2120 J. Opt. Soc. Am. A/Vol. 4, No. 11/November 1987
Fresnel-like behavior of guided waves
T. P. Shen, R. F. Wallis, and A. A. Maradudin
Department of Physics, University of California, Irvine, California 92717
G. I. Stegeman
Optical Sciences Center and Arizona Research Laboratories, University of Arizona, Tucson, Arizona 85721
Received August 16, 1985; accepted May 21, 1987
The reflection and transmission of thin-film guided waves incident at variable angles upon a variety of transversediscontinuities are analyzed. A thin film bounded by semi-infinite media is modeled by a thin film bounded byfinite media, terminated at shorting planes situated parallel to and a long distance from the film surfaces. Theresults obtained with this model are checked against previous calculations for normal incidence onto a waveguide-air endface. For nonnormal incidence upon the interface between two waveguides, various interesting phenomenasuch as guided-wave equivalents of Brewster's angle, multiple cutoff angles for both guided waves and radiationfields, and intermode conversion are investigated numerically.
1. INTRODUCTION
The reflection and transmission characteristics of guidedwaves incident upon the interface between two waveguidesor the endface of a waveguide are problems of continuinginterest in guided-wave technology. Much of the previouswork has centered on understanding (1) oscillation in semi-conductor lasersl-7 and their radiation patterns and (2) buttcoupling of waveguides to other waveguides and to fibers,8' 9
both of which involve normal incidence upon the interface.With the recent interest in bistability and similar all-opticaloperations in integrated-optics structures, the guided-waveendface reflectivity determines the finesse of a guided-wavecavity and hence the critical power required for switch-ing.10" 1 There are also other interesting questions for non-normal guided-wave incidence, specifically whether thereare guided-wave equivalents of Brewster phenomena andwhether new effects that are unique to guided waves occur.We address these questions by calculating, for a number ofcases, the reflection and transmission coefficients of theincident guided-wave mode, the fraction of power convertedinto other guided-wave modes, and the fraction of energyradiated out of the guided modes.
Similar guided-wave interface problems have been treatedfor normal incidence in the past- 7"12-22 by using a variety ofanalytical techniques. For example, for small transversediscontinuities, a simple and effective approximation hasbeen derived by Marcuse.13 For arbitrarily large transversediscontinuities for which appreciable coupling takes place toall of the guided waves of the waveguide and to the continu-ous radiation spectrum, the boundary conditions along thetransverse interface yield a set of integral equations. Oneapproximation is to solve these equations by the convention-al Neumann series.6 Another is to discretize the continuousradiation spectrum, for example, by expanding it in terms ofLaguerre functions 7 12"6 or Hermite Gaussian functions 21with an adjustable width parameter. The radiation-wavecontinuum can also be discretized by introducing bound-aries parallel to the film surface to limit the physical system
in one dimension.'7 For such a discretized mode spectrum,there are several ways to solve this set of infinite linearequations. For example, the variational technique1221 22
and the least-squares boundary-residual7 method have beenused.
In this paper we use both a boundary-residual method anda point-matching method, together with a discretized modespectrum, to examine the reflection and mode conversion ofTE and TM guided waves incident (1) at 90° upon theendface of a waveguide and (2) at variable angles of inci-dence upon an interface between two waveguides. In orderto verify the validity of this model, comparison is made firstwith previous calculations in the limited number of cases inwhich it is possible. For example, for TE modes normallyincident upon the waveguide-air interface, we obtained be-havior similar to that reported by Pudensi et al.
2 ' and Ike-gami,2 2 using variational techniques for the identical wave-guide system. However, for the TM case, the conversioncoefficients into other guided waves and the radiation losseson reflection were found to be larger than reported in thosetwo papers. We further examine the model by verifyingthat the results are only weakly dependent on the distancebetween the shorting planes.
In subsequent calculations we investigate a number ofsimilarities between plane-wave and guided-wave reflectionphenomena. These include Brewster's angle, reflectivityminima for radiation through endfaces, and Fresnel phe-nomena, including guided-wave and radiation field cutoffs.
2. THEORY
The physical system of interest is shown in Fig. 1(a), and themodel that we choose to approximate it is shown in Fig. 1(b).As discussed before, the principal difference between thetwo is the introduction of shorting planes above (z = d) andbelow (z = -d) the thin-film boundaries. For guiding to bepossible, the film index nf is chosen to be larger than that ofthe cladding (n,) and substrate (n,), and the film thickness is
Vol. 4, No. 11/November 1987/J. Opt. Soc. Am. A 2121
nc nc
x n'f
ns
(a)
z
I+d
nc
h/2nf Et-aet X
-h/2ns
-d
ns
Ic
nf
ns
,Z the x axis of the system. (This point is discussed in detailbelow.)
n c' We first expand the total field in each part (x < 0 and x >h/2 0) of the structure [Fig. 1(b)] in terms of its orthogonal
_ n_ _ x normal modes. Next, we calculate the electromagnetic-h/2 fields produced along the transverse boundary when a cho-
n sen guided wave impinges upon it from the left. A discretenumber of reflected and transmitted modes are then super-imposed at the boundary, and their amplitudes are chosen tominimize the difference in the tangential E and H fields
+ d across the interface. Finally, we interpret the results inn terms of reflection and mode-conversion coefficients for the
h/2 guided-wave fields and in terms of reflected and transmittedn --- x radiation fields. The details of the normal modes and the-h/2 analysis can be found in Appendix A.
-d(b)
Fig. 1. (a) The physical system of interest. (b) The model systemto be investigated numerically.
chosen so that at least one guided mode is above cutoff. Forgeometries in which there is a single medium on the trans-mission side of the interface, it is characterized by the dielec-tric constant Et.
The shorting planes (E = E = 0 at their surfaces) areplaced above and below the guiding film (as shown in Fig. 1)in order to discretize the radiation modes on both sides ofthe endface. The shorting planes are a sufficiently largedistance from the film boundaries so that the guided-wavefields decay effectively to zero at these planes except in theimmediate vicinity of guided-wave cutoff, and so that thereis a sufficient number of discretized radiation modes to yieldrepresentative radiation patterns. The propagating normalmodes on each side of the interface (x = 0) include (1)surface-guided waves (TE and TMm), whose field ampli-tudes decay exponentially away from the film boundaries,and (2) radiation fields, which travel parallel to the film andwhose fields exist as standing-wave resonances across eitheror both of the media bounding the film. Also included in themode expansion are nonpropagating fields that decay expo-nentially with distance along the film and whose fields arestanding waves in all three media. The mode amplitudesare evaluated by solving the boundary conditions at theinterface (x = 0) by the least-squares boundary-residualmethod and/or the point-matching method.
We used this general approach successfully before to ex-amine the reflection and transmission properties of othersurface-guided excitations, namely, surface acoustic waves2 3
and surface-plasmon polaritons.2 -26 The limits of thisshorting-plane model were discussed in some detail beforefor the surface-plasmon case. For shorting-plane separa-tions larger than a few optical wavelengths, the coefficientsof reflection, transmission, etc. were found to be indepen-dent of plate spacing, except for a few cases in which anoma-lous results were obtained for plate-separation changes inwhich a normal mode changes from an evanescent (nonprop-agating) field to a standing-wave radiation (propagating)field. Similar results are expected (and observed) here.The standing-wave nature of the radiation modes leads tocomplications in interpreting the results in terms of radia-tion patterns because the power flow is always strictly along
3. NUMERICAL RESULTS
Reflection and Radiation at a Waveguide-NonwaveguideBoundaryWe now illustrate this analytical technique with some con-crete examples based on GaAlAsl-, waveguides. In thefirst sequence of calculations we chose the refractive index ofthe film to be 3.6 and the cladding and substrate indices tobe either 3.2 or 3.4. In Figs. 2-5 it is assumed that thetransmission medium is air; i.e., Et = 1.0. In order to get areasonable number of radiated modes, the shorting planeswere placed at z = 6X, where X is the vacuum wavelength.
We have compared our results with those reported previ-ously by other investigators 2 0 '21 and have obtained goodagreement for the general trends with film thickness, etc. Adetailed comparison of our reflection coefficients with thenumerical results reported by Pudensi et al. 21 is shown inFig. 2. There are small differences, which may be due to adifference in the models or to the inclusion of evanescentfields on both sides of the transverse interface in our case.(It is not clear in the paper by Pudensi et al. whether evanes-cent fields were included on the waveguide side.)
0.7
0.6
0.51
0.4
R030.3.
0 .0 I I I I I I I I I I0.0 0.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
h / Fig. 2. The reflection coefficient for TEo - TEo versus film thick-ness for a single guided TE wave incident normally upon the endfaceof a waveguide. Circles and triangles denote our calculations andthose of Pudensi et al.,21 respectively. Here nf = 3.6, Et = 1, and d =6X. The solid and dashed curves are for n = n = 3.2 and n = n =3.4, respectively.
I I
-P S-na-V- =
0.21-
0.1
Shen et al.
2122 J. Opt. Soc. Am. A/Vol. 4, No. 11/November 1987
0.7
0.6
0.5
0.4R
0.3
0.21
0.1
0.01 l l I l l l0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
d/XFig. 3. The total (R; open circles) and guided-wave (Ri, i A or S;filled circles) reflectivities versus shorting-plane separation (2d) fora single guided TE wave incident normally upon the endface of awaveguide. Here, nf = 3.6, et = 1, h = 0.9X, and n, = n = 3.2.
We have also explored the dependence of our results onthe separation distance (2d) between the shorting planes.As shown in Fig. 3, the reflection coefficients become inde-pendent of d for d ' 5X in this case. This does not necessar-ily justify the inclusion of radiation fields in the form ofstanding waves in this model, but it does indicate that d = 6Xis sufficient for obtaining d-independent (but model-depen-dent) values for the coefficients of reflection, etc.
In Fig. 4 both the total reflectivity R (for all propagatingwaves reflected back into the medium of incidence) and theguided-wave reflectivity Ri (for i A, the antisymmetric
mode TE,; for i - S the symmetric mode TEo) for theincident guided-wave mode (TEo or TE,) are plotted asfunctions of the thickness h of the waveguide. The refrac-tive indices n (= n) were taken to be 3.2 and 3.4, whichcorrespond to An = 11 and 5.6%, respectively, where An = 1- n/nf. In the limit h - h,0, where hc. is the mode cutoffthickness (h, 0 = 0 for TEO in the symmetric waveguide), theguided waves degenerate into plane-wave fields in the sub-strate and cladding, and we find that the plane-wave limit
n, 2 - 1 2R -s - (3.1)
is obtained numerically. In the limit h - a, the guidedwaves degenerate into plane waves in the film, and
R R | nf2 1 R L-- Rs ~-_ ~~2f 2 +1I (3.2)
which agrees with the numerical results. As the film thick-ness is varied, the maxima in R and Ri for the lowest-ordersymmetric (TEO) and antisymmetric (TE,) modes occuraround h 0.3 and h 0.5XA respectively, and correspondapproximately to optimum guided-wave confinement.Note that the reflectivity can be larger than that predictedby Eq. (3.2), which indicates that quasi-plane-wave inter-pretations of guided-wave phenomena at discontinuities arenot always useful. For TEO incidence, the mode conversioninto TE, starts from zero at the TE, cutoff thickness owingto the vanishing small overlap of the TEO and TE, fields atthis cutoff.
The mode conversion into radiated fields is shown graphi-cally in Fig. 5. In a symmetric waveguide, these modesconsist of two plane-wave fields in the substrate and clad-ding, traveling at angles ±O with respect to the x axis, alongwhich the net power flow occurs. Here, we have plotted thepower flow along the x axis as a function of the angles (0)associated with the standing-wave fields. For finite d, the
0.8
0.6-
R 0.4
0.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
h/XFig. 4. The total (R; open symbols) and guided-wave (R i A or S; filled symbols) reflectivities versus film thickness for a single guided TEwave incident normally upon the endface of a waveguide. Circles and triangles denote the lowest-order symmetric (i S, TEO) andantisynmetric (i - A, TE) modes, respectively. Here ni = 3.6, et = 1, and d = 6X. The solid and dashed curves are for n, = n, = 3.2 and n, = n,= 3.4, respectively.
I I I I I I
h/X = 0.9
- ~ TEI (antisym. mode)------ TEO (sym. mode)
Shen et al.
Vol. 4, No. 11/November 1987/J. Opt. Soc. Am. A 2123
zw
0
U,
z0
0
0.1 000
zW
LU.LU.
0Uz0
z0U
0 20 40
ANGLE(a)
60 80
0.080
0.060
I-(n0.040
z
2 0.020o
0U 0.000z0Uj)
0.003z0U-
2 0.002
0.001,
0.000i
zw
UL.
00z0U,
z00
0 20 40ANGLE
(C)
60 80
0 20 40 60 80 -- 0 20 40 60 80ANGLE ANGLE
(b) (d)Fig. 5. The far-field radiation patterns, that is, the power-conversion coefficients along the x axis versus the angle of transmission, Ot, for thetraveling-wave components of the propagating standing-wave solutions with TE guided-wave incidence. Here, nf =3.6, n, =,=3.2, andd =6X. (a) TE 0 wave incidence for which h =0.025X, T =0.715, and R =0.285. (b) TE~ wave incidence for which h =0.35X, T =0.512, andR =0.488. (c) TE 0 incidence for which h =0.35X, T =0.617, and R =0.383. (d) TE2 incidence for which h =0.75X, T =0.242, and R= 0.758.
Shen et al.
0
0
0
0
0
0
. . . . . . . . . . . -
2124 J. Opt. Soc. Am. A/Vol. 4, No. 11/November 1987
0.6
0.5
0.4
N0 0.3
0.2
0.1I
0.0 L \:os
3.0 3.2 3.4 3.6 3.8n t
Fig. 6. The reflectivities R (total; open circles) and Ro (guidedwave; filled circles) versus nt (;1~) for nf = 3.51, n, = n, = 3.22, h =0.15X, and d = 6A. The solid and dashed lines represent data for theTEo and TMo modes, respectively.
pattern actually consists of a discrete number of points thatcorrespond to the individual standing-wave modes. As dincreases, the envelope of the pattern remains the same, butthe density of points increases. Although these modes arenot those for semi-infinite bounding media, they give usefulinsights into the far-field radiation patterns that would beobtained if the shorting planes were removed just after thetransverse interface. These patterns are obtained both onreflection back into the region of space that contains theguiding film (r) and on transmission into the transmissionregion (0t). The fraction of power that is mode convertedinto the reflected waves is typically smaller than 1% andreaches a peak at around 0 r = 20°.
Here we concentrate on the transmitted radiation fieldpatterns. The transmitted radiation patterns correspond-ing to incident symmetric (TEo and TE2 in this example)guided-wave fields are peaked along the forward direction,with an angular width inversely proportional to the effectiveguided-wave width, as expected. For TE1 , which corre-sponds to the first antisymmetric mode, zero power is radiat-ed along the x axis, and the radiation pattern peaks at ot 200from the axis. The behavior can be understood in terms ofeffective radiating apertures. An effective slit or aperture isassociated with each extremum in the guided-wave fielddistribution. Adjacent slits are out of phase by r with re-spect to one another. That is, for TEO, TE2 , etc. there is anodd number of effective apertures, with symmetry about the
middle one. This leads to an interference maximum alongthe axis, as well as n - 1 subsidiary maxima for the TEn case,as observed. For n = 1, 3, 5, etc. (TE 1, TE3 , TE 5, etc.) andwith the slits out of phase by 7r with respect to one another,the effective-slit radiation pattern produces a minimumalong the axis, and there are a total of n total maxima in theradiation pattern. Hence the radiation patterns predictedwith this model can be understood in terms of simple con-cepts.
The behavior for TM-mode incidence is qualitatively thesame, but of course the specific values for the coefficients aredifferent.
It is well known that the plane-wave reflection coefficientfor an interface vanishes when the index difference betweenthe two media goes to zero. An interesting question is:What happens when a guided wave of effective index / isincident upon an endface with a medium of index nt = a onthe other side? The variation in reflection coefficient withnt is illustrated in Fig. 6. When /3 = nt, the reflection coeffi-cient for the incident guided wave passes through zero.However, since this value of nt is different from the for thereflected radiated modes, some mode conversion into theseradiated modes still occurs, and the transmission coefficientis not exactly unity.
Another interesting question is whether a metallic mirrorin contact with a waveguide endface will have a reflectioncoefficient of 1 for guided-wave incidence. In Fig. 7 areshown the reflectivities for the first two symmetric modes ofa waveguide, whose material parameters are n, = n, = 3.2, nf= 3.6, h = 0.75X, and d = 6X, in contact with metals of variousdielectric constants. All the incident power of the first orsecond symmetric TE (or TM) mode is totally reflected backinto the waveguide region, as required. However, modeconversion occurs, and the guided-wave reflection coeffi-cient of the incident wave is always smaller than unity.More than 95% of the incident power is reflected back intothe same mode as the incident wave, and a larger fraction is
1.00
0.91
R
0.98
0.97
0 -2 -4 -6 -8 -10 -12 -14 -16Et
Fig. 7. The incident-wave reflectivity coefficients Ro and R2 versusthe dielectric constant for different metals when a guided waveencounters a semi-infinite metal barrier. The material parametersare n, = n = 3.2, nf = 3.6, h = 0.75X, and d = 6X. The solid anddashed lines are for TE and TM polarizations, respectively.
Shen et al.
Vol. 4, No. 11/November 1987/J. Opt. Soc. Am. A 2125
reflected without mode conversion for the lower-ordermodes (TEO and TMO) than for the higher-order modes (TE 2and TM 2). As lEtI becomes large (>>n), the mode reflectioncoefficient approaches unity.
Reflection and Transmission at a Waveguide-WaveguideBoundary for Arbitrary Angles of IncidenceVarying the angle of incidence of a guided wave at a wave-guide-waveguide boundary allows us to examine guided-wave analogs of phenomena such as total internal reflectionand Brewster's angle. The calculations shown in Figs. 8 and9 address these questions for two waveguides of the samethickness (h = 0.45X) and material parameters n, = 1.0, n, =
1.46, nf = 1.7 (waveguide A), and n, = 1.46, n, = 1.5, nf = 2.0(waveguide B), respectively. Waveguide A can support oneTEO mode and one TMo mode only, and waveguide B cansupport two TE modes (TEO, TE 1) and two TM modes (TMO,TM,) for these sets of parameters. In the numerical cacula-tions, guided-wave incidence from both A and B is exam-ined.
First we consider a TEO guided wave incident from A uponthe boundary with B [see Fig. 8(a)]. Since the mismatch inthe guided-wave effective index across the boundary is small[fl(TEO, A) = 1.5732, #(TEO, B) = 1.8680, and #(TE 1, B) =1.52301 for all cases, the net transmission coefficient is high(99% at small angles of incidence). Very little power (<4%)appears in the TE 1 transmitted mode because of the largemismatch between the incident TEO and transmitted TE1field profiles, and the transmitted TEO wave accounts formost of the transmitted power (94%). The balance of thepower is converted into reflected waves and transmittedradiation fields. The behavior of the reflection coefficientwith increasing angle is strongly reminiscent of that forplane waves, and the reflection coefficient becomes unity atan angle of incidence of 900. Although it is not clearlyevident in Fig. 8(a), the guided-wave reflection coefficientbecomes very small at an angle of 500. (We cannot say thatit totally vanishes because of the numerical nature of theresults.) Since the guided-wave field is polarized in theplane of incidence, this is a direct analog to the Brewster-angle phenomenon. Note that, in terms of the transverseinterface between the two waveguides, the TE wave acts as ap-polarized wave and the TM mode plays the role of an s-polarized wave. Calculating the Brewster angle from theguided-wave analog tan °b = fl(TEo, B)/#(TEo, A) gives Ob =
49.9°, in excellent agreement with the value found numeri-cally. Finally we note that transmission into the TE1 modegoes to zero for angles greater than 77.50 since #(TEO, A) >/3(TE1, B). For any wave to be generated on the transmis-sion side, the parallel component of the wave vector must beconserved, and hence the cutoff angle c. for the TE1 trans-mission mode is given by
sin Oco = #(TE 1 , B)/#(TEO, A), (3.3)
which corresponds to 77.5° in this case.Similar results are obtained for TMo incidence from wave-
guide A, with the exception of a few features. There is noBrewster-angle phenomenon in this case since it corre-sponds to s-polarized wave incidence when defined in termsof the transverse interface. (The reflection coefficient in-creases monotonically with increasing angle of incidence,just as in the plane-wave case.) Since the field of the inci-
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0 10 20 30 40 50 60 70 80 90
(a)
I.
0 10 20 30 40 50 60 70 80 90ei(b)
Fig. 8. The reflectivities and transmissivities versus angle of inci-dence for (a) TEo or (b) TMo guided-wave incidence from waveguideA upon the interface with waveguide B. R and T are the totalreflectivity and the total transmissivity, respectively. R is theincident-wave reflectivity, and To and T are the transmissivitiesinto the first two guided modes in waveguide B. The materialparameters are as follows: for waveguide A, n, = 1.0, ns = 1.46, andnf = 1.7; waveguide B, n, = 1.46, n, = 1.5, and nf = 2.0. h = 0.45Xand d = 6X.
RoI
Shen et al.
2126 J. Opt. Soc. Am. A/Vol. 4, No. 11/November 1987
Fig. 9. The reflectivities and transmissivities versus the angle of incidence for guided waves incident from waveguide B upon the interfacewith waveguide A. (a) TEO incidence, (b) TMo incidence, (c) TE1 incidence, and (d) TM1 incidence. T and R are the total reflectivity and thetotal transmissivity, respectively; Ro and R1 are the reflectivities into the TEO (or TMO) and TE 1 (or TM1 ) modes, respectively; and To is thetransmissivity into the lowest-order transmitted guided wave. Here, for waveguide B, n, = 1.46, n, = 1.5, and nf = 2.0; for waveguide A, nc = 1.0,n = 1.46, and nf = 1.7. For both waveguides, h = 0.45X and d = 6X.
dent TMo mode penetrates deeper into the substrate thanthe field of the TEO mode does, the conversion into thetransmitted TM, mode and into the transmitted radiationfields is larger; that is, the field mismatch is less severe inthis case. Finally, we note that the anomalous behavior of
the coefficients at angles near 630 is strongly reminiscent ofbehavior found in similar calculations using this model.24'2 5
This behavior was studied in great detail for surface-plas-mon reflection phenomena,24'25 and the oscillations wereidentified as artifacts introduced by the shorting-plane
H
DE
no -S
Shen et al.
Vol. 4, No. 11/November 1987/J. Opt. Soc. Am. A 2127
model. It occurs in some cases when changes in some vari-able (angle, shorting-plane separation, etc.) cause a changein the character of the mode from nonpropagating (evanes-cent decay along the x axis) to propagating (along the x axis).Typically, the range (angles in this case) over which thespikes shown in Fig. 8 occur is small, and smooth curves caneasily be drawn through these regions.
Possibilities for critical-angle phenomena exist when aguided wave is incident from waveguide B upon the interfacewith waveguide A. The results are shown in Fig. 9(a) for theTEo-mode case. A waveguide mode becomes cut off inwaveguide A when the angle of incidence 0 in waveguide B islarger than the cutoff angle defined by
sin ,Oo = (TEO, A)/O(TEO, B) < 1. (3.4)
Note that guided-wave cutoff does not occur for TE1 inci-dence from B because O(TEI, B) < (TEO, A), and hence theangular dependence of the reflection and transmission coef-ficients in Figs. 9(a) and 9(c) are dramatically different. ForTEo-wave incidence, the reflection coefficient goes to unityfor angles greater than 0ccO as defined by expression (3.4).Note that there is a critical angle for reflection into the TE,mode in waveguide B since fl(TEo, B) > fl(TE1 , B). Finally,there are also critical angles for conversion into the reflectedand transmitted radiation modes given by
sin Oco = nj(B)/0(TE 0, B),
sin 0O = n8(A)/fl(TE0 , B),
(3.5a)
(3.5b)
respectively.The results for TMo incidence from waveguide B are qual-
itatively the same as for the previously discussed TEo case.When the TE1 (or TM1 ) wave is incident from the wave-
guide B side of the interface, the results obtained are quali-tatively similar to those for TEo (or TMO) incidence fromwaveguide A. The reason for this similarity is that O(TE1,B) [or O(TM1 , B)] is smaller than either O(TEO, A) or O(TEo,B) [or O(TMo, A) or P(TMo, B)], and hence no critical-anglephenomena should occur. Brewster-angle behavior for theTE1 (TM1 ) reflection coefficients was noted for these cases.Since the fields associated with the incident waves are soclose to the cutoff and the fields penetrate deeply into thesubstrate, strong conversion into transmitted radiationfields occurs. Finally, the oscillatory behavior shown bysome of the curves is due to artifacts introduced by theshorting-plane model, as discussed previously.
In these calculations we explicitly assumed that there is nosignificant intermode conversion; that is, TEn TMm andvice versa. A similar question was discussed in great detailin Ref. 26 for surface-plasmon reflection problems at arbi-trary angles of incidence. It was found that when the totalpower entering the interface through the incident guidedwave was not equal to the total power leaving the interface inthe form of guided and radiation modes, then interpolariza-tion conversion was important. We assume here, withoutproof, that a similar result would occur in the present cases;that is, if interpolarization conversion were significant, wewould expect that the total guided-wave power entering theinterface would not equal numerically the total power leav-ing the interface. In fact, we found power conservation to beaccurate to at least two and usually three significant figures,which indicates that mode conversion is minimal for the
cases which we examined. However, if we had chosen wave-guides with very large changes in A between modes on thetwo sides of the interface, mode conversion could become amore serious problem and would have to be taken explicitlyinto account.
APPENDIX A: THEORETICAL DETAILS
Normal ModesAs usual, the normal modes naturally separate into TE (Es)and TM (H,) solutions in both regions. Assuming a har-monic time dependence and propagation along the x axis, wehave
E(r, t) = E(z)exp(-iflx + iwt) for TE waves (Ala)
and
H(r, t) = H(z)exp(-ix + it) for TM waves. (Alb)
TE Guided-Wave ModesFor TE modes, the nonvanishing fields are Hx, Ey, and Hz,and it is Hx and E, that are explicitly used in the boundaryconditions. When E = 0 is enforced at the two shortingplanes, it is straightforward to show that the solutions to thewave equation and boundary conditions are
h . hcos af + A sin af-
H, (z) = 2 h 2 c os a(d -z),s a d - 2) W/x o
cos caf h + A sin afhE (z) =
Hz(z) =
sin a(d -z),sin a, d- )
h .hcos f 2 + A sin af
Wosin c d h)
sin a,(d - z) (A2a)
for h/2 < z < d;
H (z) = t (sin cfz - A cos C~Z),
Ey(Z) = cos afZ + A sin fz,
Hz(z) = (cos acfZ + A sin Cafz)
for -h/2 < z < h/2; and
h hcos af -A sin f-
H(Z) = hcos aY8 (d + z),sin a, d-2j) 8
(A2b)
h . hcos ccf 2-A sin ccf 2
EY(z) = sin a,(d + z),sin a, d- )
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2128 J. Opt. Soc. Am. A/Vol. 4, No. 11/November 1987
h . hcos af - A sn af /
Hz(z) = - 2 sin as(d + z) (A2c)sin a. (d -h) - 40
2
for -d < z < h/2, where
-a, + af tan afh tan a d h)
a, tan af 2 + af tan a, d - )(A2d)
The power flow associated with the propagating TEmodes is given in the usual way by
which gives
In Eqs. (A2) the definitions of aft, a, and a, are given by
ac 2 = n,2k2 - 32,
af 2 = nt-22 - 32,
a 2= n2k2 2,
(A3a)
(A3b)
(A3c)
where k is the free-space wave vector and the allowed valuesof / are determined by the following dispersion relation:
-ac +af tanaf tan ac -
ac tan af2 + f tan c d- )
a, af tan af2tan aS (d - h)
as tan afh + af tan a (d - )
The solutions for / obtained from Eq. (A4) can be dividedinto three classes. The values are either real, which corre-spond to propagating waves, or purely imaginary, whichcorrespond to evanescent fields that decay exponentiallyaway from the transverse boundary along ax. (We notethat if one of the refractive indices is purely imaginary,which occurs, for example, for lossless metallic media thatsupport surface-plasmon polariton modes, the allowed val-ues of a can also occur in complex conjugate pairs.2 4
26 )Solutions with real values of P and imaginary values of bothac and a, correspond to guided-wave modes. Those associ-ated with real a and one of ac or a real are propagatingradiation fields. All solutions with imaginary / are evanes-cent nonpropagating solutions. Only propagating solutionscan carry power to and from the transverse boundary. Al-though the evanescent fields do not contribute to the powerflow, their inclusion in the analysis is necessary for theboundary conditions to be satisfied to some predeterminedlevel of accuracy. In general, the larger the number of eva-nescent modes that are included, the more accurately theboundary conditions are satisfied. Since decreasing valuesof 32 correspond to decreasing periodicity in the solutionfields along the transverse boundary, we order the solutionfields according to decreasing values of /2.
X
sin(a - a,*) d- h)C~c C~2
sin(ac + ac*) (d -h)
a, + a,*(A6a)
P = [(1 + Al2) h2 + (1 - 2) sin_] (A6b)L)I 2a J
cos ah -A sin a
8 2uo sin as (d -
12
[sin(as- as*) (d -
as as*
sin(a, + a*) (d - has + a,*
(A6c)
The power flow for the radiation fields ( real, and ac and/ora, real) is always along the x axis owing to the standing-wavenature of the wave solutions. The individual components ofthe standing fields propagate at angles (real a) 0/ relative tothe x axis given by tan 0i = a. We now assume that thetraveling-wave components of the solution fields would bethe same even in the absence of the shorting-plane bound-aries. This is the key assumption needed for interpretingthe radiation fields in terms of radiation patterns.
TM Guided-Wave ModesThe analysis here is similar to that for TE waves. In thiscase, the nonvanishing fields are
Px dz EH, (AS)
P = P + Pf + P,
PC= 2co/:
(A6)
Shen et al.
Vol. 4, No. 11/November 1987/J. Opt. Soc. Am. A 2129
cos af 2 + B sina 2 -
E.X(Z) = - 2 sin a (d-z),cos a d h) coEon'
The power flow associated with the propagating solutions isgiven by
P = 2, 2/3on
cos af2 + B sin afhcos a(d -z),
cos a, (c -c 2 cos Caf 2 + B sin ayf h
cos a, (d - )- s2 a,(d -z)weonc2
for h/2 < z < d;
-iaEx(Z) = ' (sin afz - B cos afz),
,wEonf2
HY(z) = cos az + B sin ag,
E,(z) = - (Cos az + B sin afz)f <On
for -h/2 < z < h/2; and
(A7a)
sin(ac - ac*) (d -ac ac*
sin(a + ae *) d h)]
ac + ac*
2( 12)2Pf = 2 1 D+ 1B + (1 -1B12) Ia 2
(A7b) P2 = 02
2cocon,
cosa n h -B sina hE. (z) = c a ( - 92 sin a((d + z),
h . hcos f -B sin f 2
HY(z) = / cos a8(d + z),
cos as (d )-2
cos fh 2-B s fh2-Ez(z) = c22n 2 _ Cos a(d + z) (7c)
cos a (d _ h -,,On.,'s2Jfor -d < z <-h/2, where
n,2 ( 2) nf2 f B=
n2 tan af 2 tan s, d
(A7d)
sin(a, -°8 s*) d )x 2
a, -a
(A9c)
sin(a +as,*) (d h
a + a,*
The ordering of the normal modes, their interpretation, etc.,are the same as for the TE solutions.
Nonguided RegionsThe analysis of endface waveguide problems also requiresthe normal modes of a uniform medium with two shortingplanes at z = id. Again, there are two possible polariza-tions, TE and TM. For TE modes, the field components aregiven by
iadnH, 0(z) = cos ad(c-
Eyn(z) = sin adn(d- z),
In Eqs. (A7), the definitions of af, a,, and a are given by Eqs.(A3), and the allowed values of /3 are determined by thefollowing dispersion relation:
n tan a (d - ) + tf tan a h
2 tan afh tan a d- 2 )- 2 tan af2
2 tan ac d h) +-2\/ f
(A8)
Hz(z) = n sin adn(d -z),CO)AO
where
(/3)2 = E, co2 n)272 (a ad 2d
(A10)
n = 1,2,....
(All)
The energy carried by the nth propagating mode is givensimply by
pn -n d.Wo
(A12)
For TM-polarized fields, the nonzero field components are
HY(z) =
Ez(z) =
(A9a)
(A9b)
- - tan af tan atc d - 2+nnc2 22 n
Shen et al.
2130 J. Opt. Soc. Am. A/Vol. 4, No. 11/November 1987
iad' E.,(z) = - sin ad'(d -z),
HY'(z) = COS ad'(d--z),
E,0(z) = adCOB adn(-Wfoet - d-z)
nrad = -, n = 0, 1, 2_.
found that interpolarization conversion is weak and can usu-ally be neglected. For the general case we write the fieldcomponents of the reflected and transmitted waves in theform
(A13)
(A14)
The energy carried by the nth propagating TM mode is givenby
pn = On d, n = 1, 2,3,...,
P° = 2° d.WfOct
(A15a)
(A15b)
Note that there is an interesting difference between theradiation modes of TE and TM polarizations. For TMpolarization, n = 0 is a valid solution and corresponds to aplane-wave field propagating along the x axis with E, - 0.However, for TE modes, the shorting-plane boundary condi-tions do not allow solutions for n = 0, and hence radiationexactly along the x axis is not allowed. This is an artifact ofthe shorting-plane approximation, and obviously radiationalong the x axis is allowed.
Boundary ConditionsIn all the calculations discussed here we assume that a guid-ed wave, for example, the lth mode, is incident from the leftonto the structure shown in Fig. 1(b) at some angle Oi withrespect to the x axis. The field components for this generalcase can be written as
Einc(r) = Einz(z)exp(-i3 cos Oi- i sin Oiy), (A16a)
Hinc(r) = Hj1 c(z)exp(-i3 cos Oi - i sin Oiy), (A16b)
where /3 is determined from Eq. (A4) [or Eq. (A8)] for the TE(or TM) case. Here,
Einc(z) = SiE(z), (A17a)
Hinc(z) = SiHl(z), (A17b)
where Si is the rotational matrix corresponding to the inci-dent angle Oi and is given by
E(r) = > rmErm(z)exp(im'x - i3o sin Oiy),m
H(r) = E rmHrm(z)exp(i'x - il sin OSy),m
E(r) = E tnEtn(z)exp(-i/3'nx - i31 sin Ojy),n
H(r) = 7 tnHtn(z)exp(-iOn'X - i sin Oiy),
where rm and tn are the mth reflection coefficient and the nthtransmission coefficient. In Eqs. (A19) and (A20) the prop-agation wave vectors along the y axis are determined bySnell's law in the plane of the surface. In Eqs. (A19), thefield components of the mth reflected mode are given by
with
Sm =
where
and gm is given by Eq. (A4) [or Eq. (A8)] for TE (or TM)modes. The field components Em(z) and Hm (z) in Eq. (A21)are defined by Eq. (A2) [or Eq. (A7)] for the mth TE [or TM]mode. In Eq. (A20), the field components of the nth trans-mitted mode are given by
cos Oi -sin Oi 0
Si = sin Oi coS i O l0 0 I
(A18)
The field components of El(z) and Hl(z) in Eqs. (A17) aredefined by Eqs. (A2) [or Eqs. (A7)] for the Ith TE (or TM)mode.
When the incident guided wave encounters the disconti-nuity at the transverse interface of the waveguides, all themodes in the waveguide(s) and/or uniform medium can beexcited. For example, if a TE guided wave is directed atnonnormal incidence upon a transverse boundary, it can inprinciple excite both TE and TM waves. (At normal inci-dence there is no polarization mixing.) In practice, we
(A24b)
where
On
Sn = /1 sin °i
On
0
-3 sin
n 0
/3n0 1
(A25)
with
(A26)
For a waveguide on the transmission side, n in Eq. (A26) isgiven by Eqs. (A4) and (A8) for TE and TM modes, respec-tively. The corresponding field components E(z) and
where
(A19a)
(A19b)
(A20a)
(A20b)
Erm (z) = SmEm (z),
Hrm (z) = SmHm(z),
(A21a)
(A21b)
(A22)
o
O
1
13m2
= 3m2
- 312 sin 2O (A23)
En(z) = SnEn(Z),
Htn(z) = SnHn(Z),
(A24a)
/n302 = n2 - i2 sin2 °i
Shen et al.
Vol. 4, No. 11/November 1987/J. Opt. Soc. Am. A 2131
Hn(z) in Eqs. (A24) are defined by Eqs. (A2) and (A7) for thenth TE and TM modes. For a uniform medium on thetransmission side, n in Eq. (A26) is given by Eq. (All) andEq. (A14) for TE and TM modes, respectively. The corre-sponding field components En(z) and Hn(z) in Eqs. (A24) aredefined by Eq. (A10) and Eq. (A13) for the nth TE and TMmodes.
The reflection (rm) and transmission (tn) coefficients in,Eqs. (Al9) and (A20) are determined by the boundary condi-tions at the interface x = 0. These are the continuities of thetangential components of the electric and magnetic fields inthe plane x = 0 between the two shorting planes (i.e., -d < z< d). We now assume that no interpolarization (i.e., TE -TM or vice versa) conversion occurs, an assumption that wewill justify numerically in terms of power conservation forthe solutions. A set of infinite linear equations results,given by the zeros of the boundary residuals, RE(z) andRH(Z):
RE(Z) = Eincy(z) + E rmEry m(z) - tnEJn(z) = 0,In n
(A27a)
RH(Z) = Hincz,(Z) + r rH~m .(z) - t tnHz(z) = 0m n
(A27b)
for TE modes, with -d < z < d, and
RE(Z) = Eic z(z) + E rmErz)- t0Et,20(Z) = 0,
m n
(A28a)
RH(Z) = Hiney(Z) + > rmry~ m (z) - tnHty n(Z) = 0m n
(A28b)
for TM modes, also along the interval -d < z < d. In Eqs.(A27) and (A28), the field components of the incident, re-flected, and transmitted waves are given by Eqs. (A16),(A19), and (A20).
Numerical MethodsIn order to solve Eqs. (A27) and (A28) it is necessary totruncate the series expansion for the reflected and transmit-ted waves and hence to limit the number of reflection andtransmission coefficients that can be evaluated to M and N,respectively. We used two numerical methods to solve forthe M + N coefficients. In the point-matching method, weset the 2(M + N) linear equations to zero at zi selected pointsbetween -d and +d, that is,
RE MN(Zi) = 0, RH MN(Z) = 0. (A29)
In the least-squares boundary-residual method, we definethe least squares of the boundary residual as
-dI= Eddz[IRE MN~(Z)12 + n2IJRHMN(Z)I1
21, (A30)
where 1 is an adjustable weighting factor introduced so thatboth terms within the brackets have the same dimensionsand the superscripts M and N denote the largest mode num-bers included in the expansions of the reflected and trans-
mitted waves. The minimization of Eq. (A30) with respectto rm and t yields a set of M + N linear equations,
a' = 0, m = 1, 2,..., M,drmn
aI-=0, n =1, 2,..., N,atn
(A31a)
(A31b)
which can be solved easily.Once the reflection (rm) and transmission (tn) coefficients
for the Ith incident guided wave have been evaluated, we cancalculate the power that is reflected and transmitted intoguided-wave modes and that is mode converted into thereflected and transmitted radiation modes. The reflectivityback into the same mode as the incident field is given by
RI = r112 (A32)
for the th mode. The conversion coefficients for the mthreflected and nth transmitted waves are given by
p mRm= Ir 2
Pincand
m = 1, 2,.. ., Mp, m 1,
Tn= Itn 12ptninc
(A33)
(A34)
where Pic and Prm are given by Eqs. (A6) [or Eqs. (A9)] andptn is given by Eq. (A12) [or Eq. (A15)] for TE (or TM)modes. The numbers of propagating modes on the reflec-tion and transmission sides of the interface are Mp and Np,respectively. Since there is no loss mechanism included inour material system, the power flow into the transverseboundary (along the x axis) must equal the total power flowout on reflection and transmission; that is,
(A35a)R + T = 1,
Mp
R = R,m=1
Np
T = 7 Tn.n=1
(A35b)
(A35c)
Equations (A35) provide a good opportunity to test thevalidity of the solutions. They should be satisfied exactlyonly if an infinite number of modes are superimposed at theboundary. When the series is truncated at a finite numberof terms, the modal reflection and transmission coefficientsare only approximate. Therefore how well Eq. (A35a) issatisfied numerically provides a useful (but not definitive)test of the accuracy of the solutions. Additional tests areuseful to increase the number of modes superimposed at the-boundary or to increase the separation between the shortingplanes and then to check for consistency in the reflectionand transmission coefficients.
ACKNOWLEDGMENT
This research was sponsored by the U.S. Army ResearchOffice under contracts DAAG-29-85-K-0026 and DAAG-29-85-K-0025.
Shen et al.
n = 2..., NP,
2132 J. Opt. Soc. Am. A/Vol. 4, No. 11/November 1987
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