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Chinese Physics B

Bistable Goos–Hänchen shift owing to leaky-mode excitation in a slab waveguide with Kerr nonlinear medium

Yuan-Ping Cai, Li 丽 Jiang 姜, Ren-Gang Wan

Published Aug 29, 2025• 0 citations
Publisher

Abstract

Abstract We investigate the nonlinear Goos–Hänchen shift of a light beam reflected from a prism-coupled leaky waveguide containing a Kerr medium. As the incident power varies, the system can switch between two states, total internal reflection and frustrated total reflection, owing to the inherent positive feedback arising from the intensity-dependent guiding mode resonance. The reflectance exhibits optical bistability; meanwhile, the lateral shift of the reflected beam also shows hysteresis behavior. It is found that the transition between the two stable states is related to the excitation of a leaky mode in the waveguide, which results from the modulation of the electric field in the nonlinear substrate. We also analyze the effects of system parameters on the bistable Goos–Hänchen shift. The thresholds as well as the width of the hysteresis curve are sensitive to the thicknesses of the gap layer and the guiding layer, which determine the resonance angle. The bistable lateral displacement in the slab waveguide may have potential applications in optical switching, beam steering, etc.

DOI
10.1088/1674-1056/ae00b3

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Abstract We investigate the nonlinear Goos–Hänchen shift of a light beam reflected from a prism-coupled leaky waveguide containing a Kerr medium. As the incident power varies, the system can switch between two states, total internal reflection and frustrated total reflection, owing to the inherent positive feedback arising from the intensity-dependent guiding mode resonance. The reflectance exhibits optical bistability; meanwhile, the lateral shift of the reflected beam also shows hysteresis behavior. It is found that the transition between the two stable states is related to the excitation of a leaky mode in the waveguide, which results from the modulation of the electric field in the nonlinear substrate. We also analyze the effects of system parameters on the bistable Goos–Hänchen shift. The thresholds as well as the width of the hysteresis curve are sensitive to the thicknesses of the gap layer and the guiding layer, which determine the resonance angle. The bistable lateral displacement in the slab waveguide may have potential applications in optical switching, beam steering, etc.
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 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.
When a three-level atomic wavepacket is obliquely incident on a "medium slab" consisting of two far-detuned laser beams, there exists lateral shift between reflection and incident points at the surface of a "medium slab", analogous to optical Goos-Hänchen effect. We evaluate lateral shifts for reflected and transmitted waves via expansion of reflection and transmission coefficients, in contrast to the stationary phase method. Results show that lateral shifts can be either positive or negative dependent on the incident angle and the atomic internal state. Interestingly, a giant lateral shift of transmitted wave with high transmission probability is observed, which is helpful to observe such lateral shifts experimentally. Different from the two-level atomic wave case, we find that quantum interference between different atomic states plays crucial role on the transmission intensity and corresponding lateral shifts.