Low-noise metasurface holographic 3D display method based on double spatial constraints

Abstract

The invention provides a low-noise metasurface holographic 3D display method based on double spatial constraints, and the method comprises the following four steps: 1, carrying out the layering processing of a 3D object, obtaining a target plane light field, calculating the corresponding complex amplitude distribution based on the angular spectrum diffraction principle, and obtaining a target plane light field; by adding double constraints of a step function and bandwidth limitation to the spatial frequency, precise calculation of holographic plane complex amplitude is realized; 2, performing back propagation on the obtained holographic plane complex amplitude and the randomly generated pure phase hologram to obtain a target complex amplitude and a calculation complex amplitude respectively; step 3, generating a loss function by comparing the difference between the amplitude and the phase in the calculated complex amplitude and the target complex amplitude, and continuously iteratively optimizing the loss function through a stochastic gradient algorithm to enable the calculated complex amplitude to approach the target complex amplitude to obtain an optimized pure phase hologram; and 4, designing a metasurface structure according to the optimized pure phase hologram, and when laser irradiates the metasurface structure, reproducing a low-noise reconstructed image of the 3D object.

Description

I. Technical Field

[0001] This invention relates to holographic display technology, and more specifically, to a low-noise metasurface holographic 3D display method based on dual spatial constraints. II. Background Technology

[0002] Holographic display technology can record and reconstruct all wavefront information of 3D objects, opening up new paths for achieving immersive and realistic 3D displays. Metasurfaces, as planar optical elements with subwavelength-level pixel periods, possess unique advantages such as wide viewing angles and high modulation degrees of freedom, providing new ideas for the development of holographic display technology and attracting widespread attention. However, due to the high coherence of lasers, the reconstructed images of 3D objects suffer from severe speckle noise. Metasurface holograms based on the Gerchberg-Saxton (GS) algorithm are widely used due to their computational simplicity, but this method has a tendency for local optimization, offering limited help in solving the speckle noise problem, and is often affected by artifacts. Some researchers have proposed a deep-depth 3D metasurface holographic display technology based on Fresnel diffraction, employing zero-filling techniques to suppress speckle noise and achieving reconstruction over a depth range of 0.95 decimeters, but at the expense of the reconstructed image resolution. In recent years, deep learning algorithms have attracted much attention due to their excellent speckle noise suppression and image quality improvement capabilities. However, the diffraction parameters related to their light field must be pre-set, limiting their flexibility and adaptability in the hologram optimization process. Furthermore, existing deep learning networks typically require time-consuming training for optimization without considering the intrinsic mechanism of speckle noise generation. How to generate low-noise metasurface holograms with subwavelength pixel periods to achieve high-resolution 3D metasurface holographic displays with a large depth range remains a pressing problem. III. Summary of the Invention

[0003] This invention proposes a low-noise metasurface holographic 3D display method based on dual spatial constraints. (See attached diagram) Figure 1As shown, the method includes four steps: First, the 3D object is processed into layers to obtain the target plane light field. Then, the corresponding complex amplitude distribution is calculated based on the principle of angular spectrum diffraction. By adding a step function and bandwidth constraint to the spatial frequency, the accurate calculation of the holographic plane complex amplitude is achieved, and speckle noise caused by high-frequency information components is suppressed. Second, the obtained holographic plane complex amplitude and the randomly generated pure phase hologram are backpropagated to obtain the target complex amplitude and the calculated complex amplitude, respectively. Third, a loss function is generated by comparing the difference in amplitude and phase between the calculated complex amplitude and the target complex amplitude. The loss function is iteratively optimized using a stochastic gradient algorithm to make the calculated complex amplitude approximate the target complex amplitude. When the loss function converges, the optimized pure phase hologram is obtained. Fourth, a metasurface structure is designed based on the optimized pure phase hologram. When the metasurface structure is illuminated by a laser, a low-noise reconstructed image of the 3D object is reproduced.

[0004] In step one, the 3D object is first layered based on depth information to obtain the target plane light field for each layer. Then, based on the principle of angular spectral diffraction, the complex amplitude distribution propagating from different planes of the 3D object to the holographic plane is calculated. The 3D object is considered to be composed of n 2D planes at different depths, with diffraction distances z1, z2, ..., z... n The corresponding target plane light fields are represented as I1(x,y), I2(x,y), ..., I n The complex amplitude distributions corresponding to (x,y) are represented as U1(x,y,z1), U2(x,y,z2), ..., U n (x,y,z n Taking the case with a diffraction distance of z1 as an example, the target plane light field is I1(x,y), and the complex amplitude distribution U1(x,y,z1) after angular spectrum diffraction is expressed as:

[0005]

[0006] in Let j be the phase distribution of the transfer function, λ be the imaginary number, λ be the wavelength of the laser, and k = 2π / λ be the wave number. and f represents the Fast Fourier Transform and the Inverse Fast Fourier Transform, respectively. ξ and f η These are the spatial frequencies in the x and y directions, respectively.

[0007]

[0008] Where N and M represent the number of sampled pixels in the horizontal and vertical directions, respectively, Δx and Δy are the sampling intervals in the horizontal and vertical directions, respectively, and a and b are integer sequences.

[0009] When N and M are even numbers When N and M are odd numbers In order to achieve For the correct calculation, the calculated value under the square root must be greater than 0. Therefore, a step function S(f) is introduced. ξ ,f η As the first constraint condition:

[0010]

[0011] Since each calculation of the diffraction process between the target plane and the holographic plane in the iterative loop involves the calculation of Fourier transform and inverse Fourier transform, taking the ξ-axis direction as an example, the case with the largest phase slope is expressed as:

[0012]

[0013] According to the Nyquist sampling theorem, to correctly resolve spatial frequencies, the maximum phase slope along the ξ-axis should be less than π / Δξ, where Δξ is the spatial sampling interval. This maximum phase slope condition must also be satisfied in η-dimensional space. Therefore, a bandwidth limiting function B(f) is introduced. ξ ,f η This serves as a second constraint to suppress speckle noise in the loop.

[0014]

[0015] Where f ξ_limit and f η_limit The common critical spatial frequency is determined by the sampling interval, propagation distance, and wavelength.

[0016]

[0017] Introducing the step function S(f) ξ ,f η ) and bandwidth limiting function B(f ξ ,f η After effectively suppressing speckle noise using dual constraints, the calculated target plane optical field I i The holographic plane complex amplitude U corresponding to (x,y) i (x,y,z i ) is represented as:

[0018]

[0019] Through U1(x,y,z1),…,U n (x,y,z n The linear superposition of the 3D objects yields the holographic plane complex amplitude U(x,y) corresponding to the object:

[0020]

[0021] In step two, when the diffraction distance is z1, the complex amplitude of the holographic plane corresponding to the 3D object is backpropagated to the target plane to obtain the target complex amplitude T(x,y,z1) at that diffraction distance:

[0022]

[0023] Meanwhile, the initial state randomly generates a phase distribution as follows: The pure phase hologram is backpropagated to a plane at a distance of -z1, and the complex amplitude C(x,y,z1) is finally calculated.

[0024]

[0025] In step three, the difference between the calculated complex amplitude and the target complex amplitude is compared, and the root mean square error is introduced as the loss function L, with the expression:

[0026] L=αMSE(arg(T),arg(C))+βMSE(abs(T),abs(C)) (13)

[0027] Where α and β are the loss weights, arg(·) represents the extraction of the phase component of the complex number, and abs(·) represents the extraction of the amplitude of the complex number. The stochastic gradient descent algorithm is used to determine whether the loss function has converged, obtaining the phase distribution after initial optimization, completing the first iteration. The initial phase distribution is replaced with the phase distribution of the first optimization to generate a new complex amplitude, and the next loop begins. The iteration loop stops when the loss function converges, and the final optimized pure phase hologram is derived.

[0028] In step four, a metasurface structure is designed based on the optimized pure phase hologram. Based on the principle of geometric phase modulation, amorphous silicon nanorods are periodically arranged on a quartz substrate, and phase modulation is achieved by rotating the angle of the rectangular amorphous silicon nanorods on the metasurface. Finally, a low-noise reconstructed image of the 3D object is displayed by irradiating the metasurface structure with a laser. IV. Description of the attached drawings

[0029] Appendix Figure 1 This is a flowchart of a low-noise metasurface holographic 3D display method based on dual spatial constraints according to the present invention.

[0030] Appendix Figure 2 These are simulation and optical experiment reconstruction results of a low-noise metasurface holographic 3D display method based on dual spatial constraints according to the present invention. (Attached) Figure 2(a)-(b) show the reconstructed effects of the letters "H" and "N" respectively when simulated using the method proposed in this invention; Appendix Figure 2 (c)-(d) show the reconstruction results of the letters "H" and "N" respectively when using the traditional GS algorithm for simulation.

[0031] It should be understood that the above figures are only schematic and are not drawn to scale. V. Detailed Implementation Methods

[0032] The following detailed embodiments of the low-noise metasurface holographic 3D display method based on dual spatial constraints proposed in this invention further illustrate the invention. It is important to note that the following embodiments are for illustrative purposes only and should not be construed as limiting the scope of protection of this invention. Any non-essential improvements and adjustments made to this invention by those skilled in the art based on the above description are still within the scope of protection of this invention.

[0033] One embodiment of the present invention involves selecting the letters "H" and "N" located at two different depth planes as 3D objects with a resolution of 2500×2500. The diffraction distances of "H" and "N" are set to 8mm and 38mm, respectively. A metasurface is fabricated using the optimized pure phase hologram obtained according to the method proposed in this invention. The fabricated metasurface is a transmission-type dielectric metasurface based on geometric phase modulation, with a resolution of 2500×2500. The height, length, and width of the metasurface structural unit are 600nm, 160nm, and 80nm, respectively. A 671nm red laser is used as the reconstructed light to illuminate the metasurface. The simulation and optical reconstruction results are attached. Figure 2 As shown. Among them, the appendix Figure 2 (a)-(b) show the reconstructed effects of the letters "H" and "N" respectively when simulated using the method proposed in this invention. Figure 2 Images (c)-(d) show the reconstructed effects of the letters "H" and "N" respectively when simulated using the traditional GS algorithm. It can be seen that the method proposed in this invention can clearly reconstruct the corresponding 3D object in the target depth plane. When the depth is 8mm, the letter "H" is clearly reconstructed, while the letter "N" appears blurred; when the depth is 38mm, the letter "N" is clearly reconstructed, while the letter "H" appears blurred. Compared to the traditional GS algorithm, the image reconstructed by the method proposed in this invention shows better noise suppression. Experimental results verify that the method proposed in this invention achieves low-noise metasurface holographic 3D display.

Claims

1. A low-noise metasurface holographic 3D display method based on dual spatial constraints, characterized in that, The main steps are as follows: First, the 3D object is processed into layers to obtain the target plane light field. Then, the corresponding complex amplitude distribution is calculated based on the principle of angular spectrum diffraction. By adding a step function and bandwidth constraint to the spatial frequency, the accurate calculation of the holographic plane complex amplitude is achieved, and speckle noise caused by high-frequency information components is suppressed. Second, the obtained holographic plane complex amplitude and the randomly generated pure phase hologram are backpropagated to obtain the target complex amplitude and the calculated complex amplitude, respectively. The third step is to generate a loss function by comparing the difference between the amplitude and phase in the calculated complex amplitude and the target complex amplitude, and to continuously optimize the loss function through the stochastic gradient algorithm so that the calculated complex amplitude approaches the target complex amplitude. When the loss function converges, the optimized pure phase hologram is obtained. The fourth step is to design a metasurface structure based on the optimized pure phase hologram. When the laser irradiates the metasurface structure, a low-noise reconstructed image of the 3D object is reproduced.

2. The low-noise metasurface holographic 3D display method based on dual spatial constraints according to claim 1, characterized in that, In step one, based on the depth information, the 3D object is processed into layers to obtain the target plane light field of each layer. Then, based on the principle of angular spectral diffraction, the complex amplitude distribution propagating from different planes of the 3D object to the holographic plane is calculated. The 3D object is considered to be composed of n 2D planes at different depths, with diffraction distances of z1, z2, ..., z... n The corresponding target plane light fields are represented as I1(x,y), I2(x,y), ..., I n The complex amplitude distributions corresponding to (x,y) are represented as U1(x,y,z1), U2(x,y,z2), ..., U n (x,y,z n Taking the diffraction distance as z1 as an example, the target plane light field is I1(x,y), and the complex amplitude distribution U1(x,y,z1) after angular spectrum diffraction is expressed as: in Let j be the phase distribution of the transfer function, λ be the imaginary number, λ be the wavelength of the laser, and k = 2π / λ be the wave number. and Let f represent the Fast Fourier Transform and the Inverse Fast Fourier Transform, respectively. ξ and f η These are the spatial frequencies in the x and y directions, respectively. Where N and M represent the number of sampling pixels in the horizontal and vertical directions, respectively, Δx and Δy are the sampling intervals in the horizontal and vertical directions, respectively, and a and b are integer sequences; In order to achieve For the correct calculation, the calculated value under the square root must be greater than 0. Therefore, a step function S(f) is introduced. ξ ,f η As the first constraint condition: Since each calculation of the diffraction process between the target plane and the holographic plane in the iterative loop involves the calculation of Fourier transform and inverse Fourier transform, taking the ξ-axis direction as an example, the case with the largest phase slope is expressed as: According to the Nyquist sampling theorem, to correctly resolve spatial frequencies, the maximum phase slope along the ξ-axis should be less than π / Δξ, where Δξ is the spatial sampling interval. This maximum phase slope condition must also be satisfied in η-dimensional space, leading to the introduction of a bandwidth limiting function B(f). ξ ,f η This serves as a second constraint to suppress speckle noise in the loop. Where f ξ_limit and f η_limit The common critical spatial frequency is determined by the sampling interval, propagation distance, and wavelength. Introducing the step function S(f) ξ ,f η ) and bandwidth limiting function B(f ξ ,f η After effectively suppressing speckle noise using dual constraints, the calculated target plane optical field I i The holographic plane complex amplitude U corresponding to (x,y) i (x,y,z i ) is represented as: Through U1(x,y,z1),…,U n (x,y,z n The linear superposition of the 3D objects yields the holographic plane complex amplitude U(x,y).

3. The low-noise metasurface holographic 3D display method based on dual spatial constraints according to claim 1, characterized in that, In step two, when the diffraction distance is z1, the complex amplitude of the holographic plane corresponding to the 3D object is backpropagated to the target plane to obtain the target complex amplitude T(x,y,z1) at that diffraction distance: Meanwhile, the initial state randomly generates a phase distribution as follows: The pure phase hologram is backpropagated to a plane at a distance of -z1, and the complex amplitude C(x,y,z1) is finally calculated. In step three, the difference between the calculated complex amplitude and the target complex amplitude is compared, and the root mean square error is introduced as the loss function L, with the expression: L=αMSE(arg(T),arg(C))+βMSE(abs(T),abs(C)) Where α and β are loss weights, arg(·) represents extracting the phase component of the complex number, and abs(·) represents extracting the amplitude of the complex number. The stochastic gradient algorithm is used to determine whether the loss function has converged, and the phase distribution after the initial optimization is obtained. The first iteration is completed. The phase distribution after the initial optimization is used to replace the initial phase distribution to generate a new complex amplitude and start the next loop. The iteration loop stops when the loss function converges, and the final optimized pure phase hologram is derived.

4. The low-noise metasurface holographic 3D display method based on dual spatial constraints according to claim 1, characterized in that, In step four, a metasurface structure is designed based on the optimized pure phase hologram; Based on the principle of geometric phase modulation, amorphous silicon nanorods are periodically arranged on a quartz substrate, and the phase is modulated by rotating the angle of the rectangular amorphous silicon nanorods on the metasurface. Finally, a low-noise reconstructed image of a 3D object is displayed by irradiating the metasurface structure with a laser.

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