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+4yIEEE 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