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Physical Review A

Coherence theory of the ring laser

Mohammad Mehdi Tehrani, L. Mandeļ

Published Feb 1, 1978• 89 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.17.677

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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 (${R}_{1}$=${R}_{2}^{\mathrm{*}}$), (ii) equal pump parameters and ``symmetric'' backscattering (\ensuremath{\Vert}${R}_{1}$\ensuremath{\Vert}=\ensuremath{\Vert}${R}_{2}$\ensuremath{\Vert}), where ${R}_{1}$ and ${R}_{2}$ 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)].

Physical Review A•Jun 1, 1989•18 citations
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.
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.
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.
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.
A unified quantum treatment is presented for bidirectional two-mode ring lasers using the approach of Haken and co-workers. Equations of motion for the reduced density matrix of the field have been derived and used to study the fluctuation properties of the radiation field in different kinds of active media. The steady-state photon number distribution for an inhomogeneously broadened gas ring laser is derived. It is shown that for equal losses the relative fluctuations and normalized cross correlation of the photon numbers approach values 13\frac{1}{3}31​ and $\ensuremath{-}\frac{1}{3}$, respectively, for large excitations. For unequal losses the laser may emit typical single-mode laser radiation in the direction of the favored mode and typical equilibrium blackbody radiation in the direction of the weaker mode. Our results generally agree with the predictions of earlier treatments when the losses are equal. However, for unequal losses we predict different behavior for large excitations. Effects of detuning, spatial, and temporal variation of atomic inversion on the fluctuation properties have been investigated in the coherent-state diagonal representation of the density matrix, and steady-state intensity distributions have been derived for various kinds of active atoms. It is shown that for inhomogeneously broadened solid-state ring lasers spatial inhomogeneities may couple the modes strongly and the statistical properties of such lasers, characterized by a double-peaked intensity distribution and large relative-intensity fluctuations, may be similar to those of a homogeneously broadened ring laser. Curves are presented to illustrate the behavior.
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)].