926 IEEE JOURNAL OF QUANTUM ELECTRONICS,SEPTEMBER 1973 while from (4) for the coupling constant and the power-exchange distance,respectively. dB=-isA exp【iBM-3.T d止 VI.PHASE MATCHING IN ELECTROOPTIC COUPLING K=名Eo (71) In general.8TM BTE even for the same-order mode so 2 that the fraction of the power exchanged in the electrooptic-coupling case described previously does not The form of(70)will apply to the general case involving exceed,according to (6),K2/(K2 +A2).If A>>K,the cou- arbitrary spatial dependence ofr and E.In that case we pling is negligible.To appreciate the importance ofthis fact, need to perform the integration in (67)to evaluate the let us use the numerical data of the example considered at coupling coefficient k. the end of Section V.We have x 1.85 cm-and Bnak The form of (70)is identical to that of(2).The solution 2.2 X 10s cm-.The exchange factor k2/(+)is thus of(70)is thus given by (6)with reduced to 0.5when△/B≈[(Bre-Brw)/Brs】~I0-s The critical importance of phase matching is thus △=8mTM-BTE (72) manifest.Since the dispersion due to the waveguide will in general be such as to make A >>k,some means for phase The transfer of power between the modes for the phase- matching are necessary.We start by considering again the matched (A 0)and A 0 case are as shown in Fig.1.A coupled-mode equations (70).reintroducing the possible z complete transfer of power between the modes thus re- dependence of k quires that A =0,i.e.,phase matching.Means for phase matching will be discussed in Section VI.For the s=-ix()B.e meantime let us assume that k >>A so that,according to 农 (6),the effects of phase mismatch can be neglected.A com- plete power transfer in this case occurs in a distance such dB。=-a)Aea that A=8TE-B.TM (74) l=/2 with or using(71) (2)=nkrz)E(2) 1E=2 入。 (73) As in the case of second-harmonic generation,we can use a spatial modulation ofr or the field E for phase matching. where Ao =2/k.The product /E is identical to the "half- Consider,for example,the case where the field E(z) wave"voltage of bulk electrooptic modulators [15].The reverses its direction periodically as with the electrode "half-voltage"in the bulk case,we recall,is the field- arrangement of Fig.5.Approximating theelectricfield in the length product which causes a 90 rotation in the plane of guiding layer by polarization of a wave incident on an electrooptic crystal. Unlike the bulk case,the coupling between the two guided modes can take place even when the electrooptic Ea)-∑45sim2g (75) 。4qm perturbation is limited to an arbitrarily small portion of the transverse dimensions [6]or when the two modes are corresponding to a field reversal between Eoand-Eevery A of different order (m). meters,we can take k(z)in (74)as To appreciate the order of magnitude of the coupling, consider a case where the guiding layer is GaAs and Ao=I um.In this case [15] =-∑2ea*aa:-eara Ko n2'krEo. H2≈3.5， m:r=59×1012m (76) V If we substitute(76)in (74)we obtain on the right-side terms with exponential dependence of the type Taking an applied field E(=10 V/m we obtain from(71) k=1.85cm 1=克=0.85cm One can choose A such that,for someq.(2q/A)=A.This resultsin a synchronous drivingterm(i.e.,one with azeroex-926 while from (4) IEEE JOURNAL OF QUANTUM ELECTRONICS, SEPTEMBER 1973 r~,~krE"' 2 K= The form of (70) will apply to the general case involving arbitrary spatial dependence of r and E''). In that case we need to perform the integration in (67) to evaluate the coupling coefficient K. The form of (70) is identical to that of (2). The solution of (70) is thus given by (6) with The transfer of power between the modes for the phase￾matched (A = 0) and A # 0 case are as shown in Fig. 1. A complete transfer of power between the modes thus re￾quires that A = 0, i.e., phase matching. Means for phase matching will be discussed in Section VI. For the meantime let us, assume that K >> A so that, according to (6),. the effects of phase mismatch can be neglected. A com￾plete power transfer in this case occurs in a distance 1 such that for the coupling constant and the power-exchange distance, respectively. VI. PHASE MATCHING IN ELECTROOPTIC COUPLING with or using (7 1) In general, pTM # pTE even for the same-order mode so that the fraction of the power exchanged in the electrooptic-coupling case described previously does not exceed, according to (6), K'/(K' + A,). If A>> K, the COU￾pling is negligible. To appreciate the importance ofthis fact, let us use the numerical data of the example considered at the end of Section V. We have K = 1.85 cm-' and p n,k = 2.2 X lo5 cm-'. The exchange factor K'/(K' + A,) is thus reduced to 0.5 when A/p = [(BTE - &&&E] - The critical importance of phase matching is thus manifest. Since the dispersion due to the waveguide will in general be such as to make A >> K, some means for phase matching are necessary. We start by considering again the coupled-mode equations (70), reintroducing the possible z dependence of K (74) (73) where A, = 27r/k. The product 1E is identical to the "half￾wave" voltage of bulk electrooptic modulators [15]. The "half-voltage'' in the bulk case, we recall, is the field￾length product which causes a 90" rotation in the plane of polarization of a wave incident on an electrooptic crystal. Unlike the bulk case, the coupling between the two guided modes can take place even when the electrooptic perturbation is limited to an arbitrarily small portion of the transverse dimensions [6] or when the two modes are of different order (1 # m). To appreciate the order of magnitude of the coupling, consider a case where the guiding layer is GaAs and X, = 1 pm. In this case [15] Taking an applied field E = loe V/m we obtain from (7 1) K = 1.85 cm-' 1 = - = 0.85 cm x 2K K(Z) = n: kr(z) Co'(z). As in the case of second-harmonic generation, we can use a spatial modulation of Y or the field E',' for phase matching. Consider, for example, the case where the field E"'(z) reverses its direction periodically as with the electrode arrangement of Fig. 5. Approximating theelectric field in the guiding layer by (75) corresponding to afieldreversal between E,and - E,every A meters, we can take K(Z) in (74) as If we substitute (76) in (74) we obtain on the right-side terms with exponential dependence of the type One can choose A such that, for some q, (2q/A) = A. This results in a synchronous driving term (Le., one with azero ex-