924 IEEE JOURNAL OF QUANTUM ELECTRONICS,SEPTEMBER 1973 Using (45)and assuming a single,say m,mode input at tion of d by taking d(z)as w/2 results in 1+之 2d sin 2a (53)】 P,,= aodd integer gT 2 Le/2, ce(i-8)c.c. corresponding to a square-wave alternation between 0 andd (47) with a period A.Instead of(37)we now have Substituting(47)into (32)we obtain dA id (/-e-savsulh) dA. d d(Bum)eaSm.十c (48) X IA(pe-1(28.-7-8)...) (54) where We can choose the period A such that for some value of g 3纪，nu/23,m./8,x)dk(49) 29+B“-282=0. (55) and This results in a synchronous term(i.e.,one with azero expo- △=(3n"rE-2Bm/)rM nent)on the right side of(54)so that For the special casem=n=I and for well-confined modes dA.) = 【A2Ps (56) we have,using(17)and(22), dz 4gr 2411.2 where the nonsynchronous terms have been neglected.A (50 comparison to (37)shows that the effective nonlinear coefficient is now reduced to Proceeding as in the previous section leads finally to detr= d Q开 (P)rE=0.72 / sin2△l/2 (△1/2 (51) and that instead of(43) (P)M p=0.72 w'den an expression identical to that obtained in(43)for TE-TE wt (57) conversion.We must recall,however,that the nonlinear coefficient din(51)is not necessarily thesameas that appear- operation based on q=I is thus most efficient,leading to a ing in(43),reflecting the differences in crystalline orientation reduction by a factor of in the conversion efficiency.We needed to achieve coupling in each case. note,however,that the factor sin2(Al/2)/(Al/2)2 is now unity,which makes it possible to take advantage of the P C.Phase Matching dependence of the conversion efficiency. It follows from(43)or(51)that a necessary condition for second-harmonic generation is Al/2<<so that the factor V.ELECTROOPTIC MODE COUPLING sin2(Al/2)/(Al/2)2 is near unity.In this case the conversion The electrooptic effect in thin-film configurations can be efficiency is proportional to P.This phase-matching condi- used in a variety of switching applications.Its use as a tion can be satisfied by using the dependence ofthe propaga- polarization switch in a GaAs waveguide at 1.15 u has been tion constants B of the various modes on the waveguide demonstrated [6].In contrast to the conventional bulk [15] dimensions [7].An alternate approach is to introduce a treatment of the electrooptic effect which relies heavily on space-periodic perturbation into the waveguide with a the concept of induced retardation,we view the process as period A satisfying that of coupling between TE and TM modes brought about by the applied low-frequency electric field. 4. 49, 9=1,2,3·. (52) The linear-electrooptic effect is conventionally defined [16]in terms of a third-rank tensor ru which relates the changes in the constants of the index ellipsoid to the applied Schemes based on waveguide corrugation and on field according to modulating the nonlinear coefficient d have been proposed [14].In this section wewill consider thecase ofdmodulation. We go back to(37)but allow explicitly for a spatial modula- (58)924 IEEE JOURNAL OF QUANTUM ELECTRONICS, SEPTEMBER 1973 Using (45) and assuming a single, say m, mode input at tion of d by taking d(z) as w/2 results in d(z) = -t d P,(r, t) = - Lyv.-, Bm\-,-,K"\ 111, -/"I t("l-2Orn*/lZ) corresponding to a square-wave alternation between 0 andd .e + C.C. (47) with a period A. Instead of (37) we now have where We can choose the period A such that for some value of q and A = (PnW)~~ - ~(PI~"~)TM. For the special casem = n = 1 and for well-confinedmodes we have, using (17) and (22), This results in a synchronous term (i.e,, onewith azero exp￾nent) on the right side of (54) so that where the nonsynchronous terms have been neglected. A (50) comparison to (37) shows that he effective nonlinear coefficient is now reduced to Proceeding as in the previous section leads finally to d de,, = - 4.rr an expression identical to that obtained in (43) for TE-TE conversion. We must recall, however, that the nonlinear P" ~ pw/2 = 0.72 (:) coefficient din (5 1) is not necessarily the same as that appear- operation based on = 1 is thus most efficient, leading to a ing in (43), reflecting the differences in crystalline orientation reduction by a factor of R2 in the conversion efficiency, We needed to achieve coupling in each case. note, however, that the factor ~in~(A1/2)/(A1/2)~ is now unity, which makes it possible to take advantage of the l2 C. Phase Matching dependence of the conversion efficiency. It follows from (43) or (51) that a necessary condition for second-harmonic generation is A1/2 << 7r so that the factor sin2 (A1/2)/(~i1/2)~ is near unity. In this case the conversion efficiency is proportional to 12. This phase-matching condi￾tion can be satisfied by using the dependence of the propaga￾tion constants 0 of the various modes on the waveguide dimensions [7]. An alternate approach is to introduce a space-periodic perturbation into the waveguide with a period A satisfying V. ELECTROOPTIC MODE COUPLING The electrooptic effect in thin-film configurations can be used in a variety of switching applications. Its use as a polarization switch in a GaAs waveguide at 1.15 p has been demonstrated [6]. In contrast to the conventional bulk [15] treatment of the electrooptic effect which relies heavily on the concept of induced retardation, we view the process as that of coupling between TE and TM modes brought about by the applied low-frequency electric field. The linear-electrooptic effect is conventionally defined [16] in terms of a third-rank tensor rijk which relates the changes in the constants of the index ellipsoid to the applied Schemes based on waveguide corrugation and on field according to modulating the nonlinear coefficient d have been proposed [14]. In this section wewillconsiderthecaseofdmodulation. We go back to (37) but allow explicitly for a spatial modula- *(+) i? = rijicEk. (58)