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