2 Experimental setup

A schematic diagram of the high-energy HG alexandrite laser is shown in Figure 1. The pump laser is a laser diode (LD) with an emission wavelength of 638 nm, offering higher quantum efficiency compared to other pump wavelengths. Owing to the polarization-dependent absorption properties of alexandrite, the pump laser is directed through a polarization beam splitter to enhance the pumping efficiency. The coupling system consists of a collimating lens (CL) and a focusing lens (FL), with focal lengths of 100 and 50 mm, respectively. The pump laser is focused through a coupling system, generating a circular spot with a diameter of 200 μm within the crystal. Both the input coupling and the output coupling are implemented using plane mirrors to maintain the beam quality of the output laser. The alexandrite crystal, with a Cr³⁺ doping concentration of 0.2%, was precision-cut to dimensions of 3 mm × 3 mm × 8 mm and finely polished. The optical path surfaces are coated with a high transmittance for a 755 nm laser. The transmittance of the output coupler (OC) is fixed at 3% and defect stripes of diverse widths and shapes were introduced, as shown in Figure 1, in order to permit the laser to pass through without being reflected, such that only the laser modes that overlap with the no-defect area can be effectively amplified. The output laser is directed through a bandpass filter to obtain high-purity HG modes, free from interference with the pump laser. The HfTe₂ absorber was coated onto the OC to achieve Q-switching and to minimize resonant cavity loss.

Figure 2 (a) Light transmittance of HfTe₂ in the wavelength range of 350–1100 nm. Inset: energy band structure and density of states of HfTe₂. (b) Saturation absorption and reverse saturation absorption results of HfTe₂ based on an open aperture Z-scan. Inset: the principle of saturation absorption and reverse saturation absorption process. (c) Surface microstructure of HfTe₂ by SEM. (d) Thickness of single-layer HfTe₂ nanosheets by AFM. Inset: the AFM height image in the area of 30 μm × 30 μm. (e) Raman spectrum of HfTe₂. Inset: EDS element analysis. (f) XRD pattern of HfTe₂.

The optical transmittance of HfTe₂ across the wavelength range of 350–1100 nm is shown in Figure 2(a), demonstrating its strong absorption of visible and near-infrared light. The band energy structure and electron density of HfTe₂ are depicted in the inset of Figure 2(a), revealing its zero-bandgap structure. Upon incidence of a laser on HfTe₂, electrons in the pristine state are excited to the excited state (Absorption I). Increasing the incident laser power causes a greater number of electrons to populate the excited state. Due to Pauli blocking, which limits the electron occupation in this state, Absorption I reaches saturation and the material ceases to absorb additional laser energy. However, under excitation from a stronger laser, electrons in the excited state will transition to a higher energy state (Absorption II). This process is known as reverse saturation absorption. With further increases in incident laser intensity, the higher energy state may also become saturated as a result of electron accumulation and the wide energy bandgap, which impedes the relaxation process from the higher state back to the excited state, thereby ceasing further laser absorption. The characteristics of saturable absorption and reverse saturation absorption under high-energy excitation are demonstrated in Figure 2(b) via the open aperture Z-scan method with an 800 nm femtosecond laser, confirming that HfTe₂ exhibits reverse saturation absorption properties. Figure 2(c) presents the surface microstructure of HfTe₂, as examined by scanning electron microscopy (SEM), clearly showing its layered structure. The height of a single-layer HfTe₂ nanosheet obtained by mechanical exfoliation, and measured by atomic force microscopy (AFM), was 16 nm, as shown in Figure 2(d). The purity of HfTe₂ was analyzed by Raman spectroscopy and an energy dispersive spectrometer (EDS), as shown in Figure 2(e) and the inset. The peaks of the Raman spectrum are located at 123.3, 142.0 and 401.3 cm^–1, corresponding to the characteristic Raman peaks of HfTe₂. The EDS results indicate that the composition consists of 35.6% Hf and 64.4% Te, with no other elements detected. The growth direction of HfTe₂ was characterized by X-ray diffraction (XRD), with diffraction peaks observed at 2θ angles of 13.4°, 26.8°, 40.8° and 55.4°, corresponding to the (1, 0, –1), (2, –1, –1), (2, 0, –2) and (3, 0, –3) growth directions, respectively, as shown in Figure 2(f).

Figure 3 (a) The far-field intensity distribution of Hermite–Gaussian lasers of different orders and the vortex laser. (b) The corresponding defect areas on the output coupling mirror.

Figure 4 (a) Output characteristics of high-energy HG₁₀, HG₂₀ laser and vortex laser. Transition state pulse of the HG₁₀ laser at (b) 10 μs; (c) 5 ms; (d) 200 μs scales.

3 Characteristics of the high-energy structured light field

The far-field intensity patterns of HG lasers of different orders and their corresponding defects on the OC are shown, respectively, in Figures 3(a) and 3(b). The defects, introduced by mechanical etching, allow the laser to pass through without undergoing oscillation. The etched defects are placed on a red-light source and observed under a microscope. The red light can pass through the defects, appearing as bright areas, while the no-defect regions remain dark due to reflection from the film, as illustrated in Figure 3(b). The intensity distribution of the HG laser is recorded by a charge-coupled device (CCD) at a distance of 5 m in the far-field, with the corresponding laser divergence angle being less than 5 mrad. With etched straight lines with a width of 20 μm in the vertical and horizontal directions, the laser is capable of generating HG₁₀ and HG₀₁ modes, respectively. As the defect width is increased to 60 μm, HG₂₀ and HG₀₂ modes can be excited. Through the orthogonal cross-etching of two defect stripes, the laser is optimized to produce the HG₁₁ mode. In addition, by etching a circular hole in the OC, the laser can produce a ring-shaped vortex laser. However, due to spot size limitations, wider defects inhibit efficient laser oscillation, preventing the generation of higher order HG modes.

Figure 5 (a) Spectrum. (b) Repetition rate. (c) Pulse width. (d) Pulse energy. (e) Peak power of the HG₁₀ mode versus absorbed pump power at the low-repetition-rate state. (f) Typical Q-switched pulse train and (g) temporal pulse shape of the HG₁₀ mode at maximum average output power.

The output pulse characteristics of the HG and vortex lasers at different modes are shown in Figure 4. Since the HG₁₀ and HG₀₁, HG₂₀ and HG₀₂, as well as HG₁₁ and vortex modes exhibit similar output power, repetition rate and pulse width, this work only presents the laser characteristics of the HG₁₀, HG₂₀ and vortex modes. The vortex laser operates in two stable states characterized by a high-repetition-rate state (>10 kHz) and a low-repetition-rate state (<300 Hz), as well as an unstable transition state, as described in Figure 4(a). The threshold absorbed pump power is 3.0 W, and it operates in the high-repetition-rate state until the absorbed pump power exceeds 6.0 W. As the absorbed pump power increases, the repetition rate increases and the pulse width decreases gradually within the high-repetition-rate state. For the HG₁₀ mode, the maximum repetition rate reaches 44 kHz with a pulse width of 1.9 μs at 6.0 W of absorbed pump power, as shown in Figure 4(a). As the absorbed pump power exceeds 6.0 W, Absorption I reaches a supersaturated state, resulting in high-repetition-rate pulses, and the laser enters an unstable transition state. During the transition state, while emitting high-energy, high-repetition-rate pulses, Absorption II also consumes the excited state electrons, causing Absorption I to no longer remain saturated. Consequently, the high-repetition-rate pulses reappear, as shown in Figure 4(a). As the pump power increases, the excited state is populated more rapidly, effectively sustaining a continuous saturation state, which results in the disappearance of the high-repetition-rate pulses. The low-repetition-rate pulse interval in the transition state is 24 ms, corresponding to a repetition rate of 42 Hz, with a pulse width of 820 ns, as depicted in Figure 4(b). The high-repetition-rate pulses in this state have a repetition rate of 31 kHz and a pulse width of 2.6 μs. Notably, the repetition rate of the high-repetition-rate pulses in the unstable transition state is lower than that in the high-repetition-rate state. This reduction is due to Absorption II decreasing the transmittance of the SA during the modulation of the high-repetition-rate pulse. The repetition rate of the low-repetition-rate pulse in the unstable transition state is also lower than that in the low-repetition-rate state, because the high-repetition-rate pulse consumes electrons in the excited state. When the absorbed pump power exceeds 6.2 W, Absorption I remains in a supersaturated state, and pulse generation depends exclusively on modulation by Absorption II, leading to low-repetition-rate pulses. Figure 4(a) illustrates a typical low-repetition-rate pulse with a repetition rate of 177 Hz and a pulse width of 480 ns, demonstrating superior stability in both pulse repetition and intensity compared to the high-repetition-rate pulses.

Figure 6 (a) Repetition rate. (b) Pulse energy. (c) Pulse width. (d) Peak power of HG₂₀ and vortex mode laser versus absorbed pump power.

The low-repetition-rate state pulse laser characteristics of the HG₁₀ mode are presented in Figure 5. Figure 5(a) shows the spectrum of the Q-switched pulse laser, with a central wavelength at 762 nm and a full width at half maximum (FWHM) of 3 nm at the maximum absorbed pump power of 8.15 W. As the absorbed pump power increases, the repetition rate increases while the pulse width decreases gradually. When the absorbed pump power was 8.15 W, the maximum repetition rate and the narrowest pulse width were 262 Hz and 454 ns, respectively, as depicted in Figures 5(f) and 5(g). The single pulse energy and peak power also increased gradually with the absorbed pump power, and the maximum pulse energy reached 5.85 mJ, corresponding to a peak power of 11.6 kW. The single pulse energy density and peak power density in the gain medium were 12.2 J/mm² and 24.3 MW/mm² _, respectively. Notably, the performance of the HfTe₂ SA is not compromised by the impact of such a high-energy pulse. Typical pulse trains and temporal pulse profile are displayed in Figures 5(f) and 5(g) in the time ranges of 0.5 s and 4 μs. Figure 5(f) demonstrates that the pulse train has good repeatability and relatively consistent peak intensity. The remarkable pulse stability originates from two key aspects. Firstly, the high saturation intensity of the excited state absorption process increases the cavity loss, which promotes the generation of stable pulses and provides substantial fault tolerance. Secondly, the intrinsic stability of the material structure combined with its well-defined energy level structure ensures consistent laser performance.

Table 1 Comparative characteristics of Q-switched high-order HG and vortex mode lasers.

Figure 7 (a) Schematic diagram of wavelength-tunable Q-switched vortex laser based on an etalon. (b) Fluorescence spectrum of alexandrite excited by a 638 nm pump laser. (c) Laser spectra and relative intensities at different wavelengths within the alexandrite wavelength tuning range. (d) Repetition rate and pulse width. (e) Pulse energy and peak power of passively Q-switched vortex laser versus wavelength.

The laser characteristics of the HG₂₀ and vortex modes are shown in Figure 6. Figure 6(a) illustrates the variation of the repetition rate with absorbed pump power in both the high- and low-repetition-rate states. In the high-repetition-rate state, the highest repetition rates were 24 kHz for the HG₂₀ mode and 28 kHz for the vortex mode, respectively, while these rates dropped sharply to 185 and 196 Hz in the low-repetition-rate mode. The corresponding pulse energy also increased from 24.8 μJ to 6.92 mJ for the HG₂₀ mode and from 24.6 μJ to 7.40 mJ for the vortex mode, respectively, as depicted in Figure 6(b), proving that Absorption II has a higher saturation flux and can accommodate more electrons due to its wider energy band. As shown in Figure 6(c), the pulse width of both the HG₂₀ and vortex modes decreased slowly with increasing absorbed pump power, even across different states. The narrowest pulse widths reached 550 ns for the HG₂₀ mode and 502 ns for the vortex mode at the maximum absorbed pump power of 8.15 W.

Figure 8 (a) Typical intensity profile of the high-energy vortex pulse laser. (b) Mach–Zehnder interferometer for characterizing the topological charge of the optical vortex. (c) Beam profile interference patterns for the vortex laser. (d) Experimentally measured transverse intensity profile and its theoretical fit.

Table 1 summarizes the characteristics of typical Q-switched high-order HG and vortex lasers. It can be seen that our experimental approach yields the highest pulse energy compared with other gain media and modulation methods. Nonetheless, it is particularly appropriate for the generation of high-energy HG pulses at short wavelengths.

4 Wavelength tuning characteristics

Since a 750–785 nm laser corresponds to the absorption peak of Rb atoms, and has a larger penetration depth in biological tissues, this wavelength range has a wide range of applications including laser medical treatment, cold atoms and laser pumping^[ Reference Huang, Su, Lin, Chiu and Huang^41– Reference Maiwald, Eppich, Ginolas, Sumpf, Erbert and Tränkle^50]. In particular, the 780 nm vortex laser has indispensable applications, such as cold atomic clocks, hot atomic steam and cold atom storage, but it can only be produced through extra-cavity modulation^[ Reference Micalizio, Levi, Calosso, Gozzelino and Godone^42, Reference Wang, Lee, Kim and Kim^43]. To expand the utility of high-energy HG and vortex lasers, an etalon was employed for wavelength tuning based on an alexandrite crystal, which has a wide emission spectrum, as shown in Figure 7(b).

The schematic diagram of a wavelength-tunable Q-switched vortex alexandrite laser is depicted in Figure 7(a). By utilizing a 0.2 mm thick etalon rotated within a tilt angle of 10°, continuous wavelength tuning can be achieved across the range of 747–787 nm. The relative intensity and FWHM at different wavelengths are shown in Figure 7(c). The FWHM of the spectrum is 3 nm at a central wavelength of 763 nm, while it narrows to 1 nm at the wavelengths of 747 and 787 nm due to the optical filtering effect of the etalon. In addition, Figure 7(d) presents the corresponding repetition rates and pulse widths at various wavelengths at an absorbed pump power of 8.15 W. As the intra-cavity power diminishes, the repetition rate decreases and the pulse width increases. The repetition rate and pulse width of the 747 nm pulsed vortex laser were 134 Hz and 708 ns, respectively, while those of the 787 nm pulsed vortex laser were 147 Hz and 760 ns, respectively. The pulse energy and peak power show a dependence on the gain during the process of wavelength tuning. As shown in Figure 7(e), the pulse energy and peak power of the 747 nm pulsed vortex laser were 6.15 mJ and 8.69 kW, respectively. Meanwhile, for the 787 nm pulsed vortex laser, the pulse energy and peak power were 6.42 mJ and 8.44 kW, respectively. The wavelength tuning range was restricted by the coating constraints of the resonant cavity mirrors.

As illustrated in Figure 8(a), the typical intensity profile of the high-energy vortex pulse laser displays a distinctive annular light intensity pattern. To characterize the phase distribution and OAM carried by the output vortex laser, a custom-built Mach–Zehnder interferometer device was employed. The output intensity distribution was collected and analyzed using a CCD, as depicted in Figure 8(b). Figure 8(e) shows the interference pattern of the vortex laser. The typical fork-shape stripe indicates that the vortex laser has a negative helical direction with a topological charge of –1. To verify the purity of the vortex mode laser, the transverse intensity profile of the vortex mode along the x-axis was measured, as shown in Figure 8(d). The experimentally measured intensity curve is theoretically fitted by using a first-order vortex function

$$\begin{align*}\frac{I}{I_0}=A\frac{2{r}^2}{\omega_0^2}\exp \left(-\frac{r^2}{\omega_0^2}\right), \end{align*}$$

where $\frac{I}{I_0}$ is the normalized intensity, A is a constant and ω₀ is the beam waist. The transverse intensity profile closely matches the theoretical fit based on the first-order vortex function, indicating excellent mode purity.