leaky waves in planar optical waveguides (*i

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Leaky waves in planar optical waveguides

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1975 Nouvelle Revue d'Optique 6 273

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Nouv. Rev. Optique, 1975, t. 6,no 5, pp. 213-284

DEPARWYTOF ELECTRICALNGINEERINGND ELECTROPHYSICSPOLYTECHNIC INSTITUTE OF NEWYORK

333 J ay St.,Brooklyn, N .Y . 11201 fU.S.A.)

LEAKY WAVESIN PLANAR OPTICAL WAVEGUIDES (*Ibv

T. T A M I RI K EYWORDSLeaky wavesIntegrated optics

M OT S CLES :Ondes de fuiteOptique integree

Ondes de fuite danslesguides optiques plans.RkSUME :On presente les proprietts de base des ondesde fuiteainsi que leurs applications, presents et futures, en optique inte-gree. Tout dabord on montre, quen plus des ondes de surface,l es champs excitts par des sources reelles disposees le long dunecouche mince comportent aussi ces ondes de fuite. Les champspeuvent se decrire en termes dondes planes inhomogenes qui sereflechissent sur l es faces de la couche mince en rayonnant delenergie vers l es mlieux exterieurs. On etudie ensuite les champsdondes de fuite gentrb par des systemes multicouches ou grio-diques et on discute leur r61e dans l es dispositifs de couplage. Pourterminer on presente certninc dispositifs a coefficient de fuite

variable permettant d Obtenlr un grand nombre de rranstormationsfaisceaux.SUMMARY :The basic properties, applications and future poten-tial uses of leaky waves in integrated optics are presented. I t is firstnoted that, in addition to surface waves, the fields excited by realisticsources along thin films also include leaky waves. These fields canbedescribed by inhomogeneous plane waves that bounce betweenthe film boundaries and radiate energy into the exterior open regions.Leaky wave fields on multilayered and periodic media are examinedand their role in beam and waveguide couplers is discussed. Finally,the construction of structures having variable leakage is describedand their capabilities in providing a wide range of beam transfor-mations are outlined.

1.- NTRODUCTION so rapidly that it may rival their use in microwaveThe subject of waves guided by open conhgurationswas studied in acoustics [l] and electromagnetics [2]already towards the end of the last century. Thesewaves played a most important role in microwavetechnology, which employed extensively waves guidedeither by closed structures (e.g., modes propagatinginside metallic waveguides) or by open configurations(e.g., surface and leaky waves on long antennas [3]).In optics, the utilization of these waves came relativelylater, with surface waves having been consideredfirst along fibers [4,51 and then in planar films [6, 71 ;leaky waves were thereafter examined in the contextof beam-to-surface-wave couplers [8-111, in lateral.beam shifts [lo, 111 and fn other related opticalprocesses [12] which occur in the presence of thindielectric films, as well as in fibers having circularcross-section [131. Currently, both surface and leakywaves fulfil important functions in the area of inte-grated optics [141, where their application increases

technology.Because of their earlier utilization and broader appli -cation, surface waves are familiar to physicists andengineers specializing in optics. By now, discussionsof their properties can befound in many recent articlesand text-books, of which references [14-17] representonly a few of the latest publications on this subject.In contrast, leaky waves are less well understood,because their theory is more complicated and theirapplication is more restricted than that of surfacewaves. The aim of this paper is therefore to review thebasic theory of leaky waves, to present their mostsignificant applications in the area of integratedoptics and to discuss some of their features whichcan possibly lead to future developments. Readersinterested in additional material concerning leakywaves should consult references [I 1-19] for furtherdetails.A lthough leaky waves have been investigated alsoin circular geometries 1131, we shall discuss here leakywaves in two-dimensickd planar configurations only,because the latter are conceptually simpler. Theextension to three-dimensiona1 non-planar situationscan thereafter be carried out: such a generalization

(*) I nvited paper presented at the Colloquium on (( the Optic ofGuided WavesD (Paris, April 8-11,1975) under the title(( leaky-wavefied configurations in optical waveguidesD


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274 T. TAMIR Nouv. Rev. Optique, 1975, t. 6, no 5involves greater mathematical complexities but nonew physical aspects. We start with some basicconsiderations in Section 2 and mention that leakywaves appear as complex solutions of boundary-value problems involving sources. We then constructleaky wave fields in Section 3 by using simple argu-ments involving inhomogeneous plane waves. The pro-perties of these fields are discussed in Section4 whichconsiders both multi-layered and periodic configu-rations. Practical applications of leaky waves arepresented in Section 5 , where beamand waveguidecouplers are discussed. The lateral displacement ofbeams incident on leaky wave structures is nexttreated in Section 6, which uses simple power f l uxarguments to explain a rather complex phenomenon.Finally, Section 7 discusses leaky wave structureshaving variable characteristics and speculates on thefuture potentialities of these structures for applicationto integrated-optics devices.2. -FIELDS IN PLANAR STRUCTURES

To put leaky waves into proper perspective, weconsider first the basic electromagnetic problem whichis posed when solving for the optical field in planarconfigurations. Such configurations usually consist ofa stratified multi-layered structure having an upperplanar boundary exposed to a semi-infinite medium,which is usually air. As shown in figure 1, we take thisupper boundary so as to coincide with the xy plane.To have an optical field, it is necessary that a sourcebe present ;this may be either a laser beam, anapertureillumination or any other form of radiant energy,which is described schematically by the three arrowsin figure 1.2.1. The fied excited by sources

In the presence of a given monochromatic sourcehaving a time variation exp(- io?) implied butsuppressed, the optical field excited by the sourcewill be found by solving the inhomogeneous Helmholtzequation(1) ( V2+k2)F = S ,whereF is any scalar componentof the electromagbeticfield, k = 2 n/A is the propagation factor in freespace and s = s(x, z) is the source distribution,which is assumed known; to restrict the discussionto two-dimensional fields, we also assume thats(x, z) is independent ofy, i.e., a/ay =0. The solutionforF = F(x, z) must, of course, satisfy field continuityconditions at the boundaries, which may includeboundaries additional to that shown at z =0 infigure 1. A radiation condition stipulating thatpower is outgoing at infinity is also needed.

iPlanarboundary

Source\!' V7 / / / / / / / / / / / /? / / I / / /Y.

FI G. .- eometry for planar Configurations.

In general, the solution of F for given s is noteasy to find. One therefore often seeks, at least asa first step, solutions of F in the absence of sources,i.e., s =0, in whioh case Eq. (1) reduces to thehomogeneous Helmholtz equation(2) ( Vz+k2)F =0,subject to the same boundary conditions as before.In this case, the solutions are in the form of guidedwaves which can be obtained by solving a transverse-resonance relation [laor by using other well knownmethods. However, these guided waves cannot actuallyappear unless a sources is present.Phrased differently, the foregoing statements implythat guided waves, as obtained by considering Eq. (2)only, represent fields which can potentially existprovided that a suitable source is brought in so asto excite such fields. To better understand what fieldsactually appear along a planar structure, it is thereforenecessary to examine the excitation problem, viz.the solution of F in Eq. (l), with s # 0, as discussedbelow.2.2. The longitudinal representationof the fields

Because the geometry shown in figure 1 is invariantwith respect to both x and y , this solution can bewritten, for all points z >0, in the formm

(3) F = J -mf(k,) exp[i(k, x +k, 41dk, 9which is basically a Fourier transform for F withrespect to k, and(4) k; +k,Z =k2,where k, is selectedso that it is positive real or positiveimaginary for k real, i.e., only lossless media areexamined. The extension to lossy media is straight-forward [19] but is not considered here.It is evident thatF n Eq. (3) consists of a continuousinfinite spectrum of waves having an amplitudedensity f(k,) exp(ik,x) and varying as exp(ik,z)along z, i.e., they progress with propagation factor k,and travel in a direction normal to the boundaryz = 0. Because this spectrum of waves is given bythe continuous real variable k,, which is itself asso-ciated with the longitudinal coordinate x, the represen-tation of F in Eq. (4) is often referred to as alongitudinal representation [19, 201. The disadvantageof this representation is that, although relativelysimple, it does not show explicitly any surface-wavefields or any other guided fidds.2.3. The transverse representationof fields

To explicitly obtain the surface waves excited byactual sources, it is necessary to modify the expressionfor F in Eq. (3). By using a contour deformation [19],the integral along the real axisinthe complex k, planecan be transformed along a path that yields an


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Nouv. Rev. Optique, 1975, t. 6, no 5 T. TAMIR 275integration along all real values of k, (instead ofreal k,). In the process, a number of pole singularitiesare captured between the original (real kJ and thedeformed (real k,) integration contours, thus obtaining

J - m

where

Now, F in Eq. (5) consists of two terms, of whichthe first represents a continuous spectrum of waveshaving an amplitude density g(z, k,) and varyingas exp(ik, x) along x, i.e., these waves progress withpropagation factor k, and travel parallel to theboundary z = 0. The second term of F in Eq. (5)stands for the summation of residues around poleslocated at k,, in the complex k, plane (or at k,,in the complex k, plane). Each of the term in thesummation represents a guided wave, usually asurface wave for which k , is positive imaginary andk , is real, so that propagation is again along x.BecauseF in Eq. ( 5 ) is given in terms of a spectrumwith respect to k,, which is associated with the trans-verse variable z, this representation is referred to asa transverse representation [19, 201.The advantage of the transverse representation isthat it exhibits surface waves explicitly in the secondterm of Eq. (5). These waves form a discrete ratherthan continuous spectrum. As f(k,) is known andthe value of k , has presumably been obtained bysolving the source-free problem of Eq. (2), the surfacewaves in the presence of the given source are readilyfound. In most practiced cases, only a few such waves(often, only one) are significant. If, in addition, thefirst (continuous spectrum) term of F is known to berelatively small, the excitation problem is then easilysolved by neglecting all field contributions except forthe few significant surface-waveterms.2.4. Theangular representationof field

The foregoing transverse representation is veryuseful in those cases where the sources excite primarilysurface waves. We recall that these surface waves arecharacterized by real values of k, =k, and bypositive imaginary values of k, =k,,, because the)propagate as exp(ik, x) parallel to the boundaryand they decay as exp(- k, I z) in a direction normalto that boundary. Thus, surface waves account forenergy flow along a polar-angle direction given by8= k 900 in figure 1. However, in certain circum-stances, a considerable amont of optical energy maytravel along an angular direction 8which is differentfrom 900. In that case, the transverse representationis no longer too helpful because a large energy would

be accounted for in the first (continuous spectrum)term of Eq. ( 5 ) , in which case this term can no longerbe neglected.To deal with such a possibility, it is then convenientto transform Eq. (3) by using a contour deformationwhich is different from thatused in arriving at Eq. (5 ) .This different deformation is often obtained byusing a steepest-descent path, as discussed in refe-rences [19-221. For the present purpose, we shallobserve only that, as a result of .such a deformation,the field F can be written as

+2 xi C f(kxn) exp[i(km x +kzn z)] 9n

whereC is a suitable contour in the complexw plane,with w being related to k, by an angular transforma-tion k, = k sinw .As obtained in Eq. (7), F has a form similar tothat in Eq. (5) because it again includes a continuousspectrum and a discrete spectrum, the latter beingthe result of capturing poles between the originalintegration path along the real k, axis and thedeformed contour C. However, there is a significantdifference in'that now the discrete spectrum containsnot only surface-wave fields (with k, real), but alsoleaky wave fields (with k,, complex). The advantageof F in Eq. (7) over F in Eq. ( 5 ) is that the formerexpression may exhibit a continuous spectrum whichis small even though the complete field contains aconsiderable amount of energy flow at an angle8# 900. Thus, this energy flow has been relocatedinto the second (discrete-spectrum) term of Eq.' (7)and is accounted by one or more complex values ofk, in the summation.Obviously, Eq. (3), (5) and (7) are all alternativerepresentations for the same quantity F. Dependingon the specific problem at hand, one of these represen-tations is more suitable than the others because, byusing discrete terms in a rapidly converging summa-tion, it puts into evidence that portion of the fieldthat may have a greater physical significance. Weshould note that the third representation of F inEq. (7), which may be referred to as the angularrepresentation, isthe most general because it includesthe longitudinal representation (for 8=0) and thetransverse representation (for 8= 900) as specialcases. However, as the angular representation involvescomplex values of k, = k,,, it is pertinent to discussthe properties of fields characterized by such complexpropagation factors.3.- ORMATION OF LEAKY-WAVE FIELDS

Although leaky waves, like surface waves, can beviewed as modes, i.e., resonant solutions of boundary-value problems [17-191, we prefer to present here amore intuitive description in terms of inhomogeneousplane waves. These plane waves are therefore discussedfirst and they are then used to construct leaky wavesby using simple superposition arguments.


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276 T. TAMIR3.1. Inhomogeneous planewaves

In an unbounded medium, solutions of thehomogeneous Helmholtz equation (2) take the form

Nouv. Rev. Optique, 1975, t. 6,no 5

(8) F , = C exp[i(k, x +k, z) ] ,where C is a constant, k, and k, are related by thedispersion formula (4) and the subscript p in F ,signifies that we are now dealing with a particularsolution of Eq. (2) rather than with a completesolution as was the case for F in Eq. (3). Hencenow k, and k, take on special values and are no longercontinuous variables.For real k, and k,, we recall that F , in Eq. (8)represents a homogeneous plane wave. However,neither k, nor k, need be real to satisfy Eq. (2);they must only be related by Eq. (4). If we let theseparameters be complex, we can write(9) k, = f l + ia,(10) k, =b+ i a.We then find [18] that F, still describes a plane wave,which is, however, no longer homogeneous. Asshown in figure2, this wave now has plane equiphase

L

FIG. .- ield of a plane inhomogeneous wave. Thesolid lines denoteequiampli tude contours; he density of these lines suggests the magni-tude of the,field ntensity. The dashed line indicate equi-phase contours,whereas the arrows denote the direction of power,f low.and equi-amplitude contours that are orthogonal toeach other. The power flux progresses parallel tothe amplitude contour at an angle Bo = tan-' (fllb)and the amplitude decays along the equi-phasecontours. We recall that this decay is not due toabsorption losses. In fact, for a =b =0, we get thepicture of a surface-wave field which is a special caseof an inhomogeneous plane wave.3.2. Reflectionand refractionof plane inhomogeneous waves

If a plane inhomogeneous wave is incident atthe interface between two media with different refrac-tion indices n, and n, , it undergoes reflection andrefraction in a manner similar to that of ahomogeneous plane wave. This is indicated in figure3wherein a wave is incident from below. Because

FIG.3.- eyection and refraction of a plane inhomogeneous waveat the interface between two different media.power transmission at the interface is predicatedsolely by the field continuity conditions at thez =0 boundary, reflection and refraction are deter-mined by the same Fresnel coefficient as in the caseof homogeneous plane waves, except that now k,andk, assume complex values.In particular, the reflection coefficient for per-pendicular polarization is given by(1 1)

k,, - k,,r(kx) =k,, +k,, b, - b, + i(a, - a,)b, +b, + i(a, +a,) '-where the subscripts 1 and 2 in the above quantitiesrefer to the two different media. Because now all ofthe parameters a,, b,, a, and b, are non-zero, it iseasily seen that I r(k,) I is always smaller than unity,so that no total reflection can occur for 8, # 0.Thus, unlike homogeneous waves, some energy mustalways be transmitted into the second medium inthe case of inhomogeneous plane waves.3.3. Guidanceof an inhomogeneous waveby a layer

Consider next an inhomogeneous plane wave whichis incident inside a dielectric layer of thickness t andrefractive index n, as shown in figure4. For simpficity,we assume that the lower boundary is a perfect mirrorwith reflection coefficient y o = & 1. Also, we mayassume that the reflection coefficient r1at the upperboundary is sufficiently large so that the wave can

FIG. .- nhomogeneous fielak guided by a planar layer. The fi eldrefracted into the upper (air) region constitutes a leaky wave whichradiates at the leaky-wave angle Bo.


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Nouv. Rev. Optique, 1975, t. 6,no 5 T. TAMIR 277bounce many times between the two boundariesbefore it loses most of its energy by refraction.Under the above conditions, the field inside thelayer will progress for a considerable distance along xbefore its magnitude decreases significantly. In parti-cular, we recall that, for surface waves, a self-consistentfield is established by means of a homogeneous planewave bouncing between the two boundaries of thelayer if we satisfy the transverse-resonance condition(12) ro r , exp(2 ik,, t,) = 1 ,where k,, denotes the phase-shift factor along zinside the layer. This self-consistency relation holdsalso for inhomogeneous plane waves provided welet the quantities r,, r, and k,, become complex.In fact, as for surface waves, the transverse-resonancecondition (12) yields those values of k,,.that produceconstructive interference between the various bouncingfields, thus setting up a mode which is guided along thelayer.Consequently, the same zig-zag mechanism of abouncing field, which is used to describe a surfacewave, can also explain the more complex guided wavedescribed schematically in figure 4. Because now thezig-zag travelling wave is an inhomogeneous planewave, the present field differs from that of a surfacewave in two important respects :

a) The field in the exterior (air) region travelsalong an oblique direction given by Bo , in contrastto a surface wave for which the flux travels parallelto the boundary. This happens because refraction isalways non-zero in the case of inhomogeneous waves.b) Due to the power flow at the angle Bo, energyleaks continuously out of the layer region and thereforethe total field must decay as it propagates along x.This decay is consistent with the fact that the pro-pagation factor k, was assumed complex in Eq. (9),with a denoting the attenuation due to energy leakage.Because complex fields guided by layers alwaysaccount for leakage out of the layer, such fields havebeen labelled (( leaky wavesD in the literature.Obviously the energy leaked out represents radiationwhich appears in the far field as a beam oriented atan angle 8, with respect to the planar structure.For surface waves, i .e., for real values of k,, onlya few solutions of the transverse-resonance rela-tion (12) exist. In contrast, an infinite set of solutionsof Eq. (12) exist for leaky waves, i.e., complex valuesof k, = p + ia. However, only a few of these satisfythe strong guidance condition

(13) a < p ,which implies that leakage per wavelength is verysmall, so that the wave extends over a long distancealong the layer. If a is comparable to /3 or larger, theenergy leaks out fast and the wave is only weaklyguided. For practical purposes, only strongly guidedleaky waves satisfying condition (13) are important.We shall therefore assume henceforth that only thiskind of leaky waves is being considered.

4. - ARIETIES OF LEAKY-WAVE FIELDSIn microwave structures, a wide range of configu-rations are known [3] to support leaky-waves. Inoptics, only two categories of such structures havebeen studied : multi-layered media and periodic(grating) configurations. These categories are discussed

below.4.1. M ulti-layered media

In general, any structure that guides surface wavescan also support leaky-waves. It can thus be shownthat leaky-waves exist on a single dielectric layerplaced in air [3], as well as on a thin film depositedon a substrate [12, 181. However, these leaky-wavesdo not usually satisfy the slow leakage condition (13)and therefore they are not very important. On theother hand, if the layer is a metal so that its permittivityat optical frequencies is well described by a plasmamedium, strongly guided leaky-waves may occur forTM modes [23, 241, which have found applications instudies of surface properties.

FIG 5.- eaky wave supported by a four-media layered structure.

In recent integrated-optics work, the largest use ofleaky-waves has been in the context of four-mediaconfigurations, as shown in figure 5. In the absence ofthe top layer with refractive index np,this configurationcan guide a surface wave along a thin film withrefractive index nf deposited on a substrate havingrefractive index n,, provided nf >n, >n, [14-171;the medium above the film is usually air, so thatn, = 1. If the top layer is brought in and its refractiveindex satisfies np>np the surface wave is modifiedinto a leaky-wave, as indicated in figure 5. By takinga coupled-mode approach, the wave leaked into thetop denser medium is often explained [8-10, 141 interms of energy crossing through the evanescent fieldin the air gap' by means of a tunnel effect.

A characteristic feature of leaky-waves in multi-layered dielectric media is that the horizontal compo-nent of the radiation field in the open (exterior)regions is oriented along the same direction as thatof the energy flow inside the principal (G surface-wave ))) ayer. For the case shown in figure 5, both ofthese components flow along the +x direction.Hence the field in this case is referred to as a forwardleaky-wave. Exceptions to this may occur only if one(or more) of the dielectric media is replaced by ametal [23], but we shall not discuss such cases here.4.2. Dielectric gratings

Leaky-waves along periodic structures were firstdiscussed by Hessel and Oliner [25] in the context ofWood's anomalies on metallic diffraction gratings.


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278 T. TAMIR Nouv. Rev. Optique, 1975, t. 6,no 5Most recently, they have been investigated in studiesof dielectric gratings for integrated-optics applica-tions [I 1, 261. Although the characteristics presentedhere also apply to metallic, gratings, we shall restrictour discussion to leaky-wavesin dielectric gratings.

- XX S I / i 1

F I G . 6.- eaky waves supported by dielectric gratings : a) Singleleakage with backward characteristics ; b) Multiple leakage withbackward and forward beams.Unlike the situation in uniform layers, a wavebouncing inside a dielectric grating does not encountera simple planar boundary on both sides. Thus, asshown in figure 6, the upper boundary is non-uniformand characterized by a periodicity d. Because of thisperiodicity, energy is scattered into the upper andlower media. If a surface wave propagating asexp(i&,x) can exist when the grating is absent,this wave will be modified by the grating to a fieldhaving the Floquet form

WF , = a,@) exp(ik,, ,, =- U3

(14)where 2VR(15) k,, = B , + iu. = B o +7 +icr !

v =0, & 1, & 2, ...Obviously, here 8 refers to the propagation factoralong x of the v-th space harmonic. When the gratingacts as a small perturbation, Bo is close to B, of thesurface wave.In the two open (air and substrate) regions, propa-gation along z is given by

2 112(16) k,, = (kj2 - k,J 3where kj denotes the plane-wave propagation factorin the ax ( j =a) on the substrate ( j =s) medium.

From Eq. (14)-(16), it is clear that k,, may be realfor some v


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Nouv. Rev. Optique, 1975, t. 6, no 5 T. TAMI R 279

I

na

- X

+ X

( d 1FIG. 7.- everal varieties of dielectric gratings. Note that thegrating inb)has an asymmetric profile.

(a ) Incidence olong +x direction

\- X

/( b ) Incidence along - x direction

FI G. .- irectional discrimination m the field supported by anasymmetri c grating. The number o ~7uxines suggests the intensityo the ield meach region. Note that most o the energy in the top casei s leaked into the substrate whereas most o the energy in the bottomcase is leaked into the air region.

5. -LEAKY WAVES IN BEAMAND WAVEGUIDE COUPLERSSo far, the widest application of leaky waves inoptics has been in the context of beam .couplers forintegrated optics. To discuss these, consider a thinfilm, as shown in figure 94 which supports a surfacewave to the left of the x =0 plane and a leaky waveto the right of thex =0plane.A surface wave incidentfrom the left is partly or fully converted into a leakywave, which then radiates energy away into the openregion (or regions) in the form of an outgoing beamat the leaky-wave angle'8,.To achieve the change of regime between a surfacewave and a leaky wave, we can use either a prismas shownin figure 9b, or a grating as shown in figure 9c.In the former case, we need to satisfy the conditionn p >nf >n, >n, already discussed in conjunctionwith figure 5 , the prism providing now the densermedium with refractive index np. In the latter case,the grating may provide one or more beams in eachopen region, as already discussed in connectionwith figure6.In the case of prim couplers, the leakage into thesubstrate is very close to the grating angle (e N 90)and its intensity is then negligible. However, the beamsleaked out by grating couplers appear in both theair and substrate regions. Thus, if energy is to be

F IG . 9.- eaky-wave interpretation of beam couplers : a) Basicstructure that converts an incoming surface wave into an outgoingleaky-wave beam; b) Implementation o beam coupler by using aprism; ) Implementation of beam coupler by using a grating.


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280 T. TAMIR Nouv. Rev. Optique, 1975, t. 6,no 5transferred from a surface wave into a single outgoingbeam, the grating coupler is generally less efficient.However, by suitably designing the grating and, inparticular, by choosing a specific profile so as toenhance the blazing effect discussed in Section 4.3,it is possible to increase the efficiency of gratingcouplers to be nearly equal to that of prism couplers.Of course, by using reciprocity arguments, it isevident that a beam incident on the configurationsshown in figure 9 can be converted into a surfacewave. This reciprocal function can be visualized bysimply reversing the direction of the arrows in figure9.It is thus seen that the operations of prism and gratingcouplers can be explained by means of a unifiedapproach in terms of leaky-wave fields. The readerinterested in applying these concepts to the design ofactual devices is referred to chapter 3 of reference [14].

%dmmers i on f l u i dAPrism"

FIG. 10.-Waveguide couplers using leaky waves : a) J unctionbetween a planar waveguide and a fiber ;b) J unction between a planarwaveguide and a linear waveguide.Although of less wide application, leaky waveshave played a role also in coupling waveguides ofdifferent shapes, two examples of which are shownin figure 10. In the first scheme, which was reportedby K ersten [28], a high-index fluid is used to provideleakage from a planar waveguide. By then introducingthe end of a fiber into the liquid and aligning the fiberalong the leakage angle e,, energy can be transferredfrom the waveguide to the fiber, or vice-versa.The second waveguide coupler, shown in figure lob,is the analog of the prism coupler reduced to a planargeometry. In this case, the beam is replaced by asurface wave travelling along a broad waveguide,which is coupled to a narrow linear waveguide (ofthe rib or strip variety) via a slanted termination. Thistermination plays the role of the prism in beam

couplers, because it is separated from the linearwaveguide by a gap. Thus, if a mode is incident fromthe right in the linear guide, leakage starts to occurwhed the gap region is reached. If the wide guide isoriented at the correct leakage angle, the energy iscoupled into this guide in a manner which is analogousto that shown in figure 9b.Leaky waves also play significant roles in the ope-ration of the Lummer-Gehreke interferometer [121and in enhancing the power carried along fibers [13].6.- EAM DISPLACEMENTAT LEAKY-WAVE STRUCTURES

A closer inspection of the behavior of beamsincident on leaky-wave structures reveals an inte-resting phenomenon, which is similar to the Goos-Haenchen shift [29]. However, leaky-wave structurescan produce a displacement of the reflected beamwhich is several orders of magnitude larger than theGoos-Haenchen shift.A simple heuristic explanation of the beam displa-cement due to leaky-waves can be given by examin-ing figure 11, which describes the profi le of an out-going beam in an output coupler. Except for thesmooth rounding off of the left-hand edge, this beamexhibits the expected exponentially decaying shapeof the leaky-wave amplitude. The rounding-off of theleft edge is due to the weak diffraction that preventsan abrupt discontinuity in the beam profile [14, 181.If the power fl ux of the entire beam is considered,we find that its center of gravity moves along thethick flux line shown in figure 11. Because the beamfield is identical to the leaky-wave in the region x >0,it is evident that energy travels horizontally for adistance I = I ' B before being completely leaked offinto the upper medium. It can be shown from simplearguments that I is closely equal to the inverse of theleakage parameter a.

- X

,Surface-wave- -L eak y- wavereg i me reg i me +

FIG.11.- eaky-wave beam produced by an output beam coupler.The thick . f lux line indicates the center of gravi ty of power .flow.

Consider now figure 12 where we assume that the xaxis represents the upper boundary of a structurewhich can support a leaky-wave having leakage atthe angle 6,. In this figure, the trajectory Z'BD infigure 12a represents the same flux as that of figure 1 1.


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Nouv. Rev. Optique, 1975, t. 6,no 5 T. TAMIR 281zc

F I G . 12.- isplacement of power flux at leaky-wave structures :a) Outgoing leaky-wave .flux I ' BD and its reciprocal incident fluxIAD' ;b) An incident leaky-wave,%.x AD' musf ultimately continueas the outgoing leaky-waveflux I ' BD ;c) Flux displacement AB n thecase of forward leakage; d) Flux displacement AB in the case ofbackward leakage.

By reciprocity, if a beam with a leaky-wave profileis incident on the structure, it will follow the pathIAD ' . The same path IAD' is repeated in figure 12b,except that it has been reflected through theyz plane.However, AD' represents energy flux flowing hori-zontally and we recall that such a horizontal flux,when described by I ' B , must ultimately leak intothe upper region alongBD. It therefore follows thatthe horizontal flux AD' generated by the incidentbeam IA must continue along a path Z'B that radiatesaway along BD. This is described in figure 12c whichsimply connects the two points D' and 1' . Hence,the beam I A re-emerges as a reflected beamBD onlyafter it has undergone the displacementAB.The construction in figures lla-l lc refers to abeam incident at the leakage angle on a leaky-wavestructure of the forward type and the beam displace-ment AB in figure 12c is therefore a forward one. Asimilar displacement occurs for incidence on structuressupporting a leaky-wave of the backward variety,such as was shown in figure6a.The construction forthis case is given in figure 124 where it is noted thatthe incident flux IA must intersect the reflected fluxBD at some point above the structure.In practice, incident beams seldom have leaky-wave profiles of the form shown in figure 11. However,if we analyze a realistic beam such as a laser beamwith Gaussian intensity distribution in its cross-

section, we find [l1, 181that it can be described as asuperposition of two fields, of which one has leaky-wave characteristics. Because this leaky-wave portionbehaves in the form suggested in figure 12, the totalbeam field undergoes a displacement. Although thenet reflected beam may exhibit a profile that is oftendistorted when compared with the incident beam,the flux displacement is quite evident and, underproper conditions, the beam shift may be nearly aslarge as the beam width.

X

X

FI G. 13.- eam shift X at leaky-wave structures : a) Forwardshift, and b) Backward Shift.As a result, beams incident on leaky-wave struc-tures are displaced as shown in figure 13, where theshift of the forward type is shown in figure 13aandthe backward type is indicated in figure 13b. Theformer beam-shift variety occurs and has beenobserved [30] in the case of multi-layered media.The backward shift may occur in the case of periodicstructures, but its experimental verification has not

been reported yet. In fact, the backward shifted beamshown in figure 13b occurs when only a single gratingorder (for v = - 1) radiates into the upper region.If more than a single grating order radiates, diffractedbeams appear and each one of themis laterally shifted.The situation is then predicted [ll]to be as indicatedin figure 14, where a specular and a diffracted beamare shown. It is noted that each one of the scatteredbeams is displaced by a different distanceX , and thatsome beams undergo a forward shift (e.g., the specu-lar beam in fig. 14) whereas others are shifted back-wards (e.g., the diffracted beam in fig. 14).Although beam shifts along dielectric gratings havenot found any applications yet, they may providethe scope for very interesting basic studies, with thebackward shift serving asan open question for expe-rimental verification.


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282 T. TAMIR Now. Rev. Optique, 1975, t. 6,no 5

I I I

FIG.14.- ateral displacement of beams scattered by a periodicstructure.7 - TRUCI'URES WITH VARIABLELEAKY-WAVE CHARACTERI STI CS

iFilm>h,,:im

Subs t ra te

( b )FIG. 5 .- ayered structures with variable leakage : a) Decreasingair gap produces a value of a(x) that increases with x ; b) Practicalimplementationof the variable air gap inaprism coupler for improvingthe coupling efficiency.

In all of the foregoing discussions, wehave assumedthat the structures considered were characterized byconstant values of /3 andE . However, structures havingvariable leaky-wave characteristics may have consi-derable potential uses in the future. In particular,if the structures vary (( slowly)) with x, they couldbe characterized by continuously varying parame-ters p ( x ) and CL(X), thus leading to leaky-wave pro-perties of a generalized nature.In the case of multi-layered media, the variationin a(x) and p(x) may be achieved by means of alayer whose thickness varies with x, as suggested infigure 15a. Such a variation has already been investi-gated by Harris and Shubert [31] and by Ulrich [32]for the purpose of increasing the efficiency of beamcouplers of the prism variety. In that case, the practicalimplementation follows the scheme outlined infigure 1% . Here the air gap is made variable in heightby manipulating the pressure produced by the clampthat holds the prism and the film waveguide together.Saad et al. [33] extended this approach to situationshaving other variations, but their theoretical conside-rations have not been verified by experimental studies.Because of their backward-wave characteristics andmultiple-beam capabilities, dielectric gratings withvariable characteristics have even greater potentialapplications. Their variation canbe achieved as shownin figure 16, where it is noted that either the gratingheight re or the periodicityd, or both, couldbechang-ing with x. It is thus conceivable, for example, thata leaky-wave supported by such gratings wouldchange from the forward to the backward varietyas it progresses along the x direction. ,Other suchvariable features can be introduced and a wide rangeof field changes may thus be achieved.By suitably controlling the phase variation p ( x )and the leakage parameter a(x), it would thus be

possible to realize leaky-wave structures that performcomplex functions within small dimensions of athin-film structure. In particular, incident beams maybe transformed into a plethora of various fields. Werecall that the beam couplers discussed in Section 5convert a beam into a surface wave. Other beamconversions and transformations could be achievedby structures with variable leaky-wave characteris-tics. Some of these possibilities are shown in figure17.Although most of these beam transformations mustyetbeexplored, it is easy to imagine what kind of varia-tion is conceptually needed to implement most ofthem. In particular, the beam-splitting operation

t g = t g ( x )

d = d ( x )

-- X

t g = t g ( x ) a nd d d ( x )FIG.16.- ielectric gratings with variable leakage.


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N ow. Rev. Optique, 1975,t. 6,no 5 T . TAMIR 283[ 21 LECHER E.).- ine Studie uber elektrische Resonanzer-

scheinungen, Ann. Phys. Chem., 1890, 41, 850.[ 31 COLLIN R. E.), ZUCKERF. J.).- d.- ntenna Theory,Part 2, McGraw Hill, New York, 1969, pp. 151-348.[ 41 SNITZERE.), OSTERBERGH.),- bserved Dielectric Wave-guide Modes in the Visible Spectrum, J. Opt. Soc. Amer.,[ 51 KAPANYN. S.), BURKEJ. J.).- iber Optics : 1X. Wave-guide Effects, J . Opt. Soc. Amer., 1961, 51, 1067-1078.[ 61 Y ~ NA.), LEITE R. C.).- ielectric Waveguide Mode ofLight Propagation inp-n Junctions, Appl. Phys. Lett., 1963,[ 71 OSTERBERGH.), SMITHL. W.).- ransmission of OpticalEnergy Along Surfaces : Part 11, Inhomogeneous Media,J . Opt. Soc. Amer., 1964, 54 , 1078-1084.[ 81 HARRISJ. H.), SHUBERTR.), POLKYJ. N.).- eam Couplingto Films, J . Opt. Soc. Amer., 1970, 60, 1007-1016.[ 91 TIEN P. K.), ULRICHR.).- heory of Prism-Film Couplerand Thin-Film Light Guides, J . Opt. Soc. Amer., 1970,60,[lo] MIDWINTERJ. E.).- vanescent Field Coupling Into aThin-Film Waveguide, ZEEE J. Quantum Electronics, 1970,[l l] TAMIR T.), BERTONI H. L.).- ateral Displacement ofOptical Beams at Multilayered and Periodic Structures,J . Opt. Soc. Amer., 1971, 61, 1397-1413.[12] ULRICH R.), PRETTLW.).- lanar Leaky Light-Guides andCouplers, Appl. Phys., 1973, I , 55-68.[13] SNY DERA. W.).- eaky-Ray Theoryof Optical Waveguidesof Circular Cross-Section,Appl. Phys., 1974, 4, 273-298.[14] TAMIR (T.), Editor.- ntegrated Optics, Springer-Verlag,Heidelberg, 1975.[15] KAPANYN. S.), BURKEJ. J.).- ptical Waveguides, Aca-demic Press, New York, 1972, pp. 1-89.[16] TAMIR T.).- nhomogeneous Wave Types at Planar Inter-faces :11Surface Waves,Optik, 1973, 37,204-228.[17] MARCUSED.).- heory of Dielectric Optical Waveguides,Academic Press, New York, 1974.[18] TAMIR T.).- nhomogeneous Wave Types at Planar Inter-faces :111 Leaky Waves,Optik, 1973, 38,269-297.[19] TAMIRT.), OLINERA. A.). Guided Complex Waves, Proc.Z.E.E.E., 1963,110, 310.[20] SHEVCHENKOV. V.).- ontinuous Transitions in OpenWaveguides, Golem Press, Boulder, Colorado, 1971.~211 ~~~~~~~~ (L. M.).-wavesn ~ ~ ~ ~ ~ eedia, &ade-mic Press, New York, 1960, pp. 245-261.[22] TYRASG.).- adiation and Propagation of Waves, Acade-mic Press, New York, 1969, pp. 118-125.[23] TAMIR T.), OLINERA. A.).- he Spectrum of Electroma-

gnetic Waves Guided by a Plasma Layer, Proc. ZEEE, 1963,[24] OTTO (A.).- treustrahlung von Silber durch Anregung vonOberffachenplasmaschwingungen, 2. Physik, 1969, 224,[251H B S ~A.), OLINER (A. A.).- new Theory of WoodsAnomalies on Optical Gratings, Appl. Optics, 1965, 4,[26] PENGS. T.), TAMIRT.), BERTONIH. L.).- heory of Perio-dic Dielectric Waveguides, IEEE Trans. Microwave TheoryTech., 1975, MTT-23, 123-133.[27] PENG S. T.), TAMIR T.).- irectional Blazing of WavesGuided by Asymmetrical Dielectric Gratings, Optics Comm.[28] KERSTENF. T.).- oupling Between Slab-Waveguide andGlass Fibers, Digest of Tech. Papers, Topical Meeting onIntegrated Optics, New Orleans, La., 1974, WB5-1-4.[29] HOROWITZB. R.), TAMIR T.).- nified Theory of TotalReflection Phenomena at a Dielectric Interface, Appl.Physics, 1973, I , 31-38.

Evanescent Wave Coupling into a Thin-Film Waveguide,Appl. Phys. Lett., 1970,16, 198-200.

1961, 51, 499-505.

2, 55-58.

1325-1337.

QE-6, 583.

FIG. 17.- eam functions realizable by means of variable leaky-wave structures.

shown in the bottom right corner has been shown [33]to be achievable by using a structure similar to thatI~ conclusion, it can be noted that leaky-waveshave already played a significant role in integrated-

optics applications. By extending the concepts alreadydeveloped to configurations having variable leakagebring about an even greater and more diversifieduse for controlling light beams by means of miniatureand reliable structures.

depicted in figure 1%. 51 , 317-332.65-73.

characteristics, it is expected that leaky-waves may 1275-1297.

1974, 11, 405-409.** *REFERENCES[ 11Lord RAY LEIGH.- n Waves Propagated Along the Plane

17, 4-11. Also in Scientific Papers, vol. 2, Cambridge Univ.Press, London, 1900, pp. 441-447.Surfaceof an Elastic Solid, proc. London Math. soc., 1885, 130 M1DWINTER (J. E.), ZERNIKE (F.1.- xperimental Studies of


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284 T. TAMIR[31] HARRISJ . A.), SHUBERTR. ).- ariable Tunneling Exci;tation of Optical Surface Waves, IEEE Trans. M icrowaveTheory Tech., 1971, MTT-19, 269-276.[32] U LR I CH R. ).- ptimum Excitation of Surface Waves,J . Opt. Soc. Amer., 1971, 61, 1467-1476.[33] SAAD A .), BER TO NIH. .), T m T.).- eam Scatteringby Nonuniform Leaky-Wave Structures, Proc. ZEEE, 1974,

62, 1552-1561.

Nouv. Rev. Optique, 1975, t. 6, no 5ACKNOWLEDGMENT.he author wishes to express his appre-ciation to Prof. C. I ~~BE R Tor facilitating his participation at theColloquium on (( the Optics o Guided Waves)) as wel as for encou-raging the presentation and subsequent preparation of this paper.This work has been supported by the U.S. OfJ ice of NavalResearch, under Contract No N00014-75-C-0421.

(Manuserit re(U le 27 mai 1975.)

leaky waves in planar optical waveguides (*i

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