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Physical Review A

Spatial effects in intrinsic optical bistability

Joseph W. Haus, Li Wang, Michael Scalora, Charles M. Bowden

Published Oct 1, 1988• 29 citations

Abstract

Using the nonlinear oscillator model as a prototype medium exhibiting intrinsic optical bistability, we investigate the inhomogeneous absorption of the electromagnetic field. The forward- and backward-field amplitudes and diffraction effects are retained in the mathematical description. Analytic results are given in the limit of plane-wave propagation under steady-state conditions. The transmitted and reflected intensity exhibit a structure that is determined by the spatial inhomogeneity of the absorption in the longitudinal direction. The transmitted intensity has a structure that is dependent on the length of the medium. The reflected intensity has an interference structure from light reflected at the front surface and the internal boundary separating a high-polarization from a low-polarization branch. A degenerate-four-wave-mixing experiment is predicted to be a very sensitive probe of the internal boundary and the interference between the forward and backward field. The phase-conjugate signal develops large oscillations as the input field is varied. Numerical results for diffraction effects are also given, and we find that the plane-wave results for the center of the beam remain reliable down to Fresnel numbers of order unity and in media that are smaller than the linear absorption length.

DOI: 10.1103/physreva.38.4043

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Transverse effects in intrinsic optical bistability

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A system of coherently driven spatially distributed two-level atoms, coupled with each other via dipole–dipole interaction, exhibits intrinsic optical bistability (IOB) at solid-state densities,1 i.e., optical bistability without optical feedback. The origin of the IOB was shown to be due to inversion-dependent renormalization of the single-atom resonance frequency caused by dipole–dipole interaction among the atoms located mainly within an atomic resonance wavelength.2 For normally incident plane waves of intensity beyond threshold, a spatially dependent discontinuity in the nonlinear refractive index was shown to exist inside the medium,1 corresponding to spatial separation of the high and low absorption phases of the IOB. We present numerical results for the time-dependent solutions of the equations of motion and dynamics of the field propagation.

Annual Meeting Optical Society of America•Jan 1, 1988•0 citations

Using the nonlinear oscillator model as a prototype medium exhibiting intrinsic optical bistability, we investigate the inhomogeneous absorption of the electromagnetic field. The forward- and backward-field amplitudes and diffraction effects are retained in the mathematical description. Analytic results are given in the limit of plane-wave propagation under steady-state conditions. The transmitted and reflected intensity exhibit a structure that is determined by the spatial inhomogeneity of the absorption in the longitudinal direction. The transmitted intensity has a structure that is dependent on the length of the medium. The reflected intensity has an interference structure from light reflected at the front surface and the internal boundary separating a high-polarization from a low-polarization branch. A degenerate-four-wave-mixing experiment is predicted to be a very sensitive probe of the internal boundary and the interference between the forward and backward field. The phase-conjugate signal develops large oscillations as the input field is varied. Numerical results for diffraction effects are also given, and we find that the plane-wave results for the center of the beam remain reliable down to Fresnel numbers of order unity and in media that are smaller than the linear absorption length.

Longitudinal spatial effects in the nonlinear medium are analyzed in a model for mirrorless optical bistability due to induced absorption that treats heat conduction, electromagnetic field propagation, and temperature-dependent absorption. We consider two specific cases: the temperature-induced shift of the absorption peak of bound I2 excitons in CdS single crystals and the temperature-induced shrinkage of the band gap in GaAs/GaAlAs quantum-well material. For both cases it is shown that the inhomogeneous spatial distribution of the nonlinear absorption profile along the direction of propagation of the electric field undergoes a first-order-like phase transition commensurate with the transition of the output intensity from one value to another for fixed input intensity. Stability conditions are presented for the steady states of the output intensity as a function of the input intensity for the cases treated. There is a multiplicity of bistability curves, and the stability analysis produced the expected branches of stable solutions.

Absorptive, dispersive, and diffractive effects are studied in a nonlinear ring-cavity configuration containing an intrinsically bistable medium modeled generically by a nonlinear oscillator. The output intensity is shown to exhibit steady-state and periodic characteristics for constant input intensity, but no chaotic solutions are found. We find that the plane-wave analysis is inconsistent with the inclusion of diffraction effects and that periodic output characteristics are affected even for Fresnel numbers of order 1000. Our results are contrasted with previously reported experimental and theoretical work.

Electromagnetic wave propagation with one transverse dimension is investigated in a medium whose response is modeled by a nonlinear oscillator. We report results with transverse effects due to diffraction, which are controlled by varying the Fresnel number. We find that diffraction does not significantly reduce the bistable intensity characteristics either for the center of the beam intensity or in the intensity integrated across the transverse coordinate, although in the latter case the contrast is reduced; bistable behavior is found to persist even for a Fresnel number of order unity, and the switching intensity is not affected by the Fresnel number. These results are explained by simple physical arguments.

A nonlinear oscillator model is used as a prototypical medium for intrinsic optical bistability.1 The phenomenon of intrinsic optical bistability is so named because the medium does not require the optical feedback of a cavity. In this model we retain dispersive effects and introduce a diffractive coupling of the electromagnetic field amplitude. The threshold of the optical bistable device is insensitive to the Fresnel number, which determines the coupling between field amplitudes at different transverse positions.2 This is useful for obtaining reproducible operating characteristics of an array of devices. The medium can be introduced inside a cavity to understand the role of dispersive and diffractive effects in a more fundamental way. The threshold intensity is again insensitive to the Fresnel number. Furthermore, we observe instabilities in the cavity configuration, but no chaotic solutions were found. The periodic instabilities are sensitive functions of the Fresnel number.

Dynamical transient response and longitudinal spatial evolution of the first-order phase transition are presented for intrinsic optical bistability of a system of interacting spatially distributed, coherently driven two-level atoms. It is shown that a discontinuity in the nonlinear dielectric function is established internal to the medium due to transient response to a coherent, externally applied, time-dependent driving field, following the attainment of a critical value for the input intensity. If the externally applied field intensity reaches the steady state, beyond the critical value, the discontinuity of the nonlinear dielectric function in the medium reaches a stable stationary condition and forms a spatial boundary between states of high and low polarization in the medium. Thus it is shown that intrinsic optical bistability corresponds to a spatial as well as a temporal first-order phase transition in light-matter interactions, far from thermodynamic equilibrium, where the spatial boundary between different phases is established by transient dynamical response to the incident field.

A system of coherently driven spatially distributed two-level atoms, coupled with each other via dipole–dipole interaction, exhibits intrinsic optical bistability (IOB) at solid-state densities,1 i.e., optical bistability without optical feedback. The origin of the IOB was shown to be due to inversion-dependent renormalization of the single-atom resonance frequency caused by dipole–dipole interaction among the atoms located mainly within an atomic resonance wavelength.2 For normally incident plane waves of intensity beyond threshold, a spatially dependent discontinuity in the nonlinear refractive index was shown to exist inside the medium,1 corresponding to spatial separation of the high and low absorption phases of the IOB. We present numerical results for the time-dependent solutions of the equations of motion and dynamics of the field propagation.