Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Quantum statistical theory of the ring laser

Surendra Singh

Published Feb 1, 1981• 14 citations
Publisher

Abstract

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 $\frac{1}{3}$ 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.

DOI
10.1103/physreva.23.837

Related papers

Statistical properties of gas ring lasers with backscattering

Luis Pesquera, R. Blanco, M. A. Rodrguez

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

Bidirectional ring laser: Stability analysis and time-dependent solutions

H. Zeghlache, Paul Mandel, N. B. Abraham +3 more

We provide a comprehensive analysis of the standard model of the bidirectional ring laser in which only one mode can be supported in each direction and in the limit that the polarization can be eliminated adiabatically. The interaction between the two counterpropagating modes can be derived and it is most naturally viewed as a coupling between them via scattering from a spatial grating formed in the population inversion. If the grating is a sufficiently small modulation of the spatial average of the population inversion, it can be approximated by a sinusoidal function. A systematic derivation of the model with only a sinusoidal grating for the homogeneously broadened case is presented that reveals and corrects errors in several previously published analyses. The stability of the steady-state solution is analyzed. The bidirectional steady-state solution is unstable and the unidirectional steady-state solutions may be stable or unstable depending on the parameters. The well-established result of bistability between the two modes when the cavity is tuned to resonance is recovered, a result that persists, in part, even when the losses between the two modes are different.With detuning, the unidirectional steady-state solutions can be destabilized, creating regions in the parameter space where only time-dependent solutions are found. For parameters characteristic of solid-state or molecular gas lasers, the instability can occur for very small detunings and even very close to the lasing threshold. Asymptotic limits of the instability boundaries for these parameters are presented. Additional results are derived formally for inhomogeneous broadening with a Lorentzian line shape and for arbitrary inhomogeneous linewidths. Explicit analytic results are presented in the limits of very small and very large inhomogeneous linewidths compared with the homogeneous linewidth. Time-dependent solutions in the homogeneously broadened case for relaxation rates appropriate to ${\mathrm{CO}}_{2}$ lasers show that there are broad regions in the parameter space of gain and detuning for which the behavior is dynamically chaotic in the form of nearly ``square-wave'' alternation between the counterpropagating modes of the laser with irregular switching times. In other regions of that parameter space we find more irregular pulsations and pulse alternations. Regions of periodic pulsations also have been found.

Physical Review A•Jan 1, 1988•90 citations

Quantum Theory of Laser Radiation. II. Statistical Aspects of Laser Light

J. A. Fleck

We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with $N$ stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.

Physical Review•Sep 9, 1966•26 citations

Quantum Theory of Radiation. II. Statistical Aspects of Laser Light

J. A. Fleck

We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with $N$ stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.

Physical Review•Mar 10, 1967•4 citations

Quantum theory of a two-mode laser with coupled transitions

Surendra Singh, Muhammad Suhail Zubairy

The quantum-statistical properties of the optical field of a two-mode laser with two coupled transitions have been studied by means of a generalization of the Scully and Lamb theory. The photon number distribution and the mode intensity distribution are obtained in the steady state for a system of homogeneously broadened atoms in resonance with the laser field. It is shown that the mode-coupling constant $\ensuremath{\xi}$ is unity and that near threshold the results of earlier treatments are recovered. Furthermore, it is shown that certain limit measures for the relative intensity fluctuations predicted by the semiclassical Fokker-Planck treatments based on the third-order theory are valid even in the limit of arbitrarily high gain levels.

Physical Review A•Jan 1, 1980•38 citations

Intensity Fluctuations and Statistics of Laser Radiation

J. A. Fleck

The steady-state intensity distribution for a laser operating somewhat above threshold is determined by solving a master equation. The master equation describes the interaction between a single radiation mode and $N$ two-level atoms, it contains the effects of pumping and radiation loss from the cavity, and it is applicable to both homogeneous and Doppler broadening. Forms of the master equation applicable to both three- and four-level lasers are considered. Solutions are obtained under the assumption that the distribution of atoms and radiation intensity is a bivariate Gaussian. Fluctuations in the output of giant-pulse lasers are also considered.

Physical Review•Dec 2, 1966•7 citations
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−13\ensuremath{-}\frac{1}{3}\ensuremath−31​, 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)].
We provide a comprehensive analysis of the standard model of the bidirectional ring laser in which only one mode can be supported in each direction and in the limit that the polarization can be eliminated adiabatically. The interaction between the two counterpropagating modes can be derived and it is most naturally viewed as a coupling between them via scattering from a spatial grating formed in the population inversion. If the grating is a sufficiently small modulation of the spatial average of the population inversion, it can be approximated by a sinusoidal function. A systematic derivation of the model with only a sinusoidal grating for the homogeneously broadened case is presented that reveals and corrects errors in several previously published analyses. The stability of the steady-state solution is analyzed. The bidirectional steady-state solution is unstable and the unidirectional steady-state solutions may be stable or unstable depending on the parameters. The well-established result of bistability between the two modes when the cavity is tuned to resonance is recovered, a result that persists, in part, even when the losses between the two modes are different.With detuning, the unidirectional steady-state solutions can be destabilized, creating regions in the parameter space where only time-dependent solutions are found. For parameters characteristic of solid-state or molecular gas lasers, the instability can occur for very small detunings and even very close to the lasing threshold. Asymptotic limits of the instability boundaries for these parameters are presented. Additional results are derived formally for inhomogeneous broadening with a Lorentzian line shape and for arbitrary inhomogeneous linewidths. Explicit analytic results are presented in the limits of very small and very large inhomogeneous linewidths compared with the homogeneous linewidth. Time-dependent solutions in the homogeneously broadened case for relaxation rates appropriate to CO2{\mathrm{CO}}_{2}CO2​ lasers show that there are broad regions in the parameter space of gain and detuning for which the behavior is dynamically chaotic in the form of nearly ``square-wave'' alternation between the counterpropagating modes of the laser with irregular switching times. In other regions of that parameter space we find more irregular pulsations and pulse alternations. Regions of periodic pulsations also have been found.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.
The quantum-statistical properties of the optical field of a two-mode laser with two coupled transitions have been studied by means of a generalization of the Scully and Lamb theory. The photon number distribution and the mode intensity distribution are obtained in the steady state for a system of homogeneously broadened atoms in resonance with the laser field. It is shown that the mode-coupling constant $\ensuremath{\xi}$ is unity and that near threshold the results of earlier treatments are recovered. Furthermore, it is shown that certain limit measures for the relative intensity fluctuations predicted by the semiclassical Fokker-Planck treatments based on the third-order theory are valid even in the limit of arbitrarily high gain levels.
The steady-state intensity distribution for a laser operating somewhat above threshold is determined by solving a master equation. The master equation describes the interaction between a single radiation mode and NNN two-level atoms, it contains the effects of pumping and radiation loss from the cavity, and it is applicable to both homogeneous and Doppler broadening. Forms of the master equation applicable to both three- and four-level lasers are considered. Solutions are obtained under the assumption that the distribution of atoms and radiation intensity is a bivariate Gaussian. Fluctuations in the output of giant-pulse lasers are also considered.