Invariant Imbedding Equations for Electromagnetic Waves in Stratified   Magnetic Media: Applications to One-Dimensional Photonic Crystals
  We derive the invariant imbedding equations for plane electromagnetic waves propagating in stratified magnetic media, where both dielectric and magnetic permeabilities vary in one spatial direction in an arbitrary manner. These equations allow us to obtain the reflection and transmission coefficients of the waves and the field amplitudes inside the media exactly for any polarization and incident angle of the incoming wave by solving an initial value problem of a small number of ordinary differential equations. We apply our results to one-dimensional photonic crystals, where the periodic variations of both dielectric and magnetic permeabilities create photonic band gaps in the frequency spectrum. 
Kihong Kim, H. Lim, and Dong-Hun Lee