A reverse optimization method for dielectric metasurfaces based on the spatial coupled-mode theory

Abstract

The present invention discloses a reverse optimization method for dielectric metasurfaces based on the spatial coupled-mode theory, which relates to the field of optics. The method includes the following steps: creating a truncated waveguide array based on the metasurface, calculating the guided modes and propagation constants of each waveguide according to the truncated waveguide array, and coupling adjacent waveguides using a coupling matrix to obtain a waveguide array; coupling an input optical field into the waveguide array, simulating the propagation of light in the waveguide based on the spatial coupled-mode theory model, and converting the near-field intensity distribution into a far-field intensity distribution using the Rayleigh-Sommerfeld diffraction theory; calculating the difference between the target intensity and the far-field intensity through a loss function, evaluating the effect of the current metasurface design, and gradually adjusting the geometric parameters of the waveguide unit through an optimization algorithm. The present invention solves the problems of efficient modeling and optimization in metasurface design through the spatial coupled-mode theory model.

Description

Technical Field

[0001] The present invention relates to the field of optics, and more specifically, to a reverse optimization method for dielectric metasurfaces based on the spatial coupled-mode theory. Background Art

[0002] A metasurface usually consists of a large number of sub-wavelength-sized scatterers, and its shape and size need to be precisely optimized to achieve specific optical functions; for this purpose, a variety of design methods have been proposed with the aim of reducing the computational cost while maintaining high precision; to accurately predict the optical response and achieve high-performance designs, researchers have developed a reverse design method for metasurfaces based on full-wave simulation; this design process generally includes two steps: forward and backward. The forward problem uses a Maxwell equation solver to predict the scattering behavior of the meta-units, and the backward problem optimizes the design of the metasurface according to its performance through numerical optimization techniques.

[0003] Full-wave simulation has high accuracy and can accurately predict the phase and amplitude responses of any metasurface without making any assumptions about the geometry and layout of the meta-units; however, the computational cost of full-wave simulation is extremely high, and as the scale of the metasurface increases, the computational burden will quickly become unbearable; to solve this problem, many methods have been proposed to reduce the computational cost.

[0004] Among them, the widely used "local periodicity" approximation (LPA) assumes that the geometric changes of the meta-units are small within a local range, so its optical response can be approximated by the scattering field of a single meta-unit; in this case, the response of the meta-unit is assumed to be the same as that of the replica of its infinite periodic array, which can conveniently use algorithms such as finite-difference time-domain (FDTD) or rigorous coupled-wave analysis (RCWA) for simulation; by selecting the meta-units with the smallest local phase error, the layout of the entire metasurface can be designed relatively easily; although the LPA method is very fast in calculation, its accuracy will decrease significantly in scenarios where large phase gradients (such as high numerical aperture lenses) need to be processed; recently, a design method based on the time-coupled mode theory (TCMT) has been proposed, which models the metasurface meta-units as tiny resonators and realizes fast and accurate numerical design by analytically calculating their coupling relationships; however, this method is only applicable to resonator-based metasurface designs and is not applicable to dielectric metasurface designs that need to consider the propagation of guided modes.

[0005] Regarding the problems in the related art, no effective solution has been proposed yet. Summary of the Invention

[0006] In view of the problems in the related art, the present invention proposes a reverse optimization method for dielectric metasurfaces based on the spatial coupled-mode theory to overcome the above-mentioned technical problems existing in the existing related technologies.

[0007] To this end, the specific technical solution adopted by the present invention is as follows:

[0008] A reverse optimization method for a dielectric metasurface based on the spatial coupled-mode theory, the optimization method comprising the following steps:

[0009] S1. Create a truncated waveguide array based on the metasurface, calculate the guided modes and propagation constants of each waveguide according to the truncated waveguide array, and couple adjacent waveguides using a coupling matrix to obtain a waveguide array;

[0010] S2. Couple the input optical field into the waveguide array, simulate the propagation of light in the waveguide based on the spatial coupled-mode theory model, and convert the near-field intensity distribution into the far-field intensity distribution using the Rayleigh-Sommerfeld diffraction theory;

[0011] S3. Calculate the difference between the target intensity and the far-field intensity through a loss function, evaluate the effect of the current metasurface design, and gradually adjust the geometric parameters of the waveguide unit through an optimization algorithm.

[0012] Further, the guided modes of the waveguide include: the electric field distribution and the magnetic field distribution of light waves in the waveguide, where

[0013] The calculation formula for the electric field distribution of light waves in the waveguide is:

[0014]

[0015] In the formula, E(x,y,z) t represents the electric field distribution in the waveguide; represents the transverse electric field component of mode i; u i (z) represents the amplitude of mode i at z; x and y represent spatial coordinates; z represents the longitudinal coordinate; t represents the transverse; N represents the total number of waveguides in the waveguide array;

[0016] The calculation formula for the magnetic field distribution of light waves in the waveguide is:

[0017]

[0018] In the formula, H(x,y,z) t represents the magnetic field distribution in the waveguide; represents the transverse magnetic field component of mode i; u i (z) represents the amplitude of mode i at z; x and y represent spatial coordinates; z represents the longitudinal coordinate; t represents the transverse; N represents the total number of waveguides in the waveguide array.

[0019] Further, the calculation formula for the coupling matrix is:

[0020]

[0021] In the formula, C represents the coupling matrix of the coupling strength between waveguides; U(z) represents the coefficient vector, U(z) = [u1(z), u2(z),... u N (z)]; K represents the coupling coefficient matrix, which is used to describe the coupling relationship between modes; B represents the diagonal matrix of the phase change characteristics during propagation, and its diagonal element B ii = β i ; β i represents the propagation constant of mode i.

[0022] Furthermore, the element c pm of the coupling matrix C is calculated by the formula:

[0023]

[0024] In the formula, c pm represents the element value of the coupling matrix C; represents the unit vector in the z longitudinal coordinate direction; represents the transverse electric field component of mode p; represents the transverse magnetic field component of mode m; x and y represent the spatial coordinates; z represents the longitudinal coordinate; t represents the transverse.

[0025] Furthermore, coupling the input optical field into the truncated waveguide array, simulating the propagation of light in the waveguide based on the spatial coupled-mode theory model, and then converting the near-field intensity distribution into the far-field intensity distribution by using the Rayleigh-Sommerfeld diffraction theory includes the following steps:

[0026] S21. Decompose the incident electromagnetic field to obtain the waveguide mode components of the input optical field;

[0027] S22. Calculate the initial mode amplitude of each waveguide based on the waveguide mode components of the input optical field;

[0028] S23. Use the spatial coupled-mode theory model to simulate the propagation process of the optical field in the waveguide, and allow the modes between adjacent waveguides to interact with each other. When the optical field propagates to the end of the waveguide, couple the optical field into free space to obtain the near-field intensity distribution of the metasurface at the waveguide exit;

[0029] S24. Propagate the near-field intensity distribution to the focal plane through the Rayleigh-Sommerfeld diffraction theory, calculate the far-field intensity distribution of the metasurface, and obtain the far-field intensity distribution of the metasurface.

[0030] Furthermore, the calculation formula for the initial mode amplitude:

[0031]

[0032] In the formula, u i(0) represents the initial mode amplitude of mode i; E incident represents the input electromagnetic field; c represents the speed of light in vacuum; represents the transverse electric field component of mode i; n0 represents the refractive index of free space; represents the effective refractive index of mode i; t represents the transverse direction.

[0033] Furthermore, the calculation formula for the far-field intensity distribution of the metasurface is:

[0034]

[0035] wherein, E far (x, y) represents the far-field intensity distribution of the metasurface; (x, y) represents the position of the electric field intensity E far (x, y) in the far-field plane; (x', y') represents the position of the electric field intensity E near (x′, y′) in the near-field plane; r represents the distance between the near-field source point (x', y') and the observation point (x, y) in the far-field; λ represents the wavelength; E near (x′, y′) represents the near-field intensity distribution at the waveguide exit; j represents the imaginary unit; x' and y' represent the spatial coordinates of the position where the electric field intensity E near (x′, y′) is located; x and y represent the spatial coordinates of the position where the electric field intensity E far (x, y) is located; z represents the longitudinal coordinate.

[0036] Furthermore, calculating the difference between the target intensity and the far-field intensity through the loss function to evaluate the effect of the current metasurface design, and gradually adjusting the geometric parameters of the waveguide unit through the optimization algorithm to make the far-field intensity distribution consistent with the target intensity distribution includes the following steps:

[0037] S31. Using the loss function, calculate the difference between the target intensity and the far-field intensity;

[0038] S32. Based on the forward simulation, use the automatic differentiation framework to calculate the gradient of the loss function with respect to the waveguide width;

[0039] S33. Through backpropagation, use the optimization algorithm of the metasurface parameters to adjust the geometric dimensions of the waveguide unit and calculate the optimized metasurface waveguide width;

[0040] S34. Based on the optimized metasurface waveguide width, obtain the far-field intensity distribution consistent with the target intensity distribution.

[0041] Furthermore, the calculation formula for the loss function is:

[0042] L = ||I far - I target|| 2 ;

[0043] In the formula, L represents the difference value between the target intensity and the far-field intensity; I far represents the far-field output intensity; I target represents the target intensity;

[0044] Among them, the calculation formula of the far-field output intensity I far is:

[0045] I far = f(w1, w2,... w n |E in );

[0046] In the formula, I far represents the far-field output intensity; w1, w2,... w n represents the waveguide width of the metasurface; E in represents the input electric field incident on the metasurface.

[0047] Furthermore, the calculation formula of the optimization algorithm for the metasurface parameters is:

[0048] w opt = argmin w ||f(E in ) - I target ||;

[0049] In the formula, w opt represents the waveguide width of the optimized metasurface; I target represents the target intensity; E in represents the input electric field incident on the metasurface; w = (w1, w2,... w n ) represents the geometric parameter vector of the metasurface design.

[0050] The beneficial effects of the present invention are:

[0051] Through the spatial coupled-mode theory model, the present invention far exceeds the traditional full-wave simulation in terms of computational efficiency, can significantly reduce the computational time while ensuring accuracy, and greatly improves the design efficiency. Secondly, compared with the local periodic approximation (LPA) method, the present invention can accurately consider the coupling effect between adjacent meta-units, thereby significantly improving the prediction accuracy and reliability in the design of complex optical systems. Especially in application scenarios such as high numerical aperture lenses, multi-wavelength focusing, and coma suppression, the spatial coupled-mode theory exhibits excellent performance. In addition, the spatial coupled-mode theory can handle the optical field design under arbitrary incident conditions, breaking through the limitations of traditional methods on incident conditions, and greatly enhancing the flexibility and adaptability of the design. Through memory optimization, the spatial coupled-mode theory has also achieved efficient calculation for large-scale metasurfaces, significantly reducing the memory occupancy, and providing strong support for the design and simulation of large-area metasurfaces. Brief Description of the Drawings

[0052] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0053] Figure 1 is a flowchart of a reverse optimization method for a dielectric metasurface based on the spatial coupled-mode theory according to an embodiment of the present invention;

[0054] Figure 2 is a flowchart of the spatial coupled-mode theory (SCMT) model in a reverse optimization method for a dielectric metasurface based on the spatial coupled-mode theory according to an embodiment of the present invention;

[0055] Figure 3 is a hybrid architecture diagram of the spatial coupled-mode theory (SCMT) model in a reverse optimization method for a dielectric metasurface based on the spatial coupled-mode theory according to an embodiment of the present invention;

[0056] Figure 4 is a comparison diagram of the focus efficiency of SCMT and LPA for the design of a high numerical aperture (NA) lens in a reverse optimization method for a dielectric metasurface based on the spatial coupled-mode theory according to an embodiment of the present invention;

[0057] Figure 5 is a comparison diagram of the focus efficiency of SCMT and LPA for the design of a multi-wavelength lens in a reverse optimization method for a dielectric metasurface based on the spatial coupled-mode theory according to an embodiment of the present invention;

[0058] Figure 6It is a comparison diagram of the side view of a lens optimized using SCMT and the light intensity distribution at different incident angles in a reverse optimization method of a dielectric metasurface based on the spatial coupled mode theory according to an embodiment of the present invention. Detailed implementation manners

[0059] To further illustrate each embodiment, the present invention provides accompanying drawings, which are part of the disclosure of the present invention. They are mainly used to illustrate the embodiments and can be combined with the relevant descriptions in the specification to explain the operating principle of the embodiments. With reference to these contents, those of ordinary skill in the art should be able to understand other possible implementation manners and the advantages of the present invention.

[0060] According to an embodiment of the present invention, a reverse optimization method of a dielectric metasurface based on the spatial coupled mode theory is provided.

[0061] Now, the present invention will be further described in combination with the accompanying drawings and detailed implementation manners. As Figure 1 shown, the reverse optimization method of a dielectric metasurface based on the spatial coupled mode theory according to an embodiment of the present invention includes the following steps:

[0062] S1. Create a truncated waveguide array based on the metasurface, calculate the guided mode and propagation constant of each waveguide according to the truncated waveguide array, and couple adjacent waveguides using a coupling matrix to obtain a waveguide array.

[0063] It should be noted that the spatial coupled mode theory (SCMT) model regards the dielectric metasurface as an array composed of truncated waveguides. Each waveguide can support a single guided mode, and this guided mode defines the electric and magnetic field distributions of light waves in the waveguide. These distributions can be expressed as the basis functions of waveguide modes, and this guided mode has transverse components and a propagation constant β.

[0064] Specifically, the guided mode of the waveguide includes: the electric field distribution and magnetic field distribution of light waves in the waveguide, where

[0065] The calculation formula for the electric field distribution of light waves in the waveguide is:

[0066]

[0067] In the formula, E(x, y, z) t represents the electric field distribution in the waveguide; represents the transverse electric field component of mode i; u i (z) represents the amplitude of mode i at z; x and y represent spatial coordinates, that is, the positions on the cross-section of the waveguide; z represents the longitudinal coordinate, that is, the longitudinal position where light propagates in the waveguide; t represents the transverse direction, that is, used to emphasize that the electric and magnetic fields are transverse components; N represents the total number of waveguides in the waveguide array;

[0068] The calculation formula for the magnetic field distribution of light waves in a waveguide is as follows:

[0069]

[0070] In the formula, H(x, y, z) t represents the magnetic field distribution in the waveguide; represents the transverse magnetic field component of mode i; u i (z) represents the amplitude of mode i at z; x and y represent the spatial coordinates, that is, the positions on the cross-section of the waveguide; z represents the longitudinal coordinate, that is, the longitudinal position where light propagates in the waveguide; t represents the transverse direction, that is, used to emphasize that the electric and magnetic fields are transverse components; N represents the total number of waveguides in the waveguide array.

[0071] It should be noted that the mutual coupling between adjacent waveguides is described by the coupling matrix C, which can capture the near-field interaction between adjacent waveguides. The elements of the coupling matrix represent the coupling strength between the modes in one waveguide and the modes in its neighboring waveguides. The coupling effect is accurately described as the overlap integral of the electric and magnetic fields between the modes. The element c pm of the coupling matrix C represents the modal overlap between the electric field (E-field) of mode p and the magnetic field (H-field) of mode m.

[0072] Specifically, the calculation formula for the coupling matrix is:

[0073]

[0074] In the formula, C represents the coupling matrix of the coupling strength between waveguides; U(z) represents the coefficient vector, U(z) = [u1(z), u2(z),... u N (z)]; K represents the coupling coefficient matrix, which is used to describe the coupling relationship between modes; B represents the diagonal matrix of the phase change characteristics during propagation, and its diagonal element B ii = β i ; β i represents the propagation constant of mode i.

[0075] Specifically, the calculation formula for the element c pm of the coupling matrix C is:

[0076]

[0077] In the formula, c pm represents the element value of the coupling matrix C; represents the unit vector in the longitudinal coordinate direction of z; represents the transverse electric field component of mode p; Represents the transverse magnetic field component of mode m; x and y represent spatial coordinates, i.e., positions on the waveguide cross-section; z represents the longitudinal coordinate, i.e., the longitudinal position where light propagates in the waveguide; t represents the transverse direction, i.e., used to emphasize that the electric and magnetic fields are transverse components.

[0078] S2. Couple the input optical field into the waveguide array, simulate the propagation of light in the waveguide based on the spatial coupled-mode theory model, and convert the near-field intensity distribution into the far-field intensity distribution using the Rayleigh-Sommerfeld diffraction theory.

[0079] It should be noted that coupling the input optical field into the waveguide array is used to simulate the propagation of light in the metasurface and the final far-field intensity distribution, and also to calculate the near-field distribution in the waveguide for a given input electromagnetic field and convert it into the far-field intensity distribution in free space.

[0080] It should be noted that the input optical field is coupled into the waveguide array, the propagation of light in the waveguide is simulated based on the spatial coupled-mode theory model, and the near-field distribution of the metasurface is recovered at the waveguide exit; subsequently, the near-field is propagated to the focal plane through the Rayleigh-Sommerfeld diffraction theory to generate the far-field intensity distribution.

[0081] It should be noted that the coupling of the input field is achieved by decomposing the incident electromagnetic field into the modes of the waveguide; this coupling is achieved by decomposing the incident field into the modes of the waveguide, and the initial mode amplitudes of each waveguide are determined by the respective mode components of the input field.

[0082] Specifically, coupling the input optical field into the truncated waveguide array, simulating the propagation of light in the waveguide based on the spatial coupled-mode theory model, and converting the near-field intensity distribution into the far-field intensity distribution using the Rayleigh-Sommerfeld diffraction theory includes the following steps:

[0083] S21. Decompose the incident electromagnetic field to obtain the waveguide mode components of the input optical field;

[0084] S22. Calculate the initial mode amplitude of each waveguide based on the waveguide mode components of the input optical field;

[0085] S23. Use the spatial coupled-mode theory model to simulate the propagation process of the optical field in the waveguide and allow mode interaction between adjacent waveguides. When the optical field propagates to the end of the waveguide, couple the optical field into free space to obtain the near-field intensity distribution of the metasurface at the waveguide exit;

[0086] S24. Propagate the near-field intensity distribution to the focal plane through the Rayleigh-Sommerfeld diffraction theory to calculate the far-field intensity distribution of the metasurface and obtain the far-field intensity distribution of the metasurface.

[0087] Specifically, the calculation formula for the initial mode amplitude:

[0088]

[0089] where \(u\) i (0) represents the initial mode amplitude of mode \(i\); \(E\) incident represents the input electromagnetic field; \(c\) represents the speed of light in vacuum; represents the transverse electric field component of mode \(i\); \(n_0\) represents the refractive index of free space; represents the effective refractive index of mode \(i\); \(t\) represents the transverse direction, i.e., used to emphasize that the electric and magnetic fields are transverse components.

[0090] Specifically, the calculation formula for the far-field intensity distribution of the metasurface is:

[0091]

[0092] where \(E\) far (x, y) represents the far-field intensity distribution of the metasurface; (x, y) represents the position of the electric field intensity \(E\) far (x, y) in the far-field plane; (x′, y′) represents the position of the electric field intensity \(E\) near (x′, y′) in the near-field plane; \(r\) represents the distance between the near-field source point (x′, y′) and the observation point (x, y) in the far-field; \(\lambda\) represents the wavelength; \(E\) near (x′, y′) represents the near-field intensity distribution at the waveguide exit; \(j\) represents the imaginary unit; \(x'\) and \(y'\) represent the spatial coordinates of the position where the electric field intensity \(E\) near (x′, y′) is located; \(x\) and \(y\) represent the spatial coordinates of the position where the electric field intensity \(E\) far (x, y) is located; \(z\) represents the longitudinal coordinate, i.e., the longitudinal position where light propagates in the waveguide.

[0093] S3. Calculate the difference between the target intensity and the far-field intensity through the loss function, evaluate the effect of the current metasurface design, and gradually adjust the geometric parameters of the waveguide unit through the optimization algorithm.

[0094] It should be noted that based on the forward simulation, the geometric dimensions of the waveguide unit are adjusted through the optimization algorithm to make the far-field intensity distribution as close as possible to the target distribution. By using the automatic differentiation framework, a computational graph is automatically generated after the forward model calculation is completed, and the gradient of the loss function with respect to all input parameters can be calculated. Through backpropagation, the hyperparameters of the metasurface can be updated using the optimizer.

[0095] Specifically, calculating the difference between the target intensity and the far-field intensity through the loss function to evaluate the effect of the current metasurface design and gradually adjusting the geometric parameters of the waveguide unit through the optimization algorithm to make the far-field intensity distribution consistent with the target intensity distribution includes the following steps:

[0096] S31. Calculate the difference between the target intensity and the far-field intensity using the loss function;

[0097] S32. Based on forward simulation, use the automatic differentiation framework to calculate the gradient of the loss function with respect to the waveguide width;

[0098] S33. Through backpropagation, use the optimization algorithm of the metasurface parameters to adjust the geometric dimensions of the waveguide unit and calculate the optimized metasurface waveguide width;

[0099] S34. Based on the optimized metasurface waveguide width, obtain the far-field intensity distribution consistent with the target intensity distribution.

[0100] Specifically, the calculation formula of the loss function is:

[0101] L = ||I far - I target || 2 ;

[0102] In the formula, L represents the difference value between the target intensity and the far-field intensity; I far represents the far-field output intensity; I target represents the target intensity;

[0103] Among them, the calculation formula of the far-field output intensity I far is:

[0104] I far = f(w1, w2,... w n |E in );

[0105] In the formula, I far represents the far-field output intensity; w1, w2,... w n represent the waveguide widths of the metasurface; E in represents the input electric field incident on the metasurface.

[0106] Specifically, the calculation formula of the optimization algorithm of the metasurface parameters is:

[0107] w opt = argmin w ||f(E in ) - I target ||;

[0108] In the formula, w opt represents the optimized waveguide width of the metasurface; I target represents the target intensity; E in represents the input electric field incident on the metasurface; w = (w1, w2,... w n ) represents the geometric parameter vector of the metasurface design.

[0109] In summary, the present invention includes the establishment of a spatial coupled mode theory model, the construction of a forward solver, the definition of an optimization objective, and gradient optimization. Among them, the spatial coupled mode theory model comprehensively describes the near-field coupling effect between adjacent elements of the metasurface by accurately calculating the coupling coefficient between waveguides. In this model, the input field is first coupled into the guided modes of each element, and then the propagation process of the optical field inside the element is simulated, while allowing the modes between adjacent elements to interact with each other. When the optical field propagates to the end of the waveguide, it is coupled back to free space, and based on the Rayleigh-Sommerfeld diffraction theory, the far-field distribution of the metasurface is further derived. In the reverse design process, an optimization objective is determined by setting a loss function, which aims to quantify the difference between the target far-field intensity of the metasurface and the intensity obtained by actual calculation. To effectively adjust the structural parameters of the element to minimize the loss, the present invention uses an automatic differentiation framework to calculate the gradient of the loss function with respect to the waveguide width and performs gradient optimization based on an optimization algorithm.

[0110] As Figure 3 shown, it is the hybrid architecture diagram of the spatial coupled mode theory (SCMT) model described in the embodiment of the present invention. The forward solver of this hybrid architecture is regarded as a non-linear function, and its output is the near-field E out , and the input is the waveguide width ω and the incident field E in ; β S is the propagation constant array of each waveguide; the matrix B is a diagonal matrix, and β is its diagonal element; C represents the coupling matrix of the coupling strength between waveguides; K represents the coupling coefficient matrix used to describe the coupling relationship between modes; by pre-training a neural network to calculate β i , c ij , k ij , the efficiency can be effectively improved and the calculation of the width gradient can be supported; the propagation of the mode in the waveguide is performed by differential operation using the Euler method, which can effectively save memory; β i represents the propagation constant of mode i; c ij represents the modal overlap between the electric field (E-field) of mode i and the magnetic field (H-field) of mode j in the coupling matrix C; k ij represents the element in the coupling coefficient matrix K.

[0111] To facilitate the understanding of the above technical solutions of the present invention, the working principle or operation method of the present invention in the actual process will be described in detail below.

[0112] To verify the effectiveness of the present invention, the method of the present invention was simulated in three application scenarios of a high numerical aperture (NA) lens, a multi-wavelength lens, and a coma suppression lens, and compared with the currently most popular full-wave simulation (FDTD) and LPA methods.

[0113] The experimental simulation content is as follows:

[0114] For high numerical aperture (NA) lenses, their edges have a rapidly varying phase profile, which reduces the accuracy of the LPA method; based on a metalens designed by the LPA method, the focusing efficiency of a radial metalens with a numerical aperture of NA = 0.8 and a wavelength of 650 nm is improved by the SCMT method; the metasurface is composed of titanium dioxide (TiO2) square waveguides, and the waveguide array is placed on a silica (SiO2) substrate with a period of 400 nm; the height of each waveguide is 800 nm, and the width ranges from 160 nm to 380 nm.

[0115] As Figure 4 shown, it is a comparison diagram of the SCMT and LPA focal efficiencies of the high numerical aperture (NA) lens design described in the embodiment of the present invention. After optimization by SCMT, the focal efficiency is increased from 35% to 40%.

[0116] Similar to large numerical aperture (NA) lenses, metalenses that focus multi-wavelength light also reduce the accuracy of the LPA method; metalenses that support simultaneous focusing of two wavelengths (650 nm and 450 nm) are designed by both SCMT and LPA methods; the NA is 0.4, the metasurface is composed of titanium dioxide (TiO2) square waveguides, the substrate is silica (SiO2), and the period is 280 nm; the height of the waveguide is 800 nm, and the width ranges from 80 nm to 240 nm.

[0117] As Figure 5 shown, it is a comparison diagram of the SCMT and LPA focal efficiencies of the multi-wavelength lens design described in the embodiment of the present invention. The lens designed by SCMT has a slightly lower efficiency at 650 nm than LPA, but at 450 nm, its efficiency is much higher than LPA.

[0118] A single-layer metasurface lens with less coma is a better choice. This requires accurate modeling of plane waves with different incident angles, which cannot be achieved by the LPA method; a single-layer metalens can be designed by using the SCMT model to maximize the focal intensity equally for different incident angles.

[0119] As Figure 6 shown, it is a comparison diagram of the side view and light intensity distribution of the lens optimized by SCMT at different incident angles described in the embodiment of the present invention. It can perfectly focus the vertically incident plane wave. However, as the incident angle increases, the coma also increases. In contrast, the lens optimized by SCMT significantly suppresses the coma when the incident angle is between plus and minus 40 degrees.

[0120] In summary, by means of the above technical solutions of the present invention, through the spatial coupled mode theory model, the present invention far exceeds the traditional full-wave simulation in terms of computational efficiency, can significantly reduce the computational time while ensuring accuracy, and greatly improves the design efficiency. Secondly, compared with the local periodic approximation (LPA) method, the present invention can accurately consider the coupling effect between adjacent elements, thus significantly improving the prediction accuracy and reliability in the design of complex optical systems. Especially in application scenarios such as high numerical aperture lenses, multi-wavelength focusing, and coma suppression, the spatial coupled mode theory exhibits excellent performance. In addition, the spatial coupled mode theory can handle the optical field design under arbitrary incident conditions, breaking through the limitations of traditional methods on incident conditions, and greatly enhancing the flexibility and adaptability of the design. Through memory optimization, the spatial coupled mode theory has also achieved efficient calculation of large-scale metasurfaces, significantly reducing the memory occupancy, and providing strong support for the design and simulation of large-area metasurfaces.

[0121] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A dielectric metasurface inverse optimization method based on spatial coupled mode theory, characterized in that: The optimization method includes the following steps: S1. Create a truncated waveguide array based on the metasurface, calculate the guided mode and propagation constant of each waveguide according to the truncated waveguide array, and couple adjacent waveguides using a coupling matrix to obtain a waveguide array; S2, coupling the input light field into the waveguide array, simulating the propagation of light in the waveguide based on the spatial coupled mode theory model, and converting the near-field intensity distribution into the far-field intensity distribution using the Rayleigh-Sommerfeld diffraction theory; S3, calculating the difference between the target intensity and the far-field intensity through the loss function, evaluating the effect of the current metasurface design, and gradually adjusting the geometric parameters of the waveguide unit through the optimization algorithm; The guided mode of the waveguide includes: the electric field distribution and magnetic field distribution of the light wave in the waveguide, wherein: The calculation formula for the electric field distribution of light waves in the waveguide is: Where E(x,y,z) t represents the electric field distribution in the waveguide; represents the transverse electric field component of mode i; u i (z) represents the amplitude of mode i at position z; x and y represent spatial coordinates; z represents the longitudinal coordinate; t represents the transverse coordinate; N represents the total number of waveguides in the waveguide array; The calculation formula for the magnetic field distribution of light waves in the waveguide is: Where H(x,y,z) t represents the magnetic field distribution in the waveguide; represents the transverse magnetic field component of mode i; u i (z) represents the amplitude of mode i at position z; x and y represent spatial coordinates; z represents the longitudinal coordinate; t represents the transverse coordinate; and N represents the total number of waveguides in the waveguide array.

2. The method for inverse optimization of dielectric metasurface based on spatial coupled mode theory according to claim 1, characterized in that: The coupling matrix calculation formula is: Where C represents the coupling matrix of the coupling strength between waveguides; U(z) represents the coefficient vector, U(z)=[u1(z),u2(z),...u N (z)]; K represents the coupling coefficient matrix, which is used to describe the coupling relationship between modes; B represents the diagonal matrix of the phase change characteristics during the propagation process, and its diagonal elements B ii =β i β i represents the propagation constant of mode i.

3. The method for inverse optimization of dielectric metasurface based on spatial coupled mode theory according to claim 2, characterized in that: The element c of the coupling matrix C pm The calculation formula is: In the formula, c pm Represents the element value of the coupling matrix C; The unit vector representing the z longitudinal coordinate direction; represents the transverse electric field component of mode p; represents the transverse magnetic field component of mode m; x and y represent spatial coordinates; z represents the longitudinal coordinate; and t represents the transverse direction.

4. The method for inverse optimization of dielectric metasurface based on spatial coupled mode theory according to claim 1, characterized in that: The method of coupling the input light field into the truncated waveguide array, simulating the propagation of light in the waveguide based on the spatial coupled mode theory model, and then converting the near-field intensity distribution into the far-field intensity distribution using the Rayleigh-Sommerfeld diffraction theory includes the following steps: S21, decomposing the incident electromagnetic field to obtain the waveguide mode components of the input light field; S22, calculating the initial mode amplitude of each waveguide based on the waveguide mode component of the input light field; S23. Use the spatial coupled mode theory model to simulate the propagation process of the light field in the waveguide, and allow the modes of adjacent waveguides to interact with each other. When the light field propagates to the end of the waveguide, couple the light field to the free space to obtain the near-field intensity distribution of the metasurface at the waveguide exit. S24. The near-field intensity distribution is propagated to the focal plane through the Rayleigh-Sommerfeld diffraction theory, and the far-field intensity distribution of the metasurface is calculated to obtain the far-field intensity distribution of the metasurface.

5. The method for inverse optimization of dielectric metasurface based on spatial coupled mode theory according to claim 4, characterized in that: The calculation formula of the initial mode amplitude is: In the formula, u i (0) represents the initial mode amplitude of mode i; E incident represents the input electromagnetic field; c represents the speed of light in vacuum; represents the transverse electric field component of mode i; n0 represents the refractive index of free space; represents the effective refractive index of mode i; t represents the transverse direction.

6. The method for inverse optimization of dielectric metasurface based on spatial coupled mode theory according to claim 4, characterized in that: The calculation formula of the far-field intensity distribution of the metasurface is: In the formula, E far (x, y) represents the far-field intensity distribution of the metasurface; (x, y) represents the electric field intensity E on the far-field plane far (x, y) represents the position of the near-field plane; (x', y') represents the electric field strength E on the near-field plane near The position of (x′, y′); r represents the distance between the near-field source point (x', y') and the observation point (x, y) in the far field; λ represents the wavelength; E near (x′, y′) represents the near-field intensity distribution at the waveguide outlet; j represents the imaginary unit; x' and y' represent the electric field intensity E on the near-field plane near (x′, y′) is the spatial coordinate of the location; x and y represent the electric field intensity E on the far field plane far (x, y) are the spatial coordinates of the location; z represents the vertical coordinate.

7. The method for inverse optimization of dielectric metasurface based on spatial coupled mode theory according to claim 1, characterized in that: The method of calculating the difference between the target intensity and the far-field intensity by using the loss function to evaluate the effect of the current metasurface design and gradually adjusting the geometric parameters of the waveguide unit by using the optimization algorithm so that the far-field intensity distribution is consistent with the target intensity distribution includes the following steps: S31, using the loss function, calculating the difference between the target intensity and the far-field intensity; S32, based on the forward simulation, using the automatic differentiation framework, calculate the gradient of the loss function with respect to the waveguide width; S33, adjusting the geometric dimensions of the waveguide unit by back propagation and using the optimization algorithm of the metasurface parameters, and calculating the optimized metasurface waveguide width; S34. Based on the optimized metasurface waveguide width, a far-field intensity distribution consistent with the target intensity distribution is obtained.

8. The method for inverse optimization of dielectric metasurface based on spatial coupled mode theory according to claim 7, characterized in that: The calculation formula of the loss function is: L=||I far -I target || 2 ; Where L represents the difference between the target intensity and the far-field intensity; I far Indicates the far-field output intensity; I target Indicates the target strength; Among them, the far-field output intensity I far The calculation formula is: I far =f(w1,w2,...w n |E in ); In the formula, I far Indicates the far-field output intensity; w1, w2, ... w n represents the waveguide width of the metasurface; E in represents the input electric field incident on the metasurface.

9. The method for inverse optimization of dielectric metasurface based on spatial coupled mode theory according to claim 7, characterized in that: The optimization algorithm calculation formula of the super surface parameters is: w opt =argmin w ||f(E in )-I target ||; In the formula, w opt represents the waveguide width of the optimized metasurface; I target Indicates target strength; E in represents the input electric field incident on the metasurface; w = (w1, w2, ... w n ) represents the geometric parameter vector of the metasurface design.

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