Non‑crossing ε = 0 barrier and Brewster switching in transverse‑magnetic waves for epsilon‑near‑zero Kerr media: analytical classification and experimental signatures

Although conservation laws for transverse-magnetic (TM) waves in Kerr nonlinear media were derived thirty years ago, a complete analytical classification has remained absent. Here we provide such a classification for epsilon-near-zero (ENZ) materials and prove a fundamental non‑crossing theorem: the point ε = 0 acts as an impassable barrier that TM waves cannot cross. This barrier splits all admissible solutions into two disjoint families – a metallic regime (ε < 0) and a dielectric regime (ε > 0) – separated by the critical intensity $\Icrit = |εl|/|\alpha|$, where εl is the linear permittivity and $\alpha$ the Kerr coefficient. Using a novel ξ-parametrization of the stationary points of the constraint polynomial, we obtain the first complete analytical taxonomy of TM wave dynamics in ENZ Kerr media (detailed derivations are given in the Supplemental Material). The barrier has direct, measurable consequences. We consider a single interface between vacuum and an ENZ Kerr half‑space. By solving the exact nonlinear boundary conditions we derive the reflection coefficient $r$ as a function of incident intensity $I_{\mathrm{inc}}$. As ε → 0 the incident intensity approaches a finite universal limit $\IC = |εl|/(16\eta_0|\alpha|\gamma_0)$, where $\gamma_0=\cos\theta_0$ and $\eta_0$ is the vacuum impedance. Because ε = 0 is singular and cannot be attained, the system must jump discontinuously from total internal reflection ($|r|=1$) to the propagating regime at $\IC$. For typical ITO parameters this critical intensity is $\sim 10^{11}$ W/m², readily accessible with femtosecond pulses. At higher intensities the reflectivity follows an S‑shaped curve characteristic of optical bistability, with two distinct Brewster points where $|r|$ vanishes completely – providing a second, clear switching signature. The two zeros correspond to perfect transmission through the nonlinear medium and can be tuned via the incident angle and the choice of the incident medium. Our work closes a gap in nonlinear waveguide theory and establishes a universal design principle for polarization‑selective ENZ photonic devices.