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 $k{l}_c{\sigma }^2$, satisfying a universal equation $k\xi \approx 1.3717{\left(k{l}_c{\sigma }^2\right)}^{-1\mathrm{/}3}$ across the entire range of parameter values. Additionally, we find that $\xi$ scales as ${\lambda }^{4\mathrm{/}3}$ 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.