Linear Ray and Wave Optics in Phase Space: Bridging Ray and Wave Optics via the Wigner Phase-Space Picture

541 Pages·2006·26.36 MB·English

by  Amalia Torre

Linear Ray and Wave Optics
in Phase Space
Bridging Ray and Wave Optics via the Wigner Phase-Space Picture
For the cover design: (cid:14)9 Ren6 Magritte, La corde sensible, BY SIAE 2005
Linear Ray and Wave Optics
in Phase Space
Bridging Ray and Wave Optics via the Wigner Phase-Space Picture
Amalia Torre
ENEA-UTS Tecnologie Fisiche A vanzate
Frasca ti (Rome), Italy
G
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To my mother
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.
This Page Intentionally Left Blank
Preface
Amo i colori, tempi di un anelito
inquieto, irresolvibile, vitale
spiegazione umilissima e sovrana
dei cosmici "perch," 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 in-
compatible models of light. Each model particularizes a specific "manifesta-
tion" of light, and then corresponds to adequate physical assumptions and
formal approxilnations, whose domain of applicability is well established. Ac-
cordingly each model comprises its own set of geometrical and dynamical pos-
tulates 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 3-space 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 Hamilto-
nian mechanics of point-particles, with "time" corresponding to the arc-length
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 3-
space dimensions and is accordingly concerned with the dynamical laws estab-
lishing how the wave function changes as the optical wave propagates through
inhomogeneous media.
viii Linear Ray and Wave Optics in Phase Space
Quantum optics recovers the grainy view of geometrical optics, picturing
the light ray as a stream of particle-like entities, the photons. Whereas geo-
metrical optics deals with the influence at a macroscopic level of the medium
on the trajectory of tile photon streams, quantum optics is typically concerned
with the wave-like 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 Coher-
e;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,r i(: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,i v(~ 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.se-
sl)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 ~ifving formalis~n. In this reslmct, elnt)hasis
is givei~ to the Li(~ alge|)ra r(~t)r(~seId,a tion 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 syst(~s.
The book is nm(le as self-c()~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 op-
tics (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 ori-
gin taken in quantum 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 phase-
space 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 func-
tion 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 space-frequency repre-
sentation 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 border-line be-
tween quantum mechanics and signal theory.
Chapters are made as self-consistent 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 connec-
tion 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 quan-
tum 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,n onical transfor-
mations and relevant representations, is elucidated in w 5.6. 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