Double-layer diffractive optical element design method based on genetic algorithm

Abstract

The invention discloses a double-layer diffractive optical element design method based on a genetic algorithm, relates to the technical field of genetic algorithms, and aims to solve the problem that a DLDOE with high diffraction efficiency in a wide band and a wide angle range is difficult to design by an existing DLDOE design method. Selecting optimization variables participating in optimization design based on a diffraction efficiency basic theory of DLDOE; 2, constructing an average value-standard deviation double-target evaluation model; step 3, establishing a fitness function containing dispersion characteristic constraints based on a workable material library; 4, constructing a genetic algorithm with a cosine annealing and cycle restart optimization strategy; and 5, matching the substrate material and updating the height of the microstructure. According to the method, the genetic algorithm in the global optimization algorithm is introduced into the DLDOE optimization design method, the optimal solution design of the DLDOE is realized, and the DLDOE with high diffraction efficiency in a broadband and wide angle range can be designed.

Description

Technical Field

[0001] The present invention relates to the technical field of genetic algorithms, and particularly to a design method of a double-layer diffractive optical element based on genetic algorithms. Background Art

[0002] Diffractive Optical Elements (DOE) is an optical element with arbitrary phase and special dispersion properties. By combining with a traditional optical system to form a refractive / diffractive hybrid system, it can correct chromatic aberration and thus improve the imaging quality of the optical system. Since Single-layer Diffractive Optical Elements (SLDOE) are difficult to achieve high diffraction efficiency in a wide wavelength range, the theory of Double-layer Diffractive Optical Elements (DLDOE) has been proposed by researchers. DLDOE can overcome the constraint of a wide wavelength range but cannot overcome the constraint of a large angle. At large angles, the diffraction efficiency drops sharply. In previous DLDOE design examples, the maximum allowable angle is less than 20°, which has great limitations in practical applications.

[0003] Genetic algorithm is an optimization algorithm that simulates the process of natural selection and biological evolution. Based on the principles of natural selection and genetics, it searches for the optimal solution to a problem by simulating the processes of reproduction, mutation, selection, and inheritance of organisms in nature. The design problem of DLDOE also belongs to a complex optimization problem with highly non-linear, multiple local extrema, and high-dimensional characteristics. Using genetic algorithms can flexibly jump out of local optimal solutions and achieve the optimal solution design of DLDOE.

[0004] Refer to the article "Optimal design of multilayer diffractive optical element in wide angle of incidence" published by Bo Zhang et al. in the journal "Optics Communications". A design method of DLDOE based on the evaluation model of Polychromatic Integral Diffraction Efficiency (PIDE) is proposed. By comparing and analyzing the characteristic angle weighted PIDE within the incident angle range, the optimal microstructure height combination of DLDOE can be obtained. The DLDOE in the example in the article achieves high diffraction efficiency within the incident angle range of 0 - 20° and the wavelength range of 0.4 - 0.7μm. However, since this design method does not use a global optimization algorithm and does not optimize both the substrate material and the microstructure height of DLDOE as variables simultaneously, it is impossible to obtain the global optimal solution and it is difficult to cope with the application scenarios in a wider wavelength band (visible light plus near-infrared light) and a larger angle range (0 - 30°). Summary of the Invention

[0005] In order to solve the problem that the existing DLDOE design method is difficult to design a DLDOE with high diffraction efficiency in a wide wavelength band and a large angle range, the present invention proposes a design method of a double-layer diffractive optical element based on a genetic algorithm.

[0006] The solution of the present invention to solve the technical problem is as follows:

[0007] A design method of a wide-wavelength-band and large-angle double-layer diffractive element based on a genetic algorithm, which includes the following steps:

[0008] Step 1, select the optimization variables participating in the optimization design based on the basic theory of the diffraction efficiency of DLDOE;

[0009] Step 2, construct a mean-standard deviation double-objective evaluation model;

[0010] Step 3, establish a fitness function containing the dispersion characteristic constraint based on the processable material library;

[0011] Step 4, construct a genetic algorithm with a cosine annealing and periodic restart optimization strategy;

[0012] Step 5, match the substrate material and update the microstructure height.

[0013] The specific content of the said Step 1 is: When the light is obliquely incident, the diffraction efficiency expression of DLDOE is:

[0014]

[0015] Among them, m is the diffraction order, λ is the wavelength, θ is the incident angle when the light arrives, and n1(λ) and n2(λ) are the relationships of the refractive index dispersion characteristics of the front and rear layer substrate materials with respect to the wavelength; H1 and H2 are the microstructural heights of the front and rear layer substrate materials respectively;

[0016] A simplified Cauchy dispersion formula is introduced to express the dispersion characteristics of the front and rear layer substrate materials of the DLDOE. The simplified Cauchy dispersion formula is:

[0017]

[0018] In the formula, a1 and b1 are the Cauchy dispersion coefficients of the front layer substrate material, and a2 and b2 are the Cauchy dispersion coefficients of the rear layer substrate material;

[0019] The Cauchy dispersion coefficients a1, a2, b1, and b2 of the front and rear layer substrate materials and the front and rear layer microstructural heights H1 and H2 are selected as the optimization variables of the genetic algorithm.

[0020] The expression of the mean-standard deviation double-objective evaluation model described in step two is:

[0021]

[0022] After expansion, it is:

[0023]

[0024] Among them, ω1 and ω2 are the weight coefficients of the mean and the standard deviation respectively, and σ(η m ) are the mean and the standard deviation of the diffraction efficiency respectively, N θ and N λ are the number of sampling points within the working angle and the wavelength respectively.

[0025] The specific process of step three is as follows: First, collect the optical material models that can be precision molded and distribute them according to the dispersion characteristics; then, fit the upper and lower bound curves on the principle that the upper and lower bound curves can cover most of the material library models, and the curves are fitted with a cubic equation of one variable; finally, obtain a fitness function with the dispersion characteristic constraints based on the processable material library, and subtract a penalty value from the fitness of the solutions that deviate from the range of the upper and lower bound curves;

[0026] The fitness function with the dispersion characteristic constraints based on the processable material library is:

[0027]

[0028] Where n and v are the refractive index and Abbe number of the base material respectively, p is the penalty value when the solution deviates from the upper and lower bounds, and upperbound and lowerbound are the upper and lower bounds of the dispersion characteristics of the precision molding processable material respectively;

[0029] When adding the dispersion characteristic constraint based on the processable material library to the fitness function during the genetic algorithm optimization iteration, individuals that deviate too far from the upper and lower bounds will be avoided in the population, and the output result can match a closer processable material model.

[0030] In step four, in addition to the basic operations of crossover, mutation, and selection, the genetic algorithm realizes flexible adjustment of the two weight coefficients of the mean and standard deviation by adding the cosine annealing and periodic restart optimization strategies. After the genetic algorithm completes the iterative optimization, the optimization result is output, which includes the Cauchy dispersion coefficients a1, a2, b1, and b2 of the front and rear layer base materials to be matched and the H1 and H2 to be updated.

[0031] After adding the cosine annealing optimization strategy, the changes of the two weight coefficients are as follows:

[0032]

[0033] Where ω(t) is the weight of the t-th generation, ω max and ω min are the upper and lower limits of the weight respectively, T is the cycle length, and t is the current generation number. During the optimization process, the mean weight gradually decreases from the maximum value in each cycle, while the standard deviation weight increases from the minimum value. The algorithm first identifies the high-efficiency population and then emphasizes stability, so as to find individuals that are both efficient and stable.

[0034] After adding the periodic restart optimization strategy, the cycle period length formula is as follows:

[0035] T i = T0·γ i ,

[0036] Where T i is the cycle period length, T i is the initial cycle length, and γ is the cycle scaling factor.

[0037] Specifically, step five is: according to the Cauchy dispersion coefficients a1, a2, b1, and b2 of the front and rear layer base materials to be matched in the output result of step four, calculate the refractive index and Abbe number of the base material of the optimization result. Taking the refractive index and Abbe number of the base material of the optimization result as the target, match the closest processable material from the material library as the front and rear layer base materials of the DLDOE.

[0038] The optimized results of the matched processable materials have slight differences in terms of dispersion characteristics. It is necessary to further update the two variables, namely the microstructure heights H1 and H2 of the front and back layers of the DLDOE, based on the mean-standard deviation dual-objective evaluation model.

[0039] Advantages of the present invention: The present invention introduces the genetic algorithm in the global optimization algorithm into the optimization design method of the DLDOE. By utilizing the ability of the genetic algorithm to search for the optimal solution in a complex multi-dimensional space, it can flexibly jump out of the local optimal solution and achieve the optimal solution design of the DLDOE.

[0040] Based on the dual-objective optimization idea, an evaluation model for the optimization design method of the DLDOE is constructed. The mean and standard deviation respectively represent the high and low diffraction efficiency and the stability of the DLDOE within the working angle-wavelength range. The weight coefficients of the two can be flexibly adjusted according to actual needs. This evaluation model has the advantages of light operation, flexible adjustability, and high robustness. This evaluation model and the dispersion characteristic constraint based on the processable material library jointly constitute the fitness function of the genetic algorithm, realizing the function of optimizing both the substrate material and the microstructure height of the DLDOE as variables.

[0041] The cosine annealing and periodic restart optimization strategies are added to the genetic algorithm. The cosine annealing optimization strategy utilizes the periodicity of the cosine function to alternately focus on high efficiency and stability during the DLDOE optimization design process, which helps the algorithm get rid of the local optimal solution and enhance the global search ability. The periodic restart optimization strategy introduces a mechanism of gradually shortening the cycle length on the basis of cosine annealing by using stochastic gradient descent with restart to further enhance the global search ability.

[0042] Based on the above characteristics, the present invention solves the problems existing in the existing DLDOE design method and can design a DLDOE with high diffraction efficiency in a wide wavelength band and a large angle range. Brief Description of the Drawings

[0043] Figure 1 It is the flowchart of a design method for a double-layer diffractive optical element based on the genetic algorithm of the present invention;

[0044] Figure 2 It is the dispersion characteristic distribution diagram of the processable material library selected by the present invention;

[0045] Figure 3 It is the effect diagram of the matching process of the processable material in the embodiment;

[0046] Figure 4 It is the relationship between the diffraction efficiency of the DLDOE and the wavelength and the incident angle in the embodiment;

[0047] Figure 5 It is the relationship between the bandwidth-integrated diffraction efficiency and the wavelength of the DLDOE at different incident angles in the embodiment. Specific Embodiments

[0048] The present invention is described in detail below with reference to the accompanying drawings.

[0049] As Figure 1 shown, a method for designing a wide-band large-angle double-layer diffraction element based on a genetic algorithm includes the following steps:

[0050] Step 1: Select optimization variables for the optimization design based on the basic theory of the diffraction efficiency of DLDOE.

[0051] When light is obliquely incident, the expression for the diffraction efficiency of DLDOE is:

[0052]

[0053] where m is the diffraction order, λ is the wavelength, θ is the incident angle when the light arrives, n1(λ) and n2(λ) are the relationships of the refractive indices of the front and rear layer substrate materials with respect to the wavelength dispersion characteristics, respectively; H1 and H2 are the microstructure heights of the front and rear layer substrate materials.

[0054] The refractive indices of the front and rear layer substrate materials of DLDOE change with the wavelength. A simplified Cauchy dispersion formula is introduced to express the dispersion characteristics of the front and rear layer substrate materials. The simplified Cauchy dispersion formula is:

[0055]

[0056] In the formula, a1 and b1 are the Cauchy dispersion coefficients of the front layer substrate material, and a2 and b2 are the Cauchy dispersion coefficients of the rear layer substrate material.

[0057] In summary, select the Cauchy dispersion coefficients (a1, a2, b1, and b2) of the front and rear layer substrate materials and the microstructure heights (H1 and H2) of the front and rear layers as the optimization variables of the genetic algorithm.

[0058] Step 2: Construct a mean-standard deviation dual-objective evaluation model.

[0059] The expression of the mean-standard deviation dual-objective evaluation model (Mean-Standard Deviation Dual-OptimizationDiffraction Efficiency, MSDODE) is:

[0060]

[0061] After expansion, it is:

[0062]

[0063] where ω1 and ω2 are the weight coefficients of the mean value and the standard deviation respectively, and σ(η m ) are the mean value and the standard deviation of the diffraction efficiency respectively, and N θ and N λ are the number of sampling points within the working angle and wavelength respectively.

[0064] Step 3: Establish a fitness function with constraints on the dispersion characteristics based on the processable material library;

[0065] The specific process is as follows: First, collect the types of optical materials that can be precision molded and distribute them according to the dispersion characteristics. The distribution diagram of the dispersion characteristics of the material library of the optical materials that can be precision molded is shown in Figure 2 . In the figure, the points of different colors represent optical materials of different brands. Then, fit the upper and lower bound curves on the principle that the upper and lower bound curves can cover most of the material library models. The curves are fitted using a cubic equation of one variable. Figure 2 In , the solid curve and the dashed curve are the upper and lower bound curves respectively. Finally, obtain a fitness function with constraints on the dispersion characteristics based on the processable material library. The fitness of the solution deviating from the range of the upper and lower bound curves is subtracted by a penalty value.

[0066] The fitness function added with the constraints on the dispersion characteristics based on the processable material library is:

[0067]

[0068] In the formula, n and v are the refractive index and Abbe number of the substrate material respectively, p is the penalty value when the solution deviates from the upper and lower bound ranges, and upperbound and lowerbound are the upper and lower bound ranges of the dispersion characteristics of the materials that can be precision molded.

[0069] When adding the constraints on the dispersion characteristics based on the processable material library to the fitness function during the genetic algorithm optimization iteration, individuals that deviate too far from the upper and lower bound ranges will be avoided in the population, and the output result can match a closer processable material model.

[0070] Step 4: Construct a genetic algorithm with a cosine annealing and periodic restart optimization strategy;

[0071] In addition to the basic operations of crossover, mutation and selection, this genetic algorithm realizes the flexible adjustment of the two weight coefficients of the mean value and the standard deviation by adding a cosine annealing and periodic restart optimization strategy. After the genetic algorithm completes the iterative optimization, the optimization result is output, including the Cauchy dispersion coefficients a1, a2, b1 and b2 of the front and rear layer substrate materials to be matched and the H1 and H2 to be updated.

[0072] After adding the cosine annealing optimization strategy, the changes of the two weight coefficients are as follows:

[0073]

[0074] where ω(t) is the weight of the t-th generation, ω max and ω min are the upper and lower limits of the weight respectively, T is the cycle length, and t is the current generation number. During the optimization process, the average weight gradually decreases from the maximum value in each cycle, while the standard deviation weight increases from the minimum value. The algorithm first identifies the high-efficiency group and then emphasizes stability, so as to find individuals that are both efficient and stable.

[0075] After adding the cycle restart optimization strategy, the cycle length formula is as follows:

[0076] T i = T0·γ i

[0077] where T i is the cycle length of the cycle, T i is the initial cycle length, and γ is the cycle scaling factor. During the optimization process, the optimization algorithm first conducts a wide search within the initial longer cycle, and then conducts a refined search within the shorter cycle, so as to quickly identify excellent individuals.

[0078] Step Five, match the substrate material and update the microstructure height;

[0079] According to the Cauchy dispersion coefficients a1, a2, b1, and b2 of the front and rear layer substrate materials in the output result of Step Four, calculate the refractive index and Abbe number of the substrate material of the optimization result. Taking the refractive index and Abbe number of the substrate material of the optimization result as the target, match the closest processable material from the material library as the front and rear layer substrate materials of the DLDOE.

[0080] There are slight differences in the optimization results of the matched processable materials in terms of dispersion characteristics. It is necessary to further update the two variables, the front and rear layer microstructure heights H1 and H2 of the DLDOE, based on the mean-standard deviation double-objective evaluation model.

[0081] Example:

[0082] Establish a design index for a DLDOE with high diffraction efficiency in a wide wavelength band and large angle range. In the specific example, the wavelength range of the DLDOE is 0.4 - 1.0 μm, which covers visible light and near-infrared light and is wider than the wavelength range of traditional DLDOE (0.4 - 0.7 μm); the incident angle range of light is 0 - 30°, which is larger than the angle range of traditional DLDOE (0 - 20°). The optimization goal is that the average diffraction efficiency of the DLDOE in the above wide wavelength band and large angle range is greater than 0.95.

[0083] Table 1 Initialization Variable Value Ranges

[0084] Variable a1, a2 b1, b2 H1 (μm) H2 (μm) Range of values 1.4～2.0 0.001～0.010 10～30 -30～10

[0085] Next, set the value ranges of 6 optimization variables and the relevant parameters of the genetic algorithm. The value ranges of the variables are shown in Table 1, and the relevant parameters of the genetic algorithm are set as follows: the population size is 5000, the number of iterations is 100, the crossover rate is 0.6, the mutation rate is 0.3, and 5 elite individuals are selected in each generation to skip crossover and mutation and directly enter the next round. The results of the global optimal solution obtained by iterative optimization using the genetic algorithm are shown in Table 2.

[0086] Table 2 Optimization Results after 200 Generations of Iteration

[0087]

[0088]

[0089] According to the processable material models corresponding to the Cauchy dispersion coefficients of the to-be-determined substrate materials, the matching process is as Figure 3 shown. Convert the obtained optimization variable parameters (a1, a2, b1, and b2) into the dispersion characteristic parameters (refractive index and Abbe number) of the substrate material. It can be obtained that the refractive index of the substrate material of the front layer of the DLDOE is 1.6558, and the Abbe number is 66.0284; the refractive index of the substrate material of the rear layer is 1.5723, and the Abbe number is 34.0498. The above two to-be-matched substrate materials are located at the blue points in the figure, which are the target values of the processable material matching process. After matching, the substrate materials of the front layer and the rear layer are K-LAFK63 and K-CD35 respectively, which are located at the dark red points in the figure. The refractive indices and Abbe numbers of these two types of materials in the candidate material library are closest to the to-be-matched substrate materials. The refractive indices and Abbe numbers of the materials at the light red points in the figure are relatively close but not as close as those of the materials at the dark red points.

[0090] Then, according to the Cauchy dispersion coefficients of these two optical materials, combine the mean-standard deviation double-objective evaluation model to update the microstructure heights of the front layer and the rear layer of the DLDOE: H1 and H2 are 22.4107μm and -23.5295μm respectively.

[0091] The relationship between the diffraction efficiency of the DLDOE in the specific embodiment and the wavelength and angle is as Figure 4 shown. It can be calculated that the average diffraction efficiency of the DLDOE reaches 0.9614 in the wavelength range of 0.4 - 1.0μm and the angle range of 0 - 30°, meeting the design goal of high diffraction efficiency (>0.95).

[0092] The relationship between the bandwidth-integrated diffraction efficiency and the wavelength of the DLDOE in the specific embodiment at different angles is as Figure 5As shown, at different incident angles, the bandwidth-averaged diffraction efficiency of the specific embodiment DLDOE fluctuates minimally, and still maintains a high efficiency of 0.9596 at a large incident angle of 30°, only decreasing by 0.08% compared to the incident angle of 0°. It effectively suppresses the angular sensitivity of the separated DLDOE structure and solves the problem that it is difficult to design a DLDOE with high diffraction efficiency in a wide wavelength band and a large angular range by traditional design methods.

Claims

1. A design method for a wide-band large-angle double-layer diffraction element based on a genetic algorithm, characterized in that It includes the following steps: Step 1: Select optimization variables participating in the optimization design based on the basic theory of the diffraction efficiency of DLDOE; Step 2: Construct a mean-standard deviation dual-objective evaluation model; Step 3: Establish a fitness function containing constraints on the dispersion characteristics based on a processable material library; Step 4: Construct a genetic algorithm with a cosine annealing and periodic restart optimization strategy; Step 5: Match the substrate material and update the microstructure height.

2. The design method of a wide-band large-angle double-layer diffraction element based on a genetic algorithm according to claim 1, wherein The specific content of Step 1 is as follows: When the light is obliquely incident, the expression of the diffraction efficiency of DLDOE is: Where, m is the diffraction order, λ is the wavelength, θ is the incident angle when the light arrives, n1(λ) and n2(λ) are the relationships of the refractive indices of the front and rear layer substrate materials with respect to the wavelength dispersion characteristics respectively; H1 and H2 are the microstructure heights of the front and rear layer substrate materials respectively; Introduce a simplified Cauchy dispersion formula to express the dispersion characteristics of the front and rear layer substrate materials of DLDOE. The simplified Cauchy dispersion formula is: In the formula, a1 and b1 are the Cauchy dispersion coefficients of the front layer substrate material, and a2 and b2 are the Cauchy dispersion coefficients of the rear layer substrate material; Select the Cauchy dispersion coefficients a1, a2, b1, and b2 of the front and rear layer substrate materials and the front and rear layer microstructure heights H1 and H2 as the optimization variables of the genetic algorithm.

3. A design method of a wide-band large-angle double-layer diffraction element based on a genetic algorithm according to claim 1, characterized in that The expression of the mean-standard deviation dual-objective evaluation model described in Step 2 is: The expression of the mean-standard deviation dual-objective evaluation model described in Step 2 is: After expansion, it is: where ω1 and ω2 are the weight coefficients of the mean value and the standard deviation respectively, and σ(η m ) are the mean value and the standard deviation of the diffraction efficiency respectively, N θ and N λ are the number of sampling points within the working angle and wavelength respectively.

4. A design method for a wide-band large-angle double-layer diffraction element based on a genetic algorithm according to claim 1, characterized in that The specific process of Step 3 is as follows: First, collect the optical material models that can be precision molded and distribute them according to the dispersion characteristics; then, fit the upper and lower bound curves based on the principle that the upper and lower bound curves can cover most of the material library models, and the curves are fitted using a cubic equation in one variable; finally, obtain a fitness function with constraints on the dispersion characteristics based on the processable material library, and subtract a penalty value from the fitness of the solutions that deviate from the range of the upper and lower bound curves; The fitness function with constraints on the dispersion characteristics based on the processable material library is: In the formula, n and v are the refractive index and Abbe number of the substrate material respectively, p is the penalty value when the solution deviates from the upper and lower bound ranges, and upperbound and lowerbound are the upper and lower bound ranges of the dispersion characteristics of the materials that can be precision molded; When adding constraints on the dispersion characteristics based on the processable material library to the fitness function during the genetic algorithm optimization iteration, individuals that deviate too far from the upper and lower bound ranges will be avoided in the population, and the output results can be matched to relatively close processable material models.

5. A design method of a wide-band large-angle double-layer diffraction element based on a genetic algorithm according to claim 1, characterized in that In addition to the basic operations of crossover, mutation, and selection, the genetic algorithm described in Step 4 realizes flexible adjustment of the two weight coefficients of the mean and standard deviation by adding a cosine annealing and periodic restart optimization strategy; after the genetic algorithm completes the iterative optimization, the optimization results are output, including the Cauchy dispersion coefficients a1, a2, b1, and b2 of the front and rear layer substrate materials to be matched and the H1 and H2 to be updated; After adding the cosine annealing optimization strategy, the changes of the two weight coefficients are as follows: where ω(t) is the weight of the t-th generation, ω max and ω min are the upper and lower limits of the weight respectively, T is the cycle length, and t is the current generation; during the optimization process, the average weight gradually decreases from the maximum value in each cycle, while the standard deviation weight increases from the minimum value. The algorithm first identifies the high-efficiency group and then emphasizes stability, so as to find individuals that are both efficient and stable. After adding the periodic restart optimization strategy, the cycle period length formula is as follows: T i = T0·γ i , Where, T i is the cycle length of the cycle, T i is the initial cycle length, and γ is the cycle scaling factor.

6. A design method of a wide-band large-angle double-layer diffraction element based on a genetic algorithm according to claim 5, characterized in that, Step 5 specifically includes: According to the Cauchy dispersion coefficients a1, a2, b1, and b2 of the front and rear layer substrate materials to be matched in the output result of Step 4, calculate the refractive index and Abbe number of the substrate material of the optimization result. Taking the refractive index and Abbe number of the substrate material of the optimization result as the target, match the closest processable material from the material library as the front and rear layer substrate materials of the DLDOE; There are slight differences in the optimization results of the matched processable materials in terms of dispersion characteristics. It is necessary to further update the two variables, the front and rear layer microstructural heights H1 and H2 of the DLDOE, based on the mean-standard deviation dual-objective evaluation model.

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