This study is devoted to generating second harmonic (SH) vortex laser beams from a pump Gaussian beam. We propose that a Gaussian beam obliquely incident on a nonlinear fork grating can achieve SH vortex outputs at more wavelengths and angles compared to normal incidence. Additionally, the energy of the light wave under oblique incidence concentrates on one side for SH generation, resulting in higher conversion efficiencies. The oblique incidence scheme also helps obtain higher-order vortex beams with uniform radial intensity distribution, which is crucial for applications leveraging optical orbital angular momentum.

Vortex light has a helical phase front with a phase singularity at its center, where the light field intensity is zero, forming a dark core. The vortex beams, also known as orbital angular momentum beams, carries orbital angular momentum and have significant applications in optical tweezers, optical communications, and quantum engineering. Optical frequency conversion is an important method to broaden the spectral range of vortex beams. The most common approach is to direct an incident Gaussian beam to a nonlinear fork grating (Fig. 1a), imprinting the grating's twisted phase (quasi-angular momentum) onto the SH wave excited by the Gaussian beam, thus producing a vortex beam with a wavelength half that of the incident light (second harmonic).

Many studies have reported on Gaussian beams normally incident on nonlinear fork gratings, producing SH vortex laser beams symmetrically distributed about the incident light (Fig. 1b). This type of SH process is also referred to as nonlinear diffraction. In the normal incidence scheme, the Gaussian beam's energy is split into multiple SH channels, resulting in low conversion efficiency for a single channel. To improve efficiency, periodic nonlinear fork structures are widely used. Introducing longitudinal periods compensates for phase mismatches in the light propagation direction, achieving more effective SH conversion. However, longitudinal periods also limit the wavelengths and angles for efficient SH generation, hindering the practical application of vortex light.

To address these issues, Professor Sheng Yan's team at Ningbo University, in collaboration with researchers from the Australian National University and Xidian University, proposed an oblique incidence scheme for interactions between Gaussian beams and fork-shaped nonlinear photonic gratings, achieving SH vortex beams at more wavelengths, more angles, and higher efficiency. The related results were published in the Chinese Optics Letters, 2024, Volume **, Issue *.

In this study, femtosecond laser-induced ferroelectric domain inversion was used to prepare a periodic nonlinear grating structure with a transverse period of 2.30 μm and a longitudinal period of 22.82 μm in a barium calcium niobate crystal (Figs. 1c and 1d). When a Gaussian beam was normally incident, two symmetrically distributed SH vortex beams were observed at a wavelength of 1270 nm. When the Gaussian beam was obliquely incident at an angle of 1.27 degrees, SH vortex beams were observed at wavelengths of 1260 nm and 1275 nm on the left and right sides, respectively, with different exit angles. The energy flows more to the second harmonic vortex beam, indicating higher conversion efficiency than that of normal incidence. Similar experimental phenomena were observed in other orders of nonlinear diffraction.

Our results indicate that periodic nonlinear fork gratings are excellent media for generating SH vortex beams, and the oblique incidence scheme leads to second harmonic resonances at more wavelengths and angles. It also brings benefit to the generation of highly divergent vortex light (e.g., higher-order Laguerre-Gaussian) with uniform radial intensity distribution. This work is a successful combination of experimental exploration and theoretical conception, inspiring researchers to delve deeper into the theoretical mechanisms in the preparation and application of nonlinear micro-structured materials, fostering more efficient practical applications. In the future, the team will focus on the interaction between femtosecond lasers and nonlinear optical materials and apply these studies to advanced light sources and optical information processing to address critical technological challenges.

Figure 1. Generation of second harmonic vortex laser beams based on nonlinear fork gratings. (a) Nonlinear fork grating; (b) Gaussian beam normally incident producing symmetrically distributed SH vortex light; (c, d) Periodic nonlinear fork grating prepared using femtosecond laser-induced ferroelectric domain inversion; (e) Experimentally observed asymmetric, multi-wavelength SH vortex light output.