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Physical Review A

Polarization instability in lasers. I. Model and steady states of neodymium-doped fiber lasers

H. Zeghlache, Arnaud Boulnois

Published Nov 1, 1995• 34 citations
Publisher

Abstract

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)].

DOI
10.1103/physreva.52.4229

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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)].
In a previous paper [Phys. Rev. A 52, 4229 (1995)] a vectorial model has been derived to describe the dynamics of an optical fiber laser output. In this semiclassical theory, the time-dependent behaviors of the two field polarizations are deduced through the interaction between the light and the active atoms which supposes the presence of a vectorial atomic polarization. This paper is devoted to the analysis of the dynamics presented by such a model using the bifurcation theory. An analytical study investigates the parameter space as far as possible and numerical simulations show the presence of various types of behaviors displayed by the two output field polarizations in the domains where the steady states have lost their stability. The influence of the pump polarization is emphasized. The results are in accordance with the experimental observations.
Beams of homogeneously broadened two-state sodium atoms in a confocal ring resonator have been employed to investigate the steady-state characteristics of absorptive [1] and dispersive optical bistability. Also studied in the same system is the dynamical behavior due to the intrinsic single-longitudinal-mode instability [2] originating in the alteration of the atomic absorption profile by a strong field as in the Lorenz-type instability in the homogeneously broadened laser. All experimental parameters are carefully measured and corrected for transit broadening, finite absorption, and residual Doppler broadening in order to be able to make an absolute comparison with the single-Gaussian-transverse-mode theory [3-6]. Despite the frequency degeneracy of the transverse modes of the resonator, the agreement with this theory for absorptive bistability is remarkably good. The dispersive bistability data are more difficult to interpret, but they seem to be in reasonable agreement with the theory; the same can be said of the results for the domain and oscillation frequency of the instability, at least within a certain region of parameter space. Outside this region, multiple unstable states appear for various intracavity intensities; recent observations suggest that these may be due in part to instabilities involving higher-order transverse modes of the resonator.
A theoretical model that allows numerical simulations of optical bistability phenomena in CW-pumped erbium-doped fiber lasers is presented. Results of the simulations match correctly with previous experimental data. In order to simulate this phenomenon, the population inversion and optical powers longitudinal distributions along the active fiber have been considered. If this fiber is long enough, it has been found that the transition between the non-lasing and the lasing states involves a dramatic redistribution of the population inversion along the active medium. This fact makes that non-lasing and lasing states constitute different enough initial conditions which determine different final responses to the same control parameter, that is to say, bistability. On the other hand, a theoretical justification is found for a previous experimental result which pointed out that the bistability range becomes reduced if an external signal is launched to the cavity. Finally, the bistability range width dependence on the active fiber length is analyzed.
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.
The polarization and spatiotemporal dynamics of coherent pulses propagating in a doped birefringent nonlinear fiber are modeled with the inclusion of the local Lorentz field. The numerical solutions describe the full set of equations for circularly polarized components of a coherent optical pulse coupled to the inhomogeneously ensemble of doped resonance atoms. The local field is shown to degrade the stability of the $4\ensuremath{\pi}$-decay process. The development of a steady-state signal from the series of multiple photon echoes generated in active fiber is observed in simulations.
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.