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Physical Review A

Nonlinear transverse modes of large-aspect-ratio homogeneously broadened lasers: II. Pattern analysis near and beyond threshold

Joceline C. Lega, Per Kristen Jakobsen, Jerome V. Moloney, Alan C. Newell

Published May 1, 1994• 58 citations

Abstract

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 ${\mathit{k}}_{\mathit{c}}$.

DOI: 10.1103/physreva.49.4201

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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

{\mathit{k}}_{\mathit{c}}

.

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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.

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Transverse effects in the laser threshold, originated by electronic-vibrational coupling in the active centers, are analyzed theoretically by means of the semiclassical two-level Maxwell-Bloch equations. A single longitudinal mode is considered. It is found that the first laser threshold suffers modifications depending on the electronic-vibrational coupling strength. This coupling imposes certain conditions for the selection of a particular transverse spatial state and provides the minimum wavelength that can appear in the transverse pattern. The nature of the bifurcation and the stability of the homogeneous and critical traveling waves are analyzed.

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Transverse effects in the laser dynamics due to near dipole-dipole interactions are studied considering the presence of permanent electric dipole moments. The semiclassical two-level Maxwell-Bloch equations are used and a single longitudinal mode is assumed. A traveling wave is selected at threshold when the sum of the cavity detuning and the near dipole-dipole parameter is larger than zero. As a consequence of this, transverse pattern can occur even when the laser frequency is larger than the frequency of the atomic transition. Also, a cutoff in the laser field spectrum arises. We found that the near dipole-dipole interactions significantly modify the stability picture of the traveling waves. Numerical simulations have been carried out and the effect of the near dipole-dipole on the pattern formation is addressed.