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