Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2024; 8(2): 151-155
Published online April 25, 2024 https://doi.org/10.3807/COPP.2024.8.2.151
Copyright © Optical Society of Korea.
Yohan Kim^{1} , Theo Nam Sohn^{2}, Cheong Soo Seo^{1}, Jin Young Sohn^{1}
Corresponding author: ^{*}yhkim2@sdoptics.com, ORCID 0009-0005-0239-7202
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Optical systems often suffer from optical aberrations caused by imperfect hardware, which places significant constraints on their utility and performance. To reduce these undesirable effects, a comprehensive understanding of the aberrations inherent to optical systems is needed. This article presents an effective method for aberration detection using Zernike polynomials. The process involves scanning the object plane to identify the optimal focus and subsequently fitting the acquired focus data to Zernike polynomials. This fitting procedure facilitates the analysis of various aberrations in the optical system.
Keywords: Focus measure, Optical aberration, Wavefront error, Zernike polynomials
OCIS codes: (010.7350) Wave-front sensing; (080.1010) Aberrations (global); (100.2550) Focal-plane-array image processors; (110.0110) Imaging systems; (120.4640) Optical instruments
The performance of optical systems can be compromised due to errors in design, manufacturing, and measurement, leading to deviations from their intended results. While design and manufacturing errors can be addressed with optical simulations and tolerance analysis, predicting measurement errors proves to be a more challenging task. Consequently, it is difficult to discern the exact causes of the final performance of an optical system due to the interplay of these errors.
To address this issue and enhance the performance of optical systems, it is crucial to grasp the factors affecting the end outcomes. Specifically, optical aberrations [1, 2] are the most common. This necessitates the measurement of these aberrations in optical systems. The conventional method of assessing aberrations entails gauging the contribution of each aberration by evaluating the wavefront [3] and resolving the results with Zernike polynomials [4, 5]. Currently, the Shack-Hartmann sensor [6] is the most widely employed wavefront sensor for this purpose. It is a challenging task to integrate these sensors into existing optical systems without changing the optical design.
In this paper, we introduce a straightforward and efficient alternative. This method involves using the data obtained from an optical system equipped with a micromirror array lens system (MALS^{TM}) and analyzing it with Zernike polynomials. This approach acquires images by adjusting the focal length of the object plane (with concurrent adjustments to the image plane’s position) and allows for a straightforward analysis of aberrations that affect the overall performance of the system.
The images used for optical system analysis were acquired with WiseScope, a MALS^{TM}-based standalone 3D digital microscope (SD Optics Inc., Seoul, Korea). MALS^{TM} is a micro-electro-mechanical system (MEMS)-based reflective type variable focusing lens designed to rapidly and reliably change its focus. This innovative technology allows for swift adjustments to the focal length.
The optical layout of WiseScope is shown in Fig. 1. WiseScope is equipped with a coaxial illumination system and an optical imaging system. The optical imaging system gives a detailed image of the sample. Moreover, as previously noted, MALS^{TM} enables the adjustment of the optical system’s focal length. This feature allows for scanning of the object plane to provide three-dimensional information about the sample.
The chosen target for aberration analysis was equipped with circular patterns (see Fig. 2) for focus measurement across the entire field. This design choice enables the assessment of focus variations throughout the field for the identification and quantification of specific aberrations present in the optical system.
The target images were obtained using a monochrome industrial camera with a resolution of 2,048 pixels by 2,048 pixels by adjusting the object plane at 7.5 µm intervals in the range from −375 µm to 375 µm.
The sum-modified Laplacian (SML) method, introduced by Nayar [7], was used to determine the best focus [8] in the captured images. This method is based on a modified Laplacian (ML), which avoids the cancellation of the second derivatives along the horizontal and vertical directions. The expression for the discrete approximation of the ML is defined in Eq. (1):
To accommodate potential differences in the sizes of texture elements, Nayar introduced a flexible pixel spacing (step) while computing the ML.
The focus measure at a given point (i, j) is determined by summing the values of the ML in a localized window centered on the interest point. Only Laplacian values that exceed a defined threshold are included in this summation [Eq. (2)].
where the parameter N determines the kernel size used to compute the focus measure.
The SML determines the clarity and contrast numerically for each area among different focused images. Selecting the highest values helps identify the best focus for the corresponding area.
When the best focus is identified for all circles using the SLM method, the aberrations of the optical system can be found using Zernike polynomials.
Zernike polynomials are a set of mathematical functions used to describe aberrations in a plane, particularly in the context of optics and wavefront analysis. Zernike polynomials are defined in Eq. (3):
where n is a non-negative integer, m is an integer, and n − m ≥ 0 is even. The radial polynomials are defined as:
Wavefront aberrations refer to deviations from a perfect wavefront, which is a theoretical ideal wavefront of light. These aberrations can lead to image distortions and reduce the quality of optical systems. Zernike polynomials provide a systematic way to analyze and quantify these aberrations.
Zernike polynomials are typically used to represent the wavefront error as a sum of different polynomials [9], each corresponding to a specific type of aberration. Each polynomial is associated with a specific Zernike mode, which represents a particular type of aberration. These modes include defocus, astigmatism, coma, spherical aberration, etc.
Zernike polynomials have several useful properties, such as orthogonality over a circular aperture [10, 11], which makes them well-suited for representing wavefront aberrations that are rotationally symmetric.
Using the methods above, an algorithm was developed in MATLAB to determine the values of aberrations in an optical system. The algorithm diagram is depicted in Fig. 3. This algorithm scans a stack of images stored in a designated folder. For each circle in the images, it precisely determines the best focus using the SML method, then plots the surface based on the best focus values of each circle. The resulting surface data is then fitted with Zernike polynomials for a detailed analysis of the aberrations present in the optical system.
Figure 4 displays the best focus of circles as determined by the algorithm. The presence of aberrations causes the circles with the best focus to be at different heights.
As the variation in heights on the object plane can be regarded as a wavefront error, using this data allows for the approximation of the Zernike curve and the extraction of aberration values.
Zernike curve fitting is done up to the fifth order. The values of the Zernike coefficients are depicted as a histogram in Fig. 5. The dominant terms are x tilt and y tilt, with coefficients of −3.2059 and 3.8205, respectively. Lower order terms such as piston, x tilt, and y tilt were excluded from the analysis because they were influenced by the physical orientation of the object, which should be corrected by the proper placement of the setup. Analysis of the measurement results shows that the predominant aberration in this optical system manifests as field curvature, with a coefficient of −1.4471. Field curvature can significantly affect image quality and lead to blurring toward the edges, challenges in achieving the best focus, distortion of object shapes, and reduced resolution at the edges. Additional optical elements or improved lens designs can be used to overcome these problems.
All coefficients of Zernike polynomials up to the fifth order are given in Table 1. These coefficients provide valuable information about the specific aberrations present in the optical system. Careful examination of the coefficients and their implications can help inform future optimization efforts to ensure that the optical system operates at peak performance.
TABLE 1 Coefficients of Zernike polynomials
Z ^{m}n | Polar Form | Cartesian Form | Coefficient | Classical Name |
---|---|---|---|---|
Z ^{0}0 | 1 | 1 | −0.1940 | Piston |
Z −^{1}1 | r sin θ | x | −3.2059 | y Tilt |
Z ^{1}1 | r cos θ | y | 3.8205 | x Tilt |
Z −^{2}2 | r^{2} sin 2θ | 2xy | 0.5558 | 45° Primary Astigmatism |
Z ^{0}2 | 2r^{2} − 1 | −1 + 2x^{2} + 2y^{2} | −1.4471 | Defocus/Field Curvature |
Z ^{2}2 | r^{2} cos 2θ | −x^{2} + y^{2} | −0.1985 | 0° Primary Astigmatism |
Z −^{3}3 | r^{3} sin 3θ | −x^{3} + 3xy^{2} | 0.4843 | - |
Z −^{1}3 | (3r^{3} − 2r) sin θ | −2x + 3x^{3} + 3xy^{2} | −0.2328 | Primary y Coma |
Z ^{1}3 | (3r^{3} − 2r) cos θ | −2y + 3y^{3} + 3x^{2}y | 0.3233 | Primary x Coma |
Z ^{2}3 | r^{3} cos 3θ | y^{3} − 3x^{2}y | −0.2048 | - |
Z −^{4}4 | r^{4} sin 4θ | −4x^{3}y + 4xy^{3} | −0.0732 | - |
Z −^{2}4 | (4r^{4} − 3r2) sin 2θ | −6xy + 8x^{3}y + 8xy^{3} | −0.9126 | 45° Secondary Astigmatism |
Z ^{0}4 | 6r^{4} − 6r2 + 1 | 1 − 6x^{2} − 6y^{2} + 6x^{4} + 12x^{2}y^{2} + 6y^{4} | −0.3310 | Primary Spherical Aberration |
Z ^{2}4 | (4r^{4} − 3r2) cos 2θ | 3x^{2} − 3y^{2} − 4x^{4} + 4y^{4} | 0.0024 | 0° Secondary Astigmatism |
Z ^{4}4 | r^{4} cos 4θ | x^{4} − 6x^{2}y^{2} + y^{4} | 0.0731 | - |
Z −^{5}5 | r^{5} sin 5θ | x^{5} − 10x^{3}y^{2} + 5xy^{4} | 0.3346 | - |
Z −^{3}5 | (5r^{5} − 4r3) sin 3θ | 4x^{3} − 12xy^{2} − 5x^{5} + 10x^{3}y^{2} + 15xy^{4} | 0.6260 | - |
Z −^{1}5 | (10r^{5} − 12r3 + 3r) sin θ | 3x − 12x^{3} − 12xy^{2} + 10x^{5} + 20x^{3}y^{2} + 10xy^{4} | 0.1548 | Secondary y Coma |
Z ^{1}5 | (10r^{5} − 12r3 + 3r) cos θ | 3y − 12y^{3} − 12x^{2}y + 10y^{5} + 20x^{2}y^{3} + 10x^{4}y | −0.0020 | Secondary x Coma |
Z ^{3}5 | (5r^{5} − 4r3) cos 3θ | − 4y^{3} + 12x^{2}y + 5y^{5} − 10x^{2}y^{3} − 15x^{4}y | 0.1972 | - |
Z ^{5}5 | r^{5} cos 5θ | y^{5} − 10x^{2}y^{3} + 5x^{4}y | 0.0106 | - |
The results of Zernike polynomial fitting have enabled the identification and quantification of aberrations and provided valuable insights for the improvement of the optical system. The integration of WiseScope with MALS^{TM} has demonstrated the feasibility of efficient tests and measurements for optical systems. MALS^{TM} employs a rapid object (image) plane scanning method and has proven to be a valuable tool in this context.
Measurement capabilities can be expanded with WiseTopo (SD Optics), an optical system designed to convert a conventional 2D microscope into a 3D-capable system. WiseTopo relays the image planes of the optical system for in-depth analysis of the image plane under consideration.
This paper has introduced a novel and expedited method for measuring aberrations with object (image) plane scanning using MALS^{TM}. The quantification of aberrations using Zernike polynomials has been successfully demonstrated. The versatility of this method is also highlighted by its adaptability to any imaging system with the use of a relaying optics configuration to enable the measurement of aberrations in the image plane of the specific optical system of interest.
The authors received no financial support for the research, authorship, and publication of this article.
The authors declare no conflict of interest.
Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.
Curr. Opt. Photon. 2024; 8(2): 151-155
Published online April 25, 2024 https://doi.org/10.3807/COPP.2024.8.2.151
Copyright © Optical Society of Korea.
Yohan Kim^{1} , Theo Nam Sohn^{2}, Cheong Soo Seo^{1}, Jin Young Sohn^{1}
^{1}SD Optics, Inc., Seoul 06752, Korea
^{2}Stereo Display, Inc., Anaheim, CA 92801, USA
Correspondence to:^{*}yhkim2@sdoptics.com, ORCID 0009-0005-0239-7202
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Optical systems often suffer from optical aberrations caused by imperfect hardware, which places significant constraints on their utility and performance. To reduce these undesirable effects, a comprehensive understanding of the aberrations inherent to optical systems is needed. This article presents an effective method for aberration detection using Zernike polynomials. The process involves scanning the object plane to identify the optimal focus and subsequently fitting the acquired focus data to Zernike polynomials. This fitting procedure facilitates the analysis of various aberrations in the optical system.
Keywords: Focus measure, Optical aberration, Wavefront error, Zernike polynomials
The performance of optical systems can be compromised due to errors in design, manufacturing, and measurement, leading to deviations from their intended results. While design and manufacturing errors can be addressed with optical simulations and tolerance analysis, predicting measurement errors proves to be a more challenging task. Consequently, it is difficult to discern the exact causes of the final performance of an optical system due to the interplay of these errors.
To address this issue and enhance the performance of optical systems, it is crucial to grasp the factors affecting the end outcomes. Specifically, optical aberrations [1, 2] are the most common. This necessitates the measurement of these aberrations in optical systems. The conventional method of assessing aberrations entails gauging the contribution of each aberration by evaluating the wavefront [3] and resolving the results with Zernike polynomials [4, 5]. Currently, the Shack-Hartmann sensor [6] is the most widely employed wavefront sensor for this purpose. It is a challenging task to integrate these sensors into existing optical systems without changing the optical design.
In this paper, we introduce a straightforward and efficient alternative. This method involves using the data obtained from an optical system equipped with a micromirror array lens system (MALS^{TM}) and analyzing it with Zernike polynomials. This approach acquires images by adjusting the focal length of the object plane (with concurrent adjustments to the image plane’s position) and allows for a straightforward analysis of aberrations that affect the overall performance of the system.
The images used for optical system analysis were acquired with WiseScope, a MALS^{TM}-based standalone 3D digital microscope (SD Optics Inc., Seoul, Korea). MALS^{TM} is a micro-electro-mechanical system (MEMS)-based reflective type variable focusing lens designed to rapidly and reliably change its focus. This innovative technology allows for swift adjustments to the focal length.
The optical layout of WiseScope is shown in Fig. 1. WiseScope is equipped with a coaxial illumination system and an optical imaging system. The optical imaging system gives a detailed image of the sample. Moreover, as previously noted, MALS^{TM} enables the adjustment of the optical system’s focal length. This feature allows for scanning of the object plane to provide three-dimensional information about the sample.
The chosen target for aberration analysis was equipped with circular patterns (see Fig. 2) for focus measurement across the entire field. This design choice enables the assessment of focus variations throughout the field for the identification and quantification of specific aberrations present in the optical system.
The target images were obtained using a monochrome industrial camera with a resolution of 2,048 pixels by 2,048 pixels by adjusting the object plane at 7.5 µm intervals in the range from −375 µm to 375 µm.
The sum-modified Laplacian (SML) method, introduced by Nayar [7], was used to determine the best focus [8] in the captured images. This method is based on a modified Laplacian (ML), which avoids the cancellation of the second derivatives along the horizontal and vertical directions. The expression for the discrete approximation of the ML is defined in Eq. (1):
To accommodate potential differences in the sizes of texture elements, Nayar introduced a flexible pixel spacing (step) while computing the ML.
The focus measure at a given point (i, j) is determined by summing the values of the ML in a localized window centered on the interest point. Only Laplacian values that exceed a defined threshold are included in this summation [Eq. (2)].
where the parameter N determines the kernel size used to compute the focus measure.
The SML determines the clarity and contrast numerically for each area among different focused images. Selecting the highest values helps identify the best focus for the corresponding area.
When the best focus is identified for all circles using the SLM method, the aberrations of the optical system can be found using Zernike polynomials.
Zernike polynomials are a set of mathematical functions used to describe aberrations in a plane, particularly in the context of optics and wavefront analysis. Zernike polynomials are defined in Eq. (3):
where n is a non-negative integer, m is an integer, and n − m ≥ 0 is even. The radial polynomials are defined as:
Wavefront aberrations refer to deviations from a perfect wavefront, which is a theoretical ideal wavefront of light. These aberrations can lead to image distortions and reduce the quality of optical systems. Zernike polynomials provide a systematic way to analyze and quantify these aberrations.
Zernike polynomials are typically used to represent the wavefront error as a sum of different polynomials [9], each corresponding to a specific type of aberration. Each polynomial is associated with a specific Zernike mode, which represents a particular type of aberration. These modes include defocus, astigmatism, coma, spherical aberration, etc.
Zernike polynomials have several useful properties, such as orthogonality over a circular aperture [10, 11], which makes them well-suited for representing wavefront aberrations that are rotationally symmetric.
Using the methods above, an algorithm was developed in MATLAB to determine the values of aberrations in an optical system. The algorithm diagram is depicted in Fig. 3. This algorithm scans a stack of images stored in a designated folder. For each circle in the images, it precisely determines the best focus using the SML method, then plots the surface based on the best focus values of each circle. The resulting surface data is then fitted with Zernike polynomials for a detailed analysis of the aberrations present in the optical system.
Figure 4 displays the best focus of circles as determined by the algorithm. The presence of aberrations causes the circles with the best focus to be at different heights.
As the variation in heights on the object plane can be regarded as a wavefront error, using this data allows for the approximation of the Zernike curve and the extraction of aberration values.
Zernike curve fitting is done up to the fifth order. The values of the Zernike coefficients are depicted as a histogram in Fig. 5. The dominant terms are x tilt and y tilt, with coefficients of −3.2059 and 3.8205, respectively. Lower order terms such as piston, x tilt, and y tilt were excluded from the analysis because they were influenced by the physical orientation of the object, which should be corrected by the proper placement of the setup. Analysis of the measurement results shows that the predominant aberration in this optical system manifests as field curvature, with a coefficient of −1.4471. Field curvature can significantly affect image quality and lead to blurring toward the edges, challenges in achieving the best focus, distortion of object shapes, and reduced resolution at the edges. Additional optical elements or improved lens designs can be used to overcome these problems.
All coefficients of Zernike polynomials up to the fifth order are given in Table 1. These coefficients provide valuable information about the specific aberrations present in the optical system. Careful examination of the coefficients and their implications can help inform future optimization efforts to ensure that the optical system operates at peak performance.
TABLE 1. Coefficients of Zernike polynomials.
Z ^{m}n | Polar Form | Cartesian Form | Coefficient | Classical Name |
---|---|---|---|---|
Z ^{0}0 | 1 | 1 | −0.1940 | Piston |
Z −^{1}1 | r sin θ | x | −3.2059 | y Tilt |
Z ^{1}1 | r cos θ | y | 3.8205 | x Tilt |
Z −^{2}2 | r^{2} sin 2θ | 2xy | 0.5558 | 45° Primary Astigmatism |
Z ^{0}2 | 2r^{2} − 1 | −1 + 2x^{2} + 2y^{2} | −1.4471 | Defocus/Field Curvature |
Z ^{2}2 | r^{2} cos 2θ | −x^{2} + y^{2} | −0.1985 | 0° Primary Astigmatism |
Z −^{3}3 | r^{3} sin 3θ | −x^{3} + 3xy^{2} | 0.4843 | - |
Z −^{1}3 | (3r^{3} − 2r) sin θ | −2x + 3x^{3} + 3xy^{2} | −0.2328 | Primary y Coma |
Z ^{1}3 | (3r^{3} − 2r) cos θ | −2y + 3y^{3} + 3x^{2}y | 0.3233 | Primary x Coma |
Z ^{2}3 | r^{3} cos 3θ | y^{3} − 3x^{2}y | −0.2048 | - |
Z −^{4}4 | r^{4} sin 4θ | −4x^{3}y + 4xy^{3} | −0.0732 | - |
Z −^{2}4 | (4r^{4} − 3r2) sin 2θ | −6xy + 8x^{3}y + 8xy^{3} | −0.9126 | 45° Secondary Astigmatism |
Z ^{0}4 | 6r^{4} − 6r2 + 1 | 1 − 6x^{2} − 6y^{2} + 6x^{4} + 12x^{2}y^{2} + 6y^{4} | −0.3310 | Primary Spherical Aberration |
Z ^{2}4 | (4r^{4} − 3r2) cos 2θ | 3x^{2} − 3y^{2} − 4x^{4} + 4y^{4} | 0.0024 | 0° Secondary Astigmatism |
Z ^{4}4 | r^{4} cos 4θ | x^{4} − 6x^{2}y^{2} + y^{4} | 0.0731 | - |
Z −^{5}5 | r^{5} sin 5θ | x^{5} − 10x^{3}y^{2} + 5xy^{4} | 0.3346 | - |
Z −^{3}5 | (5r^{5} − 4r3) sin 3θ | 4x^{3} − 12xy^{2} − 5x^{5} + 10x^{3}y^{2} + 15xy^{4} | 0.6260 | - |
Z −^{1}5 | (10r^{5} − 12r3 + 3r) sin θ | 3x − 12x^{3} − 12xy^{2} + 10x^{5} + 20x^{3}y^{2} + 10xy^{4} | 0.1548 | Secondary y Coma |
Z ^{1}5 | (10r^{5} − 12r3 + 3r) cos θ | 3y − 12y^{3} − 12x^{2}y + 10y^{5} + 20x^{2}y^{3} + 10x^{4}y | −0.0020 | Secondary x Coma |
Z ^{3}5 | (5r^{5} − 4r3) cos 3θ | − 4y^{3} + 12x^{2}y + 5y^{5} − 10x^{2}y^{3} − 15x^{4}y | 0.1972 | - |
Z ^{5}5 | r^{5} cos 5θ | y^{5} − 10x^{2}y^{3} + 5x^{4}y | 0.0106 | - |
The results of Zernike polynomial fitting have enabled the identification and quantification of aberrations and provided valuable insights for the improvement of the optical system. The integration of WiseScope with MALS^{TM} has demonstrated the feasibility of efficient tests and measurements for optical systems. MALS^{TM} employs a rapid object (image) plane scanning method and has proven to be a valuable tool in this context.
Measurement capabilities can be expanded with WiseTopo (SD Optics), an optical system designed to convert a conventional 2D microscope into a 3D-capable system. WiseTopo relays the image planes of the optical system for in-depth analysis of the image plane under consideration.
This paper has introduced a novel and expedited method for measuring aberrations with object (image) plane scanning using MALS^{TM}. The quantification of aberrations using Zernike polynomials has been successfully demonstrated. The versatility of this method is also highlighted by its adaptability to any imaging system with the use of a relaying optics configuration to enable the measurement of aberrations in the image plane of the specific optical system of interest.
The authors received no financial support for the research, authorship, and publication of this article.
The authors declare no conflict of interest.
Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.
TABLE 1 Coefficients of Zernike polynomials
Z ^{m}n | Polar Form | Cartesian Form | Coefficient | Classical Name |
---|---|---|---|---|
Z ^{0}0 | 1 | 1 | −0.1940 | Piston |
Z −^{1}1 | r sin θ | x | −3.2059 | y Tilt |
Z ^{1}1 | r cos θ | y | 3.8205 | x Tilt |
Z −^{2}2 | r^{2} sin 2θ | 2xy | 0.5558 | 45° Primary Astigmatism |
Z ^{0}2 | 2r^{2} − 1 | −1 + 2x^{2} + 2y^{2} | −1.4471 | Defocus/Field Curvature |
Z ^{2}2 | r^{2} cos 2θ | −x^{2} + y^{2} | −0.1985 | 0° Primary Astigmatism |
Z −^{3}3 | r^{3} sin 3θ | −x^{3} + 3xy^{2} | 0.4843 | - |
Z −^{1}3 | (3r^{3} − 2r) sin θ | −2x + 3x^{3} + 3xy^{2} | −0.2328 | Primary y Coma |
Z ^{1}3 | (3r^{3} − 2r) cos θ | −2y + 3y^{3} + 3x^{2}y | 0.3233 | Primary x Coma |
Z ^{2}3 | r^{3} cos 3θ | y^{3} − 3x^{2}y | −0.2048 | - |
Z −^{4}4 | r^{4} sin 4θ | −4x^{3}y + 4xy^{3} | −0.0732 | - |
Z −^{2}4 | (4r^{4} − 3r2) sin 2θ | −6xy + 8x^{3}y + 8xy^{3} | −0.9126 | 45° Secondary Astigmatism |
Z ^{0}4 | 6r^{4} − 6r2 + 1 | 1 − 6x^{2} − 6y^{2} + 6x^{4} + 12x^{2}y^{2} + 6y^{4} | −0.3310 | Primary Spherical Aberration |
Z ^{2}4 | (4r^{4} − 3r2) cos 2θ | 3x^{2} − 3y^{2} − 4x^{4} + 4y^{4} | 0.0024 | 0° Secondary Astigmatism |
Z ^{4}4 | r^{4} cos 4θ | x^{4} − 6x^{2}y^{2} + y^{4} | 0.0731 | - |
Z −^{5}5 | r^{5} sin 5θ | x^{5} − 10x^{3}y^{2} + 5xy^{4} | 0.3346 | - |
Z −^{3}5 | (5r^{5} − 4r3) sin 3θ | 4x^{3} − 12xy^{2} − 5x^{5} + 10x^{3}y^{2} + 15xy^{4} | 0.6260 | - |
Z −^{1}5 | (10r^{5} − 12r3 + 3r) sin θ | 3x − 12x^{3} − 12xy^{2} + 10x^{5} + 20x^{3}y^{2} + 10xy^{4} | 0.1548 | Secondary y Coma |
Z ^{1}5 | (10r^{5} − 12r3 + 3r) cos θ | 3y − 12y^{3} − 12x^{2}y + 10y^{5} + 20x^{2}y^{3} + 10x^{4}y | −0.0020 | Secondary x Coma |
Z ^{3}5 | (5r^{5} − 4r3) cos 3θ | − 4y^{3} + 12x^{2}y + 5y^{5} − 10x^{2}y^{3} − 15x^{4}y | 0.1972 | - |
Z ^{5}5 | r^{5} cos 5θ | y^{5} − 10x^{2}y^{3} + 5x^{4}y | 0.0106 | - |