light rays and imaging in wave optics - ifpan.edu.pl · light rays and imaging in wave optics...

PHYSICAL REVIEW E, VOLUME 64, 066610

Light rays and imaging in wave optics

Władysław ZakowiczInstitute of Physics, Polish Academy of Sciences, Al. Lotniko´w 32/46, 02-668 Warsaw, Poland

~Received 10 May 2001; published 20 November 2001!

An interpretation of focusing and image formation on scattering of electromagnetic waves by a dielectriccylinder~a cylindrical lens! is proposed on the basis of the full Maxwell theory. It is centered on analysis of thebehavior of integral curves of the Poynting vector here called wave rays. These wave rays cannot intersect sothat the focusing and imaging spots are identified with regions of high flow concentration. Two-dimensionalexamples of wave rays and wave fronts in the scattering of plane and cylindrical electromagnetic waves as wellas of Gaussian beams by a dielectric cylinder derived from rigorous solution of the Maxwell equations forincident waves perpendicular to and uniform along the scatterer are given. Their qualitative comparison withgeometrical and diffraction approximations are provided. Fixed points and vortex structure of the Poyntingflow are investigated. An example of~Gaussian-beam! scattering with transparent multiple internal reflectionsand multiple wave splitting is given.

DOI: 10.1103/PhysRevE.64.066610 PACS number~s!: 41.20.Jb, 42.25.Bs, 42.25.Fx, 42.30.Va

foahtises

yso

ticligs

merblnteusicn

f

thgghmtinst

ererc

firdrfe

l atl isp-bemedull

g-ellther

orpro-he

tums.theics,r-he

ingntshusicaltheityunt

ven

ple,ed.

ar-tor

re

I. INTRODUCTION

This paper attempts to discuss wave propagation andcusing effects with the help of exactly solvable problemclassical electrodynamics of the scattering of an electromnetic wave by a lossless perfect dielectric cylinder. Tmonochromatic electromagnetic fields that are used saexactly the corresponding Maxwell equations with all necsary boundary conditions.

Formation and properties of optical images in optical stems belong to the most extensively studied problemsclassical electrodynamics. Elementary geometrical-optheory, based on the ray tracing methods, describes thefocusing and formation of images of illuminated objects uing few rules of rectilinear propagation of rays in uniformedia and simple laws of refraction and reflection at diffent media interfaces. This very simple theory is not only ato explain the basic principles of most optical instrumebut also, in its more developed form, enables one to dwith distortions and aberrations of optical systems. Discsions of these problems can be found in all books on optin particular, one can refer to the monographs by Born aWolf @1# and Sommerfeld@2#. Mathematical aspects ogeometrical-optics are discussed in@3# while numerous ex-amples of applications can be found in@4#.

Beside purposely achieved focusing and imaging,light very often exhibits unintended or ‘‘natural’’ focusinproperties occurring in various forms of conspicuous brilines and spots in reflected or transmitted light. In the fraof geometrical-optics and ray tracing methods these disguished effects correspond to caustics, i.e., envelopebunches of rays that are basically smooth and stable withexception of certain points at which they change their pattin discontinuous way forming cusps. The range of such vcommon effects has been systematically covered withintastrophe theory by Nye@5#.

To define the rays in theories based on wave physicsan eikonal, defined as an optical path and closely relatethe phase of the wave, has been introduced by Sommeand Runge@6# and discussed in@1,2#. The rays are identified

1063-651X/2001/64~6!/066610~14!/$20.00 64 0666

o-fg-efy-

-fsht-

-esal-s,d

e

te-of

henya-

sttold

with lines that are tangent to the gradient of the eikonaeach point. It is to be noted that the concept of eikonastrictly connected with the short-wave approximation in otics and diffraction theory. The eikonal approximation canuseful in analysis of light propagation in weakly nonuniformedia. At first, the concept of eikonal had been introducfor scalar waves this scalar function were applied in a fvector wave theory based on Maxwell’s equations@1,3#.

Although our discussion is restricted to the electromanetic wave propagation problems described by Maxwequations many properties of these problems occur in owave theories. The most important example is quantumwave mechanics. Just as the eikonal approximation wasposed to link wave optics with geometrical optics, so thydrodynamic approximation to the Schro¨dinger equationwas proposed, as early as in the beginnings of quantheory @7#, to link wave functions with classical trajectorie

However, these asymptotic theories are insufficient ifgeometrical rays are intersecting. In the geometrical optthe definition of intensity of light at a given point is detemined by its distance from the center of curvature of twave front passing through this point@8#. The centers ofcurvature of the wave fronts lie on surfaces, which accordto @8#, can be identified with caustics. Therefore, at the poilying on the caustics the intensity becomes infinite, and tcan be determined only by wave theory. Geometro-optfoci themselves are distinguished points forming cusps ofcaustics. To deal with the apparent singularities of intenson caustics and foci, better theories, which take into accothe wave properties of radiation are necessary.

Wave properties of light, not restricted by the short-waapproximation, are usually considered within diffractiotheories. These theories started with the Huygens princiwhile later findings of Young and Fresnel led to thHuygens-Fresnel formulation followed by the Kirchhoff anKirchhoff-Helmholtz diffraction integral representationThese early formulations of the diffraction theory were cried out for scalar waves. The generalizations to the vecwave fields have been discussed in@9–12#.

Quantitative applications of these diffraction theories a

©2001 The American Physical Society10-1

onensarls

muaas

tntaee

esiffi,y

lyoth

gdi

thisd

an

rie

edreac.g

.

wnginhueaonsihcmrre

ieia

callect-hede-

di-y-s it

essel

thisn.

on-dsromer

lyhepar-ntin

elluan-an

n ofs-or

ac-thect,

ec-al

re

rmeralb-ein

rgyof

esw

gionp-bep-

WŁADYSŁAW Z AKOWICZ PHYSICAL REVIEW E 64 066610

difficult as the solutions are given in terms of integralsrapidly oscillating functions and thus many special functiowere defined and asymptotic methods of ‘‘steepest desc~called also ‘‘stationary phase’’ or ‘‘saddle points’’ method!were developed. The dominant contribution to the integrcan be evaluated by deformation of the integration contouthe complex plane. In addition, the diffraction integramethods require the knowledge of field distribution on soreference surface. Very often ‘‘a reasonable guess’’ coprovide this distribution or it could be derived usinggeometrical-optics approach. The geometrical-opticsproach was applicable in multicomponent optical systemwhich the diffraction-based calculations were restrictedthe last stage of wave beam propagation. The comparisovarious diffraction approximations, presentation of computional methods in the studies of focusing problems have bdiscussed by Stamnes@13#.

As mentioned above, diffraction problems lead to exprsions containing various complicated integrals. Severe dculties in computation of these integrals were overcomeparticular, by Pearcey@14# still before computers were reallavailable.

Different properties of the diffraction integrals, usualstudied with the help of stationary phase methods, were cnected with various image patterns and interpreted inframe of catastrophe theory@15,5,16#. The wave fronts de-rived in these diffraction-based theories exhibit phase sinlarities at points or along lines where the wave amplituvanishes. Nye and Berry compared these singularities wdislocations occurring in crystals@17#. This paper stimulateddiscussions of many optical effects, e.g., airy rings inwave diffracted by a circular aperture, in terms of wave dlocations @18,19#. The relations of phase singularities andislocations of waves with geometrical rays, caustics,catastrophes are presented in@16#.

The observed optical phenomena are so complex andthat approximate theories to describe them become vcomplicated. Many approximations have only a limitrange of applicability and any claims for universality arather problematic. Therefore, the desire a solvable diffrtion model has been expressed many times, e@13,5,20,21#. In this context, Mie’s scattering@1# of a planewave by a dielectric sphere was mentioned most oftenfact, Khare and Nussenzveig@22# attempted to apply rigor-ous Mie approach to the rainbow theory. They found, hoever, an extremely fine structure of solutions, rapidly chaing with the sphere radius, as well as the necessity of takfor their parameters, several thousand partial waves. Tthat approach was not recommended any further. Instapproximate but analytical summation of partial waves ctribution, based on the Watson transformation, was conered advantageous@23#. However, this analytical approacleads to an extremely complicated formalism involving funtions analytically extended into the complex plane of coplex angular momenta, as well as complex wave vectok,and still is only approximated as it requires data on theflection and transmission coefficients of Debye terms.

The problem studied in this paper is similar to the Mproblem as the solution is expressed in the form of part

06661

fst’’

lsin

eld

p-inoof-n

--

in

n-e

u-eth

e-

d

chry

-.,

In

--g,s,d,-

d-

--

-

l-

wave expansion. However, it is simpler since the cylindripartial waves are simpler than the spherical ones, and seing incident waves to be normal to and uniform along tcylinder the configuration chosen can be completelyscribed on a two-dimensional~2D! plane.

The scattering of plane electromagnetic waves by aelectric cylinder was treated for the first time by Lord Raleigh @24# and has long been of great theoretical interest aadmits exact solution~see, e.g.,@25# for a more recent refer-ence!. This solution can be written in terms of partial wavthat can be expressed in terms of the well known Besfunctions. With the present state of the art computationssolution can be handled with almost arbitrary precisioWhile most papers dealing with scattering on a cylinder csider the incident field to be a plane wave, similar methocan be used to describe the scattering of waves emitted fwell-localized sources placed in the vicinity of the cylindand for incident Gaussian beams@26–28#. A linear long an-tenna placed near an infinite cylinder, while being a highidealized source of isotropic cylindrical wave, provides tsimplest model of a point source located near a lens. Inticular, it allows for a reconstruction of the image formatiofrom the point of view of the rigorous Maxwell theory. Leus notice that the solvability of the model has been utilized@29# where the complete system of solutions to the Maxwequations has been determined and applied to the field qtization and description of spontaneous emission fromatom located near the cylinder.

The present paper attempts to provide an interpretatiothe image formation within the framework of the rigorouMaxwell theory without explicit reference to traditional optical interpretations based either on geometrical opticseven supplemented by wave-optics effects including diffrtion and the Huygens principle. That does not mean thattraditional interpretations of optical phenomena are incorrehowever, it may be of some interest to show how the eltromagnetic fields themselves deal with very rich opticphenomena.

In this work we only consider incident waves that aperpendicular to the cylindrical scatterer axis~taken as thezaxis of the Cartesian or cylindrical coordinate system!. Inaddition, we assume that the incident waves are unifoalong the cylinder. With these two assumptions the genscattering problem simplifies to the two-dimensional prolem with the relevant field components dependent on thxand y variables and the Poynting vector field remainingthe x-y plane.

Wave-optical rays associated with the transport of eneare consequently defined as integral curves of the field~time-averaged! Poynting vector and show how the imagare formed by focusing of rays. In fact, such energy flolines were introduced by Braunbek and Laukien@30# ~in theirinvestigation of the diffraction by a half-plane! and byBoivin, Dow, and Wolf @11#, in their study of a focusedbeam. The second analysis was restricted to the focal reonly and the fields were evaluated within the diffraction aproximation. In the present model the wave light rays canderived for the entire space including the interior of the otical system~cylindrical lens in this case!.

0-2

ththiontic

t

veua

isthdiIne

seincinckifyhr

ofecinan

tele

rinrrrereoreed

thinin

arteh

bu

er

ruf

wi

msta-hisalf-

ctedavea-netter-thee istam-nd

fied2Dret

ve

IIasicthetheionaysro-

ryna-iorof

VIIer.sIX

an

tare

non-

erhe

he

ibed

LIGHT RAYS AND IMAGING IN WAVE OPTICS PHYSICAL REVIEW E 64 066610

A simple consequence of mathematical properties ofdifferential equations describing the light wave rays isexclusion of the ray intersections as well as their divisand termination except at absorbers, conductors, and parlar discrete~for a given finite cylinder radius! set of trajec-tories, discussed further.

Wave fronts or equiphase surfaces are represented inpresent simple 2D model by curves in thex-y plane orthogo-nal to the set of wave rays. This set of equiphase curwave fronts, can be derived from a set of differential eqtions similar to that for the rays.

Both sets of equations for light rays and wave fronts dtinguish certain characteristic points. At these pointstime-averaged Poynting vectors vanish. A more detailedcussion of these stationary points is given in Sec. IV D.particular, there are stationary points of ‘‘vortex’’ type, in thvicinity of which the Poynting flow circulates and equiphalines are attracted, and those of ‘‘saddle point’’ type thatgeneric cases repel both integral curves. Some very sperays can either terminate at the ‘‘saddle points,’’ or, startat these points, run to infinity or, forming a loop, return bato the same ‘‘saddle point.’’ These critical curves specseparatrices isolating Poynting flows of different types. Tgeneric wave light rays can either start at the source andto infinity or form a close vortex line. The existenceclosed Poynting flows is an important property of the eltromagnetic radiation flux. Such lines occur very oftenshadow regions forbidden for the geometrical rays andimportant for the descriptions of radiation beam splitting aintersection of two separate beams.

In the framework of quantum scattering theory the ingral curves tangent to the quantum current field, calstreamlines, were proposed in@31# to illustrate fluid featuresof the wave function in the scattering process. The scatteof plane scalar waves by spherical potential wells and baers were considered. Although the scatterers were thdimensional spheres the axial symmetry of scatteringduced this discussion to the 2D problem. The formationthe streamlines vortices were shown. However, the scattewere rather small and not too many partial waves were nessary. Therefore, many characteristic optical featurescussed in the present paper were not present.

The vortex-type fixed points in the theories that takewave fronts as principal objects correspond to singular poof phase. However, there is nothing singular in these powhen the fields are taken as fundamental variables.

In contrast to the light rays that in the generic casesindestructible, the equiphase lines can terminate and stathe vortex points. The number of fixed points strongly dpends on the radius of the cylinder, the position of the ligsource, and on the type of incident beam. Perhaps the adance of such points for the system investigated in@22#caused the rapid variation of the scattered radiation patt

As was noticed in@32,19,18# the structure of the focuswas partly influenced by diffraction effects at the apertuboundaries or stops supporting the focusing lenses. Forsupported lenses and for Gaussian-beam illumination thecusing patterns simplify. Similar features have been foundthe investigated system for the incident Gaussian beam

06661

ee

u-

he

s,-

-es-

ificg

eun

-

red

-d

gi-e--frs

c-is-

etsts

reat

-tn-

n.

en-o-inth

the width smaller than the cylinder diameter. For such beathe rays propagate very smoothly through the lens. All stionary points are shifted outside the incident beam. In tcase one can find the reflected rays going back to the hspace of the incident beam. There were no such reflerays in the other two cases, i.e., those of incident plane wor incident cylindrical wave. For the plane-wave illumintion, all rays seem to flow to the forward direction and ocan ask what happened with evidently present backscaing. Backscattered rays can emerge only at the wings ofGaussian beams where the usually weak scattered wavnot dominated~‘‘overshined’’! by the much stronger incidenbeam. When a Gaussian beam with nonzero ‘‘impact pareter’’ is applied, several internal reflections are visible acan be studied in detail. Nussenzveig’s@20# classes ofmultiple-reflected and refracted rays can be easily identiand interpreted as scattered bundles of wave rays in ourscattering model. This result makes it possible to interpthe 2D analogue of rainbow within the framework of waoptics in a Cartesian-like way.

The rest of the paper is organized as follows. Sectiondefines the mathematical models and contains the bequations. Section III is devoted to the presentation ofglobal properties of wave rays and the comparison withstandard geometrical-optics picture of ray propagatthrough a lens. The discussion of the fine structure of rand equal-phase contours as well as of fixed points is pvided in Sec. IV. An investigation of rays and stationapoints specifically for the case of Gaussian-beam illumition is given in Sec. V. Section VI demonstrates the behavof intensity and phase along the optical axis. The problemimage formation of extended objects is elucidated in Sec.by analysis of radiation of two point sources near a cylindSection VIII is devoted to the multiple internal reflectionand refractions of rays by and inside the lens. Sectioncontains several final remarks.

II. THE MODEL AND ITS SOLUTION

In this work we consider the problem of scattering ofincident Einc stationary wave~dependent on time ase2 ivt)propagating in thex-y plane and with such polarization thathe electric field is orthogonal to this plane. The wavesscattered by a dielectric infinite cylinder of radiusa. Thedielectric is assumed to be homogeneous, lossless, anddispersive. Its refraction index isnd . We place the origin ofthe cylindrical coordinate system in the center of the cylindwith the z axis directed along the cylinder. In this case twhole electromagnetic field is determined from theEz com-ponent of the electric field. The magnetic field stays in tx-y plane.

The electromagnetic field outside the sources is descrby the Helmholtz equation

¹2Ez~r ,f!1n2~r !k2Ez~r ,f!

5S ]2

]r 21

1

r

]

]r1

1

r 2

]2

]f21n2~r !k2D Ez~r ,f!50,

~1!

0-3

nd

rca, aree

i-

-

e-

ex-ethe

esebe

nite

rizedude

on

nsgedrgyto


where k5v/c, n(r )5nd for r ,a, and n(r )51 for r .a.The solution of Eq.~1! can be written as

Ez~r ,f!5Ezinc~r ,f!1 (

m52`

`

bmeifmHm(1)~kr !, r .a

Ez~r ,f!5 (m52`

`

ameifmJm~ndkr !, r ,a, ~2!

whereJm denote the Bessel functions of the first kind aorderm, while Hm

(1) is the Hankel functions of the first kindand orderm.

Three different incident waves corresponding to a souat infinity, an infinitely long and infinitely thin linear antennat finite distance from the cylinder, and a Gaussian beamconsidered in this paper. The electric fields in these thcases, expressed in terms of partial cylindrical waves, arfollows:

I. Plane-wave propagating in thek(a)5k$cosa,sina,0%direction

Ezinc~r !5E0eik"r5 (

m52`

`

i mJm~kr !eim(f2a). ~3!

II. Cylindrical wave emitted by an antenna placed atr0 or$r 0 ,f0% if polar coordinates are used

Ezinc5AH0

(1)~kur2r 0u!

5AH0(1)@kAr 21r 0

222rr 0 cos~f2f0!#, ~4!

Ezinc~r ,f!5AH0

(1)~kur2r 0u!

5A (m52`

`

Jm~kr !Hm(1)~kr0!eim(f2f0),

r ,r 0 . ~5!

III. Gaussian beam

Ezinc~r !5E0E daP~a!eik(a)•(r2r0)

5E0 (m52`

`

i mJm~kr !eimfE daP~a!e2 imae2 ik(a)•r0,

~6!

where P(a)5(w/Ap)e2w2a2, r0 is the position of the

beam waist, andw is the width of the beam waist~also speci-fying its angular spread,D51/w).

The coefficientsam andbm are determined by the contnuity conditions of the tangent electric (Ez) and magneticfields (Hf}] rEz) at the boundary of the cylinder. These amplitudes are equal to

am5Gm

Jm8 ~ka!Hm(1)~ka!2Jm~ka!Hm

(1)8~ka!

ndHm(1)~ka!Jm8 ~ndka!2Jm~ndka!Hm

(1)8~ka!,

~7!

06661

e

reeas

bm5Gm

Jm8 ~ka!Jm~ndka!2ndJm~ka!Jm8 ~ndka!

ndHm(1)~ka!Jm8 ~ndka!2Jm~ndka!Hm

(1)8~ka!,

~8!

where in the three cases considered here we have

Gm5E0i me2 ima, plane wave,

Gm5Ae2 imf0Hm(1)~kr0!, cylindrical wave, ~9!

Gm5E0i mE daP~a!e2 ik(a)•r0e2 ima Gaussian beam.

A small simplification may be achieved in Eq.~7! by notingthat the numerator contains the Wronskian of theJm andHm

(1)

functions.Finding the solution for the electric field one can imm

diately get the corresponding magnetic field

Hx52i

vm0

]Ez

]y, Hy5

i

vm0

]Ez

]x, Hz50,

and the time-averaged Poynting vector field:S&5 1

2 ReeEÃH!.In the case of a Gaussian beam the incident waveEz

inc andthe expansion coefficients for the cylindrical partialGm re-quire the integration over the anglesa. While for a givenbeam and cylinder configuration all relevant partial-wavepansion coefficientGm must be calculated only once and thscattered part of the wave can be evaluated as fast as inother two cases, the fields of the incident wave requireaintegration at each position pointr . This leads to some in-conveniences when one looks for light rays. To avoid thinconveniences two computational approximations canmade. In the first one the beam is represented by a finumber of plane waves witha j5 j D/Ne , $ j 52Ne ,2Ne11, . . . ,Ne21,Ne%. The second approximation is valid fobeams that are not very narrow and therefore characterby very small angular spread. In this case one can inclonly the quadratic expansion terms ink(a)'$12 1

2 a2,a,0%and all integrations with the Gaussian distribution functiP(a) can be done analytically. Thus, one gets

Ezinc~x,y!'E0

2w

A4w212ik~x2x0!eik(x2x0)

3expS 2k2~y2y0!2

4w212ik~x2x0!D , ~10!

Gm'E0i m2w

A4w222ikx0

e2 ikx0expS 2~m1ky0!2

4w222ikx0D .

~11!

Light rays in the theory based on the Maxwell equatiocan be defined as the lines tangential to the time-averaPoynting vector. Such lines are sometime called ‘‘the eneflow lines’’ @1#. They can be determined from the solutionthe equation

0-4

e

she

nalor-

u

th

nabalaE

tintet

te

o

yidane

r

it

i-

yo

hexeslensting

tedn is

pic-mes ininveld,

lf,re-heonve

ve


dr lr

dt5^S@r lr ~t!#&. ~12!

It is to be stressed that wedefinethe wave-optical light raysas the integral curves of Eq.~12!.

There exist specific points for which the time-averagPoynting vector vanishes,

^S~r lr !&50 ~13!

so thatdr lr /dt50 at these points. They are usually~in thetheory of differential equations and dynamical system!called stationary or fixed. They play a crucial role in tcharacterization of fields in an optical system.

The equal-phase manifolds are in fact two-dimensiosurfaces. However, since our system is homogeneous athe cylinder~z! axis, we can restrict ourselves to the intesection of these surfaces with any plane forz5const. Suchintersections define curves described by the following eqtion

drf

dt5 z3^S@rf~t!#&. ~14!

The curves specified above will henceforth be called‘‘equal-phase contours.’’

III. GLOBAL STRUCTURE OF LIGHT RAYS:COMPARISON WITH GEOMETRICAL OPTICS

Although the expressions provided in Sec. II give the alytical solutions to the electromagnetic field scattering prolem, they are fairly complicated and require numerical trement for more specific exposure of the results. In particusuch numerical treatment is necessary to find solutions of~12! that are used to draw the lines tangent to the Poynvector interpreted in this paper as wave light rays. The ingrations have been performed using a routine based onRunge-Kutta method with an automatically adjustable s~the Merson’s scheme!.

The Bessel functions were computed with the help@33#.

The number of terms in the field expansion necessarreach the required accuracy depends basically on the wof the scattering cylinder and the separation of the wiretenna~line source! if this case is considered. Although thpartial-wave expansion of fields in Eq.~2! runs formallyfrom minus to plus infinity, for a fixed radius of the cylindethere is only a finite numberN of terms for which thean andbn coefficients take significant values. We estimate this crcal numberN to be 10a/l.

In our example of a dielectric cylinder of radiusa515l,the emitting antenna atr 0540l, and on using 23150 ~i.e.,2150<m<150) partial waves the field continuity condtions are fulfilled with relative accuracy of 10213.

A. Comparison with geometrical optics

Figures 1 and 2 show the continuation of such light racalculated as lines tangent to the Poynting vector and th

06661

d

lng

a-

e

--t-r,q.g-hep

f

toth-

i-

sse

obtained according to the rules of geometrical optics. Tlight rays are sampled in such a way that the Poynting flubetween the two consecutive rays passing through theare equal. In the case of an incident plane wave the starpoints should be far to the left at equalDy intervals. In thecase of point antenna initial points for rays can be selecjust as easily, since near this antenna radiation emissioisotropic.

One can find general agreement in the correspondingtures, i.e., the incident plane wave is focused in the saregions and the wave emitted from the point source resultsimilar images. However, there are striking differencesboth theories. The light rays defined according to the watheory, being the integral curves of a continuous vector fiecannot cross each other~nor can any one of them cross itseof course! as is shown in the top right segments of the psented figures. This property is completely different from tbehavior of rays in geometrical optics. This observatileads to the reinterpretation of the image formation in waoptics.

FIG. 1. Focusing of a plane wave by a cylindrical lens in waand geometrical light ray optics. Normal wave incidence,a515l, nd5A1.7.

0-5

asomns

thae

aalhesb

evs

thapulTothasingr-

anulheig

er-htnd-n,. 4.

ndin

if-ll.

riesandthedel

ng

elfne

thers ae

s tothe

ngr is

ated

jec-ray-

ve-

yn-


In wave optics the flow of electromagnetic flux as wellthe density of the field energy become more dense in sregions of the space. Such regions, where the separatiolight rays are very small, can be identified as the imagesthe sources located on the other side of the lens. The factthe focus is not a point but a rather extended region in spis evident as well. There are some other differences betwgeometrical and wave rays.

One of the most ubiquitous and beautiful effects appeing in optical instruments are caustics, analyzed usuwithin the geometrical optics and/or theory of catastrop@5,34,16#. In our pictures of the wave-optical rays, the cautics behind the cylinder are visible as well. They cannotdefined as the envelopes of bundles of crossing rays sincwave optical rays do not cross. The caustics appear, howeas the regions of enlarged density of rays at the boundariethe beam of rays, see Figs. 1 and 2.

Near the boundaries and light concentration regionswave light rays may become quite complicated, lookingparently erratic, though in geometrical optics the light shopropagate along straight lines in homogeneous media.wave ray flow approaching the focal region of image spwith all its wiggles, is much different from the rather smoodeparture flow. The behavior of fields in this region wintensively studied within diffraction-based theories startfrom the numerical calculation of a diffraction integral peformed by Pearcey and co-workers@14,34,5#. These resultshave usually been illustrated by drawing the magnitudephase contours of the electric field. To compare our reswith those earlier works, we include figures showing tcontours of equal magnitude of the Poynting vector, see F

FIG. 2. Focusing and image formation of a cylindrical waemitted at the distancer 0 from a cylindrical lens in wave and geometrical light ray optics. Normal wave incidence,a515l, nd

5A2, r 0540l.

06661

eof

ofat

ceen

r-lys-etheer,of

e-

dhet,

dts

s.

3 and 4. Figure 3 illustrates rather well the focusing propties of the cylindrical lens with the shadow region and brigand dark speckles distributed inside the lens along its bouary. A magnified part of Fig. 3 covering the focal regiotogether with an adjacent part of the lens is shown in FigIt evidently resembles analogous figures~as well as photo-graphs! obtained with the help of the Pearcey integral adescribed as the cusp diffraction catastrophe shown@34,15,5,16#, although our approach does not use either dfraction integrals or any approximate diffraction theory at a

Another interesting region is located near the boundawhere the rays approach the cylinder at grazing anglesgroups of rays split into two bundles, one passing bycylinder and the other one refracted by the lens. Our moallows for detailed study of the mechanism of this splittiprocess.

B. Rays in the shadow region

We observe that the beam of wave-optical rays splits itsin some regions close to the boundary of the cylinder. Oset of rays enters the cylinder and is refracted, while anoflows round the lens. Between these two parts there iregion of low intensity.~Both groups of rays eventually comagain together at a large distance from the cylinder.!

The standard geometrical optics does not allow the rayenter the shadow regions. However, from the solutions toMaxwell equations it follows that there exist non-vanishifields in these regions. The time-averaged Poynting vectononzero as well, and the wave-optical rays can be evaluand studied.

Figure 5 shows a sample of several characteristic tratories near the upper boundary of the lens where the

FIG. 3. Shaded contour plot of constant modulus of the Poting vectoruSu ~contours themselves are removed! for a cylindere52, a530l, and plane-wave illumination.

0-6

a

me

be

ioio

litt

tiath

bleryre-

of

ichnotnoit.

rayst

ud-rynefor

ryes

eiveen-illthation

rys

es-

rtingnds

n

ion

edre-of


bundle splitting takes place. Some of these trajectoriesclearly closed. There are other curves that, while startingthe source and going to infinity, meander and wind in a coplex manner. The following section will be devoted to a dtailed investigation and interpretation of such complexhavior.

Let us remark that in spite of sometimes erratic behavof the flow lines mentioned above, the numerical integratis stable and on reversing the integration~i.e., performing it‘‘back in time’’ ! one follows the same trajectory.

The flow lines can neither intersect, nor join nor spapart. The latter fact is a mathematical consequence ofunique dependence of solutions of a system of differenequations on initial conditions. Indeed, the vector field of

FIG. 4. Details of Fig. 3 in the focal and internal caustic regioIntensity pattern equivalent to a cusp diffraction catastrophe of@34#outside the cylinder lens is visible.

FIG. 5. Details of the upper part of Fig. 2 in the shadow regwith families of meandering and bounded rays.

06661

reat---

rn

hel

e

time-averaged Poynting vector is continuously differentia~even real analytical! in the whole space except the boundaof the lens. Thus, it satisfies the Lipshitz condition. Thefore, the Picard theorem implies that the integral curvesthe Poynting vector cannot intersect~cf., e.g., @35#!. Thereare specific points—the saddle points of the phase—at whtwo or more wave-optical rays seem to meet. This doesmean, however, that the rays actually intersect. There ispossibility for one ray to reach a saddle point and to leaveThe saddle points cannot be reached by the wave-opticalfor any finite value of the parametert. There are rays thaapproach these points asymptotically to cross att→6`.~Such rays called separatrices will be more thoroughly stied below.! In other words, one might say that a trajectoattempting to cross itself at a point does not contain opiece only. There are at least two pieces that can join onlyinfinite pseudotimet.

IV. PROPERTIES OF LIGHT RAYS,EQUAL-PHASE CONTOURS, AND FIXED POINTS

It is clear from Fig. 5 that the flow of rays can be vecomplicated. We may expect that the ray pattern becomsimpler for smaller radius of the cylinder. In what follows wshall consider in more detail a cylinder of the same refractindex but smaller radius to grasp more efficiently the esstial properties of the light structure. The analysis of rays wbe completed by investigation of equal-phase surfacesare more frequently used in the studies of light propagatthrough optical instruments and other optical systems@1#.

A. Islands in the stream of Poynting curvesin the shadow region

For a cylinder with radius 5l ~i.e., three times smallethan in Fig. 5!, we have plotted a very dense beam of ra~the source of light is still the line source! passing close tothe boundary of the cylinder in Fig. 6. We observe the prence of ‘‘empty’’ or ‘‘white’’ islands with well-definedboundaries. They are inaccessible to the trajectories stafrom the source. To investigate what happens in these isla

.

FIG. 6. Flow of an initially very dense bundle of rays—selectfrom all the rays emitted by the point source—in the shadowgion; the overall picture of the bundle splitting and formationvoid islands is provided; the parameters area55l, nd5A2, r 0

512l.

0-7

aao

oh

oxis

tyne

scyonrai

tathowdiy

wotpl

nta

inAa

de

o-

on-

-lues.butheaseterse,rtex

edes,

notended

-s a

sthe

gand

’’

g.


we need to analyze additionally the equal-phase contoursthe stationary points of the flow. Let us note that the equphase contours and stationary points are the very basicjects of important works dealing with the fine structurelight using the eikonal approximations and catastroptheory @5#.

In Fig. 7 the wave-optical rays and the equal-phase ctours are plotted for the half-plane above the optical aThe phase contours are obtained by integration of Eq.~14!. Itis clear that these contours~dashed lines! are perpendicularto the rays~solid lines!, as should be the case. In the viciniof the source they are almost circles, but in the cylinder abehind it, its form is very different and sometimes windy. Whave started integration of Eq.~14! from the optical axis~inall cases we have checked that the phase along them ideed constant, which has been another test of the accuraour solutions!. It turns out that in many cases the integratiterminates and it is not possible to continue it further. Staing integration from the top toward the bottom results insimilar termination of some contours at certain points. Itinteresting that all the equal-phase curves starting in cerinterval can terminate at the same point. We have foundsuch points are actually the stationary points of the fldefined by Eq.~13!. Most of the stationary points are locatein the regions empty of rays coming from the source. Itimportant, however, that in these regions the fields, the Poting vector does not, in general, vanish and the phase isdefined with the exception of some points. Therefore, bthe wave-optical rays and the equal-phase lines can beted. The structure of flows in the transient~‘‘ray beam-splitting’’ ! region will be analyzed below.

B. Rays, phase contours, and fixed pointsfor small lenses and line source—transient region

Figure 8 provides a picture of both unbounded abounded rays in the transient region. The former ones sfrom the source and escape to infinity, although they wand meander when passing through the transient region.sociated equal-phase contours and stationary points areplotted.

Figure 8 indicates that the stationary points can be diviinto two classes, the first one consists of the vortices~V! andthe second one are the saddle points (S). We can also distin-guish vortices that correspond to clockwise (V1) andcounter-clockwise (V2) vortex flow. One can associate a t

FIG. 7. Wave-optical rays~solid lines!, equal-phase contour~dashed lines!, and stationary points in the upper half-space forsame system as in Fig. 6.

06661

ndl-b-

fe

n-.

d

in-of

t-

sinat

sn-ellhot-

drt

ds-lso

d

pological charge to each vortex, and the equal-phase ctours can join only the vortices with different charge~cf.,e.g.,@36#!.

It is visible in Fig. 8 that the vortices form attractive basins for equiphase curves in certain ranges of phase vaThere are rays that start initially very close to each otherone of them just goes without rounding a vortex, while tother one flows round it. It might be expected that the phrelation along these two rays is completely broken afrounding the vortex by one of them. This is not the cahowever, since the change of phase after rounding the vois equal to an integer multiple of 2p.

C. Bounded structures of the flux

The apparently void island seen in Fig. 6 is displayagain in Fig. 9 but with the bounded rays, equiphase curvand stationary points. It is clear that the island is actually‘‘void’’; the solutions of the Maxwell equations show that thfield intensities and the Poynting vector do not vanish athe wave-optical rays exist. The rays form, however, closloops rather than unbounded curves.

Thus the ‘‘void’’ island is full of bounded, periodic trajectories, and plenty of stationary points, and is contained a

FIG. 8. Detailed structure of the flow in the bundle-splittinregion for the same parameters as in Fig. 6. Both boundedunbounded rays~solid lines! are shown. Black (V2) and white(V1) small circles denote the stationary points~corresponding tovortices! of opposite topological charge. Crosses~S! denote saddlepoints. The vortices are attractors for equiphase lines~dashed lines!.The equiphase lines can join only vortices of opposite ‘‘charge.

FIG. 9. Detailed structure of the flow in the void region of Fi6. Nested families of bounded rays are shown.

0-8

ngiin

th

ddtoo

h

pocthq

hip

ha

ranopo

eearowrrmly

ddino

stricow

i

seces.iffer-thatlso

ara-

ann-Byosern,

entlinesrayof

Eq.

c-pa-pro-ces.igh-

ofs-

thee

rtexres.


whole within a large-scale periodic ray. Some vortices teto group in pairs of stationary points with opposite topolocal charges, such pairs are assisted by pairs of saddle poHowever, some single vortices are present as well sototal topological charge of the island is nonzero.

D. Ray separatrices

The plethora of wave-optical rays shown in Figs. 8 anseem to avoid the stationary points, especially the sadpoints. There are, however, some special rays that asympcally approach the saddle points. These rays separate flof different kinds. In the theory of dynamical systems@35#they are called separatrices. The separatrices can reacsaddle points for infinite values of the ‘‘pseudotime’’t. Twoor more such separatrices that approach a single saddleseem to intersect each other. They are in fact disconneand there is no single dynamical trajectory passing througsaddle point, which could be obtained by integration of E~12!. Indeed two or three independent integrations of tequation are needed to obtain the full collection of ray seratrices attached to saddle points~three for unbounded rayseparatrices and two for bounded!. They have to start frompoints slightly displaced from the saddle points and oneto integrate both forward and backward int. The shape ofany remaining ray is determined by two separatrices embing it. Like all the other rays, the ray separatrices docross each other so that they can meet at one saddleonly ~at t56`).

All ray separatrices and stationary points that have bfound in a region close to the boundary of the cylindershown in Fig. 10. This figure as well as Figs. 8 and 9 shthat every saddle point can be associated with the cosponding vortex point. Not all data presented in Fig. 7 seeto confirm this property. However, this property is true; ondue to the scale of the figure some vortex points and sapoints overlap. Figure 11 shows the vortex and saddle potogether with equiphase lines and separatrices for the mright-hand vortex dot of Fig. 7. This figure looks almoidentical to Fig. 11 of@17# that was considered as a geneexample of a wave edge dislocation. Our discussion shthat such simple pattern of the energy flow may occur onlythe vicinity of an isolated vortex point—saddle point pair.

FIG. 10. Collection of all separatrices for an upper part ofcylindrical lens of Fig. 6. The lens is illuminated by a point sourc

06661

d-ts.at

9leti-ws

the

inteda.sa-

s

c-tint

ne

e-s

letsst

sn

In addition to the light ray separatrices and equiphalines this figure shows an object—equiphase separatriThese equiphase separatrices determine boundary for dent basins of the equiphase lines. This example showsthe saddle point, similarly as for the ray separatrices, is aa meeting point for the equiphase separatrices.

The following subsection discusses the equiphase septrices in a region with many stationary points.

The number and location of stationary points is notintrinsic property of the cylinder lens. It depends on the etire optical system that includes the sources of radiation.moving the point source one can change the position of thpoints, hence the form of separatrices, which, in tuchanges the shape of the rays.

E. Equiphase separatrices

The equiphase contours can be divided into differclasses by the equiphase separatrices. The equiphaseand separatrices play a similar role to the wave rays andseparatrices that characterize the light flow. An exampleequiphase separatrices~bold dashed lines! with backgroundof ray separatrices~thin lines! is shown in Fig. 12.

All equiphase separatrices are obtained integrating~14! in the forward and backward directions oft, startingtfrom two points displaced infinitesimally in opposite diretions from the saddle points. This way four equiphase seratrices attached to each saddle point can be found. Thecedure is similar to that used in the case of ray separatri

These equiphase separatrices extend either to the neboring vortex points or to the points on the optical axesy50. The end points on the optical axis can lie to the rightthe light source atx.x0 or, for equiphase lines encompasing the light source, atx,x0.

.

FIG. 11. Separatrices and equiphase lines for the isolated vosaddle points pair. The same system as in the four previous figu

0-9

n

rgthp

y.

lyina

i00eoae

e-ithhete

be

thethe

xis.aysby

nd itilar

ostintopsanyoseur-llerh

In

llu-ofi-ing

illu-

a

ofd

urs,e as

m as


The equiphase lines are always staying in the correspoing basins enclosed by the equiphase separatrices. Theyextend either between the vortex points of opposite chaor between the two points on the optical axis placed ontwo sides of the source. The second case occur, for examfor equiphase circles surrounding the source in its vicinit

F. Fixed points for the large cylinder

On having obtained the necessary experience by anaing small cylinders we can reconsider the light propagatalong a bigger system, to the one presented in Figs. 1, 2,5 ~radiusa equal to 15l). The number of fixed points rapidlygrows with the radius. In the upper half-plane displayedFig. 13, the number of stationary points is larger than 30Most of the singularities are concentrated in the annulus nthe surface of the cylinder. In addition, there is a familyfixed points along the rim of the focused bundle of rays,shown in Fig. 14. There is a clear correspondence betwthe positions of~groups of! stationary points in Fig. 13 andthe form of the rays from Fig. 5. Also, the irregularities in thrays near the ‘‘entrance’’~front! surface and ‘‘departure’’ surface of the cylinder in Fig. 2 are evidently connected wthe region in which there is plenty of stationary points. Tstationary points close to the optical axis in Fig. 13 en

FIG. 12. Equiphase and ray separatrices in the multistationpoints region. The same parameters as in the last figures.

FIG. 13. All stationary points and their influence on the flowthe rays for a larger lens for the set of parameters of Figs. 2 an

06661

d-cane,ele,

z-gnd

n.

arfsen

r

deeper towards the center of the cylinder and may thusassociated with a cusp of a caustic~compare Fig. 17 below!.It is interesting that the stationary points are present onoptical axis as well, but there are no such points close toposition of the image of the source~see Fig. 14!. At theimage distance the stationary point is moved out of the a

As in the case of the smaller lens, the shapes of rpropagating through the optical system are determinedseparatrices. The number of separatrices is very large adoes not seem useful to show them all. Their shape is simto that shown for the smaller system. The loop part of mof them is of very small size, such that only one vortex pois included. There are, however, separatrices with large loembracing several tens of stationary points as well as mbounded ‘‘internal’’ separatrices. Figure 15 presents thlargest separatrices located at the top of the cylindrical sface. It is interesting to notice that, as in the case of smaseparatrices~Figs. 8–10!, the large ones can form pairs witopposite vorticity of members of the pair.

V. RAY STRUCTURE AND FIXED POINTSFOR GAUSSIAN BEAMS

It has been observed by Karman and co-workers@19,18#,that the structure of stationary points~phase singularities inhis language! is strongly dependent on the type of source.particular, they investigated in detail~both theoretically andexperimentally! the case of a truncated Gaussian-beam imination and the light structure in the focal region. Onetheir conclusions was that both the ‘‘creation’’ and ‘‘annihlation’’ of phase singularities can happen when proceedfrom uniform to Gaussian illumination.

Let us now consider the case of the Gaussian-beam

ry

5.

FIG. 14. Details of the structure of rays, equal-phase contoand stationary points in the focal region; parameters are the samin Figs. 2 and 5.

FIG. 15. Selected large-scale separatrices for the same systein Fig. 5.

0-10

sehe


FIG. 16. Wave-optical rays and equal-phacontours for the Gaussian-beam illumination. Tparameters area515l, e52, w52.

ron

idtod. Ireat

re

yowanrtpai

si

llyl.wsergcoia

ar-Fig.rayson

thenarypartamlla-dd itherensre-

oth

e

is as.riesn-isriesve

hin am

mination. This can be a good model for light beams pduced by many laser sources. In order to find out effectsvisible in the case of point source and plane wave, the ww of the Gaussian beam cannot be either too large orsmall. Indeed, ifw were too large, the whole picture woulbe very similar to the case of the plane-wave illuminationw were too small, the beam would spread out very fastsembling the point source pattern. But for an intermedivalue of w, i.e., lr 0 /(2a),w,a, the entire incident beampasses through the cylinder. A large portion of the scattelight stays outside the incident beam and, therefore, canvisible. The general pattern of the light scattered by the cinder, including the rays and equal-phase contours, is shin Fig. 16. The parameters of the position of the sourceof the lens are the same as in Fig. 2. The light rays are stato keep equal fluxes between the subsequent lines. Coming Fig. 16 with Fig. 2 we observe that the rays flow invery regular, smooth way while the equal-phase contoursside the beam resemble those usually drawn for Gausbeams. Most of them are terminated just out of the bundlethe rays of Fig. 16 since the intensity falls off exponentiaand the right-hand side of Eq.~14! becomes extremely smal

In Fig. 16, one very distinguished feature is present. Treflected rays appear, scattered at large angles. In the cathe other two sources considered in this paper, these laangle scattered rays are hidden due to the overwhelmingtribution of the incident beams. But the incident Gauss

FIG. 17. Geometric-optics rays starting at a point source. Trays that had been internally reflected one time are visible includthose contributing to the rainbow.

06661

-ottho

f-e

dbel-ndedar-

n-anof

oof

e-n-n

fields decay rapidly thus making possible the clear appeance of the reflected rays. The reflected rays shown in16 correspond to a multiple-reflected geometro-opticalcontributing to the rainbow, as can be seen by compariwith geometrical-optics drawing of Fig. 17.

The smooth and regular flow of rays in the case ofGaussian beam does not mean that there are no statiopoints. They are, however, located outside the dominantof the beam, cf. Fig. 18. A part of them bounds the bebefore it enters the cylinder. There is an interesting oscitory structure in the vicinity of the reflected ray. Lots of fixepoints are present within the cylinder outside the beam anseems that behind the lens and close to the optical axis tis a similar group of these points as in the other illuminatioshown in Fig. 14. One cannot, however, consider thesesults to be complete.

VI. INTENSITY AND PHASE PROFILESALONG THE OPTICAL AXIS

Our approach enables us to show the properties of bthe intensity and phase of the fields~which are often dis-cussed in a similar context! along the optical axis. The phasis specified asF5arg(Ez).

For the case of the point source, Fig. 19, the intensityvery rapidly oscillating function of position along the axiThe change in the oscillation pattern close to the boundaof the cylinder clearly correspond to the structure of statioary points shown in Fig. 13. A similar change of patternevident in the position dependence of the phase, which varapidly in the same regions. Beyond the cylinder, we ha

eg FIG. 18. Stationary points of the flow for the Gaussian-beillumination. All the parameters are as in Fig. 15.

0-11

oin

ios

betiehe

iio

raow

tedtri-of

ve-e

ter

is-in

atfre-aretheust

edre-

ig.asr-

riseayse tosero

dingeo-

n-ityderbe-

in

i-l

tim-


found only one point of the phase anomaly close to the px/l;33 ~see Fig. 14 for a comparison!. To our surprise, wehave not detected any phase jump close to the image reg

For the case of Gaussian beam, the analogous intenoscillations are much smaller as shown in Fig. 20. Thehavior of phase is very regular except the small irregulariat x/l;7 that one may possibly associate with tgeometro-optical cusp occurring in Fig. 17. Again, thereno phase jump either at the source or at the image posit

VII. IMAGE FORMATION OF EXTENDED OBJECTSIN WAVE OPTICS

We have stressed several times that the wave-opticalas defined in this paper cannot intersect. But then the foll

FIG. 19. Intensity and phase along the optical axis for the posource case.

FIG. 20. ~a! Intensity of the field for the Gaussian-beam illumnation along the optical axis.~b! Phase distribution along the opticaaxis for the Gaussian-beam illumination.

06661

t

n.ity-s

sn.

ys-

ing question arises: how is it possible to get the inverimages of objects, which is usually obtained in the geomecal optics by a construction that requires the crossingrays? Another important question is, how to get the waoptical picture of multiple reflection inside the cylinder? Thformer one will be discussed in this section, while the latin the following one.

To answer the first question in the simplest way, we dcuss the wave-optical rays from two point sources locatedthe vicinity of the cylinder. Let us for a moment assume ththe sources are coherent—they oscillate with the samequency and the phase difference is fixed. The total fieldssuperpositions of the fields irradiated by each source, andinterference effects clearly have to be present. The rays mnot be, of course, simply ‘‘added,’’ but should be obtainfrom the total electric and magnetic fields and the corsponding Poynting vector.

The flow of the wave-optical rays case is displayed in F21~a!. The rays start at both the emission points. If there wno lensing cylinder, the density of rays would form interfeence fringes around the system. The cylinder itself givesto the image of the sources in a very peculiar way. The rstarting at the more distant source do not have any chancreach the cylinder at all. Only the rays starting at the closource enter the cylindrical lens. In spite of this fact, twdistinct ray concentration spots are apparent corresponthe source images as is clear by comparing with the gmetrical construction shown beneath@Fig. 21~b!#. Thus, theinformation on the second source is ‘‘recorded’’ and cotained in the rays originating at the closer one. The visibilof images of the sources becomes better for larger cylinsize as it gathers more radiation rays. Thus the contrast

t

FIG. 21. ~a! Wave-optical rays for radiation from two poinsources scattered by the cylindrical lens. Formation of invertedage by the enlarged concentration of rays is shown.~b! The sameimage formation according to the geometrical optics.

0-12

ttennah

etrihensi2

ro

n-

thsa

atoin-e

aanid

ot

ndw-i-

ralt sovenof

ellob-y atse-ar-on,eo-

heedalitycalion

ing

hatetheare

ynesl-th

a-Thed—

re-


tween the image spots and the surrounding regions is beFigure 21~a! has been drawn for zero difference betwephases of the two sources. In the case of nonzero differethe individual rays could look differently, but the concentrtion spots would stay at the same places so that the coence of two sources assumed above does not matter.

VIII. MULTIPLE-REFLECTED AND REFRACTED RAYS

To address the second important question posed abovconsider multiple reflections inside the cylinder. In geomecal optics such multiple reflections obviously lead to t~multiple! ray intersections. Now, our wave-optical rays canot cross. To solve this puzzle, let us consider a Gausbeam entering the cylindrical lens noncentrally. Figureshows the modulus of the Poynting vector~not yet the rays!inside and close to the lens. The shades of gray ranging fwhite to black are proportional to loguSu and cover the rangeof 40 dB of relative intensities to highlight the small intesities of multiple-reflected beams.

The presented exact solution shows in a natural waymultiple reflection and transmission~corresponding to Clas1 - Class 5 of rays in@20#!. There is no need for any extrassumptions or constructions. Parts of the reflected beamter two and three internal reflections are very importantthey contribute to the rainbow. Every reflection takes inaccount the curvature of the reflecting cylindrical surfacecluding the focusing properties~such as in a composite optical system!. However, Fig. 22 itself does not yet solve th‘‘intersection’’ puzzle.

Only a picture of wave-optical rays themselves can leto the solution. It is shown in Fig. 23, again for the Gaussibeam illumination entering the lens from the left-hand swith a nonzero impact parameter. In the entering bundlerays there is a splitting into sub-bundles corresponding

FIG. 22. Shaded contour plot of constant modulus of the Poting vector (log10uSu) ~light intensity shades; contours themselvare removed! for the noncentral Gaussian-beam illumination. Mutiple refraction and reflections are apparent. The position ofwaist of the beam~units of l) is x050, y0525, the width of thebeamw54, the radius of the cylindera/l530.

06661

er.

ce,-er-

we-

-an2

m

e

af-s

-

d-

efo

direct reflection, direct transmission, as well as to single adouble internal reflections. The picture is not complete, hoever, unless we draw the ‘‘circulating’’ rays that do not orignate in the entering bundle. The bundle of rays of Class 3~cf.@20#! consists of the circulating sub-bundle and a periphepart of the incident bundle. The fact that these two parts fiwell is truly astonishing. These two examples show that ein the situation in which we could expect the crossingtrajectories, they find their way to avoid any intersection.

IX. FINAL REMARKS

This discussion shows how the solutions of Maxwequations for various electromagnetic radiation beam prlems or external source problem and their scattering bdielectric cylindrical lens may illustrate all optical effecinvolving light propagation. It is possible to account for rflections, refractions, focusing and image formation, appeance of caustics and foci, waves’ interference and diffractietc. Usually such effects are treated in the approximate thries referring either to geometrical or diffractive optics. Tlong history of research in optics resulted in well developand understood accurate analytical methods and the quof optical instruments is the best proof of the value of optitheories. However, I do believe that the present discussusing explicit solutions for the fields has revealed interestproperties of classical electromagnetic radiation.

Taking these findings into account, we may conclude tthe concept of rays is still very well defined in the full wavtheory. What is more, these rays may be used to findimages of objects formed by optical devices. The images

-

e

FIG. 23. Wave-optical rays for the same illumination and prameters as in Fig. 22. It is clear that the rays do not intersect.rays corresponding to the double internal reflection are formeinside the lens—by the ‘‘circulating’’~periodic! rays, which are thusnecessary to obtain the full picture of scattering with multipleflections.

0-13

rgth-s

thi-

ldauthom

lfth

ewcer a

av

prghys

ting. In

be-ther-soto

eves ofandct,hes.

rhis-le


regions in space where the density of rays becomes laAlso, we argue that the ray concept is not less useful infull Maxwell theory than in the geometrical optics. The images of objects provided by optical devices can be rigorouobtained by tracing the rays being integral curves ofPoynting vector. It is to be noted that the tracing of ‘‘physcal’’ rays, that is, the integration of the Poynting vector fiemust be performed with extreme care. Indeed, the raysnot segments of straight lines, and they can even be qerratic. This is obviously connected with the properties ofphase of electromagnetic field that can be an extremely cplicated function of a position in space.

It is to be noted that the geometrical rays are sedetermined and can be plotted without any knowledge offields. In fact, they can be specified with the help of a fsimple rules and equations even for multicomponent sourOn the other hand, the wave rays discussed in this papenot self-defined and require first the solution for the field ingiven problem. Only after these fields are known the warays can be determined. Thus, the wave rays allow to resent and illustrate in full richness the wave properties of liand can be compared to the properties of geometrical ra

,

-

06661

e.e

lye

,reitee

-

-e

s.re

ee-t.

Concluding this discussion one can pose some interesand fundamental questions to be addressed in the futureparticular, when the rays are not straight lines but cancome quite complicated curves, what is the meaning of‘‘optical path’’ in wave optics? Moreover, how can we fomulate a wave-optical version of the Fermat principle,that the curves of energy flow would provide the solutiona corresponding variational problem? Besides, we belithat the above discussion can be useful for investigationmore complicated systems, as, e.g., a dielectric spherereal electromagnetic fields requiring 3D analysis. In faeven a dielectric cylinder would lead to 3D problems if tincident beams were finite in the two transverse direction

ACKNOWLEDGMENT

I would like to thank very much Dr Maciej Janowicz fohis contributions to the problems discussed above andhelp in writing this paper. In particular I would like to acknowledge his contribution in raising the question of the roof stationary points of the Poynting flow.

t-

.

c.

@1# M. Born and E. Wolf,Principles of Optics~Cambridge Uni-versity Press, Cambridge, 1998!.

@2# A. Sommerfeld,Optics ~Academic Press, New York, 1954!.@3# R. K. Luneburg,Mathematical Theory of Optics, Polish trans-

lation ~PWN-Polish Scientific, Warsaw, 1995!.@4# E. Hecht,Optics, 3rd ed.~Addison-Wesley, Amsterdam, 1998!.@5# J. F. Nye,Natural Focusing and Fine Structure of Light~Insti-

tute of Physics Publishing, Bristol, 1999!.@6# A. Sommerfeld and I. Runge, Ann. Phys.~Leipzig! 35, 277

~1911!.@7# E. Madelung, Z. Phys.40, 322 ~1926!.@8# L. Landau and E. Lifshitz,The Classical Theory of Fields

~PWN-Polish Scientific, Warsaw, 1958!.@9# B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A253,

358 ~1959!.@10# A. Boivin and E. Wolf, Phys. Rev.138, B1561~1965!.@11# A. Boivin, J. Dow, and E. Wolf, J. Opt. Soc. Am.57, 1171

~1967!.@12# D. Jackson,Classical Electrodynamics~PWN-Polish Scien-

tific, Warsaw, 1982!.@13# J. J. Stamnes,Waves in Focal Regions~Institute of Physics

Publishing, Bristol, 1986!.@14# T. Pearcey, Philos. Mag.37, 311 ~1947!.@15# M. V. Berry and C. Upstill, inProgress in Optics, edited by E.

Wolf, Vol. XVIII ~1980!, p. 259.@16# M. V. Berry, in Physics of Defects, Proceedings of the Les

Houches Summer School, Session 35, edited by R. BalianKleman, and J-P. Poirier~North Holland, Amsterdam, 1981!,pp. 454–543.

@17# J. F. Nye and M. V. Berry, Proc. R. Soc. London, Ser. B336,165 ~1974!.

@18# G. P. Karman, M. W. Beijersbergen, A. van Duijl, D. Bouw

L.

meester, and J. P. Woerdman, J. Opt. Soc. Am. A15, 884~1998!.

@19# G. P. Karman and J. P. Woerdman, J. Opt. Soc. Am. A15, 2862~1998!.

@20# H. M. Nussenzveig, Sci. Am.236, 116 ~1977!.@21# H. M. Nussenzveig,Diffraction Effects in Semiclassical Sca

tering ~Cambridge University Press, Cambridge, 1992!.@22# V. Khare and H. M. Nussenzveig, Phys. Rev. Lett.33, 973

~1974!.@23# H. M. Nussenzveig, J. Math. Phys.10, 82 ~1969!.@24# Lord Rayleigh, Philos. Mag.36, 365 ~1918!; 12, 81 ~1881!;

John William Strutt-Baron RayleighScientific Papers~Cam-bridge University Press, Cambridge, 1899!, Vol. 1, p. 533.

@25# D. S. Jones,Acoustic and Electromagnetic Waves~ClarendonPress, Oxford 1986!.

@26# S. Kozaki, J. Appl. Phys.53, 7195~1981!.@27# E. Zimmermann, R. Da¨ndliker, N. Souli, and B. Krattiger, J

Opt. Soc. Am. A12, 398 ~1995!.@28# J. P. Barton, J. Opt. Soc. Am. A16, 160 ~1999!.@29# W. Zakowicz and M. Janowicz, Phys. Rev. A62, 013820

~2000!.@30# W. Braunbek and G. Laukien, Optik~Stuttgart! 9, 174 ~1952!.@31# J. O. Hirschfelder and K. T. Tang, J. Chem. Phys.65, 470

~1976!.@32# J. F. Nye, J. Mod. Opt.38, 743 ~1991!.@33# D. E. Amos, amos.lib, http://www.netlib.org.@34# M. V. Berry, J. F. Nye, and F. J. Wright, Philos. Trans. R. So

London291, 453 ~1979!.@35# V. I. Arnold, Ordinary Differential Equations~PWN-Polish

Scientific, Warsaw, 1975!.@36# I. Freund and N. Shvartsman, Phys. Rev. A50, 5164~1994!.

0-14

light rays and imaging in wave optics - ifpan.edu.pl · light rays and imaging in wave optics...

Documents