The determination of secondary ray-aberration coefficients for axis-symmetrical optical systems

Abstract

The ray-aberrations in axis-symmetrical systems are conventionally derived from wavefront functions or characteristic functions using classical approximate partial derivatives. However, the resulting aberrations typically have fifth-order errors, as described by Restrepo et al. (2017) [1]. Accordingly, in the present study, the secondary ray-aberration coefficients for object placed at finite distance are determined using the fifth-order Taylor series expansion of a skew ray. Notably, the derived expressions are exact since they are determined without any approximations. It is found that some of the aberration coefficients are not constants, but are functions of the polar angle of the entrance pupil. It is additionally found that, once the required derivative matrices have been generated, determination of the secondary aberration coefficients is straightforward without iteration, and incurs only a low computational cost.

Taylor series expansion; Derivative matrices; Axis-symmetrical system; Ray-aberrations

1. Introduction

To explore the monochromatic ray-aberrations of an axis-symmetrical system, the object ${\bar{\mathrm{P}}}_0={\left[\begin{array}{ccc}\hfill 0\hfill & \hfill {\mathrm{h}}_0\hfill & \hfill {\mathrm{P}}_{0\mathrm{z}}\hfill \end{array}\right]}^{\mathrm{T}}$ is assumed to be located on the meridional plane at height ${\mathrm{h}}_0$, as shown in Fig. 1. For a ray originating from ${\bar{\mathrm{P}}}_0$ and passing through the entrance pupil at a point with polar coordinates ρ and ϕ, the intersection point ${\bar{\mathrm{P}}}_n$ of the ray on the image plane is given by (p. 63 of [2])

$${\bar{\mathrm{P}}}_{\mathrm{n}}=\left[\begin{array}{ccc}\hfill {\mathrm{P}}_{\mathrm{nx}}\hfill & \hfill {\mathrm{P}}_{\mathrm{ny}}\hfill & \hfill {\mathrm{P}}_{\mathrm{nz}}\hfill \end{array}\right]={\left[\begin{array}{ccc}\hfill \mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}}\hfill & \hfill {\mathrm{A}}_2{\mathrm{h}}_0+\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}}\hfill & \hfill {\mathrm{P}}_{\mathrm{nz}}\hfill \end{array}\right]}^{\mathrm{T}},$$ (1)

where ${\mathrm{A}}_2$ is the lateral magnification and $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}}$ and $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}}$ are the transverse aberrations of the incidence point in the ${\mathrm{x}}_{\mathrm{n}}$- and ${\mathrm{y}}_{\mathrm{n}}$-directions, respectively, and are given by

$$\begin{gathered} \mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}}={\mathrm{A}}_1\rho \mathrm{S}\varphi +{\mathrm{B}}_1{\rho }^3\mathrm{S}\varphi +{\mathrm{B}}_2({\mathrm{h}}_0{\rho }^2)\mathrm{S}(2\varphi )+({\mathrm{B}}_3+{\mathrm{B}}_4)({\mathrm{h}}_0^2\rho )\mathrm{S}\varphi \\ +{\mathrm{C}}_1{\rho }^5\mathrm{S}\varphi +{\mathrm{C}}_3({\mathrm{h}}_0{\rho }^4)\mathrm{S}(2\varphi )+({\mathrm{C}}_5+{\mathrm{C}}_6{\mathrm{C}}^2\varphi )({\mathrm{h}}_0^2{\rho }^3)\mathrm{S}\varphi \\ +{\mathrm{C}}_9({\mathrm{h}}_0^3{\rho }^2)\mathrm{S}(2\varphi )+{\mathrm{C}}_{11}({\mathrm{h}}_0^4\rho )\mathrm{S}\varphi , \end{gathered}$$ (2a)

$$\begin{gathered} \mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}}={\mathrm{A}}_1\rho \mathrm{C}\varphi +{\mathrm{B}}_1{\rho }^3\mathrm{C}\varphi +{\mathrm{B}}_2({\mathrm{h}}_0{\rho }^2)(2+\mathrm{C}(2\varphi ))+(3{\mathrm{B}}_3+{\mathrm{B}}_4)({\mathrm{h}}_0^2\rho )\mathrm{C}\varphi \\ +{\mathrm{B}}_5{\mathrm{h}}_0^3+{\mathrm{C}}_1{\rho }^5\mathrm{C}\varphi +({\mathrm{C}}_2+{\mathrm{C}}_3\mathrm{C}2\varphi )({\mathrm{h}}_0{\rho }^4)+({\mathrm{C}}_4+{\mathrm{C}}_6{\mathrm{C}}^2\varphi )({\mathrm{h}}_0^2{\rho }^3)\mathrm{C}\varphi \\ +[{\mathrm{C}}_7+{\mathrm{C}}_8{\mathrm{C}}^2\varphi ]({\mathrm{h}}_0^3{\rho }^2)+{\mathrm{C}}_{10}({\mathrm{h}}_0^4\rho )\mathrm{C}\varphi +{\mathrm{C}}_{12}{\mathrm{h}}_0^5, \end{gathered}$$ (2b)

where C and S denote cosine and sine, respectively. (Note that the term ${\mathrm{C}}_7+{\mathrm{C}}_8\mathrm{C}(2\varphi )$ of [2] is replaced by ${\mathrm{C}}_7+{\mathrm{C}}_8{\mathrm{C}}^2\varphi$ in Eq. (2b) in order to satisfy the Buchdahl-Rimmer formulae [3], [4], [5].) The aberrations, $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}}$ and $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}}$, represent the distance by which the ray misses the ideal image point ${\left[\begin{array}{ccc}\hfill 0\hfill & \hfill {\mathrm{A}}_2{\mathrm{h}}_0\hfill & \hfill {\mathrm{P}}_{\mathrm{nz}}\hfill \end{array}\right]}^{\mathrm{T}}$ on the image plane, as determined by the paraxial raytracing equation. It is noted that Eqs. (2a) and (2b) contain two first-order terms (referred to as A coefficients), five third-order terms (referred to as B coefficients), and twelve fifth-order terms (referred to as C coefficients).

Figure 1.

Ray originating from an object, passing through an aperture, and then intersecting the image plane.

Smith [2], Zemax [6] and Johanson [7] use different terminologies for describing ray-aberrations. For example, Smith [2] utilizes ${\mathrm{B}}_1$, ${\mathrm{B}}_2$, ${\mathrm{B}}_3$, ${\mathrm{B}}_4$ and ${\mathrm{B}}_5$ to refer to primary spherical aberration, coma, astigmatism, field curvature and distortion, respectively. By contrast, Zemax [6] and Johanson [7] use ${\mathrm{B}}_1{\rho }_{\max }^3$, ${\mathrm{B}}_2{\mathrm{h}}_0{\rho }_{\max }^2$, ${\mathrm{B}}_3{\mathrm{h}}_0^2{\rho }_{\max }$, ${\mathrm{B}}_4{\mathrm{h}}_0^2{\rho }_{\max }$ and ${\mathrm{B}}_5{\mathrm{h}}_0^3$, where ${\rho }_{\max }$ is the maximum entrance pupil radius, as the coefficients of spherical aberration, coma, astigmatism, field curvature and distortion, respectively. The aberration coefficients given in [6], [7] are defined as the aberrations at $\rho ={\rho }_{\max }$ for a given ${\mathrm{h}}_0$ since the aberrations at any other value of ρ can be simply determined using an appropriate scaling factor (i.e., ($\rho /{\rho }_{\max }$)). The present study adopts the notations employed in [6], [7]. Consequently, ${\mathrm{C}}_1{\rho }_{\max }^5$ is the secondary spherical aberration coefficient; ${\mathrm{C}}_2{\mathrm{h}}_0{\rho }_{\max }^4$ and ${\mathrm{C}}_3{\mathrm{h}}_0{\rho }_{\max }^4$ are the linear coma coefficients; ${\mathrm{C}}_4{\mathrm{h}}_0^2{\rho }_{\max }^3$, ${\mathrm{C}}_5{\mathrm{h}}_0^2{\rho }_{\max }^3$ and ${\mathrm{C}}_6{\mathrm{h}}_0^2{\rho }_{\max }^3$ are the oblique spherical aberration coefficients; ${\mathrm{C}}_7{\mathrm{h}}_0^3{\rho }_{\max }^2$, ${\mathrm{C}}_8{\mathrm{h}}_0^3{\rho }_{\max }^2$ and ${\mathrm{C}}_9{\mathrm{h}}_0^3{\rho }_{\max }^2$ are the elliptical coma coefficients; ${\mathrm{C}}_{10}{\mathrm{h}}_0^4{\rho }_{\max }$ and ${\mathrm{C}}_{11}{\mathrm{h}}_0^4{\rho }_{\max }$ are the Petzval and astigmatism coefficients; and ${\mathrm{C}}_{12}{\mathrm{h}}_0^5$ is the distortion coefficient.

Aberrations result in significant blurring or distortion of the image. Consequently, effective methods for quantifying their effects during the optical design stage are of paramount importance. The literature contains many approaches for determining the Seidel primary aberration coefficients [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. One of the most widely used methods in optical software (e.g., Zemax [6]) is that proposed by Buchdahl [8]. However, an alternative approach was recently proposed by Lin and Johnson [23], [24] based on Taylor series expansion of a skew ray. The numerical results of the primary ray-aberrations were shown to be in remarkable agreement with those obtained from Zemax simulations. Consequently, the present study extends this approach to determine the secondary ray-aberration coefficients of an axis-symmetrical system when the object is placed at finite distance. In doing so, a general skew ray ${\bar{\mathrm{R}}}_{\mathrm{i}}$ is expanded with respect to the source ray variable vector ${\bar{\mathrm{X}}}_0$ by a Taylor series expansion up to the fifth order. Notably, the obtained equations are exact formulae without approximation. The validity of the proposed method is demonstrated by comparing the computation results with those obtained from Zemax simulations and raytracing, respectively.

2. Variable vectors of source ray

Fig. 2 shows an illustrative axis-symmetrical system with n boundaries when the object is placed at finite position. In accordance with convention, the label i=0 is assigned to the source ray ${\bar{\mathrm{R}}}_0={\left[\begin{array}{cc}\hfill {\bar{\mathrm{P}}}_0\hfill & \hfill {\bar{\ell }}_0\hfill \end{array}\right]}^{\mathrm{T}}$ originating from the object ${\bar{\mathrm{P}}}_0$ with height ${\mathrm{h}}_0$, i.e.,

$${\bar{\mathrm{P}}}_0={\left[\begin{array}{ccc}\hfill {\mathrm{P}}_{0\mathrm{x}}\hfill & \hfill {\mathrm{h}}_0\hfill & \hfill {\mathrm{P}}_{0\mathrm{z}}\hfill \end{array}\right]}^{\mathrm{T}}.$$ (3)

The unit directional vector of ${\bar{\mathrm{R}}}_0$ can be defined as

$${\bar{\ell }}_0={\left[\begin{array}{ccc}\hfill \mathrm{S}{\alpha }_0\mathrm{C}{\beta }_0\hfill & \hfill \mathrm{S}{\beta }_0\hfill & \hfill \mathrm{C}{\alpha }_0\mathrm{C}{\beta }_0\hfill \end{array}\right]}^{\mathrm{T}}.$$ (4)

From Eqs. (3) and (4), the independent variable vector of ray ${\bar{\mathrm{R}}}_0$ is given by

$${\bar{\mathrm{X}}}_0={\left[\begin{array}{ccccc}\hfill {\mathrm{P}}_{0\mathrm{x}}\hfill & \hfill {\mathrm{h}}_0\hfill & \hfill {\mathrm{P}}_{0\mathrm{z}}\hfill & \hfill {\alpha }_0\hfill & \hfill {\beta }_0\hfill \end{array}\right]}^{\mathrm{T}}.$$ (5)

Figure 2.

Illustrative axis-symmetrical optical system comprising six elements and eleven boundaries.

In conventional aberration determination, the unit directional vector, ${\bar{\ell }}_0$, is defined in terms of the in-plane coordinates ${\left[\begin{array}{cc}\hfill {\mathrm{x}}_{\mathrm{a}}\hfill & \hfill {\mathrm{y}}_{\mathrm{a}}\hfill \end{array}\right]}^{\mathrm{T}}$ of the entrance pupil (Fig. 3). This representation causes the independent variable vector to be different from that given in Eq. (5), i.e., (Eq. (5) of [24]):

$${\bar{\mathrm{Y}}}_0={\left[\begin{array}{ccccc}\hfill {\mathrm{P}}_{0\mathrm{x}}\hfill & \hfill {\mathrm{h}}_0\hfill & \hfill {\mathrm{P}}_{0\mathrm{z}}\hfill & \hfill {\mathrm{x}}_{\mathrm{a}}\hfill & \hfill ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)\hfill \end{array}\right]}^{\mathrm{T}}.$$ (6)

Figure 3.

Entrance pupil with Cartesian coordinates (x_a,y_a) and polar coordinates (ρ,ϕ).

In exploring the aberrations of object ${\bar{\mathrm{P}}}_0={\left[\begin{array}{ccc}\hfill 0\hfill & \hfill {\mathrm{h}}_0\hfill & \hfill {\mathrm{P}}_{0\mathrm{z}}\hfill \end{array}\right]}^{\mathrm{T}}$ in Fig. 2, a general skew ray ${\bar{\mathrm{R}}}_{\mathrm{i}}$ should be expanded with respect to the ray originating from point ${\bar{\mathrm{P}}}_{0,\mathrm{axis}}={\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill {\mathrm{P}}_{0\mathrm{z}}\hfill \end{array}\right]}^{\mathrm{T}}$ with unit directional vector ${\bar{\ell }}_{0,\mathrm{axis}}={\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]}^{\mathrm{T}}$ (see Fig. 4). The two independent variables of ${\bar{\ell }}_{0,\mathrm{axis}}$ are thus given as ${\alpha }_0={\beta }_0={\mathrm{x}}_{\mathrm{a}}={\mathrm{y}}_{\mathrm{a}}=0$. For convenience, this on-axis ray ${\left[\begin{array}{cc}\hfill {\bar{\mathrm{P}}}_{0,\mathrm{axis}}\hfill & \hfill {\bar{\ell }}_{0,\mathrm{axis}}\hfill \end{array}\right]}^{\mathrm{T}}$ can be denoted by the following vector:

$${\bar{0}}_{\mathrm{axis}}\equiv {\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill {\mathrm{P}}_{0\mathrm{z}}\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]}^{\mathrm{T}}.$$ (7)

It was shown in the Appendix of [24] that the conversion between Eqs. (5) and (6) for the ray defined by Eq. (7) can be achieved by Eq. (8)

$$\frac{{\partial }^{\mathrm{f}+\mathrm{g}+\mathrm{h}}{\bar{\mathrm{R}}}_{\mathrm{i}}}{\partial {\mathrm{h}}_0^{\mathrm{f}}\partial {\mathrm{x}}_{\mathrm{a}}^{\mathrm{g}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^{\mathrm{h}}}=\frac{{\partial }^{\mathrm{f}+\mathrm{g}+\mathrm{h}}{\bar{\mathrm{R}}}_{\mathrm{i}}}{\partial {\mathrm{h}}_0^{\mathrm{f}}\partial {\alpha }_0^{\mathrm{g}}\partial {\beta }_0^{\mathrm{h}}}{\left(\frac{\partial {\alpha }_0}{\partial {\mathrm{x}}_{\mathrm{a}}}\right)}^{\mathrm{g}}{\left(\frac{\partial {\beta }_0}{\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}\right)}^{\mathrm{h}},$$ (8)

where $\partial {\alpha }_0/\partial {\mathrm{x}}_{\mathrm{a}}=\partial {\beta }_0/\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)=4.502409965\times {10}^{-3}$. (Note that all lengths and angles given in this study have units of mm and degrees, respectively.)

Figure 4.

Ray with source ray variable vector ${\bar{0}}_{\mathrm{axis}}$.

3. Taylor series expansion of skew ray

In geometrical optics, a general ray ${\bar{\mathrm{R}}}_{\mathrm{i}}$ is a composite function of ${\bar{\mathrm{Y}}}_0$ (or ${\bar{\mathrm{X}}}_0$). The most elegant solution in mathematics for hard problems of this type is the Taylor series expansion since it can provide the polynomial expression of ${\bar{\mathrm{R}}}_{\mathrm{i}}={\bar{\mathrm{R}}}_{\mathrm{i}}({\bar{\mathrm{Y}}}_0)$ in terms of ${\bar{\mathrm{Y}}}_0-{\bar{0}}_{\mathrm{axis}}$, i.e.,

$$\begin{gathered} {\bar{\mathrm{R}}}_{\mathrm{i}}({\bar{\mathrm{Y}}}_0)\cong {\bar{\mathrm{R}}}_{\mathrm{i}}({\bar{0}}_{\mathrm{axis}})+\mathrm{\Delta }{\bar{\mathrm{R}}}_{\mathrm{i}}={\bar{\mathrm{R}}}_{\mathrm{i}}({\bar{0}}_{\mathrm{axis}})+\mathrm{\Delta }{\bar{\mathrm{R}}}_{i/1\mathrm{st}}+\mathrm{\Delta }{\bar{\mathrm{R}}}_{i/2\mathrm{nd}}+\mathrm{\Delta }{\bar{\mathrm{R}}}_{i/3\mathrm{rd}} \\ +\mathrm{\Delta }{\bar{\mathrm{R}}}_{i/4\mathrm{th}}+\mathrm{\Delta }{\bar{\mathrm{R}}}_{i/5\mathrm{th}}. \end{gathered}$$ (9)

Equation (9) indicates that general ray

$${\bar{\mathrm{R}}}_{\mathrm{i}}({\bar{\mathrm{Y}}}_0)={\left[\begin{array}{cccccc}\hfill {\mathrm{P}}_{\mathrm{ix}}({\bar{\mathrm{Y}}}_0)\hfill & \hfill {\mathrm{P}}_{\mathrm{ix}}({\bar{\mathrm{Y}}}_0)\hfill & \hfill {\mathrm{P}}_{\mathrm{iz}}({\bar{\mathrm{Y}}}_0)\hfill & \hfill {\ell }_{\mathrm{ix}}({\bar{\mathrm{Y}}}_0)\hfill & \hfill {\ell }_{\mathrm{iy}}({\bar{\mathrm{Y}}}_0)\hfill & \hfill {\ell }_{\mathrm{iz}}({\bar{\mathrm{Y}}}_0)\hfill \end{array}\right]}^{\mathrm{T}}$$

is expanded up to the fifth order. In aberration coefficient determination, the first term in Eq. (9), i.e., ${\bar{\mathrm{R}}}_{\mathrm{i}}({\bar{0}}_{\mathrm{axis}})$, should be evaluated using the ray defined by ${\bar{0}}_{\mathrm{axis}}$. Furthermore, in aberration exploration, only the two coordinates of the incidence point on the image plane (i.e., ${\mathrm{P}}_{\mathrm{nx}}({\bar{\mathrm{Y}}}_0)$ and ${\mathrm{P}}_{\mathrm{ny}}({\bar{\mathrm{Y}}}_0)$) are needed. They can be obtained from the first and second components of Eq. (9) by setting i=n, to have

$${\mathrm{P}}_{\mathrm{nx}}({\bar{\mathrm{Y}}}_0)\cong {\mathrm{P}}_{\mathrm{nx}}({\bar{0}}_{\mathrm{axis}})+\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/1\mathrm{st}}+\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/3\mathrm{rd}}+\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/5\mathrm{th}},$$ (10a)

$${\mathrm{P}}_{\mathrm{ny}}({\bar{\mathrm{Y}}}_0)\cong {\mathrm{P}}_{\mathrm{ny}}({\bar{0}}_{\mathrm{axis}})+\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/1\mathrm{st}}+\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/3\mathrm{rd}}+\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/5\mathrm{th}}.$$ (10b)

Note that $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/2\mathrm{nd}}=\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/2\mathrm{nd}}=\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/4\mathrm{th}}=\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/4\mathrm{th}}=0$ can be set in Eqs. (10a) and (10b) due to the axis-symmetrical nature of the considered system. The explicit expansions of $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/1\mathrm{st}}$, $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/1\mathrm{st}}$, $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/3\mathrm{rd}}$ and $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/3\mathrm{rd}}$ are given in Eqs. (9a)-(11b) of [24], respectively. Terms ${\mathrm{P}}_{\mathrm{nx}}({\bar{0}}_{\mathrm{axis}})$ and ${\mathrm{P}}_{\mathrm{ny}}({\bar{0}}_{\mathrm{axis}})$ in Eqs. (10a) and (10b) are the offsets between the origins of coordinate frame ${(\mathrm{xyz})}_{\mathrm{n}}$ on the image plane and the world coordinate frame ${(\mathrm{xyz})}_0$, respectively. As such, they have no connection with ray-aberrations, and can hence be ignored. Terms $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/1\mathrm{st}}$ and $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/1\mathrm{st}}$ determine the defocus aberration coefficient ${\mathrm{A}}_1$ and transverse magnification ${\mathrm{A}}_2$, respectively, while $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/3\mathrm{rd}}$ and $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/3\mathrm{rd}}$ can be used to explore the B aberration coefficients [24]. The complete expression of $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/5\mathrm{th}}$ in Eq. (10a), (10b)(a) has the form

$$\begin{gathered} \mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/5\mathrm{th}}=\frac{1}{120}[\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^5}{\mathrm{h}}_0^5+5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^4\partial {\mathrm{x}}_{\mathrm{a}}}{\mathrm{h}}_0^4{\mathrm{x}}_{\mathrm{a}}+5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}{\mathrm{h}}_0^4({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)+10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^3\partial {\mathrm{x}}_{\mathrm{a}}^2}{\mathrm{h}}_0^3{\mathrm{x}}_{\mathrm{a}}^2+20\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^3\partial {\mathrm{x}}_{\mathrm{a}}\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}{\mathrm{h}}_0^3{\mathrm{x}}_{\mathrm{a}}({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)+10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}{\mathrm{h}}_0^3{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2+10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^3}{\mathrm{h}}_0^2{\mathrm{x}}_{\mathrm{a}}^3 \\ +30\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^2\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}{\mathrm{h}}_0^2{\mathrm{x}}_{\mathrm{a}}^2({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)+30\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}{\mathrm{h}}_0^2{\mathrm{x}}_{\mathrm{a}}{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2 \\ +10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}{\mathrm{h}}_0^2{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3+5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^4}{\mathrm{h}}_0{\mathrm{x}}_{\mathrm{a}}^4 \\ +20\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^3\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}{\mathrm{h}}_0{\mathrm{x}}_{\mathrm{a}}^3({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)+30\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}{\mathrm{h}}_0{\mathrm{x}}_{\mathrm{a}}^2{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2 \\ +20\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}{\mathrm{h}}_0{\mathrm{x}}_{\mathrm{a}}{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3 \\ +5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}{\mathrm{h}}_0{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^5}{\mathrm{x}}_{\mathrm{a}}^5 \\ +5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}{\mathrm{x}}_{\mathrm{a}}^4({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)+10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}{\mathrm{x}}_{\mathrm{a}}^3{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2 \\ +10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}{\mathrm{x}}_{\mathrm{a}}^2{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3+5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}{\mathrm{x}}_{\mathrm{a}}{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4 \\ +\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5}{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5]. \end{gathered}$$ (11)

Meanwhile, the complete expression of $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/5\mathrm{th}}$ in Eq. (10b) can be obtained from Eq. (11) by simply substituting ${\mathrm{P}}_{\mathrm{ny}}$ for ${\mathrm{P}}_{\mathrm{nx}}$.

Due to the axis-symmetrical nature of the considered system, various terms in Eq. (11) disappear (as confirmed by our numerical results), and hence the expressions of $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/5\mathrm{th}}$ and $\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/5\mathrm{th}}$ become

$$\begin{gathered} \mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/5\mathrm{th}}=\frac{1}{120}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^5}{\mathrm{x}}_{\mathrm{a}}^5+10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}{\mathrm{x}}_{\mathrm{a}}^3{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2 \\ +5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}{\mathrm{x}}_{\mathrm{a}}{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4 \\ +20\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^3\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}{\mathrm{h}}_0{\mathrm{x}}_{\mathrm{a}}^3({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)+20\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}{\mathrm{h}}_0{\mathrm{x}}_{\mathrm{a}}{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3 \\ +10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^3}{\mathrm{h}}_0^2{\mathrm{x}}_{\mathrm{a}}^3+30\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}{\mathrm{h}}_0^2{\mathrm{x}}_{\mathrm{a}}{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2 \\ +20\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^3\partial {\mathrm{x}}_{\mathrm{a}}\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}{\mathrm{h}}_0^3{\mathrm{x}}_{\mathrm{a}}({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)+5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^4\partial {\mathrm{x}}_{\mathrm{a}}}{\mathrm{h}}_0^4{\mathrm{x}}_{\mathrm{a}}), \end{gathered}$$ (12a)

$$\begin{gathered} \mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/5\mathrm{th}}=\frac{1}{120}(5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}{\mathrm{x}}_{\mathrm{a}}^4({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)+10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}{\mathrm{x}}_{\mathrm{a}}^2{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3 \\ +\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5}{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5+5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^4}{\mathrm{h}}_0{\mathrm{x}}_{\mathrm{a}}^4 \\ +30\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}{\mathrm{h}}_0{\mathrm{x}}_{\mathrm{a}}^2{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2 \\ +5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}{\mathrm{h}}_0{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4 \\ +30\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^2\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}{\mathrm{h}}_0^2{\mathrm{x}}_{\mathrm{a}}^2({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)+10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}{\mathrm{h}}_0^2{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3+10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^3\partial {\mathrm{x}}_{\mathrm{a}}^2}{\mathrm{h}}_0^3{\mathrm{x}}_{\mathrm{a}}^2 \\ +10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}{\mathrm{h}}_0^3{({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2 \\ +5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}{\mathrm{h}}_0^4({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^5}{\mathrm{h}}_0^5). \end{gathered}$$ (12b)

As shown, Eqs. (12a) and (12b) are both expressed with respect to the object height ${\mathrm{h}}_0$ and in-plane Cartesian coordinates (${\mathrm{x}}_{\mathrm{a}},{\mathrm{y}}_{\mathrm{a}}$) of the entrance pupil. However, aberration equations are usually expressed in terms of the polar coordinates of the entrance pupil, i.e., ($\rho ,\varphi$) (Fig. 3). Hence, it is necessary to replace (${\mathrm{x}}_{\mathrm{a}},{\mathrm{y}}_{\mathrm{a}}$) in Eqs. (12a) and (12b) with ($\rho \mathrm{S}\varphi ,\rho \mathrm{C}\varphi$). The following equations are thus obtained:

$$\begin{gathered} \mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/5\mathrm{th}}={\mathrm{f}}_{\mathrm{x}/0}{\rho }^5\mathrm{S}\varphi +{\mathrm{f}}_{\mathrm{x}/1}({\mathrm{h}}_0{\rho }^4)\mathrm{S}2\varphi +{\mathrm{f}}_{\mathrm{x}/2}({\mathrm{h}}_0^2{\rho }^3)\mathrm{S}\varphi \\ +{\mathrm{f}}_{\mathrm{x}/3}({\mathrm{h}}_0^3{\rho }^2)\mathrm{S}(2\varphi )+{\mathrm{f}}_{\mathrm{x}/4}({\mathrm{h}}_0^4\rho )\mathrm{S}\varphi , \end{gathered}$$ (13a)

$$\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/5\mathrm{th}}={\mathrm{f}}_{\mathrm{y}/0}{\rho }^5\mathrm{C}\varphi +{\mathrm{f}}_{\mathrm{y}/1}({\mathrm{h}}_0{\rho }^4)+{\mathrm{f}}_{\mathrm{y}/2}({\mathrm{h}}_0^2{\rho }^3)\mathrm{C}\varphi +{\mathrm{f}}_{\mathrm{y}/3}({\mathrm{h}}_0^3{\rho }^2)+{\mathrm{f}}_{\mathrm{y}/4}({\mathrm{h}}_0^4\rho )\mathrm{C}\varphi +{\mathrm{f}}_{\mathrm{y}/5}{\mathrm{h}}_0^5.$$ (13b)

The leading coefficients of Eqs. (13a) and (13b) are given respectively by

$${\mathrm{f}}_{\mathrm{x}/0}=\frac{1}{120}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^5}{\mathrm{S}}^4\varphi +5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}{\mathrm{C}}^4\varphi +10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}{\mathrm{S}}^2\varphi {\mathrm{C}}^2\varphi ),$$ (14a)

$$\begin{gathered} {\mathrm{f}}_{\mathrm{x}/1}=\frac{-1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}){\mathrm{C}}^2\varphi \\ -\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^3\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}){\mathrm{S}}^2\varphi , \end{gathered}$$ (14b)

$$\begin{gathered} {\mathrm{f}}_{\mathrm{x}/2}=\frac{1}{4}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}-2\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}){\mathrm{C}}^2\varphi \\ +\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}-2\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^3\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^3}){\mathrm{S}}^2\varphi , \end{gathered}$$ (14c)

$$\begin{gathered} {\mathrm{f}}_{\mathrm{x}/3}=\frac{1}{12}(-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}+3\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}-3\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2} \\ +\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^3\partial {\mathrm{x}}_{\mathrm{a}}\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}), \end{gathered}$$ (14d)

$$\begin{gathered} {\mathrm{f}}_{\mathrm{x}/4}=\frac{1}{24}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}-4\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}+6\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2} \\ -4\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^3\partial {\mathrm{x}}_{\mathrm{a}}\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^4\partial {\mathrm{x}}_{\mathrm{a}}}), \end{gathered}$$ (14e)

$$\begin{gathered} {\mathrm{f}}_{\mathrm{y}/0}=\frac{1}{120}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5}{\mathrm{C}}^4\varphi +5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}{\mathrm{S}}^4\varphi \\ +10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}{\mathrm{S}}^2\varphi {\mathrm{C}}^2\varphi ), \end{gathered}$$ (15a)

$$\begin{gathered} {\mathrm{f}}_{\mathrm{y}/1}=\frac{1}{24}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^4}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}){\mathrm{S}}^4\varphi \\ +\frac{1}{4}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}){\mathrm{S}}^2\varphi {\mathrm{C}}^2\varphi \\ +\frac{1}{24}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5}){\mathrm{C}}^4\varphi , \end{gathered}$$ (15b)

$$\begin{gathered} {\mathrm{f}}_{\mathrm{y}/2}=\frac{1}{4}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}-2\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^2\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}){\mathrm{S}}^2\varphi \\ +\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5}-2\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}){\mathrm{C}}^2\varphi , \end{gathered}$$ (15c)

$$\begin{gathered} {\mathrm{f}}_{\mathrm{y}/3}=\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^3\partial {\mathrm{x}}_{\mathrm{a}}^2}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}+3\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2} \\ -3\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^2\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}){\mathrm{S}}^2\varphi \\ -\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}+3\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3} \\ -3\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}){\mathrm{C}}^2\varphi , \end{gathered}$$ (15d)

$$\begin{gathered} {\mathrm{f}}_{\mathrm{y}/4}=\frac{1}{24}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5}-4\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}+6\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3} \\ -4\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}), \end{gathered}$$ (15e)

$$\begin{gathered} {\mathrm{f}}_{\mathrm{y}/5}=\frac{1}{120}(-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5}+5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}-10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3} \\ +10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}-5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^5}). \end{gathered}$$ (15f)

Equations (10a) and (10b) are derived independently from other aberration theory. However, Eqs. (2a) and (2b) are obtained from the wavefront aberration function $\mathrm{W}({\mathrm{h}}_0,\rho ,\varphi )$ by using the following approximate partial derivatives [1]:

$$\mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}}\approx \frac{{\mathrm{R}}_{\mathrm{reference}}}{{\xi }^{\prime }}\frac{\partial \mathrm{W}}{\partial {\mathrm{x}}_{\mathrm{a}}},$$ (16a)

$$\mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}}\approx \frac{{\mathrm{R}}_{\mathrm{reference}}}{{\xi }^{\prime }}\frac{\partial \mathrm{W}}{\partial {\mathrm{y}}_{\mathrm{a}}},$$ (16b)

where ${\mathrm{R}}_{\mathrm{reference}}$ and ${\xi }^{\prime }$ are the radius and refractive index of the reference sphere, respectively. As discussed by Restrepo et al. [1], Eqs. (16a) and (16b) inevitably induce errors as a result of their approximate nature. Furthermore, Sasián pointed out that these errors are of the fifth order (p. 102 of [17]). Figure 5, Figure 6 compare the values of ${\mathrm{P}}_{\mathrm{nx}}$ and ${\mathrm{P}}_{\mathrm{ny}}$ obtained from Eqs. (1), (10a) and (10b) for the system shown in Fig. 2 (its parameters are given in Table 1) with those obtained from raytracing. The values of the A, B and C aberration coefficients computed by Zemax [6] for the considered system are listed in Table 2 of the present study for the case where the object is placed at ${\bar{\mathrm{P}}}_0=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 17\hfill & \hfill -200\hfill \end{array}\right]$ and the maximum entrance pupil radius is ${\rho }_{\max }=21$. The computed curves in Figure 5, Figure 6 show the interception coordinates on the Gaussian image plane of rays originating from ${\bar{\mathrm{P}}}_0$ and striking the entire rim of the entrance pupil (i.e., $0^{\circ }\le \varphi <{360}^{\circ }$) with ${\rho }_{\max }=21$. Curves ${\mathrm{P}}_{\mathrm{nx}}({\bar{\mathrm{Y}}}_0)$ and ${\mathrm{P}}_{\mathrm{ny}}({\bar{\mathrm{Y}}}_0)$ are obtained from raytracing, and are thus marred by higher-order aberrations. By contrast, the computed curves from Eq. (1), Eqs. (10a) and (10b) consist only of the zero- to fifth-order series. It is seen in Fig. 5 that the curves obtained from Eq. (1) and Eq. (10a) are almost overlaid for $0^{\circ }\le \varphi <{360}^{\circ }$. Similarly, in Fig. 6, the curves obtained from Eq. (1) and Eq. (10b) are indistinguishable except for the region near $\varphi ={180}^{\circ }$. Consequently, it is reasonable to infer that the secondary ray-aberration coefficients obtained using the proposed method in the present study (see the following section) will not deviate too much with those obtained from Zemax.

Figure 5.

Comparison of results obtained from Eqs. (1) and (10a) for P_nx with those obtained from raytracing.

Figure 6.

Comparison of results obtained from Eqs. (1) and (10b) for P_ny with those obtained from raytracing.

Table 1.

Specification of illustrative rotationally-symmetric optical system shown in Fig. 2.

Surface	Radius	Separation	Refractive index
0 (object)	∞	200.00000	1.00000
1	38.22190	15.84960	1.65000
2	-56.08570	5.96900	1.71736
3	-590.68200	3.02260	1.00000
4 (aperture)	∞	14.02080	1.00000
5	-41.79570	2.514600	1.52583
6	29.34460	7.924800	1.00000
7	63.56350	6.09600	1.65000
8	-56.86550	92.088474	1.00000
9 (image plane)	∞

Table 2.

Values of aberration coefficients obtained from Zemax simulations.

A1 = 0.000000	A2 = −0.660413	${\mathrm{B}}_1{\rho }_{\max }^3=-1.533486$
${\mathrm{B}}_2{\mathrm{h}}_0{\rho }_{\max }^2=-0.112776$	${\mathrm{B}}_3{\mathrm{h}}_0^2{\rho }_{\max }=8.16154\times {10}^{-3}$	${\mathrm{B}}_4{\mathrm{h}}_0^2{\rho }_{\max }=-0.033624$
${\mathrm{C}}_1{\rho }_{\max }^5=-0.075594$	${\mathrm{C}}_2{\mathrm{h}}_0{\rho }_{\max }^4=-0.090476$	${\mathrm{C}}_3{\mathrm{h}}_0{\rho }_{\max }^4=-0.067381$
${\mathrm{C}}_4{\mathrm{h}}_0^2{\rho }_{\max }^3=0.090356$	${\mathrm{C}}_5{\mathrm{h}}_0^2{\rho }_{\max }^3=0.033730$	${\mathrm{C}}_6{\mathrm{h}}_0^2{\rho }_{\max }^3=0.067592$
${\mathrm{C}}_7{\mathrm{h}}_0^3{\rho }_{\max }^2=-0.010459$	${\mathrm{C}}_8{\mathrm{h}}_0^3{\rho }_{\max }^2=-0.032328$	${\mathrm{C}}_9{\mathrm{h}}_0^3{\rho }_{\max }^2=-0.009668$
${\mathrm{C}}_{10}{\mathrm{h}}_0^4{\rho }_{\max }=0.001088$	${\mathrm{C}}_{11}{\mathrm{h}}_0^4{\rho }_{\max }=0.001406$	${\mathrm{C}}_{12}{\mathrm{h}}_0^5=0.0006322$

4. Results of secondary ray-aberration coefficients

In this section, the values of the secondary ray-aberration coefficients are determined for the axis-symmetrical system shown in Fig. 2 (with the object positioned at ${\mathrm{P}}_{0\mathrm{z}}=-200$, the image plane coinciding with the Gaussian image plane, and the maximum radius of the entrance pupil being ${\rho }_{\max }=21$). The symbols in Eqs. (2a) and (2b) are still used under the assumption that they are functions of ϕ. One has the following equations from Eqs. (2a) and (2b) by taking only the terms with C coefficients:

$$\begin{gathered} \mathrm{\Delta }{\mathrm{P}}_{\mathrm{nx}/5\mathrm{th}}={\mathrm{C}}_1{\rho }^5\mathrm{S}\varphi +{\mathrm{C}}_3({\mathrm{h}}_0{\rho }^4)\mathrm{S}(2\varphi )+({\mathrm{C}}_5+{\mathrm{C}}_6{\mathrm{C}}^2\varphi )({\mathrm{h}}_0^2{\rho }^3)\mathrm{S}\varphi \\ +{\mathrm{C}}_9({\mathrm{h}}_0^3{\rho }^2)\mathrm{S}(2\varphi )+{\mathrm{C}}_{11}({\mathrm{h}}_0^4\rho )\mathrm{S}\varphi , \end{gathered}$$ (17a)

$$\begin{gathered} \mathrm{\Delta }{\mathrm{P}}_{\mathrm{ny}/5\mathrm{th}}={\mathrm{C}}_1{\rho }^5\mathrm{C}\varphi +({\mathrm{C}}_2+{\mathrm{C}}_3^{\prime }\mathrm{C}2\varphi )({\mathrm{h}}_0{\rho }^4)+({\mathrm{C}}_4+{\mathrm{C}}_6^{\prime }{\mathrm{C}}^2\varphi )({\mathrm{h}}_0^2{\rho }^3)\mathrm{C}\varphi \\ +[{\mathrm{C}}_7+{\mathrm{C}}_8{\mathrm{C}}^2\varphi ]({\mathrm{h}}_0^3{\rho }^2)+{\mathrm{C}}_{10}({\mathrm{h}}_0^4\rho )\mathrm{C}\varphi +{\mathrm{C}}_{12}{\mathrm{h}}_0^5. \end{gathered}$$ (17b)

It is noted that coefficients ${\mathrm{C}}_3^{\prime }$ and ${\mathrm{C}}_6^{\prime }$ are used in Eq. (17b) to take account of possible differences in the expressions and possible errors arising from the approximation of Eqs. (16b). One thus has the following equations by comparing Eqs. (13a) and (13b) with Eqs. (17a) and (17b), respectively:

$${\mathrm{C}}_1={\mathrm{f}}_{\mathrm{x}/0},$$ (18a)

$${\mathrm{C}}_3={\mathrm{f}}_{\mathrm{x}/1},$$ (18b)

$${\mathrm{C}}_5+{\mathrm{C}}_6{\mathrm{C}}^2\varphi ={\mathrm{f}}_{\mathrm{x}/2},$$ (18c)

$${\mathrm{C}}_9={\mathrm{f}}_{\mathrm{x}/3},$$ (18d)

$${\mathrm{C}}_{11}={\mathrm{f}}_{\mathrm{x}/4},$$ (18e)

$${\mathrm{C}}_1={\mathrm{f}}_{\mathrm{y}/0},$$ (19a)

$${\mathrm{C}}_2+{\mathrm{C}}_3^{\prime }\mathrm{C}2\varphi ={\mathrm{f}}_{\mathrm{y}/1},$$ (19b)

$${\mathrm{C}}_4+{\mathrm{C}}_6^{\prime }{\mathrm{C}}^2\varphi ={\mathrm{f}}_{\mathrm{y}/2},$$ (19c)

$${\mathrm{C}}_7+{\mathrm{C}}_8{\mathrm{C}}^2\varphi ={\mathrm{f}}_{\mathrm{y}/3},$$ (19d)

$${\mathrm{C}}_{10}={\mathrm{f}}_{\mathrm{y}/4},$$ (19e)

$${\mathrm{C}}_{12}={\mathrm{f}}_{\mathrm{y}/5}.$$ (19f)

Since Eqs. (16a) and (16b) are not exact equations and possess fifth-order errors [1], it is possible that the values obtained in this study are different from those of Zemax. Therefore, the following percentage difference metric is introduced to compare the results obtained using the proposed method with those obtained from Zemax simulations:

$${\mathrm{C}}_{\mathrm{k}}\text{\%}=100\left|\frac{({\mathrm{C}}_{\mathrm{k}/\mathrm{this}}-{\mathrm{C}}_{\mathrm{k}/\mathrm{ze}\max })}{{\mathrm{w}}_{\mathrm{k}/\mathrm{ze}\max }}\right|,$$ (20)

where ${\mathrm{C}}_{\mathrm{k}/\mathrm{this}}$ and ${\mathrm{C}}_{\mathrm{k}/\mathrm{ze}\max }$ of Eq. (20) are the kth (k=1-12) aberration coefficient values obtained from the proposed method and Zemax simulations, respectively.

(1) ${\mathrm{C}}_1$ coefficient: The spherical aberration is the only fifth-order term which does not depend on ${\mathrm{h}}_0$. From Eqs. (18a) and (14a), ${\mathrm{C}}_1$ can be expressed as

$${\mathrm{C}}_1={\mathrm{C}}_{1\mathrm{zero}}+{\mathrm{C}}_{1\mathrm{two}}{\mathrm{S}}^2\varphi +{\mathrm{C}}_{1\mathrm{four}}{\mathrm{S}}^4\varphi ,$$ (21a)

where

$${\mathrm{C}}_{1\mathrm{zero}}=\frac{1}{24}\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4},$$ (21b)

$${\mathrm{C}}_{1\mathrm{two}}=\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}),$$ (21c)

$${\mathrm{C}}_{1\mathrm{four}}=\frac{1}{120}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^5}+5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}-10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}).$$ (21d)

Meanwhile, from Eqs. (19a) and (15a), ${\mathrm{C}}_1$ can also be expressed as

$${\mathrm{C}}_1={\mathrm{C}}_{1\mathrm{zero}}+{\mathrm{C}}_{1\mathrm{two}}{\mathrm{C}}^2\varphi +{\mathrm{C}}_{14}{\mathrm{C}}^4\varphi ,$$ (21e)

where

$${\mathrm{C}}_{1\mathrm{zero}}=\frac{1}{24}\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)},$$ (21f)

$${\mathrm{C}}_{1\mathrm{two}}=\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}),$$ (21g)

$${\mathrm{C}}_{1\mathrm{four}}=\frac{1}{120}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5}+5\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}-10\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}).$$ (21h)

If ${\mathrm{C}}_1$ is determined from Eq. (21a) by setting $\varphi ={90}^{\circ }$, or from Eq. (21e) by setting $\varphi =0^{\circ }$, the following value is obtained:

$${\mathrm{C}}_1={\mathrm{C}}_{1\mathrm{zero}}+{\mathrm{C}}_{1\mathrm{two}}+{\mathrm{C}}_{1\mathrm{four}}=-218.65267\times {10}^{-10}.$$ (21i)

The value of the secondary spherical aberration coefficient can be obtained from Eq. (21i) as

$${\mathrm{C}}_{1/\mathrm{this}}={\mathrm{C}}_1{\rho }_{\max }^5=-0.089300.$$ (21j)

This value differs by ${\mathrm{C}}_1\text{\%}=18.1\text{\%}$ from the value obtained by Zemax simulations (${\mathrm{C}}_{1/\mathrm{ze}\max }=-0.075594$).

(2) ${\mathrm{C}}_2$ and ${\mathrm{C}}_3$ coefficients: Equations (18b) and (14b) show that ${\mathrm{C}}_3$ can be expressed as

$${\mathrm{C}}_3={\mathrm{C}}_{3\mathrm{zero}}+{\mathrm{C}}_{3\mathrm{two}}{\mathrm{S}}^2\varphi ,$$ (22a)

where

$${\mathrm{C}}_{3\mathrm{zero}}=\frac{-1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}),$$ (22b)

$${\mathrm{C}}_{3\mathrm{two}}=\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3})-\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^3\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}).$$ (22c)

Furthermore, from Eqs. (19b) and (15b), ${\mathrm{C}}_3^{\prime }$ can be expressed as the following function of ϕ:

$${\mathrm{C}}_3^{\prime }={\mathrm{C}}_{3\mathrm{two}}^{\prime }{\mathrm{C}}^2\varphi +{\mathrm{C}}_{3\mathrm{four}}^{\prime }{\mathrm{C}}^4\varphi .$$ (22d)

From Eqs. (15b), (19b) and (22d), the expressions for ${\mathrm{C}}_2$, ${\mathrm{C}}_{3\mathrm{two}}^{\prime }$ and ${\mathrm{C}}_{3\mathrm{four}}^{\prime }$ are obtained as

$${\mathrm{C}}_2=\frac{1}{24}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^4}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}),$$ (22e)

$${\mathrm{C}}_{3\mathrm{two}}^{\prime }=-\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^4}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)})+\frac{1}{4}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}),$$ (22f)

$${\mathrm{C}}_{3\mathrm{four}}^{\prime }=\frac{1}{24}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^4}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^4\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)})-\frac{1}{4}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3})+\frac{1}{24}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5}).$$ (22g)

The values of the linear coma aberration coefficient are obtained from Eq. (22e) as ${\mathrm{C}}_{2/\mathrm{this}}={\mathrm{C}}_2{\rho }_{\max }^4{\mathrm{h}}_0=-0.018219$. The other two linear coma aberration coefficients are found from Eq. (22a) with $\varphi ={90}^{\circ }$ and Eq. (22d) with $\varphi =0^{\circ }$ to be ${\mathrm{C}}_{3/\mathrm{this}}=({\mathrm{C}}_{3\mathrm{zero}}+{\mathrm{C}}_{3\mathrm{two}}){\rho }_{\max }^4{\mathrm{h}}_0=-0.062674$ and ${\mathrm{C}}_{3/\mathrm{this}}^{\prime }=({\mathrm{C}}_{3\mathrm{two}}^{\prime }+{\mathrm{C}}_{3\mathrm{four}}^{\prime }){\rho }_{\max }^4{\mathrm{h}}_0=-0.108368$, respectively. These values deviate by ${\mathrm{C}}_2\text{\%}=79.9\text{\%}$, ${\mathrm{C}}_3\text{\%}=7.0\text{\%}$ and ${\mathrm{C}}_3^{\prime }\text{\%}=60.8\text{\%}$ from the corresponding Zemax results (i.e., ${\mathrm{C}}_{2/\mathrm{ze}\max }=-0.090476$ and ${\mathrm{C}}_{3/\mathrm{ze}\max }={\mathrm{C}}_{3/\mathrm{ze}\max }^{\prime }=-0.067381$).

(3) ${\mathrm{C}}_4$, ${\mathrm{C}}_5$ and ${\mathrm{C}}_6$ coefficients: From Eqs. (18c) and (14c), one has the following analytical expressions for ${\mathrm{C}}_5$ and ${\mathrm{C}}_6$:

$${\mathrm{C}}_5=\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}-2\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^3\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^3}),$$ (23a)

$${\mathrm{C}}_6=\frac{1}{4}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}-2\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2})-\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{x}}_{\mathrm{a}}^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}-2\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^3\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{nx}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^3}).$$ (23b)

Moreover, the following expressions for ${\mathrm{C}}_4$ and ${\mathrm{C}}_6^{\prime }$ are obtained from Eqs. (19c) and (15c):

$${\mathrm{C}}_4=\frac{1}{4}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}-2\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^2\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}),$$ (23c)

$${\mathrm{C}}_6^{\prime }=-\frac{1}{4}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}-2\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^2\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)})+\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5}-2\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}+\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}).$$ (23d)

From Eqs. (23c), (23a), (23b) and (23d), coefficients ${\mathrm{C}}_4$, ${\mathrm{C}}_5$, ${\mathrm{C}}_6$ and ${\mathrm{C}}_6^{\prime }$ have values of ${\mathrm{C}}_4=313.038676\times {10}^{-10}$, ${\mathrm{C}}_5=128.793685\times {10}^{-10}$, ${\mathrm{C}}_6=188.76106\times {10}^{-10}$ and ${\mathrm{C}}_6^{\prime }=194.629533\times {10}^{-10}$, respectively. Consequently, the oblique spherical aberration coefficients have values of ${\mathrm{C}}_{4/\mathrm{this}}={\mathrm{C}}_4{\mathrm{h}}_0^2{\rho }_{\max }^3=0.083783$, ${\mathrm{C}}_{5/\mathrm{this}}={\mathrm{C}}_5{\mathrm{h}}_0^2{\rho }_{\max }^3=0.034471$, ${\mathrm{C}}_{6/\mathrm{this}}={\mathrm{C}}_6{\mathrm{h}}_0^2{\rho }_{\max }^3=0.050521$ and ${\mathrm{C}}_{6/\mathrm{this}}^{\prime }={\mathrm{C}}_6^{\prime }{\mathrm{h}}_0^2{\rho }_{\max }^3=0.052091$. These values deviate from the corresponding Zemax results (i.e., ${\mathrm{C}}_{4/\mathrm{ze}\max }=0.090356$, ${\mathrm{C}}_{5/\mathrm{ze}\max }=0.033730$ and ${\mathrm{C}}_{6/\mathrm{ze}\max }=0.067592$) by ${\mathrm{C}}_4\text{\%}=7.3\text{\%}$, ${\mathrm{C}}_5\text{\%}=2.2\text{\%}$, ${\mathrm{C}}_6\text{\%}=25.3\text{\%}$ and ${\mathrm{C}}_6^{\prime }\text{\%}=22.9\text{\%}$, respectively.

(4) ${\mathrm{C}}_7$, ${\mathrm{C}}_8$ and ${\mathrm{C}}_9$ coefficients: The expressions of ${\mathrm{C}}_7$ and ${\mathrm{C}}_8$ are obtained from Eqs. (19d) and (15d), respectively, as

$$\begin{gathered} {\mathrm{C}}_7=\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^3\partial {\mathrm{x}}_{\mathrm{a}}^2}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}+3\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2} \\ -3\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^2\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}), \end{gathered}$$ (24a)

$$\begin{gathered} {\mathrm{C}}_8=-\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^3\partial {\mathrm{x}}_{\mathrm{a}}^2}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3}+3\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {\mathrm{x}}_{\mathrm{a}}^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2} \\ -3\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {\mathrm{x}}_{\mathrm{a}}^2\partial ({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)})-\frac{1}{12}(\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^5}-\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^3\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^2}+3\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0^2\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^3} \\ -3\frac{{\partial }^5{\mathrm{P}}_{\mathrm{ny}}}{\partial {\mathrm{h}}_0\partial {({\mathrm{y}}_{\mathrm{a}}-{\mathrm{h}}_0)}^4}). \end{gathered}$$ (24b)

From Eqs. (24a), (24b) and (18d), coefficients ${\mathrm{C}}_7$, ${\mathrm{C}}_8$ and ${\mathrm{C}}_9$ have values of ${\mathrm{C}}_7=-36.93595\times {10}^{-10}$, ${\mathrm{C}}_8=-144.98793\times {10}^{-10}$ and ${\mathrm{C}}_9=-36.48465\times {10}^{-10}$, respectively. Moreover, the elliptical coma coefficients have values of ${\mathrm{C}}_{7/\mathrm{this}}={\mathrm{C}}_7{\mathrm{h}}_0^3{\rho }_{\max }^2=-0.008003$, ${\mathrm{C}}_{8/\mathrm{this}}={\mathrm{C}}_8{\mathrm{h}}_0^3{\rho }_{\max }^2=-0.031413$ and ${\mathrm{C}}_{9/\mathrm{this}}={\mathrm{C}}_9{\mathrm{h}}_0^3{\rho }_{\max }^2=-0.007906$, and thus deviate by ${\mathrm{C}}_7\text{\%}=23.5\text{\%}$, ${\mathrm{C}}_8\text{\%}=2.8\text{\%}$ and ${\mathrm{C}}_9\text{\%}=18.2\text{\%}$ from the corresponding Zemax results (i.e., ${\mathrm{C}}_{7/\mathrm{ze}\max }=-0.010459$, ${\mathrm{C}}_{8/\mathrm{ze}\max }=-0.032328$ and ${\mathrm{C}}_{9/\mathrm{ze}\max }=-0.009668$).

(5) ${\mathrm{C}}_{10}$ and ${\mathrm{C}}_{11}$ coefficients: From Eqs. (19e) and (18e), ${\mathrm{C}}_{10}$ and ${\mathrm{C}}_{11}$ can be obtained as

$${\mathrm{C}}_{10}=12.00188\times {10}^{-10},$$ (25a)

$${\mathrm{C}}_{11}=10.04826\times {10}^{-10}.$$ (25b)

From Eqs. (25a) and (25b), the Petzval coefficient and astigmatism coefficients have values of ${\mathrm{C}}_{10/\mathrm{this}}={\mathrm{C}}_{10}{\mathrm{h}}_0^4{\rho }_{\max }=0.002105$ and ${\mathrm{C}}_{11/\mathrm{this}}={\mathrm{C}}_{11}{\mathrm{h}}_0^4{\rho }_{\max }=0.001762$, respectively. These values deviate by ${\mathrm{C}}_{10}\text{\%}=93.5\text{\%}$ and ${\mathrm{C}}_{11}\text{\%}=25.4\text{\%}$ from the corresponding Zemax results (i.e., ${\mathrm{C}}_{10/\mathrm{ze}\max }=0.001088$ and ${\mathrm{C}}_{11/\mathrm{ze}\max }=0.001406$).

(6) ${\mathrm{C}}_{12}$ coefficient: From Eq. (19f), coefficient ${\mathrm{C}}_{12}$ is obtained as

$${\mathrm{C}}_{12}=4.55416\times {10}^{-10}.$$ (26)

The fifth-order distortion coefficients determined from Eq. (26) and Zemax simulations, respectively, are ${\mathrm{C}}_{12/\mathrm{this}}={\mathrm{C}}_{12}{\mathrm{h}}_0^5=6.46626\times {10}^{-4}$ and ${\mathrm{C}}_{12/\mathrm{ze}\max }=6.3220\times {10}^{-4}$. In other words, the two values differ by just ${\mathrm{C}}_{12}\text{\%}=2.3\text{\%}$. The values of C aberration coefficients are summarized in Table 3.

Table 3.

Values of C aberration coefficients obtained from the proposed method.

${\mathrm{C}}_1{\rho }_{\max }^5=-0.089300$	${\mathrm{C}}_2{\rho }_{\max }^4{\mathrm{h}}_0=-0.018219$	$({\mathrm{C}}_{3\mathrm{zero}}+{\mathrm{C}}_{3\mathrm{two}}){\rho }_{\max }^4{\mathrm{h}}_0=-0.062674$
$({\mathrm{C}}_{3\mathrm{two}}^{\prime }+{\mathrm{C}}_{3\mathrm{four}}^{\prime }){\rho }_{\max }^4{\mathrm{h}}_0=-0.108368$	${\mathrm{C}}_4{\mathrm{h}}_0^2{\rho }_{\max }^3=0.083783$	${\mathrm{C}}_5{\mathrm{h}}_0^2{\rho }_{\max }^3=0.034471$
${\mathrm{C}}_6{\mathrm{h}}_0^2{\rho }_{\max }^3=0.050521$	${\mathrm{C}}_6^{\prime }{\mathrm{h}}_0^2{\rho }_{\max }^3=0.052091$	${\mathrm{C}}_7{\mathrm{h}}_0^3{\rho }_{\max }^2=-0.008003$
${\mathrm{C}}_8{\mathrm{h}}_0^3{\rho }_{\max }^2=-0.031413$	${\mathrm{C}}_9{\mathrm{h}}_0^3{\rho }_{\max }^2=-0.007906$	${\mathrm{C}}_{10}{\mathrm{h}}_0^4{\rho }_{\max }=0.002105$
${\mathrm{C}}_{11}{\mathrm{h}}_0^4{\rho }_{\max }=0.001762$	${\mathrm{C}}_{12}{\mathrm{h}}_0^5=6.46626\times {10}^{-4}$

5. Conclusions

The aberrations of an optical system can be described in terms of the wavefront aberration and ray-aberration and are commonly related using the approximations given in Eqs. (16a) and (16b). However, the obtained expressions for the ray-aberrations, given in Eqs. (2a) and (2b), have errors of a fifth order. Restrepo et al. [1] found that Eqs. (2a) and (2b) thus yield large errors for systems with large numerical apertures. Accordingly, in previous studies [23], [24], the present group proposed a new method for determining the aberration coefficients of the primary ray based on the Taylor series expansion of a skew ray. In the present study, this method has been extended to determine the secondary ray-aberration coefficients when the object is placed at finite position. The Taylor series expansion decomposes the aberrations into different orders without approximation, and hence the resulting expressions are exact. It has been shown that some of the secondary ray-aberration coefficients are functions of the polar angle of the entrance pupil. It is further noted that, once the derivative matrices have been generated, determination of the aberration coefficients is straightforward without iteration, and hence incurs only a low computational cost.

Declarations

Author contribution statement

Psang Dain Lin: Conceived and designed the analysis; Analyzed and interpreted the data; Contributed analysis tools or data; Wrote the paper.

Funding statement

Professor Psang Dain Lin was supported by Ministry of Science and Technology, Taiwan (MOST) (111-2221-E-006-138-).

Data availability statement

Data included in article/supp. material/referenced in article.

Declaration of interests statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.