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Physical Review A

Mean switching time of bistable two-mode lasers

X. W. Wang, D. L. Lin, F. T. Hioe

Published Aug 1, 1986• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.34.1251

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Physical Review A•Oct 1, 1983•23 citations
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.
The problem of first passage time is mathematically equivalent to the tunnelling of a particle out of a potential well and hence can be treated by the multi-dimensional WKB technique. In general, the most probable tunnelling path (MPTP) is curved and is difficult to determine. The authors discuss a model potential for which the curved MPTP can be solved analytically. It is shown that the rate of tunnelling calculated along the straight path as usually assumed is unreliable. Numerical results of the mean switching time for the bistable two-mode laser and comparison with experiments and other theories are also discussed.
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 $\ensuremath{\xi}$, 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.
A two-mode laser operating in the bistable region switches spontaneously from one mode to the other under the influence of quantum fluctuations. We present here a Monte Carlo simulation of the coupled Langevin equations describing the dynamics of a two-mode laser. We present results on the variation of average switching time with pump parameter. We show that if fluctuations in the pump parameter are incorporated, the Monte Carlo estimates agree well with measured switching times. To further check this observation we reanalyze the results of a recent analytical study and show that by including the same pump parameter fluctuations, the agreement with measurements becomes much better.
Using the Bonifacio-Lugiato mean field model of an absorptive bistable device, we discuss in some detail the long time dynamics and the influence of quantum noise on the metastable states of a bi-stable system. We develop our analysis along two seemingly different lines: first, by means of a suitable generalization and reinterpretation of a method due to Kramers, we obtain an explicit analytic expression for the escape time out of the metastable state, which is valid not only near steady state, but through the long time evolution of the system as well; next, we calculate the same escape time by the so-called mean first passage time method. The two results are found to differ by a numerical factor of two. In our discussion we have made an effort to explore details of both the original Kramers' scheme and of the first passage time method that needed clarification. We have also related the global long time evolution to the escape processes out of the metastable states.
As a result of quantum fluctuations, a bistable system will undergo spontaneous transitions from one state to the other. For a bistable two-mode laser we present a theoretical analysis of the switching statistics by treating it as a one-dimensional first-passage---time (FPT) problem for the intensity of one of the modes. The methodology is also applicable to other bistable systems, such as in optical bistability. Several new asymptotic results for the small- and large-time behavior of various FPT distributions are given. The switching-time distribution is introduced as a weighted integral over FPT distributions and asymptotic expressions for this distribution are given. Several results of Monte Carlo computer simulations of the FPT problem are presented.