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Extension of a fifth-order intrinsic aberration for a soft x-ray and vacuum ultraviolet optical system from a one- to two-dimension field light source

YiQing Cao and ZhiJuan Shen

Open Access Open Access

Abstract

Based on the fifth-order intrinsic aberration of a soft x-ray and vacuum ultraviolet (XUV) optical system with a one-dimension field light source, it is extended to the case of two-dimension in this paper, and the corresponding aberration calculation method is studied. Firstly, we derived sixth-order intrinsic wave aberration expressions of these kinds of optical systems with the aperture-ray coordinates on the reference exit wavefront; and then, derived the calculation expressions for fifth-order intrinsic aberration on the image plane caused by the contributions of wave aberration and defocus, respectively. Finally, the aberration expressions derived in this paper are applied to calculate the aberration distributions of two design examples of these kinds of optical systems, and these calculation results are compared with the ray-tracing program Shadow to validate the aberration expressions. The study shows that the calculation accuracy of aberration expressions is very satisfactory.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Soft x-ray and vacuum ultraviolet (XUV) optical system is widely used for microscope, telescope and synchrotron radiation light, etc., especially the optical system with two-dimension field light source [1]. To this kind of optical systems, in order to gain sufficiently high optical transmission, it usually needs to adopt ray with grazing-incidence impinged optical surface, and it has the imaging performance of plane-symmetric optical system [2]. Aberration analysis method is an important measure for imaging evaluation and design of optical system and still an interesting research subject. And Sasian extends the wave aberration theory of axially symmetric systems to develop aberration function of a paraxial optical system [3,4]; although it also belongs to a plane-symmetric optical system, but it mainly focuses on the small field angle (usually not larger than ∼30°). Therefore, it is not suitable to study the aberration of XUV optical system.

In recent years, some aberration analysis methods for plane-symmetric optical system have been developed by many researchers, and are mainly as follows: light-path function [5,6], analytic formulas of ray-tracing spot diagram [7,8], lie optics [9–11], and wavefront aberration (WFA) [12,13]. For multi-element optical system, the total wave aberration of optical system is a sum of that of each of its element, and thus WFA method is a classical method for studying the aberration of this kind of optical systems. Reference [14] adopted a torodial surface as a reference wavefront to develop the sixth-order wave aberration theory for ultrawide-angle optical system, and applied it to design fisheye lens system [14]. To XUV optical system with one-dimension field light source, Lu studied third-order intrinsic aberration calculation method of the optical system and derived the corresponding aberration expressions with aperture-ray coordinates on the optical surface [13]. However, to imagining of the optical system with large acceptance aperture, the effect of high-order aberration is not negligible. Therefore, study of high-order aberration is very important in the design of this kind of optical systems. In Ref. [15], the authors studied fifth-order intrinsic aberration calculation method for XUV optical system and but it was only used to handle the case of one-dimension field light source [15]. Nevertheless, to a kind of optical system with a large acceptance aperture and two-dimension field light source, there is still no high-order aberration analysis method to analyze its imaging, and mainly relied on the ray-tracing program Shadow [16]. To an optical system with a single optical element, it only contains the intrinsic aberration, and but we need to consider the effect of additional extrinsic aberration in the case of multi-element optical system. The intrinsic aberration is the basis for studying the imaging of optical system. Therefore, based on the previous research works, this paper firstly proposes fifth-order intrinsic aberration calculation method for XUV optical system with two-dimension field light source.

In next section, we introduce the definition of a plane-symmetric optical system with two-dimension field light source; section 3 firstly derives the sixth-order wave aberration expressions with aperture-ray coordinates on the optical surface of XUV optical system with two-dimension field light source, and then fits the second-order mapping relationship of aperture-ray coordinates between the optical surface and reference exit wavefront for two-dimension field light source, and we obtained sixth-order wave aberration expressions with the ones on the reference exit wavefront in this case; section 4 studies fifth-order intrinsic aberration calculation method caused by wave aberration and defocus, respectively, and derives the corresponding aberration expressions; finally, they are applied to calculate aberration distributions for two design examples of XUV optical system with large acceptance aperture and two-dimension field light source, and these calculation results are compared with the ones obtained by ray-tracing program Shadow.

2. Definition of a plane-symmetric optical system with two-dimension field light source

XUV optical system with two-dimension field light source has the imaging performance of plane-symmetric optical system, and thus its reference coordinate systems and rays should be defined. As shown in Fig. 1 [17], a XUV optical system consists of a single optical element with an off-axis object point S0, the optical surface is symmetrical with respect to the plane $\chi Oz^{\prime}$, and O is the vertex of the optical surface. The angles of incidence and reflection of the base ray $O^{\prime}OO^{\prime\prime}$ are α and β, respectively. The principal ray ${S_\textrm{0}}\bar{P}{S_1}$, emitted from source S0 and passing through the centre of entrance pupil, and intersects optical surface at $\bar{P}$, which is stipulated to be the origin of the pupil coordinate system $xyz$. $S_{0}^{\ast }$ and $S_1^\mathrm{\ast }$ are the projection points of ${S_0}$ and ${S_1}$ on the symmetry plane, u and v are the sagittal and meridional field angles in object space, and they are included in between ${S_0}O$ and $S_0^{\ast} O$, $S_0^{\ast} O$ and $O^{\prime}O$, respectively; similarly, $u^{\prime}$ and $v^{\prime}$ are the sagittal and meridional field angles in image space.

 figure: Fig. 1.

Fig. 1. Optical scheme of a plane-symmetric optical system consisting of a single optical element with two-dimension field light source.

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For plane-symmetric optical system, the shape of wave front will deviate from the sphere significantly, and the focusing geometry of a light beam in the meridional plane will differ from that in the sagittal one. In order to accurately obtain the information of meridional and sagittal planes of optical system, this paper adopted a toroidal surface as a reference wave front [14,18].

The general form of a plane-symmetric surface can be expressed in the vertex coordinate system of $\chi \eta \mathrm{z^{\prime}}$ by the expression [14]

$$z^{\prime} = \sum\limits_{i = 0}^\infty {\sum\limits_{j = 0}^\infty {{c_{i,j}}{\chi ^i}} } {\eta ^j}, {\quad}{c_{0,0}} = {c_{1,0}} = 0;{\qquad}j\textrm{ = even}\textrm{.}$$

The sixth-order wave aberration coefficient expressions are studied in this paper, and thus the power series of Eq. (1) needs to kept to sixth order, the figure equation is then denoted by

$$\begin{aligned} z^{\prime} &= {c_{2,0}}{\chi ^2} + {c_{0,2}}{\eta ^2} + {c_{3,0}}{\chi ^3} + {c_{1,2}}\chi {\eta ^2} + {c_{4,0}}{\chi ^4} + {c_{2,2}}{\chi ^2}{\eta ^2} + {c_{0,4}}{\eta ^4}\\ &+ {c_{5,0}}{\chi ^5} + {c_{1,4}}\chi {\eta ^4} + {c_{3,2}}{\chi ^3}{\eta ^2} + {c_{6,0}}{\chi ^6} + {c_{0,6}}{\eta ^6} + {c_{2,4}}{\chi ^2}{\eta ^4} + {c_{4,2}}{\chi ^4}{\eta ^2}, \end{aligned}$$
with the coefficients ${c_{i,j}}$ of a toroidal surface are as follows:
$$\begin{array}{l} {c_{2,0}} = \frac{1}{{2R}},\textrm{ }{c_{0,2}} = \frac{1}{{2\rho }},\textrm{ }{c_{3,0}} = 0,\textrm{ }{c_{1,2}} = 0,\textrm{ }{c_{4,0}} = \frac{1}{{8{R^3}}},\textrm{ }{c_{0,4}} = \frac{1}{{8{\rho ^3}}},\textrm{ }{c_{2,2}} = \frac{1}{{4{R^2}\rho }},\\ {c_{5,0}} = 0,\textrm{ }{c_{1,4}} = 0,\textrm{ }{c_{3,2}} = 0,\textrm{ }{c_{6,0}} = \frac{1}{{16{R^5}}},\textrm{ }{c_{0,6}} = \frac{1}{{16{\rho ^5}}},\textrm{ }{c_{2,4}} = \frac{{R + 2\rho }}{{16{R^3}{\rho ^3}}},\textrm{ }{c_{4,2}} = \frac{3}{{16{R^4}\rho }}. \end{array}$$
where R and $\rho$ are the major and minor curvature radii of the toroid, respectively.

3. Sixth-order wave aberration expressions for two-dimension field light source

Sixth-order wave aberration of XUV optical system with one-dimension field light source (i.e., the sagittal field angle) only has been expounded in Ref. [15]; and we now extend the wave aberration from one-dimension to two-dimension (i.e., including the meridional and sagittal field angle) field light source. Similar to the axially symmetric optical systems, the wave aberration of plane-symmetrical optical systems is related to the pupil-position parameters in the meridional and sagittal plane, lm, ls, respectively; and thus the deviation processes are divided two steps: the entrance pupil at optical surface (lm = ls = 0) and not at optical surface (lm≠0 or ls≠0).

The wave aberration can be expressed in the form,

$$W = \sum\limits_{ijk}^6 {w_{ijk}^\# {x^i}} {y^j}{u^k},$$
where the notation (#) in the wave aberration coefficients above is used to distinguish the new ones for two-dimension field light source, ${w_{ijkh}}$, with h = 0; and the additional fourth subscript h refers to the order of power of meridional field component v on which the wave aberration depend. Then
$$w_{ijk}^\# = {w_{ijk0}},$$

For the first step, we assume the entrance pupil to be positioned at the optical surface, and then the wave aberration coefficients in Eq. (4) can be expressed by

$$w_{ijk}^\# = M_{ijk}^\# ({\alpha ,{r_m},{r_s},0} )+ {({ - 1} )^k}M_{ijk}^\# ({\beta ,{{r^{\prime}}_m},{{r^{\prime}}_s},0} ),$$

In Eq. (6), the wave aberration coefficients of object pencil, $M_{ijk}^\# ({\alpha ,{r_m},{r_s},0} )$, are given by Eqs. (6)-(45) in Ref. [15]. The parameters ${r_m}$, ${r_s}$, ${r^{\prime}_m}$, ${r^{\prime}_s}$ represent the meridional and sagittal focal distance in the object and image space, respectively.

As shown in Fig. 1, the wave aberration of two-dimension field light source can be derived according to the same model as in the case of previous one-dimension field light source, but the substitution of the parameters in calculation of wave aberration should be made using the approximation relation [17],

$$\begin{array}{l} \alpha \to \alpha + v,{\qquad}{r_m} \to {{{r_m}} / {\cos v}},{\qquad}{r_s} \to {{{r_s}} / {\cos v}},\\ \beta \to \beta + v',{\quad}{{r'}_m} \to {{{{r'}_m}} / {\cos v'}},{\quad}{{r'}_s} \to {{{{r'}_s}} / {\cos v'}}, \end{array}$$

On making the substitution of the above parameters in every $M_{ijk}^\# ({\alpha ,{r_m},{r_s},0} )$, the wave aberration of object pencil $\tilde{W}_{obj}^0$, can be expanded into a power series x, y, u, v up to the sixth order

$$\tilde{W}_{obj}^0 = \sum\limits_{ijkh}^6 {M_{ijkh}^0{x^i}{y^j}{u^k}{v^h}} .$$

The wave aberration can be then be denoted by

$${\tilde{W}^0} = \sum\limits_{ijkh}^6 {w_{ijkh}^0{x^i}{y^j}{u^k}{v^h}} .$$
with
$$w_{ijkh}^0\textrm{ = }M_{ijkh}^0({\alpha ,{r_m},{r_s}} )+ {({ - 1} )^{k + h}}{\left( {\frac{{\cos \alpha }}{{\cos \beta }}} \right)^h}M_{ijkh}^0({\beta ,{{r^{\prime}}_m},{{r^{\prime}}_s}} ),$$

In Eq. (10), the following relation between the field angle of object space and the one of image space are used [17],

$$u' = - u,{\qquad }v' = - \frac{{\cos \alpha }}{{\cos \beta }}v.$$
where the wave aberration and its coefficients are marked with a “0” upper index to denote that the position of entrance pupil is placed at the optical surface, $M_{ijkh}^0({\alpha ,{r_m},{r_s}} )$, are the wave aberration coefficients of object pencil with the case of the position of entrance pupil is placed at the optical surface (i.e., lm = ls = 0), and the corresponding expressions are derived and are given in Supplement 1.

In the above discussion, the wave aberration calculation expressions with aperture-stop position at optical surface are derived. However, for a plane-symmetric optical system, the focusing equation in the meridional and sagittal plane is different, it is reasonable to adopt a separate pupil functions in the meridional and sagittal plane independently, and thus people usually use two independent one-dimension aperture stops to define the acceptance angle of light beam in the meridional and sagittal plane. If the aperture stops are placed at different position along the base ray in the meridional and sagittal plane (i.e., lm≠0 or ls≠0), and then the origin of the pupil coordinate system will change; and thus we will discuss the wave aberration calculation with aperture-stop position not at optical surface in the following. Applying the ray geometrical relationship, the pupil-position displacements in the meridional and sagittal plane using linear approximation are given by [17]

$${\bar{x}_p} = \frac{{{l_m}v}}{{\cos \alpha }},{\qquad}{\bar{y}_p} = {l_s}u.$$
however, the wave aberration and ray transverse aberration remain unchanged. Therefore, we can obtain the calculation expression of the wave aberration with ${l_m} \ne 0,{l_s} \ne 0$ based on the case of ${l_m}\textrm{ = }{l_s}\textrm{ = }0$ as follows:
$${\tilde{W}_{obj}}\textrm{ = }\sum\limits_{ijkh}^6 {{M_{ijkh}}{x^i}{y^j}{u^k}{v^h} = } \textrm{ }\sum\limits_{ijkh}^6 {M_{ijkh}^0{{\left( {x + {{\bar{x}}_p}} \right)}^i}{{\left( {y + {{\bar{y}}_p}} \right)}^j}{u^k}} {v^h}{\qquad}\left( {i + j + k + h \le 6} \right),$$
where $M_{ijkh}^{}$ are the wave aberration coefficients of the object pencil with the aperture stop displaced from optical surface.

On substituting the expressions of $M_{ijkh}^0$ derived in Eq. (8) into Eq. (13), and then we can obtain the relation expressions between ${M_{ijkh}}$ and $M_{ijkh}^0$. From Eq. (13), the wave aberrations of on-axis terms (k = h = 0) are unrelated to the pupil-position parameter, and thus the expressions of ${M_{ijkh}}$ are coincident with those of $M_{ijkh}^0$ in this case; in addition, the relation expressions between ${M_{ijkh}}$ and $M_{ijkh}^0$ for off-axis terms (k≠0 or h≠0) are listed in Supplement 1.

In terms of the symmetry of the geometry of the light pencil, the wave aberration coefficient expressions of the image pencil, ${M^{\prime}_{ijkh}}\textrm{ = }{M_{ijkh}}({\beta ,{{r^{\prime}}_m},{{r^{\prime}}_s},{{l^{\prime}}_m},{{l^{\prime}}_s}} )$, can be obtained by substituting $\alpha$, ${r_m}$, ${r_s}$, ${l_m}$, ${l_s}$ in the corresponding ${M_{ijkh}}$ with $\beta$, ${r^{\prime}_m}$, ${r^{\prime}_s}$, ${l^{\prime}_m}$, ${l^{\prime}_s}$. The parameters ${l_m}$, ${l^{\prime}_m}$, ${l_s}$ and ${l^{\prime}_s}$ represent the pupil-position parameters, and they are related to the displacement of the pupil from the optical surface in the meridional and sagittal plane in the object and image space, ${t_m}$, ${t^{\prime}_m}$, ${t_s}$ and ${t^{\prime}_s}$, the relation of them are given by [17]

$${l_m} = \frac{{{t_m}{r_m}}}{{{t_m} - {r_m}}},{\qquad}{l_s} = \frac{{{t_s}{r_s}}}{{{t_s} - {r_s}}},{\qquad}{l'_m} = \frac{{{{t'}_m}{{r'}_m}}}{{{{t'}_m} - {{r'}_m}}},{\qquad}{l'_s} = \frac{{{{t'}_s}{{r'}_s}}}{{{{t'}_s} - {{r'}_s}}}.$$

And the relation of the pupil-position parameters between the object and image space is

$${l'_m} = - \frac{{{{\cos }^2}\beta }}{{{{\cos }^2}\alpha }}{l_m},{\quad \quad}{l'_s} = - {l_s}.$$

The sixth-order wave aberration expressions derived in Ref. [15] based on the light path function; and thus if the optical element is grating, the groove function of a grating will also contribute to the wave aberration. Reference [15] gives the groove function for holographic and mechanically ruled grating as follows,

$$\begin{aligned} n = &\frac{\chi }{\sigma } + \frac{\Gamma }{\sigma }\left( {\frac{{{n_{20}}}}{2}{\chi^2} + \frac{{{n_{02}}}}{2}\eta {}^2 + \frac{{{n_{30}}}}{2}{\chi^3}} \right. + \frac{{{n_{12}}}}{2}\chi \eta {}^2 + \frac{{{n_{40}}}}{8}{\chi ^4} + \frac{{{n_{22}}}}{4}{\chi ^2}\eta {}^2 + \frac{{{n_{04}}}}{8}\eta {}^4 + \frac{{{n_{50}}}}{8}{\chi ^5} + \frac{{{n_{32}}}}{4}{\chi ^3}\eta {}^2\\ {\kern 0.8cm}&\textrm{ + }\frac{{{n_{14}}}}{8}\chi \eta {}^4 + \frac{{{n_{60}}}}{{16}}{\chi ^6} + \frac{{{n_{42}}}}{{16}}{\chi ^4}\eta {}^2 + \frac{{{n_{24}}}}{{16}}{\chi ^2}\eta {}^4 + \left. {\frac{{{n_{06}}}}{{16}}\eta {}^6} \right),\textrm{ } \end{aligned}$$
where $\sigma$ is the groove spacing of grating at the vertex, and $\Gamma $ and ${n_{ij}}$ are given by Eqs. (20)–(22) and (A.5) of Ref. [7]. In addition, if the point object is placed on the off axis, the pupil coordinate transformation $\chi \textrm{ = }x + \frac{{{l_m}}}{{\cos \alpha }}$, $\eta \textrm{ = }y + {l_s}u$ should be substituted into Eq. (16), n can be expressed as
$$n = \sum\limits_{ijkh}^6 {({{\Gamma / \sigma }} ){N_{ijkh}}{x^i}{y^j}{u^k}{v^h}} ,$$
where the coefficients ${N_{ijkh}}$, as function of ${n_{ij}}$, are given in Supplement 1.

The contribution from groove of grating to the optical path length is nmλ, and according to wave aberration calculation expression given in Eq. (9), and combining the relationship of the field angle in the meridional and sagittal plane between the object and image space given in Eq. (11). Therefore, the wave aberration calculation of the optical surface can be calculated by

$$W = \sum\limits_{ijkh}^6 {{{\tilde{w}}_{ijkh}}{x^i}{y^j}{u^k}{v^h}} {\quad \quad \quad \quad}\left( {i + j + k + h \le 6} \right),$$
and the wave aberration coefficients are
$${\tilde{w}_{ijkh}} = {M_{ijkh}}({\alpha ,{r_m},{r_s},{l_m},{l_s}} )+ {({ - 1} )^{k + h}}{\left( {\frac{{\cos \alpha }}{{\cos \beta }}} \right)^h}{M_{ijkh}}({\beta ,{{r^{\prime}}_m},{{r^{\prime}}_s},{{l^{\prime}}_m},{{l^{\prime}}_s}} )+ \Lambda {N_{ijkh}},$$
where $\Lambda \textrm{ = }({{{m\lambda } / \sigma }} )\Gamma $ and the last term is an additional optical path difference brought by the groove of grating.

According to the above discussion, the wave aberration coefficients calculation needs to know the parameters of the base ray and principal ray, $\alpha$, $\beta$, ${l_m}$, ${l_s}$, ${l^{\prime}_m}$, ${l^{\prime}_s}$, and ${r_m}$, ${r_s}$, ${r^{\prime}_m}$, ${r^{\prime}_s}$ of each optical surface. Similar to Gaussian optics, the first- and second-order wave aberrations should be zero to define the aberrations of a plane-symmetric optical system [15,17]. First, the base ray, like the optical axis of axially symmetric systems, is defined by ${w_{1000}}\textrm{ = }0$; this leads to grating equation (for non-grating system, m and $\Lambda $ are zero):

$$\sin \alpha + \sin \beta = {{m\lambda } / \sigma }.$$

Second, the principal ray (or pupil position) can be determined with additional condition ${w_{1001}}\textrm{ = }0$ and ${w_{0110}}\textrm{ = }0$, respectively:

$$\begin{array}{l} \frac{{{{l^{\prime}}_m}\cos \alpha }}{{\cos \beta }}\left( {\frac{{\cos \beta }}{{{{r^{\prime}}_m}}} - 2{c_{2,0}}} \right) - {l_m}\left( {\frac{{\cos \alpha }}{{{r_m}}} - 2{c_{2,0}}} \right) = \frac{{\Lambda {n_{20}}{l_m}}}{{\cos \alpha }},\\ {{l^{\prime}}_s}\left( {\frac{1}{{{{r^{\prime}}_s}}} - 2{c_{0,2}}\cos \beta } \right) - {l_s}\left( {\frac{1}{{{r_s}}} - 2{c_{0,2}}\cos \alpha } \right) = \Lambda {n_{02}}{l_s}, \end{array}$$

Finally, the position of the image plane along the base ray on the meridional and sagittal plane are determined by ${w_{2000}}\textrm{ = }0$ and ${w_{0200}}\textrm{ = }0$, respectively:

$$\begin{array}{l} 2{c_{2,0}}({\cos \alpha + \cos \beta } )- \left( {\frac{{{{\cos }^2}\alpha }}{{{r_m}}} + \frac{{{{\cos }^2}\beta }}{{{{r^{\prime}}_m}}}} \right) = \Lambda {n_{20}},\\ 2{c_{0,2}}({\cos \alpha + \cos \beta } )- \left( {\frac{1}{{{r_s}}} + \frac{1}{{{{r^{\prime}}_s}}}} \right) = \Lambda {n_{02}}. \end{array}$$

Once $\alpha$, $\beta$, ${l_m}$, ${l_s}$, ${l^{\prime}_m}$, ${l^{\prime}_s}$, and ${r_m}$, ${r_s}$, ${r^{\prime}_m}$, ${r^{\prime}_s}$ are determined, the image aberration contributed by the remaining third- to sixth-order wave aberrations:

$$\begin{aligned} \tilde{W}&= {{\tilde{w}}_{3000}}{x^3} + {{\tilde{w}}_{4000}}{x^4} + {{\tilde{w}}_{5000}}{x^5} + {{\tilde{w}}_{6000}}{x^6} + {{\tilde{w}}_{1200}}x{y^2} + {{\tilde{w}}_{2200}}{x^2}{y^2} + {{\tilde{w}}_{3200}}{x^3}{y^2} + {{\tilde{w}}_{4200}}{x^4}{y^2}\\&\textrm{ } + {{\tilde{w}}_{0400}}{y^4} + {{\tilde{w}}_{1400}}x{y^4} + {{\tilde{w}}_{2400}}{x^2}{y^4} + {{\tilde{w}}_{0600}}{y^6} + {{\tilde{w}}_{1110}}xyu + {{\tilde{w}}_{2110}}{x^2}yu + {{\tilde{w}}_{3110}}{x^3}yu\\&\textrm{ } + {{\tilde{w}}_{4110}}{x^4}yu + {{\tilde{w}}_{0310}}{y^3}u + {{\tilde{w}}_{1310}}x{y^3}u + {{\tilde{w}}_{2310}}{x^2}{y^3}u + {{\tilde{w}}_{0510}}{y^5}u + {{\tilde{w}}_{1020}}x{u^2} + {{\tilde{w}}_{2020}}{x^2}{u^2}\\&\textrm{ } + {{\tilde{w}}_{3020}}{x^3}{u^2} + {{\tilde{w}}_{4020}}{x^4}{u^2} + {{\tilde{w}}_{0220}}{y^2}{u^2} + {{\tilde{w}}_{1220}}x{y^2}{u^2} + {{\tilde{w}}_{2220}}{x^2}{y^2}{u^2} + {{\tilde{w}}_{0420}}{y^4}{u^2}\\&\textrm{ } + {{\tilde{w}}_{0130}}y{u^3} + {{\tilde{w}}_{1130}}xy{u^3} + {{\tilde{w}}_{2130}}{x^2}y{u^3} + {{\tilde{w}}_{0330}}{y^3}{u^3} + {{\tilde{w}}_{1040}}x{u^4} + {{\tilde{w}}_{2040}}{x^2}{u^4} + {{\tilde{w}}_{0240}}{y^2}{u^4}\\&\textrm{ } + {{\tilde{w}}_{0150}}y{u^5} + {{\tilde{w}}_{2111}}{x^2}yuv + {{\tilde{w}}_{1201}}x{y^2}v + {{\tilde{w}}_{0202}}{y^2}{v^2} + {{\tilde{w}}_{1311}}x{y^3}uv + {{\tilde{w}}_{1221}}x{y^2}{u^2}v + {{\tilde{w}}_{2004}}{x^2}{v^4}\\&\textrm{ } + {{\tilde{w}}_{1111}}xyuv + {{\tilde{w}}_{1112}}xyu{v^2} + {{\tilde{w}}_{1113}}xyu{v^3} + {{\tilde{w}}_{1131}}xy{u^3}v + {{\tilde{w}}_{0221}}{y^2}{u^2}v + {{\tilde{w}}_{3021}}{x^3}{u^2}v\\&\textrm{ } + {{\tilde{w}}_{3201}}{x^3}{y^2}v + {{\tilde{w}}_{0311}}{y^3}uv + {{\tilde{w}}_{4001}}{x^4}v + {{\tilde{w}}_{2002}}{x^2}{v^2} + {{\tilde{w}}_{3001}}{x^3}v + {{\tilde{w}}_{1002}}x{v^2} + {{\tilde{w}}_{1003}}x{v^3} + {{\tilde{w}}_{1004}}x{v^4}\\&\textrm{ } + {{\tilde{w}}_{1005}}x{v^5} + {{\tilde{w}}_{2112}}{x^2}yu{v^2}\textrm{ + }{{\tilde{w}}_{0402}}{y^4}{v^2} + {{\tilde{w}}_{0204}}{y^2}{v^4} + {{\tilde{w}}_{3111}}{x^3}yuv + {{\tilde{w}}_{2001}}{x^2}v + {{\tilde{w}}_{1202}}x{y^2}{v^2}\\&\textrm{ } + {{\tilde{w}}_{1021}}x{u^2}v + {{\tilde{w}}_{1022}}x{u^2}{v^2} + {{\tilde{w}}_{1023}}x{u^2}{v^3} + {{\tilde{w}}_{0201}}{y^2}v + {{\tilde{w}}_{0203}}{y^2}{v^3} + {{\tilde{w}}_{1041}}x{u^4}v + {{\tilde{w}}_{1401}}x{y^4}v\\&\textrm{ } + {{\tilde{w}}_{1203}}x{y^2}{v^3} + {{\tilde{w}}_{2022}}{x^2}{u^2}{v^2} + {{\tilde{w}}_{2202}}{x^2}{y^2}{v^2} + {{\tilde{w}}_{3003}}{x^3}{v^3} + {{\tilde{w}}_{4002}}{x^4}{v^2} + {{\tilde{w}}_{2201}}{x^2}{y^2}v\\&\textrm{ } + {{\tilde{w}}_{2021}}{x^2}{u^2}v + {{\tilde{w}}_{0312}}{y^3}u{v^2} + {{\tilde{w}}_{0222}}{y^2}{u^2}{v^2} + {{\tilde{w}}_{5001}}{x^5}v + {{\tilde{w}}_{0401}}{y^4}v + {{\tilde{w}}_{2003}}{x^2}{v^3} + {{\tilde{w}}_{3002}}{x^3}{v^2}. \end{aligned}$$

To calculate the high-order wave aberration and aberration of a plane-symmetric optical system using the aperture-ray coordinates on the optical surface, their calculation expressions are exceedingly complex, if we use the aperture-ray coordinates on the reference exit wavefront instead of the ones on the optical surface, the expressions will be relatively simple. Therefore, the mapping relationships of the aperture-ray coordinates between on the optical surface and the reference exit wavefront are required.

For a plane-symmetric optical system with one-dimension field light source, the sixth-order mapping relationships of aperture-ray coordinates between on the optical surface and on the reference exit wavefront are given in Eq. (60) of Ref. [15]; and in this paper, the mapping relationships still use sixth order in the wave aberration transformation with the subscript h = 0; if we assume that aperture-stops is placed on the optical surface (i.e., lm = 0 and ls = 0) in the above derivation process, and but the above sixth-order mapping relationship expressions for an optical system with two-dimension field light source would be very complicated, and thus we adopt second-order mapping relationship in this case to obtain the wave-aberration coefficients expressions with the subscript h≠0 with aperture-ray coordinates on the reference exit wavefront, and the relationships can be obtained:

$$x = \sum\limits_{i = 0}^{2} {\sum\limits_{j = 0}^{2} {a_{ij}^{\prime 0}x_{0}^{\prime i}y_{0}^{\prime j}} } ,{\quad \quad}y = \sum\limits_{i = 0}^{2} {\sum\limits_{j = 0}^{2} {b_{ij}^{\prime 0}x_{0}^{\prime i}y_{0}^{\prime j}} } .$$

For this case of two-dimension field light source, the substitution of the parameters, $\beta$, ${r^{\prime}_m}$ and ${r^{\prime}_s}$, in the second-order mapping relationship coefficients $a^{\prime 0}_{ij}$ and $b^{\prime 0}_{ij}$ of Eq. (24), should be made using the approximation relation given in Eq. (7). However, the aperture-stops are not on the optical surface in the meridional and sagittal plane (i.e., lm≠0 or ls≠0), the displacements of the origin of the pupil coordinate system in the both directions are should be considered, and thus the second-order mapping relationships in this case are derived,

$$x = \sum\limits_{i = 0}^{2} {\sum\limits_{j = 0}^{2} {{{\tilde{a}{'_{ij}}}}{x'_{0}}^{i}{y'_{0}}^{j}}} ,{\quad \quad}y = \sum\limits_{i = 0}^{2} {\sum\limits_{j = 0}^{2} {{{\tilde{b}{'_{ij}}}}{x'_{0}}^{i}{y'_{0}}^{j}}} .$$

Substituting Eq. (25) into Eq. (23), the wave aberration expressions with aperture-ray coordinates on the reference wavefront can be expressed by

$$\tilde{W}' = \sum\limits_{ijkh}^6 {{{\tilde{w'}}_{ijkh}}\tilde{x}_{0}^{\prime i}\tilde{y}_{0}^{\prime j}{u^{k}}{v^{h}},{\quad \quad \quad \quad}\left( {i + j + k + h \le 6} \right)} .$$

The relationship expressions between ${\tilde{w}^{\prime}_{ijkh}}$ and ${\tilde{w}_{ijkh}}$ with the subscript h≠0 can be derived, and they are given in Supplement 1. In addition, the relationship expressions between ${\tilde{w}^{\prime}_{ijkh}}$ and ${\tilde{w}_{ijkh}}$ with the subscript h = 0 are given by Eqs. (S1)-(S40) in Supplementary material of Ref. [15].

Satisfying the conditions of Eqs. (20)–(22), the remaining third- and sixth-order wave aberrations in the case of the aperture-ray coordinates on the reference exit wavefront will be

$$\begin{aligned} \tilde{W}^{\prime}& = {{\tilde{w}^{\prime}}_{3000}}x^{\prime 3}_0 + {{\tilde{w}^{\prime}}_{4000}}x^{\prime 4}_0 + {{\tilde{w}^{\prime}}_{5000}}x^{\prime 5}_0 + {{\tilde{w}^{\prime}}_{6000}}x^{\prime 6}_0 + {{\tilde{w}^{\prime}}_{1200}}{{x^{\prime}}_0}y^{\prime 2}_0 + {{\tilde{w}^{\prime}}_{2200}}x^{\prime 2}_0y^{\prime 2}_0 + {{\tilde{w}^{\prime}}_{3200}}x^{\prime 3}_0y^{\prime 2}_0 + {{\tilde{w}^{\prime}}_{4200}}x^{\prime 4}_0y^{\prime 2}_0\\&\textrm{ } + {{\tilde{w}^{\prime}}_{0400}}y^{\prime 4}_0 + {{\tilde{w}^{\prime}}_{1400}}{{x^{\prime}}_0}y^{\prime 4}_0 + {{\tilde{w}^{\prime}}_{2400}}x^{\prime 2}_0y^{\prime 4}_0 + {{\tilde{w}^{\prime}}_{0600}}y^{\prime 6}_0 + {{\tilde{w}^{\prime}}_{1110}}{{x^{\prime}}_0}{{y^{\prime}}_0}u + {{\tilde{w}^{\prime}}_{2110}}x^{\prime 2}_0{{y^{\prime}}_0}u + {{\tilde{w}^{\prime}}_{3110}}x^{\prime 3}_0{{y^{\prime}}_0}u\\&\textrm{ } + {{\tilde{w}^{\prime}}_{4110}}x^{\prime 4}_0{{y^{\prime}}_0}u + {{\tilde{w}^{\prime}}_{0310}}y^{\prime 3}_0u + {{\tilde{w}^{\prime}}_{1310}}{{x^{\prime}}_0}y^{\prime 3}_0u + {{\tilde{w}^{\prime}}_{2310}}x^{\prime 2}_0y^{\prime 3}_0u + {{\tilde{w}^{\prime}}_{0510}}y^{\prime 5}_0u + {{\tilde{w}^{\prime}}_{1020}}{{x^{\prime}}_0}{u^2} + {{\tilde{w}^{\prime}}_{2020}}x^{\prime 2}_0{u^2}\\&\textrm{ } + {{\tilde{w}^{\prime}}_{3020}}x^{\prime 3}_0{u^2} + {{\tilde{w}^{\prime}}_{4020}}x^{\prime 4}_0{u^2} + {{\tilde{w}^{\prime}}_{0220}}y^{\prime 2}_0{u^2} + {{\tilde{w}^{\prime}}_{1220}}{{x^{\prime}}_0}y^{\prime 2}_0{u^2} + {{\tilde{w}^{\prime}}_{2220}}x^{\prime 2}_0y^{\prime 2}_0{u^2} + {{\tilde{w}^{\prime}}_{0420}}y^{\prime 4}_0{u^2}\\&\textrm{ } + {{\tilde{w}^{\prime}}_{0130}}{{y^{\prime}}_0}{u^3} + {{\tilde{w}^{\prime}}_{1130}}{{x^{\prime}}_0}{{y^{\prime}}_0}{u^3} + {{\tilde{w}^{\prime}}_{2130}}x^{\prime 2}_0{{y^{\prime}}_0}{u^3} + {{\tilde{w}^{\prime}}_{0330}}y^{\prime 3}_0{u^3} + {{\tilde{w}^{\prime}}_{1040}}{{x^{\prime}}_0}{u^4} + {{\tilde{w}^{\prime}}_{2040}}x^{\prime 2}_0{u^4} + {{\tilde{w}^{\prime}}_{0240}}y^{\prime 2}_0{u^4}\\&\textrm{ } + {{\tilde{w}^{\prime}}_{0150}}{{y^{\prime}}_0}{u^5} + {{\tilde{w}^{\prime}}_{2111}}x^{\prime 2}_0{{y^{\prime}}_0}uv + {{\tilde{w}^{\prime}}_{1201}}{{x^{\prime}}_0}y^{\prime 2}_0v + {{\tilde{w}^{\prime}}_{0202}}y^{\prime 2}_0{v^2} + {{\tilde{w}^{\prime}}_{1311}}{{x^{\prime}}_0}y^{\prime 3}_0uv + {{\tilde{w}^{\prime}}_{1221}}{{x^{\prime}}_0}y^{\prime 2}_0{u^2}v\\&\textrm{ } + {{\tilde{w}^{\prime}}_{2004}}x^{\prime 2}_0{v^4} + {{\tilde{w}^{\prime}}_{1111}}{{x^{\prime}}_0}{{y^{\prime}}_0}uv + {{\tilde{w}^{\prime}}_{1112}}{{x^{\prime}}_0}{{y^{\prime}}_0}u{v^2} + {{\tilde{w}^{\prime}}_{1113}}{{x^{\prime}}_0}{{y^{\prime}}_0}u{v^3} + {{\tilde{w}^{\prime}}_{1131}}{{x^{\prime}}_0}{{y^{\prime}}_0}{u^3}v + {{\tilde{w}^{\prime}}_{0221}}y^{\prime 2}_0{u^2}v\\&\textrm{ } + {{\tilde{w}^{\prime}}_{3021}}x^{\prime 3}_0{u^2}v + {{\tilde{w}^{\prime}}_{3201}}x^{\prime 3}_0y^{\prime 2}_0v + {{\tilde{w}^{\prime}}_{0311}}y^{\prime 3}_0uv + {{\tilde{w}^{\prime}}_{4001}}x^{\prime 4}_0v + {{\tilde{w}^{\prime}}_{2002}}x^{\prime 2}_0{v^2} + {{\tilde{w}^{\prime}}_{3001}}x^{\prime 3}_0v + {{\tilde{w}^{\prime}}_{1002}}{{x^{\prime}}_0}{v^2}\\&\textrm{ } + {{\tilde{w}^{\prime}}_{1003}}{{x^{\prime}}_0}{v^3} + {{\tilde{w}^{\prime}}_{1004}}{{x^{\prime}}_0}{v^4} + {{\tilde{w}^{\prime}}_{1005}}{{x^{\prime}}_0}{v^5} + {{\tilde{w}^{\prime}}_{2112}}x^{\prime 2}_0{{y^{\prime}}_0}u{v^2}\textrm{ + }{{\tilde{w}^{\prime}}_{0402}}y^{\prime 4}_0{v^2} + {{\tilde{w}^{\prime}}_{0204}}y^{\prime 2}_0{v^4} + {{\tilde{w}^{\prime}}_{3111}}x^{\prime 3}_0{{y^{\prime}}_0}uv\\&\textrm{ } + {{\tilde{w}^{\prime}}_{2001}}x^{\prime 2}_0v + {{\tilde{w}^{\prime}}_{1202}}{{x^{\prime}}_0}y^{\prime 2}_0{v^2} + {{\tilde{w}^{\prime}}_{1021}}{{x^{\prime}}_0}{u^2}v + {{\tilde{w}^{\prime}}_{1022}}{{x^{\prime}}_0}{u^2}{v^2} + {{\tilde{w}^{\prime}}_{1023}}{{x^{\prime}}_0}{u^2}{v^3} + {{\tilde{w}^{\prime}}_{0201}}y^{\prime 2}_0v + {{\tilde{w}^{\prime}}_{0203}}y^{\prime 2}_0{v^3}\\&\textrm{ } + {{\tilde{w}^{\prime}}_{1041}}{{x^{\prime}}_0}{u^4}v + {{\tilde{w}^{\prime}}_{1401}}{{x^{\prime}}_0}y^{\prime 4}_0v + {{\tilde{w}^{\prime}}_{1203}}{{x^{\prime}}_0}y^{\prime 2}_0{v^3} + {{\tilde{w}^{\prime}}_{2022}}x^{\prime 2}_0{u^2}{v^2} + {{\tilde{w}^{\prime}}_{2202}}x^{\prime 2}_0y^{\prime 2}_0{v^2} + {{\tilde{w}^{\prime}}_{3003}}x^{\prime 3}_0{v^3} + {{\tilde{w}^{\prime}}_{4002}}x^{\prime 4}_0{v^2}\\&\textrm{ } + {{\tilde{w}^{\prime}}_{2201}}x^{\prime 2}_0y^{\prime 2}_0v + {{\tilde{w}^{\prime}}_{2021}}x^{\prime 2}_0{u^2}v + {{\tilde{w}^{\prime}}_{0312}}y^{\prime 3}_0u{v^2} + {{\tilde{w}^{\prime}}_{0222}}y^{\prime 2}_0{u^2}{v^2} + {{\tilde{w}^{\prime}}_{5001}}{{x^{\prime}}_0}^5v + {{\tilde{w}^{\prime}}_{0401}}y^{\prime 4}_0v + {{\tilde{w}^{\prime}}_{2003}}x^{\prime 2}_0{v^3} + {{\tilde{w}^{\prime}}_{3002}}x^{\prime 3}_0{v^2}. \end{aligned}$$

4. Fifth-order intrinsic aberration for two-dimension field light source

In order to make the aberration expressions of optical system with two-dimension field light source more concise, similar to one-dimension field light source, we will separately discussed the aberration contributed by the wave aberration and defocus aberration. Firstly, similar to the axially symmetric optical system, the angular deviation of the actual ray from the reference ray in the meridional and sagittal directions, respectively,

$${\theta '_x} = \frac{{d\tilde{W}'}}{{d{{x'}_0}}},{\quad \quad \quad \quad \quad \quad}{\theta '_y} = \frac{{d\tilde{W}'}}{{d{{y'}_0}}}.$$

The aberration on the image plane at a distance ${r^{\prime}_0}$ from the optical surface will be

$$\bar{X} = {r'_0}{\theta '_x} = {r'_0}\frac{{d\tilde{W}'}}{{d{{x'}_0}}},{\quad \quad \quad \quad \quad \quad}\bar{Y} = {r'_0}{\theta '_y} = {r'_0}\frac{{d\tilde{W}'}}{{d{{y'}_0}}}.$$

Substituting Eq. (27) into Eq. (29), the fifth-order aberration expressions can be obtained,

$\tilde{X}=\tilde{d}_{2000}^{\prime} x_{0}^{\prime 2}+\tilde{d}_{3000}^{\prime} x_{0}^{\prime 3}+\tilde{d}_{4000}^{\prime} x_{0}^{\prime 4}+\tilde{d}_{5000}^{\prime} x_{0}^{\prime 5}+\tilde{d}_{0200}^{\prime} y_{0}^{\prime 2}+\tilde{d}_{1200}^{\prime} x^{\prime}{ }_{0} y_{0}^{\prime 2}+\tilde{d}_{2200}^{\prime} x_{0}^{\prime 2} y_{0}^{\prime 2}+\tilde{d}_{3200}^{\prime} x_{0}^{\prime 3} y_{0}^{\prime 2}+\tilde{d}_{0400}^{\prime} y_{0}^{\prime 4}$ $+\tilde{d}_{1400}^{\prime} x_{0}^{\prime} y_{0}^{\prime 4}+\tilde{d}_{0110}^{\prime} y_{0}^{\prime} u+\tilde{d}_{1110}^{\prime} x_{0}^{\prime} y_{0}^{\prime} u+\tilde{d}_{2110}^{\prime} x_{0}^{\prime 2} y_{0}^{\prime} \tilde{d}_{3110}^{\prime} x_{0}^{\prime 3} y_{0}^{\prime} u+\tilde{d}_{0310}^{\prime} y_{0}^{\prime 3} u+\tilde{d}_{1310}^{\prime} x_{0}^{\prime} y_{0}^{\prime 3} u+\tilde{d}_{0020}^{\prime} u^{2}$ $+\tilde{d}_{1020}^{\prime} x_{0}^{\prime} u^{2}+\tilde{d}_{2020}^{\prime} x_{0}^{\prime 2} u^{2}+\tilde{d}_{3020}^{\prime} x_{0}^{\prime 3} u^{2}+\tilde{d}_{0220}^{\prime} y_{0}^{\prime 2} u^{2}+\tilde{d}_{1220}^{\prime} x_{0}^{\prime} y_{0}^{\prime 2} u^{2}+\tilde{d}_{0130}^{\prime} y_{0}^{\prime} u^{3}+\tilde{d}_{1130}^{\prime} x_{0}^{\prime} y_{0}^{\prime} u^{3}+\tilde{d}_{0040}^{\prime} u^{4}$ $+\tilde{d}_{1040}^{\prime} x_{0}^{\prime} u^{4}+\tilde{d}_{1111}^{\prime} x_{0}^{\prime} y_{0}^{\prime} u v+\tilde{d}_{0201}^{\prime} y_{0}^{\prime 2} v+\tilde{d}_{0311}^{\prime} y_{0}^{\prime 3} u v+\tilde{d}_{0221}^{\prime} y_{0}^{\prime 2} u^{2} v+\tilde{d}_{1004}^{\prime} x_{0}{ }_{0} v^{4}+\tilde{d}_{0111}^{\prime} y_{0}^{\prime} u v+\tilde{d}_{0112}^{\prime} y_{0}^{\prime} u v^{2}$ $+\tilde{d}_{0113}^{\prime} y_{0}^{\prime} u v^{3}+\tilde{d}_{0131}^{\prime} y_{0}^{\prime} u^{3} v+\tilde{d}_{2021}^{\prime} x_{0}^{\prime 2} u^{2} v+\tilde{d}_{2201}^{\prime} x_{0}^{\prime 2} y_{0}^{\prime 2} v+\tilde{d}_{3001}^{\prime} x_{0}^{\prime 3} v+\tilde{d}_{1002}^{\prime} x_{0}^{\prime} v^{2}+\tilde{d}_{2001}^{\prime} x_{0}^{\prime 2} v+\tilde{d}_{0002}^{\prime} v^{2}$ $+\tilde{d}_{0003}^{\prime} v^{3}++\tilde{d}_{0004}^{\prime} v^{4}+\tilde{d}_{0005}^{\prime} v^{5}+\tilde{d}_{1112}^{\prime} x_{0}^{\prime} y_{0}^{\prime} u v^{2}+\tilde{d}_{2111}^{\prime} x_{0}^{\prime 2} y_{0}^{\prime} u v+\tilde{d}_{1001}^{\prime} x_{0}^{\prime} v+\tilde{d}_{0202}^{\prime} y_{0}^{\prime 2} v^{2}+\tilde{d}_{0021}^{\prime} u^{2} v$ $+\tilde{d}_{0022}^{\prime} u^{2} v^{2}+\tilde{d}_{0023}^{\prime} u^{2} v^{3}+\tilde{d}_{0041}^{\prime} u^{4} v+\tilde{d}_{0401}^{\prime} y_{0}^{\prime 4} v+\tilde{d}_{0203}^{\prime} y_{0}^{\prime 2} v^{3}+\tilde{d}_{1022}^{\prime} x_{0}^{\prime} u^{2} v^{2}+\tilde{d}_{1202}^{\prime} x_{0}^{\prime} y_{0}^{\prime 2} v^{2}+\tilde{d}_{2003}^{\prime} x_{0}^{\prime 2} v^{3}$
$+\tilde{d}_{3002}^{\prime} x_{0}^{\prime 3} v^{2}+\tilde{d}_{1201}^{\prime} x_{0}^{\prime} y_{0}^{\prime 2} v+\tilde{d}_{1021}^{\prime} x_{0}^{\prime} u^{2} v+\tilde{d}_{4001}^{\prime} x_{0}^{\prime 4} v+\tilde{d}_{1003}^{\prime} x_{0}^{\prime} v^{3}+\tilde{d}_{2002}^{\prime} x_{0}^{\prime 2} v^{2}$ $\tilde{Y}=\tilde{h}_{1100}^{\prime} x_{0}^{\prime} y_{0}^{\prime}+\tilde{h}_{2100}^{\prime} x_{0}^{\prime 2} y_{0}^{\prime}+\tilde{h}_{2200}^{\prime} x_{0}^{\prime 2} y_{0}^{\prime 2}+\tilde{h}_{3200}^{\prime} x_{0}^{\prime 3} y_{0}^{\prime 2}+\tilde{h}_{0300}^{\prime} y_{0}^{\prime 3}+\tilde{h}_{1300}^{\prime} x_{0}^{\prime} y_{0}^{\prime 3}+\tilde{h}_{2300}^{\prime} x_{0}^{\prime 2} y_{0}^{\prime 3}+\tilde{h}_{0500}^{\prime} y_{0}^{\prime 5}$ $+\tilde{h}_{1010}^{\prime} x_{0}^{\prime} u+\tilde{h}_{2010}^{\prime} x_{0}^{\prime 2} u+\tilde{h}_{3010}^{\prime} x_{0}^{\prime 3} u+\tilde{h}_{4010}^{\prime} x_{0}^{\prime 4} u+\tilde{h}_{0210}^{\prime} y_{0}^{\prime 2} u+\tilde{h}_{1210}^{\prime} x_{0}^{\prime} y_{0}^{\prime 2} u+\tilde{h}_{2210}^{\prime} x_{0}^{\prime 2} y_{0}^{\prime 2} u+\tilde{h}_{0410}^{\prime} y_{0}^{\prime 4} u$ $+\tilde{h}_{0120}^{\prime} y_{0}^{\prime} u^{2}+\tilde{h}_{1120}^{\prime} x_{0}^{\prime} y_{0}^{\prime}{ }_{0} u^{2}+\tilde{h}_{2120}^{\prime} x_{0}^{\prime 2} y_{0}^{\prime} u^{2}+\tilde{h}_{0320}^{\prime} y_{0}^{\prime 3} u^{2}+\tilde{h}_{0030}^{\prime} u^{3}+\tilde{h}_{1030}^{\prime} x_{0}^{\prime} u^{3}+\tilde{h}_{2030}^{\prime} x_{0}^{\prime 2} u^{3}+\tilde{h}_{0230}^{\prime} y_{0}^{\prime 2} u^{3}$
$+\tilde{h}_{0140}^{\prime} y_{0}^{\prime} u^{4}+\tilde{h}_{0050}^{\prime} u^{5}+\tilde{h}_{2011}^{\prime} x_{0}^{\prime 2} u v+\tilde{h}_{1101}^{\prime} x_{0}^{\prime} y_{0}^{\prime} v+\tilde{h}_{0102}^{\prime} y_{0}^{\prime} v^{2}+\tilde{h}_{1211}^{\prime} x_{0}^{\prime} y_{0}^{\prime 2} u v+\tilde{h}_{1121}^{\prime} x_{0}^{\prime} y_{0}^{\prime} u^{2} v$ $+\tilde{h}_{1011}^{\prime} x_{0}^{\prime} u v+\tilde{h}_{1012}^{\prime} x_{0}^{\prime} u v^{2}+\tilde{h}_{1013}^{\prime} x_{0}^{\prime}{ }_{0} u v^{3}+\tilde{h}_{1031}^{\prime} x_{0}^{\prime} u^{3} v+\tilde{h}_{0121}^{\prime} y_{0}{ }_{0} u^{2} v+\tilde{h}_{3101}^{\prime} x_{0}^{\prime 3} y_{0}^{\prime} v+\tilde{h}_{0211}^{\prime} y_{0}^{\prime 2} u v$ $+\tilde{h}_{2012}^{\prime} x_{0}^{\prime 2} u v^{2}+\tilde{h}_{0302}^{\prime} y_{0}^{\prime 3} v^{2}+\tilde{h}_{0104}^{\prime} y_{0}^{\prime} v^{4}+\tilde{h}_{3011}^{\prime} x_{0}^{\prime 3} u v+\tilde{h}_{1102}^{\prime} x_{0}^{\prime} y_{0}^{\prime} v^{2}+\tilde{h}_{0101}^{\prime} y_{0}^{\prime} v+\tilde{h}_{0103}^{\prime} y_{0}^{\prime} v^{3}$ $+\tilde{h}_{1301}^{\prime} x_{0}^{\prime} y_{0}^{\prime 3} v+\tilde{h}_{1103}^{\prime} x_{0}^{\prime} y_{0}^{\prime} v^{3}+\tilde{h}_{2102}^{\prime} x_{0}^{\prime 2} y_{0}^{\prime} v^{2}+\tilde{h}_{2101}^{\prime} x_{0}^{\prime 2} y_{0}^{\prime} v+\tilde{h}_{021}^{\prime} y_{0}^{\prime 2} u v^{2}+\tilde{h}_{0122}^{\prime} y_{0}^{\prime} u^{2} v^{2}+\tilde{h}_{0301}^{\prime} y_{0}^{\prime 3} v$
where
$${\tilde{d}'_{ijkh}} = \left( {i + 1} \right){r'_0}{\tilde{w}'_{\left( {i + 1} \right)jkh}},{\qquad}{\tilde{h}'_{ijkh}} = \left( {j + 1} \right){r'_0}{\tilde{w}_{i\left( {j + 1} \right)kh}},{\qquad}\left( {i + j + k + h \le 6} \right).$$

Equation (31) represents the aberration of optical systems contributed by the wave aberration and do not consider the defocus aberration. If the image plane is not coincident with the focal plane, the defocus aberration needs to be considered. Reference [15] derived the fifth-order defocus aberration calculation expressions of optical system using aperture-ray coordinates on the reference exit wavefront, they are given as follows:

$$\begin{array}{l} \Delta x^{\prime} = {{\bar{d}^{\prime}}_{1000}}{{x^{\prime}}_0} + {{\bar{d}^{\prime}}_{3000}}x^{\prime 3}_0 + {{\bar{d}^{\prime}}_{1200}}{{x^{\prime}}_0}y^{\prime 2}_0 + {{\bar{d}^{\prime}}_{5000}}x^{\prime 5}_0 + {{\bar{d}^{\prime}}_{3200}}x^{\prime 3}_0y^{\prime 2}_0 + {{\bar{d}^{\prime}}_{1400}}{{x^{\prime}}_0}y^{\prime 4}_0,\\\Delta y^{\prime} = {{\bar{h}^{\prime}}_{0100}}{{y^{\prime}}_0} + {{\bar{h}^{\prime}}_{2100}}x^{\prime 2}_0{{y^{\prime}}_0} + {{\bar{h}^{\prime}}_{0300}}y^{\prime 3}_0 + {{\bar{h}^{\prime}}_{4100}}x^{\prime 4}_0{{y^{\prime}}_0} + {{\bar{h}^{\prime}}_{0500}}y^{\prime 5}_0 + {{\bar{h}^{\prime}}_{2300}}x^{\prime 2}_0y^{\prime 3}_0, \end{array}$$
where the defocus aberration coefficients, ${\bar{d}^{\prime}_{ijkh}}$ and ${\bar{h}^{\prime}_{ijkh}}$, are given by
$$\begin{array}{l} {{\bar{d}^{\prime}}_{1000}}\textrm{ = }{\Lambda _m},\textrm{ }{{\bar{d}^{\prime}}_{3000}}\textrm{ = }\frac{{{\Lambda _m}}}{{2r^{\prime 2}_m}},\textrm{ }{{\bar{d}^{\prime}}_{1200}}\textrm{ = }\frac{{{\Lambda _m}}}{{2{{r^{\prime}}_m}{{r^{\prime}}_s}}},\textrm{ }{{\bar{d}^{\prime}}_{5000}}\textrm{ = }\frac{{3{\Lambda _m}}}{{8r^{\prime 4}_m}},\textrm{ }{{\bar{d}^{\prime}}_{3200}}\textrm{ = }\frac{{3{\Lambda _m}}}{{4r^{\prime 3}_m{{r^{\prime}}_s}}},\textrm{ }{{\bar{d}^{\prime}}_{1400}}\textrm{ = }\frac{{3{\Lambda _m}}}{{4{{r^{\prime}}_m}r^{\prime 2}_s}}\left( {\frac{1}{{{{r^{\prime}}_m}}} + \frac{1}{{2{{r^{\prime}}_s}}}} \right),\\{{\bar{h}^{\prime}}_{0100}}\textrm{ = }{\Lambda _s},\textrm{ }{{\bar{h^{\prime}}}_{2100}}\textrm{ = }\frac{{{\Lambda _m}}}{{2{{r^{\prime}}_m}{{r^{\prime}}_s}}},\textrm{ }{{\bar{h^{\prime}}}_{0300}}\textrm{ = }\frac{{{\Lambda _s}}}{{2r^{\prime 2}_s}},\textrm{ }{{\bar{h^{\prime}}}_{4100}}\textrm{ = }\frac{{3{\Lambda _m}}}{{8r^{\prime 3}_m{{r^{\prime}}_s}}},\textrm{ }{{\bar{h^{\prime}}}_{0500}}\textrm{ = }\frac{{3{\Lambda _s}}}{{4r^{\prime 4}_s}},\textrm{ }{{\bar{h^{\prime}}}_{2300}}\textrm{ = }\frac{{{\Lambda _m}}}{{2{{r^{\prime}}_m}r^{\prime 2}_s}}\left( {\frac{1}{{{{r^{\prime}}_m}}} + \frac{1}{{2{{r^{\prime}}_s}}}} \right). \end{array}$$
with ${\Lambda _m} = 1 - \frac{{{{r^{\prime}}_0}}}{{{{r^{\prime}}_m}}}$ and ${\Lambda _s} = 1 - \frac{{{{r^{\prime}}_0}}}{{{{r^{\prime}}_s}}}$.

Equations (30) and (32) represent the aberration of optical systems contributed by the wave aberration and defocus aberration. However, in the calculation of the ray coordinates on the image plane, the displacement of the principal ray from the base ray should be considered; in the case of the first-order approximation, the ray coordinates need to add the aberration components,

$${\Delta }x'' = \left( { - \frac{{{{r'}_0}\cos \alpha }}{{\cos \beta }} + \frac{{{{\Lambda }_m}{l_m}\cos \beta }}{{\cos \alpha }}} \right)v,{\quad \quad \quad \quad }{\Delta }y'' = \left( { - {{r'}_0} + \left( {1 - \frac{{{{r'}_0}}}{{{{r'}_s}}}} \right){l_s}} \right)u.$$

According to the above discussions, the final coordinates of ray on the image plane should be the sum of Eqs. (30), (32) and (34),

$$\begin{array}{l} x^{\prime} = \tilde{X} + \Delta x^{\prime} + \Delta x^{\prime\prime},\\y^{\prime} = \tilde{Y} + \Delta y^{\prime} + \Delta y^{\prime\prime}. \end{array}$$

5. Numerical validation

To validate the aberration formulae derived above, we now apply them to calculate the imagings of two design examples of XUV optical system consisting of a single optical element with a large acceptance aperture, and compare them with the ray-tracing program obtained by Shadow. The definition of the intrinsic wave aberration is based on a model of single optical element, and thus applying the optical system with a single element to validate the fifth-order intrinsic aberration expressions is very persuasive in the following.

Optical system I consists of a single spherical mirror, and its optical scheme as shown in Fig. 2. The optical system accepts the diverging angle the source a light beam is assumed to be $2{\theta _v} \times 2{\theta _h} = 20 \times 30\textrm{ }mra{d^2}$, and its working wavelength is 4.4 nm and entrance pupil positioned on the spherical mirror; in addition, to prove that the aberration expressions derived in this paper is applicable to aspheric surface, adding an optical system II consisting of a single toroidal mirror, and its diverging angle the source a light beam and working wavelength are consistent with that of optical system I, and but its entrance pupil is assumed at a distance of ${t_m} = {t_s} = 30\textrm{mm}$ before the toroidal mirror in the meridional and sagittal directions. The other optical parameters including the imaging plane distances of the optical system I and II are listed in Table 1, and in the following aberration calculation of optical system I and II given in the above, the point light source with the field angles are assumed to be ${u_1} = {0.2^0},\textrm{ }{v_1} = {0.2^0}$. The ray spot diagrams on the image plane using the aberration expressions of the optical system with two-dimension field light source derived in this paper and ray-tracing program Shadow are shown in Fig. 3 (a) and (b), respectively; and the calculation results given in the first row are the aberration distributions of optical system I, whereas those from the second row are the aberration distributions of optical system II.

 figure: Fig. 2.

Fig. 2. Optical scheme of optical system I.

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 figure: Fig. 3.

Fig. 3. Ray spot diagrams for optical system I and II, the first and second row correspond to calculation results of optical system I and II obtained with a point source with the field angle ${u_1} = {0.2^0},\textrm{ }{v_1} = {0.2^0}$, respectively; (a) and (b) are the calculation results using fifth-order intrinsic aberration calculation expressions derived in this paper and the ray-tracing program Shadow, respectively.

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Tables Icon

Table 1. Optical parameters of optical system I and II [units: mm (unless otherwise stated)].

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According to spot diagrams shown in Fig. 3, compared with the calculation results obtained with the ray-tracing program Shadow, the calculation accuracy of the fifth-order aberration expressions for two-dimension field light source derived in this paper is satisfactory both in size and shape of the spot diagrams. However, there is a small deviation between them, mainly because of the contribution of the higher-order aberration and higher-order components of mapping relationship between on the optical surface and the reference exit wavefront of aperture-ray coordinates in the transformation of the coordinates of wave aberration calculation.

6. Conclusions

In this paper, we studied fifth-order intrinsic aberration calculation method for XUV optical system with two-dimension field light source; and the corresponding aberration expressions with the aperture-ray coordinated on the reference exit wavefront of optical element has been extended and derived from pervious one-dimension to two dimension field light source. And we applied the resultant aberration formulae and the ray-tracing program Shadow to calculate the ray spot diagrams of two design examples of XUV optical system consisting of a single optical element, respectively, and compared with their calculation results and obtained that the aberration expressions have satisfactory calculation accuracy. They will provide an analytical measure and will be helpful in imaging analysis, determination of initial structure parameters, and optimization of optical parameters of XUV optical systems with a large acceptance aperture for two-dimension field light source.

Funding

Introductionof Talent Research Start-up Fee Project of Putian University of China (2019010); Science and Technology Planning Project of Putian City of China (2020GP004); Natural Science Foundation of Fujian Province (2020J01916).

Acknowledgments

The author would like to thank Lijun Lu for helpful discussions and useful comments.

Disclosures

The author declares that there are no conflicts of interest related to this article.

Data availability

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

Supplemental document

See Supplement 1 for supporting content.

References

1. P. Wilde and L. Hesselink, “Modeling of an X-ray grating-based imaging interferometer using ray tracing,” Opt. Express 28(17), 24657–24681 (2020). [CrossRef]  

2. Y. Cao and L. Lu, “Aberrations of soft x-ray and vacuum ultraviolet optical system with orthogonal arrangement of elements,” J. Opt. Soc. Am. A 34(3), 299–307 (2017). [CrossRef]  

3. J. Saian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33(6), 2045–2061 (1994). [CrossRef]  

4. J. Saian, “Theory of sixth-order wave aberrations,” Appl. Opt. 49(16), D69–D95 (2010). [CrossRef]  

5. H. Beutler, “The Theory of the concave grating,” J. Opt. Soc. Am. 35(5), 311–350 (1945). [CrossRef]  

6. H. Noda, T. Namioka, and M. Seya, “Geometrical theory of the grating,” J. Opt. Soc. Am. 64(8), 1031–1036 (1974). [CrossRef]  

7. T. Namioka, M. Koike, and D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33(31), 7261–7274 (1994). [CrossRef]  

8. S. Masui and T. Namioka, “Geometric aberration theory of double-element optical systems,” J. Opt. Soc. Am. A 16(9), 2253–2268 (1999). [CrossRef]  

9. K. Goto and T. Kurosaki, “Canonical formulation for the geometric optics of concave gratings,” J. Opt. Soc. Am. A 10(3), 452–465 (1993). [CrossRef]  

10. C. Palmer and W. Mckinney, “Imaging equations for spectroscopic systems using Lie transformations: I. Theoretical foundations,” Proc. SPIE 3450, 55–66 (1998). [CrossRef]  

11. C. Palmer, B. Wheeler, and W. Mckinney, “Imaging equations for spectroscopic systems using Lie transformations: II. Multielement systems,” Proc. SPIE 3450, 67–77 (1998). [CrossRef]  

12. M. P. Chrisp, “Aberrations of holographic toroidal grating systems,” Appl. Opt. 22(10), 1508–1518 (1983). [CrossRef]  

13. L. Lu, “Aberration theory of plane-symmetric grating systems,” J. Synchrotron Radiat. 15(4), 399–410 (2008). [CrossRef]  

14. L. Lu and Y. Cao, “Sixth-order wave aberration theory of ultrawide-angle optical systems,” Appl. Opt. 56(30), 8570–8583 (2017). [CrossRef]  

15. Y. Cao and Z. Shen, “Fifth-order intrinsic aberration calculation method for soft x-ray and vacuum ultraviolet optical systems,” Appl. Opt. 60(11), 3242–3249 (2021). [CrossRef]  

16. M. Sanchez del Rio, N. Canestrari, F. Jiang, and F. Cerrina, “SHADOW3: a new version of the synchrotron x-ray optics modeling package,” J. Synchrotron Radiat. 18(5), 708–716 (2011). [CrossRef]  

17. L. Lu and G. Zhu, “Extension of aberration theory of grating systems from one- to two-dimension extended source,” Optik 123(2), 157–166 (2012). [CrossRef]  

18. J. Caron and S. Bäumer, “Aberrations of plane-symmetrical mirror systems with freeform surfaces. Part I: generalized ray-tracing equations,” J. Opt. Soc. Am. A 38(1), 80–89 (2021). [CrossRef]  

Supplementary Material (1)

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Supplement 1       Extension of fifth-order intrinsic aberration for soft X-ray and vacuum ultraviolet optical system from one- to two-dimension field light source: supplementary material

Data availability

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

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Figures (3)
Fig. 1.
Fig. 1. Optical scheme of a plane-symmetric optical system consisting of a single optical element with two-dimension field light source.

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Fig. 2.
Fig. 2. Optical scheme of optical system I.

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Fig. 3.
Fig. 3. Ray spot diagrams for optical system I and II, the first and second row correspond to calculation results of optical system I and II obtained with a point source with the field angle ${u_1} = {0.2^0},\textrm{ }{v_1} = {0.2^0}$ , respectively; (a) and (b) are the calculation results using fifth-order intrinsic aberration calculation expressions derived in this paper and the ray-tracing program Shadow, respectively.

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Tables (1)

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Table 1. Optical parameters of optical system I and II [units: mm (unless otherwise stated)].

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Equations (33)

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z ′ = ∑ i = 0 ∞ ∑ j = 0 ∞ c i , j χ i η j , c 0 , 0 = c 1 , 0 = 0 ; j  = even .
z ′ = c 2 , 0 χ 2 + c 0 , 2 η 2 + c 3 , 0 χ 3 + c 1 , 2 χ η 2 + c 4 , 0 χ 4 + c 2 , 2 χ 2 η 2 + c 0 , 4 η 4 + c 5 , 0 χ 5 + c 1 , 4 χ η 4 + c 3 , 2 χ 3 η 2 + c 6 , 0 χ 6 + c 0 , 6 η 6 + c 2 , 4 χ 2 η 4 + c 4 , 2 χ 4 η 2 ,
c 2 , 0 = 1 2 R ,   c 0 , 2 = 1 2 ρ ,   c 3 , 0 = 0 ,   c 1 , 2 = 0 ,   c 4 , 0 = 1 8 R 3 ,   c 0 , 4 = 1 8 ρ 3 ,   c 2 , 2 = 1 4 R 2 ρ , c 5 , 0 = 0 ,   c 1 , 4 = 0 ,   c 3 , 2 = 0 ,   c 6 , 0 = 1 16 R 5 ,   c 0 , 6 = 1 16 ρ 5 ,   c 2 , 4 = R + 2 ρ 16 R 3 ρ 3 ,   c 4 , 2 = 3 16 R 4 ρ .
W = ∑ i j k 6 w i j k # x i y j u k ,
w i j k # = w i j k 0 ,
w i j k # = M i j k # ( α , r m , r s , 0 ) + ( − 1 ) k M i j k # ( β , r ′ m , r ′ s , 0 ) ,
α → α + v , r m → r m / cos ⁡ v , r s → r s / cos ⁡ v , β → β + v ′ , r ′ m → r ′ m / cos ⁡ v ′ , r ′ s → r ′ s / cos ⁡ v ′ ,
W ~ o b j 0 = ∑ i j k h 6 M i j k h 0 x i y j u k v h .
W ~ 0 = ∑ i j k h 6 w i j k h 0 x i y j u k v h .
w i j k h 0  =  M i j k h 0 ( α , r m , r s ) + ( − 1 ) k + h ( cos ⁡ α cos ⁡ β ) h M i j k h 0 ( β , r ′ m , r ′ s ) ,
u ′ = − u , v ′ = − cos ⁡ α cos ⁡ β v .
x ¯ p = l m v cos ⁡ α , y ¯ p = l s u .
W ~ o b j  =  ∑ i j k h 6 M i j k h x i y j u k v h =   ∑ i j k h 6 M i j k h 0 ( x + x ¯ p ) i ( y + y ¯ p ) j u k v h ( i + j + k + h ≤ 6 ) ,
l m = t m r m t m − r m , l s = t s r s t s − r s , l m ′ = t ′ m r ′ m t ′ m − r ′ m , l s ′ = t ′ s r ′ s t ′ s − r ′ s .
l m ′ = − cos 2 β cos 2 α l m , l s ′ = − l s .
n = χ σ + Γ σ ( n 20 2 χ 2 + n 02 2 η 2 + n 30 2 χ 3 + n 12 2 χ η 2 + n 40 8 χ 4 + n 22 4 χ 2 η 2 + n 04 8 η 4 + n 50 8 χ 5 + n 32 4 χ 3 η 2  +  n 14 8 χ η 4 + n 60 16 χ 6 + n 42 16 χ 4 η 2 + n 24 16 χ 2 η 4 + n 06 16 η 6 ) ,  
n = ∑ i j k h 6 ( Γ / σ ) N i j k h x i y j u k v h ,
W = ∑ i j k h 6 w ~ i j k h x i y j u k v h ( i + j + k + h ≤ 6 ) ,
w ~ i j k h = M i j k h ( α , r m , r s , l m , l s ) + ( − 1 ) k + h ( cos ⁡ α cos ⁡ β ) h M i j k h ( β , r ′ m , r ′ s , l ′ m , l ′ s ) + Λ N i j k h ,
sin ⁡ α + sin ⁡ β = m λ / σ .
l ′ m cos ⁡ α cos ⁡ β ( cos ⁡ β r ′ m − 2 c 2 , 0 ) − l m ( cos ⁡ α r m − 2 c 2 , 0 ) = Λ n 20 l m cos ⁡ α , l ′ s ( 1 r ′ s − 2 c 0 , 2 cos ⁡ β ) − l s ( 1 r s − 2 c 0 , 2 cos ⁡ α ) = Λ n 02 l s ,
2 c 2 , 0 ( cos ⁡ α + cos ⁡ β ) − ( cos 2 α r m + cos 2 β r ′ m ) = Λ n 20 , 2 c 0 , 2 ( cos ⁡ α + cos ⁡ β ) − ( 1 r s + 1 r ′ s ) = Λ n 02 .
W ~ = w ~ 3000 x 3 + w ~ 4000 x 4 + w ~ 5000 x 5 + w ~ 6000 x 6 + w ~ 1200 x y 2 + w ~ 2200 x 2 y 2 + w ~ 3200 x 3 y 2 + w ~ 4200 x 4 y 2   + w ~ 0400 y 4 + w ~ 1400 x y 4 + w ~ 2400 x 2 y 4 + w ~ 0600 y 6 + w ~ 1110 x y u + w ~ 2110 x 2 y u + w ~ 3110 x 3 y u   + w ~ 4110 x 4 y u + w ~ 0310 y 3 u + w ~ 1310 x y 3 u + w ~ 2310 x 2 y 3 u + w ~ 0510 y 5 u + w ~ 1020 x u 2 + w ~ 2020 x 2 u 2   + w ~ 3020 x 3 u 2 + w ~ 4020 x 4 u 2 + w ~ 0220 y 2 u 2 + w ~ 1220 x y 2 u 2 + w ~ 2220 x 2 y 2 u 2 + w ~ 0420 y 4 u 2   + w ~ 0130 y u 3 + w ~ 1130 x y u 3 + w ~ 2130 x 2 y u 3 + w ~ 0330 y 3 u 3 + w ~ 1040 x u 4 + w ~ 2040 x 2 u 4 + w ~ 0240 y 2 u 4   + w ~ 0150 y u 5 + w ~ 2111 x 2 y u v + w ~ 1201 x y 2 v + w ~ 0202 y 2 v 2 + w ~ 1311 x y 3 u v + w ~ 1221 x y 2 u 2 v + w ~ 2004 x 2 v 4   + w ~ 1111 x y u v + w ~ 1112 x y u v 2 + w ~ 1113 x y u v 3 + w ~ 1131 x y u 3 v + w ~ 0221 y 2 u 2 v + w ~ 3021 x 3 u 2 v   + w ~ 3201 x 3 y 2 v + w ~ 0311 y 3 u v + w ~ 4001 x 4 v + w ~ 2002 x 2 v 2 + w ~ 3001 x 3 v + w ~ 1002 x v 2 + w ~ 1003 x v 3 + w ~ 1004 x v 4   + w ~ 1005 x v 5 + w ~ 2112 x 2 y u v 2  +  w ~ 0402 y 4 v 2 + w ~ 0204 y 2 v 4 + w ~ 3111 x 3 y u v + w ~ 2001 x 2 v + w ~ 1202 x y 2 v 2   + w ~ 1021 x u 2 v + w ~ 1022 x u 2 v 2 + w ~ 1023 x u 2 v 3 + w ~ 0201 y 2 v + w ~ 0203 y 2 v 3 + w ~ 1041 x u 4 v + w ~ 1401 x y 4 v   + w ~ 1203 x y 2 v 3 + w ~ 2022 x 2 u 2 v 2 + w ~ 2202 x 2 y 2 v 2 + w ~ 3003 x 3 v 3 + w ~ 4002 x 4 v 2 + w ~ 2201 x 2 y 2 v   + w ~ 2021 x 2 u 2 v + w ~ 0312 y 3 u v 2 + w ~ 0222 y 2 u 2 v 2 + w ~ 5001 x 5 v + w ~ 0401 y 4 v + w ~ 2003 x 2 v 3 + w ~ 3002 x 3 v 2 .
x = ∑ i = 0 2 ∑ j = 0 2 a i j ′ 0 x 0 ′ i y 0 ′ j , y = ∑ i = 0 2 ∑ j = 0 2 b i j ′ 0 x 0 ′ i y 0 ′ j .
x = ∑ i = 0 2 ∑ j = 0 2 a ~ i j ′ x 0 ′ i y 0 ′ j , y = ∑ i = 0 2 ∑ j = 0 2 b ~ i j ′ x 0 ′ i y 0 ′ j .
W ~ ′ = ∑ i j k h 6 w ′ ~ i j k h x ~ 0 ′ i y ~ 0 ′ j u k v h , ( i + j + k + h ≤ 6 ) .
W ~ ′ = w ~ ′ 3000 x 0 ′ 3 + w ~ ′ 4000 x 0 ′ 4 + w ~ ′ 5000 x 0 ′ 5 + w ~ ′ 6000 x 0 ′ 6 + w ~ ′ 1200 x ′ 0 y 0 ′ 2 + w ~ ′ 2200 x 0 ′ 2 y 0 ′ 2 + w ~ ′ 3200 x 0 ′ 3 y 0 ′ 2 + w ~ ′ 4200 x 0 ′ 4 y 0 ′ 2   + w ~ ′ 0400 y 0 ′ 4 + w ~ ′ 1400 x ′ 0 y 0 ′ 4 + w ~ ′ 2400 x 0 ′ 2 y 0 ′ 4 + w ~ ′ 0600 y 0 ′ 6 + w ~ ′ 1110 x ′ 0 y ′ 0 u + w ~ ′ 2110 x 0 ′ 2 y ′ 0 u + w ~ ′ 3110 x 0 ′ 3 y ′ 0 u   + w ~ ′ 4110 x 0 ′ 4 y ′ 0 u + w ~ ′ 0310 y 0 ′ 3 u + w ~ ′ 1310 x ′ 0 y 0 ′ 3 u + w ~ ′ 2310 x 0 ′ 2 y 0 ′ 3 u + w ~ ′ 0510 y 0 ′ 5 u + w ~ ′ 1020 x ′ 0 u 2 + w ~ ′ 2020 x 0 ′ 2 u 2   + w ~ ′ 3020 x 0 ′ 3 u 2 + w ~ ′ 4020 x 0 ′ 4 u 2 + w ~ ′ 0220 y 0 ′ 2 u 2 + w ~ ′ 1220 x ′ 0 y 0 ′ 2 u 2 + w ~ ′ 2220 x 0 ′ 2 y 0 ′ 2 u 2 + w ~ ′ 0420 y 0 ′ 4 u 2   + w ~ ′ 0130 y ′ 0 u 3 + w ~ ′ 1130 x ′ 0 y ′ 0 u 3 + w ~ ′ 2130 x 0 ′ 2 y ′ 0 u 3 + w ~ ′ 0330 y 0 ′ 3 u 3 + w ~ ′ 1040 x ′ 0 u 4 + w ~ ′ 2040 x 0 ′ 2 u 4 + w ~ ′ 0240 y 0 ′ 2 u 4   + w ~ ′ 0150 y ′ 0 u 5 + w ~ ′ 2111 x 0 ′ 2 y ′ 0 u v + w ~ ′ 1201 x ′ 0 y 0 ′ 2 v + w ~ ′ 0202 y 0 ′ 2 v 2 + w ~ ′ 1311 x ′ 0 y 0 ′ 3 u v + w ~ ′ 1221 x ′ 0 y 0 ′ 2 u 2 v   + w ~ ′ 2004 x 0 ′ 2 v 4 + w ~ ′ 1111 x ′ 0 y ′ 0 u v + w ~ ′ 1112 x ′ 0 y ′ 0 u v 2 + w ~ ′ 1113 x ′ 0 y ′ 0 u v 3 + w ~ ′ 1131 x ′ 0 y ′ 0 u 3 v + w ~ ′ 0221 y 0 ′ 2 u 2 v   + w ~ ′ 3021 x 0 ′ 3 u 2 v + w ~ ′ 3201 x 0 ′ 3 y 0 ′ 2 v + w ~ ′ 0311 y 0 ′ 3 u v + w ~ ′ 4001 x 0 ′ 4 v + w ~ ′ 2002 x 0 ′ 2 v 2 + w ~ ′ 3001 x 0 ′ 3 v + w ~ ′ 1002 x ′ 0 v 2   + w ~ ′ 1003 x ′ 0 v 3 + w ~ ′ 1004 x ′ 0 v 4 + w ~ ′ 1005 x ′ 0 v 5 + w ~ ′ 2112 x 0 ′ 2 y ′ 0 u v 2  +  w ~ ′ 0402 y 0 ′ 4 v 2 + w ~ ′ 0204 y 0 ′ 2 v 4 + w ~ ′ 3111 x 0 ′ 3 y ′ 0 u v   + w ~ ′ 2001 x 0 ′ 2 v + w ~ ′ 1202 x ′ 0 y 0 ′ 2 v 2 + w ~ ′ 1021 x ′ 0 u 2 v + w ~ ′ 1022 x ′ 0 u 2 v 2 + w ~ ′ 1023 x ′ 0 u 2 v 3 + w ~ ′ 0201 y 0 ′ 2 v + w ~ ′ 0203 y 0 ′ 2 v 3   + w ~ ′ 1041 x ′ 0 u 4 v + w ~ ′ 1401 x ′ 0 y 0 ′ 4 v + w ~ ′ 1203 x ′ 0 y 0 ′ 2 v 3 + w ~ ′ 2022 x 0 ′ 2 u 2 v 2 + w ~ ′ 2202 x 0 ′ 2 y 0 ′ 2 v 2 + w ~ ′ 3003 x 0 ′ 3 v 3 + w ~ ′ 4002 x 0 ′ 4 v 2   + w ~ ′ 2201 x 0 ′ 2 y 0 ′ 2 v + w ~ ′ 2021 x 0 ′ 2 u 2 v + w ~ ′ 0312 y 0 ′ 3 u v 2 + w ~ ′ 0222 y 0 ′ 2 u 2 v 2 + w ~ ′ 5001 x ′ 0 5 v + w ~ ′ 0401 y 0 ′ 4 v + w ~ ′ 2003 x 0 ′ 2 v 3 + w ~ ′ 3002 x 0 ′ 3 v 2 .
θ x ′ = d W ~ ′ d x ′ 0 , θ y ′ = d W ~ ′ d y ′ 0 .
X ¯ = r 0 ′ θ x ′ = r 0 ′ d W ~ ′ d x ′ 0 , Y ¯ = r 0 ′ θ y ′ = r 0 ′ d W ~ ′ d y ′ 0 .
+ d ~ 1020 ′ x 0 ′ u 2 + d ~ 2020 ′ x 0 ′ 2 u 2 + d ~ 3020 ′ x 0 ′ 3 u 2 + d ~ 0220 ′ y 0 ′ 2 u 2 + d ~ 1220 ′ x 0 ′ y 0 ′ 2 u 2 + d ~ 0130 ′ y 0 ′ u 3 + d ~ 1130 ′ x 0 ′ y 0 ′ u 3 + d ~ 0040 ′ u 4
d ~ i j k h ′ = ( i + 1 ) r 0 ′ w ~ ( i + 1 ) j k h ′ , h ~ i j k h ′ = ( j + 1 ) r 0 ′ w ~ i ( j + 1 ) k h , ( i + j + k + h ≤ 6 ) .
Δ x ′ = d ¯ ′ 1000 x ′ 0 + d ¯ ′ 3000 x 0 ′ 3 + d ¯ ′ 1200 x ′ 0 y 0 ′ 2 + d ¯ ′ 5000 x 0 ′ 5 + d ¯ ′ 3200 x 0 ′ 3 y 0 ′ 2 + d ¯ ′ 1400 x ′ 0 y 0 ′ 4 , Δ y ′ = h ¯ ′ 0100 y ′ 0 + h ¯ ′ 2100 x 0 ′ 2 y ′ 0 + h ¯ ′ 0300 y 0 ′ 3 + h ¯ ′ 4100 x 0 ′ 4 y ′ 0 + h ¯ ′ 0500 y 0 ′ 5 + h ¯ ′ 2300 x 0 ′ 2 y 0 ′ 3 ,
d ¯ ′ 1000  =  Λ m ,   d ¯ ′ 3000  =  Λ m 2 r m ′ 2 ,   d ¯ ′ 1200  =  Λ m 2 r ′ m r ′ s ,   d ¯ ′ 5000  =  3 Λ m 8 r m ′ 4 ,   d ¯ ′ 3200  =  3 Λ m 4 r m ′ 3 r ′ s ,   d ¯ ′ 1400  =  3 Λ m 4 r ′ m r s ′ 2 ( 1 r ′ m + 1 2 r ′ s ) , h ¯ ′ 0100  =  Λ s ,   h ′ ¯ 2100  =  Λ m 2 r ′ m r ′ s ,   h ′ ¯ 0300  =  Λ s 2 r s ′ 2 ,   h ′ ¯ 4100  =  3 Λ m 8 r m ′ 3 r ′ s ,   h ′ ¯ 0500  =  3 Λ s 4 r s ′ 4 ,   h ′ ¯ 2300  =  Λ m 2 r ′ m r s ′ 2 ( 1 r ′ m + 1 2 r ′ s ) .
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