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The Schmidt decomposition is exploited to study the spatial entanglement of laser transverse modes analogous to quantum Lissajous states. Based on the inverse Fourier transform, the stationary Lissajous state can be analytically derived as a coherent superposition of degenerate Hermite–Gaussian eigenmodes. With the derived stationary state, the Schmidt modes and the participation number N can be employed to evaluate the spatial localization and the quantum entanglement. The larger the participation number, the more localized is the stationary coherent state on the Lissajous figure. Moreover, the larger the participation number, the higher is the spatial entanglement.

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Quantum entanglement by a beam splitter (BS) is shown to be analogous to laser mode transformation by an astigmatic mode converter (AMC). Schmidt decomposition is used to characterize the entanglement by an AMC for generating orbital angular momentum and by a BS for creating quantum photon interference. The probability distributions of Schmidt decomposition are calculated to manifest the sameness and difference between AMC and BS in generating entanglement. Finally, the theoretical patterns of mode transformations by an AMC are confirmed with experimental results to validate the present analysis.

The coupled rate equations with the spatial overlap effect for four-level passively Q-switched lasers are fully considered. A transcendental equation is derived for the residual fraction of the inversion density after the finish of the Q-switched pulse. Comprehensive calculations for the transcendental equation were executed to attain an analytical function for precisely fitting the residual fraction of the inversion density. With the fitting function, a pedagogical model with the correction for high output coupling is developed to straightforwardly analyze the output pulse energy and peak power. Detailed experiments are carried out to validate the model.