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 radiation in dielectric waveguides is cast in the coupled-mode formalism. This approach 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 optical modulators which are promised by this approach A variety of theoretical ad hoc formalisms have been utilized to datein treating thevarious phenomena ofguidedwave 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 perturbations; 3) electrooptic switching and modulation; 4) photoelastic switching and modulation; and 5) optical filtering and reflection by a periodic perturbation. [51-[71. 11. THE COUPLED-MODE FORMALISM We will employ, in what follows, the coupled-mode formalism [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 unperturbed medium so that they represent propagating disturbances 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, California 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 situation 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 convenient 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 expressed 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 acoustoopd 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 COUPLEDMODE 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≤x<m E.,,)=84=足5现,x2-m we Oz e C[cos (hx)-(q/h)sin (hx)1, 主0H 8,(x)= -t≤x≤0 E,(x,z,t)=- (23) we dx C[cos (ht)+(q/h)sin (ht)]exp [p(x 1)], -0<x≤-1(17) The transverse function C,(x)is taken as which,applying(15)to regions 1,2,3,yields cos (ht)+sin (hr) p+1) x<一t h=(n22k2-B2)1va 3C,(x)= cos (hx)+sin (hx) 一1<x<0 9=(82-m:k)a p=(82-n32k32 h Ce x>0.(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 corFig. 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 = thevariouspropagationconstantsobeytheeigenvalueequaand 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 interest to us we can represent the perturbation as a distributed polarization source Ppert(r,t), which accounts for the deviation 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 components 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 integration 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 following 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 secondnarmonic 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,ontheposition r. A. Case I: TEinpUt-TEoutpUt Without going, at this point, into considerations involving 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 distance 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 converFig. 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