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. 
Kihong Kim, Dong-Hun Lee, H. Lim