Stable pulsed operation of Lissajous structured beams by Nd:YVO4/Cr4+:YAG laser in a concave-convex resonator.

A Nd:YVO4/Cr4+:YAG laser with a symmetric concave-convex cavity ensuring strong intracavity beam focusing on the absorber is designed for stable pulsed operation of Lissajous structured modes with transverse patterns as Lissajous figures. Setting the cavity length to fulfill the criterion for efficient passive Q switching (PQS), as well as to meet the accidental degenerate conditions, Lissajous pulsed beams with well-defined structures and good temporal stability are created under two-dimensional off-axis pumping. Although the multi-transverse-mode oscillation inevitably induces asynchronous pulsation and leads the short-term pulse profiles to reveal parasitic effects, the overall long-term behavior of Lissajous pulses can be kept regular with amplitude fluctuations ≤15% and pulse-to-pulse timing jitter ≤5%. With the maximum peak power exceeding 500 W at a pump power of 4.5 W, the PQS Lissajous modes are further transformed into trochoidal pulsed beams to realize high-order and high-peak power structured vortex fields.

High-peak-power structured beams by an Nd:YAG/Cr4+:YAG laser in a near-hemispherical cavity.

An Nd:YAG/Cr4+:YAG passively Q-switched (PQS) laser in a near-hemispherical cavity is exploited to generate high-order structured pulsed fields. Under tightly focused on-axis pumping, radial-order Laguerre-Gaussian (LG) modes with controllable mode orders by the input pump power are realized to exhibit quite stable temporal behavior. The pulse repetition rates of radial-LG modes can reach up to 78 kHz with an average output power of 0.57 W and peak power beyond 300 W under a 5-W pump level. Furthermore, by introducing 1D off-axis pumping into the PQS laser, various structured pulsed fields with transverse morphologies as high-order Ince-Gaussian (IG) modes are further created. With clean and well-defined beam structures, the IG pulsed fields can be nicely reconstructed by the resonant modes of the inhomogeneous Helmholtz equation for spherical cavities. More importantly, these high-order PQS IG modes reveal highly regular pulse trains with the maximum pulse repetition rate beyond 20 kHz and overall peak power higher than 1.5 kW.

Generating high-power Lissajous structured modes and trochoidal vortex beams by an off-axis end-pumped Nd:YVO4 laser with astigmatic transformation.

High-power structured beams with the transverse morphologies as the Lissajous figures are generated by an Nd:YVO4 laser under two-dimensional off-axis pumping. By fine-tuning the cavity length in the neighborhood around the condition of longitudinal-transverse coupling, different cases of accidental degeneracies from the intracavity astigmatism are achieved to lead the output emission to be various Lissajous modes with different transverse frequency ratios. The generated Lissajous modes reveals good power performance with slope efficiency up to 47% and optical-to-optical conversion efficiency to be higher than 37.5% at a pump power of 16 W. Moreover, by applying beam transformation via a single-lens astigmatic mode converter, the generated Lissajous modes are further converted into structured vortex beams with transverse patterns localized on the trochoidal curves. The transformed trochoidal vortex beams are confirmed to preserve well-defined mode structures even when the average output power has been scaled up to be higher than 4 W.

Exploring the formation of thermally detuned transverse patterns in a broad-area square VCSEL.

The formation of thermally detuned transverse patterns of a broad-area square-aperture vertical-cavity surface-emitting laser (VCSEL) via cryogenic cooling is explored. It is found that the transverse wave vector gradually rotates from the horizontal or vertical direction to the diagonal direction of the square boundary as the transverse mode order increases. A model based on the quantum billiards with a finite potential well is developed to emulate the transition behavior of lasing modes. Combining the effective modal gain analysis with the response wave function of driven finite potential billiards, all experimental lasing patterns under different operation temperatures are well reconstructed.

Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter.

The generalized geometric mode with several high-order Hermite-Gaussian (HG) beams localized on ray periodic orbits in the degenerate resonator is generated by an off-axis pumped Nd:YVO4 laser, by performing beam transformation via an astigmatic mode converter, the generalized geometric modes are found that can be converted into the multi-axis vortex beams with the bundled-rings structures. Experimental results reveal that the generated multi-axis vortex beams can preserve quite stable beam structures even under high-power operation. Moreover, the radius of the bundled rings for the multi-axis vortex beams can be flexibly adjusted by the off-axis pumping to lead to vortex structures with easily controlled orbital angular momentum distribution. The good agreement between the experimental and theoretical results of propagation evolution for the astigmatic transformation of generalized geometric modes further verify the feasibility of using the proposed system to realize various high-powered, multi-center vortex beams with good reliability and predictability.

Point-driven modern Chladni figures with symmetry breaking.

Point-driven modern Chladni figures subject to the symmetry breaking are systematically unveiled by developing a theoretical model and making experimental confirmation in the orthotropic brass. The plates with square shape are employed in the exploration based on the property that the orientation-dependent elastic anisotropy can be controlled by cutting the sides with a rotation angle with respect to the characteristic axes of the brass. Experimental results reveal that the orientation symmetry breaking not only causes the redistribution of resonant frequencies but also induces more resonant modes. More intriguingly, the driving position in some of new resonant modes can turn into the nodal point, whereas this position is always the anti-node in the isotropic case. The theoretical model is analytically developed by including a dimensionless parameter to consider the orientation symmetry-breaking effect in a generalized way. It is numerically verified that all experimental resonant frequencies and Chladni patterns can be well reconstructed with the developed model. The good agreement between theoretical calculations and experimental observations confirms the feasibility of using the developed model to analyze the modern Chladni experiment with orientation symmetry breaking. The developed model is believed to offer a powerful tool to build important database of plate resonant modes for the applications of controlling collective motions of micro objects.

Extracting trajectory equations of classical periodic orbits from the quantum eigenmodes in two-dimensional integrable billiards.

The trajectory equations for classical periodic orbits in the equilateral-triangular and circular billiards are systematically extracted from quantum stationary coherent states. The relationship between the phase factors of quantum stationary coherent states and the initial positions of classical periodic orbits is analytically derived. In addition, the stationary coherent states with noncoprime parametric numbers are shown to correspond to the multiple periodic orbits, which cannot be explicable in the one-particle picture. The stationary coherent states are further verified to be linked to the resonant modes that are generally observed in the experimental wave system excited by a localized and unidirectional source. The excellent agreement between the resonant modes and the stationary coherent states not only manifests the importance of classical features in experimental systems but also paves the way to manipulate the mesoscopic wave functions localized on the periodic orbits for applications.

Characterizing classical periodic orbits from quantum Green's functions in two-dimensional integrable systems: Harmonic oscillators and quantum billiards.

A general method is developed to characterize the family of classical periodic orbits from the quantum Green's function for the two-dimensional (2D) integrable systems. A decomposing formula related to the beta function is derived to link the quantum Green's function with the individual classical periodic orbits. The practicality of the developed formula is demonstrated by numerically analyzing the 2D commensurate harmonic oscillators and integrable quantum billiards. Numerical analyses reveal that the emergence of the classical features in quantum Green's functions principally comes from the superposition of the degenerate states for 2D harmonic oscillators. On the other hand, the damping factor in quantum Green's functions plays a critical role to display the classical features in mesoscopic regime for integrable quantum billiards, where the physical function of the damping factor is to lead to the coherent superposition of the nearly degenerate eigenstates.

Modelling end-pumped passively Q-switched Nd-doped crystal lasers: manifestation by a Nd:YVO4/Cr4+:YAG system with a concave-convex resonator.

A theoretical model for the passively Q-switched (PQS) operation which includes the spatial overlapping between the pump and lasing modes under the thermal lensing effect is developed to give a transcendental equation that can directly determine the critical parameters such as pulse energy, pulse repetition rate, and pulse width for the PQS performance. More importantly, an analytical function which gives the approximate solution for the transcendental equation as well as a specific critical criterion for good PQS operation are derived for practical analyses and design. A Nd:YVO4/Cr4+:YAG system with a concave-convex resonator which can achieve fairly stable PQS pulse trains even at a high pump level is further exploited to manifest the proposed spatially dependent model. The good agreement between the experimental results and the theoretical predictions is verified to show the feasibility of the proposed model for designing high-power PQS lasers with high accuracy.

Exploiting concave-convex linear resonators to design end-pumped solid-state lasers with flexible cavity lengths: Application for exploring the self-mode-locked operation.

The characteristics of a convex-concave linear resonator under the thermal lensing effect are theoretically analyzed to find an analytical model for designing end-pumped solid-state lasers with flexible cavity lengths. By exploiting the design model, the power scaling for continuous-wave operation under strong thermal lensing can be easily achieved in the proposed resonator with different cavity lengths. Furthermore, the proposed resonator is applied to explore the exclusive influence of cavity length on the self-mode-locked (SML) operation. It is discovered that the lasing longitudinal modes will split into multiple groups in optical spectrum to lead to a multi-pulse mode-locked temporal state when the cavity length increases. Finally, a theoretical model is derived to reconstruct the experimental results of SML operation to deduce a simple relationship between the group number of lasing modes and the cavity length.

Exploiting broad-area surface emitting lasers to manifest the path-length distributions of finite-potential quantum billiards.

Broad-area vertical-cavity surface-emitting lasers (VCSELs) with different cavity sizes are experimentally exploited to manifest the influence of the finite confinement strength on the path-length distribution of quantum billiards. The subthreshold emission spectra of VCSELs are measured to obtain the path-length distributions by using the Fourier transform. It is verified that the number of the resonant peaks in the path-length distribution decreases with decreasing the confinement strength. Theoretical analyses for finite-potential quantum billiards are numerically performed to confirm that the mesoscopic phenomena of quantum billiards with finite confinement strength can be analogously revealed by using broad-area VCSELs.

High-peak-power optically-pumped AlGaInAs eye-safe laser with a silicon wafer as an output coupler: comparison between the stack cavity and the separate cavity.

An intrinsic silicon wafer is exploited as an output coupler to develop a high-peak-power optically-pumped AlGaInAs laser at 1.52 µm. The gain chip is sandwiched with the diamond heat spreader and the silicon wafer to a stack cavity. It is experimentally confirmed that not only the output stability but also the conversion efficiency are considerably enhanced in comparison with the separate cavity in which the silicon wafer is separated from other components. The average output power obtained with the stack cavity was 2.02 W under 11.5 W average pump power, corresponding to an overall optical-to-optical efficiency of 17.5%; the slope efficiency was 18.6%. The laser operated at 100 kHz repetition rate and the pulse peak power was 0.4 kW.

Cryogenically monolithic self-Raman lasers: observation of single-longitudinal-mode operation.

A cryogenically monolithic Nd:YVO4 self-Raman laser is experimentally explored and theoretically analyzed. The variation of the stimulated Raman scattering (SRS) threshold on the temperature is found to be nonlinear because the reduction of thermal lensing enlarges the cavity mode size. In spite of the nonlinear variation of the SRS threshold on the temperature, the overall SRS output power can be efficiently increased from 0.78 to 1.36 W for temperature decreasing from 285 to 80 K at an absorbed power of 17.2 W. More interestingly, the single-longitudinal-mode operation is experimentally achieved when temperature is lower than 125 K.

24-W cryogenically cooled Nd:YAG monolithic 946-nm laser with a slope efficiency >70.

A high-power efficient monolithic Nd:YAG 946-nm laser is demonstrated at the cryogenic temperature. By exploring the absorption and the fluorescence spectra of the Nd:YAG crystal, it reveals the fact that the absorption bandwidth at 808 nm is narrowing and the fluorescence intensity at 1061 nm is significant enhanced when the temperature is decreased. The temperature dependence of the lasing threshold at 946 nm is found to display a minimum value near a temperature of 170 K. At an incident pump power of 34.5 W, the local heating leads the optimum temperature to be approximately 120 K and the maximum output power can reach 24.4 W with the conversion efficiency of 71% as well as the slope efficiency up to 75%.

Exploring the resonant vibration of thin plates: Reconstruction of Chladni patterns and determination of resonant wave numbers.

The Chladni nodal line patterns and resonant frequencies for a thin plate excited by an electronically controlled mechanical oscillator are experimentally measured. Experimental results reveal that the resonant frequencies can be fairly obtained by means of probing the variation of the effective impedance of the exciter with and without the thin plate. The influence of the extra mass from the central exciter is confirmed to be insignificant in measuring the resonant frequencies of the present system. In the theoretical aspect, the inhomogeneous Helmholtz equation is exploited to derive the response function as a function of the driving wave number for reconstructing experimental Chladni patterns. The resonant wave numbers are theoretically identified with the maximum coupling efficiency as well as the maximum entropy principle. Substituting the theoretical resonant wave numbers into the derived response function, all experimental Chladni patterns can be excellently reconstructed. More importantly, the dispersion relationship for the flexural wave of the vibrating plate can be determined with the experimental resonant frequencies and the theoretical resonant wave numbers. The determined dispersion relationship is confirmed to agree very well with the formula of the Kirchhoff-Love plate theory.

Manifesting the evolution of eigenstates from quantum billiards to singular billiards in the strongly coupled limit with a truncated basis by using RLC networks.

The coupling interaction between the driving source and the RLC network is explored and characterized as the effective impedance. The mathematical form of the derived effective impedance is verified to be identical to the meromorphic function of the singular billiards with a truncated basis. By using the derived impedance function, the resonant modes of the RLC network can be divided into the open-circuit and short-circuit states to manifest the evolution of eigenvalues and eigenstates from closed quantum billiards to the singular billiards with a truncated basis in the strongly coupled limit. The substantial differences of the wave patterns between the uncoupled and strongly coupled eigenmodes in the two-dimensional wave systems can be clearly revealed with the RLC network. Finally, the short-circuit resonant states are exploited to confirm that the experimental Chladni nodal-line patterns in the vibrating plate are the resonant modes subject to the strong coupling between the oscillation system and the driving source.

Exploring the influence of boundary shapes on emission angular distributions and polarization states of broad-area vertical-cavity surface-emitting lasers.

We design the stadium-shaped and rectangular vertical-cavity surface-emitting lasers (VCSELs) to investigate the influence of boundary shapes on the emission angular distributions and polarization states. For the stadium-shaped VCSELs, the emission angular distribution prefers to be almost omnidirectional because the lasing mode with purely scarred structure is seldom to be excited. On the contrary, the rectangular VCSELs usually generate dominant lasing modes with the morphology of quasi-periodic linear ridges, which can make emission angular distribution to be concentrated on the certain direction. From the polarization-resolved light-current curves, the stadium-shaped VCSEL is quite prone to exhibit numerous abrupt changes (kinks) associated with polarization switching with increasing current, whereas for rectangular VCSEL there is no conspicuous kink to be seen during a wide range of current changing from near to far above lasing threshold.

Exploring the distinction between experimental resonant modes and theoretical eigenmodes: from vibrating plates to laser cavities.

Experimentally resonant modes are commonly presumed to correspond to eigenmodes in the same bounded domain. However, the one-to-one correspondence between theoretical eigenmodes and experimental observations is never reached. Theoretically, eigenmodes in numerous classical and quantum systems are the solutions of the homogeneous Helmholtz equation, whereas resonant modes should be solved from the inhomogeneous Helmholtz equation. In the present paper we employ the eigenmode expansion method to derive the wave functions for manifesting the distinction between eigenmodes and resonant modes. The derived wave functions are successfully used to reconstruct a variety of experimental results including Chladni figures generated from the vibrating plate, resonant patterns excited from microwave cavities, and lasing modes emitted from the vertical cavity.

Generation of pseudonondiffracting optical beams with superlattice structures.

We demonstrate an approach to generate a class of pseudonondiffracting optical beams with the transverse shapes related to the superlattice structures. For constructing the superlattice waves, we consider a coherent superposition of two identical lattice waves with a specific relative angle in the azimuthal direction. We theoretically derive the general conditions of the relative angles for superlattice waves. In the experiment, a mask with multiple apertures which fulfill the conditions for superlattice structures is utilized to generate the pseudonondiffracting superlattice beams. With the analytical wave functions and experimental patterns, the pseudonondiffracting optical beams with a variety of structures can be generated systematically.

Level statistics and eigenfunctions of square torus billiards: manifesting the transition from regular to chaotic behaviors.

We thoroughly analyze the level statistics and eigenfunctions in concentric as well as nonconcentric square torus billiards. We confirm the characteristics of quantum and classical correspondence and the existence of scarred and superscarred modes in concentric square torus billiards. Furthermore, we not only verify that the transition from regular to chaotic behaviors can be manifested in nonconcentric square torus billiards, but also develop an analytical distribution to excellently fit the numerical level statistics. Finally, we intriguingly observe that numerous eigenstates commonly exhibit the wave patterns to be an ensemble of classical diamond trajectories, as the effective wavelengths are considerably shorter than the size of internal hole.