Nonlinear optics of circular-grating distributed-feedback semiconductor lasers

Abstract

This dissertation investigates the nonlinear optics of circular-grating distributed-feedback (CGDFB) semiconductor lasers. Included are gain saturation, index saturation, and self- and cross-phase modulation third-order nonlinearities. After a brief review of the historical and technical background needed to understand our results, a numerical model is developed for gain saturation. This model includes a radially-varying nonlinear gain and a uniformly-distributed grating loss in the solution of the coupled-mode equations. The results show that lossy, high-power operation results in an optimum coupling strength for efficient conversion of pump power into useful output pourer. Results also show a multi-mode spectrum for large coupling strengths, a consequence of mode selection governed by a spatially-varying gain distribution. Single-mode selection entails operating at approximately the optimum coupling coefficient determined for efficient pumping. These results are extended by including the gain/index coupling described by the linewidth enhancement factor. A unique feature of this coupling is the possibility of above-threshold, single-mode operation over a limited power range, even for the case of large coupling coefficients. Similar results are obtained for the circular-grating distributed-Bragg-reflector (CGDBR) laser. The excess spontaneous emission rate associated with the nonuniform CGDFB radial (longitudinal) field profiles is also calculated. The resulting above-threshold linewidth closely follows the inverse-power dependence predicted by the Schawlow-Townes relation. To include third-order nonlinearities, we derive coupled-mode equations which describe self- and cross-phase modulation effects via an intensity-dependent refractive index. It is then shown that the circular-grating structure acts as an all-optical switch. We also find that an additional pi/2 phase shift at the center of the grating permits the possibility of self-pulsing cylindrical gap solitons. For a positive nonlinearity (n2 it is shown numerically that these solitons are not physically allowable. That is, for a passive structure, time-dependent self-pulsing behavior is damped by the 1/beta r factor in the self- and cross-phase modulation terms. This damping can be compensated for by the addition of gain. In this case, self-pulsing with an excellent contrast ratio is obtained. The numerical methods used to obtain both steady-state and time-dependent solutions are also described. The steady-state results are obtained using a multi-dimensional Newton-Raphson technique known as the "shooting" method. Time-dependent data use a fourth-order predictor-corrector technique. The stability of the time-dependent solutions to the exact coupled-mode equations is reviewed. Coupled-mode equations based on a large-radius approximation for the Hankel functions are found to be stable over a wider range of variables. Numerical tests used to verify the time-dependent software are described.