Coupled mode theory

Coupled mode theory

Coupled mode theory (CMT) is a paradigm that allows one to solve physical problems involving systems in vibration of different kinds: mechanical, optical, electrical and others. The system is made of different "resonating sub-components" that interact together. The coupled mode theory allows to get solutions for the oscillating and propagating waves.

History

The first mention of the CMT was made in papers from Miller and Pierce in the domain of the microwave transmission lines[1] and in electron beams[2]. They put in place the mathematical foundations for the actual formulation. In 1985, the subject of integrated optics have seen a new interest in this theoretical approach, specially by introducing the notion of coupling between modes [3].

Formulation

The formulation of the coupled mode theory is based on the development of the solution to an electromagnetic problem into modes. Most of the time it is eigenmodes which are taken in order to form a complete base. The choice of the base and the adoption of certain hypothesis like parabolic approximation differs from formulation to formulation. The classification proposed by [4] of the different formulation is as follows:

1. The choice of starting differential equation. some of the coupled mode theories are derived directly from the Maxwell differential equations [3] [5] ( here) although others use simplifications in order to obtain an Helmholtz equation.
2. The choice of principle to derive the equations of the CMT. Either the reciprocity theorem [3] [5] or the variational principle have been used.
3. The choice of orthogonality product used to establish the eigenmode base. Some references use the unconjugated form [3] and others the complex-conjugated form[5].
4. Finally, the choice of the form of the equation, either vectorial [3] [5] or scalar.

References

1. ^ S.E.Miller,"Coupled wave theory and waveguide applications.",Bell Syst.Tech. J.,1954
2. ^ J.R. Pierce, "Coupling of modes of propagations", Journal of Applied Physics, 25, 1954
3. ^ a b c d e Hardy and Streifer, "Coupled mode theory of parallel waveguides", Journal of Lightwave Technology,1985
4. ^ Barybin and Dmitriev,"Modern Electrodynamics and Coupled-mode theory",2002
5. ^ a b c d A. W. Snyder and J. D. Love, "Optical waveguide Theory",Chapman and Hall, 1983

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