A high-precision solution method for the structural parameters of an optical system

Abstract

The present invention discloses a method for accurately solving the structural parameters of an optical system. The steps include: 1) Denote the N structural parameters of the optical system as x(x1,…,xN); Denote the aberration F defined by the optical system, and the aberration function as f; 2) According to the definition in step 1), transform the optical design problem of the optical system into solving a non-linear equation set F = f(x); 3) Calculate the aberration values F0 corresponding to the initial structural parameters x0; Then, in turn, add a step size to each of the N structural parameters, and recalculate the aberration values F* and the corresponding aberration increments ΔF; 4) Obtain a difference quotient matrix A according to each obtained ΔF; 5) Convert F = f(x) into a linear equation set AΔx = ΔF; 6) Solve AΔx = ΔF by a linear solver based on QR decomposition to obtain the values of the N structural parameters of the optical system. The present invention can obtain the structural parameters of the optical system with better numerical stability and higher accuracy.

Description

Technical Field

[0001] The present invention belongs to the technical field of optical automatic design, and particularly relates to a method for accurately solving the structural parameters of an optical system. Background Art

[0002] Optical automatic design is an optimization technology that obtains an optimized design meeting the requirements by changing the structural parameters of an optical system (such as the curvature radius of a lens, the central thickness of a lens, etc.). The existing optical automatic design technology mainly obtains an overdetermined or underdetermined linear system by linearly expanding a non-linear aberration function, solves this overdetermined or underdetermined linear system to obtain the increment of the structural parameters of the optical system, and then solves the new structural parameters of the optical system.

[0003] Disadvantages of the prior art:

[0004] (1) The increment of the structural parameters of the optical system is mainly obtained by using Gaussian elimination method. However, from the perspective of numerical calculation, Gaussian elimination method is essentially a non-stable numerical calculation method and cannot meet the high-precision requirements of the optical system.

[0005] (2) To improve the calculation accuracy, currently, the method of partial pivoting or complete pivoting is mostly used in the solution of the structural parameters of the optical system to improve the calculation accuracy. However, even so, the Gaussian elimination method with partial pivoting or complete pivoting is still an unstable solution method in terms of mathematical essence, which poses a potential threat to the optical automatic design with extremely high requirements for calculation accuracy.

[0006] (3) Theory and practice have proved that the matrix condition number in optical automatic design is extremely poor. Using the traditional Gaussian elimination method, even if a set of structural parameters of the optical system can be calculated from a "mathematical" perspective, from the perspective of optical automatic design, this set of optimized structural parameters of the optical system deviates from the physical meaning and only has theoretical significance from a mathematical perspective. Summary of the Invention

[0007] Aiming at the problems existing in the prior art, the purpose of the present invention is to provide a method for accurately solving the structural parameters of an optical system. The present invention can obtain the structural parameters of the optical system with better numerical stability and higher accuracy.

[0008] The problem to be solved in optical optimization design is the aberration design problem, that is, according to the optical characteristics and imaging quality requirements for each lens group given by the calculation of the external dimensions, determine the structural parameters of the lens group, including the glass material used, the curvature radius of each spherical surface, and the thickness interval between lenses, etc. This topic will focus on the intelligent problem in the optimization design, improve the intelligence of the optimization, and reduce the dependence of software users on experience.

[0009] The optimized physical-mathematical model can be described as follows: various structural parameters of the optical system are denoted as x(x1, …, x N ), indicating that there are N independent variables, which include c, d, n, δn, and a (aspherical coefficient). The required aberrations are denoted as F(F1, …, F m ), indicating that there are m aberration requirements, and the aberration function is denoted as f(f1, …, f m ). Then the optical design problem becomes solving a system of nonlinear equations, that is

[0010]

[0011] Or written in vector form, that is

[0012] F = f(x)

[0013] First, give the structural parameters x0 of an initial system, and calculate its various aberration values F0 = (F 01 , …, F 0m ) T , and then sequentially add a small quantity δx j to each independent variable: x 0j + δx j , and recalculate the aberration The corresponding aberration increment is ΔF = F * - F0 = (δF1, …, δF m ) T .

[0014] The difference quotient is obtained as If each independent variable takes an increment, then according to ΔF, the difference quotient matrix A can be obtained, that is, the coefficient matrix A of the system of nonlinear equations is

[0015]

[0016] If δx j is taken small enough and the calculation has sufficient accuracy, then the difference quotient matrix can be used to approximately replace the first-order partial derivative matrix, that is

[0017]

[0018] Using the Taylor expansion of a multivariate function, the system of nonlinear equations F = f(x) can be replaced by the following system of linear equations, that is

[0019]

[0020] If we let

[0021]

[0022] Then the above system of equations can be expressed in the following form as

[0023] AΔx = ΔF

[0024] Solve the linear equation system Δx = A -1 ΔF according to a certain algorithm. Different methods correspond to different algorithms and programs.

[0025] The technical solution of the present invention is as follows:

[0026] A method for accurately solving the structural parameters of an optical system, the steps of which include:

[0027] 1) Denote the N structural parameters of the optical system as x(x1,..., x N ); x N represents the Nth structural parameter of the optical system; Denote the aberrations defined by the optical system as F(F1,..., F m ), and the aberration function as f(f1,..., f m ), F m represents the mth aberration requirement of the optical system, and the corresponding aberration function for F m is f m ;

[0028] 2) According to the definition in step 1), transform the optical design problem of the optical system into solving a non - linear equation system F = f(x);

[0029] 3) Initialize the structural parameters x of the optical system to obtain an initial structural parameter x0; Calculate the aberration values F0=(F 01 ,..., F 0m ) T , F 0m [[ID=4)]]is the mth aberration value of the optical system when the initial structural parameter is x0; Then, successively add a step size to each of the N structural parameters, and recalculate the aberration values and the corresponding aberration increment ΔF = F * - F0=(δF1,..., δF m ) T ; is the mth aberration value of the optical system when a step size is added to the current structural parameter, and δF m is the increment of the mth aberration value of the optical system after adding a step size to the current structural parameter;

[0030] 4) Obtain a difference quotient matrix according to each obtained ΔF

[0031] 5) Use the Taylor expansion of a multivariate function to transform the non - linear equation system F = f(x) into a linear equation system x 0Nis the initial value of the Nth structural parameter in the initial structural parameter x0; then Δx = x - x0, ΔF = F - F0, to obtain the system of equations AΔx = ΔF;

[0032] 6) Solve AΔx = ΔF using a linear solver based on QR decomposition to obtain the values of the N structural parameters of the optical system.

[0033] Furthermore, the method for solving AΔx = ΔF using a linear solver based on QR decomposition is as follows: First, perform Householder reflection transformation calculation on the difference quotient matrix A to obtain an orthogonal matrix Q, decompose matrix A such that A = QR, where R is a non-singular upper triangular matrix; then calculate the intermediate value y = Q T b, where b = ΔF; then solve the upper triangular linear system RΔx = y using the backward propagation method.

[0034] Furthermore, given the calculated column vector Δx, the orthogonal matrix Q = I - γuu is obtained through Householder reflection transformation T ; where, ||u||2 = 1, I is the identity matrix, and QΔx = [-τ 0 … 0] T ,

[0035] Furthermore, the Householder reflection transformation starts from calculating the infinity norm of the column vector Δx, that is, β = ||Δx|| ∞ = max 1≤i≤N |Δx|; if β is 0, then take ||Δx||2 = 0, otherwise take that is, perform scaling calculation.

[0036] A server, comprising a memory and a processor, the memory stores a computer program, the computer program is configured to be executed by the processor, and the computer program includes instructions for executing the steps in the above method.

[0037] A computer-readable storage medium, on which a computer program is stored, characterized in that the steps of the above method are implemented when the computer program is executed by a processor.

[0038] The advantages of the present invention are as follows:

[0039] 1) The present invention can prevent upper overflow and lower overflow of the structural parameter reflection vector;

[0040] 2) The present invention proposes an efficient calculation method for the structural parameter orthogonalization matrix;

[0041] Conventional orthogonalization methods update one column of the optical matrix at a time. In view of the characteristics of the optical matrix, the present invention accumulates multiple column updates into one update while maintaining numerical accuracy and stability unchanged.

[0042] 3) The present invention provides a high-precision solution mechanism for structural parameters;

[0043] So far, there is no clear linear solution method using QR decomposition in commercial and open-source software packages. The present invention is original in the automatic design of optical systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 is a flowchart of the method of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0045] The present invention will be further described in detail below with reference to the accompanying drawings. The examples given are only used to explain the present invention and are not intended to limit the scope of the present invention.

[0046] The essence of the present invention to solve the structural parameters of the optical system with high precision lies in the use of the stability of mathematical methods and several high-precision and high-stability techniques in numerical calculations.

[0047] The essence of the present invention is a linear solver based on QR decomposition, that is, to solve AΔx = ΔF. The main steps are as follows:

[0048] (1) Use the Householder reflection transformation to calculate the orthogonal matrix Q, decompose the matrix A so that A = QR, where Q is an orthogonal vector group and R is a non-singular upper triangular matrix;

[0049] (2) Calculate the intermediate value y = Q T b, where b = ΔF;

[0050] (3) Use the backward propagation method to solve the upper triangular linear system RΔx = y; after solving Δx, x can be obtained according to Δx = x - x0; where x0 is a known variable.

[0051] QR decomposition is mainly used in the field of eigenvalue calculation. As far as the author's ability and experience are concerned and after consulting experts in relevant mathematical fields, no method of using QR decomposition to solve unknown elements has been found.

[0052] Let Use the QR decomposition linear solver to solve The key lies in decomposing the matrix A to make A = QR. The matrix Q = I - γuu T can be obtained through the Householder reflection transformation, where ||u||2 = 1, I is the identity matrix. In order to obtain optical structure parameters with good stability and high precision, during the process of calculating the orthogonal matrix Q using the Householder reflection transformation, considering the characteristics of IEEE floating-point number storage, numerical measures are taken specifically for floating-point errors such as floating-point overflow (overflow / Underflow) and floating-point catastrophic cancellation (Cancellation) to eliminate these effects and obtain theoretically accurate numerical values. See Algorithm 1. Assume that the given computational column vector Algorithm 1 will calculate the exact numerical values of τ, γ, and u, satisfying

[0053]

[0054] As can be seen from Algorithm 1, the method of avoiding floating-point overflow in the Householder reflection transformation starts from calculating the infinity norm of the column vector That is If β is 0, then take γ = 0, otherwise let We can get Thus, for any i, we have Such a scaling calculation process avoids the overflow situation that may occur in the actual calculation process.

[0055] It can be seen from Algorithm 1 that the situation of floating-point catastrophic cancellation will not occur because the algorithms operate on positive numbers. For example is greater than 0, which effectively avoids the occurrence of catastrophic cancellation; another example is line 11 of the algorithm makes and τ have the same sign, which also avoids the occurrence of catastrophic cancellation.

[0056] In summary, the calculations of τ, γ, and u are all "accurate", so the matrix Q obtained is precise, thereby improving the accuracy of solving AΔx = ΔF.

[0057] Although specific embodiments of the present invention are disclosed for illustrative purposes, which are intended to help understand the content of the present invention and implement it accordingly, those skilled in the art can understand that: without departing from the spirit and scope of the present invention and the appended claims, various substitutions, changes, and modifications are possible. Therefore, the present invention should not be limited to the content disclosed in the best embodiments, and the scope of protection required by the present invention is defined by the scope of the claims.

Claims

1. A method for accurately solving the structural parameters of an optical system, the steps of which include: 1) Denote the N structural parameters of the optical system as x(x1, …, x N ); x N represents the Nth structural parameter of the optical system; Denote the aberrations defined by this optical system as F(F1, …, F m ), the aberration function as f(f1, …, f m ), F m represents the mth aberration requirement of this optical system, and the aberration function corresponding to F m is f m ; 2) According to the limitation in step 1), transform the optical design problem of the optical system into solving a non-linear equation set F = f(x); 3) Initialize the structural parameter x of the optical system to obtain an initial structural parameter x0; calculate each aberration value F0 = (F 01 , …, F 0m ) T , F 0m is the m-th aberration value of the optical system when the initial structural parameter is x0; then sequentially add a step size to each of the N structural parameters, and recalculate each aberration value and the corresponding aberration increment ΔF = F * - F0 = (δF1, …, δF m ) T ; is the m-th aberration value of the optical system when a step size is added to the current structural parameter, and δF m is the increment of the m-th aberration value of the optical system after adding a step size to the current structural parameter; 4) Obtain a difference quotient matrix based on each obtained ΔF 5) Use the Taylor expansion of a multivariate function to convert the nonlinear equation system F = f(x) into a linear equation system x 0N is the initial value of the Nth structural parameter in the initial structural parameter x0; then Δx = x - x0, ΔF = F - F0, and the equation system AΔx = ΔF is obtained; 6) Solve AΔx = ΔF by a linear solver based on QR decomposition to obtain the values of N structural parameters of the optical system.

2. The method according to claim 1, wherein The method for solving \(A\Delta x=\Delta F\) using a QR decomposition-based linear solver is as follows: First, perform a Householder reflection transformation on the difference quotient matrix \(A\) to calculate the orthogonal matrix \(Q\), and decompose matrix \(A\) such that \(A = QR\), where \(R\) is a nonsingular upper triangular matrix; then calculate the intermediate value \(y = Q T b\), where \(b=\Delta F\); Then solve the upper triangular linear system RΔx = y by the back propagation method.

3. The method according to claim 1 or 2, characterized in that, Given the computed column vector Δx, the orthogonal matrix Q = I - γuu is obtained through Householder reflection transformation T ; where ||u||2 = 1, I is the identity matrix, and QΔx = [-τ 0…0] T , 4. The method according to claim 3, wherein The Householder reflection transformation starts from calculating the infinity norm of the column vector Δx, that is, β = ||Δx|| ∞ = max 1≤i≤n |Δx|; if β is 0, then take ||Δx||2 = 0, otherwise take 5. A server, characterized in that, It includes a memory and a processor, the memory stores a computer program, the computer program is configured to be executed by the processor, and the computer program includes instructions for executing the steps in any one of claims 1 to 4.

6. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, the steps of any one of claims 1 to 4 are implemented.

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