NONLINEAR TRANSVERSE-MODES OF LARGE-ASPECT-RATIO HOMOGENEOUSLY BROADENED LASERS .2. PATTERN-ANALYSIS NEAR AND BEYOND THRESHOLD

Complex order-parameter equation descriptions of pattern evolution in large-aspect-ratio two-level and Raman lasers are derived systematical ly as solvability conditions in a multiple-scales asymptotic expansion of the original Maxwell-Bloch laser equations in powers of a small pa rameter. These amplitude equations, although strictly valid near thres hold for lasing, are shown to capture the essential features of patter n instability and evolution well beyond lasing threshold. A technical difficulty that can arise in the Raman laser, namely, subcriticality o f the bifurcation near the critical wave number, is not addressed in t he present paper and the order-parameter equations as derived are vali d only when this situation does not arise. Analytical expressions for long-wavelength phase instabilities of the underlying traveling-wave p attern, which appears as the natural nonlinear lasing mode when the de tuning 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 equation s, the complex order-parameter equations and the phase equation, are s hown to be consistent for all cases studied with the exception of the case when a subcritical bifurcation approaches the critical wave numbe r k(c).