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Physical Review A

Space-time dynamics of wide-gain-section lasers

Per Kristen Jakobsen, Jerome V. Moloney, Alan C. Newell, Robert A. Indik

Published Jun 1, 1992• 221 citations
Publisher

Abstract

The space-time behavior of a wide-gain-section, single-longitudinal-mode laser is investigated. Analysis of the full Maxwell-Bloch model in one transverse spatial dimension identifies a traveling-wave solution beyond the first laser threshold, for positive detuning from resonance, which appears to be a globally attracting state of the laser. On the negative-detuning side we observe an initial bifurcation to a homogeneous plane-wave state. A second instability threshold can be identified by linearizing about the respective solutions for both positive and negative detuning. We also find that the standard approach to adiabatic elimination, whereby the polarization equation is adiabatically eliminated, can lead to a spurious high-transverse-wave-number instability in the reduced equations. The derivation of an adiabatically reduced set of equations to remove the instability on the positive-detuning side is a nontrivial matter.

DOI
10.1103/physreva.45.8129

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The space-time behavior of a wide-gain-section, single-longitudinal-mode laser is investigated. Analysis of the full Maxwell-Bloch model in one transverse spatial dimension identifies a traveling-wave solution beyond the first laser threshold, for positive detuning from resonance, which appears to be a globally attracting state of the laser. On the negative-detuning side we observe an initial bifurcation to a homogeneous plane-wave state. A second instability threshold can be identified by linearizing about the respective solutions for both positive and negative detuning. We also find that the standard approach to adiabatic elimination, whereby the polarization equation is adiabatically eliminated, can lead to a spurious high-transverse-wave-number instability in the reduced equations. The derivation of an adiabatically reduced set of equations to remove the instability on the positive-detuning side is a nontrivial matter.
Two-level and Raman lasers with wide-gain sections undergo spontaneous symmetry breaking beyond the first laser threshold resulting in an off-axis emission that, if stable, would correspond to a ring in the far field. The angle subtended by the emission to the laser axis depends on the laser detuning, polarization decay time, and size of the laser cross-section. Optical vortices, propagating fronts, and complex spatio–temporal patterns are predicted to emerge at the first and second thresholds. The intensity dependent detuning and pump laser transverse profile in the Raman laser offers a fascinating array of pattern forming possibilities involving coexisting dark and bright waves. When the polarization decays much faster than the cavity field and medium inversion, one usually adiabatically eliminates this variable from the Maxwell-Bloch equations. Inclusion of transverse effects in this limit, however, leads to spurious transverse instabilities that can be mistaken for numerical grid oscillations. Decoupling of the polarization variable from the electric field allows arbitrarily high transverse wavenumbers to grow.
Complex order-parameter equation descriptions of pattern evolution in large-aspect-ratio two-level and Raman lasers are derived systematically as solvability conditions in a multiple-scales asymptotic expansion of the original Maxwell-Bloch laser equations in powers of a small parameter. These amplitude equations, although strictly valid near threshold for lasing, are shown to capture the essential features of pattern instability and evolution well beyond lasing threshold. A technical difficulty that can arise in the Raman laser, namely, subcriticality of the bifurcation near the critical wave number, is not addressed in the present paper and the order-parameter equations as derived are valid only when this situation does not arise. Analytical expressions for long-wavelength phase instabilities of the underlying traveling-wave pattern, which appears as the natural nonlinear lasing mode when the detuning of the laser from the gain peak is positive, are obtained from the coefficients of a Cross-Newell phase equation. Phase and amplitude instability boundaries, when computed via the original laser equations, the complex order-parameter equations and the phase equation, are shown to be consistent for all cases studied with the exception of the case when a subcritical bifurcation approaches the critical wave number kc{\mathit{k}}_{\mathit{c}}kc​.
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.
A complex order parameter description of the spatiotemporal dynamics close to threshold for a single-longitudinal-mode, large-aspect-ratio laser with a weak injected signal is presented in one transverse spatial dimension and for a longitudinal cavity frequency smaller than the atomic resonance frequency. It is shown that, when the frequency of the external signal is tuned close to the atomic resonance frequency, the spatiotemporal dynamics of the laser may be described by two coupled Ginzburg-Landau equations with parametric terms analogous to those derived in hydrodynamics to study excitation of parametric waves in oscillatory convection. As a result of the parametric terms, which physically arise by means of a four-wave-intercation process mediated by the external signal, the bifurcating modes which are superimposed to the forced (homogeneous) mode may be either traveling or standing waves. In particular, it is shown that for small frequency detunings of the injected signal standing waves are preferred to traveling waves close to threshold. Stability analysis of traveling and standing waves beyond threshold within the amplitude equations reveals the emergence of both phase (Eckhaus) and amplitude instabilities.
The processes of the formation of optical field structures in the transverse profile of wide-aperture laser radiation are studied within the model described by the system of Maxwell-Bloch equations. The main types of instabilities appearing above the second lasing threshold and corresponding mechanisms controlling the laser dynamics at negative frequency detuning are considered. The characteristic lasing modes and conditions of the spontaneous formation of transverse optical structures are determined.
Pattern formation in large-aspect-ratio single-mode inhomogeneously broadened lasers is studied by means of the integro-differential Maxwell-Bloch equations. As the inhomogeneous linewidth increases, the neutral stability curve shows a relaxation in the traveling wave selection allowing structures with different sizes to grow. We have performed numerical simulations with a simplified model based on a few discrete groups of atoms at different resonant frequencies to observe transverse dynamics above threshold for good and bad cavity configurations. We obtain, in general, that close to threshold the inhomogeneous broadening leads to a more complex pattern in comparison to the homogeneous broadening case. At higher pumpings, a stabilization in the number of spatial frequencies taking part in the transverse pattern is found. The influence of the laser aperture on temporal dynamics is also studied. It is shown how the typical self-pulsing regime present in single-mode inhomogeneously broadened lasers is destroyed for a wide enough aperture.