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Optimization of optical system for reducing tolerance sensitivity based on the high-order nodal aberration theory

Zhang Jipeng, Shi Guangwei, Huang Yuqi, Zhao Shangnan, Liu Mingxin, and Zhang Xin

Open Access Open Access

Abstract

This paper proposes an optimization method aimed at addressing the challenge of balancing system imaging performance and tolerance performance. The method is based on the high-order nodal aberration theory. The calculation method for high-order aberration coefficients is derived, along with the change in aberrations after introducing tolerance perturbation. The evaluation system includes both the imaging performance of the system and its tolerance factors. A comprehensive merit function is created for optimization. Three aspherical system examples were designed for verification, and the optimization results of this method show significantly lower tolerance sensitivity compared to the general optimization results and the SAB function optimization results of CODE V.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

A design with excellent tolerance performance can significantly reduce the difficulty of installation, improve machining yields, and consequently lower costs. However, the optimization process is often carried out independently of the evaluation of tolerance performance. A system with high-quality imaging may require redesign if the tolerance performance is poor, leading to significant time and effort consumption. Many methods have been tried to optimize imaging and tolerance performance. Jeffs described methods to improve completion performance by adding new terms related to tolerance sensitivity in the merit function, such as incident angle and aberration coefficients [1]. Rogers [2,3] and Isshiki [4] used global optimization methods to identify the structure with the lowest tolerance sensitivity through extensive calculations and analysis. The analysis method by Meng et al [5,6]. for off-axis reflection systems can reduce tolerance sensitivity by controlling surface aberration contributions. Leticia [7] proposed a stepwise reduction of aspherical terms and a multi-configuration method to desensitize systems with high-order aspherics. CODE V software uses the SAB function with a wavefront differentiation algorithm, while Zemax offers two methods: high yield optimization for controlling incident angle and TOLR operand. Some of these methods are time-consuming or lack a theoretical basis.

The nodal aberration theory (NAT), proposed by Shack and refined by Buchroeder [8], Thompson [9] et al, was applied early on to evaluate the misalignment performance of optical systems and is now often used for alignment [10–12]. By introducing the notion of an optical axis ray (OAR), a local coordinate system (LCS) paraxial ray tracing is implemented, and the off-axis aberration field shift vector $\vec{\sigma }$. This enables the calculation of aberrations in optical systems that are out of alignment.

Gu et al [13]. developed a model for evaluating tolerance sensitivity using nodal aberration theory, which successfully reduced tolerance sensitivity. The model was compared with SAB of CODE V and TOLR of Zemax, demonstrating its effectiveness. Nevertheless, its model relies on third-order aberration, which restricts its applicability. The demand for imaging has been on the rise, leading to a growing complexity of optical systems. For instance, the three-mirror anastigmatic (TMA) structure employed in the James Webb Space Telescope (JWST) corrects third-order coma and astigmatism, resulting in mainly higher-order residual astigmatism. The third-order aberration theory is unable to provide an accurate assessment of the effects of aspherics, necessitating the introduction of higher-order aberration theory.

This paper is based on Sasian's [14,15] high-order aberration theory and Thompson's [16] discourse on nodal aberration theory. In Section 2, the coefficients for high-order aberrations are converted into vectors, and the impacts of intrinsic aberrations, extrinsic aberrations, and aspherical aberrations are individually examined. The variation of aberration coefficients following the introduction of tolerance perturbation is determined. Section 3 delineates the decenter and tilt tolerance of the lens, formulates a new comprehensive evaluation index (merit function), and accomplishes precise assessment of the performance of misaligned optical systems, thereby improving the optimization of low tolerance sensitivity optical systems. Section 4 validated this method with three examples and compared it with the design results of general optimization and SAB optimization of CODE V.

2. High-order nodal aberration theory

The aberration of an optical system can be expressed using the following expansion:

$$\begin{aligned} W &= {W_{\textrm{040}}}{\textrm{(}\vec{\rho } \cdot \vec{\rho }\textrm{)}^\textrm{2}} + {W_{\textrm{131}}}\textrm{(}\vec{H} \cdot \vec{\rho }\textrm{)(}\vec{\rho } \cdot \vec{\rho }\textrm{)} + {W_{\textrm{220}}}\textrm{(}\vec{H} \cdot \vec{H}\textrm{)(}\vec{\rho } \cdot \vec{\rho }\textrm{)} + {W_{\textrm{222}}}{\textrm{(}\vec{H} \cdot \vec{\rho }\textrm{)}^\textrm{2}}\\ &+ {W_{\textrm{311}}}\textrm{(}\vec{H} \cdot \vec{H}\textrm{)(}\vec{H} \cdot \vec{\rho }\textrm{)} + {W_{\textrm{060}}}{\textrm{(}\vec{\rho } \cdot \vec{\rho }\textrm{)}^\textrm{3}} + {W_{\textrm{151}}}\textrm{(}\vec{H} \cdot \vec{\rho }\textrm{)(}\vec{\rho } \cdot \vec{\rho }{\textrm{)}^\textrm{2}} + {W_{\textrm{240}}}\textrm{(}\vec{H} \cdot \vec{H}\textrm{)(}\vec{\rho } \cdot \vec{\rho }{\textrm{)}^\textrm{2}}\\ &+ {W_{\textrm{242}}}{\textrm{(}\vec{H} \cdot \vec{\rho }\textrm{)}^\textrm{2}}\textrm{(}\vec{\rho } \cdot \vec{\rho }\textrm{)} + {W_{\textrm{331}}}\textrm{(}\vec{H} \cdot \vec{H}\textrm{)(}\vec{H} \cdot \vec{\rho }\textrm{)(}\vec{\rho } \cdot \vec{\rho }\textrm{)} + {W_{\textrm{333}}}{\textrm{(}\vec{H} \cdot \vec{\rho }\textrm{)}^\textrm{3}}\\ &+ {W_{\textrm{420}}}{\textrm{(}\vec{H} \cdot \vec{H}\textrm{)}^\textrm{2}}\textrm{(}\vec{\rho } \cdot \vec{\rho }\textrm{)} + {W_{\textrm{422}}}\textrm{(}\vec{H} \cdot \vec{H}\textrm{)(}\vec{H} \cdot \vec{\rho }{\textrm{)}^\textrm{2}} + {W_{\textrm{511}}}{\textrm{(}\vec{H} \cdot \vec{H}\textrm{)}^\textrm{2}}\textrm{(}\vec{H} \cdot \vec{\rho }\textrm{)} + \ldots \end{aligned}$$
among them, $\vec{\rho }$ is the normalized pupil vector, $\vec{H}$ is the normalized field vector, W is the aberration of the optical system, ${W_{040}}$, ${W_{131}}$, ${W_{220}}$, ${W_{222}}$ and ${W_{311}}$ are the third-order aberration (fourth-order wave aberration) coefficients, while ${W_{060}}$, ${W_{151}}$, ${W_{240}}$, ${W_{242}}$, ${W_{331}}$, ${W_{333}}$, ${W_{420}}$, ${W_{422}}$ and ${W_{511}}$ represent the fifth-order aberration (sixth-order wave aberration) coefficients. Generally, aberrations of fifth-order and higher are referred to as high-order aberrations.

High-order aberrations can be categorized into two parts: intrinsic aberrations and extrinsic aberrations. Assuming the incident beam is ideal and free from aberrations, the aberrations solely generated by the optical surface are termed intrinsic aberrations. The aberrations resulting from the transfer of aberration from one system to another are termed extrinsic aberrations. This represents the most conspicuous difference between high-order aberration theory and third-order aberration theory.

$${W^6} = W_I^6 + W_E^6$$

${W^6}$ represents the contribution of fifth-order aberrations, $W_I^6$ represents the intrinsic part of fifth-order aberrations, and $W_E^6$ represents the extrinsic part of fifth-order aberrations. The total aberration contribution is the sum of these two parts.

Thompson provided a detailed introduction to his insights into nodal aberration theory [17–19], demonstrating the nodal characteristics of nodal aberration and the following relationship between aberration coefficients:

$$\begin{aligned} {W_{220M}} &= {W_{220}} + \frac{1}{2}{W_{222}}\\ {W_{240M}} &= {W_{240}} + \frac{1}{2}{W_{242}}\\ {W_{331M}} &= {W_{331}} + \frac{3}{4}{W_{333}}\\ {W_{420M}} &= {W_{420}} + \frac{1}{2}{W_{422}} \end{aligned}$$
where the subscript M represents the middle image plane at the minimum wavefront RMS. In addition, ${W_{040}}$ and ${W_{060}}$ are spherical terms that remain constant despite misalignment disturbances, while ${W_{311}}$ and ${W_{511}}$ are distortion terms that sole shift the position of the reference wavefront in the image without impacting the image quality. Therefore, after introducing a tolerance disturbance, only six terms ${W_{131}}$, ${W_{222}}$, ${W_{151}}$, ${W_{242}}$, ${W_{333}}$, and ${W_{422}}$ need to be considered. The two third-order terms ${W_{131}}$ and ${W_{222}}$ have been fully discussed, and this paper focuses on the remaining four fifth-order terms. The study on vectorization of intrinsic aberrations is referenced from Thompson's work, while the study on vectorization of extrinsic and aspherical aberrations represents a new derivation.

2.1 Intrinsic aberrations

As mentioned earlier, the intrinsic aberration is solely related to the properties of the optical surface itself, and therefore follows the vector expansion form derived by Thompson. After introducing tolerance perturbation, the field of view offset factor $\vec{\sigma }$ was incorporated to obtain various aberration changes. Taking ${W_{151}}$ as an example:

$$\begin{aligned} {W_I} &= {W_{151}}_I[(\vec{H} - \vec{\sigma }) \cdot \vec{\rho }]{(\vec{\rho } \cdot \vec{\rho })^2}\\ &= {W_{151}}_I(\vec{H} \cdot \vec{\rho }\textrm{)}{(\vec{\rho } \cdot \vec{\rho })^2} - {W_{151}}_I(\vec{\sigma } \cdot \vec{\rho }\textrm{)}{(\vec{\rho } \cdot \vec{\rho })^2} \end{aligned}$$
thus obtaining the change in ${W_{151I}}$
$$\Delta {W_{151}}_I ={-} {W_{151}}_I(\vec{\sigma } \cdot \vec{\rho }\textrm{)}{(\vec{\rho } \cdot \vec{\rho })^2}$$

The derivation of the changes in the other three fifth-order terms can be found in Appendix A. Table 1 summarizes the changes in the four fifth-order terms.

Tables Icon

Table 1. Changes in Fifth Order Intrinsic Aberrations after Introducing Tolerance Perturbations

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2.2 Extrinsic aberrations

The extrinsic part of the fifth-order aberrations is more complex, as it is influenced not only by the properties of the surface itself but also by the aberrations transfer from the preceding system. Examine the analytical models of high-order extrinsic aberration theory and nodal aberration theory, as depicted in Fig. 1 and Fig. 2. In Fig. 1, the exit pupil of the preceding system A coincides with the entrance pupil of the subsequent system B, denoted by ${\bar{y}_A}$ and ${\bar{y}_B}$ as the heights of the two system plane principal rays, and ${y_{EA}}$ and ${y_{EB}}$ as the exit pupil sizes of the two systems. The theory of high-order extrinsic aberrations involves normalizing the exit pupil of the latter system. Due to the existence of pupil aberration, the normalized exit pupil of the previous system becomes a displacement vector perpendicular to the exit pupil plane, and it has an approximate first-order partial derivative relationship with the pupil aberration.

$$\Delta {\vec{\rho }_B} ={-} \frac{1}{\varPsi }{\vec{\nabla }_H}\bar{W}_B^4(\vec{H},\vec{\rho }) + {O^{(5)}}$$
where $\varPsi $ represents the Lagrangian invariant of the system and $\bar{W}_B^4$ represents the third-order pupil aberration of system B. The extrinsic aberrations can be determined
$$\begin{aligned} W_E^6 &= W_A^4(\vec{H},\vec{\rho } + \Delta {{\vec{\rho }}_B}) - W_A^4(\vec{H},\vec{\rho })\\ &\approx {{\vec{\nabla }}_\rho }W_A^4(\vec{H},\vec{\rho }) \cdot \Delta {{\vec{\rho }}_B} \\ &= {{\vec{\nabla }}_\rho }W_A^4(\vec{H},\vec{\rho }) \cdot {{\vec{\nabla }}_H}\bar{W}_B^4(\vec{H},\vec{\rho }) \end{aligned}$$

 figure: Fig. 1.

Fig. 1. High-order extrinsic aberrations analysis model.

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 figure: Fig. 2.

Fig. 2. Nodal aberration theory LCS paraxial tracing analysis model.

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The required fifth-order extrinsic aberration coefficients can be obtained as shown in Table 2.

Tables Icon

Table 2. Fifth-Order Extrinsic Aberration Coefficients

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The theory of nodal aberration is based on LCS paraxial ray tracing. The reference for ray tracing is shifted from the mechanical axis (MCA) to the OAR. $\delta {Q^\# }$ represents the offset of the object point after introducing disturbance, $y_{OAR}^\#$ stands for the height of the OAR, and $\delta {E^\# }$ denotes the offset of the pupil center.

Both theories are based on the assumption that the pupil surface is perpendicular to the MCA. After misalignment, only the center position of the pupil is shifted, and the first-order deviation relationship with the pupil aberration still holds. Hence, vectorize and rewrite Eq. (7)

$${W_E} ={-} \frac{1}{\varPsi }{\vec{\nabla }_\rho }{W_A}(\vec{H} - {\vec{\sigma }_A},\vec{\rho }) \cdot {\vec{\nabla }_H}{\bar{W}_B}(\vec{H} - {\vec{\sigma }_B},\vec{\rho })$$

Taking ${W_{151}}$ item as an example:

$${W_{151E}} ={-} \frac{1}{\varPsi }(3W_{131}^A\bar{W}_{311}^B + 8W_{040}^A\bar{W}_{220}^B + 8W_{040}^A\bar{W}_{222}^B)$$

The three terms in parentheses are respectively vector expanded and represented by ${W_{151E1}}$, ${W_{151E2}}$ and ${W_{151E3}}$.

$$\begin{aligned} {W_{151E1}} &= 3W_{131}^A\bar{W}_{311}^B{(\vec{\rho } \cdot \vec{\rho })^2}[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_A})]\\ &= 3W_{131}^A\bar{W}_{311}^B[{(\vec{\rho } \cdot \vec{\rho })^2}(\vec{\rho } \cdot \vec{H}) - {(\vec{\rho } \cdot \vec{\rho })^2}(\vec{\rho } \cdot {{\vec{\sigma }}_A})] \end{aligned}$$
$$\begin{aligned} {W_{151E2}} &= 8W_{040}^A\bar{W}_{220}^B{(\vec{\rho } \cdot \vec{\rho })^2}[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_B})]\\ &= 8W_{040}^A\bar{W}_{220}^B[{(\vec{\rho } \cdot \vec{\rho })^2}(\vec{\rho } \cdot \vec{H}) - {(\vec{\rho } \cdot \vec{\rho })^2}(\vec{\rho } \cdot {{\vec{\sigma }}_B})] \end{aligned}$$
$$\begin{aligned} {W_{151E3}} &= 8W_{040}^A\bar{W}_{222}^B{(\vec{\rho } \cdot \vec{\rho })^2}[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_B})]\\ &= 8W_{040}^A\bar{W}_{222}^B[{(\vec{\rho } \cdot \vec{\rho })^2}(\vec{\rho } \cdot \vec{H}) - {(\vec{\rho } \cdot \vec{\rho })^2}(\vec{\rho } \cdot {{\vec{\sigma }}_B})] \end{aligned}$$

The variation of ${W_{151E}}$ after introducing tolerance disturbance can be obtained, and the calculation of the variation for the other three items is shown in Appendix B. The final results are shown in Table 3.

Tables Icon

Table 3. Fifth-Order Extrinsic Aberration Coefficients

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2.3 Aspherical aberrations

The introduction of aspherical surfaces introduces new complexity to the computation of changes in aberration coefficients. In the context of aberration theory, an aspherical surface can be conceptualized as a composite structure consisting of a spherical substrate and a correction plate that does not contribute to the optical power. The intrinsic aberrations conform to the pattern observed in Table 1. The extrinsic aberrations can be categorized into two parts: those resulting from the combination of a spherical base and a correction plate, and extrinsic aberrations generated by the front and rear systems like spherical surfaces.

For the former extrinsic aberrations, we have

$${W_{E1}} ={-} \frac{1}{\varPsi }{\vec{\nabla }_\rho }{W^{cap}}(\vec{H},\vec{\rho }) \cdot {\vec{\nabla }_{\rm H}}{\bar{W}^{sphere}}(\vec{H},\vec{\rho })$$
among them, ${W^{cap}}(\vec{H},\vec{\rho })$ represents the aberration of the aspherical correction plate, while ${\bar{W}^{sphere}}(\vec{H},\vec{\rho })$ represents the pupil aberration of the spherical base. After vectorization processing, the result can be obtained
$${W_{E1}} ={-} \frac{1}{\varPsi }{\vec{\nabla }_\rho }{W^{cap}}(\vec{H} - {\vec{\sigma }_{as}},\vec{\rho }) \cdot {\vec{\nabla }_{\rm H}}{\bar{W}^{sphere}}(\vec{H} - {\vec{\sigma }_{sphere}},\vec{\rho })$$

It has become an expression similar to formula (8), following the same pattern as Table 3.

For the latter extrinsic aberrations, assuming the calculation of the j-th surface, we have

$$\begin{aligned} {W_{E2,sj}} &= {-} \frac{1}{\varPsi }[{{\vec{\nabla }}_\rho }W_{s1}^{sphere}(\vec{H},\vec{\rho }) + {{\vec{\nabla }}_\rho }W_{s1}^{cap}(\vec{H},\vec{\rho }) + {{\vec{\nabla }}_\rho }W_{s2}^{sphere}(\vec{H},\vec{\rho }) + {{\vec{\nabla }}_\rho }W_{s2}^{cap}(\vec{H},\vec{\rho }) + \ldots \\ &+ {{\vec{\nabla }}_\rho }W_{sj - 1}^{sphere}(\vec{H},\vec{\rho }) + {{\vec{\nabla }}_\rho }W_{sj - 1}^{cap}(\vec{H},\vec{\rho })] \cdot [{{\vec{\nabla }}_{\rm H}}\bar{W}_{sj}^{sphere}(\vec{H},\vec{\rho }) + {{\vec{\nabla }}_{\rm H}}\bar{W}_{sj}^{cap}(\vec{H},\vec{\rho })] \end{aligned}$$

After vectorization processing, it can be obtained that

$$\begin{aligned} {W_{E2,sj}} &= {-} \frac{1}{\varPsi }[{{\vec{\nabla }}_\rho }W_{s1}^{sphere}(\vec{H} - {{\vec{\sigma }}_{sphere,s1}},\vec{\rho }) + {{\vec{\nabla }}_\rho }W_{s1}^{cap}(\vec{H} - {{\vec{\sigma }}_{as,s1}},\vec{\rho }) + {{\vec{\nabla }}_\rho }W_{s2}^{sphere}(\vec{H} - {{\vec{\sigma }}_{sphere,s2}},\vec{\rho })\\ &+ {{\vec{\nabla }}_\rho }W_{s2}^{cap}(\vec{H} - {{\vec{\sigma }}_{as,s2}},\vec{\rho }) + \ldots + {{\vec{\nabla }}_\rho }W_{sj - 1}^{sphere}(\vec{H} - {{\vec{\sigma }}_{sphere,sj - 1}},\vec{\rho })\\ &+ {{\vec{\nabla }}_\rho }W_{sj - 1}^{cap}(\vec{H} - {{\vec{\sigma }}_{as,sj - 1}},\vec{\rho })] \cdot [{{\vec{\nabla }}_{\rm H}}\bar{W}_{sj}^{sphere}(\vec{H} - {{\vec{\sigma }}_{sphere,sj}},\vec{\rho }) + {{\vec{\nabla }}_{\rm H}}\bar{W}_{sj}^{cap}(\vec{H} - {{\vec{\sigma }}_{as,sj}},\vec{\rho })]\\ &={-} \frac{1}{\varPsi }[{{\vec{\nabla }}_\rho }W_{s1}^{sphere}(\vec{H} - {{\vec{\sigma }}_{sphere,s1}},\vec{\rho }) \cdot {{\vec{\nabla }}_{\rm H}}\bar{W}_{sj}^{sphere}(\vec{H} - {{\vec{\sigma }}_{sphere,sj}},\vec{\rho }) + {{\vec{\nabla }}_\rho }W_{s1}^{sphere}(\vec{H} - {{\vec{\sigma }}_{sphere,s1}},\vec{\rho })\\ &\cdot {{\vec{\nabla }}_{\rm H}}\bar{W}_{sj}^{cap}(\vec{H} - {{\vec{\sigma }}_{as,sj}},\vec{\rho }) + {{\vec{\nabla }}_\rho }W_{s1}^{cap}(\vec{H} - {{\vec{\sigma }}_{as,s1}},\vec{\rho }) \cdot {{\vec{\nabla }}_{\rm H}}\bar{W}_{sj}^{sphere}(\vec{H} - {{\vec{\sigma }}_{sphere,sj}},\vec{\rho })\\ &+ {{\vec{\nabla }}_\rho }W_{s1}^{cap}(\vec{H} - {{\vec{\sigma }}_{as,s1}},\vec{\rho }) \cdot {{\vec{\nabla }}_{\rm H}}\bar{W}_{sj}^{cap}(\vec{H} - {{\vec{\sigma }}_{as,sj}},\vec{\rho }) + \ldots + {{\vec{\nabla }}_\rho }W_{sj - 1}^{sphere}(\vec{H} - {{\vec{\sigma }}_{sphere,sj - 1}},\vec{\rho })\\ &\cdot {{\vec{\nabla }}_{\rm H}}\bar{W}_{sj}^{sphere}(\vec{H} - {{\vec{\sigma }}_{sphere,sj}},\vec{\rho }) + {{\vec{\nabla }}_\rho }W_{sj - 1}^{sphere}(\vec{H} - {{\vec{\sigma }}_{sphere,sj - 1}},\vec{\rho }) \cdot {{\vec{\nabla }}_{\rm H}}\bar{W}_{sj}^{cap}(\vec{H} - {{\vec{\sigma }}_{as,sj}},\vec{\rho })\\ &+ {{\vec{\nabla }}_\rho }W_{sj - 1}^{cap}(\vec{H} - {{\vec{\sigma }}_{as,sj - 1}},\vec{\rho }) \cdot {{\vec{\nabla }}_{\rm H}}\bar{W}_{sj}^{sphere}(\vec{H} - {{\vec{\sigma }}_{sphere,sj}},\vec{\rho })\\ &+ {{\vec{\nabla }}_\rho }W_{sj - 1}^{cap}(\vec{H} - {{\vec{\sigma }}_{as,sj - 1}},\vec{\rho }) \cdot {{\vec{\nabla }}_{\rm H}}\bar{W}_{sj}^{cap}(\vec{H} - {{\vec{\sigma }}_{as,sj}},\vec{\rho })] \end{aligned}$$

Each sub term in parentheses follows the form of aberration multiplied by pupil aberration, similar to formula (8) and following the same pattern as Table 3.

3. Merit function

The preceding section has concluded the derivation of nodal high-order aberration theory, which enables the prediction of changes in high-order aberration coefficients after the introduction of tolerance perturbations. The primary objective is to achieve a balance between imaging quality and tolerance performance. This section introduces a comprehensive evaluation index designed to characterize the overall performance of optical systems.

The decenter and tilt tolerances are two of the most commonly encountered types of tolerances in lens. LCS paraxial ray tracing requires providing the decenter and tilt of the optical surface, as opposed to the lens itself. The decenter and tilt tolerances of the lens are modeled and decomposed, and characterized as the decenter and tilt of a single surface.

The model of decenter and tilt of the lens is illustrated in Fig. 3. When a decenter tolerance of YDE size is introduced on the surface j, it is equivalent to introducing a decenter tolerance of YDE size on both the surface j and the surface j + 1. When a tilt tolerance of ADE size is introduced on the surface sj, it is equivalent to introducing a inclination tolerance of ADE size for both the surface sj and the surface sj + 1 respectively, and introducing a YDE decenter tolerance for the surface sj + 1, with a size of

$$YDE = THI \times ADE$$

THI represents the thickness of the lens or the thickness of the surface sj. Additionally, a minor decenter tolerance along the axis is also introduced on the surface sj + 1, although it may be ignored.

 figure: Fig. 3.

Fig. 3. Model of lens decenter and tilt. (a)Lens decenter model. (b)Lens tilt model.

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The merit function for imaging quality of optical systems can use the variance of the optical path

$$\varPhi = \frac{1}{N}\sum\limits_i^N {OPD_i^2}$$

OPD represents the optical path difference of the sampled rays, where N stands for the number of sampled rays, and $\varPhi $ denotes the merit function. When introducing factors that disturb tolerance

$$\varPhi = \frac{1}{N}\sum\limits_i^N {OPD_i^2} + {\varPhi _{tol}}$$

${\Phi _{tol}}$ represents the deviation caused by tolerance disturbance, which is constructed as the variance of each aberration

$$\begin{aligned} {\varPhi _{tol}} &= {E_{131}} + {E_{222}} + {E_{151}} + {E_{242}} + {E_{333}} + {E_{422}}\\ &= \frac{1}{8}{(\Delta {W_{131}})^2} + \frac{1}{6}{(\Delta {W_{222}})^2} + \frac{1}{{12}}{(\Delta {W_{151}})^2} + \frac{1}{{10}}{(\Delta {W_{242}})^2} + \frac{1}{8}{(\Delta {W_{333}})^2} + \frac{1}{6}{(\Delta {W_{422}})^2} \end{aligned}$$

The coefficients in front of each item are normalized coefficients, obtained by integrating the circular pupil. In general, a system may have multiple lenses, and each lens may have decenter and tilt in the X and Y directions. Therefore, the final merit function is

$$\varPhi = \frac{1}{N}\sum\limits_i^N {OPD_i^2} + {\varPhi _{lens1,Xtile}} + {\varPhi _{lens1,Ytilt}} + {\varPhi _{lens1,Xdecenter}} + {\varPhi _{lens1,Ydecenter}} + {\varPhi _{lens2,Xtilt}} + \ldots $$

The final merit function has been obtained through the above steps. Next, the damping least squares algorithm is utilized to optimize the initial structure and achieve a system with low tolerance sensitivity.

4. Verification of the cases

Three cases: a three-piece lens, a wide-angle lens, and a TMA lens are designed to validate our method. Our method (hereinafter referred to as NAT optimization) is compared with general optimization and the SAB optimization of CODE V. Tolerance analysis is conducted on the optimization results and utilized the Monte Carlo method. The calculation takes a long time, but the results are the most accurate. The performance evaluation index is the diffraction Modulation Transfer Function (MTF) at 30 line pairs per millimeter (lp/mm) in the tangential direction. We employ the user-defined merit function in the commercial optical design software CODE V and utilize the damping least squares method to implement our method. The above simulations are all implemented on a personal computer, with an AMD R7-4800 H CPU. The lens data before and after optimization for three cases can be found in Appendix C.

4.1 Three-piece lens

Using a three-piece flat panel system as the initial structure, the pupil diameter is 22.2222 mm, the half field angle is 15°, and the wavelength is 587.6 nm. Among them, surfaces 2, 3, 4, and 6 utilize up to sixth-order aspherical surfaces. In optimization, the effective focal length is constrained to 100 mm to prevent the system excessively long. Add a lens decenter tolerance of 0.025 mm and a lens tilt tolerance of 0.001 rad without a compensator.

Figure 4 depicts a two-dimensional graph illustrating three different results. It is evident that general optimization and SAB optimization yield similar structures, while NAT optimization generates a distinct new structure. Figure 5 and Fig. 6 show the MTF and tolerance performance of the three results, which exhibit similar MTF performance but different tolerance performance. Figure 7 illustrates that after introducing tolerances, the optimization results have a 50%, 84.1%, 97.7%, and 99.9% probability of leading to a decrease in MTF performance. Under the same probability, a smaller decrease leads to better tolerance performance. Under four different feature probabilities, the optimization result of NAT for three fields consistently demonstrates the lowest MTF decrease. In addition, the optimization result of SAB also exhibits an improvement in tolerance performance compared to general optimization, although not as significant as the improvement seen with NAT optimization.

 figure: Fig. 4.

Fig. 4. 2D diagram of initial structure and optimization results. (a) Initial structure. (b)General optimization result. (c) SAB optimization result. (d) NAT optimization result.

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 figure: Fig. 5.

Fig. 5. MTF performance of three optimization results. (a) General optimization result. (b) SAB optimization result. (c) NAT optimization result.

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 figure: Fig. 6.

Fig. 6. Monte Carlo tolerance analysis of three optimization results. (a) General optimization result. (b) SAB optimization result. (c) NAT optimization result.

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 figure: Fig. 7.

Fig. 7. Tolerance analysis under four feature probabilities. (a) 50% probability. (b) 84.1% probability. (c) 97.7% probability. (d) 99.9% probability.

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4.2 Wide-angle lens

The lens consists of 7 glass elements, with an F-number of 2.8, a maximum half field angle of 37.84°, and a wavelength of 587.6 nm. Among them, surface 14 utilizes up to sixth-order aspherical surface. The effective focal length of the constraint in optimization is 28 mm. Add a lens decenter tolerance of 0.005 mm and a lens tilt tolerance of 0.0005 rad without a compensator.

Figure 8 depicts a two-dimensional graph of the initial structure and three optimization results. Similarly, SAB optimization yields a structure similar to that produced by general optimization, while NAT optimization generates entirely a new structure that differs from the initial structure. Figure 9 and Fig. 10 display the MTF performance and tolerance performance of three optimized results. It is evident that they exhibit similar imaging performance but differ in tolerance performance. Figure 11 illustrates the MTF reduction for three different fields of view, each with distinct optimization results, across four feature probabilities. In a 0° field, the MTF reduction from NAT optimization is greater than the other two. With a 50% probability, the NAT optimization result at 19.65° is also greater than the other two, although they are both small. In other probabilities, NAT optimization results in a minimal decrease, while SAB optimization consistently yields a smaller decrease than general optimization.

 figure: Fig. 8.

Fig. 8. 2D diagram of initial structure and optimization results. (a) Initial structure. (b)General optimization result. (c) SAB optimization result. (d) NAT optimization result.

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 figure: Fig. 9.

Fig. 9. MTF performance of three optimization results. (a) General optimization result. (b) SAB optimization result. (c) NAT optimization result.

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 figure: Fig. 10.

Fig. 10. Monte Carlo tolerance analysis of three optimization results. (a) General optimization result. (b) SAB optimization result. (c) NAT optimization result.

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 figure: Fig. 11.

Fig. 11. Tolerance analysis under four feature probabilities. (a) 50% probability. (b) 84.1% probability. (c) 97.7% probability. (d) 99.9% probability.

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4.3 TMA lens

There are three mirrors in total, with an entrance pupil diameter of 200 mm. The central field is 10°, and the field range is 9° to 11°, and a wavelength of 550 nm. The primary mirror and the third mirror (surfaces 2 and 4) utilize up to sixth-order aspherical surfaces. In the optimization process, the effective focal length is constrained to −800 mm, and suitable constraints are established to prevent light occlusion. When using the primary mirror as the reference, the secondary and tertiary mirrors are adjusted to have an decenter tolerance of 0.025 mm and an tilt tolerance of 0.0002 rad, without the use of compensators.

Figure 12 depicts a two-dimensional graph of the initial structure and three optimization results. It can be observed that general optimization and SAB optimization still yield similar structures, while NAT produces a different one. Figure 13 and Fig. 14 show the MTF performance and tolerance performance of three optimized results. These results have similar imaging performance but different tolerance performance. Figure 15 illustrates the MTF reduction for various fields and optimization methods across four feature probabilities. At a 50% probability, the 9° and 11° field SAB exhibits a greater decrease in MTF, while in other probabilities, it is smaller than the general optimization. The optimization result of NAT always yields the minimum MTF decrease.

 figure: Fig. 12.

Fig. 12. 2D diagram of initial structure and optimization results. (a) Initial structure. (b)General optimization result. (c) SAB optimization result. (d) NAT optimization result.

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 figure: Fig. 13.

Fig. 13. MTF performance of three optimization results. (a) General optimization result. (b) SAB optimization result. (c) NAT optimization result.

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 figure: Fig. 14.

Fig. 14. Monte Carlo tolerance analysis of three optimization results. (a) General optimization result. (b) SAB optimization result. (c) NAT optimization result.

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 figure: Fig. 15.

Fig. 15. Tolerance analysis under four feature probabilities. (a) 50% probability. (b) 84.1% probability. (c) 97.7% probability. (d) 99.9% probability.

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4.4 Analysis

Among the three-piece lens, the NAT optimization result has consistently demonstrated the best tolerance performance. In wide-angle lens, the performance of NAT optimization is not as effective as SAB optimization and general optimization in a small field. However, the differences are small, and the tolerance performance is significantly improved in a large field. In TMA lens, there are a few cases where SAB is not as effective as general optimization, and the NAT optimization result still exhibits the lowest tolerance sensitivity. From the optical system's structure, the SAB optimization results of CODE V tend to be similar to general optimization results but exhibit lower tolerance sensitivity. Based on high-order NAT optimization methods, there is a tendency to search for novel structures. In summary, the high-order NAT optimization results indicate that this method has the lowest tolerance sensitivity, demonstrating its ability to reduce the tolerance sensitivity of optical systems.

5. Summary

To address the challenge of general optical design processes struggling to balance imaging performance and tolerance performance, we propose an optimization method based on high-order NAT to reduce tolerance sensitivity. This method can effectively improve tolerance performance while avoiding the universal issue of third-order NAT when dealing with aspherical systems. Vectorization is applied to the high-order aberration coefficients of scalars, and this paper discusses the effects of intrinsic aberrations, extrinsic aberrations, and aspherical aberrations. The changes in aberration coefficients after introducing tolerance perturbations are obtained. Based on the actual processing and assembly, a comprehensive merit function is developed for optimization. Three instance validations are conducted to compare the general optimization results, the SAB optimization results of CODE V, and our proposed high-order NAT optimization results. When the imaging performance is similar, the high-order NAT optimization results show significantly lower tolerance sensitivity.

However, this method also has limitations, as high-order aberrations are only derived up to the fifth order. If higher-order aberrations dominate a system, such as when using a high-order aspherical mobile phone lens or a freeform surface system, this method may not effectively reduce sensitivity and may require the derivation of higher-order aberrations or the exploration of new methods.

Appendix A: Derivation of fifth-order intrinsic aberration variation

The expansion of the remaining three nodal fifth-order intrinsic aberrations is listed, and the variation after tolerance perturbation is extracted:

${W_{242}}$:

$$\begin{aligned}{W_I} &= \frac{1}{2}{W_{242I}}[{(\vec{H} - \vec{\sigma })^2} \cdot {{\vec{\rho }}^2}](\vec{\rho } \cdot \vec{\rho })\\ &= \frac{1}{2}{W_{242}}[(\vec{H} - 2\vec{H}\vec{\sigma } + {{\vec{\sigma }}^2}) \cdot {{\vec{\rho }}^2}](\vec{\rho } \cdot \vec{\rho }) \end{aligned}$$
$$\Delta {W_{242I}} ={-} {W_{242I}}(\vec{H}\vec{\sigma } \cdot {\vec{\rho }^2})(\vec{\rho } \cdot \vec{\rho })$$

${W_{333}}$:

$$\begin{aligned} {W_I} &= \frac{1}{4}{W_{333I}}[{(\vec{H} - \vec{\sigma })^3} \cdot {{\vec{\rho }}^3}]\\ &= \frac{1}{4}{W_{333I}}[({{\vec{H}}^3} - 3{{\vec{H}}^2}\vec{\sigma } + 3\vec{H}{{\vec{\sigma }}^2} - {{\vec{\sigma }}^3}) \cdot {{\vec{\rho }}^3}] \end{aligned}$$
$$\Delta {W_{333I}} ={-} \frac{3}{4}{W_{333I}}{\vec{H}^2}\vec{\sigma } \cdot {\vec{\rho }^3}$$

${W_{422}}$:

$$\begin{aligned} {W_I} &= \frac{1}{2}{W_{422I}}[(\vec{H} - \vec{\sigma }) \cdot (\vec{H} - \vec{\sigma })]{(\vec{H} - \vec{\sigma })^2} \cdot {{\vec{\rho }}^2}\\ &= \frac{1}{2}{W_{422I}}\{ [(\vec{H} \cdot \vec{H}) - 2(\vec{H} \cdot \vec{\sigma }) + (\vec{\sigma } \cdot \vec{\sigma }\textrm{)}]({{\vec{H}}^2} - 2\vec{H}\vec{\sigma } + {{\vec{\sigma }}^2})\} \cdot {{\vec{\rho }}^2}\\ &= \frac{1}{2}{W_{422I}}[(\vec{H} \cdot \vec{H}){{\vec{H}}^2} - 2(\vec{H} \cdot \vec{H})\vec{H}\vec{\sigma } + (\vec{H} \cdot \vec{H}){{\vec{\sigma }}^2}\\ &- 2(\vec{H} \cdot \vec{\sigma }){{\vec{H}}^2} + 4(\vec{H} \cdot \vec{\sigma })\vec{H}\vec{\sigma } - 2(\vec{H} \cdot \vec{\sigma }){{\vec{\sigma }}^2}\\ &+ (\vec{\sigma } \cdot \vec{\sigma }\textrm{)}{{\vec{H}}^2} - 2(\vec{\sigma } \cdot \vec{\sigma }\textrm{)}\vec{H}\vec{\sigma } + (\vec{\sigma } \cdot \vec{\sigma }\textrm{)}{{\vec{\sigma }}^2}] \cdot {{\vec{\rho }}^2} \end{aligned}$$
$$\Delta {W_{422I}} ={-} {W_{422I}}(\vec{H} \cdot \vec{H})(\vec{H}\vec{\sigma } \cdot {\vec{\rho }^2}) - {W_{422I}}(\vec{H} \cdot \vec{\sigma })({\vec{H}^2} \cdot {\vec{\rho }^2})$$

Appendix B: Derivation of fifth-order extrinsic aberration variation

The expansion of the remaining three nodal fifth-order extrinsic aberrations is listed, and the variation after tolerance perturbation is extracted:

${W_{242}}$:

$${W_{242E}} ={-} \frac{1}{\varPsi }(2W_{222}^A\bar{W}_{311}^B + 4W_{131}^A\bar{W}_{220}^B + 6W_{131}^A\bar{W}_{222}^B + 8W_{040}^A\bar{W}_{131}^B)$$

The four terms in parentheses are respectively vectorized and defined as ${W_{242E1}}$, ${W_{242E2}}$, ${W_{242E3}}$ and ${W_{242E4}}$. The derivation process uses vector multiplication identities

$$2(\vec{A} \cdot \vec{B})(\vec{A} \cdot \vec{C}) = (\vec{A} \cdot \vec{A})(\vec{B} \cdot \vec{C}) + {\vec{A}^2} \cdot \vec{B}\vec{C}$$
$$\begin{aligned} {W_{242E1}} &= 2W_{222}^A\bar{W}_{311}^B(\vec{\rho } \cdot \vec{\rho }){[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_A})]^2}\\ &= W_{222}^A\bar{W}_{311}^B{(\vec{\rho } \cdot \vec{\rho })^2}[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_A})] + W_{222}^A\bar{W}_{311}^B(\vec{\rho } \cdot \vec{\rho })[{{\vec{\rho }}^2} \cdot {(\vec{H} - {{\vec{\sigma }}_A})^2}]\\ &= W_{222}^A\bar{W}_{311}^B[\ldots ] + W_{222}^A\bar{W}_{311}^B[(\vec{\rho } \cdot \vec{\rho })({{\vec{H}}^2} - 2\vec{H}{{\vec{\sigma }}_A} + {{\vec{\sigma }}_A}^2)] \cdot {{\vec{\rho }}^2} \end{aligned}$$
where $[...]$ represents items unrelated to $\cos 2\theta $ and same as below.
$$\Delta {W_{242E1}} ={-} 2W_{222}^A\bar{W}_{311}^B(\vec{\rho } \cdot \vec{\rho })\vec{H}{\vec{\sigma }_A} \cdot {\vec{\rho }^2}$$
$$\begin{aligned} {W_{242E2}} &= 4W_{131}^A\bar{W}_{220}^B(\vec{\rho } \cdot \vec{\rho })[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_A})][\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_B})]\\ &= 2W_{131}^A\bar{W}_{220}^B{(\vec{\rho } \cdot \vec{\rho })^2}[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_B})] + 2W_{131}^A\bar{W}_{220}^B(\vec{\rho } \cdot \vec{\rho }){{\vec{\rho }}^2} \cdot [(\vec{H} - {{\vec{\sigma }}_A})(\vec{H} - {{\vec{\sigma }}_B})]\\ &= 2W_{131}^A\bar{W}_{220}^B[\ldots ] + 2W_{131}^A\bar{W}_{220}^B[(\vec{\rho } \cdot \vec{\rho })({{\vec{H}}^2} - \vec{H}{{\vec{\sigma }}_A} - \vec{H}{{\vec{\sigma }}_B} + {{\vec{\sigma }}_A}{{\vec{\sigma }}_B})] \cdot {{\vec{\rho }}^2} \end{aligned}$$
$$\Delta {W_{242E2}} ={-} 2W_{131}^A\bar{W}_{220}^B(\vec{\rho } \cdot \vec{\rho })(\vec{H}{\vec{\sigma }_A} + \vec{H}{\vec{\sigma }_B}) \cdot {\vec{\rho }^2}$$
$$\begin{aligned} {W_{242E3}} &= 6W_{131}^A\bar{W}_{222}^B(\vec{\rho } \cdot \vec{\rho })[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_A})][\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_B})]\\ &= 3W_{131}^A\bar{W}_{222}^B{(\vec{\rho } \cdot \vec{\rho })^2}[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_B})] + 3W_{131}^A\bar{W}_{222}^B(\vec{\rho } \cdot \vec{\rho })[(\vec{H} - {{\vec{\sigma }}_A})(\vec{H} - {{\vec{\sigma }}_B})] \cdot {{\vec{\rho }}^2}\\ &= 3W_{131}^A\bar{W}_{222}^B[\ldots ] + 3W_{131}^A\bar{W}_{222}^B[(\vec{\rho } \cdot \vec{\rho })({{\vec{H}}^2} - \vec{H}{{\vec{\sigma }}_A} - \vec{H}{{\vec{\sigma }}_B} + {{\vec{\sigma }}_A}{{\vec{\sigma }}_B})] \cdot {{\vec{\rho }}^2} \end{aligned}$$
$$\Delta {W_{242E3}} ={-} 3W_{131}^A\bar{W}_{222}^B(\vec{\rho } \cdot \vec{\rho })(\vec{H}{\vec{\sigma }_A} + \vec{H}{\vec{\sigma }_B}) \cdot {\vec{\rho }^2}$$
$$\begin{aligned}{W_{242E4}} &= 8W_{040}^A\bar{W}_{131}^B(\vec{\rho } \cdot \vec{\rho }){[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_B})]^2}\\ &= 4W_{040}^A\bar{W}_{131}^B{(\vec{\rho } \cdot \vec{\rho })^2}[(\vec{H} - {{\vec{\sigma }}_B}) \cdot (\vec{H} - {{\vec{\sigma }}_B})] + 4W_{040}^A\bar{W}_{131}^B(\vec{\rho } \cdot \vec{\rho })[{{\vec{\rho }}^2} \cdot {(\vec{H} - {{\vec{\sigma }}_B})^2}]\\ &= 4W_{040}^A\bar{W}_{131}^B[\ldots ] + 4W_{040}^A\bar{W}_{131}^B[(\vec{\rho } \cdot \vec{\rho })({{\vec{H}}^2} - 2\vec{H}{{\vec{\sigma }}_B} + {{\vec{\sigma }}_B}^2)] \cdot {{\vec{\rho }}^2} \end{aligned}$$
$$\Delta {W_{242E4}} ={-} 8W_{040}^A\bar{W}_{131}^B(\vec{\rho } \cdot \vec{\rho })\vec{H}{\vec{\sigma }_B} \cdot {\vec{\rho }^2}$$

${W_{333}}$:

$${W_{333E}} ={-} \frac{1}{\varPsi }(4W_{131}^A\bar{W}_{131}^B + 4W_{222}^A\bar{W}_{222}^B)$$

The two terms in parentheses are respectively vectorized and defined as ${W_{333E1}}$ and ${W_{333E2}}$. The derivation process uses vector multiplication identities

$$4(\vec{A} \cdot \vec{B})(\vec{A} \cdot \vec{C})(\vec{A} \cdot \vec{D}) = {\vec{A}^3} \cdot \vec{B}\vec{C}\vec{D} + (\vec{A} \cdot \vec{B})(\vec{A}\vec{C} \cdot \vec{A}\vec{D}) + (\vec{A} \cdot \vec{C})(\vec{A}\vec{B} \cdot \vec{A}\vec{D}) + (\vec{A} \cdot \vec{D})(\vec{A}\vec{B} \cdot \vec{A}\vec{C})$$
$$\begin{aligned} {W_{333E1}} &= 4W_{131}^A\bar{W}_{131}^B[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_A})]{[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_B})]^2}\\ &= W_{131}^A\bar{W}_{131}^B[{{\vec{\rho }}^3} \cdot (\vec{H} - {{\vec{\sigma }}_A}){(\vec{H} - {{\vec{\sigma }}_B})^2}]\\ &+ W_{131}^A\bar{W}_{131}^B[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_A})]\{ [\vec{\rho }(\vec{H} - {{\vec{\sigma }}_B})] \cdot [\vec{\rho }(\vec{H} - {{\vec{\sigma }}_B})]\} \\ &+ 2W_{131}^A\bar{W}_{131}^B[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_B})]\{ [\vec{\rho }(\vec{H} - {{\vec{\sigma }}_A})] \cdot [\vec{\rho }(\vec{H} - {{\vec{\sigma }}_B})]\} \\ &= W_{131}^A\bar{W}_{131}^B{[\ldots ]_1} + 2W_{131}^A\bar{W}_{131}^B{[\ldots ]_2} + W_{131}^A\bar{W}_{131}^B[{{\vec{H}}^3} - 2{{\vec{H}}^2}{{\vec{\sigma }}_B}\\ &+ \vec{H}{{\vec{\sigma }}_B}^2 - {{\vec{H}}^2}{{\vec{\sigma }}_A} + 2\vec{H}{{\vec{\sigma }}_A}{{\vec{\sigma }}_B} - {{\vec{\sigma }}_A}{{\vec{\sigma }}_B}^2] \cdot {{\vec{\rho }}^3} \end{aligned}$$
$$\Delta {W_{333E1}} = W_{131}^A\bar{W}_{131}^B[( - 2{\vec{H}^2}{\vec{\sigma }_B} - {\vec{H}^2}{\vec{\sigma }_A})] \cdot {\vec{\rho }^3}$$
where ${[...]_1}$ and ${[...]_2}$ represent items unrelated to $\cos 3\theta $ and same as below.
$$\begin{aligned} {W_{333E2}} &= 4W_{222}^A\bar{W}_{222}^B[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_B})]{[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_A})]^2}\\ &= W_{222}^A\bar{W}_{222}^B[{{\vec{\rho }}^3} \cdot (\vec{H} - {{\vec{\sigma }}_B}){(\vec{H} - {{\vec{\sigma }}_A})^2}]\\ &+ W_{222}^A\bar{W}_{222}^B[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_B})]\{ [\vec{\rho }(\vec{H} - {{\vec{\sigma }}_A})] \cdot [\vec{\rho }(\vec{H} - {{\vec{\sigma }}_A})]\} \\ &+ 2W_{222}^A\bar{W}_{222}^B[\vec{\rho } \cdot (\vec{H} - {{\vec{\sigma }}_A})]\{ [\vec{\rho }(\vec{H} - {{\vec{\sigma }}_B})] \cdot [\vec{\rho }(\vec{H} - {{\vec{\sigma }}_A})]\} \\ &= W_{222}^A\bar{W}_{222}^B{[]_1} + 2W_{222}^A\bar{W}_{222}^B{[]_2} + W_{222}^A\bar{W}_{222}^B[{{\vec{H}}^3} - 2{{\vec{H}}^2}{{\vec{\sigma }}_A}\\ &+ \vec{H}{{\vec{\sigma }}_A}^2 - {{\vec{H}}^2}{{\vec{\sigma }}_B} + 2\vec{H}{{\vec{\sigma }}_A}{{\vec{\sigma }}_B} - {{\vec{\sigma }}_A}^2{{\vec{\sigma }}_B}] \cdot {{\vec{\rho }}^3} \end{aligned}$$
$$\Delta {W_{333E2}} = W_{131}^A\bar{W}_{131}^B[( - 2{\vec{H}^2}{\vec{\sigma }_A} - {\vec{H}^2}{\vec{\sigma }_B})] \cdot {\vec{\rho }^3}$$

${W_{422}}$:

$${W_{422E}} ={-} \frac{1}{\varPsi }(2W_{311}^A\bar{W}_{222}^B + 4W_{220}^A\bar{W}_{131}^B + 6W_{222}^A\bar{W}_{131}^B + 8W_{131}^A\bar{W}_{040}^B)$$

Similarly, define the four items as ${W_{422E1}}$, ${W_{422E2}}$, ${W_{422E3}}$ and ${W_{422E4}}$.

$$\begin{aligned}{W_{422E1}} &= 2W_{311}^A\bar{W}_{222}^B[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_A})][(\vec{H} - {{\vec{\sigma }}_A}) \cdot \vec{\rho }][(\vec{H} - {{\vec{\sigma }}_B}) \cdot \vec{\rho }]\\ &= W_{311}^A\bar{W}_{222}^B[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_A})](\vec{\rho } \cdot \vec{\rho })[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_B})]\\ &+ W_{311}^A\bar{W}_{222}^B[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_A})][(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_B})] \cdot {{\vec{\rho }}^2}\\ &= W_{311}^A\bar{W}_{222}^B[\ldots ] + W_{311}^A\bar{W}_{222}^B[(\vec{H} \cdot \vec{H}){{\vec{H}}^2} - (\vec{H} \cdot \vec{H})\vec{H}{{\vec{\sigma }}_A} - (\vec{H} \cdot \vec{H})\vec{H}{{\vec{\sigma }}_B}\\ &+ (\vec{H} \cdot \vec{H}){{\vec{\sigma }}_A}{{\vec{\sigma }}_B} - 2(\vec{H} \cdot {{\vec{\sigma }}_A}){{\vec{H}}^2} + 2(\vec{H} \cdot {{\vec{\sigma }}_A})\vec{H}{{\vec{\sigma }}_A} + 2(\vec{H} \cdot {{\vec{\sigma }}_A})\vec{H}{{\vec{\sigma }}_B}\\ &- 2(\vec{H} \cdot {{\vec{\sigma }}_A}){{\vec{\sigma }}_A}{{\vec{\sigma }}_B} + ({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_B}){{\vec{H}}^2} - ({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_B})\vec{H}{{\vec{\sigma }}_A} - ({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_B})\vec{H}{{\vec{\sigma }}_B}\\ &+ ({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_B}){{\vec{\sigma }}_A}{{\vec{\sigma }}_B}] \cdot {{\vec{\rho }}^2} \end{aligned}$$
$$\Delta {W_{422E1}} = W_{311}^A\bar{W}_{222}^B[ - (\vec{H} \cdot \vec{H})\vec{H}{\vec{\sigma }_A} - (\vec{H} \cdot \vec{H})\vec{H}{\vec{\sigma }_B} - 2(\vec{H} \cdot {\vec{\sigma }_A}){\vec{H}^2}] \cdot {\vec{\rho }^2}$$
Where $[...]$ represents items unrelated to $\cos 2\theta $ and same as below.
$$\begin{aligned} {W_{422E2}} &= 4W_{220}^A\bar{W}_{131}^B[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_A})]{[(\vec{H} - {{\vec{\sigma }}_B}) \cdot \vec{\rho }]^2}\\ &= 2W_{220}^A\bar{W}_{131}^B[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_A})](\vec{\rho } \cdot \vec{\rho })[(\vec{H} - {{\vec{\sigma }}_B}) \cdot (\vec{H} - {{\vec{\sigma }}_B})]\\ &+ 2W_{220}^A\bar{W}_{131}^B[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_A})]{(\vec{H} - {{\vec{\sigma }}_B})^2} \cdot {{\vec{\rho }}^2}\\ &= 2W_{220}^A\bar{W}_{131}^B[\ldots ] + 2W_{220}^A\bar{W}_{131}^B[(\vec{H} \cdot \vec{H}){{\vec{H}}^2} - 2(\vec{H} \cdot \vec{H})\vec{H}{{\vec{\sigma }}_B} + (\vec{H} \cdot \vec{H}){{\vec{\sigma }}_B}^2\\ &- 2(\vec{H} \cdot {{\vec{\sigma }}_A}){{\vec{H}}^2} + 4(\vec{H} \cdot {{\vec{\sigma }}_A})\vec{H}{{\vec{\sigma }}_B} - 2(\vec{H} \cdot {{\vec{\sigma }}_A}){{\vec{\sigma }}_B}^2\\ &+ ({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_A}){{\vec{H}}^2} - 2({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_A})\vec{H}{{\vec{\sigma }}_B} + ({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_A}){{\vec{\sigma }}_B}^2] \cdot {{\vec{\rho }}^2} \end{aligned}$$
$$\Delta {W_{422E2}} = 2W_{220}^A\bar{W}_{131}^B[ - 2(\vec{H} \cdot \vec{H})\vec{H}{\vec{\sigma }_B} - 2(\vec{H} \cdot {\vec{\sigma }_A}){\vec{H}^2}] \cdot {\vec{\rho }^2}$$
$$\begin{aligned} {W_{422E3}} &= 2W_{222}^A\bar{W}_{131}^B[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_B})]{[(\vec{H} - {{\vec{\sigma }}_A}) \cdot \vec{\rho }]^2}\\ &+ 4W_{222}^A\bar{W}_{131}^B[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_B})][(\vec{H} - {{\vec{\sigma }}_A}) \cdot \vec{\rho }][(\vec{H} - {{\vec{\sigma }}_B}) \cdot \vec{\rho }] \end{aligned}$$

Two items are named respectively ${W_{422E31}}$ and ${W_{422E32}}$.

$$\begin{aligned} {W_{422E31}} &= 2W_{222}^A\bar{W}_{131}^B[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_B})]{[(\vec{H} - {{\vec{\sigma }}_A}) \cdot \vec{\rho }]^2}\\ &= W_{222}^A\bar{W}_{131}^B[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_A})](\vec{\rho } \cdot \vec{\rho })[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_B})]\\ &+ W_{222}^A\bar{W}_{131}^B[{(\vec{H} - {{\vec{\sigma }}_A})^2} \cdot {{\vec{\rho }}^2}][(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_B})]\\ &= W_{222}^A\bar{W}_{131}^B[\ldots ] + W_{222}^A\bar{W}_{131}^B[(\vec{H} \cdot \vec{H}){{\vec{H}}^2} - (\vec{H} \cdot {{\vec{\sigma }}_A}){{\vec{H}}^2} - (\vec{H} \cdot {{\vec{\sigma }}_B}){{\vec{H}}^2} + ({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_B}){{\vec{H}}^2}\\ &- 2(\vec{H} \cdot \vec{H})\vec{H}{{\vec{\sigma }}_A} + 2(\vec{H} \cdot {{\vec{\sigma }}_A})\vec{H}{{\vec{\sigma }}_A} + 2(\vec{H} \cdot {{\vec{\sigma }}_B})\vec{H}{{\vec{\sigma }}_A} - 2({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_B})\vec{H}{{\vec{\sigma }}_A}\\ &+ (\vec{H} \cdot \vec{H}){{\vec{\sigma }}_A}^2 - (\vec{H} \cdot {{\vec{\sigma }}_A}){{\vec{\sigma }}_A}^2 - (\vec{H} \cdot {{\vec{\sigma }}_B}){{\vec{\sigma }}_A}^2 + ({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_B}){{\vec{\sigma }}_A}^2] \cdot {{\vec{\rho }}^2} \end{aligned}$$
$$\Delta {W_{422E31}} = W_{222}^A\bar{W}_{131}^B[ - (\vec{H} \cdot {\vec{\sigma }_A}){\vec{H}^2} - (\vec{H} \cdot {\vec{\sigma }_B}){\vec{H}^2} - 2(\vec{H} \cdot \vec{H})\vec{H}{\vec{\sigma }_A}] \cdot {\vec{\rho }^2}$$
$$\begin{aligned} {W_{422E32}} &= 4W_{222}^A\bar{W}_{131}^B[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_B})][(\vec{H} - {{\vec{\sigma }}_A}) \cdot \vec{\rho }][(\vec{H} - {{\vec{\sigma }}_B}) \cdot \vec{\rho }]\\ &= 2W_{222}^A\bar{W}_{131}^B{[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_B})]^2}(\vec{\rho } \cdot \vec{\rho })\\ &+ 2W_{222}^A\bar{W}_{131}^B[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_B})][(\vec{H} - {{\vec{\sigma }}_A})(\vec{H} - {{\vec{\sigma }}_B})] \cdot {{\vec{\rho }}^2}\\ &= 2W_{222}^A\bar{W}_{131}^B[\ldots ] + 2W_{222}^A\bar{W}_{131}^B[(\vec{H} \cdot \vec{H}){{\vec{H}}^2} - (\vec{H} \cdot \vec{H})\vec{H}{{\vec{\sigma }}_A} - (\vec{H} \cdot \vec{H})\vec{H}{{\vec{\sigma }}_B}\\ &+ (\vec{H} \cdot \vec{H}){{\vec{\sigma }}_A}{{\vec{\sigma }}_B} - (\vec{H} \cdot {{\vec{\sigma }}_A}){{\vec{H}}^2} + (\vec{H} \cdot {{\vec{\sigma }}_A})\vec{H}{{\vec{\sigma }}_A} + (\vec{H} \cdot {{\vec{\sigma }}_A})\vec{H}{{\vec{\sigma }}_B} - (\vec{H} \cdot {{\vec{\sigma }}_A}){{\vec{\sigma }}_A}{{\vec{\sigma }}_B}\\ &- (\vec{H} \cdot {{\vec{\sigma }}_B}){{\vec{H}}^2} + (\vec{H} \cdot {{\vec{\sigma }}_B})\vec{H}{{\vec{\sigma }}_A} + (\vec{H} \cdot {{\vec{\sigma }}_B})\vec{H}{{\vec{\sigma }}_B} - (\vec{H} \cdot {{\vec{\sigma }}_B}){{\vec{\sigma }}_A}{{\vec{\sigma }}_B}\\ &+ ({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_B}){{\vec{H}}^2} - ({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_B})\vec{H}{{\vec{\sigma }}_A} - ({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_B})\vec{H}{{\vec{\sigma }}_B} + ({{\vec{\sigma }}_A} \cdot {{\vec{\sigma }}_B}){{\vec{\sigma }}_A}{{\vec{\sigma }}_B}] \cdot {{\vec{\rho }}^2} \end{aligned}$$
$$\Delta {W_{422E32}} = 2W_{222}^A\bar{W}_{131}^B[ - (\vec{H} \cdot \vec{H})\vec{H}{\vec{\sigma }_A} - (\vec{H} \cdot \vec{H})\vec{H}{\vec{\sigma }_B} - (\vec{H} \cdot {\vec{\sigma }_A}){\vec{H}^2} - (\vec{H} \cdot {\vec{\sigma }_B}){\vec{H}^2}] \cdot {\vec{\rho }^2}$$
$$\begin{aligned} {W_{422E4}} &= 8W_{131}^A\bar{W}_{040}^B[(\vec{H} - {{\vec{\sigma }}_B}) \cdot (\vec{H} - {{\vec{\sigma }}_B})]{[(\vec{H} - {{\vec{\sigma }}_A}) \cdot \vec{\rho }]^2}\\ &= 4W_{131}^A\bar{W}_{040}^B[(\vec{H} - {{\vec{\sigma }}_B}) \cdot (\vec{H} - {{\vec{\sigma }}_B})](\vec{\rho } \cdot \vec{\rho })[(\vec{H} - {{\vec{\sigma }}_A}) \cdot (\vec{H} - {{\vec{\sigma }}_A})]\\ &+ 4W_{131}^A\bar{W}_{040}^B[(\vec{H} - {{\vec{\sigma }}_B}) \cdot (\vec{H} - {{\vec{\sigma }}_B})]{(\vec{H} - {{\vec{\sigma }}_A})^2} \cdot {{\vec{\rho }}^2}\\ &= 4W_{131}^A\bar{W}_{040}^B[\ldots ] + 4W_{131}^A\bar{W}_{040}^B[(\vec{H} \cdot \vec{H}){{\vec{H}}^2} - 2(\vec{H} \cdot \vec{H})\vec{H}{{\vec{\sigma }}_A} + (\vec{H} \cdot \vec{H}){{\vec{\sigma }}_A}^2\\ &- 2(\vec{H} \cdot {{\vec{\sigma }}_B}){{\vec{H}}^2} + 4(\vec{H} \cdot {{\vec{\sigma }}_B})\vec{H}{{\vec{\sigma }}_A} - 2(\vec{H} \cdot {{\vec{\sigma }}_B}){{\vec{\sigma }}_A}^2\\ &+ ({{\vec{\sigma }}_B} \cdot {{\vec{\sigma }}_B}){{\vec{H}}^2} - 2({{\vec{\sigma }}_B} \cdot {{\vec{\sigma }}_B})\vec{H}{{\vec{\sigma }}_A} + ({{\vec{\sigma }}_B} \cdot {{\vec{\sigma }}_B}){{\vec{\sigma }}_A}^2] \cdot {{\vec{\rho }}^2} \end{aligned}$$
$$\Delta {W_{422E4}} = 4W_{131}^A\bar{W}_{040}^B[ - 2(\vec{H} \cdot \vec{H})\vec{H}{\vec{\sigma }_A} - 2(\vec{H} \cdot {\vec{\sigma }_B}){\vec{H}^2}] \cdot {\vec{\rho }^2}$$

Appendix C: Lens data of three cases

Detailed lens data for three validation cases are listed.

  • a. Three-piece lens

    Table 4 presents the initial structure of the lens, while Table 5 and Table 6 provide the lens data and aspherical coefficients for three optimization results.

  • b. Wide-angle lens

    Table 7 presents the initial structure of the lens, while Table 8 and Table 9 provide the lens data and aspherical coefficients for three optimization results.

  • c. TMA lens

    Table 10 presents the initial structure of the lens, while Table 11 and Table 12 provide the lens data and aspherical coefficients for three optimization results.

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Table 4. Lens Data for Initial Structure

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Table 5. Lens Data for Three Optimization Results

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Table 6. Aspherical Coefficients for Three Optimization Results

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Table 7. Lens Data for Initial Structure

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Table 8. Lens Data for Three Optimization Results

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Table 9. Aspherical Coefficients for Three Optimization Results

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Table 10. Lens Data for Initial Structure

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Table 11. Lens Data for Three Optimization Results

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Table 12. Aspherical Coefficients for Three Optimization Results

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Funding

National Natural Science Foundation of China (62005271).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. M. Jeffs, “Reduced manufacturing sensitivity in multi-element lens systems,” Proc. SPIE 4832, 104 (2002). [CrossRef]  

2. J. R. Rogers, “Using global synthesis to find tolerance-insensitive design forms,” Proc. SPIE 6342, 63420M (2006). [CrossRef]  

3. J. Rogers, “Global optimization and desensitization,” Proc. SPIE 9633, 96330S (2015). [CrossRef]  

4. M. Isshiki, L. Gardner, and G. G. Gregory, “Automated control of manufacturing sensitivity during optimization,” Proc. SPIE 5249, 343 (2004). [CrossRef]  

5. Q. Meng, H. Wang, W. Wang, et al., “Desensitization design method of unobscured three-mirror anastigmatic optical systems with an adjustment-optimization-evaluation process,” Appl. Opt. 57(6), 1472 (2018). [CrossRef]  

6. Z. Qin, X. Wang, C. Ren, et al., “Design method for a reflective optical system with low tilt error sensitivity,” Opt. Express 29(26), 43464 (2021). [CrossRef]  

7. L. Carrión-Higueras, A. Calatayud, and J. Sasian, “Improving as-built miniature lenses that use many aspheric surface coefficients with two desensitizing techniques,” Opt. Eng. 60(5), 051208 (2021). [CrossRef]  

8. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation, (University of Arizona, 1976).

9. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation, (University of Arizona, 1980).

10. M. Yeşiltepe, A. Bauer, Ö. Karcı, et al., “Sigma vector calculations in nodal aberration theory and experimental validation using a Cassegrain telescope,” Opt. Express 31(25), 42373 (2023). [CrossRef]  

11. Ö. Karcı, M. Yeşiltepe, E. Arpa, et al., “Experimental investigation in nodal aberration theory (NAT): separation of astigmatic figure error from misalignments in a Cassegrain telescope,” Opt. Express 30(7), 11150 (2022). [CrossRef]  

12. Ö. Karcı, E. Arpa, M. Ekinci, et al., “Experimental investigation of binodal astigmatism in nodal aberration theory (NAT) with a Cassegrain telescope system,” Opt. Express 29(13), 19427 (2021). [CrossRef]  

13. Z. Gu, Y. Wang, and C. Yan, “Optical system optimization method for as-built performance based on nodal aberration theory,” Opt. Express 28(6), 7928 (2020). [CrossRef]  

14. J. Sasian, “Sixth-order wavefront deformations: the coefficients and insights into wavefront propagation in optical systems,” in J. Bentley, A. Gupta, and R. N. Youngworth, eds. (2010), p. 765205.

15. J. Sasián, “Theory of sixth-order wave aberrations,” Appl. Opt. 49(16), D69 (2010). [CrossRef]  

16. K. P. Thompson, T. Schmid, O. Cakmakci, et al., “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503 (2009). [CrossRef]  

17. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090 (2009). [CrossRef]  

18. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490 (2010). [CrossRef]  

19. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821 (2011). [CrossRef]  

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (15)
Fig. 1.
Fig. 1. High-order extrinsic aberrations analysis model.

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Fig. 2.
Fig. 2. Nodal aberration theory LCS paraxial tracing analysis model.

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Fig. 3.
Fig. 3. Model of lens decenter and tilt. (a)Lens decenter model. (b)Lens tilt model.

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Fig. 4.
Fig. 4. 2D diagram of initial structure and optimization results. (a) Initial structure. (b)General optimization result. (c) SAB optimization result. (d) NAT optimization result.

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Fig. 5.
Fig. 5. MTF performance of three optimization results. (a) General optimization result. (b) SAB optimization result. (c) NAT optimization result.

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Fig. 6.
Fig. 6. Monte Carlo tolerance analysis of three optimization results. (a) General optimization result. (b) SAB optimization result. (c) NAT optimization result.

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Fig. 7.
Fig. 7. Tolerance analysis under four feature probabilities. (a) 50% probability. (b) 84.1% probability. (c) 97.7% probability. (d) 99.9% probability.

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Fig. 8.
Fig. 8. 2D diagram of initial structure and optimization results. (a) Initial structure. (b)General optimization result. (c) SAB optimization result. (d) NAT optimization result.

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Fig. 9.
Fig. 9. MTF performance of three optimization results. (a) General optimization result. (b) SAB optimization result. (c) NAT optimization result.

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Fig. 10.
Fig. 10. Monte Carlo tolerance analysis of three optimization results. (a) General optimization result. (b) SAB optimization result. (c) NAT optimization result.

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Fig. 11.
Fig. 11. Tolerance analysis under four feature probabilities. (a) 50% probability. (b) 84.1% probability. (c) 97.7% probability. (d) 99.9% probability.

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Fig. 12.
Fig. 12. 2D diagram of initial structure and optimization results. (a) Initial structure. (b)General optimization result. (c) SAB optimization result. (d) NAT optimization result.

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Fig. 13.
Fig. 13. MTF performance of three optimization results. (a) General optimization result. (b) SAB optimization result. (c) NAT optimization result.

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Fig. 14.
Fig. 14. Monte Carlo tolerance analysis of three optimization results. (a) General optimization result. (b) SAB optimization result. (c) NAT optimization result.

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Fig. 15.
Fig. 15. Tolerance analysis under four feature probabilities. (a) 50% probability. (b) 84.1% probability. (c) 97.7% probability. (d) 99.9% probability.

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Tables (12)

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Table 1. Changes in Fifth Order Intrinsic Aberrations after Introducing Tolerance Perturbations

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Table 2. Fifth-Order Extrinsic Aberration Coefficients

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Table 3. Fifth-Order Extrinsic Aberration Coefficients

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Table 4. Lens Data for Initial Structure

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Table 5. Lens Data for Three Optimization Results

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Table 6. Aspherical Coefficients for Three Optimization Results

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Table 7. Lens Data for Initial Structure

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Table 8. Lens Data for Three Optimization Results

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Table 9. Aspherical Coefficients for Three Optimization Results

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Table 10. Lens Data for Initial Structure

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Table 11. Lens Data for Three Optimization Results

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Table 12. Aspherical Coefficients for Three Optimization Results

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Equations (55)

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W = W 040 ( ρ → ⋅ ρ → ) 2 + W 131 ( H → ⋅ ρ → )( ρ → ⋅ ρ → ) + W 220 ( H → ⋅ H → )( ρ → ⋅ ρ → ) + W 222 ( H → ⋅ ρ → ) 2 + W 311 ( H → ⋅ H → )( H → ⋅ ρ → ) + W 060 ( ρ → ⋅ ρ → ) 3 + W 151 ( H → ⋅ ρ → )( ρ → ⋅ ρ → ) 2 + W 240 ( H → ⋅ H → )( ρ → ⋅ ρ → ) 2 + W 242 ( H → ⋅ ρ → ) 2 ( ρ → ⋅ ρ → ) + W 331 ( H → ⋅ H → )( H → ⋅ ρ → )( ρ → ⋅ ρ → ) + W 333 ( H → ⋅ ρ → ) 3 + W 420 ( H → ⋅ H → ) 2 ( ρ → ⋅ ρ → ) + W 422 ( H → ⋅ H → )( H → ⋅ ρ → ) 2 + W 511 ( H → ⋅ H → ) 2 ( H → ⋅ ρ → ) + …
W 6 = W I 6 + W E 6
W 220 M = W 220 + 1 2 W 222 W 240 M = W 240 + 1 2 W 242 W 331 M = W 331 + 3 4 W 333 W 420 M = W 420 + 1 2 W 422
W I = W 151 I [ ( H → − σ → ) ⋅ ρ → ] ( ρ → ⋅ ρ → ) 2 = W 151 I ( H → ⋅ ρ → ) ( ρ → ⋅ ρ → ) 2 − W 151 I ( σ → ⋅ ρ → ) ( ρ → ⋅ ρ → ) 2
Δ W 151 I = − W 151 I ( σ → ⋅ ρ → ) ( ρ → ⋅ ρ → ) 2
Δ ρ → B = − 1 Ψ ∇ → H W ¯ B 4 ( H → , ρ → ) + O ( 5 )
W E 6 = W A 4 ( H → , ρ → + Δ ρ → B ) − W A 4 ( H → , ρ → ) ≈ ∇ → ρ W A 4 ( H → , ρ → ) ⋅ Δ ρ → B = ∇ → ρ W A 4 ( H → , ρ → ) ⋅ ∇ → H W ¯ B 4 ( H → , ρ → )
W E = − 1 Ψ ∇ → ρ W A ( H → − σ → A , ρ → ) ⋅ ∇ → H W ¯ B ( H → − σ → B , ρ → )
W 151 E = − 1 Ψ ( 3 W 131 A W ¯ 311 B + 8 W 040 A W ¯ 220 B + 8 W 040 A W ¯ 222 B )
W 151 E 1 = 3 W 131 A W ¯ 311 B ( ρ → ⋅ ρ → ) 2 [ ρ → ⋅ ( H → − σ → A ) ] = 3 W 131 A W ¯ 311 B [ ( ρ → ⋅ ρ → ) 2 ( ρ → ⋅ H → ) − ( ρ → ⋅ ρ → ) 2 ( ρ → ⋅ σ → A ) ]
W 151 E 2 = 8 W 040 A W ¯ 220 B ( ρ → ⋅ ρ → ) 2 [ ρ → ⋅ ( H → − σ → B ) ] = 8 W 040 A W ¯ 220 B [ ( ρ → ⋅ ρ → ) 2 ( ρ → ⋅ H → ) − ( ρ → ⋅ ρ → ) 2 ( ρ → ⋅ σ → B ) ]
W 151 E 3 = 8 W 040 A W ¯ 222 B ( ρ → ⋅ ρ → ) 2 [ ρ → ⋅ ( H → − σ → B ) ] = 8 W 040 A W ¯ 222 B [ ( ρ → ⋅ ρ → ) 2 ( ρ → ⋅ H → ) − ( ρ → ⋅ ρ → ) 2 ( ρ → ⋅ σ → B ) ]
W E 1 = − 1 Ψ ∇ → ρ W c a p ( H → , ρ → ) ⋅ ∇ → H W ¯ s p h e r e ( H → , ρ → )
W E 1 = − 1 Ψ ∇ → ρ W c a p ( H → − σ → a s , ρ → ) ⋅ ∇ → H W ¯ s p h e r e ( H → − σ → s p h e r e , ρ → )
W E 2 , s j = − 1 Ψ [ ∇ → ρ W s 1 s p h e r e ( H → , ρ → ) + ∇ → ρ W s 1 c a p ( H → , ρ → ) + ∇ → ρ W s 2 s p h e r e ( H → , ρ → ) + ∇ → ρ W s 2 c a p ( H → , ρ → ) + … + ∇ → ρ W s j − 1 s p h e r e ( H → , ρ → ) + ∇ → ρ W s j − 1 c a p ( H → , ρ → ) ] ⋅ [ ∇ → H W ¯ s j s p h e r e ( H → , ρ → ) + ∇ → H W ¯ s j c a p ( H → , ρ → ) ]
W E 2 , s j = − 1 Ψ [ ∇ → ρ W s 1 s p h e r e ( H → − σ → s p h e r e , s 1 , ρ → ) + ∇ → ρ W s 1 c a p ( H → − σ → a s , s 1 , ρ → ) + ∇ → ρ W s 2 s p h e r e ( H → − σ → s p h e r e , s 2 , ρ → ) + ∇ → ρ W s 2 c a p ( H → − σ → a s , s 2 , ρ → ) + … + ∇ → ρ W s j − 1 s p h e r e ( H → − σ → s p h e r e , s j − 1 , ρ → ) + ∇ → ρ W s j − 1 c a p ( H → − σ → a s , s j − 1 , ρ → ) ] ⋅ [ ∇ → H W ¯ s j s p h e r e ( H → − σ → s p h e r e , s j , ρ → ) + ∇ → H W ¯ s j c a p ( H → − σ → a s , s j , ρ → ) ] = − 1 Ψ [ ∇ → ρ W s 1 s p h e r e ( H → − σ → s p h e r e , s 1 , ρ → ) ⋅ ∇ → H W ¯ s j s p h e r e ( H → − σ → s p h e r e , s j , ρ → ) + ∇ → ρ W s 1 s p h e r e ( H → − σ → s p h e r e , s 1 , ρ → ) ⋅ ∇ → H W ¯ s j c a p ( H → − σ → a s , s j , ρ → ) + ∇ → ρ W s 1 c a p ( H → − σ → a s , s 1 , ρ → ) ⋅ ∇ → H W ¯ s j s p h e r e ( H → − σ → s p h e r e , s j , ρ → ) + ∇ → ρ W s 1 c a p ( H → − σ → a s , s 1 , ρ → ) ⋅ ∇ → H W ¯ s j c a p ( H → − σ → a s , s j , ρ → ) + … + ∇ → ρ W s j − 1 s p h e r e ( H → − σ → s p h e r e , s j − 1 , ρ → ) ⋅ ∇ → H W ¯ s j s p h e r e ( H → − σ → s p h e r e , s j , ρ → ) + ∇ → ρ W s j − 1 s p h e r e ( H → − σ → s p h e r e , s j − 1 , ρ → ) ⋅ ∇ → H W ¯ s j c a p ( H → − σ → a s , s j , ρ → ) + ∇ → ρ W s j − 1 c a p ( H → − σ → a s , s j − 1 , ρ → ) ⋅ ∇ → H W ¯ s j s p h e r e ( H → − σ → s p h e r e , s j , ρ → ) + ∇ → ρ W s j − 1 c a p ( H → − σ → a s , s j − 1 , ρ → ) ⋅ ∇ → H W ¯ s j c a p ( H → − σ → a s , s j , ρ → ) ]
Y D E = T H I × A D E
Φ = 1 N ∑ i N O P D i 2
Φ = 1 N ∑ i N O P D i 2 + Φ t o l
Φ t o l = E 131 + E 222 + E 151 + E 242 + E 333 + E 422 = 1 8 ( Δ W 131 ) 2 + 1 6 ( Δ W 222 ) 2 + 1 12 ( Δ W 151 ) 2 + 1 10 ( Δ W 242 ) 2 + 1 8 ( Δ W 333 ) 2 + 1 6 ( Δ W 422 ) 2
Φ = 1 N ∑ i N O P D i 2 + Φ l e n s 1 , X t i l e + Φ l e n s 1 , Y t i l t + Φ l e n s 1 , X d e c e n t e r + Φ l e n s 1 , Y d e c e n t e r + Φ l e n s 2 , X t i l t + …
W I = 1 2 W 242 I [ ( H → − σ → ) 2 ⋅ ρ → 2 ] ( ρ → ⋅ ρ → ) = 1 2 W 242 [ ( H → − 2 H → σ → + σ → 2 ) ⋅ ρ → 2 ] ( ρ → ⋅ ρ → )
Δ W 242 I = − W 242 I ( H → σ → ⋅ ρ → 2 ) ( ρ → ⋅ ρ → )
W I = 1 4 W 333 I [ ( H → − σ → ) 3 ⋅ ρ → 3 ] = 1 4 W 333 I [ ( H → 3 − 3 H → 2 σ → + 3 H → σ → 2 − σ → 3 ) ⋅ ρ → 3 ]
Δ W 333 I = − 3 4 W 333 I H → 2 σ → ⋅ ρ → 3
W I = 1 2 W 422 I [ ( H → − σ → ) ⋅ ( H → − σ → ) ] ( H → − σ → ) 2 ⋅ ρ → 2 = 1 2 W 422 I { [ ( H → ⋅ H → ) − 2 ( H → ⋅ σ → ) + ( σ → ⋅ σ → ) ] ( H → 2 − 2 H → σ → + σ → 2 ) } ⋅ ρ → 2 = 1 2 W 422 I [ ( H → ⋅ H → ) H → 2 − 2 ( H → ⋅ H → ) H → σ → + ( H → ⋅ H → ) σ → 2 − 2 ( H → ⋅ σ → ) H → 2 + 4 ( H → ⋅ σ → ) H → σ → − 2 ( H → ⋅ σ → ) σ → 2 + ( σ → ⋅ σ → ) H → 2 − 2 ( σ → ⋅ σ → ) H → σ → + ( σ → ⋅ σ → ) σ → 2 ] ⋅ ρ → 2
Δ W 422 I = − W 422 I ( H → ⋅ H → ) ( H → σ → ⋅ ρ → 2 ) − W 422 I ( H → ⋅ σ → ) ( H → 2 ⋅ ρ → 2 )
W 242 E = − 1 Ψ ( 2 W 222 A W ¯ 311 B + 4 W 131 A W ¯ 220 B + 6 W 131 A W ¯ 222 B + 8 W 040 A W ¯ 131 B )
2 ( A → ⋅ B → ) ( A → ⋅ C → ) = ( A → ⋅ A → ) ( B → ⋅ C → ) + A → 2 ⋅ B → C →
W 242 E 1 = 2 W 222 A W ¯ 311 B ( ρ → ⋅ ρ → ) [ ρ → ⋅ ( H → − σ → A ) ] 2 = W 222 A W ¯ 311 B ( ρ → ⋅ ρ → ) 2 [ ( H → − σ → A ) ⋅ ( H → − σ → A ) ] + W 222 A W ¯ 311 B ( ρ → ⋅ ρ → ) [ ρ → 2 ⋅ ( H → − σ → A ) 2 ] = W 222 A W ¯ 311 B [ … ] + W 222 A W ¯ 311 B [ ( ρ → ⋅ ρ → ) ( H → 2 − 2 H → σ → A + σ → A 2 ) ] ⋅ ρ → 2
Δ W 242 E 1 = − 2 W 222 A W ¯ 311 B ( ρ → ⋅ ρ → ) H → σ → A ⋅ ρ → 2
W 242 E 2 = 4 W 131 A W ¯ 220 B ( ρ → ⋅ ρ → ) [ ρ → ⋅ ( H → − σ → A ) ] [ ρ → ⋅ ( H → − σ → B ) ] = 2 W 131 A W ¯ 220 B ( ρ → ⋅ ρ → ) 2 [ ( H → − σ → A ) ⋅ ( H → − σ → B ) ] + 2 W 131 A W ¯ 220 B ( ρ → ⋅ ρ → ) ρ → 2 ⋅ [ ( H → − σ → A ) ( H → − σ → B ) ] = 2 W 131 A W ¯ 220 B [ … ] + 2 W 131 A W ¯ 220 B [ ( ρ → ⋅ ρ → ) ( H → 2 − H → σ → A − H → σ → B + σ → A σ → B ) ] ⋅ ρ → 2
Δ W 242 E 2 = − 2 W 131 A W ¯ 220 B ( ρ → ⋅ ρ → ) ( H → σ → A + H → σ → B ) ⋅ ρ → 2
W 242 E 3 = 6 W 131 A W ¯ 222 B ( ρ → ⋅ ρ → ) [ ρ → ⋅ ( H → − σ → A ) ] [ ρ → ⋅ ( H → − σ → B ) ] = 3 W 131 A W ¯ 222 B ( ρ → ⋅ ρ → ) 2 [ ( H → − σ → A ) ⋅ ( H → − σ → B ) ] + 3 W 131 A W ¯ 222 B ( ρ → ⋅ ρ → ) [ ( H → − σ → A ) ( H → − σ → B ) ] ⋅ ρ → 2 = 3 W 131 A W ¯ 222 B [ … ] + 3 W 131 A W ¯ 222 B [ ( ρ → ⋅ ρ → ) ( H → 2 − H → σ → A − H → σ → B + σ → A σ → B ) ] ⋅ ρ → 2
Δ W 242 E 3 = − 3 W 131 A W ¯ 222 B ( ρ → ⋅ ρ → ) ( H → σ → A + H → σ → B ) ⋅ ρ → 2
W 242 E 4 = 8 W 040 A W ¯ 131 B ( ρ → ⋅ ρ → ) [ ρ → ⋅ ( H → − σ → B ) ] 2 = 4 W 040 A W ¯ 131 B ( ρ → ⋅ ρ → ) 2 [ ( H → − σ → B ) ⋅ ( H → − σ → B ) ] + 4 W 040 A W ¯ 131 B ( ρ → ⋅ ρ → ) [ ρ → 2 ⋅ ( H → − σ → B ) 2 ] = 4 W 040 A W ¯ 131 B [ … ] + 4 W 040 A W ¯ 131 B [ ( ρ → ⋅ ρ → ) ( H → 2 − 2 H → σ → B + σ → B 2 ) ] ⋅ ρ → 2
Δ W 242 E 4 = − 8 W 040 A W ¯ 131 B ( ρ → ⋅ ρ → ) H → σ → B ⋅ ρ → 2
W 333 E = − 1 Ψ ( 4 W 131 A W ¯ 131 B + 4 W 222 A W ¯ 222 B )
4 ( A → ⋅ B → ) ( A → ⋅ C → ) ( A → ⋅ D → ) = A → 3 ⋅ B → C → D → + ( A → ⋅ B → ) ( A → C → ⋅ A → D → ) + ( A → ⋅ C → ) ( A → B → ⋅ A → D → ) + ( A → ⋅ D → ) ( A → B → ⋅ A → C → )
W 333 E 1 = 4 W 131 A W ¯ 131 B [ ρ → ⋅ ( H → − σ → A ) ] [ ρ → ⋅ ( H → − σ → B ) ] 2 = W 131 A W ¯ 131 B [ ρ → 3 ⋅ ( H → − σ → A ) ( H → − σ → B ) 2 ] + W 131 A W ¯ 131 B [ ρ → ⋅ ( H → − σ → A ) ] { [ ρ → ( H → − σ → B ) ] ⋅ [ ρ → ( H → − σ → B ) ] } + 2 W 131 A W ¯ 131 B [ ρ → ⋅ ( H → − σ → B ) ] { [ ρ → ( H → − σ → A ) ] ⋅ [ ρ → ( H → − σ → B ) ] } = W 131 A W ¯ 131 B [ … ] 1 + 2 W 131 A W ¯ 131 B [ … ] 2 + W 131 A W ¯ 131 B [ H → 3 − 2 H → 2 σ → B + H → σ → B 2 − H → 2 σ → A + 2 H → σ → A σ → B − σ → A σ → B 2 ] ⋅ ρ → 3
Δ W 333 E 1 = W 131 A W ¯ 131 B [ ( − 2 H → 2 σ → B − H → 2 σ → A ) ] ⋅ ρ → 3
W 333 E 2 = 4 W 222 A W ¯ 222 B [ ρ → ⋅ ( H → − σ → B ) ] [ ρ → ⋅ ( H → − σ → A ) ] 2 = W 222 A W ¯ 222 B [ ρ → 3 ⋅ ( H → − σ → B ) ( H → − σ → A ) 2 ] + W 222 A W ¯ 222 B [ ρ → ⋅ ( H → − σ → B ) ] { [ ρ → ( H → − σ → A ) ] ⋅ [ ρ → ( H → − σ → A ) ] } + 2 W 222 A W ¯ 222 B [ ρ → ⋅ ( H → − σ → A ) ] { [ ρ → ( H → − σ → B ) ] ⋅ [ ρ → ( H → − σ → A ) ] } = W 222 A W ¯ 222 B [ ] 1 + 2 W 222 A W ¯ 222 B [ ] 2 + W 222 A W ¯ 222 B [ H → 3 − 2 H → 2 σ → A + H → σ → A 2 − H → 2 σ → B + 2 H → σ → A σ → B − σ → A 2 σ → B ] ⋅ ρ → 3
Δ W 333 E 2 = W 131 A W ¯ 131 B [ ( − 2 H → 2 σ → A − H → 2 σ → B ) ] ⋅ ρ → 3
W 422 E = − 1 Ψ ( 2 W 311 A W ¯ 222 B + 4 W 220 A W ¯ 131 B + 6 W 222 A W ¯ 131 B + 8 W 131 A W ¯ 040 B )
W 422 E 1 = 2 W 311 A W ¯ 222 B [ ( H → − σ → A ) ⋅ ( H → − σ → A ) ] [ ( H → − σ → A ) ⋅ ρ → ] [ ( H → − σ → B ) ⋅ ρ → ] = W 311 A W ¯ 222 B [ ( H → − σ → A ) ⋅ ( H → − σ → A ) ] ( ρ → ⋅ ρ → ) [ ( H → − σ → A ) ⋅ ( H → − σ → B ) ] + W 311 A W ¯ 222 B [ ( H → − σ → A ) ⋅ ( H → − σ → A ) ] [ ( H → − σ → A ) ⋅ ( H → − σ → B ) ] ⋅ ρ → 2 = W 311 A W ¯ 222 B [ … ] + W 311 A W ¯ 222 B [ ( H → ⋅ H → ) H → 2 − ( H → ⋅ H → ) H → σ → A − ( H → ⋅ H → ) H → σ → B + ( H → ⋅ H → ) σ → A σ → B − 2 ( H → ⋅ σ → A ) H → 2 + 2 ( H → ⋅ σ → A ) H → σ → A + 2 ( H → ⋅ σ → A ) H → σ → B − 2 ( H → ⋅ σ → A ) σ → A σ → B + ( σ → A ⋅ σ → B ) H → 2 − ( σ → A ⋅ σ → B ) H → σ → A − ( σ → A ⋅ σ → B ) H → σ → B + ( σ → A ⋅ σ → B ) σ → A σ → B ] ⋅ ρ → 2
Δ W 422 E 1 = W 311 A W ¯ 222 B [ − ( H → ⋅ H → ) H → σ → A − ( H → ⋅ H → ) H → σ → B − 2 ( H → ⋅ σ → A ) H → 2 ] ⋅ ρ → 2
W 422 E 2 = 4 W 220 A W ¯ 131 B [ ( H → − σ → A ) ⋅ ( H → − σ → A ) ] [ ( H → − σ → B ) ⋅ ρ → ] 2 = 2 W 220 A W ¯ 131 B [ ( H → − σ → A ) ⋅ ( H → − σ → A ) ] ( ρ → ⋅ ρ → ) [ ( H → − σ → B ) ⋅ ( H → − σ → B ) ] + 2 W 220 A W ¯ 131 B [ ( H → − σ → A ) ⋅ ( H → − σ → A ) ] ( H → − σ → B ) 2 ⋅ ρ → 2 = 2 W 220 A W ¯ 131 B [ … ] + 2 W 220 A W ¯ 131 B [ ( H → ⋅ H → ) H → 2 − 2 ( H → ⋅ H → ) H → σ → B + ( H → ⋅ H → ) σ → B 2 − 2 ( H → ⋅ σ → A ) H → 2 + 4 ( H → ⋅ σ → A ) H → σ → B − 2 ( H → ⋅ σ → A ) σ → B 2 + ( σ → A ⋅ σ → A ) H → 2 − 2 ( σ → A ⋅ σ → A ) H → σ → B + ( σ → A ⋅ σ → A ) σ → B 2 ] ⋅ ρ → 2
Δ W 422 E 2 = 2 W 220 A W ¯ 131 B [ − 2 ( H → ⋅ H → ) H → σ → B − 2 ( H → ⋅ σ → A ) H → 2 ] ⋅ ρ → 2
W 422 E 3 = 2 W 222 A W ¯ 131 B [ ( H → − σ → A ) ⋅ ( H → − σ → B ) ] [ ( H → − σ → A ) ⋅ ρ → ] 2 + 4 W 222 A W ¯ 131 B [ ( H → − σ → A ) ⋅ ( H → − σ → B ) ] [ ( H → − σ → A ) ⋅ ρ → ] [ ( H → − σ → B ) ⋅ ρ → ]
W 422 E 31 = 2 W 222 A W ¯ 131 B [ ( H → − σ → A ) ⋅ ( H → − σ → B ) ] [ ( H → − σ → A ) ⋅ ρ → ] 2 = W 222 A W ¯ 131 B [ ( H → − σ → A ) ⋅ ( H → − σ → A ) ] ( ρ → ⋅ ρ → ) [ ( H → − σ → A ) ⋅ ( H → − σ → B ) ] + W 222 A W ¯ 131 B [ ( H → − σ → A ) 2 ⋅ ρ → 2 ] [ ( H → − σ → A ) ⋅ ( H → − σ → B ) ] = W 222 A W ¯ 131 B [ … ] + W 222 A W ¯ 131 B [ ( H → ⋅ H → ) H → 2 − ( H → ⋅ σ → A ) H → 2 − ( H → ⋅ σ → B ) H → 2 + ( σ → A ⋅ σ → B ) H → 2 − 2 ( H → ⋅ H → ) H → σ → A + 2 ( H → ⋅ σ → A ) H → σ → A + 2 ( H → ⋅ σ → B ) H → σ → A − 2 ( σ → A ⋅ σ → B ) H → σ → A + ( H → ⋅ H → ) σ → A 2 − ( H → ⋅ σ → A ) σ → A 2 − ( H → ⋅ σ → B ) σ → A 2 + ( σ → A ⋅ σ → B ) σ → A 2 ] ⋅ ρ → 2
Δ W 422 E 31 = W 222 A W ¯ 131 B [ − ( H → ⋅ σ → A ) H → 2 − ( H → ⋅ σ → B ) H → 2 − 2 ( H → ⋅ H → ) H → σ → A ] ⋅ ρ → 2
W 422 E 32 = 4 W 222 A W ¯ 131 B [ ( H → − σ → A ) ⋅ ( H → − σ → B ) ] [ ( H → − σ → A ) ⋅ ρ → ] [ ( H → − σ → B ) ⋅ ρ → ] = 2 W 222 A W ¯ 131 B [ ( H → − σ → A ) ⋅ ( H → − σ → B ) ] 2 ( ρ → ⋅ ρ → ) + 2 W 222 A W ¯ 131 B [ ( H → − σ → A ) ⋅ ( H → − σ → B ) ] [ ( H → − σ → A ) ( H → − σ → B ) ] ⋅ ρ → 2 = 2 W 222 A W ¯ 131 B [ … ] + 2 W 222 A W ¯ 131 B [ ( H → ⋅ H → ) H → 2 − ( H → ⋅ H → ) H → σ → A − ( H → ⋅ H → ) H → σ → B + ( H → ⋅ H → ) σ → A σ → B − ( H → ⋅ σ → A ) H → 2 + ( H → ⋅ σ → A ) H → σ → A + ( H → ⋅ σ → A ) H → σ → B − ( H → ⋅ σ → A ) σ → A σ → B − ( H → ⋅ σ → B ) H → 2 + ( H → ⋅ σ → B ) H → σ → A + ( H → ⋅ σ → B ) H → σ → B − ( H → ⋅ σ → B ) σ → A σ → B + ( σ → A ⋅ σ → B ) H → 2 − ( σ → A ⋅ σ → B ) H → σ → A − ( σ → A ⋅ σ → B ) H → σ → B + ( σ → A ⋅ σ → B ) σ → A σ → B ] ⋅ ρ → 2
Δ W 422 E 32 = 2 W 222 A W ¯ 131 B [ − ( H → ⋅ H → ) H → σ → A − ( H → ⋅ H → ) H → σ → B − ( H → ⋅ σ → A ) H → 2 − ( H → ⋅ σ → B ) H → 2 ] ⋅ ρ → 2
W 422 E 4 = 8 W 131 A W ¯ 040 B [ ( H → − σ → B ) ⋅ ( H → − σ → B ) ] [ ( H → − σ → A ) ⋅ ρ → ] 2 = 4 W 131 A W ¯ 040 B [ ( H → − σ → B ) ⋅ ( H → − σ → B ) ] ( ρ → ⋅ ρ → ) [ ( H → − σ → A ) ⋅ ( H → − σ → A ) ] + 4 W 131 A W ¯ 040 B [ ( H → − σ → B ) ⋅ ( H → − σ → B ) ] ( H → − σ → A ) 2 ⋅ ρ → 2 = 4 W 131 A W ¯ 040 B [ … ] + 4 W 131 A W ¯ 040 B [ ( H → ⋅ H → ) H → 2 − 2 ( H → ⋅ H → ) H → σ → A + ( H → ⋅ H → ) σ → A 2 − 2 ( H → ⋅ σ → B ) H → 2 + 4 ( H → ⋅ σ → B ) H → σ → A − 2 ( H → ⋅ σ → B ) σ → A 2 + ( σ → B ⋅ σ → B ) H → 2 − 2 ( σ → B ⋅ σ → B ) H → σ → A + ( σ → B ⋅ σ → B ) σ → A 2 ] ⋅ ρ → 2
Δ W 422 E 4 = 4 W 131 A W ¯ 040 B [ − 2 ( H → ⋅ H → ) H → σ → A − 2 ( H → ⋅ σ → B ) H → 2 ] ⋅ ρ → 2
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