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Physical Review A

Correlations, transients, bistability, and phase-transition analogy in two-mode lasers

F. T. Hioe, Surendra Singh

Published Oct 1, 1981• 37 citations
Publisher

Abstract

The general time-dependent problem associated with a two-mode-laser Fokker-Planck equation has been considered. Analytic expressions for the eigenvalues and the eigenfunctions have been derived for the below- and above-threshold operation of the system. It is shown that depending on the values of the pump parameters ${a}_{1},{a}_{2}$ and the coupling constant $\ensuremath{\xi}$, a wide variety of dynamical behavior may be encountered. Thus for $\ensuremath{\xi}<1$ interesting threshold effects may appear in the behavior of correlation times and linewidths as a function of excitation which may change their behavior whenever a mode passes through its threshold of oscillation. For $\ensuremath{\xi}=1$ the correlation time of the stronger mode may pass through a maximum before it starts to decrease with increasing excitation. The weaker mode may show particularly interesting behavior for $\ensuremath{\xi}=1$. Both the correlation time and the linewidth may approach a constant value independent of the excitation. For $\ensuremath{\xi}>1$ the system may show bistability. Our analysis provides a deeper mathematical understanding of the onset of bistability and the analogy to the thermodynamic phase transition. Expressions for the switching time have been derived using several different procedures. We have also been able to derive closed-form expressions for the two-time joint probability densities. It should be possible to test some of these predictions in photon-counting experiments. Finally, we mention that our results and method should also be applicable in other similar problems, e.g., to the classical theory of coupled fields.

DOI
10.1103/physreva.24.2050

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Physical Review A•Aug 1, 1986•3 citations
The general time-dependent problem associated with a two-mode-laser Fokker-Planck equation has been considered. Analytic expressions for the eigenvalues and the eigenfunctions have been derived for the below- and above-threshold operation of the system. It is shown that depending on the values of the pump parameters a1,a2{a}_{1},{a}_{2}a1​,a2​ and the coupling constant \ensuremathξ\ensuremath{\xi}\ensuremathξ, a wide variety of dynamical behavior may be encountered. Thus for \ensuremathξ<1\ensuremath{\xi}<1\ensuremathξ<1 interesting threshold effects may appear in the behavior of correlation times and linewidths as a function of excitation which may change their behavior whenever a mode passes through its threshold of oscillation. For \ensuremathξ=1\ensuremath{\xi}=1\ensuremathξ=1 the correlation time of the stronger mode may pass through a maximum before it starts to decrease with increasing excitation. The weaker mode may show particularly interesting behavior for \ensuremathξ=1\ensuremath{\xi}=1\ensuremathξ=1. Both the correlation time and the linewidth may approach a constant value independent of the excitation. For \ensuremathξ>1\ensuremath{\xi}>1\ensuremathξ>1 the system may show bistability. Our analysis provides a deeper mathematical understanding of the onset of bistability and the analogy to the thermodynamic phase transition. Expressions for the switching time have been derived using several different procedures. We have also been able to derive closed-form expressions for the two-time joint probability densities. It should be possible to test some of these predictions in photon-counting experiments. Finally, we mention that our results and method should also be applicable in other similar problems, e.g., to the classical theory of coupled fields.
Intensity correlations in a two-mode laser oscillating on two coupled transitions have been studied by means of the Fokker-Planck equation. It is shown that they can be expressed in terms of the eigenvalues and eigenfunctions of a one-dimensional Schr\"odinger-type equation. It is found that near threshold intensity correlations cannot be approximated by a single exponential. For moderately large excitations, however, a single exponential dominates. Approximate analytic expressions for large excitations have been obtained which show that the correlation time increases with increasing excitation.
Introduction The transient behavior of nonlinear optical resonators which display bistability and its reaction to fluctuations of system parameters are of extreme importance in all possible applications of these devices. Moreover, however, studies on these processes may give insight into some more general aspects of the behavior of nonlinear dynamical systems. In fact it has been emphasized in the early stages of the development of the field that optically bistable systems, in a suitable model, may exhibit a first-order phase transition (Bonifacio and Lugiato, 1978). Since the threshold behavior of a laser can be described as an example of a second-order phase transition (Haken, 1964,1975; see also Haken, 1977), optical bistability in (passive) nonlinear resonators complements nicely lasers as nonlinear dissipative systems in optics. Both types of optical systems are easily accessible to measurements, and the time scales coming into play are convenient for detailed experimental studies of the dynamics. It is worth mentioning that there is a close connection between laser operation and optical bistability in nonlinear (passive) resonators (see Lugiato, 1984). The standard single-mode laser model with distributed losses (Haken, 1970; Sargent, Scully and Lamb, 1974) can be generalized to include the effect of an external field that continuously drives the atoms within the optical resonator. The standard model of absorptive bistability, the Bonifacio–Lugiato model, is a special case of this generalized model. It represents the situation of a pumping mechanism being absent.
The laser--phase-transition analogy is considered for a system of identical emitters interacting with a cavity mode. As in the case of equilibrium phase transitions, the thermodynamic limit is shown to be an essential condition for the appearance of an abrupt transition, with a well-defined threshold. The proof is discussed first in the simpler formalism of the rate equation and then at the higher level of the master equation for probabilities. The latter is derived from the von Neumann--Lindblad evolution of the density operator. In the thermodynamic limit and for two-level emitters the master equation becomes a two-dimensional dynamic flow with two possible fixed points, related to the normal and the lasing regimes, respectively. The master equation analysis confirms the threshold value deduced from the rate equation, but also provides further evidence of lasing by showing that all photonic autocorrelation functions converge to unity.
We study the properties of the laser field in the case where the cavity contains two cells, one which amplifies and one which absorbs the radiation. We adopt the simplest possible model, in which there is only one running mode in the cavity and all atoms are fixed and homogeneously distributed in space. We first set up a semiclassical description. In the frame of the stationary semiclassical (SC) theory the most striking feature is the occurrence of a domain in which three solutions coexist. A linear-stability analysis shows that two of these solutions are stable. This gives rise to an hysteresis cycle. Most of the results in this section are analytic. In the frame of a purely quantum description, we derive a generalized Fokker-Planck equation which is exactly solved in the stationary case. This yields a stationary quasiprobability distribution function for the field. In the domain where the SC theory predicts three solutions, the field-distribution function has two maxima and one minimum whose positions are precisely given by the SC intensities. With this distribution function we study the intensity and the intensity fluctuations: the intensity shows a rather abrupt increase when crossing the narrow transition region; in this region the intensity fluctuations are extremely high. By means of an approximate method to solve the time-dependent Fokker-Planck equation we study the linewidth and show that it drastically decreases above threshold. Finally we consider the linearized and quasilinearized Fokker-Planck equations which can be solved analytically. We show that the domain in which neither of these equations fits the results of the generalized Fokker-Planck equation is extremely small. The properties of the metastable states are analyzed in terms of the generalized and asymptotic Fokker-Planck equations.
Intensity fluctuations for a single-mode laser near threshold are calculated from the Scully-Lamb equation of motion for the reduced density matrix of the radiation field. The magnitude and correlation time of the intensity fluctuations are contrasted with the results obtained from the usual Fokker-Planck analysis of the rotating-wave, Van-der-Pol (RWVP) model. When the photon number at threshold is not large, these calculations predict measurable differences from the RWVP Fokker-Planck results. Two methods are used to calculate the dynamics. One is a direct eigenfunction expansion of the Green's function of the equation of motion. The other is based on the calculation of the initial derivatives of the correlation function.
Mean switching time of a two-mode laser is calculated in two different approaches: the first-passage-time treatment by the generalized Kramers method and the lifetime approach by the two-dimensional WKB method. In both cases, the most probable path is generally curved. Numerical study in the WKB calculation reveals that the general curved most probable path can be replaced with high accuracy by the path specified by the minimum potential for many cases of practical interest, that is, when the mode-coupling constant \ensuremath{\xi} is within the range 1<\ensuremath{\xi}<5. Thus, we are able to carry out calculations of the mean switching time that are accurate (within the framework of the WKB approximation) throughout the entire range, above threshold, of the pump parameter a. Compared with the analytical formula which is valid only for large a, our result yields perfect agreement in the large-a limit, but shows that the previously used formula consistently overestimates the mean switching time when a is not large where the experiments are performed.