928 IEEE JOURNAL OF QUANTUM ELECTRON]CS,SEPTEMBER 1973 and the resulting TE mode is shifted up in frequency to% which is similar to k of(71)except that the photoelastic constant p replaces r,the electrooptic constant,and a fac- WTE =w+1. tor of 2 appears in the denominator.The latter is due to the fact that the sound strain was taken as a time-har- Since the sign of BrE and BTM is the same,the coupling is monic field while,in the electrooptic case,the modulation codirectional.This is the case which we consider in detail field E was taken as a dc field.The solution of(89)is below.Since K/B =(c/vs)(2/w),where vs is the sound given by (6)and illustrated by Fig.1.Complete power velocity,it is possible for reasonable values of the sound transfer can take place only when A =0,i.e.,when frequency n to have K28.In this case the second term on the right side of(84)represents a polarization wave K=BTE -BTM (91) traveling in the-z direction with a phase velocity-@/(K -B)(-w/B).This wave is capable of coupling to the Since K =R/es,this condition can be fulfilled by adjusting backward TE (or TM)mode.In this case we have the sound frequency R.Under phase-matched conditions we have,according to (6) BTE=BTM-K<0 A2=B-(0)12 sin2 Kz wTE=w十且. (86) B(2=B(0)2 cos2 Kz (92) Another possibility exists when the sound wave travels op- positely to the input TM mode.In this case we merely with complete power exchange in a distance reverse the sign of K in(84).Codirectional coupling is now provided by the second term on the right side of(84)with 1=T/2x (93) BrB=BrM十K It is of interest to estimate the acoustic power needed to satisfy the switching condition(93).Solving (93)for the wTE=w一0 (87) strain using(90)gives where the fact that now wre <w can be understood by noting that for each photon removed by the interaction 入22 s-Tpm from the input TM mode one new (negative traveling) phonon and one new TE photon are generated.Con- The corresponding acoustic intensity /(W/m2)can be ob- tradirectional coupling can take place due to the first term tained using the relation I=[(pv,3S2)/2]where p is the when mass density.The result is BrE=BrM一K<0 Jawitching= 入g2pe3 =2M 21p'n (94) wTB=④十 (88) Returning to the codirectional-coupling case represented where M=np2/ov,3 is the acoustic figure of merit [18]. by (85),we obtain,following the same steps leading to In a GaAs crystal,as an example,using the following (70), data:M10-13,I=5 mm,and an optical wavelength Ao I um,we get d A(/dz ix B (e dB(/dz =-iA(e lawitehing 20 W/cm2. (89) △=K一(BTg-BrM) The corresponding strain amplitude is where we assumed w>>n.In the case of well-confined Sm兰2.3×10-6 modes and of a photoelastic medium filling uniformly the guiding region 2,the coupling constant,following the where we used procedure leading to (71),is found to be p=5.34g/cm3and0s=5.15×103m/s. k=TpS(hng (90) 2入0 VIlI.COUPLING BY A SURFACE CORRUGATION Consider an idealized dielectric waveguide such as that A quantum mechanical analysis of this phenomenon [19]shows that in Fig.3.Let us next perturb the spatial distribution of n2 in the section of the waveguide in which the TE mode grows,phonons combine with TM photons on a one to one basis to generate TE photons slightly from that shown in the figure.If the perturbation so that wrE WTM +2. is small it is useful to consider its effect in terms of cou-928 IEEE JOURNAL OF QUANTUM ELECTRONICS, SEPTEMBER 1973 which is similar to K of (71) except that the photoelastic constant p replaces r, the electrooptic constant, and a fac￾tor of 2 appears in the denominator. The latter is due to the fact that the sound strain was taken as a time-har￾monic field while, in the electrooptic case, the modulation field Eo' was taken as a dc field. The solution of (89) is given by (6) and illustrated by Fig. 1. Complete power transfer can take place only when A = 0, i.e., when and the resulting TE mode is shifted up in frequency to3 UTE ='w + R. Since the sign of PTE and PTM is the same, the coupling is codirectional. This is the case which we consider in detail below. Since K/P = (c/v,)(?/w), where us is the sound velocity, it is possible for reasonable values of the sound frequency Q to have K 2p. In this case the second term on the right side of (84) represents a polarization wave traveling in the -z direction with a phase velocity -w/(K - 0) = (-w/P). This wave is capable of coupling to the backward TE (or TM) mode. In this case we have Another possibility exists when the sound wave travels op￾positely to the input TM mode. In this case we merely reverse the sign of K in (84). Codirectional coupling is now provided by the second term on the right side of (84) with where the fact that now wTE < w can be understood by noting that for each photon removed by the interaction from the input TM mode one new (negative traveling) phonon and one new TE photon are generated. Con￾tradirectional coupling can take place due to the first term when Returning to the codirectional-coupling case represented by (85), we obtain, following the same steps leading to (70), where we assumed w >> Q. In the case of well-confined modes and of a photoelastic medium filling uniformly the guiding region 2, the coupling constant, following the procedure leading to (71), is found to be A qua,ntum mechanical analysis of this phenomenon 1191 shows that in the section of the waveguide in which the TE mode grows, phonons combine with TM photons on a one to one basis to generate TE photons Since K = Q/vs, this condition can be fulfilled by adjusting the sound frequency s2. Under phase-matched conditions we have, according to (6) with complete power exchange in a distance It is of interest to estimate the acoustic power needed to satisfy the switching condition (93). Solving (93) for the strain using (90) gives The corresponding acoustic intensity I (W/mz) can be ob￾tained using the relation I = [(pvS3S2)/2] where p is the mass density. The result is where M = n6pz/pvs3 is the acoustic figure of merit [18]. In a GaAs crystal, as an example, using the following data: M = 1O-I3, I = 5 mm, and an optical wavelength X, = 1 pm, we get The corresponding strain amplitude is S(R' Y 2.3 x where we used p = 5.34 g/cmS and us = 5.15 x los m/s. VIII. COUPLING BY A SURFACE CORRUGATION Consider an idealized dielectric waveguide such as that in Fig. 3. Let us next perturb the spatial distribution of n2 slightly from that shown in the figure. If the perturbation so that wTE = uTM + 0. is small it is useful to consider ts effect in terms of COU-