Ultra-dense moving cascaded metasurface holography by using a physics-driven neural network

Hongqiang Zhou; Xin Li; He Wang; Shifei Zhang; Zhaoxian Su; Qiang Jiang; Naqeeb Ullah; Naqeeb Ullah; Xiaowei Li; Yongtian Wang; Lingling Huang

doi:10.1364/OE.463104

1. Introduction

As a generic ultra-thin flat optics component, metasurface has attracted great enthusiasm due to its powerful multi-dimensional optical manipulation capabilities [1–9]. Due to the design freedom of meta-structure, the fundamental properties of light, such as amplitude, phase, polarization, wavelength, and spin/orbital angular momentum, can be independently or simultaneously modulated [10–15]. As an outstanding representative of optical information processing, metasurface holography demonstrates multi-dimensional conspicuous information-carrying and encrypting capacity. Braiding metasurfaces and computer-generated hologram algorithms can achieve vectorial holography, colourful holography, stereo holography, and dynamic holography [16–23]. Metasurface holography provides higher spatial bandwidth products, a larger diffraction angle, and integration capabilities compared to traditional optical elements.

Deep learning is an inevitable topic in the big data and artificial intelligence era because of its robust feature learning and systemic optimization capabilities [24,25]. As implemented in the computational optics field, deep learning neural networks mainly include two types, that is, data-driven and physics-driven neural networks, respectively [26–30]. The former one is mainly to build a training neural network model with a large amount of raw data for training, convergence, and generalization. This situation mainly shows the ability to learn the characteristics from the stereotype data or models [31–34]. While the latter one can be achieved by inserting a series of analytical physical/mathematical models. Such a physics-driven neural network is more suitable for known theoretical models because it has a faster convergence speed and low computational cost without collecting massive prior data. Therefore, it has excellent computational imaging and optical diffraction performance, especially in real-time computing and ultra-high information density application scenarios. Thereafter, with the help of deep learning neural network, the design limitations of few-layer metasurface holography can be eliminated to some extent, and the functionalities can be further improved [35–40].

Here we propose and demonstrate a cascaded metasurface holography strategy for multiple images reconstruction using a physical-driven neural network. And the cascaded metasurfaces consist of two complex amplitude profiles, where the second piece is a moving component along the forward propagation path. In detail, the first metasurface produces a diffractive pattern that accumulates different angular spectrum along the propagation path. The region of interest (ROI) within the diffractive pattern can generate enough differences by tuning the distance and meet the second decrypted metasurface, and then a completely different holographic image can be reconstructed in the far-field. Meanwhile, we built a physics-driven neural network model to inverse design the two cascaded complex amplitude distributions. On the other hand, the supercell in each layer for possessing complex-amplitude modulation is composed of 2 × 2 anisotropic nanofins based on the Pancharatnam−Berry (PB) phase principle. Hence the encoding of each metasurface can be achieved by using the data from a neural network. In this way, the information density and the flexibility of holographic display can be strikingly improved compared to traditional computer holographic technologies. And the information security and privacy can be guaranteed. In addition, the cascaded metasurface holography with one moving piece of metasurface may open new frontiers for optical encryption, data storage, dynamic display, and other applications.

2. Methods and results

In the following, we demonstrate the strategy of cascaded metasurface holography based on a physical-driven neural network illustrated in Fig. 1. By constructing an optical diffraction neural network model, the plane wave modulates by the front metasurface A (MA) and hits the second metasurface B (MB) after diffractive propagation within a separation distance. One cannot observe any useful information when illuminating the circularly polarization light upon the former MA. While another new piece of MB is cascaded on a forward diffractive path at 1 mm distance, the Table Tennis image is reconstructed in the far-field. When MB moves forward along the propagation axis with known step size, Athletics, Equestrian, Gymnastics Artistic, Taekwondo, Hockey images will appear in sequence, and even more images will be reconstructed moving MB forward further. By using the data generated from the neural network model, the cascaded metasurfaces can be inversely designed. Each supercell is composed of 2 × 2 nanofins to form the complex amplitude modulation. The structural dimensions of the four nanofins structures are the same, but their orientation angles are different. That is, we have found a set of holographic solutions in which two-phase profiles from both metasurfaces are cascaded through Fresnel diffraction propagation to form meaningful holographic reconfiguration.

Fig. 1. Schematic diagram of ultra-dense cascaded metasurface holography. A physics-driven neural network is used to inverse design the metasurface distribution. The separation between the two layers is the main variable. Each movement of a different distance produces a different holographic reconstruction image.

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Based on the above strategy, we construct a neural network with movable distances, as shown in Fig. 2. The separated distances (D₁, D₂, D₃…) between physical diffractive layer 1 (PDL1) and layer 2 (PDL2) are set as input parameters. The connection between both layers is constructed by the angular spectrum diffraction theory. The propagation of light satisfies the condition of a linear space-invariant system, which can be expressed as follows:

(1)$$U({x,y,d} )= {{\boldsymbol F}^{ - 1}}\left\{ {{\boldsymbol F}[{{U_A}({{x_0},{y_0}} )} ]\times \textrm{exp} \left( {jkd\sqrt {1 - {{({\lambda {f_x}} )}^2} - {{({\lambda {f_y}} )}^2}} } \right)} \right\}$$

where, $U({x,y,d} )$ is the diffraction distribution along propagation direction. ${\boldsymbol F}$ and ${{\boldsymbol F}^{ - 1}}$ is 2-D Fourier transform and inverse operation. ${U_A}({{x_0},{y_0}} )= A({{x_0},{y_0}} )\textrm{exp} ({j{\varphi_a}} )$ represents the optical field distribution of PDL1. $({{f_x},{f_y}} )$ is frequency coordinate. j is an imaginary unit. k is wave vector with $k = {{2\pi } / \lambda }$. d is propagation distance between PDL1 and PDL2 with $d = {d_0} + n\Delta d$, where ${d_0}$ is the initial distance between the two layers. $n\Delta d$ means (n+1)^th step distance. Note the moving step distance is not necessarily the same. To facilitate the demonstration, an evenly spaced propagation distance interval is selected. Then, the light field distribution (${U_{B + }}({x,y} )$) passing through the PDL2 can be expressed as:

(2)$${U_{B + }}({x,y} )= U({x,y,d} )\times {U_B}({x,y} )$$

where ${U_B}({x,y} )= B({x,y} )\textrm{exp} ({j{\varphi_b}} )$ represents the optical field distribution of PDL2. The distance is an important factor that determines the light field distribution in the angular spectrum diffraction theory. And this is a variable parameter for the cascaded system. From PDL2 to the output layer, one gets the reconstructed intensity distribution that satisfies the Fourier transform relationship in the far-field region.

(3)$$I({x,y} )= {|{{\boldsymbol F}[{{U_{B + }}({x,y} )} ]} |^2}$$

Then, we constructed the loss function (mean square error, MSE) of the network which connects the output and the ground truth ($G({x,y} )$):

(4)$$loss = {|{I({x,y} )- G({x,y} )} |^2}$$

Fig. 2. Schematic diagram of a physics-driven neural network model. We take the interval between the two layers and the complex amplitudes of the two layers as the input of the model. A loss function is constructed by comparing the holographic reconstruction image and the ground truth. After finite training, the complex amplitude distributions of the system are obtained.

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The Adam (Tensorflow 2.0) is chosen as the optimizer. The learning rate is set as 0.0001. The complex-amplitude matrix is 100 × 100. After a finite number of iterative neural network training, the neural network converged. The complex-amplitude distribution of PDL1 and PDL2 can be output. To build a complex-amplitude metasurface array, a supercell as complex amplitude pixel is designed as prototype with 2 × 2 nanofins. And the supercell is transformed into nanofins’ rotation angles according to an inverse dual-pole model solution. The output complex-response of the supercell comes from averaging the complex amplitude of single nanofin in the supercell. The complex amplitude of supercell can be expressed as follows:

(5)$$E = \textrm{exp} (j2\sigma {\theta _2}) + \textrm{exp} (j2\sigma {\theta _1})$$

where, θ₁ and θ₂ represent the azimuth angle of the diagonal (N1) and anti-diagonal (N2) orientation of the corresponding nanofin, respectively. σ means the helicity of the incident beam, which is defined as ±1, respectively. Finally, one can encode the azimuth matrix according to Eq. (5) to form the complex amplitude distributions of both layers accordingly.

To build the complex-amplitude metasurface, silicon nanofin is the selected as nano-antenna material, shown in Fig. 3(a). We used particle swarm optimization (PSO) to optimize an anisotropic nanofin using the finite differential time difference (FDTD) method. The lattice constant is fixed at 340nm, and the height is 500nm. The length and width of each nanofin are 260nm and 120nm, respectively. The broadband transmission curve of single nanofin is verified using rigorous coupled wave analysis method, shown in Fig. 3(b). The orthogonal handedness polarization conversion efficiency at broadband 720–1050nm exceeds 80%. According to Eq. (5), the theoretical and FDTD calculated intensity distribution are obtained and shown in Fig. 3(c–d). Those are not related to incident wavelength due to the dispersionless property of the PB phase. Therefore, we choose the incident wavelength of 845nm to verify the transmitted light field distribution of nanofin with the FDTD method. The simulated amplitude and phase distribution are demonstrated in Fig. 3(e–f). Comparing the FDTD simulation results and theoretical results, the distribution of amplitude and phase are almost the same.

Fig. 3. The design of metasurface. (a) Supercell structure. (b) The transmittance curve of each nanofin by using rigorous coupled wave analysis method. (c-d) The theoretical and FDTD calculated intensity distribution. (e-f) The intensity and phase distribution by using full-wave simulation.

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Further, we verified the cascaded metasurface holography using FDTD simulations in Fig. 4. Six propagation distances are set as discrete controllable parameters. Correspondingly, six different holographic images resulting from such a cascaded system can be reconstructed in the far-field. The distance between the two metasurfaces ranges from 1mm to 1.5mm, with a separation of 0.1mm. After 12000 epoch training, the loss function reaches the threshold of convergence. The complex amplitude and rotation profile distribution of the two-layer metasurface is calculated as shown in Fig. 4(a–b). We have obtained six holographic reconstruction images in the far-field using our physics driven neural network: the digits and letters ‘6’, ‘8’, ‘B’, ‘X’, ‘E’, ‘S’ in Fig. 4(c). Even though there is some noise between those images, the whole set is still recognizable with satisfactory qualities.

Fig. 4. FDTD simulation verification results. (a) The amplitude and phase and rotation profile distribution of the first layer. (b) The amplitude and phase and rotation profile distribution of the second layer. (c) The holographic reconstructed image is calculated by using the neural network. (d) Full-wave simulation results using the FDTD method by importing the realistic metasurface array.

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In order to quantitatively evaluate the image quality of holographic reconstruction, we establish evaluation functions. The mean square error (MSE) and the peak signal-to-noise ratios (PSNR) are expressed:

(6)$$\begin{array}{l} {L_{MSE}} = \frac{1}{{mn}}\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {{{[{O({i,j} )- G({i,j} )} ]}^2}} } \\ {L_{PSNR}} = 10{\log _{10}}\left( {\frac{{{{255}^2}}}{{{L_{MSE}}}}} \right) \end{array}$$

O(i,j) and G(i,j) represent the pixel value of holographic reconstruction and ground truth, respectively. m and n represent pixel size of images. Meanwhile, the cascaded metasurface arrays are imported into FDTD solutions for simulation. We use two steps for such cascaded systems to reduce the cost of computational resources. The light field generated by the first metasurface is diffracted propagation to a preset plane, and then the second metasurface layer is superimposed. The expected images are then reconstructed in k-space, in Fig. 4(d). The normalized MSEs (L_MSE/(255)²) of the reconstructed images with full wave simulations are 0.0353, 0.0333, 0.0396, 0.0293, 0.0291 and 0.0332, respectively. While the PSNRs of the reconstructed images are 14.52dB, 14.78dB, 14.02dB, 15.33dB, 15.36dB and 14.791dB, respectively. Note the phase noise may deteriorate the reconstructed images in the full-wave simulations compared to neural network output results. This is mainly due to the sampling rate for such a big metasurface array in the numerical simulation and matching error between the pixels when the light propagate from the first layer to the second layer. Nevertheless, the image qualities can be improved with larger optimized array and high computing power.

In fact, our proposed method enables more holographic storage with limited cascaded metasurface size with varying distances. After 20,000 iterations of training, we obtained a holographic reconstruction image that changed 26 distances, as shown in Fig. 5. The complex amplitude distribution of the neural network output cascaded system is shown in Fig. 5(a–b). We achieve the reconstruction of 26 images shown in Fig. 5(c). The distance between the two layers of metasurfaces ranges from 1mm to 2.25mm, with a step of 0.05mm. The simulated wavelength is 845nm. The pixel size is 0.68µm. We still get holographic reconstructions with recognizable features. Moreover, in the case of this configuration, there is still expanded room for improvement in the holographic information capacity. We select two kinds of different patterns, the digits, and capital letters. In the case of 26 hologram capacity, both their PSNR and structural similarity remain relatively uniform levels. Their mean normalized MSE and PSNR are 0.017, 18.15dB, respectively. For the pattern of digits and letters, the structural similarity (SSIM) of the holographic reconstruction images are high. The method for calculating SSIM is:

(7)$$SSIM({O,G} )= \frac{{({2{\mu_O}{\mu_G} + {C_1}} )({2{\sigma_{OG}} + {C_2}} )}}{{({\mu_O^2 + \mu_G^2 + {C_1}} )({\sigma_O^2 + \sigma_G^2 + {C_2}} )}}$$

where µ_O and µ_G represent the mean value of holographic reconstruction (O) and ground truth(G), respectively. σ_O and σ_G represent the variance of images O and G, respectively. And σ_OG represents the covariance of images O and G. C₁, C₂ are default constants of (0.01 × 255)² and (0.03 × 255)², respectively. Furthermore, in order to test the performance of the holographic storage capacity of this method, we calculated the holographic reconstruction results of 6 to 50 images, respectively, with the normalized mean square error (MSE) and SSIM as the evaluation shown in Fig. 6. From the calculated curve results, as the number of holograms increases, the reconstructed image's quality will gradually deteriorate as shown in Fig. 6(a). The normalized MSE between reconstruction and ground-truth is becoming increasing. This is mainly because the information capacity of two cascaded limited-size metasurfaces is fixed by pixel numbers. The storage of more holograms will lead to the goal of convergence only by weighing the average loss of all outputs in the calculation process of the neural network. In addition, we also tested the effect of the separation distance between the two metasurfaces on the holographic reconstruction in Fig. 6(b). It is also concluded that as the separation distance may also affect the reconstruction qualities a bit. Furthermore, we calculated the best crosstalk-free separated distance at 50µm in case of 20 pictures capacity. Those are consistent with theoretical expectations.

Fig. 5. Neural network output results. (a) The amplitude and phase and rotation profile distribution of the first layer. (b) The amplitude and phase distribution and rotation profile of the second layer. (c)The simulated holographic reconstructions based on cascades 100 × 100 complex-amplitude metasurfaces. The separated distance between two layers is varied from 1 mm to 2.25 mm, step 0.05 mm.

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Fig. 6. The metrics of cascaded complex-amplitude metasurface reconstructions. (a) Metric curve with information capacity enhancement. (b) Metric curve with separated distance moving.

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3. Summary

In summary, we propose and demonstrate a holographic encryption strategy using a moving cascaded complex-amplitude metasurface. We construct a physical-driven neural network to inverse design the cascaded holographic system with a precise and easier reliable method. Compared with the traditional computer-generated holograms, the holographic encryption information capacity and storage density can be strikingly increased while ensuring the image qualities by utilizing the limited space bandwidth product. The in-plane alignment error is on the order of one pixel, while the axial distance error is about 2 µm in the case of 6 hologram capacity. In the case of misalignment, holographic reconstruction quality can be significantly reduced. Therefore, the alignment required accuracy of this method is relatively high. The holograms stored in the cascaded metasurfaces are strictly encrypted, while the single layer has no information displayed. Apart from that, the experimental challenges can be alleviated by increasing the array size and pixel size. High-precision alignment is achieved through high-precision fabrication and experimental setups for high quality experimental detection. Therefore, Meanwhile, such cascaded metasurfaces can be used for massive information storage, optical camouflage, and dynamic holographic display. This further enriches many practical applications in other fields such as active modulation, dynamic holographic display, encryption, and storage.

Funding

National Key Research and Development Program of China (2021YFA1401200); National Natural Science Foundation of China (U21A20140, 92050117, 12104046, 62005017); Beijing Outstanding Young Scientist Program (BJJWZYJH01201910007022); Fok Ying Tung Education Foundation (161009); Administrative Commission of Zhongguancun Science Park (Z211100004821009).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Ultra-dense moving cascaded metasurface holography by using a physics-driven neural network

Abstract

1. Introduction

2. Methods and results

3. Summary

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (7)

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