Optical vortex laser beams possess a phase singularity and a helical wavefront that twists in the beam’s direction of propagation. Every photon of a vortex beam carries orbital angular momentum (OAM) of $\ell h$, which can be expressed as a phase factor $\exp (i\ell \phi )$ (where $\phi$ is an azimuthal angle).1^–3 The unique characteristics of these beams have enabled new techniques in fields including optical tweezing,4^,5 optical manipulation,6^,7 optical communication,8 nano/microfabrication of helical structured materials,9 and spectroscopy.10^,11 Laguerre–Gaussian (LG) modes are eigensolutions of the Helmholtz equation under cylindrical symmetry, which can be directly generated from a laser cavity. LG modes are examples of vortex beams. High-order LG modes that have spatial profiles with petal-like features across their azimuth are an important subset of LG modes. These modes have garnered significant attention for their use in manipulation of microparticles,12^,13 spatial measurement,14 and quantum communication.15 Therefore, significant research effort has focused on developing methods whereby high-order LG modes with petal-like spatial profiles that can be selectively generated from a laser system.

One approach for generating these types of LG modes is the coherent superposition of two optical vortex modes with opposite azimuthal order.16 These modes can be generated directly within a laser cavity by controlling the gain/loss of specific cavity modes using intracavity phase modulating elements17^–19 and mode-selection approaches.20^,21 High-order LG modes with petal-like spatial profiles can also be generated by pumping a laser cavity with a ring-shaped pump beam that is slightly offset from the axis of the laser cavity. This method selectively promotes mode overlap between LG modes and the ring-shaped pump beam.22^,23 Another approach is to make use of defect spots and patterns within the laser cavity to force the oscillation of high-order LG modes.24 More recently, researchers have used spherical aberration from a spherical lens to generate ultrahigh-order LG beams with $\ell$ order beyond 300 from an end-pump Nd: ${\mathrm{YVO}}_4$ laser at 1064 nm; this is the highest-order LG mode with a petal-like spatial profile demonstrated from an experimental laser system.25^,26

Nonlinear optical processes, such as stimulated Raman scattering, can be used to induce the generation of high-order LG modes.27^,28 It should be noted that there have been rather few reports of high-order LG mode generation in the mid-infrared wavelength region. The capacity to generate such modes in this wavelength range would be significant and have potential utility in next-generation optical trapping, quantum optics, and materials processing applications.

Optical parametric oscillation (OPO) is a nonlinear process, which has proven effective for the generation of optical vortex beams in the near-infrared and mid-infrared wavelength ranges.29^–34 There have however, been few reports concerning the generation of high-order LG petal-like modes in the mid-infrared wavelength range using OPOs. Recently, our group has demonstrated the generation of an off-axis vortex beam with noninteger OAM states from an OPO with a half-spherical cavity, resulting from the coherent superposition of Gaussian and vortex modes inside the cavity.35

In this paper, we report the first demonstration (to the best of our knowledge) of the generation of OAM-tunable high-order LG petal-like modes in the mid-infrared region from an idler-resonant ${\mathrm{KTiOAsO}}_4$ (KTA)-OPO pumped by a nanosecond optical vortex source (${\ell }_p=1-3$). This cavity enables the selective generation of high-order ${\mathrm{LG}}_{0,\pm \ell }$ ($\ell =0-10$) petal-like modes (with controllable OAM) in the mid-infrared region by adjusting the cavity length. Our experimental work is supported by theoretical modeling that analyzes the spatial overlap efficiency of the beam waists of the pump and resonant idler fields within the center of the KTA crystal.

The experimental setup of the mid-infrared OPO generating high-order LG modes with petal-like spatial profiles is shown in Fig. 1. A flash lamp pumped Q-switched Nd:YAG laser that had a Gaussian output beam, pulse width of 25 ns, pulse repetition frequency of 50 Hz, and emission wavelength of 1064 nm was used as the pump source. A half-wave plate, thin-film polarizer, and Faraday rotator (isolator) were used to control the power of the pump beam and prevent backreflection into the laser cavity. Multiple spiral phase plates (SPPs) (azimuthally divided into 16 parts with an $n\pi /8$ phase shift) were combined to convert the pump beam into an optical vortex beam. Using different arrangements of SPP, the topological charge $\ell$ of the pump beam could be changed from 1 to 3. A half-wave plate was used to adjust the polarization of the pump vortex beam so as to optimize phase matching within the OPO. Using a lens with a focal length $f=750\mathrm{mm}$, the pump vortex beam was focused to a spot with a waist of $380\mu \mathrm{m}$ (${\ell }_p=1$), $474\mu \mathrm{m}$ (${\ell }_p=2$), and $510\mu \mathrm{m}$ (${\ell }_p=3$) at the center of the KTA crystal.

The idler-resonant KTA-OPO had a hemispherical cavity comprising a concave input mirror (IM) with a radius of curvature $R=100\mathrm{mm}$ and a plane output coupler (OC). The IM was antireflection-coated for $1.064\mu \mathrm{m}$ and high-reflection-coated for 1.3 to $5\mu \mathrm{m}$ ($R=99\%$); the OC was coated partially reflecting ($R\approx 80\%$) for the idler beam and high-transmitting for the pump and signal beams. A type II noncritically phase-matched KTA crystal ($x$-cut, $\theta =90\mathrm{deg}$, $\phi =0\mathrm{deg}$) with dimensions of $5\mathrm{mm}\times 5\mathrm{mm}\times 30\mathrm{mm}$ was used in the OPO. Both sides of the crystal were antireflection-coated for the pump, signal, and idler fields in order to minimize loss. The OC was mounted on a one-dimensional translation stage, and the geometrical length of the cavity ($L$) was varied in the range of 32 to 90 mm by moving the OC along the optical axis. The IM and KTA crystal were then fixed. A Ge filter was used to separate the idler output from the undepleted pump and signal outputs.

In these experiments, we first generated high-order ${\mathrm{LG}}_{0,\pm \ell }$ modes with tunable topological charge by varying the resonant cavity length of the OPO. The spatial intensity profile of the pump beam was recorded using a CCD camera (Spiricon BGS-USB-SP620) and that of the signal and idler beams using a pyroelectric camera (Spiricon Pyrocam III, spatial resolution of $75\mu \mathrm{m}$). When the OPO was pumped with a first-order optical vortex ${\ell }_p=1$ [spatial profile shown in Fig. 2(a)], the signal field was generated with a similar spatial profile, as shown in Fig. 2(c). The OAM characteristics of the pump and signal beams were examined using the tilted lens method.36 The spatial profiles of the pump and signal beams after being focused through the tilted lens are shown in Figs. 2(b) and 2(d), respectively. In both cases, the presence of two lobes at the focal plane confirm the order of both the pump and the signal vortex beams to be $\ell =1$. This result is in agreement with the conservation of OAM within an OPO wherein the OAM of the pump beam is transferred to the nonresonant signal beam. Generally, the idler beam should exhibit a Gaussian spatial profile, however, this was not always observed in this work. The spatial profile of the idler beam (as recorded using the pyroelectric camera) for a range of resonant cavity lengths is shown in Figs. 2(e1)–2(e6). It can be seen that the spatial intensity profile of the idler beam changes with the cavity length. Here, the order of the idler beam LG mode (and the number of petals in the spatial profile) can be increased by decreasing the length of the resonant cavity. The theoretically modeled spatial intensity profiles of the generated LG modes are shown for comparison in Figs. 2(f1)–2(f6). There have been a number of publications reporting the generation of similar mode shapes directly from a laser cavity.37^,38 These mode shapes can arise from the coherent superposition of LG modes with opposite azimuthal order oscillating within the laser cavity (such modes can freely oscillate within the cavity if no additional mode-control elements are used).

To further investigate the generation of high-order LG petal-like modes, the idler-resonant OPO was pumped using vortex beams with ${\ell }_p=2$ and 3. When the OPO was pumped with a second-order vortex beam ($\ell =2$) [free space and tilted lens focused spatial intensity profiles shown in Figs. 3(a) and 3(b), respectively], the generated signal beam was observed to have a doughnut-shaped intensity profile and carried an OAM of $\ell =2$. The free-space and tilted lens-focused spatial intensity profiles of the generated signal beam are shown in Figs. 3(c) and 3(d), respectively. The results show that the signal field takes on the same OAM characteristics as the pump field; this characteristic was observed for all examined lengths of resonant cavity. The idler fields were generated in the form of high-order LG petal-like modes, and its OAM could be tuned from 0 to $\pm 8$ by varying the length of the resonant cavity. The experimentally and theoretically and theoretically obtained spatial intensity profiles of the generated idler beams are shown in Fig. 3 for a range of resonant cavity lengths.

When the OPO was pumped using an optical vortex of order ${\ell }_p$ of 3 [free-space and the tilted lens-focused spatial intensity profiles shown in Figs. 3(g) and 3(h), respectively], the generated signal beam was observed to have a doughnut-shaped intensity profile and carried an OAM of $\ell =3$; this characteristic was observed for all investigated lengths of the resonant cavity. The free-space and tilted lens-focused spatial intensity profiles of the generated signal beam are shown in Figs. 3(i) and 3(j), respectively. By adjusting the length of the resonant cavity, the idler field could be generated in the form of high-order LG petal-like modes, and its OAM could be tuned from 0 to $\pm 10$. The experimentally obtained spatial intensity profiles of the generated idler fields are shown in Fig. 3 for a range of resonant cavity lengths.

The above results show that high-order LG modes with petal-like spatial profiles and tunable OAM can be generated directly from an idler-resonant KTA-OPO by adjusting the length of its resonant cavity. In fact, the handedness ($\pm \ell$) of LG mode with the same intensity profile cannot be well distinguished in the laser cavity, thereby resulting in the generation of petal-like modes formed of the coherent superposition of annular ${\mathrm{LG}}_{0,+\ell }$ and ${\mathrm{LG}}_{0,-\ell }$ modes. To better understand how these high-order LG modes were produced from this system, we performed theoretical modeling, wherein we examined the spatial overlap of the pump field with the cavity modes of the idler field within the resonant cavity of the OPO and the KTA crystal. The spatial overlap efficiency $\eta$ of the pump intensity $I_p(r,z)$ and the cavity mode intensity $I_c(r,z)$ was modeled using the following equation: $$\eta =\iint I_p(r,z)I_c(r,z)\mathrm{d}r\mathrm{d}z.$$(1)

The normalized intensity distribution of an ${\mathrm{LG}}_{0,\ell }$ mode can be expressed as39$$I(r,\phi ,z)=\frac{2}{\pi w_0^{2}dl!}\left[\frac{2r^2}{w_0^2}{]}^{|l|}{\left[L_p^{|l|}\right(\frac{2r^2}{w_0^2}\left)\right]}^2\exp \right(\frac{-2r^2}{w_0^2}){\left\{\begin{array}{c}\sin l\phi \\ \cos l\phi \end{array}\right\}}^2,$$(2)where $L_p^{|\ell |}(·)$ is the generalized Laguerre polynomial, and $p$ and $\ell$ are the radial and azimuthal indices, respectively. $d$ is the length of the gain medium, and $w_0$ is the cavity mode size, which is assumed to be constant in the laser crystal (along the axis of the laser).

If we consider a focused doughnut-shaped vortex pump beam, we can define the beam radius $r_{\text{outer}}$ as the radius of a circle that contains 86.5% of the total energy of the beam, and the ring radius $r_{\text{inner}}$ as the radius of the circle that contains 13.5% the total energy. The equation determining the spatial overlap efficiency can then be modified to40$$\eta =\sum _{n=0}^l\frac{1}{(l-n)!}\left[{\left(\frac{2r_{\text{inner}}^2}{w_0^2}\right)}^{l-n}\exp \right(-\frac{2r_{\text{inner}}^2}{w_0^2})-(\frac{2r_{\text{outer}}^2}{w_0^2}{)}^{l-n}\exp (-\frac{2r_{\text{outer}}^2}{w_0^2})],$$(3)where the waist ($w_0$) of the fundamental cavity mode of a plano–concave linear cavity can be determined using41$$w_0=\sqrt{\frac{L_{\text{eff}}\lambda }{\pi }}{\left[\frac{g_1{g}_2(1-g_1{g}_2)}{(g_1+g_2-2g_1{g}_2{)}^2}\right]}^{1/4},$$(4)where $L_{\mathrm{eff}}=L-(n_r-1)d/n_r$ is the effective resonator length, $\lambda$ is the wavelength of the lasing mode, $n_r$ is the refractive index of the gain medium, $g_i=(1-L_{\mathrm{eff}})/R_i$ is the g-factor of the resonator, and $R$ is the radius of curvature of the cavity mirrors. The values of $r_{\text{inner}}$ and $r_{\text{outer}}$ were determined directly from our experiment. Here, the values of $r_{\text{inner}}$ and $r_{\text{outer}}$ for the pump vortex beams with ${\ell }_p=1-3$ were 35.6 and $330\mu \mathrm{m}$; 92.1 and $410\mu \mathrm{m}$; and 132.8 and $440\mu \mathrm{m}$, respectively. Using these values along with the above equations, we theoretically modeled the spatial overlap efficiency ($\eta$) as a function of the resonant cavity mode order ($\ell$). These results are plotted in Fig. 4 for pump vortex beams with ${\ell }_p=1-3$.

Figure 4 shows the calculated spatial overlap efficiency $\eta$ between the vortex pump mode and high-order LG modes within the resonant cavity of the OPO at various cavity lengths ($32-90\mathrm{mm}$). When pumping with a vortex beam with order ${\ell }_p=1$, the spatial overlap efficiency (red square) decreases monotonically as $\ell$ increases. When pumping with vortices of order, ${\ell }_p=2$ and 3, the spatial overlap efficiency first increases and then decreases as the value of $\ell$ increases. The high-order LG mode, which is generated from the resonant cavity, is a function of both the intracavity loss of that mode and the spatial overlap of the pump field with that mode. Both of these factors are a function of the length of the resonant cavity. The output energies of the signal and idler beams were measured as a function of the pump energy when pumping with vortex beams of order ${\ell }_p=1-3$ and for a range of resonant cavity lengths. These plots are shown in Fig. 5.

The plot of Fig. 5(a) shows that when using a pump field with ${\ell }_p=1$ and an energy of 20.2 mJ, for cavity lengths of $L=32$, 55, and 90 mm, the maximum output energies of the signal and idler fields were 2.9, 2.6, 2.1, and 0.8, 0.67, and 0.54 mJ, respectively. The corresponding slope efficiencies were 23.5%, 28.9%, 22.9%, and 6.4%, 7.4%, 6.2%, respectively. Figure 5(b) shows that the maximum output energies of the signal and idler fields when using a pump field with ${\ell }_p=2$ for cavity lengths of $L=32$, 55, and 90 mm were 2.5, 2.3, 1.8, and 0.53, 0.45, 0.38 mJ, respectively. Figure 5(c) shows that the maximum output energies of the signal and idler fields when using a pump field with ${\ell }_p=3$ for cavity lengths of $L=32$, 55, and 90 mm were 1.4, 1, 0.54, and 0.37, 0.27, 0.18 mJ, respectively. The corresponding to slope efficiencies were 18.7%, 12.8%, 7.2%, 4.9%, 3.6%, and 2.4%, respectively. It should be noted that the oscillation threshold of high-order petal-like modes increases, as the extending of cavity length from 32 to 90 mm, which is due to the severe cavity loss. The measured oscillation threshold of OPO pumped by the optical vortex with order of, ${\ell }_p=1-3$, were ranged within 7.7–10.2, 7.9–11.2, and $11.2–14.4\mathrm{mJ}$, respectively. From these results, it is clear that pumping with the lowest order vortex mode yields the highest signal and idler output energies for a given pump energy and resonant cavity length.

The spectra of the signal and idler outputs (measured to be 1535 and 3468 nm, respectively) were recorded using a scanning monochromator (Spectra Pro HRS-500, $300\text{lines}/\mathrm{mm}$, aperture size of $30\mu \mathrm{m}$, spectral resolution of 0.3 to 0.4 nm in the wavelength range of 1000 to 5000 nm) and are plotted in Fig. 6. The spectral bandwidths of the signal and idler outputs were measured to be $\mathrm{\Delta }{\lambda }_s\sim 0.32\mathrm{nm}$ (ca. $1.36{\mathrm{cm}}^{-1}$) and $\mathrm{\Delta }{\lambda }_i\sim 0.67\mathrm{nm}$ (ca. $0.56{\mathrm{cm}}^{-1}$)

We have demonstrated the generation of high-order ${\mathrm{LG}}_{0,\pm \ell }$ mode idler field vortex beams with $\ell$ tunable from 0 to $\pm 10$ using an idler-resonant KTA-OPO. The order of the output idler beam could be varied by adjusting the length of the resonant cavity and by changing the order of the vortex pump beam. A theoretical investigation of the spatial overlap efficiency of the pump vortex and resonant idler beams was undertaken to glean insight into the mechanism by which these high-order LG modes were generated. When pumped with vortex beams with $\ell =1$, 2, and 3, due to the increase of spatial overlap efficiency, the highest-order idler beams that could be generated from the system were ${\mathrm{LG}}_{0,\pm 5}$, ${\mathrm{LG}}_{0,\pm 8}$, and ${\mathrm{LG}}_{0,\pm 10}$, respectively. This work demonstrates the relative ease by which high-order LG laser modes in the mid-infrared wavelength range can be generated directly from an OPO system. We anticipate that such laser beams will have utility for applications in precise spatial measurements, optical trapping and tweezing, and free-space optical/quantum communications.

Yuxia Zhou is a PhD in the School of Chemistry and Chemical Engineering, Xinjiang Normal University, China, and also belongs to the Xinjiang Key Laboratory for Luminescence Minerals and Optical Functional Materials.

Xining Yang received her PhD in testing and measurement technology and instruments from Harbin University of Science and Technology, China, in 2020. She is currently a professor at the Xinjiang Key Laboratory for Luminescence Minerals and Optical Functional Materials, Xinjiang Normal University, China. Her research focuses on solid lasers.

Jianqiang Ye received his BS degree in applied physics from Jiangsu University of Science and Technology, China, in 2022. He is currently an MS student at the Xinjiang Key Laboratory for Luminescence Minerals and Optical Functional Materials, Xinjiang Normal University, China. His research focuses on optical vortices, optical parametric oscillators, and amplifiers.

Yuanyuan Ma received her PhD in optical engineering from the Graduate School of Advanced Integration Science, Chiba University, Japan, in 2020. She is currently a postdoctoral research fellow in the Graduate School of Engineering at Chiba University. Her research area focuses on structured-light beam generation and solid-state Raman lasers.

Ying Wan received her PhD in communication and information systems from Shanghai University, China, in 2022. She is currently a lecturer at Nanjing University of Information and Technology, China. Her research interests focus on optical fiber and nonlinear frequency conversion.

Jianxiang Wen received his PhD in communication and information systems from Shanghai University, Shanghai, China, in 2011. Currently, he is an associate professor at Shanghai University. His current research interests include special optical fiber structure design, research, and preparation.

Taximaiti Yusufu received his PhD in optical engineering from the Graduate School of Advanced Integration Science, Chiba University, Japan, in 2014. He is currently a professor at the Xinjiang Key Laboratory for Luminescence Minerals and Optical Functional Materials, Xinjiang Normal University, China. His current research interests include optical vortices, OAM, structured light, nonlinear optics, optical parametric oscillators, and amplifiers.