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Journal of Physics A Mathematical and General

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

Gerhard Dangelmayr, M. Neveling

Published May 7, 1989• 6 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/22/9/018

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•Mar 2, 2018•0 citations
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.
We study the semiclassical equations for a laser with a saturable absorber in the mean-field limit, assuming homogeneously broadened two-level atoms, for a set of parameters where the system displays optical bistability and time-periodic solutions. In the first part the bifurcation diagram for stationary and periodic solutions is obtained by numerical integration. Two different classes of stable periodic solutions arise: small-amplitude solutions and passive QQQ switching. We observe hysteresis domains involving up to three solutions (stationary and/or periodic). We also discuss the validity of some standard approximations and show that even in the absence of detuning the phases play an important role. We also discuss the influence of the initial conditions whose symmetry properties induce important modifications of the bifurcation diagram. In the second part we introduce an alternative adiabatic elimination scheme which allows us to construct the small-amplitude periodic solutions over nearly their whole range of existence. We then study these solutions near the Hopf bifurcation from which they emerge and derive analytic conditions for their stability. When they are stable, $\stackrel{\mathrm{\ifmmode\acute\else\textasciiacute\fi{}}}{\mathrm{w}}$e also give the conditions under which a secondary Hopf bifurcation will occur, leading to quasiperiodic solutions.
We study the properties of the laser field in the case where the cavity contains two cells, one which amplifies and one which absorbs the radiation. We adopt the simplest possible model, in which there is only one running mode in the cavity and all atoms are fixed and homogeneously distributed in space. We first set up a semiclassical description. In the frame of the stationary semiclassical (SC) theory the most striking feature is the occurrence of a domain in which three solutions coexist. A linear-stability analysis shows that two of these solutions are stable. This gives rise to an hysteresis cycle. Most of the results in this section are analytic. In the frame of a purely quantum description, we derive a generalized Fokker-Planck equation which is exactly solved in the stationary case. This yields a stationary quasiprobability distribution function for the field. In the domain where the SC theory predicts three solutions, the field-distribution function has two maxima and one minimum whose positions are precisely given by the SC intensities. With this distribution function we study the intensity and the intensity fluctuations: the intensity shows a rather abrupt increase when crossing the narrow transition region; in this region the intensity fluctuations are extremely high. By means of an approximate method to solve the time-dependent Fokker-Planck equation we study the linewidth and show that it drastically decreases above threshold. Finally we consider the linearized and quasilinearized Fokker-Planck equations which can be solved analytically. We show that the domain in which neither of these equations fits the results of the generalized Fokker-Planck equation is extremely small. The properties of the metastable states are analyzed in terms of the generalized and asymptotic Fokker-Planck equations.
This chapter concludes the discussion about temporal instabilities and self-oscillations, considering the instabilities that arise in the systems analyzed in Chapter 13, with the exception of optical bistability, which has been discussed separately in Chapter 11. Section 25.1 focuses on the laser with an injected signal. As already mentioned in Section 13.1, in this system self-pulsing instabilities are rather ubiquitous, and on exploring the parameter space one meets an extremely rich variety of oscillatory behaviors. For example, under extremely high-gain conditions, breathing, spiking and chaos have been identified over various ranges of the driving field strength [294]. In this section we show some results that arise under parametric conditions that are more accessible experimentally. We illustrate especially the coexistence, for the same parametric values, of different oscillatory behaviors that are reached by starting from different initial conditions. On the other hand, Section 25.2 deals with a laser with a saturable absorber. In this system even the full scenario of stationary intensity solutions in the single-mode model is very complex. For the sake of simplicity, in this section we focus on a simpler rate-equation model that is obtained from the single-mode model of Eqs. (13.13)–(13.17) by adiabatically eliminating the atomic polarization of both active and passive medium. As in Chapter 13, we assume that the atomic transition frequencies are equal and also coincide with a cavity frequency. In this way all the variables in the model can be assumed real, and the linearization around the stationary solution described in Section 13.2 leads to a cubic eigenvalue equation. The spontaneous oscillations which arise from this instability are called repetitive passive Q-switching in the literature. Section 25.3 considers the stability of the stationary solutions of the degenerate optical parametric oscillator described in Section 13.4 and outlines the scenario of period doubling and chaos that arises under appropriate parametric conditions.
This paper presents new results for the steady states of a detuned ring laser with a saturable absorber. We employ a semiclassical model which assumes homogeneously broadened two-level atoms. We proceed by solving the Maxwell-Bloch equations for the longitudinal dependence of the steady states of this system, and then simplify our solution by use of the uniform-field approximation. We present uniform-field results for squared electric field versus operating frequency, and for each of these versus cavity tuning and laser excitation. Various cavity linewidths and both resonant and nonresonant amplifier and absorber line-center frequencies are considered. The most notable finding is that cavity detuning breaks the degeneracies found in the steady-state solutions of the fully tuned case. This leads to the prediction that an actual system will bifurcate from the zero-intensity solution to a steady-state solution as laser excitation increases from zero, rather than to the small-amplitude pulsations found for the model with exactly resonant tuning of the cavity and the media line centers. Other phenomena suggested by the steady-state results include tuning-dependent hysteresis and bistability, and instability in both intensity and frequency due to the appearance of one or more new steady-state solutions as tuning is varied. These effects of detuning are being tested by a linearized stability analysis whose results will be reported separately.
The aim of this paper is to provide a vectorial model which can describe the dynamics of the two transverse field polarizations presented by an optical fiber laser. These dynamics are supposed to take their origin from the interaction between the propagating field and the atoms: an assumption based on the existence of a transverse distribution of dipole moments and which leads to a vectorial atomic polarization. Also, and as we have considered a frequency for each field polarization, a longitudinal dependence of the population inversion must be taken into account. A double Fourier expansion (transverse and longitudinal) of the material variables is then needed and the resulting model is analyzed using the bifurcation theory: the existence of all stationary states is fully expressed in the parameter space. The stability of these states and the system dynamic is studied in the following paper [Phys. Rev. A 52, 4243 (1995)].
We study the dynamics of the solution of a non-linear quantum master equation describing a simple laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and a set of two level atoms that are coupled to two reservoirs. First, we establish the existence of a unique regular stationary state for the non-linear evolution equation under consideration. Second, we examine the free interaction solutions, i.e., the solutions to the non-linear quantum master equation that coincide with unitary evolutions generated by the Hamiltonian resulting from neglecting the interactions between the laser mode, atoms and the bath. We obtain that a family of non-constant free interaction solutions is born at the regular stationary state as a relevant parameter, which is denoted by $C_\mathfrak{b} $, passes through the critical value 111. These free interaction solutions yield the periodic solutions of the Maxwell Bloch equations modeling our physical system in the framework of the semiclassical laser theory. Third, in case $ C_\mathfrak{b} < 1$ we deduce that the system converges exponentially fast to the equilibrium, and so the regular stationary state is a globally asymptotically stable equilibrium solution. Thus, the quantum system has a Hopf bifurcation at $C_\mathfrak{b} =1 $.