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Physical Review A

Correlation functions of the two-mode ring laser

F. T. Hioe, Surendra Singh, L. Mandeļ

Published May 1, 1979• 31 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.19.2036

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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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ of pump parameters in terms of the solutions for \ensuremathϵ=0\ensuremath{\epsilon}=0\ensuremathϵ=0. It is found that the cross correlations are unchanged to the first order in \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, whereas the autocorrelations both of the light amplitude and of the light intensity are modified. Curves are presented to illustrate the behavior.
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 light intensities of the two counter-rotating modes of a ring laser have been investigated by photoelectric counting measurements near the laser threshold and for various detunings. The results are found to be in good quantitative agreement with the theory. They confirm that, at the line center, competition effects prevent the light output in one mode from growing and becoming coherent as the pump parameter is increased. Instead the relative intensity fluctuations of this mode become unity, as for thermal light. Negative cross correlations between the intensity fluctuations of the two modes are also measured. The competition effects virtually disappear when the laser is detuned three natural linewidths from line center, and an intermediate situation is found for smaller detuning.
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 fluctuations and correlations of the light intensities associated with the two counterrotating modes of a ring laser have been investigated by measurement of the photoelectric counting statistics. With the laser tuned to line center, it is found that the weaker mode intensity does not grow with increasing pump parameter above threshold and that its relative intensity fluctuations do not die out, as in a conventional laser, but become thermal instead. The cross-correlation coefficient tends toward a constant negative value. The results are all in substantial quantitative agreement with the theory.
Exact formulas for the effective eigenvalue characterizing the initial decay of intensity correlation functions are given in terms of stationary moments of the intensity. Spontaneous emission noise and nonwhite pump noise are considered. Our results are discussed in connection with earlier calculations, simulations, and experimental results for single-mode dye lasers, two-mode inhomogeneously broadened lasers, and two-mode dye ring lasers. The effective eigenvalue is seen to depend sensitively on noise characteristics and symmetry properties of the system. In particular, the effective eigenvalue associated with cross correlations of two-mode lasers is seen to vanish in the absence of pump noise as a consequence of detailed balance. In the presence of pump noise, the vanishing of this eigenvalue requires equal pump parameters for the two modes and statistical independence of spontaneous emission noise acting on each mode.
The two-mode inhomogeneously broadened ring laser with spontaneous-emission noise is investigated in the presence of backscattering. Using a linear transformation it is shown that this problem in the absence of detuning is equivalent to that of the ring laser without backscattering in the two following cases: (i) in-phase backcoupling case (R1{R}_{1}R1​=R2∗{R}_{2}^{\mathrm{*}}R2∗​), (ii) equal pump parameters and ``symmetric'' backscattering (\ensuremath{\Vert}R1{R}_{1}R1​\ensuremath{\Vert}=\ensuremath{\Vert}R2{R}_{2}R2​\ensuremath{\Vert}), where R1{R}_{1}R1​ and R2{R}_{2}R2​ are complex backscattering coefficients. We recover in this way the stationary probability density obtained by Christian and Mandel [Phys. Rev. A 34, 3932 (1986)]. Exact expressions for the mean values, fluctuations, and correlations of the two-mode intensities are derived. Our results show that the relative intensity fluctuations of both modes die out well above threshold. Mode-competition effects are reduced by symmetric backscattering. Exact formulas for the effective eigenvalue characterizing the initial decay of intensity correlation functions are given. These results are compared with the experimental data measured by Singh [Phys. Rep. 108, 217 (1984)].