Grating enhanced all-optical switching in a Mach Zehnder interferometer

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1 15 October 1999 Optics Communications Grating enhanced all-optical switching in a Mach Zehnder interferometer Hans-Martin Keller a,), Suresh Pereira b, J.E. Sipe b a Laser-Laboratorium Gottingen e.v., 3777 Gottingen, Germany b Department of Physics, UniÕersity of Toronto, Toronto, M5S 1A7, Canada Received 1 March 1999; received in revised form 31 May 1999; accepted 5 August 1999 Abstract We consider a Mach Zehnder interferometer with a nonlinear Bragg grating in one arm, and numerically simulate the time dependent behaviour of the all-optical switch. Compared with the same device without a grating the switching threshold is reduced by a factor of 4, while low pulse distortion is maintained for a pulse width of 8.8 ps. Bloch function theory leads to a simple model that allows one to estimate the enhancement factor for an arbritrary grating structure. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 4.65.Tg; 4.65.Wi; 4.79.Ta; 4.8.Gw Keywords: All-optical switching; Bragg gratings; Kerr medium; Nonlinear guided waves; Integrated optics 1. Introduction It has been demonstrated theoretically and experimentally that a nonlinear distributed Bragg grating can be used for all-optical switching w1 8 x. Typically the emphasis is on what might be called the nonlinear aspects of the amplitude response: the stop-gap is detuned at high intensities due to a nonlinear index change, making the device transparent at frequencies originally in the gap. In this work we demonstrate that the phase response of a grating can be used instead if the device, operated just outside the stop- ) Corresponding author. Tel.: q ; fax: q , hkeller@gwdg.de gap where no reflection takes place, is integrated into an interferometer structure. As a model system we use a Mach Zehnder interferometer wx 9 with a nonlinear Bragg grating, consisting of the polymer polyž 4-BCMU., in one arm; see Fig. 1a. This polymer has interesting properties for integrated optics applications because of its large Kerr nonlinearity and its small one- and twow1 x. To compensate photon absorption coefficients the strong dispersion normally introduced by a Bragg reflector a two-grating assembly is used, where the first grating is detuned to the red and the second to the blue side as indicated in Fig. 1b. In addition, both gratings are apodized with a quartic profile to reduce back-scattering. In time-dependent numerical 3-418r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S

2 36 ( ) H.-M. Keller et al.roptics Communications Fig. 1. The grating enhanced Mach Zehnder interferometer with a nonlinear Bragg grating in one path Ž. a. The shaded area in Ž. b indicates frequencies within the local stop gap; the device is operated at the carrier frequency indicated by the dotted line, just at the lower band edge of one grating and the upper band edge of the other. The dispersion relation Ž. c has a position dependent gap y1 width Dv z rpc with a maximum value of cm. simulations the switching characteristics on the picosecond time-scale is investigated.. Coupled mode equations To calculate the envelope functions A Ž z,t. " of the forward and backward traveling waves, we solve the weak-grating coupled mode equations wx 6 E Aq i E Aq i q qk A E z Õ E t y qg < A < q< A < A s, Ž 1. q y q E Ay i E Ay yi q qk A E z Õ E t q qg < A < q< A < A s, Ž. y q y for each grating separately, where Õ is the group velocity of the waveguide mode, G characterizes the self and cross-phase modulation, and kž z. is the grating coefficient. The envelope functions are normalized such that < A Ž z,t.< " represents the peak intensity of the forward and backward going waves in the center of the waveguide mode. The polymer polyž 4-BCMU. has a bulk index of n s 1.55 and a nonlinear index of n s 5 = 1 y14 cm rw at the vacuum wavelength l s 13 nm w1 x, at which we perform our simulations. To take into account mode structure effects in waveguides with thicknesses on the order of a micron on typical substrates, we assume a mode group velocity of Õ scr1.59 and a nonlinear parameter Gs1.69 =1 y9 cmrw. We consider a Bragg grating extending from zsya to zsa with an apodized profile k z sk 1y zra, 3 a maximum coupling strength of k s1 mm y1, and a length a s 15r8 mm chosen to ensure an a effective length L sh kž z. eff ya d zrk s1 mm. This corresponds to an index profile nž z. s n q d nzcos Ž p zrd. where d, the grating spacing, is 435 nm; n, the background index, is 1.55; and d nzrn s.31= 1y zra Ž. is the amplitude of the index modulation. This leads to a stop-gap width of DvrpcsÕkrpcs cm y1 at the center of the grating. 3. Numerical results We have solved Eqs. Ž. 1 Ž. in both time-dependent and time-independent integration schemes. Fig. shows the phase and amplitude CW response of a single apodized grating in the linear regime ŽG s, A sa Ž z. expž yiv t.. " ", as a function of the detuning V from the center of the stop gap. The transmis- Ž. Ž. Fig.. a Phase and b amplitude response of a single linear Bragg grating as a function of detuning from the center of the stop gap. The phase shift is calculated relative to the response of the bare waveguide.

3 ( ) H.-M. Keller et al.roptics Communications Fig. 3. Intensity-dependent phase shift of a single grating for the pure Ž dot-dashed. and the grating enhanced waveguide at different y1 detunings V rpc given in cm Ž solid lines.. The negative intensities represent the results for negative n. The solid horizontal line indicates the phase shift of p needed for switching. The circles represent the approximate result of the Bloch function theory for V rpcs1 cm y1 and the dashed line is its linear approximation. sion spectrum Ž. b exhibits a symmetric stop-gap with a width of Dvrpcs cm y1. Due to the quartic shape of the grating profile Ž. 3 there are essentially no transmission oscillations outside the stop gap, and there the device is highly transparent. The phase response Ž. a shows its maximum derivative close to the band edges; this derivative is identical to the transmission group delay due to the grating, df ts. Ž 4. dv The enhancement of the nonlinear phase response can be associated, in a simple picture, with this group delay: at high intensities, the band gap as well as the phase spectrum of Fig. a is shifted to lower frequencies Ž for positive n. inducing a phase shift at a given detuning V approximately proportional to dfrd V. In Fig. 3 we present the numerical results of a calculation of the CW nonlinear phase shift versus incoming intensity for a single apodized grating. Without a grating an intensity of about 1 GWrcm is needed to induce a phase shift of p. This switching threshold can be reduced to.1 GWrcm, without significant reflection losses, if a grating is used and operated just outside the band gap at Vrpcs 1 cm y1. The second derivative, d FrdV, becomes important if the response to pulses is considered. It characterizes the dispersion of the grating, and reaches a maximum value close to the band edges, where the highest nonlinear enhancement is expected. Therefore, to avoid pulse distortion while achieving a large enhancement, we use a two-grating device where the first grating is tuned to the red and the second to the blue Ž Fig. 1b.. The dispersion of the two gratings cancels each other to first order, while the group delays sum up. Higher order dispersion still exists and is implicitly included in the coupled mode equations Ž. 1 and Ž. and thus in the numerical simulations we present below; the success of the approximate treatment presented in Section 4 shows that, for the parameters considered here, it has no qualitative effect on device performance. The performance of the two-grating Mach Zehnder device was investigated in simulations of the time-dependent response, where the gratings were Fig. 4. Response of the two-grating Mach Zehnder device to a 8.8 ps pulse Ž solid lines.. The output of the grating branch is shown for reference Ž dotted lines.. While the device is Ž a. opaque at low intensities, it becomes Ž. b completely transparent at the peak intensity of.3 GWrcm.

4 38 ( ) H.-M. Keller et al.roptics Communications detuned "14 cm y1 from the operating frequency. The length of the reference path was adjusted to compensate the group delay of the grating path and to ensure a phase difference of p at low intensities. The response to a Gaussian pulse of 8.8 ps FWHM is shown in Fig. 4. At low intensities the output signals of the two arms cancel each other and the device is opaque to the input pulse Ž. a. If the peak intensity reaches the switching threshold of about.3 GWrcm the pulse is transmitted without significant distortion Ž. b. The small oscillations that are present are due to modulation instabilities w11 x, not dispersion, and can be understood following the discussion of Eggleton et al. w1 x. We performed similar simulations of the one grating device and observed strong pulse broadening, due to dispersion, leading to a low switching contrast. 4. Bloch function theory In the following we present a simple model to explain the observed phase response in terms of Bloch function theory. We initially assume a uniform grating, characterized by k, and seek a plane wave solution A sa expžž i QzyV t.. " ", with detuning V, and wave number Q. Introducing this ansatz into Eqs. Ž.Ž. 1 3 yields the equations Ž rys. aqqk ays, Ž 5. k aqq Ž rqs. ays, Ž 6. where 3 rsvrõq G ÕW, Ž 7. 1 ssqq G S Ž 8. are the effective detuning and the effective wave number. These quantities are shifted from their low intensity limits, VrÕ and Q, by the energy density W s Ž< a < q < a <. rõ and flux S s Ž< a < q y q y < a <. y, respectively. A solution exists only if the well-known dispersion relation, ' rs" s qk, 9 is fulfilled, showing a band gap between r s"k Fig. 1c. Each of these solutions characterizes a Bloch function of the grating structure wx 6, in the presence of an energy density W and flux S. All these solutions a of Eqs. Ž 5 6. " satisfy the energy conservation rule, 1 r Ws S, Ž 1. Õ s which can be cast in a more familiar form with the help of the group velocity expression EV dr s Õgs sõ sõ Ž 11. E Q ds r S,W obtained from Eq. Ž. 9, yielding WsSrÕ g. For a non-uniform grating we assume that the incoming light follows the upper Ž or lower. Bloch state adiabatically for frequencies close to the upper Ž or lower. band edge. This implies that no light is back-scattered by the device, and that the energy flux S is equal to the incident energy flux. The quantity k acquires a z-dependence as given by the apodization profile Ž. 3, and in this approximation the overall phase shift can be calculated as H a fž V,S. s QŽ V,S, z. d z, Ž 1. ya where QŽ V,S, z. can be found from the nonlinear set of Eqs. Ž 7. Ž 1.. Substituting Eq. Ž 1. into expression Ž. 7 for r, and using relation Ž. 9 we find 3r r Ž V,S, z. svrõ q G S. Ž 13. ' r yk Eq. Ž 13. leads to a fourth order polynomial for r which can be solved either analytically or by direct iteration of Eq. Ž 13., starting with rsvrõ. Once r is known, we immediately determine Q from Eqs. Ž. 8 and Ž. 9 as 1 ' Qs" r yk y G S. Ž 14. In Fig. 3 the overall phase shift of one apodized grating, as calculated by the nonlinear coupled mode equations, is compared with the result presented above. Clearly this description is in very close agreement with the full numerical solution. To extract a general sense of how effectively the grating can enhance the nonlinear phase shift, we

5 ( ) H.-M. Keller et al.roptics Communications now present a further approximation of this calculation by considering the phase shift to lowest order in the intensity. From Eq. Ž 13. we find r Ž V,S, z. svrõ q3r G Sq..., Ž 15. where r V s VrÕ and s V, z s( ry k Ž z. are the low intensity limits of r and s, respectively. Using this result, we can immediately determine Q, and hence the low intensity limit of the nonlinear phase response E a 3yr Ž z. FŽ V,Ss. sgh d z, Ž 16. ES ya r Ž z. where rž z. is the local ratio of the linear group velocity Õ Ž Ss. g to the linear group velocity Õ in the absence of a grating. From Eq. Ž 11. we have rž z. ss Ž z. rr. The dashed line in Fig. demonstrates the validity of this first order approximation. 5. Discussion The right hand side of Eq. Ž 16. is in agreement with a spatial integral of the local nonlinear phase shift one would find by using the nonlinear Schrodi- nger equation to describe the transmission response of an apodized grating in the simplest adiabatic limit w13 x; it thus relates to the enhanced nonlinearity in a grating structure, close to the bandgap, which has been studied in other contexts w14 x. However, for typical operating conditions this approximation quickly breaks down at frequencies close to the band edge. In our model system, for detunings above y1 11 cm the lowest order result 16 predicts the p phase shift point within 5%, while for our operating detuning of 14 cm y1 it is off by 13%. Nonetheless, using this lower order result we can approximate an enhancement factor, F, that compares the phase shift of the grating device to the same length of a bare waveguide made of the same material. The response is enhanced by a factor EF EFref 1 a 3yr Ž z. Fs s H d z. Ž 17. ES ES L ya r Ž z. The local enhancement factor fž z. s Ž3 y r Ž z.. rž r Ž z.. can be thought of as the product of two factors. The first, Ž3 y r Ž z.. r, arises because r s the Bloch function acquires some standing wave character for r/1, and the resulting spatially varying intensity leads to a larger effective nonlinearity for the envelope function modulating the Bloch funcw x y tion 15. The second, r Ž z., is associated with the reduced group velocity of the envelope function itself and can be attributed to two effects: First, although in the adiabatic approximation the flux in the pulse remains constant as it passes into the grating, the energy density, to which the nonlinear change in effective index is essentially proportional, is enhanced because the pulse in the grating is travelling more slowly. Second, the nonlinear phase shift acquired by the pulse in its passage through the grating depends on the time the pulse can sample the nonlinearity, and that increases as the group velocity decreases. Eq. Ž 17. shows that for the largest enhancement the group velocity should be as low as possible, i.e. the device should be operated as close as possible to the stop gap where the group velocity tends to zero. However, the center frequency of the incident pulse must be sufficiently detuned from the band edge to avoid unwanted reflection resulting from the smearing out of the effective grating band edge due to the finite device length, and from the finite spectral width of the pulse. In the simulations of our model system we found that, to achieve a good switching constrast, we had to adopt a detuning of at least Vrpcs4cm y1, yielding a maximum enhancement factor of Fs Conclusion We have demonstrated that a Bragg grating can improve the performance of an all-optical interferometer without introducing significant dispersion or back-scattering. This effect can be described quantitatively in terms of Bloch function theory leading to a simple expression for the enhancement factor of a given layout. Acknowledgements This work was supported in part by the German Federal Ministry for Education and Research BMBF

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