Wave-mechanical phenomena in optical coupled-mode structures

Abstract

We derive a formal mapping between Schrödinger equations and certain classes of Maxwell equations describing the classical electromagnetic wave’s propagation inside coupled-modes waveguides. This mapping reveals a phenomenon, which is not visible in the original form of Maxwell equations: multiple solutions occur which satisfy same boundary conditions but correspond to different eigenvalues of a certain operator; the latter is analogous to Hamiltonian operators which occur in quantum systems. If one deals with normalized state vectors then a proper analogy with the conventional wave mechanics is established: solutions form a Hilbert space which is somewhat similar to that in the quantum mechanics. Therefore, coupled-mode configurations should possess certain wave-mechanical features, which can be formally studied using a formalism of quantum mechanics or, at least, its mathematical part. We notice also that the occurring Hamiltonian operators always possess a skew-adjoint part if one deals with normalized state vectors – even if permittivity and permeability are real-valued. This leads to the “dressing” effect of propagation constants, which indicates presence of additional gain or loss processes in the coupled-mode systems.