A method and system for calculating Seidel coefficients using Zernike polynomials of wavefront phase

Abstract

This application belongs to the field of computational optics and specifically discloses a method and system for calculating Seidel coefficients using Zernike polynomials of wavefront phase. By precisely defining the transformation process and strictly matching the conversion formula, this application achieves accurate and rapid conversion of Zernike coefficients to Seidel coefficients. This method is applicable to conversions of Zernike polynomials of different orders and different types of Seidel aberrations. Furthermore, the Zernike polynomial representation based on wavefront distortion is converted into a representation based on Seidel aberrations, making optical design and analysis more intuitive and convenient.

Description

Technical Field

[0001] The present application belongs to the field of computational optics, and more specifically, to a method and system for calculating Seidel coefficients using Zernike polynomials of wavefront phase. Background Art

[0002] The description of wavefront distortion is crucial for understanding and correcting imaging defects in optical systems. With a precise description, appropriate measures can be taken to improve imaging quality. In optical system design and analysis, wavefront distortion is typically described using two methods: Zernike polynomials and Seidel aberrations. Zernike polynomials are a set of orthogonal polynomials defined on the unit circle that are used to describe the wavefront distortion of an optical system. They can decompose the wavefront error into multiple coefficients, each corresponding to a different aberration term. Due to their numerical stability and orthogonality, these polynomials are widely used to describe wavefront distortion at circular apertures, particularly in the fields of wavefront correction and image processing. Seidel aberrations are a set of parameters that describe the wavefront distortion of an optical system, including spherical aberration, coma, astigmatism, field curvature, and distortion. They provide a more intuitive and specific description of various aberrations (such as spherical aberration and astigmatism) used in traditional optical design. In certain situations, including but not limited to correcting or optimizing the primary aberrations of an optical system, it is necessary to convert the wavefront distortion described by Zernike polynomials into Seidel aberrations for further analysis and processing.

[0003] Zernike polynomials represent the standard orthogonal basis for circular regions under constant weight functions, while Seidel aberrations describe primary and higher-order aberrations. Wavefront errors derived from Zernike polynomial expansions can be converted to Seidel aberrations by linking Zernike coefficients with Seidel aberration parameters. However, optical software typically calculates Seidel aberrations and Zernike representations separately. Seidel aberration calculations rely on ray models derived from complex ray tracing models. Direct conversion methods from Zernike polynomials to Seidel aberrations are limited, and an efficient and accurate method is urgently needed to achieve this conversion. Summary of the Invention

[0004] In response to the defects of the existing technology, the purpose of this application is to provide a method and system for calculating Seidel coefficients using Zernike polynomials of wavefront phase, aiming to solve the current problem of lack of efficient and accurate methods to convert Zernike polynomial expansion of wavefront aberrations into Seidel aberrations.

[0005] To achieve the above objectives, in a first aspect, the present application provides a method for calculating Seidel coefficients using Zernike polynomials of wavefront phase, comprising:

[0006] Obtain the order of the radial term in the Zernike polynomial, the wavefront phase to be analyzed, and the type of aberration to be determined;

[0007] According to the order of radial terms, the wavefront phase to be analyzed is expanded using Zernike polynomials to obtain a set of Zernike coefficients corresponding to the wavefront phase to be analyzed;

[0008] According to the type of aberration to be determined, match the power series corresponding to the Seidel coefficient in the Seidel expansion;

[0009] After expanding the multiple angle cosine terms in the Zernike polynomial expansion using the first kind Chebyshev polynomials, the coefficients of all terms in the power series containing the corresponding Seidel coefficients are combined to obtain the conversion formula between Zernike coefficients and Seidel coefficients.

[0010] Substituting a set of Zernike coefficients corresponding to the phase of the wavefront to be analyzed into the above conversion formula, the Seidel coefficients corresponding to the phase of the wavefront to be analyzed are obtained.

[0011] Preferably, the Seidel expansion is as follows:

[0012]

[0013] Where W(H,ρ,θ) represents the wavefront aberration, H is the normalized field height of the object, ρ is the normalized radial radius of the image, θ is the angle between the two, and S i is the corresponding Seidel coefficient, which takes values ​​of 1, 2, 3, 4, 5, .... The coefficients of the first 5 items represent spherical aberration, coma, astigmatism, field curvature and distortion aberration respectively.

[0014] Preferably, the multiple angle cosine terms in the Zernike polynomial expansion include cos(mθ) and sin(mθ).

[0015] Preferably, the method further comprises:

[0016] According to the obtained Seidel coefficient and the Seidel expansion form, the corresponding Seidel aberration is obtained.

[0017] Preferably, when the method is applied to the spherical aberration of a single-lens optical system, the order of the azimuthal terms is within 6 orders, that is, the total number of Zernike coefficients is 0 to 38 terms.

[0018] To achieve the above objectives, in a second aspect, the present application provides a system for calculating Seidel coefficients using Zernike polynomials of wavefront phase, comprising at least one processor and at least one memory;

[0019] The at least one memory is for storing computer instructions;

[0020] The at least one processor is configured to execute at least part of the computer instructions to implement the method described in the first aspect.

[0021] To achieve the above objectives, in a third aspect, the present application provides a computer-readable storage medium, which stores computer instructions. When a computer reads the computer instructions in the storage medium, the computer executes the method described in the first aspect.

[0022] It can be understood that the beneficial effects of the second to third aspects mentioned above can be found in the relevant description of the first aspect mentioned above, and will not be repeated here.

[0023] In general, the above technical solutions conceived by this application have the following beneficial effects compared with the existing technologies:

[0024] The present application provides a method and system for calculating Seidel coefficients using Zernike polynomials of wavefront phase. By precisely defining the transformation process and strictly matching the conversion formula, accurate and rapid conversion of Zernike coefficients to Seidel coefficients is achieved. The method is applicable to conversions of Zernike polynomials of different orders and different types of Seidel aberrations. The method further converts the wavefront distortion representation based on Zernike polynomials into a representation based on Seidel aberrations, making optical design and analysis more intuitive and convenient. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 This is a flow chart of a method for calculating Seidel coefficients using Zernike polynomials of wavefront phase, provided in an embodiment of the present application.

[0026] Figure 2 This is a schematic diagram of the single lens structure provided in an embodiment of the present application.

[0027] Figure 3 This is a schematic diagram of the Zernike coefficient error provided in an embodiment of the present application.

[0028] Figure 4 This is a schematic diagram of the structure of an electronic device provided in an embodiment of the present application. DETAILED DESCRIPTION

[0029] In order to make the purpose, technical solutions and advantages of this application more clear, the following further describes this application in detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0030] The term "and / or" as used herein describes an association between related objects, indicating that three possible relationships exist. For example, "A and / or B" can represent: A exists alone, A and B exist simultaneously, or B exists alone. The symbol " / " as used herein indicates that the related objects are in an "or" relationship, for example, A / B means either A or B.

[0031] The terms "first" and "second" in this specification and claims are used to distinguish different objects rather than to describe a specific order of objects. For example, "first response message" and "second response message" are used to distinguish different response messages rather than to describe a specific order of response messages.

[0032] In the embodiments of this application, words such as "exemplary" or "for example" are used to indicate examples, illustrations, or descriptions. Any embodiment or design described as "exemplary" or "for example" in the embodiments of this application should not be interpreted as being preferred or advantageous over other embodiments or designs. Rather, the use of words such as "exemplary" or "for example" is intended to present the relevant concepts in a concrete manner.

[0033] In the description of the embodiments of the present application, unless otherwise specified, "multiple" means two or more, for example, multiple processing units means two or more processing units, etc.; multiple elements means two or more elements, etc.

[0034] Next, the technical solutions provided in the embodiments of this application are introduced.

[0035] like Figure 1 As shown, the present application provides a method for calculating Seidel coefficients using Zernike polynomials of wavefront phase, including:

[0036] Obtain the order of the radial term in the Zernike polynomial, the wavefront phase to be analyzed, and the type of aberration to be determined;

[0037] According to the order of radial terms, the wavefront phase to be analyzed is expanded using Zernike polynomials to obtain a set of Zernike coefficients corresponding to the wavefront phase to be analyzed;

[0038] According to the type of aberration to be determined, match the power series corresponding to the Seidel coefficient in the Seidel expansion;

[0039] After expanding the multiple angle cosine terms in the Zernike polynomial expansion using the first kind Chebyshev polynomials, the coefficients of all terms in the power series containing the corresponding Seidel coefficients are combined to obtain the conversion formula between Zernike coefficients and Seidel coefficients.

[0040] Substituting a set of Zernike coefficients corresponding to the phase of the wavefront to be analyzed into the above conversion formula, the Seidel coefficients corresponding to the phase of the wavefront to be analyzed are obtained.

[0041] Preferably, the Seidel expansion is as follows:

[0042]

[0043] Where W(H,ρ,θ) represents the wavefront aberration, H is the normalized field height of the object, ρ is the normalized radial radius of the image, θ is the angle between the two, and S i is the corresponding Seidel coefficient, which takes values ​​of 1, 2, 3, 4, 5, .... The coefficients of the first 5 items represent spherical aberration, coma, astigmatism, field curvature and distortion aberration respectively.

[0044] Preferably, the multiple angle cosine terms in the Zernike polynomial expansion include cos(mθ) and sin(mθ).

[0045] Preferably, the method further comprises:

[0046] According to the obtained Seidel coefficient and the Seidel expansion form, the corresponding Seidel aberration is obtained.

[0047] Preferably, when the method is applied to the spherical aberration of a single-lens optical system, the order of the azimuthal terms is within 6 orders, that is, the total number of Zernike coefficients is 0 to 38 terms.

[0048] Example

[0049] The first step is to obtain the order of the radial term in the Zernike polynomial, the wavefront phase to be analyzed, and the type of aberration to be determined.

[0050] In this embodiment, the wavefront phase to be analyzed is a single lens optical system. Figure 2 As shown, the single-lens optical system has an aperture of 10 mm, a curvature radius of 20 mm on both the front and rear surfaces, a fixed material refractive index of 1.65, and an operating wavelength of 0.588 μm. The aberration to be determined is spherical aberration, and the radial term order n is within 6, i.e., between 0 and 6. These parameters can be specified by the user.

[0051] The second step is to use Zernike polynomials to expand the wavefront phase to be analyzed according to the order of the radial term to obtain a set of Zernike coefficients corresponding to the wavefront phase to be analyzed.

[0052] The expansion form of the Zernike polynomial is:

[0053]

[0054] Where, represents the Zernike characterization, represents the radial polynomial, ρ is the image-square normalized radial radius, is the azimuth angle, m is the order of the azimuth term, and n is the order of the radial term.

[0055] The selection of n and m follows the following rules: (1) n and m are either both odd or both even, that is, nm is an even number; (2) Always holds true.

[0056] In this embodiment, the value of n is first determined, and then the value of m is determined according to the above rules.

[0057] Take the first 38 terms of the Zernike expansion, which is shown in the second column of Table 1:

[0058]

[0059]

[0060] Table 1

[0061] In Table 1, z i That is, the Zernike coefficient corresponding to the i+1th term. The phase distribution output by the single-lens system is expanded by the Zernike polynomial shown in Table 1, and the corresponding first 38 Zernike coefficients are shown in the third column of Table 1.

[0062] The third step is to match the power series corresponding to the Seidel coefficient in the Seidel expansion according to the type of aberration to be determined.

[0063] Seidel expansion represents the wavefront by ray tracing, and the wavefront is expanded by Seidel sums. There is an obvious one-to-one correspondence between the coefficients of the expanded polynomials and the geometric aberrations and power series expansion polynomials.

[0064]

[0065] Where W(H,ρ,θ) represents the wavefront aberration, H is the normalized field height of the object, ρ is the normalized radial radius of the image, θ is the angle between the two, and S i are the corresponding Seidel coefficients, taking values ​​of 1, 2, 3, 4, 5, .... In the above formula, the coefficients of the first five terms represent spherical aberration, coma, astigmatism, field curvature, and distortion, respectively. The expanded form does not have a clear transformation relationship with the Zernike expansion described in the first step, especially with respect to the sine and cosine terms of the multiple angles.

[0066] In this embodiment, ρ 4 The coefficient corresponding to the term The spherical aberration data corresponding to the geometric aberration is widely distributed in terms 14 to 36 in the Zernike expansion described in the first step, and there is no obvious transformation relationship, especially the sine and cosine terms of the multiple angles.

[0067] The fourth step is to use the first kind of Chebyshev polynomials to expand the multiple angle cosine terms in the Zernike polynomial expansion, and then combine the coefficients of all terms of the power series containing the corresponding Seidel coefficients to obtain the conversion formula between Zernike coefficients and Seidel coefficients.

[0068] Use the first kind of Chebyshev polynomials to expand the cosine terms of multiple angles, and then perform ρ on the first 38 Zernike polynomials. 4 That is, extract all the items containing ρ 4 , obtain the sum of all coefficients of the extracted items, and get the corresponding ρ 4 The conversion formula is:

[0069]

[0070] Step 5: Substitute a set of Zernike coefficients corresponding to the wavefront phase to be analyzed into the above conversion formula to obtain the Seidel coefficients corresponding to the wavefront phase to be analyzed.

[0071] The wave optics algorithm is used to calculate the wavefront distribution of the light field when it is transmitted to a specific surface. The data obtained is generally the phase. Usually, the phase distribution of this surface is expressed as a Zernike polynomial to obtain the corresponding Zernike coefficients of different terms. According to the conversion process, the conversion formula for converting the first n Zernike coefficients into Seidel coefficients is obtained. Substituting the known Zernike coefficients into the conversion formula, the Seidel coefficients corresponding to the phase distribution can be obtained.

[0072] According to the Zernike coefficients of the first 38 terms of the single lens, we substitute them into the above conversion formula to get the corresponding The coefficient is 0.02992. It should be noted that the conversion formula is applicable to all cases with less than 38 terms. It is only necessary to set the coefficients of higher-order terms to zero. Therefore, the above conversion formula can also be used to demonstrate the influence of the number of expansion coefficients on the accuracy of the conversion result, such as Figure 3 As shown, it can be seen that for this single-lens system, the accuracy of the result obtained by taking the first 32 items is the best, and the coefficient of the 35th item can be ignored subsequently.

[0073] Step 6: Based on the obtained Seidel coefficient and the Seidel expansion form, the corresponding Seidel aberration is obtained.

[0074] According to the Seidel aberration coefficient converted from the calculated Zernike coefficient, the power series expansion coefficient corresponding to different aberrations is compared with ρ 4 For example, its coefficient corresponds to the primary spherical aberration. This spherical aberration data is converted by precise wave optics algorithm. Compared with the spherical aberration data obtained by traditional ray tracing calculation, it includes the possible influence of diffraction effect and has richer information.

[0075] The spherical aberration obtained by the method described in this application was compared with the spherical aberration data obtained by ray tracing using ZEMAX software. Compared with ZEMAX's 0.02182, the relative error in the number of convergence terms of the conversion method described in this application was about 5%, verifying the reliability of the method described in this application.

[0076] It is understandable that the detailed functional implementation of each of the above units / modules can be found in the introduction of the aforementioned method embodiment, and will not be repeated here.

[0077] It should be understood that the above-mentioned device is used to execute the method in the above-mentioned embodiment. The implementation principle and technical effect of the corresponding program module in the device are similar to those described in the above-mentioned method. The working process of the device can refer to the corresponding process in the above-mentioned method and will not be repeated here.

[0078] Based on the method in the above embodiment, Figure 4 As shown, an embodiment of the present application provides an electronic device, which may include: a processor, a communications interface, a memory, and a communication bus, wherein the processor, the communications interface, and the memory communicate with each other via the communication bus. The processor may call logic instructions in the memory to execute the method of the above embodiment.

[0079] In addition, the logical instructions in the above-mentioned memory can be implemented in the form of a software functional unit and can be stored in a computer-readable storage medium when sold or used as an independent product. Based on this understanding, the technical solution of the present application, or the part that contributes to the prior art, or the part of the technical solution, can be embodied in the form of a software product, which is stored in a storage medium and includes a number of instructions for enabling a computer device (which can be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the method described in each embodiment of the present application.

[0080] Based on the method in the above embodiment, an embodiment of the present application provides a computer-readable storage medium, which stores a computer program. When the computer program runs on a processor, the processor executes the method in the above embodiment.

[0081] Based on the method in the above embodiment, an embodiment of the present application provides a computer program product. When the computer program product runs on a processor, the processor executes the method in the above embodiment.

[0082] It is understood that the processor in the embodiments of the present application may be a central processing unit (CPU), or may be other general-purpose processors, digital signal processors (DSP), application-specific integrated circuits (ASIC), field programmable gate arrays (FPGA), or other programmable logic devices, transistor logic devices, hardware components, or any combination thereof. The general-purpose processor may be a microprocessor or any conventional processor.

[0083] The method steps in the embodiments of the present application can be implemented by hardware or by a processor executing software instructions. The software instructions can be composed of corresponding software modules, which can be stored in random access memory (RAM), flash memory, read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), registers, hard disks, mobile hard disks, CD-ROMs or any other form of storage medium known in the art. An exemplary storage medium is coupled to the processor so that the processor can read information from the storage medium and write information to the storage medium. Of course, the storage medium can also be a component of the processor. The processor and the storage medium can be located in an ASIC.

[0084] In the above embodiments, it can be implemented in whole or in part by software, hardware, firmware or any combination thereof. When implemented using software, it can be implemented in whole or in part in the form of a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, the process or function described in the embodiment of the present application is generated in whole or in part. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted via the computer-readable storage medium. The computer instructions can be transmitted from one website, computer, server or data center to another website, computer, server or data center via a wired (e.g., coaxial cable, optical fiber, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) method. The computer-readable storage medium can be any available medium that a computer can access or a data storage device such as a server or data center that includes one or more available media integrated. The available medium can be a magnetic medium (e.g., a floppy disk, a hard disk, a tape), an optical medium (e.g., a DVD), or a semiconductor medium (e.g., a solid state drive (SSD)).

[0085] It will be understood that the various numerical numbers involved in the embodiments of the present application are merely distinctions for the convenience of description and are not intended to limit the scope of the embodiments of the present application.

[0086] It is easy for those skilled in the art to understand that the above is only a preferred embodiment of the present application and is not intended to limit the present application. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present application should be included in the scope of protection of the present application.

Claims

1. A method for calculating Seidel coefficients using Zernike polynomials of wavefront phase, characterized in that: include: Obtain the order of the radial term in the Zernike polynomial, the wavefront phase to be analyzed, and the type of aberration to be determined; According to the order of radial terms, the wavefront phase to be analyzed is expanded using Zernike polynomials to obtain a set of Zernike coefficients corresponding to the wavefront phase to be analyzed; According to the type of aberration to be determined, match the power series corresponding to the Seidel coefficient in the Seidel expansion; After expanding the multiple angle cosine terms in the Zernike polynomial expansion using the first kind Chebyshev polynomials, the coefficients of all terms in the power series containing the corresponding Seidel coefficients are combined to obtain the conversion formula between Zernike coefficients and Seidel coefficients. Substituting a set of Zernike coefficients corresponding to the phase of the wavefront to be analyzed into the above conversion formula, the Seidel coefficients corresponding to the phase of the wavefront to be analyzed are obtained.

2. The method according to claim 1, wherein The Seidel expansion is as follows: Where W(H,ρ,θ) represents the wavefront aberration, H is the normalized field height of the object, ρ is the normalized radial radius of the image, θ is the angle between the two, and S i is the corresponding Seidel coefficient, which takes values ​​of 1, 2, 3, 4, 5, .... The coefficients of the first 5 items represent spherical aberration, coma, astigmatism, field curvature and distortion aberration respectively.

3. The method according to claim 2, wherein The multiple angle cosine terms in the Zernike polynomial expansion include cos(mθ) and sin(mθ).

4. The method according to any one of claims 1 to 3, wherein The method further includes: According to the obtained Seidel coefficient and the Seidel expansion form, the corresponding Seidel aberration is obtained.

5. The method according to any one of claims 1 to 3, characterized in that When this method is applied to the spherical aberration of a single-lens optical system, the order of the azimuthal terms is within 6, that is, the total number of Zernike coefficients is 0 to 38.

6. A system for calculating Seidel coefficients using Zernike polynomials of wavefront phase, characterized in that: comprising at least one processor and at least one memory; The at least one memory is for storing computer instructions; The at least one processor is configured to execute at least part of the computer instructions to implement the method according to any one of claims 1 to 5.

7. A computer-readable storage medium, characterized in that The storage medium stores computer instructions. When a computer reads the computer instructions in the storage medium, the computer executes the method according to any one of claims 1 to 5.

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