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

摘要

The interplay between Anderson localization and total internal reflection ofelectromagnetic waves incident near the critical angle on randomly-stratifieddielectric media is investigated theoretically. Using an exact analyticalformula for the localization length for the Schr\"odinger equation with aGaussian $\delta$-correlated random potential in one dimension, we show thatwhen the incident angle is equal to the critical angle, the localization lengthfor an incident $s$ wave of wavelength $\lambda$ is directly proportional to$\lambda^{4/3}$ throughout the entire range of the wavelength, for any value ofthe disorder strength. This result is different from that of a recent studyreporting that the localization length at the critical incident angle for abinary multilayer system with random thickness variations is proportional to$\lambda$ in the large $\lambda$ region. We also discuss the characteristicbehaviors of the localization length or the tunneling decay length for allother incident angles. Our results are confirmed by an independent numericalcalculation based on the invariant imbedding method.