Design method of reconfigurable multi-beam forming antenna based on cascaded metasurfaces

Abstract

The invention discloses a design method of a reconfigurable multi-beam forming antenna based on cascaded metasurfaces, and belongs to the technical field of beam forming. The method comprises the steps of firstly building a cascade metasurface system, then preparing a training data set, then building a forward transfer model of the N layers of cascade metasurface systems, carrying out backward gradient descent iterative optimization through the forward transfer model and the training data set, and finally obtaining the cascade metasurface system according to an optimized phase modulation coefficient. According to the antenna, a mechanical reconfigurable N-layer cascade metasurface system is adopted, a passive metasurface serves as a hardware carrier to achieve reconfigurable beam forming, compared with a traditional phased array, a complex feed system and a radio frequency front-end device are not needed, complex digital signal processing algorithm design is not needed, and the antenna has the advantages of being low in cost, simple in system and convenient to apply.

Description

Technical Field

[0001] The present invention relates to the technical field of beamforming, and particularly relates to a design method of a reconfigurable multi-beamforming antenna based on cascaded metasurfaces. Background Art

[0002] Beamforming is a technology that optimizes communication and signal processing performance by controlling the propagation direction of signals in space. Its core objective is to use signal superposition and phase regulation technologies to form high-gain beams in the target direction while suppressing interference or noise in non-target directions. This technology was initially used in radar and sonar systems, and in recent years, with the rapid growth of communication demands, it has been widely applied in fields such as wireless communication, satellite communication, and the Internet of Things.

[0003] Array signal processing algorithms based on phased arrays have been widely studied to implement beamforming algorithms in various scenarios. These methods receive signals through antenna array elements at different positions, and then perform amplitude and phase weighted synthesis on the array signals to achieve high-gain beam pointing in one or more directions. Although array signal processing algorithms can effectively implement beamforming in the specified direction, due to their dependence on phased array systems composed of a large number of antenna array elements, each element requires an independent radio frequency link to achieve amplitude and phase regulation of the received signals. Therefore, an increase in the element scale will significantly increase the system cost and hardware complexity. In recent years, the emerging artificial material - metasurface, as a two-dimensional material composed of sub-wavelength scale artificial structures, has shown unique potential in manipulating electromagnetic waves in an unconventional way. These artificial structures can precisely regulate the refraction, reflection, and transmission behaviors of light waves, thereby achieving electromagnetic characteristics beyond the capabilities of natural materials. Compared with traditional phased array antennas, metasurfaces have the advantages of simple structure, easy manufacturing, relatively low cost, and strong scalability. Therefore, metasurfaces have great potential for implementing new low-cost multi-beamforming antennas. Summary of the Invention

[0004] Based on the above analysis, the present invention proposes a design method of a reconfigurable multi-beamforming antenna based on cascaded metasurfaces for realizing multi-beamforming; it is compact in design, flexible in structure, can efficiently realize multi-beam formation, and is suitable for multi-target communication systems and wireless transmission requirements in complex environments.

[0005] To achieve the above object, the present invention provides the following technical solutions:

[0006] A design method of a reconfigurable multi-beamforming antenna based on cascaded metasurfaces, comprising the following steps:

[0007] Step 1, build an N-layer cascaded metasurface system to obtain W mechanically multiplexed channels;

[0008] Step 2: Input a plane wave through the N - layer cascaded metasurface system, and the output is the target beams scattered in A j directions. A j is the number of directions of the target beam in the j - th channel, j = 1, …, W. Model it with an all - one matrix, and obtain the Fourier spectrum diagram according to the target beam pointing:

[0009]

[0010]

[0011] where, are the elevation angle and azimuth angle of the i - th target beam pointing in the j - th channel respectively, i = 1, …, A j , are the abscissa and ordinate of the Fourier spectrum diagram corresponding to the i - th target beam pointing respectively, is the working wavelength of the antenna; in the Fourier spectrum diagram, set the value at the position of ( ) to 1 and the values at other positions to 0. The obtained two - dimensional Fourier spectrum diagram is the rectangular coordinate system description of the antenna target pattern; finally, use the two - dimensional Fourier spectrum diagrams corresponding to the target beam pointings of W channels as the training data set;

[0012] Step 3: The incident plane wave is first modulated by the first - layer metasurface, and the modulated outgoing field is transmitted to the second - layer metasurface through the diffraction process of electromagnetic waves, and then successively passes through the modulation and diffraction of each layer of metasurface, and finally exits from the N - th layer of metasurface. Perform a Fourier transform on the outgoing field to obtain the antenna pattern. The diffraction process is expressed by the Rayleigh - Sommerfeld diffraction formula as follows:

[0013]

[0014] where, represents the diffraction connection relationship between two adjacent layers of metasurfaces, is the central coordinate of a super - cell located in the (n - 1) - th layer, is the central coordinate of a super - cell located in the n - th layer, represents the diffraction connection relationship between two super - cells whose central coordinates in two adjacent layers of metasurfaces are located at and respectively, , represents the Euclidean distance between two super - cells;

[0015] Step 4: In a mechanical configuration, the forward - transfer model expression of the incident plane wave in the cascaded metasurface is as follows:

[0016]

[0017] Among them, represents the incident plane wave, represents the Fourier spectrum of the output field of the corresponding channel under this mechanical configuration, and the matrix element values are determined by the Rayleigh-Sommerfeld diffraction formula in step 3. represents the transmission coefficient modulation effect matrix of the nth layer of metasurface on the incident electromagnetic wave under this mechanical configuration;

[0018] Step 5: Repeat step 4 to obtain the forward transfer models of W channels;

[0019] Step 6: Use the mean square error between the Fourier spectrum amplitude of the forward output field of the cascaded metasurface and the target Fourier spectrum amplitude distribution as the loss function, and use the gradient descent algorithm to train the phase modulation coefficients of the meta-units of each layer of the N-layer cascaded metasurface system to minimize this loss function; the optimization expression of the gradient descent algorithm is as follows:

[0020]

[0021] Among them, is the phase modulation coefficient of the meta-unit, respectively represent the Fourier spectrum amplitude of the output field of the jth channel of the forward transfer model and the target Fourier spectrum amplitude; after the forward transfer model and the backward gradient descent algorithm are both constructed, use the backward gradient descent algorithm to iteratively optimize the phase modulation coefficients until the set number of iteration rounds is reached and stop, and obtain the phase modulation coefficients of the meta-units of each layer of the metasurface;

[0022] Step 7: Obtain the N-layer cascaded metasurface system according to the obtained phase modulation coefficients of the meta-units of each layer of the metasurface, input the plane wave into the N-layer cascaded metasurface system under different mechanical configurations, and obtain the antenna radiation patterns with different beam directions to achieve reconfigurable beamforming.

[0023] Furthermore, the process of building the N-layer cascaded metasurface system in step 1 is specifically as follows: The antenna uses an N-layer cascaded metasurface system, which includes N layers of metasurfaces, which are sequentially placed at intervals along the propagation direction, and the metasurfaces are perpendicular to the propagation direction;

[0024] The rotation operation forms multiple rotation angles by rotating the angle of each layer of the metasurface, and the exchange operation forms multiple permutation orders by exchanging the order of each layer of the metasurface. Each rotation angle and permutation order constitute a mechanical configuration. The rotation operation and the exchange operation have W mechanical configurations through the Cartesian product. Each mechanical configuration corresponds to a mechanically multiplexed channel, so as to obtain W mechanically multiplexed channels.

[0025] Further, in step 1, the N layers of metasurfaces of the N-layer cascaded metasurface system are placed at equal intervals along the propagation direction, and the interval distance d is 10 to 30 operating wavelengths of the antenna.

[0026] Further, in step 1, N is a natural number between 2 and 5.

[0027] Further, in step 1, each layer of metasurface is square, and each layer of metasurface includes M×M meta-units, where M is a natural number ≥ 30.

[0028] Further, in step 1, four rotation angles are formed by rotating each layer of metasurface at angles of 0°, 90°, 180°, or 270° respectively. The swapping operation is to form a total of N! permutation orders by swapping the order of each layer of metasurface. The rotation operation and the swapping operation obtain 4 N-1 ×N! mechanically multiplexed channels, that is, W = 4 N-1 ×N!.

[0029] Further, in step 2, A j is a natural number ≥ 1.

[0030] Further, in step 5, the rotation operation is achieved by rotating the transmission coefficient modulation matrix , and the swapping operation is achieved by changing the order of the transmission coefficient modulation matrix in the formula; the combination of the rotation operation and the swapping operation will result in W different transformations.

[0031] Further, in step 6, an adaptive moment estimation optimizer is adopted.

[0032] Further, in step 6, a suitable learning rate is set between 0.1 and 0.005.

[0033] Further, in step 6, the set number of iteration rounds is 500 to 1200.

[0034] The technology of the present invention has the following advantages:

[0035] 1. The antenna adopts a mechanically reconfigurable N-layer cascaded metasurface system, and uses a passive metasurface as a hardware carrier to achieve reconfigurable beamforming. Compared with traditional phased arrays, it does not require a complex feeding system and radio frequency front-end devices, nor does it require complex digital signal processing algorithm design, and has the advantages of low cost, simple system, and convenient application;

[0036] 2. The mechanically reconfigurable operation based on rotation and swapping is convenient for practical implementation, does not require the introduction of additional system hardware, nor additional algorithm iterations, and can quickly switch the beam pointing;

[0037] 3. By increasing the number of supercells, the scale of the system's trainable parameters can be enhanced to increase the number of multi-beams as much as possible, meeting the actual requirements of high-capacity communication scenarios;

[0038] 4. Only using an N-level cascaded metasurface system, it has a high integration level and has the advantages of simplicity and efficiency compared with traditional digital signal processing solutions. The frequency band scalability of the system solution is strong and can be used in multiple frequency bands from GHz to THz communication. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 It is a flowchart of the design method of the reconfigurable multi-beam shaping antenna based on cascaded metasurfaces of the present invention; Figure 2 It is the target spectrum distribution diagram obtained according to an embodiment of the design method of the reconfigurable multi-beam shaping antenna based on cascaded metasurfaces of the present invention; Figure 3 It is the Fourier spectrum and elevation angle mapping relationship of the design method of the reconfigurable multi-beam shaping antenna based on cascaded metasurfaces of the present invention; Figure 4 It is the Fourier spectrum and azimuth angle mapping relationship of the design method of the reconfigurable multi-beam shaping antenna based on cascaded metasurfaces of the present invention; Figure 5 It is a schematic diagram of the N-level cascaded metasurface system according to an embodiment of the design method of the reconfigurable multi-beam shaping antenna based on cascaded metasurfaces of the present invention; Figure 6 It is a schematic diagram of the mechanically reconfigurable channel according to an embodiment of the design method of the reconfigurable multi-beam shaping antenna based on cascaded metasurfaces of the present invention;

[0040] Figure 7 It is the output Fourier spectrum diagram of the first channel according to the design method of the reconfigurable multi-beam shaping antenna based on cascaded metasurfaces of the present invention Figure 8 It is the cross-sectional field distribution diagram of the first channel according to the design method of the reconfigurable multi-beam shaping antenna based on cascaded metasurfaces of the present invention; Figure 9 It is the output Fourier spectrum of the second channel according to the design method of the reconfigurable multi-beam shaping antenna based on cascaded metasurfaces of the present invention; Figure 10 It is the sum cross-sectional field distribution diagram of the second channel according to the design method of the reconfigurable multi-beam shaping antenna based on cascaded metasurfaces of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0041] The present invention will be further described below with reference to the drawings through specific embodiments.

[0042] AsFigure 1 As shown, the design method of the reconfigurable multi-beam shaping antenna based on cascaded metasurfaces in this embodiment includes the following steps:

[0043] Step 1, build an N-layer cascaded metasurface system:

[0044] The N-layer cascaded metasurface system includes two layers of metasurfaces, that is, N = 2, which are placed at intervals of 30 times the operating wavelength along the propagation direction. The metasurfaces are perpendicular to the propagation direction. The operating frequency point of this embodiment is set to 3 GHz. The input plane wave is discretized with a sampling interval of 1 wavelength according to the side length of the square supercell, and the size is the same as that of the metasurface. Each layer of metasurface is square, and each layer of metasurface includes 300×300 supercells, that is, M = 300; a total of four rotation angles are formed by rotating the angles of each layer of metasurface by 0°, 90°, 180°, or 270°. The exchange operation is to form a total of 2! permutation orders by exchanging the order of each layer of metasurface. The rotation operation and the exchange operation obtain 8 mechanically multiplexed channels through the Cartesian product, that is, W = 4 2-1 ×2!; Each rotation angle and permutation order constitute a mechanical configuration; the more the number of supercells, the more parameters that can be trained in the model, and the more beams that can be realized;

[0045] Step 2, prepare the training dataset:

[0046] The input is a plane wave, and the output is the target beam scattered in A j directions. A j is the number of directions of the target beam in the j-th channel, j = 1,…, W. The all-one matrix is used for modeling. From the two-dimensional space Fourier transform formula in the rectangular coordinate system, the mapping relationship between the target beam pointing and the Fourier spectrum diagram is as follows:

[0047]

[0048]

[0049] Among them, are the elevation angle and azimuth angle of the i-th target beam pointing in the j-th channel respectively, i = 1,…, A j , are the abscissa and ordinate of the Fourier spectrum diagram corresponding to the i-th target beam pointing respectively, the operating wavelength of the antenna; in the Fourier spectrum diagram, the value at the position of ( ) is set to 1, and the values at other positions are set to 0. The two-dimensional Fourier spectrum diagram obtained in this way is the rectangular coordinate system description of the antenna target direction diagram;

[0050] Use the two-dimensional Fourier spectrum corresponding to the target beam pointing of 8 channels as the training dataset for the iterative optimization of the subsequent backpropagation gradient descent algorithm;

[0051] The target beam directions of the eight channels are respectively set to [(10°, 20°), (10°, 30°), (0°, 45°), (0°, -30°)], [(25°, 20°)], [(15°, -45°), (15°, 30°)], [(25°, -20°)], [(60°, -25°), (35°, 30°)], [(40°, 30°), (50°, -30°)], [(15°, -45°)], [(25°, 50°), (70°, -30°)], [(30°, 50°)], [(60°, -12°), (44°, 32°)]. Figure 2 A typical target spectrum distribution diagram is shown. The points corresponding to the small squares are the Fourier spectrum points corresponding to the target beam pointing directions. The amplitude of this point is set to 1, and the amplitudes of other points are all 0; Figure 3 and Figure 4 respectively show the Fourier spectrum of the output field and the mapping relationship between the elevation angle and the azimuth angle.

[0052] Step 3, build the forward propagation model:

[0053] Each layer of the N-layer cascaded metasurface system is a transmissive phase modulation metasurface, and the phase modulation coefficient of each meta-unit of each layer of the metasurface is the parameter to be optimized;

[0054] The propagation process of a plane wave in one channel in the N-layer cascaded metasurface system is as follows: The incident plane wave is first modulated by the first layer of the metasurface, and the modulated output field is transmitted to the second layer of the metasurface through the diffraction process of electromagnetic waves, and then successively passes through the modulation and diffraction of each layer of the metasurface, and finally exits from the Nth layer of the metasurface. The Fourier transform of the output field is used to obtain the antenna pattern. According to the Rayleigh-Sommerfeld diffraction formula in Fourier optics theory, the diffraction process is expressed as follows:

[0055]

[0056] Among them, represents the diffraction connection relationship between adjacent two layers of the metasurface, is the central coordinate of a meta-unit located on the (n - 1)th layer, is the central coordinate of a meta-unit located on the nth layer, represents the diffraction connection relationship between two meta-units whose central coordinates in adjacent two layers of the metasurface are respectively located at and ​ , represents the Euclidean distance between two supercells;

[0057] Step 4, in a mechanical configuration, the forward propagation model expression of the incident plane wave in the cascaded metasurface is as follows:

[0058]

[0059] where, represents the incident plane wave, represents the Fourier spectrum of the output field of the corresponding channel in this mechanical configuration, and the matrix element values are determined by the above Rayleigh–Sommerfeld diffraction formula, represents the transmission coefficient modulation matrix of the \(n\)th layer metasurface in this mechanical configuration for the incident electromagnetic wave; the convolution operation represented by ‘*’ is derived from the Rayleigh–Sommerfeld diffraction formula, which describes the diffraction propagation process of the wave between layers, and the dot product operation represented by ‘ ’ is modeled by the element-by-element modulation of the incident wave by the metasurface;

[0060] Step 5, Step 4 describes the propagation process of a corresponding channel in a mechanical configuration. Through rotation operations and exchange operations, the propagation processes of all 8 channels can be obtained based on the above formula; among them, the rotation operation is achieved by rotating the transmission coefficient modulation matrix , and the exchange operation is achieved by changing the order of the transmission coefficient modulation matrix in the formula; the combination of the rotation operation and the exchange operation will cause the above formula to produce 8 different transformations for the joint optimization of the subsequent gradient descent algorithm;

[0061] Step 6, backward gradient descent iterative optimization:

[0062] The mean square error between the Fourier spectrum amplitude of the forward output field of the cascaded metasurface and the target Fourier spectrum amplitude distribution is used as the loss function, and the gradient descent algorithm is used to train the phase modulation coefficients of the supercells of each layer of the \(N\)-layer cascaded metasurface system to minimize this loss function; according to statistical definitions, the expression of the target mean square error optimized by the gradient descent algorithm is as follows:

[0063]

[0064] where, is the phase modulation coefficient of the supercell, respectively represent the Fourier spectrum amplitude of the output field of the j-th channel of the forward propagation model and the target Fourier spectrum amplitude; after both the forward propagation model and the backward gradient descent algorithm are constructed, the Adaptive Moment Estimation optimizer (Adam) is used, the learning rate is set to 0.01, and the backward gradient descent algorithm is used to iteratively optimize the phase modulation coefficient until the set number of iteration rounds is reached and stop, so as to obtain the phase modulation coefficient of the meta-units of each layer of the metasurface;

[0065] Figure 3 shows a schematic diagram of the system structure of the present invention. The incident plane wave is successively modulated layer by layer through two cascaded metasurfaces. The two metasurfaces are A and B respectively, and the output field forms the target pattern of the directional beam after far-field propagation; Figure 4 shows a total of eight mechanically reconfigured channels constructed by combining two types of mechanical operations. The arrow is used to indicate the current rotation angle; according to the diffraction formula, the transmission modulation effect and the fast Fourier transform algorithm, a forward propagation model and a backward gradient descent optimization model of two cascaded metasurfaces are built; the distance between the two metasurfaces is set to 30 times the working wavelength; the update method of the network weights is based on the loss update of the full dataset, that is, the plane wave is incident on the different channels obtained by eight mechanical reconstructions respectively, and then the loss between the output of each channel and the target spectrum diagram is calculated, and the average of this loss is used as the final loss; finally, based on the final loss, a gradient descent is performed once to update the weights, and this is repeated iteratively until the network converges;

[0066] Step 7, mechanically reconfigurable beamforming:

[0067] According to the phase modulation coefficient of the meta-units of each layer of the metasurface obtained in step 6, an N-layer cascaded metasurface system is obtained. When a plane wave is input into one of the eight mechanical configurations, a gain beam in the specified direction can be obtained on the far-field pattern, and there is no output in other directions; through rotation operations and switching operations, the rapid switching of eight groups of different multi-beams can be conveniently realized. Figures 7 to 10 respectively show the beamforming results of the first channel and the second channel in the eight channels, where Figure 7 and Figure 9 are the output Fourier spectrum distributions, Figure 8 and Figure 10 are the two-dimensional plane field distributions after the output field transmits 30 wavelengths.

[0068] Finally, it should be noted that the purpose of publishing the embodiments is to help further understand the present invention. However, those skilled in the art can understand that: within the spirit and scope of the present invention and the appended claims, various substitutions and modifications are possible. Therefore, the present invention should not be limited to the content disclosed in the embodiments, and the scope of protection claimed by the present invention is subject to the scope defined by the claims.

Claims

1. A design method of a reconfigurable multi-beam shaping antenna based on cascaded metasurfaces, characterized in that It includes the following steps: Step 1: Build an N - layer cascaded metasurface system to obtain W mechanically multiplexed channels; Step 2: Input a plane wave through the N-level cascaded metasurface system, and the output is the target beam scattered in A j directions. A j is the number of directions of the target beam in the j-th channel, j = 1, …, W. It is modeled using an all-ones matrix, and the Fourier spectrum diagram is obtained according to the target beam pointing; the obtained two-dimensional Fourier spectrum diagram is the rectangular coordinate system description of the antenna target direction diagram; finally, the two-dimensional Fourier spectrum diagrams corresponding to the target beam pointings of W channels are used as the training data set; Step 3: The incident plane wave is first modulated by the first - layer metasurface, and the modulated output field is transmitted to the second - layer metasurface through the diffraction process of electromagnetic waves, and then successively passes through the modulation and diffraction of each layer of metasurface, and finally exits from the N - th layer metasurface. The Fourier transform of the output field is performed to obtain the antenna pattern; Step 4: In a mechanical configuration, the forward - propagation model expression of the incident plane wave in the cascaded metasurface is as follows: ； Among them, represents the incident plane wave, represents the Fourier spectrum of the output field of the corresponding channel under this mechanical configuration, and the matrix element values are determined by the Rayleigh-Sommerfeld diffraction formula in step 3, represents the transmission coefficient modulation matrix of the nth layer of metasurface on the incident electromagnetic wave under this mechanical configuration; Step 5: Repeat Step 4 to obtain the forward - propagation models of W channels; Step 6: Use the mean - square error between the Fourier - spectrum amplitude of the forward - output field of the cascaded metasurface and the target Fourier - spectrum amplitude distribution as the loss function, and use the gradient - descent algorithm to train the phase - modulation coefficients of the meta - units of each layer of the N - layer cascaded metasurface system to minimize this loss function; after both the forward - propagation model and the backward gradient - descent algorithm are constructed, use the backward gradient - descent algorithm to iteratively optimize the phase - modulation coefficients until the set number of iteration rounds is reached and stop, so as to obtain the phase - modulation coefficients of the meta - units of each layer of the metasurface; Step 7: Obtain the N - layer cascaded metasurface system according to the obtained phase - modulation coefficients of the meta - units of each layer of the metasurface. Input the plane wave into the N - layer cascaded metasurface system under different mechanical configurations to obtain the antenna patterns with different beam directions, and realize reconfigurable beamforming.

2. The design method of a reconfigurable multi-beam shaping antenna based on cascaded metasurfaces according to claim 1, characterized in that, The process of building the N - layer cascaded metasurface system in Step 1 is as follows: The antenna uses an N - layer cascaded metasurface system, which includes N layers of metasurfaces, which are placed at intervals in sequence along the propagation direction, and the metasurfaces are perpendicular to the propagation direction; The rotation operation forms multiple rotation angles by rotating the angles of each layer of metasurface, and the exchange operation forms multiple permutation orders by exchanging the sequence of each layer of metasurface. Each rotation angle and permutation order constitute a mechanical configuration. The rotation operation and the exchange operation have W mechanical configurations through the Cartesian product. Each mechanical configuration corresponds to a mechanically multiplexed channel, so as to obtain W mechanically multiplexed channels.

3. A design method of a reconfigurable multi-beam shaping antenna based on cascaded metasurfaces according to claim 1, characterized in that, In Step 1, the N layers of metasurfaces of the N - layer cascaded metasurface system are placed at uniform intervals along the propagation direction, and the interval distance d is 10 - 30 working wavelengths of the antenna.

4. The design method of a reconfigurable multi-beam shaping antenna based on cascaded metasurfaces according to claim 1, wherein, In Step 1, N is a natural number between 2 and 5.

5. A design method of a reconfigurable multi-beam shaping antenna based on cascaded metasurfaces according to claim 1, characterized in that, In Step 1, each layer of metasurface is square, and each layer of metasurface includes M×M meta - units, where M is a natural number greater than or equal to 30.

6. The design method of a reconfigurable multi-beam shaping antenna based on cascaded metasurfaces according to claim 1, characterized in that, In step 1, a total of four rotation angles are formed by rotating each layer of the metasurface at angles of 0°, 90°, 180°, or 270°. The swapping operation forms a total of N! permutation orders by swapping the order of each layer of the metasurface. The rotation operation and the swapping operation obtain 4 N-1 ×N! mechanically multiplexed channels, i.e., W = 4 N-1 ×N!.

7. A design method of a reconfigurable multi-beam shaping antenna based on cascaded metasurfaces according to claim 1, characterized in that, In Step 2, obtain the Fourier - spectrum diagram according to the target beam direction: ； ； wherein, are respectively the elevation angle and azimuth angle of the i-th target beam pointing of the j-th channel, i = 1, …, A j , A j is a natural number ≥ 1, are respectively the abscissa and ordinate of the Fourier spectrogram corresponding to the i-th target beam pointing, the operating wavelength of the antenna; In the Fourier spectrum diagram, set the value at the position of ( ) to 1 and the values at other positions to 0.

8. The design method of a reconfigurable multi-beam shaping antenna based on a cascaded metasurface according to claim 1, characterized in that In Step 3, the diffraction process is expressed by the Rayleigh - Sommerfeld diffraction formula as follows: ； Among them, represents the diffraction connection relationship between two adjacent layers of metasurfaces, is the central coordinate of a meta - unit located on the (n - 1)th layer, is the central coordinate of a meta - unit located on the nth layer, represents the diffraction connection relationship between two meta - units whose central coordinates in two adjacent layers of metasurfaces are respectively located at and respectively, , represents the Euclidean distance between two meta - units.

9. A design method of a reconfigurable multi-beam shaping antenna based on cascaded metasurfaces according to claim 1, characterized in that In step 5, a rotation operation is achieved by rotating the transmission coefficient modulation action matrix , and a swapping operation is achieved by changing the order of the transmission coefficient modulation action matrix in the formula; the combination of the rotation operation and the swapping operation will result in W different transformations.

10. The design method of a reconfigurable multi-beam shaping antenna based on cascaded metasurfaces according to claim 1, characterized in that, In Step 6, the optimization expression of the gradient - descent algorithm is as follows: ； Among them, is the phase modulation coefficient of the supercell, respectively represent the Fourier spectrum amplitude of the output field of the j-th channel of the forward transfer model and the target Fourier spectrum amplitude. Adaptive Moment Estimation optimizer is used.

11. A design method of a reconfigurable multi-beam shaping antenna based on cascaded metasurfaces according to claim 1, characterized in that, In Step 6, set an appropriate learning rate between 0.1 and 0.

005.

12. The design method of a reconfigurable multi-beam shaping antenna based on a cascaded metasurface according to claim 1, wherein, In Step 6, the set number of iteration rounds is 500 - 1200.

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