[-i] indicates the integer that is equal to or smaller thanv/2. The above expression solves our problem becausethe sum over k vanishes. The following proof for thevanishing of the inner sum over 1 has been invented byMammel. Consider the expression
k= k1 (n k)k Iw-12
Next we take the n - 1 derivative of this expression:
d fen l(l (n) ( 1) (v+ n-1-2k)1 Xd'n-q X X - = k1(,n-k)l(v-2k)1
For x = 1 the right-hand side of this expressionbecomesidentical to n1 times the inner sum of 4,o. The left-handside of the n - 1 derivative vanishes for x =1, proving thatthe sum does not contribute to Iw. We thus have finally,
1,0 = (2/iy) e-"'(I/ 2) etva
'M. A. Miller and V. I. Talanov, "Electromagnetic SurfaceWaves Guided by a Boundary with Small Curvature," Zh.
Tekh. Fiz. 26, 2755 (1956).2 E. A. J. Marcatili, "Bends in Optical Dielectric Guides,"
Bell Syst. Tech. J. 48, 2103-2132 (1969).3 L. Lewin, "Radiation from Curved Dielectric Slabs and
Fibers," IEEE Trans. Microwave Theory Tech. MTT-22,718-727 (1974).
4J. A. Arnaud, "Transverse Coupling in Fiber Optics Part III:Bending Losses," Bell Syst. Tech. J. 53, 1379-1394 (1974).
5A. W. Snyder (private communication).6A. W. Snyder, 1. White, and D. J. Mitchell, "Radiation from
Bent Optical Waveguides," Electron. Lett. 11, 332-333(1975).
7 V. V. Shevehenko, "Radiation Losses in Bent Waveguides forSurface Waves," Radiophys. Quantum Electron. 14, 607-614 (1973) (Russian original 1971).
8 D. C. Chang and E. F. Kuester, "General Theory of Surface-Wave Propagation on a Curved Optical Waveguide of Arbi-trary Cross Section," Scientific Report No. 11, Electromag-netics Laboratory, Dept. Electr. Eng., Univ. of Colo.,Boulder, Colo.; also, IEEE J. Quantum Electron. QE-11,903-907 (1975).
9 D. Marcuse, Light Transmission Optics (Van Nostrand,Princeton, 1972), 398-406.
"1D. Marcuse, Theory of Dielectric Optical Waveguides, (Aca-demic, New York, 1974).
12Reference 11, Eq. (2.2-23), p. 65, and Eq. (2.2-25), p. 66.t 3 Reference 9, Eqs. (8.2-7) through (8.2-10), p. 290.14M. Abramovitz and I. A. Stegun, Handbook of Mathematical
Functions, Eq. 9. 2.4, p. 364. U. S. Department of Com-merce, National Bureau of Standards, Appl. Math. Ser., 55.
t5 Reference 11, Eq. (2.2-38), p. 68.t tReference 11, Eq. (2.2-69), p. 73.t7I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Se-ries and Products (Academic, New York, 1965).
t8Reference 17, Eq. 8.468, p. 967.19Reference 14, Eq. 6.1. 18, p. 256.
Bragg diffraction of Gaussian beams by periodically modulated media*
Ruey-Shi ChuCommunication Systems Division, GTE Sylvania Inc., Needham Heights, Massachusetts 02194
Theodor TamirDepartment of Electrical Engineering and Electrophysics, Polytechnic Institute of New York, Brooklyn, New York 11201
(Received 3 October 1975)
Analytical and numerical results are given on the diffraction of optical beams by a periodically modulateddielectric medium, which represents a thick hologram or an acoustic column. By using a rigorousrepresentation for the field of a realistically bounded beam incident at a Bragg angle, we examine both therefracted beam and the beam due to Bragg scattering inside the periodic medium. The Bragg-scattered beam isformed by continuous coupling of energy from the refracted beam into the Bragg-scattered wave. As theBragg-scattered wave also couples part of its energy back to the refracted wave, the continuous coupling ofenergy back and forth between these two waves results in a diffusion of energy over a wide region. For asufficiently thick modulated layer, both the refracted and the Bragg-scattered beams split into two beams.Because of this distortion of the beam profiles, the diffraction efficiency is found to be smaller than thataccounted for by previous approaches using a single incident plane-wave model.
The scattering of optical waves by a periodically modu-lated layer has been studied extensively in the contextof light diffraction by sound'- 5 and in image reconstruc-tion by holograms. 4- llowovor, most theoretical in-vestigations have dealt with the fields due to a single in-cident plane wave of infinite extent rather than with real-
220 J. Opt. Soc. Am., Vol. 66, No. 3, March 1976
istic beams having bounded cross sections. By con-sidering the scattering of a Gaussian beam incident on aperiodic slab of arbitrary width, the present work showsthat the diffraction process exhibits features that arenot evident from the simpler (single plane-wave) analy-sis. In particular, the efficiency of diffraction into the
Bragg-reflected wave is found to be dependent on thebeam width, its value being generally lower than thatpredicted by the plane-wave model. Also, the profileof the scattered beams may be quite different from thatof the incident beam because of field distortion producedby the periodic medium.
The diffraction of a bounded beam by a periodicallymodulated medium has already been considered in a fewprevious studies of light scattering by an ultrasoniccolumn. 9-12 However, those past studies did not deter-mine how the optical beams progress inside the modu-lated medium; furthermore, they have simplified theanalysis by using assumptions (e. g., that the periodicitylength d is much larger than the light wavelength X),which are not always satisfied in more recent applica-tions. By concentrating on the field inside the modu-lated medium and by allowing the light wavelength to becomparable in magnitude to the periodicity of the layer,our analysis clarifies the fine structure of the diffrac-tion process and thereby reveals the limitation it im-poses on the transmitted field.
For incidence in a Bragg regime, we find that the ex-pected Bragg-scattered wave is formed by continuouscoupling of energy from the refracted wave, as pre-dicted by the plane-wave theory. However, this cou-pling process forces the fundamental (zero-order) trans-mitted beam and the Bragg-scattered beam to evolve aprofile which is no longer Gaussian. In particular, ifthe modulated layer is sufficiently thick, each one ofthe two beams develops two well separated peaks, thusexhibiting a beam-splitting phenomenon that has recent-ly been observed by Forshaw. 7 As a result of this beamdistortion, only partial energy conversion can occurfrom the incident beam into the Bragg-scattered beam.
To determine the diffracted fields for incidentbounded beams, we recall in Sec. I the analysis of thefirst-order Bragg regime for the case of a single inci-dent plane wave. We then apply these results in Sec. IIto obtain an integral representation for the field of a
zAIR REGION MODULATED REGIONg, 6 ,r 6o[VMCOS2 dzt],0)
2wo - -
: e ann 3
Gaussian beam and express this field in terms of sever-al components having different physical characteristics.By utilizing numerical results for Bragg scattering inSec. III, we discuss the two-beam phenomenon, showhow the beam energy diffuses inside an angular region,and present quantitative aspects of this process.
I. PLANE-WAVE INCIDENCE IN THE FIRST BRAGGREGIME
As shown in Fig. 1, we consider a bounded beam in-cident on a periodically-modulated dielectric region,which is characterized by a permittivity of the form
,E(z, t) =, ,1 - Mcos[(27r/d)(z - t)]} 1
where Er is the relative permittivity in the absence ofmodulation, M is the modulation index, d is the period-icity, and v is the speed of the sound wave. For thecase of a hologram we set v = 0.
The first Bragg angle OB for a space-time periodicmedium is defined by13
sinGB =(X/2d)+ (ErV/C)[1 -(v/2c)X/d] , (2)
where X and c are the wavelength and the speed of theincident light wave in the exterior (air) region. Becausec >> v, we can neglect the second term in Eq. (2), so thatwe shall assume henceforth
sinO = X/2d . (3)To measure the deviation of the incident angle from
the Bragg angle, we use a parameter s defined as
s=(2d/X) (sinO- sin O)=go3d/vr- 1,
where 0 is the incidence angle and
30 = ko sinO, with ko = 2 7r/X .
Incidence in the first-order Braggif Is I< 2q, with
q = 2(d/X) 2 ME«<< 1 .
(4)
(5)
regime then occurs5
(6)
Refracted beam
FIG. 1. Geometry of ax Gaussian beam incident on a
periodically modulatedlayer.
Diffracted beam
x=O x=L
221 J. Opt. Soc. Am., Vol. 66, No. 3, March 1976 R.-S. Chu and T. Tamir 221
For plane-wave incidence in the first-order Bragg
regime, Chu and Tamir have shown 5 that only two waves
are significant in the modulated region, namely, the re-
fracted wave and the Bragg-scattered wave. For an in-
cident plane wave of the form exp [ik,(x cosO +z sine)
- w0 t] and for perpendicular polarization, the electric
field of the refracted wave is then given by
(7)
II. INTEGRAL REPRESENTATION FOR BEAMS IN THEMODULATED REGION
As is well known, 14 the electric field of an incident
beam can be represented by
Ei~(Xz Z; t) = I G(p0 ) exp [i(tox+ 03- z - co0 t)] doo , (19)
where, for a beam with Gaussian profile, we have
whereas the electric field of the Bragg-scattered wave
is given by
9(x, z ; t) = Vl,(x) exp[i( P1z - w-1 t)] -
Here w0 is the angular frequency of the incident wave
wno= w0 + 2nvrv/d,
3 =Igo+ 2n7T/d ,
(8)
(9)
(10)
and n=0, ±1, ±2, . Of course, wn is close to w0
for small Ink.
When a wave is incident on the left-hand boundary of
the periodically modulated layer, energy is refracted
into the layer and it subsequently reaches the right-
hand boundary, which causes reflections. In order to
examine the energy flow, we can neglect these reflec-
tions by assuming that the right-hand boundary recedes
towards x-0 0 so that the layer becomes infinitely thick.
If desired, the effect of the right-hand boundary can
later be assessed by simple considerations.
The neglect of reflections simplifies the derivation of
the "voltage" amplitudes V0(x) and V .,(x). These can be
derived by setting the reflection coefficients rn at the
boundary equal to zero in Ref. 5, thus obtaining
'0(x) o O(I30) [c s s) - iy ) sin(
l(x)= - ii7o(0 o) [(7r/Q) sin(Qx/D,)] exp(it 0 x)
where
Fo(po) = 2 o/Ato + W0)
Q = (i/q) (q2 + 4s 2 )1/2
D= (2d/q) Zo d/l,
To=(rk2Uo3)"2,E _ ( 7 /d k ] /2 = o) v
Zo =[kor _ (/)]=t o/d
(11)
(12)
G(3o) = (wo1/{) exp [ - (IBM - b)2 w0
b = ko sin 0 ,
(20)
(21)
and 00 is the angle of incidence of the beam axis. Thus,
for incidence at exactly the Bragg angle, we have b= ir/d.
The integral representation (19) and Eq. (20) de-
scribe an incident Gaussian beam of width 4wo mea-
sured along the z axis. This beam appears here as a
linear superposition of plane-wave spectral components
of the form
exp[i(Q0 x+ 0z - wo)]
with amplitude G(g0). For each plane-wave component
incident upon the periodic layer, the field in the modu-
lated region has been given in the previous section;
thus, the electric field within the modulated region due
to the incident beam of Eq. (19) is merely a linear field
superposition given by
(22)
where
z Z; t) = X G(p0) V0(x; i0) exp [i(3o z - c 0 t)] d1o , (23)
E 1 (x, z; t) =fJ G( 0))V, (x; i0) exp[i(3, z - w-1 t)] dpo 0
(24)
Because of the meaning of the amplitudes f'0(x) and
(13) V l(x), the quantity Eo represents the fundamental field,
(14) which we denote as the refracted beam, whereas the
quantity E-1 represents the first-order diffracted field,
(15) which we denote as the Bragg-scattered beam.
(16) It is now convenient to further separate the refracted
/. _\ beam Eo into two components,( 1 8)
(18)
In obtaining Eqs. (11) and (12), terms of order q2
or smaller were neglected. The above results are con-
sistent with those obtained by others, as described in
the literature.',2 ' It can be seen from Eqs. (11) and (12)
that when x=(2m- 1) D, (with m= 1,2,3,...) and for
incidence at exactly the first Bragg angle (i. e., s = 0),
we have f50(x) = 0 and V l1(x) = Fo(v/d). Because the re-
fracted wave vanishes under those conditions, complete
conversion of energy from the refracted wave into the
Bragg-scattered-wave is theoretically possible in the
case of plane-wave incidence. However, we shall see
in Sec. III that complete conversion of energy betweenthese two waves can no longer be expected in the case
of bounded-beam incidence.
222 J. Opt. Soc. Am., Vol. 66, No. 3, March 1976
fo(x, Z ; t) = ffo(x, Z ; t) + f,(x, z; ) (25)
where Eo denotes the refracted beam that would appear
in the absence of the periodic modulation, i. e., when
M= 0. Hence Et then represents the field perturbation
due to the presence of periodicity. Obviously, E., is
also a field perturbation which appears whenever My 0
because E..1 must vanish in the absence of periodicity,
i. e., E,1 - , = 0 as M- 0. By virtue of these defini-
tions, we have
zo(x z; t) = f G(j3o)0 O(0 ) exp [i( x + go z - w0 t)] d3o .
(26)
The perturbation fields E' and E., may, after some
rather lengthy manipulations, be cast into the followingforms:
R.-S. Chu and T. Tamir. 222
_i�o(x, z ; t) = flo(x) exp [i(po z - Wo 01 ,
f(X, Z ; t) = fo(X, Z ; t) + t_,(X,,Z ; t) 9
'(x, z ; t) = VO Io(x, z) exp {i [ Z0 x + (7/d)z - coo t] } ,
E -(xz z;t) = - i V0I.l(x, z) exp{i[ 0 x - (7r/d)z - co- t]}
(28)where
Vo= Vo (,I d) n(29)
-' (X, 2= ho (x,z)* (27T/d*F(z), (30)
with
°( ) 2 (xtan - z)
X11 7qr (X2tanS _ Z2)1/2) U (x tan0B - z ,(3 1)
h-,(x, Z) = 2Tq Jo(- (x2 tanB - Z2) 2
xU(xtan0B- IZ|), (32)tan0B = 7T/Zo d . (33)
Here 0 B is the Bragg angle inside the periodic mediumand Jn(u) is the nth-order Bessel function of argumentu; U(W) is the unit step function, which is zero forW< 0 and unity for W> 0. The asterisk multiplicationin Eq. (30) denotes the convolution operation defined by
f(z)*g(z)= f(z- z)g(z)dz= ff(zt)g(zgz-.z')dz'
(34)whereas the last function in Eq. (30) is given by
F(z) = f GQ0 ) exp [i (i0 d - 70) z/d] d~o
=f r (s)exp(iwTsz/d)ds, (35:
so that F(z) is the function describing the profile of theincident beam in the plane x=0, and 6(s)=(d/w)G(P,).
The above representations of fields in the form ofconvolution integrals are suitable for numerical inte-grations for any given incident beam-profile F(z). Fora Gaussian beam incident at an angle 00 with respect tothe x axis, the function G(l 00) is given by Eq. (20). Bysubstituting this into Eq. (35), we obtain
F(z) = exp [- (z/2wo)2] . (36)
The term given by Eq. (26), i. e., the refractedGaussian beam in the dielectric region when the modu-lation vanishes, can be evaluated by a saddle-point in-tegration. If we are interested only in the paraxial re-gion, such an evaluation leads15 to the asymptotic re-sult
____ (z - x tan~_o)2 '
.Eo(x, z w; t) W( w) V -exp ( 4w'(x )
x exp [i'.FE7 ko(x cos 0 + z sinS0 )] exp(- iRo t)
(37)where Do is the angle of refraction defined by Snell'slaw
sinO0 = ' sinj 0 (38)and
223 J. Opt. Soc. Am., Vol. 66, No. 3, March 1976
(39)
The form of Eo is a Gaussian beam with an effectivebeam width of 41 w(x; 70) I measured along a constant xplane. From Eq. (39), we restrict the well collimatedregion to x<LccosO0 , where L0 is the distance alongthe beam axis for which w(x; 00) has equal real and im-aginary parts, i. e.,
Lc = T(2wo cosZ0 )2
X/V-, (40)
In most practical cases, the observation point x satis-fies x<< L, cos80 so that Eq. (37) can then be well ap-proximated by
z - tano )2]E0 (x, z ; t) - ~V0 exp[P( -tano)
x exp[ iVk, (x cosS0 + z sinS0 ) - iRo t] . (41)
To examine Eo of Eq. (27), i. e., the perturbation ofthe refracted beam, we introduce Eq. (36) into Eqs.(27)-(35) to find
Eo(x, z ; t) = - V exp [(0 x + dT - COOt)
x tanB qf(Z - z)(xtan0B + Z "1/2-_X taOB4 xtan6B - z)
XJi(7jf (X2tan2EB _ z 2)1/2) dz',
where
f exp (Z- Z) sin(27rqz'/d) dz'f. , ?. ) fz'/d
(42)
(43)
A similar derivation for E ,(x,z ; t) of Eq. (28), i.e.,for the Bragg-scattered beam, yields
E i(x, z; t) = - iVo exp [i( 0 x .. - w-1t)]
X x taneB qf(Z -Z')fx tanG
5
xJi 'T0 (x 2 tan2 B - z1 2)1 /2) dz' . (44)
Equations (42) and (44) are the integral representationsthat are suitable for the numerical evaluation of the re-fracted and the Bragg-scattered beams.
III. BEHAVIOR OF BEAMS INSIDE THE MODULATEDREGION
To examine the progression of the fields in the in-terior of the periodic medium, we have carried out nu-merical evaluations of Eqs. (42) and (44) by means of acomputer program using the Gaussian formula. In or-der to discuss the principal features of the diffractionprocess, we present here a typical illustration forwhich we have chosen Er = 2.25 (quartz), X = 0. 6328 gm(wavelength of He-Ne laser), d= 1 gm, and M=10-4 .Thesequantitiesyield q=1.156x10n3 , and Bragg anglesOB= 18. 170 and WB = 12. 00°. For convenience, a value ofwo= 7wwo/d= 1000 was selected, which implies that thebeam width 4wo is 1. 27 mm.
R.-S. Chu and T. Tamir 223
(27) (1 . 1 X 1/2W(X;,jo) = Wo + Z - -
2 �,E, ko w ' cos'Wo )0
Modulated Region
FIG. 2. Characteristic re-gions showing the refracted
X and the Bragg-scatteredbeams for incidence at theBragg angle OB-
X=B
X= 8I-'
X= Di4 30i
Di
Computed results for the field amplitudes have been
calculated along four different locations, namely,
x= 8D1 , 4D1, 3D1 , and 2D 1, as illustrated in Fig. 2. As
mentioned in Sec. I, the physical significance of the
quantity D1 is that, for plane-wave incidence at the first
Bragg angle, complete conversion of energy from the
refracted wave into the Bragg-scattered wave happens
when x= (2m - 1) 2D1 , with m = 1, 2, 3, ... . The field
amplitudes at these four locations are shown succes-
sively in Figs. 3(a)-3(d), where the normalized quan-
tities IE 0/V 0I, IE 1 /Vo1 I1E 0/VoI, and IEo/VoI are
plotted.
By examining the curves in Fig. 3, we observe that
the fields are confined mostly inside a region between
the two planes z = ± (xtan0B + 2w 0 ), which are indicated
in Fig. 2. Outside this truncated wedge region, the
field intensities are relatively small. Analytically,
this behavior is confirmed by the two functions ho and
h- 1, which appear in Eqs. (31) and (32). These func-
tions include a unit-step function that confines their do-
main within a region given by I zI <xtanOB. Because ho
and h-1 convolve with the Gaussian-profile function hav-
ing an effective half-width of 2wo, the appearance of
fields having negligible intensities outside the above
wedge region is expected.
Another significant feature of the diffracted fields is
that the shapes of the refracted and Bragg-scattered
beams are quite different from that of the incident
Gaussian beam. The distortion of the beam profile and
the field confinement inside the domain discussed above
can be explained in terms of the classical Bragg-scat-
tering mechanism, which has in the past been employed,
however, strictly in the context of a single plane wave
of infinite extent. This mechanism implies that, as the
refracted wave enters inside the modulated region, its
energy is scattered by the periodicity (striation) planes
at z = nd that describe the periodic variation of the re-
fractive index. Due to constructive interference, the
scattering process is strongest for incidence at the
Bragg angles 0B, with the scattered field progressing
224 J. Opt. Soc. Am., Vol. 66, No. 3, March 1976
along the direction - jB. However, this scattered field
undergoes a secondary Bragg-scattering process, which
channels energy back along the + WB direction. Of
course, this energy is again scattered and the scatter-
ing process continues but is subject to constructive in-
terference only along the two ± OB directions.
The above Bragg-scattering mechanism has been in-
vestigated rigorously by Singer and Tamir-Berman, 16
who examined the field produced by a small localized
source embedded inside a periodic medium of the same
form as that discussed here. The situation for such a
case is described in Fig. 4, where the source is as-
sumed to be at the origin and the periodic striations are
indicated by dashed lines. Energy that emerges at the
Bragg angle 0B travels along the plane z = xtanOB and is
scattered along planes parallel to z = - xtanOB. Simi-
larly, energy emerging at the Bragg angle - jB is scat-
tered along planes parallel to z = xtaniB. As this pro-
cess occurs at every point, it is clear that the energy
flow can be described by all the criss-crossing flux
lines in Fig. 4. A simple geometric consideration
makes it evident that these flux lines are confined to the
interior of the wedge region of angular extent 2 DB shown
in Fig. 4. If we now apply these considerations to the
incident beam shown in Fig. 2, we may regard all
points on the z axis to be localized sources. However
only those points that satisfy - 2wo <z <2wo represent
sources having a significantly large amplitude. Hence,
the field in this case is effectively confined within an
irradiated region bounded by z = ± (xtanjB + 2wo) rather
than within the smaller domain z= xtanOB shown in
Fig. 4.
Because of the scattering mechanism described in
Fig. 4, energy traveling along the + jB direction is con-
tinuously coupled into energy traveling along the - oB
direction. Thus, for the incident beam shown in Fig.
2, which refracts energy along the + WB direction, a
Bragg-scattered beam is gradually produced along the
- JB direction. However, the continuous coupling of en-
ergy between the two beams produces diffusion of the
tirely into the Bragg-scattered beam, as discussed atthe end of Sec. I. However, Fig. 3(d) shows that IE../
o I has a peak of about 0.3, whereas the peak of JE 0 /VoI reaches close to 0.7. Judging from these values,the conversion efficiency is well below 50% instead ofthe 100% efficiency predicted by a model involving asingle plane wave.
The principal effect produced by the distortion in thebeam profile is that the two (refracted and Bragg-scat-ter beam) are ultimately modified so that each one ofthem splits into two separate portions. Such a beam-splitting phenomenon is clearly seen in the case of therefracted beam, which consists of two separate portionsin Figs. 3(b)-3(d). These two portions are in antiphaseand their peaks approach each other in magnitude as thebeam progresses inside the modulated medium. Al-though less pronounced, an analogous beam-splitting ef-fect also occurs for the Bragg-scattered beam. Thus,in Fig. 3(d), the Bragg beam exhibits already two dis-tinct peaks. Although not shown here, the minimum be-tween the two peaks decreases for x> D1 and the Bragg-scattered beam is expected to have a null similar to theone shown by the refracted beam. It is this beam-splitting phenomenon that has been observed most re-cently by Forshaw. 7
To summarize the above discussion, we observe thatthe diffraction of a light beam by a periodically modu-lated dielectric layer is accompanied by distortion inthe profile of the diffracted beams. The distortion isdue to coupling of energy back and forth between the re-fracted and Bragg-scattered beams, which results in abroadening and, ultimately, in a splitting of thesebeams. This process is accompanied by a loss in theefficiency for converting the energy of the incident beaminto that of the Bragg-scattered beam. Consequently,the theoretical 100% efficiency predicted by models us-ing a plane-wave incident cannot be realized in the caseof realistic beams of bounded extent.
- dd
d-T-
0 1000 2000 13000 4000 (7rz/d)xtanUB 9Z
FIG. 3. Field intensities of the refracted and Bragg-scatteredbeams at various planes inside the modulated region.
energy over the entire irradiated region. This diffu-fusion in turn causes a distortion of the Gaussian pro-file of the refracted beam. As a result, the Bragg-scattered beam also has a profile that is quite differentfrom Gaussian. Another important effect of this diffu-sion process is that complete energy conversion fromone beam to another does not occur. Thus, at x=2D1 ,a single plane wave would have its energy converted en-
225 J. Opt. Soc. Am., Vol. 66, No. 3, March 1976
z - X tanOx
Z =-xtanOB
FIG. 4. Mechanism of Bragg scattering for energy emergingfrom a small source embedded inside a modulated medium.
R.-S. Chu and T. Tamir 225
* -2w0 f 2w0 1
I , I
I ----
-3000 -2000-xtangB
- - - - - - - -
*Work supported by the Joint Services Electronics Program
under Contract No. F44620-69-C-0047. Dr. Chu was at the
Polytechnic Institute while the work was carried out.1M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon,
New York, 1965), Chap. 12.2M. V. Berry, The Diffraction of Light by Ultrasound (Aca-
demic, New York, 1966).3W. R. Klein and B. D. Cook, IEEE Trans. SU-14, 123 (1967).4 H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).5R. S. Chu and T. Tamir, IEEE Trans. MTT-18, 486 (1970).6 C. B. Burckhardt, J. Opt. Soc. Am. 56, 1502 (1966).7M. R. B. Forshaw, Opt. Commun. 12, 279 (1974).
8S. F. Su and T. K. Gaylord, J. Opt. Soc. Am. 65, 59 (1975).9M. G. Cohen and E. I. Gordon, Bell Syst. Tech. J. 44, 693
(1965).1 0L. E. Hargrove, J. Acoust. Soc. Am. 43, 847 (1968).11R. J. Hallermeier and W. G. Mayer, J. Appl. Phys. 41,
3664 (1970).12D. H. McMahon, IEEE Trans. SU-16, 41 (1969).13R. S. Chu and T. Tamir, Proc. IEE 119, 797 (1972).14 L. M. Brekhovskikh, Waves in Layered Media (Academic,
New York, 1960), Sec. 8, p. 100.15R. S. Chu, The Diffraction of Bounded Electromagnetic
Beams by Periodically-Modulated Media, Polytechnic Insti-
tute of Brooklyn, N. Y. Ph. D. Dissertation, 1971 (Univ. Mi-
crofilms No. 71-29042, Ann Arbor, Mich.).16B. Singer and T. Tamir-Berman, J. Opt. Soc. Am. 60,
1640 (1970).
Effects of a thin overlying film on optical waveguides and couplers*
Denis Vincent and John W. Y. LitLaboratoire de Recherches en Optique et Laser, Universit6 Laval, Qu6bec, P.Q., Canada GIK7P4
(Received 31 May 1975)
The effects of a thin film on the propagation of waves inside an optical waveguide are examined. The cases
considered are planar guides, rectangular embedded guides, and rectangular ridged guides. Couplers formed by
such guides are also studied. Analytical results together with some numerical and experimental results are
presented.
The effects of a second thin film deposited on an optical
thin-film waveguide has been attracting the attention of
a number of research workers. Some used the effects
to make filters, 1 couplers, 2 and waveguides. 3 Most
recently, Tada and Hirose used them to modulate a
light beam. 6 Tien used an overlaying thin film to inter-
connect two planar waveguides.
In this article, we report a preliminary study of the
effects of such a film on the propagation of waves in-
side a planar or rectangular optical guide, and also on
the coupling between two optical guides. The effects of
this thin film will be studied as functions of its thick-ness for given refractive indices. A few numerical
examples will be given for planar guides, rectangularembedded guides and rectangular ridged guides to illus-
trate the effects. More complete results will be pre-sented later.
THEORY
A cross-sectional diagram of the system studied isgiven in Fig. 1. The three cases are shown in Fig. 2.
In order to calculate the fields and their wave vectors
inside the guides, we use the classical Maxwell equa-
tions, assuming no source. The optical dielectric con-
stant is n2 (x, y), given by the constant refractive in-
dices in each of the optical regions considered. The
media are assumed to be lossless and nonmagnetic.
By assuming the existence of a mode with wave-vectorcomponent k, and field harmonics A(x, y) expj(wt- kz),
we can derive the well-known wave equation
where v denotes the region in which the field is being
considered. To obtain the TE and TM modes that can
be supported by the planar guides [Figs. 2(a) and 2(b)],we put, respectively, E,, and HI equal to zero. To ob-
tain the E', and El', modes [Figs. 2(c)-2(f)] of rectan-
gular guides, 8 we put EVY and H,,, respectively, equalto zero. Having done this we can express all other
wave components of E, and f,, in terms of one single
component, if the Maxwell equations are used
(2)v X =
=V i con1 nV
v0- = | . (3)
We suppose that, inside the guides, the fields in the x
and v directions are sinusoidal; thus
E(x, v, z, t) - sin(k, x + a) sin(k, y + A)
(4)x expj(wt - kz) .
Outside the guide, in the surrounding media, the eva-
nescent waves can be represented by
(exp (- pV x) sin(ky + 8))
E(x) y, z, t) c( exp(- pV y) ) (sin(k~x+ a))
x expj(wt - kz) . (5)
2 2 2 = °,
226 J. Opt. Soc. Am., Vol. 66, No. 3, March 1976
By use of the conditions of continuity of the tangential(1) components of E and H at the boundary surfaces, we ob-
tain the wave-vector components kx and ky given by the
