Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 18, Number 1, 2022



Article Number  3  
Number of page(s)  4  
DOI  https://doi.org/10.1051/jeos/2022004  
Published online  06 July 2022 
Short Communication
It is a sufficient condition only, not a necessary and sufficient condition, for decomposing wavefront aberrations
Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan
^{*} Corresponding author: pdlin@mail.ncku.edu.tw
Received:
14
March
2022
Accepted:
22
June
2022
The classic equation for decomposing the wavefront aberrations of axissymmetrical optical systems has the form,$$W({h}_{0},\rho ,\varphi )=\sum _{j=0}^{\propto}\mathrm{}\sum _{p=0}^{\propto}\mathrm{}\sum _{m=0}^{\propto}\mathrm{}{C}_{\left(2j+m\right)\left(2p+m\right)m}\left({h}_{0}{)}^{2j+m}\right(\rho {)}^{2p+m}(\mathrm{cos}\varphi {)}^{m}$$where j, p and m are nonnegative integers, ρ and ϕ are the polar coordinates of the pupil, and h_{0} is the object height. However, one nonzero component of the aberrations (i.e., C_{133}h_{0}ρ^{3}cos^{3}ϕ) is missing from this equation when the image plane is not the Gaussian image plane. This implies that the equation is a sufficient condition only, rather than a necessary and sufficient condition, since it cannot guarantee that all of the components of the aberrations can be found. Accordingly, this paper presents a new method for determining all the components of aberrations of any order. The results show that three and six components of the secondary and tertiary aberrations, respectively, are missing in the existing literature.
Key words: Aberrations / Imaging systems / Wavefront
© The Author(s), published by EDP Sciences, 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The wavefront and ray aberrations of axissymmetrical systems have attracted significant attention in the literature [1–12]. The usual equation for decomposing the monochromatic wavefront aberration W(h_{0}, ρ, ϕ) into different orders and components is given as (e.g., Eq. (3.31b) of [5]),$$W({h}_{0},\rho ,\varphi )=\sum _{j=0}^{\propto}\mathrm{}\sum _{p=0}^{\propto}\mathrm{}\sum _{m=0}^{\propto}\mathrm{}{C}_{\left(2j+m\right)\left(2p+m\right)m}\left({h}_{0}{)}^{2j+m}\right(\rho {)}^{2p+m}(\mathrm{cos}\varphi {)}^{m}$$(1)where j, p and m are nonnegative integers, ρ and ϕ are the polar coordinates of the pupil, and h_{0} is the object height. The sum of the powers of h_{0} and ρ gives the order of the related component. That is,$$2q=2\left(j+p+m\right)$$(2)
For example, if the piston term is included, the primary (i.e., fourthorder W_{4th}) aberrations are obtained from equation (1) with 2q = 4 as,$${W}_{4\mathrm{th}}={C}_{400}{h}_{0}^{4}+{C}_{040}{\rho}^{4}+{C}_{131}{h}_{0}{\rho}^{3}\mathrm{cos}\varphi +{C}_{222}{h}_{0}^{2}{\rho}^{2}(\mathrm{cos}\varphi {)}^{2}+{C}_{220}{h}_{0}^{2}{\rho}^{2}+{C}_{311}{h}_{0}^{3}\rho \mathrm{cos}\varphi $$(3)where the six components of the equation represent the piston term and the spherical, coma, astigmatism, field curvature, and distortion aberrations, respectively. The composition ability of the primary aberrations from equation (1) is echoed by Buchdahl [1], who computed the Buchdahl aberration coefficients to determine the wavefront and ray aberrations of axissymmetrical systems. However, the question arises as to whether equation (1) provides all the components of the various order wavefront aberrations in an axissymmetrical optical system.
2 Decomposition of wavefront aberrations
Equation (1) is based on the fact that the aberration function W(h_{0}, ρ, ϕ) must satisfy the following three equations related to the fundamental axissymmetrical nature of axissymmetrical systems (e.g., p. 154 of [5]):$$W\left(0,\rho ,\varphi \right)=W\left(0,\rho ,\varphi \right)$$(4) $$W\left({h}_{0},\rho ,\varphi \right)=W\left({h}_{0},\rho ,\varphi \right)$$(5) $$W\left({h}_{0},\rho ,\varphi \right)=W\left({h}_{0},\rho ,\pi +\varphi \right)=W\left({h}_{0},\rho ,\pi \varphi \right)$$(6)
Equation (4) indicates that the aberration function of an onaxis object must be radially symmetric, and hence implies that the components of W(h_{0}, ρ, ϕ) that do not depend on h_{0} should vary as ρ^{2} (or its integer power). Equation (5) states that W(h_{0}, ρ, ϕ) must be a function of cosϕ. Finally, equation (6) shows that W(h_{0}, ρ, ϕ) must equal W(−h_{0}, ρ, π + ϕ) for an object with height h_{0} above the optical axis and W(−h_{0}, ρ, π − ϕ) for an object with height h_{0} below the optical axis. Hence, those terms that depend on ϕ should be a function of h_{0}ρcosϕ. Combining this with ϕindependent terms, it follows that W(h_{0}, ρ, ϕ) must consist of terms containing ${h}_{0}^{2}$, ρ^{2} and h_{0}ρcosϕ factors, to have a sufficient condition given by equation (1). Note that a sufficient condition is taken here to mean that any term generated by equation (1) is a component of an aberration.
The present group recently proposed a method for determining the aberrations of axissymmetrical optical systems [12]. It is shown in Figure 1 that when the image plane is not the Gaussian image plane, a nonzero component (i.e., C_{133}h_{0}ρ^{3}(cosϕ)^{3}) is missing from equation (1), even though it satisfies equations (4)–(6). This implies that equation (1) alone does not guarantee that all of the components of the aberrations can be found. In other words, equation (1) is not a necessary and sufficient condition for determining all the components of the aberrations in an axissymmetrical optical system.
Fig. 1 The variation of W_{133} = C_{133}h_{0}ρ^{3}/λ (where h_{0} = 17 mm, ρ = 21 mm, and λ = 550 µm) versus the separation of image plane V_{image} for the optical system of [12]. The Gaussian image plane of this system is located at V_{image} = 92.088474 mm when the object is placed at P_{0z} = −200 mm. This figure shows that C_{133}h_{0}ρ^{3}(cosϕ)^{3} has nonzero value when the image plane is not the Gaussian image plane. It also shows that C_{133} = 0 when V_{image} = 92.088474 mm, indicating C_{133}h_{0}ρ^{3}(cosϕ)^{3} is the defocus component of primary aberrations. 
Thus, a second question arises as to how all of these components may be found. To address this question, it is first necessary to realize that the power of cosϕ should be nonnegative. That is,$$m\ge 0$$(7)
Without any loss of generality, the power of h_{0} can be confined to a nonnegative integer value in order to have 2j + m ≥ 0. That is,$$j\ge m/2$$(8)
Mathematically, the power of ρ should be greater than or equal to the power of cosϕ, i.e., 2p + m ≥ m, which yields,$$p\ge 0$$(9)
One then has the following inequality from the sum of q = j + p + m (see Eq. (2)) and equation (8):$$p\le qm/2$$(10)
The intersection of equations (9) and (10) defines the possible range of p. That is,$$0\le p\le qm/2$$(11)
Equation (11) shows that the integer index p starts at p = 0 and ends at,$${p}_{\mathrm{max}}=\langle qm/2\rangle $$(12)where 〈q − m/2〉 is the maximum nonnegative integer value of p for a given m and q. In other words, index p belongs to the following set:$$p\in \left\{\mathrm{0,1},2,...,{p}_{\mathrm{max}}\right\}$$(13)
Furthermore, from equation (11), the possible upper limit of m which yields 0 ≤ q − m/2 is,$$m\le 2q$$(14)
The intersection of equations (7) and (14) then shows the possible domain of integer m for a given q. That is,$$0\le m\le 2q$$(15)
Given the preceding derivations, it is possible to obtain all components of any order (say, the (2q)th order) wavefront aberration for an axissymmetrical optical system by the following equation when j = q − (p + m) is used:$${W}_{\left(2q\right)\mathrm{th}\mathrm{order}}({h}_{0},\rho ,\varphi )=\sum _{p=0}^{p={p}_{\mathrm{max}}}\mathrm{}\sum _{m=0}^{m=2q}\mathrm{}{C}_{\left(2q2pm\right)\left(2p+m\right)m}\left({h}_{0}{)}^{2q2pm}\right(\rho {)}^{2p+m}(\mathrm{cos}\varphi {)}^{m}$$(16)
Note that, as shown in equation (4), components of the aberrations that do not depend on h_{0} should vary as ρ^{2}, or as its integer power only. In other words, if 2q − 2p – m = 0 and $m\ne 0$, that component generated from equation (16) does not exist.
Consider Table 1 below, which shows all the components of the secondary aberration (q = 3) of an axissymmetrical optical system for illustration purposes. The entries of the first and second columns are the values of m and p obtained from equations (15) to (13), respectively. Meanwhile, the entries of the third column denote the sequence (2j + m, 2p + m, m) = (2q − 2p − m, 2p + m, m) for each value of m. The fourth column shows the aberration component for each value of m (if it exists). Comparing Table 1 with the existing literature, it is found that three components in Table 1 (i.e., C_{244} ${h}_{0}^{2}$ρ^{4}(cosϕ)^{4}, C_{153}h_{0}ρ^{5}(cosϕ)^{3} and C_{155}h_{0}ρ^{5}(cosϕ)^{5}) are not included among the secondary aberrations given in the literature despite satisfying equations (4)–(6). In order to validate Table 1, the methodology proposed in [12] was extended to determine the values of all the secondary aberrations listed in the righthand column of the table [3]. The results confirmed that all of the secondary aberrations possessed nonzero values.
Components of secondary aberrations with q = 3 in axissymmetrical system.
The method in this study was further applied to determine all the components of the tertiary aberrations (q = 4) (see Table 2). Comparing the results in Table 2 with those in Table 3–3 of [5], it is found that six components (i.e., C_{173}h_{0}ρ^{7}(cosϕ)^{3}, C_{264} ${h}_{0}^{2}$ρ^{6}(cosϕ)^{4}, C_{355} ${h}_{0}^{3}$ρ^{5}(cosϕ)^{5}, C_{175}h_{0}ρ^{7}(cosϕ)^{5}, C_{266} ${h}_{0}^{2}$ρ^{6}(cosϕ)^{6} and C_{177}h_{0}ρ^{7}(cosϕ)^{7}) are missing from equation (1).
Components of tertiary aberrations with q = 4 in axissymmetrical system.
3 Conclusions
The wavefront aberrations W(h_{0}, ρ, ϕ) of axissymmetrical systems are generally decomposed into their various components using equation (1). However, the numerical results presented in [12] show that this equation cannot guarantee that all of the components of the primary aberrations can be found. In other words, the equation is a sufficient condition only, not a necessary and sufficient condition.
Accordingly, this study has presented a method for determining the possible domains of the nonnegative integer indices, m and p, in equation (1) such that all of the components of the aberrations can be found. It has been shown that the index j computed from equation (2) may be negative. Furthermore, three and six new components of the secondary and tertiary aberrations of an axissymmetrical system have been found, where these components all satisfy the equations describing the fundamental axissymmetrical nature of axissymmetrical systems. Overall, the method proposed in this study provides a systematic and robust approach for ensuring that all of the components of any order wavefront aberration in an axissymmetrical system can be found.
Conflict of interest
The author declares no conflicts of interest.
Funding
Ministry of Science and Technology, Taiwan (MOST) (1092221E006045).
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All Tables
All Figures
Fig. 1 The variation of W_{133} = C_{133}h_{0}ρ^{3}/λ (where h_{0} = 17 mm, ρ = 21 mm, and λ = 550 µm) versus the separation of image plane V_{image} for the optical system of [12]. The Gaussian image plane of this system is located at V_{image} = 92.088474 mm when the object is placed at P_{0z} = −200 mm. This figure shows that C_{133}h_{0}ρ^{3}(cosϕ)^{3} has nonzero value when the image plane is not the Gaussian image plane. It also shows that C_{133} = 0 when V_{image} = 92.088474 mm, indicating C_{133}h_{0}ρ^{3}(cosϕ)^{3} is the defocus component of primary aberrations. 

In the text 
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