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Journal of Physics A Mathematical and General

Static-critical phenomenon analogy in absorptive optical bistability

Jian-M Liu

Published Nov 21, 1992• 0 citations
Publisher

Abstract

The critical phenomenon analogy in an absorptive optical bistability system is studied. The asymptotic form of the family of potential functions for the system near threshold is obtained by using catastrophe theory. The well-known scaling hypothesis in the general homogeneous function form in critical phenomena is shown to be a characteristic of the asymptotic family. Four threshold exponents beta , delta , gamma , alpha and their four accompanied threshold amplitudes B, D, Gamma , A on both bistability and monotonic regions are estimated. The threshold exponents obey the same scaling laws as those in critical phenomena, while the threshold amplitudes obey the definite relations between exponents and amplitudes.

DOI
10.1088/0305-4470/25/22/017

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The critical phenomenon analogy in an absorptive optical bistability system is studied. The asymptotic form of the family of potential functions for the system near threshold is obtained by using catastrophe theory. The well-known scaling hypothesis in the general homogeneous function form in critical phenomena is shown to be a characteristic of the asymptotic family. Four threshold exponents beta , delta , gamma , alpha and their four accompanied threshold amplitudes B, D, Gamma , A on both bistability and monotonic regions are estimated. The threshold exponents obey the same scaling laws as those in critical phenomena, while the threshold amplitudes obey the definite relations between exponents and amplitudes.
The theory of critical phenomena in systems at equilibrium is reviewed at an introductory level with special emphasis on the values of the critical point exponents α, β, γ,..., and their interrelations. The experimental observations are surveyed and the analogies between different physical systems - fluids, magnets, superfluids, binary alloys, etc. - are developed phenomenologically. An exact theoretical basis for the analogies follows from the equivalence between classical and quantal `lattice gases' and the Ising and Heisenberg-Ising magnetic models. General rigorous inequalities for critical exponents at and below T c are derived. The nature and validity of the `classical' (phenomenological and mean field) theories are discussed, their predictions being contrasted with the exact results for plane Ising models, which are summarized concisely. Padé approximant and ratio techniques applied to appropriate series expansions lead to precise critical-point estimates for the three-dimensional Heisenberg and Ising models (tables of data are presented). With this background a critique is presented of recent theoretical ideas: namely, the `droplet' picture of the critical point and the `homogeneity' and `scaling' hypotheses. These lead to a `law of corresponding states' near a critical point and to relations between the various exponents which suggest that perhaps only two or three exponents might be algebraically independent for any system.