Effects of radiation on circular-grating DFB lasers. I. Coupled-mode equations

We derive near-threshold coupled-mode equations that include the effects of radiation for TE fields in circular-grating surface-emitting distributed feedback (DFB) lasers with second-order gratings. The analysis uses an exact description of the periodic grating that is valid for any grating shape and depth. For low-order azimuthal modes, approximate equations are found that are the same as those for a linear DFB laser, though the boundary conditions differ. We solve the coupled-mode equations for a weakly guiding laser numerically for low- and higher-order radial and azimuthal modes and compare these numerical results to previous experimental observations and to the results found for linear gratings. For sufficiently strong coupling, radiative losses have a significant impact on the mode structure of the laser, breaking the radial (longitudinal) mode symmetry otherwise seen for the lowest order azimuthal modes and improving mode selectivity