With the development of computer and Internet technology, cameras and image sensors have increasingly higher requirements for three-dimensional (3D) imaging that contain depth information^[1]. 3D imaging solutions typically use time-of-flight (ToF) and structured light technique^[2,3]. The structured light is widely used in motion perception and face recognition because of its high spatial resolution for short distance ranging^[4]. The generation of structured light mainly depends on the diffractive optical element (DOE)^[5]. The surface of traditional DOE is etched into different depths to form different phases. Dammann grating is a typical binary-phase DOE^[6,7]. However, Dammann grating has the problems of low efficiency and small field of view (FOV) angle because of its binary phase modulation and large pixels^[8,9]. Multiple phase levels, equivalent to multiple etching depths, are essential to improve efficiency of Dammann grating. However, multiple etching depths usually increase the complexity of fabrication. Meanwhile, large pixels lead to poor performance in a large FOV, including the uniformity and low efficiency of the generated beam^[10].

A metasurface is a kind of artificial ultrathin planar optical nanostructure^[11–13] that has been widely used in recent years to control the polarization^[14,15], phase, and amplitude of incident light on the subwavelength scale^[16–18]. A variety of applications can be realized for better performance by using a metasurface, including color filters^[19,20], holographic displays^[21,22], beam shapers^[23], and beam steering^[23,24]. The metasurface can modulate the phase of incident light at subwavelength scale, solving the large pixel problem of traditional DOE and effectively improving diffraction efficiency at a large FOV. By using various nanopillar structure sizes instead of multiple etching depths, metasurfaces can achieve multiple phase level modulation without complex fabrication.

In recent years, some works on using metasurfaces to generate structured light have been reported. Li et al. produced wafer-level metasurfaces based on 12-inch immersion lithography technology, proving the feasibility of large-scale production based on metasurface point projection^[25]; Wang et al. demonstrated that DOE-based metasurfaces can also be highly integrated with on-chip light sources, opening new perspectives for the design of structured light systems with compactness, light weight, and scalability^[26–28]. Many structured light projections have been proposed. Zhang et al. designed a dot array based on vector diffraction theory with high overall diffusion efficiency and low efficiency error of target orders^[29], which can be used as a beam splitting device. Ni et al. also designed the structured light projection based on vector diffraction theory^[30], which can achieve an FOV angle of 120° with an efficiency of 59.1%. However, the first two devices have a limited number of diffraction orders and the area covered by the dot array is too small to be applied to depth perception. Using fast Fourier transform (FFT), designed structured light projection can generate more points. Kim et al. made a structured light projection with an FOV angle approaching 180°^[31], resulting in a dot array of about 10,000 dots covering the entire transmitted space. Hsu et al. made a system to project approximately 45,700 infrared (IR) dots from a compact 297-µm-dimention metasurface^[32]. Jing et al. used a structured light projection with 1201 points to reconstruct 3D information^[33]. However, a large FOV structured light projection designed using the Fresnel approximate diffraction algorithm, which only uses FFT to calculate far-field patterns, has the problems of pattern distortion and uneven intensity due to violation of the paraxial approximation.

In this paper, we try to overcome the problems that occur when designing large FOV structured light projection with FFT. We use the Rayleigh–Sommerfeld (R-S) integral instead of FFT to calculate the far-field diffraction of light to optimize the typical Gerchberg–Saxton (G-S) algorithm. We present the demonstration of two metasurface-based structured light projections that project dot arrays with two FOV angles of 60° and 144°, operating at near-infrared 940 nm suitable for 3D imaging. The polarization-independent metasurface consists of a silicon cylinder on a fused silica substrate. We use the optimized G-S algorithm to calculate the phase distribution of a single supercell. Then, the supercells are periodically arranged along the $x$ and $y$ directions, forming the whole metasurface. On the basis of regular dot array, we encode the dots to generate a pseudo-random pattern, which can improve the anti-interference ability. We experimentally measured structured light projections with an efficiency of 61.04% and 42.96% with a root-mean-square error (RMSE) of 0.45 and 0.75 in a 60° FOV sample and a 144° FOV sample, respectively.

We use the diffraction effect of a single scatterer and the interference effect caused by the periodic arrangement of scatterers to generate a dot array^[34–36]. According to the Huygens effect, each position in the wavefront can be considered as a point wave source that generates spherical secondary waves, and every subsequent wavefront can be considered as the envelope of these secondary waves. If the interaction between each unit cell is not considered, the light source passing through a metasurface composed of $n\times n$ unit cells will form $n\times n$ sub-wave sources. By designing the unit cells that make up the metasurface, each sub-wave source has the appropriate phase and amplitude to form the desired diffraction pattern. Considering the application field of structured light, we choose near-infrared, which is invisible to the human eye, as the working wavelength. In the near-infrared range, the vertical cavity surface-emitting laser (VCSEL) technology near 940 nm is relatively mature. So, the metasurface is designed for the wavelength of 940 nm; we choose amorphous Si (a-Si) as the material of the cylinder and silicon dioxide (${\mathrm{SiO}}_2$) as the material of the substrate, as shown in Fig. 1(a). Figure 1(b) shows the top view of the partial metasurface. The height $H$ of the unit cell is fixed at 820 nm, and the period $P$ is fixed at 460 nm. Using the finite-difference time-domain (FDTD) method, we simulate the relationship between the radius $R$ of the unit cell and the transmission phase difference of light passing through the unit cell, as shown in Fig. 1(c).

By using an iterative discrete 2D Fourier transform, i.e., the G-S algorithm, the phase required to generate the target diffraction pattern can be calculated. The far-field diffraction pattern used for calculating light in typical G-S algorithms can be represented by the following equation: $$U_o(x_o,y_o)=\frac{1}{j\lambda z}\exp (jkz)\iint U_i(x_i,y_i)\times \exp \left\{j\frac{k}{2z}[{(x_o-x_i)}^2+{(y_o-y_i)}^2]\right\}\mathrm{d}{x}_i\mathrm{d}{y}_i.$$(1)

In Eq. (1), $k=2\pi /\lambda$ is the number of waves in free space ($\lambda$ is the light wavelength), and $z$ is the diffraction distance. $U_o$ and $U_i$ are the electric field distributions on the imaging plane and on the metasurface plane, respectively. $(x_o,y_o)$ and $(x_i,y_i)$ are the corresponding coordinates. However, paraxial approximation is made when FFT is used for calculation, making it only applicable for small FOV angles calculations. For large FOV angles calculation, it may cause significant errors, including pattern distortion and uneven intensity^[10]. Therefore, we have made some optimization, using R-S diffraction instead of FFT to calculate the electric field distribution on the imaging plane, as shown in Fig. 2(a). R-S diffraction integral can be expressed as $$U_o(x_o,y_o)=\frac{e^{ikr}}{i\lambda z}\frac{z^2}{r^2}\iint U_i(x_i,y_i)\times e^{\frac{-i2z}{dz}(x_i\frac{x_{o}z}{r}+y_i\frac{y_{o}z}{r})}\mathrm{d}{x}_i\mathrm{d}{y}_i,$$(2)where $r$ is the distance between sample points on the imaging plane and the metasurface plane, $$r=\sqrt{{(x_o-x_i)}^2+{(y_o-y_i)}^2+z^2}.$$(3)

Equation (2) can be used to solve the diffraction problems at large angles because no paraxial approximation is performed. Compared to calculations using FFT, Eq. (2) can be more accurate^[37]. We obtain the phase required to generate far-field diffraction patterns by using the optimized G-S algorithm. The layout of the supercell, including $101\times 101$ unit cells, is generated by placing the unit cell in corresponding transmission phase. We make periodic arrangement of the designed supercells in the $x$ and $y$ directions, which is described by a 2D Dirac comb function. The far-field pattern is the product of the supercell diffraction effect and the 2D Dirac comb function after the Fourier transform. By designing the target diffraction pattern of a single supercell, each dot in the regular dot array can be encoded to produce a pseudo-random dot array, as shown in Figs. 2(b) and 2(c).

To verify the effectiveness of the R-S diffraction theory, we simulated two structured light projections for the same target pattern based on different theories. The target pattern is a square with a length and width of 0.8 m at a distance of 0.5 m from the metasurface. Figures 2(e) and 2(f) show simulated diagrams of the normalized light intensity distribution at a distance of 0.5 m generated by the structured light projections based on FFT (pattern A) and generalized R-S diffraction theory (pattern B), respectively. It can be clearly seen that pattern A has significant shape distortion, more like a cross. Pattern B is more like a square, with only a slight deformation at the four corners of the square. The reason for the slight deformation in the results may be that the selected metasurface units do not match the calculated phase well, as the phase corresponding to the metasurface units is simulated by a single unit in periodic boundary conditions. A single metasurface unit placed within the entire metasurface device may not satisfy its periodic boundary conditions, resulting in slight deformation. We take the light intensity from $-0.4\mathrm{m}$ to 0.4 m in the $x$ direction and $y$ direction as the target region, for efficiency and uniformity analysis. The efficiency and uniformity of pattern A are 44% and 0.78, the efficiency and uniformity of pattern B are 53% and 0.66, where efficiency is defined as the ratio of the sum of intensities in the target region to the sum of intensities across the entire region. Uniformity is defined as $$\frac{1}{I_{\text{mean}}}\sqrt{\frac{1}{M}\sum _{i=1}^M{(I_i-I_{\text{mean}})}^2}.$$(4)

Here, $M$ is the total number of points used for calculation, $I_i$ is the $i$th intensity, and $I_{\text{mean}}$ is the average intensity in target region.

We also conducted an experimental comparison on the improvement of distortion between R-S diffraction and the FFT algorithm. We designed a square with a diffraction angle of 30°. In order to obtain a full view of the projection pattern, we first photographed the far-field pattern on the side at a distance of 6.5 cm, as shown in Figs. 3(a) and 3(b). Figure 3(a) is designed by FFT and Fig. 3(b) is designed by R-S diffraction. We reduced the impact of zeroth order diffraction on photo shooting by making a hole in the screen to allow zeroth-order diffraction to pass through. In order to make the edges of the square clear enough, we adjusted the exposure time of the camera, so we did not calculate the efficiency of these two images. We judge the distortion situation by the angle of the square in the far-field pattern. We shot one corner of the square from the front, as shown in Figs. 3(c) and 3(d). The angle of the square designed by FFT and R-S theory methods ${\theta }_1$ is 60° and 84°, respectively. The target angle of the square ${\theta }_0$ is 90°, and distortion is defined as $1-{\theta }_1/{\theta }_0$. The distortion of DOE designed by FFT and R-S theory is 33.3% and 6.7%, respectively. Compared to the FFT method, the structured light projection based on the R-S diffraction theory shows improved shape, higher efficiency, and more uniform intensity.

The fabrication process is shown in Fig. 4(a). We grew an 820-nm-thick a-Si thin film on a 500 µm fused silica substrate using plasma-enhanced chemical vapor deposition (PECVD). Subsequently, a resist layer was spin-coated on the a-Si film. After electron beam lithography (EBL), the resist was removed and the patterns were transferred to the 20-nm-thick chromium (Cr) as a hard mask layer. With a mixture of ${\mathrm{C}}_4{\mathrm{F}}_8$ and ${\mathrm{SF}}_6$, we used inductively coupled plasma-reactive ion etching (ICP-RIE) to etch a-Si layer to form a-Si nanopillars. After etching the sample, we removed the hard mask and obtained the final structure. The scanning electron microscopy (SEM) images of the samples are shown in Figs. 4(b) and 4(c). The height of the unit cell is fixed at 820 nm, with a diameter range of 100 to 300 nm; the maximum aspect ratio is 8.2, and the minimum gap is 120 nm.

We evaluate the device using efficiency and RMSE. Efficiency is defined as the sum of the intensities converted into spots normalized to the intensity of the incident laser. RMSE is defined as $$\frac{1}{1/N}\sqrt{\frac{1}{N}\sum _{i=1}^N{(I_i-\frac{1}{N})}^2},$$(5)where $N$ is the total number of designed spots, and $i$ is the $i$th-order diffraction intensity. $N=360$ for the 60° FOV sample and $N=600$ for the 144° FOV sample. Vectorial electromagnetic calculation, executed by the commercial software Lumerical-FDTD solutions (Lumerical Solutions, Vancouver, Canada), is utilized to simulate the structures. We obtain an efficiency of 64.9% and 37.2% with an RMSE of 0.27 and 0.46 in a single supercell of the 60° FOV sample and the 144° FOV sample, respectively. For experimental measurement of intensity of diffracted beams, we set up an experimental apparatus shown in Fig. 5(a) to test the metasurface sample. The 940 nm laser (LD-PD PL-NL-940-A-A81-PA) is collimated through a collimator, and then an aperture is used to limit the spot size of the incident light. An IR camera (Allied Vision Goldeye G-030 TEC1) is used to capture diffraction patterns projected onto a paper screen. The diffraction patterns of light transmitted through the metasurfaces projected onto the paper screen are shown in Figs. 5(b) and 5(d). Figures 5(c) and 5(e) are the target diagrams of the metasurfaces. Because the near-infrared camera used for shooting is not on the optical axis, we performed a simple stretching process on Fig. 4(b) to make the projection range circular. We measured an efficiency of 61.04% and an RMSE of 0.45 for the 60° FOV sample and an efficiency of 42.96% and an RMSE of 0.75 for the 144° FOV sample. The efficiency mentioned above includes the zeroth-order efficiency; the zeroth-order efficiency is 12.04% for the 60° FOV sample and 17.86% for the 144° FOV sample of the incident light. One of the reasons for the high zeroth-order diffraction efficiency is that the radius of the incident light is greater than the radius of the sample, as increasing the number of supercell arrangements results in a decrease in the zeroth-order diffraction efficiency.

Table 1 lists a detailed comparison of this work with previous works, mostly designed by FFT or FDTD. Works that are designed based on FDTD have a limited number of points because the number of unit cells that make up the supercell is limited by computational complexity. Works that are designed based on FFT have the problems of pattern distortion and uneven intensity because of violating the paraxial approximation. Using R-S diffraction to design structured light projection can accurately generate a large quantity of dots at a large FOV without high computational complexity.

In summary, we have designed the metasurfaces working at 940 nm based on generalized R-S diffraction theory to project a pseudo-random dot array for 3D imaging. Designed based on generalized R-S diffraction theory, the large FOV diffractive pattern can be closer to the design pattern and the intensity can be more uniform. The low efficiency of the two samples may be attributed to the fabrication errors and the limited number of unit cells that make up a supercell, which can be further improved by adding unit cells in a single supercell. The proposed metasurface can be widely applied in motion perception, autonomous vehicles, and other structured light applications. The metasurface designed based on generalized R-S diffraction theory may have the advantage in large FOV structured light projection and holographic projection.