Scaling behavior of the localization length for TE waves at critical incidence on short-range correlated stratified random media

We theoretically investigate the scaling behavior of the localization length for $s$-polarized electromagnetic waves incident at a critical angle on stratified random media with short-range correlated disorder. By employing the invariant embedding method, extended to waves in correlated random media, and utilizing the Shapiro-Loginov formula of differentiation, we accurately compute the localization length $\xi$ of $s$ waves incident obliquely on stratified random media that exhibit short-range correlated dichotomous randomness in the dielectric permittivity. The random component of the permittivity is characterized by the disorder strength parameter $\sigma^2$ and the disorder correlation length $l_c$. Away from the critical angle, $\xi$ depends on these parameters independently. However, precisely at the critical angle, we discover that for waves with wavenumber $k$, $k\xi$ depends on the single parameter $kl_c\sigma^2$, satisfying a universal equation $k\xi\approx 1.3717\left(kl_c\sigma^2\right)^{-1/3}$ across the entire range of parameter values. Additionally, we find that $\xi$ scales as ${\lambda}^{4/3}$ for the entire range of the wavelength $\lambda$, regardless of the values of $\sigma^2$ and $l_c$. We demonstrate that under sufficiently strong disorder, the scaling behavior of the localization length for all other incident angles converges to that for the critical incidence.