《光波导理论与技术 Optical Waveguides Principles and Technologies》课程教学资源（参考文献）Coupled-Mode Theory for Guided-Wave Optics

IEEE JOURNAL OF QUANTUM ELECTRONICS,VOL.QE-9,NO.9,SEPTEMBER 1973 919 Coupled-Mode Theory for Guided-Wave Optics AMNON YARIV Abstract-The problem of propagation and interaction of optical radia- bz,x,）=Betu6x) (1) tion in dielectric waveguides is cast in the coupled-mode formalism.This ap- proach is useful for treating problems involving energy exchange between with A and B constant. modes.A derivation of the general theory is followed by application to the specific cases of electrooptic modulation,photoelastic and magnetooptic In the presence of a perturbation which,as an example, modulation,and optical filtering.Also treated are nonlinear optical can take the place ofa periodicelectricfield,a sound wave,or applications such as second-harmonic generation in thin films and phase a surface corrugation,power is exchanged between modes a matching. and b.The complex amplitudes A and B in this case are no longer constant but will be found to depend onz.They willbe shown below to obey relations of the type I.INTRODUCTION GROWING BODY of theoretical and experimental dA work has been recently building up in the area of d =Kas Be guided-wave optics,which may be defined as the study and dB utilization of optical phenomena in thin dielectric (2) d =Kha Aetia waveguides [1],[2].Some of this activity is due to the hopes for integrated optical circuits in which a number of optical where the phase-mismatch constant A depends on the functions will be performed on small solid substrates with propagation constants Ba and B.as well as on the spatial the interconnections provided by thin-film dielectric variation of the coupling perturbation.The coupling waveguides [3],[4].Another reason for this interest is the coefficients Ka and Koa are determined by the physical situa- possibility of new nonlinear optical devices and efficient op- tion under consideration and their derivation will take up a tical modulators which are promised by this approach major part of this paper.Before proceeding,however,with [5]-[7]. the specific experimental situations,let us consider some A variety of theoretical ad hoc formalisms have been general features of the solutions of the coupled-mode utilized to date in treating the various phenomena ofguided- equations. wave optics.In this paper we present a unified theory cast in the coupled-mode form to describe a large number of A.Codirectional Coupling seemingly diverse phenomena.These include:1)nonlinear optical interactions;2)phase matching by periodic pertur- We take up,first,the case where modes a and b carry bations;3)electrooptic switching and modulation;4) (Poynting)power in the same direction.It is extremely con- photoelastic switching and modulation;and 5)optical filter- venient to define A and B in such a way that |A(z)3 and ing and reflection by a periodic perturbation. B(z)2correspond to the power carried by modes a and b, respectively.The conservation of total power is thus ex- II.THE COUPLED-MODE FORMALISM pressed as We will employ,in what follows,the coupled-mode for- malism [8]to treat the various phenomena listed in Section I. 是G4+明=0 (3) Before embarking on a detailed analysis it will prove beneficial to consider some of the common features of this which,using(2),is satisfied when [9] theory.Consider two electromagnetic modes with,in general,different frequencies whosecomplex amplitudes are Kab=一Kba* (4④) A4 and B.These are taken as the eigenmodes of the unper- turbed medium so that they represent propagating distur- If boundary conditions are such that a single mode,say b,is bances incident at z=0 on the perturbed region z>0,we have a(z,x,）=Aewi.x) b(0)=Bo,a(0)=0 (5) Manuscript received March 9,1973.This research was supported in Subject to these conditions the solutions of(2)become part by the National Science Foundation and in part by the Advanced Research Projects Agency through the Army Research Office,Durham. N.C. The author is with the Department of Electrical Engineering.Califor- nia Institute of Technology,Pasadena.Calif.91109. 4@=B4千sin(4+4y

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-9, NO. 9, SEPTEMBER 1973 919 Coupled-Mode Theory for Guided-Wave Optics AMNON YARIV Absrruct-The problem of propagation and interaction of optical radia￾tion in dielectric waveguides is cast in the coupled-mode formalism. This ap￾proach is useful for treating problems involving energy exchange between modes. A derivation of the general theory is followed by application to the specific cases of electrooptic modulation, photoelastic and magnetooptic modulation, and optical filtering. Also treated are nonlinear optical applications such as second-harmonic generation in thin films and phase matching. I. INTRODUCTION A GROWING BODY of theoretical and experimental work has been recently building up in the area of guided-wave optics, which may be defined as the study and utilization of optical phenomena in thin dielectric waveguides [l], [2]. Some of this activity is due to the hopes for integrated optical circuits in which a number of optical functions will be performed on small solid substrates with the interconnections provided by thin-film dielectric waveguides [3], [4]. Another reason for this interest is the possibility of new nonlinear optical devices and efficient op￾tical modulators which are promised by this approach A variety of theoretical ad hoc formalisms have been utilized to datein treating thevarious phenomena ofguided￾wave optics. In this paper wepresent a unified theory cast in the coupled-mode form to describe a large number of seemingly diverse phenomena. These include: 1) nonlinear optical interactions; 2) phase matching by periodic pertur￾bations; 3) electrooptic switching and modulation; 4) photoelastic switching and modulation; and 5) optical filter￾ing and reflection by a periodic perturbation. [51-[71. 11. THE COUPLED-MODE FORMALISM We will employ, in what follows, the coupled-mode for￾malism [X] to treat the various phenomena listed in Section I. Before embarking on a detailed analysis it will prove beneficial to consider some of the common features of this theory. Consider two electromagnetic modes with, in general, different frequencies whosecomplex amplitudes are A and B. These are taken as the eigenmodes of the unper￾turbed medium so that they represent propagating distur￾bances Manuscript received March 9, 1973. This research was supported in part by the National Science Foundation and in part by the Advanced Research Projects Agency through the Army Research Office, Durham, N.C. The author is with the Department of Electrical Engineering, Califor￾nia Institute of Technology, Pasadena, Calif. 91109. with A and B constant. In the presence of a perturbation which, as an example, can take the place ofaperiodicelectricfield, asoundwave, or a surface corrugation, power is exchanged between modes a and 6. The complex amplitudes A and B in this case are no 1ongerconstantbutwillbefoundtodependonz.Theywillbe shown below to obey relations of the type where the phase-mismatch constant A depends on the propagation constants Pa and Pb as well as on the spatial variation of the coupling perturbation. The coupling coefficients K~~ and Kba are determined by the physical situa￾tion under consideration and their derivation will take up a major part of this paper. Before proceeding, however, with the specific experimental situations, let us consider some general features of the solutions of the coupled-mode equations. A. Codirectional Coupling We take up, first, the case where modes a and b carry (Poynting) power in thesame direction. It is extremely con￾venient to define A and B in such a way that IA(z)( and I B(z)l correspond to the power carried by modes a and b, respectively. The conservation of total power is thus ex￾pressed as t 3) which, using (2), is satisfied when [9] If boundary conditions are such that a single mode, say b, is incident at z = 0 on the perturbed region z > 0, we have b(O)=B,, a(O)=O. (5) Subject to these conditions the solutions of (2) become

920 IEEE JOURNAL OF QUANTUM ELECTRONICS,SEPTEMBER 1973 兰 (o)q 8o】 IB(z)2 IA(z)2 ertur bati 2*0 ZL Fig.I.The variation of the mode power in the case of codirectional Fig.2.The transfer of power from an incident forward wave B(z)to a coupling for phase-matched and unmatched operation. reflected wave A(z)in the case of contradirectional coupling. B(2)=Bea/2{cos[(4x2十△2)] the space betweenz=0andz=L.Sincemodeais generated by the perturbation we have a(L)=0.With these boundary △ -14+a7isin(4+△)' (6) conditions the solution of(11)is given by where k2=K2.Under phase-matched condition A =0,a A(z)=B(0) 2ike sinh -A sinh头+i5cosh SL S(-L) complete spatially periodic power transfer between modes a and b takes place with a period /2k. eita12) a(z)=Bae sin (z) B(z)=B(0) -△sinh SL+is coshS 2 b(z,1)=Boe)cos (x2). (7) {as[e-]+sco咖[g-]} (12) A plot of the mode intensities a2and b2is shown in Fig. 1.This figure demonstrates the fact that for phase mismatch A>>Ka the power exchange between the modes is negligi- 5=V4k2-△2， KKh (13) ble.Specific physical situations which are describable in terms of this picture will be discussed further below. Under phase-matching conditions A =0 we have B.Contradirectional Coupling 4(z)=B(0) sinh[w(e-L)】 cosh (KL) In this case the propagation in the unperturbed medium is described by Ba)=BO)cosh-L】 (14) cosh (KL) a Aei(wt+) b=Be(st-Bue) (8) A plot of the mode powers B(z)2and4()2for this case is shown in Fig.2.For sufficiently large arguments of the where A and B are constant.Mode a corresponds to a left cosh and sinh functions in (14),the incident-mode power (-z)traveling wave whileb travels to the right.A time-space decays exponentially along the perturbation region.This periodic perturbation can lead to power exchange between decay,however,is due not to absorption but to reflection of the modes.Conservation of total power can be expressed as power into the backward traveling mode a.This case will be considered in detail in following sections,where acoustoop- 是-a的-0 tic,electrooptic,and spatial index perturbation will be (9) treated.The exponential-decay behavior of Fig.2 will be shown in Section VIII to correspond to the stopband region which is satisfied by(2)if we take of periodic optical media. Kab=Koa (10) III.ELECTROMAGNETIC DERIVATIONS OF THE COUPLED- MODE EQUATIONS so that A.TE Modes dA dB (11) d =Kal Be-i dz =Kat*ede Consider the dielectric waveguide sketched in Fig.3.It consists of a film of thickness t and index of refraction na In this case we take the mode b with an amplitude B(0)to be sandwiched between media with indices n and na.Taking incident at z =0 on the perturbation region which occupies (a/ay)=0,this guide can,in the general case,support a

920 IEEE JOURNAL OF QUANTUM ELECTRONICS, SEPTEMBER 1973 z=O 2.L Fig. 1. The variation of the mode power in the case of codirectional Fig. 2. The transfer of power from an incident forward wave B(z) to a coupling for phase-matched and unmatched operation. reflected wave A(z) in the case of contradirectional coupling. B(z) = BoeiA2/'{ cos [9(4~' + A2)'''zI the space btweenz = 0 andz = L. Sincemodeaisgenerated by the perturbation we have a(L) = 0. With these boundary A - 2 (4K + A ) sin [4(4,(' + ~')"~~z]j (6) conditions the solution of (1 1) is given by where K~ = I K=,,I 2. Under phase-matched condition A = 0, a A(Z) = B(O) SL SL complete spatially periodic Ijower transfer between modesa -A sinh - f is cosh - and b takes place with a period ir/2K. 2 2 2iK,be-i(4.2/2) sinh [f (z - L)] -i(Az/2) B(z) = B(0) - c -A sinh -- + is cosh - SL SL 2 2 b(=, t) = ~~~~(~b~-@b~) cos (KZ). (7) -{A sinh [$ (z - L)] + is cosh A plot of the mode intensities 1 a1 and 1 bJ is shown in Fig. 1. This figure demonstrates the fact that for phase mismatch A >> I K~,,[ the power exchange between the modes is negligi- s E 44K2 - A', K E ]K,bI. (1 3) ble. Specific physical situations which are describable in terms of this picture will be discussed further below. Under phase-matching conditions A = 0 we have B. Contradirectional Coupling In this case the propagation in the unperturbed medium is described by A(z) = B(0) tf) - sinh [x(z - L)] cosh (KL) cosh [K(Z - L)] cash (KL) B(z) = B(0) - (1 4) A plot of the mode powers 1 B(z)J and 1 A(z)J for this case (') is shown in Fig. 2. For sufficiently large arguments of the where A and B are constant. Mode a corresponds to a left (-z) traveling wave whileb travels to the right. A time-space periodic perturbation can lead to power exchange between the modes. Conservation of total power can be expressed as cosh and sinh functions in (14), the incident-mode power decays exponentially along the perturbation region. This decay, however, is due not to absorption but to reflection of power into the backward traveling mode a. This case will be considered in detail in following sections, where acoustoop￾d tic, electrooptic, and spatial index perturbation will be - (1 AI2 - IBI') = 0 dz (9) treated. The exponential-decay behavior of Fig. 2 will be shown in Section VI11 to correspond to the stopband region which is satisfied by (2) if we take of periodic ptical media. (10) 111. ELECTROMAGNETIC DERIVATIONS OF THE COUPLED￾MODE EQUATIONS so that A. TE Modes - dA = K,bBe-iAz dB = K,b*Ae"z dz dz (11) Consider the dielectric waveguide sketched in Fig. 3. It consists of a film of thickness t and index of refraction nz In this case we take the mode b with an amplitude B(0) to be sandwiched between media with indices n, and n,. Taking incident at z = 0 on the perturbation region which occupies (a/ay) = 0, this guide can, in the general case, support a

YARIV:COUPLED-MODE THEORY 921 power flow of 2W/m.The normalization condition is thus n2 propagation (20) -x-t n3 where the symbolm denotes the mth confined TE mode cor- Fig.3.The basic configuration of a slab dielectric waveguide. responding to mth eigenvalue of (19). Using (17)in (20)we determine finite number of confined TE modes with field components Ey,Hx,and H:,andTM modes with components Hy,Ex,and 71/2 E2.The"radiation"modes of this structure which are not Cn=2hm ⊙ (21) confined to the inner layer are not considered in this paper 18-1+ and will be ignored.The field component Ey of the TE modes,as an example,obeys the wave equation Since the modes 8,(m)are orthogonal we have 6-g0,1=23 (15) 8,8,md= 201.m B (22) J-o We take E(x,z,t)in the form B.TM Modes E(x,z,)=8(x)et-. (16) The field components are The transverse function &(x)is taken as H,(c,2,)=50(xeu-a [C exp (-qx), 0≤x0.(24) kw/c. (18) From the requirement that E,and H be continuous atx=0 The continuity of Hy and Ez at the interfaces requires that and x =-t,we obtain! the various propagation constants obey theeigenvalueequa- tion tan (ht)=- 9十p (19) 1-) tan(h创)=+到 -pq (25) where This equation in conjunction with(18)is used to obtain the eigenvalues B of the confined TE modes. The constant Cappearing in(17)isarbitrary.Wechooseit ng3 D 9 in such a way that the field 8(x)in(17)corresponds to a power flow of I W(per unit width in the y direction)in the The normalization constant C is chosen so that the field mode.A mode whose E=48(x)will thus correspond to a represented by(23)and(24)carries 1 W per unit width in the y direction. The assumed form of E,in (17)is such that 8 and 3C=(i/wu) a8,/ax are continuous atx =0 and that 8,is continuous at x =-f.All that is left is to require continuity of as/ax at x =-t.This leads to (19). A,e*x=是四k=1 2w J-o E

YARIV: COUPLED-MODE THEORY 92 1 power flow of 1 A I W/m. The normalization condition is nl thus n2 - propagation n3 x=-t where the symbol m denotes themth confined TE mode cor￾Fig. 3. The basic configuration of a slab dielectric waveguide. responding to mth eigenvalue of (19). Using (1 7) in (20) we determine finite number of confined TE modes with field components E,, H,, and Hz, andTMmodeswithcomponents H,, E,, and E,. The "radiation" modes of this structure which are not cM = 2hm y2. (21) and will be ignored. The field component E, of the TE modes, as an example, obeys the wave equation Since the modes are orthogonal we have confined to the inner layer are not considered in this paper [P., (t + - 11 + --)(hm~ + qmz), 4m Pm We take E,(x, z, t) in the form B. TM Modes Ey(x,z,l) =&y(x)e"wt-flz'. (16) I The field components are The transverse function &,,(x) is taken as H,(x, z, t) = Xy(x)ei(Wt-iBZ) COS (hx) - (q/h) sin (hx)], &"(X) = -t<xIO which, applying (15) to regions 1, 2, 3, yields The continuity of H, and E, at the interfaces requires that From the requirement that and Hz becontinuous at x = thevariouspropagationconstantsobeytheeigenvalueequa￾and x = -t, we obtain' tion tan (ht) = 4+P . (1 9) tan (hi) = htP + 4) h(l - y) ha - (25) where This equation in conjunction with (18) is used to obtain the eigenvalues p of the confined TE modes. 2 2 Theconstant Cappearingin (17) isarbitrary. Wechooseit ji G -sp, n2 n q -4j 4. in such a way that the field &,(x) in (17) corresponds to a n3 nl power flow of 1 W (per unit width in they direction) in the The normalization constant C is chosen so that the field mode. A mode whose EN = A& .(x) will thus correspond to a represented by (23) and (24) carries 1 W per unit width in the y direction. The assumed form of E, in (17) is such that E, and X, = (i/wp) a o,/ax are continuous at x = 0 and that E, is continuous at x = -I. All that is left is to require continuity of aE,/ax at x = -I. This leads to (1 9). 1 HUEx* dx = !.-/rn x,"o dx = 1 2 -m 2u -m E

922 IEEE JOURNAL OF QUANTUM ELECTRONICS,SEPTEMBER 1973 or using n2=/ Multiplying (31)by 8y(m(x),and integrating and making use of the orthogonality relation(22)yields dx (26) n2(x) B. dA《 ewt+dA -ewt-+c.c. dz dz This condition determines the value of Cm as [10] 部Re,k,w = (32) p2十 where Am is the complex normal mode amplitude of the negative traveling TEmode while Am+isthat ofthe positive 02 方2+9十h1 g+h2nq one.Equation(32)is the main starting point for the follow- (27) ing discussion in which we will consider a number of special cases. C.The Coupling Equation The wave equation obeyed by the unperturbed modes is IV.NONLINEAR INTERACTIONS In this section we consider the exchange of power between 7gk,)=μea (28) three modes of different frequencies brought about through the nonlinear optical properties of the guiding or bounding We will show below that in most of the experiments of in- layers.The relevant experimental situations involve second- narmonic generation,frequency up-conversion,and optical terest to us we can represent the perturbation as a distributed parametric oscillation.To be specific we consider first the polarization source Ppert(r,t),which accounts for the devia- tion of the medium polarization from that which accompanies case of second-harmonic generation from an input mode at w/2 to an output mode at w.The perturbation polarization is the unperturbed mode.The wave equation for the perturbed taken as case follows directly from Maxwell's equations if we take D =6oE+P. P,m红，)=P,weu-)十c.c.l. (33) 72E(c,t）=4e [P) (29) The complex amplitude of the polarization is with similar equations for the remaining Cartesian com- P)=dun(r)EE (34) ponents of E. We may taketheeigenmodes of(28)asan orthonormal set whered is an element of the nonlinear optical tensor and in which to expand Ey and write summation over repeated indices is understood.We have allowed,in(34),for a possible dependence ofdi on the posi- E,=∑42&，“o-n十cc tion r. 2 A.Case I:TEinput-TEoutput A(@et-8,(x)d的 (30) Without going,at this point,into considerations in- 一k 11<有： volving crystalline orientation,let us assume that an optical field parallel to the waveguide y direction will generate a where extends over the discrete set of confined modes and second-harmonic polarization along the same direction includes both positive and negative traveling waves.The in- tegration over B takes in the continuum of radiation modes, and c.c.denotes complex conjugation.Our chiefinterest lies P)=dE E) (35) in perturbations which couple only discrete modes so that,in what follows,we will neglect the second term on theright side where P and E represent complex amplitudes,and d of(30).Problems of coupling to the radiation modes arise in corresponds to a linear combination of dus which depends connection with waveguide losses [11]and grating couplers on the crystal orientation.In this special case an input TE [121. mode at w/2 will generate an output TE mode at w.Using Substituting(30)into(29),assuming"slow"variation so (30)in (35)gives that dAm/dz2<<Bm dAm/dz,and recalling that 8,(m)(x) el-obeys the unperturbed waveequation(28),gives P,,0=d)∑∑Ano/2Ae/8,e8,m Xelu-(8c.c. (36) We consider a case of a single mode input,say n.In that case (31) the double summation of(36)collapsesto asingle termn=p

922 IEEE JOURNAL OF QUANTUM ELECTRONICS, SEPTEMBER 1973 Multiplying (31) by &y(m)(x), and integrating and making use of the orthogonality relation (22) yields 2W€o dx = - Pn This condition determines the value of C, as [lo] I (27) C. The Coupling Equation The wave equation obeyed by the unperturbed modes is a2E at V2E(r, t) = pe -3 . We will show below that in most of the experiments of in￾terest to us we can represent the perturbation as a distributed polarization source Ppert(r,t), which accounts for the devia￾tion of the medium polarization from that which accompanies the unperturbed mode. The wave equation for the perturbed case follows directly from Maxwell's equations if we take D = coE + P. with similar equations for the remaining Cartesian com￾ponents of E. We may take theeigenmodes of (28) as an orthonormal set in which to expand E, and write where 1 extends over the discrete set of confined modes and includes both positive and negative traveling waves. The in￾tegration over /3 takes in the continuum of radiation modes, and C.C. denotes complex conjugation. Our chiefinterest lies in perturbations which couple only discrete modes so that, in what follows, we will neglect the second term on the right side of (30). Problems of coupling to the radiation modes arise in connection with waveguide losses [ 1 11 and grating couplers Substituting (30) into (29), assuming "slow" variation so that d2Am/dz2 << Dm dAm/dz, and recalling that &ycml (x) P21. ei(wt - Omz) obeys the unperturbed wave equation (28), gives where A m(-j is the complex normal mode amplitude of the negative traveling TE mode while A m(+) is that of the positive one. Equation (32) is the main starting point for the follow￾ing discussion in which we will consider a number of special cases. IV. NONLINEAR INTERACTIONS In thisection we consider the exchange of power between three modes of different frequencies brought about through the nonlinear optical properties of the guiding or bounding layers. The relevant experimental situations involve second￾narmonic generation, frequency up-conversion, and optical parametric oscillation. To be specific we consider first the case of second-harmonic generation from an input mode at w/2 to an output mode at w. The perturbation polarization is taken as The complex amplitude of the polarization is where dijk("') is an element of the nonlinear optical tensor and summation over repeated indices is understood. We have allowed, in (34), for apossible dependence ofdij,ontheposi￾tion r. A. Case I: TEinpUt-TEoutpUt Without going, at this point, into considerations in￾volving crystalline orientation, let us assume that an optical field parallel to the waveguide y direction will generate a second-harmonic polarization along the same direction where P and E represent complex amplitudes, and d corresponds to a linear combination of dijk which depends on the crystal orientation. In this special case an input TE mode at w/2 will generate an output TE mode at w. Using (30) in (35) gives We consider a case of a single mode input, say n. In that case the double summationf (36) collapses to asingle term n = p

YARIV:COUPLED-MODE THEORY 923 If we then use P,(r,t)as [Ppert(r,t)]y in (32)we get 【001 dA.( =-de【4um12ta.-."Sa dz 4 (37) with 8(8(fx)dx (38) where we took d(r)=d(z)fx). Fig.4.The orientation of a 43m crystal for converting a TM input at In the interest of conciseness let us consider the case where w/2 to a TE wave at w.x.y.z are the dielectric-waveguide coordinates. while 1.2.and 3 are the crystalline axes.Top surface is(100). the inner layer 2 is nonlinear and where both the input and output modes are well confined.We thus have gm,Pm>>hm and hmd.From (17)and (21)we get sion results when the phase-matching condition 4=8-28/=0 (44) 8nm.一2N8. μ sin mnx -t≤x≤0. is satisfied.In this case the factor sin2(Al/2)(Al/2)2 is The overlap integral Syn.n.m)is maximum for n =m=1,i.e., unity.Phase-matching techniques will be discussed later. fundamental mode operation both at w and /2.For this case the overlap integral becomes B.Case I1:TMinput-TEoutput The anisotropy of the nonlinear optical properties can be used in such a way that the output at w is polarized orthogonally to the field of the input mode at w/2.To be specific,we consider the case of an input TM mode and an 8,428,./28,1dk output TE mode.If,as an example,the guiding layer(or one of the bounding layers)belongs to the 43m crystal class =1.2v2u (39) (GaAs,CdTe,InAs),it is possible to have a guide geometry Vi(B)B2 as shown in Fig.4.x.y,z is the waveguide coordinate system as defined in Fig.4,while 1,2,and 3 are the conventional and (37)can be written as crystalline axes.For input TM mode with Ex we have =-2X12日2wn E=E1= E d 4 Vi(8g (4 √/2 (40) The nonlinear optical properties of 43m crystals are with described by [13] 4=8-28/2 (41) P1=2diasE2E3 and where the,now-superfluous,mode-number subscripts P2=2diEE3 have been dropped.Integrating(40)over theinteraction dis- P3=2disEEa tance gives 14o°=24r3 so that (41/2)9 (42) 83"(8y: Py=P:=dimE:. (45) The normalization condition(20)was chosen so thatA2 is the power per unit width in the mode.We can thus rewrite Taking (42)as H,=3∑Be,"”(xeua-w+c.c. sin"(Al/2) (△1/2) (43) and using (aHy/8z)=-iwe Ex gives where we used Bwvue,e/eo=n2.Note that (P/wt)is theintensity(watts/square meter)of the input mode.Except for a numerical factor of 1.44,this expression is similar to that derived for the bulk-crystal case [13].Efficient conver- (46)

YARIV: COUPLED-MODE THEORY 923 If we then use Py(r,t) as [Ppert(r, t)Jy in (32) we get with where we took d(r) = d(z)Ax), In the interest of conciseness let us consider the case where the inner layer 2 is nonlinear and where both the input and output modes are well confined. We thus have qm,pm >> h, and h,d = T. From (17) and (21) we get 8- The overlap integral S(n,n,m) is maximum for n = m = 1, i.e., fundamental mode operation both at w and 0/2. For this case the overlap integral becomes (39) and where the, now-superfluous, .mode-number subscripts have been dropped. Integrating (40) over the interaction dis￾tance 1 gives The normalization condition (20) was chosen so that I A I is the power per unit width in the mode. We can thus rewrite (42) as where we used ,P adz, E/€,, = n2. Note that (P12/wt) is the intensity (watts/square meter) of the input mode. Except for a numerical factor of 1.44, this expression is similar to that'derived for the bulk-crystal case'[ 131. Efficient conver￾Fig. 4. The orientation of a 43m crystal for converting a TM input at w/2 to a TE wave at w. x, y, z are thedielectric-waveguide coordinates, while I, 2, and 3 are the crystalline axes. Top surface is (100): sion results when the phase-matching condition is satisfied. In this case the factor sin2 (Al/2)(A1/2l2 is unity. Phase-matching techniques will be discussed later. B. Case Ii: TMingut-TEoutput The anisotropy of the nonlinear optical properties can be used in such a way that the output at w is polarized orthogonally to the field of the input mode at w/2. To be specific, we co'nsider thecase of an input TM mode and an output TE mode. If, as an example, theguiding layer (or one of the bounding layers) belongs to the 43m crystal class (GaAs, CdTe, InAs), it is possible to have a guide geometry as shown in Fig. 4. x,y,z is the waveguide coordinate system as defined in Fig. 4, while 1,'2, and 3 are the conventional crystalline axes. For input TM mode with E I I x we have The nonlinear optical properties of a3m crystals are described by [13] so that Taking and using (8Hy/8z) = -iwt E, gives

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)

YARIV:COUPLED-MODE THEORY 925 It follows from(58)that an alternativeandequivalent defini- Using (22)the Ex component of a single forward- tion would betospecify thechanges ofthe dielectrictensor traveling TM mode is given by as C (65) △eg=SrEo (59) E,”红，0=2e8 where the normalization(26)is such that|B2is the power where the (0)superscript denotes a"low"frequency,i.e.,a per unit width in the mode.From(64)and(65)we obtain frequency well below the crystal's Reststrahl band.Using the relations Pc,）=ex,Eo t)B,(+c.c. (66) D=E+P Substitution of (66)into the wave equation (32)leads to D=E and choosing a principal coordinate system so that d4cxp(-8.)-4ep限.到 d止 c(Eo*0)=6i,+△e rx,2E"c2BB30,(x8,)dk leads to E(x)Eo Xexp(-iB,TM2)十c.c. (67) P () )+(-(o)8 E t) (60) Equation(67)is general enough to apply to a large variety where we used the convention e=e.The perturbation of cases.The dependence of E and r(x,z)on x allows for polarization to be used in (32)is that part of P which is coupling by electrooptic material in the guiding or in the proportional to the "low"-frequency electric field,i.e., bounding layers.The z dependence allows for situations where E or r depend on position.To be specific,we con- [P()=E [wn+ee sider first the case where the guiding layer -t>h and the expressions(17)for 8,(m(x)and (24)for corresponds to the output TEand [Ppert],is theycomponent cm(x)in the guiding layer become of the polarization(61)induced by the x(and z*)electric- field components of the input TM mode.Using(61)we get 8mx)→ 4oμ IB TE PttbluEE (62) 2\1/2 Eo 3C,m(x)→ 18.TM sin x (68) where the Is are direction cosines.Defining where for well-confined mode B.TMBmT=8=kna.In ctualiyljsEx=erE) (63) this case the overlap integral becomes (62)becomes P.(='rg(o) -E ( (64) (69) where P,is the complex amplitude of the polarization. Having chosen the case of a uniform Eto and r,the In most cases of practical interest the choice of crystal only z dependence on the right side of(67)is that of the orientation and the field E is such as to simplify (63)to exp (-i8,TMz)factor.Since BTMmTE(I=m)we may a simple form resembling(64);an example is provided at neglect the term involving Am.The coupling thus involves the end of Section VI.In any case,the definition of (63) only the forward TE and TM modes.Using(69).(67) applies to the most general case. becomes The E,component ofa TM mode can also cause coupling but this will typically be a smaller effect,since E.<<E. 会=-kB.ep-0.m-.(0

YAKIV: COUPLED-MODE THEORY It follows from (58) that an alternativeand equivalent defini￾tion would be to specify thechanges of the dielectric tensor ciJ as 925 where the (0) superscript denotes a "low" frequency, i.e., a frequency well below the crystal's Reststrahl band. Using the relations and choosing a principal coordinate system so that where we used the convention ci = cii. The perturbation polarization to be used in (32) is that part of Picw1 which is proportional to the "low"-frequency electric field, i.e., To bespecific, weassume that theinput isaTM modewith E(W1 I I a, which is coupled by the electrooptic properties of the bounding media or theguiding layer to theTEmodewith ElW) 1 1 aY., The starting point is again (32) where the modem corresponds to the output TEand [PpertIyis they component of the polarization (61) induced by the x (and z*) electric￾field components of the input TM mode. Using (61) weget py(w) = e.t.y.. z 1 r1k I. tv 1. 12 Ek(0) €0 where the I's are direction cosines. Defining (62) becomes where Py(W) is the complex amplitude of the polarization. In most cases of practical interest the choice of crystal orientation and the field is such as to simplify (63) to a simple form resembling (64); an example is provided at the end of Section VI. In any case, the definition of (63) applies to the most general case. The E, component of a TM mode can also cause coupling but this will typically be a smaller effect, since E, > h and the expressions (1 7) for E~(~)(X) and (24) for X $"'(x) in the guiding layer become where for well-confined mode PLTM = OmTE P = kn,. In this case the overlap integral becomes lt X!,'"'(x)&,'"'(x) dx = -- 4w dig mm dx = 2 Having chosen the case of a uniform E'O' and Y, the only z dependence on the right side of (67) is that of the exp (-iPITMz) factor. Since DmTM = PmTE (I = rn) we may neglect the term involving Am-. The coupling thus involves only the forward TE and TM modes. Using (69), (67) becomes __ - - - id, exp [- i@m'rM - 6, "')z] (70) dz

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-

YARIV:COUPLED-MODE THEORY 927 P isthephotoelastictensor.Comparing(79)to(58)wecan apply the results of Section V directly.Taking the strain field in the form of guiding layer Sk:@G,)=&S四eo-Ka》十c.c. (80) we obtain in a manner similar to (61) 【PorG,tl,=D迪【S,@Eeew+8t-a+ 4e0 Fig.5.An interdigital-electrode structure for applying a spatially S(-E(eto-mt-(8-k)]c.c. (81) modulated electric field in electrooptic phase matching.x,y,and z are the waveguide coordinates,while 1,2,and 3 refer to the cubic [100] for the polarization wave arising from the nonlinear mixing axes of a 43m crystal. of an electric field ponent).To be specific,let us choose eftut-8)+c.c. (82) 2领=△ (77) and a sound strain wave (80). A To be specific,we will assume again that the input optical field is a TM mode and will derivetheequation governingthe and keeping only the synchronous term,obtain from(74) evolution of the TE mode due to the coupling.In a manner dA地= similar to(63)we abbreviate the information relating to crystal symmetry and orientation by defining -为4 epS=PSliy (83) (78) dz and instead of(81)use This corresponds to phase-matched operation with an effec- tive coupling coefficient reduced by a/2 relative to phase- [Per红，tle matched operation with a uniform field Et(z)=Eo.The solution of(78)is given by (7). 年6psE,ep{e+r-Bru+0 We close this section by considering,again,the use of 43m crystals for the phase-matching scheme just discussed.The +S(-E()exp (il(@-2)t -(Brxt -K)z]]+c.c. nonvanishing elements of the rux tensor are [15]raa rai2= In a manner identical to that leading to (67)we obtain r12.From(61)it follows directly that a 43m crystal oriented, as in Fig.5,so that its cubic 1,2,3axes coincide,respectively, dA (+ with thex,y,zdirections ofthe waveguide,is optimal sincein da exp[i(rEt-8re2刃 this case 、d4f) deexp (i(ret+Bz)] P,w=兰nnE,oE,e P,oi=至naE,oE, -看9aa定，"eg E(x)Eo thus coupling the TE mode(E,()to the TM(Ex(4),and vice versa,in the presence of a longitudinal dc field E,(0. .[exp (i[(+2)t -(8TM+K)z] expi(@-2)t -(BTM -K)z]]] (84) VII.PHOTOELASTIC COUPLING The possibility of coupling dielectric-waveguide optical A few comments may be in order here.Each of the two modes through the intermediary of sound waves has been terms on the right-hand side of(84)represents a traveling demonstrated [17].In this section we will treat this class ofin- polarization wave.Both input waves,i.e.,S and E(,we teractions using the coupled-mode formalism. recall,are taken as traveling in the +z direction.Or- The photoelastic effect is defined by relating the effect of dinarily,BrE is close to,but slightly larger than,BrM.In this case the coupling is via the first term on the right side strain Sw on the constants of the index ellipsoid through [18] of(84)and the wavelength of the sound wave is adjusted so that =Piik SL. (79) BTE =BTM+K (85)

YARIV: COUPLED-MODE THEORY 927 Pijklisthephotoelastictensor.Comparing(79)to(58)wecan apply the results of Section V directly. Taking the strain field in the form of w guiding layer we obtain in a manner similar to (61) Fig. 5. An interdigital-electrode structure for applying aspatially modulated electric field in electrooptic phase matching. x, y, and z are the waveguide coordinates, while 1, 2, and 3 refer to the cubic [IOO] axes of a $3171 crystal. ponent). To be specific, let us choose (77) and keeping only the synchronous term, obtain from (74) This corresponds to phase-matched operation with an effec￾tive coupling coefficient reduced by ~/2 relative to phase￾matched operation with a uniform field E(O)(z) = Eo. The solution of (78) is given by (7). We close this section by considering, again, the use of43m crystals for the phase-matching scheme just discussed. The nonvanishing elements of the iijk tensor are [ 151 raZ1 = rslz = r123. From (61) it follows directly that a43m crystal oriented, as in Fig. 5, so that its cubic 1,2,3 axescoincide, respectively, with thex,y,zdirectionsofthewaveguide,isoptimalsincein this case thus coupling theTE mode(E,'"') to theTM (Ex(")), andvice versa, in the presence of a longitudinal dc field E,('). for the polarization wave arising from the nonlinear mixing of an electric field and a sound strain wave (80). To be specific, we will assume again that the input optical field is a TM mode and will derive the equation governing the evolution of the TE mode due to the coupling. In a manner similar to (63) we abbreviate the information relating to crystal symmetry and orientation by defining and instead of (81) use VII. PHOTOELASTIC COUPLING A few comments may be in order here. Each of the two The possibility Of dielectric-waveguide Optical terms on the right-hand side of (84) represents a traveling modes through the intermediary Of sound has been polarization wave. Both input waves, i.e., S(0) and Ex(@), we demonstrated [ 171. In thissectionwewill treat thisclassofin- recall, are taken as traveling in the +z direction. Or￾teractions using the coupled-mode formalism. dinarily, PTE is close to, but slightly larger than, PTM. In The photoelastic effect is defined by relating the effect of this the coupling is via the first term on the right side on theconstants oftheindexellipsoid through [I81 of (84) and the wavelength of the sound wave is adjusted so that (79) PTE = PTM + K

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>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 > 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-