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Journal of Physics B Atomic Molecular and Optical Physics

Third-order harmonic-expansion analysis of the Casperson instability

Belkacem Meziane

Published Aug 15, 2007• 3 citations
Publisher

Abstract

We revisit the integro-differential ‘Maxwell–Bloch’ equations that describe the dynamic properties of single-mode inhomogeneously broadened lasers, with a strong harmonic expansion analysis carried up to the third order in field amplitude. In particular, we extract a closed form expression of its long-term eigenpulsation that naturally yields an iterative algorithm to build, analytically , the regular pulsing solutions of the system. Despite the complicated nature of the equations due to the presence of an integral over the polarization, the method allows for the disclosure of quite the same instructive information as those given by the much simpler Laser–Lorenz case.

DOI
10.1088/0953-4075/40/17/003

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We revisit the integro-differential ‘Maxwell–Bloch’ equations that describe the dynamic properties of single-mode inhomogeneously broadened lasers, with a strong harmonic expansion analysis carried up to the third order in field amplitude. In particular, we extract a closed form expression of its long-term eigenpulsation that naturally yields an iterative algorithm to build, analytically , the regular pulsing solutions of the system. Despite the complicated nature of the equations due to the presence of an integral over the polarization, the method allows for the disclosure of quite the same instructive information as those given by the much simpler Laser–Lorenz case.
This paper gives a full account of the dynamical properties of single-mode inhomogeneously broadened lasers. The study divides itself into two parts. First, the integro-differential ``Maxwell-Bloch'' equations, which describe such systems, are numerically solved for a large scan of values of the various control parameters that govern the dynamical evolution of self-pulsing lasers. The instability hierarchies, delimited with respect to the excitation parameter, reveal that, along with the various decay rates of the interacting variables, the chosen number of velocity components, which divides the polarization integral into discrete intervals, also plays an important role in the dynamical evolution of the system, especially in the self-pulsing regime of operation. The second part of this paper concerns another approach to modeling single-mode inhomogeneously broadened laser dynamics that largely circumvents the infinite-dimensional complexity of the medium's polarization. A hierarchy of low-dimensional models, with increasing accuracy with respect to the infinite-dimensional ``Maxwell-Bloch'' equations, is constructed and adapted to describe correctly most of the properties of single-mode inhomogeneously broadened lasers.
We study numerically the effects of local-field corrections originating from near dipole—dipole interactions on the dynamics of a single mode, inhomogeneously broadened laser. Our analysis is based on a set of generalized Maxwell–Bloch equations in which the inhomogeneous nature of gain broadening is accounted for by introducing two new dynamical variables and a single parameter that governs the extent of inhomogeneous broadening. Our results show that local-field effects occurring in a dense gain medium reduce the range of continuous wave operation and lead to instabilities and chaos at much lower pumping levels.
Summary form only given. Despite the large number of reports that were devoted to the investigation of the dynamic properties of single-mode inhomogeneously broadened (SMIB) lasers, some of their operative characteristics still demand further investigation. In connection with physical understanding of laser instabilities, we demonstrate that the sideband approach is insufficient to explain the distortion that occurs in the (initially) Gaussian profile when the output field undergoes large departures from stable operation. The report focuses on both numerical and analytical handling of the integro-differential 'Maxwell-Bloch' equations. An original approach, presented for the first time, consists in tracking the gain contour when a self-pulsing regime sets in. We demonstrate the occurrence of a lateral hole-burning effect that has never been revealed before. These findings lend themselves to the recognition of the fact that the unstable state initiates lateral saturation (an effect that has always been ignored in the treatment of SMIB laser dynamics). We show that lateral hole-burning cannot be described with the usual small-sideband analysis. In addition, we prove that, to first order, the role of population pulsations is masked by a DC component, distinct from the steady-state value, and responsible for the gain distortion. Leaning on obvious hints related to the occurrence of an amplitude field-switching around a zero mean-value, we derive analytical expressions that give a remarkably good account of the dynamic profiles, obtained likewise by numerical integration of the integro-differential equations.
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.
We discuss a new way of solving the optical Maxwell–Bloch equations governing the interaction of two-level inhomogeneously broadened atoms with a laser by expanding the components of the Bloch vector in power series of the detuning. This method leads to a set of infinitely many coupled ordinary differential equations in which the detuning no longer appears explicitly. These equations are solved numerically, reproducing some analytic solutions reported previously and obtaining a variety of other shape-preserving steady-state solutions. These new solutions, however, are neither factorable nor simply periodic. In this connection we discuss the implications of the factorization assumption, which appears to be the only condition leading to closed-form analytic solutions.
This paper aims at demonstrating that the dynamic gain contour of an inhomogeneously broadened laser tuned to line center undergoes a lateral saturation effect in its unstable regime of operation. Such an effect is shown to be contained in the numerical solutions of the integro-differential Maxwell-Bloch equations that describe the system. After delimiting the domain of applicability of the usual small-sideband approach in describing dynamic gain profiling, we apply a strong harmonic-expansion method to these same equations to obtain an analytical expression that shows an excellent fit with the results of the numerical simulations.