An optical beam shaping metasurface device and a method for manufacturing the same

Abstract

This invention discloses a beam-shaping metasurface device and its fabrication method. The metasurface device includes a substrate and a metasurface located on the substrate. The fabrication method is as follows: First, set the characteristic parameters of the metasurface unit, the characteristic parameters of the input and output beams, and the iterative optimization parameters; then calculate the amplitude distribution of the input surface, the target amplitude distribution of the device surface, the target amplitude distribution of the output surface, and the initial phase distribution; next, perform selective amplitude constraint iterative optimization to obtain the optimal phase distribution; then calculate the pre-collimated phase distribution based on the characteristic parameters of the input and output beams, and calculate the metasurface compensation phase distribution based on the optimal phase distribution and the pre-collimated phase distribution; finally, determine the geometric parameters of each metasurface unit based on the correspondence between the metasurface compensation phase distribution and the geometric parameters-phase of the metasurface unit, and fabricate the metasurface device. This invention improves the beam-shaping quality of the metasurface device.

Description

Technical Field

[0001] This invention relates to the field of beam shaping technology, and in particular to a beam shaping metasurface device and its fabrication method. Background Technology

[0002] Beam shaping technology generates beams with specific energy distribution patterns by controlling the intensity and phase distribution of light, and has been widely used in industry, military, medical, and scientific research. Traditional beam shaping is mainly achieved through aspherical lens groups, microlens arrays, diffractive optical elements, and spatial light modulators, which are bulky and complex, hindering the miniaturization of optical systems. Metasurfaces are artificial subwavelength micro / nanostructures; carefully designed micro / nanostructures can achieve various beam shaping methods, greatly simplifying the structure of beam shaping devices.

[0003] Calculating the compensated phase distribution is an important step in the metasurface design process, and the classic GS algorithm is usually used for calculation. The traditional GS algorithm directly replaces the modulus of the complex amplitude distribution of the output surface with the target amplitude distribution of the output surface, and the amplitude constraint is relatively strict. On the one hand, this strict constraint condition is prone to large fluctuations in the phase distribution, resulting in phase singularities on the device surface, and thus generating speckle noise on the output surface. On the other hand, the strict constraint condition limits the algorithm's search range for phase, which can easily cause the algorithm to get stuck in a local optimum and stagnate, resulting in the inability to further improve the shaping quality. In 1996, Harald Aagedal et al. (Aagedal H, Schmid M, Beth T, et al. Theory of speckles indiffractive optics and its application to beam shaping[J]. Journal of modern optics, 1996, 43(7):1409-1421.) of the University of Karlsruhe, Germany, derived and proved the relationship between phase singularities and speckle, and proposed an improved algorithm with a spherical initial phase distribution and relaxed amplitude constraint conditions, which eliminates speckle by suppressing phase singularities. Under relaxed constraints, the output surface is divided into a signal region with light intensity and a background region without light intensity based on whether there is light intensity at each target amplitude sampling point. Relaxed constraints are applied to the amplitude distribution within the signal region, and the relaxation of the constraints is adjusted by the amplitude constraint coefficient, ensuring that the constrained amplitude distribution is influenced by both the target amplitude distribution and the current amplitude distribution of the output surface. No constraints are applied to the amplitude distribution within the background region. Under relaxed constraints, the phase distribution fluctuation is smaller, effectively improving the problems of speckle noise and algorithm stalling.

[0004] On the other hand, the GS algorithm has insufficient sampling accuracy for light intensity, resulting in a significant difference between the actual beam shaping effect and the numerical simulation results. In 2002, Tan Qiaofeng et al. (Tan Qiaofeng, Yan Yingbai, Jin Guofan, et al. Refined design of diffractive optical beam smoothing devices [J]. Chinese Journal of Lasers, 2002, 29(1):29-32. DOI:10.3321 / j.issn:0258-7025.2002.01.009.) modified the one-dimensional design formula for diffractive optical beam smoothing devices and proposed an oversampling method. Without changing the sampling accuracy of the device surface, the number of spatial frequency samples was increased, reducing the frequency domain sampling interval of the complex amplitude distribution of the output surface obtained by discrete Fourier transform. The improved method satisfied the light intensity sampling accuracy requirement, and the actual beam shaping effect matched the numerical simulation results. Kexuan Liu et al. from Tsinghua University (Liu K, He Z, Cao L. Double amplitude freedom Gerchberg–Saxton algorithm for generation of phase-only hologram with speckle suppression[J]. Applied Physics Letters, 2022, 120(6).) integrated several improved methods, including applying a spherical initial phase, using relaxed constraints, and oversampling, to propose the "double amplitude degree of freedom GS algorithm," which significantly improved the beam shaping effect compared to the traditional GS algorithm. While the algorithm continues to improve, researchers are also applying the improved algorithm to the design of beam-shaping metasurfaces. Yanling Sun et al. from Yangtze Normal University (Sun Y, He D, Liu Y, et al. Design of beam shaping and focusing metasurface device based on GS algorithm[J]. Optical Materials, 2020, 109: 110247.) proposed a beam-shaping metasurface designed based on the improved GS algorithm in 2020, with a theoretical diffraction efficiency exceeding 90%.

[0005] Besides the constraints themselves, the constraint range also significantly impacts beam shaping quality. However, the constraint range of the dual-amplitude degree-of-freedom GS algorithm is not adjustable, leading to persistent stagnation issues and generally poor beam shaping results. To improve the beam shaping quality of metasurface devices, a fabrication method for metasurface devices with adjustable constraint ranges is needed. Summary of the Invention

[0006] The purpose of this invention is to provide a beam-shaping metasurface device and its fabrication method, so as to improve the shaping quality of the beam-shaping metasurface device.

[0007] The technical solution for achieving the objective of this invention is: a method for fabricating a beam-shaping metasurface device, comprising the following steps:

[0008] Step 1: Set parameters: Set metasurface unit characteristic parameters, input and output beam characteristic parameters, and iterative optimization parameters;

[0009] Step 2: Model building: Based on the metasurface unit characteristic parameters and the input and output beam characteristic parameters, calculate the input surface amplitude distribution, the device surface target amplitude distribution, the output surface target amplitude distribution, and the initial phase distribution;

[0010] Step 3: Perform selective amplitude constraint iterative optimization: Based on the iterative optimization parameters, the target amplitude distribution on the device surface, the target amplitude distribution on the output surface, and the initial phase distribution, perform selective amplitude constraint iterative optimization to obtain the optimal phase distribution;

[0011] Step 4: Calculate the metasurface compensation phase distribution: Calculate the pre-collimated phase distribution based on the input and output beam characteristic parameters, and calculate the metasurface compensation phase distribution based on the optimal phase distribution and the pre-collimated phase distribution;

[0012] Step 5: Determine the geometric parameters of each metasurface unit: Based on the correspondence between the metasurface compensation phase distribution and the geometric parameters-phase of the metasurface unit, determine the geometric parameters of each metasurface unit and fabricate the metasurface device.

[0013] Further, the metasurface unit characteristic parameters mentioned in step 1 include the number of metasurface units in the first direction, the number of metasurface units in the second direction, the period of metasurface units in the first direction, the period of metasurface units in the second direction, the refractive index of the substrate material, and the substrate thickness; the first direction represents a direction in which the metasurface units are arranged; the second direction represents a direction perpendicular to the first direction; and both the first direction and the second direction are perpendicular to the beam propagation direction.

[0014] The input and output beam characteristic parameters include beam wavelength, Gaussian beam waist radius, input surface to beam waist distance, reference distance, super-Gaussian beam waist radius, and super-Gaussian beam order.

[0015] The iterative optimization parameters include the number of iterations, the number of zero-filling parameters, the amplitude constraint coefficient, and the minimum selected amplitude. The minimum selected amplitude is the criterion for selecting the condition in the selective amplitude constraint. It represents the critical value of the target amplitude of the output surface to which the constraint is applied and determines the size of the constraint range. In the dual-amplitude degree of freedom GS algorithm, the minimum selected amplitude is fixed to 0.

[0016] Furthermore, the model described in step 2 includes an input surface, a device surface, and an output surface;

[0017] The input surface represents the position where the metasurface receives the Gaussian beam, and is located on the plane where the bottom surface of the metasurface is located;

[0018] The device surface represents the position where the Gaussian beam is output after being modulated by the metasurface, and is located on the plane of the top surface of the metasurface;

[0019] The output surface represents the position where the shaping result is observed, located in a far-field plane parallel to the input surface and the device surface;

[0020] The input surface phase distribution is obtained by metasurface modulation to obtain the device surface phase distribution. The input surface amplitude distribution is the same as the device surface amplitude distribution. The output surface complex amplitude distribution is obtained by performing a Fourier transform on the device surface complex amplitude distribution.

[0021] Furthermore, the input surface target amplitude distribution mentioned in step 2 is a Gaussian distribution, and the input surface amplitude distribution satisfies:

[0022]

[0023] Among them, |E input (x,y)| represents the amplitude distribution of the input surface; {.} normalization The expression represents the normalization operation; x represents the position in the first direction; y represents the position in the second direction; z represents the distance from the input surface to the beam waist; ω0 represents the beam waist radius of the Gaussian beam; λ represents the beam wavelength; R(z) is the wavefront curvature radius, and satisfies:

[0024]

[0025] Φ(z) is the phase factor of the Gaussian beam, and satisfies:

[0026]

[0027] The target amplitude distribution on the device surface is Gaussian, and the target amplitude distribution on the device surface is the same as the target amplitude distribution on the input surface.

[0028] The output surface target amplitude distribution is a super-Gaussian distribution, and the output surface target amplitude distribution satisfies:

[0029]

[0030] Among them, |E target (f x f y | represents the target amplitude distribution on the output surface; {.} normalization z represents the normalization operation. f Indicates the reference distance; f x f represents the spatial frequency in the first direction; yω represents the spatial frequency in the second direction. s The beam waist radius represents the super-Gaussian beam; n represents the order of the super-Gaussian beam.

[0031] The initial phase distribution refers to the phase distribution of the device surface during the first iteration, and satisfies:

[0032]

[0033] Among them, Ф origin (n x n y ) represents the initial phase distribution; n x Indicates the element number in the first direction; n y The number of elements in the second direction is represented by ; N represents the number of zero-fill elements; and n1 represents the number of metasurface elements in the first direction.

[0034] Furthermore, the iterative optimization parameters mentioned in step 3 include the number of iterations, the number of zero-filling parameters, the amplitude constraint coefficient, and the minimum selected amplitude; wherein, the minimum selected amplitude is the criterion for judging the selection condition in the selective amplitude constraint, and represents the critical value of the target amplitude of the output surface to which the constraint is applied.

[0035] Furthermore, the selective amplitude constraint iterative optimization described in step 3 involves performing selective amplitude constraint iterative optimization based on the iterative optimization parameters, the target amplitude distribution on the device surface, the target amplitude distribution on the output surface, and the initial phase distribution to obtain the optimal phase distribution, as detailed below:

[0036] Step 3.1: Fill the target amplitude distribution and initial phase distribution on the device surface with zeros according to the zero-filling quantity, and generate the complex amplitude distribution on the device surface;

[0037] Step 3.2: Perform a discrete Fourier transform on the complex amplitude distribution of the device surface and normalize the result to obtain the complex amplitude distribution of the output surface.

[0038] Step 3.3: Calculate the root mean square error of the output surface light intensity distribution based on the output surface light intensity distribution and the target light intensity distribution on the output surface;

[0039] Step 3.4: Fill the target amplitude distribution of the output surface with zeros according to the zero-filling quantity, and apply constraints to the output surface amplitude corresponding to the sampling points that meet the selection conditions according to the amplitude constraint coefficient. Do not apply constraints to the output surface amplitude corresponding to the sampling points that do not meet the selection conditions, and obtain the constrained complex amplitude distribution of the output surface.

[0040] Step 3.5: Perform an inverse discrete Fourier transform on the constrained output surface complex amplitude distribution to obtain the constrained device surface complex amplitude distribution, and replace the device surface amplitude distribution with the device surface target amplitude distribution as the device surface complex amplitude distribution for the next cycle;

[0041] Step 3.6: Repeat steps 3.2 to 3.5 until the number of iterations reaches the number of iterations for optimization. The device surface phase distribution corresponding to the minimum root mean square error is taken as the optimal phase distribution.

[0042] Furthermore, the complex amplitude distribution of the output surface described in step 3.2 satisfies:

[0043]

[0044] Among them, E output (f x f y ) represents the complex amplitude distribution of the output surface; E input (x, y) represents the complex amplitude distribution of the input surface; Represents the Discrete Fourier Transform; {.} normalization f represents the normalization operation; x f represents the spatial frequency in the first direction; y y represents the spatial frequency in the second direction; x represents the position in the first direction; y represents the position in the second direction.

[0045] The root mean square error described in step 3.3 satisfies:

[0046]

[0047] Wherein, RMSE represents the root mean square error; Indicates the light intensity distribution on the output surface; Indicates the target light intensity distribution on the output surface; f x f represents the spatial frequency in the first direction; y Indicates the spatial frequency in the second direction;

[0048] The selection condition mentioned in step 3.4 is that the target amplitude of the output surface is greater than the minimum selected amplitude;

[0049] The complex amplitude distribution of the output surface after constraint as described in step 3.4 satisfies:

[0050]

[0051] Among them, E′ output (f x f y ) represents the complex amplitude distribution of the output surface after constraint; E output (f x f y ) represents the complex amplitude distribution of the output surface; |E target (f x f y | represents the target amplitude distribution on the output surface; E selectIndicates the minimum selectable amplitude; φ output (f x f y ) represents the phase distribution of the output surface; f x f represents the spatial frequency in the first direction; y The frequency in the second direction is represented by α; α represents the amplitude constraint coefficient.

[0052] The constrained complex amplitude distribution of the device surface described in step 3.5 satisfies:

[0053]

[0054] Among them, E′ input (x, y) represents the constrained complex amplitude distribution of the device surface; E′ output (f x f y () represents the complex amplitude distribution of the output surface after constraint; f represents the inverse discrete Fourier transform; x f represents the spatial frequency in the first direction; y y represents the spatial frequency in the second direction; x represents the position in the first direction; y represents the position in the second direction.

[0055] Furthermore, the pre-collimated phase distribution in the calculation of the metasurface compensation phase distribution in step 4 satisfies:

[0056]

[0057] Where x represents the position in the first direction; y represents the position in the second direction; λ represents the beam wavelength; and d is the equivalent distance, satisfying:

[0058]

[0059] Where z represents the distance from the input surface to the waist; δ sub Indicates substrate thickness; n sub Indicates the refractive index of the substrate material;

[0060] The metasurface compensation phase distribution satisfies:

[0061] Ф meta (x, y) = Ф collimator (x, y) + Ф best (x, y)

[0062] Among them, Ф meta (x, y) represents the metasurface compensation phase distribution; Ф collimator (x, y) represents the pre-collimated phase distribution; Φ best(x, y) represents the optimal phase distribution; x represents the position in the first direction; y represents the position in the second direction.

[0063] Furthermore, the geometric parameter-phase correspondence of the metasurface unit mentioned in step 5 is the relationship between the bottom edge length of the metasurface unit and the generated phase delay.

[0064] A beam-shaping metasurface device is fabricated using the aforementioned method for fabricating beam-shaping metasurface devices, wherein the beam-shaping metasurface device comprises a substrate and a metasurface located on the substrate;

[0065] The substrate is a fused silica glass sheet;

[0066] The metasurface is an array of metasurface units arranged in a lattice, and the metasurface unit is a silicon square prism.

[0067] Compared with the prior art, the present invention has the following significant advantages: (1) by introducing the minimum selection amplitude to further relax the constraint range, the phase retrieval range is expanded to obtain a better solution, thus improving the stagnation problem of the dual-amplitude degree of freedom GS algorithm; (2) the beam shaping quality of metasurface devices is improved. Attached Figure Description

[0068] Figure 1 This is a functional schematic diagram of a beam-shaping metasurface device according to the present invention.

[0069] Figure 2 This is a schematic diagram of the fabrication process of a beam-shaping metasurface device according to the present invention.

[0070] Figure 3 This is a schematic diagram of the target amplitude distribution on the input surface and the amplitude distribution on the device surface in an embodiment of the present invention.

[0071] Figure 4 This is a schematic diagram of the target amplitude distribution on the output surface in an embodiment of the present invention.

[0072] Figure 5 This is a schematic diagram of the initial phase distribution in an embodiment of the present invention.

[0073] Figure 6 This is a schematic diagram of the selective amplitude constraint iterative optimization process in an embodiment of the present invention.

[0074] Figure 7 This is a schematic diagram of the optimal phase distribution in an embodiment of the present invention.

[0075] Figure 8 This is a schematic diagram of the pre-collimated phase distribution in an embodiment of the present invention.

[0076] Figure 9 This is a schematic diagram of the metasurface compensation phase distribution in an embodiment of the present invention.

[0077] Figure 10 This is a schematic diagram of the correspondence between the bottom edge length and phase of the metasurface in an embodiment of the present invention.

[0078] Figure 11 This is a schematic diagram of the bottom edge length distribution of the metasurface unit in an embodiment of the present invention.

[0079] Figure 12 This is a schematic diagram of the actual metasurface device and its SEM image in an embodiment of the present invention.

[0080] Figure 13 This is a schematic diagram of the test optical path for verifying the shaping effect of metasurface devices in an embodiment of the present invention;

[0081] Figure 14 This diagram illustrates the variation of the root mean square error of the metasurface unit base length distribution and numerical simulation results with the number of iterations in the metasurface device fabrication method and the dual-amplitude degree-of-freedom GS algorithm obtained in this embodiment of the invention.

[0082] Figure 15 The figure shows a comparison of numerical simulation results, full-wave simulation results, and experimental test results of the metasurface device fabrication method and the beam-shaping metasurface device designed based on the dual-amplitude degree-of-freedom GS algorithm in the embodiments of the present invention. Detailed Implementation

[0083] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0084] Combination Figure 1 The present invention provides a beam shaping metasurface device, comprising a substrate and a metasurface located on the substrate;

[0085] The substrate is a fused silica glass sheet;

[0086] The metasurface is an array of metasurface units arranged in a lattice, and the metasurface unit is a silicon square prism.

[0087] Combination Figure 2 The present invention discloses a method for fabricating a beam-shaping metasurface device, comprising the following steps:

[0088] Step 1: Set parameters: Set metasurface unit characteristic parameters, input and output beam characteristic parameters, and iterative optimization parameters;

[0089] Furthermore, the metasurface unit characteristic parameters include the number of metasurface units in the first direction, the number of metasurface units in the second direction, the period of metasurface units in the first direction, the period of metasurface units in the second direction, the refractive index of the substrate material, and the substrate thickness; the first direction represents a direction in which the metasurface units are arranged; the second direction represents a direction perpendicular to the first direction; and both the first direction and the second direction are perpendicular to the beam propagation direction.

[0090] The input and output beam characteristic parameters include beam wavelength, Gaussian beam waist radius, input surface to beam waist distance, reference distance, super-Gaussian beam waist radius, and super-Gaussian beam order.

[0091] The iterative optimization parameters include the number of iterations, the number of zero-filling parameters, the amplitude constraint coefficient, and the minimum selected amplitude. The minimum selected amplitude is the criterion for selecting the condition in the selective amplitude constraint. It represents the critical value of the target amplitude of the output surface to which the constraint is applied and determines the size of the constraint range. In the dual-amplitude degree of freedom GS algorithm, the minimum selected amplitude is fixed to 0.

[0092] Step 2: Model building: Based on the metasurface unit characteristic parameters and the input and output beam characteristic parameters, calculate the input surface amplitude distribution, the device surface target amplitude distribution, the output surface target amplitude distribution, and the initial phase distribution;

[0093] Furthermore, the model includes an input surface, a device surface, and an output surface;

[0094] The input surface represents the position where the metasurface receives the Gaussian beam, and is located on the plane where the bottom surface of the metasurface is located;

[0095] The device surface represents the position where the Gaussian beam is output after being modulated by the metasurface, and is located on the plane of the top surface of the metasurface;

[0096] The output surface represents the position where the shaping result is observed, located in a far-field plane parallel to the input surface and the device surface;

[0097] The input surface phase distribution is obtained by metasurface modulation to obtain the device surface phase distribution. The input surface amplitude distribution is the same as the device surface amplitude distribution. The output surface complex amplitude distribution is obtained by performing a Fourier transform on the device surface complex amplitude distribution.

[0098] Furthermore, the amplitude distribution of the input surface target is Gaussian, and the amplitude distribution of the input surface satisfies:

[0099]

[0100] Among them, |E input (x, y)| represents the amplitude distribution of the input surface; {.} normalizationThe expression represents the normalization operation; x represents the position in the first direction; y represents the position in the second direction; z represents the distance from the input surface to the beam waist; ω0 represents the beam waist radius of the Gaussian beam; λ represents the beam wavelength; R(z) is the wavefront curvature radius, and satisfies:

[0101]

[0102] Ф(z) is the phase factor of the Gaussian beam, and satisfies:

[0103]

[0104] The target amplitude distribution on the device surface is Gaussian, and the target amplitude distribution on the device surface is the same as the target amplitude distribution on the input surface.

[0105] The output surface target amplitude distribution is a super-Gaussian distribution, and the output surface target amplitude distribution satisfies:

[0106]

[0107] Among them, |E target (f x f y | represents the target amplitude distribution on the output surface; {.} normalization z represents the normalization operation. f Indicates the reference distance; f x f represents the spatial frequency in the first direction; y ω represents the spatial frequency in the second direction. s The beam waist radius represents the super-Gaussian beam; n represents the order of the super-Gaussian beam.

[0108] The initial phase distribution refers to the phase distribution of the device surface during the first iteration, and satisfies:

[0109]

[0110] Among them, Ф origin (n x n y ) represents the initial phase distribution; n x Indicates the element number in the first direction; n y The number of elements in the second direction is represented by ; N represents the number of zero-fill elements; and n1 represents the number of metasurface elements in the first direction.

[0111] Step 3: Perform selective amplitude constraint iterative optimization: Based on the iterative optimization parameters, the target amplitude distribution on the device surface, the target amplitude distribution on the output surface, and the initial phase distribution, perform selective amplitude constraint iterative optimization to obtain the optimal phase distribution;

[0112] Furthermore, the iterative optimization parameters include the number of iterations, the number of zero-filling parameters, the amplitude constraint coefficient, and the minimum selected amplitude; wherein, the minimum selected amplitude is the criterion for judging the selection condition in the selective amplitude constraint, and represents the critical value of the target amplitude of the output surface to which the constraint is applied.

[0113] Furthermore, the selective amplitude constraint iterative optimization process described in step 3 is as follows:

[0114] Step 3.1: Fill the target amplitude distribution and initial phase distribution on the device surface with zeros according to the zero-filling quantity, and generate the complex amplitude distribution on the device surface;

[0115] Step 3.2: Perform a discrete Fourier transform on the complex amplitude distribution of the device surface and normalize the result to obtain the complex amplitude distribution of the output surface.

[0116] The output surface complex amplitude distribution satisfies:

[0117]

[0118] Among them, E output (f x f y ) represents the complex amplitude distribution of the output surface; E input (x, y) represents the complex amplitude distribution of the input surface; Represents the Discrete Fourier Transform; {.} normalization f represents the normalization operation; x f represents the spatial frequency in the first direction; y y represents the spatial frequency in the second direction; x represents the position in the first direction; y represents the position in the second direction.

[0119] Step 3.3: Calculate the root mean square error of the output surface light intensity distribution based on the output surface light intensity distribution and the target light intensity distribution on the output surface;

[0120] The root mean square error satisfies:

[0121]

[0122] Wherein, RMSE represents the root mean square error; Indicates the light intensity distribution on the output surface; Indicates the target light intensity distribution on the output surface; f x f represents the spatial frequency in the first direction; y Indicates the spatial frequency in the second direction;

[0123] Step 3.4: Fill the target amplitude distribution of the output surface with zeros according to the zero-filling quantity, and apply constraints to the output surface amplitude corresponding to the sampling points that meet the selection conditions according to the amplitude constraint coefficient. Do not apply constraints to the output surface amplitude corresponding to the sampling points that do not meet the selection conditions, and obtain the constrained complex amplitude distribution of the output surface.

[0124] The selection condition is that the target amplitude of the output surface is greater than the minimum selected amplitude;

[0125] The constrained complex amplitude distribution of the output surface satisfies:

[0126]

[0127] Among them, E′ output (f x f y ) represents the complex amplitude distribution of the output surface after constraint; E output (f x f y ) represents the complex amplitude distribution of the output surface; |E target (f x f y | represents the target amplitude distribution on the output surface; E select Indicates the minimum selectable amplitude; φ output (f x f y ) represents the phase distribution of the output surface; f x f represents the spatial frequency in the first direction; y The frequency in the second direction is represented by α; α represents the amplitude constraint coefficient.

[0128] Step 3.5: Perform an inverse discrete Fourier transform on the constrained output surface complex amplitude distribution to obtain the constrained device surface complex amplitude distribution, and replace the device surface amplitude distribution with the device surface target amplitude distribution as the device surface complex amplitude distribution for the next cycle;

[0129] The constrained amplitude distribution of the device surface satisfies:

[0130]

[0131] Among them, E′ input (x, y) represents the constrained complex amplitude distribution of the device surface; E′ output (f x f y () represents the complex amplitude distribution of the output surface after constraint; f represents the inverse discrete Fourier transform; x f represents the spatial frequency in the first direction; y y represents the spatial frequency in the second direction; x represents the position in the first direction; y represents the position in the second direction.

[0132] Step 3.6: Repeat steps 3.2 to 3.5 until the number of iterations reaches the number of iterations for optimization. The device surface phase distribution corresponding to the minimum root mean square error is taken as the optimal phase distribution.

[0133] Step 4: Calculate the metasurface compensation phase distribution: Calculate the pre-collimated phase distribution based on the input and output beam characteristic parameters, and calculate the metasurface compensation phase distribution based on the optimal phase distribution and the pre-collimated phase distribution;

[0134] Furthermore, the pre-collimated phase distribution in the calculation of the metasurface compensation phase distribution satisfies:

[0135]

[0136] Where x represents the position in the first direction; y represents the position in the second direction; λ represents the beam wavelength; and d is the equivalent distance, satisfying:

[0137]

[0138] Where z represents the distance from the input surface to the waist; δ sub Indicates substrate thickness; n sub Indicates the refractive index of the substrate material;

[0139] The metasurface compensation phase distribution satisfies:

[0140] Φ meta (x, y) = Ф collimator (x, y) + Ф best (x, y)

[0141] Among them, Ф meta (x, y) represents the metasurface compensation phase distribution; Ф collimator (x, y) represents the pre-collimated phase distribution; Φ best (x, y) represents the optimal phase distribution; x represents the position in the first direction; y represents the position in the second direction.

[0142] Step 5: Determine the geometric parameters of each metasurface unit: Based on the correspondence between the metasurface compensation phase distribution and the geometric parameters-phase of the metasurface unit, determine the geometric parameters of each metasurface unit and fabricate the metasurface device;

[0143] Furthermore, the geometric parameter-phase correspondence of the metasurface unit is the relationship between the bottom edge length of the metasurface unit and the generated phase delay.

[0144] Example 1

[0145] Figure 1This is a functional schematic diagram of a beam-shaping metasurface device provided in this embodiment. A Gaussian beam is incident directly on the metasurface and becomes a flat-top beam after being shaped by the metasurface. In this embodiment of the invention, the substrate is a fused silica glass sheet with a length of 20 mm, a width of 20 mm, and a thickness of 0.5 mm. The metasurface is an array of metasurface units arranged in a lattice. Each metasurface unit is a silicon square prism with a base length between 0.25 μm and 0.482 μm and a height of 1 μm.

[0146] Figure 2 This embodiment provides a flowchart of a method for fabricating a beam-shaping metasurface device, which includes the following steps:

[0147] Step 1: Set parameters. The metasurface element characteristic parameters, input / output beam characteristic parameters, and iterative optimization parameters are set as follows: number of metasurface elements in the first direction: 300; number of metasurface elements in the second direction: 300; metasurface element period in the first direction: 0.75 μm; metasurface element period in the second direction: 0.75 μm; substrate material refractive index: 1.44; substrate thickness: 500 μm; beam wavelength: 1.55 μm; Gaussian beam waist radius: 4.91 μm; input surface to beam waist distance: 1100 μm; reference distance: 4910 μm; super-Gaussian beam waist radius: 110 μm; super-Gaussian beam order: 20; number of iterations: 100; zero-fill quantity: 45; amplitude constraint coefficient: 0.5; minimum selected amplitude: 0.5.

[0148] Step 2: Model Building. Both the input surface amplitude distribution and the target surface amplitude distribution are Gaussian distributions, such as... Figure 3 As shown. The output surface target amplitude distribution is a 20th-order supergaussian distribution, as... Figure 4 As shown. A spherical phase distribution is used as the initial phase distribution, as... Figure 5 As shown.

[0149] Step 3: Perform selective amplitude constraint iterative optimization. Figure 6 The flowchart for selective amplitude constraint iterative optimization is as follows. Selective amplitude constraint iterative optimization includes the following steps:

[0150] Step 3.1: Pad zeros to the target amplitude distribution and initial phase distribution of the device surface according to the zero-padding amount and generate a complex amplitude distribution of the device surface. In this embodiment of the invention, zero-padding is performed on the target amplitude and initial phase distribution of the device surface, expanding the original 300×300 target amplitude distribution matrix and initial phase distribution matrix to 13500×13500, which is 45 times the size of the original matrix.

[0151] Step 3.2: Perform a Discrete Fourier Transform on the complex amplitude distribution of the device surface and normalize the result to obtain the output complex amplitude distribution. For example... Figure 6As shown in step two.

[0152] Step 3.3: Calculate the root mean square error of the output surface light intensity distribution based on the output surface light intensity distribution and the target light intensity distribution on the output surface.

[0153] Step 3.4: Fill the target amplitude distribution of the output surface with zeros according to the zero-filling quantity, and apply constraints to the output surface amplitude corresponding to the sampling points that meet the selection conditions according to the amplitude constraint coefficient. Do not apply constraints to the output surface amplitude corresponding to the sampling points that do not meet the selection conditions, and obtain the constrained complex amplitude distribution of the output surface.

[0154] In this embodiment, zero-padding is performed on the target amplitude distribution of the output surface, expanding the target amplitude distribution matrix from 300×300 to 13500×13500, which is 45 times the size of the original matrix. The minimum selected amplitude is set to 0.5; therefore, the regions in the target amplitude distribution matrix with amplitudes greater than 0.5 are designated as constraint regions. Constraints are applied to the output surface amplitude distribution within these constraint regions, while no control is applied to the output surface amplitude distribution outside the constraint regions, resulting in the constrained complex amplitude distribution of the output surface. The constrained complex amplitude distribution of the output surface satisfies:

[0155]

[0156] Among them, E′ output (f x f y ) represents the complex amplitude distribution of the output surface after constraint; E output (f x f y ) represents the complex amplitude distribution of the output surface; |E target (f x f y | represents the amplitude distribution of the target on the output surface; φ output (f x f y ) represents the phase distribution of the output surface; f x f represents the spatial frequency in the first direction; y Indicates the spatial frequency in the second direction;

[0157] Step 3.5: Perform an inverse discrete Fourier transform on the constrained output surface complex amplitude distribution to obtain the constrained device surface complex amplitude distribution. Replace the device surface amplitude distribution with the zero-filled target device surface amplitude distribution as the device surface complex amplitude distribution for the next iteration. Repeat steps two to five until the number of iterations reaches the required number of iterations for optimization. The device surface phase distribution corresponding to the minimum root mean square error is taken as the optimal phase distribution. In this embodiment of the invention, the optimal phase distribution after 100 iterations is as follows: Figure 7 As shown.

[0158] Step 4: Calculate the metasurface compensation phase distribution. Based on the input surface distance to the beam waist, substrate thickness, and substrate material refractive index, the equivalent distance is calculated to be 947.222 micrometers. The pre-collimated phase distribution is as follows: Figure 8 As shown. Adding the pre-collimated phase to the optimal phase distribution yields the metasurface-compensated phase distribution, as shown. Figure 9 As shown.

[0159] Step 5: Determine the geometric parameters of each metasurface unit. The correspondence between the metasurface unit structure and the base length-phase is as follows: Figure 10 As shown. The distribution of the base edge length of the metasurface unit is as follows. Figure 11 As shown. The physical object and SEM image of the beam-shaped metasurface device fabricated using electron beam lithography are shown below. Figure 12 As shown, the designed metasurface is located within the dashed circle in the figure.

[0160] To verify the actual shaping effect of the beam-shaping metasurface device provided in this embodiment, a structure was constructed as follows: Figure 13 The test optical path shown is used to test the metasurface device. The laser output wavelength is 1550.12 nm. The metasurface device is fixed by a specially designed fixture, which is then mounted on a displacement stage. The shaped light spot is observed using an infrared camera. A metasurface is also designed and fabricated using the dual-amplitude degree-of-freedom (GS) algorithm as a control. The difference between the two design methods is that the minimum selected amplitude in the dual-amplitude GS algorithm is set to 0, while the minimum selected amplitude in this invention is set to 0.5.

[0161] The changes in root mean square error of the metasurface element base length distribution and numerical simulation results obtained by the two design methods with the number of iterations are as follows: Figure 14 As shown, a and b represent the bottom edge length distribution of the metasurface unit obtained by the dual-amplitude degree-of-freedom GS algorithm and the present invention, respectively, and c and d represent the root mean square error of the numerical simulation results obtained by the dual-amplitude degree-of-freedom GS algorithm and the present invention as a function of the number of iterations. Figure 14 The results show that the metasurface unit obtained by the method of the present invention has a smoother bottom edge length distribution at the edge, and the root mean square error of the numerical simulation results is lower.

[0162] Figure 15 This paper summarizes the numerical simulation, full-wave simulation, and experimental results of the dual-amplitude degree-of-freedom GS algorithm and the metasurface designed in this invention. In the figures, a and b represent the numerical simulation results of the dual-amplitude degree-of-freedom GS algorithm and the metasurface designed in this invention, respectively; c and d represent the full-wave simulation results of the dual-amplitude degree-of-freedom GS algorithm and the metasurface designed in this invention, respectively; and e and f represent the experimental test results of the dual-amplitude degree-of-freedom GS algorithm and the metasurface designed in this invention, respectively. Figure 15It can be seen that the intensity distribution at the top of the metasurface designed by the dual-amplitude degree of freedom GS algorithm shows obvious fluctuations in numerical simulation and full-wave simulation results, while the intensity distribution in the top region of the metasurface designed in this invention is more uniform.

[0163] The beam shaping quality of the two design methods was evaluated using two metrics: root mean square error and non-uniformity. A smaller non-uniformity indicates a more uniform shaping result, and the non-uniformity must satisfy the following conditions:

[0164]

[0165] Among them, E output (f x ,f y ) represents the amplitude distribution of the output surface, and W represents the region where the target amplitude distribution of the output surface is equal to 1. f represents the average light intensity within W, where n is the number of sampling points within W, and f x f represents the spatial frequency in the first direction; y This represents the spatial frequency in the second direction.

[0166] The root mean square error and inhomogeneity of the dual-amplitude degree-of-freedom GS algorithm and the metasurface shaping results designed in this invention are shown in Table 1.

[0167] Table 1. Root mean square error and inhomogeneity of the dual-amplitude degree-of-freedom GS algorithm and the metasurface shaping results designed in this invention.

[0168]

[0169] Depend on Figure 14 , Figure 15 As shown in Table 1, the present invention exhibits lower root mean square error and non-uniformity in numerical simulation, full-wave simulation, and experimental results. The root mean square error is 10% lower than that of the dual-amplitude degree-of-freedom GS algorithm, and the non-uniformity is 40% lower. These data demonstrate that the method for fabricating a beam-shaping metasurface device provided by the present invention can improve the stalling problem of the dual-amplitude degree-of-freedom GS algorithm and obtain a superior solution. The beam-shaping quality of the provided metasurface device is superior to that of metasurface devices designed based on the dual-amplitude degree-of-freedom GS algorithm.

Claims

1. A method for fabricating a beam-shaping metasurface device, characterized in that, Includes the following steps: Step 1, Setting Parameters: Set the metasurface unit characteristic parameters, input and output beam characteristic parameters, and iterative optimization parameters; the iterative optimization parameters include the number of iterations, the number of zero-filling parameters, the amplitude constraint coefficient, and the minimum selected amplitude; the minimum selected amplitude is the criterion for selecting the condition in the selective amplitude constraint, representing the critical value of the target amplitude of the output surface to which the constraint is applied, and determining the size of the constraint range; Step 2: Model building: Based on the metasurface unit characteristic parameters and the input and output beam characteristic parameters, calculate the input surface amplitude distribution, the device surface target amplitude distribution, the output surface target amplitude distribution, and the initial phase distribution; Step 3: Perform selective amplitude constraint iterative optimization: Based on the iterative optimization parameters, the target amplitude distribution on the device surface, the target amplitude distribution on the output surface, and the initial phase distribution, perform selective amplitude constraint iterative optimization to obtain the optimal phase distribution; Step 4: Calculate the metasurface compensation phase distribution: Calculate the pre-collimated phase distribution based on the input and output beam characteristic parameters, and calculate the metasurface compensation phase distribution based on the optimal phase distribution and the pre-collimated phase distribution; Step 5: Determine the geometric parameters of each metasurface unit: Based on the correspondence between the metasurface compensation phase distribution and the geometric parameters-phase of the metasurface unit, determine the geometric parameters of each metasurface unit and fabricate the metasurface device.

2. The method for fabricating the beam-shaping metasurface device according to claim 1, characterized in that, The metasurface unit characteristic parameters mentioned in step 1 include the number of metasurface units in the first direction, the number of metasurface units in the second direction, the period of metasurface units in the first direction, the period of metasurface units in the second direction, the refractive index of the substrate material, and the substrate thickness; the first direction represents a direction in which the metasurface units are arranged; the second direction represents a direction perpendicular to the first direction; and both the first direction and the second direction are perpendicular to the beam propagation direction. The input and output beam characteristic parameters include beam wavelength, Gaussian beam waist radius, input surface to beam waist distance, reference distance, super-Gaussian beam waist radius, and super-Gaussian beam order.

3. The method for fabricating the beam-shaping metasurface device according to claim 1, characterized in that, The model described in step 2 includes an input surface, a device surface, and an output surface; The input surface represents the position where the metasurface receives the Gaussian beam, and is located on the plane where the bottom surface of the metasurface is located; The device surface represents the position where the Gaussian beam is output after being modulated by the metasurface, and is located on the plane of the top surface of the metasurface; The output surface represents the position where the shaping result is observed, located in a far-field plane parallel to the input surface and the device surface; The input surface phase distribution is obtained by metasurface modulation to obtain the device surface phase distribution. The input surface amplitude distribution is the same as the device surface amplitude distribution. The output surface complex amplitude distribution is obtained by performing a Fourier transform on the device surface complex amplitude distribution.

4. The method for fabricating the beam-shaping metasurface device according to claim 1, characterized in that, The input surface target amplitude distribution mentioned in step 2 is a Gaussian distribution, and the input surface amplitude distribution satisfies: in, Indicates the amplitude distribution of the input surface; This represents the normalization operation; Indicates the position in the first direction; Indicates the position in the second direction; This indicates the distance between the input surface and the waistband; This represents the waist radius of the Gaussian beam; Indicates the wavelength of the light beam; Let be the radius of curvature of the wavefront, and satisfy: Let be the phase factor of the Gaussian beam, and satisfy: The target amplitude distribution on the device surface is Gaussian, and the target amplitude distribution on the device surface is the same as the target amplitude distribution on the input surface. The output surface target amplitude distribution is a super-Gaussian distribution, and the output surface target amplitude distribution satisfies: in, Indicates the target amplitude distribution on the output surface; This represents the normalization operation; Indicates a reference distance; Indicates the spatial frequency in the first direction; Indicates the spatial frequency in the second direction; This represents the waist radius of the super-Gaussian beam; Indicates the order of the super-Gaussian beam; The initial phase distribution refers to the phase distribution of the device surface during the first iteration, and satisfies: in, Indicates the initial phase distribution; Indicates the unit number in the first direction; Indicates the unit number in the second direction; Indicates the number of zero-padding items; This indicates the number of surface units in the first direction.

5. The method for fabricating the beam-shaping metasurface device according to claim 1, characterized in that, The iterative optimization parameters mentioned in step 3 include the number of iterations, the number of zero-filling parameters, the amplitude constraint coefficient, and the minimum selected amplitude; wherein, the minimum selected amplitude is the criterion for judging the selection condition in the selective amplitude constraint, and represents the critical value of the target amplitude of the output surface to which the constraint is applied.

6. The method for fabricating the beam-shaping metasurface device according to claim 1, characterized in that, Step 3 describes selective amplitude constraint iterative optimization: Based on the iterative optimization parameters, the target amplitude distribution on the device surface, the target amplitude distribution on the output surface, and the initial phase distribution, selective amplitude constraint iterative optimization is performed to obtain the optimal phase distribution, as detailed below: Step 3.1: Fill the target amplitude distribution and initial phase distribution on the device surface with zeros according to the zero-filling quantity, and generate the complex amplitude distribution on the device surface; Step 3.2: Perform a discrete Fourier transform on the complex amplitude distribution of the device surface and normalize the result to obtain the complex amplitude distribution of the output surface. Step 3.3: Calculate the root mean square error of the output surface light intensity distribution based on the output surface light intensity distribution and the target light intensity distribution on the output surface; Step 3.4: Fill the target amplitude distribution of the output surface with zeros according to the zero-filling quantity, and apply constraints to the output surface amplitude corresponding to the sampling points that meet the selection conditions according to the amplitude constraint coefficient. Do not apply constraints to the output surface amplitude corresponding to the sampling points that do not meet the selection conditions, and obtain the constrained complex amplitude distribution of the output surface. Step 3.5: Perform an inverse discrete Fourier transform on the constrained output surface complex amplitude distribution to obtain the constrained device surface complex amplitude distribution, and replace the device surface amplitude distribution with the device surface target amplitude distribution as the device surface complex amplitude distribution for the next cycle; Step 3.6: Repeat steps 3.2 to 3.5 until the number of iterations reaches the number of iterations for optimization. The device surface phase distribution corresponding to the minimum root mean square error is taken as the optimal phase distribution.

7. The method for fabricating the beam-shaping metasurface device according to claim 6, characterized in that, The complex amplitude distribution of the output surface described in step 3.2 satisfies: in, Indicates the complex amplitude distribution of the output surface; Indicates the complex amplitude distribution of the input surface; Represents the Discrete Fourier Transform; This represents the normalization operation; Indicates the spatial frequency in the first direction; Indicates the spatial frequency in the second direction; Indicates the position in the first direction; Indicates the position in the second direction; The root mean square error described in step 3.3 satisfies: in, Indicates the root mean square error; Indicates the light intensity distribution on the output surface; This indicates the target light intensity distribution on the output surface; Indicates the spatial frequency in the first direction; Indicates the spatial frequency in the second direction; The selection condition mentioned in step 3.4 is that the target amplitude of the output surface is greater than the minimum selected amplitude; The complex amplitude distribution of the output surface after constraint as described in step 3.4 satisfies: in, This represents the complex amplitude distribution of the output surface after constraints. Indicates the complex amplitude distribution of the output surface; Indicates the target amplitude distribution on the output surface; Indicates the minimum selectable amplitude; This represents the phase distribution of the output surface; Indicates the spatial frequency in the first direction; Indicates the spatial frequency in the second direction; Indicates the amplitude constraint coefficient; The constrained complex amplitude distribution of the device surface described in step 3.5 satisfies: in, This represents the complex amplitude distribution of the device surface after constraint. This represents the complex amplitude distribution of the output surface after constraints. This represents the inverse discrete Fourier transform. Indicates the spatial frequency in the first direction; Indicates the spatial frequency in the second direction; Indicates the position in the first direction; Indicates the position in the second direction.

8. The method for fabricating the beam-shaping metasurface device according to claim 6, characterized in that, The pre-collimated phase distribution in step 4, which calculates the metasurface compensation phase distribution, satisfies: in, Indicates the position in the first direction; Indicates the position in the second direction; Indicates the wavelength of the light beam; For the equivalent distance, and satisfying: in, This indicates the distance between the input surface and the waistband; Indicates substrate thickness; Indicates the refractive index of the substrate material; The metasurface compensation phase distribution satisfies: in, This indicates the metasurface-compensated phase distribution; This indicates the pre-collimated phase distribution; Indicates the optimal phase distribution; Indicates the position in the first direction; Indicates the position in the second direction.

9. The method for fabricating the beam-shaping metasurface device according to claim 1, characterized in that, The geometric parameter-phase correspondence of the metasurface unit mentioned in step 5 is the relationship between the bottom edge length of the metasurface unit and the generated phase delay.

10. A beam-shaping metasurface device, characterized in that, The fabrication process employs the fabrication method of the beam-shaping metasurface device according to any one of claims 1 to 9, wherein the beam-shaping metasurface device comprises a substrate and a metasurface located on the substrate; The substrate is a fused silica glass sheet; The metasurface is an array of metasurface units arranged in a lattice, and the metasurface unit is a silicon square prism.

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