We employ the inhomogeneous Helmholtz equation to explore the influence of the fractional degeneracy and the pump distdbution on the resonant lasing mode. Theoretical analyses clearly reveal the relationship between the fractional degeneracy and the emergence of the ray-wave duality. Furthermore, we perform thorough laser experiments to confirm the theoretical exploration that the resonant modes near the degenerate cavities are well localized on the ray trajectories under the condition of the off-axis pumping. We also exploit the dedved wave functions to calculate the resonant strengths that can noticeably manifest the enhancements of the output powers in the degenerate cavities.

It is theoretically demonstrated that the planar geometric mode with a π/2 mode converter, so called the circularly geometric mode, can be solved from the inhomogeneous Helmholtz equation by considering the pump distribution on the lasing mode. Theoretical analysis clearly reveal that the vortex structures of circularly geometric modes are determined by the minimum order of transverse lasing modes, the total number of transverse lasing modes and the degenerate condition in the cavity. Moreover, we experimentally manifest that the circularly geometric mode can be generated from the selective pumped solid-state laser with an external π/2 mode converter. To explore the vortex structures of the generated geometric modes, the interference patterns are performed by an experimental apparatus consisting of a Mach-Zehnder interferometer. The good agreement between experimental observations and numerical calculations confirms the analysis of vortex structures is reliable.

An end-pumped Nd:YVO laser under selective pumping is used to excite lasing modes with transverse patterns performed to exhibit the characteristics of multiple spots arranged on elliptical features near degenerate cavities. The spatial distribution of elliptical lasing modes is clearly revealed to be localized on the nonplanar ray orbits, so-called nonplanar elliptical modes, which possess large fractional orbital angular momentum. Moreover, temporal dynamics for the output emission of nonplanar elliptical modes are verified to obtain self-mode-locked operation. We further numerically manifest not only the influence of radial-asymmetry distributions on the vortex structures of nonplanar elliptical modes, but also the vector field of transverse lasing modes altered with twisting phase structures in the propagation direction.

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We report for the first time on simultaneously longitudinal and transverse self-mode locking in a diode-pumped Nd:YVO 4 laser to achieve a pulse train traveling along the zigzag multi-pass trajectories in the degenerate cavity. At a pump power of 2.5 W, the average output powers in the mode-locked operation are 490 mW and 520 mW for the M-mode and Z-mode, respectively. The pulse widths are measured to be approximately 22.2 and 21.1 ps for the M-mode and Z-mode, respectively.

We theoretically verify that the symmetry breaking in spherical resonators can result in a fractal frequency spectrum that is full of numerous new accidental degeneracies to cluster around the unperturbed degenerate cavity. We further experimentally discover that the fractal frequency spectrum excellently reflects the intimate connection between the emission power and the degenerate mode numbers. It is observed that the wave distributions of lasing modes at the accidental degeneracies are strongly concentrated on three-dimensional (3D) geometric topology. Considering the overlapping effect, the wave representation of the coherent states is analytically derived to manifest the observed 3D geometric surfaces.