ABSTRACT
Orbital angular momentum densities in the astigmatic transformation of Lissajous geometric laser modes are originally and systematically investigated. The quantum theory of the coherent state is exploited to derive an analytical wave representation for the transformed output beams. The derived wave function is further employed to numerically analyze the propagation dependent orbital angular momentum densities. The parts of the negative and positive regions in the orbital angular momentum density are found to rapidly change in the Rayleigh range behind the transformation.
ABSTRACT
The propagation-dependent spatial entanglement for the structured laser beams generated by an arbitrary incident Hermite-Gaussian (HG) mode passing through an astigmatic mode converter (AMC) is theoretically explored. The structured output beams are analytically decomposed into the expansion of HG modes for any given rotation angle of the AMC. Based on the Schmidt decomposition, the propagation-dependent spatial entanglements of the structured output modes are quantified with the von Neumann entropy. To manifest the propagation-dependent entropy, the probability distribution of the expanded HG modes in the structured output beam is quantitatively analyzed.
ABSTRACT
Lissajous structured beams emerging from a spherical laser cavity subject to the birefringent effect of the laser crystal are quantitatively analyzed. The analysis reveals that the birefringent effect leads to numerous frequency degeneracies at the cavity lengths near an ideal degenerate cavity. By using a diode-pumped Nd:GdVO4 laser, the emergence of Lissajous structured modes relevant to frequency degeneracies is precisely quantified by comparing experimental results with numerical analyses. The present quantitative analysis provides an important guideline for the generation of structured transverse modes related to the ray-wave correspondence.
ABSTRACT
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.
ABSTRACT
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.
ABSTRACT
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.
ABSTRACT
In this Letter, we experimentally explore the propagation-dependent evolution of generating the pseudo-nondiffracting quasi-crystalline (crystalline) beams based on the multibeam interference. We originally derived an analytical formula to exactly manifest the propagation evolution of interfering multiple beams. With the analytical formula, the formation of quasi-crystalline structures in the focal plane can be explicitly verified. Furthermore, the distance of the effective propagation-invariant region can be verified in terms of experimental parameters. More importantly, we employed the developed formula to confirm the formation of kaleidoscopic vortex lattices by means of numerically computing the propagation-dependent phase singularities.
ABSTRACT
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.
ABSTRACT
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.
ABSTRACT
The integral representation of the Zernike radial functions is well approximated by applying the Riemann sums with a surprisingly rapid convergence. The errors of the Riemann sums are found to averagely be not exceed 3 ×10-14, 3.3×10-14, and 1.8×10-13 for the radial order up to 30, 50, and 100, respectively. Moreover, a parallel algorithm based on the Riemann sums is proposed to directly generate a set of radial functions. With the aid of the graphics processing units (GPUs), the algorithm shows an acceleration ratio up to 200-fold over the traditional CPU computation. The fast generation for a set of Zernike radial polynomials is expected to be valuable in further applications, such as the aberration analysis and the pattern recognition.
ABSTRACT
The parametric equations for periodic rays in degenerate astigmatic laser resonators are derived in a generalized way. The derived parametric equations clearly reveal the formation of Lissajous geometric modes with phase variation along the propagation. Using the representation of a quantum coherent state, a generalized wave-packet formula is derived to connect with the periodic rays of geometric modes. The derived wave-packet formula is not only validated through comparing with various experimental patterns, but also is directly exploited to manifest the spatial asymmetries of Lissajous lasing modes due to a highly transmissive output coupler.
ABSTRACT
An analytical wave representation is derived to express the asymmetric elliptic vortex beams generated by using an astigmatic mode converter to transform the asymmetric Hermite-Gaussian modes. The effect of the spatial asymmetry on the elliptic vortex beam can be straightforwardly considered in the analytical wave representation. A detailed comparison with experimental intensity patterns and phase structures is made. The good agreement between experimental results and theoretical analyses confirms the characterization and generation of the asymmetric elliptic vortex beams.
ABSTRACT
A general integral formula is analytically derived to represent the wave function of elliptical vortex modes. The derived formula can be straightforwardly employed to take account of spatial damping to explore vortex structure. Spatial damping is found to cause the central degenerate singularity of the circular vortex mode to be split into several singularities with different topological charges. For the non-circular vortex beam, the influence of spatial damping on the distribution of singularities is not significant. Theoretical analyses are confirmed to agree very well with the experimental measurements.
ABSTRACT
The transverse structures of geometric modes in degenerate laser resonators with large astigmatism are systematically investigated by starting from the Gaussian wave packet formulation with ray-wave connections. The overlapping degree D between Gaussian wave packets is derived as an analytic expression that manifests the critical overlapping with D=1. The transverse patterns are confirmed to display a structure of continuous curves or dotted spots, depending on D>1 or D<1, respectively. A thorough comparison between experimental results and numerical calculations is performed to validate the theoretical analysis.
ABSTRACT
We employ an off-axis pumped Nd:YVO4 laser and control the reflectance of output coupler to directly generate asymmetrical Hermite-Gaussian (HG) modes with various transverse orders. By using an astigmatic mode converter, the generated asymmetrical HG modes are straightforwardly transformed into asymmetrical Laguerre-Gaussian (LG) modes with a crescent-like shape. The average output power of all the crescent-shaped LG modes can exceed 1W at the pump power of 4W. Furthermore, experimental results are theoretically verified by resonant modes derived from the inhomogeneous Helmholtz equation with the localized source distribution. Theoretical resonant modes are also used to explore the dependence of the phase structures of LG modes on the system loss. As the loss increases, it is found that the singularities at the origin will be rearranged and new singularities are formed in the outside region with average orbital angular momentum remaining unchanged.