t,l~(' r(d(wmd, Imral~(q,('~I'S, l)r()vi(l(~s ill(' g(~('ral (~()~(liI,i()x~ ~u~(l('a' whi('l~ (le('()~l)()siti()~ (5.4.34) is fl'~a,sit)l('~. W(~ ()l)tai~, i~ fa(:t,
(~()s" 0  A D ,
(5.4.35)
() dz  '
(5.5.12)
which represent the definite form of Ehrenfest's theorem. Evidently, although Eqs. (5.5.12) reselnble Hamilton's equations of geometrical optics, one can not say that the expectation values (~} and (~) follow the laws of ray optics, unless the following identities (5.5.13) can be stated. It is easy to verify that the above identities hold true only if the Hamiltonian operator is a quadratic function of ~ and ~, and hence only in the paraxial approximation. In that case only, the expectation values of the position and momentum operators follow tilt "classical" motion.
Further czploring the correspondence between the ray and wave optics pictures. II
5.5.1
Continuing the line of the presentation in ~ 4.4.4, we reconsider again the evolution equations for the Heisenberg operators ~(z) and ~(z), we rewrite in the slightly different form dz
d D(z)  [D, 7~], dz A
A
with 7  I   i k o H .
(5.5.14)
Linear Ray and Wave Optics in Phase Space
260
R e p h r a s i n g the presentation of w 1.7.1, concerning the q u a d r a t i c polynomials and the associated m a t r i x representatives under Poisson brackets, we (:onsider a.s a g(;~eral H a m i l t o n i a n o p e r a t o r the t)ilinear form in ~ and ~: (5.5.15) A
A
T h e Lie t)ra,('k(;ts [~, ~] a,,~(1 [~, ~] ('()m(; t,() t)(', li,~(;a,r in ~ a,~(l ~, t,('~ing, in fa.(:t,
[~, 7~]  a~  (' ~,
(5.5.16)
wlfi(:h a,ll()ws ~s I,()i(l(u~tify 1,1~('~,xm.t,rix r(',l)r('s(',~l,ativ(, ()f ~ a,s
H(
b) '  a(: (:
(5.5.17)
(?l('arly t,ll('~ tra,(:(;h:ss llm,tI'ix H 1)(',l()~s t() ~,1~('~sy~q)l(',:ti(: alg(,~t)ra .,q)(2, R). TI~(,~ tt(:is(:~fi)(:rg (:(l~al,i(n~s (5.5.14) ('a.~ l,l~(~ 1)('~r('~(:ast i~l,()a ~m,l,rix r('la,l,i(n~ si~ila,r t()!';(I. (1.7.4) fl)r tl~('~ ray v(;(:t()r u; l~m~(:ly, d dz~(Z)

HN(z),
w(zi)~r+
(5.5.1S)
i,,v()lvi,~g tt,,: ,,l,,,ra, t()rval,,(:(l v(','.t()r ~ ( z )  (~(z), ~(z)) T (s,:(; !i 4.4.4). It, ix illll(',r(;llt i~l tll('~ a,1)()V(; ('*tlm,ti()ll a, lill('ar trmlsf()r~lla,ti()ll f()r~ a,n(t ~, wlfi('ll ret)r()(llu:('s tlm.t f()r t,ll('~ cznlnfi)(;r ray va,rial)l(:s q ml(! p. Ill fa,('t, siil(',e the (:hang(; ()ftll(', v(','t()r ~ ( z ) ()ver a,n infilfit('si~lm.l (tistml('('~ d z fr()Ill ttl(; initia,1 l)lan(; zi ix (;t[(~('t(',(l t)y t,ll(; 2 x 2 symt)l(;('ti(" Illatrix ~'H(:')'l:a,('(:()r(liilg I,()
(5.5.19)
'~r(zi Jr dz)  ('H(zi)dz'vr
the evoluti(nl ()f @(z) ()ver tll(; finite distml(:(; z  zi ix (:ffi;(:t(;(l t)y a , l l infinite ordered string ()f 2 x 2 nm,tri('(;s like tha,t, a,t)()v(;, whi(:ll ev(mtlm.lly y M d s for the opera,t()rs (~, ~) the, sa,1~(,, finite linear symt)le(:ti(" tra,nsf()rmati()n ruling the evolution of the rayva,ria,)les (q, p). Interestingly, if w(; ('()nv(',niently define the ot)erators A
A
/C  ik0K
A
,
A
A
/C+   i k 0 K + ,
A
/Ca  'ikoK:~
(5.5.20)
we can rewrite the comInuta.tion relations (5.2.2) a,s [/C+,/C_]  2/Ca,
[/C3,/C•
 :F/C+,
(5.5.21)
which, of course, p e r t a i n to tile metaplectic Lie algebra mp(2, R), and exactly mirror the c o m m u t a t i o n relations (2.3.8) of the sp(2, R) m a t r i x basis
261
1D FirstOrder Optical Systems: The HuygensFresnel Integral
{K_, K+, K s }. Also, by rule (5.5.16) the expected correspondences between the rap(2, IR) operator basis, and the sp(2, N) matrix basis are recovered, being
]~_K_ (0 1), J~+~K+ (~176
]~3 ~Ka 1(01 ~
The inverse a,rrows, from matrices to operators, work through the operator form of (1.7.11), which writes ~(~, ~) _ 1 ( H e ,
J@)
 I@THTJ@.
(5.5.23)
An isomorphism can evidently be established between the symplectic and metaplectic a,lgebras sp(2, R) and rap(2, R). We will see later that the respective groups are isomorphic up to a sign.
5.6
Waveoptical propagators as unitary representations of linear canonical transformations i
It is noteworthy that the waveoptical propagators M(z, zi) in their integral transform representation (5.3.5) arise as unitary operators acting oil tile Hilbert space s of wave functions, which correspond to unita,ry tra,nsformations mapping the HeisenbergWeyl algebra operators ~ and ~ into real linear combinations of themselves [10]. As said in w4.4.4, the propagation of a wave fieht through an optical system described by the propagator M reflects into the the similarity transformations (by Ml) of the Heisenberg operators ~(z) and ~(z), which then evolve from tile SchrSdinger operators (5.1.2) according to
q(z)
Ml(z, zi)q(zi)M(z, zi),
p(z)  Ml(z, zi)P(zi)M(z, zi),
(5.6.24)
where, of course, ~(zi)~ and ~ ( z i )  ~. On the other hand, guided by the considerations developed in the previous section, showing the formal identity of the evolution laws for the qnumber pair (~(z), ~(z)) to those of the cnumber variables (q(z),p(z)), we may relate the propagated operators ~(z) and ~(z) to the input operators ~(zi) and ~(zi) through the optical matrix M of rayoptics. So, we explicitly write
"~(z)  A(z, zi)'~(zi) + B ( z , ~(~)  C(~, ~ ) ~ ( ~ ) + D(~,
zi)~(zi), ~)~(~).
(5.6.25)
The unimodularity of M guarantees that the evolved operators ~(z) and b(z), as ~(zi) and b(zi), obey the Lie bracket (5.1.3)" [~(z),~(z)]  [~(z~), ~(z~)] i ~ .
Linear Ray and Wave Optics in Phase 5pace
262
Th(~ question arises about the st)e('ific flmctionM form of the tra,nsfer op
erator M(z, zi), which through (5.6.24), leads to the linear ca,nonical transforina, tion (5.6.25) for q a,n(t p, nam(;ly
MI(~,
~)~(,~)M(,, ~)
~r
zi)p(z,i)~C~(Z, Zi)
_ A(~, ,~)~(,~) + 

~(z, ,~)~(~),
C(Z, Zi)q(Zi) J D(z, z,i)~(z,i) .
(5.G.2G)
Ill ()tlmr words, wc ar(" tryillg t()(t('till(' a ~mt)l)illg ()f sylllt)le('ti(', illatri('~es M into llnitary ()t)(~ra,t()rs M" t..
M E Hp(2, I~)
~ M C ]Ilp(2, IR),
(5.6.27)
t)a,se(l ()~ the r(',(lllil'(~llleld, tla.t t,ll(', ()l)(~I'al,()rs q(z), p(z) f()lh)w tll(~ S~t,lll( ~, ~m,trix law ()f tim ray varial)h~s q(z), p(z), wlml~ tim wave li(;l(l l)r()I)aga,t(~s tl~i'()~gt~ tim systelll, a u(1 h(ul(:e tlm ('()rresl)(nl(li~lg wave fiUl(:ti()ll is a,(:t(~(l )y M. Evi(h;ntly, siu(:e 1)()th M al~(l s (ud,(:r (5.6.26), ally ()t)(u'at()r I~l, satisfying (5.6.26), is (tetilm,I)h~ llI) t()all a,rl)itra,ry t)lm,se fa,(%()r. ()llr alla,lysis t)eh)w is ai~lm(t at (~xt)h)I'il~g tt~(; l)()ssil)ility t() ~u~i(t~mly (l(~fi~(' m~('l~ a I)lm.s(~ i~ a,(:(:()r(t witt~ tlm (tesir(;(l I)r(:s(:rw~ti(n~ ()f tl~(~ gr()~ t) str~(:t~u'(' ()ftlm s(:t ()f M's. We will see that the t)lm,se il~(t(~t(:n~i~a~'y (:m~ 1)e r(~(h~(:c(l ~l) t()a sig~. As s~lggeste(t t)y (5.1.8), w(' a s s ~ ( ' tl~at M(z. z~) ('()~('r(,tiz('s i~t,() tim integral tra~sf()I'~
M(z, zi)r
zi) 
/. dq'G(q, q', z, zi)r
zi),
(5.6.28)
s() tlm,t the prolJ(u~ ()f (l('.t(~r~lli~i~g M is t~u'~(',(l i~t,() tl~a.t ()f (l(~t,er~i~fi~g the functioIml form ()f th(~ k(~'~(~l G(q, q', z, z~) a,ss~ri~g tl~at r(da.ti(n~s (5.6.25) hoht. We rewrite rela.ti()~s (5.6.26) ill tile e(t~fiwde~d, f~n'~
~ ^M  M(A~~.+ B~), ~M  M(C~ + D~),
(5.~.~9)
the dependence of the vari()~s fim(:tions and ot)erat()rs Oil tim axial (:()ordinate z being implie(t. Acting (n~ tim wave flm(:tion ~(q) t)y tim ()t)(;I'a,tors on both sides of (5.6.29), we ot)ta,in the i(l(u~tities
f dq'G(q, q')(A~ + B~)~(q')  ~ f dq'(E(q, q')~(q'), y J
(5.6.30)
On account of (5.1.2), we may integrate by parts the leftha,nd sides of both
equations. Then, from the resulting identities, that must hold for any 7:(q), we obtain the following set of linear partial differential equations for G(q, q')" i B o )G(q, q')  q G ( q , q'), (Aq'+ ~ ~ o ~(q, q,)
(cq' + ~oD ~ )C(q, 4)
~o~
(5.6.31)
1D FirstOrder Optical Systems: The HuygensFresnelIntegral
263
We may propose a solution in the form
G(q, q')
(5.6.32)
 7 e i(#q2+~'q'2+rqq'),
with the constant parameters (i.e. independent from q and q~) 7, P, L,, 7 being to be appropriately defined. Indeed, inserting (5.6.32) into (5.6.31), we obtain for p, u and 7 the definite expressions / ]   ]go a
P   k02B
7

(5.6.33)
k o 1

In order to specialize ~,, we resort to relation (4.3.21) (adapted to G), which reflects the unitarity of M. With (5.6.32) and (5.6.33), Eq. (4.3.21) yields 712
k~
(5.6.34)
2 IBI' by which 7 comes to be defined up to a phase factor: 7  ei~
27r[BI 9
Accordingly, we can establish for G(q, q~) the flmctional form k0B C~" 2" ~ (Aq'2bDq22qq') G(q, q')=c i( i 2~ri
(5.6.35)
with an undefined phase ~. It, couht seem (:onvenient, in order to form a onetoone mat)t)ing of symplectic matrices M into operators M, to envisage a rule to assign a unique phase 0(M) to every matrix M. With such a choice, however, the resulting set, of operators M(M) (to not close under multit)lication, an(t hence it, is not a B1) and M 2 _ (A2 group , a s it would be desirable 9 In fact, let M ,  (A1 c 1 D1 C2 D2 h
\
\
/
B2) \
2"
be two syinplectic matrices, which are imaged respectively into i 1 aIl(] i 2 , with r e p r e s e n t i I l g kernels (~1 a n d C__g2 . Let M  (A B) denote the product matrix M = M 1M~; of course, ]
\
J
( A1A2+ M

B1C2
A1B2~
B1D2)
A 2 C 1 Jr C 2 D 1 B 2 C 1 Jl D 1 D 2
(5.636) "
Finally, let M12 and M respectively denote the operator resulting from the product M~2  MIM 2 and that corresponding to the product matrix M through (5.6.35). From (5.6.28) it follows that the representative kernels G 1 and G 2 of i I &rid M 2 compose as (see also Eq. (4.3.18)) 1.
A
G12 (q, qt)
_ / dqttG1 (q, qtt)(~2 (qtt qt).
(5.6.37)
In order to verify that the product of operators exactly reproduces the multiplication of matrices according to A
M(M)
t".
A
 M 1 (M1)M2 (M2) ,
(5.6.38)
264
Linear Ray and Wave Optics in Phase Space A
A
for a.ny choice of M 1 a.n(t M.,, we will work out the explicit expression of G~2(q, q') in terms of the entries of the product m a t r i x (5.6.36), a.n(t thenA we will comI)are it with the exI)ression of the representing kernel G(q, q') ()f M. By (5.6.35), the integral in (5.6.37) yiehts G~2( q qt) __ r162
_
_
I"0
,'t~ 2 ('42
~.
2
/__~.q2)
u2q' + "~
k~ (,i~l (,i~'2 r ~sg~(B,HB:t ) r 2n"/%13 (' i 21~
is t)()sitiv(: ()r ~(:gal,iv(' (s(:(; fi)()l,~()t(; (n~ I). 226). ()n tl~(', ()tl~(:r lm.~(t, tl~(; r(',t)r(',s(u~t,i~g k(;r~(~,l G(q, r / G(q q')  (,i~ ~/ l,.o
dq"('
ikO[ !~ qtt22(,___g__F,_51~__)] '2 " ~ ' 2
"~
"'2
+l)q 2 2qq'),
()f M is
i kO ( Aq~'2 + D q 2 _ 2 q q ~)
Evi(h',~tly, fl)r t,l~(', i(l(',~tity (5.6.38) t() l~()ht, (n~(', sl~()~l(l lmv( ~
7r u ) ((M)  ( ( M , ) + ( ( M ~ ) +  ~ s g , ~ ( u, n~ 9
71C(M)   7~sg~(B),
(5.6.41)
(5.6.42)
1)y wlli(:ll We evi(h',lltly fi)rlll a (nl(',t,(~(nlC a,ssr 1)etweell syllll)h;cti(: rim,trices zul(l zllt,'tat)h;t:ti(: ()t)erat(n's. Bllt witll tiffs ct~oi(:c tl~c t)llase ()f tim l)rottuct era, tors 1)y the, ternz
)] (5.6.43) which in(h;t;(1, as tilt; r(;a(tt;r may v(;rify t)y direct cht'.t:k, tlu'xls i~lto 0 or +re according t() the signs ()f tllc involve(t Bentries. This mea.ns tha,t the set of opera.tors M ( M ) we obtain with this choice do not form a grout), since on multiplying two opera,t(ws, M~(M~) a.n(t M,,(M~), it (:ould hat)t)en that, depending on M l and M2, the result be  M ( M ) , which is not in the set. Therefore in or(ler to obtMn a. group, wel.mllst let t h e c o r r e s p o n d e n c e (5.6.27) be not onetoone, allowing tha,t both M ( M ) and  M ( M ) be a,ssociable to each symple('tic m a t r i x M . The resulting set of operators closes under multiplication, a.nd yields a twovalued representation (r of Sp(2, IR), imaging linear e Note that this is not strictly a representation, which is intended to admit only one "image" corresponding to possibly more than one "object".
1D FirstOrder Optical Systems: The HuygensFresnel Integral
265
canonical transformations of the HeisenbergWeyl algebra operators ~, ~ into transformations of the nilbcrt space s of wave functions ~(q). The integral kernel G(q, q') for the operator M corresponding to M is then, k0 5~ G(q, q')  t~/ 27riBr (Aq'2+Dq22qq'),
(5.6.44)
thus evidencing that M is uifiquely identified by M along with the choice of the sign. The phase of the square root is taken within the interval (  ~ , ~]. It is a rather common convention to write the kernel C, without the sign, letting implied the possibility that two unitary operators, M(M) and M(M), may correspond to every sytnplectic Inatrix M. Evidently, the line spread function gABCD(q, q') in (5.3.5) has been written in conformity with this convention. In the next paragraph, we illustrate an example of this sign ambiguity. As a conclusion, we may say that, whilst an isomorphism holds between the symplectic and metaplectic algebras sp(2, R) and rap(2, R), the respective groups Sp(2, R) and Mp(2, R) are globally homomorphic (precisely, isomorphic up to a sign).
An example Let us consider the optical Fourier transform, which is represented in ray optics by the symplectic matrix F(f)
1/f 0 '
and in wave optics by the unitary operator FI, discussed in w 5.4.1. We have seen in w 3.6.1 that tile Fourier transform matrix is cyclic by 4. A cascade of four identical Fourier transformers compose to the identity; namely,
F(f)F(f)F(f)F(f) I
(5.6.46)
In contrast, the operator Ff is not cyclic by 4 but by 8, thus differing from the "mathematical" operator 9c, which is indeed cyclic by 4. Evidently the different periodicity of the "optical" operator Ff with respect to .7 arises from the phase factor c  i 4 , which basically distinguishes between tile optical and the mathematical Fourier transforms. In fact, propagating an optical signal ~(q) through the cascade (5.6.46), one obtains at the intermediate planes, firstly (q)
_
_
y
and then (q)

l(q)

(5.6.47)
266
Linear Ray and Wave Optics in Phase Space
aga,in leads to
So, ot)tically F o u r i e r t r a n s f o r m i n g
~.~ (q)
F f ~2 (q)  

(5.6.49)
i/@(~q)"
F i n a l l y after a, fi~rther tra,nsfiwm wc (',u(t ~t) a,t the. exit t)la,lw, witll ~., (q)  F / ~ : , (q)   r
~Fhis nl(',mls ,lm.t
#} 
(5.6.50)
i,
a,n(111(;~1(:(;, wil,l~i~ tl~(' f()rl~mlisl~ ()f 1,11(; a,l)()v(' l)aragral)l~, tim t,
IVI(F)IVI(F)IVl(F)M(F)  IVI(F '1),
(5.6.52)
1,11(; nlim~s sigl~ sig~lifyi~lg a. l)l~ase sllift 1)y 7r ()f 1,11(; ()l)l,i(:al wav(; g()i~lg l,ln'()~@l 1,11(' 4F()~ri('r l,ra.~sfi)r~l(;rs (:as(:a(l(; [,().~()]. Tl~('r(;fi)r(', ()l~ ()~(; si(h' r(;la, ti()~ (5.(i.4(i) sig~ifi('s l,lm,t a, 271 r(),ati()~ i~ l)tlas(; l)lml(; s~lfi('('s t() l,~n~ l,l~(' ray v(;(:t()r i~m) its(:lf. ()~ t,ll(' ()tlwr si(h:, r('la,ti()n (5.(i.51) sig~ili('s l,lm,l, a, 47r r()l,a,l,i()~ is ~(;('(h'(l 1,() r(;(:()v(:r tl~(: i(l('~tity ()l)(:ra,t()r I, anl()~ml,i~lg 1,() a~l (;igl,l~ t)()w(:r ()f 1,11(;~mta,l)h;('ti(: ( )l )(:ra t( )r F / . Tll(; ~mtal)l(:(:ti(: gr()~ 1) 3Ip(2. IR) r(;aliz(:s a (l()~l)l(; (:()v('r ()f tl~(, sy~q)h;('l,i(' gr(),~ l) Np(2, ll~) [2].
5.7
Summary
W(' lmv(; ('la.1)()ra,l,(,(i tll(' wav(;()t)ti(:a,l r('l)r(:s(:lltati()ll ()f tll(' 1)asi(' ()l)ti(:al sys1 2, t(;lllS, i(h;lltili(:(l wil, llill tll(; ray()t)ti(:s (:()lll,('Xl, I)y l,ll('~ (tlm,(lra,l,i(: lll()ll()lllia,ls ~l) _1q22 a,n(l lqp, ill t,(;nlls ()f llllit,ary. ()t)('ra.t()rs 1)(;l()l~gil~g 1.() l,ll(; lll(;l,a,I)h;('l,i(' gr()ut) AIp(2, N). Tllis is syl~th(;siz('(l ill th(; tal)h, l)('h)w, wlli(:ll (:()lllI)l(;t('S Fig. 5.1 t ) a r a l M i n g tll(; ray()l)ti(:s r(;l)r(;s(;llta,ti()n ()f [irst()r(l('r ()l)ti(:a,l sysl,(;llls in t(;rms of mfim()(llfla.r 2 x 2 llm,tri(:(,s M , with tll(; wav(; ()t)ti(:s r(;t)r(;s(;lltati(nl iu t(;rms ()f u n i t a r y ()t)('rat()rs M, )()tl~ (h't)a,rting fi'()x~ th(; (pu)t(;(1 ~()x~()~ia,ls" Ray Optics
4:=
@(2,~) T(d)
(1 r
Quadratic Monomials as generators of
Wave
Optics
1~lp(2,lk~)
=~
0
freemediuml propagation
T (d )  ~
lenstransferl 2
"L(f )  ci~q2
positive scaling
A
p2
L(f)
( _~/f )01
S(IYI, )  
((;n
lO/.,n
5q
2qPl
S('m) __ ~1c
(tql (',ia'2~d(qq')2
 ln('m) q~q
T h e opera, tor representation of an a,rbitra,ry A B C D s y s t e m has then been obtained in the form of an integral transform w h o s e kernel is d e t e r m i n e d by
1D FirstOrder Optical Systems: The HuygensFresnel Integral
267
the entries of the raymatrix of the system, thus yielding the operator t" __ k0 / "._~0__[Aq'2+Dq2 2qq'] M(M) V/2~iB dq'e ~2B
.____+
M
A
A
with an implicit possibility that both M(M) a,nd  M ( M ) are associable to each matrix M. This means that the symplcctic and the metaplcctic groups, Sp(2, R) and Mp(2, R), are isomporphic up to a sign. In the forthcoining chapters we will describe the Wigner representa, tion, which composes both the ray and the wave optics approaches to firstorder optical systems, retaining the wave optical description in its kinematic structure and displaying the ray optical features in its dynamics. Problems 1. Prove t h a t the fractional Fourier t r a n s f o r m of
(a) the spaceshifted flmction ~ ( x 
x,, ) is X0) }  e  i x t , ( x  : ~ 2
.~c~ { ( r
and of
cosqS) s i n r (~c~ ( X 
X 0 COS r
(b) the frequencyshifted function e~*,, ~ ( x ) is
b o t h manifesting an c~dependent
shift variance.
2. Write down the evolution law ml(ler ~ ~ of the HeisenbergWeyl operators c ~p and c ~ax, a and b being real parameters, which are known to produce respectively space and frequency shifts of the wave fimction. 3. Prove t h a t the fractional Fourier transform of the convolution
~(~)  / ~1 (x')~2 (~  ~')dx', A
.)L'c~ {(r
=
eigt
..... /,
/
r 12
( i ~ z _  tanc~[~_c~(~2](X/)(~?1
I,".... /,I
(xx t )dx,
.
It obtains chirpmodulating the fractional Fourier transform of one fllnction, convolving with the scaled version of tile other function and finally imlltiplying by a chirpfimction and a A scale factor. (Hint: use the result of P r o b l e m 2 to specialize the effect of .7 ~ on ~2 (x  x'). 4. T h e complex amplitude of a Gaussian b e a m is typically written as ~ (q, z)  
2(~(z) e  < ( z ) , A(z)e ~k'/L
where r denotes the Gouy phase and Q(z) is the complex b e a m p a r a m e t e r , depending on the wave front radius curvature and the b e a m width; k is the light wavenumber in the medium. Verify t h a t on propagating t h r o u g h an A B C D system the Gaussian b e a m remains exactly Gaussian, and t h a t specifically the complex p a r a m e t e r changes to
AQ(z~) + B Q(zo) 
CQ(z~) + D '
provided t h a t the same m e d i u m is in the object and image spaces.
Linear Ray and Wave Optics in Phase 5pace
268
References [1]
S o m e basic titles c o n c e r n e d w i t h t h e flwmalization of p a r a x i a l wave t)I)t, ics by a (tuantizati()nlikc t)ri)ccss r tmraxial ray optics are listc(t in [9] ()f (:ll. 4. As an altcI'llativc to tim (lllaldizati()ll l)r()cc(lurc, tlm direct gr(mt) tlm(w(~tit:al a t)t)r()ach t() tim t)ara])()li(: wave e(tllati()ll is l)l'(~s(~lg(~(t ill [1.1]
l). St()lcr, "OI)crat()r lImt, ll()(l ill I)llysi(:M ()I)ti(:s", ,]. OI)t. S()('. Atli. 71, 3:t433.()
(t.(~sl).
[2]
[1.2]
II. l/acry ml(l El. (:a(lillm(', "hlctaI)hwti(: gr()lq) all(l F()lu'icr ()I)ti('s", Pllys. I{cv. A 23, 253:~253(i ( 1.981 ).
[1.3]
II. Ba('ry, "(;r()llp tllt',()ry ml(I paraxial ()pries", ill Pro(;. XIII tj' IllI,r C()II. ()n Gr()lq) 'l'ltr N,lctli, ill l)ltys., W.W. Za,clm.ry (c(l.) (Worl(l So. l)lil)l., Silga, l)()re, 1,()84), t)P. 215224.
[1.4]
.I. Ojc(la(',~L~tafic(la mltl A. N()yr "Ditt'urcld, ial ()pcrat()r for s('alar wave cqlm.ti()lF, .]. ()l)t. Soc. Azll. A 5, 16()516().() (1.988).
[1.5]
(l. l)at, t,()li, S. S()lilll('ll() ml(l A. T()rr(;, "Algcl)raic view ()f tim ()l)tical l)r()l)aga, ti()ll il~
[1.6]
(]. I)attr
[1.7]
(I. I)attoli ;tl~(] A. 'l'()rr(', "A g('~(;ral view t() I~i(' alg(;1)raic ~('l,]~()(ls i~ al)l)li('(1 ~atl~e~mti(:s, ()I)tics m~(l trm~sl)()rt ,syst(,~s f()r ('l~argc(l I)em~ a('('clcrat()rs", i~ l)y'namir Sy'm,mct'~c.s and (:h.aoti~ lh'h.a~riour in l)h.ysical Systems, (;. lklai~(), I~. l'u m~(1 M. P,.~ti,i~fi ((:(Is.) (W()rl(l So. l'~fl)l., Si~gap()r(', l!)!)l), pp. 218,!).
.l. (',. (lallm'r m~r A. 'l'(wrc, "A~ algol)talc' view tr tim r orttcri~g m~r it,s applicati()~s to optics", Riv. N~mvo (',i~. 11, 11, 17!) (1.()88).
F()ra l)asi(' l'('allll('lli ()f ilm sy~l)l(~(:li(' gr()~q) ill llm get,oral (:()lll(;xt ()f ll(~ gl'()ll t) tlm()ry scc [9] ()f ('1~. 1; sI)e('iti(' i~v(~sligali()~s arc i~ [l l] ()f ('l~. 2. Sl~a(li(,s ()~ /tw, x~ctal)l(~(:ti(: r(~l)r('sc~lali()~, firslly i~lr()(l~w(;(l I)y Weil [2.1], all(l ()ll ils r(4ati()n t() the sy~ll)le('ti(' g(;()~(',try ar(, l)l'(',sc~t(',(1 i~ tim titles 1)cl()w. We ,sig~mlize the (mlightcning l)i'(',s(',l~lal,i()l~S i~ [2.6], an(l, withix~ tim ()t)ti(:al (:()~t(',xt, i~ [2.4] an(t [2.1]
A. Wcil, "S~lr ccrtai~s gro~q)cs (l'opdratc~rs ~fitaircs", Acta Math. 111, 143211 (1964); also ill Collected l'apcrs, Vol. 3 (SpriIlgcrVcrlag, Hci(h;lbcrg, 1.()8()), pp. 169.
[2.2]
S. Stcrnberg, "Sodom record, rcs~llts o~ the ~netaplectic rcprcs(;~d, atio~", in Group Theoretical Methods in Physics, P. KraIncr m~(l A. Rieckcrs (c(ls.) (SpringcrVerlag, Berlin, 1978), pp. 117143.
[2.3]
V. Guillcinin and S. Stenfl)erg, "The mctaplcctic representation, Weyl operators, and spectral theory", J. F~H~ctioiml Anal. 42, 129225 (1981).
[2.4]
V. Guillemin and S. Sternberg, Symplcctic Techniques in Physics (Cambridge Un. Press, New York, 1984), oh. 1, pp. 1104.
[2.5]
M. Garc/aBull6, W. Lassner and K.B. Wolf, " T h e metaplectic group within the HcisenbergWeyl ring", J. Math. Phys. 27, 2936 (1986).
1D FirstOrder Optical Systems: The HuygensFresnel Integral
269
[2.6] R. G. Littlejohn, "Tile semiclassical evolution of wave packets", Phys. Rep. 138, 198291 (1986). [2.7] R. Simon and N. Mukunda, "The twodimensional symplectic and metaplectic groups and their universal cover", in Symmetries in Science VI, B. Gruber (ed.) (Plenum Press, New York, 1993), pp. 659689. [2.8] R. Simon and N. Mukunda, "Optical phase space, Wigner representation, and invariant quality parameters", J. Opt. Soc. Am. A 17, 24402463 (2000). [2.9] H. ter Morsche and P.J. Oonincx, "On the integral representations for metaplectic operators", J. Fourier Anal. & Appl. 8, 245257 (2002). [3]
For the diffraction theory the reader is directed to [3] of ch. 4, and to [6] below.
[4]
The link between the raymatrix and waveoperator formulations of firstorder optics through the HuygensFresnel integral, firstly recognized by Collins [4.1], is clearly elucidated in the textbook by Siegman [4.2]. [4.1] S.A. Collins, "Diffractionintegral written in terms of matrixoptics", J. Opt. Soc. Am. 60, 11681177 (1970). [4.2] A.E. Siegman, Lasers (Univ. Science Book, Mill Valley, 1986), ch. 20, pp. 777782.
[5]
For a discussion of the concept of duality in relation with optical systems, see [5.1] A. Papoulis, "Dual optical systems", J. Opt. Soc. Am. 58, 653654 (1968). [5.2] H.J. Butterweck, "General theory of linear, coherent, optical dataprocessing systerns", J. Opt. Soc. Am. 67, 6070 (1977). [5.3] H.J. Butterweck, "Principles of optical dataprocessing',' in Progress in Optics, Vol. XIX, E. Wolf (ed.) (NorthHolland, Amsterdam, 1981), oh. 4, pp. 211280. [5.4] A. Lohmann, "Ein neues Dualit/itsprinzip in der Optik',' Optik 11, 478488 (1954). For the English translation, see: A. Lohmann, "Duality in optics',' Optik 89, 9397 (1992).
[6]
For a mathematics and/or physicsoriented treatment of the Fourier transform the reader may consult the textbooks in [15] and [16] of oh. 4. In particular, for a discussion of the Fresnel and Fraunhofer diffraction regimes see: [6.1] J.W. Goodman, Introduction to Fo'ur~er" Optics (McGrawHill, New York, 1968), ch. 4, pp. 5776. [6.2] A. Papoulis, Systems and Transforms with Applications in Optics (McGrawHill, New York, 1968), ch. 9, pp. 315345. [6.3] J.D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley & Sons, New York, 1978), pp. 361385.
[7]
An essential bibliography about the fractional Fourier transform is in [8] of ch. 3. We quote below tile paper by Nalnias, in which the fractional Fourier transform is introduced through an eigenvalue equation like (5.4.42), and those by Agarwal and Simon, and Dattoli, Torre and Mazzacurati, the former dealing with an interpretation of the fractional Fourier transform in terms of the time evolution operator of the quantum oscillator, and the latter with an extension of such an interpretation to an in general quadratic Hamiltonian. Papers concerned with fractional Fourier transforms of complex order and generalized forms of the fractional Fourier transform are also suggested. For a discussion of the fractional Fourier transform within the general context of real and complex linear canonical transforms, see [10].
270
Linear Ray and Wave Optics in Phase 5pace
[7.1] S.V. Namias, " T h e fractional order Fourier transforin and its application to quantum mechanics", J. Inst. Maths. Applics. 25, 241265 (1980). [7.2] G.S. Agarwal and R. Simon, "A simple realization of fractional Fourier transformation and relation to harInonic oscillator Green's fllnction',' Opt. Coimn. 110, 2326 (1994). [7.3] G. Dattoli, A. Torre all(1 G. Mazzacllrati, "All altcnlative point of view to the theory of fractional Follricr trailsfonn", IMA ,l. Appl. Mattl. 60, 215224 (1998). [7.4] C.C. Stfitl, "Ot)tical illt,crl)rctatioll of a colIlI)lcxor(lcr Foln'icr trmlsfiml?', Opt. Lctt. 20, 11781180 ( 1.().()5). [7.5] L.M. Bcnmr(lo a.ii(l O.O.I). Soarcs, "()ptical fl'a(:ti()llal t)r(h'r l"olu'icr trmlsforllls wittl complex or(lcrs", AppI. Opt. 35, 31(i331(i(i (1.9.()(i). [7.6] ,]. Hua, L. I~ill mlr (I. IA, "Exl, cll(lcd fl'actioilal Follricr trmlsforlll", ,I. Opt. Soc. Ain. A 14, 331(i3322 (19!}7). [s]
A (;a~ssim~ l)(:a~ relale(l I)il)li()grat)l~y (:al~ ])c f(nl~(l i~ [13] ()f (:l~. 2.
scc, tl~c tcxtl)()()ks i~ [l()] ()f r ,1. 'l'l~(; I)rcsc.~dali()~ i~ !i 5.5 is i~sl)irc(l l)y t h a t in [!).l] D. MarctlSC, Light 7}'ansmis.~ion Optics (Vm~ Nostrm~(lll.ci~l~ohl, l'ri~ceto~, 1972), cir. 3, pp. 1()71()9. [1()] F()llowi~g tt~(, s~gg(;sli()~ i~ [~().l], l l~(; l)r(;s(;~flali()~ i~ !i 5.6 r(;l)l~ras(;s witl~i~a tl~(; ()t)ti(:al (:()nt('xt t l~(; g(,~(,ral ('()~si(l(;l'ati()l~S (:()ll(:('rllillg lll('~ illlcgral lrallSf()rln as1)cots ()f tl~(; ('(nlstn~('ti()~ ()f ~l~litary r(,l)res(;~ltati()~s ()f li~(,ar ('a~()~li('al trmisf()rl~mti()ns ()f ll~(, H(,is(,~d)(,r~Wcyl algel)ra ()t)(,l'ai()rs q, p ()~ il~(' tIill)rl'l si)a(:e t)cl()w. Sccals() [2.4]. A~ (;~ligl~lc~i~g (;xl)laI~ali()~ ()f I l~(', 7rl)lmse sl~ifl, s~ff'crc(1 t)y a light wave I)a,ssing tl~r(~gl~ a ('()~j~gatc I)()i~d is given i~ II().l()]. [10.1] M. Nazara.i,l~y m~(l ,]. Slm~ir, "Firstor(ler optics a (:mm~fi('al ()l)erat()r reprcsci~tat,i()~: lossless syst, c~s", .l. Opt. S()(:. A~n. 72, 35(i3(i4 (1.982). [10.2] L. IIffcl(l m~(l ,]. I)lcl)m~ski, "()~ it certain class of ~fit, ary trm~sform~tio~s",Acta Phys. Polon. 14, 4175 (1.q54). [10.3] V. Bargmmu~, "O~ a Hill)err Sl)aCe of m~alyti(: fl~('ti()~s m~(! m~ ~mso('iatc(l integral transform, r', ( ; o x ~ . lhm,~ & AI)t)I. Matt~. 14, 187214 (1.()61); II. A fiunily of related fllnction spaces aPl)li(:ati()~ t()(listril)~d, ion theory, i)i(l., 20, 1101 (1967). [10.4] M. Moshi~sky m~(l C. Q~leS~c, "Li~ear (:anonical transf()r~mtiox~s mul tlmir unitary representatio~s", J. Matl~. Pt~ys. 12, 17721780 (1971); C. Q~ms~c an(l M. Moshinsky, "Canonical tral~sformat, i()~s and matrix elements", ibid., 17801783. [10.5] M. Moshinsky, "Cm~o~dcal transformations an(l quantllln ~nechanics", SlAM J. Appl. Math. :15, 17721780 (1973). [10.6] K.B. Wolf, "Canonical transforlns. I. Complex linear transfortns", ,J. Math. Phys. 15, 12951301 (1974); II. Complex radial transforms", ibid., 21022111. [10.7] P. Kramer, M. Moshinsky and T.H. Seligman, "Complex extensions of canonical transformations and qllalltlllIl inechanics", in Group Theory and Its Applications, Vol. 3, E.M. Loebl (ed.) (Academic Press, New York, 1975), pp. 249332. [10.8] K.B. Wolf, Integral Transforms in Science and Engineerin 9 (Plenum Press, New York, 1979), ch. 9, pp. 381416. [10.9] K.B. Wolf, Geometric Optics On Phase Space (Springer, Berlin, 2004), pp. 277282. [10.10] V. Guillemin and S. Sternberg, Geometric Asymptotics (Am. Math. Sot., Providence, 1977), ch. 1, pp. 119.
6 The VVigner Distribution Function: Analytical Evaluation
6.1
Introduction
The Wigner distribution function was originally introduced by Wigner [1] as the simplest quantum analogue of the classical phase space distribution function permitting to find probabilities and expectation values of quantum operators using as much classical language and methods as allowed. The Wig:mr method establishes a rule to associate a cimln)er function in phasespace with every operator being a fimction of position and momentum operators. Thus, the expectation vahms of quantum mechanical observables can be calculated in the same mathematical form as the averages of tile classical statistical nmchanics rather than through tile operator fornmlism of quantum inechanics. As an illustrative example, we consider a particle moving along a line. In classical mechanics the dynamical state of the particle at any instant of time is fully described by the position q(t) and the momentum p(t) of the particle [2]. The motion of the particle is generated by the cla,ssica,1 Hamiltonia,n p2
H(q, p)  2m + V(q),
(6.1.1)
according to which the classical equations of motion obtain as
dq dt
p m
,
dp dt

V'(q)
(6.1.2)
m denoting the mass of the particle and V(q) the potential it experiences (V'(q)  dV/dq). If the initial conditions q0  q(O) and Po  p(O) arc known, the evolution of the state of the particle is uniquely determined by the solution (q(t), p(t)) of (6.1.2), which individualizes as well a definite trajectory in the particle phasespace. It may happen however that the initial state of the particle be not specified through the definite values q0 and Po of the position and momentum of the
272
Linear Ray and Wave Optics in Phase Space
particle at t = 0, but r a t h e r t h r o u g h the .joint probability distribution of q and p at t = 0. In t h a t case, the d y n a m i c s of the particle will not be determined by Ha nfilton's equations (6.1.2) for q(t) and p(t), t)ut by Liouville's equa,tion for the t)ha,sespa(:e. (tistributi(m flm(:ti(m Pc(q, P; t),
0 p 0 V' 0 otp(;( q, p; t)  {p(,, H } . . . 'mOq . P(" + (q) ~ p(,.
(6.1.3)
Clea.rly, p(,(q, p; t) lll(~.tl.ll,S 1,11{;l)r()lm.1)ility (lellsit,y fin" tilt; tmrt, it:le I,() 1)e a,t the t)]m.sr l)r (q, p) at t,i,,l(: I. It,:,,(',;, p(,(q, p; l) >_ ()m,(l Ji/'P,,(q, P; t)dqdp 1 a,t a.lLy t, tlu; ilm;gra,ti(nL l~r (~w;r tlLr ilL1Lr l)lLas(; st)a(:r r tlLr I)article. If tlle t)rr r ir r t,hr (lylm,nfica,1 wu'ia,lfles q a,llr p a,t the initial t,i~l~; is k~lr p,,(q, p; ()) = p,,(q, p), t11r lw(~tm,l~ility r p(,(q, p; t) a.t a.lly slll~sr162162 t,ilLLr is lllLir (lr 1)y t, lLr sr i(~]L r Er (6.1.3). hi ()tiLer W()l'(ls, ill (:la,ssi(:al ~Le(:lm~Li(:s t,lle illf()rllla,t,i()li ('(nitaille(l Lilt,() the (:ml(nli(',ally (:~)ll.j~lg~at,(; wu'ialfl~;s q mL(l p, r(;l)r(;s(;~lt, i~Lg: t,lL(; lmrt,ir st,a,te as a, t)(~i~lt, i~1 lfllaseSlm.('e, r t,ra.~lsf~;rre~l int,()t, ll(; (list,riln~t,itnl fllll(:l,i()ll p(,(q, p; t), ret)res(;nt, ing the t)r()lm.I)ility fl)r every t)()int in t)hase st)a(:(; t,() ln'~ a. l)(~ssil)l(; st, a.l,c fin" 111(; l)arti('l(;. Ar 1,11(;(;v()l~lt,i()11 ()f 1,11(; Imrti(:l(' st.at(, ('i.lll ~zli(l~z(;ly 1)e a.(:('(nnm;(l f()l" ])y t,ll~; l~l()l,i~)ll ()f t,l,; l)lm,se l)la.ll(; I,~,i~lt, (q(t),p(t)), ,)r 1)y the ew)l~t,i(nL ()f t.lL~' (list,ril)~t,i~n~ fl~L('t,i~nL p(;(q, p; t). W~; 1L()t,(; t,lLa,t I,i()~vill~:'s (;(l~m,t,i,)~ ILas t)(;(;~L (l(:riw;(l ill !i 2.2.1 wit,lli~L t,ll(; ()f ligllt, rays, t,ll(: relewud, (ly~lmlli(:al flnl(:t,i(~l p(q,p; z) l~('~i~lg i~d,erl)r(;t(;(t a,s the l)lmsesl)a,(:e ray ~lensit,y a,t every ref(;re~l(:e z t)lmu;, hi efIi;(:l,, p(q, p; z)dqdp yiehls 1,11e ~n~td)(~r ()f rays a,t z izl l,ll(; (;le~lexd,a,l v()l~l(; dqdp ar, nn11(l 1,11(;t)ha,sest)a,(:e t)()i~lt (q,p); l,m,(:e a,t ea,(:tl z l)la,ne .]i['vp(q,p;z)dqdp = N" (1,11(; total lnn~fl)(;r ()f rays i~1 l,lle l)ea~tl), i,t1(; izltegration inv()Ivi~lg tlm i~lw~,rimlt t)ha,se st)a,(:e ret)rese~d, ative v()h~11e 12 of 1,11(; n~1(:11 ()f rays. It is (;vi(l(;~lt tlmt, within the me(:ha,~fi(:al()t)ti(:a,l t)a,si(: (:()rr(;st)(nl(t(;n(:(; ()f the temt)()ra.1 (:()(w(lina.t,e t to the axia.1 (:o()rdina, te z, the (listrit)~d,i(m fim(:tions p(:(q,p; t) a,n(l p(q,p; z) are equiwdent, a,t)art fi'()I~1 the relative n()rIna.liza, tion settings. T h e I)r()l)a,bilistic picture inllerent in p(;(q, p; t) i~lt)li(:itly involves, in N,('t,, the e~lsmlfl)le picture of a hug(; nm~d)(;r ()f i(tenti(:a,1 systenls, i.e. t)artMcs with mass 7n, (tistributed at ea,ch tinle t ()vet" all the possible (lyna,mi(:a,1 states (q,p) a,(:(:()r(ling to the releva,nt weighting function Pc (q, P; t). E(t~m,tion (6.1.3)(:orrest)on(ts to (6.1.2) when a statistical ensemble picture of the system d y n a m i c s is concerned. Accordingly, the average value of any dynami(:al filnction f(q, p) is calculated by the integral ( / ) c (t)  . f / p(: (q, p; t)f(q,p)dqdp,
(6.1.4)
the integration involving the entire phase space d o m a i n p e r t a i n i n g to the syst e m [2].
The Wigner Distribution Function: Analytical Evaluation
273
Integral (6.1.4) formally expresses the nature of the statistical description of classical mechanics, where the mea,n values of dynamical variables are obtained by averaging over uniquely determined processes. This clea,rly mirrors tile fact that in classical mechanics the position and IIlOilleil[,lliil of tile particle a,s well as any function of q and p, a,nd hence the probability distribution Pc(q, P; t), can exactly be known at any time, the relevant time evolution being uniquely determined by deterministic laws of evolution; the whole uncertainty in classical mechanics is indeed contained in the form of the initial probability distribution of the dytmmical variables specifying the given system. In contrast, in quantum mechanics the simultaneous specification of the position and monlentunl of the particle with arbitrary precision is impossible. The dynamical state of the particle is described by specifying at each time t the wa,ve function ~(q,t), which determines the probability density of the position coordinate according to [3] OQ (q, t)  I~(q, t) 2,
(6.1.5)
and separately the probability density of the momentum coordinate through the Fourier tra,nsform ~/,(p, t) according to q3q (p, t)  I ~(p, t)12.
(6.1.6)
In addition, the knowledge of '(.,(q,t) enables us to find the expectation value of any observable .f(~, ~) in the state ~/~ by the scalar product
If)q(t)
 (~/~(q,t)]f(~,~)lt/~(q,t))
 ]~/~*(q,t)f(~,~)~/~(q,t)dq.
(6.1.7)
In particular, if f depends only on the coordinate operator ~, i.e., f(~, ~) h.(~), we have
( f )q (t) 
~Q (q, t)h(q)dq,
(6.1.8)
whereas, if f depends only on the momentum operator ~, namely: f(~, ~) ~(~), one obtains
( f }q (t) 
j 9i3q (p, t)9(p)dp,
(6.1.9)
where h ( q ) a n d g(P) arc the classical functions corresponding to the quantum operators h(~) and ~(~). The evolution of the dynamical state of the particle is determined by the SchrSdinger equation for the wave function ~h(q, t),
ih Ot O__~h(q, t)  H(~, ~ ~)~h(q, t),
(6.1.10)
Linear Ray and Wave Optics in Phase 5pace
274
with H(~, ~)  ~ 2 / 2 m + 0 ( ~ ) being the q u a n t u m Hamiltonian operator corrcspon(ting to the classical function (6.1.1) [3]. As II()t(;(t above, a, classi(:al joint probability distriblltion p(,(q, p; t) can be (teviser whose tra,nsfornmtion with tinlc is I'111c(t by Liollvillc's equa,ti()n (6.1.3), tllllS a.lh)wi~lg for a, flflly t)r())a,1)ilisti(:likc a.t)t)roa('ll to ('la.ssi('al dynmlli(:al t)robh;nls witll()ld, cxpli('itly res()rting to tll(; (t(;s(:rit)tir in torlllS of the; t)osition all( lllt)lllt;lltlllll c()()r(lilm,t(;s, a ll(l h(;ll(:(; t() IImllilt()ll'S (;(tlla,ti()liS. It, is llal.llral t,() illv(,stiga,tt, t,lw. t)()ssil)ility ()f cla.l)()ratillg a (:lassi('allik(; (l(;s(Til)l,i(nl ()f l,ll(' r sta,l,('. ()f tll(; (lllalltlllll systr (t,h(; I)()illl, I)ai'ti(:l(; in l,ll(; t)rt;st;ld, (list:llssit)ll) a ll(l its ('~vr i()ll wil,ll()lll, (;xl)li(:itly rt;s()rl,ill~ t() tll(; wav~
Ea.(:h siglml (:()llll)()llt:llt ~, (:(nltrilnlt~:s t(~ 142~ witll its (~wll siglml te.rxll ]/V~ , 9
j
a n(t (;a,(:ll Imir (~f siglml (:~nlq)~nlel~l,s (,~ , ~ ) l)r,~(l~l(:es a (:(n'r(~sl)~(ii~g (:r()ss t(;rn~, giv(;l~ )y tl~(; r(;a,1 l)a.rt (~f tt~(: (:r()ss ~Vig~(;r (listrinlti(~ fi~(:ti(~ W%, ~ ()f siglml (6.3.15) will (lisl)lay. N sigl~a.1 t(,.l'lllS a,ll(l (N2) (:r()ss tt;l'~s ((:~lst()~m,i'ily. l,('n~(;(l i~ l,l~; sig~a,1 tl~(;()ry (:()x~t('xt as i'nte'tfi:'rence I,(WIlIS).
the s(t~a,re(l l~l(~(t~l~s ()f 1,1~; (:r(~ss Wig~mr (tistril)~ti()l~ fiu~(:ti(n~ ]/Y~ ,~ (q, h;) is cxt)ressil~l~; i~ l,~'~rl~S()f 1,1~; m~l,(~ Wig~mr (listrit~ti(n~ fi~(:ti(n~s ~f tl~(', i~wolve(t signals, i.('~., 142~j a,n(t 142~. Ext)li(:itly, after s(~nl(; a,lg(;ln'a (~n(~ fi~l(ls II/V~,., ~, (q,h;)l 2 
'
W~ ( q  ~
2~
.
hb/
~
h:ff)l/V~(q+~
/~/
~
h'+ff)dq'dh;'
~
(6.3 17) 9
giving W ~ , ~t a,s the st)a,(:~; a,n(t fl'c(tlmn(:y (:(mvohlti(m ()f the Wign(;r (tistribulions of tim in(tividua,1 signals. Tile 2D integral on the right rcpro(hl(:es with respect to tim space and frcqucn(:y wl,riablcs the structure of the crossWigncr intcgra,1 (6.3.12) at zero frequcn(:y. Hcn('c, resorting to tile hingelike picture of the selfWigner integral (6.2.2) (:la,borate(t in w 6.2, it (:an bc visualized a,s arising at any point (q, ~) in the Wigner plane by summing up the contributions of )/V~j and )/V~ respe(:tively at points lying symmetrically to tile chosen point (q, ~;). Thus, if )d;~j and 142~ sharp around the points (ql,/~1 ) and (q2, ~2) ii1 tile (q, ~) plane, the cross contribution 142~j, ~t (q' t~) develops mainly in the surroundings of the midway point (qm Nm) with qm '
~
~
ql +% and /~m 2
~1+~2 2 "
The
WignerDistribution Function: Analytical Evaluation
287
F I G U R E 6.4. G e o m e t r y in the W i g n e r plane of the self and cross t e r m s in the W i g n e r distribution flmction of the stun of two identical signals centered on (ql, ~,) and (%, t%).
Specifically, let us consider a signal ~h(q) composed by two identical copies of a model signa,1 p(q). We suppose that p(q) is mainly concentrated around the origin of the spacefrequency plane, and that the two signal replicas are shifted in space and frequency to the points (q,, ~,) and (%, ~ ) . So we write 

ql ) c
Jr

q2 ) C'iK2q ,
(6.3.18)
a,nd accordingly,
)/Vr (q, h;)  ~;~,(qql,~  /"~'1) "31)"V~(qq2,~ ~2) 112 COS[/'~d (q  qm)  qd (h;  h;m) + h;dqn,]~V ~ (q  qn,, t~  h;m).
(6.3.19)
Here, 142 is the Wigner distribution flxnction of the model signal ~; qm and /~m are as defined before, whilst qd' /fi~dare tile difference coordinates: qd  ql q2 a n d / ~ d = /5~11~2" The crossterm above has been obtained from an appropriate manipula,tion of the crossWigner integral (6.3.12) explicitly involving the two spatially and spatial frequency shifted versions of the model signal. The geometry of the terms entering (6.3.19) is illustrated in Fig. 6.4. As expected, the two signa,1 contributions in W e are located a,t the regions in the spacefrequency plane where the relative signals are located, whilst the cross contribution lies midway between the signal terms. Moreover, the envelope of the cross term shapes just as the Wigner distribution W~ of the model signal with a double amplitude, but is modulated by an oscillation occurring transversely to the line through the centres of the self terms, with a frequency governed by the separation between those centres. Although established in the specific case of a signal like (6.3.18), i.e., composed by two identical waveforms, the behaviour exemplified in Fig. 6.4 is quite generally displayed by the self
Linear Ray and Wave Optics in Phase Space
288
and (:rossterms of the W i g n e r distribution function of a composed signal, even when the signal terms have not the same form. Specifically, the g r a p h in Fig. 6.4 refers to a G a u s s i a n  s h a p e d model signal (see w 6.4.5). It is evi(tent fl'onl (6.3.19) t h a t even t h o u g h the siglla,1 ternls are t)ositive everywhere thr(mgh th(,~ st)a.(:efl'e(tlu'~n(:y t)la.ne, as it, ()(:(:lu's in the (rely (:ase ()f a Gmlssimlstm.t)e(t sigilal, the ()s(:illa,ti~lg )ehavioln" ()f tlle (:ross ternl unav 0 and ~ 432 > 0. Also, note that again f f G(q, ,c)dqd~ = 1. The reader may verify the above assertion retracing the steps of the proof previously elaborated in regard to the kernel (6.3.36). In particular, if the parameters in (6.3.45) are chosen to satisfy the equality in (6.3.46), the resulting 2D Gaussian function is interpretable as a Wigner distribution function; specifically, the Wigner (tistribution fimction of the comt)lex Gaussian amt)litude _q2 (~ pg(q)
_
(~_~)1/4e ~(1
i~).
(6.3.47)
Evidently, one may regard (6.3.38) as the amplitude profile of a Gaussian beam at the waist with beam radius equal to v ~ , and accordingly (6.3.47) as the amplitude profile of a Gaussian beam at a plane different froin the waist, where specifically the wave front curvature is ~ and the beam radius is V/~. The smoothing by a minimum uncertainty Gaussian has been originally introduced by Husimi in his extensive investigation of the formal properties of the density matrix [10.13]. Accordingly, the phasespace distribution resulting
296
Linear Ray and Wave Optics in Phase Space
from the convohltion process (6.3.35) by a, m i n i m u n i u n c e r t a i n t y Ga,ussia,n is frequently r e p o r t e d as the H~simi distribution flmction. It is one of the basic t)hasest)a,(:e~ (tistril)llti(ms of (tua.ntunl opti(:s, (:orr(;st)()n(ting to the n o r m a l ordering of the })oson (:rea,/,ion and a,nnihila,ti(m ot)erat, ors. T h e s m o o t h i n g t)r()(:(;(ture imt)lie(t by (6.3.35) t,ra,nslat,(;s in an (;l(;ga,nt, and a,(tat)t,a})l(; fornl I,}l(; naive i(t(;a, of (:()a,rsely r(;t)la,(:ing t,ll(; wdll(: ()f the W i g n e r (lisl, ril)lll, i(nl fllll(:l,i()ll a,I, (;a,('} l)()illI, ill t,}l 0, the Wigner distribution 142 (q, ~) still comes to be the frequency domain convolution of W,1 and I/F,2 , the arguments of the latter being appropriately scaled according to (6.3.59), thus giving .
W
(q,

~/
(6.3.6s)
(~1(t~  t~')~ 2(j) dt~'
(6.3.69)
1
Note that the Fourier spectrunl of (6.3.67) reads 0(/.@)
1
v~l
Expressions (6.3.63) a.nd (6.3.65) of the Wigner distributions W, (q, a~) and We (q, ~) relative to tile convolutions (6.3.60) a,nd (6.3.66) make evident that the convolution is transmitted from the functions to the corresponding Wigner distributions, and involves the space or frequency dimension according to whether the functions or the respective Fourier transforms are convolved. Interestingly, the convolutions (6.3.63) and (6.3.65) may be interpreted as the spacefrequency representations of the transformations described by products (6.3.61) and (6.3.64). Thinking in physical terms, the latters may describe filtering operations respectively in the frequency and space domains, with one function representing the signal and the other the filter function. Thus, both the frequency and spatial filtering manifest in the q~ plane through the convolution of the Wigner distributions of the signal and the filter function along the space and frequency dimension, respectively. Within the optical context, relations (6.3.60) and (6.3.67) may be seen as describing the transfer properties of spaceinvariant and diffractionfree optical systems, which, as noted in w 4.6.3, act as frequency and spatial filters for optical signals. Convolution (6.3.60) and
Linear Ray and Wave Optics in Phase Space
302
p r o d u c t (6.3.64) express therefore the transforma, tion la,ws for optical signals un(ter t)rot)agation t h r o u g h a, h o m o g e n e o u s m e d i u m and a, thin lens, one of the flm(:titms representing the int)ut wa,veform, the other one the line spread functioI~ of the me(tim~ or ('orrest)ondingly the t)ha,sc n~o(lulation fl~n(:tion of the lens. Ac(:or(til~gly, (:(revolutions (6.3.63) an(1 (6.3.65) nmy t)(; rega,r(ted as the transfer la,ws ()f the Wig~(;r (listributioi~ fl~I~('ti()~ f()r fl'eet)r()t)a,ga.tion and lens transfer, a,s they relate the Wig~mr (tisl,ritnd,i()~ fiu~(:(,i()~ ()f the int)~t a n(t t,h('~ ()~l(,t)lll, siglia,ls. W(; will r('(:()vt',r l,h('lii ill !i 8.3.1 wll(;n illv('stigatint4 the t,rm~sfer laws f()r t,l~(' Wig]~(,r (listri)~ti()~ flll~(:(,i()l~ llI~(l(;r l)r()i)a,gati()~ t,hrougt~ a fl'(','=n~e(li~m~ S(;(:(,i()ll a,ll(l a, l,llill l(',llS.
l~,ep'l'esen,tability
6.3.10
As a. na,t,lira.l (:()ll(:llisi()li t() tliis s(;('.ti(nl, w(; (:()liilil(;llt ()li ill(; 'l'~'p'l'~'m"ntability ()f i,lle Wigll('r (lisi,rii)ilt,i(nl filll('li()ll. Evi(t(;lltly, ll()t ('~v(',ry filll('t,i()ll ()f Sl)a,(:(; a.ll(1 fi'e(tlU;ll(:y (:ml 1)(; 1111(Iersto()d as a. Wiglmr (list, rilnlti()ll filll('t,i()ll, l)e('a,llse tllere lllay ~()l, (:xist, a sig~al, fl'()~ wl~i(:l~ it, ~igl~t, l)e ge~(;rate(I. 'l'l~s, a 21) fllll(%i()ll ()f Sl)m'(' m~(l fI'(;(lll(;l~(:y is sa.i(t t,() t)e Wig~wr 'rt'p't't',s~'n.tabh' ()r 't'calizabh', if it is ~(;~erat(;(l I)y s()~(' signal. A~ exl)lit:it (:()~(lili()~ ('m~ l)('~ ('stal)lisl~('(l as a l~('a~ I() as('t;rtai~ wl~ether ()r ~()t, [4.13, (i.7, 1o.21]. Basit:a,lly, it grt)~na(ls ()n tl~e (:()~t)a, tilfility ()f rela, tions (6.3.23), (6.3.25) a,~(l (6.3.28); ilk(lee(I, l()()kil~g at (6.3.23) m~(l (6.3.25), wc sec t,l~at a, fiu~(:ti()~ ()f q m~(1 t," is a W i g n e r (listril)~ti()~ if m~(l ()i~ly if l,t~e i~d,egrals on the rightlm,~(l si(les res~lt i~d,() the Sel)aral)le fl)n~s al)l)eari~g in the leftlmn(1 si(l('s. St)('(:iti(:ally, w('~ (:a~ I)hrase t,l~(' ll('~(:(;sSa.l'y a l~(t s~ffti('ie~d, (:()l~(titi(m for a flmt:ti()~ ()f the F()~riert'.onj~gate variM)les 142(q, ~') l,() )(, a r(;t)resentat)le~ W i g n e r (tistri)~d,i(ni ill t(',l'lllS ()f the fl~n(:ti(n~ itself l, hr()~gl~ th(; r('Ja.ti()~ [4.13, 6.7]
W(q,, _
_


1 27r.
/
2
+ u'
+
W ( ( ] o Jr ~ ' I'," o Jr  ~ ) W ( q

'2,1 t,;~} 
~)
o  ~ ' H,o  ~)('.i(ml'qtc~)d(]tdh;t
9
which m~st h()l(t fi)r any % and t% (see P r o b l e m 5). In pra.ctica,1 terms, in order to establish w h e t h e r a fimction is representable or not, we a,ss~m~e the flmcti(m, we are dealing with, be a W i g n e r distribution fl~nction of sonm signal, wtfi(:h can be found from the inverse formula (6.3.28). Then, we calculate the W i g n e r d i s t r i b u t i o n of the derived signM and compare it with the originM flmction. Evidently, if we recow;r the same function, the d i s t r i b u t i o n at h a n d is a, representable W i g n e r distribution function. In w 6.4.6 we will illustrate an example of a flmction of q and t~:, which is not representable W i g n e r d i s t r i b u t i o n function. As we will see, the features of the p o s s i b l e g e n e r a s signal, derived from the inverse f o r m u l a (6.3.28), are
The Wigner Distribution Function: Analytical Evaluation
303
in contrast with relations (6.3.23) and (6.3.26). Also, we know that the Wigner distribution function attains always negative values somewhere in the qec plane, except when it is generated by a Gaussian signal. Thus, any nonnega,tive 2D function can immediately be recognized as a non representable Wigner distribution function, unless it is in the Gaussian form (6.3.45) with the relevant parameters satisfying the equality in (6.3.46).
6.4
The Wigner distribution function of light signals further examples
We will further illustrate the potential of the Wigner distribution function to act as a local frequency spectrum, simultaneously displaying the space and frequency features of the signal. We will consider typical waveforms (e) and derive the analytical expression of the corresponding Wigner distribution functions and/or show on graphs the 3D images of it [5, 10]. We will start by considering signals limited within a, finitelength interval in space or frequency. The mathematical representation for such waveforms is the rectangular function, whose dual representation is the sinefunction. Then, resorting to the prime Fouriertransform pair, i.e., the monofrequency and &like signals, discussed in w 6.2.1, we will firstly consider a cosine signal, arising as the sum of two st)atial lmrlnoni('s of equal an(t ()t)posite frequencies. The obvious generalization is evidently the infinite (numera, ble) sum of pure spatia,1 harlnoni(:s, whi(:h for suitable coeffi(:ients (:a.n t)e rcgar(ted as tile Fourier series of spa,tially periodic wavefields. As a basic example of periodic signal, we will consider the comb function. It is of great relevance in connection with the analysis of optical effects, like, for instance, the Talbot and Lau effects, produced by proper illumination of suitably arranged periodic structures. Frequencymodulated and amplitudemodulated signals will then be COilsidered. In the first case, we will start describing the pure quadratic phase signals, which are the basic waveforms of firstorder wave optics, na,turally arising in fact in the Fresnel approximation as model function for the response function of optical systems as well as for the phase front of the incoming disturbance. As a natural continuation, the pure polynomialphase signals are briefly coinmented on, being seen as a tool for investigations of the optical propagation properties beyond the Fresnel approximation. As a primary example of amplitudemodulated signal, we will consider the real Gaussian amplitude which provides, for instance, the model function for the transverse profile of the fundamental mode of the field oscillations in a stable resonator laser as well as of the quantum mechanical harmonic oscillator. c A detailed description of many typical waveforms of relevance in optics is [14].
Linear Ray and Wave Optics in Phase 5pace
304
T h e H e r m i t e  G a u s s i a n b e a m s will also be discussed a,s well as the more recently i n t r o d u c e d HcrmitccoshGaussia, n modes. Wc will give a numerical a,ccount of the supcrGa, ussian envelope wa,vcforms, whi(:h m a y rnodcl the la,tcral profile of b e a m s froi~l highga,in unsta,ble resonator lasers slfital)ly e(01ipped with sut)crGaussia,~l transnlitta.ncc or refle('tivity mirrors. It is worth noting t h a t , sill(:(; the W i g n c r (tistrilmtion flln(:tion is (tcfincd 1)y a,n iIlfinite integral of a sllt)t)ose(lly (:ontimlolls flul(:tion, its cxa(:t cvalllal,i()ll is ill g(;n(;ral imt)()ssil)h'. In(l(;(;(l, f()r a,llll()Sl, all ()f l,ll(; wav('~f()rnls, w(; will (:()llsi(l(;l" ill wlmt f()ll()ws, a,ll (;xa,(:l, (;xt)r(;ssi(nl ()f l,ll(; l"(;st)(;(:l,iv(; Wigll(;r (tistrit)lll, i(nl fllll('ti()ll (:il,ll l)(; ()t)l,a.ill(:(l. Tllis is s(), (;xt:eI)t ill l,ll(; (:as(; ()f t,llc (tuarti(: ml(1 (llfixlti(: t)()lyll()lllia.l I)llas('~ siglmls, mi(l ill l,llc (:a.s(; ()f l, ll(', slq)(;rGaalssian a.lllt)lil,~dc, ft)r wlli(:l~ a, ~nn~t'rit:al a,t:(:t)~t is givtnl. ()f (:()lll'S(~., tiffs inlt)lies a, (tis('rt;l,iza,titnl t)f tilt: sig~ml m~tl st) a (lis(:rt;l,iza, titnl ()f tilt; rt:s~ll,i~g (tistril)~ti()l~. W(; slmll ~l()l, tlwt;ll (hi tilt; (l~U;sl,itms rclatt'~tt t,() tilt; tlt:til~ititnl ()f a (tiscrctc v(~,rsi()~l ()f l,l~(; Wigll(;r (lisl,ril)~l,i()~l fi~(:l,i()~ a~(1 (hi t]~(; (:()llS(;(tll(~,ll(:(; ()f this (lis(:rel,iza,ti()ll, fi)r wl~i(;l~avi K,
which evidently has tile same behaviour with rest)ect to the space va,riable a.s (6.4.3) with rest)cot to the frequency va,riable. Figure 6.6 shows the (q, ~:)contour images of the distributions (6.4.3) and (6.4.7) for the same numericaJ va,lue of (~) and K; we :nay see the characteristic structure of nested symmetric hyperbolas, which axe rota,ted by 571" one to the other, being confined respectively within a, vertical and a horizontaJ strip. Rectangle and sinc signaJs relate to each other by the Fourier integral, and accordingly the respective Wigner charts in the qt~ plane relate to each other by a,~rotation. Resorting to the results of w 6.2.1, we note that the same ~rota,tion in the q~ plane rela,tes the Wigner distribution functions of the impulse and the spatia,1 ha,rmonic, which axe a, Fouriertransform pair a,s well.
Modulated and shifted rectangular signal As noted in w 6.3.2, a modulation of the recta,ngula,r signal (6.4.1) by the harmonic factor ci~0q according to F(q)  rect(2~)c i~0q
(6.4.8)
produces a, frequency shift of the corresponding Wigner distribution (6.4.3) as
w ~ (q, ~) = w ~ (q, ~  ~0).
(6.4.9)
Linear Ray and Wave Optics in Phase Space
308
Tit(; ab()ve b e h a v e s a,s (6.4.3), a p a r t fl'om a, shift of the center of g r a v i t y fl'om the t)lm.sct)la.ne t)oint ((), 0) to the point (0, ~:o)Trm~slating signal (6.4.8) a,hmg the q line t)y qo as h;a(ls t()
~(q)  r e t : t ( ~ ) ,
'i%q,
W~(q, ~:)  W.,(q  q,,, ~:  t,:,,),
(6.4.10) (6.4.11)
wl~()se ])('~l~avi()llr agai]~ r(;l)li('at('s tl~al ()f (tisl,l'i)~]ti()]l (6.4.3) witl~ 11~' ('e]d,r()i(t t)(;illg sl~it'l,('~(l l,() l,ll(; l)()il~l, (q~,, t~,~) i~l 1,11(; Sl)a,(:t;fl'c(t~l(;~l(:y l)lall(',. Sig~lals (6.4.8)a~(l (6.d.l()) l~av('~ l,l~(' sa~('~ sill(:2lik(; 1)()wt;r Sl)(;(:l,l'lllll. Ill fa(:l,, l,ll(;ir fl'(;(lll(;ll(:y Sl)(;(:l,ra,, r(;sl)(;t:l,iv(;ly giv('ll ])y sill(:[(2(t~" t~',,)]r 'i(~ ~"")q", (6.4.12) si~t)ly (litt'('r fin" l,ll(; l)l~as(', fl~l(:l,i(n~ ~,i(,,.,,.~,)%. ~l'l~(' (:(n'r(;sl)(n~(lil~g s(pm,rc ~mg~lit~t~h:s slml)~' l,ll~:r~;fi)r~: fi)r 1)~)1,1~ sig~mls as l,ll~; si~('2fi~('ti~n~ w(t;) 
sill(:[(2(t~" t~',,,)],
~(t~') 
I~(u)l 2  2(a2sil,(:2[Q)(t; t;,,,)],
(6.d.13)
71"
whit:ll evi~h'~lltly t:()ll,a.illS lit) illf~)l'llla,,i~)ll al)t~ltt l,llt; h~t:atitnl ill Sl~a~'t: ~f tlle (;ll(;rgy (h;llsily. I~l (~a.tial
d
~
b ( q  'n=,). '1'1~;~, ('xl,l()iti~g t,t~,; wi'()t)(;ri,y ~~~I'~~o/~(qqo)J=  %(q)12[@~(~,)l2.
(6.4.54)
T h e elliptical contours of W~ (q, r~) axe traced in Fig. ??, the relevant equation being ~22 ( q   q o )
2 +ifw~ [/~; /~o  fl(q  qo)] 2   : .
(6.4.55)
The Wigner Distribution Function: Analytical Evaluation
323
FIGURE 6.16. The elliptical contours of the Wigner distribution function of the chirped and shifted Gaussian signal (6.4.53) may be seen as resulting from a translation and a ~cshear of the elliptical contours of the Wigner distribution function of the basic Gaussian amplitude (6.4.50) (see Fig. 6.15). In the graph, w = 2, q~, = 2, ~, = 4. A compa.rison with tile ellipses relevant to the Wigner distribution flmction (6.4.51), also traced in the figure, Mlows one to recognize a translation of the ellipse center from tile origin to the t)hasepla,ne point (qo, ~',0), a,nd a h',shcar, (tctcrinined by the curvature t)a,ra,Inetcr/~. W i t h increasing the value of w the (:hirp (tomiim, tes over the Ga,ussian amt)litu(te, which widens a.lmost uniformly over the qaxis. On graphs, with w ~ + o c the phasct)la,IlC ellipse (:ollapscs into the ret)resentative line of the chirp signal, i.e. the line delta, 5(~;~; o  [ J ( q  q o ) ) Likewise, for small va,hles of w the Ga,ussian is almost like a, (5flm('tion, and so the representative pha,seplane ellipse collapses into the vertical line ~ ( q  %) proper of a point source. Notably, distributions (6.4.51) and (6.4.54) are positive t h r o u g h the whole phase plane, in accord with the Hudson theorem [10.6], which, a.s mentioned in w 6.3.6, identifies the Gaussia,n signals as the only wavcforms whose W i g n c r distribution function is positive everywhere in the spacefrequency plane. S u m o f two G a u s s i a n s i g n a l s
As a filrther example of the non linearity of the W i g n e r distribution fimction a,nd accordingly of the beha,viour of the crossterms entering the expression of W(q, ~c) when the superposition of signals is involved, we consider a signa,1 composed by two identical Gaussian signals, which are however shifted one to the other in b o t h space and frequency. So, we consider the a m p l i t u d e 1 1 q2(q)  ( ~ 2 ) l / 4 e   s ) 2+i~lq + (~_~w2 (6.4.56)
)l/4~~~(qq~)9+i~2q.
Linear Ray and Wave Optics in Phase Space
324
l;l(;tIl{l,] 6.17.31) i~mgc m~(l rclcvm~t Sl)aC(.,fr(:q~c~wy c(nmnu" ph)ts ()f tl~c Wiggler (listrib~l(,i()~ fi~c(,i()~ ()f (,l~(' sig~al (6.4.5(i), c()s~sp()s(,(l ()f (,w() (;m~ssias~ aSS~l)lit,~t(l(:s wi(,l~ (a)')v  1.5, q~  q., 1.5, +,') ~'.,  1.5, m~(! ( b ) q,  q.,  2.5, tl~ys. Rel). 104, :~473.()1 (1.()~;4). [4.1()] M. Hillery, ll..l". ()'(',()~mll, M.(). S(:~flly m~(l E.! ). Wig~mr, "l)istril)~ti()~ fln~ctio~s i~ I)t~ysi(:s: flu~(lm~m~tals", Plws. ReI). 106, 12 l 167 (1984). [4.11] D. Drag()~m~, "I>lms(: Sl)a('(' ('()rr('.st)()~(le~(:e 1)etwee~ ('lassi('al ()l)ti('s m~(l q~ml~t~H~ ~eclm~fi(:s", i~ l)r~jrc.,~.,~ in Optic.~, V()l. 43, E. W()lf (e(I.) (Els(,vier, A~stcr(lmn, 2()()2), (:t~. 5, I)I). 4334.()(i. [4.12] Y.S. K i ~ m~(! NI.E. Noz, l'ha.,~r b'pacc l'icturc r Q'u,.'n,tum Mr (Worl(l Sci('~gifi(', Si~gal)()r(;, 1.9.(tl). [.1.13] S. l/,. l)c (~r()()t m~(l l~.(]. S~t,t.()rl), bb'wn,datio'n..s of l'Jh,ct.'rodyna'm.ics (N()rtl~H()llm~(l, A~ster(lm~, 1!172), ('1~. VIVII, l)l ). 3174()(i. 'l'i~(: l)r()l)(:rti(:s ()f tim W(:yl l,rm~sfor~ m~(l tl~e Wiggler (listril)~ti(,~ fl~('ti()~ are sp(,('ifi('ally (l(,s('ril)(,(I ie~ l~(, al)l)e~(lix ()f (:l~. VI, pp. 3413(i4. [5]
A I'(:w (i(.h:s (:()~('('r~('(l wi(l~ (I)(' \u r(:l)r('s('~(a(i()~ i~ r('lali()~ (() (l~(: sig~ml (.l~(:()ry at(' list(:(l l)(:r('. S('e als()[.l][(i] ()f (:I~. 7 i~ r(,la(i())) (()(l)(, W~l'i()us a Sl)(:(:ts ()f th(,. j()in( sig~ml l'('l)r(:S('ll al i()ll (l~(:()ry. [5.1] ,l. Ville, "'l'lld()rie et al)l)li('ati(,ls (h: la. llol, i()ll (h: siglml mlalil, i(llle '', (~al)l(:s el, TrmlsIifissiolls 2A, (il74 (1948). [5.2] A.W.R.itmczek, "Siglmi elmrgy (list,ril)lltioll ill l,iIIle ml(l freqllellcy", IEEE Trans. Inf. Theory, I T  14, 3(i,()374 (1.(t68). [5.a] T.C.A.M. (;lmlseil ml(l W.F.G. Mecklellbr/iilker, "Tim Wiglmr (lisl, riblltiozl a tool for tilnefreqlltuwy sipglml mlalysis. I: contiimollstilne sigluds", Plfilips ,J. R,es. 35, 217250 (1980); II: r iiile signals", ibid., 27630(); Ill: relatiollS wittl other timefrequelmy sigzml trazlsfonnations", ibid., 372389. [5.4] F. PeyriIl ml(1 Ii,. l)r()st, "A ~nified defilfitim~ for the (liscrel, el,i~ne, discretefrequency, m~(l (liscretetiIne/frequency Wigner distril)~tio~ls", IEEE Trans. on Aco~st., Speech an(t Sig~ml Process. 34, 858867 (1986). [5.5] L. Cohen, "Tilnefreq~mncy distributionsA Review", Proc. IEEE 77, 941981
(1989).
[5.6] F. Hlawatsch an(1 G.F. Bou(treauxBartels, "Linear an(t quadratic Limefrequency signal representations," IEEE Signal Process. Mag., 2127, April 1992. [5.7] L. Cohen, TimeFrequency Analysis (PrenticeHall, Englewoo(1 Cliffs, 1995). [5.8] The Wigner Distrv;bution: Theory and Applications in, Signal Processing, W. Mecklenbrguker and F. Hlawatsch (eds.) (Elsevier, Amsterdam, 1997).
[6]
T h e pioneering p a p e r s by B a s t i a a n s i n t r o d u c i n g the W i g n e r r e p r e s e n t a t i o n in optics and investigating m a n y related theoretical and applicative issues are q u o t e d below. We also address the reader to the review p a p e r s [6.8] and [6.9], and to the J O S A issue [6.10].
The Wigner Distribution Function: Analytical Evaluation
337
[6.1]
M.J. Bastiaans, "Wigner distribution flmction applied to optical signals and systems", Opt. Comm. 25, 2630 (1978). [6.2] M.J. Bastiaans, "The Wigner distribution flmction and Hamilton's characteristics of a geometricaloptical system", Opt. Comm. 30, 321326 (1979). [6.a] M.J. Bastiaans, "Wigner distribution flmction and its application to firstorder optics", J. Opt. Soc. Am. 69, 17101716 (1979). [6.4] M.J. Bastiaans, "The Wigner distribution flmction and its applications to optics", in @tics in Four Dimensions1980, AIP Conf. Proc. 65, M.A. Machado and L.M. Narducci (eds.) (AIP, New York, 1981), ch. 9, pp. 292312. [6.5] M.J. Bastiaans, "Signal description by means of a local frequency spectrum", ill Transformations in Optical Signal Processing, W.T. Rhodes, J.R. Fienup and B.E.A. Saleh (eds.) SPIE 373, 4962 (1981). [6.6] M.J. Bastiaans, "Use of tile Wigner distribution function ill optical problems", in 198~ European Conference in Optics, Optical Systems and Applications, SPIE 492, 251262 (1984). [G.7] M.J. Bastiaans, "Application of the Wigner distribution function in optics", in The WiDner Distribution: Theory and Applications in Signal Processing, W. Mecklenbrguker and F. Hlawatsch (eds.) (Elsevier, Amsterdam, 1997), pp. 375426. [~.s] G. Cristdbal, C. Gonzalo and J. Bescds, "hnage filtering an(t analysis through the Wigner distribution", in Advances in Electronics and Electron Physics, Vol. 80, P.W. Hawkes (ed.) (Academic Press, San Diego, 1991), pp. 309397. [6.9] D. Dragoman, "The Wigner distribution flmction in Optics and Optoelectroidcs", in ProDress in Optics, Vol. XXXVII, E. Wolf (e(t.) (Elsevier, Amsterdam, 1997), oh. 1, pp. 156. [6.10] Wigner Distribution and Phase Space in Optics, G.W. Forbes, V.I. Man'ko, H.M. Ozaktas, R. Simon and K.B. Wolf (e(ts.), J. Opt. Soc. Am. A 17 (12), 22732542
(2000). [7]
The W i g n e r distribution flmction is one of the basic t()ol of q u a n t u m optics, from which the whole class of sparameterized distribution functions can be obtained. We signalize the seminal papers by Cahill and Glauber [7.1] and some fimdamcntal textbooks on q u a n t u m optics, dealing as well with the Wigner representation, specifically addressed in the quoted pages. [7.1] K.E. Cahill and R.J. Glaubcr, "Ordcred expansions in boson amplitude operators", Phys. Rev. 177, 18571881 (1969); "Density operators and quasiprobability distributions", ibid., 18821902. [7.2] W.H. Louisell, Quantum Statistical Properties of Radiation (John Wiley & Sons, New York, 1973), ch. 3, pp. 168176. [7.a] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 541542. [r.4] U. Leonardt, Measuring the Quantum State of Light (Cambridge U. Press, New York, 1997), ch. 3, pp. 3754. [7.5] W.P. Schleich, Quantum Optics in Phase Space (WileyVCH, Berlin, 2001), ch. 3, pp. 6798.
[8]
Presentations concerned with the application of the Wigner representation to electron optics in b o t h synchrotron radiation and microscope transfer function related issues are in [s.1] K.J. Kim, "Brightness, coherence and propagation characteristics of synchrotron radiation", Nucl. Instr. & Meth. in Phys. Res. A 246, 7176 (1986).
338
Linear Ray and Wave Optics in Phase Space
[S.2] V.M. Castafio, "A generalized display of tile contrast transfer fimction in transmission electron microscopy", Optik 81, 3537 (1988). [s.a] G. Dattoli, C. Marl ail(t A. Torre, "A note oil Wigilcrphase space and electron beam transport", ENEA R T / I N N / 9 1 / 3 0 . [S.4] V.M. C~stafio, P. Smltiago, A. G&Ilez, "linage (tisplacciileilts ill trmlslldssion dectron lliicroscopy: a pliasc space approacll", Optik 95, :~133 (1993). Is.5]
F. Ciocci, (]. Dattoli, A. Torrc ml~l A. Rcliicri, hlsertioll I)eviccs for SylmhrotroIl Ra~liat, ic)ll a,ll~! lq'ee Eh;ctr~,l I.~Lser (W~wl~l S('., Sixlgal)~We, 2()()()), oil. 5, t)I). 167178.
[!)] (',ellm'alizal,i~)ll ~)f 1]1~' l)llasCSlm('c fiWllnllali~ll ~f (lllallllllll lll~;~'lla~i~'s 1~ a~y set t)f llt)ll r C~lljllgalt' t)l)Cl'nl~l'S al'~' ~'la])t)l'al~'{l ill lilt lml)Crs l)cl~)w. Sr l il h;s ~:~('~;r~er wi111 s i ~ i l a r ~'lal~wal i~n~s will~il~ 11~; ('r r sigl~al I lm(~ry (:a~
I~; f i ~ { l i~ [5] ~f ~'1~.7. [!~.1] kl.(). S(uflly ~tll~l l,. (:()}ICll, "Q~siln'~flm,lfility {lisl,riln~l,it~s fin" arl)il,l';try ~l)erators '', i~ 7'h.c l'h.ysics of l'h.a.~c b'pa~:~', Y.S. K i ~ m~l W.W. Z~u'l~al'y (e~ls.) (Spri~gerVcrlag, New Ycwk, 1.q87), 1)!). 25326(). [9.2] A. ]~i~,~l)ale a~r lkl. ]{azavy, "Q~lalltllllllllc('llallical l)ll;~S~:space: a gt~'~cralizal,io~ of Wig~mr 1)lmscspacc fi~r~n~lal,i{~ I,~ al'l)itl'al'y c()r162 ~ systclllS", l~l~ys. Rev. 38,
A 41, 515(i6()54 (1.q.gt)). crlics ~f 1l~t; W i g ~ ' r (lisl rii)~ll i(~ fit,~('l i ~ , f i ' ~ 1)~11~ l l~'()r~,l i~'al a~ul al)t)li~:alivc vi,;wl)(,i~ls, (:a,, i),, f i , ~ , l i~ [1], [4], [8] a,~r il~ l l~t; lml)e,'s 1,,;l,,w. A,l,liti,,~,al tith;s (:an l)e fr i~ l l~' f()l'ill('()~i~g ('lml)i(;rs. co~(liti~s f(~r ~iq~u,~ess", l~t~ys, l~ett. A 83, 14514~, (1.981). [10.2] I~. Com~i, "()1~ a fi~(la~w~tal l)r~)pcrty ~f tl~c Wig~wr (listriln~ti{)~C, IEEE Trans. o~ Ac(~st., Sl)cec}~, mul Sig~ml Pro('cssi~g, ASSP35, 55.%561 (1987). [10.a] F. Hlawatsch, "l~tc'rfcre~mc l,e r ~ s i~ tim Wig~mr ~lisl.ri])~l,i~)~", i~ Digital Signal l'rocessing84, V. (',appellilli alibi A.G. Collstalltillir (e~ls.) (Elsevier, Alllster(laln, 1.984), pp. 363367. [10.4] A.,I.E.M. Jansscn, "O,~ thr loc~s a~{l sprea(l of pse~ulor timefrequency I)la~c", PlfiliI)s ,1. Res. 37, 79110 (1982).
fin~ctio~is in the
[10.5] L. Cohen, " W h a t is a ~mdticompo~e]~t signal?", Proc. I E E E , I(',ASSP .92 5, 113116 (1992). [10.6] Hudson, R.L., "Wlmn is tl~e Wigner q~msiprobability (lensity ~o~>~mgative?", Rep. Math. Phys. 6, 249252 (1974). [10.7] L. Cohen and Y.I. Zaparovmmy, "Positive quantum joint distributions", J. Math. Phys. 21, 794796 (1980). [10.8] R.F. O'Com~ell an(t E. P. Wigimr, "Some properties of nonx~egative quantmnmechanical distribution fimction ", Phys. Lett. A 85, 121126 (1981). [10.9] F. Soto and P. Claveric, "When is the Wigner function of multidimensional systems nonnegative?", J. Math. Phys 24, 97100 (1983). [10.10] A.J.E.M. Janssen, "A note on Hudson's theorem about fimctions with nonnegative Wigner distributions", SIAM J. Math. Anal. 15, 170176 (1984). [10.11] A.J.E.M. Janssen, "Positivity properties of phaseplane distribution functions", J. Math. Phys. 25, 22402252 (1984).
The Wigner Distribution Function: Analytical Evaluation
339
[10.12] A.J.E.M. Jansscn and T.A.C.M. Claasen, "On positivity of timefrequency distributions", IEEE Trans. Acoust., Speech, Signal Processing, ASSP32, 8194 (1985). [10.13] K. Husimi, "Some s properties of the density matrix", Proc. Phys. Math. Soc. Japan 22, 264314 (1940). [10.14] N.D. Cartwright, "A nonnegative Wigncrtype distribution", Physica 83A, 210212 (1976). [10.15] P. Bertrand, J.P. Doremus, B. Izrar, V.T. Nguyen and M.R. Feix, "Obtaining nonnegative quantum mechanical distribution function", Phys. Lett. A 94, 415417 (1983). [10.16] T.B. Smith, "A comment on the smoothed Wigner function", Phys. Lett. A 127, 7578 (1988). [10.17] D. Dragoman, "Tile origin of negative values of tile Wigner distribution function", Optik 111, 179183 (2000). [10.18] R. Jagannathan, R. Simon, E.C.G. Sudarshan and R. Vasudevan, "Dynamical maps and nonnegative phasespace distribution flmctions in quantum mechanics", Phys. Lett. A 120, 161164 (1987). [10.19] F.J. Narcowich, "Conditions for the convolution of two Wigner distributions to be itself a Wigner distribution", J. Math. Phys. 29, 20362041 (1988). [10.20] H.H. Szu and J.A. Blodgett, "Wigner distribution and mnbiguity flmction", in Optics in Four Dimensions1980, AlP Conf. Proc. 65, M.A. Machado and L.M. Narducci (eds.) (ALP, New York, 1981), pp. a s s  a s 1 [10.21] F.J. Narcowich and R.F. O'Connell, "Necessary and sufficient conditions for a phasespace flmction to be a Wigner distribution", Phys. Rev. A 34, 16 (1986). [10.22] Y. Aharonov, D.Z. Albert and C.K. Au, "New interpretation of the scalar product in Hilbert space", Phys. Rev. Lett. 47, 10291031 (1981). [10.22] R. F. O'Connell and A.K. Rajagopal, "New interpretation of the scalar product in Hilbert space", Phys. Rev. Lett. 48, 525526 (1981). [10.24] V. Buzek and P.L. Knight, "Quantum interference, superpositioll states of light, and nonclassical effects", in Progress in Optics, Vol. XXXIV, E. Wolf (ed.) (Elsevier, Amsterdam, 1995), ch. 1, pp. 1158. [10.25] D. Onciul, "Eflqciency of light launching into waveguides", Optik 96, 2024 (1994). [10.26] D. Onciul, "Waveguide launching efficiency for multimode and partially coherent light sources", Optik 97, 7577 (1994). [10.27] D. Dragoman, "Phase space representation of modes in optical waveguides", J. Mod. Opt. 42, 18151823 (1995). [10.28] D. Dragoman, "Wigner distribution function representation of the coupling coefficient", Appl. Opt. 35, 67586763 (1995). [10.29] M. Testorf and J. O.jedaCastafieda, "Fractional Talbot effect: analysis in phase space", J. Opt. Soc. Am. A 13, 119125 (1996). [10.30] J. OjedaCastafieda and E.E. Sicre, "Quasi rayoptical approach to longitudinal periodicities of free and bounded wavefields", Opt. Acta 32, 1726 (1985). [10.31] K.H. Brenner and A.W. Lohmann, "Wigner distribution function display of complex 1D signals", Opt. Comm. 42, 310314 (1982). [10.32] A.W. Lohmann, J. OjedaCastafieda and N. Streibl, "The influence of wave aberrations on the Wigner distribution", Opt. Appl. 13,465471 (1983). [11] The concept of the m u t u a l coherence function was firstly introduced in the seminal paper by Wolf [11.1]. Some representative t e x t b o o k s are listed here. See also [7.2] and [7.3].
340
Linear Ray and Wave Optics in Phase Space
[11.1] E. Wolf, "A macroscopic theory of interference and diffraction of light from finite sources. II. Fiehts with a spectral range of arbitrary width", Proc. R,oy. Soc. (Londo,~) 230, 246265 (1955). [11.2] M. Born and E. Wolf, Pri'n,ciple,s of Optics (Pergamon, Oxford, 6th,1980), ch. 10, pp. 491555. [11.3] M.,J. Berall ml(l (].B. l'arrent, ,Jr., 7'hi'or!/ of l'artial Cohere~,ce (l'reilticeHall, Ellglewwo(1 CIiIN, 1964). [11.4] .l.W. (loor Statistical Optics (.lo}~ Wiley & So~s, New York, 1985). [1 1.5] B.E.A. Salctl ml~l lkl.(:. Tci('ll, Ftm,da'ntt"n,/als of Ptm/,o'n,i~:s (.It~lHl Wiley & Sculls, New York, 1991), r 1(1, 1)1). :H2383. [12] F'~r tll~' Wigll~'rn'l)rt's~'lltatit~ll ~f l)artially 2.
(7.5.36)
378
Linear Ray and Wave Optics in Phase Space
FI(~IJI1E 7.14. (a) l)lmutr l)lot of tl~c finwti()~ ~r.Ap,,('u,,~,) rcl)rcsc~ti~g tl~c ol)tical traI~sfcr fi~(:ti()~ ()f tt~e syste~ witl~ w~ryi,~g (lef()('~s parm~eter ~. (b) I)r()tiles ()f ~r~4p,,(u, v) alo~g tl~e straigld, lix~c v  2bu for (leli~fitc wd~tes ()f b  L'r (l~sl)irc~! i)y [7.5].) Cl(~a,rly, (hi l'(;la,t,illN .,4p,~ t() 1t ill(, r('.a(till~ ()f Mp,, ill ill(' l)(~l'till(;llt (tt, t,) I)lane llnlSt 1)c l)erfi)r~;(l a.l(~g t,l~e straigld, lilac t ,  2~tt. '1'1~' wd~w, ()f .Ap ~ a.t tl~c
(5 = L:c, als() ()l)tai~(;(l gral)l~i(:ally as tll(; tmig(;~lt ()f 1,11(; 1)()lar a.llgl(; /;/ ()f th(; line t ,  2(~u t() tl~(' u axis: 2b  l,m~/~, h~ t)arti('l~lar i~ tl~(' ('()l~(liti(m ()f f()(:lts (7.5.28), a~()~l~d,i~g t() b  (), tl~(' vahws ()f Ap,, a l()~g l,l~(, tt axis yiel(t the va.l~w,s ()f tt~(; il~f()('~s ()t)ti(:a,1 tra~sfer fit~(:ti()~ [7.3]. Als(), tl~e i~t(,rs(~,(:ti()ns ()f the straigtd, line ~ ,  2(5u witli tl~(', s y ~ c t I ' i ( " t~yl)(;rt)()la.s ,'
21I .
II
.
< 2, ' n . . .
1 2,
(7.5.37)
whi(:tl as notc(t ill w 7.4.4 are tile zero wdlu; lo(:i ()f flnl(:ti(nl (7.5.36), (tetermine the points a,t which the ()I)ti(:a,1 transfer fimctioxl II(u; ~) va,xlishcs. In Fig. 7.14.a,) the t)la.lm,r plot of 7rApo (u, ~,) is rct)ortc(t, with a straight line '~ = 2~'u, for a (:h()scll vallw, of (5, t)eing showll a.l()~lg whi(:tl the reading of the ()t)ti(:a.1 tra.nsfcr flln(:ti(m for t h a t va.lue of the (tcf()(:lls nlllst t)c t)crformed. Fina.lly, the profiles of 7rAp ~ (u, ~,) a.long the lines ~,  2(~u for different values of the dcfocusing t)a.ramct(;r b, i.e., c, are shown in Fig. 7.14.b). Ilm;rcstingly, if P0(~c) is mo(tellc(1 by a symmctri(:al flm(:ti(m, a,n equivalent (tcs(:rit)tion in t e r m s of the W i g n c r distribution flmction can be elaborated in virtue of relation (7.4.29). Finally, it is w o r t h m e n t i o n i n g t h a t the a m b i g u i t y flmctionbased account of the optical transfer function can effectively be used to obtain the polychro
The Wigner Distribution Function: Optical Production
379
matic optical transfer function. In fact, since longitudinal chromatic aberrations, which affect imaging with polychromatic light, amount to a wavelengthdependent defocus parameter c(A), planar plots like that in Fig. 7.14.a) can be interpreted as polar displays of all the monochromatic optical transfer functions of the system, and hence used to calculated the relevant polychromatic optical transfer function.
7.6
Optical production of the Wigner distribution function general considerations
The optical production of the Wigner distribution function is a particular case of the more general issue concerning the realization of coherent optical computing systems. Noncoherent optical techniques for signal processing have been developed as well. As noted before, coherent illuminated optical processors operate on typically complex amplitudes, whereas incoherent illuminated optical processors operate on intensities, and so on inherently real signals. Conforming with the limited purpose of the present analysis, we consider here coherently illuminated optical processors. Optical processors may be thought of as analog computers, which by use of optical architectures are capable of performing a certain class of operations [10]. Opti('al systems can inherently handle 2D data, displayed on the surface of the object or image plane. They can therefore operate simultaneously and independently oil two separate channels as opposed to serially as in digita.1 processing, whi(:h employs electronic devices having only time as an independent variable. The Fresnel and Fourier transforms are prime examples of matheInatical operations that can be implemented by very simple optical systems. As noted in w 5.2, the propagatioil of coherent light between two planes in a homogeneous and isotropic medium provides a means to perform the 2D Fresnel transform. Similarly, the propagation between the focal planes (front to back) of a positive sphericM lens provides a means to perform the 2D Fourier transform. In both cases, obviously, the respective optical setups must satisfy the Fresnel approximation, demanding for small extensions of the light distribution in both the object and observation regions with respect to the propagation distance. The Fresnel and Fourier transformations are specific realizations of a general integral transformlike relation, expressing through the mathematical form of a superposition integral the relation between the input and output signals for linear systems. We recall, in fact, from the introductory considerations of w 4.3.2 that for a 2D linear system the input ~i(ql, q2) and the output (I)o(ql, q~) are linked through the 2D superposition integral
~o(ql,q2)  //e(ql,q2;qtl,qt~)~i(qtl,qt2)dqtldqt2, JJ
(7.6.1)
Linear Ray and Wave Optics in Phase Space
380
.
!
the t)ointsprea,d function O(q~, %, q'~, q2) b e i n g tile mathema,ti('a,1 model for the t)hysi(:al t r a n s m i s s i o n t)r()t)erties of the s y s t e m . As in w 4.3.2, q~ and q2 (ten()te Ca,rtesia, n (:oor(tinates in the referen('e t)la,nes, tra,usverse t() the optical z axis, the sa,nlc synlt)()ls t)(;ing ~se(t for b()th the int)~t ml(t (n~tt)~t (:()()r(tinates. Th(: t)()int sI)r(:a,(l fim('(,i()n O(q,, (12;q'~, q;) signifies th(: r(:st)()ns(: ()f th(: s y s t e m a,(, t)()sil,i(m (q~, q2) i~ (,t~(: ()~(,t)~(, t)lm~(: t ( ) a (l(:l(,a sig~m,1 gS(q" q'~)b(q~'  (1~), aI)I)li(:(1 at I)()sil,i(nl (q',, q ~ ) i ~ (,l~(: inI),~(, t)lal~(:. A sI)(:('ial ('as(' is wl~(:~ t,l~(' r(:sI)()~s(: fiu~(:t,i()~ O(q,, q.~; q',q~) fa('t()rs as .
I
l
l
O(q,, '12, '1,, %)  g~ (ql' q, )g2('12'
I
'12
)'
(7.6.2)
(list)layi~g i~l(l(:l)(ul(l('.~t l'(:Sl)()llS(:s ()f tl~(: syst(ul~ i~ ()I'I,l~()g()~al (lir(:('ti()~s. Tiffs ha.s 1)(:(ul 1,11(: ('(:~I,ral l~yl)()tl~(:sis ()f 1,11(: (:()usi(h:rati()l~S (l(:v(:l()l)(:(l i~ l,ll(: t)r(:vi()~s (:l~al)t(:rs, l~a,vi~g (:~ml)h:(l ~s t()I,~rl~ fl'(n~ a 21) I,()a l I) (h:s(:l'il)I,i()ll (i.(:., i~l I,(:r~s ()f a si~gl(: l)r()til(' (:()()r(li~mt(:) ()f I.l~(' ligl~I, l)I'()t)a,gaI.i()~l. As u()t('(t a, ~U~l~d)(:r ()f I,i~(:s, fa('i,()riza(,i()~s lik(: (7.6.2) al)l)ly I,() sl, ig~m, ti(: m~(t six~q)l(: a,sl,ig~mti(: sysl,(:IIlS. (',h:a.rly, (,l~(: a rra.~g(:l~(:l~tS fi)r (,I~(: ()l)ti('al i~l~l)l(:~(:~tati()~ ()f (,I~(: l:r(:Sl~(:l (,rm~sf()rl~, ~m~(:ly, a s(,ra,igl~(, s(:(~l,i(n~ ()f a. l~()~()g(u~(:()~s ~(:(li~l~, a.s well as ()f (,I~(: l"()~ri(,r (,rm~sf()r~, i.(:., a sl)l~('ri('al l('~s ()r a s('(, ()f ('yli~(Irical l(:~s(:s, alh)w f()r (I~(' (l(,('()~q)()si(i(n~ ()f (,l)(: I)()i~( sl)r('a(l fi~('(,i()~ O(q,, q.~; q',, q~) i~(,() tw() li~(: sl)r(:a(l fi~('(,i(n~s g(q, q') a,s ix~ (7.6.2). A('('()r(li~gly, f()r (:a(:l)()f (,I~(: (,w() (',a,rt(:sia,~l (lir(:(:l,i()l~S w(: l~my wri(,(:, iu (,l~(: lh'(:s~(:I a,l)l)r()xil~m,(,i()n, gF ..... I,( q, q~,)
V/ k
( i fft, 9t,.t ( q   q t )2
II(:l'(:, ]~: (l(ul()(,(:s (,ll(: Wa,V(:llllllfi)(:r ill (,ll(: lll(:(lilllll, a,S d ml(l .f Sl)(:(:ify l'(:st)(:(:(,iv(:ly (,ll(: s(:(:ti()n l(ulg(,ll a.ll(l (,lm l(:lls f()(:a,1 l(:ngth ('). Als() int(:r(:stillg, a,u(l l)(:r(,ill(Ult t() th(: f()rth(:(nlfing (lis(:llssi(nl, is (,ll(: imaging (,rmlsf()rnm,ti()ll, wlli(:tl is (l(;s(:ril)(:(l t)y a, set)ara.l)h: illlI)llls(' r(:sl)()lls(: i~) (,ll(: fornl a.s (7.6.2), ea,(:h (lir(:(:t,i()n l)(:i~lg a(:t(:(1 a,(:(:or(tixlg (,()
(q, r

Mq),
(7.G.4)
w h e r e M d e n o t e s the ilmgllifica.tion fa,ctoI'. Tim ot)tical inq)lellmnta,tion of a,n ilna,ging t r a n s f o r m m a y (:(nlsist of a simple twolens system; a sequence of two singlelens FOUl'i(',r tra,ilsforxners (a, pure ina,gxlifier), for iilsta,ii(:e, performs an i m a g i n g t r a n s f o r m a t i ( m , with m a g n i f i c a t i o n I)rot)ortional t,() tt~e ra,tio of the lens f(,(:a,1 lengths: A4 = f,2/.fl (see w 3.4.3 and Figs. 3.910). T h e st)rea(t fi~n('ti(ms gFour~er(q' q') a n d g~m~g~.g(q,q') a)()ve t)(;rta,in to ot)tical setlH)s t h a t realize a,n (;xa,(:t Fourier t r a n s f o r m i n g or i m a g i n g relation b e t w e e n ~ We evidence that the expressions of the line spread flmctions for tile Fresnel and Fourier transforinations given in w167 5.2.12 contain the wavenumber k() in the vacuum space, and accordingly the reduced section and focal lengths. Of course, they are the same as those in (7.6.3), where the refractive index is absorbed into the wavenumber k in the medium, and hence d and f signify the physical section and focal lengths.
The Wigner Distribution Function: Optical Production
381
the input and output planes. In general a nonexact Fourier transforming or imaging relation between the input a,nd output planes gives rise to a spatially varying quadratic phase factor in the output signM that is not present in the input signal. If, for instance, the signal is out of the front focal plane, the lens produces the Fourier tra,nsform of the input with a, quadratic pha,se factor depending on the signal misloca, tion. Similarly, if in the doublelens arramgcmcnt of an imaging system the back focM plane of the first lens is not perfectly superimposed on the front focal plane of the second, the imaged distribution presents a, quadratic pha,se fa,ctor depending on the misloca,tion of the two planes relatively one to the other. 7.6.1
The Wigncr transform as an optical 2D linear transformation
It has been already empha, sized that the Wigner integrals represent a, bilinear transformation of 1D signals. The general expression for a bilinear [11.1] transformation of the signal ~(q) to ~/~(q) has the form of the double integra,1 ~* (q:)~(q;)L~(q; q'~, q; )dq'1dq;,
~/~(q)  ~ !
(7.6.5)
!
where the kernel Q(q; ql'q2) may t)e interprete(t as tile response of the system to two impulses applie(t a,t positions q'l and q~. Although not cxpli('itly cvi(tcncc(t in the notation, the kernel ~(q; qtl' (t~) may (tet)en(t (m s()m(; other pa,ramcters a,nd hence also tlle flmction 't/,(q) will. Evidently the Wigncr distriblltion and ambiguity flm(:tion arc in conformity with the scheme (7.6.5), t)oth converting a 1D flmction of spa,ce or fl'equency into a, 2D flmction of spa.ce and frequency together. It is a.lso evident tha,t self(:orrela,tion and selfconvolution arc fllrthcr examples of bilincar transformations. Moreover, we mention that a bilincar transformation like (7.6.5) relates tile optica,1 intensity ~h(q) in the image plaile to the optica,1 transmitta,nce p(q) of a tra,nsparency illuminated by partia, lly coherent light, the kernel accounting then for both the response of the system and the coherence degree of the illuminating light. Relation (7.5.20) with ql = q2 illustrates the possible bilinea,rity of the response of a,n optical system to pa,rtially coherent illumina,tion. Indeed, it has been evidenced in w167 6.3.5 and 7.5.2 that the Wigncr distribution naturally finds an eminent place within the partial coherence theory, also due to its intrinsic bilincarity. As observed iI1 [11.1], Eq. (7.6.5) describes more optica,1 systems a,nd processes than the linear transformation equation (7.6.1), as a consequence of the quadratic relation between the optical intensity and the optical field amplitude. The reader may readily verify that the Wigner transform is recovered from the general expression (7.6.5) with ~vv(q;
qll~ q12)  l7r~ ( q l
~ q2 
2q) ~i~(qlq2)
(7.G.G)
382
Linear Ray and Wave Optics in Phase Space
wherea.s the ambiguity function is obtained setting t
t
1
,i
L~A (q; ql' q2)  2~a(ql  q2 + (]) cTn(q' +q2).
(7.6.7)
The spatial fre(t~mncies h: and ;: enter the above kernels a,s t)araniete, rs on which the Wigner (tistriblltion a.n(1 the ambiguity flmction (:onie then to depend. The flul(:tiolial fornis of L~w ml(t ~A Ilia,nifestly stl()w the relati()li (ff the st)a,ce va.ria,t)l(;s q a.ll(] (l ()f)42 a,li(t A t,() tle (:(;lit(;r (:o(ir(tiliate" q '#~+q~ a.n(t t() i,li(: (lilI'(:r(;li(:(: (:()()r(lilla,(,(:: O = q~  q l , r(:sl)(:(:i'iv(:lY" W(; ilivi(,(: (,lle r(:adcr to (h;(lll(:(; i, ll(: ali,(;rlia.i, iv(; (;xl)r(;ssi()li ()f (,]l(; (,ra,lisfi)rliia,(,i()li (7.6.5) ill (,(;rills ()f (,lie siglla,l sI)(:(:trlllll ~(/;,), a,ll(l a,(:(',()r( l i l lgly t,]l(: fr(:(lll(:ll(:y(l()lllaili (:Xl)l'(:ssi()ns ()f i,li(; k(;rn(;ls (7.6.6)a,li(l (7.6.7). It is easy (,() r(:(:()gniz(: (,]i(: I)()ssil)ility ()f r(:la.tillg (,lle l)ilili(:ar (,rallsf()rma,tion (7.0.5) (,()a. 2D lil~(:a,r ()l)(:ra.i,i()~, wl~()s(: ~m.(,l~(:~m.ti(:a.l f()r~ is giv(:,~ l)y (7.6.1). Ill fa.(:(,, w(: lilay a.rra.lig(: a. 21) sys(,(;lii wh()s(,, sl)r(;a.(l fllll(:(,i()li O(qi, q~, qi' q~) t)e 9
+(q, q; q',, q;) Tii(;li,
~(q)
,s,, #
#
#
(7.s.s)
#
l(;t,t, ilig t, ll(; 21) ()t),i(:('t, siglia,l Oi(qi, q.2) !)(; g(;li(;l'at(;(l tlu'()llgli (,lie l)I'()(t~l(:(,
I)y t, ll(; 1 l) wa.vcf()rni
~i(q,, q2)  ~o*(q,),,c(q.~),
(7.6.9)
wc nia.y ()l)serv(; tile finl(:ti(ni (,(q) a.l()lig tll(; lille q~ = q~ ill tll(; t)lltt)llt t)la.nc ()f th(; sysl,(;lli, lla,lll(;ly tl,(q) = r q). (7.6.10)
Evi~tei~t, ly, ()tll('x (:ll(fices fin" tile s.vst('aIl r(;sl)(nlse a.ll(1 llell(:(: fin" ill(; geometry of the olltt)llt (h;te(:tioli are a,lh)we(1. Th~ls, fin" iIlsi, a,il('e,, witli #
#
#
(); q,, %) we nla.y observe ()(q) t41Olig the line q2

#
q,, %), 
(7.6.11)
0 in t,tie ollt, t)llt t)la.ll(;, t)eiiig
('(q)  (Po(q, ()).
(7.6.12)
In w 7.5.1 we proved tha,t tile Wigner distribution and tile a,nfl)iguity function can both t)e interpreted a,s 1D Fourier tra, nsforms of the 2D product function Try(q, q) ~ ( q  ~)~o(q + ~), (7.6 13) the former a,s Fourier tra.nsfi)rm with respect to the difference, space coordinate q and the latter as Fourier transform with respect to the mean spa,ce coordinate q (see Fig. 7.12). Here we conveniently rewrite Eqs. (7.5.4) and (7.5.7) in the slightly different but equivalent forms
W(q, t~)
1
x/~ i i l_l_
t
~t
it~q
rr ~"(q ' q )c t ~t
JJ
~ "
i~q ~
/
5(q ""t
q)dq'dq',
(7.6.14)
The Wigner Distribution Function: Optical Production
383
Accordingly, by comparison with (7.6.1), it becomes evident that 142(q, ~) or A(~, ~) (:an be computed optically by a 2D linear transformation where the product function Try(q, ~) be assigned as the 2D object signal and the optical processor be designed to ha.ve the response function
r
(q, i~; q' , i]')
 ~ 1 c  i ~ q ' ~ ( q '  q ) ,
(7.6.15)
to compute the Wigner distribution, or (~A (q, q; q' , q')

 ~1c
i
t~ q'~(Ol
~),
(7.6.16)
to compute the ambiguity function. Clearly, tile space variables q and ~ come to acquire the role as the orthogonal coordinates of the Cartesian frames in the input and output planes. It is evident that ~5w describes an optical system capable to perform the Fourier transform in the q direction and the imaging transformation in the ~ direction, and similarly for ~A with the role of the two directions interchanged. The optical implementation of the Wigner and/or ambiguity integrals requires therefore the following two processes: i) a multiplica.tion process to form the 2D product flmction 7r,(q, ~) from the 1D signal ~(q), and then ii) a. para.llel imaging/Fourier transforming process involving the two orthogonal directions of the object 7r.~(q,q); specifically, the qaxis direction is imaged whilst q is Fourier tra.nsformed in order to display 1A2 and vice versa,, q is Fourier tra,nsformed whilst O is imaged to display A. It is worth emphasizing that since the ambiguity function is in general complex whilst the Wigner distribution function is always real, when the detection scheme of the display setup makes use of devices responding only to light intensity (like, for instance, a photographic film), information regarding the phase of the ambiguity function atll(t correst)otl(tiilgly the negative regions of the Wigner distribution function are got lost. The optical realization of the processes i) and ii) is illustrated in the following section. Within this respect, we mention that, due to its relevance to radar applications, the generation aond display of the ambiguity function have received a great attention [8]. Since the Wigner and ambiguity integrals sireply differ in the direction along which the signalgenerated 2D product function 7rs(q, @ is Fourier transformed, the optical implementation of the Wigner transform has greatly benefited from the optical arrangements for the ambiguity function display [ 1 2 . 1 ] ; also, optical setups operating as both Wigner distribution and ambiguity function processors have been arranged. In the considerations to follow, we shall refer specifically to the Wigner transform.
384
7.7
Linear Ray and Wave Optics in Phase Space
Wigner processor for I D real signals: basic configurations
Since ()in' x~m,in c o n c e r n is to ilhistrate how the "ma,thenm,ti(:s" of the W i g n e r integral ('oul(t t)e iInt)h;l~e~te(t t)y ()t)ti('a,1 ~ma,~s, we (les(:ri])c ttw, tm,si(: s(:he~nes ()f ttw, (list)la,y sct~q)s witl~ (u~q)lm,sis ()x~ tl~e (:(m(:et)ts i'a,tl~(;r tila.ll ()11 tlw, t,e(:h~i(:al (l(;tails. 'I't~(' illl,(:rt's,('.(] I'('.a,(lt',r is (lir(;(:l,(;(l t() l,l~(: lil,(',l'a,l, llI'(; f()r a. ~()re ('()]~q)l(',l,(' a]~([ ~l)([a,l,(' a('t:()~]]d, ()f l,l~(; s~]l)j(;('l,, ('()]~('('n~(',(1 a ls() wit,]~ l t~(; (ligil,al a.l~(l l~yl)ri(l i~t)l(:~(;~d,a.l,i(n~s ()f l,l~(~ Wig~(;r (lisl,ril)~ll,i(n~ a,s well as with l,]~c ()l)l,it:al vislm,lizal,i()~s l,]~al, ]~av(; ])('~(q~ l)l'()l)()s('(l 1,()(lisl)lay 1,1~(',,1/) Wigl~(;r (tisl, ril)~l,i()~ fl~(:l,i()~ ()f 21) sig~m,ls [12]. W(; I)(;gi~ (:()l~si(l(',ri]~g ~(n~('ga.l,iv(; r('a,1 sig~als, s() l,]m,l, 1,1~('~]~n~ll,il)li(:al, iv(; t)r()(:css I,() fl)n~ 1,1~(; 21) l)r()(l~u:t f~u~(:l,i(n~ 7r.~(q,~) i]~v()lv(,,s 1,1~('~sy~l~(;l,ri(:a,lly shifl,e(l a,~(! r(;a.l v(;rsi(n~s, ~ ( q  '}) a,~(l ,c(qt~) a l'(; t)ri('tly (:sl)lil,l,(;r (:lfl)e as a, t)lu'c st){fla.rize~l l)em~. Ih'.~(:~; il, ~u~('.rg~s fr~)~ 1,1~(~l)(',a,ll~sl)littcr at 90 ~ with respect 1,~) tim ret, m'~d~g tma,l~ a,~(l witl~ a,h~r the sa,n~e i~l)~t, t)ower a,s tlm r ])ea,~. ()f (:(~rs~;, l,l~e rigl~l, a,~gh: se(:l,i~n~ r l,l~: ~l)l,i(:al s(:lmn~e is (tesigne(1 t~ l)~;rf{n'~l~ 1,1~; Im.ra.ll{'J it~m.gi~g/b'r l,ra,~sf~r~l~i~g ~imra, t,ion, wtfi(:h serves t{~ g(umra.t,c tim Wigx~er r icn~ fil~r ~f tim 1D ()t).ject ln~(tcr (',xa,nfi~a,ti~n~ a,s I)revi(n~sly (tcs(:ril)e~l. F i n a l l y we ~1,~; tl~a,t tl~e si~@ei~tnlt s~;l,~l) is h;ss tl~;xit~l~; l,lm,~ 1,1~('~(toubh;i~t)ut a,rra,l~g(u~m~d, slime its ~lsc is lil~fil, er I,~ the r ~f tim selfWigner (tistrimtion fln~(:ti(n~ (and selfm~fl)ig~fity flu~(:t,i(n~). CrossWig~u" (listribution a n d (:rossam)ig~fity fl~m:ti~n~ r re(l~fire, {)f (:()~rs(;, l,w{) tra,~st)a,relmies ret)r(;s(;nting the two multit)lying signals (;nl,(;ring th(; t)(;rtin(;~t integrals.
7.8
Wigner processor for 1 D complex signals basic configurations
In m a n y practical s i t u a t i o n s 1D (or 2D) comph',x signals are encountered. E v i d e n t l y w h e n dealing w i t h c o m p l e x signa,ls one needs to p r o d u c e the complex c o n j u g a t e version of the signal in order to m o d e l the p r o d u c t function
7r,(q, ~) o< ~*(q ~)~(q + ~2) We describe here the basics of s o m e of t h e proposed m e t h o d s to effect process i); of course, process ii) can be effected by the
The Wigner Distribution Function: Optical Production
395
sa,ine optical arra, ngements, described in the foregoing paragraphs. 7.8.1
rc~productfunction modelling setups for 1D complex signals
A certain nutnber of %function modelling a,rrangenmnts for coinplex signals have been proposed and implemented [12]. A proposed method makes use of a (possibly computer generated) hologram of the signal, replacing the tra,nsparency at the object plane (OBJ) in the optical setup of Fig. 7.21.a) [12.3]. The hologram conta,ins informa,tion a,bout both the signal and its complex conjugate. Thus, by use of suitable masks, it is possible to select the informa, tion about p or p*, a,s desired. We may suppose, for insta,nce, tha,t under coherent illumination in the first pa,ssage through the hologram (a,t)prot)riately rotated according to the t)revious (tiscussion) the reconstructed image of ~ ( ~ ) is transmitted, which image is returned back as p(q+q) whilst in the second passage the complex conjugate p * ( ~ ) is v~' delivered. The obtained product function p * ( ~ ) ~ ( ~ ) i s then input into the Wigner distribution flmction processor on the right of the object plane. Alterna, tively, an interference ba.sed method ca,n be used to effect the wanted phase conjugation of the complex signal. In fact, the interfcren(:e pattern of the sum of two signals when yiehted by an intensity recording ma,teria,1 like, for instance, a holographic fihn, always contains, at its + /  first order, the t)roduct of the amplitude of one signa,1 a.nd the complex conjuga, te amplitude of the other and vice versa,. Using therefore a non linear ma.terial, one can produce the required product function to be used for the Wigner distribution display of the original signal. Also, illuminating the complex object at a certain a.ngle may improve the visibility of the final Wigner distribution function display. Referring to Fig. 7.23.a), inspired by [12.3], we may suppose to illuminate the complex object representing the rota,ted signa,1 ~(~q2) by a monochromatic plane wave at a certain angle Oi with respect to the axis, thus producing the signal ~(qqq ~)C ikq s i n Oi behind the object plane. The signal is then separated in "
two parts at 90 ~ apart by a bea, m splitter. One part is reflected back to the beamsplitter without change by a mirror. The other is reflected back a.s well, but inverted with respect to the vertical direction by a roof top prism; it is then formally represented as (~ ( qq ~ )e ikq sin0i The two parts are again combined and added on some recording material yielding the modulus squared, '.
ikqsin Oi q~9(~22 )GikqsinOi12
(7.8.18)
Evidently it contains the selfterms ~ ~(q+q ' v~)l 2 and  2 yielding two ~mirrored contributions proportional to the Fourier transform of the instantaneous power ~(q)l 2 at frequency ! +  v/2k}, modulated by a qdepending
Linear Ray and Wave Optics in Phase Space
396
complex object
mirror
q
at angle (h
~
order
I1
transparency ( p (  ~ )
n. r 
x oc ~
[[
q+q "~
I t beamsplitter
roof top prism
~tl
D
.
q.,)(q + q )el]~~i .s'JnO,lO, [[ I
/
2k, s ~
,~)
2k. ~ m ~
/
~
0~ = 0
,,,,, ' ~~ ,
~,
q~'( q q )~, i~q .sin(),
~ / 0 +
I
I I recording material
aO
1 order
(h)
l,'l(;lllll,; 7.23. (a) Sclu,~mtic fintlu, Wiggler (iisplay ()f a 11) c()~ph,x sig~a.l. (h)Tl~e (lelt,a lilies t)r()(l~u'e(l ()~ tim ()~tl)~l. l)lmm I)y l.l~e + / ()r(ler l,en~s wril, te~ ()~ tim re('()r(li~lg til~ii i~l l,l~(, Wig~wr l)r()c('ss()r sclu,~w of l)arl (a,)i~ tl~(, cas(,()f a h'~slik(' (l~a(lralic l)lu~s(' wav('fi)r~l.
sil'e(l ('(n~ll)h;x ln'i)il~u'l, fi~l~l'l,i()li l~n~ll,il)lil;(l 1)y all I'Xl)i)~li'~l,ial. 'l'l~s, (;a,(:li l,(;l'lll ~ll(h'r (~xa.l~ilia.l,i(ni. l)ri;i:is(;ly, l,li(; { / 
,,* ~ a,il(1
),t2 ( ~
til'sl, (n'(hu" l,(~r~ls ri;sl)(;(:l,iw'ly yi('~l(l
)(,2il,'q si,, Oi
~( ~,/5, );* ( '~,/5, )"'';*"; ..i,, o,
2k
, W.(_%
(7.8,.19)
_ , +'_,,...,, o,
As ll()l,e(l, l,lle Sl)a('illg t)etweell tile t,vr (tistrit)llti(ms is (:(ml,r()lh;(l t)y the illlmfixmti()ll a llgh' Oi. In Fig. 7.23.1)) the ext)('(:te(l vi('w ()f the two) ild,(;rfere~n('ca,risi~g tra('('s i~ tll(' (nltI)~lt l)lmie is sll()w~l ill tll~' ca s(, ()f a 1D (tlm(lrati(:I)hase wa,vef()nl~ a,s l,lm,l, l)r()(l~t:t;tl t)y a, r162 a,1)(;rra,ti()l~fre(; t,l~i~ h;~s. It can be ct)~q)a,rt;tl witl~ tl~e Wigncr dist)lay of a, (:ylil~tll'i(:al lens, sl~t)w~ in Fig. 7.24.t)), wl~ir has t)ee~ ot)tained l~sing the setlq) sketr in Fig. 7.24.a), b o t h rel)r()(hu:e(t fi'()m [12.3]; here, a, holograt)hi(: film ha,s )(;(;n enq)l()yed ~ a recording device. In t)a,rt ((:) of the figure the Wigner distribution fi~n(:tion of a third orr t)hase ot).jc(:t t)ro(tuced with the sa,nw~ setup is shown. Tlw~ notation in the figure is a,s described in relation to Fig. 7.21. Moreover, as r in [12.11], optical phase r via, a nonlinear fourwave mixing process can be used to model the p r o d u c t of the signal and its complex conjugate. To this end, a, collimated monochronlatic plane wave is s e p a r a t e d into three beams, which are then suitably redirected to a t h i r d  o r d e r optical nonlinear material. As a result of the interaction of the three beams inside the m~terial, ~ fourth b e a m is generated which carries the
The Wigner Distribution Function: Optical Production
397
FIGURE 7.24. Wigner dispay for truly complex signals. M: mirror, BS: beam splitter, R: roof top prism. Wigner distribution flmction of (b) a cylindrical lens had (c) a third order phase function, both produced with the setup in (a). (Reprinted with permission from [12.3].) product of the three ongoing t)ealns, one of them entering the t)rodu('t through its comt)lex conjuga.t,e version. Referring to the s(:henl(~ of Fig. 7.25, a,da,t)te(t from [12.11] to the t)resent, discussion, we see that, the (',omt)lex object is inserted b(;twccn t,hc )cam st)litt,('~rs B 1 and B 2 . It, nla,y )C sut)t)()scd to be a,t)prot)ria,tcly oriented in the input, plane of the processor in or(ter t() produce the 7rsflmction factor ~ ( ~ ) upon illumination from the left by the undeflected t)ortion of the ongoing monochromat, ic plane wave. The reflected portion of the incoming wave at B 1 is made to follow an appropriate path up to the nonlinear material. On the other hand, the signal ~ ( ~ ) is separated into two identical copies by the beamsplitter B 2. One copy is imaged without change to the nonlinear material along the path B2B3NLM. The other copy is conveyed through the path B2M~NI,M, along which the required coordinate inversion is effected (by a Dove prism), thus reaching the non linear material in counterpropagating with respect to the unaltered illumination signal. The signal s(q, ~ produced by the nonlinear material carries just the product
q) travelling opposedly to the signal copy ~ ( ~ ) . It can then be input into a strictly Wigner distribution function processor. The details of the optical setup, described in [12.11], have been disregarded in the basic scheme of Fig. 7.25. The reader is directed to the quoted paper for a view of the complete scheme of the optical setup used to display the Wigner
Linear Ray and Wave Optics in Phase Space
398
/
M~ / ~e W~gner
display
1:/\/
colkmated
__
monochromatic
p~aRowave
(
~
/
B1
....
: t
tI ~ t I
%
XXXXJX /
B2
(P( q q )
M2
Input object
FI(~tIF/t,; 7.25. Scl~t,~,uttic ~f tim Wig~,,r l~roct~'ssor sectim~ (~lescrilu,~! i,~ [12.11]) for~fi~g the rr.~i)ro(l~ct fl~ct, io~ for a co~ph,x sig~ml. (lisl,ril)lll,i{)n fllllr
7.9
a,~l wil,l~ sligld, ~{lilir
l,l~: a l~lfig~il,y f]lll(:ti()n.
The smoothed Wigner distribution function and the crossambiguity
function
optical p r o d u c t i o n
Following the s~lgg(',sl,i()ll ()f [12.2], we (h;s(:ril)e tll(', ()t)ti(:a,1 illll)lt;lll(;lll,~tI, i()ll of loll(', smoothed Wigxlcr (list, l'il)llti()zl filll(:t,i(nl ill t)a.rl,i(:lflar t)y l,ll(', Wigller (list, ribul,ion function of a Ga,llssiml siglml. The rclater (lisr will filrther (:onfirm tlle link, ba,sica,lly roote(l ill tile Moyal I'ela, ti()ll, a,lllollg a,ll tim (:(nmet)ts intro(tuced and illllstrate(t iIl tlu', I)revious se(:tiolls, i.e. tim st)(;(:trogram, the (smoothed) Wig,mr (tist, riln~ti()I~ f, umtion and tim (cross) m,flfig,fity function. We resort to the C, al)()r elellmnta,ry signal (7.2.16), whose Wigner (tistribution fllnction, a,s showil in w 6.2.1, sha,pes as a, 2D rea,1 Gmlssia,n centered at the point (q0, n0) in the spa,(:efrequeimy plane with rela,tiw; n n s w i d t h s ~ and 1 v% in t,he q and ~: directions, respectively. Nanmly,
1 W~ (q, ~:)  7e
(qqo )2 a2
 a 2 (K,~,0)2
.
(7.9.1)
The overlap of any Wigner distribution flmction W~ with ]/VG, i.e.,
(
ff dqd~W~(q,~)WG(q,
h;),
(7.9.2)
is evidently equivalent to the convolution of W~ with the at the origin centered
The Wigner Distribution Function: Optical Production q2 _a2~ 2
2D Gaussian 1 (~ a2
. In
fact, S/
(
(q,
qo,

399
1
dqdtd42~(q, ~)c 
(qq0)2
~2
 a2 (a;a;0)2 ,
(7.9.3)
where the dependence on the parameters of the Gabor signal q0,~0 and a is explicitly displayed. Typically tile convolution of ldd~ with another appropriately chosen Wigncr distribution function is reported as the smoothed Wigncr distribution function. The smoothing has been suggested to overcome the problem of the in general non positiveness of the Wigner distribution function (see 6.3.6) as well as that of the crossterms contributions (see w 6.3.4). In accord with the Moyal formula, the integral in Eq. (7.9.3) is related to the inner product of the signal p(q) with the Gabor function PG" Namely,
1/
( )N~ )~G (q' t~; qo, t%, a)  ~
2
~(q)Fa (q)dq
.
(7.9.4)
Using the explicit expression of ~G(q), a trivial manipulation of the integral above yields the enlightening relation, (W~}~G ( q , h ; ; q o , t , ; o , a ) 
27rlA~G,~(qo, h:o)l 2,
(7.9.5)
between the smoothed Wigner distribution flmction and the modulus squared of tile crossambiguity function A~G,~ of the Gabor signal and the signal under investigation, nleasured at the space and frequency mean values, qo and h:o, of ]~G(q)] 2 and its st)ectrum 193G(t~;)l2. This result conforms with the analysis of .~ 7.2.3. Also, it is a particular case of the general result linking the convolution of two crossWigner distribution functions to the product of two crossambiguity flmctions, the involved signals in the former being crossly coupled in the latter (see [6.4] and Problem 10). As noted in w 7.2.3, the smoothed Wigner distribution function (7.9.2) can also be interpreted as a windowed Wigner distribution function. Therefore, the phasespace concentration of (I/F~}~G is controlled by the windowing distribution 142G,and hence the position resolution is determined by the rms width of FG and similarly the frequency resolution is deterinined by tile corresponding diffraction angle. On account of relation (7.9.5), the implementation of the smoothed Wigner distribution function can be realized by an ambiguity function processor, and so by any one of the previously described optical setups with the roles of the q and ff coordinates being interchanged. Also, two different signals are written on the two transparencies, which are rotated by 90 ~ one to the other, as previously described. The signal under study is indeed written on one transparency, whilst for the Gaussian amplitude an appropriately rotated Gaussian transmission mask can be used or as well the Gaussian profile output from a laser. Figure 7.26 reproduces from [12.2] the smoothed Wigner distribution runetion of a monofrequency signal of finite duration and a linearly increasing frequency signal. The symbols in these figures are as in Fig. 7.21.
Linear Ray and Wave Optics in Phase Space
400
FI(;I;III'; 7.26. S~_ 0 everywhere t h r o u g h the (q,p) plane the (:ross distri)lltion q3r m u s t vanish, thus ensuring for q3~ the definitely n o n n e g a t i v e expression: q3, (q, p)  q3% + q3%.
(8.2.43)
1D FirstOrder Optical Systems: Transfer Laws for the Wigner Distribution Function 419
It is easy to verify that the above relation is in contrast with the assumption that g3~ yields the proper marginal distributions when either one of the variables is integrated over. In fact, on the ground of such an hypothesis, integrating g3.e(q,p) over q we should have
(.)I'+
the last expression being in accord with (8.2.40) and_ the linearityof the Fourier
transform operation, which implies that ~h(p)  ~/)1( P ) + @2(P) On the other hand, according to (8.2.43) the integration of g3~(q, p) with respect to q gives
,/9t3~(q,p)dq
1'~1(p)12 _~_[,f2(p)[2
(8.2.45)
since both q3~1 and g1% are supposed to yield the correct marginals. In order that (8.2.44) and (8.2.45) be equal, the product ~/,~(p)~,(p)should vanish. Such a condition could be satisfied only if the Fourier transforms ~/)1(P) and ~2(P) were_ nonzero over two nonoverlapping bounded intervals. But, being ~/)1 and ~/h the Fourier transforms of functions defined over bounded intervals, they can not be limited to finite intervals as well. As a conclusion, we can state that phase spa,(:(; distribution functions involving the state bilinearly and having correct marginals must locally take on negative values for (:ertain states. This raises the question about the interpretation of tile g3.e,"s, whi(:h clearly. cannot strictly represent the probability of fin(ling the t)article at q with motnentum p. The non positiveness of the (bilinear) quantum phasespa(:e distribution flmctions is a plain inanifestation of the departure of the joint position momentum description of quantum inechanics frotil the classical behaviour [5]. It is evident that not every phase space distribution flmction is realizable in the quantum mechanical case. As an example, we may consider a phaseplane point, described by tile product of delta functions ~(q  qo)~(p Po). Such a distribution is classically realizable, representing a single particle at qo with momentum P0, which moves according to Hamilton's equations of motion, and thereby individualizes a definite trajectory in phase plane. Delta functions are evolved into delta functions by the classical Liouville equation (8.2.33). In contrast, the realizability of the pointlike distribution function is inhibited within the quantum mechanical context by the uncertainty principle. Rephrasing, for example, properties (6.3.1) and (6.3.53) for the quantum distribution (8.2.32), we see that for any normalized pure state function @ the corresponding Wigner distribution W(q, p) must satisfy the quantum constraints [W(q,p)[ < 
/[W(q,
1 71"h
~

1 27rh.
(8.2.46)
Linear Ray and Wave Optics in Phase 5pace
420
which evidently prevent W(q, p) from being peaked on one single point of the (q, p) plane. Indeed, they yield a, value of the order of It, for the ndnimal area in the (q, p) t)la,nc over whi('h the W i g n e r (tistribution flm(:tion st)r(;a,(ts out. 8.2.3
Wigne'r distribution function and Weyl t'ran,s'fo'r'm
As a.h'('a(ly ll()t(;(l, i,ll(; .j()iill, I)()siti()IIlll()lll(;ld, lUil (tr ()f a, (llm,ld,llln systClll is (Ii(:ta.t('.([ 1)y tilt; (Icsir(; ()f tr(;a,tillt4 tile (tlla,lltlllll tll(;()ry witllill a, fra.lllt;w()I'k silililar as tllat ()ftll(: (:lassi(:al statisti(:al Illr tllllS ~Imkillg f(;a,sit)lc, f()r illsl, ml('(:, l,ll(; ('alt'ltlal,i()ll ()f l,ll(' (;Xl)t't'l,al,i()ll valltt;s ()f (llla,lll,lllll
()l)s('~I'Va,1)lt'~s
t)y a (q, p)illlt;gra, titni illv()lvillg lunllt;rit:alvalll(;(l fllllt:l,it)llS ()f q ml(l p ra,tht'~r tlla.~l t)y a qillt,(;gral,i()ll illv()lvilitZ ()I)(;ra.ttwvalll(; 1 or m < 1. In Fig. 8.4.b) the magnifying effect of a, Galileanlike telescope on the beam radius of a Gaussian bealn is sketched. Evidently, with the limiting values m + oc and m ~ O, we recover the monofrequency and impulsive signals respectively, and accordingly the ellipses in Fig. 8.4.a) should crush over the horizontal and vertical axis, respectively. Pheosespace and physical space views of the transformation undergone by a Gaussian signal under scale transform, the latter being exemplified by passage of a Gaussian beam through a telescopic system. 8.3.2
The ray spread f u n c t i o n
We deduce now the transformation law for the Wigner distribution function under propagation through a linear optical system within a general context.
Linear Ray and Wave Optics in Phase Space
430
Let }/V(q,t~:,Zo) = VVo(q, t,:) and }/V(q, t,:, z,) = }iVy(q, t,.)denote the Wigner distribution fimctions of the signa,ls ~ (q) and ~o (q) respectively entering and emerging fr(m~ the ()t)tica,1 system. As ~.~ (q) a.nd 7:~,(q) are linked by the space (tonm,in transfer relation, 0
~,,(q) 
g(q,q')~.~(q')dq',
(8.3.17)
il, is t)~ssilflr t,r r162 1,1~; Wig~:r (lisl,rit~l,i~ fi~(:l,ir ~V,, (q, ~.) r l,l~r (mttmt sig~m,1 i~ t,~'.r~s ~)t"l,l~e li~eSl)I'ea(l flu~(:t,io~ g(q, q') ~)f l,l~{; sysl,r m~(l t,l~e int)~d, sig~ml a.n~I)lil,~ul{' r (q). I~ fa.~:t,, a.(:(t(w(i~g I,{~ the q(l~m.i~ ~l~;ti~il,i~m (6.2.2) of l,l~c Wig~wr r fin~{:l,i~m, we; write
w,,(q,
1
l'/.'/",
iml'g*

(u
,
4')g(q
,
,/")< (q")r (,/")d,/d,/'dq'" *
.
[Jsi~g r~;lal,i~ ((i.3.23) I,~ r~:(:~w~'r l,l~: Wig~(:r ~lisl,ril~l,i~ fll~('l,i~ r l,l~; i~i)~t sig~,~,l fl"(llll l, ll{ ' [)l'{){lllf'(,., "* (q")r (q'"), a ~ l s~italfiy a,rrm~gi~g l,l~; va.rir i~l,egrals a.~{I wn'ia])h:s cqderi~g l.l~, rr i~g exl)rr w~: r ~I) wil,l~ a (lcti~il, e rela.ti~)n ~)f }/V(q. h') I,~ }IV (q, ~') i~ 1,1~: f ~ ' ~ ~)f 1,1~: S~ll)~;rl)~sil.i~ i~d,r
W,,(q. ~') 
//
G(q, ~, q', ~")I/V, (q'. t;')dq'd~".
(8.3.19)
~I'1~(; kenl(,1 G(q, ~,', q', ~,") (:xl)li(:itly writ(:s a.s G(q,~,;,q',t;')
~l
g* ( q  ~2
q'" 2 )g(q t q'' 2 " q'
, q'
t
u"' 2 )~i(,q",~'q'")dq"dq'"
,
(s.a.20) a.ll(l is SeCll t,r lJ~: tile' syzlll)lr162
D)ln'ir
t,ra.llsfr
r l,llr t)rr162
g*(q ~2 ,q'
2 )g(q + ~ 2 ' q ' + q"')2 rr162 as a t'~r162 eft q" m~r q'" Naturally the l,rm~sfer relation fi)r t,l~e Wigner (listrit)~ti(m fl~n(:tion can also })e w()rk~;~l ~n~t wil, l~i~ l,t~e fl'(;(tll(;II(:y ~t~m.in rel)res~;~d,a,l,i~)~ ~f th(; system, ~sing the t)crtix~r i~t)~tr relations,
j h(~:, h;')~ (h:') d~;'. 0
~o (h;) 
(8.3.21)
In fact, on a,:co,l,~t of deft,fit, ion (6.2.3) of Wo (q, ~c) in tern,s of the Fourier spectrum ~o(~:) a,n(t relatio~ (6.3.25) for the product of two spectra, a,t different frequencies, we obtain the superposition integral (8.3.19) with the kernel G(q, n, q', ~,:') being expressed in terms of the wavest,tea(1 function h(t~, n') in the same form as (8.3.20), namely
// h * ( K  Z ~" , n' ~ c
G(q, n,q', n')
__
~1
,.i,.!
__
/~// ~ " ' ) h ( n + Z, T
nl
/~ttt + g)c i(q~'' q/ /~/t/)dn"dn"'
(8.3.22)
1D FirstOrder Optical Systems: Transfer Laws for the Wigner Distribution Function 431
The equMity of the two expressions on the right of (8.3.20) and (8.3.22), implied by the use of the same symbol G(q, ~, q~, ~ ) , can rigorously be proved making use of the relations (4.6.20) of g(q, q') to h(~;, t~') or (4.6.21) of h(t~, ~c') to g(q, q'). The dual correspondence of the two expressions in (8.3.20) and (8.3.22) is evident. Accordingly, G(q, ~, q', ~') can also be interpreted as the inverse symp]ectic Fourier tra,nsform of the product h*(~ 5,  ~)h(~ + /~/// ~" ~ + ) with respect to ~:" and ~"~. The function G(q, t~, q~, h;~) is completely determined by the optical system through the pointspread function g(q, q') or the wavespread function h(n, n'); it provides a characterization of the system appropriate to the spacefrequency represent.atio~ of light signals conveyed by the Wigner representation. As the reader may verify, the hybrid inputoutput relations for the optical system, where the input and output signals arc described in different domains, yield forlnM expressions for the kernel G(q,t~,q~,~c~), which involve the hybrid response functions in the same way a.s g(q, q') or h(~;, ~;') in (8.3.20) and (8.3.22), tile integrations being obviously perforined over both the q and ~; variables, and not exclusively in the space or frequency domain as in (8.3.20) and (8.3.22). Notably, G(q, h',, q~, h;') can be understood as the response of the system to a signal represented by the Wigner distribution fimction 142~(q, ~) = (5(qq~)~(t~ t~;~), for which the transfer relation (8.3.19) would give 1A2o(q,t~,) G(q, h;, q~, h',~). ()f (:ours(;, this interpretation has only a formal significance, since there does not exist any real optical signal, whose Wigner distribution function is a point delta in the (q, t~) plane, i.e., 142~(q, t~) = (~(q q~)~(h; t~ ). However, taking a,c(:omlt of the geometrical optics representation of light rays, we inay consider the nonret)resentable distribution lNi ((t, ~;) = ~(qtL )(~(~; ~;~) as the mathematical representation of a single ray, entering the system at the position q~ with the optical momentum (i.e., direction) p~, and hence with the frequency ~,~  k0p;. The kernel G(q, ~c, q~, ~ ) is accordingly reported in the literature as the rayspread fltnction or the ray response of the system [2]. The ray spread function G(q, ~, q~, ~ ) represents a sort of trait d'union between the ray and wave optical description of light propagation in the inherent paraxial and Frcsnel approximations. In fact, on one side it marks the transformation law of the Wigner distribution function, which is defined in terms of the signal w~veform p(q), and on the other side it reproduces the rayopticM picture of propagation, which is based on tile transfer matrix formalism.
A little more about the rayspread function It is interesting to note that expressions (8.3.20) and (8.3.22) of G(q, ~c, q', ec') have the structure of a "double" Wigner distribution function; hence, the function G(q, t~, q~, n ~) has all the properties of a Wigner function.
Linear Ray and Wave Optics in Phase Space
432
In fiwt, G is real. Also, it is e a s y to d e d u c e t h e r e l a t i o n s
g(q,q')g*(q",q
ttt )  ~1 ./z./.Gz.
t.." q' +q'' ( ~ 2 ,'~', 2 , t~') ci~(qq'')ci~;'(q'q''')dmtt~;',
" ~t~'")(.iq(t~h;")(~iq'(t~'t.,'")dqdq, ' h(t,;, ~',')h*(~:", h,,,, ) = ~1 / 7._. ._. G(q, ~;~t~" 2 ,q', ~'2
(s.:~.~:~) wt~i(:l~ h~t~k very si~fila,r t(~ (6.3.24) a.~(t (0.3.25). I~(h'e.(t, we als()~t)taiu
/*
]g(q, q,) 1'2  2~'
G(q. ~. , q' .,')d~,Z~' " Ih( ~;,
')
~") I
 2~ ~
//
G(q , ~", q' , t~"' )dqdq', (s.:~.~)
~qll(l
g(q, q')
/7'

,.,
~
~ ~;,)(,i(t~q t 2, arc then
n~
__
1 ~o
/
qr
7P(q)dq,
m~
__
~1
/
(qml)
r
T)(q)dq.
(9.2.5)
In particular, the central moment of order r relate to tile ordinary moInents of order zero up to 7". In fact, using tile binonfial expansion ( q  m l ) r E~=0 (~)()~hqhm[ h in the expression for m~ and integrating term by term, we obtain m r 
L
(rh)()rhIlhm rh I ,
(9.2.6)
h=0
where specifically n o = 1 and n 1 = m 1. W i t h r = 2 we regain the relation of m 2 to n 2 and ml, as shown in (9.2.4). Also, we note t h a t by an appropriate translation of the coordinate origin to m I the central moments (:an be made to coincide with the ordinary moments. The symbol q used here to denote the independent variable does not strictly relate to the space coordinate of optics, signifying in general a real quantity. Thus, the definitions above can be applied to the complex amplitude and the frequency spectrum of an optical signal as well. In that case, the pertinent 7)function identifies as the absolute square IP(q)12 or I@(t~)l2, which actually have both the conceptual valence and the m a t h e m a t i c a l properties to be considered as density functions. Besides, as we will see in the next section, the moments of Ip(q)l 2 and I@(n)l 2 can be interpreted in terms of the expectation values of the position aim frequency operators, and their powers, in the "state" p.
Averages in the space and frequency domains as expectation values of the operators ~ and ~ and their powers We show that the moments of Ip(q)l 2 and I~(n)l 2 are interpretable as expectation values of the position and spatialfrequency operators ~ and ~, and their powers in the "state" p. This conforms to the parallelism between the geometric/wave and classical/quantum correspondences. Also, the seemingly abstract operator picture, we have used in connection with the waveoptical description of the light propagation properties of paraxial optical systems, gains both calculational and physical merit revealing a profitable tool for the characterization of optical signals and systems.
468
Linear Ray and Wave Optics in Phase Space
According2 to the coiilIilOn definition, tile ext)ectation, or average, value of an o p e r a t o r O in a given "state" is i n t e r p r e t e d in the sense of tim inner product
(0)(,0.).
(9.2.7)
T h e dots can inter(:ha.ngea,t)ly t)e ret)laced t)y p(q)r 75(h') witt~ tlie involved ()t)erat()r, i.e. ~ anr ~:,, t)r ac(:(mlingly giv('.ll ill tllr q r rr fi)rlils. Ttle illtr162 illll~lir1621)y (9.2.7) will l.llr 1~' r t llr q r h,(h)llmill. It is evi,hn~t tl~a,t tl~,', zc'r,, ,,rch~r ~,,u~r ,,f I,;(q)l 2 a~,l I,~(~)l ~ i(lentify the 11()I'111 II~l[ ~ I[~ I'2, wliir lllay, c,,,lveifielitly 1,,; set ,',tilal t,, ,,lie. Theli h't its (:~lill~lit~; tllr lil~:ali wfi~u; ~ (~f tlie Slm tia.1 fr~:(llU:ll(W. Tlic (letinilig relati(m (9.2.3), wlii(:li sl)~;~:ii(:ally writes a.s ~, 
is ~'xtm'ssilfl~: a,s
/
~: I@(,,)1.2 d~:,
(9.2.s)
~,  ( ~ , ~ ) .
(9.2.9)
The; ('.ah'.~fla,ti~,~ ,,f (!).2.!t) r~,~l~irr t,l~ k~,w'h,~lg;~, ,,f t l~'~ 1;~n~ri~,r Sl,~,~:tr~m~ ~(~). Ih~wew~r, it, ('a.ii a.lsr t~e exl)ressr162 ill teniis r tllr siglm.1 a.liltflitllr ~(q) in a very enligl~i.~:~i~g fl)rlli. II~ far ~mfr i,lie S~l~mrr ~)~l~h~s [~?(h~)l2 = i,~tr we: r rr162 (9 2.8) as Wa*(h;)v?(h:) re,el ~akil~g ,,sr ~,f tl~e l"r
/
F,
~?*(h)h'~.(h')d~"  77
///
dh;dqdq'2*(q)~(q' ), i'~qh','i'~"/dq'.
(9.2.10)
9
d idT/c
Ile(:~gnizing t,lmt ~.~.i~,/

h; 9
/7'
r
ih'q ~
, w~'~ ~:a.li ilitx,gra.t~, t~y Imrt witli r~'~st)e(:t
d dT/,~(q')]i~(q'  q)dqdq' 
/'
~ * ( q ) [  i ~ 2 d( q ) ] d
q.
(9.2.11)
()n a,('(:olmt of tlie rel~reselml.ti(m (4.6.57) ()f ~., the a.t)~w~ ilitegral i(hmtifies the inner t)r(~(hl(:t ~,  ( ~ , ~,~). (9.2.12) T h e first ortter mon~ei~t ()f [~(h:)[2 is then interpretat)le a.s the ext)e(:tation value of the spatialfrequency ot)erator a,, i.e., @).
T h e explicit calculation t r u m @(~) according to takes the form  i ~ or Likewise the average
~
(9.2.13)
involves the wave flmction p(q) or the frequency specour convenience, and correspondingly the operator h;. value h;~ of the n  t h power h;" can be calculated as 
]
~
I@(~)12 d,~ 
(@, ~@),
(9.2.14)
1D FirstOrder Optical Systems: Moments of the Wigner Distribution Function
469
and alternatively as
t~n _ f ~ , ( q ) ( _ i ~ ) r ~ ( q ) d q _ (~, ~n~).
(9.2.15)
Therefore, the nth order moment of 1~5(~)[2 can be brought into relation with the operator 9~ through the average value of ?~, namely ~ = (~),
(9.2.16)
the choice for the wave flmction and operator representation being a matter of convenience in accord with the problem at hand. Finally, one can straightforwardly infer that the average f(~) of a frequency function f ( a ) i n the state ~(q), defined by the integral f(n)  f f(n)I@(~)12
 (@, f(?~)@),
(9.2.17)
whose inherent innerproduct interpretation has been displayed, is also obtainable a,s the integra,1
f(t~)  J ~*(q)f(s
 (~, f(?~)~),
(9.2.18)
and hence interpretable as the expectation value of the operator function f(~) in the state ~(q): f(a) = (f(~)). (9.2.19) It has implicitly been assumed that the function f(h;) be expressible through the Taylor series (x)
f(g) _ ~
f~gn _ f0 + fi n + f2 h;2 +"",
(9.2.20)
nO
with fn = fn(O)/n!. Then, applying result (9.2.16) to the series above, which amounts to substituting the operator ?~ for ~c in each occurrence, we end up with the operator function f(~), oo
f(~)  ~
fn ~ ,
(9.2.21)
n0
in both (9.2.17) and (9.2.18), the appropriate operator forms being ~ and  i ~ , respectively. The computational relevance of relations (9.2.16) and (9.2.19) is apparent. The calculation of the average ~ , or, more in general, of the average f(~) of a frequency function, does not necessarily require the knowledge of the spectrum ~5(~). It can alternatively be calculated using the signal amplitude ~(q), which
Linear Ray and Wave Optics in Phase Space
470
is indeed simply ma, nipulated t h r o u g h a p p r o p r i a t e differentiations and then inserted into the integraJs implied by (9.2.16), or (9.2.19). Clea,rly, simila,r results hold a,s well for the average of the spa, tia,1 variable q and, in general, of a.ny spatial function 9(q). In fa,ct, sta,rting from the definiti()~s q" q" I~(q)l 2 dq, .q(q) 9(q) I~(q)l 2 dq, (9.2.22) ,
/
/
t,ll(; a lt('rlmtiv(; (:xl)r(;ssi()llS (:a,ll t)('~ worke(l ()~l(, ,
d
)'n.
__
"
q "  . / r (~)(i= ~(~)d,~ (~,~"r :~(q) ./ r
(9.2.2:~)
 (~, .q(~)~),
1,() 1,11(: ( : X l ) I ' ( : s s i ( ) l l ()f ~ i l l (,ll(; h', r(:l)r(:s(:lll,al,i(n1" ~  i ? ~ .o. Ttl(;r(;f~)r(; r(;lati()~is ((,).2.16) m~(l (9.2.19) f()r tl1(: fr(:(l~U:~1(:y av(:ra,g(:s a.r(: l)aralh:h:(l )y
a,(:(:()I'(liIlg
q,V_ (~,,),
:/(q) (:/(~)),
(9.2.24)
fin" (,11('. Sl)a.l,ial a.v('rag(;s. Tluls, a.s r('gar(ls 1.11(: ('a.l('lllat.i()ll ()f sl)a(,ial a.vera,g(;s 1,11(:frt:(llU:11(:y Sl)(:('l,nIII1 ,~(h;) l)lays 1,11(:sa.111(: r()h: as 1,11('Sl)a,l,ial m~11)lil,~1(le r wl1(:ll (l(;alillg wi(,ll (,If(: ('ah'llla(,i()ll ()f fr(:(lll('11('y av(;rag('s. Fi~mlly, 11()1,(' flint, if (:v(ul l)()w(:rs a r(: illv()Iv(:(l i~l (!).2.15), 1,11(' fillal (:xl)r(:ssi(nl is fiu'(,11(:r si1111)lili(:(l. Ill flu:l,, wi(,ll l ~  21)~, (!).2.15) yi('l(Is ' :* ( q ) (  i ~ a ) 2,, :(q),tq.
~"~'" 
(9.2.25)
whi(:h l)y a,~l '~fl)l(l ill(,(:gra(,i()n l)y l)ar(,s r(:(:asl,s i~11,() h;2.m, __
9
__
'
d
m
2
,~
.
In t)arti(:ula,r, th(; avera,g(', sr
frequem:y (:an 1)(; (:al('~fla,l,('sl t)y (,ll(; integral h;2   . / I ~ q ~ ( q ) 12 dq ,
(9.2.27)
simply inv(flving the first()r(ter deriva, tive of the spatial (tist, ritn~tir The correspon(ting exlm:ssion for q~" is, of course, '
"'
d ~ ;  IIq''@ll ~.
(9.2.28)
In w 4.6.4 we have discussed the well known uncerta,inty inequality 1
CrqCL > ~
(9.2.29)
for the p r o d u c t of the space and frequency widths of the optical wave function C~q, c~, which, according to the accepted variance measures (4.6.35), are just
1D FirstOrder Optical Systems: Moments of the Wigner Distribution Function
471
the secondorder moments of I~(q)l 2 and ~5(n)l 2, and hence the average values (p, ~2~) and (~, ~2qp). In effect, (9.2.29) is a consequence of the relation holding between the operators ~ and ~ as expressed by the Lie bracket" [~, ~]  i. It is a particular case of a more general theorem holding for any pair of variables, whose associated operators do not commute, as it is for ~ and s The proof of the uncertainty relation on the ground of the operator representation is elegant and simple; also it evidences the generality of this principle which applies to any pair of non commuting observables (see, for instance, [3.10], pp. 216218).
9.2.2
Moments of a twodimensional function
Moment notions apply as well to a 2D real function 13(q, n); of course, q and n denote now in general real independent quantities, without any close reference to the Fourier pair of optics. Evidently, the variety of possibilities is greatly enriched as both local and global moments can be defined. The formers depend on q or n, as the latter, being constant, characterize the function 13(q, t~) in a global sense. It is not required that B(q, ~) be nonnegative over the relevant domain, but only that the local and global averages be positive. The global average is obtained by integrating B(q, t~) over the whole q~ plane to the consta,nt value
//s(q, t~;)dqdh; > O.
moo
(9.2.30)
In turn, the local averages are determined by partial integration of B(q, ~) along the ~ or q line, thus yielding the q and ~ depending quantities o(e) 
f
./B(q,~)dq > O.
> 0,
It is apparent t h a t
(9.2.31)
/,
s
moo  I [~o(q)dq  I qo (t~)dt~. d
(9.2.32)
d
Local moments In the light of the averages (9.2.31), one defines the local moments of B(q, ~) with respect to n and q simply reproducing the previously introduced formulae. Thus, the local firstorder moments are given by the integrals
1~1(q)

l/~B(q,~)dn tt0(q)
'
ql(~ ) 
i f ql(q,
q0(~ )
(9.2.33)
and respectively interpreted as the ~average value of B at q and the qaverage value of B at ~. In general, the local higherorder moments with respect to ~ and q write as 1
t~r (q)  t~0 (q)
][/~{~l(q)]rB(q,
~)d~
'
qr(~)
1 q o~,~)
/
[q
ql ( ~ ) ] r ~ ( q ' ~)dq.
(9.2.34)
Linear Ray and Wave Optics in Phase Space
472
Global m o m e n t s Similarly, following (9.2.30), the global firstorder moments are obtained by integration over the whole qh: plane as mo
i
_~
1
m oo
jj 'h:B(q, h:)dqdt~:,
mr0 ~_
1
ITI O0
/'/'
q]3(q, h:)dqd~,.
(9 2.35)
Evi(teId, ly, tll(;y st)c(:ify t,lle l()(:atioIl of the ceid, r()i(t ()f B(q, h'), i.e. ()f the point (re,o, m,,, ) ()11 t,l~(,,st)a.(:efl'cq~le]~(:y t)la,Im (q, h',) a,t whicl~ B, t,l~t)~lgld, of as a, ma.ss (tistril)ld, i()ll ()vet tim (q, h:) lfla.lm, is ill lm,la.ll(:e. Th(:ll, lillfil,iIlg ()~n's(:lvt:s (,()(:(msi(h:r tile (:md,ral lll()lll(:ll(,S, it is ()l)vi(nlS tha,t the se(:(n~(l()r(ter m()lllellts write too2
__ 1 mi)(}
m II

~2o
 m o O
(h"
' .
(q
1 1
(q
WI{)()
mot
~
)2~(q,
H,)dq(tH,,
m,,))(t,"  m,),)/~(q,' t;)dqdh"
(9.2.36)
m,())2~(q, t~,)dqdt~,,
as a lm,(,~ral (:xt(:nsi()~ of m~ i~ (9.2.4), whi(:l~ in(e(:( is f()l(h:(] i~t() tl~ree quantiti(:s i~ ()r(h:r t()(:l~ara(:t(:riz(: (,l~(: (:xt(:~si(n~ ()f/3 wi(,l~ r(:sl)(:(:(, t() q a~d t,: a,s well as its ()ri(:~d,a.(,i()~ iI~ tlm (q, ~,) I)la.~e. hi g(umral, tim gl()l)a,1 x~()~(:~(,s a,r(; ~m.rk(:(t t)y tw() i~t,(:g(:I" ~n~x~fi)(:rs, being lT~rl)

' m()()
//(q
~
m,() )'(t;

too, )hB(q , t;)dqdt; .
(9.2.37)
I.I
As th(: (:()()r(linat(: ()rigi~ > zxt th(; b e a m is similar t() a spheri(:al wave creating from a t)oint s(nlr(:e at z  0, t)eing in(teed 1 The two regioils z > z,t are generally known Q(1z ) z>> zr~ n(1z ) ~" ;. as nearfield region and farfield region respectively. A n o t h e r interesting p a r a m e t e r related to the shape and the p r o p a g a t i o n features of the b e a m (9.5.1) is the b e a m divergence angle O(z)(tefiim(t by
O(z)
w(z) Z
z2 w~ V/ 1 + z2 Z
T h e farfield divergence angle 0~ of the b e a m radius dependence; p r o p a g a t i n g towards infinity. In are reciprocal p a r a m e t e r s , which
R
z>>zs
O~ = ~ ./~
(9.5.8)
71Wo
represents the slope of the oblique a s y m p t o t e it describes the spreading of the b e a m when accord with the width principle, 0~ and w 0 m u l t i p l y to the z  i n d e p e n d e n t value A/Tr.
1D FirstOrder Optkal Systems: Moments of the Wigner Distribution Function
495
FIGURE 9.3. (a) Intensity of a Gaussian beam, (b) corresponding beam width w(z) all( wave front parabolic profiles, and (c) Guoy phase shift, as functions of the axial variable z.
F i n a l l y we recall t h a t tile G u o y p h a s e z ), ( ( z )  t a n  l ( z~
(9.5.9)
which adds to the planewave p h a s e kz to form the l o n g i t u d i n a l p h a s e of the b e a m , a c c o u n t s for the excess delay of the b e a m wave front in c o m p a r i s o n w i t h a plane or a spherical wave. It ranges from  ~71" at z   o c to 571" at z  oc, t h u s yielding a t o t a l excess delay of 7r as the b e a m travels from  o c to + o c . Figure 9.3 depicts the g e o m e t r y of a p r o p a g a t i n g G a u s s i a n b e a m . 9.5.2
Gaussian beams: the matrix of second m o m e n t s
T h e W i g n e r d i s t r i b u t i o n function of (9.5.1) is r e a d i l y c a l c u l a t e d to 2q2 _ ~ 2 ( z ) [ n _ R~)] 2
_
w (q,

_
~.2(z)
7I"
2
.
(9.5.1o)
T h e zero a n d firstorder global m o m e n t s are of course moo(Z)

1,
rI~lo(Z ) tirol (z)   0 ,
(9.5.11)
Linear Ray and Wave Optics in Phase Space
496
in accord with the n o r m a t i o n condition and the onaxis p r o p a g a t i o n of (9.5.1). Interestingly, for the global secondorder m o m e n t s we find
w~(z) 4 '
m~o(Z )
m~l (z)

4R(z)
1

2z R
m,,~ (z)  ,,,,(:) + 4n~(:)
As (~xI)e(:tr
t,llr sl)a,('e sr162
ilitI'ilisi(: t)em~l t)r()I)r i.r wave inlllfl)ei', wlfi(:li is i]l(ler
(9.5.12)
'
~:2~1~2( Z)
/,:2 0o ~
1
"w()2 
4
~ll()~ll(~]lt, m~o(z ) r(;lat(:s t,()t,l~(: h)(:a,l I)r
ra,(ti~s
l,llr fa.r tir (tiw~rgr 0 , l,lirr l,llr light t,lie 1)l'()l)(~I" "(:()listmit" t,()l,~mi fl'(nxi a l@es t()
st)a,tial fr(~(t~u',n('ies. Fi~lally, wr see; tlm.t a.t t,tie i~ll)tfl, t)la,llr z  {1, m2~~({)) . 4 whi(:li t,ll('ai linflt, il)lies wit, li m.2(()) t,(~ tlic lililii]lnl]~l alh)wr 1)y tile; ()t)ti(:al 1 ~m('(;rl,ail,ty r(;lati(,,l (9.2.29)" m2,,(l))m.~ (1)) a" It is (:()liv(;ni(;lll, l,() a,rra,llgr tim a])(we, lil()llient, s i~ll,() a, ~latrix s(:lm~lie (h;filfing tilt; 'l~a'l'iit'lt(:(', (()r, .,~c('oltd "nto'nl~"ltt ) "ntall"i:r E a s
ml, (z)
mo2(z )
kw2 (z)
1
L.2.w2(z )
.
(9.5.13)
A(:(:(n'(tingly tim s(~(:()ll(l Ill(nlielll,s will a,lterIm,tivr t)~; (l~llr a,ls() a,s m2o ~~qq, 11102  }~t,:t,; ~l.II(l m,~  Eq~=. N(~r t(~ say, E is (h'~tillr a.s a, rc;a,1 symlllr (all(l llr llr llm, t,rix, wll()se (~tf(liagr r162162 S are (:lea,rly ,~ith~ss wliilst t,l~r (lia,gonal (rues have (limensio~s rcslm(:tiw:ly (~f h;l~gt, h ~ a,n(1 h;x~gtli 2. T h e (teternfinaJlt r IE, wt~i(:t~ is easily seen to l)c 1 (let ]E(z)  ~,
(9.5.14)
comes to t)e an iIlvaria,nt ()f the nlotion. T h e inverse of E is ttmil 1 ~']l(z)  ( t e t ~
(mo2(Z) roll(z)
m~(z) ) m2o(Z )
(9.5.15) "
Also, it is w o r t h m e n t i o n i n g t h a t ]E is positive definite. Note t h a t a, H e r m i t i a n m a t r i x A is said to be positive definite if for &Ily vector u it results t h a t [8] (u, A u ) > 0.
(9.5.16)
It is a, well e s t a b l i s h e d result of m a t r i x t h e o r y t h a t the necessary and sutficient condition for a H e r m i t i a n m a t r i x A to be positive is t h a t all leading principal
1D FirstOrder Optical Systems: Moments of the Wigner Distribution Function
497
minors of A are positive. Since rn0~(z ) > 0 (and as well m~0(z ) > 0) and det E(z) > 0, the variance matrix E is evidently positive definite. Notably, relation (9.5.14) implies the optical uncertainty relation (9.2.29). In fact, writing the explicit expression of the determinant of E, i.e., 2 Crq2Cr~
1

2 _ ~, Crq~

(9.5.17)
we obtain for the spacebandwidth product the relation O'qO'~   ~1V/ 1
+ 4cr2~ >  2 '1
(9.5.18)
which at a primary level confirms inequality (9.2.29). At a deeper level, it yields a more precise relation for the space and frequency signal widths product being linked to the beamdepending mixed moment Crq~r (see also Problem 4.8). 9.5.3
Gaussian beams: propagation law of the variance rnatriz
The utility of introducing the variance matrix is conveyed by the evident possibility of expressing the Wigner distribution function in a compact and elegant form in terms of E. In fact, one may note that the exponent of (9.5.10) directly involves the entries of E, i.e., tile second moments, being 2q2 q_w22(z)[N
~(z)
kq 2
1 T[~'](Z)]I
R(z)]  ~u
u,
(9.5.19)
where as usual u = (q, h;)T. Therefore, according to the normalization of Ga,ussian functions (a) we can finally write
}&g(q,n;z)


1
2rrv/det ~
e2
l u T [Y](Z)]I
u
(9.5.20)
thus relating the Wigner distribution function of the Gaussian beam (9.5.1) to the relevant variance matrix E. Remarkably, this allows us to obtain tile propagation law for E and hence for the second moments. In fact, the paraxial transfer law for the Wigner distribution function signifies that 1426(u; Zo)  ] / V G ( M  I u ; zi),
(9.5.21)
where M is the raymatrix connecting the planes at zi and Zo between which the beam (9.5.1) is propagating. Recall that when dealing with the Wignerplane pair (q, ~) instead of the geometricaloptics phaseplane pair (q, p), the offdiagonal entries of the optical matrix are scaled respectively by ~0 and k 0. Recall that for an ndimensional Gaussian function G(u)  e  l u T Au, with A an n (2~)n/2 positive definite matrix, one has f ~ ( u ) d u  d,/~A a
x
n
Linear Ray and Wave Optics in Phase Space
498
In terms of the variaIme inatrices ~(zi) a,nd X(Zo) at the endplanes, relation (9.5.21) mea,ns 1 ( ;   ~ l u T [Y](Zo)]  1 U __ _1 C  1 [ M 7r 71"
1 u ] T [~](Zo)]  1 [ M  1 u]
whi(:h imt)lies of (:ourse t,l~(; i(te~tity ()f tl~(; ext)()nent. H(;~(:(;, w(; (;n(1 ~t) with
~(zo)  M ~ ( z i ) M T,
(9.5.22)
whi(:h is t,tl(; (',entral r(;la.ti(ni ()f ()llr mlalysis. Alth(nlgtl it, lm,s t)(:(;ii inf(;rrc(t ill (:()nXl(;(:(,i()ll with tll(; ('~asily llm,lm,g(;a,1)h; Wigll(;r (lisi,ril)ll(,i()ll fiU~(:ti()n ()f the signal (9.5.1), it h()l(ls tr~u~ fi)r m~y Wiglmr (listrit)~ti(n~ fi~(:ti()~, m~(l s()()t)ti(:a.1 (tist~lrl)m~(:(; (i~ 1,t~(; 1)a,raxia,1 a,l)l)r()xi~m,ti()~). ()~(:e ~fi)l(l(:(l, it, yicl(ls the (;xt)li(:it (;v()l~ti(n~ laws fi)r (;a(:l~ ()f tl~(; s(:(:(n~(l ()r(l(;r ~()~(;~ts, tln~s (:stat)lislfix~g tt~(; r~l(,s, l)y wl~i('l~ 1,t~(' s(:('()~(l~()~m~t t)a,s('(l I)l~as(~l)lm~(: r(:l)r(:s('~tativ(; regi(n~ a,ss()(:iat('~(l witt~ tt~(', I)(;a,~ (:lm,~g(;s (l~ri~g I)r()l)a,gati()~, witl~ (,1~(',rel(;va,i~t a.r(:a r(:~m,i~i~g (:()IIS(,a.II(,, ~IS S(,a.[,(:(I l)y tlm i~va.ria,~(:(: ()f (h:t ]E. Wc st~a.ll (lis(:~lSS r('la(,i()~ (9.5.22) lal,(:r i~ a ~()re g('~(;ral (,(rim;x(,. th're we ex~q)ha,size (,l~a,(, it (l()es yM(l tl~e AI:1(71) law (9.5.5) fi)r (,Ira (',()~l)l(;x 1)eax~ l)ara,~(:t(:r Q(z). N()t(:, i~ fa('(,, (,Ira(, (2 vn'it,(:s i~ [,(~,l'IIIS ()f (,l~(: E(,~t,ri(:s a.s
•
z,,. + iv/,l(,t E
(9.5.23)
or, c(tlfivah;lltly
c2  L,(z,,.  i v / , l , ; t
(,9.5.24)
()f collrs(;, v/(let E  89 ill (,lie (:a,s(; ()f (,ll(; filll(la,lllClltal (la,llssia,ll 1)cam we are (tealing with, howew;r l(;tl,ing v/(let E lm(lcfinite i~ (,h(,~ a.I)()w; exl)ressions confer thein a certain gelmrality, that will revea,1 usefill f(w filtllre sllggestions. Inferring f()rm (9.5.22) the r(4ati(m ()f the m a t r i x (',h'mcxlts Eq~ and E ~ between the en(tpla,nes, i.e., Y]q~(Zo)

ACkoEqq(Zi) + (AD + BC)Eq,~(zi) + ~. BDE,r 2 C 2 k)Eqq(Zi ) + 2koCDEq,~(zi ) + D~E,~,,~(zi),
and noting from (9.5.24) t h a t 2/,:Eqr Q, E~ = Q +
2ikv/det E E~ = Q
k2Eqq Q*'
E~
, QQ ,
(9.5.26)
after some algebra we recover the A B C D transfer law (9.5.5) for the complex beam parameter under propagation from the plane at zi to the plane at Zo being connected by the ray m a t r i x M  (A ~). \v /
\
1D FirstOrderOptical Systems: 9.5.4
Moments of the
WignerDistribution Function
499
Gaussian beams: the phaseplane ellipse of second moments
Sections of the Wigner distribution function (9.5.20) at planes parallel to the (q, ~) plane individualize elliptical contours defined by the equation (b) luT
[E(z)]  1 u 
c.
(9.5.27)
[E(z)]  1 u 
1,
(9.5.28)
In particular, the ellipse (c = 2) 1u T
occurs at, an height which is 0.135 of the maximum height of the 3 9 plot of the Wigner distribution function. Of course, any one of tile ellipses (9.5.27) can be taken as a phaseplane representative region of the corresponding Wigner distribution function (and Ga,ussian beam) and hence used to inspect the modifica,tion suffered by the beam under propagation through the systen under investigation. Mantaining the correspondence with the analysis in previous chapters, we specifically consider the phaseplane ellipse identified by (9.5.28), which explicitly writes }~.#~t~(z)q2


2Y]q~(Z)qh; Jr Y]qq(Z)#~2  1.
(9.5.29)
For a, descriptiom of the geometrical properties of (9.5.29), the reader is diretted to w 9.6.4, where the topic is illustrated in relation with the general case 1 of an a,rbitra,ry nonGaussian Wigner distribution flmction (v/det E r ~).
9.6
Propagation laws for the moments of the Wigner distribution function in firstorder optical systems
Here we reformulate tile propagation laws for tile secon(] nnoinents of the Wigner distribution function in firstorder optical systems just described in connection with Gaussian beams, within the general context pertaining to arbitrary non Gaussian optical signals [4.1, 4.2, 4.4]. Firstly we consider the firstorder moments ~, g. In conformity to the vector notation u  (q,ec)T, we may concisely write (9.4.44) and (9.4.50) into one form U(z)  N
u W ( u ; z)du,
(9.6.1)
where, of course, f f ..du = f f ..dqd~. As we know, the propagation of the signal through an ABCD system implies a transformation of the firstmoment vector (9.6.1), explicitly expressed by
U(Zo)


[l JJ
U}/~2(U;
zo)du


if JJ
u~2(MXu;zi)
du.
b N a t u r a l l y the conic g1 u T[r~(z)]i u is recognized as an ellipse since det ~ > 0.
(9.6.2)
Linear Ray and Wave Optics in Phase Space
500
C h a n g i n g the variables from u to v  M  l u , h)r which d u due to the u n i m o d u l a r i t y of M , we end up w i t h
~(Zo) M f /" vl/V(v; zi)dv
det M d v  d v
(9.6.3)
M~(zi).
1,!
As ext)e(:t(',(t, the, first()r(ter m()me, nts ()f the, W i g n e r (tistrit)llti()n fim(:ti()n evolve j u s t as the ra,yva,ria,t)les ()f t)a,raxial ray()t)ti(:s, in a,(:(:()r(l evi(lently with tim reslllt, s (~stal)lish(~(t in ~ 5.2 regar(lillg l,lle l)r()l)a,ga,ti()ll law ()f l,h('~ m e a n va,hms, ~ ml(t ~ ()ftll(; l)()sil, i()ll ml(t fl'e(lll(;ll('y ()l)(~I'a,l,()rs. In ()r(ler t()(:()n(:isely write tim se('()ll(t()r(ler lll()lllents, We f()rlll tlm Kr()lm('ker I)r()(llu:t ()f tlle Wigll(:rI)lmle ve('t()r u witll l,ll(~ trmlsl)()s(~ ve('t()r u T, tluls ()t)taining tim 2 x 2 syIiulmtri(: llmtrix(") qh ,~.2
9
(9.(i.4)
A('(:()r(lil@y, t)y (9.6.1) l,ll(, l)r()(lll('t ii C4 ii T l)r()(llu'('s l,tm lilatrix ,,
.
'l'lle (l(~i)eli(l(~1i(:(' ()ll tile l)r()l)a,gati()ll wu'ia,1)h~ z is ill~l)li(~(l il~ (9.6.4) all(t (9.6.5). Tlm wu'im~('(~ ~m.trix E , i~l,r()(l~l(:(~(l i~ ((.).5.13) i~ ('.()~m(:l,i()~ witl~ Gm~ssia,n })ea,~s, is tl~er(~fi)re se(u~ a.s a, l)arti(:~la,r (:ase ()f l,l~(', g(~era,1 (l(~fi~il, i()~ E(z) 1.!
TyI)i('ally it is a s s ~ ( ; ( t tt~at ~  (), tlu~s si~al)lifying tl~e w()rki~g f()r~mlae. h~ or(ter t() i~fer tt~e trm~sfi,r law f()r 1,1~(, varia.~(:(' l~mtrix E ( z ) i~ firstorder ()t)ti(:aJ systen~s, we a,t)t)ly tlm t)rot)agati()~ law for tlm W i g n e r (tistrit)ution function, c h a n g e tim integrati()n varia.t)h~s m~(l tt~ex~ ext)loit tim, sy~q)le(:ti(:ity of t h e rayma,trix M in t h e s a m e way as ()n (leriving (9.6.3). We fi~m.lly fin(t
~(Zo)
 / / U 9

M
(~ UT]&(U; zo)du  / / u j/.
~ uT]A;(Mlu;
zi)du (9.6.7)
I,!
v  v T W ( v ; z i ) d v M T,
~The Kroneckcr (or direct) product of an n x m matrix E and an n' x rn' matrix F produces tile nn' x ram' matrix E  F by tile rule [8] el: F
999 el,,, F '~
. . . . .
E
e,,i F
.~i
) ~
e ...... F
Evidently, the Kronecker product is not commutative. It satisfies, among the others, the properties" (a) E  (E174 E174 (b) E   E  1 7 4 (c) (E  F ) ( H  G)  E H  FG, provided that the ordinary products E H and F G may be formed; (d) ( E Q F ) T  E T  (f) ( E Q F )  1  E 1  1, i f E and F are nonsingular.
1D FirstOrder Optical Systems: Moments of the Wigner Distribution Function
501
which, after identifying the double integral as E ( z i ) , yields E(Zo)  M E ( z i ) M T,
(9.6.8)
reproducing, as expected, relation (9.5.22). Relation (9.6.8) represents a congruence transformation (a). It implies however a sitnilarity transformation for the matrix E ( z ) J , for which, in fact, we firstly obtain E(Zo)J  M E ( z i ) M T J , (9.6.9) and then, being M T J 
J M 1, ]E(Zo)J  M ] E ( z i ) J M 1
(9.6.10)
Similarity transformations have the notable property of preserving eigenvalues and traces. Therefore, Eq. (9.6.10) may provide a useful supply of invariants. The interested reader may consult the related literature [4.2, 4.4, 4.7, 4.5]; here, we only detail the basic invariant yielded by the determinat of E. It is, in fact, evident that, due to the unimodularity of the optical matrix M, det E(Zo)  det E(zi),
(9.6.11)
as in tile prevously described case of Gaussian Wigner distribution functions. 9.6.1
The optical uncertainty
relation. H
The explicit form of (9.6.11) gives EqqEnn  E2q~  const.
(9.6.12)
In particular, choosing a transfer matrix M such that E comes to be diagonal: Eq~ = 0, we have d e t E Y]qq}]t~n, (9.6.13) which according to the width principle yields 1 d e t E _ > ~.
(9.6
14)
Since, whichever be the optical matrix M, the variance matrix resulting from (9.6.8) and obeyng (9.6.14) corresponds to a physical signal, i.e., the signal transformed by some other signal by propagation through the system represented by M , relation (9.6.14) should hold for all variance matrices. This d Given a matrix A and a matrix B, the product BAB T realizes the congruence transformation of the matrix A by B, whilst, if B is nonsingular, the product BAB 1 realizes the similarity transformation of the matrix A by B.
Linear Ray and Wave Optics in Phase Space
502
establishes a b o u n d for the invariai~t value in (9.6.12), which then can be written as 1 2 1 E q q E ~ _> ~(1 + 4Eq~) > '4 (9.6.15) It ret)rcsents the con~t)h'te form of the ot)tical ~ln(:(;rl,a,ild,y relal, i()i~, il~volving a t)re(:isc signa.l(h;t)cn(h;nt t(;n~ (relate(t to tim mixe(1 ~(nn(;~d, Eq~,) t)csi(h;s the 1 signa,li~(let)e~(le~t valll(; 4" 9.6.2
l~vpagation la,~s for the seco'ndorde.'r mommy'his
Unfr tt~(', trm~sfi.'r law (9.6.8) fi)r 1,1~('~w~,ria,~(:(; ~ a t r i x E ( z ) , (n~(: gets the exI)li(:il, i~l)~l,()~l,l)~t, r(;la.ti()l~S fin' 1,1~(;se('()l~r ~()l~(;~d,s, l,ha.l, (:a,n as well l)e a,rra.l~ge(1 i~d,() a ~m.l,rix st:l~(;l~e. I~ fa,(:l,, il, is sl, ra.igl~l,fi)rwa,r(l 1,~)wI'il, e
Eq,,.
(z,,)
Es:Sl
k,,A(7
AD + BC
/,:2(.,2
2/% (71)
HI) 1) 2
Eq,~
(zi).
We will st)c(:iti(:a.lly exm~i~w, tl~e eflb,('l, r s{n~m r162 tra.~sfr l~l(n~r la,l,~;r i~ r~:lal,ir witl~ 1,1~r lflm,s~;l)lm~e s~'.('~r162162 9.6.3
(9.6.16)
i{)~s ()~ tl~e ~:llil)Sr
Positive definiteness of E
Firstly, let 1is ~l~t,~' tlm,t r~:lati~m (9.6.16) ('ml }~e lls~'~t t~ l)r~v~; tlla.t l,lle va,ria,n(:e m a t r i x E is t)~sitiw; ~h;tiIdt~:. Tim tirst r~w, fin" izlstml(:e, writes
2
+ v.
+
1 /3 2

> o.
(,9.6.17)
Evidently ttlq U
v=M
lu
,
Mv ~ vTM T ~ Mv.
(9.7.4)
I{e,t)(',a,te,(t a.t)l)li(:ati()lls ()f the (lire(:t I)ro(lu(:t t)r()t)erty (see f()()tll()te, ()Il t). 500)
(E r F ) ( H r G)  E H r FG,
(9.7.5)
M v ~ v T M T ~ M v  ( M  M ) v ~ v n  v M T.
(9.7.6)
leads to the nice form
It is therefl)re evi(tc, llt tha,t the prot)a,ga, tion la,w for E 3 writes as E3(Zo )  ( M @ M ) E a ( z i ) M
T.
(9.7.7)
Tile t)rocedure can be iterated, thus yielding for the n  t h order m o i n e n t m a t r i x E~ the average of the nvector direct p r o d u c t
E. (u
T174 n  t l rues
T
(97.8)
1D FirstOrder Optical Systems: Moments of the Wigner Distribution Function
513
with u and u T being alternatingly involved into the sequence of products. It is straightforwardly, although a little lengthy, using the properties of the Kronecker product, to verify that the matrix E n propagates according to En(Zo ) (M174
(9.7.9)
Here, [~s yields tile closest integer to ~, and (M
=M  M 174
M
(9.7.10)
htimes
denotes the hth direct product of the ray matrix M. Needless to say, with n  2 we regain the propagation law (9.6.8) for the variance matrix E, which indeed within the formalism of this section could be identified as E  E 2. Let us separately consider the cases of n even and odd. For n even" n  2j, relation (9.7.9) yields the symmetric equation
E2j(Zo) (M@j)E2j(zi)(MT@j).
(9.7.11)
Note that E2j is a 2j • 2 j matrix. It (:an also bc seen as arising from ~2j (]E2
(9.7.12)
Multiplying both sides by the 2J x 2J matrix (J using the relation M n = J M  1 J 1 and repeatedly applying (9.7.5), we obtain on the righthand side
[(M~2(zi)JMlj1)@J](J@J)  [ ( M E 2 ( z i ) J M  1 j  1 j ) @ J ] [(ME~(zi)JM1)  (M174174
(9.7.13)
Finally, on due account of the further property of the direct product
(E @ F) 1  E 1  F 1,
(9.7.14)
( M  I @ j )  (M@j) 1,
(9.7.15)
by which,
the propagation law (9.7.11) is seen to be equivalent to the similarity transformation
E~j(Zo)(J
 (M174174
~,
(9.7.16)
of the 2J x 2J symmetric matrix E2j (zi)(J by the 2J x 2J matrix ( M  ). A similar result, even though in a less direct form, can be worked out for the case of n odd. Indeed, with n  2j + 1 in (9.7.9) we obtain
~~2j+l( Z O )

for the 2 j+l • 2j matrix E~j+ 1.
(M@jt1)Y]2j+I(Zi)(gT @j),
(9.7.17)
Linear Ray and Wave Optics in Phase Space
514
Multiply b o t h sides by ( J 
~]~2j+~(Zo)(J(/~)j)

a,nd r e p e a t e d l y use (9.7.5), to fina,lly write
(M@j+I)~2j+I ( z i ) ( J @ j ) ( M  l @ j ) ,
(9.7.18)
t)()th si(tes mea,ning a 2 j+l • 2 j matrix. C()rrest)on(tingly, taking the tra,nspose of (9.7.17) a,n(t multit)lyi**g t)y ( J  we hav(;
Er.,j,l (z,,) (Jc4j+ l )  (Mc~j)E,T +, (zi)(Jc~j+,)(M 1 c~j+l ),
(9.7.19)
t)()tl, si(l(;s i**q)lyi**g ()f (:(n,rs(; a, 2 j x 2j+ l**m,trix. Finally, 1,1,(; l)r()(l~u:l, ()f (9.7.19) l,i***(;s (9.7.18) r(;s~,lts i**l,()
NT2 j
! 1
(z,,)(Jc4j+ l)N
2j { 1
(z,,)(Jc4j)
= (Mc+j)[IE~,, (zi)(J~j+l)~.~j+, (zi)(a(/oj)l(McAj) 1 ,
(9.7.20)
wl,i('l, (:a.,~ I)(: r(:('()g,~iz(:(l as tl,(; si**filarity tra.**sf()nlm.ti(),~ ()f ttm 2 j+l • 2 j+l **m.trix ]E 2.i1 q 1 (z,,)(Jc4j+~)lE.~j  + 1 (z,,)(Jc4j) l)y t,l,(; (tire('t l)r()(l~u:t l,m.trix (Me+j) As (;arli('.r n()l,e(l, si**filarity tra,**sf()r,lm,l,i()**s are very ('()nV(:ld(;nt 1,() w()rk wil,l,, si**(:(', l,l,('.y l)r('.s('rv(: 1)()I,1~ (:ig(:~wal~u:s m~(l tra,('(:s. 'l'l,(:r(;f()r(;, 1,1,(, i(l(;**tifi(:a.ti()n ()f si~ila,rity trm~sfi)n~m,ti()l~s ill r(;lati()~ witl~ s(;(:(n~(l()r(l(;r as well as l~igl~(;r()r(h'.r ~()~(:~l,s ()f ligl~l, (lisl,rit)l~t,i()~s t)r()vi(l(; m~ (;tli'.(:tiv('~ t()()l t() in(tivi(t~la,liz(; i~wa.rim~t (l~mlity l)a.ra,~(:t('rs i~ t(;r~ ()f wl~i(:l~ ()t)ti(:al sig~a,ls (:a,~ s~fita,l)ly t)(; (:lmra.('t(;riz(;(l a.l~(l a.(:(:()r(ti~gly ()l)ti(:al sysl,(;~s ('.tfi;('tiv('ly (l(;sign(;(t.
9.8
Summary
After illtr()(llu'illg t,ll(' l)asi(: II()ti()llS ()f lll()lll(;Ilts ()f 1D ml(1 2D flnl(:ti()ns, w(; have fo(:lls(;(t (nlr a,l,teIlti()ll ()ll the first a n(t s('r lll()21u'nts of the Wigner (tistrit)llti(nl fllll(:ti()21, fi)r wlfi(:tl a variety ()f siNlitir relati(ms have been esta,blishe(1 ill gmlera,1 aald then specia, lize(t to tlu; case of tim, Wigner distribution fllncti()Ii ()f a Gm~ssian signal, which is il~(tee(t 2D Gaussian. 2D Gaussia,n (tistributio~ flumtion have relevance a,s well within the context of pa, rtial coherence tt~eory. In particular, a, suita, bh; m a t r i x scheme has )een envisaged ena,bling a compact a,n(t elegant f()rmula,tion of the t)r()paga,tioI, laws for the secondorder m o m e n t s ~,n(ter t)rot)agati()n ()f the signal t h r o u g h A B C D ()t)tical systeins. A similarity tra,nsformati()n ha,s been individualize(t, whi(:h involves the secondm o m e n t ma,trix a,nd the ray transfer ma,trix of geometri(:a,1 optics. From it a va,riety ()f interesting a.n(t 1,s(;fl,1 relations can t)e (t(',(t~,('(;(t. Of p a r t i c u l a r releva,n(:e is the inva,riance of the determina,nt of the variance matrix, interpretable as the area of the s e c o n d  m o m e n t phaseplane ellipse. This has suggested the identification of suitable p a r a m e t e r s for a meaningful definition (and operative p r o c e d u r e for the measure) of the " b e a m quality", as well as possible generalizations to partially coherent beams.
1D FirstOrder Optical Systems: Moments of the Wigner Distribution Function
515
Problems
1. (a) Prove the general relation rn,.~  27ri ~+s 0 ~+ "~A(q',~) 9
0~0~
~
q, v ; = 0
'
holding between the central moments of the Wigner distribution function and the derivatives of the ambiguity function at the origin. It is implied above t h a t the ambiguity plane origin has been translated at the mean values point of the Wigner distribution function. (b) Prove t h a t the ambiguity function can be constructed by knowledge of the Wigner distribution function moments as i lm, A(~, ~)  ~ ~ m ....., q ~ , t,rn=0
thus also implying the reconstruction of the Wigner distribution function by knowledge (possibly gained by experimental observations) of its moments. n 2. Verify that, the application of the energy operator g to a signal of the form (9.4.5), under the assumption t h a t b o t h the a m p l i t u d e f(q) and the local frequency t~1 (q)  d_~ dq do not vary too rapidly, results into ~f~l, i.e. the product of the squared amplitude and the squared (local) frequency of the signal. 3. Prove t h a t for the Gaussian b e a m (9.5.1), the local uncertainty product, defined in (9.4.91), takcs the explicit zdepending expression 2 ZR
q~(~)~
(q) 
4 ( z~ + z ~ )
C o m m e n t oil it in the light of the considerations presented in w (9.4.4). 4. Prove that in order that, under propagation through an A B C D system, the secondorder moments of the light distribution do not change with respect to the respective input values, it must be
B ~qq ( Zi ) 
]%
~
E~,,~z, )
Eq,~(z,)sin/~'
sin/3'
e sin/~
AD ~
'
where cos/3 = A+D It is implied, of course, that A + D < 2 (stability condition) 2 " 5. Consider a light b e a m propagating t h r o u g h a quadratic index mediuln (see w 3.9). (a) Prove that the b e a m radius, defined by (9.6.32) evolves according to the Pinney equation d2w
dz 2
16e 2 n2 _2~2,.,3 + w  0. t ~0 r~o vv
(Hint: use Eq. (9.6.8) for 2E and Eq. (1.7.18) for M to deduce the paraxial equations of motion for the secondorder moments in a general form. Specialize the result to the case of propagation in a quadratic index medium.) (b) In particular, verify t h a t with n 2 = 0 (free propagation) an evolution law for w(z) is obtainable in the similar form as for Gaussian beams, i.e. Z2
w2(z) Wo2(1 + 22)' ZR
demanding for Z R 
in accord with relation (9.6.34).
ZR M2
'
516
Linear Ray and Wave Optics in Phase 5pace
References [1] Only a h~w titles on the nlonlent m c t hods in relation wittl inmge characterization, evahmti()II an(1 nlallitnllal, i()II, (:OmpI'isillg tlm t)ioneeriIlg t)at)er 1)y tlll and the subsequent investigatioils by Teaguc, are listed here. The. reader m a y find an cxhallstive t)resentati(m in the text,1)t)t)k l)y Mukml(lan an(l liamakrishlmn. [1.1] M.K. Hll, "Vislml pattenl recogidtioll W inolneilt iilvarimlts", IIi E 'IYmls. I~ffo. Theory, IT8, 179187 (1962). [1.2] M.R. Teague, "Optical ('al('lllatioll of irrar InoInmlts", AppI. OI)t. 19, 13531356
(l:)s0). [1.3] M.IL 'IL'aglm, "linage mlalysis via tim gmleral tlmory of IllOIlmllts", .1. Opt. Soc. Am. 70, 92()93() (198()). [1.4] l{. rVl,lk,nl{lml ml{l K.I{. llmlmkrislllmll, Moment ],;wn,ctions in Image Analy.~is: Theory and Appli~ntions (W()rl(l So., Siilgap()rc, 1.().()8). [2] Tlu: l)r(:s(;~m~ti()~ i~ w 9.2 is ])(,a~ltif~lly (:()vet(;(1 i~ tim (l~l()t(,(I l)ag(;s ()f [2.1]. S()lll(; r(;l)r('sc~m~tiv(', titles ('()n(:cr~m(l witl~ tim ttm()ry ()f g('~mral slalisti('s ar(; s~gg(;st, e(t.
[2.1] I,. Mm~(lel m~(l E. Wolf, Optical (7ohcrcncc
and Quantum Optics ((:mn)ri(lge
UIliv.,
N(',w Y()rk, l .().()5), oh. l, pp. 140.
[2.2] II. (;rm~6r, Matlwmatical Mcthod,s ~g Statistics (1)ri~a('(;l,()~ I,J~fiv. l)rcss, l)ri~meto~,
1:)4(~). [2.3] M. Kc~(tall ai~(l A. St~mrt, Tk,c Advancc, d Thco'rv of Stati.~tics (Griffins, l~()~(l()~, 1958). [2.4] B.IL Martin, Stati.~t,ic.~ for l)hy.sicist.s (Aca(lc~nic Press, I~()~(l()~, l.()Tl). [3] (~()Im(;I)tS m~(t (lefi~iti()~s (:()~(:(;r~(;(l witt~ tl~e i~()it~(,,~l,s (l()('al m~(l gl()t)al)()f a ligl~(, (lisl, ril)uti()~ ti~ul (,lmir (:()~(,(;rl)ar(, il~ tim (',()ll(,Cxl, ()f sig~al th(;()ry withil~ the t)cr(i~mld, tiI~Cfl'(~,(l~U;~(:y (l()~mi~. S()~(; I)('~rtil~(;nt (i(h;s arc sugg(:s(e(l I)clow. In t)ar(i(:ular wc signaliz(; th(; l)ar( I ()f (h(; thr(;(;t)ar( Imt)(;r l)y ('laascn a~(1 3l(;(:klcnl)riiukcr [3.1], wlmr(; tim ~()~(;~(,s of (,t~c Wigner (lisl, ril)~ti()~ f~('ti()~ m~(1 t)crtin(',xd, in(;(tualities at(; (:()l~si(l(',re(1. W(;llk~()wn tcxt)()()ks (:()~m(;i'~m(l witl~ l,l~(', signal th(;ory and the r(;levaI~t tin~(;fre(tucn(:y rct)r(;seI~tati()~ issue(; at(; als() listed. [3.1] T.C.A.M. Claasci~ ai~(t W.F.G. Mccklenbr/4uker, "The Wigner (listribution a tool for timefrequcncy signal m~alysis. I: continuoustiIne sig~mls", Philit)s ,]. Res. 35, 217250 (1980). [3.2] S.C. Pohlig, "Signal duration and the Fourier transforin", Proc. IEEE 68, 629 (1980). [3.3] L. Cohen, "InstaI~taImO~m mean qum~titics in tiinefreq~mncy analysis", Proc. IEEE ICASSP88, 21882191 (1988). [3.4] L. Cohen and C. Lee, "InstaI~taimous frequency, its standard deviation and multicomponent signals", in Advanced Algorithms and Architectures for Signal Processing. III SPIE Vol. 975, 191208 (1988). [3.6] P.,]. Loughlin and B. Tacer, "Instantaneous frequency and the conditional mean frequeimy of a signal", Signal Process. 60, 153162 (1997).
1D FirstOrder Optical Systems: Moments of the Wigner Distribution Function
517
[3.7] B. Boashash, "Estimating and interpreting the instantaneous frequency of a signal. I: Pundamentals", Proc. IEEE 80, 520538 (1992); "II: Algorithms and applications", ibid. 540568. [3.8] L.E. Franks, Signal Theory (Prentice Hall, London, 1971). [3.9] A. Papoulis, Signal Analysis (McGraw Hill, New York, 1977). [3.10] L. Cohen, TimeFrequency Analysis (PrenticeHall, Englewood Cliffs, 1995). [4] The papers by Bastiaans [4.1, 4.2], concerned with the definition of the secondorder moments of the Wigner distribution function, relative propagation laws and related invariants, have been basic to subsequent investigations and generalizations [4.4, 4.5], also in relation with partia,lly coherent beams [4.3]. Tile paper by Sheppard and Larkin [4.6] presents a variety of interesting relations involving the secondmoments, some of which reproduced in w 9.4.3. The fascinating review by Simon and Mukunda [4.8] elucidates the relation between the ray, wave and Wigner optics formalisms, in relation as well to the partial coherence theory. The link of the secondmoments of the Wigner distribution function to the second derivatives of the ambiguity function has been evidenced by Papoulis [4.9]. [4.1] M.J. Bastiaans, "Propagation laws for the secondorder moments of the Wigner distribution fimction in firstorder optical systems", Optik 82, 173181 (1989). [4.2] M.J. Bastiaans, "Secondorder moments of the Wigner distribution function in firstorder optical systems", Optik 88, 163168 (1991). [4.3] M.J. Bastiaans, "ABCD law for partially coherent Gaussian light, propagating through firstorder optical systems", Opt. Quantum Electron. 24, $1011$1019 (1992). [4.4] D. Onciul, "Invariance properties of general astigmatic beams through firstorder optical systems", J. Opt. Soc. Am. A 10, 295299 (1993). [4.5] D. Dragoman, "Higherorder moments of the Wigner distribution fimction in firstorder optical systems", J. Opt. Soc. Am. A 11, 26432646 (1994). [4.6] C.J.R. Sheppard and K. G. Larkin, "Focal shift, optical transfer function, and phase space representation", J. Opt. Soc. Am. A 17, 772779 (2000). [4.7] V.V. Dodonov and O.V. Man'ko, "Universal invariants of quantummechanical and optical systems", J. Opt. Soc. Am. A 17, 24032410 (2000). [4.8] R. Simon and N. Mukunda, "Optical phase space, Wigner representation, and invariant quality parameters", J. Opt. Soc. Am. A 17, 24402463 (2000). [4.9] A. Papoulis, "Ambiguity flmction in Fourier optics", J. Opt. Soc. Am. 64, 779788, (1974). [5] Tile non linear energy operator g was originally developed by Teager and then systematically used by Kaiser. In the context of signal analysis, it has revealed an efficient tool for extracting amplitude and frequency modulation information in signals of the form s(t) = a(t)cos(r As representative papers, we suggest [5.1] H.M. Teager, "Some observations on oral air flow during phonation", IEEE Trans. Acoust., Speech, Signal Process., ASSP28,599601 (1980). [5.2] H.M. Teager and S.M. Teager, "Evidence for nonlinear sound production mechanisms in the vocal tract", in Speech Production and Speech Modelling, J. Hardcastle and A. Marchal (eds.) (Kluwer, Boston, 1990), pp. 241261. [5.3] J.F. Kaiser, "Some useful properties of Teager's energy operator", IEEE ICASSP93, Vol. 3, 149152 (1993).
Linear Ray and Wave Optics in Phase Space
518
[5.4] P. Maragos, J.F. Kaiser and T. F. Quatieri, "On amplitude and frequency demodulation using energy operators", IEEE Trails. Signal Process. 41, 15321550 (1993). [6] T h e richness of inequalities (:on(:erne, d with the nlonmnts of the W i g n e r distributi(nl flUl(',ti(nl is well c(nlve, ye(t )y (I(; Bi'lfi.j~l's t)apers [6.1, 6.2], the se('()n(1 offering one ()f tim first, malh(;mat, i(:al f()rmalizalion ()f the W i g n e r ret)resentation. T h e t)res(;Idati()ll ixl !i 9.,1.,1 is illst)ire(t ])y I.lm l)aI)er ])y Ix)llglllill a Il(l (',()lmll [6.3]. [6.1] N.G. (le Brlfi.jli, "UI(:erl,~dIlty t)rilwil)le ill Follrier mmlysis", ill ['ncq'ualilic,s, O. Shisha (e(l.) (A(:~(h;lIfi(" l)ress, New York, 1.()(i7), pp. 5771. [6.2] N.G. (te Bruijll, "A tlmory of gelmralize(l flulcti(nlS, wittl applicatiolls to Wigner (listribld, ion an(l Weyl c()rresl)ollth;llCe," Niellw Arclfief vt)or Wisklultle 21(3), 2()52s0 (1:}73). [6.3] I)..I. I~ollgllill ml, l I~. (;()lleIl, "Tim Iillcer,a.illty prillciple: glol)al, local, or bottf, z'', IEEE 'IYmls. SigI~al lh'(wess. 52, 12181227 (2()()4).
[7] A r('a(li~g list (,~ (;a~ssia~ I)(:a~s is i~ [131 ()f (:1~. 2. Iler,, we lis~ a f('w t i t l e s reh'va~! t ( ) t i m I)rese~tati()~ i~ !i!i 9.6.69.6.7. a~(l 1~)(l,'ti~ili, n~s. al)l)licali(n~s a~r ~ e a s ~ r e ~ m ~ t s ()f relew~('e t,) tim i~(livi(l~mlizati()~ (~f s~ital)le i)ara~mters f()r laser 1)ea~s/(levi(:(~s (:lmrat)racket preservation property, 72 pro(hlct I)reservatioll property, 7() qlm(h'atic l)olyIloznials, gelmrat('.(l I)y, 26 l~illest)rea(1 flilwti()ll, 183 l,i(nlville tlmoreln, 67 Liollville's eqlmti(nl, 62, 65, 67, 455 M
M 2 factor, 507 MagIlilic'at,i()li, 125 ray ('()(n'(litmt(', 46 ray lIl()lIleltttull, 46 Meri(li(nml l)lmm (or zneri(liml), 35 Mometlts, 463 1D fllIl('t,i()II, 466 2D fllIlcti(nl, 471 2D filxmtioIl, global, 472 2D fimcti(nl, local, 471 Moment~Hn operator coor(linate rel)reseIltati(nl , 187 eigenfllxlctiolis of, 1.()6 lnomentllln representation, 214 Moyal fi)rmllla, 2.(}7 Mlltlml illtensity finlct, ioIl, 372 ambigllity fimction, and, 374 Wigner distriblltion flnlction, and, 374 N
Nodal planes, 50 O Optical computing Fourier transform, 380 Fresnel transform, 380 Wigner transform, 384399 Optical SchrSdinger equation, 189 Optical transfer flmction, 205 Orthogonal optical systems, 32, 33
Subject Index P
Paraxial approximation Helmholtz equation, in the, 171 Path length geometrical, 4 optical, 4 Phase space beam analyzer, 509510 ellipse of 2ridorder moment, 503 geometrical optical, 2, 7, 99 mechanical, 2 ray density distribution, 63, 78 trajectory, 62 volume, 62, 65, 69, 504 wave optical, 8, 176, 211 Phase space picture fractional Fourier transformation, 136 free propagation, 99 refraction, 101 scale transformation (positive), 102 Phase space rotations, 136 generating matrices of, 150 Phase space squeezes, 147 generating matrices of, 150 Pointspread function, 180 Poisson brackets, 1518 Position operator coordinate representation, 186 eigenflmctions of, 195 frequency representation, 214 moinentum representation, 214 Wigner representation, 452 Principal plane primary, 48 secondary, 48
Q Quadratic gradedindex medium, 43, 158 defocusing, raymatrix for, 161 focusing, ray matrix for, 159 Fourier transformer, and, 160 fractional Fourier transformer, and, 160 Quadrupole lens, 508 R
Radiance (or, brightness), 80 definition, of, 81 phase space function, as a, 83 phasespace ray density, and, 84 Radiant intensity, 85
525
Radon transform, 443, 447 projectionslice theorem, 445 RadonWigner transform, 443 fractional Fourier transform, and, 443 Ray (light), 1, 175 axial, 41 coordinates, 9 equation, 6 field, 41 meridional, 35 momenta, 6, 9 paraxial, 30 paraxial, coordinates of, 36 skew, 35 Rayspread flmction, 431 Raytransfer matrix, 40, 60, 179 singleexponential form, of the, 42, 179 Raytransfer operator, 22,177 singleexponential form, of the, 22 Refracting power, 44, 96 S Shift(or, space)invariant systems, 181,204 Simple astigmatic systems, 33, 61 Skewness, 35 Slidingwindow Fourier transform, 343 ambiguity function, and, 352 inversion formula, 344 Wigner distribution flmction, and, 352 Spacebandwidth product, 210 Spacefrequency distribution flmctions, 354 correlative classes of, 361 energetic classes of, 361 inversion formula, 357 kernels, of the, 356 marginals, 356 Spatial harmonic, 196 Spectrogram, 344 crossterms, 350 smoothed Wigner distribution function, as, 353 Spreadless systems, 205 Squeezer, ray matrix, 145 Stigmatic systems, 34, 61 Symplectic condition, 12 transformation, 11 unit matrix, 11 Symplectic matrices canonical representation, 154
526
Linear Ray and Wave Optics in Phase Space group of real, 12 LSF '~ synthesis, 152 LST synthesis, 132 LTL factorization, 121 non,canonical representations, 154 optical ~natrices, as, 130 TLT factorization, 121
T 'l'(q(,s('()pi(' systemics, sy~ti~esis ()f, 122 Tlai(:k le~s, 114 'l'l~i~ le~s, 114 (:()~vei'gi~g, ()r p()sitive, 115 (livergi~g, ()r ~egative, 115 i~t)~tol~t,l)~d, relati()~, ~(l()~mi~, 232 i~I)~t,()~lt,I)~ll. relati()~, q(h)~nai~, 231 liI~e spr(:a(l fluwt.io~, 231 ray i~mtrix, 115 md)gr()~q) of ~mtri('es fi)r, l l[i wave sprea(l fiu~(:ti()~, 232 ~lbI~ograplw (ol)tical), 4,1.1 U Uiwertai~ty relations, 2()7, 4.()(), 5()1 gh)l)al, 490 lo('al, 4.90 V Varim~ce matrix, 5()() Gm~ssiaI~ sig~ml, 496 transfer laws, 50()
W Wave trm~sfer operator, 222 unitary represex~tatiox~ of lixmar cax~o~ical transformatioim, as, 261265 WeiNorman (tecompositio~ metaplectic operators, 240 symplectic Inatrices, 132 Wcyl transform, 420 Window function, 343 Wigner distribution flmction, 271 chirp, 314 315 combsignal, 312 condition for uifiqueness, 411 convolutio~ property, 300 cosinesigIml, 308 crossterms, 286 cubicphase signal, 317 cylin(lrical wave, 281 :ifimctioi~, 280
finite support property, 284 l"'or(ler global InoIneIlts, 483484 l"tor(ter local moment, h;wtrial)le, 477 l'~tor(ler local lIlomellt, qwtriable, 480 Follrier trallsfi)rnl, all(l, 43.() fractioiml Follrier trmmfi)nn, ml(l, 441 (lallssiml siglml, 321:122 (;mmsiml SlIl()ot.llillg, 2.94 l l(;riifitec()sll(Imlssiall siglml, 32!) lt(,rlifit(,(lallssiml siglml, :127 lligller()r(l(;r lll()llleIlts, 512514 illversioll forlmd~Ls, 2.()() llmrgilml (list,ril)llti()llS, 28!) IIudtil)licatioll l)rop(wty, 301 ll()ll lillearity, 284 ll()lll)()sitiv(:lmSS, 2.92, 417 ()l)ti('s, il~, 277 l)artially ('()lmr(;~t sig~mls, 2.91 r ~ne('ha~fi('s, i~, 275 r('alit.y l)r()l)(,rty , 283 r(:('tm~gh,sig~al, 305 r('l)resentability, 3()2,332 s('ali~g l)r()perty, 2!)!) 2"'L~)r(ler h)('al ~()~e~t, hw~rial)le, 479 2"d()r(h'r l()cal ~o~nei~l., qw~riable, 482 selfter~ns, 286 sii~(:sig~al, 3()7 si~u~soi(lal plume sig~ml, 320 spatial harmo~fic, 280 s~nn of two (~mmsim~ sig~mls, 324 s~un ()f two i~nl)~lses, :11() symmetry property, 283 trm~sfer laws, ABCD syste~, 433 transfer laws, free ~ne(ti~m~, 425 transfer laws, positive ~nagnifier, 429 trm~sfer laws, tl~ix~ lex~s, 427 transport eq~mtionr, 454 Talbot effect, and, 436 Wigner phase plane, 304 Wig~mr processor Gaussiax~ s~noothed WigImr (listrib~tion fiH~ctio~, 398 1D real signals, 384 1D coinplex sigImls, 394