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Journal of Mathematical Physics

Correlations in an N-mode laser

F. T. Hioe

Published Jun 1, 1978• 29 citations
Publisher

Abstract

Analytic expressions for the correlations of the light emitted by an N-mode laser are given. We study in particular a special case of strongest couplings and show, for example, that at steady state, the second order cross intensity correlation tends toward a negative value, −1/(N−1), for large pump parameters. We also study the time dependent solutions, and express the second order amplitude and intensity correlations of the emitted light in terms of the eigenvalues and eigenfunctions of a Schrödinger-type differential equation which reduces to that studied by Risken for N=1 and to that studied by M-Tehrani and Mandel for N=2. Analytic expressions which approximate the eigenvalues of this differential equation for a general N are given.

DOI
10.1063/1.523818

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Analytic expressions for the correlations of the light emitted by an N-mode laser are given. We study in particular a special case of strongest couplings and show, for example, that at steady state, the second order cross intensity correlation tends toward a negative value, −1/(N−1), for large pump parameters. We also study the time dependent solutions, and express the second order amplitude and intensity correlations of the emitted light in terms of the eigenvalues and eigenfunctions of a Schrödinger-type differential equation which reduces to that studied by Risken for N=1 and to that studied by M-Tehrani and Mandel for N=2. Analytic expressions which approximate the eigenvalues of this differential equation for a general N are given.
The intensity and amplitude correlation functions of the optical field of a ring laser at line center are calculated, under conditions when the two pump parameters corresponding to the counter-rotating traveling wave modes are not necessarily equal. This represents a generalization of earlier treatments by M-Tehrani and Mandel (1978) and Hioe (1978). The laser is assumed to be at rest. A perturbative technique is used to express the correlation functions for a small difference $\ensuremath{\epsilon}$ of pump parameters in terms of the solutions for $\ensuremath{\epsilon}=0$. It is found that the cross correlations are unchanged to the first order in $\ensuremath{\epsilon}$, whereas the autocorrelations both of the light amplitude and of the light intensity are modified. Curves are presented to illustrate the behavior.
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.
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.
The steady-state fluctuations of an inhomogeneously broadened two-mode laser with both additive and multiplicative white noise are investigated theoretically. A Fokker-Planck equation of the intensity-distribution function is presented. Analytic expressions of the autocorrelation and cross correlation of the steady-state two-mode intensities are derived for equal pump parameters of the two modes. Large intensity fluctuations occur when the laser is operated below threshold. However, when the laser is operated far above threshold, there is almost no difference between the laser models with and without multiplicative noise.
Expressions for the fluctuations and correlations of the light emitted by a ring laser with counter-rotating traveling wave modes are derived theoretically. It is shown that the competition of the two modes for the excited atomic population leads to negative correlations between their intensity fluctuations, whose magnitude depends on the detuning of the laser cavity from the atomic line center. As a result of the mode competition, the relative intensity fluctuations do not die out in general as the working point of the laser is raised above threshold at line center, and the emitted light does not become fully coherent as in a conventional laser. The two-time amplitude and intensity correlation functions of the light are calculated at line center, and are shown to be expressible in terms of the eigenfunctions and eigenvalues of a certain Schr\"odinger equation, in complete analogy with the theory of the single-mode laser. However, the intensity correlation time increases with pump parameter above threshold, unlike that for the single-mode laser.
The probability distribution p(n)p(n)p(n) for the detection of nnn photons emitted by a single-mode dye laser in a short time at various excitations is measured, together with the distribution of time intervals between pairs of detected photons. The first measurement yields the probability density p(I)\mathcal{p}(I)p(I) and the moments of the light intensity III, and the second one the two-time intensity-correlation function. Measurements are also performed at three different wavelengths. p(I)\mathcal{p}(I)p(I) is found to have a two-component structure, and the relative intensity fluctuations $\frac{〈{(\ensuremath{\Delta}I)}^{2}〉}{{〈I〉}^{2}}$ grow, apparently without limit, as the excitation tends to zero, so that no thermal state is reached. A possible explanation in terms of pumping fluctuations is discussed. The correlation function has the form of a sum of three exponential functions, and the amplitudes and decay constants of the three components are derived. The effective decay-time constant exhibits a thirteenth-power-law dependence on frequency of the laser light.