Due to the degree of freedom of orbital angular momentum, vortex beams have become a research hotspot in the field of spatially structured light fields in recent years, which are widely used in micro-particle manipulation, optical imaging, optical measurement, and optical communication. The wide application of this novel beam has also greatly stimulated researchers to explore and understand the structure of light fields. With the in-depth study of the new vortex field, researchers are no longer limited to the study of a single optical vortex, but focus on multiple optical vortices arranged according to a certain rule, namely the optical vortex lattice. In recent years, optical vortex lattices have been applied to optical measurements and show the promise of ultra-cold atoms trapping. Therefore, the generation and regulation of optical vortex lattice are of great scientific significance in related optical fields. So far, optical vortex lattices are usually generated by the superposition of two or more specific optical beams. Researchers have produced several optical vortex lattices via the superposition of Gaussian beams, Ince-Gaussian (IG) beams, Hermite-Gaussian beams, and Laguerre-Gaussian beams. Among these approaches, optical vortex arrays generated by the superposition of IG beams have additional abundant structures owing to the rich transverse intensity distribution of the IG beams. However, the influence of beam waist on the optical vortex lattice generated by IG beam superposition is not further studied. Therefore, it is necessary to study the characteristics of optical vortex lattice generated by IG superposition with different beam waist radii, so as to enrich the spatial pattern distribution of optical vortex lattice.

In this paper, a double-layer flower-shaped optical vortex lattice is proposed by superimposing odd and even mode IG beams with different waist radii based on computational holography. The optical vortex lattice is successfully generated by single optical path interference method, and the existence of vortex phase in the generated optical vortex lattice is verified by spherical wave interference. The experimental setup is shown in Fig. 2. The 532 nm beam generated by Nd∶YAG laser after frequency doubling passes through the pinhole filter and convex lens L1 (f₁=100 mm) to obtain an expanded beam. The expanded beam is split into two beams after it passes through the beam splitter. One beam is illuminated on the spatial light modulator (SLM) used to load the phase mask (HOLOEYE,PLUTO-VIS-016, pixel size is 8 μm×8 μm). The phase mask records the phase and amplitude information of the double-layer flower-shaped optical vortex lattice. The beam modulated by the SLM is screened out by the 4f (f₂=200 mm,f₃=200 mm) system of aperture A2, and the required +1 order diffraction beam is finally recorded by CCD camera (Basler acA1600-60gc,pixel size is 4.5 μm×4.5 μm) placed in the back focal plane of lens L3. Another beam is converted into a spherical wave through the lens L4 (f₄=75 mm).

The optical vortex lattices are double-layer distribution, and the number of vortices N meets the relationship of N=4m. And the topological charge values of the inner and outer vortices equal, but the signs are opposite (Fig. 3). The spherical wave interferes with the optical vortex lattice. According to the properties of the optical vortex, when it interferes with the spherical wave, a fork filament appears at the dark core, therefore verifying the vortex phase of the producing optical vortex lattice (Fig. 4). In order to research the effect of waist radius on the light intensity distribution of the optical vortex lattice, the odd mode waist radius w_o=4 mm is controlled. When the even mode IG beam waist radius increases from 3 mm to 4 mm with a step of 0.2 mm, the inner and outer edge bulges gradually flatten and the intensity distribution intends to be circular. As the waist radius gap between the odd and even mode IG beams decreases, the vortex distribution gradually changes from double layer to single layer, and the topological charge values and signs remain unchanged. When the waist radii equal, the surrounding dark cores in optical vortex lattice disappear, and only the central large dark core exists (Fig. 5). To investigate the properties of initial phase difference of DFOVL, the phase difference φ is added to the even mode IG beams, forming a completely isolated light flap when the initial phase difference is equal to π/2, which the dark cores and phase singular dots completely disappear. When the initial phase difference increases to π, the intensity mode of optical vortex lattice restores, but the signs of the inner and outer vortexes in the optical vortex lattice change (Fig. 6).

We have experimentally produced a double-layer flower-shaped optical vortex lattice, which is generated by coaxial superposition of odd-mode and even-mode Ince-Gaussian beams with different waist radius. The distributions of vortices in optical vortex lattice are double-layer, the number of vortices N meets the relationship of N=4m, and the topological charge values of the inner and outer vortices equal, but the signs are opposite. The effects of waist radius and phase difference on the distribution characteristics of light intensity and phase are analyzed. The results show that: when the phase difference φ between the odd and even mode IG beams is an integer multiple of π, the vortex sign changes; when φ is an odd multiple of π/2, the vortex disappears. Therefore, the generation and disappearance of vortex or the modulation of the vortex sign can be realized by changing the phase difference. When the waist radius gap between the odd mode and even mode IG beams gradually decreases, the light intensity distribution of the double-layer optical vortex lattice generated by superposition gradually intends to a concentric annulus, and the distribution of the vortex dark core changes from double-layer to single-layer. The research results greatly enrich the spatial mode distributions of optical vortex lattice and have potential applications in micro-particle manipulation.