The excitation of high-order Laguerre–Gaussian (LG) modes in a neodymium-doped yttrium aluminum garnet (Nd:YAG) laser resonator was presented by applying the diffraction of a second-order circular Dammann grating (CDG) for annular pumping. In the study, the 808 nm pump light was shaped into a double-ring structure by the CDG and matched spatially with that of an ideal LG₁₁ mode. As a result, the laser resonator generated an LG₁₁ vortex mode, and the laser power reached 204 mW with 14.5% slope efficiency. Also, when the cavity mirror was tilted, the laser output could switch to the LG₀₁ vortex mode. The results showed the convenience and versatility of CDG in an annular-pumped vortex laser.