A high-transmittance grating topology optimization method for enhancing light field uniformity

Abstract

The application belongs to the technical field of information science, and relates to a high-transmittance grating topology optimization method for enhancing light field uniformity, which comprises the following steps: 1) establishing a physical optical control equation, combining with construction of a mathematical function expression of grating field uniformity, so as to realize uniform distribution of the light field; 2) establishing a constraint function, so as to meet the transmittance requirement of the grating design; 3) filtering the design variable by using a Helmholtz filter equation, so as to smooth the material distribution of the grating topology structure; 4) solving the gray problem introduced by filtering by adding a Heaviside function for projection operation, so that the design variable gradually tends to { -1, 1}; 5) processing the design variable interpolation as a material function of the grating; and 6) establishing an optimization design model. The application converts the grating design problem into a continuity optimization problem of material topology by constructing a reasonable parameterized model, is suitable for being solved by using a gradient-based algorithm, and thus an optimal design result balancing between transmittance and field uniformity is found.

Description

Technical Field

[0001] This invention belongs to the field of information science and technology, and relates to a method for optimizing the topology of a high-transmittance grating to enhance the uniformity of the light field. Background Technology

[0002] As a crucial optical element, the grating plays a vital role in optical systems. Its working principle involves controlling and modulating light through the diffraction and interference effects of its periodic structure. With the rapid development of technology and the continuous expansion of optical applications, the performance requirements for gratings are becoming increasingly stringent. In many high-end optical systems, such as high-precision measurement, high-resolution imaging, laser processing, holographic projection, and 3D reconstruction, the demands for image quality are becoming increasingly stringent. These systems require gratings with high light transmittance and excellent optical field uniformity. High light transmittance ensures that more light waves can pass through the grating smoothly, while excellent optical field uniformity guarantees that the transmitted light is evenly distributed in space, which is crucial for improving the imaging quality and overall performance of the optical system.

[0003] However, the design of high-transmittance uniform field gratings is a difficult design optimization problem, the difficulty of which lies in:

[0004] 1) Accurate modeling of the physical field, which involves accurately constructing an analytical model of the light field, as well as reasonably predicting and analyzing the impact of complex optical behavior in the grating structure on light transmission performance;

[0005] 2) When designing gratings using traditional methods, it is difficult to simultaneously achieve high light transmittance and uniform light field. Improving light transmittance requires increasing the grating aperture or reducing the material's absorbance, but this may sacrifice the uniformity of the light field.

[0006] Traditional methods struggle to find the optimal balance between these two factors. Topology optimization, as an emerging approach, offers significant design freedom and optimization capabilities, providing new insights for grating design. Through topology optimization, the structure of the grating can be precisely designed to achieve specific optical properties. Summary of the Invention

[0007] This invention provides a method for optimizing the topology of high-transmittance gratings to enhance optical field uniformity. This method utilizes the beam envelope method to construct a reasonable parametric model, transforming the grating design problem into a continuous optimization problem of material topology. It is applicable to both optical field physics analysis and gradient-based algorithms. In this method, transmittance is introduced as a constraint function, and the optimization objective function characterizing optical field uniformity is solved, thereby finding the optimal design result that balances transmittance and field uniformity.

[0008] To achieve the above-mentioned objectives, the technical solution of the present invention is as follows:

[0009] A method for optimizing the topology of a high-transmittance grating to enhance optical field uniformity includes the following steps:

[0010] Step 1: Determine the physical field analysis model of the grating to be designed. Define the dependent variable E1 to be solved, called the envelope function. Use the beam envelope method to establish the physical optics control equations as shown below:

[0011]

[0012] Where E1 represents amplitude, φ1 represents phase, and μ r Represents the relative permeability of the optical transmission medium, ∈ r The relative permittivity of the optical transmission medium, ∈0, is the permittivity of vacuum: 8.854187817 × 10⁻⁶. -12 F / m, σ represents the conductivity of the optical transmission medium, ω represents the angular frequency, and k0 is the electromagnetic wave number;

[0013] Step 2: To achieve the optimized design goal, it is necessary to construct a mathematical function expression characterizing the uniformity of the grating field. The phasor representation of the light field is as follows: Define the design variable θ and the grating design domain Ω. design , and θ∈Ω design By establishing an objective function that minimizes the difference in the light vector field at each point of the transmitted light. Achieving a uniform distribution of the light field:

[0014]

[0015] Where E(θ) represents the electric field at each point of the transmitted light from the grating, E avg (θ) represents the average electric field at each point of the transmitted light from the grating;

[0016] Step 3: The design variable θ in Step 2 takes a continuous value between 0 and 1, where 0 indicates that the position should be designed with air material and 1 indicates that the position should be designed with grating medium material.

[0017] Step 4: To meet certain light transmittance requirements, a constraint function needs to be set to limit the luminous flux before and after the light passes through the grating. Based on the Poynting vector of the electromagnetic field, the luminous flux is expressed as: The constraint equations can then be expressed as:

[0018]

[0019] Among them, l down This represents the lower limit of the transmittance constraint, l up P represents the upper limit of the transmittance constraint. out (θ) represents the luminous flux output through the grating, P in(θ) represents the luminous flux entering the grating. In this context, B(θ) represents the magnetic field at each point, E(θ) represents the electric field at each point, and μ0 represents the free magnetic permeability (4π × 10⁻⁶). -7 N / A 2 ;

[0020] Step 5: Establish the Helmholtz filtering equation to filter the design variable θ, such as... Figure 1 As shown, the filtered design variable θ is obtained by smoothing the material distribution of the grating topology. f for:

[0021]

[0022] Among them, R min This is expressed as the filtration radius, typically taken as a value. r mes h represents the minimum radius of the design variable grid;

[0023] Step 6: In step 5, the addition of the Helmholtz filter will introduce a new design variable θ. f Taking intermediate values ​​outside {0,1} will result in design outcomes that do not meet actual physical meaning. Therefore, a Heaviside function is added to the objective function to project the design variables, forcing θ to... f During the optimization design process, the new design variable θ gradually approaches {0,1}. p for:

[0024]

[0025] Where β represents the projection slope, and adjusting β can change the speed at which the design variable tends to {0,1}, and η represents the projection threshold, which is generally taken as 0.5;

[0026] Step 7: The design variable θ after processing in steps 5 and 6. p The material function of the grating needs to be interpolated for optimization. The interpolated material parameter is the relative permittivity ∈ r Then the interpolation function ∈ r (θ p )for:

[0027] ∈ r (θ p ) = tan -1 (∈ r,air )+(tan -1 (∈ r,silicon )-tan -1 (∈ r,air ))×θ p (6)

[0028] Where, ∈ r,air The relative permittivity of air is 1; ∈ r,silicon This represents the relative permittivity of the grating material;

[0029] Step 8: Incorporate the processing of design variables from steps 5-7 into the optimization formula. In the corresponding optimization problem, define the objective function as minimizing the variable and the constraint condition as the design variable satisfying the upper and lower limits. The optimization model is established as follows:

[0030]

[0031] Where, θ v This represents the relative permittivity inserted into the physical field for analytical calculation;

[0032] Step 9: Calculate the first derivative of the objective function with respect to the design variables, update the design variables using the MMA algorithm, and complete the design of the grating.

[0033] Furthermore, in step 1, i and j are imaginary units.

[0034] Furthermore, in step 2, the design domain Ω design The proposed grating design domain is defined in 3D space.

[0035] Furthermore, in step 4, the luminous flux refers to the radiant power of the transmitted light, which can be represented by the Poynting vector P(θ) of the electromagnetic field.

[0036] Furthermore, in step 4, the transmittance is the ratio of the luminous flux transmitted through the grating to its incident luminous flux, i.e.

[0037] Furthermore, in step 6, the projection slope β in the design process increases from 2, and the strategy for increasing β is p = 2. i (i = 1, ..., m), where i is the number of loops to be optimized and m is the maximum number of loops to be optimized. The value of m is different depending on the problem. Generally, m ≤ 8.

[0038] Furthermore, in step 8, since the relative permittivity obtained in step 7 is based on the actual physical value, tan -1 The transformed value therefore needs to be tan-ized when inserted into the physics calculation, i.e., the design variable θ needs to be inserted. v =tan(∈ r (θ p )).

[0039] Unlike existing methods, the method of this invention can balance light field uniformity and light transmission performance in grating design. It uses a gradient algorithm to solve the problem and can obtain a high-performance grating unit cell configuration. Attached Figure Description

[0040] Figure 1 This is a schematic diagram of the Helmholtz filter in this invention.

[0041] Figure 2 This is a schematic diagram of the basic configuration and design domain of the cross-shaped grid in Example 1.

[0042] Figure 3 This is a schematic diagram of the physical field analysis model of the grating to be designed in Example 1.

[0043] Figure 4 This is a table showing the target results of grating topology optimization in Example 1. Detailed Implementation

[0044] The present invention will now be further described in conjunction with the accompanying drawings and specific embodiments.

[0045] Example 1

[0046] This implementation case considers designing a grating unit cell configuration that balances light field uniformity and high transmittance. The design variable is the presence or absence of material; the interpolation material parameter is chosen as the relative permittivity, and other material parameters are not included in the design. The objective function is set to minimize the difference in the light field at various points through the grating. To ensure electrical connectivity, a cross-shaped grid grating is used as the initial configuration of the grating, and its configuration structure diagram and design domain are shown below. Figure 2 As shown. Meanwhile, to reduce computational load, one-quarter of the domain is used for topology optimization design, and then mapped to the other three design domains according to certain rules.

[0047] Step 1: Determine the physical field analysis model of the grating to be designed and establish... Figure 3 The physical field model shown employs periodic boundary conditions on all four sides, with the upper and lower surfaces serving as the incident and transmitted light receiving ports, respectively. The dependent variable E1 to be solved is defined as the envelope function. The physical optics governing equations are established using the beam envelope method as shown below:

[0048]

[0049] Where E1 represents amplitude, φ1 represents phase, and μ r Represents the relative permeability of the optical transmission medium, ∈ r The relative permittivity of the optical transmission medium is 3.7, and the permittivity of vacuum (∈0) is 8.854187817 × 10⁻⁶. -12 F / m, σ represents the conductivity of the optical transmission medium, ω represents the angular frequency, and k0 is the electromagnetic wave number;

[0050] Step 2: To achieve the optimized design goal, it is necessary to construct a mathematical function expression characterizing the uniformity of the grating field. The phasor representation of the light field is as follows: Define the design variable θ and the grating design domain Ω. design , and θ∈Ω design The number of design variables is 1563. An objective function is established. Minimize the difference in the light vector field at each point of the transmitted light to achieve a uniform distribution of the light field:

[0051]

[0052] Where E(θ) represents the electric field at each point of the transmitted light from the grating, E avg (θ) represents the average electric field at each point of the transmitted light from the grating;

[0053] Step 3: The design variable θ in Step 2 takes a continuous value between 0 and 1, where 0 indicates that the position should be designed with air material and 1 indicates that the position should be designed with grating medium material.

[0054] Step 4: To meet certain light transmittance requirements, a constraint function needs to be set to limit the luminous flux before and after the light incident on the grating. Based on the Poynting vector of the electromagnetic field, the luminous flux is expressed as: The constraint equations can then be expressed as:

[0055]

[0056] Among them, l down This represents the lower limit of the transmittance constraint, with a value of 80%. up This represents the upper limit of the transmittance constraint, with a value of 100%, P. out (θ) represents the luminous flux output through the grating, P in (θ) represents the luminous flux entering the grating. In this context, B(θ) represents the magnetic field at each point, E(θ) represents the electric field at each point, and μ0 represents the free magnetic permeability (4π × 10⁻⁶). -7 N / A 2 ;

[0057] Step 5: Establish a Helmholtz filtering equation to filter the design variable θ to smooth the material distribution of the grating topology. The resulting filtered design variable θ f for:

[0058]

[0059] Where, r mes h represents the minimum radius of the design variable mesh, with a value of 240 nm; R min This is expressed as the filtration radius, typically taken as a value. Approximately 69nm;

[0060] Step 6: In step 5, the addition of the Helmholtz filter will introduce a new design variable θ. f When intermediate values ​​other than {0,1} are taken, the design results will exhibit grating designs that do not meet the actual physical meaning. Therefore, a Heaviside function is added to the objective function to project the design variables, forcing θ to... f During the optimization design process, the design gradually approaches {0,1}. The resulting new design variable θ p for:

[0061]

[0062] Here, β represents the projection slope, and adjusting the value of β can affect how quickly the design variables approach the {0,1} interval. The initial value of β is set to 2, and it is doubled every 20 calculation steps. This growth process continues until the calculation reaches 100 steps, at which point the optimization stopping condition is met, and the final value of β reaches 64. η represents the projection threshold, with a value of 0.5.

[0063] Step 7: The design variable θ after processing in steps 5 and 6. p The material function of the grating needs to be interpolated for optimization. The interpolated material parameter is the relative permittivity ∈ r Then the interpolation function ∈ r (θ p )for:

[0064] ∈ r (θ p ) = tan -1 (∈ r,air )+(tan -1 (∈ r,silicon )-tan -1 (∈ r,air ))×θ p (13)

[0065] Where, ∈ r,air The relative permittivity of air is 1; ∈ r,silicon The relative permittivity of the grating material is 3.7.

[0066] Step 8: Incorporate the processing of design variables from steps 5-7 into the optimization formula. In the corresponding optimization problem, define the objective function as minimizing the variable and the constraint condition as the design variable satisfying the upper and lower limits. The optimization model is established as follows:

[0067]

[0068] Where, θ v θ represents the relative permittivity inserted into the physical field for analytical calculation, with an initial value of 0;

[0069] Step 9: Calculate the first derivative of the objective function with respect to the design variables, update the design variables using the MMA algorithm, and complete the grating design. The grating topology optimization results during the design process are shown in Table 1, and the final topology configuration is as follows. Figure 4 As shown, the black part represents the grating medium material, and the white part represents the air material.

[0070]

Claims

1. A method for optimizing the topology of a high-transmittance grating to enhance the uniformity of the light field, characterized in that: Includes the following steps: Step 1: Determine the physical field analysis model of the grating to be designed; define the dependent variable to be solved. This is called the envelope function; the following physical optics control equations are established using the beam envelope method: （15） in, The imaginary unit, Indicates amplitude, Indicates phase, Represents the relative permeability of the optical transmission medium. This represents the relative permittivity of the optical transmission medium. The dielectric constant of vacuum is 8.854187817 × 10⁻⁶. -12 F / m, Indicates the conductivity of the optical transmission medium. Represents angular frequency. The wave number of electromagnetic waves; Step 2: To achieve the optimization design goal, it is necessary to construct a mathematical function expression characterizing the uniformity of the grating field; the phasor representation of the light field is as follows: Define design variables and grating design domain ,and By establishing the objective function To minimize the difference in the light vector field at each point of the transmitted light, a uniform distribution of the light field is achieved: （16） in, This represents the electric field at various points along the transmitted light path of the grating. This represents the average electric field at various points in the transmitted light from the grating. Step 3: Design variables in step 2 The value ranges continuously from 0 to 1, where 0 indicates that the location should be designed with air material and 1 indicates that the location should be designed with grating medium material. Step 4: To meet certain light transmittance requirements, a constraint function needs to be set to limit the luminous flux before and after the light incident on the grating; according to the Poynting vector of the electromagnetic field, the luminous flux is expressed as: Then the constraint equation can be expressed as: （17） in, This represents the lower limit of the transmittance constraint. This represents the upper limit of the transmittance constraint. This indicates the luminous flux output through the grating. This represents the luminous flux entering the grating. middle Indicates the magnetic field at each point. This represents the vacuum permeability of 4π × 10⁻⁶. -7 N / A 2 ; Step 5: Establish the Helmholtz filtering equation for the design variables Filtering is performed to smooth the material distribution of the grating topology, resulting in filtered design variables. for: （18） in, Represented as the filtration radius, with a value of , Represented as the minimum radius of the design variable grid; Step 6: The design variables filtered by Helmholtz in Step 5. Received When the values ​​are not intermediate, the design results will exhibit grating designs that do not meet the actual physical meaning. Therefore, a Heaviside function is added to the objective function to project the design variables, forcing... Gradually tending towards optimization design process The new design variables obtained for: （19） in, Indicates the projection slope, which is adjusted by... It can change the tendency of design variables. speed, This represents the projection threshold, with a value of 0.

5. Step 7: The new design variables after processing in Steps 5 and 6. The material function of the grating needs to be interpolated for optimization; the interpolated material parameter is the relative permittivity. Then the interpolation function for: （20） in, The relative permittivity of air is represented by a value of 1; This represents the relative permittivity of the grating material; Step 8: Incorporate the treatment of design variables from Steps 5 to 7 into the optimization formula. In the corresponding optimization problem, define the objective function as minimizing the variable and the constraint condition as the design variable satisfying the upper and lower limits. The optimization model is established as follows: （21） in, This represents the relative permittivity inserted into the physical field for analytical calculation; Step 9: Calculate the first derivative of the objective function with respect to the design variables, update the design variables using the MMA algorithm, and complete the design of the grating.

2. The method for optimizing the topology of a high-transmittance grating to enhance optical field uniformity as described in claim 1, characterized in that: In step 2, the design domain The proposed grating design domain is defined in 3D space.

3. The method for optimizing the topology of a high-transmittance grating to enhance optical field uniformity as described in claim 2, characterized in that: In step 4, the luminous flux refers to the radiant power of light, measured by the Poynting vector of the electromagnetic field. It can be represented accordingly.

4. The method for optimizing the topology of a high-transmittance grating to enhance optical field uniformity as described in claim 1, characterized in that: In step 4, the transmittance is the ratio of the luminous flux transmitted through the grating to its incident luminous flux, i.e. .

5. The method for optimizing the topology of a high-transmittance grating to enhance optical field uniformity as described in claim 4, characterized in that: In step 6, the projection slope Starting from 2 and increasing, The incremental strategy is , To optimize the number of loops, To optimize the maximum number of loops, .

6. A method for optimizing the topology of a high-transmittance grating to enhance optical field uniformity as described in claim 1, 3, or 5, characterized in that: In step 8, since the relative permittivity obtained in step 7 is based on the actual physical values... The value after that, therefore, needs to be taken when inserting it into the physics calculation. Value, that is .

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