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Topical Meeting on Optical Bistability (OB)

Absorptive Optical Bistability in a bad cavity

Paul Mandel, Thomas Erneux

Published Jan 1, 1983• 2 citations
Publisher

Abstract

In a theoretical discussion of optical bistability, it is often necessary to resort to some approximation scheme to discuss time-dependent properties. The standard method consists in assuming a very good cavity (κ«γ⊥, γ11) Or a very bad cavity (γ11 <<κ1, γ⊥ ) and to perform an adiabatic elimihation of the fast varying variables. In the good cavity limit /1/ the atomic variables are eliminated and one is left with an equation for the field. In the bad cavity limit /2/ the field and the atomic polarization are eliminated and one is left with an eouation for the atomic inversion.

DOI
10.1364/obi.1983.fa5

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In a theoretical discussion of optical bistability, it is often necessary to resort to some approximation scheme to discuss time-dependent properties. The standard method consists in assuming a very good cavity (κ«γ⊥, γ11) Or a very bad cavity (γ11 <<κ1, γ⊥ ) and to perform an adiabatic elimihation of the fast varying variables. In the good cavity limit /1/ the atomic variables are eliminated and one is left with an equation for the field. In the bad cavity limit /2/ the field and the atomic polarization are eliminated and one is left with an eouation for the atomic inversion.
A quantum theory describing N two-level atoms in a coherently driven single-mode optical ring cavity is presented. The usual approximation that one may adiabatically eliminate either the cavity or atomic variables is not made. Hence the results are valid for a range of relative cavity and atomic relaxation rates. The theory is based on the linearized equations derived in the positive P representation. A simple analytical solution is presented, including the effect of detunings, for the steady-state transmitted spectrum. The transmitted intensity and squeezing spectra are discussed.
The analysis of experiments on bistability and instability in nonlinear optical resonators depends critically on an understanding of the properties of the corresponding empty (linear) resonators. Both real and ideal cavities exhibit behavior which can lead to a misinterpretation of experimental results, and in both cases this behavior becomes more pronounced with increasing finesse. As an example of the first case, consider a real cavity which is nonideal in that different methods of measurement of the cavity loss give different values for it and hence for the cooperativity C. A simple but realistic model for such a cavity leads to the derivation of a state equation for bistability which differs in functional form from the usual state equation derived assuming an ideal cavity. In such a nonideal cavity, one might find its loss by measuring the decay time of the transmitted light after rapidly cutting off the incident light or by scanning the length of the cavity to measure the finesse. If these two different values of loss vary by about a factor of two, as they have in certain of our investigations, then making either one or the other measurement and applying the usual state equation would cause one to overestimate or underestimate the value of C necessary for the critical onset of bistability by as much as 40%.
The relaxation rates κ , γ ⊥ and γ ∥ which appear in the single-mode model are irrelevant for the stationary state. Instead, their values become critical when we consider dynamical, i.e. time-dependent, phenomena. In particular, they control the possibility of eliminating adiabatically some of the variables in play, thus reducing the complexity of the problem. In this chapter we first illustrate, in Section 19.1, the useful laser classification introduced by Arecchi. Next, in Section 19.2, we discuss the dynamical behavior of the solutions of a model obtained from the single-mode model (derived in Chapter 12) by performing the adiabatic elimination of the atomic variables, which procedure is called the good-cavity limit in the literature. Section 19.2.1 is devoted to the case of the free-running laser ( class-A laser ) and discusses in particular, at the end, the cubic single-mode laser model which is the simplest model capable of describing a laser. On the other hand, in Section 19.2.2 we discuss the case of optical bistability in the good-cavity limit. If we eliminate adiabatically the atomic polarization only, we obtain, instead, the single- mode rate equations which are discussed in Section 19.3. In particular, Section 19.3.1 concerns the case of class-B lasers and focusses especially on the phenomenon of relaxation oscillations. Special attention is paid to demonstrating the equivalence of the model which describes class-B lasers to the rate-equation laser model derived in Chapter 1 and the semiconductor laser model obtained in Chapter 16. In Section 19.3.2 we describe the generation of a giant pulse in class-B lasers, a phenomenon which is called active Q- switching . The final section, Section 19.4, deals with the model obtained by adiabatically eliminating the field variable in the so-called bad-cavity limit . Classification of lasers Arecchi [243] identifies three especially important kinds of laser, which he calls class-A, class-B and class-C lasers, respectively. (1) Class-A lasers are characterized by the condition κ ≪ γ ⊥ , γ ∥ . In this case the atomic variables can be adiabatically eliminated. The He–Ne laser, argon laser and quantum-cascade laser belong to this class.
We specialize the quantum-statistical theory of optical bistability developed in "Optical Bistability. II" to a low-QQQ cavity. From a linearized Fokker-Planck equation derived in the positive PPP representation, the steady-state correlation matrix and spectrum of fluctuations are calculated. Expressions which normally assume a positive-definite diffusion matrix are shown to be valid for nonpositive-definite diffusion. For absorptive bistability we find agreement with results derived previously by quite different methods. More generally we extend previously published results by including atomic and cavity detunings. We compare the transmitted spectrum for dispersive bistability with that for absorptive bistability, and photon antibunching in dispersive bistability is discussed.