The selective pumping and the spatial hole burning are thoroughly considered to determine the excited transverse and longitudinal eigenmodes. By using the mode locking of the excited eigenmodes, an analytical wave representation is derived to manifest the ray-wave duality of the optical wave packet dynamics in the laser resonator. The derived wave function is employed to extract the parametric formula for the periodic ray orbits that reveal the subtle origin between the hyperbolic caustics and linear trajectories. Moreover, the temporal dynamics of the output beam for the total mode-locked state is theoretically demonstrated by combining with the parametric formula and the derived wave function. More importantly, an experiment based on the self-mode-locked laser is performed to make a comparison with theoretical analysis.
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