In this section, details would cover the SHG, SFG, and HHG. SHG is a χ ( 2 ) nonlinear effect. Under the phase-matching condition, a photon at frequency ω will become a photon of frequency 2 ω . For conventional beams, the phase-matching condition is achieved through birefringence phase-matching and quasi-phase matching. For birefringence phase-matching, there were mainly two techniques—angle-tuned critical phase matching and temperature-tuned non-critical phase matching (NCPM) [93–95]. The well-studied nonlinear crystals include KTP, BBO, LBO, and so on. For quasi-phase matching, periodical crystals such as PPKTP and PPLN were studied in a number of reports [96, 97]. Phase-matching in conventional cases is governed mainly by two rules—the conservation of linear momentum ( n 1 ω 1 + n 2 ω 2 = n 3 ω 3 ) and the conservation of energy ( ω 1 + ω 2 = ω 3 ). For the χ ( 2 ) process of a vortex beam, conservation of the angular momentum should also be considered in the phase match condition ( l 1 + l 2 = l 2 ).

As early as 1996, Padgett et al. [98], who were among the earliest pioneers who studied optical vortices, reported SHG of an LG mode vortex beam by deploying a pair of cylindrical lenses as the mode converter. The fundamental HG mode was converted into the LG mode which contains the OAM, at the work of the π / 2 -mode-converter. Padgett’s theoretical simulations, as well as extra-cavity experiments (by making use of gadgets such as the q-plate, spiral phase plate, and so on) have confirmed that during an SHG process. The topological charge of the corresponding LG mode will also be doubled, signifying the law of conservation of angular momentum. In 1997, Courtial et al. [99] reported the SHG of an LG beam again, where their experimental configuration is depicted in Figure 3A. The topological charge was doubled for the SHG process, as shown in Figures 4A,B. In 1998, Berzanskis et al. [100] investigated the influence of walk-off on sum-frequency mixing of optical vortex lasers in type-I critical phase-matching cut KDP crystals. Two zeroes in the vortex field distribution were observed, as shown in Figures 4C,D , and the separation between the cores of two vortices was larger in the longer crystal. In 2011, Fabio et al. [101] employed a BBO crystal for the SHG experiment. The experimental configuration is shown in Figure 3B. Two femtosecond beams with opposite sign fractional topological charges served as the fundamental, which were achieved through the spiral phase plate. Each fundamental carried an OAM equivalent of 1/2  ℏ , while the SHG fields remain zero. This also proved that the conservation law remains true for vortex beams obtained through half-integer SPP.

Photonic crystals were suggested to produce twisted light by Bloch in 2012 [102]. In 2014, Zhou et al. [103] conducted an SHG experiment using periodically poled KTP (PPKTP) and successfully produced an ultraviolet beam with the OAM higher to 100  ℏ . Later the same year, Zhou et al. [104] proposed to improve the power performance by proper alignment of the quasi-phase-matching crystal and pumping it with the vortex beam. PPKTP with the dimension of 1 mm × 2 mm×10 mm was used. A conversion efficiency as high as 10.3% and SHG-power of 22.5 mW were achieved. The experimental configuration is shown in Figure 3C. The benefit of quasi-phase matching is the devoid of the walk-off effect and having a large acceptance angle. In 2018, Xu et al. [105] achieved the third harmonic generation (THG) with tunable topological orders by the acute use of quasi-phase matching. They demonstrated the control of OAM by using two orthogonally polarized beams at the same frequency by carrying different topological charges to pass through the periodically poled LiTaO₃ optical superlattice. After a cascaded nonlinear process in the crystal, the obtained beam carried a topological charge identical to that of the input OAM-carrying LG beam. Wang et al. [106] designed nonlinear fork gratings with supercell structures in lithium niobate crystals. The development of lithium niobate photonic crystals and their application in wavefront modulation have suggested a possible route to generate the second harmonic wave. They combined the fork pattern and constructed them in the LiNbO₃ crystal and produced a vortex beam with the OAM range from l = -3 to l =3.

In 2012, Ding et al. [107] experimentally displayed an infrared light with the OAM being up converted into visible light through four-wave-mixing via a ladder-type configuration in ⁸⁵Rb atoms in a vapor cell. Two pump beams were imprinted with the OAM by a computer-generated hologram. Regardless of the different topological charges the two pump beam carries, all results remain identical apart from the conversion efficiency and the temperature of the vapor cell. In 2017, Zhao et al. [108] observed the frequency doubling effect of the topological charge with a composite vortex beam. A 2-f system was built while a laser at the wavelength 1,064 nm after alignment and magnification were separated by a beam split, then to form a composite vortex beam after passing through a vortex phase plate. Two beams were then converged to the KTP crystal for frequency doubling. A filter (T_532nm > 85%, T_1064nm < 1%) was used to filter out the fundamental light. They observed that for an input composite vortex, the frequency-doubled beam was also composite (coherent superposition of the pure LG mode). Tung et al. [109] also reported a frequency-doubled vortex beam in 2018, which they designated as the bottle beam. The experimental configuration is in Figure 3D. The KTP crystal was placed inside the fundamental cavity, with some small distance d from the output coupler. As d varied from 1.8 to 3 mm, the recorded transverse pattern changes drastically. The green bottle beam was generated with a narrow 3-D dark core. This suggested that shortening the distance between the SHG-crystal and the output coupler can be a possible route toward mode tunability. It was interesting to observe that while the wave remained perfectly a central-dark spatial profile in the near field, for the far field, it propagated; however, it resembles a circular beam with a shining spot in the middle. Such a beam was said to have a unique advantage when both the doughnut shaped beam and the ordinary Gaussian beam are required because they can be tailored, respectively.

In 2021, Gui et al. [110] demonstrated the SHG of a beam with spatiotemporal orbital angular momentum (ST-OAM). ST-OAM is a form of structured electromagnetic field recently identified. It features a space-time spiral phase structure and a transverse OAM which were perpendicular to the energy-density flow. They observed the modification of the OAM pulse by complex spatiotemporal astigmatism, in conjunction with multiple phase singularities. A 400-nm second-harmonic and an ST-OAM pulse with undetermined charge l were obtained. A temporal OAM beam can have a short pulse width and high peak power. This is useful in super-resolution imaging. The Airy beam could contain vortex characteristics. Li et al. [111] observed the Airy vortex beam when the fundamental wave modulated by SLM interacted with nonlinear media. SHG of the vortex Airy beam was observed. In 2021, Liu et al. [112] manages to control vortex characteristics in the Airy beam. Loading of OAM is controlled electrically, which showed great potential in information processing.

In 2017, Gauther et al. [113] studied HHG for producing the XUV beam with tunable OAM. Their configuration is shown in Figure 5A, where a femtosecond pump beam at the wavelength of 805 nm is frequency-doubled by a BBO crystal; then, they were separated, and the 2 ω -beam was introduced a topological charge by a spiral phase plate. Both the fundamental ω -beam and the 2 ω -beam were converged in the same gas cell for HHG. They have demonstrated that for HHG of order q, it follows q = n 1 + 2 n 2 , while n 1 , n 2 are the number of photons absorbed from the ω and 2 ω beams. The topological charge of the HHG-output follows l = n 1 l ω + n 2 l 2 ω . In 2017, Kong et al. [114] also reported the HHG experiment based on the gas jet. Two arms of beam at wavelength 800 nm with different topological charges ( l 1 = 0 , l 2 = 1 ) were reported. Extreme ultraviolet with order l = ± 1 ,   0 was produced. Details are illustrated in Figure 5B and Figure 5C. In 2020, Fan et al. [115] reported the generation of the polarized vortex beam in the waveband of extreme ultraviolet based on HHG of a gas-jet. HHG is an effective and efficient approach for achieving the generation of EUV, and it is also a promising scheme to achieve a tunable high OAM beam since the resultant topological charge of the HHG wave was l n t h = n ⋅ l f , where l f is the topological charge of the fundamental wave, and n is the order.

Overall, we have summarized the important results concerning SHG, SFG, and HHG. First, extensive works have explicitly confirmed that during such process, the change of the topological charge abided by the laws of conservation. Second, the walk-off on frequency conversion with critical phase-matching leads to the collapse of vortex field distribution; therefore, the noncritical phase matching and quasi-phase matching crystals are excellent candidates for SHG/SFG of the vortex beam. Finally, SHG, THG, and HHG are efficient to obtain visible and ultraviolet beam with the OAM. For n-order high harmonic generation, if the OAM of the fundamental beam is 1, the HHG beam should naturally have n ⋅ l ℏ OAM.

Stimulated Raman Scattering, or SRS in short, is a third-order nonlinear effect. Many materials with Raman configuration were used for Stokes output, such as vanadates, tungstates, molybdates, and KTP family crystals [116–121]. Self-Raman experiments are most popular because it can ensure the compactness and attribute to versatile wavelength operation.

Vanadate laser crystals have a unique structure that enables the relatively low threshold output. In 2014, Lee et al. [60] used Nd:GdVO₄ self-Raman to achieve the first Stokes line at 1,173 nm with the experimental configuration shown in Figure 6A, corresponding to the Raman shift of 882 cm⁻¹. At the incident pump power of 6.8 W, the fundamental and the Stokes output were both produced with the output power at 400 and 380 mW, respectively. The LG_0,1 mode was observed, and the transfer of its topological charge to the Stokes field was also recorded. The vortex output was attributed to the annular gain profile and the Fresnel diffraction loss by the defect spot mirror. To improve the compactness, a microchip laser might be the choice. In 2015, Lee et al. reported a YVO₄/Nd:YVO₄ self-Raman laser with off-axis pumping for the high-order Hermite–Gaussian beam generation. The HG mode order of the first Stokes wave can be up to TEM_28,0. The output power was higher than 1.0 W from TEM_0,0 to TEM_3,0 at a pump power of 18.6 W [121]. In 2018, Dong et al. [122] designed a Raman microchip laser with the combination of the Yb:YAG and YVO₄ crystals; the overall length was 2.2 mm. The corresponding design is shown in Figure 6B. The optical vortices realized by tilting the laser crystal with off-axis pumping geometry. Different frequency combs were obtained as the pump power increased, and it remained perfectly a doughnut-shaped intensity profile. The vortex characteristics were identified by using an interferogram. In 2020, Ma et al. [123] came up with the annular pumping scheme by an axicon. Figure 6C is a portrait of their experimental design. Nd: GdVO₄ was used to serve both as the laser medium and the Raman medium. The annular beam output carried zero OAM for both the first Stokes of 1,108 nm and 1,173 nm were achieved. Figure 7 shows the spatial intensity profiles of the fundamental and Stokes waves, as well as the interferogram of the Stokes waves. Compared between the spatial intensity profiles of the fundamental and Stokes waves, obvious beam-cleaning phenomena could be observed. More recent, Miao et al. reported the petal-like Raman laser in a Yb:YAG/YVO₄ microchip laser with annular pumped with the setup shown in Figure 6D. The LG_0,n mode petal-like lasers with order of n tuning from 3 to 11 had been obtained. [124].

The aforementioned self-Raman lasers had expanded the optical vortices in the near-infrared waveband. If the Raman combined with SHG and SFG, the optical vortices in the visible waveband also could be expanded. In 2014, Lee et al. [125] also combined this vortex Raman laser design with the LBO crystal for the SHG of both the fundamental and the first Stokes waves to produce the 532/586 nm laser. The experimental configuration is depicted in Figure 8A. Figure 9 shows the fork interferogram of the first Stokes wave and its SHG with yellow light. The OAM has been transferred to Stokes field in the Raman process and then doubled in the SHG process. In 2016, the SFG of the fundamental and the first Stokes wave for 559 nm vortex laser generation was also reported by his team [126] with the experimental configuration shown in Figure 8B. The jointing of self-Raman effects and an LBO crystal, along with the defect spot mirror cavity design, has enabled the multi-wavelength operation. If the wavelength selection mechanism based on vanadate crystals be introduced [127, 128], it will put forward the application of a selective multi-wavelength vortex laser. It is natural to include our future work in the areas of vortex Raman operation, particularly to introduce vortex characteristics into the visible band laser.

During the Raman process, it is evident to see that the transfer of a topological process occurs as a result of the superposition of fundamental modes, HG_1,0, HG_{0, 1}. It is modeled by Lee et al. [60] They also observed an interesting phenomenal for the SFG field: it possesses a +2 or 0 topological charge while an annular profile is retained in the near field but only to degrade into a bright spot in the far field. The SFG field was different in the near and far fields. They explained this because the SFG field was not resonated by the laser cavity and not an eigenmode of the resonator.

In addition to expanding visible optical vortices through SHG of the Raman laser, the cascade Raman excited by the 532 nm laser also had been reported to fill the gap in the visible waveband of optical vortices. In 2021, Omatsu et al. [129] realized the cascaded Raman output with a 5 mm × 5 mm×50 mm Ba(NO₃)₂ (Raman shift 1,046 cm-1). A 10 ns green pulse at wavelength 532 nm served as the pump beam. A topological charge of 2 was introduced to the pump via a q-plate. The design is shown in Figure 8C. Wavelengths of first, second, and third Stokes at 563 nm, 599 nm, and 639 nm, respectively, had been achieved.

Table 1 lists performance data of aforementioned experiments for Raman vortex generation. According to the research progress of the Raman vortex laser, intracavity Raman is popular and has been widely reported. Compact configuration can be easily achieved as the fundamental laser and the Raman output shared the same cavity. It is also concluded as follows: first, the topological charge used to transfer to the Stokes waves in the Raman process according to the published work. Second, the Raman vortex laser was ideal to maintain a better beam quality than the fundamental laser, due to the Raman beam cleanup effect. Third, stimulated Raman scattering and its combination with second-order nonlinear optical frequency conversion can expand the wavelength of vortex light in infrared and visible bands.

Optical parametric oscillators, or OPO in short, are a useful nonlinear technique known for the application of tunable and mid-infrared laser generation [130–132]. For a given pump frequency ω p , a signal frequency ω s and an idler frequency ω i were produced. It should satisfy the phase-matching condition—the conservation of linear momentum ( n p ω p = n s ω s + n i ω i ) and the conservation of energy ( ω p = ω s + ω i ). If the pump is a vortex beam, conservation of angular momentum ( l p = l s + l i ) also should be satisfied. Many reports have discussed the extension of the wavelength-tunable range in the OPO laser; however, little is known for tuning OAM-related properties. The transfer of the topological charge is also a problem worth looking into.

One interesting property to be investigated during an OPO process of a vortex beam is the possible mechanism for the topological charges to be transferred from the pump beam to the signal beam. In 1999, Maleev et al. [133] reported the violation of the conservation law for topological charge transfer in parametric down conversion. No further reports have followed that conclusion. Usually speaking, a L G 0 l mode excited OPO will produce a signal wave and an idler wave, both with the topological order split in half to be l / 2 . However, this fact was first proved unsolid by Yusufu et al. [134] in their study published in 2012 where a 30 mm KTP crystal with θ = 51.4° was employed as the nonlinear crystal for the OPO. Figure 10A is their experimental configuration. According to the self-interference fringes of both signal and idler waves in Figure 10B, the topological order of the pump beam was transferred to the signal wave solely without sharing its value with the resultant idler wave, and the idler wave was a Gaussian profile without any phase singularity. A tunable vortex parametric laser source with a wavelength-tunable range from 1.953 to 2.158 μm has been achieved. They also give the walk-off effect a thorough account as to their influence for vortex optical parametric source. In fact, they showed that it is indeed the plano-concave cavity and the walk-off effect that prevented the lasing of the idler wave even in the Gaussian mode; therefore anisotropic transfer of the topological charge was possible. To avoid the walk-off effect of critical phase matching, an efficient OPO vortex source could be obtained by compensating the walk-off effect with two cascaded KTP with experimental configuration, as shown in Figure 11A [135]. Without the cascaded crystal, the signal and the idler were respectively limited to 1923–1955 nm and 2,376–2,335 nm. After difference-frequency generation by a ZGP crystal, tunable vortex output at wavelength 6.3–12 μm was achieved.

Apart from compensating the walk-off effect with two cascaded crystals, the quasi-phase matching and non-critical phase matching without walk-off were widely used. For the quasi-phase matching, Yusufu et al. [136] reported a tunable vortex laser based on the parametric source with a periodically poled crystal MgO: PPLN. A 1064 nm Nd: YAG laser was chosen as the seed light, and a spiral phase plate was adopted for tailing the pump light into a vortex beam. They obtained the milli-joule-level 3 μm output (2.14 mJ maximum output with a conversion efficiency of 10.2%). They observed the transfer of the topological charge from the pump to the idler. The experimental configuration is shown in Figure 11B. In 2019, Sharma et al. [137] reported the vortex mode at order l = 1 with doubly resonating the optical parametric oscillator based on MgO-doped periodic poled lithium tantalate. At the green pump beam, the same order and sign were achieved at the frequency tunable across 970–1,174 nm. The vortex dipole is generated by controlling the beam overlap. In 2020, Tong et al. [138] reported the femtosecond vortex beam generated from an OPO based on a MgO:PPLN crystal with a polarization period of 31 μm. The pulse duration was variable from ∼400 fs to ∼1.1 ps. 1∼ sixth-order femtosecond vortex beam were realized after mode converting by cylindrical lens. More recent, in 2022, they also demonstrated the tunable high-order optical vortices with topological charge transfer to the non-resonant idler beam using a picosecond optical parametric oscillator [139]. The mid-infrared vortex beams of orders l = 1 − 5 with tunability across the 2,493–4,035 nm spectral range was achieved.

For the non-critical phase matching, Mamuti et al. [140] reported the superposition of laser modes that carried orbital angular momentum by a parametric process with the use of NCPM-LBO, which is illustrated in Figure 11C The tunability of the topological charge was realized mainly by two mechanisms: 1) the shortening of the laser cavity provided a change in mode spatial overlapping, which caused the lasing conditions favorable for certain modes. 2) the non-critical phase matching not only avoided large the walk-off effect, but it also provided the favorable mode-overlapping condition; therefore, the tunable topological charges were realized at different frequencies. Depicted in Figure 12A is the spatial profile and the interference fringes of the signal and the idler wave from a compact cavity. The selective OAM transferred to either the signal or the idler was also realized simply by shortening the extended cavity. Since the mode radius at the plano-concave cavity changes while the cavity length extended, the spatial overlapping thus changes as well. This combined with the NCPM-LBO for tunable frequencies create a condition for separating the signal wave from the idler wave. When the cavity length was suited for the lasing condition of the signal or the idler wave, the topological charge will be transferred to it simultaneously. From Figure 11A a∼f) is the spatial profile, and the corresponding interference fringes of the signal wave at the lasing signal frequency of 930 nm; g)∼l) are the spatial profile and the corresponding interference fringes of the idler wave at frequency 1,243 nm. The tunable OAM ranging from l = − 2 to l = 3 was reported. In 2019, they reported wheel-mode generation (seen Figure 12B) using the same configuration as in Figure 11C. They suggested the pump source with a short coherence time for producing larger OAM-carrying beam [141]. l s = 2 ∼ 4 was realized. The wheel-shaped modes are actually the coherent superposition of multiple OAM states. In 2021, Ababaike et al. [142] reported the milli-joule level vortex optical parametric source, with an NCPM-KTA crystal and a singly resonant linear plane-parallel cavity. The experimental configuration is shown in Figure 11D. In this report, selective topological transfer to the signal or to the idler output was achieved by changing the output coupler. In the low-Q cavity with partial reflective coated for the signal wave, the OAM was transferred to the signal (1.535 μm), while in the high-Q cavity with high reflective coated for the signal wave, it was manifested as the 3.468  μ m   idler output. This OAM switching effect between the idler and the signal wave could be explained with intracavity photon lifetime. Moreover, noncritical phase-matching KTA is also a potential candidate for special wavelength vortex laser generation; based on cascaded frequency conversion [143–145], the cascaded OPO could fill the wavelength gap in the mid-infrared band.

According to the above results, all the reported OPO vortex lasers were extra-cavity pumped. Compensating the walk-off effect with two cascaded crystals, quasi-phase matching crystals and non-critical phase-matching crystals were used to avoid the walk-off effect. More importantly is the selective transfer of the topological charge and the tunability of OAM and wavelength, which have been reported. OPO has been used for mid-infrared widely tunable laser generation. The high-order vortex beam and anisotropic transfer of the topological charge to signal and idler waves were possible in OPO with a reasonable design.