The power distribution as a function of propagation distance in a network of coupled optical waveguides is determined for several interesting cases. An electrically controllable coupler is proposed and analyzed in detail. High efficiency coupling and decoupling between two optical guides can be accomplished with the use of an electrooptically generated dynamic channel, of finite length, located in between the two guides

Elachi, C.

Yeh, C.

English

Caltech Authors - Main

Distribution Networks and Electrically Controllable
Couplers for Integrated Optics
C. Elachi and C. Yeh
The power distribution as a function of propagation distance in a network of coupled optical waveguides
is determined for several interesting cases. An electrically controllable coupler is proposed and analyzed
in detail. High efficiency coupling and decoupling between two optical guides can be accomplished with
the use of an electrooptically generated dynamic channel, of finite length, located in between the two
guides.
1. Introduction
Recently, a number of researchers" 2 have reported
the development of thin film and channel waveguide
optical couplers for use in the emerging field of inte-
grated optics. Applications of these couplers in op-
tical networks, modulators, and multiplexers/demul-
tiplexers, would be drastically increased if the cou-
pling is dynamically controllable by an electric sig-
nal. Electrooptic substrates can be used to control
the coupling coefficient between two waveguides, but
such a scheme would be inefficient because of the
upper limitation on the change of the refractive
index of existing materials that could be achieved
with reasonable voltage. In this communication, we
study the power distributions as a function of the
distance from the input plane in a network of N par-
allel guides. Then we discuss a number of functions
that could be achieved using coupled optical wave-
guides, and we will study in detail a scheme for an
electrically controllable coupler.
I. Symmetric and Nonsymmetric Optical Networks
Let us consider N identical optical waveguides
with K being the coupling coefficient between two
neighboring guides. The field En in the nth guide is
determined by the system of equations:
(dEn)/(dz) = -iKEn+l - iKEnj for 2 < n < N - 1,
(dE1)/(dz) =-iKE 2,
(dEN)/(dz) =-iKEN-l,
where we have neglected the direct coupling between
C. Elachi is with the California Institute of Technology, Pasa-
dena, California 91109; C. Yeh is with the University of Califor-
nia, Los Angeles, California 90026.
Received 21 September 1973.
nonneighboring guides. If the input light is fed into
the mth guide, the normalized initial condition is
En(O) = 1 for n = m.Et0) ifor n -#m.
The above system of equations can
matrix form as follows:
(d/(d4)E = M-E
E(O) = c,
where t = Kz and
E = 2
be written in a
(1)
c = i (- mth element
0 1 0 .... 0
/ 0 .... 0\
M=-i 01 0-)
0 1 .0
Equation (1) is a well known differential equation 3
that can be solved by determining the eigenvalues
and eigenvectors of the matrix M.
For N - -, the solution of Eq. (1) is the well
known Bessel functions:
En = (-i)n-m n~m,(20).
For N finite, the solution can be determined in a
straightforward manner with a digital computer. In
Fig. 1 we present the power Pn = EnEn* for a num-
ber of configurations (N = 2, 3, and 5) with different
input conditions. These various configurations can
be used to perform a number of functions in optical
networks. Some of the possible applications are
shown in Fig. 2 and .discussed below.
1372 APPLIED OPTICS / Vol. 13, No. 6 / June 1974
3( (e) - 2
P 3
n/2 2
I
-3
1
n/ 2
n
I 6/9 
'3
Fig. 1. Power distribution in coupled optical networks. P is
the power in the nth guide as a function of the propagation dis-
tance z. K is the coupling constant between neighboring guides.
Nonneighboring guides are assumed to be uncoupled. For the
case where there is more than one input, these inputs are as-
sumed to be in phase.
Figure 2(a) shows an energy transfer function con-
figuration. Complete transfer occurs if the coupling
length is an odd integer of 7r/2K [see Fig. 1(b)]. Fig-
ure 2(b) shows an energy divider configuration. The
input energy is equally and completely divided be-
tween the two outputs if the coupling length is 7r/
2K/2 [see Fig. 1(c)]. An elementary ligic system
is shown in Fig. 2(c). The two inputs A and B are in
phase, and the output is given by the truth table.
Figures 2(d) and 2(c) correspond to energy transfer
from two inputs to two outputs, where the useful in-
formation is the amplitude or the frequency of the
signal (assuming that the two frequencies w and w2.
are not very different so that K is approximately the
same for both). In Figs. 2(f) and 2(g) we present a
possible configuration for an electrically controllable
coupler or switch that is discussed in the next sec-
tion.
111. Electrically Controllable Coupler
The basic configuration for an electrically control-
lable coupler is shown in Figs. 2(f) and 2(h). The
two permanent channel guides are imbedded at the
surface of an electrooptic substrate. They can be
formed by proton bombardment,4 ion implantation,
diffusion, or other techniques. The two guides are
located such that the direct coupling is very weak.
In the region between the two guides, a third chan-
nel of finite length is dynamically generated by
applying a voltage to the two electrodes shown in the
figure. The resulting electric field generates a local
change in the refractive index. The feasibility of
such an electrooptically generated channel guide was
recently reported by Channin.5 This controllable
channel plays the role of a bridge between the two
permanent guides. The electrodes should be located
such that the cross section of the dynamic guide is
similar to the cross section of the two permanent
guides.
The power distribution as a function of the propa-
gation distance in a three guides system, where the
energy is fed in the first guide, is shown in Fig. 1(d)
and is given by [from Eq. (1)]:
P1(z) = 4[cos(KV2z) + 1]2,
P 2(z)= 2[sin(Kf2z)]2,
P3 (z) = 1[cos(KV2z) - 1]2,
where we assumed that the direct coupling coeffi-
cient K' between the two permanent guides is <K.
Complete energy transfer between the two perma-
nent guides occurs if the controllable channel has a
length L = 7r/KV/2.
In the absence of the bridge channel, the power
distribution in the two permanent guides is
P1'(z) = cos2(K'z),
P2'(z) = sin2(K'z),
and the energy transferred over the length L is
AP = sjn2[(r/f2)(K'/K)].
Therefore the dynamic efficiency of the coupler can
be defined as
= 1 - AP = cos2[(7/12)(K'/K)].
The analytic expression of the coupling coefficients
was derived by Marcatilli 6 as
c,/2K 
I E T
ENERGY TRANSFER
(b) -/2_ K
'- 1/2
l t__ -1/2
ENERGY DIVIDER
(C) A I A 0 C
_ _ _ C 1 0 1
B - 0 1 2
ir1/2I/2 K 
LOGIC NETWORK
(d) -/K1 (e) w2 /K
1_ -1 -2 -2
_1 .1
(1) 7r/,/2 K
I - 0
0 _ 
(h)
GUIDE 1GUIDE 2
G EECTROTO TICALLD
GENERATED GUIDE
(9) / K
41 2
-2 .1
Fig. 2. Different configurations of optical network that could be
used in energy transfer, energy distribution, controlled
switching (see text).
June 1974 / Vol. 13, No. 6 / APPLIED OPTICS 1373
(0)
II
rl/2 K,
I
11 ~~~~~ 3
4/92
41/319 2
4/9
n1/ 3
(b) - I
2
n/2
P2
-33
1/2, 2
1' 
P3
11/2 3
ni +~-
- .-t n(l(+Y) a
-I-_. nlil+Y) I a
D n1 f
a)
ELECTRO-OPTIC X/
LAYER
SUBSTRATE
PERMANENT
CHANNELS
/ CHANNEL\
+ 
I1il r R
I
Fig. 3. (a) Simplified (ECC) for thin film
guides. is the percentage change of the index
of refraction in the permanent guides. -y is also
taken as the percentage change due to the elec-
trooptic effect. (b) Another possible configura-
tion for an ECC.
7 ELECTRODES
b)
0 1 2 3 4 5 6 7
a/
Fig. 4. Dynamic efficiency and effective length
of the ECC shown in Fig. 3(a), as a function of
a/X, for different values of no and -y. The value
of D/a is taken equal to 1.5.
K = [(2s2b)/(s2 + 2)] exp[-b(D - a)]}l(Ka)
and
K' = [(2S26)/(S 2 + 62)] exp[-6(2D -a)]I(Ka),
where a is the width of each channel, D is the dis-
tance between the center lines, K and s are, respec-
tively, the propagation constants along and perpen-
dicular to the propagation direction in the coupler
plane, and 3 is the exponential full off between the
guides. The above expressions were derived for
well-confined modes, but they may be used as a good
approximation in the general use. Using the above
expressions of K and K' we can express L and 77 as
L = [7r/(2V2)][(s2 + 6 2 )/(s 2 6)]Ka exp[6(D - a)],
77 = cos2[(r/V2) exp(-3D)].
We carried out a numerical study of the simplified
coupler scheme shown in Fig. 3(a), where the guides
INTEGRATED
DETECTOR
E.C. GUIDE …==_=
1 2
LrL
r~~ -
1
_'1n_
2
ni -
_ _ …_ 
12 3 4
-J Lru1 _ _
L 
3 4JiL
AIN CHANNELMl
3
- r -
4
-ni
Fig. 5. Possible configuration for a 4-channel optical demulti-
plexer. The dashed guides are electrically controllable.
1374 APPLIED OPTICS / Vol. 13, No. 6 / June 1974
10uu
1600
1400l-
1200[
1000
N
or--.. I . . ..
- ni = 1.5 bb)
___ -nj = 3.5
-= 00.005 = -a /I'0.0 )O.001
I I
- I I
I I I
I I
I / / =0.001
- I A 
I 
- I
I /
- - I I I I I
800 
600 
400 
200
0
. . .
-_-_---_-
L * *-L
are thin film layers. n is the index of refraction of
the substrate, and y is the percentage increase gen-
erated when a voltage is applied. y is also taken as
the percentage index increase in the permanent
guides. In Fig. 4 we plotted i7 and L/X as a function
of a/X for a fixed value of D/a, where is the optical
wavelength in vacuum. It is clear that high efficien-
cy (77 90%) is possible at high frequencies, and
larger values of n1 or y lead to a wider region of high
efficiency. On the other hand, the value of L/X in-
creases with a/X. To illustrate, let us choose n =
3.5, = 0.001, and X = 1.15 A. For an efficiency of
90%,
a = 2.3 , D = 3.45 u,andL = 304 Ii.
If we want to increase the efficiency to 99%,
a = 3.4 , D = 5.1 , and L = 920 p.
For shorter wavelengths, the above numbers are pro-
portionally smaller. In Fig. 3(b) another scheme for
an electrically controllable coupler is shown.
IV. Conclusion
The above results show that some simple function
and efficient dynamic switching are possible over rel-
atively short distances. The dynamic coupler may
play a central role in complex optical networks and
multiplexers/demultiplexers. A scheme for a 4-
channel multiplexer/demultiplexer is shown in Fig.
5.
C. Elachi is also with the University of California
at Los Angeles and the Jet Propulsion Laboratory.
This research was funded partly by NECL.
References
1. A. Ihaya, H. Furuta, and H. Noda, Proc. IEEE 60, 470 (1972).
2. S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G.
Hunsperger, Appl. Phys. Lett. 22, 46 (1973).
3. R. Bellman, Introduction to Matrix Algebra (McGraw-Hill, New
York, 1970).
4. E. Garmire, H. Stoll, A. Yariv, and R. G. Hunsperger, Appl.
Phys. Lett. 21, 87 (1972).
5. D. J. Channin, Appl. Phys. Lett. 19, 128 (1971).
6. E. A. J. Marcatilli, Bell Syst. Tech. J. 48, 2071 (1969).
7. H. F. Taylor, J. Appl. Phys., 44, 3257 (1973).
June 1974 / Vol. 13, No. 6 / APPLIED OPTICS 1375


Distribution networks and electrically controllable couplers for integrated optics

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Distribution networks and electrically controllable couplers for integrated optics

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