An integral equation based method for designing and optimizing surface impedance of finite-size anomalous reflection metasurfaces
By combining integral equations with the method of moments, the surface impedance value is discretized, and the design method is optimized, solving the problems of complex design, high cost and low efficiency in the existing technology, and realizing the design of efficient large-angle anomalous reflective metasurfaces under finite size conditions.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- XIDIAN UNIV
- Filing Date
- 2026-04-28
- Publication Date
- 2026-07-24
AI Technical Summary
Existing metasurface designs for anomalous reflections suffer from high design difficulty, high cost, large computational load, and low computational efficiency. In particular, edge effects are significant under finite size conditions, making it difficult to achieve anomalous reflections at large angles.
A design method based on integral equations, the method of moments, and impenetrable impedance surfaces is adopted. The surface impedance is discretized into a finite number of values. By optimizing the surface impedance value, arbitrarily large-angle anomalous reflections can be achieved. The calculation is performed by combining integral equations and matrix equations, which simplifies the algorithm and improves the computational efficiency.
This technology enables the efficient and low-cost design of arbitrarily large-angle anomalous reflective metasurfaces under finite size conditions, reducing computational complexity and cost while improving design efficiency and accuracy.
Smart Images
Figure CN122451964A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of anomalous reflection metasurface technology, specifically relating to a finite-size anomalous reflection metasurface design and surface impedance optimization method based on integral equations. Background Technology
[0002] Currently, the most mainstream approach in metasurface design is to first design and optimize the unit structure, and then conduct simulation analysis based on the ideal condition of infinite periodic arrangement. However, in practical engineering applications, the size of the metasurface is inevitably finite. Therefore, directly truncating the infinite periodic model to match the actual finite size will inevitably lead to a series of practical problems such as edge effects.
[0003] With the deepening penetration of electromagnetic modulation technology into fields such as communications and new energy, the demand for precise and flexible control of electromagnetic wave propagation direction is becoming increasingly urgent. Metasurfaces, as a type of artificial electromagnetic structure with unique electromagnetic modulation capabilities, can overcome the traditional constraints of Snell's law through ingenious structural design and optimization, achieving efficient reflection of incident electromagnetic waves in any predetermined direction. Their core advantages, such as low profile, lightweight, easy integration, and wide frequency response, make them one of the key supporting technologies for next-generation electromagnetic applications. Through precise manipulation of the reflected beam, including directional control, scattering suppression, and phase compensation, anomalous reflective metasurfaces can achieve multiple functions such as radar stealth, communication beamforming, electromagnetic interference suppression, and solar energy concentration, providing revolutionary technological paths and development opportunities for wireless communication, new energy utilization, and precision detection.
[0004] Integrating metasurfaces capable of anomalous reflection onto the surfaces of various carriers can overcome the functional limitations and application bottlenecks of traditional electromagnetic devices, constructing customizable and adaptive intelligent electromagnetic environments. This will bring a new application paradigm to the future development of electromagnetic technology and has the potential to become a core direction for breakthroughs in fundamental original innovation in the field of electromagnetic control, while also leading the technological upgrading and large-scale development of related industrial chains.
[0005] However, most current methods for achieving anomalous reflection using metasurfaces rely on multi-bit phase to expand the control dimension, which is complex and limited by the inherent electromagnetic response of the unit structure. In Zhang XG, Tang WX, Jiang WX, et al. Light‐controllable digital coding metasurfaces[J]. Advanced Science, 2018, 5(11): 1801028, a 1-bit light-controlled digital coding metasurface was designed. Although the designed metasurface can achieve anomalous reflection of electromagnetic waves, its structure and DC-powered circuit design are complex, and it uses a large number of varactor diodes and light-controlled diodes, resulting in high cost.
[0006] To optimize the design, researchers proposed using an impenetrable impedance surface model to achieve anomalous reflection. Kwon DH. Lossless scalar metasurfaces for anomalous reflection based on efficient surface field optimization[J]. IEEE Antennas and Wireless Propagation Letters, 2018, 17(7): 1149-1152, proposed a surface impedance design method based on field optimization: successfully achieving anomalous reflection, but with significant drawbacks: the proposed algorithm has a complex structural design, and the mathematical form of the objective optimization function is cumbersome, leading to a significant increase in overall computational load.
[0007] In Wang X, Díaz-Rubio A, Tretyakov S A. Independent control of multiple channels in metasurface devices[J]. Physical Review Applied, 2020, 14(2): 024089, a method for designing the Fourier expansion coefficients of the surface impedance distribution function was proposed. Compared with the method in Kwon D H. Lossless scalar metasurfaces for anomalous reflection based on efficient surface field optimization[J]. IEEE Antennas and WirelessPropagation Letters, 2018, 17(7): 1149-1152, the objective function of this method is simpler, but it still has limitations: if the mathematical form of the surface impedance function is complex, its corresponding impedance matrix will exhibit dense characteristics, resulting in a significant increase in computational cost; at the same time, the truncation process when performing the Fourier series expansion of the surface impedance function will directly affect the final solution accuracy of the algorithm.
[0008] In summary, the following shortcomings still exist: Existing metasurface arrays with anomalous reflections use specific resonant structures to modulate electromagnetic waves, which requires optimization of the patch size, dimensions, and dielectric material design, making the design highly challenging.
[0009] Existing metasurface arrays with anomalous reflection require a large number of electronic control devices such as PIN diodes and varactor diodes, resulting in high manufacturing costs and hindering the large-scale industrial application of metasurfaces.
[0010] Existing reflective metasurface design algorithms based on impenetrable impedance surface models are complex in structure, computationally intensive, require a large amount of computer memory, and have low design efficiency. Summary of the Invention
[0011] To overcome the shortcomings of the existing technology, the present invention aims to provide a finite-size anomalous reflective metasurface design and surface impedance optimization method based on integral equations. This method combines integral equations, the method of moments, and impenetrable impedance surfaces, which has the advantages of simple modeling and low computational cost. At the same time, in view of practical engineering needs, the impedance surface proposed by this method can be designed for any large-angle anomalous reflection and any target size, which greatly improves its practicality in future application scenarios.
[0012] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A method for designing and optimizing the surface impedance of a finite-size anomalous reflective metasurface based on integral equations includes the following steps; Step 1: Determine the required operating frequency, incident angle, and reflection angle of the abnormal reflection based on the actual engineering conditions, and determine the size of the abnormal reflection surface unit; Step 2: Determine the number of elements in the finite size direction using the finite size and element size of the anomalous reflective surface; Step 3: For each unit, divide it into impedance plates of equal length along a finite dimension direction; Step 4: Divide each small impedance sheet into equal sub-units of equal length along the finite dimension direction; Step 5: In MATLAB, simulate the electromagnetic wave incident at 0° in the form of a plane wave using code, and generate a surface current that excites a scattering field. The scattered field is related to the surface impedance through the Huygens principle integral equation of the scalar field, forming an analytical expression. The two are interconnected, so by optimizing the surface impedance, the scattered field can be made to reach an ideal state in the target direction.
[0013] In step 1, the anomalous reflective surface unit is square, and its dimensions are calculated using the following formula: ; In the formula Let be the side length of the unit. For wavelength, The target reflection angle; The angle of incidence of the target.
[0014] In step 2, each anomalous reflective surface unit is arranged in a single row along a finite size direction to obtain a metasurface array, which is 1 × the number of units.
[0015] In step 3, each unit is divided into 6 impedance plates of equal length in a finite dimension direction. Each impedance plate represents a different impedance value. This operation realizes the process of discretizing continuous impedance. Since computers can only process discrete values, 6 discrete values can already achieve the ideal result. Discretizing into more values does not improve the result much, but will increase the complexity of the algorithm.
[0016] In step 4, each impedance sheet is divided into 10 equal-length sub-units along a finite dimension direction.
[0017] In step 5; ; ; The above equations are analytical expressions for the scattering electric and magnetic fields, respectively, formed by relating the scalar field Huygens principle integral equation to the impedance of the anomalous reflecting surface. For the scattered electric field, For scattering magnetic field, Surface impedance, For free space Green's function, For wave number, For free space wave impedance, , For surface current, Both are position vectors in cylindrical coordinates. For the integral curve, For the infinite space outside the integral curve; First, an initial impedance value is set for the impedance surface. This impedance value is taken from a set of impedance values that cause abnormal reflection of the impedance surface under periodic boundary conditions (infinite size) studied by predecessors. However, under finite size, the grating lobes of this set of impedance values will deteriorate (the grating lobe is the non-target direction, and the deterioration means that the energy in this direction is raised). Therefore, this set of impedance values needs to be optimized. The specific optimization steps are as follows: Since computers can only process discrete values, the continuous impedance is first discretized into multiple discrete impedances. Then, based on the Huygens principle integral equation for scalar fields, an analytical expression for the relationship between the scattered electric / magnetic field and the surface impedance is constructed. This is achieved by controlling the surface impedance Z... s The value of Z can be adjusted to change the scattered electric / magnetic field. In MATLAB, the "fmincon" function can find the minimum value of a function in the region near the independent variable given a known function expression; it is a local minimum. After obtaining the analytical expression, Z is... sAs the independent variable, the grating lobe of the scattered electric / magnetic field, as a function value, can be changed by altering Z. s The effect of reducing the scattered electric / magnetic field grating lobe, thereby increasing the main lobe (target angle); According to Floquet's theorem, for a reflecting surface, its tangential beam is defined as: ; in For free space wavenumber, Angle of incidence Let be the Flokai mode number, and D be the metasurface period.
[0018] The calculation formula is: : In the formula For wavelength, For the target reflection angle; if If the mode is evanescent, its amplitude decays along the surface normal direction. If so, then the pattern is a propagable pattern.
[0019] For an anomalous reflective surface, the frequency range of a single frequency wave is applicable between 8-12 GHz, and both the angle of incidence and the angle of reflection must be less than or equal to 70°.
[0020] When the angle of incidence is determined =0°, reflection angle =70° and electromagnetic wave frequency After 8GHz, only three harmonic modes can be propagated: n=-1 (-70°), n=0 (0°), and n=1 (70°). If the target reflection angle is 70°, then the grating lobes of the modes n=0 and n=-1 need to be reduced. If the actual application scenario requires a center frequency of 8 GHz, then the metasurface operates at 8 GHz; this enables the metasurface to achieve the requirement of 0° incident and 70° anomalous reflection; the corresponding finite-size anomalous reflection metasurface based on integral equations is composed of 1×10 metasurface units arranged periodically.
[0021] The desired optimization objective is to reduce the mode grating lobes at n=0 and n=-1. The Euclidean norm of the mode grating lobes at n=0 and n=-1 is selected as the optimization function, with the discrete impedance as the independent variable. Local optimization is performed using the "fmincon" function in MATLAB to minimize the field in the two desired directions locally, thus completing the optimization.
[0022] In step 3, the metasurface unit is designed based on the impenetrable impedance surface model, that is, only the field value above the surface needs to be considered in the analysis. The specific structure includes an impedance layer, a dielectric layer and a metal ground plane. The top of the metasurface unit is an impedance layer, the value of which is adjusted in the simulation to meet the required impedance. The bottom of the unit is a metal ground plane to simulate the effect of the field being impermeable. The dielectric layer is positioned between the impedance layer and the metal ground plane.
[0023] For the case of TM polarized wave incidence, the boundary condition of the impenetrable impedance surface takes the following form: (1) In the formula To represent the induced current density on the surface, we introduce the Huygens principle integral equation for the scalar field: (2) In the formula For the incident wave, Let be the total wave function in space. The Green's function in free space is expressed as: (3) In the formula, For zero-order Hankel function of the second kind; Substituting equation (1) into equation (2), we obtain the electric field integral equation satisfied by the impenetrable impedance surface under TM polarized wave incidence; (4) The scattered electric field Represented as: (5); Radial vector in cylindrical coordinates; Total electric field; : Incident electric field; : Incident electric field; The surface current density induced on the conductor surface by the incident field; Tangential magnetic field; Free space wavenumber; Free-space wave impedance; : The boundary of the surface; Surface impedance; The space outside the surface; For the case of TE polarized wave incidence, the boundary conditions are: (6) The corresponding magnetic field integral equation is written as: (7) The scattered magnetic field Represented as: (8).
[0024] Based on the impermeable impedance surface model, the impermeable impedance surface model will shield the field located below the surface; Discretize the continuous boundaries in equations (4) and (8) as follows: A with constant current density The lengths are all Line elements are used to transform integral equations into matrix equations using the point matching method. (9) For the case of TM polarized wave incidence, the parameters in equation (9) are written in the following form: (10) (11) For the case of TE polarized wave incidence, the parameters in equation (9) are written in the following form: (12) (13) The induced current density is obtained through matrix operations. Thus, the strength of the corresponding induced electric field and induced magnetic field can be obtained; Scattering width is used to describe a target's scattering ability in a specific direction. Scattering width is defined as follows: (14) In the formula , , , These are the scattered electric field, the incident electric field, the scattered magnetic field, and the incident magnetic field, respectively. The beneficial effects of this invention are: This invention presents a design and optimization method for a one-dimensional finite-size anomalous reflective impedance surface based on an integral equation. The unit cell is based on an impenetrable impedance surface model and consists of three layers: the first layer is an impedance layer, whose impedance value needs to be discretized and optimized to achieve the target anomalous reflection; the second layer is a dielectric substrate layer; and the third layer is a metal ground plane, simulating the impenetrable effect of the field. Only the discrete impedance values need to be designed and optimized, eliminating the need for complex unit cell structures. First, the electromagnetic wave is assumed to be incident on the metasurface as a plane wave. By combining the boundary conditions of the impenetrable impedance surface with the Huygens principle integral equation of the scalar field, the electric field integral equation satisfied by the metasurface is obtained. The integral equation is then transformed into a matrix equation using the method of moments, facilitating matrix operations for computer solution. After obtaining the unknowns in the equation, the scattered field is represented by these unknowns, and the scattered width is represented by the scattered field to characterize the anomalous reflection effect. The scattered width at the corresponding angle (-70°, 0°) of the desired grating lobe and the scattered width at angles within ±10° of it are selected to form a new vector. The Euclidean norm, minimum value, average value, or a value considering all weights of this vector is used as the objective function.
[0025] The "fmincon" function from the MATLAB toolbox is used to find the local minimum of the function, with the discrete impedance values as the independent variable. Therefore, by optimizing the discrete impedance values, the objective function can be minimized, causing energy to concentrate primarily in the direction of the target's anomalous reflection. Multiple optimization metrics can be set to match different application scenarios, making it more versatile. Attached Figure Description
[0026] Figure 1 This is a schematic diagram of the overall mathematical model of an impermeable impedance surface.
[0027] Figure 2 This is a three-dimensional schematic diagram of the impedance surface unit structure.
[0028] Figure 3 This is a schematic diagram of the scattering width in different directions of the reflecting surface.
[0029] Figure 4 This is a schematic diagram of the impedance optimization results under a single optimization index.
[0030] Figure 5 This is a schematic diagram showing the optimization results of the scattering width under a single optimization index.
[0031] Figure 6 This is a schematic diagram showing the impedance optimization results under comprehensive optimization indicators.
[0032] Figure 7 This is a schematic diagram showing the optimized scattering width under comprehensive optimization indicators.
[0033] Figure 8 A schematic diagram showing the scattering width of the reflective surface in each direction after changing and optimizing the starting point.
[0034] Figure 9 This is a schematic diagram of impedance optimization results under a single optimization index after changing the optimization starting point.
[0035] Figure 10This is a schematic diagram showing the scattering width optimization results under a single optimization index after changing the optimization starting point. Detailed Implementation
[0036] The present invention will now be described in further detail with reference to the accompanying drawings.
[0037] Example: like Figure 1 As shown, this invention proposes a finite-size anomalous reflection metasurface design and surface impedance optimization method based on integral equations. The impedance surface is composed of 1×10 metasurface units arranged periodically. The surface impedance value is analyzed and designed. The frequency of anomalous reflection surface design supported by the algorithm is between 8GHz and 12GHz. However, since anomalous reflection is for a specific frequency, for the convenience of algorithm demonstration, the working frequency of the metasurface is set to 8GHz. At the same time, the anomalous reflection surface designed by this algorithm supports anomalous reflection with large incident angle (-70°~70°) and large emission angle (-70°~70°). In the algorithm demonstration, the incident angle is assumed to be 0° and the reflection angle is assumed to be 70°. The metasurface operates at a frequency of 8GHz, so that the metasurface can achieve the requirement of 0° incident and 70° anomalous reflection.
[0038] The designed metasurface unit structure is as follows Figure 2 As shown. The dielectric layer thickness of the unit cell structure in the non-penetrating impedance model is 1.575 mm, the dielectric constant is 2.2, and the unit cell period is... .
[0039] The top of the unit is an impedance layer, whose value can be flexibly adjusted during simulation to meet the required impedance.
[0040] To simplify and improve the accuracy of the algorithm, a continuous surface impedance is discretized. A single cell is divided into six equally spaced intervals, each corresponding to a different surface impedance value. ; The surface impedance value of each interval is randomly generated by the random algorithm in MATLAB along a finite dimension, and used as the initial value for optimization.
[0041] It is not necessary to divide it into 6 intervals. The algorithm can achieve ideal results with 6 discrete intervals. The more discrete intervals there are, the more accurate it will be, but the more computational it will be. To improve the accuracy of the algorithm results, each interval is further divided into 10 sub-intervals along a finite dimension. These 10 impedance intervals correspond to the same discrete impedance value. Subdividing the intervals allows the discretization to be closer to the continuous value, while avoiding problems such as non-convergence and distortion in the calculation.
[0042] To facilitate subsequent derivations, the parameter notation and the representation of the vector field are given here: Radial vector in cylindrical coordinates; Total electric field; : Incident electric field; : Incident electric field; The surface current density induced on the conductor surface by the incident field; Tangential magnetic field; Free space wavenumber; Free-space wave impedance; : The boundary of the surface; Surface impedance; The space outside the surface; For the case of TM polarized wave incidence, the boundary condition of the impenetrable impedance surface takes the following form: (1) In the formula To represent the induced current density on the surface, we introduce the Huygens principle integral equation for the scalar field: (2) In the formula For the incident wave, Let be the total wave function in space. The Green's function in free space is expressed as: (3) In the formula, For zero-order Hankel function of the second kind; Substituting equation (1) into equation (2), we obtain the electric field integral equation satisfied by the impenetrable impedance surface under TM polarized wave incidence; (4) The scattered electric field It can be represented as: (5) Similarly, for the case of TE-polarized wave incidence, the boundary conditions are: (6) The corresponding magnetic field integral equation is written as: (7) The scattered magnetic field It can be represented as: (8).
[0043] This invention is based on the impermeable impedance surface model, which shields the field located below the surface. Therefore, this invention only focuses on the electromagnetic field distribution above the surface.
[0044] Since computers struggle to handle continuous integral equations, the continuous boundaries in equations (4) and (8) are discretized here. A with constant current density The lengths are all Line elements are used to transform integral equations into matrix equations using the point matching method. (9) For the case of TM polarized wave incidence, the parameters in equation (9) are written in the following form. (10) (11) For the case of TE polarized wave incidence, the parameters in equation (9) are written in the following form. (12) (13) The induced current density is obtained through matrix operations. This allows us to obtain the strength of the induced electric and magnetic fields.
[0045] This invention uses scattering width to describe the scattering capability of a target in a specific direction. The scattering width is defined as follows: (14) In the formula , , , These are the scattered electric field, the incident electric field, the scattered magnetic field, and the incident magnetic field, respectively.
[0046] The dielectric layer thickness of the unit cell structure in the non-penetrating impedance model is 1.575 mm, the dielectric constant is 2.2, and the unit cell period is [not specified]. .
[0047] At the same time, according to Flocay's theorem, when the angle of incidence is determined... =0°, reflection angle =70° and electromagnetic wave frequency After 8GHz, only three harmonic modes can propagate: n=-1 (-70°), n=0 (0°), and n=1 (70°). If the target reflection angle is 70°, then the grating lobes of the modes n=0 and n=-1 need to be reduced.
[0048] The initial six discrete values of the surface impedance of the element are shown in the table below: Table 1. Initial discrete values of surface impedance 1 (unit: ) The unit structure composed of this set of impedance values, arranged in a 1×10 reflective surface, exhibits the following behavior when facing a TM-polarized wave incident at 0°: Figure 3 As shown, the scattering width reaches 23.508dB in the n=1 mode, and 10.043dB and 11.436dB in the n=0 and n=-1 modes, respectively. Based on the standard of -12dB relative to the scattering width in the n=1 mode, the other two modes are required to be reduced to at least 1.4833dB.
[0049] The surface impedance optimization method of this invention includes the following steps; For anomalous reflective surfaces of finite size, the operating frequency, incident angle, and reflection angle of the required anomalous reflection must first be determined based on the actual engineering conditions. These three parameters can determine the size of the anomalous reflective surface unit. ; Secondly, based on the finite size requirements of the actual project, the number of metasurface units in the finite size direction is obtained by "finite size / metasurface unit size", forming a metasurface array of "1 × number of units"; Subsequently, for each individual unit, it is divided into 6 impedance plates of equal length in a finite dimension direction. Each impedance plate represents a different impedance value. This operation realizes the process of discretizing continuous impedance. Since computers can only process discrete values, 6 discrete values can already achieve the ideal result. Discretizing into more values does not improve the result much, but will increase the complexity of the algorithm. Then, each small impedance sheet is divided into 10 equal-length sub-units along the finite size direction, in order to decompose the algorithm network and increase the accuracy of the algorithm.
[0050] In MATLAB, electromagnetic waves are simulated as plane waves incident at 0° on a metasurface array. The plane wave generates a surface current on the metasurface, which in turn excites a scattered field. The scattered field can be connected to the surface impedance through the Huygens principle integral equation for scalar fields, forming an analytical expression. Since the two are interrelated, by optimizing the surface impedance, the scattered field can be ideally positioned in the target direction.
[0051] According to Floquet's theorem, for a reflecting surface, its tangential beam is defined as: ; in For free space wavenumber, Angle of incidence Let D be the Flokai mode number, and D be the metasurface period. Calculated; In the formula For wavelength, The target reflection angle. If If the mode is evanescent, its amplitude decays along the surface normal direction. If so, then the pattern is a propagable pattern.
[0052] When the angle of incidence is determined =0°, reflection angle =70° and electromagnetic wave frequency After 8GHz, only three harmonic modes can propagate: n=-1 (-70°), n=0 (0°), and n=1 (70°). If the target reflection angle is 70°, then the grating lobes of the modes n=0 and n=-1 need to be reduced.
[0053] In summary, the desired optimization objective can be determined as reducing the mode grating lobes at n=0 and n=-1. The Euclidean norms of the mode grating lobes at n=0 and n=-1 are chosen as the optimization function, with discrete impedance as the independent variable. Local optimization is performed using the "fmincon" function in MATLAB to minimize the field in the two desired directions locally, thus completing the optimization.
[0054] This invention allows for flexible selection of optimization parameters to optimize surface impedance. Figure 4 and Figure 5 The results show the optimization results using the Euclidean norm of the scattering width as an indicator for n=0 and n=-1. The scattering width corresponding to n=1 (70°) is 23.5086dB, while the scattering width is reduced to 1.204dB and -0.123dB in the n=0 and n=-1 modes, respectively, achieving the target optimization effect.
[0055] Figure 6 and Figure 7 The results show that the Euclidean norm, peak value, and average value of the scattering width for n=0 and n=-1 are comprehensively considered by weighting factors. The scattering width is reduced to 0.395dB and -1.331dB in the n=0 and n=-1 modes, respectively, and the optimization effect is better than the result of optimization of a single optimization index.
[0056] To verify that the optimization effect has no loss of generality, the optimization starting point for changing the surface impedance was still optimized using the algorithm proposed in this invention. The initial discrete impedance values are shown in Table 2, and the scattering width before optimization is as follows: Figure 8 As shown. The Euclidean norm of the scattering width in the n=-1 and n=0 modes was selected as the optimization index, and the optimization results are as follows. Figure 9 and Figure 10 As shown, the optimization effect is obvious.
[0057] Table 2. Initial discrete values of surface impedance 2 (unit: ) To simplify and improve the accuracy of the algorithm, a continuous surface impedance is discretized. A single cell is divided into six equally spaced intervals, each corresponding to a different surface impedance value. To improve the accuracy of the algorithm results, each interval is further divided into 10 sub-intervals. These 10 impedance intervals correspond to the same discrete impedance value. Subdividing the intervals allows the discretization to be closer to the continuous value, while avoiding problems such as non-convergence and distortion in the calculation.
[0058] After obtaining the scattering width of the metasurface at different angles, the scattering widths at the corresponding angles (-70°, 0°) of the desired grating lobe to be reduced, as well as the scattering widths at angles within ±10° of that lobe, are selected to form a new vector. The Euclidean norm, minimum, average, or a combination of weighted values of this vector are used as the objective function. The "fmincon" function in the MATLAB toolbox is used to find the local minimum of this function, with the previously discrete impedance values as the independent variable. Therefore, by optimizing the discrete impedance values, the objective function can be minimized, causing the energy to be concentrated mainly in the direction of the target's anomalous reflection.
[0059] A finite-size anomalous reflective metasurface can be considered as an electromagnetic device. When a plane wave is incident on this metasurface, its incident field, scattered field, surface impedance, and surface current density satisfy an analytical integral equation. In this invention, the unknown in the integral equation is the surface current density, and other parameters can be expressed using the surface current density. The goal is to solve this integral equation to obtain the surface current density, and then use the surface current density to represent the scattered field. By optimizing the field value of the scattered field, the anomalous reflection effect is achieved, thus realizing a "finite-size anomalous reflective metasurface."
[0060] The relationship between the design structural characteristics of a finite-size anomalous reflective metasurfaces and their surface impedance is as follows: an impenetrable impedance surface is an ideal model of a metasurface, and the impedance value on the impedance surface is called the surface impedance. By designing the value of the surface impedance, the metasurface can achieve anomalous reflection at an ideal angle within a finite size.
[0061] This invention is implemented through algorithms, involving numerical calculations, without specific structural design. The model is based on an impermeable impedance surface design, with the top layer being an impedance layer whose value is directly controlled and optimized using algorithms. The middle layer is a dielectric layer, and the bottom layer is a metal ground plane, which can simulate the effect of an impermeable field.
Claims
1. A method for designing finite-size anomalous reflective metasurfaces and optimizing surface impedance based on integral equations, characterized in that, Includes the following steps; Step 1: Determine the required operating frequency, incident angle, and reflection angle of the abnormal reflection based on the actual engineering conditions, and determine the size of the abnormal reflection surface unit; Step 2: Determine the number of elements in the finite dimension direction using the finite dimensions and element dimensions of the anomalous reflective surface; Step 3: For each unit, divide it into impedance plates of equal length along a finite dimension direction; Step 4: Divide each small impedance sheet into equal sub-units of equal length along the finite dimension direction; Step 5: In MATLAB, simulate the electromagnetic wave incident at 0° in the form of a plane wave. The resulting surface current excites a scattered field. The scattered field is connected to the scalar field Huygens principle integral equation and the anomalous reflection surface impedance to form an analytical expression. By optimizing the surface impedance, the scattered field is made to reach an ideal state in the target direction.
2. The method for designing and optimizing surface impedance of a finite-size anomalous reflective metasurfaces based on integral equations according to claim 1, characterized in that, In step 1, the anomalous reflective surface unit is square, and its dimensions are calculated using the following formula: ； In the formula Let be the side length of the unit. For wavelength, The target reflection angle; The angle of incidence of the target.
3. The method for designing and optimizing surface impedance of a finite-size anomalous reflective metasurfaces based on integral equations according to claim 1, characterized in that, In step 2, each anomalous reflective surface unit is arranged in a single row along a finite size direction to obtain a metasurface array.
4. The method for designing and optimizing surface impedance of a finite-size anomalous reflective metasurfaces based on integral equations according to claim 1, characterized in that, In step 3, each unit is divided into 6 impedance plates of equal length in a finite dimension direction, and each impedance plate represents a different impedance value. In step 4, each impedance sheet is divided into 10 equal-length sub-units along a finite dimension direction.
5. The method for designing and optimizing surface impedance of a finite-size anomalous reflective metasurfaces based on integral equations according to claim 1, characterized in that, In step 5 The above equations are analytical expressions for the scattering electric and magnetic fields, respectively, formed by relating the scalar field Huygens principle integral equation to the impedance of the anomalous reflecting surface. For the scattered electric field, For scattering magnetic field, Surface impedance, For free space Green's function, For wave number, For free space wave impedance, , For surface current, Both are position vectors in cylindrical coordinates. For the integral curve, For the infinite space outside the integral curve; By controlling the surface impedance Z s The value of Z is adjusted to scatter the electric / magnetic field. In MATLAB, the fmincon function finds the local minimum of the function in the region near the independent variable given the function expression. After obtaining the analytical expression, Z is... s As the independent variable, the grating lobe of the scattered electric or magnetic field is used as a function value, which is changed by altering Z. s This reduces the effect of the scattered electric / magnetic field grating lobe, thereby increasing the main lobe height. According to Floquet's theorem, for a reflecting surface, its tangential beam is defined as: ； in For free space wavenumber, Angle of incidence Let ρ be the Flokai mode number, and D be the metasurface period. The formula for calculating Flokai is: : In the formula For wavelength, For the target reflection angle; if If the mode is evanescent, its amplitude decays along the surface normal direction. If so, then the pattern is a propagable pattern.
6. The method for designing and optimizing surface impedance of a finite-size anomalous reflective metasurfaces based on integral equations according to claim 1, characterized in that, The anomalous reflective surface is applicable to a single frequency wave with a frequency range between 8-12 GHz, and both the angle of incidence and the angle of reflection must be less than or equal to 70°.
7. The method for designing and optimizing surface impedance of a finite-size anomalous reflective metasurfaces based on integral equations according to claim 1, characterized in that, In step 3, the metasurface unit is designed based on the impenetrable impedance surface model, that is, only the field value above the surface needs to be considered in the analysis. The specific structure includes an impedance layer, a dielectric layer and a metal ground plane. The top of the metasurface unit is an impedance layer, whose value is adjusted in the simulation to meet the required impedance. The bottom of the unit is a metal ground plane, which is used to simulate the effect of the field being impenetrable. The dielectric layer is positioned between the impedance layer and the metal ground plane.
8. The method for designing and optimizing surface impedance of a finite-size anomalous reflective metasurfaces based on integral equations according to claim 7, characterized in that, For the case of TM polarized wave incidence, the boundary condition of the impenetrable impedance surface takes the following form: (1) In the formula To represent the induced current density on the surface, we introduce the Huygens principle integral equation for the scalar field: (2) In the formula For the incident wave, Let be the total wave function in space. The Green's function in free space is expressed as: (3) In the formula, For zero-order Hankel function of the second kind; Substituting equation (1) into equation (2), we obtain the electric field integral equation satisfied by the impenetrable impedance surface under TM polarized wave incidence; (4) The scattered electric field Represented as: (5)； Radial vector in cylindrical coordinates; Total electric field; : Incident electric field; : Incident electric field; The surface current density induced on the conductor surface by the incident field; Tangential magnetic field; Free space wavenumber; Free-space wave impedance; : The boundary of the surface; Surface impedance; The space outside the surface.
9. The method for designing and optimizing surface impedance of a finite-size anomalous reflective metasurfaces based on integral equations according to claim 8, characterized in that, For the case of TE polarized wave incidence, the boundary conditions are: (6) The corresponding magnetic field integral equation is written as: (7) The scattered magnetic field Represented as: (8)。 10. The method for designing and optimizing surface impedance of a finite-size anomalous reflective metasurfaces based on integral equations according to claim 9, characterized in that, Based on the impermeable impedance surface model, the impermeable impedance surface model will shield the field located below the surface; Discretize the continuous boundaries in equations (4) and (8) as follows: A with constant current density The lengths are all For line elements, the integral equation is transformed into a matrix equation using the point matching method: (9) For the case of TM polarized wave incidence, the parameters in equation (9) are written in the following form: (10) (11) For the case of TE polarized wave incidence, the parameters in equation (9) are written in the following form. (12) (13) The induced current density is obtained through matrix operations. Thus, the strength of the corresponding induced electric field and induced magnetic field can be obtained; Scattering width is used to describe a target's scattering capability in a specific direction. Scattering width is defined as follows: (14) In the formula , , , These are the scattered electric field, the incident electric field, the scattered magnetic field, and the incident magnetic field, respectively.