The propagation of light in one-dimensional inhomogeneous deterministic media where the refractive index varies on a wavelength scale is numerically evaluated without approximations. To this end, the field is written in amplitude and phase variables leading to a nonlinear amplitude differential equation. The numerical solutions to this Ermakov-type equation for five different refractive index profiles are presented. The amplitude oscillations are construed in terms of opposite phase or counter-propagating waves. The reflectivity is then evaluated for different interface thicknesses. The discontinuities in the first-and second-order derivatives are shown to produce an enhanced reflectivity even if the functions are continuous and monotonic.
Surfaces within a dielectric material, where the derivatives of a continuous and real refractive index profile are discontinuous, are shown to enhance reflection. To this end, the amplitude and phase representation of electromagnetic waves is used to model light propagating normally through a transparent medium with a continuous refractive index profile that varies only in one spatial direction. The amplitude equation is solved under the slowly varying refractive index approximation (SVRI). To isolate the effect of a single surface where the refractive index derivatives are discontinuous, an n(z) profile is proposed that is analytical, smooth and slowly varying except for a single piecewise junction. At this junction, n(z) is continuous but some of the mth order derivatives are not. Two different SVRI approximated solutions are joined at the discontinuity plane and, by demanding that boundary conditions are satisfied, a general complex reflection coefficient is obtained. By categorizing profiles according to the lowest order discontinuous derivative at the junction, a simple expression for the reflection coefficient can be written. Results are compared favorably with previous numerical solutions. Furthermore, a conjecture by the authors in a previous paper, 'for a C m−1 refractive index profile type, where m stands as the order of the lowest order discontinuous derivative, phase change upon reflection at the discontinuity plane is (2 − m) π 2 for an increasing lowest order discontinuous derivative and −m π 2 for the decreasing case', is proved here.
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