Stratified media: nonlinear ODE is better

摘要

Propagation of light in stratified media is described with Maxwell's partial differential equations (PDE). Separation of variables allow to decouple the linear PDE's to obtain second order non autonomous linear ODEs for the electric and magnetic fields. In the last decades, the problem has been tackled with matrices whose elements are linearly independent solutions of the fields.In our approach, although counter-intuitive, the linear differential equations are transformed into a non-linear ODE. To this end, the field is written in terms of amplitude and phase variables. An Ermakov invariant then permits the decoupling of the amplitude and phase nonlinear equations. The amplitude or Milne nonlinear equation is then solved numerically. This method has important advantages: i) initial or final conditions are easily imposed, ii) important physical quantities such as the reflectivity are readily obtained, iii) no further approximations have to be made iv) complex profiles can be modeled with arbitrary degree of precision. The abrupt and adiabatic limits are obtained but most importantly, intermediate more realistic cases can also be tackled, for example, adsorption between thin film layers. Novel effects are addressed such as enhanced reflectivity at derivative discontinuities where the refractive index is continuous.