In a recent study, the present group proposed a methodology for determining the Seidel primary aberration coefficients in terms of the polar coordinates of the source ray for an object placed at a finite distance from the entrance pupil [P. D. Lin and R. B. Johns, Opt. Express 27, 19712 (2019)]. However, that model will be failed for an object placed at infinity. It is also found that all existing works in the optics field use in-plane coordinates of the entrance pupil (i.e., Xa and ya) to investigate the aberration coefficients. Accordingly, the present study revisits the problem once again using a Taylor series expansion of a ray in terms of the object height h0 and coordinates ${\left[ {\begin{array}{cc} {\textrm{x}_{\textrm{a}}}&{\textrm{y}_{\textrm{a}}} \end{array}} \right]^{\textrm T}}$. In the proposed methodology, the independent variables of the optical system are identified and the intercept coordinates of the skew ray on the image plane are then expanded with respect to these variables. It is shown that the expressions of the Seidel primary aberration coefficients are very concise and the corresponding numerical results are in good agreement with those obtained from Zemax simulations. Notably, the method proposed in this study is also valid for objects lying at infinity provided that the collimated rays emerging from the object are incident on the entrance pupil. Moreover, the methodology can also be extended to have the numerical values of the higher-order ray aberration coefficients for axis-symmetrical systems.
1. Introduction
The most widely recognized study of monochromatic ray aberrations is that of Philip Ludwig von Seidel [1], in which he extended Gaussian theory of axis-symmetrical systems to describe all ray aberrations up to the third-order. In his study, objects located on the meridional plane of an axis-symmetrical system were described as ${\bar{\textrm P}_0} = {\left[ {\begin{array}{ccc} 0&{\textrm{h}_{0}}&{{\textrm{P}_{0\textrm{z}}}} \end{array}} \right]^{\textrm{T}}}$, where $\textrm{h}_{0}$ is the height of the object and ${\textrm{P}_{0\textrm{z}}}$ is object distance (see Fig. 1 of [2]). For a ray originating from point ${\bar{\textrm P}_0}$ and passing through the entrance pupil at polar coordinates ${\mathrm{\rho}}$ and ${\mathrm{\phi}}$ (Fig. 1), the incidence point of the ray on the image plane located at ${\textrm{P}_{\textrm{nz}}}$ is given by (p.63 of [3])
(1)$${\bar{\textrm P}_{\textrm{n}}} = {\left[ {\begin{array}{ccc} {{\textrm{P}_{\textrm{nx}}}}&{{\textrm{P}_{\textrm{ny}}}}&{{{\textrm P}_{\textrm{nz}}}} \end{array}} \right]^{\textrm T}} = {\left[ {\begin{array}{ccc} {\Delta {\textrm{P}_{\textrm{nx}}}}&{{{\textrm A}_2}{{\textrm h}_0} + \Delta {\textrm{P}_{\textrm{ny}}}}&{{{\textrm P}_{\textrm{nz}}}} \end{array}} \right]^{\textrm T}}, $$
where $\textrm{A}_{2}$ is the lateral magnification on the Gaussian image plane; and $\Delta {\textrm{P}_{\textrm{nx}}}$ and $\Delta {\textrm{P}_{\textrm{ny}}}$ are the transverse aberrations of the incidence point on the image plane and are given respectively by (2a)$$\Delta {\textrm{P}_{\textrm{nx}}} = \textrm{A}_{1}{\mathrm{\rho}} \textrm{S}{\mathrm{\phi}} + {\textrm{B}_{1}}{{\mathrm{\rho}} ^3}\textrm{S}{\mathrm{\phi}} + {\textrm{B}_{2}}\textrm{h}_{0}{{\mathrm{\rho}} ^2}\textrm{S}(2{\mathrm{\phi}} ) + ({\textrm{B}_{3}} + {\textrm{B}_{4}})\textrm{h}_0^2{\mathrm{\rho}} \textrm{S}{\mathrm{\phi}} \textrm{ ,}$$
(2b)$$\Delta {\textrm{P}_{\textrm{ny}}} = \textrm{A}_{1}{\mathrm{\rho}} \textrm{C}{\mathrm{\phi}} + {\textrm{B}_{1}}{{\mathrm{\rho}} ^3}\textrm{C}{\mathrm{\phi}} + {\textrm{B}_{2}}{\textrm{h}_0}{{\mathrm{\rho}} ^2}[{\textrm{2 + }\textrm{C}(2{\mathrm{\phi}} )} ]+ (3{\textrm{B}_{3}} + {\textrm{B}_{4}})\textrm{h}_0^2{\mathrm{\rho}} \textrm{C}{\mathrm{\phi}} + {\textrm{B}_{5}}\textrm{h}_0^3.$$
For reasons of simplicity, in this paper we use C and S to denote cosine and sine, respectively. The aberrations in Eqs. (2a) and (2b) represent the distance by which the ray misses the ideal image point ${\left[ {\begin{array}{ccc} 0&{\textrm{A}_{2}\textrm{h}_{0}}&{{\textrm{P}_{\textrm{nz}}}} \end{array}} \right]^{\textrm{T}}}$ on the image plane, as determined by paraxial ray-tracing equations. It is noted that Eqs. (2a) and (2b) each contain a single first-order term (referred to hereafter as the A coefficient) and five third-order terms (referred to hereafter as the B coefficients). ${\textrm{A}_1}$ is the transverse defocus aberration coefficient, while ${\textrm{B}_{1}}$, ${\textrm{B}_{2}}$, ${\textrm{B}_{3}}$, ${\textrm{B}_{4}}$ and ${\textrm{B}_{5}}$ are the Seidel primary ray aberration coefficients of spherical aberration, coma, astigmatism, field curvature and distortion, respectively.The design and development of optical systems relies on a thorough theoretical understanding of optical aberrations [4]. Determining the Seidel primary ray aberration coefficients is a challenging task, even when using computational tools, since optical systems invariably contain multiple aberrations simultaneously. Nonetheless, many approaches for deriving the equations required to compute the Seidel primary aberration coefficients have been proposed [5–19]. One of the most widely used methods in optical software (e.g., Zemax [20]) is that proposed by Buchdahl [5], in which the marginal paraxial ray and chief ray are traced using the first-order raytracing method and the results are then used to estimate the third-order Buchdahl aberration coefficients (denoted as ${{\mathrm{\sigma}} _{\textrm j}}$, j=1-5). His method can generate aberration coefficients of any order although ninth order was the practical limit due to the enormous algebraic complexity.
This paper proposes an alternative method for determining the values of A and B coefficients of the Seidel primary ray aberrations for an object placed at a finite distance from the entrance pupil in an axis-symmetrical system. The method is based on the relationship between Eqs. (2a)–(2b) and the Taylor series of a skew ray passing through the system. It is shown that the expressions derived for the aberration coefficients are very concise and the numerical results are in good agreement with those obtained from Zemax simulations. Notably, the method is valid not only for objects placed at a finite distance from the entrance pupil, but also those lying at infinity, provided that its collimated rays emerging from the object are incident on the entrance pupil. Moreover, the proposed method can be easily extended to derive the higher-order ray aberration coefficients for axis-symmetrical systems. To explore the aberrations of non-axially-symmetrical systems, the references given in [21–25] are important work.
For analytical convenience, the fourth component of the homogeneous coordinate notation (Chap. 1 of [26]) is deliberately omitted in this study since the first three components are sufficient to address the considered problem. In addition, the vectors of interest are expressed in the form of column-wise matrixes (e.g., ${\left[ {\begin{array}{ccc} {\textrm{x}_{\textrm{a}}}&{\textrm{y}_{\textrm{a}}}&{{z_{\textrm{a}}}} \end{array}} \right]^{\textrm T}}$), while the lengths and angles are given in units of mm and degrees, respectively.
2. Ray aberration polynomial for object at finite distance
Figure 2 shows the axis-symmetrical system considered in the present study. As shown, the system comprises six elements (k=6) and nine boundary surfaces (n=9). The parameters of the system are listed in Table 1 with the exception of the image plane position. In accordance with convention, the label i=0 is assigned to the source ray ${\bar{\textrm R}_0} = {\left[ {\begin{array}{cc} {{{\bar{\textrm P}}_0}}&{{{\bar{\ell }}_0}} \end{array}} \right]^{\textrm T}}$ originating from an object point ${\bar{\textrm P}_0}$. Since the system is symmetrical about the optical axis, the object point ${\bar{\textrm P}_0}$ placed at ${\textrm{P}_{0\textrm{z}}}$ can be positioned with no loss of generality on the meridional plane with a height $\textrm{h}_{0}$, i.e.,
(3a)$${\overline {\textrm P}_0} = {\left[ {\begin{array}{ccc} 0&{\textrm{h}_{0}}&{{\textrm{P}_{0\textrm{z}}}} \end{array}} \right]^{\textrm T}}. $$
In general, the unit directional vector $\overline {\ell } {}_0$ of the skew ray ${\bar{\textrm R}_0}$ emitted from the object point ${\bar{\textrm P}_0}$ can be expressed in terms of two independent variables, e.g., ${\left[ {\begin{array}{cc} {{{\mathrm{\alpha}}_0}}&{{{\mathrm{\beta}}_0}} \end{array}} \right]^{\textrm T}}$, i.e., (3b)$$\overline {\ell } {}_0 = {\left[ {\begin{array}{ccc} {{\ell_{\textrm{0x}}}({{\mathrm{\alpha}}_0},{{\mathrm{\beta}}_0})}&{{\ell_{\textrm{0y}}}({{\mathrm{\alpha}}_0},{{\mathrm{\beta}}_0})}&{{\ell_{\textrm{0z}}}({{\mathrm{\alpha}}_0},{{\mathrm{\beta}}_0})} \end{array}} \right]^{\textrm T}}. $$
Table 1. Specification of illustrative rotationally-symmetric optical system shown in Fig. 2. Note that entrance pupil is located at distance of $22.103275\,\textrm{mm}$ from vertex of the first surface.
However, most existing works on geometrical optics define $\overline {\ell } {}_0$ in terms of the intercept point ${\left[ {\begin{array}{ccc} {\textrm{x}_{\textrm{a}}}&{\textrm{y}_{\textrm{a}}}&{{\textrm{v}_{\textrm{entrance}}}} \end{array}} \right]^{\textrm T}}$ of the ray on the entrance pupil (see Fig. 3). Referring to Fig. 3, the incidence point ${\left[ {\begin{array}{ccc} {\textrm{x}_{\textrm{a}}}&{\textrm{y}_{\textrm{a}}}&{{\textrm{v}_{\textrm{entrance}}}} \end{array}} \right]^{\textrm T}}$ of the skew ray ${\bar{\textrm R}_0}$ on the entrance pupil can be formulated as
(4a)$$\left[ {\begin{array}{c} {\textrm{x}_{\textrm{a}}}\\ {\textrm{y}_{\textrm{a}}}\\ {{\textrm{v}_{\textrm{entrance}}}} \end{array}} \right] = {\bar{\textrm P}_0} + {\mathrm{\lambda}} {\bar{\ell }_0} = \left[ {\begin{array}{c} 0\\ {\textrm{h}_{0}}\\ {{{\textrm P}_{\textrm{0z}}}} \end{array}} \right] + {\mathrm{\lambda}} \left[ {\begin{array}{c} {{\ell_{\textrm{0x}}}({{\mathrm{\alpha}}_0},{{\mathrm{\beta}}_0})}\\ {{\ell_{\textrm{0y}}}({{\mathrm{\alpha}}_0},{{\mathrm{\beta}}_0})}\\ {{\ell_{0z}}({{\mathrm{\alpha}}_0},{{\mathrm{\beta}}_0})} \end{array}} \right], $$
where ${\textrm{v}_{\textrm{entrance}}}$ is the position of the entrance pupil measured from the origin of the system frame ${(\textrm{xyz})_0}$. From Eq. (4a), in-plane coordinates ${\left[ {\begin{array}{cc} {\textrm{x}_{\textrm{a}}}&{\textrm{y}_{\textrm{a}}} \end{array}} \right]^{\textrm T}}$ can be obtained mathematically as (4b)$$\textrm{x}_{\textrm{a}} = ({\textrm{v}_{\textrm{entrance}}} - {{\textrm P}_{\textrm{0z}}})\frac{{{\ell _{\textrm{0x}}}({{\mathrm{\alpha}} _0},{{\mathrm{\beta}} _0})}}{{{\ell _{0z}}({{\mathrm{\alpha}} _0},{{\mathrm{\beta}} _0})}},$$
(4c)$$\textrm{y}_{\textrm{a}} = \textrm{h}_{0} + ({\textrm{v}_{\textrm{entrance}}} - {{\textrm P}_{\textrm{0z}}})\frac{{{\ell _{\textrm{0y}}}({{\mathrm{\alpha}} _0},{{\mathrm{\beta}} _0})}}{{{\ell _{\textrm{0z}}}({{\mathrm{\alpha}} _0},{{\mathrm{\beta}} _0})}}. $$
Equations (4b) and (4c) not only give the incidence point ${\left[ {\begin{array}{cc} {\textrm{x}_{\textrm{a}}}&{\textrm{y}_{\textrm{a}}} \end{array}} \right]^{\textrm T}}$ of ray ${\bar{\textrm R}_0} = {\left[ {\begin{array}{cc} {{{\bar{\textrm P}}_0}}&{{{\bar{\ell }}_0}} \end{array}} \right]^{\textrm T}}$ on the entrance pupil, but also provide a useful basis for exploring the ray aberrations. It is noted from Eq. (4c) that $(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})$ not $\textrm{y}_{\textrm{a}}$ is the independent variable since $\textrm{y}_{\textrm{a}}$ is a function not only of ${{\mathrm{\alpha}} _0}$ and ${{\mathrm{\beta}} _0}$, but also of $\textrm{h}_{0}$. Consequently, the present study defines (5)$${\overline {\textrm X} _{\textrm{Seidel}}} = {\left[ {\begin{array}{ccc} {\textrm{h}_{0}}&{\textrm{x}_{\textrm{a}}}&{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})} \end{array}} \right]^{\textrm T}}$$
as the Seidel variable vector to determine the Seidel primary ray aberration coefficients. Mathematically, the coordinates of incidence point ${\bar{\textrm P}_{\textrm{n}}} = {\left[ {\begin{array}{ccc} {{{\textrm P}_{\textrm{nx}}}}&{{{\textrm P}_{\textrm{ny}}}}&{{{\textrm P}_{\textrm{nz}}}} \end{array}} \right]^{\textrm T}}$ on the image plane are functions of ${\bar{\textrm X}_{\textrm{Seidel}}}$, i.e., (6)$${\bar{\textrm P}_{\textrm{n}}}({\bar{\textrm X}_{\textrm{Seidel}}}) = \left[ {\begin{array}{ccc} {{{\textrm P}_{\textrm{nx}}}({\textrm h}_{0},{\textrm x}_{\textrm a},{\textrm y}_{\textrm a} - {\textrm h}_{0})}&{{{\textrm P}_{\textrm{ny}}}({\textrm h}_{0},{\textrm x}_{\textrm{a}},{\textrm y}_{\textrm a} - {\textrm h}_{0})}&{{{\textrm P}_{\textrm{nz}}}} \end{array}} \right].$$
The following two equations are thus obtained when ${\textrm{P}_{\textrm{nx}}}$ and ${\textrm{P}_{\textrm{ny}}}$ of Eq. (6) are expanded as Taylor series with respect to ${\bar{\textrm X}_{\textrm{Seidel}}}$: (7a)$$\begin{array}{l} {\textrm{P}_{\textrm{nx}}}(\textrm{h}_{0},\textrm{x}_{\textrm{a}},\textrm{y}_{\textrm{a}} - \textrm{h}_{0}) = {\textrm{P}_{\textrm{nx}}}(0,0,0) + \left( {\frac{{\partial {\textrm{P}_{\textrm{nx}}}}}{{\partial {{{\bar{\textrm X}}}_{\textrm{Seidel}}}}}} \right){{\bar{\textrm X}}_{\textrm{Seidel}}}\\ + \frac{\textrm{1}}{\textrm{2}}\bar{\textrm X}_{\textrm{Seidel}}^{\textrm{T}}\ \left( {\frac{{\partial {}^2{\textrm{P}_{\textrm{nx}}}}}{{\partial \bar{\textrm X}_{\textrm{Seidel}}^{2}}}} \right){{\bar{\textrm X}}_{\textrm{Seidel}}} + \frac{1}{6}\bar{\textrm X}_{\textrm{Seidel}}^{\textrm{T}}\bar{\textrm X}_{\textrm{Seidel}}^{\textrm{T}}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \bar{\textrm X}_{\textrm{Seidel}}^{3}}}} \right){{\bar{\textrm X}}_{\textrm{Seidel}}} + ...\\ \equiv 0 + \Delta {\textrm{P}_{\textrm{nx/1st}}} + \Delta {\textrm{P}_{\textrm{nx/2nd}}} + \Delta {\textrm{P}_{\textrm{nx/3rd}}} + ... \end{array}, $$
(7b)$$\begin{array}{l} {\textrm{P}_{\textrm{ny}}}(\textrm{h}_{0},\textrm{x}_{\textrm{a}},\textrm{y}_{\textrm{a}} - \textrm{h}_{0}) = {\textrm{P}_{\textrm{ny}}}(0,0,0) + \left( {\frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial {{{\bar{\textrm X}}}_{\textrm{Seidel}}}}}} \right){{\bar{\textrm X}}_{\textrm{Seidel}}}\\ + \frac{\textrm{1}}{\textrm{2}}\bar{\textrm X}_{\textrm{Seidel}}^{\textrm{T}}\left( {\frac{{\partial {}^2{\textrm{P}_{\textrm{ny}}}}}{{\partial \bar{\textrm X}_{\textrm{Seidel}}^{2}}}} \right){{\bar{\textrm X}}_{\textrm{Seidel}}} + \frac{1}{6}\bar{\textrm X}_{\textrm{Seidel}}^{\textrm{T}}\bar{\textrm X}_{\textrm{Seidel}}^{\textrm{T}}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \bar{\textrm X}_{\textrm{Seidel}}^{3}}}} \right){{\bar{\textrm X}}_{\textrm{Seidel}}} + ...\\ \equiv 0 + \Delta {\textrm{P}_{\textrm{ny/1st}}} + \Delta {\textrm{P}_{\textrm{ny/2nd}}} + \Delta {\textrm{P}_{\textrm{ny/3rd}}} + ... \end{array}. $$
It should be noted that terms ${{\partial {}^{\textrm{m}}{\textrm{P}_{\textrm{nx}}}} \mathord{\left/ {\vphantom {{\partial {}^{\textrm{m}}{\textrm{P}_{\textrm{nx}}}} {\partial \bar{\textrm X}_{\textrm{Seidel}}^{\textrm{m}}}}} \right.} {\partial \bar{\textrm X}_{\textrm{Seidel}}^{\textrm{m}}}}$ and ${{\partial {}^{\textrm{m}}{\textrm{P}_{\textrm{ny}}}} \mathord{\left/ {\vphantom {{\partial {}^{\textrm{m}}{\textrm{P}_{\textrm{ny}}}} {\partial \bar{\textrm X}_{\textrm{Seidel}}^{\textrm{m}}}}} \right.} {\partial \bar{\textrm X}_{\textrm{Seidel}}^{\textrm{m}}}}$ (m=1-3) in Eqs. (7a) and (7b) must be evaluated at the ray originating from object ${\overline {\textrm P} _0} = {\left[ {\begin{array}{ccc} 0&0&{{\textrm{P}_{0\textrm{z}}}} \end{array}} \right]^{\textrm{T}}}$ and traveling along the optical axis. In other words, they are evaluated by (8)$${\overline {\textrm X} _{\textrm{Seidel/optical axis}}} = {\left[ {\begin{array}{ccc} 0&0&0 \end{array}} \right]^{\textrm{T}}}. $$
${\textrm{P}_{\textrm{nx}}}(0,0,0) = 0$ and ${\textrm{P}_{\textrm{ny}}}(0,0,0) = 0$ in Eqs. (7a) and (7b) indicate that both Taylor series are centered at ${\left[ {\begin{array}{ccc} {{\textrm{P}_{\textrm{nx}}}}&{{\textrm{P}_{\textrm{ny}}}}&{{\textrm{P}_{\textrm{nz}}}} \end{array}} \right]^{\textrm{T}}} = {\left[ {\begin{array}{ccc} 0&0&{{\textrm{P}_{\textrm{nz}}}} \end{array}} \right]^{\textrm{T}}}$, i.e., the intercept point of the optical axis and the image plane. In addition, $\Delta {\textrm{P}_{\textrm{nx/1st}}}$ and $\Delta {\textrm{P}_{\textrm{ny/1st}}}$, $\Delta {\textrm{P}_{\textrm{nx/2nd}}}$ and $\Delta {\textrm{P}_{\textrm{ny/2nd}}}$, and $\Delta {\textrm{P}_{\textrm{nx/3rd}}}$ and $\Delta {\textrm{P}_{\textrm{ny/3rd}}}$ are the corresponding first-, second-, and third-order Taylor series expansions, respectively. Their full explicit expansions are given by
(9a)$$\Delta {\textrm{P}_{\textrm{nx/1st}}} = \frac{{\partial {\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}}}\textrm{h}_{0} + \frac{{\partial {\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}}}\textrm{x}_{\textrm{a}} + \frac{{\partial {\textrm{P}_{\textrm{nx}}}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}(\textrm{y}_{\textrm{a}} - \textrm{h}_{0}), $$
(9b)$$\Delta {\textrm{P}_{\textrm{ny/1st}}} = \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}}}\textrm{h}_{0} + \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}}}\textrm{x}_{\textrm{a}} + \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}(\textrm{y}_{\textrm{a}} - \textrm{h}_{0}), $$
(10a)$$\begin{aligned} \Delta {\textrm{P}_{\textrm{nx/2nd}}} =& \frac{\textrm{1}}{\textrm{2}}\left( {\frac{{{\partial^2}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_0^2}}\textrm{h}_0^2 + 2\frac{{{\partial^2}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}}}\textrm{h}_{0}\textrm{x}_{\textrm{a}}} \right. + 2\frac{{{\partial ^2}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{h}_{0}(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\\ &\left. { + \frac{{{\partial^2}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}^2}}\textrm{x}_{\textrm{a}}^2 + 2\frac{{{\partial^2}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{x}_{\textrm{a}}(\textrm{y}_{\textrm{a}} - \textrm{h}_{0}) + \frac{{{\partial^2}{\textrm{P}_{\textrm{nx}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}{{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}} \right), \end{aligned}$$
(10b)$$\begin{aligned} \Delta {\textrm{P}_{\textrm{ny/2nd}}} = &\frac{\textrm{1}}{\textrm{2}}\left( {\frac{{{\partial^2}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^2}}\textrm{h}_0^2 + 2\frac{{{\partial^2}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}}}\textrm{h}_{0}\textrm{x}_{\textrm{a}}} \right. + 2\frac{{{\partial ^2}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{h}_{0}(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\\ & \left. { + \frac{{{\partial^2}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2}}\textrm{x}_{\textrm{a}}^2 + 2\frac{{{\partial^2}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{x}_{\textrm{a}}(\textrm{y}_{\textrm{a}} - \textrm{h}_{0}) + \frac{{{\partial^2}{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}{{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}} \right), \end{aligned}$$
(11a)$$\begin{aligned} \Delta {\textrm{P}_{\textrm{nx/3rd}}} = &\frac{1}{6}\left( {\frac{{{\partial^3}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_0^3}}\textrm{h}_0^3} \right. + 3\frac{{{\partial ^3}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_0^2\partial \textrm{x}_{\textrm{a}}}}\textrm{h}_0^2\textrm{x}_{\textrm{a}} + 3\frac{{{\partial ^3}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_0^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{h}_0^2(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\\ & + 3\frac{{{\partial ^3}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}^2}}\textrm{h}_{0}\textrm{x}_{\textrm{a}}^2 + 6\frac{{{\partial ^3}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{h}_{0}\textrm{x}_{\textrm{a}}(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\\ & + 3\frac{{{\partial ^3}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}\textrm{h}_{0}{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})^2} + \frac{{{\partial ^3}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}^3}}\textrm{x}_{\textrm{a}}^3 + 3\frac{{{\partial ^3}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{x}_{\textrm{a}}^2(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\\ &\left. { + 3\frac{{{\partial^3}{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}\textrm{x}_{\textrm{a}}{{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2} + \frac{{{\partial^3}{\textrm{P}_{\textrm{nx}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}}{{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}} \right), \end{aligned}$$
(11b)$$\begin{aligned} \Delta {\textrm{P}_{\textrm{ny/3rd}}} = &\frac{1}{6}\left( {\frac{{{\partial^3}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^3}}\textrm{h}_0^3} \right. + 3\frac{{{\partial ^3}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^2\partial \textrm{x}_{\textrm{a}}}}\textrm{h}_0^2\textrm{x}_{\textrm{a}} + 3\frac{{{\partial ^3}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{h}_0^2(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\\ & + 3\frac{{{\partial ^3}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}^2}}\textrm{h}_{0}\textrm{x}_{\textrm{a}}^2 + 6\frac{{{\partial ^3}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{h}_{0}\textrm{x}_{\textrm{a}}(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\\ & + 3\frac{{{\partial ^3}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}\textrm{h}_{0}{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})^2} + \frac{{{\partial ^3}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^3}}\textrm{x}_{\textrm{a}}^3 + 3\frac{{{\partial ^3}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{x}_{\textrm{a}}^2(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\\ &\left. { + 3\frac{{{\partial^3}{\textrm{P}_{\textrm{nx}}}y}}{{\partial \textrm{x}_{\textrm{a}}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}\textrm{x}_{\textrm{a}}{{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2} + \frac{{{\partial^3}{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}}{{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}} \right). \end{aligned}$$
Due to the axis-symmetrical nature of the considered system, various terms in Eqs. (9a)–(11b) disappear [8], and hence Eqs. (9a) and (11b) become
(12a)$$\Delta {\textrm{P}_{\textrm{nx/1st}}} = \frac{{\partial {\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}}}\textrm{x}_{\textrm{a}}, $$
(12b)$$\Delta {\textrm{P}_{\textrm{ny/1st}}} = \left( {\frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}}} - \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}} \right)\textrm{h}_{0} + \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{y}_{\textrm{a}}, $$
(13a)$$\Delta {\textrm{P}_{\textrm{nx/2nd}}} = 0, $$
(13b)$$\Delta {\textrm{P}_{\textrm{ny/2nd}}} = 0, $$
(14a)$$\begin{aligned} \Delta {\textrm{P}_{\textrm{nx/3rd}}} = &\frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_0^2\partial \textrm{x}_{\textrm{a}}}}\textrm{x}_{\textrm{a}}\textrm{h}_0^2 + \frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{h}_{0}\textrm{x}_{\textrm{a}}(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\\ & + \frac{1}{6}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}^3}}\textrm{x}_{\textrm{a}}^3 + \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}\textrm{x}_{\textrm{a}}{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})^2} \end{aligned}, $$
(14b)$$\begin{aligned} \Delta {\textrm{P}_{\textrm{ny/3rd}}} = &\frac{1}{6}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^3}}\textrm{h}_0^3 + \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{h}_0^2(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\\ & + \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}^2}}\textrm{h}_{0}\textrm{x}_{\textrm{a}}^2 + \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}\textrm{h}_{0}{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})^2}\\ & + \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\textrm{x}_{\textrm{a}}^2(\textrm{y}_{\textrm{a}} - \textrm{h}_{0}) + \frac{1}{6}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}}{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})^3} \end{aligned}. $$
Equations (12a) and (12b) determine the transverse defocus aberration coefficient $\textrm{A}_{1}$ and lateral magnification $\textrm{A}_{2}$. Equations (13a) and (13b) indicate that the second-order Tylor series expansions do not have any contribution to the ray aberrations.
3. Primary aberration coefficients for object at finite distance
Consider the entrance pupil shown in Fig. 1 having ${(\textrm{xyz})_{\textrm{a}}}$ as its coordinate frame. Let ${\mathrm{\rho}}$ ($0 \le {\mathrm{\rho}}$) be the length parameter along the $\textrm{y}_{\textrm{a}}$ axis measured from the origin of ${(\textrm{xyz})_{\textrm{a}}}$. The in-plane Cartesian coordinates, $\textrm{x}_{\textrm{a}}$ and $\textrm{y}_{\textrm{a}}$, and polar coordinates, ${\mathrm{\rho}}$ and ${\mathrm{\phi}}$ ($- \pi < {\mathrm{\phi}} \le \pi$), are related as
(15a)$${\textrm{x}_{\textrm{a}}} = {\mathrm{\rho}} \textrm{S}{\mathrm{\phi}}, $$
(15b)$${\textrm{y}_{\textrm{a}}} = {\mathrm{\rho}} \textrm{C}{\mathrm{\phi}}. $$
Substituting Eqs. (15a) and (15b) into Eqs. (12a) to (14b), the following expressions for ${\textrm{P}_{\textrm{nx}}}$ and ${\textrm{P}_{\textrm{ny}}}$ are obtained: (16a)$$\begin{aligned} {\textrm{P}_{\textrm{nx}}} = &\Delta {\textrm{P}_{\textrm{nx/1st}}} + \Delta {\textrm{P}_{\textrm{nx/2nd}}} + \Delta {\textrm{P}_{\textrm{nx/3rd}}}\\ \textrm{ = }&\frac{{\partial {\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}}}{\mathrm{\rho}} \textrm{S}{\mathrm{\phi}} + \frac{1}{6}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}^3}}{\textrm{S}^2}{\mathrm{\phi}} + 3\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}{\textrm{C}^2}{\mathrm{\phi}} } \right){{\mathrm{\rho}} ^3}\textrm{S}{\mathrm{\phi}} \\ & + \frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} - \frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}} \right)\textrm{h}_{0}{{\mathrm{\rho}} ^2}\textrm{S}2{\mathrm{\phi}} \\ & + \frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}} - 2\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} + \frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_0^2\partial \textrm{x}_{\textrm{a}}}}} \right)\textrm{h}_0^2{\mathrm{\rho}} \textrm{S}{\mathrm{\phi}} \end{aligned}, $$
(16b)$$\begin{aligned} {\textrm{P}_{\textrm{ny}}} = &\Delta {\textrm{P}_{\textrm{ny/1st}}} + \Delta {\textrm{P}_{\textrm{ny/2nd}}} + \Delta {\textrm{P}_{\textrm{ny/3rd}}}\\ = &\left( {\frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}}} - \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}} \right)\textrm{h}_{0} + \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}{\mathrm{\rho}} \textrm{C}{\mathrm{\phi}} \\ & + \frac{1}{6}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}}{\textrm{C}^2}{\mathrm{\phi}} + 3\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}{\textrm{S}^2}{\mathrm{\phi}} } \right){{\mathrm{\rho}} ^3}\textrm{C}{\mathrm{\phi}} \\ & + \frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}^2}} - \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}} \right){\textrm{h}_0}{{\mathrm{\rho}} ^2}{\textrm{S}^2}{\mathrm{\phi}} \\ & + \frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}} - \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}}} \right){\textrm{h}_0}{{\mathrm{\rho}} ^2}{\textrm{C}^2}{\mathrm{\phi}} \\ & + \frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}} - 2\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}} + \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}} \right)\textrm{h}_0^2{\mathrm{\rho}} \textrm{C}{\mathrm{\phi}} \\ & + \frac{1}{6}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^3}} - 3\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} + 3\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}} - \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}}} \right)\textrm{h}_0^3. \end{aligned}$$
Comparing Eqs. (16a) and (16b) with the components of Eq. (1), the following equations are obtained: (17a)$$\textrm{A}_{1} = \frac{{\partial {\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}}}, $$
(17b)$$\textrm{A}_{1} = \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}, $$
(18)$$\textrm{A}_{2} = \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}}} - \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}, $$
(19a)$${\textrm{B}_{1}} = \frac{1}{6}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}^3}}{\textrm{S}^2}{\mathrm{\phi}} + 3\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}{\textrm{C}^2}{\mathrm{\phi}} } \right), $$
(19b)$${\textrm{B}_{1}} = \frac{1}{6}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}}{\textrm{C}^2}{\mathrm{\phi}} + 3\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}{\textrm{S}^2}{\mathrm{\phi}} } \right), $$
(20a)$${\textrm{B}_{2}} = \frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} - \frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}} \right), $$
(20b)$$\begin{array}{l} {\textrm{B}_{2}} = \frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}^2}} - \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}} \right)\frac{{{\textrm{S}^2}{\mathrm{\phi}} }}{{\textrm{(3} - 2{\textrm{S}^2}{\mathrm{\phi}} )}}\\ + \frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}} - \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}}} \right)\frac{{{\textrm{C}^2}{\mathrm{\phi}} }}{{\textrm{(3} - 2{\textrm{S}^2}{\mathrm{\phi}} )}} \end{array}, $$
(21)$${\textrm{B}_{3}} + {\textrm{B}_{4}} = \frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}} - 2\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} + \frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial \textrm{h}_0^2}}} \right), $$
(22)$$3{\textrm{B}_{3}} + {\textrm{B}_{4}} = \frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}} - 2\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}} + \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}} \right), $$
(23)$${\textrm{B}_{5}} = \frac{1}{6}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^3}} - 3\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} + 3\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}} - \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}}} \right). $$
It is observed from Eqs. (17a)–(23) that the A and B coefficients are independent of the object height $\textrm{h}_{0}$.
4. Numerical results of primary aberration coefficients for object at finite distance
It is noted that Eqs. (17a)–(23) can determine the transverse aberrations not only for any meridional or sagittal rays, but also for any skew rays. However, for simplicity, in the present study, the aberration coefficients are determined only along the $\textrm{x}_{\textrm{a}}$ and $\textrm{y}_{\textrm{a}}$ axes of the illustrative system shown in Fig. 2. Moreover, in the discussions which follow, the object is assumed to be positioned at ${{\textrm P}_{\textrm {0z}}} ={-} 200$ and the image plane is the Gaussian image plane (i.e., separation $92.088474$). Finally, the maximum opening radius of the entrance pupil is assumed to be ${{\mathrm{\rho}} _{\max }} = 21$. The values of the required derivatives for Eqs. (17a)–(23) are provided in Appendix A.
(1) Defocus aberration coefficient
$\textrm{A}_{1}$: When the object is positioned at
${{\textrm P}_{\textrm {0z}}} ={-} 200$ and the image plane is the Gaussian image plane, the defocus aberration coefficient is found from Eqs. (
17a) and (
17b) to be
$\textrm{A}_{1} = \textrm{0}\textrm{.000000}$, while the transverse magnification is found from Eq. (
18) to be
$\textrm{A}_{2} ={-} \textrm{0}\textrm{.660413}$. For the case where the image plane is not the Gaussian image plane, e.g.,
${\textrm{v}_6} = 85.088474$, the numerical results show that
$\textrm{A}_{1} ={-} \textrm{0}\textrm{.047723}$ and
$\textrm{A}_{2} ={-} \textrm{0}\textrm{.625178}$.
(2) Coefficient
${\textrm{B}_{1}}$: The primary spherical aberration is the only third-order term when the object is positioned on the optical axis. Although Eqs. (
19a) and (
19b) are functions of
${\mathrm{\phi}}$, the numerical results show that the values evaluated by Eqs. (
19a) and (
19b) for all marginal rays with
$- 180 \le {\mathrm{\phi}} < {180^ \circ }$ are very similar. Therefore,
${\textrm{B}_{1}}$ can be determined from Eqs. (
19a) and (
19b) by setting
${\mathrm{\phi}} = {0^ \circ }$ or
${\mathrm{\phi}} = {90^ \circ }$, respectively, to give
(24)$$\begin{array}{l} {\textrm{B}_{1}} = \frac{1}{6}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}^3}} = \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}} = \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}\\ = \frac{1}{6}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}} ={-} 1.655506 \times {10^{ - 4}}. \end{array}$$
The spherical aberration coefficient can then be computed by $\textrm{B}_{1}{\mathrm{\rho}} _{\max }^{\textrm{ 3}} ={-} 1.533164$, which is the spherical aberration at the maximum opening radius of the entrance pupil. The spherical aberration at any other value of ${\mathrm{\rho}}$ can be determined by $\textrm{B}_{1}{{\mathrm{\rho}} ^3}$. The value of $\textrm{B}_{1}{\mathrm{\rho}} _{\max }^{\textrm{ 3}} ={-} 1.533164$ is essentially identical to that obtained from Zemax, namely ${{\mathrm{\omega}} _1} ={-} 1.533486$ (determined by dividing the Buchdahl coefficient ${{\mathrm{\sigma}} _1} ={-} 0.439103$ by $- 2{{\mathrm{\xi}} _0}{{\mathrm{\mu}} _{\textrm{k}}}$, where ${{\mathrm{\xi}} _0} = 1$ is the refractive index of air and ${{\mathrm{\mu}} _{\textrm{k}}} ={-} 0.143171$ is the angle between the exit marginal ray and the optical axis when the object height is $\textrm{h}_{0} = 17$).
(3) Coefficient
${\textrm{B}_2}$: The
${\textrm{B}_2}$ can be determined from Eq. (
20a) as
(25a)$${\textrm{B}_{2}} = \frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} - \frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}}} \right) ={-} 1.504828 \times {10^{ - 5}}. $$
Equation (20b) is a function of ${\mathrm{\phi}}$. However, the numerical simulations show that the values evaluated by Eq. (20b) for all marginal rays with $- {180^ \circ } \le {\mathrm{\phi}} < {180^ \circ }$ are almost the same. Consequently, ${\textrm{B}_2}$ can be determined from Eq. (20b) by setting ${\mathrm{\phi}} = {0^ \circ }$ or ${\mathrm{\phi}} = {90^ \circ }$, to give
(25b)$$\begin{array}{l} {\textrm{B}_{2}} = \frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}^2}} - \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}} \right)\\ = \frac{1}{6}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}} - \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^3}}}} \right) ={-} 1.504828 \times {10^{ - 5}}. \end{array}$$
The coma aberration coefficient is thus obtained as ${\textrm{B}_2}\textrm{h}_{0}{\mathrm{\rho}} _{\max }^{\textrm{ 2}} ={-} 0.112817$. This value is again in good agreement with the Zemax result, i.e., ${{\mathrm{\omega}} _2} ={-} 0.112776$ (obtained by dividing the Buchdahl coefficient ${{\mathrm{\sigma}} _2} ={-} 0.032296$ by $- 2{{\mathrm{\xi}} _0}{{\mathrm{\mu}} _{\textrm{k}}}$).
(4) Coefficient
${\textrm{B}_{3}}$: From Eqs. (
21) and (
22), the following relations are obtained for the astigmatism and field curvature aberration coefficients:
(26)$${\textrm{B}_{3}} + {\textrm{B}_{4}} ={-} \textrm{4}\textrm{.194156} \times {10^{ - 6}}, $$
(27)$$3{\textrm{B}_{3}} + {\textrm{B}_{4}} ={-} \textrm{1}\textrm{.503640} \times {10^{ - 6}}. $$
Coefficient ${\textrm{B}_{3}}$ can thus be determined from Eqs. (26) and (27) as ${\textrm{B}_{3}} = 1.345258 \times {10^{ - 6}}$. The transverse astigmatism coefficient can then be computed as ${\textrm{B}_{3}}\textrm{h}_0^2{{\mathrm{\rho}} _{\max }} = 8.164373 \times {10^{ - 3}}$. It is noted that this value is in good agreement with that of ${{\mathrm{\omega}} _3} = 8.161541 \times {10^{ - 3}}$ determined from Zemax simulations (computed by dividing the Buchdahl coefficient ${{\mathrm{\sigma}} _3} = 0.002337$ by $- 2{{\mathrm{\xi}} _0}{{\mathrm{\mu}} _{\textrm{k}}}$).
(5) Coefficient
${\textrm{B}_4}$: The
${\textrm{B}_4}$ is found from Eqs. (
26) and (
27) to be
(28)$${\textrm{B}_4} ={-} 5.539414 \times {10^{ - 6}}. $$
The coefficient of the field curvature aberration is thus obtained as ${\textrm{B}_4}\textrm{h}_0^2{{\mathrm{\rho}} _{\max }} ={-} 0.033619$, which is very close to that of ${{\mathrm{\omega}} _4} ={-} 0.033624$ obtained from Zemax simulations (determined by dividing the Buchdahl coefficient ${{\mathrm{\sigma}} _4} ={-} 0.009628$ by $- 2{{\mathrm{\xi}} _0}{{\mathrm{\mu}} _{\textrm{k}}}$).
(6) Coefficient
${\textrm{B}_5}$: Substituting Eqs. (
47d), (
47g), (
47j), and (
47l) in the
Appendix into Eq. (
23), the
$\textrm{B}_{5}$ is determined to be
(29)$$\textrm{B}_{5} ={-} \textrm{2}\textrm{.309201} \times {10^{ - 6}}. $$
The distortion coefficient is hence obtained as $\textrm{B}_{5}\textrm{h}_0^3 ={-} 0.011345$. This value is almost identical to that obtained from Zemax simulations, i.e., ${{\mathrm{\omega}} _1} ={-} 0.011345$ (determined by dividing the Buchdahl coefficient ${{\mathrm{\sigma}} _5} ={-} 0.003248$ by $- 2{{\mathrm{\xi}} _0}{{\mathrm{\mu}} _{\textrm{k}}}$). Notably, Eq. (23) indicates that there are four terms contributing toward the primary distortion. This finding differs from conventional knowledge, which traditionally holds that ${{\partial {}^3{\textrm{P}_{\textrm{ny}}}} \mathord{\left/ {\vphantom {{\partial {}^3{\textrm{P}_{\textrm{ny}}}} {\partial \textrm{h}_0^3}}} \right.} {\partial \textrm{h}_0^3}}$ is the only source of primary distortion.
5. Primary aberration coefficients for object at infinity
It is impossible to accurately determine the A and B coefficients for a system having an object lying at infinity by simply assigning a large number to ${{\textrm P}_{\textrm {0z}}}$ in the model above since the developed numerical methods for the ray derivatives will fail. However, for an object placed at infinity, all of the rays coming from the object are collimated rays. Thus, to investigate the ray aberrations, one can simply describe the collimated rays at the entrance pupil by Fig. 4. Given the assumption that each object, located at ${\left[ {\begin{array}{cc} {\textrm{x}_{\textrm{a}}}&{\textrm{y}_{\textrm{a}}} \end{array}} \right]^{\textrm{T}}}$, emits a single ray with a fixed unit directional vector $\overline {\ell } {}_0 = {\left[ {\begin{array}{ccc} 0&{\textrm{S}{{\mathrm{\beta}}_0}}&{\textrm{C}{{\mathrm{\beta}}_0}} \end{array}} \right]^{\textrm{T}}}$, the Seidel variable vector for an object laying at infinity can be expressed as
(30)$${\overline {\textrm X} _{\textrm{Seidel}}} = {\left[ {\begin{array}{ccc} {{{\mathrm{\beta}}_0}}&{\textrm{x}_{\textrm{a}}}&{\textrm{y}_{\textrm{a}}} \end{array}} \right]^{\textrm{T}}}$$
Equations (7a)–(7b) and (9a)–(11b) are still valid provided that $(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})$ is first substituted by $\textrm{y}_{\textrm{a}}$ and $\textrm{h}_{0}$ is then replaced by ${{\mathrm{\beta}} _0}$. Due to the intrinsic nature of axis-symmetric lens systems, various terms of the new equations vanish [8]. Thus, the following equations for the in-plane coordinates of incidence point on the image plane are obtained when Eqs. (15a) and (15b) are used: (31a)$$\begin{array}{l} {\textrm{P}_{\textrm{nx}}}\textrm{ = }\frac{{\partial {\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}}}{\mathrm{\rho}} \textrm{S}{\mathrm{\phi}} + \frac{1}{6}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}^3}}{{(\textrm{S}{\mathrm{\phi}} )}^2} + \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial \textrm{y}_{\textrm{a}}^2}}{{(\textrm{C}{\mathrm{\phi}} )}^2}} \right){{\mathrm{\rho}} ^3}\textrm{S}{\mathrm{\phi}} \\ + \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial {{\mathrm{\beta}} _0}\partial \textrm{x}_{\textrm{a}}\partial \textrm{y}_{\textrm{a}}}}{{\mathrm{\beta}} _0}{{\mathrm{\rho}} ^2}\textrm{S}2{\mathrm{\phi}} + \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial {\mathrm{\beta}} _0^2\partial \textrm{x}_{\textrm{a}}}}{\mathrm{\beta}} _0^2{\mathrm{\rho}} \textrm{S}{\mathrm{\phi}} , \end{array}$$
(31b)$$\begin{array}{l} {\textrm{P}_{\textrm{ny}}}\textrm{ = }\frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial {{\mathrm{\beta}} _0}}}{{\mathrm{\beta}} _0} + \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{y}_{\textrm{a}}}}{\mathrm{\rho}} \textrm{C}{\mathrm{\phi}} \\ + \frac{1}{6}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{y}_{\textrm{a}}^3}}{{(\textrm{C}{\mathrm{\phi}} )}^2} + 3\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial \textrm{y}_{\textrm{a}}}}{{(\textrm{S}{\mathrm{\phi}} )}^2}} \right){{\mathrm{\rho}} ^3}\textrm{C}{\mathrm{\phi}} + \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{\mathrm{\beta}} _0}\partial \textrm{x}_{\textrm{a}}^2}}{{\mathrm{\beta}} _0}{{\mathrm{\rho}} ^2}{\textrm{S}^2}{\mathrm{\phi}} \\ + \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{\mathrm{\beta}} _0}\partial \textrm{y}_{\textrm{a}}^2}}{{\mathrm{\beta}} _0}{{\mathrm{\rho}} ^2}{C^2}{\mathrm{\phi}} + \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {\mathrm{\beta}} _0^2\partial \textrm{y}_{\textrm{a}}}}{\mathrm{\beta}} _0^2{\mathrm{\rho}} \textrm{C}{\mathrm{\phi}} + \frac{1}{6}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {\mathrm{\beta}} _0^3}}{\mathrm{\beta}} _0^3. \end{array}$$
Comparing Eqs. (31a) and (31b) with the components of Eq. (1) (with $\textrm{h}_{0}$ being substituted by ${{\mathrm{\beta}} _0}$), the following equations are obtained: (32a)$$\textrm{A}_{1} = \frac{{\partial {\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}}}, $$
(32b)$$\textrm{A}_{1} = \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{y}_{\textrm{a}}}}, $$
(33)$$\textrm{A}_{2} = \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial {{\mathrm{\beta}} _0}}}, $$
(34a)$${\textrm{B}_{1}} = \frac{1}{6}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}^3}}{{(\textrm{S}{\mathrm{\phi}} )}^2} + 3\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial \textrm{y}_{\textrm{a}}^2}}{{(\textrm{C}{\mathrm{\phi}} )}^2}} \right), $$
(34b)$${\textrm{B}_{1}} = \frac{1}{6}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{y}_{\textrm{a}}^3}}{{(\textrm{C}{\mathrm{\phi}} )}^2} + 3\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial \textrm{y}_{\textrm{a}}}}{{(\textrm{S}{\mathrm{\phi}} )}^2}} \right), $$
(35a)$${\textrm{B}_{2}} = \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial {{\mathrm{\beta}} _0}\partial \textrm{x}_{\textrm{a}}\partial \textrm{y}_{\textrm{a}}}}, $$
(35b)$${\textrm{B}_{2}} = {{\frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{\mathrm{\beta}}_0}\partial \textrm{x}_{\textrm{a}}^2}}{\textrm{S}^2}{\mathrm{\phi}} + \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{\mathrm{\beta}}_0}\partial \textrm{y}_{\textrm{a}}^2}}{\textrm{C}^2}{\mathrm{\phi}} } \right)} \mathord{\left/ {\vphantom {{\frac{1}{2}\left( {\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{\mathrm{\beta}}_0}\partial \textrm{x}_{\textrm{a}}^2}}{\textrm{S}^2}{\mathrm{\phi}} + \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{\mathrm{\beta}}_0}\partial \textrm{y}_{\textrm{a}}^2}}{\textrm{C}^2}{\mathrm{\phi}} } \right)} {\textrm{(3} - 2{\textrm{S}^2}{\mathrm{\phi}} )}}} \right. } {\textrm{(3} - 2{\textrm{S}^2}{\mathrm{\phi}} )}}, $$
(36)$${\textrm{B}_{3}} + {\textrm{B}_{4}} = \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial {\mathrm{\beta}} _0^2}}, $$
(37)$$3{\textrm{B}_{3}} + {\textrm{B}_{4}} = \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {\mathrm{\beta}} _0^2\partial \textrm{y}_{\textrm{a}}}}, $$
(38)$${\textrm{B}_{5}} = \frac{1}{6}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {\mathrm{\beta}} _0^3}}. $$
Consider the system shown in Fig. 2 once again for illustration purposes. Assume that the object is placed at infinity with ${{\mathrm{\beta}} _0} = {4.377^ \circ }$. Assume also that the image plane is the Gaussian image plane (i.e., ${\textrm{v}_{\textrm{6/Gaussian}}} = 36.36323$) and the opening radius of the entrance pupil is ${{\mathrm{\rho}} _{\max }} = 21$. Coefficients $\textrm{A}_{1}$ and $\textrm{A}_{2}$ are obtained from Eqs. (32a), (32b), and (33) as $\textrm{A}_{1} = 0$ and $\textrm{A}_{2} = \textrm{84}\textrm{.379297}$, respectively. Equations (32a)–(38) are valid for all rays intercepted by the entrance pupil. Therefore, ${\textrm{B}_{1}}$ can be obtained from Eqs. (34a) and (34b) by setting ${\mathrm{\phi}} = {0^ \circ }$ or ${\mathrm{\phi}} = {90^ \circ }$ to have the following four equations:
(39)$$\textrm{B}_{1} = \frac{1}{6}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}^3}} = \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial \textrm{y}_{\textrm{a}}^2}} = \frac{1}{6}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{y}_{\textrm{a}}^3}} = \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial \textrm{y}_{\textrm{a}}}}\textrm{ = } - \textrm{1}\textrm{.006099} \times \textrm{1}{\textrm{0}^{ - 4}}. $$
Equation (39) indicates that the four equations yield the same numerical value for ${\textrm{B}_{1}}$ when the result is evaluated up to six decimal places. Similarly, three equations can be obtained from Eqs. (35a) and (35b) to determine $\textrm{B}_{2}$, namely (40)$$\textrm{B}_{2} = \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial {{\mathrm{\beta}} _0}\partial \textrm{x}_{\textrm{a}}\partial \textrm{y}_{\textrm{a}}}} = \frac{1}{2}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{\mathrm{\beta}} _0}\partial \textrm{x}_{\textrm{a}}^2}} = \frac{1}{6}\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {{\mathrm{\beta}} _0}\partial \textrm{y}_{\textrm{a}}^2}}\textrm{ = 8}\textrm{.642519} \times \textrm{1}{\textrm{0}^{ - 4}}. $$
${\textrm{B}_{3}}$ and ${\textrm{B}_4}$ are determined from Eqs. (36) and (37) to be $\textrm{B}_{3}\textrm{ = } - 0.177544$ and $\textrm{B}_{4}\textrm{ = } - 0.157195$, respectively. Similarly, ${\textrm{B}_5}$ is found from Eq. (38) to be $\textrm{B}_{5}\textrm{ = 72}\textrm{.028142}$. Table 2 lists the coefficient values of transverse spherical aberration, coma, astigmatism, field curvature and distortion computed as $\textrm{B}_{1}{\mathrm{\rho}} _{\max }^{\textrm{ 3}}$, ${\textrm{B}_2}{{\mathrm{\beta}} _0}{\mathrm{\rho}} _{\max }^{\textrm{ 2}}$, ${\textrm{B}_3}{\mathrm{\beta}} _0^2{{\mathrm{\rho}} _{\max }}$, ${\textrm{B}_4}{\mathrm{\beta}} _0^2{{\mathrm{\rho}} _{\max }}$ and ${\textrm{B}_5}{\mathrm{\beta}} _0^3$, respectively. The corresponding results obtained from Zemax simulations are also listed for comparison purposes. It is seen that the coefficients of transverse spherical aberration, coma, astigmatism and field curvature are all in good agreement with the Zemax results. The distortion coefficient deviates significantly from the Zemax value. One has to note that the image height of the proposed method is given by (41a)$${\textrm{P}_{\textrm{ny}}} = \textrm{A}_{2}{{\mathrm{\beta}} _0} + \textrm{B}_{5}{\mathrm{\beta}} _0^3, $$
since $\textrm{A}_{2}\tan {{\mathrm{\beta}} _0}$ is replaced by ${{\mathrm{\beta}} _0}$ in Taylor series expansion. However, Zemax keeps $\tan {{\mathrm{\beta}} _0}$ to estimate its image height by (41b)$${\textrm{P}_{\textrm{ny}}} = \textrm{A}_{2}\tan {{\mathrm{\beta}} _0} + \textrm{DIST}, $$
where ${\textrm{DIST}}$ is its primary distortion coefficient. By equating above two equations, we can transfer the values of distortion coefficient from the proposed method to those of Zemax: (42)$$\textrm{DIST} = \textrm{A}_{2}({{{\mathrm{\beta}}_0} - \tan {{\mathrm{\beta}}_0}} )+ \textrm{B}_{5}{\mathrm{\beta}} _0^3. $$
Table 2. Seidel aberration coefficients for object lying at infinity.
In Table 3 we compare the numerical results of Eq. (42) with the primary distortion coefficients from Zemax. It is shown in the 4th column of Table 3 that the percentage errors are acceptable for small value of ${{\mathrm{\beta}} _0}$. In fact it is found that Buchdahl’s equation is a complex function of the angles of the chief ray, leading its numerical value deviates gradually from that of Eq. (42) for larger angle ${{\mathrm{\beta}} _0}$.
Table 3. Distortion coefficients from Eq. (42) and Zemax.
6. Conclusions
Determining the A and B coefficients of the Seidel primary ray aberrations and their associated aberrations is extremely challenging since optical systems generally contain many aberrations (including higher order aberrations) simultaneously. Accordingly, the present study has proposed a methodology for determining the numerical values of these coefficients based on the relationship between the ray aberration equations and the Taylor series expansion of a skew ray. In the proposed approach, the coefficients of transverse magnification, $\textrm{A}_2$, and defocus aberration, $\textrm{A}_1$, are determined accurately using the first-order ray derivative matrices, while the aberration coefficients (i.e., ${\textrm{B}_{\textrm j}}$, j=1-5) are estimated using the third-order ray derivative matrices. Numerical results have shown that the Seidel primary ray aberration coefficients computed using the proposed method agree remarkably with these obtained from Zemax when the object is placed at finite distance. Consequently, their corresponding primary ray aberrations can be determined accurately.
The proposed methodology can also determine the A and B coefficients for object lying infinity by placing the collimated rays at the entrance pupil. The obtained ${\textrm{B}_1}$, ${\textrm{B}_2}$, ${\textrm{B}_3}$, and ${\textrm{B}_4}$ agree well with these from Zemax. However, the difference between the values of ${\textrm{B}_5}$ is significant. We found that Zemax determines the image height by triangular function $\textrm{A}_{2}\tan {{\mathrm{\beta}} _0}$, leading us to transfer the obtained primary distortion coefficient by Eq. (42) for compare. Its values agree well with these from Zemax for small value of ${{\mathrm{\beta}} _0}$. The proposed method can be extended to the derivation for the coefficients of primary wave aberrations and for higher order ray aberrations.
Appendix
As described in a previous study by the present group [26], the unit directional vector of a ray can be described in spherical coordinates ${\left[ {\begin{array}{cc} {{{\mathrm{\alpha}}_0}}&{{{\mathrm{\beta}}_0}} \end{array}} \right]^{\textrm{T}}}$ as
(43)$$\overline {\ell } {}_0 = {\left[ {\begin{array}{cccc} {\textrm{S}{{\mathrm{\alpha}}_0}\textrm{C}{{\mathrm{\beta}}_0}}&{\textrm{S}{{\mathrm{\beta}}_0}}&{\textrm{C}{{\mathrm{\alpha}}_0}\textrm{C}{{\mathrm{\beta}}_0}}&0 \end{array}} \right]^{\textrm{T}}}. $$
The incidence point
${\left[ {\begin{array}{cc} {\textrm{x}_{\textrm{a}}}&{\textrm{y}_{\textrm{a}}} \end{array}} \right]^{\textrm{T}}}$ of the ray on the entrance pupil can be obtained by substituting Eq. (
43) into Eqs. (
4b) and (
4c), to give
(44)$$\left[ {\begin{array}{c} {\textrm{x}_{\textrm{a}}}\\ {\textrm{y}_{\textrm{a}}} \end{array}} \right] = \left[ {\begin{array}{c} 0\\ {\textrm{h}_{0}} \end{array}} \right] + \frac{{({\textrm{v}_{\textrm{entrance}}} - {\textrm{P}_{\textrm{0z}}})}}{{\textrm{C}{{\mathrm{\alpha}} _0}\textrm{C}{{\mathrm{\beta}} _0}}}\left[ {\begin{array}{c} {\textrm{S}{{\mathrm{\alpha}}_0}\textrm{C}{{\mathrm{\beta}}_0}}\\ {\textrm{S}{{\mathrm{\beta}}_0}} \end{array}} \right]. $$
From Eq. (
44), angles
${{\mathrm{\alpha}} _0}$ and
${{\mathrm{\beta}} _0}$ of the ray originating from
${\overline {\textrm P} _0} = {\left[ {\begin{array}{ccc} 0&{\textrm{h}_{0}}&{ - 200} \end{array}} \right]^{\textrm{T}}}$ can be obtained as
(45a)$${{\mathrm{\alpha}} _0} = \textrm{actan} \left( {\frac{{\textrm{x}_{\textrm{a}}}}{{(200 + {\textrm{v}_{\textrm{entrance}}})}}} \right), $$
(45b)$${{\mathrm{\beta}} _0} = \textrm{actan} \left( {\frac{{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\textrm{C}{{\mathrm{\alpha}}_0}}}{{(200 + {\textrm{v}_{\textrm{entrance}}})}}} \right). $$
The following values of
${{\partial {{\mathrm{\alpha}} _0}} \mathord{\left/ {\vphantom {{\partial {{\mathrm{\alpha}}_0}} {\partial \textrm{x}_{\textrm{a}}}}} \right.} {\partial \textrm{x}_{\textrm{a}}}}$,
${{\partial {{\mathrm{\alpha}} _0}} \mathord{\left/ {\vphantom {{\partial {{\mathrm{\alpha}}_0}} {\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}} \right.} {\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}$,
${{\partial {{\mathrm{\beta}} _0}} \mathord{\left/ {\vphantom {{\partial {{\mathrm{\beta}}_0}} {\partial \textrm{x}_{\textrm{a}}}}} \right.} {\partial \textrm{x}_{\textrm{a}}}}$ and
${{{\partial {{\mathrm{\beta}} _0}} \mathord{\left/ {\vphantom {{\partial {{\mathrm{\beta}}_0}} {\partial (\textrm{y}_{\textrm{a}} - h}}} \right.} {\partial (\textrm{y}_{\textrm{a}} - \textrm{h}}}_0})$, obtained from Eqs. (
45a)–(
45b) and evaluated at
${\overline {\textrm X} _{\textrm{Seidel/optical axis}}} = {\left[ {\begin{array}{ccc} 0&0&0 \end{array}} \right]^{\textrm{T}}}$, are needed:
(46a)$$\frac{{\partial {{\mathrm{\alpha}} _0}}}{{\partial \textrm{x}_{\textrm{a}}}} = \frac{{(200 + {\textrm{v}_{\textrm{entrance}}})}}{{{{(200 + {\textrm{v}_{\textrm{entrance}}})}^2} + {{(\textrm{x}_{\textrm{a}})}^2}}} = 4.502409965 \times {10^{ - 3}}, $$
(46b)$$\frac{{\partial {{\mathrm{\alpha}} _0}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} = 0, $$
(46c)$$\frac{{\partial {{\mathrm{\beta}} _0}}}{{\partial \textrm{x}_{\textrm{a}}}} = \frac{{ - (200 + {\textrm{v}_{\textrm{entrance}}})(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\textrm{S}{{\mathrm{\alpha}} _0}}}{{{{(200 + {\textrm{v}_{\textrm{entrance}}})}^2} + {{({(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\textrm{C}{{\mathrm{\alpha}}_0}} )}^2}}}\frac{{\partial {{\mathrm{\alpha}} _0}}}{{\partial \textrm{x}_{\textrm{a}}}} = 0, $$
(46d)$$\frac{{\partial {{\mathrm{\beta}} _0}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} = \frac{{(200 + {\textrm{v}_{\textrm{entrance}}})\textrm{C}{{\mathrm{\alpha}} _0}}}{{{{(200 + {\textrm{v}_{\textrm{entrance}}})}^2} + {{({(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})\textrm{C}{{\mathrm{\alpha}}_0}} )}^2}}} = 4.502409965 \times {10^{ - 3}}. $$
It is noted from Eqs. (46b) and (46c) that ${{\partial {{\mathrm{\alpha}} _0}} \mathord{\left/ {\vphantom {{\partial {{\mathrm{\alpha}}_0}} {\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}} \right.} {\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} = 0$ and ${{\partial {{\mathrm{\beta}} _0}} \mathord{\left/ {\vphantom {{\partial {{\mathrm{\beta}}_0}} {\partial \textrm{x}_{\textrm{a}}}}} \right.} {\partial \textrm{x}_{\textrm{a}}}} = 0$. Consequently, the process of determining the required derivatives for the proposed method is greatly simplified, and gives
(47a)$$\frac{{\partial {\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}}} = \frac{{\partial {\textrm{P}_{\textrm{nx}}}}}{{\partial {{\mathrm{\alpha}} _0}}}\frac{{\partial {{\mathrm{\alpha}} _0}}}{{\partial \textrm{x}_{\textrm{a}}}} = \textrm{8}\textrm{.612854} \times {10^{ - 8}}, $$
(47b)$$\frac{{\partial {\textrm{P}_{\textrm{nx}}}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} = \frac{{\partial {\textrm{P}_{\textrm{ny}}}}}{{\partial {{\mathrm{\beta}} _0}}}\frac{{\partial {{\mathrm{\beta}} _0}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} = \textrm{8}\textrm{.612854} \times {10^{ - 8}}, $$
(47c)$$\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_0^2\partial \textrm{x}_{\textrm{a}}}} = \frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_0^2\partial {{\mathrm{\alpha}} _0}}}\frac{{\partial {{\mathrm{\alpha}} _0}}}{{\partial \textrm{x}_{\textrm{a}}}} ={-} \textrm{ 3}\textrm{.99682666} \times {10^{ - 4}}, $$
(47d)$$\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} = \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^2\partial {{\mathrm{\beta}} _0}}}\frac{{\partial {{\mathrm{\beta}} _0}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} ={-} \textrm{1}\textrm{.176890} \times {10^{ - 3}}, $$
(47e)$$\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} = \frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{h}_{0}\partial {{\mathrm{\alpha}} _0}\partial {{\mathrm{\beta}} _0}}}\frac{{\partial {{\mathrm{\alpha}} _0}}}{{\partial \textrm{x}_{\textrm{a}}}}\frac{{\partial {{\mathrm{\beta}} _0}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} ={-} \textrm{3}\textrm{.611977} \times {10^{ - 4}}, $$
(47f)$$\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial \textrm{x}_{\textrm{a}}^2}} = \frac{{{\partial ^3}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial {\mathrm{\alpha}} _0^2}}{\left( {\frac{{\partial {{\mathrm{\alpha}}_0}}}{{\partial \textrm{x}_{\textrm{a}}}}} \right)^2} ={-} \textrm{3}\textrm{.611978} \times {10^{ - 4}}, $$
(47g)$$\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0}){}^2}} = \frac{{{\partial ^3}{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_{0}\partial {\mathrm{\beta}} _0^2}}{\left( {\frac{{\partial {{\mathrm{\beta}}_0}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}} \right)^2} ={-} \textrm{1}\textrm{.083593} \times {10^{ - 3}}, $$
(47h)$$\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}\partial {{(\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}^2}}} = \frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial {{\mathrm{\alpha}} _0}\partial {\mathrm{\beta}} _0^2}}\frac{{\partial {{\mathrm{\alpha}} _0}}}{{\partial \textrm{x}_{\textrm{a}}}}{\left( {\frac{{\partial {{\mathrm{\beta}}_0}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}} \right)^2} ={-} \textrm{3}\textrm{.311012} \times {10^{ - 4}}, $$
(47i)$$\frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial \textrm{x}_{\textrm{a}}^3}} = \frac{{\partial {}^3{\textrm{P}_{\textrm{nx}}}}}{{\partial {\mathrm{\alpha}} _0^3}}\left( {\frac{{\partial {{\mathrm{\alpha}}_0}}}{{\partial \textrm{x}_{\textrm{a}}}}} \right){}^3 ={-} \textrm{9}\textrm{.933036} \times {10^{ - 4}}, $$
(47j)$$\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0}){}^3}} = \frac{{\partial {}^2{\textrm{P}_{\textrm{ny}}}}}{{\partial {\mathrm{\beta}} _0^3}}\left( {\frac{{\partial {{\mathrm{\beta}}_0}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}}} \right){}^3 ={-} \textrm{9}\textrm{.933036} \times {10^{ - 4}}, $$
(47k)$$\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{x}_{\textrm{a}}^2\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} = \frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial {\mathrm{\alpha}} _0^2\partial {{\mathrm{\beta}} _0}}}{\left( {\frac{{\partial {{\mathrm{\alpha}}_0}}}{{\partial \textrm{x}_{\textrm{a}}}}} \right)^2}\frac{{\partial {{\mathrm{\beta}} _0}}}{{\partial (\textrm{y}_{\textrm{a}} - \textrm{h}_{0})}} ={-} \textrm{3}\textrm{.311012} \times {10^{ - 4}}, $$
(47l)$$\frac{{\partial {}^3{\textrm{P}_{\textrm{ny}}}}}{{\partial \textrm{h}_0^3}} ={-} \textrm{1}\textrm{.287049} \times {10^{ - 3}}. $$
The required values of ${{\partial {\textrm{P}_{\textrm{ny}}}} \mathord{\left/ {\vphantom {{\partial {\textrm{P}_{\textrm{ny}}}} {\partial \textrm{h}_{0}}}} \right.} {\partial \textrm{h}_{0}}}$, ${{\partial {\textrm{P}_{\textrm{nx}}}} \mathord{\left/ {\vphantom {{\partial {\textrm{P}_{\textrm{nx}}}} {\partial {{\mathrm{\alpha}}_0}}}} \right.} {\partial {{\mathrm{\alpha}} _0}}}$, ${{\partial {\textrm{P}_{\textrm{nx}}}} \mathord{\left/ {\vphantom {{\partial {\textrm{P}_{\textrm{nx}}}} {\partial {{\mathrm{\beta}}_0}}}} \right.} {\partial {{\mathrm{\beta}} _0}}}$, ${{\partial {}^3{\textrm{P}_{\textrm{nx}}}} \mathord{\left/ {\vphantom {{\partial {}^3{\textrm{P}_{\textrm{nx}}}} {\partial {\mathrm{\alpha}}_0^3}}} \right.} {\partial {\mathrm{\alpha}} _0^3}}$, ${{\partial {}^3{\textrm{P}_{\textrm{nx}}}} \mathord{\left/ {\vphantom {{\partial {}^3{\textrm{P}_{\textrm{nx}}}} {\partial {{\mathrm{\alpha}}_0}\partial {\mathrm{\beta}}_0^2}}} \right.} {\partial {{\mathrm{\alpha}} _0}\partial {\mathrm{\beta}} _0^2}}$, ${{\partial {}^3{\textrm{P}_{\textrm{nx}}}} \mathord{\left/ {\vphantom {{\partial {}^3{\textrm{P}_{\textrm{nx}}}} {\partial \textrm{h}_{0}\partial {{\mathrm{\alpha}}_0}\partial {{\mathrm{\beta}}_0}}}} \right.} {\partial \textrm{h}_{0}\partial {{\mathrm{\alpha}} _0}\partial {{\mathrm{\beta}} _0}}}$, ${{\partial {}^3{\textrm{P}_{\textrm{nx}}}} \mathord{\left/ {\vphantom {{\partial {}^3{\textrm{P}_{\textrm{nx}}}} {\partial \textrm{h}_0^2\partial {{\mathrm{\alpha}}_0}}}} \right.} {\partial \textrm{h}_0^2\partial {{\mathrm{\alpha}} _0}}}$, ${{\partial {}^3{\textrm{P}_{\textrm{ny}}}} \mathord{\left/ {\vphantom {{\partial {}^3{\textrm{P}_{\textrm{ny}}}} {\partial {\mathrm{\beta}}_0^3}}} \right.} {\partial {\mathrm{\beta}} _0^3}}$, ${{\partial {}^3{\textrm{P}_{\textrm{ny}}}} \mathord{\left/ {\vphantom {{\partial {}^3{\textrm{P}_{\textrm{ny}}}} {\partial {\mathrm{\alpha}}_0^2\partial {{\mathrm{\beta}}_0}}}} \right.} {\partial {\mathrm{\alpha}} _0^2\partial {{\mathrm{\beta}} _0}}}$, ${{\partial {}^3{\textrm{P}_{\textrm{ny}}}} \mathord{\left/ {\vphantom {{\partial {}^3{\textrm{P}_{\textrm{ny}}}} {\partial \textrm{h}_{0}\partial {\mathrm{\alpha}}_0^2}}} \right.} {\partial \textrm{h}_{0}\partial {\mathrm{\alpha}} _0^2}}$, ${{\partial {}^3{\textrm{P}_{\textrm{ny}}}} \mathord{\left/ {\vphantom {{\partial {}^3{\textrm{P}_{\textrm{ny}}}} {\partial \textrm{h}_{0}\partial {\mathrm{\beta}}_0^2}}} \right.} {\partial \textrm{h}_{0}\partial {\mathrm{\beta}} _0^2}}$, ${{\partial {}^3{\textrm{P}_{\textrm{ny}}}} \mathord{\left/ {\vphantom {{\partial {}^3{\textrm{P}_{\textrm{ny}}}} {\partial \textrm{h}_0^2\partial {{\mathrm{\beta}}_0}}}} \right.} {\partial \textrm{h}_0^2\partial {{\mathrm{\beta}} _0}}}$ and ${{\partial {}^3{\textrm{P}_{\textrm{ny}}}} \mathord{\left/ {\vphantom {{\partial {}^3{\textrm{P}_{\textrm{ny}}}} {\partial \textrm{h}_0^3}}} \right.} {\partial \textrm{h}_0^3}}$ for Eqs. (47a)–(47l) are listed in Table 2 of [2].
Funding
Ministry of Science and Technology, Taiwan (106-2221-E-006-091-MY3).
Acknowledgments
The support, motivation and encouragement offered by my colleague, Professor R. Barry Johnson (Alabama Agricultural & Mechanical University) throughout the study of ray aberrations is greatly appreciated.
Disclosures
The author declare no conflicts of interest.
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