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

Many-mode stability analysis in the mean-field limit for absorptive bistability

H. J. Carmichael

Published Jul 1, 1983• 10 citations
Publisher

Abstract

The eigenvalue equation for absorptive bistability in a ring cavity is analyzed in the mean-field limit. Perturbative solutions are obtained in two distinct limits: $\ensuremath{\kappa}\ensuremath{\ll}{\ensuremath{\gamma}}_{\ensuremath{\parallel}},{\ensuremath{\gamma}}_{\ensuremath{\perp}}$ and $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}_{\ensuremath{\parallel}},{\ensuremath{\gamma}}_{\ensuremath{\perp}}$, where $\ensuremath{\kappa}$ and ${\ensuremath{\gamma}}_{\ensuremath{\parallel}}$,${\ensuremath{\gamma}}_{\ensuremath{\perp}}$ are field and atomic damping rates, respectively. For $\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}_{\ensuremath{\parallel}},{\ensuremath{\gamma}}_{\ensuremath{\perp}}$, the eigenvalues satisfy a cubic equation, derived previously in the theory of self-pulsing in absorptive bistability. For $\ensuremath{\kappa}\ensuremath{\ll}{\ensuremath{\gamma}}_{\ensuremath{\parallel}},{\ensuremath{\gamma}}_{\ensuremath{\perp}}$, while the "dressed-field-mode" eigenvalues are solutions to this cubic equation, the "dressed-atomic-mode" eigenvalues are not.

DOI
10.1103/physreva.28.480

Related papers

Dressed-mode theory of optical bistability

Luigi A. Lugiato

We devise a procedure to obtain an analytical treatment of self-pulsing in optical bistability, which is parallel to that of the dressed-mode theory but does not require evaluation of the eigenvalues of the linearized equations. Our analysis shows that the uniform---stationary-field approximation does not forbid our arriving at the correct final equations in the mean-field limit.

Physical Review A•Jul 1, 1983•3 citations

Off-resonant-mode instabilities in mixed absorptive and dispersive optical bistability

M. L. Asquini, L. A. Lugiato, H. J. Carmichael +1 more

We analyze instabilities in mixed absorptive and dispersive optical bistability in the rate-equation approximation and the mean-field and good-cavity limits. Our starting point is a set of multimode equations derived from the Maxwell-Bloch equations for ring-cavity boundary conditions. We obtain analytic expressions for the instability conditions. In a plane-wave analysis, we find that a portion of the lower transmission branch can be unstable in addition to the upper-branch instability found in purely absorptive bistability. Also, a new disconnected region of instability can exist on the upper branch. Our analysis becomes particularly simple for equal and opposite cavity detunings and we explore this case in detail. We extend our treatment to include a Gaussian transverse intensity profile and show that the instabilities remain in the presence of Gaussian averaging. We also show that many of the results obtained in the rate-equation approximation hold when the atomic linewidth and atomic decay rate are of the same order.

Physical Review A•Jan 1, 1986•14 citations

The generalized dynamic equation of optical bistability, and the laser and its stability analysis

Fa Ou, Yong Qiang Cai

A single-mode dynamic equation is obtained from the Maxwell equation with the slowly-varying amplitude approximation, and the boundary conditions of the cavity. Here the mean-field limit and the good-quality cavity case have been used. This equation is useful in describing the dynamic behavior of intrinsic optical bistable systems and/or lasers with injected signals for any kinds of materials, including two-level atoms. Furthermore, according to the linear stability analysis, a generalized instability criterion is derived.

Acta Physica Sinica•Jul 1, 1989•0 citations

Temporal aspects of absorptive optical bistability

Thomas Erneux, Paul Mandel

We study the semiclassical mean-field theory of absorptive optical bistability in the limit of a fully developed hysteresis cycle, i.e., in the limit of large values of the bistability parameter $C$. We show by multiple time-scale perturbation analysis that it is possible to describe the transition between the two stable branches by simple equations. Near the low-transmission branch, the behavior of the system is governed by an equation for the atomic population, whereas, near the high-transmission branch, it is an equation for the field amplitude which determines the long-time evolution of the system. We then present an analytic study, completed by numerical results whenever necessary, of the time-dependent response of the system when the input field is swept across the bistable region. We study the influence of the initial conditions, the sweeping velocity, and the ratio of the atomic relaxation times on the dynamical hysteresis as well as its relation with the stationary hysteresis.

Physical Review A•Aug 1, 1983•44 citations

Dispersive optical bistability: Stability of the steady states

Thomas Erneux, Paul Mandel

We study the semiclassical mean-field theory of single-mode dispersive optical bistability in the limit of a fully developed hysteresis cycle (i.e., in the limit of large values of the bistability parameter C) and when all decay rates have comparable magnitude. We locate (i) on the upper branch instabilities leading to time periodic regimes, (ii) on the lower branch instabilities leading to a switching before the limit point is reached. None of these instabilities are observed in the case of purely absorptive optical bistability.

Physical Review A•Mar 1, 1986•8 citations

Beyond the mean-field theory of dispersive optical bistability

Rajarshi Roy, Muhammad Suhail Zubairy

The problem of dispersive optical bistability has so far been treated only in the mean-field approximation. A rigorous justification of the mean-field theory can only be obtained from exact solutions of the steadystate Maxwell-Bloch equations which retain the spatial dependence of the field. In this paper exact analytic solutions are presented for these equations. The authors demonstrate that the mean-field equation connecting the input and the output fields follows naturally from these solutions in the limits $T\ensuremath{\rightarrow}0$, ${\ensuremath{\delta}}_{F}\ensuremath{\rightarrow}0$, and $\ensuremath{\alpha}L\ensuremath{\rightarrow}0$ for the mirror transmission coefficient, the detuning of the field from the cavity resonance, and the linear absorption, respectively, with $\frac{\ensuremath{\alpha}L}{2T}$ and $\frac{{\ensuremath{\delta}}_{F}L}{2cT}$ remaining finite. The results are illustrated with the help of graphs showing the output versus input intensity for different values of the relevant parameters. The effect of these parameters on the phase shift of the output field is also displayed.

Physical Review A•Jan 1, 1980•27 citations
The eigenvalue equation for absorptive bistability in a ring cavity is analyzed in the mean-field limit. Perturbative solutions are obtained in two distinct limits: \ensuremathκ\ensuremath≪\ensuremathγ\ensuremath∥,\ensuremathγ\ensuremath⊥\ensuremath{\kappa}\ensuremath{\ll}{\ensuremath{\gamma}}_{\ensuremath{\parallel}},{\ensuremath{\gamma}}_{\ensuremath{\perp}}\ensuremathκ\ensuremath≪\ensuremathγ\ensuremath∥​,\ensuremathγ\ensuremath⊥​ and \ensuremathκ\ensuremath∼\ensuremathγ\ensuremath∥,\ensuremathγ\ensuremath⊥\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}_{\ensuremath{\parallel}},{\ensuremath{\gamma}}_{\ensuremath{\perp}}\ensuremathκ\ensuremath∼\ensuremathγ\ensuremath∥​,\ensuremathγ\ensuremath⊥​, where \ensuremathκ\ensuremath{\kappa}\ensuremathκ and \ensuremathγ\ensuremath∥{\ensuremath{\gamma}}_{\ensuremath{\parallel}}\ensuremathγ\ensuremath∥​,\ensuremathγ\ensuremath⊥{\ensuremath{\gamma}}_{\ensuremath{\perp}}\ensuremathγ\ensuremath⊥​ are field and atomic damping rates, respectively. For \ensuremathκ\ensuremath∼\ensuremathγ\ensuremath∥,\ensuremathγ\ensuremath⊥\ensuremath{\kappa}\ensuremath{\sim}{\ensuremath{\gamma}}_{\ensuremath{\parallel}},{\ensuremath{\gamma}}_{\ensuremath{\perp}}\ensuremathκ\ensuremath∼\ensuremathγ\ensuremath∥​,\ensuremathγ\ensuremath⊥​, the eigenvalues satisfy a cubic equation, derived previously in the theory of self-pulsing in absorptive bistability. For \ensuremathκ\ensuremath≪\ensuremathγ\ensuremath∥,\ensuremathγ\ensuremath⊥\ensuremath{\kappa}\ensuremath{\ll}{\ensuremath{\gamma}}_{\ensuremath{\parallel}},{\ensuremath{\gamma}}_{\ensuremath{\perp}}\ensuremathκ\ensuremath≪\ensuremathγ\ensuremath∥​,\ensuremathγ\ensuremath⊥​, while the "dressed-field-mode" eigenvalues are solutions to this cubic equation, the "dressed-atomic-mode" eigenvalues are not.
We analyze instabilities in mixed absorptive and dispersive optical bistability in the rate-equation approximation and the mean-field and good-cavity limits. Our starting point is a set of multimode equations derived from the Maxwell-Bloch equations for ring-cavity boundary conditions. We obtain analytic expressions for the instability conditions. In a plane-wave analysis, we find that a portion of the lower transmission branch can be unstable in addition to the upper-branch instability found in purely absorptive bistability. Also, a new disconnected region of instability can exist on the upper branch. Our analysis becomes particularly simple for equal and opposite cavity detunings and we explore this case in detail. We extend our treatment to include a Gaussian transverse intensity profile and show that the instabilities remain in the presence of Gaussian averaging. We also show that many of the results obtained in the rate-equation approximation hold when the atomic linewidth and atomic decay rate are of the same order.
A single-mode dynamic equation is obtained from the Maxwell equation with the slowly-varying amplitude approximation, and the boundary conditions of the cavity. Here the mean-field limit and the good-quality cavity case have been used. This equation is useful in describing the dynamic behavior of intrinsic optical bistable systems and/or lasers with injected signals for any kinds of materials, including two-level atoms. Furthermore, according to the linear stability analysis, a generalized instability criterion is derived.
We study the semiclassical mean-field theory of absorptive optical bistability in the limit of a fully developed hysteresis cycle, i.e., in the limit of large values of the bistability parameter CCC. We show by multiple time-scale perturbation analysis that it is possible to describe the transition between the two stable branches by simple equations. Near the low-transmission branch, the behavior of the system is governed by an equation for the atomic population, whereas, near the high-transmission branch, it is an equation for the field amplitude which determines the long-time evolution of the system. We then present an analytic study, completed by numerical results whenever necessary, of the time-dependent response of the system when the input field is swept across the bistable region. We study the influence of the initial conditions, the sweeping velocity, and the ratio of the atomic relaxation times on the dynamical hysteresis as well as its relation with the stationary hysteresis.
We study the semiclassical mean-field theory of single-mode dispersive optical bistability in the limit of a fully developed hysteresis cycle (i.e., in the limit of large values of the bistability parameter C) and when all decay rates have comparable magnitude. We locate (i) on the upper branch instabilities leading to time periodic regimes, (ii) on the lower branch instabilities leading to a switching before the limit point is reached. None of these instabilities are observed in the case of purely absorptive optical bistability.
The problem of dispersive optical bistability has so far been treated only in the mean-field approximation. A rigorous justification of the mean-field theory can only be obtained from exact solutions of the steadystate Maxwell-Bloch equations which retain the spatial dependence of the field. In this paper exact analytic solutions are presented for these equations. The authors demonstrate that the mean-field equation connecting the input and the output fields follows naturally from these solutions in the limits $T\ensuremath{\rightarrow}0$, ${\ensuremath{\delta}}_{F}\ensuremath{\rightarrow}0$, and $\ensuremath{\alpha}L\ensuremath{\rightarrow}0$ for the mirror transmission coefficient, the detuning of the field from the cavity resonance, and the linear absorption, respectively, with $\frac{\ensuremath{\alpha}L}{2T}$ and $\frac{{\ensuremath{\delta}}_{F}L}{2cT}$ remaining finite. The results are illustrated with the help of graphs showing the output versus input intensity for different values of the relevant parameters. The effect of these parameters on the phase shift of the output field is also displayed.