Due to its peculiar phase and intensity distribution, the vortex beam has attracted extensive attention in particle manipulation and communication. Interference of a vortex beam with a plane wave can generate a helix beam with peculiar helical intensity distribution, providing a potential research platform for studying nonlinear topological edge solitons and anomalous topological phases. Multiple vortex and helix beams arrange in a specific distribution to form the field of the optical vortex array (OVA) and optical helix array (OHA). The array has multiple phase singularities compared to a single beam, which has essential applications in multi-particle manipulation and multi-channel communication. The wide-ranging applications of array fields rely on generating high-quality optical fields. Currently, various methods have been proposed to generate OVA and OHA, such as using fractional Tabor effect, grating diffraction, or direct adoption of vortex lasers. In these methods, by employing the fractional Talbot effect, the field with the best contrast can only be obtained at a specific distance. The intensity distribution of the OVA generated by grating diffraction is not uniform, and the vortex laser suffers from low energy efficiency. Compared to the above-mentioned methods, the field generated by the multi-beam interference features propagation invariance and high efficiency and becomes one way to generate the OVA and OHA. Therefore, the adoption of multi-beam interference to generate OVA and OHA is of potential research significance.

Based on the principle of multi-beam interference, a periodic orthogonal binary phase plate is designed for generating square optical vortex array (SOVA) and square optical helix array (SOHA) fields. After filtering the spectrum of the phase plate, four symmetric spots in the central region of the spectrum and eight symmetric spots in the subcentral region of the spectrum are modulated separately. Then, the corresponding beams of these spots are obtained by the Fourier transform, and they interfere with each other to generate a square beam array (SBA) and a SOVA. The interference superposition of SBA and SOVA results in SOHA.

The designed binary phase plate has the same period and structure in two orthogonal directions. The difference in the phase modulation quantities of adjacent rectangular phase modulation units is π (Fig. 1). The central direct component of its spatial spectrum is 0. After filtering the spectrum, four spots of the central region and eight spots of the subcentral region are preserved (Fig. 2). First, phase modulation is performed on the four-point sources located in the central region (Fig. 3). After phase modulation, the SBA can be generated by four-point sources (Fig. 4). In the SBA, the beam is distributed in a checkerboard shape, and the phase difference between adjacent beams is π. Then, phase modulation is performed on eight points located in the subcentral region (Fig. 5). The SOVA can be generated by the phase-modulated eight points (Fig. 6). There are two kinds of staggered vortex beams with topological charge l=±1 in the array. The SOHA can be obtained by interfering with the SOVA and SBA, and the design parameters of the binary phase plate should meet b/a=1/6 (Fig. 7) to obtain SOHA with the best interference effect. Under such conditions, the SOVA and SBA have the same transverse distribution period. At the maximum amplitude of the vortex beam, the beams in SBA have the same amplitude value (Fig. 8). In this case, the phase change direction of adjacent helix beams in the SOHA obtained is opposite (Fig. 9). With the SOHA propagation, the intensity of helix beams presents a spiral distribution along the optical axis. The rotation directions of adjacent helix beams are opposite (Fig. 10). Finally, we build a 4f optical path for experimental verification (Fig. 11) and obtain experimental results consistent with the theoretical results (Fig. 12).

In conclusion, we propose a method of generating SOVA and SOHA fields by utilizing a periodic orthogonal binary phase plate. The phase modulation of each unit of the phase plate is 0 or π, and the central direct component of the spectrum of the phase plate is 0. By filtering and phase modulation of the phase plate spectrum, the SBA and the SOVA fields with propagation invariant characteristics can be generated by four spots in the central region and eight spots in the subcentral region respectively. There are two kinds of vortex beams with topological chargel=±1 in the array. The two square array fields have the same transverse spatial period, and the wave vectors of the two square array fields in the optical axis direction are different. Therefore, the SOHA in which intensity distribution rotates with the changing transmission distance can be obtained by the interference superposition of the SBA and the SOVA. The SOHA has two kinds of helix beams with opposite rotation directions. When the parameter of the phase plate is b/a=1/6, the SOHA with the best contrast can be generated. Simulation and experimental results demonstrate the feasibility of the proposed method.