We report the emission properties of concentric-circle-grating, surface-emitting (CCGSE), distributed-feedback semiconductor lasers oscillating in single high-order azimuthal spatial modes. The evolution of the spatial profile of the beam emitted by such lasers with increasing excitation depends upon the spatial gain distribution and the depth of the circular waveguide grating. The polarization of the high-order spatial modes is predominantly radial. ©1998 American Institute of Physics.