Reflected wave atypical phase change at a boundary

摘要

According to Fresnel formulae, at normal incidence on an abrupt interface, the reflected wave has a phase difference of zero or w, if the second medium has a lower or larger refractive index than the first. However, what happens if the refractive indices of two media are the same at the interface but the derivative of the refractive index varies abruptly? Since the two media are not homogeneous because the refractive index derivative is finite, the problem cannot be tackled with the Fresnel formalism. In order to deal with this problem the amplitude and phase representation of plane electromagnetic waves is used. An invariant is obtained that permits the decoupling of the amplitude and phase equations, both of which, are nonlinear. The amplitude equation is then solved numerically. No approximations are made regarding how slow or fast refractive index varies compared to the wavelength. Interpretation of the amplitude equation solutions reveal that surfaces where any of the derivatives of the refractive index profile is discontinuous, do enhance reflection. At normal incidence, the reflected wave thus generated will have a phase difference that may be a multiple of π/2, apparently contradicting the Fresnel equations.