Coupled-Mode Theory HERMANN A.HAUS AND WEIPING HUANG,MEMBER,IEEE Invited Paper The principal features of coupled mode theory are reviewed and fiber optics in recent years are described.We then con- in particular as applied to passive structures,such as coupled sider lossless coupling of two modes in time.Two coupled resonators and coupled waveguides.The former are examples of coupling of modes in time,the latter are examples of coupling of resonance circuits,or two coupled microwave or optical modes in space.Active structures are considered briefly insofar resonators,are the physical examples.Energy conservation as they obey conservation laws in terms of positive and negative requires that the eigenfrequencies be real.The start-up of a energies.The complications caused by nonorthogonality of the parametric oscillator is another example.Then we look at mode energies,or powers,are brought up.These were the topic of recent publications. the formal derivation of coupled mode theory and consider The coupling of modes formalism has intuitive appeal in that the the more general case when the modes are not energy- coupling equations in their simplest form can be written down by orthogonal and the energies are not necessarily positive. inspection.When "fine points"as,for instance,cross talk are at A more detailed account of the nonorthogonal coupled issue,then more formal derivations are required.The one presented mode theory developed in the last five years for optical here is based on a variational principle. waveguides is given.When one resonator stores negative energy,the other positive energy,the frequencies appear in I.INTRODUCTION complex-conjugate pairs.A resonator containing an active In this paper we review the principal features of coupled medium,the latter treated in a linearized approximation,is mode theory.The literature on the subject is vast and it an example. would be difficult to do justice to all aspects of coupled This study is followed up by showing how the coupled mode theory.When the first author,H.A.Haus,was asked mode formalism can be derived from a variational principle to contribute this invited paper,he accepted on the grounds for the frequencies of the system.If a trial solution is that aspects of coupled mode theory had intrigued him introduced for the electric field in a lossless electromagnetic throughout his career,his first papers on the subject having system that is the linear superposition of modes,the coupled been written in the 1950's.Recent work done with W.P. mode formalism is the result.We do not consider the formal Huang,one of his doctorate students,prompted him to ask derivation for more complicated,active systems,they can him to collaborate on this invited paper.Hence the present be found in the literature.Indeed,the very virtue of the cou- paper is an overview of coupled mode theory through the pling of modes formalism lies in its intuitive appeal.Much “lenses”of two“afficionados'of coupled mode theory.The physical insight can be obtained by inspection,without the review is not intended to be exhaustive due to the imposed need for a formal derivation limit on the length of the paper.It represents the view of Next,we consider coupling of modes in space.Here the the authors based on their own experience and knowledge initial conditions in time,relevant to coupling of modes in We first give a brief historic perspective of the coupled time,are replaced by boundary conditions in space.The mode theory.The development and applications of the sign of the group velocity of a wave indicates at which theory in microwaves in early years and in optoelectronics "end"of a structure a wave is excited.The sign of its energy indicates whether it is passive (positive energy)or active Manuscript received August 2,1990.revised December 28.1990.This (negative energy).On the basis of such simple arguments work was supported in part by the National Science Foundation under one may distinguish the operation of a Bragg reflector, Grant EET8700474 and by the Joint Services Electronics Program under Contract DAAL03-89-C-0001. traveling wave tube,or backward wave oscillator H.A.Haus is with the Department of Electrical Engineering and The formal derivation of the coupled mode formalism in Computer Science and Research Laboratory of Electronics,Cambridge, space is again restricted to the case of passive electromag- MA02139. W.P.Huang is with the Department of Electrical Engineering,Univer. netic structures.We show how the formalism results from sity of Waterloo,Waterloo,OntCanada. a variational principle.The variational principle provides a IEEE Log Number 9103551. refined method for the evaluation of coupling coefficients, 0018-9219/91/01.00©1991IEEE PROCEEDINGS OF THE IEEE,VOL 79,NO.10,OCTOBER 1991 1505
some aspects of which are easily missed by the intuitive then the best value obtainable for the propagation constant approach.The refinements are necessary for an understand- results from the coupled mode equations ing of cross talk in optical waveguide couplers.It was in The coupled mode theory for optical waveguides was this context that the need for a more formal derivation of developed by Marcuse [12],[13],Snyder [14],[15],Yariv coupled mode theory reasserted itself. and Taylor [161,[17],and Kogelnik [18],[19]in the early 1970's.It has been successfully applied to the modeling and analysis of various guided-wave optoelectronic and fiber II.HISTORICAL PERSPECTIVE optical devices,such as optical directional couplers made The concept of coupled modes in electromagnetics may of thin film and channel waveguides [20]-[23]and optical be traced back to the early 1950's.The application was fibers [24]-[28],multiple waveguide lenses [29],[30], initially to microwaves and developed gradually through phase-locked laser arrays [31],distributed feedback lasers the contributions of many people.In 1954,Pierce applied [31]-[37]and distributed Bragg reflectors (38]-[40],grating the coupled mode theory to the analysis of microwave waveguides and couplers [41]-[45],nonparallel and tapered traveling-wave tubes [1].Later followed the work of Gould waveguiding structures [46]-[55],Y-branch waveguides [2]on the backward-wave oscillators.The coupled mode 56]-[58],TE/TM polarization converters [59]-(63],polar- theory was then employed to treat parametric amplifiers, ization rotation in optical fibers [64]-(66],mode conversion oscillators,and frequency converters [3]. and radiation loss in slab waveguides and fibers [67]-(71], A parallel development happened in microwave waveg- residual coupling among scalar modes [72].[73].It has uides and devices.Miller [4]first introduced the coupled also been used to study the wave coupling phenomena mode theory to the analysis and design of microwave in nonlinear media such as harmonic generation in bulk waveguides and passive devices.The theory was soon 74]and guided-wave devices [75],[76],nonlinear pulse or generalized by Louisell [5]to treat tapered waveguide soliton propagation[77],[78]and the modulation instability structures,where the coupling coefficients depend on the [79]in optical fibers,and nonlinear coherent couplers length a.In the 1960's,the coupled mode theory was further [80]-[86.Many of these applications are well documented developed to describe mode conversions due to various and summarized in [87]-(93]. irregularities in microwave waveguides [6],[7]and periodic An assumption made in the conventional coupled mode waveguide structures [8],[9]. theory is that the modes of the uncoupled systems are The early approach to the coupled mode theory was orthogonal to each other.This may be true if the modes rather heuristic.The modes of the "uncoupled systems" belong to the same reference structures.In studying the were identified and the coupled mode equations obeyed mode coupling in coupled systems,however,one often by the mode amplitudes were determined from power chooses the modes of the isolated systems as the basis considerations. for the mode expansion and these modes may not be A rigorous derivation of coupled mode theory was car- orthogonal.The orthogonal coupled mode theory (OCMT) ried out by Schelkunoff (10].By expanding the unknown is not correct for the description of the mode-coupling electromagnetic fields of a coupled system in terms of the process in this case.The effect of nonorthogonality between known modes of an uncoupled system,he obtained a set waveguide modes on cross talk in optical couplers was first of generalized telegraphist's equations (a different version recognized by Chen and Wang [94]and then studied by of the coupled mode equations)directly from Maxwell's Haus and Whitaker [95]in a proposal to eliminate the equations.The coupling coefficients are determined unam- cross talk due to this effect.Later on,several formula- biguously once the modes of the uncoupled system are tions of the nonorthogonal coupled mode theory (NCMT) defined.The coupled mode equations are equivalent to were developed by Hardy and Streifer [96],Haus,Huang. Maxwell's equations as long as a complete set is assumed Kawakami,and Whitaker [97],and Chuang [98].The new for the mode expansion.For most applications,however, nonorthogonal coupled mode theory (NCMT)is shown only a limited number of modes (usually two)is used to yield more accurate dispersion characteristics and field in the expansion.Therefore,the coupled mode theory patterns for the modes of the coupled waveguides.It also remains an approximate,yet insightful and often accurate calls for a modification of the description of the power mathematical description of electromagnetic oscillation and exchange between the waveguides. wave propagation in a coupled system.In order to put the There were some discrepancies among the different for- approach on a more formal mathematical footing,one of mulations at the early stage of the development.Some were the authors (HAH)showed in 1958 that the coupled mode superficial and soon resolved by reformulation [99];some theory was derivable from a variational principle set up are more subtle [97],[99]-101].Snyder,Ankiewicz,and for the propagation constant of the coupled system [111. Altintasl [102]showed that the nonorthogonal formulations Because of the stationary nature of the variational principle, could lead to erroneous results for the coupling length of the the errors made in an incomplete expansion do not lead to TM modes of parallel slabs when the index discontinuity a dramatic deterioration of the accuracy for the propagation is large.The origin of the error is apparent in this case constants calculated from the coupled mode theory.If one since the waveguide modes used as the trial solution in decides to approximate the fields of the coupled system as a the coupled mode theory are subject to serious error when linear superposition of the fields of the uncoupled systems, the index steps are large.But they also demonstrated in PROCEEDINGS OF THE IEEE,VOL 79,NO.10,OCTOBER 1991
the same example that the conventional orthogonal coupled is important only when ww2.Then,the derivative of mode theory based on the same trial solution gives excellent a2 is approximately equal to jw x a2,its integral equal prediction about the coupling length.This unexpected result to 1/jw2 x a2.In the coupling term of (2.3),the coupling triggered a series of debates in the field [103]-[110].It was coefficient is small compared with both w and w2,and later resolved by Haus,Huang,and Snyder [111]. thus the replacement of the derivative or integral by its Despite the controversies,there have been intense re- approximate values causes an error of higher order in the search activities in the past few years in developing and coupling and can be ignored.This is why the coupling of applying the nonorthogonal coupled mode theory in the modes formalism can be set up so simply in the limit of area of optoelectronics and fiber optics.Simplified scalar weak coupling versions that may be applied to weakly guiding struc tures [112][115]and a modified vector version for the A.Coupling of Energy Orthogonal Modes of Positive Energy strongly guided structures [111],[115]were developed. Generalizations of the theory to multiwaveguide and/or We think of energy,generally,as a positive quantity.This multimode structures [116]-[121],anisotropic media [122] is the case we shall address first.There are many perfectly [123],periodic and tapered structures [124]-[128],and non realistic situations in which the energy must be considered linear couplers [129]were attempted.The nonorthogonal negative.We shall come back to such cases later on. coupled mode theory has been employed in the analysis and We concentrate first on energy orthogonal modes for design of optical guided-wave devices [130]-[138]and fiber which the energy can be written optical couplers [139]-(146].The experimental verification of the theory was carried out by Marcatili [147]and Syms W=la12+a22 (2.5) [148}-[150] even in the presence of the coupling.Here we have nor- malized the electric field amplitudes a1 and a2 so that III.COUPLING OF MODES IN TIME their squares are equal to the energies in the modes.No We shall first address coupling of modes in time starting cross terms are assumed to exist,the modes are energy with a few intuitively obvious postulates.From these we orthogonal.If the coupling is lossless,the only case we can develop a formalism that describes coupling of two shall treat here,energy must be conserved: resonators,parametric amplification and oscillation.Then we shall show how the coupling of modes of passive d electromagnetic structures can be derived from a varia- (l+laa)=jmajaz+jaa tional principle.This variational principle provides rules for -jki2a1a2-jK21a2a1=0(2.6) the evaluation of the coupling coefficients when intuitive arguments are not sufficiently precise to provide these where we have used (2.3)and (2.4).Because the initial coefficients. conditions can be picked arbitrarily,(2.6)can only be Consider two weakly coupled lossless resonators.Denote obeyed when the amplitude with the time dependence exp(jwit)in one resonator by a1,the amplitude in the other resonator K12=K21=K (2.7 with the time dependence exp(jwat)by a2.These are the positive frequency components of the electric field This is the constraint on the coupling coefficients of two amplitudes.They obey the differential equations: modes coupled in a lossfree way.When (2.7)is intro- duced into (2.3)and(2.4),a time dependence of the form dai dt jw1a1 (2.1) exp(jwt)is assumed and the determinantal equation is solved one finds the two roots for w: da2 =1w202. (2.2) w1+w2 w= W1-W2 When the two resonators are coupled,their time dependence +2 (2.8) 2 changes.When the coupling is weak,it has to be of the form: The two frequencies are real,as they must be when two day modes of positive energy couple and energy is to be dt =jw1a1+jK1202 (2.3) conserved.Figure 1 shows the roots ws (lower curve) daz and wa(upper curve)of the symmetric and antisymmetric dt jw2a2 jK21a1 (2.4) normal modes for the case when one of the resonance frequencies (w)is changed,the other is kept fixed.This is At first it may not be obvious why the coupling should the kind of diagram familiar also from quantum mechanical be proportional to the amplitude of the other resonator, perturbation theory.When an energy level crossing occurs rather than to the time derivative,or the time integral of in a coupled system,the "eigenvalues"split,crossing is the amplitude,or any complicated operator operating on prevented.At the crossover point,for real the solutions the amplitude.However,if the coupling is weak,the time are the symmetric and antisymmetric combinations of a dependence is perturbed only weakly and the coupling term and a2.Farther from the crossover point the solutions HAUS AND HUANG:COUPLED-MODE THEORY 1507
1 、+。 (3 2 28 。+61 Fig.1.The resonance frequencies of the symmetric (lower curve) and antisymmetric(upper curve)mode of a coupled resonator as a function of detuning 1. acquire the character of the mode to whose frequency the eigenvalue is the closest. It is of interest to ask for the implications of this simple coupled mode formalism.Consider first the case when the two resonators have equal frequencies.Then,if resonator 1 is excited at t=0,a=1,and resonator 2 is unexcited,a2 =0,the initial conditions are matched by a superposition of equal amounts of the symmetric and antisymmetric solutions.The phases of the two solutions evolve at different rates and after the time t=/2 all of the excitation will have been transferred to the other resonator.The excitation oscillates back and forth between the two resonators.When the frequencies of the Fig.2.Coupled metallic rectangular waveguide resonators that permit exact analysis. two uncoupled resonators are not equal,and initially only one resonator is excited,the transfer is not complete. by setting B.A Simple Example 1 Figure 2 shows a simple example that permits both a rigorous analysis and yields easily to the coupled mode for- malism.In this way one may compare the exact answer with We use energy arguments to derive the coupling.Consider the approximate solution.The two resonators are metallic the time rate of change of the energy in mode (1): rectangular waveguides partially filled with a dielectric so d that the empty waveguides are below cutoff at the resonance a1P=jN12aia2-jxza1o时. (2.13) frequency.The uncoupled waveguides are each terminated in infinite air-filled waveguides.The question is as to how This energy change must be equal to the power fed into one may find the coupling coefficient for this structure,and mode (1)by mode(2).Mode(2)finds the perturbation of thus the value of the beat frequency of the two eigenmodes. dielectric constant e within resonator (1)and drives a Denote the spatial distribution of the dielectric constant that polarization current density through that perturbation that forms cavity (1)by is equal to Eo +6E1 2.9) jwP12 jwde1a2e2 (2.14) and the one that forms resonator (2)by The power fed into mode (1)by this current density is Eo +662. (2.10) equal to The actual distribution is 1 4 juwP12·ajeidv+c.c. e=e。+6e1+6e2, (2.11) =jk12aja2-jki2a1a2 Denote the electric field patterns of modes(1)and(2)by 1 er and e2,respectively.The energy in mode (1)is 4 jwbeie2ejdvaja2 +c.c.(2.15) cleil2dv a12 (2.12) and must be responsible for the rate of growth in time of the energy in resonator (1)due to the presence of mode (2) 1508 PROCEEDINGS OF THE IEEE,VOL 79,NO.10.OCTOBER 1991
15 where W is a square matrix of nth rank and may be positive 一k12 definite,if all wave energies are positive,or indefinite,if there are wave energies of either sign.The dagger indicates a Hermitian transpose.Then the equations of motion of the .09 modes can be written 形6 dt jHa (2.18) .06 where H is the coupling matrix that incorporates the (small) 03 frequency differences and the coupling.If energy is to be conserved we have 0.0 .0 1.5 2.0 0- da'Wa=jla'Ha-a'H'al. (2.19) d t/a This puts a constraint on the matrix H.Because the initial conditions are arbitrary,the matrix a is arbitrary and one must have H=H (2.20) The energy nonorthogonality may appear naturally as a consequence of the coupling,as we shall see later on.Hence c-+h d+-c+ it is of importance to understand its consequences.We shall look at these consequences in greater detail when we Fig.3.The coupling coefficients12 (solid line)and (dashed line)as functions of the resonator separation.The parameters of address coupling of modes in space,in the context of which the waveguides are a/b 1,c/b=1 and e/o 2.25. power nonorthogonality [96]-98]was first recognized as an important effect.However,the formalism can always be cast into a form that is energy orthogonal.Let us briefly Since the excitation amplitudes are arbitrary,the equality look at the orthogonalization.Because the energy W is must hold: real,the matrix W is Hermitian.A Hermitian matrix can w∫6e1ei·e2du always be diagonalized by a unitary transformation.Denote K12= (2.16) 支∫cel2do the matrix that does it by U.Then, UWUt=P (2.21) where we have written the result independent of the nor- malization.Figure 3 shows a plot of the coupling coefficient where P is a diagonal matrix.It may be indefinite,possess- as a function of normalized resonator separation ing both positive and negative diagonal elements.With no loss of generality one may assume that the elements are all C.Nonorthogonal Modes of Positive and Negative Energy either +1 or-1,because the original amplitudes a:of the All modes need not have positive energy.An example modes can be properly normalized for that purpose.Then, of negative energy is a moving electron beam or plasma of course, If the equations of this system are linearized,and wave P'P=PP=I solutions are found,the energy of these waves can be (2.22) negative.A negative energy simply means that the energy where I is the identity matrix.Now define the new ampli- of the system is lower when the wave is excited than when tude matrix b: it is not:A moving electron beam stores positive kinetic energy and electromagnetic energy.The excitation of the b=Ua. (2.23) wave decreases this energy. For the present discussion we need not assume that there Multiplying both sides of(2.18)by U we obtain an equation are only two coupled waves.Suppose that there are n modes for the amplitude matrix b: of interest,their n amplitudes are arranged in a column vector of nth order a.In the absence of coupling,the aib=jMb (2.24) modes of different resonance frequencies must be energy orthogonal.If they were not,if there were cross terms of where energy,they would vary with time,and the energy would M=P'UHUT (2.25) not be independent of time.If the modes are of the same frequency,they can always be orthogonalized.The energy The new coupling matrix has to obey the constraint imposed W of the system can then be written by energy conservation.Indeed let us study the matrix PM: W=aWa (2.17) PM =UHU (2.26) HAUS AND HUANG:COUPLED-MODE THEORY 1509
and thus a growing mode and a decaying mode.One should point PM=M'P. (2.27) out that "synchronism,"i.e.,w=w2,in the case of the parametric oscillator,occurs when (2.28)is strictly obeyed. We have thus derived a coupling of modes formalism that is The growth and decay found in the present case is the energy orthogonal and allows for both positive and negative signature of an active unstable system.Positive energy energy modes.In the case when P is positive-definite,or the (or photon number)in one mode grows or decays in identity matrix,we see that M is Hermitian.A Hermitian consonance with the growth or decay of the (negative) matrix can have only real eigenvalues,i.e,all frequencies of energy (or photon number)in the other mode. the coupled oscillators must be real.When P is indefinite, M is not Hermitian,the eigenvalues can be complex.One IV.THE VARIATIONAL PRINCIPLE FOR POSITIVE can show that the complex eigenvalues appear in complex- ENERGY MODES conjugate pairs;for every solution growing in time one Thus far we developed the intuitive approach to coupled finds a solution that decays in time. mode theory.Clearly,if the theory is indeed physically Now that we have developed the general formalism for correct,then it ought to be possible to derive it from coupling of modes that do not all have positive energy,it the fundamental equations of the system.This is indeed is of interest to look at a special case of two modes,one possible.Several approaches are possible.In the case of of which has positive energy,the other negative energy.As a purely electromagnetic system,Schelkunoff derived the mentioned earlier,this could be the case of an electromag- coupling of modes formalism from an expansion of the netic mode of a cavity through which moves a plasma with field in terms of a complete set of modes [10].Another a wave that carries negative energy.Clearly the coupling approach is to derive the coupled mode formalism from a of modes in time implies that the two respective modes variational principle.This has been done for the case of an change uniformly over all of space.This means that the electron beam in a microwave structure [11],and can be plasma may have to be reentrant.A more realistic model is done rather easily for the case of a purely electromagnetic a parametric oscillator,in which a signal mode at frequency structure with a positive definite energy matrix W.We shall wa,and an idler mode at frequency wi are coupled by a concentrate on this case here because it is simple and gives pump of frequency p with insight into the procedure. up=wg十 (2.28) A variational principle provides an approximate value for an eigenvalue of a differential equation that is more This case is not strictly one of energy conservation,but accurate than the trial solution used in its evaluation.This rather the case of a system obeying the Manley-Rowe is an important aspect of the use of a variational principle. relations [151].The Manley-Rowe relations were derived Indeed,when one assumes that the excitation of a coupled originally from classical considerations,and can be deduced system is given by a linear superposition of the modes from the Hamiltonian of the (lossless nonlinear)system of an uncoupled system one uses field patterns that have [152].However,they may be stated in terms of photon been obtained in the absence of coupling,field patterns that number"conservation"[153].In a parametric oscillator,the ignore the coupling.Thus,how does one expect to obtain pump is stimulated to emit into the idler and into the signal a reliable prediction of the time evolution of the system,if mode.There is a simultaneous generation of signal photons the fields employed in the evaluation ignore the coupling? and idler photons.Thus if2 stands for the number of The variational principle lays this criticism to rest. signal photons,22 for the number of idler photons,then The equation for the electric field in a medium of the relation holds: dielectric tensor a function of position,is P=盖a d (2.29) 7×(7×E)=w2h元·E. (3.1) But this looks like an energy conservation relation with one Suppose further that the system is enclosed in an enclosure mode having negative energy.Now that we have stated the consisting of perfect electric and/or magnetic conductors. type of system we are analyzing,let us look at the solution Then,on the enclosure the boundary conditions are satis- of the determinantal equation.The equations are of the form fied: of (2.3)and(2.4),except that the ai's have to be replaced n×(7×E)=0 (3.2a) by b;'s,and that,according to (2.25): on the magnetic conductor,and K12=-21=K. (2.30) n×E=0 (3.2b) The solution of the determinantal equation is now on the electric conductor.Finally,consider the constraint W= w1十w2 1-w2 -2 (2.31) on the tensor imposed by the losslessness of the medium. 2 2 It is well known that the e tensor written in Cartesian coor- dinates forms a Hermitian matrix obeying the relationship: When the two modes have nearly the same frequency, wI-w2<2,the eigenvalues are complex,there is = (3.3) 1510 PROCEEDINGS OF THE IEEE,VOL.79,NO.10.OCTOBER 1991
Here,the dagger indicates,as usual,the complex-conjugate When this trial solution is introduced into (3.4)one obtains transpose of the tensor.Let us now prove the variational character of (2.18).By dot-multiplying (3.1)by E",inte- w2= atKa aiWa (3.9) grating over the volume enclosed by the enclosure,and by integration by parts,using the boundary conditions (3.2) where one obtains w2=P×E)(xE)dw eg.元，e,dw (3.10) 。∫E*·e·Ed (3.4) and Equation(3.4)is a variational expression for the frequency w.Indeed suppose that we substitute into it a field E+E, hoK=/(T×e)·(7xe)d where E is an error,a deviation from the exact solution. Then we can show that w2 will be unaffected to first order eijejdv. (3.11) in 6E.Taking a perturbation of (3.4)we find Clearly,both W and K are Hermitian matrices.Because both W and K are also positive definite the value of w2 is real and positive,as it ought to be.An stationary value is +w2h。E..Edw found for the eigenvaluew by differentiating the right hand side of (3.9)with respect to the amplitudes and phases of the ai's.This can be done formally by differentiating (3.9) =/(×6E)(×E)dw with respect to the a's,keeping the a;'s constant.The Appendix shows that this is indeed legitimate.The result is +/(×E*)-(V×6E)dw (3.5) w2Wa=Ka (3.12) Because of(3.3),the order of the vectors on the left-hand We shall now show that the above expression can be side of(3.5)may be reversed.Further,integration by parts written as a coupling of modes equation.Indeed,assume with the aid of the boundary conditions (3.2)gives that perturbation theory is valid,that all frequencies cluster 6w2E"..Edv around a typical value wo.Then one may write w2-w2≥2uo6w (3.13) dr6E.[w2uoe.E-7×(V×E】 where ow is the deviation from this frequency.However, d6E.w2h”.E*-7×(7×E)】=0. jow can be interpreted as a time derivative,and (3.12)can be written with the help of (3.13): (3.6) wda=j(K-w3W)a. (3.14) The integrands in the last two expressions of (3.6)vanish t=2。 and thus 82 must be zero to first order in 6E.This This is clearly in the form of a coupled mode equation with completes the proof of the variational character of(3.4). a Hermitian energy matrix W and a Hermitian coupling We shall now apply the formalism to a multimode matrix of the formH(K-w2W)/2w.The formalism resonator.The first part of the analysis will be general, has not only led to the coupled mode equations,but with no specific situation in mind.Then we shall specialize also provided a recipe for the evaluation of the coupling the analysis to the case of two dielectric cavities that are coefficients. coupled by their fringing fields. The field patterns we use for the trial solution consist of B.Application to Coupled Dielectric Cavities the fields in some spatially varying dielectric medium,in We now turn to the coupled mode analysis of the coupled general a different medium for every mode.We denote the cavities analyzed previously by the simple orthogonal mode modes by e;and the dielectric constant distribution by Ei. coupling theory.Here we handle the case more carefully, The dielectric tensors E;are assumed to be those of lossless allowing for the nonorthogonality introduced by the cou- media.The modes are assumed to obey the wave equation pling,which changes the equations somewhat.We compare 7×(V×e)=w2o·e (3.7) this improved coupled mode analysis with the previous one, and also with the exact analysis that is not difficult in this where w;is the resonance frequency of the mode.The simple case. modes obey the proper boundary conditions over the surface We introduce two dielectric distributions associated with enclosing the volume of the system.We assume as the trial the two individual waveguides in the absence of the other field the linear superposition of these two field patterns: (see Fig.2). B=aiei (3.8) 元=(co+6e1)I (3.15a) 2=(eo+c2)l. (3.15b) HAUS AND HUANG:COUPLED-MODE THEORY 1511
一Ezact 1.10 -Orthoganl CMT =-a-aos(2笑a: d (4.2) All the arguments as to the form of the coupling made in connection with the coupled mode formalims in time .00 apply to the present case.The modes may carry positive or negative power.Negative power may occur because positive energy is transported in the -z direction (the group velocity is negative)or negative energy is transported forward (positive group velocity).In this connection one 9001.0 15 2.0 may ask as to how is it possible to have two waves with t/a phase velocities of the same sign possess group velocities of opposite sign.This property is common in periodic structures that possess many "Bloch waves,"some of which Fig.4.The resonance frequencies of the symmetric (lower curves) and antisymmetric(upper curves)modes of the coupled resonators have phase and group velocities of opposite sign.It is in Fig.2.Solid:Exact solution;Dash-dot:Nonorthogonal coupled coupling with some of these "backward"Bloch waves that mode;Dashed:Orthogonal coupled mode.The parameters are the gives rise to the physical situations to be discussed below. same as in Fig.3. Define the power matrix P diag(1,1)where the signs correspond to the sign of the power flow of the two The resonance frequencies are equal and can be set equal waves,so that with proper normalization,the power can to wo.The coupled mode equation becomes be written wd a=jHa Power at Pa. (4.3) (3.16) with the coupling matrix Suppose that the space harmonic with the propagation constant 2/A+B2 is close to synchronism with B1.Then = e,6ejdv. (3.17) using the ansatz The coupling coefficients were obtained previously from a=4即-2+ (4.4) physical arguments.What was not obtained ab initio,when energy orthogonality was taken for granted,were the self 2 (4.5) terms on the right-hand side.They are corrections that affect we can write the value of the eigenvalues as shown in Fig.4. dA1 V.COUPLING OF MODES IN SPACE dz 2 ）A1-j12A2 (4.6) Consider two waves in a linear system,with the implicit dA2 dz +T)A2-j21A1. (4.7) time dependence exp(jwt)that,uncoupled,have the spatial 2 dependences exp(-j2)and exp(-jB22),respectively Just as in the case of coupling of modes in time,the The two waves are coupled in space,either by a uniform coupling matrix M,which in the present case is of the form: structure,or by some periodic structure,like a grating,or M=- ,-开 K12 a helix (in a backward oscillator)or some other periodic (4.8) structure;or by a periodic space-time phenomenon,like an K21 ,+天】 2 optical pump in a parametric amplifier.In the latter case, has to obey the power conservation law (compare(2.27)): the common frequency w specified above is,in fact,the signal frequency wa of mode (1)on one hand,and the PM=M'P. (4.9) frequency p-w:for mode(2),on the other hand,which is equal to ws at synchronism.In a uniform structure,the The power matrix may be positive-definite or indefinite propagation constants B and 32 must be of same sign and The physical reasons for this are more varied than in the approximately equal,if the waves are to affect each other case of coupling of modes in time.Thus we may have the If the coupling structure is periodic with the period A,then simple situation of two waves with colinear group velocities coupling among space harmonics(Bloch waves or Brillouin and positive energies.Such waves carry power in the same components)becomes possible.Suppose,for example,that direction and have a positive definite P matrix.The two the coupling coefficient is of the form:2K12 cos(2x2/A). waves may have oppositely directed group velocities,but The coupled mode equations in space,then read energies of same sign.Then P is indefinite.On the other hand,the energies may be of opposite signs,the group d (4.1) velocities codirectional,and then P is also indefinite. We shall concentrate here on the case of two synchronous and waves when the two modes are phase matched.For the 1512 PROCEEDINGS OF THE IEEE,VOL.79.NO.10,OCTOBER 1991
A()=1 Fig.5.Schematic of a dielectric waveguide coupler. Fig.6. Schematic of a backward wave microwave oscillator. uniform structures where the period A-oo,the phase- matching requires that B=2;for the periodic structures, circuit ipout it leads to B1-B2 =2x/A.Thus,coupling can occur,even when the propagation constants are widely different,but circuit output the periodicity of the structure makes up for the difference. In this way,waves with opposite phase velocities can be made to couple to each other. beam output Under phase-matching,the coupled mode equations as- e beam input sume the simpler form: dA =-jkA? dz (4.9) dA2 da =干jk*A1 (4.10) Fig.7.A photograph of physical backward wave oscillator. and the gain of the system is where =K12.The two signs correspond to equal or opposite directions of power flow.The solutions are A1(012 1 A1(0) cos2 KI. (4.15) A1=A+e-r:+Ae✉ (4.11a) A2 =Is [Ase-jlelA_ell-] (4.11b) The gain goes to infinity when=/2.The "backward wave"amplifier becomes an oscillator.Another physical for P positive(or negative)definite example is a parameter amplifier with phase-matched signal and idler waves. A1 Are-lxl:+A_elxl= (4.12a) Finally,look at the case of exponential solutions.The A2 =jl(A+e-lsl-A_elml-) passive case,where two waves of positive energy and (4.12b) opposite group velocities couple,the forward wave A excited from the left,the backward wave A2 unexcited from for P indefinite. the right,leads to the solution: Consider first the periodic solutions (4.11).If the system is passive,the energies of the waves are positive,the two A2 Ao sinh(z-1) (4.16) group velocities must be codirectional.Suppose the system A1=-J Acosh(). (4.17) is excited from the left in mode (1).After a distance /2 the excitation is all in mode (2).This is analogous This solution applies to a grating coupler that couples two to coupling of modes in time discussed earlier (see Fig. counterpropagating waves (see Fig.8).The coupler has the 5).Examples of structures employing this behavior are reflectivity: waveguide couplers used in microwaves and optics. However,active structures with opposite group velocities A2 have the same solution.Such is a backward wave amplifier tanh2 I. (4.18) Suppose wave (1)is the circuit wave with negative group velocity.It is excited at the far end of the structure of length Conversely,if one wave has negative energy and both I(see Figs.6 and 7).The active wave has positive group have positive group velocities,then both waves have to velocity and enters at z =0,presumably unexcited.The be excited from the left.The negative energy wave A2 is solutions are the beam wave in a TWT,the positive energy wave A1 is the circuit wave.The gain of the system is A2=A sin (4.13) |41(U)2 (4.19) A1=A。cos 2 (4.14) 14 (0)=cosh? HAUS AND HUANG:COUPLED-MODE THEORY 1513
associated with the vector mode in the ith guide.For 几几n几几几几几n几几HA)=o simplicity,only one guided-mode is assigned to each guide. A(0)=1 More than one guided mode for each guide and even the radiation modes may be included in the summation without much modification of the derivation. Fig.8.Schematic of a distributed Brag reflector In forming the linear superposition of the modes,we expand the total fields (both transverse and longitudinal) such that Thus far we looked at the range of different physical device characteristics covered by the simple coupling of E=∑as(e)e(红，) (5.6a) modes formalism for coupling in space.Next we shall H=∑a(a)h:(c,y) (5.6b) look at the derivation of the formalism from a variational principle.We shall confine the study to the passive case where e;and hi are the total electric and magnetic fields of and consider some issues that have arisen in connection the ith individual guide.Substitute the trial solution(5.6) with this formalism. into the variational expressions (5.2). ∑0H4 VI.VECTOR NONORTHOGONAL COUPLED-MODE THEORY 入=jB=j ∑ia畦Pa吲 (5.7) Consider a typical optical waveguide structure with an index distribution that is a function of the transverse coor- where dinates only.The Maxwell equations for the structure can Hij Pisej+Kis (5.8) be written V:XE+jw4H=A2×E (5.1a Vt×H-jwE.E=入2×H (5.1b) B=d/egxh+e,xh1a (5.9) where a is the dielectric tensor of the medium and 8 V:=元 =4e时·-eeda. (5.10) +. Multiply (5.1a)by H,(5.1b)by E",subtract,and Next we use the stationary property of the expression for A and differentiate with respect to aor [141]. integrate over the entire cross section.Solving for A one obtains A∑Pa=∑H5 (5.11) A=H*.(V:×E+jwpoH) Since -E.(V:×H-jwe,Eda 正=-8=-A d .{[E×H+E×Hda-1. (5.2) The coupled mode equations results by replacing-by For a lossless medium, d/dz: =元 (5.3) ∑P装=-∑ (5.12) Expression(5.2)can be shown to be stationary if bothxE ornx H are continuous where n is the normal unit vector Since the waveguides are lossless,power conservation to the boundaries across the index discontinuities,nxE or imposes a constraint on Pj: x vanish on an external boundary or at infinity. 是∑aRa=a (5.13)） A.Nonorthogonal Coupled Mode Formulations To derive the coupled mode theory from the variational Making use of (5.12),it follows from(5.13)that principle,we use the vector waveguide modes as trial H=H为 (5.14) solutions.The vector waveguide modes obey the following equations: which may be further reduced to tt×ei+jwuh:=j,2×e (5.4) P(B-月)=K-Kt (5.15) Vt×h:-jwee:=jB:2×hi (5.5) We note that the matrix is Hermitian only when synchronism prevails,=Bj. where e;is the dielectric constant distribution that includes Figure (9a)and (9b)shows the dispersion characteristics the dielectric constant of the ith guide,which is assumed to of TE and TM modes for parallel slabs.The index differ- be lossless and isotropic.B is the propagation constant ence (n1 -n2)/ni is assumed to be small (5.6%),but the 1514 PROCEEDINGS OF THE IEEE,VOL 79,NO.10,OCTOBER 1991