Theory of the propagation of coupled waves in arbitrarily- inhomogeneous stratified media

– We generalize the invariant imbedding theory of the wave propagation and derive new invariant imbedding equations for the propagation of arbitrary number of coupled waves of any kind in arbitrarily-inhomogeneous stratified media, where the wave equations are effectively one-dimensional. By doing this, we transform the original boundary value problem of coupled second-order differential equations to an initial value problem of coupled first-order differential equations, which makes the numerical solution of the coupled wave equations much easier. Using the invariant imbedding equations, we are able to calculate the matrix reflection and transmission coefficients and the wave amplitudes inside the inhomogeneous media exactly and efficiently. We establish the validity and the usefulness of our results by applying them to the propagation of circularly-polarized electromagnetic waves in one-dimensional photonic crystals made of isotropic chiral media. We find that there are three kinds of bandgaps in these structures and clarify the nature of these bandgaps by exact calculations. Introduction. – The phenomena of the coupling of two or more wave modes in inhomogeneous media and mode conversion between them are ubiquitous in various branches of science, including plasma physics, optics, condensed matter physics and electrical engineering [1–6]. In this Letter, we develop a generalization of the powerful invariant imbedding method [6–13] to the case of several coupled waves in stratified media. Starting from a very general wave equation of a matrix form, we derive a new version of the invariant imbedding equations for calculating the reflection and transmission coefficients and the field amplitudes. By doing this, we transform the original boundary value problem of coupled second-order differential equations to an initial value problem of coupled first-order differential equations. This makes the numerical solution of the coupled wave equations much easier. Furthermore, our equations have a great advantage that there is no singular coefficient even in the cases where the material parameters change discontinuously at the boundaries and inside the inhomogeneous medium. We check the validity and the usefulness of our invariant imbedding equations by applying them to the propagation of electromagnetic waves in stratified chiral media. By calculating () E-mail: [email protected]