Dual-parameter optimization method based on multi-dimensional weighted cost function and metasurface design method

Abstract

The application discloses a double-parameter optimization method and a super surface design method based on a multi-dimensional weighted cost function, and belongs to the technical field of electromagnetic super surfaces. The method comprises the following steps: performing electromagnetic full-wave simulation on a super surface unit with two adjustable geometric parameters to obtain amplitude and phase response of each parameter combination in a two-dimensional parameter space; constructing a multi-dimensional weighted cost function comprising a phase deviation term, an amplitude threshold segmented penalty term, a neighborhood amplitude smoothness penalty term and a neighborhood phase smoothness penalty term; for each discrete target phase, traversing the parameter space to select the optimal parameter combination that minimizes the total cost function; and outputting the optimal unit size and the corresponding amplitude and phase information. The application can simultaneously meet the design requirements of low energy loss, wide phase modulation range, high phase accuracy and high processing tolerance, and provides an efficient and reliable optimization strategy for super surface design.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic metasurface technology, and in particular to a two-parameter optimization method and metasurface design method based on a weighted multidimensional cost function. Background Technology

[0002] Electromagnetic metasurfaces are composed of subwavelength structural units arranged periodically or aperiodically in a two-dimensional plane. Each metasurface unit is essentially a phase-tunable resonant structure, its geometric parameters related to its phase response. By arranging units with different phase responses, the desired phase distribution can be achieved, thereby modulating the radiation direction of spatial electromagnetic waves. Metasurfaces possess significant application potential in modern wireless communication due to their advantages of low profile, lightweight design, and flexible beam control. Using metasurface units with 360° full phase control capability for arraying enables continuous and precise control of spatial beams, effectively eliminating sidelobes and distortions caused by phase deviations and significantly improving beam pointing accuracy. However, in existing metasurface designs, traditional unit selection methods face difficulties in balancing phase and amplitude. For example, units with a wide phase tuning range are often difficult to ensure consistency with low amplitude loss. Furthermore, abrupt changes in phase and amplitude between adjacent dimensions can increase sensitivity to manufacturing errors, easily leading to significant discrepancies between simulation and test results. Therefore, using the multidimensional weighted cost function approach to optimize the selection process of geometric parameters can further improve the efficiency and reliability of element optimization. With the continuous development of metasurface technology in the future, this design method will provide an efficient, reliable, low-cost and low-power optimization strategy for metasurface design. Summary of the Invention

[0003] This invention aims to solve the technical problems existing in the design methods of metasurface units, such as the difficulty in balancing phase and amplitude, the low processing tolerance caused by abrupt changes in the response of adjacent size units, and the difficulty in simultaneously achieving low loss and wide phase tuning range. To address the above problems, this invention provides an optimization method and metasurface design method based on a multidimensional weighted cost function. By constructing a multidimensional weighted cost function that includes phase deviation, amplitude threshold constraint, and neighborhood smoothness, the optimal combination of geometric parameters with electromagnetic performance is selected from the dual-parameter scanning simulation data of metasurface unit size. The optimization condition is that the final generated metasurface unit needs to simultaneously meet the electromagnetic performance requirements of multiple dimensions: (1) amplitude loss is lower than the set threshold; (2) the phase tuning range covers the entire 360° cycle as much as possible; (3) the amplitude loss and phase abrupt changes of similar size units are minimized.

[0004] As one feasible embodiment, suppose the metasurface unit has two adjustable geometric parameters, the values ​​of which are denoted as follows: and . Dimensional parameter set location set :

[0005]

[0006] in, Starting point As the endpoint, To determine the step size for scanning parameters, These are the scan parameters.

[0007] Dimensional parameter set location set :

[0008]

[0009] in, Starting point As the endpoint, To determine the step size for scanning parameters, These are the scan parameters.

[0010] Therefore, the parameter space generated by the dual geometric parameter scanning of the metasurface unit is:

[0011]

[0012] Among them, parameter space Any set of parameters Mapping the geometry of the actual unit For any set of parameters The amplitude response obtained from its full-wave electromagnetic simulation is (Unit: dB), phase response is (Unit: degree).

[0013] To ensure accurate coverage of the full phase angle, the target phase is defined. (Unit: degrees) Discrete and linearly covering full-phase principal values ​​( ):

[0014]

[0015] Among them, the phase start point End point , To determine the step size, This represents the total number of points taken.

[0016] Define the total cost function as follows:

[0017]

[0018] The meanings of each item are as follows:

[0019] (1) Phase deviation term This item quantifies the deviation between the control phase provided by the cell at the current size and the target phase. The target phase selected in this embodiment... It changes linearly; however, the phase response of the metasurface element simulation is linear. Since it is nonlinear, its value does not necessarily correspond perfectly to the target phase; therefore, this term is introduced to take the phase deviation into account. Considering the periodicity of the phase, a periodic normalization operator is introduced. Take the principal phase value ( This is to correctly calculate the minimum phase difference.

[0020]

[0021] (2) Amplitude threshold segmentation penalty term This item introduces a minimum amplitude threshold. By applying distinctly different weighting coefficients to filter parameter points within a threshold, the metasurface is ensured to achieve low reflection or transmission loss performance. Since the reflection / transmission amplitude (dB) of the metasurface is negative, this penalty term optimizes the region exceeding the minimum amplitude threshold, applying normal weighting coefficients. The region with amplitude less than the minimum threshold is considered the out-of-limit region, and a penalty weighting coefficient is applied accordingly. ,and :

[0022]

[0023] (3) Neighborhood amplitude smoothness penalty term This parameter characterizes the sensitivity of the unit's amplitude response to changes in geometric dimensions. It is used to determine whether a parameter point is near a point of drastic amplitude change, i.e., high geometric sensitivity. By applying a weighting coefficient, such points can be avoided, thus improving the processing tolerance of metasurface design. Definition Define the neighborhood magnitude gradient operator as the magnitude smoothness weight coefficient. For the current parameter point Maximum amplitude deviation within its 3×3 neighborhood:

[0024]

[0025] (4) Neighborhood phase smoothness penalty term This term characterizes the sensitivity of the unit phase response to changes in geometric dimensions, and its effect is the same as the neighborhood phase smoothness penalty term. Definition Define the neighborhood phase gradient operator as the phase smoothness weight coefficient. For the current parameter point Maximum phase deviation within its 3×3 neighborhood:

[0026]

[0027] Weighting coefficients , , , It can be flexibly adjusted according to specific design goals to balance the priorities between phase accuracy, amplitude loss and processing tolerance.

[0028] For each discrete target phase The algorithm traverses the entire parameter space. Find the total cost function Minimize the combination of parameters :

[0029]

[0030] The final generated The sequence is the optimal cell size mapping table. This mapping table contains the optimal cell sizes mapped. and the corresponding amplitude response and phase response The optimal cell can have phase modulation capability covering the entire cycle as much as possible, and the deviation between the actual phase and the target phase is very small. The amplitude response of each cell size is basically greater than the preset minimum amplitude threshold, and the amplitude and phase abrupt changes of adjacent cell sizes are minimal. Thus, a metasurface design with low energy loss, wide phase modulation range, high phase accuracy and high processing tolerance is achieved.

[0031] Compared with the prior art, the present invention has the following beneficial effects:

[0032] First, this invention constructs a multidimensional weighted cost function that includes a phase deviation term, an amplitude threshold segmentation penalty term, a neighborhood amplitude smoothness penalty term, and a neighborhood phase smoothness penalty term, thereby achieving a multidimensional comprehensive evaluation of the electromagnetic performance of metasurface units. Compared with traditional screening methods that only consider phase matching or a single amplitude threshold, this invention can simultaneously take into account phase accuracy, amplitude loss, and geometric continuity during the optimization process, effectively overcoming the shortcomings of existing design methods in balancing phase and amplitude.

[0033] Second, this invention introduces a segmented penalty term based on an amplitude threshold and sets differentiated weighting coefficients. and ,and This mechanism enables the priority selection of low-loss regions and the mandatory penalty for exceeding the limit. It ensures that all ultimately selected metasurface units meet the preset low-loss requirements (e.g., transmission loss below 3dB), providing reliable amplitude performance assurance for high-performance metasurface design.

[0034] Third, by introducing neighborhood amplitude smoothness penalty terms and neighborhood phase smoothness penalty terms, this invention quantifies the sensitivity of the element response to geometrical changes, effectively avoiding parameter points in regions of drastic response jumps. This design significantly improves the processing tolerance of metasurface elements, reduces the impact of actual processing errors on electromagnetic performance, and effectively minimizes the deviation between simulation and test results.

[0035] Fourth, this invention employs a two-parameter traversal optimization strategy, independently selecting the optimal parameter combination for each discrete target phase, which can generate a parameter combination covering the entire phase range. to The optimal cell size mapping table is provided. This mapping table can be directly used for the layout design of metasurface arrays, achieving a balance between wide phase tuning range, high phase accuracy, and low energy loss, providing a high-performance cell foundation for applications such as beamforming and holographic imaging.

[0036] Fifth, the two-parameter optimization algorithm proposed in this invention is characterized by its strong versatility, high computational efficiency, and ease of implementation. This method is not only applicable to transmissive metasurfaces but can also be extended to the design of reflective metasurfaces and other two-parameter tunable electromagnetic structures, demonstrating significant engineering application value and promising prospects for wider application. Attached Figure Description

[0037] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the following description of the embodiments taken in conjunction with the accompanying drawings, wherein:

[0038] Figure 1 This embodiment provides a flowchart of a two-parameter optimization algorithm based on a multidimensional weighted cost function;

[0039] Figure 2 This embodiment presents a "receive-transmit" three-layer structure for a transmissive metasurface unit;

[0040] Figure 3 These are two unit metal patches with different electrical lengths in this embodiment;

[0041] Figure 4 The transmission amplitude response distribution diagram is shown, where (a) and (b) are the transmission amplitude thermogram and contour plot corresponding to the optimal parameter combination of a transmission metasurface unit in this embodiment, respectively.

[0042] Figure 5 This is a transmission phase thermogram corresponding to the optimal parameter combination of a transmission metasurface unit in this embodiment;

[0043] Figure 6 This is a bi-vertical line graph showing the amplitude and phase of 181 phase-tuning units obtained by the optimization algorithm for a transmissive metasurface unit in this embodiment. Detailed Implementation

[0044] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0045] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings. This embodiment provides an optimization method and metasurface design method based on a multidimensional weighted cost function. Specifically, this embodiment will construct a multidimensional weighted cost function that includes phase deviation, amplitude threshold constraint and neighborhood smoothness to select the parameter combination with the best electromagnetic performance from the dual geometric parameter scanning data of the transmissive metasurface unit. The optimization condition is that the final transmissive metasurface unit simultaneously meets the electromagnetic performance requirements of multiple dimensions: (1) transmission loss is less than 3 dB; (2) the phase modulation range covers the entire period of 360° as much as possible; (3) the amplitude loss and phase abrupt change of adjacent size units are minimized.

[0046] Specifically, Figure 1 This is a flowchart of a two-parameter optimization method based on a multidimensional weighted cost function provided in this embodiment.

[0047] like Figure 1 As shown, the two-parameter optimization algorithm based on a multidimensional weighted cost function includes the following steps:

[0048] Step 1: Model a transmissive metasurface unit using the electromagnetic full-wave simulation software ANSYS Electronics HFSS. Figure 2 This demonstrates the "receive-emitter" three-layer structure of the transmissive metasurface unit, with a unit period... The metasurface unit comprises three metal patches and two dielectric substrates. The substrate 1 is made of Rogers RO4350 (relative permittivity...). Loss tangent ),thickness Its top surface is covered with a metal patch 1, and its bottom surface is covered with a metal patch 2; the material of the adhesive layer 1 is Rogers RO4450 (relative permittivity). Loss tangent ),thickness It is used to press the upper and lower substrates together; the material of substrate 2 is Rogers RO4350, with a thickness of... Its bottom surface is covered with a metal patch 3. Metal patches 1, 2, and 3 are all metal resonant structures of metasurface units, initially shaped as rectangular patches. The lengths of metal patches 1 and 3 are... The length of metal patch 2 is The width is And satisfy: if The metal patch is rectangular; otherwise, two rectangular patches of equal width and symmetrical extension about the center are added to each end of the rectangular patch, with a length of [length missing]. This ensures that the metal patch can continue to increase its electrical length without changing the cell period, thereby achieving a wider range of phase modulation. Figure 3 Two different electrical lengths of the unit metal patch are shown.

[0049] Step 2: Perform a two-parameter scan of the unit cell, establish a parameter library, and introduce a two-dimensional parameter space. In this embodiment, the metasurface unit has two adjustable geometric parameters, namely the total length of the outer patches (metal patch 1 and metal patch 3). The total length of the inner patch (metal patch 2) Their values, expressed in millimeters, are denoted as follows: and . The set of parameters is defined as a set. :

[0050]

[0051] Among them, the starting point End point , sweeping step length Total scan parameters .

[0052] The set of parameters is defined as a set. :

[0053]

[0054] Among them, the starting point End point , sweeping step length Total scan parameters According to the set and ANSYS Electronics HFSS was used to perform element parameter scanning and full-wave electromagnetic simulation to obtain the transmission amplitude response corresponding to each parameter combination. and transmission phase response The simulation results are stored in the form of a two-dimensional matrix. The two-dimensional parameter space of this optimization algorithm is defined as follows:

[0055]

[0056] Among them, parameter space Any set of parameters Mapping the geometry of the actual unit Therefore, for any set of parameters... Each set of parameters The amplitude and phase information included are denoted as: amplitude response is (Unit: dB), phase response is (Unit: degree).

[0057] Step 3: Establish a multi-dimensional weighted cost function and set optimization conditions. In this embodiment, the optimization algorithm defines the total cost function. as follows:

[0058]

[0059] The meaning of each item is given, and the optimization conditions set in this embodiment are as follows:

[0060] (1) Phase deviation term This item quantifies the deviation between the control phase provided by the cell at the current size and the target phase. The target phase selected in this embodiment... It changes linearly; however, the phase response of the metasurface element simulation is linear. Since the phase is nonlinear, its value does not necessarily correspond perfectly to the target phase. Therefore, this term is introduced to take the phase deviation into account. The smaller this term is, the smaller the deviation between the selected simulated phase and the target phase, and the closer it is to the phase requirement of the optimization condition; conversely, the larger the term is, the less it meets the optimization condition. Considering the periodicity of the phase, a periodic normalization operator is introduced. Take the principal phase value ( This is to correctly calculate the minimum phase difference.

[0061]

[0062] In this embodiment, in order to ensure accurate coverage of the full phase angle, a target phase is defined. (Unit: degrees) Discrete and linearly covering full-phase principal values ​​( ):

[0063]

[0064] Among them, the phase start point End point Step size Total number of points Therefore, the target phase set can be written as .

[0065] (2) Amplitude segmentation penalty item This item introduces a minimum amplitude threshold. By applying distinctly different weighting coefficients to filter parameter points within a threshold, the metasurface is ensured to achieve low reflection or transmission loss performance. Since the reflection / transmission amplitude (dB) of the metasurface is negative, this penalty term optimizes the region exceeding the minimum amplitude threshold, applying normal weighting coefficients. The region with amplitude less than the minimum threshold is considered the out-of-limit region, and a penalty weighting coefficient is applied accordingly. ,and :

[0066]

[0067] In this embodiment, a minimum transmission amplitude threshold is set. Normal weighting coefficient Penalty weight coefficient So, the amplitude segmented penalty item It can be rewritten as:

[0068]

[0069] This penalty will ultimately rigorously screen out transmission amplitude responses. The parameter combination satisfies the requirement that the transmission loss of the target is less than 3 dB.

[0070] (3) Neighborhood amplitude smoothness penalty term This parameter characterizes the sensitivity of the unit's amplitude response to changes in geometric dimensions. It is used to determine whether a parameter point is near a point of drastic amplitude change, i.e., high geometric sensitivity. By applying a weighting coefficient, such points can be avoided, thus improving the processing tolerance of metasurface design. Definition Define the neighborhood magnitude gradient operator as the magnitude smoothness weight coefficient. For the current parameter point Maximum amplitude deviation within its 3×3 neighborhood:

[0071]

[0072] In this embodiment, in order to accurately filter out the maximum amplitude deviation within the neighborhood, an amplitude smoothness weighting coefficient is set. .

[0073] (4) Neighborhood phase smoothness penalty term This term characterizes the sensitivity of the unit phase response to changes in geometric dimensions, and its effect is the same as the neighborhood phase smoothness penalty term. Definition Define the neighborhood phase gradient operator as the phase smoothness weight coefficient. For the current parameter point Maximum amplitude deviation within its 3×3 neighborhood:

[0074]

[0075] In this embodiment, in order to accurately filter out the maximum phase deviation within the neighborhood, a phase smoothness weighting coefficient is set. .

[0076] Step 4, for each discrete target phase The algorithm traverses the entire parameter space. Filter and find the total cost function Minimize the combination of parameters :

[0077]

[0078] The final generated The sequence is the optimal cell size mapping table. This mapping table contains the optimal cell size combinations for mapping. The sequence, and the corresponding transmission amplitude response and transmission phase response If the optimization result meets the engineering requirements, then output the parameter combination and the corresponding transmission amplitude and phase data; otherwise, remodel the transmissive metasurface unit, and adjust the substrate material, thickness, or structure until the optimization result meets the engineering requirements.

[0079] This embodiment uses the two-parameter optimization algorithm based on the multidimensional weighted cost function to select a series of optimal parameter combinations for the transmissive metasurface unit. Figure 4 Figure (a) shows the thermal diagram of transmission amplitude corresponding to the optimal parameter combination. Figure 4 (b) is the corresponding amplitude contour plot, with the −3 dB contour line also marked. Figure 5 The transmission phase thermograms corresponding to the optimal parameter combination are shown. Figure 6 The diagram shows a bi-axis line graph of the 181 phase modulation units obtained through optimization, with respect to their corresponding transmission phase and transmission amplitude.

[0080] In summary, the two-parameter optimization algorithm provided in this embodiment establishes four weighted cost functions, simultaneously considering the requirements of low amplitude and phase loss, wide phase tuning range, and high precision in metasurface unit design, and achieving rapid optimization. Based on the optimization algorithm provided in this embodiment, a high-performance transmissive metasurface unit can be realized, which possesses 340° phase tuning capability, transmission amplitude loss of less than 3 dB, and minimal amplitude and phase abrupt changes among adjacent unit sizes. This achieves a metasurface design with low energy loss, wide phase tuning range, high phase accuracy, and high processing tolerance.

[0081] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this application. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0082] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this application, "N" means at least two, such as two, three, etc., unless otherwise explicitly specified.

Claims

1. A two-parameter optimization method based on a multidimensional weighted cost function, characterized in that, Includes the following steps: Step 1: Perform electromagnetic full-wave simulation on a metasurface unit with two adjustable geometric parameters to obtain the amplitude response and phase response of each parameter combination in the two-dimensional parameter space. Step 2: Based on the amplitude response and phase response obtained in Step 1, construct a multidimensional weighted cost function. The cost function includes a phase deviation term, an amplitude threshold segmentation penalty term, a neighborhood amplitude smoothness penalty term, and a neighborhood phase smoothness penalty term. Step 3: Based on the multidimensional weighted cost function constructed in Step 2, for each discrete target phase, traverse the entire two-dimensional parameter space to find the parameter combination that minimizes the multidimensional weighted cost function. Step 4: Based on the optimal parameter combination obtained in Step 3, output the optimal unit geometry and its corresponding amplitude and phase responses.

2. The two-parameter optimization method based on a multidimensional weighted cost function according to claim 1, characterized in that, In step 1, the two-dimensional parameter space The construction method is as follows: Let the two adjustable geometric parameters be respectively and The parameter sets are defined as follows: ； ； The parameter space is then ,in Mapping actual geometric dimensions The corresponding magnitude response is The phase response is , , The first indivual Parameter values ​​and the first indivual Parameter value, , x and The starting point, To determine the step size for scanning parameters, , x and The parameters of the dimensional scan.

3. The two-parameter optimization method based on a multidimensional weighted cost function according to claim 1, characterized in that, The target phase Discrete and linearly covering full-phase principal values Defined as: ； in , , To determine the step size, For the total number of points, For the first Each target phase value.

4. The two-parameter optimization method based on a multidimensional weighted cost function according to claim 1, characterized in that, The multidimensional weighted cost function Defined as: ； in For phase deviation term, This is a segmented penalty term for the amplitude threshold. This is the amplitude smoothness weighting coefficient. For neighborhood magnitude gradient operators, This is a penalty term for the smoothness of the neighborhood amplitude. This is the phase smoothness weighting coefficient. For neighborhood phase gradient operators, This is a neighborhood phase smoothness penalty term. The target phase.

5. The two-parameter optimization method based on a multidimensional weighted cost function according to claim 2, characterized in that, The phase deviation term Defined as: ； in This is a periodic normalization operator used to normalize phase difference values ​​to the principal value interval. , The target phase.

6. The two-parameter optimization method based on a multidimensional weighted cost function according to claim 2, characterized in that, The amplitude threshold segmentation penalty term Defined as: ； in Minimum amplitude threshold, These are normal weighting coefficients. The penalty weight coefficient, and .

7. The two-parameter optimization method based on a multidimensional weighted cost function according to claim 2, characterized in that, The neighborhood amplitude smoothness penalty term Neighborhood magnitude gradient operator Defined as the current parameter point In its Maximum amplitude deviation within the neighborhood: ； m and n are the neighborhood offsets. ,and .

8. The two-parameter optimization method based on a multidimensional weighted cost function according to claim 2, characterized in that, The neighborhood phase smoothness penalty term Neighborhood phase gradient operator Defined as the current parameter point In its Maximum phase deviation within the neighborhood: ； m and n are the neighborhood offsets. ,and .

9. The two-parameter optimization method based on a multidimensional weighted cost function according to claim 2, characterized in that, In step 3, for each discrete target phase Optimal parameter combination Obtained through the following formula: 。 10. A metasurface design method based on the two-parameter optimization method according to any one of claims 1 to 9, characterized in that, The optimal parameter sequence is applied to the arrangement of metasurface arrays to obtain metasurfaces that simultaneously meet the requirements of low energy loss, wide phase tuning range, high phase accuracy and high processing tolerance.

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