Exact localization length for s-polarized electromagnetic waves incident at the critical angle on a randomly-stratified dielectric medium

The interplay between Anderson localization and total internal reflection of electromagnetic waves incident near the critical angle on randomly-stratified dielectric media is investigated theoretically. Using an exact analytical formula for the localization length for the Schr\"odinger equation with a Gaussian $\delta$-correlated random potential in one dimension, we show that when the incident angle is equal to the critical angle, the localization length for an incident $s$ wave of wavelength $\lambda$ is directly proportional to $\lambda^{4/3}$ throughout the entire range of the wavelength, for any value of the disorder strength. This result is different from that of a recent study reporting that the localization length at the critical incident angle for a binary multilayer system with random thickness variations is proportional to $\lambda$ in the large $\lambda$ region. We also discuss the characteristic behaviors of the localization length or the tunneling decay length for all other incident angles. Our results are confirmed by an independent numerical calculation based on the invariant imbedding method.