Phase functions for potential scattering in optics

Equations for the phase functions are presented in a form convenient for analyzing the solutions of both the Schrödinger equation and the problem of electromagnetic wave propagation in inhomogeneous media. It is shown how the problem of electromagnetic wave propagation in media with a real-valued dielectric constant (and, hence, the problem of above- and below-barrier transmission) reduces to one of calculating the phase for the potential scattering. One can then obtain not only such integrated characteristics as the coefficients of transmission and reflection but also exact expressions for the solution of the wave equation, in the form of quadratures containing the current (“instantaneous”) value of the phase. The problem simplifies substantially for a layer symmetric with respect to the coordinate. It is shown that the method of phase functions gives an exact description of two opposite limiting cases, viz., the short-wavelength limit and Fresnel reflection, through a single simple analytical formula. A detailed discussion is given for above- and below-barrier reflection near the edge of the barrier.