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 frequency degeneracy induced by the astigmatism in a nearly hemispherical cavity is originally exploited to generate vortex array laser modes with the output power up to 300 mW. The inhomogeneous Helmholtz equation is employed to derive the wave function for manifesting the characteristics of the lasing modes. The theoretical wave function explicitly reveals the role of the Gouy phase in the formation of vortex arrays. Numerical analyses are further performed to confirm that the thermal lensing effect in the laser crystal assists the lasing transverse order to increase with increasing pump power. It is believed that the high efficiency enables the present laser modes to be useful in the applications of structured vortex beams.

The distributions of resonant frequencies in an astigmatic cavity are theoretically confirmed to be analogously equivalent to the quantum energy structures of two-dimensional commensurate harmonic oscillators. In the first part [Opt. Lett. 45, 4096 (2020)OPLEDP0146-959210.1364/OL.399251] of this two-part series study, the lasing modes were verified to reveal a variety of vortex array structures. Here, in the second part of this two-part series study, the lasing modes are confirmed to agree very well with the quantum Green’s functions that correspond to a bundle of Lissajous figures in the high-order regime.

Based on the birefringent effect of the gain medium, a diode-pumped Nd-doped vanadate laser with nearly hemispherical cavity is exploited to emulate the quantum Green functions of two-dimensional commensurate harmonic oscillators. By matching the theoretical calculations to the far-field patterns of lasing modes, the resonant transverse frequencies can be accurately determined up to extremely high orders. The Shannon entropy is further employed to calculate the spatial entanglement of the quantum Green function as a function the transverse frequency. From the resonant transverse frequencies, all lasing modes are confirmed to be in excellent agreement with the maximum entropy states. This discovery implies that the formation of lasing modes is relevant to the coupling interaction between the pump source and the laser cavity.