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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Region of phase transition in optical bistable systems

Iurii Gudyma

Published Aug 19, 1998• 0 citations
Publisher

Abstract

The threshold bistable system was investigated near a threshold of switching between phases. It was shown, that in spite of the fact that the equation, that describes this system does not have the stable localized structures among the solutions, envelopes of these solutions can form solitons of a nonlinear Schrodinger equations. Dynamic critical index was calculated near this point in the second order (epsilon) . It characterizes cross-section of scattering on fluctuation modes.

DOI
10.1117/12.320981

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The threshold bistable system was investigated near a threshold of switching between phases. It was shown, that in spite of the fact that the equation, that describes this system does not have the stable localized structures among the solutions, envelopes of these solutions can form solitons of a nonlinear Schrodinger equations. Dynamic critical index was calculated near this point in the second order (epsilon) . It characterizes cross-section of scattering on fluctuation modes.
The purpose of this work is to study the behavior of atomic fluctuations and the spectrum of transmitted light by a nonlinear medium with absorption and dispersion. The authors make use of the system-size expansion and apply it to the quantum-mechanical Langevin's equations for the atomic operators and obtain the characteristic curves for bistable behavior, macroscopic atomic expectation values, and the spectrum of the atomic fluctuations. The calculated spectrum of transmitted light exhibits line narrowing near bistable thresholds, discontinuous formation of bands along the high-transmission bistable branch, and hysteresis effects, etc., for detuning $|\ensuremath{\delta}|$ less than a critical value ${\ensuremath{\delta}}_{c}$. For $|\ensuremath{\delta}|>{\ensuremath{\delta}}_{c}$, the discontinuous character of all the physical variables of the system vanishes; the system no longer exhibits bistable behavior. The system, however, still shows cooperative and single-atom behavior separated by a critical value of the intensity of the incident coherent pump at which a critical slowing down of the medium occurs, exhibiting a phase transition.
Abstract Mean field and beyond mean field model calculations of Bose–Einstein condensates trapped in optical lattices have shown that initially homogeneous condensates can evolve into self-trapped, strongly localized states in the presence of weak boundary dissipation, a phenomenon called self-localization . A dynamical phase transition from extended to localized states can be observed when the effective nonlinearity exceeds a critical threshold Λ eff c . We investigate this phase transition to self-localization in the mean field approximation of the discrete nonlinear Schrödinger equation. We quantitatively characterize the properties of the discrete breathers , i.e. the nonlinear localized solutions, at the phase transition. This leads us to propose and numerically verify an analytical lower bound Λ eff L for the critical nonlinearity based on the idea of self-induced Anderson localization.
We present herein a phenomenological approach to the critical phenomena in the dissipative systems, including the optical bistable system, the laser, etc., which satisfies the principle of detailed balance. The substantial discussion is started from the standpoint that there are two kinds of critical points: stable and unstable, associated with the continuous and discontinuous transitions, respectively. In several senses this work would be a development of Landau's second order phase transition theory in equilibrium thermodynamics.
Taking the optical system as a prototype, this paper demonstrates that a quasi-thermodynamic model for dissipative systems, satisfying the principle of detailed balance, may be developed, which is parallel to the traditional thermodynamics. The present systems are described by the generalized dynamical equation of nonlinear optical cavities, introduced in previous work of Ou Fa et al. In view of the quasi-thermodynamic model, the critical phenomena and phase transitions in optical bistability are discussed systematically and generally. By taking the advantage of Ginzburg-Landau's model, the fluctuations and correlations near the (second kind of) critical point of optical bistability are studied also. So the Landau's theory of phase transitions in equilibrium thermodynamics is generalized to the bistable dissipative systems far from equilibrium.
In this paper a quantum mechanics model for describing multiphoton optical bistability is given. Based on this molel, the state equation of multi-photon option1 bistability is obtained. The method which for calculating the threshold of multi-photon optical bistability is also given. After analyzing the potential function, we obtain the statics stability criterion and the explanation of switching characteristics. The stable region and the inferior stable region in the bistability characteristic curve are separated. In fact, multiphoton optical stable states are actually multi-photon optical stable-inferior stable states. We also discuss the phase transition problem of multi-photon optical bistability and multi-photon lasers.