Splitting methods and invariant imbedding for time‐independent wave propagation in focusing media and wave guides
For time‐independent wave propagation in focusing media or wave guides, backscattering and coupling between propagation modes are caused by deterministic or random variations of the refractive index in the distinguished (x) direction of propagation. Various splittings of the wave field into forward and backward traveling components, which lead to coupled equations involving abstract operator coefficients, are presented. Choosing a natural explicit representation for these operators immediately yields a coupled mode form of these equations. The splitting procedure also leads naturally to abstract transmission and reflection operators for slabs of finite thickness (a≤x≤b), and abstract invariant imbedding equations satisfied by these. The coupled mode form of these equations, together with such features as reciprocity (associated with an underlying symplectic structure) are also discussed. The example of a square law medium is used to illustrate some of these concepts.