Resonance effects in electromagnetic propagation using a modified J.W.K.B. approach

We consider the propagation of electromagnetic waves through a medium of arbitrarily varying density that can give rise to evanescent behavior. We solve the Helmholtz equation using a phase integral approximation and show that this leads to the J.W.K.B. method. However, the conventional J.W.K.B. approach for the transmission of electromagnetic waves deals with evanescent characteristics only and does not consider the medium beyond the turning points or caustics that give rise to resonance/interference effects. By the use of two-dimensional matrix propagators, we modify the conventional J.W.K.B. formulation to include a correction term that allows us to calculate all regions of interest. The modified J.W.K.B. method we present is used to calculate the transmission probability (τ) and reflection probability (ρ) exactly and without the need to use the ´connectivity´ property of the waves across the turning points which can otherwise give rise to singularities in the solutions.