Effect of end reflectivity on statistical properties of light generated by circular grating DBR laser

Considerable attention has been given recently to the surface-emitting DBR lasers with circular grating. These structures are of interest primarily because they can deliver high-power and low-divergence beam of circular symmetry. They consists of active region and first-order grating DBR resonator surrounded by independent outcoupling grating which focuses emitting light. The grating outcoupler is described by the complex reflectivity at the end of first-order grating responsible for feedback. We present the analytical analysis of the statistical properties of light generated by circular grating DBR laser with integrated focusing outcoupler. Especially, the light coherence is discussed. The coherence properties of laser light are determined by phase and amplitude fluctuations of the electromagnetic field, which are added by the spontaneous emission to the coherent optical field established inside the cavity by stimulated emission. To characterize the influence of the complex end reflectivity on coherence properties of laser light, the longitudinal excess spontaneous emission factor as a consequence of the nonorthogonal nature of the cylindrical laser modes, is included in the stochastic theory based on the Fokker-Planck equation. We derive the exact analytic expressions relating mean laser intensity, intensity fluctuations, and the correlation function of the amplitude that determines the laser linewidth to the real parameters of the circular grating DBR laser. With the help of these relations the laser characteristics revealing the effect of the complex reflectivity introduced by integrated focusing outcoupler are obtained.