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Journal of the Optical Society of America B

TE wave reflection from a Kerr interface: bistability, critical transitions, and thermodynamic analogies

Nyamsuren Ochirbat, Ochirbat Guseelee

Published Aug 26, 2026• 0 citations
Publisher

Abstract

We analyze transverse-electric (TE) wave reflection at a planar interface between a linear medium and a Kerr nonlinear half-space. Using an amplitude–phase formalism, we derive exact expressions for the reflection coefficient and the incident intensity as functions of the interface permittivity ε . For total internal reflection, we obtain the spatial evolution equation for the permittivity, which decays monotonically toward the linear value ε l . The shape of S inc ( ε ) is examined, yielding the condition for a maximum on the evanescent branch: 2 β 2 > ε 1 + ε l , where β is the conserved tangential wavenumber. When this condition holds, an intensity interval ( S c , S max ) exists where three steady states coexist (two evanescent and one propagating), giving rise to optical bistability. The propagating branch exhibits a square-root singularity at ε = β 2 , leading to a non-analytic derivative d S inc / d ε ; the reflectivity slope d R / d S inc remains finite. An alternative representation in terms of the normalized decay constant μ reveals a folded curve, indicating internal bistability within the total internal reflection regime. Both self-focusing ( α >0) and self-defocusing ( α <0) Kerr nonlinearities are treated. The propagation constant β appears as a free design parameter, allowing the switching intensity to be tuned via waveguide geometry. The two switching thresholds exhibit distinct critical behavior: a first-order jump at S max and a second-order (divergent derivative) transition at S c , the latter suggesting analogies with thermodynamic critical points. A Landau-type expansion of the effective potential is presented heuristically to support this connection. The results may be useful for all-optical switching and sensing applications.

DOI
10.1364/josab.605962

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Applied Physics Letters•Dec 1, 1979•121 citations
We analyze transverse-electric (TE) wave reflection at a planar interface between a linear medium and a Kerr nonlinear half-space. Using an amplitude–phase formalism, we derive exact expressions for the reflection coefficient and the incident intensity as functions of the interface permittivity ε . For total internal reflection, we obtain the spatial evolution equation for the permittivity, which decays monotonically toward the linear value ε l . The shape of S inc ( ε ) is examined, yielding the condition for a maximum on the evanescent branch: 2 β 2 > ε 1 + ε l , where β is the conserved tangential wavenumber. When this condition holds, an intensity interval ( S c , S max ) exists where three steady states coexist (two evanescent and one propagating), giving rise to optical bistability. The propagating branch exhibits a square-root singularity at ε = β 2 , leading to a non-analytic derivative d S inc / d ε ; the reflectivity slope d R / d S inc remains finite. An alternative representation in terms of the normalized decay constant μ reveals a folded curve, indicating internal bistability within the total internal reflection regime. Both self-focusing ( α >0) and self-defocusing ( α <0) Kerr nonlinearities are treated. The propagation constant β appears as a free design parameter, allowing the switching intensity to be tuned via waveguide geometry. The two switching thresholds exhibit distinct critical behavior: a first-order jump at S max and a second-order (divergent derivative) transition at S c , the latter suggesting analogies with thermodynamic critical points. A Landau-type expansion of the effective potential is presented heuristically to support this connection. The results may be useful for all-optical switching and sensing applications.
We analyze transverse-electric (TE) wave reflection at a planar interface between a linear medium and a Kerr nonlinear half-space. Using an amplitude-phase formalism, we derive exact expressions for the reflection coefficient and the incident intensity as functions of the permittivity at the interface. Under total internal reflection, the spatial evolution of the permittivity is described by a first-order differential equation and decays monotonically to the linear value. The incident intensity on the evanescent branch is shown to possess a maximum when 2𝛽2 > 𝜀1 +𝜀𝑙, where 𝛽 is the conserved propagation constant, 𝜀1 the linear permittivity, and 𝜀𝑙 the low-intensity permittivity of the Kerr medium. For intensities between the threshold 𝑆𝑐 and the maximum 𝑆max, three steady states coexist: two evanescent (TIR) solutions (one stable, one unstable) and one propagating solution. The propagating branch rises with an infinite slope at 𝜀 = 𝛽2 (square-root singularity), leading to an ultra-sharp switching transition. The relation between the normalized decay constant 𝜇 and the incident intensity exhibits a folded curve, revealing an “internal” bistability in the TIR regime. Both self-focusing (𝛼 > 0) and self-defocusing (𝛼 < 0) Kerr nonlinearities are treated. The propagation constant 𝛽 appears as a free design parameter, allowing the switching intensity to be tuned through waveguide geometry. These results provide a unified picture of optical bistability at a nonlinear interface, with potential applications in all-optical switching and sensing.
We analyze transverse-electric (TE) wave reflection at a planar interface between a linear medium and a Kerr nonlinear half-space. Using an amplitude-phase formalism, we derive exact expressions for the reflection coefficient and the incident intensity as functions of the permittivity at the interface. Under total internal reflection, the spatial evolution of the permittivity is described by a first-order differential equation and decays monotonically to the linear value. The incident intensity on the evanescent branch is shown to possess a maximum when 2𝛽2 > 𝜀1 +𝜀𝑙, where 𝛽 is the conserved propagation constant, 𝜀1 the linear permittivity, and 𝜀𝑙 the low-intensity permittivity of the Kerr medium. For intensities between the threshold 𝑆𝑐 and the maximum 𝑆max, three steady states coexist: two evanescent (TIR) solutions (one stable, one unstable) and one propagating solution. The propagating branch rises with an infinite slope at 𝜀 = 𝛽2 (square-root singularity), leading to an ultra-sharp switching transition. The relation between the normalized decay constant 𝜇 and the incident intensity exhibits a folded curve, revealing an “internal” bistability in the TIR regime. Both self-focusing (𝛼 > 0) and self-defocusing (𝛼 < 0) Kerr nonlinearities are treated. The propagation constant 𝛽 appears as a free design parameter, allowing the switching intensity to be tuned through waveguide geometry. These results provide a unified picture of optical bistability at a nonlinear interface, with potential applications in all-optical switching and sensing.
We develop an approximate analytical theory for the reflection of a finite-width square-wave light beam at a nonlinear interface and make use of the results to assess the effect of the finiteness of the incident beam on the bistability of the signal reflected from a triple boundary interface under conditions for excitation of a waveguide mode. The nonlinearity of the interface is assumed to result from a substrate medium that has a Kerr-type quadratic dependence on the local electric-field amplitude. Our results indicate that the finiteness of the width of the incident beam substantially reduces the magnitude of the reflectance jump at the bistable switching point in comparison with that predicted by an exact plane-wave theory. Nevertheless, we conclude that a bistable reflectivity should still exist for the geometry that we consider at the switching intensity predicted by the plane-wave theory.
We study the behavior of light waves at an interface between linear and photorefractive media, where the incident wave propagates from the linear side. We find bistable behavior of the reflection coefficient as a function of the incident angle at incident wave directions in the vicinity of the critical angle for total internal reflection. The analysis is based on approximations in the photorefractive-process equations, for which an analytical plane-wave solution for the boundary problem is found. In this regime an exponentially decaying evanescent field induces a uniform change of the refractive index. The limitation of this solution is also studied by a numerical analysis.