Three nonlinear barriers and generalized reflectivity for TM waves in Kerr media

We analyze transverse-magnetic (TM) wave propagation in Kerr media using a phase-amplitude formalism. From the algebraic constraints imposed by the conserved quantities, we identify three distinct nonlinear barriers: a material barrier at ε =0 intrinsic to epsilon-near-zero (ENZ) media, a propagation threshold at ε = β 2 (with β the normalized propagation constant), and a geometrical barrier at ε =2 β 2 arising from waveguide boundary conditions. We prove that, within the stationary lossless model, ε =0 and ε =2 β 2 are impassable: continuous solutions cannot cross these values. For the ε =2 β 2 barrier we derive a switching intensity I switch =|2 β 2 − ε l |/| α |, valid for both self-focusing ( α > 0) and self-defocusing ( α < 0) Kerr nonlinearities, and for arbitrary linear permittivity ε l . We also extend Kaplan’s reflectivity theory to TM polarization, obtaining exact expressions for the reflection coefficient and incident intensity at a linear–nonlinear interface. This yields a generalized Snell’s law and identifies intensity-dependent Brewster conditions where reflection vanishes. The two thresholds at ε = β 2 and ε =2 β 2 provide complementary switching mechanisms, both tunable via waveguide geometry. The results offer an analytical basis for designing polarization-selective photonic devices.