Nonlinear directional couplers with integrating nonlinearities

Volume 61, number 4

OPTICS COMMUNICATIONS

15 February 1987

N O N L I N E A R D I R E C T I O N A L COUPLERS W I T H I N T E G R A T I N G N O N L I N E A R I T I E S G.I. STEGEMAN 1, C.T. SEATON Optical Sciences Center, UniversityofArizona, Tucson, AZ85721, USA

A.C. WALKER Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK

and C.N. IRONSIDE Department of Electronics and Electrical Engineering, The Universityof Glasgow, GlasgowGI28QQ, Scotland, UK

Received 6 November 1986

The response of a nonlinear directional coupler excited by pulses shorter in duration than the nonlinearity response time is analysed for one- and two-photon nonlinearities exhibiting both loss and saturation. Large pulse distortion is predicted in a number of cases.

When one of the media inside a coherent directional waveguide coupler exhibits an intensitydependent refractive index, the operational characteristics of the device become power-dependent [ 1 ]. The basic phenomenon has been demonstrated [ 2 ] in strain-induced MQW waveguides in a time regime in which the nonlinearity can be considered instantaneous and Kerr-law. However, many of the interesting applications of such a nonlinear directional coupler (NLDC) are to all-optical devices for ultrafast signal processing [1-10]. To date, all of the treatments of this phenomenon have been limited to media in which the nonlinearity responds instantaneously to the applied optical field, and for which the optically-induced change in refractive index can increase indefinitely with increasing power. In this paper we examine the time response of a NLDC in the limit in which optical pulses are used whose temporal width ztt is much shorter than the nonlinearity relaxation time z. This case is interesting for a number of reasons. The nonlinear response This research was performed during a sabbatical leave at Heriot-Watt and GlasgowUniversities. 0 030-401/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

of many waveguide devices is dominated by thermal effects which have long relaxation times, of the order of microseconds [ 11 ] and it is important to recognize the device response for this case. Furthermore, for semiconductors the nonlinearity relaxation times range from a few picoseconds [12-14] to 100's of nanoseconds. Hence it is relatively easy to excite NLDCs based on these nonlinearities with pulses whose duration is short relative to the relaxation time. Furthermore, since at high powers two-photon effects can become important [ 15], we examine device response for nonlinearities based on both one- and two-photon effects. Because highly nonlinear media are absorptive in nature and because there is a limit to the optically induced change in refractive index, both saturation and attenuation effects are explicitly included in order to realistically simulate device response. Coupled mode theory has previously yielded many valuable insights into the operation ofNLDCs [ 1-8 ]. Although it has recently been shown [ 10 ] that this formalism can break down for powers comparable to those required for all-optical switching, we use cou277

Volume 61, number 4

OPTICS COMMUNICATIONS

pied mode theory because the results from beam propagation studies [ 9,10] have confirmed that the qualitative features predicted by coupled mode theory are still valid. The equations which describe the interaction between the modes guided by two parallel channel waveguides [ 1 ] were modified to include integrating nonlinearities, saturation and attenuation to give: - i ( d / d z ) a l ( z , t ) = x a 2 ( z , t)

15 February 1987

respectively. Although this model neglects spatial diffusion, we feel it is justified because we consider pulses short compared to the diffusion time and hence the optically induced index change is "frozen in" with respect to both spatial and temporal diffusion. Furthermore, n2 is the usual intensity-dependent refractive index coefficient ( p = 1) defined by n = no + n21, and n4 (in n = no + n4I 2) for two-photon absorption ( p = 2 ) can be shown [ 15] to be given by n4 = ( dn/dN) ry2/2hoJ

(la)

+ Afl,(z, t) koal( z, t) + iaa~( z, t) , - i ( d / d z ) a z ( z , t ) = x a l ( z , t) +3f12(z, t) koa2(z, t)+iola2(z, t) ,

(lb)

where we have made the usual assumption that the nonlinear cross-channel mixing term can be neglected. Here x is the inter-channel coupling coefficient which is related to the linear cross-over coupling distance lc by x = r~/2lc. (When only one channel is excited, lc is the shortest linear liar(0, t)l--,0] device length for which complete transfer to the neighbouring channel is obtained.) Also ko=og/c, Jar(z, t) 12 is the guided wave power in the ith channel, and ot is the amplitude absorption coefficient. We have introduced saturation in a very simple way by writing

(4 )

in the usual limit that the index change is dominated by the photo-generated carriers. Here N is the optically induced carrier density and Y2 is the usual twophoton absorption coefficient. Thus in the limit that the pulse width At is much less r, that is an integrating nonlinearity, eq. (3) can be written as t

An(r, t ) =

f dr' n2p(r)-12p(r, t') . d -

(5)

T

c9C~

The power-dependent waveguide parameter Aft',(z, t) is now obtained by averaging eq. (5) over the waveguide fields. This gives Afl'i( z, t ) = F~

i

dt' laAz, t')l 2p

(6a)

where

A#i(z, t) =Afloat[ 1 - e x p ( -Afl',(z, t)]zlflsat)]

,

where Aflsat is the maximum allowed change in the waveguide effective index. The parameter Afl; (z, t) depends on the details of the nonlinearity and waveguide, as well as the temporal shape and power of the optical pulse. The time response of the nonlinear opticallyinduced change in the refractive index, An, is modelled here by a simple relaxation equation. We write

(3)

where r is the turn-off (relaxation) time for the nonlinearity, I is the local intensity and p = 1, 2 correspond to one- and two-photon absorption 278

×

d y n 2 v ( x , y ) [Ei(x,y) r 2p+2

dx .-~

dy I E , ( x , y ) l 2

(6b)

--oo

Here we have assumed the following form for the guided wave field: Ei(r, t) = ~$E,(x, y) ×ai(z, t) exp[i( ~ot-flikoz)] + c . c ,

( d/dt)zJn(r, t) = [nzv(r)/r]I2V(r, t ) - , J n ( r , t ) / r ,

dx

(2)

(7)

where//, is the effective mode index for the ith channel, i is the optical field unit vector and E,(x, y) describes the cross-sectional field distribution for the ith mode. The model discussed above has been used to

Volume 61, number 4

OPTICS COMMUNICATIONS

15 February 1987

(b)

(o)

5C

>-

4C

,,\ P~(Lc)

(=

03

i J f

Pq(Lc)

Z

~

l

I

o J

2(Lc) I.d

i L t

j

)

,

f

o

>I-Z

N zc

Z

IO

L,n

TIM E--~"

Fig. 1. The pulse envelope obtained after one coupling length at both the input and cross-channels for three successive pulses with initial excitation of the input channel. A one-photon nonlinearity was assumed, along with the parameters given in the text.

examine the time response of a nonlinear directional coupler under different levels of excitation. For a onephoton nonlinearity, we assumed semiconductordoped glass [ 16,17 ] as the nonlinear material: in particular n2/r = 10 cm2/Ws and waveguides with a cosine variation across each cross-sectional dimension and channel area of 10 #m 2 were assumed. In order to illustrate the time response, we assumed 5 ps pulses (fwhm, gaussian shape). It was assumed in all cases that only one channel is excited (1) at the input end and that both channels (1 and 2) are sampled after a propagation distance of one coupling length (1c=0.4 cm). We note that, although the assumed cosine field distributions correspond to very strong field confinement and lead to underestimates for the switching power in a realistic NLDC, they do provide simple analytical formulae for the required averaging over the wavegnide cross-section (overlap integrals) of the fields raised to various powers. Hence, we expect that the pulse distortions discussed later will actually occur at higher power levels than those quoted in the figures. Shown in fig. 1 is the time evolution of the output signal for both the incident (1) and neighbouring (2) channels for three successive pulses whose total separation and duration in time is short relative to the nonlinearity relaxation time. For these calculations, a Kerr-law nonlinearity (no saturation) with a = 1.5 cm- ' and a pulse energy of 1 picojoule were assumed.

._1 t~ or

J

TIME'-'-"-

Fig. 2. The pulse envelope obtained after one coupling length at both the input and cross-channels for a single pulse applied to the input channel. A one-photon nonlinearity was assumed, along with the parameters given in the text. For (a), no saturation is assumed, and for (b) a saturation index change of 3× 10 -4 (solid lines) and 1 × 10 -4 (dashed lines) were assumed.

Although the transfer efficiency between the two channels changes during the duration of each pulse, the pulse shapes are quite symmetric with respect to time. The integrating nature of the nonlinearity manifests itself by an increase in the power in channel (1) and a decrease in channel (2) output power with increasing pulse number. This behaviour occurs because the change in index accumulates in time at every point along the coupler and hence the transfer efficiency changes with time. Furthermore, for the channels into which the transfer is increasing and decreasing with time, the peaks of the pulses trail and precede respectively the incident pulse peak (channels (1) and (2) respectively in fig. 1 ). Such results are obtained when changes of less than ---20% occur in transfer efficiency per pulse. Severe pulse distortion can be obtained by increasing the pulse energy so that the transfer efficiency changes dramatically during the duration of a single pulse. An example is shown in fig. 2 for which the incident pulse energy has been increased numerically by a factor of ten over the previous case to 10 picojoules. The output in both channels is severely distorted because large phase differences are introduced between the channels by the combination of the large pulse energy and integrating nonlinearity. A similar effect has been observed [ 11 ] in distrib279

Volume 61, number 4

OPTICS COMMUNICATIONS

uted coupling to thin film waveguides excited by pulses shorter than the nonlinearity relaxation time. Note that the distortion is asymmetric with respect to the peak o f the incident pulse, in contrast to the symmetric distortions [ 4] which can be produced in a N L D C governed by Kerr-law nonlinearities and excited by intense pulses with At,> r. The effect of saturation in the optically-induced change in refractive index is shown for this case in fig. 2b. For Zlnsat-----3 × 10 4, the general features are the same as in the absence of saturation, but with some o f the temporal pulse distortion smoothed out. Furthermore, there is a long tail to the trailing edge of the crossed-channel output pulse due to saturation. If the saturation value of the index is decreased further to dn~t = 10-4, the N L D C essentially behaves in a linear fashion and almost no power can be obtained in the incidence channel. Therefore a strong saturation has completely changed the nonlinear properties of the directional coupler and it now behaves essentially like a linear coupler. This occurs because the m a x i m u m (saturation) change in refractive index has been achieved at every point along the coupler before all-optical switching occurs and hence the response is no longer power-(or energy-) dependent. The next case which we consider is that of nonlinearities due to two-photon absorption [ 15 ]. N o w the absorption also depends on the local guided wave intensity, and this effect must be taken explicitly into account. This local absorption is given by OL=ab q - ~ 2 I ,

(8)

where O~b is the background absorption coefficient. Since the absorption is small over an optical wavelength, the waveguide absorption can be obtained by averaging the intensity dependent absorption over the waveguide fields, that is

oq=otu+72 ; dx ; dylE(x,y) l 4

X[ ~ dx i dylE(x,y) 12] 2 -oo

× lai(z, 012 280

oo

(9)

15 February 1987 (b)

(Q)

7-

i

p2(Lc:)

z

O2 TI M E - . . ~

Fig. 3. The pulse envelope obtained after one coupling length at both the input and cross-channelsfor two (a) and one (b) pulse(s ) applied to the input channel. A two-photon nonlinearity was assumed, along with the parameters given in the text. For (aL the pulse energy and duration were 3 nJ and 30 ps respectively. In (b), the energy and pulse duration were 10 nJ and 300 ps respectively. Numerical calculations were made for a nonlinear directional coupler in GaAs [15] with two-photon nonlinearities (72 = 23 c m / G w ) . Assuming an effective waveguide area of 10¢zm 2, Cgb= 0.5 cm 1.2 = 1.06 /zm, l~.=0.1 cm, d r = 3 0 ps (fwhm, gaussian shape). the results shown in fig. 3a were obtained for two successive 3 nJ pulses. Here the intensity-dependent attenuation is sufficiently strong to produce an intensity m i n i m u m at the center of the pulse. This clearly occurs because the center (high intensity region) of the pulse encounters a larger absorption coefficient (C~,'>C~b) than the wings (low intensity region, c~, ~ c~b)) of the pulse. In addition, the pulses are asymmetric due to the integrating nonlinearity which changes the inter-channel transfer efficiency with time, just as in the single-photon nonlinearity case. In the second example, shown for a single 10 nJ pulse with At= 300 ps, the peak intensity is reduced substantially so that there is no m i n i m u m in the output pulse intensity. Note however, that the total number of carriers produced at the entrance face of the N L D C is the same as in the previous case. The result o f the integrating nonlinearity now is that asymmetrically distorted signal pulses are obtained at both channel outputs, displaced with respect to each other in time. Furthermore, the modulation of the pulse envelopes indicates large nonlinear phase shifts due to the integrating nonlinearity.

Volume 61, number 4

OPTICS COMMUNICATIONS

In summary, the effect of integrating nonlinearities is to produce asymmetric pulse distortion. W h e n saturation effects are included, a n d the saturation change in the refractive index is small, the N L D C behaves as a linear directional coupler. In addition, for two-photon absorption, high peak intensities can result in output pulses characterized by m i n i m u m intensities at times corresponding to peak intensities of the i n p u t pulse. This research was supported by the National Science F o u n d a t i o n (ECS-8501249), the Army Research Office (DAAG29-8 5-K-017 3) and the Joint Services Program of ARO a n d AFOSR, and the Science a n d Engineering Research Council, UK.

References [ 1] S.M. Jensen, IEEE J. Quant. Electron. QE-18 (1982) 1580. [2] P. Li Kam Wa, J.E. Sitch, N.J. Mason, J.S. Roberts and P.N. Robson, Electron. Lett. 21 (1985) 26; P. Lie Kam Wa, J.H. Marsh, P.N. Robson, J.S. Roberts and N.J. Mason, Proc. 2'nd Conf. on Integrated optical engineering, Cambridge, SPIE, ed. S. Sriram 578 (1985) 110.

15 February 1987

[3] Recently reviewed in G.I. Stegeman and C.T. Seaton, AppliedPhysicsReviews (in J. Appl. Phys.) 58 (1985) R57. [4] K. Kitayama and S. Wang, Appl. Phys. Lett. 43 (1983) 17. [5] R. Hoffe and J. Chrostowski, Optics Comm. 57 (1986) 34. [6] A.A. Maier, Sov. J. Quantum Electron. 12 (1982) 1490; 14 (1984) 101. [7] B. Daino, G. Gregori and S. Wabnitz, J. Appl. Phys. 58 (1985) 4512. [8] H.G. Winful, Optics Lett. 11 (1986) 33. [9] L. Thylen, E.M. Wright, G.I. Stegeman and C.T. Seaton, Optics Lett., in press. [10] S. Wabnitz, E.M. Wright, C.T. Seaton and G.I. Stegeman, Appl. Phys. Lett., in press. [ 11] R.M. Fortenberry, R. Moshrefzadeh, G. Assanto, Xu Mai, E.M. Wright, C.T. Seaton and G.L Stegeman, Appl. Phys. Lett., in press. [ 12] S.S. Yap, C. Karaguleff,A. Gabel, R. Fortenberry, C.T. Seaton and G.I. Stegeman,Appl. Phys. Lett. 46 (1985) 801. [ 13] G.R. Olbright and N. Peyghamberian,Appl. Phys. Lett. 48 (1986) 1184. [ 14] D. Cotter, Electron. Lett. 22 (1986) 693. I15] E.W. van Stryland, H. Vanherzeele, M.A. Woodall, M.J. Soileau,A.L. Smirl, S. Guha and T.F. Boggess,Opt. Eng. 24 (1985) 613. [ 16] C.N. Ironside, T.J. Cullen, J.F. Duffy, R.H. Hutchins, W.C Banyai, C.T. Seaton and G.I. Stegeman, Proc. 2'nd Conf. on Integrated optical engineering,Cambridge,ed. S. Sriram 578 (1985) 162. [ 17] R.K. Jain and R.C. Lind, J. Opt. Soc. Am. 73 (1983) 647.

281