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