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

Codimension-two bifurcations in single-mode optical bistable systems

Guang Hu, Cun-Zheng Ning, Hermann Haken

Published Mar 1, 1990• 13 citations
Publisher

Abstract

The codimension-two bifurcation set of single-mode optical bistable systems is investigated. The entire set is classified into three classes, a cusp bifurcation set, a degenerate Hopf bifurcation set, and the intersection of a Hopf bifurcation and a saddle-node bifurcation. Based on the specification of degenerate Hopf bifurcation, super- and subcritical Hopf bifurcations can be identified. It is found that, in the subcritical Hopf bifurcation region, an attractor of time-dependent motion may coexist with the stable stationary solution when the cavity is filled by a passive medium. Moreover, the coexistence of three attractors is observed for certain parameter combinations.

DOI
10.1103/physreva.41.2702

Related papers

Distribution of subcritical Hopf bifurcations and regular and chaotic attractors in optical bistable systems

Guang Hu, Cun‐Zheng Ning, Hermann Haken

The distributions, in parameter space, of subcritical Hopf bifurcations and the various attractors of time-dependent motions of optical bistable systems are investigated systematically. The main emphasis is placed on the influence of the variations of the control parameters on the distributions. We have observed the tristability of large-amplitude regular or chaotic motions, periodic pulsations, and the stationary state for the parameters where no instability of the stationary solution exists at all over the complete range of the external field. The problem of how to find the various attractors and how to reveal chaotic motions by adjusting the control parameters is discussed according to the numerical results.

Physical Review A•Apr 1, 1990•11 citations

Periodic attractors in two-photon processes

Paul Mandel, Nicolas Pettiaux, Wang Kaige +2 more

We consider three nonlinear optical models that display a two-photon process and exhibit a Hopf bifurcation corresponding to a phase instability. The three models describe second-harmonic generation, two-photon optical bistability, and degenerate four-wave mixing in resonant cavities. In all three cases we consider only the resonant configuration and analyze the attractors which emerge when the driving field is increased beyond the Hopf bifurcation threshold. At a finite distance from threshold, we find a hysteresis domain that involves a pair of periodic attractors. For all three models, each of the two solutions has the same ``field portrait'' i.e., the same representation in a plane whose axes are the imaginary and the real parts of the electric field. This suggests that the scenario is generic for this class of systems. On the other hand, the models for two-photon optical bistability and degenerate four-wave mixing in resonant cavities exhibit additional nongeneric attractors when the driving field is further increased.

Physical Review A•Jan 1, 1991•16 citations

UNSTABLE REGION AND BIFURCATIONS AND CHAOS IN SINGLE-MODE BISTABLE OPTICAL SYSTEMS

MA Wen-qi, Guojian Yang, Gang Xiang Hu +1 more

Based on the instability criterion of semi-classical Maxwell -Bloch equations of single-mode optical systems, the global distribution of instability region is depicted in △-θ parameter space (△ detuning of the atoms, θ mistuning of the cavity). In the instability region predicted, various bifurcations and chaos are disclosed. We find that chaos may appear in both upper and lower branches of one bistable curve Moreover, a new time-dependent solution coexisting with the stable stationary solution is found.

Acta Physica Sinica•Jan 1, 1989•0 citations

Bifurcation analysis of nonlinear coupled photonic nanocavities

Neil G. R. Broderick, Bernd Krauskopf, Soizic Terrien +4 more

Recently, various research groups have studied spontaneous symmetry breaking and bistable behaviour in both active and passively driven coupled photonic nanocavities[1]. This work is of interest both in itself and because it naturally extends to coupled nonlinear quantum systems described by the Bose-Hubbard model [2]. Here we present a theoretical study of the dynamics in such a system by considering the bifurcation behaviour as the different parameters are varied. In normalised units the model can be written as: dA/dt = i(δ τ + |A|2)A-A+(iκ + γ)τB+f (1) dB/dt = i(δ τ + |B|2)B-B+(iκ + γ)τA+f (2) where A and B are the slowly varying normalised amplitudes of the fields in each nano-cavity. Further, δ is the detuning from the cavity resonance, τ is the photon lifetime, κ and γ describe the linear coupling between the cavities (if γ = 0 and κ < 0 the model is identical to the Bose-Hubbard model [2] in the semiclassical regime), while f represents a coherent driving term. In deriving the above equations it is assumed that the cavities are identical, and so a focus is the switching between the symmetrical state [A(t) = B(t)] and asymmetrical states, which must come in pairs. Fig. 1(a) shows a typical boundary curve dividing the (δ, f) plane into the regions where the symmetric and asymmetric solutions exist. Further analysis shows that more complicated dynamics arise from the asymmetric state. Fig. 1(b) shows the field intensity for the symmetric and asymmetric steady-states as a function of f, where stable solutions are represented by solid and unstable ones by dashed curves. Hopf bifurcations, marked by H, cause the asymmetric solution to change stability. A typical bifurcating periodic solution is shown in Fig. 1(c) for f = 6.

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)•Jun 1, 2017•0 citations

Codimension-two bifurcations and interactions between differently polarised fields for the laser with saturable absorber

Gerhard Dangelmayr, M. Neveling

The equations for the laser with a saturable absorber in the mean-field limit and exact resonance are considered. For critical values of the number of active atoms in the laser and the absorber the equations exhibit a codimension-two degeneracy of the Takens-Bogdanov type, where a stationary bifurcation and a Hopf bifurcation with infinite period coalesce. The envelopes of the electric field and the atomic polarisations are treated as complex variables. This allows one to take differently polarised laser fields into account. The authors compute the relevant normal form coefficients and divide the space of the physical parameters into a number of regions that give rise to qualitatively different bifurcation scenarios. These include hysteretic and continuous stability exchanges between circularly polarised and modulated linearly polarised laser fields.

Journal of Physics A Mathematical and General•May 7, 1989•6 citations

Single- and two-mode models

Luigi A. Lugiato, Franco Prati, Massimo Brambilla

In the first section we discuss the stationary solutions for a laser with an injected signal, the active counterpart of the optical bistability analyzed in Chapter 11. Next, in Section 13.2 we consider another very interesting system, namely a laser with a saturable absorber, a free-running laser in which the optical cavity contains a passive medium in addition to the active medium. The single-mode model for this system can be constructed by combining the single-mode models for the free-running laser and for optical bistability. The discussion is then focussed on its stationary states. In Section 13.3 we return to the single-mode model for optical bistability and, by introducing an appropriate “cubic limit” on the parameters in play, we derive a cubic model for dispersive optical bistability with two-level atoms. The stationary equation for such a model formally coincides with that derived and discussed for a Kerr medium in Section 11.2. In the last section we start from the equations for a quadratic nonlinear medium in the degenerate configuration derived in Chapter 6 and combine them with the appropriate boundary conditions for the two field envelopes in play, assuming that the quadratic medium is contained in a ring cavity as happens in an optical parametric oscillator or in second-harmonic generation in a cavity. Next, by introducing the low-transmission limit and the single-mode limit we derive a two-mode model that describes the dynamics of a degenerate optical parametric oscillator or of second-harmonic generation in a cavity. Finally, for the case of a parametric oscillator we illustrate extensively the stationary solutions of the equations. A laser with an injected signal From a technical point of view, this system is interesting because of the possibility of stabilizing the frequency and the phase of separate lasers by taking advantage of the known affinity of nonlinear oscillators to lock onto a periodic forcing element. In 1972 Spencer and Lamb [125] predicted the possibility of bistable behavior in a laser with an injected signal before the surge of interest in optical bistability in the passive configuration.

Cambridge University Press eBooks•Feb 28, 2015•0 citations
The codimension-two bifurcation set of single-mode optical bistable systems is investigated. The entire set is classified into three classes, a cusp bifurcation set, a degenerate Hopf bifurcation set, and the intersection of a Hopf bifurcation and a saddle-node bifurcation. Based on the specification of degenerate Hopf bifurcation, super- and subcritical Hopf bifurcations can be identified. It is found that, in the subcritical Hopf bifurcation region, an attractor of time-dependent motion may coexist with the stable stationary solution when the cavity is filled by a passive medium. Moreover, the coexistence of three attractors is observed for certain parameter combinations.
The distributions, in parameter space, of subcritical Hopf bifurcations and the various attractors of time-dependent motions of optical bistable systems are investigated systematically. The main emphasis is placed on the influence of the variations of the control parameters on the distributions. We have observed the tristability of large-amplitude regular or chaotic motions, periodic pulsations, and the stationary state for the parameters where no instability of the stationary solution exists at all over the complete range of the external field. The problem of how to find the various attractors and how to reveal chaotic motions by adjusting the control parameters is discussed according to the numerical results.
We consider three nonlinear optical models that display a two-photon process and exhibit a Hopf bifurcation corresponding to a phase instability. The three models describe second-harmonic generation, two-photon optical bistability, and degenerate four-wave mixing in resonant cavities. In all three cases we consider only the resonant configuration and analyze the attractors which emerge when the driving field is increased beyond the Hopf bifurcation threshold. At a finite distance from threshold, we find a hysteresis domain that involves a pair of periodic attractors. For all three models, each of the two solutions has the same ``field portrait'' i.e., the same representation in a plane whose axes are the imaginary and the real parts of the electric field. This suggests that the scenario is generic for this class of systems. On the other hand, the models for two-photon optical bistability and degenerate four-wave mixing in resonant cavities exhibit additional nongeneric attractors when the driving field is further increased.
Based on the instability criterion of semi-classical Maxwell -Bloch equations of single-mode optical systems, the global distribution of instability region is depicted in △-θ parameter space (△ detuning of the atoms, θ mistuning of the cavity). In the instability region predicted, various bifurcations and chaos are disclosed. We find that chaos may appear in both upper and lower branches of one bistable curve Moreover, a new time-dependent solution coexisting with the stable stationary solution is found.
Recently, various research groups have studied spontaneous symmetry breaking and bistable behaviour in both active and passively driven coupled photonic nanocavities[1]. This work is of interest both in itself and because it naturally extends to coupled nonlinear quantum systems described by the Bose-Hubbard model [2]. Here we present a theoretical study of the dynamics in such a system by considering the bifurcation behaviour as the different parameters are varied. In normalised units the model can be written as: dA/dt = i(δ τ + |A|2)A-A+(iκ + γ)τB+f (1) dB/dt = i(δ τ + |B|2)B-B+(iκ + γ)τA+f (2) where A and B are the slowly varying normalised amplitudes of the fields in each nano-cavity. Further, δ is the detuning from the cavity resonance, τ is the photon lifetime, κ and γ describe the linear coupling between the cavities (if γ = 0 and κ < 0 the model is identical to the Bose-Hubbard model [2] in the semiclassical regime), while f represents a coherent driving term. In deriving the above equations it is assumed that the cavities are identical, and so a focus is the switching between the symmetrical state [A(t) = B(t)] and asymmetrical states, which must come in pairs. Fig. 1(a) shows a typical boundary curve dividing the (δ, f) plane into the regions where the symmetric and asymmetric solutions exist. Further analysis shows that more complicated dynamics arise from the asymmetric state. Fig. 1(b) shows the field intensity for the symmetric and asymmetric steady-states as a function of f, where stable solutions are represented by solid and unstable ones by dashed curves. Hopf bifurcations, marked by H, cause the asymmetric solution to change stability. A typical bifurcating periodic solution is shown in Fig. 1(c) for f = 6.
The equations for the laser with a saturable absorber in the mean-field limit and exact resonance are considered. For critical values of the number of active atoms in the laser and the absorber the equations exhibit a codimension-two degeneracy of the Takens-Bogdanov type, where a stationary bifurcation and a Hopf bifurcation with infinite period coalesce. The envelopes of the electric field and the atomic polarisations are treated as complex variables. This allows one to take differently polarised laser fields into account. The authors compute the relevant normal form coefficients and divide the space of the physical parameters into a number of regions that give rise to qualitatively different bifurcation scenarios. These include hysteretic and continuous stability exchanges between circularly polarised and modulated linearly polarised laser fields.
In the first section we discuss the stationary solutions for a laser with an injected signal, the active counterpart of the optical bistability analyzed in Chapter 11. Next, in Section 13.2 we consider another very interesting system, namely a laser with a saturable absorber, a free-running laser in which the optical cavity contains a passive medium in addition to the active medium. The single-mode model for this system can be constructed by combining the single-mode models for the free-running laser and for optical bistability. The discussion is then focussed on its stationary states. In Section 13.3 we return to the single-mode model for optical bistability and, by introducing an appropriate “cubic limit” on the parameters in play, we derive a cubic model for dispersive optical bistability with two-level atoms. The stationary equation for such a model formally coincides with that derived and discussed for a Kerr medium in Section 11.2. In the last section we start from the equations for a quadratic nonlinear medium in the degenerate configuration derived in Chapter 6 and combine them with the appropriate boundary conditions for the two field envelopes in play, assuming that the quadratic medium is contained in a ring cavity as happens in an optical parametric oscillator or in second-harmonic generation in a cavity. Next, by introducing the low-transmission limit and the single-mode limit we derive a two-mode model that describes the dynamics of a degenerate optical parametric oscillator or of second-harmonic generation in a cavity. Finally, for the case of a parametric oscillator we illustrate extensively the stationary solutions of the equations. A laser with an injected signal From a technical point of view, this system is interesting because of the possibility of stabilizing the frequency and the phase of separate lasers by taking advantage of the known affinity of nonlinear oscillators to lock onto a periodic forcing element. In 1972 Spencer and Lamb [125] predicted the possibility of bistable behavior in a laser with an injected signal before the surge of interest in optical bistability in the passive configuration.