novel ways for tracking rays

954 J. Opt. Soc. Am. A/Vol. 2, No. 6/June 1985 Leopold B. Felsen

Novel ways for tracking rays

Leopold B. Felsen

Department of Electrical Engineering and Computer Science/Microwave Research Institute, PolytechnicInstitute of New York, Farmingdale, New York 11735

Received October 8, 1984; accepted October 30, 1984

Rays no longer just represent the trajectories traversed by a beam of light but they describe the general transportproperties of high-frequency fields. Viewed within the framework of wave spectra, ray fields are localized spectralobjects synthesized by constructive spectral interference. The spectral flesh within these objects can be trimmedto the bare bones for the simplest type of ray field, a local plane wave, but must be augmented in ray transition re-gions near caustics, shadow boundaries, diffraction centers, etc., where more-intricate wave phenomena occur.Thus ray tracing along ray paths generally involves a succession of bare-bones and fleshed-out (uniformized) spec-tral entities that can furnish in a highly effective manner the amplitude and the phase of the high-frequency fieldin a complex environment. Relevant spectral concepts are discussed, including those for complex rays that de-scribe evanescent fields and Gaussian beams and those leading to collective treatment of multiple reflected or dif-fracted contributions.

A ray used to be just a ray,But who can define it today?

INTRODUCTION

With the evolution of the wave theory of light, rays were in-troduced as the geometrical trajectories along which lighttravels from one point to another. Light waves were consid-ered in the limit of zero wavelength, and the wave process aslocally plane, with the wave front perpendicular to the ray (atleast, in an isotropic medium). Rays are straight lines in ahomogeneous medium and smooth curves in an inhomo-geneous medium, and they change direction abruptly whenimpinging upon a boundary across which the optical proper-ties of the medium vary discontinuously. The resulting re-flected and refracted rays follow Snell's law. Optical lengthsalong rays are extrema, and, by calculating these lengths alongthe trajectories, one may construct the wave fronts corre-sponding to various ray congruences and thereby determinethe properties of an optical system. By invoking the law ofconservation of energy in a tube of rays, one may construct theamplitude of the local plane-wave field along a ray as long asthe energy density remains finite, i.e., the ray-tube cross sec-tion does not shrink to zero.'

The preceding scenario can be given a rigorous foundationby considering solutions of the full wave equation, which re-duces in the limit X - 0, where X is the wavelength, to the ei-konal and transport equations of geometrical optics. Failuresof this theory occur in caustic or focal regions of the geomet-rical ray system, where the ray-tube cross section becomesvanishingly small. Phenomena in these regions are wave-length dependent and are referred to as diffraction. Theyrequire a refined theory to account for this complication.Diffraction also plays a role when a smooth finite object castsa shadow because across the tangent rays, which define theshadow boundaries, the geometric optical field is discontin-uous, being equal to the incident field on the illuminated sideand equal to zero on the shadow side. In the absence ofstructural discontinuities across the shadow boundaries, such

behavior is inconsistent with the continuity requirements onthe full wave field. If structural discontinuities such as edgesor corners do exist, they create further complications since thescattered field caused by them cannot be predicted by thereflection and refraction laws.

Some of these shortcomings of geometrical ray theory canbe removed by enlarging the category of ray fields to includenot only the incident, reflected, and refracted rays but alsorays generated by scattering centers such as edges or corners(edge or corner diffracted rays), by critical refraction andshedding back into the host medium (lateral rays), and byglancing incidence on smoothly curved boundaries (creepingrays) (Fig. 1). In regular regions, away from scattering centersand critical boundaries, these diffracted ray fields behave likelocal plane waves and therefore obey the simple geometricalrules, but determination of the initial optical field values alongthese rays in terms of the incident ray field values requiresseparate consideration of an appropriate diffraction problem.This enlarged theory, which incorporates results from wave-length-dependent diffraction theory locally into the wave-length-independent geometrical ray theory, is called thegeometrical theory of diffraction (GTD).2 Although theamplitude ratio of ray fields in different categories (for ex-ample, the ratio of the incident to an edge-diffracted fieldamplitude) may be frequency dependent, the field in eachcategory behaves according to the wavelength-independentgeometrical rules. Thus, by the GTD, one may construct theoptical field in a rather general system involving reflection,refraction, and (or) diffraction by adding contributions fromall ray fields reaching the observer, the ray fields in each cat-egory being computed from wavelength-independent geo-metrical principles but with ray fields in different categoriespossibly weighted by a frequency-dependent factor. Theseproperties are consistent with the leading asymptotic be-havior, in the zero-wavelength limit, of the full wave equationsubject to initial and boundary conditions.

The GTD-constructed ray fields are valid as long as eachray field arrives at the observer as a distinct local plane wave.However, in critical or transition regions near caustics, foci,

0740-3232/85/060954-10$02.00 © 1985 Optical Society of America

Vol. 2, No. 6/June 1985/J. Opt. Soc. Am. A 955

causticOf

rays passing through a caustic

lateral rayn2 < n r

n 7, _ .creeping rays on a critically reflected rayconvex scatterer

Fig. 1. Various ray types in GTD. Shaded regions are ray transitionregions. Solid lines: bare-bones rays (local plane waves); hatchedlines: fleshed-out rays (enlarged spectral objects).

and shadow boundaries (Fig. 1), ray fields in different cate-gories interact so violently as to destroy the independent localplane-wave character of each. The field behavior in transitionregions is wavelength dependent, with a different dependencefor different transition phenomena. Smooth and continuousfield tracking across these regions requires solutions of thewave equation that retain more sophisticated wave objectsthan local plane waves but that reduce to the relevant localplane waves outside the transition domains. This process isoften referred to as uniformization, and the resulting theoryas uniform asymptotic ray theory. Uniformization can beeffected by frequency-dependent scaling of the coordinatesin the transition region (boundary-layer technique) andsolving the resulting reduced wave equation in the short-wavelength limit.3 It can also be done by fleshing out the localplane-wave spectral skeleton through spectral enhancementexpressed as a continuum of constructively interferingplane-wave fields.4 This spectral approach, which has afoundation of broad generality and physical content, shallconcern us here. The resulting spectral objects describe thetransport properties of short-wavelength fields localized tothe vicinity of the geometrical ray trajectories. These objectsmay therefore still be referred to as rays, but in a vastly gen-eralized sense. As will be seen, generalized rays are a far cryfrom the conventional geometrical rays that gave rise to them.The more fleshed out the spectral skeleton, the more gener-alized is the resulting ray field. In certain instances, fortransient wave fields, a generalized ray may describe correctlyeven the low-frequency spectral content at late observationtimes far behind the wave front, thereby significantly ex-tending the wave-front approximation that results from in-version of the high-frequency GTD ray field. 5

Generalized ray theory is one of the most effective methodsfor studying high-frequency propagation and scattering inenvironments that are complicated but not so chaotic as torequire statistical treatment. Although constructed around

the zero-wavelength limit, it can account for wave phenomenaon a scale in which the (X/d) ratio is an appreciable fractionof unity. The critical dimension d here denotes a length overwhich physical properties of a scatterer or environment changeappreciably. Because of the nonvanishing (A/d), generalizedray theory reveals fine structural detail of wavefields that islost in the (A/d) = 0 limit. In a variety of fields of applicationbased on microwave and millimeter-wave electromagnetics,6-8acoustic waves,9 elastic waves,10 and waves appearing in thegeophysical environment,"l' 3 the (Aid) scales are such as torequire the spectral ray refinements. In optics, the (X/d) ratioin traditional applications has been so small as to render dif-fractive corrections unimportant. Yet, with the advent ofcoherent sources of light, ever more sophisticated measure-ment techniques, and new device applications,1416 thefleshed-out spectral theory can be expected to play an in-creasingly important role. The discussion to follow is in-tended to provide some insight into the spectral maze and thevariety of ray objects that can emerge from it.

THE SPECTRAL MAP

Full and Contracted SpectraDeficiences of simple ray theory have traditionally been re-paired by seeking that spectral enhancement that addresseseach individual shortcoming. Referring to the spectralskeleton, this implies putting flesh only on the affected skel-etal regions while leaving the other bones bare. Uniformi-zation is an austere discipline. The trick is to use the leastamount of flesh that will repair the disease, in order not tocomplicate the modified ray objects beyond what is absolutelyrequired. As may be expected, the process need not be uniquebecause various spectral enhancements, while yielding thesame leading asymptotic behavior, may have differenthigher-order terms. Which enhancement is simplest and bestis often decided on the basis of analytical and numericalconvenience and by comparison with independently derivednumerical results for canonical prototype problems. More-over, since uniformization is phenomenon dependent (forexample, caustic regions behave differently from shadowboundaries), it is difficult to relate the spectral flesh for oneto that for another.

Therefore, to provide an overview, we shall take an oppositeapproach: to start from the fully fleshed spectral body andto explore what spectral objects are generated by selectivespectral contraction, i.e., by removal of flesh in varyingamounts from various regions. For a space-time-dependentfield G(r, t), r = (x, y, z), the full spectrum is synthesized byspectral objects U(r, t; k, w), where k = (kr, ky, k,) is thevector spatial spectral variable (the spatial vector wave num-ber, corresponding to r) and w is the temporal spectral variable(the temporal wave number, corresponding to t). The syn-thesis then involves a fourfold integration over (kr, ky, k,, co).In an unbounded homogeneous medium, the spectral objectsare expressed in plane-wave form as

U(r, t; k, co) = A(k, w) exp(i k r - it), (1)

where A is an amplitude function. Thus G is represented asa fourfold Fourier integral

G(r, t) = f dzo f dkU(r, t; k, c) (2)

I

/

edge diffraracted rays

conventionaln2 raysz7

n_

/ -shadow- boundary

_ _ sa

Leopold B. Felsen

/4,1-.

956 J. Opt. Soc. Am. A/Vol. 2, No. 6/June 1985 Leopold B. Felsen

z

(a) (b)

(c)

Fig. 2. Spectral decomposition into alternative harmonic or transientwave types. If the c integration is done after the spatial spectralintegrations, the wave types are harmonic. If the X integration is donebefore the spatial spectral integrations (possible for nondispersivephenomena), the wave types are transient. (a) Plane waves: (f d)f dk, f dky, k = k [k. k, ()]; (b) conical waves: ( d) dk,, kt= k[k, ()]; (c) spherical waves: ( d); k = k[(cv)].

wherein the spectral variables can generally assume complexvalues.

For impulsive excitation at t = 0 by a point source at r = 0,G satisfies a linear wave equation

l0 dyd tG 6 (r) 6(t), (3)ax ay az at where 0 is the wave operator. When Eq. (2) is substitutedinto Eq. (3), one finds that

U[k, cv]

By expressions (1), (2), and (4), the impulsive point source hasbeen smeared out into a continuum of harmonic plane-wavefields.

Contraction of the full spectrum in Eq. (2) can be achievedby various means. First, the dispersion equation

U[k, w] = 0 (5)

implies an interdependence of the spectral variables that canbe used to eliminate one of them by residue calculus appliedto the pole singularities at the zeros of When solutions ofEq. (5) are expressed as

X = c(k), (6)

one obtains the contracted spectrum (dk) in terms of a spatialplane-wave continuum with constrained temporal oscillations(resonances) (k), whereas by

k, = k(ky) kz ),ky = ky(k., k, c), k, = kz(kx, ky, c),

one obtains the contracted spectrum (dktdcv) in terms of aharmonic transverse plane-wave continuum with constrainedspatial oscillations (resonances) k (kt, c), etc., along a pre-ferred spatial coordinate z, etc. (see Fig. 2). Further con-tractions are possible by taking advantage of symmetries thatmay exist in the field. Thus, in an isotropic homogeneous

medium, by eliminating the (kr, k,) variables from the full(dcvdk) spectrum, for example, one expresses the impulsivepoint-source field as a contracted (dcdk,) continuum ofharmonic conical waves with axis along z. Each conical wavemay be regarded as being generated by superposition of atwo-dimensional continuum of plane waves. Finally, byelimination of (k) from the full (dcdk) spectrum, the impul-sive point source field is represented in terms of harmonicspherical waves. Each spherical wave is a superposition ofa three-dimensional continuum of plane waves.

When dispersion can be neglected so that cv(k) kc, wherec is the wave propagation speed, the co contraction can beperformed before contraction of the spatial constituents.This leads to a representation of the impulsive point-sourcefield as a spatial continuum of transient plane or conical wavesor as a single discrete spherical wave front.

Asymptotic Contractions

Bare-Bones Spectral SkeletonThe spectral contractions enumerated above have been per-formed globally; i.e., a two-parameter plane-wave spatialcontinuum has been converted into a one-parameter coni-cal-wave continuum, etc. However, at high frequencies, wavetransport properties become localized because of constructiveinterference of the spectral plane waves near the direction ofthe radius vector r linking source and observer. Thus theimportant spectral components are those for which the wavevector k is approximately parallel to r, and the resulting localplane-wave field may then be synthesized compactly from anyof the spectral decompositions in Fig. 2. The mathematicaljustification of this shrinkage is the reduction of the spectralintegrals in any of the formulations by the method of sta-tionary phase. 6 The resulting local plane-wave field, a singleharmonic spectral object, is the geometrical ray field17

O(r, t; k, co) A(r, k, c) exp (ik r - ict), (8)

where ks is the wave vector parallel to r and A is the spatiallydependent ray-field amplitude synthesized by constructiveinterference of the spectral plane-wave bundle surroundingthe stationary value k,. When dispersion is absent, thetime-dependent field is expressed as in expression (8) but interms of a local transient plane wave. Even for dispersivewave processes, the local transient plane-wave approximationis valid near the wave front, where the wave field has high-frequency spectral content and dispersive effects are usuallynegligible. Further behind the wave front, where the spectralcontent of the wave field encompasses lower frequencies, theresulting local plane-wave transient field propagates like awave packet.' 7

By the drastic shrinkage performed above, almost all thespectral flesh has been stripped away to reveal the bare skel-eton in expression (8) defined by the single spectral value k.When more-complicated spectral processes are operative,more flesh must be retained. The indication to that effect isusually an anomaly (for example, divergence) in the simpleray-field amplitude A. Retaining more spectral flesh may beregarded as a smoothing operation whereby the burden oftraversing a region that is critical for a strongly localized singlespectral object described by k is transferred to a bundle ofbetter-behaved (smeared-out) spectral objects in a spectral


interval surrounding k,. The mathematical machinery forsystematic determination of how much of the spectrum canbe removed is again the method of stationary phase. A

bare-bones spectrum is possible when stationary points,singularities, and end points of integration intervals (if any)in the spectral integrals are separated from one another. Each

of these critical points contributes a local plane-wave (simple

ray) field in a particular category that, in the corresponding

regular regions in physical space, adequately describes thefield structure. In transition regions in physical space, two

or more of the critical spectral points approach one another,and the retained spectral interval must now be large enoughto accommodate their simultaneous presence.18"19 Systematic

asymptotic procedures are available to reduce the spectralintegrands to the simplest possible (canonical) form thatachieves the desired uniformization for various arrangements

of critical points. 6 When the objects U in Eq. (2) are regarded

as defined in a multidimensional coordinate-wave-numberphase space, contraction is obtained by selective projectioninto submanifolds. The original highly mathematical ma-chinery attached to this viewpoint2 0 has subsequently beenadapted to uniformization of ray fields near caustics in ho-mogeneous and inhomogeneous media 21 22

Selectively Fleshed-Out Skeleton-UniformizationThe observations above have been predicated on the as-

sumption that the desired field can be represented by super-position of plane-wave spectra or of the more-compact spectral

objects sketched in Fig. 2. This is strictly true in unbounded

homogeneous media. Under more general conditions, in in-

homogeneous media, near diffraction centers, etc., the fleshing

out of the spectrum around its local plane-wave value k,proceeds by recourse to canonical analytically tractable

spectral solutions, which adequately model the essential local

properties in the vicinity of a ray. In these spectral integrals,

the true plane waves in Eq. (1) are typically replaced by WKB

plane waves, which propagate through the medium locally like

true plane waves but change direction because of refractive

inhomogeneities. These local WKB plane waves differ from

the single local plane wave in the ray field in that they are

global objects in the spectral space. They reduce to the ray-field local plane wave by the constructive interference dis-

cussed earlier. Spectral representations in terms of WKBplane waves differ from those in terms of true plane waves in

that the spectral wave numbers kh, k,, and k, are generallycoordinate dependent, and propagation functions such as k.xin an x -stratified medium, for example, are replaced by phase

integrals Xx kx (x')dx'. 2 3

The preceding discussion illustrates how the simple rayfields can be generated by the most drastic spectral contrac-tion and how fields valid more generally in critical regions can

be generated by retention of canonical spectral intervals thatare enlarged, but only to the minimal extent required to cope

with the particular transition or diffraction effects. Thiseffective localization of the spectrum around the central value

k, justifies the use of formally infinite intervals in the spectral

integration. Since the enlarged (uniformized) spectral objects

are designed to reduce to the simple ray fields outside thecritical regions, they can be regarded as rays in a generalized

sense. Accordingly, one speaks of generalized rays or of un-

iformized rays. As was noted earlier, generalized rays are not

unique since leading-order asymptotic equivalence can be

complex source point(compact)

complex rays

equiv. real source

(smeared out)

L real space trajectoriesof constant amplitudeevanescent waves

Fig. 3. Gaussian-beam generation by complex rays from a sourcepoint at a complex location (two dimensional). The (x, y) planeschematizes the physical real observation plane. Observable fieldsare generated at the real-space intersections of the complex rays.

achieved by different spectral objects. In what follows, weshall discuss various procedures that have been employed toachieve a desired fleshing out of the bare ray spectral skeleton.High-frequency field evaluation in rather general propagationand scattering environments is then accomplished by trackinga sequence of ordinary and generalized ray fields, with thelatter switched in when the former fail.

Beams and Evanescent Fields as Spectra of Complex RaysThe same spectral route can be followed when the elementary

spectral objects are not ordinary or WKB-type propagatingplane waves (with real phase) but evanescent plane waves(with complex phase). The latter are objects in a complexspectral space. They follow the rules of complex ray theory,which is an analytic extension of ordinary ray theory to acomplex-coordinate space.24 This extension includesboundary surfaces as well as the ray trajectories. Complexrays play an essential role in the description of Gaussianbeams (Fig. 3), of fields on the dark side of real ray caustics,

and of surface fields diffracted around a smooth scatteringobject.2 5 The bare-bones complex ray field can be fleshed out

in its critical regions by the same procedures as employed forreal rays, but here the spectral enhancement is unnecessarywhen the critical regions, as often occurs, are located in thecomplex-coordinate space, thereby making the real physicalspace behavior regular. Thus a Gaussian beam, even at itsnarrowest portion (the waist), can be represented by bare-bones complex rays.

Complex ray tracing of beam fields, even initially paraxial

beams, is preferable to tracing the corresponding real plane-wave spectra when the environment strongly perturbs theparaxial behavior (for example, scattering by an edge ortransmission through a thick, strongly curved lens) becauseplane-wave passage through a nonplanar environment andsubsequent spectral recombination are not easily accom-plished. Complex rays, on the other hand, are traced directlyby the rules of ordinary (or uniformized, if necessary) complexGTD. Paraxial approximations, when appropriate, can bederived systematically from the complex ray construction by

perturbation of the on-axis real ray field.

Collective Effects-Hybrid FormsWhen the propagation or scattering environment is such asto produce multiple ray species by repeated reflection among

Leopold B. Felsen

958 J. Opt. Soc. Am. A/Vol. 2, No. 6/June 1985

Fig. 4. Spectral reordering options in multiple-re-flection (guiding) environment. Filled circles: fqcrosses, Fp. The options provide for treating groups ofdiscrete fq elements (say, rays) collectively as groups ofdiscrete Fp elements (say, modes), and vice versa. Thereordering of the underlying continuous spectrum isaccomplished by the Poisson transformation. 2 7 A.Total conversion of fq into Fp. B. Partial conversion.A continuous-spectrum interval filled completely withfq elements is generally filled incompletely with Fp el-ements. The voids near the edges (shaded) are filledwith spectral remainders. This is the general form ofan fq - Fp equivalent. C. Hybrid. Alternate spectralintervals are filled with fq or Fp, plus remainders.

A. Total

Q PZfq= ZF pqCI- P.1

B. Partial

Q2 P2

Z fq= FpqaQ P.P

Spectral Reordering Options

e ae _ _q discrete

I x x x x x xXx xx x x xx

: I<QQ 2<Q,

(remainders)

I

continuous

p discrete

I<.P<P< P

... 0 020Q.ctral voids

XX XX X X X

pectral voids -

fq F equivalent-

C. Hybrid: G= fq+ X Fp +qSl P

remainders

R(QI) + R( PQ2)

Q3 P4+ f + F +....Q2 P3

* **0Q0 1 Q2 Q3 Q40 0 T 9009 90!

evanescentA leaky

5. o z p

P P21

0, 02,S*< n2

n, >n2

8 ' - _ _ _82

PoissMany rays / Few modes

Summation

Fig. 5. Collective summation of ray fields for propagation in a di-electric layer. Many multiple totally reflected and leaky rays fromthe source S to the observer P are summed collectively (by the Poissonsum formula; see Ref. 15) into trapped and leaky modes. Twotrapped modes with characteristic angles 0 and 02 are shown. Totallyreflected rays have real, and leaky rays have complex, lateral shiftson the boundaries. The same considerations apply to curved ormultiple layers.

several boundaries, repeated reflection along a single concaveboundary, repeated excursions because of confinement bymedium inhomogeneities as in a graded-index optical fiber,or repeated diffraction among scattering centers (for example,between edges), then direct summation of the ray fields maybecome impractical, and it is desirable to develop new spectralobjects, which account for these multiple effects collectively.Such collective regrouping can likewise be regarded as an al-ternative, more efficient contraction of the underlying con-tinuous spectrum (Fig. 4). For propagation in layered mediaor along concave boundaries, this rearrangement expressesinconvenient multiple ray species compactly as guided[trapped and (or) leaky] modes (Fig. 5). For propagationthrough or reflection from layered media, inconvenientmultiple internally reflected rays are clustered into a singlecollective ray field whose amplitude is weighted by a com-posite transmission or reflection coefficient (Fig. 6). These

xY x ,I P3

schemes work for general layer configurations, even those withcurved and nonparallel interfaces. In the latter case, for ex-ample, in a tapered dielectric waveguide, the multiple re-flected ray spectra are converted into local intrinsic modes,2 6

which adapt smoothly to the changing environment and canbe fleshed out to account continuously for the transition fromtrapped to radiating when propagating into a narrowing taper(Fig. 7). Phrased most generally as a bilateral ray-modeequivalent, a spectral interval contracted in terms of rays(modes) can alternatively be contracted in terms of modes(rays) plus a collective remainder:

Q2 rays P2 {modes\ ( rays 1E = E + collective)q=Q1 modes/ p=pi I rays [ modes

For propagation in the layered medium, where guided modes

p p2

/ / ~~~~~n2

n.

S

Fig. 6. Collective summation of ray fields for propagation througha dielectric layer. Source at S. For observer P, all internal multiplereflections are compacted into a single collective ray (double line),which follows the trajectory of the transmitted direct ray but with acollective transmission coefficient. For observer P2, the direct rayis retained intact, but all other internal reflections are compacted intoa collective ray following the trajectory of the first ray included in thisgroup. The same considerations apply to curved or multiplelayers.

-

, . ,ISA V>i -- 1 .. ., _

r ' ' ' r _ _ _

Leopold B. Felsen

I

n,

Leopold B. Felsen

modeJr nrnfila

evanescent

propa-gating

(a)transition

j :utZcutoff Ctf

local mode congruences mode profiles contour plot

(b)

Fig. 7. Guided propagation in plane-parallel and tapered dielectriclayers on a perfectly conducting ground plane. E polarization.Typical mode shapes and constant-amplitude contours are shown.Here, n1 and n2 are refractive indices, f and f, are the operating andlocal cutoff frequencies, respectively, and a is the taper angle. (a)Plane-parallel layer (conventional bare-bones trapped mode). Modalfield in the layer is synthesized by two self-consistent parallel raycongruences (one upgoing and one downgoing) inclined at the con-stant mode angle 0p. Total reflection at the upper boundary givesrise to evanescent waves (complex ray fields) outside. (b) Taperedlayer (fleshed-out intrinsic mode). Field in the layer synthesized bycontinuous superposition of self-consistent angular plane-wavespectra passes uniformly through cutoff from the trapped to the ra-diating regime.

are strongly excited, the collective remainders play a minorrole. For propagation through or reflection from layers, thecollective ray predominates because the guided modes areweakly excited. The complete field can then be expressed inhybrid form, in which successive spectral intervals are com-pacted either in terms of rays or in terms of modes, plus re-

mainders, with the arrangement chosen so as to retain the


most desirable groupings (see Fig. 4). The hybrid formulationhas been carried out for layer materials that support only asingle-wave species as well as those that, because of aniso-tropy, elasticity, etc., support multiple-wave species (for acomprehensive review, see Ref. 27). It should be emphasizedthat the (oscillatory) modal fields, while they are compactspectral objects, express global properties of the propagationprocess since they fill the entire waveguide cross section, incontrast with the (traveling-wave) ray fields, which samplethe environment locally along their trajectories. These con-siderations apply as well when the incident illumination is aGaussian beam or some other type of highly collimated illu-mination represented by complex rays.

Under transient conditions, the ray fields generate fieldsnear the wave front that, by spectral enhancement, can beextended to later observation times (see below). Successivewave-front arrivals that are due to multiple interaction cannow be reexpressed collectively as global (oscillatory) resonantmodes that encompass the propagation or scattering envi-ronment as a whole. A resulting hybrid representation, withwave fronts dominant at early times and resonances at latertimes, provides an effective format for analysis of transientpropagation and scattering.2 8 For the early time portion,where high frequencies predominate and dispersive effectsare therefore minimal, one has a further option based on thedecomposition of U into a nondispersive part Zi and a dis-persive remainder. 2 9- 3' By performing co inversion before k

inversion, one can get closed-form results for the contributionfrom U., which are valid beyond the wave-front approxima-tion of GTD. The strategy motivating this decomposition issummarized in Fig. 8. It should be noted here that dispersivespectral effects may arise not only from physical propertiesof the ambient medium (true dispersion) but also from thechoice of spectral representation selected for nonplanarboundaries, inhomogeneities, etc. in media that are physicallynondispersive (virtual dispersion).

true (medium losses, particle resonances, etc.)

Dispersion -- virtual (structural inhomogeneities, curved boundaries, etc.)

Spectral properties::WI t,'

High frequencies (weakly dispersive) -- G-G Am A) I

1: Observation times near wavefront arrivals Local pl(WKB)KLow frequencies (strongly dispersive)

Observation times far behind wavefront arrivals

lane wave

highly oscillatoryStrategy (num. inefficient)

fdwG d= dw Goo

rdkX fd= .- do in closed form

weakly oscillatory(num. efficient)

+ fdw(G-G)do numerically, if necessary

Fig. 8. Weakly dispersive decomposition.

For "resonant" structures (layers , finite targets, etc.):I~oalino.l

Mixed ("hybrid") representation - - G = [omne} E fw.vefrontsJ

effective atearly times

IGobal info. Rome

resonancesl

effective atlate times

I


Spectral Strategy

Because of the immense variety of ray spectral options (Fig.9) that are at the user's disposal, it is essential to devise astrategy that is well suited to the problem at hand. Thestrategy is influenced not only by the environmental condi-tions but also by the observables, which are of interest. If sidelobes in diffraction are important, the strategy differs fromthat confined to main beam effects; if fields transmittedthrough layers, but not the fields inside the layers, are re-quired, the latter can be treated by collective summation, etc.Compromises may also be struck among computability, pre-cision, numerical economy, and other factors that are peculiarto one organizational structure and not to another. The typeof computational facilities plays a major role in this regard.There is a delicate balance between using numerically efficientasymptotic approximations of restricted validity and brute-force numerical integration everywhere of the full spectralintegrals that generate them; by the latter option, one tradesoff the need for a decision-making process, which switches to

Leopold B. Felsen

the asymptotic form when applicable, against a less efficientbut decision-free routine. Thus, in a complicated environ-ment, the calculation of the high-frequency or transient fieldby the ray method involves a sequence of bare-bones ray fieldsbetween source and observer, fleshed out in transition regions,collectivized in multiple interaction regions, renormalized asordinary or intrinsic modes in guiding regions, and trackedalong complex trajectories when the incident illumination isfrom a collimated beam.

Numerous applications in the literature on electromag-netic, 3 2 acoustic, 33 seismic,34 and other types of fields 3 5 de-scribed by a linear wave equation attest to the effectivenessof the strategy outlined above. These applications includetime-harmonic and transient propagation in complicatedenvironments (with or without guiding properties), scatteringby complicated targets, radiation from large-aperture andarray antennas (including those with multiple reflectors), anda host of others, both for direct and inverse analysis and forwide-angle as well as narrow-angle (beam-type) excitation.

S1EClRHAL OPTONS

Space-Time Green's Function: G(r, t)

Spectral Domain,- k, t-W

1>

f d f dkia i; rd G ; f dk tiGp;I I _ I I

f dk. Gip

f dwfdk G(r,t;kt,w)

f dufdk tiCi;

f dk ti d . f d i;I

f dki d. r iI

L

…11l--- ----__ - - - _

I

Partial spectrum

Alternative Spectral Orderings constructive interference; spectral invariants|I

C o m plex _ _sin le__ _ _ _ su g r u pq a Ben m s Coul 1

' , \ ' X | _J I->J / \ j~~ Nrml] - 1 Uniform Modes __/ \ ' I ~~~~~~~~~(intrinsic) I

Parabolic Rasle Cupigx-

lPerfect (separable) waveguides. Imperfect (non-separable) waveguides.Fig. 9. Spectral map. The full spectrum at the top is contracted according to the various options schematized in Fig. 2, involving conical orspherical, time-harmonic or transient (Gp) constituents. Drastic contraction by constructive interference leads to single-species skeletal fields,such as ray fields, beam fields, and modal fields. Hybrid combinations result from alternate spectral groupings (see Fig. 4). A uniformized,better-fleshed skeleton can be structured either by less drastic spectral stripping originally or by spectral restoration. Since rays and modeshave different spectral flesh, replacement of transitional bare-bones rays by a cluster of modes, as in the hybrid scheme, is an option to achievethe required uniformization.

G (rt) =

I II

- - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I

Full s pe ctrum

I

IIII


.2 .

200

400

hlrnl

TIRANSMISSION LOSSMODE: 2

Soo PARAMETERS : a-1.55, f25 Hz

v* -1500 /sec

i v1.1704

.5 /sec

Source at x' -0, z' -180m; H(0) -200.

(a) (b)

Fig. 10. Second intrinsic mode for pressure field in a wedge-shaped ocean model with penetrable bottom (corresponds to E- polarized mode

in a grounded dielectric wedge with refractive index nl/n2 = vl/v). H(x) is the local thickness of the wedge, and x and z are range and depth

coordinates, respectively. Note that the transverse (depth) coordinate has been scaled by a factor of 103 with respect to the longitudinal (range)

coordinate to reveal details. (a) Decibel contour plot; after cutoff, mode energy is beamed into the bottom exterior. (b) Mode profile.

1.0

'M 2\ I 0o

:I \x

0.0 I I28.30 5a30

4,NEAR FIELD

5.

0.88.3'

I I M- .LMI.

40

410

FAR FIELD

Fig. 11. Near-field (p = 31Xo) and far-field (p = 104,O) patterns for electric line source at a complex location in the presence of a circular cylindrical

layer, plotted versus observation angle q5, for various source parameters and values of refractive index n1. The patterns are normalized with

respect to the maximum field, on the beam axis, in the absence of the layer. The origin of the (p, 0) coordinate system is at the center of the

layer. The inner radius of the layer is al = 2OXo. The outer radius a2 is adjusted so that the layer thickness equals one-half wavelength (in

medium ni) for a normally incident plane wave; i.e., the layer is matched for normal incidence along the beam axis. All length coordinates are

normalized with respect to the free-space wavelength (this is equivalent to setting Xo = 1). The complex source point is located at (x, 5') =

(Xr + ib sin ab, Yr + ib cos ab). The coordinates (xr, yr) define the center of the equivalent aperture in real space, b determines the effective

profile width, and ab is the angle of inclination of the beam axis with respect to the y axis. The sketch in the insert shows the equivalent aperture

plane and effective field distribution drawn to scale with respect to the inner-layer wall. The half-interior-wavelength distance to the outer

wall has been exaggerated in the sketch. ) (dotted curve): without layer; (l) (solid curve): reference solution. M = 0: direct ordinary ray.

M = 1: direct plus once internally reflected ordinary rays. M = 2: direct, once, and twice internally reflected ordinary rays. = 0: direct

collective ray. M = 1: direct ordinary plus once internally reflected collective rays. ni = I, a2 = 20.32XO, b = 10OXo, x' = 0 + ib/, 5'= lOXo + ibA/.

No attempt has been made here to cite all that is relevant. Itis hoped that a few examples below, admittedly biased by thewriter's own interest but with implications in optics, illustrate

how some of the concepts presented in this paper can be uti-lized.

First, we examine the ray method for propagation in, ortransmission through, a single dielectric layer. When source

and observer are located inside the layer, collective summation

converts the multiple ray fields into guided modes, as sche-matized in Fig. 5. When all the rays are treated in this man-ner, the field in the waveguide is expressed as a sum over allof the (trapped and leaky) modes. When only certain clusters

of rays are treated collectively, as in expression (9), the field

is expressed in hybrid form. The collective ray remaindersin expression (9) are insignificant here except when one of the

modes approaches the end points of a spectral i nterval occu-pied by rays (see Fig. 4). In a tapered layer [Fig. 7(b)], the

collective treatment of the ray fields yields local modes that,when the taper narrows, transform from the trapped into theradiating regime after passing through cutoff. Before cutoff,

the local modes are of the bare-bones type, but they may befleshed out into intrinsic modes to accommodate the cutofftransition. Mode shapes and contour plots for the lowest-order intrinsic mode in a grounded tapered layer are sketched

490

Leopold B. Felsen

45'


Dielectric waveguide

A B

Thick lensD

Aperture

Leopold B. Felsen

Curvedmirror

Opticalresonator

E F

Fig. 12. Ray analysis of optical system. Region A: ray congruencesfor lowest incident mode [see Fig. 7(a)]. Region B: intrinsic mode

- Gaussian profile [see Fig. 7(b)]. Model by complex source point(see Fig. 3). Region C: complex ray tracing. Include collectivecomplex rays to account for multiple internal reflections (see Fig. 6),if required. Region D: complex ray tracing through the aperture.Real ray tracing of edge-diffracted rays (see Fig. 1). Region E: realray tracing of multiple reflected edge-diffracted rays. Collectivetreatment of higher-order multiples (see Ref. 39).

in Fig. 7(b)36 and to scale for the second mode in an oceanacoustic model in Fig. 10.

When source and observer are on opposite sides of the layer,collective treatment of multiple internal reflections leads byexpression (9) to the collective rays schematized in Fig. 6. Thetrapped and leaky modes are insignificant here unless sourceand observer are quite near, but at locations widely separatedalong, the layer boundaries. Collective treatment in termsof a collective transmission coefficient works best when thetrajectories of the compacted multiples have nearly the samedirection (i.e., for internal incidence angles near vertical).This is the case for observer P in Fig. 6. For observer P 2, itis preferable to retain as many conventional rays individuallyas is necessary to validate the near-vertical incidence condi-tion; collective treatment is applied thereafter. In Fig. 6, itis assumed that retention of one conventional ray is sufficient.The validity and the utility of the collective-ray concept havebeen demonstrated in applications to plane-parallel, tapered,and curved layers, for excitation by a line source37 as well asby a Gaussian beam. The latter case is treated by the complexray method.3 8 A typical result from the complex ray tracingis shown in Fig. 11.

The collective ray concept has also been applied to deter-mining mode shapes and resonant frequencies in unstableoptical resonators. Here, diffracted rays from the mirroredges undergo reflections inside the resonator, which can becompacted for the higher-order multiples. The totality oflower-order reflected rays plus the collective ray is thensubjected to a self-consistency condition that synthesizes theresonances. 3 9

Complex rays have recently been used for an integratedoptical system in which a single-mode fiber excites a filmstructure that may contain a thick lens or a curved grating.The incident mode profile strongly resembles a Gaussianbeam, but conventional paraxial-beam optics is inapplicablehere because the beam is strongly divergent and the curvatureof the lens or grating contours is so large as to invalidate theparaxial assumption. Accordingly, the performance of thissystem is analyzed by complex ray tracing.40 Attention mayalso be called to complex ray tracing applied to beams incidentin graded-index layers,4 1 in optical fibers,42 and in circulardomains 4 3 useful for delay lines.

Putting together the various canonical portions, one maycontemplate ray tracing for determining the amplitude andthe phase of the optical field in a complicated optical systemsuch as the one schematized in Fig. 12.

CONCLUSIONS

High-frequency wave propagation from a source to an ob-server is localized around trajectories that connect these pointsaccording to the laws of geometrical optics as extended by thegeometrical theory of diffraction.4 4 Spectral concepts canserve to elucidate the wave processes that are taking place.45

Lowest-order ray theory is applicable for computation of thephase and the amplitude of the high-frequency field wheneverthe ray fields are distinct so that each can be approximatedby a local plane wave. This is representative of the strongest(bare-bones) contraction of the full wave spectrum. Morespectral flesh must be retained in transition regions where thewave field along a ray is distorted substantially away from itslocal plane-wave form. The variety of options for accom-plishing this uniformization, depending on such factors asaccuracy, computability, and numerical economy, is broughtto bear not only on new transitional effects but also on dif-ferent approaches to effects already understood. The ma-chinery assembled in this manner, consisting of bare-bonesand fleshed-out real and complex spectra tracked along raytrajectories individually, collectively, or by selective conver-sion into modal ray congruences, offers an effective means foranalyzing high-frequency propagation and diffraction incomplicated environments.

ACKNOWLEDGMENTS

This paper is based on research sponsored in part by the U.S.Office of Naval Research under contracts N-00014-79-C-0013and N-00014-83-K-0214, by the U.S. Army Reseach Officeunder contract DAAG 29-82-K-0097, and by the Joint ServicesElectronics Program under contract F-49620-82-C-0084.

REFERENCES

1. G. A. Deschamps, "Ray techniques in electromagnetics," Proc.IEEE 60, 1022-1035 (1972).

2. R. C. Hansen, ed., Geometric Theory of Diffraction, IEEE PressSelected Reprint Series (Institute of Electrical and ElectronicsEngineers, New York, 1981).

3. V. M. Babich and N. Y. Kirpichnikova, The Boundary LayerMethod in Diffraction Problems, Vol. 3 of Series in Electro-physics (Springer-Verlag, New York, 1979).

4. R. Mittra and Y. Rahmat-Samii, "A spectral domain analysis ofhigh frequency diffraction problems," in ElectromagneticScattering, P. L. Uslenghi, ed. (Academic, New York, 1978), pp.121-183.

5. L. B. Felsen, "Transient solutions for a class of diffractionproblems," Q. Appl. Math. 23, 151-169 (1965).

6. L. B. Felsen and N. Marcuvitz, Radiation and Scattering ofWaves (Prentice Hall, Englewood Cliffs, N. J., 1973), Chap. 4.

7. K. G. Budden, Radio Waves in the Ionosphere (Cambridge U.Press, Cambridge, 1961).

8. L. M. Brekhovskikh, Waves in Layered Media (Academic, NewYork, 1980).

9. J. B. Keller and J. S. Papadakis, ed., Wave Propagation andUnderwater Acoustics, (Springer-Verlag, New York, 1977).

10. B. A. Auld, Acoustic Waves in Solids (Wiley, New York,1973).

11. K. Aki and P. G. Richards, Quantitative Seismology (Freeman,San Francisco, 1980).

12. B. L. N. Kennett, Seismic Wave Propagation in Stratified Media(Cambridge U. Press, Cambridge, 1983).

13. V. Cerveny, I. A. Molotkov, and I. Psencik, Ray Method inSeismology (Charles U. Press, Prague, Czechoslovakia, 1977).

14. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic,New York, 1974).


15. J. A. Arnaud, Beam and Fiber Optics (Academic, New York,1976).

16. A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, New York, 1983).

17. Ref. 6, Secs. 1.6 and 1.7.18. J. M. Arnold, "Oscillatory integral theory for uniform represen-

tation of wave functions," Radio Sci. 17, 1181-1191 (1982).19. J. M. Arnold, "A global geometrical theory of diffraction," J. Phys.

A (to be published).20. V. P. Maslov, Perturbation Theory and Asymptotic Methods

(originally in Russian) (Dunod, Paris, 1972; in French).

21. R. W. Ziolkowski and G. A. Deschamps, "Asymptotic evaluation

of high-frequency fields near a caustic: an introduction toMaslov's method," Radio Sci. 19, 1001-1025 (1984).

22. C. H. Chapman and R. Drummond, "Body wave seismograms in

inhomogeneous media using Maslov asymptotic theory," Bull.

Seismol. Soc. Am. 72, 277-317 (1982).23. L. B. Felsen, "Hybrid ray-mode fields in inhomogeneous wave-

guides and ducts," J. Acoust. Soc. Am. 69, 352-361 (1981).

24. P. D. Einziger and L. B. Felsen, "Evanescent waves and complexrays," IEEE Trans. Antennas Propag. AP-30, 594-605 (1982).

25. E. Heyman and L. B. Felsen, "Evanescent waves and complex rays

for modal propagation in curved open waveguides," SIAM J.Appl. Math. 43, 855-884 (1983).

26. J. M. Arnold and L. B. Felsen, "Intrinsic modes in a nonseparable

ocean waveguide," J. Acoust. Soc. Am. 76, 850-860 (1984).

27. L. B. Felsen, "Progressing and oscillatory waves for hybrid syn-

thesis of source excited propagation and diffraction," IEEETrans. Antennas Propag. AP-32, 775-796 (1984).

28. E. Heyman and L. B. Felsen, "Traveling wave and oscillatoryformulations of scattering problems," in Handbook on AcousticElectromagnetic and Elastic Wave Scattering-Theory andExperiment (Acadamic, New York, to be published).

29. A. Ezzeddine and J. A. Kong, "Time response of a vertical electricdipole over a two-layer medium by the double deformationtechnique," J. Appl. Phys. 53, 813-822 (1982).

30. E. F. Kuester and A. G. Tijhuis, "Two-dimensional transientscattering of an arbitrary electromagnetic field by a stratified

dielectric and conducting region," presented at the InternationalUnion of Radio Science Symposium 1983, Santiago de Compos-tela, Spain, August 1983.

31. E. Heyman and L. B. Felsen, "Nondispersive approximations for

transient ray fields in an inhomogeneous medium," in Hybrid

Formulation of Wave Propagation and Scattering, L. B. Felsen,

ed. (Nijhoff, The Hague, 1984); "Nondispersive closed form ap-

proximations for transient propagation and scattering of rayfields," Wave Motion (to be published).

32. Such articles appear for example, in IEEE Transactions on An-tennas and Propagation, Radio Science, Electromagnetics, andRadio Engineering and Electronic Physics (Radiotekhnika iElektronika).

33. Such articles appear for example, in Journal of the AcousticalSociety of America and Soviet Physics-Acoustics (Akusti-cheskii Zhurnal).

34. Such articles appear, for example, in Bulletin of the Seismological

Society of America and Geophysical Journal of the Royal As-tronomical Society.

35. Such articles appear, for example, in Wave Motion, Journal ofApplied Physics, Journal of Mathematical Physics, and Journalof Applied Mechanics.

36. L. B. Felsen, "Numerically efficient spectral representations for

guided ocean acoustics," Comput. Math. Appl. (to be pub-lished).

37. P. D. Einziger and L. B. Felsen, "Ray analysis of two-dimensionalradomes," IEEE Trans. Antennas Propag. AP-31, 870-884(1983).

38. X. J. Gao and L. B. Felsen, "Complex ray analysis of beamtransmission through two-dimensional radomes," IEEE Trans.Antennas Propag. (to be published).

39. S. H. Cho and L. B. Felsen, "Ray method for unstable resonan-tors," Appl. Opt. 19, 3506-3517 (1980).

40. J. Jacob and H. G. Unger, "Gaussian beam transformationthrough lenses and by curved gratings," in Hybrid Formulationof Wave Propagation and Scattering, L. B. Felsen, ed. (Nijhoff,The Hague, 1984.

41. L. B. Felsen and S. Y. Shin, "Rays, beams and modes pertainingto the excitation of dielectric waveguides," IEEE Trans. Micro-wave Theory Tech. MIT-23, 150-161 (1975).

42. S. Y. Shin and L. B. Felsen, "Beam evolution along a multimodeoptical fiber," presented at the AGARD Symposium on OpticalFibers, Integrated Optics and Their Military Applications,London, May 15-21, 1977.

43. S. Y. Shin and L. B. Felsen, "Multiply reflected Gaussian beamsin a circular cross section," IEEE Trans. Microwave Theory Tech.MTT-26, 845-851 (1978).

44. S. Cornbleet, "Geometrical optics reviewed," Proc. IEEE 71,471-502 (1983).

45. L. B. Felsen, ed., Hybrid Formulation of Wave Propagation andScattering (Nijhoff, The Hague, 1984).

Leopold B. FelsenLeopold B. Felsen received the Ph.D.degree in 1952 from the Polytechnic In-

stitute of Brooklyn (now the PolytechnicInstitute of New York), where he has

been Professor of Physics and Dean ofEngineering and is now Institute Profes-

S / sor. His interests are in wave propaga-tion and diffraction in various physicaldisciplines with emphasis on high-fre-quency phenomena. He is the author or

editor of several textbooks and has morethan 190 technical publications. He isa member of the National Academy of

Engineering, has an honorary doctorate from the Technical University

of Denmark, and received the Balthasar van der Pol Gold Medal of

the International Union of Radio Science. He has been a Guggen-

heim Fellow and received the Humboldt Foundation Senior Scientist

Award. Dr. Felsen is a Fellow of the Optical Society of America and

of the Institute of Electrical and Electronics Engineers.

Leopold B. Felsen

novel ways for tracking rays

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