Optical Sciences Center, University of Arizona, Tucson, Arizona 85721
Received December 30, 1981
We analyze the operation of a tunable device that is composed of two channel waveguides that are fabricated ona nonlinear substrate and of two adjacent electrodes, where the evanescent fields of the two guided modes overlap.The optical power-dependent propagation constant of the guided modes is modified locally as the power leaks fromone channel waveguide to the other, resulting in a continuous change in the local phase matching. We present ageneral expression for the operation of the device together with a numerical example that demonstrates that thedevice acts as an all-optical gate, discriminator, and limiter, having a speed that is limited only by the transit timealong the channels and the response time of the material.
INTRODUCTION
Directional couplers composed of line-channel optical wave-guides play an important role in integrated-optics devices.The tightly confined optical power in each of the two adjacentchannels enables one to obtain efficient electro-optic modu-lation of the propagation constants of the two guided modesby means of surface electrodes that are fabricated next to thechannels.1"2 The two channels exchange power as the guidedmodes propagate along the channels. One can construct aMach-Zehnder interferometer, for example, by merging thetwo channels at both ends, and one can control the interfer-ence pattern at a point where the two channels merge, thusobtaining efficient electro-optic modulation of the outputpower. 3' 4
One can also envision a subpicosecond gating device thatutilizes a directional coupler fabricated on an electro-opticsubstrate interacting with a traveling wave.5' 6 Various otherdirectional-coupler configurations have been reported in theliterature in the past years that feature efficient modulationand frequency selectivity. 7-9 Such devices were used recentlyto obtain bistable operation by feeding the output from onechannel into a photodetector and an amplifier, which in turndrives the electrodes controlling the propagation constant ofthe other channel.10" l These devices are the integrated-optics hybrid version of intrinsic focused-beam bistable de-vices,12 and they operate as a switch or as a multivibrator,having a speed determined by the properties of the nonlinearmedium and the external electronic circuits.
In all these devices the propagation constant of eachchannel was controlled externally by a pair of electrodes andwas constant all along the channel or was modified locally byan rf traveling wave.5 A discussion of a nonlinear coherentcoupler was recently given by Jensen,'13 who presented hissolution in terms of elliptical integrals. The characteristicsof his device were determined by the geometry of the twochannels and the material constants and could not be adjustedafter fabrication.
In this paper we describe a new type of tunable fast intrinsicdevice consisting of a directional coupler composed of a
power-dependent refractive-index material, which utilizes apair of electrodes that serve as a tuning element. We haverecently developed a theory that yields the power-dependentpropagation constant as a function of the geometry of achannel waveguide and the power that it carries.' 4 Thetheory, which applies to cases in which the nonlinearity is inthe channel, in the substrate, or in the cover, and was used toanalyze the bistable operation of a ring-channel waveguide,15can also be used to characterize the nonlinearity of the twochannel waveguides of the directional coupler that is treatedin this paper.
We consider two single-mode parallel channel waveguidesthat are close together so that the evanescent fields of theguided modes overlap. The local power carried by eachchannel changes continuously along the propagation directionby virtue of the coupling between the two channels, and thenonlinearity creates a local change in both the propagationconstant and the phase matching. We present the generaltheory of operation of this tunable device and obtain aclosed-form expression for the output of each channel in termsof the two inputs, taking into account possible absorption andnonreciprocal coupling between the two channels. We thengive a numerical example that demonstrates the behavior ofa typical tunable device as an all-optical gate, discriminator,or limiter, having a speed that is determined by the transittime along the channels and the response time of the nonlinearmaterial.
THEORY OF OPERATION
The geometry of the device is shown in Fig. 1. The twochannel waveguides are fabricated onto a power-dependentrefractive-index, electro-optic substrate as two photoresistchannels, or they are embedded inside the substrate as twoin-diffused channels. The dimensions of the channels arechosen to support only a single mode, and the distance be-tween the channels is chosen so that the evanescent tails ofthe guided modes overlap to produce a coupling constant k= 7r/2L, where L is the length of the channels. Two elec-
The geometry of the tunable, nonlinear, directional cou-
trodes, fabricated on top of the channels, are biased by a dcexternal field, so that the relative propagation constant of thetwo channels can be controlled externally (a Cobra configu-ration).
Let the substrate, the waveguides, or both have a localpower-dependent refractive index nj defined by
ni = n + n2,iP/area, (1)
where P is the optical power carried by a plane wave havinga given area. From the general theory of power-dependentrefractive-index phenomena in optical waveguides,1 4 oneobtains an expression for the change in the propagation con-stant KAi3 of a guided mode as a function of the total powerthat it carries, for cases in which the nonlinearity is either inthe waveguide itself or in one of the media bounding thewaveguide. For any single-mode waveguide and, in particu-lar, a channel-waveguide geometry, the change in the propa-gation constant can be written as
KAf3 = (K/3C2EO2/4P) Sf n(x, y)n 2,j(x, y)jEj 4dxdy. (2)
By using Eq. (2), one can define an effective nonlinear coef-ficient n' 2 ,1 as
KAi3 = Kn' 2 ,IP/X 2 . (3)
Here K = 27r/X, and X, Eo, and c are the optical wavelength, thedielectric constant, and the speed of light in free space, re-spectively.
The general coupled-mode equations are given by 16
CHANNEL WAVEGU IDES
Here
a3 = (a 2 + aol* + 2iM)/2,
a = (a2 - a,* - 2ib)/2,
and
W = ( 2 - kk2*)1/2.
We now neglect the absorption and consider the case inwhich the two coupling coefficients are the same. Conse-quently, the coupled-mode equations for the two channelsreduce to
and
dA,(z)/dz = ikA2(z)e2i6z
dA2(z)/dz = ikAl(z)e 2i6z.
(6a)
(6b)
Here Pi(z) = IA,(z)j 2 and P 2 (2) = 1A2(z)l2 are the power car-ried by channels 1 and 2, respectively.
The solution to these coupled-mode equations is givenby1 6
where 42 = 1 + (3/k) 2.Equations (7a) and (7b) are general in that they describe
the evolution of the power in the two waveguides for two ar-bitrary complex input amplitudes. (We assume, of course,that the two inputs have the same optical wavelength.)
We now consider the case in which 6 is power dependent,so that it varies as a function of z according to
6(z) = Kn'2,I [P(Z)A - P(Z)B - PC]/X2. (8)
Here Kn' 2 ,IPc/X2 represents the externally induced phasemismatch introduced by biasing the two electrodes. We nowscale the field amplitudes in Eqs. (7a) and (7b) and define
and
a(z) = a *(z)(Kn'2,/X2)1/2
b(z) = A2(Z)(Kn'2,I/X2)1/2.
(9)
(10)
dal*/dz = -(.a,* + 2ib)al* + iklA2 ,
dA2/dz = -a 2 A2 + ik2*al*.
(4a)
(4b)
Here ai* and a 2 are the absorption coefficients in each of thetwo channels, respectively, k, and k 2* are the coupling coef-ficients between channels 1 and 2, and 2 and 1, respectively,a,* = Ale 2 z'z, 3 is the difference between the propagationconstants of the channels, and Al and A2 are the amplitudesof the two guided modes, respectively.
The phase mismatch, Eq. (8), is therefore given by
6 = a(Z) 2 - b(z) 2 - c 2, (11)
and our scaled results become independent of any materialconstant, wavelength, and geometry.
To solve Eqs. (6a) and (6b) for the case in which 3 is opti-cally power dependent, we divide the channels into n segmentsand assume that in each segment 3 is constant, so that Eqs.(7a) and (7b) can be used. The calculation starts with a (0)= Pijl2 and b (0) = 0. We find the power in each channel atthe end of a segment and calculate the value of 3 with Eq. (12).Next, the resulting values of a(z) and b(z) are inserted as thenew a (0) and b(O) into Eqs. (7a) and (7b), producing the new
=n+n 2,1 /A Pout
(5b)
D. Sarid and M. Sargent III
Vol. 72, No. 7/July 1982/J. Opt. Soc. Am. 837
values of a (0) and b (0) for the next segment. By choosing nlarge enough that the phase shift 3 is small along each segment,one can ensure that the result of the calculation becomes in-dependent of the choice n.
NUMERICAL EXAMPLES
Figure 2 shows | b(z)/a(0)12 as a function of z for b(0) = 0 andfor la(0)12 = c2 for several values of input power. We assumedthat X = 1 m, L 1 cm, and kL = 7r/2. Note that in each casethe biasing is chosen to phase match the two channels at z =0. For a small input power, the power-dependent phase shiftis negligible, and, as expected, the optical power in channel2 evolves as sin2(ktz). As the input power increases, a phasemismatch develops along the propagation direction, and onlypart of the input power in channel 1 leaks into channel 2.Both the amplitude and the spatial period of the oscillationof the output power decrease as the input power increases. In
N..0
1.0
0.8
0.6
0.4
0.2
0 0.2 0.4 0.6 0.8 1.0
z
Fig. 2. Ib(z)/a(0)12 as a function of z with an initial phase matchingat z = 0 for the following values of the scaled input power I a (0)1 2: 1,10-6; 2, 10-5; 3, 5 X 10-5, 4 10-4; atid 5, 2 X 10-4.
1.0
0.8
Ci4
a.0
0.6
0.4
0.2
input
Fig. 3. lb(L)/a(0)1 2 as a function of la(0)12 for 1, c2 = 10-4 and L =
5 mm, and 2, c2 = 2 X 10- 4 K and L = 3.88 mm.
TIME
w 0.5
"I-
Fig. 4. Pout(t)/Pin(t) for a phase-matching bias of 10-4 at the peakof the Gaussian pulse which is also 10-4.
10
INPUT
Li 5
OUTPUT
0 -2 0 +2
TIME
Fig. 5. Same as in Fig. 4 but the input power is 10 times larger.
INPUT TRAIN OF PULSES
TIME
OUTPUT TRAIN OF PULSES
Fig. 6. The response of the device to a train of input boxcarpulses.
Fig. 3 we adjust the length of the channels to maximize I b(L)12
for a given input power. One observes that the system be-haves as a gate that opens when the input power approaches
D. Sarid and M. Sargent III
n IT- _n___
838 J. Opt. Soc. Am./Vol. 72, No. 7/July 1982
the biasing power and closes as the input power exceeds thatbiasing power.
Figures 4-6 show the response of the system to Gaussianpulses and to a train of boxcar pulses. In Fig. 4 the biasingis adjusted to give phase matching for Pi, = 10-4 at z = 0, andthe output from channel 2 is a smaller, narrower pulse. In Fig.5 the input pulse is 10 times larger than in the previous case,and two pulses appear at the output of channel 2, each for thepoint where the input power reaches the value of the biasing.In Fig. 6 one obtains two short output pulses for each inputpulse having a magnitude exceeding the biasing. Further-more, the magnitude of the output pulses is clipped. The de-vice behaves therefore both as a discriminator and, for largeinput pulses, as a gate and as a limiter.
CONCLUSIONS
We have analyzed the operation of a new tunable devicecomposed of two parallel channel waveguides that are fabri-cated on an electro-optic material with a power-dependentrefractive index. By placing two electrodes in the vicinity ofthe channels, one can externally adjust the phase-matchingconditions at the input of the two channels. By solving thecoupled-mode equations, we are able to show that the deviceacts as an all-optical gate, discriminator, and limiter. We arecurrently extending our investigation to include optical ab-sorption and channels that vary in thickness and distancefrom each other, and we plan on fabricating and testing suchdevices.
ACKNOWLEDGMENTS
The authors would like to thank H. M. Gibbs for helpful dis-cussions. This work is supported by the Air Force Office ofScientific Research, U.S. Air Force, and the Army ResearchOffice, U.S. Army, under contract no. F49620-80-C-0022.
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