© The Author(s), published by EDP Sciences, 2025

1 Introduction

Many approaches for deriving the primary wavefront aberrations of an axially symmetrical system have been proposed (e.g., [1–5]). One of the most widely used methods is that developed by Buchdahl [1]. In this method, the marginal and chief paraxial rays at a single refractive boundary are traced to compute the Buchdahl aberration coefficients, and these coefficients are then converted to the image space by applying the Gaussian imaging equation sequentially, surface by surface. The final wavefront aberrations are expressed in terms of the object height h₀ and coordinates (x_e, y_e) in the image space as W(h₀, x_e, y_e). When the values of the wavefront aberrations are much smaller than the reference sphere radius R_s, the ray aberration (ΔP_nx, ΔP_ny) can be estimated as ([2, 6], p. 13 of [3], [7–10])$$\left[\begin{array}{c}\Delta P_{\mathrm{nx}}\\ \Delta P_{\mathrm{ny}}\end{array}\right]\approx \frac{R_s}{{\xi }_s}\left[\begin{array}{c}\partial W(h_0,x_e,y_e)/\partial x_e\\ \partial W(h_0,x_e,y_e)/\partial y_e\end{array}\right],$$(1)where ξ_s is the refractive index of the reference sphere. Notably, the partial derivatives in equation (1) must be performed in coordinate frames in the image space. One frequently used coordinate frame is (xyz)_e attached to the exit pupil (Fig. 1). The need to utilize coordinate frames in the image space arises from the fact that, by doing so, the normal vector of the wavefront can be determined directly by the gradient of the wavefront expression to determine ray aberrations (see Eq. (1.11) of [3], [11, 12]). Equation (1) is given in a Cartesian coordinate frame (xyz)_e. If the wavefront aberration is expressed in terms of polar coordinates as W(h₀, ρ_e, ϕ_e), its transformation equation to (ΔP_nx, ΔP_ny) can be obtained from equation (1) simply by using (x_e, y_e) = (ρ_e sinϕ_e, ρ_e cosϕ_e) (see Fig. 1), to yield$$\left[\begin{array}{c}\Delta P_{\mathrm{nx}}\\ \Delta P_{\mathrm{ny}}\end{array}\right]\approx \frac{R_s}{{\xi }_s}\left\{\left[\begin{array}{c}\sin {\varphi }_e\\ \cos {\varphi }_e\end{array}\right]\frac{\partial W}{\partial {\rho }_e}+\frac{1}{{\rho }_e}\left[\begin{array}{c}\cos {\varphi }_e\\ -\sin {\varphi }_e\end{array}\right]\frac{\partial W}{\partial {\varphi }_e}\right\}.$$(2)

Fig. 1 Four coordinate frames are frequently used in geometrical optics: (xyz)0, (xyz), (xyz)e, and (xyz)n. These frames are referred to as the world coordinate frame, coordinate frame attached to the entrance pupil, coordinate frame attached to the exit pupil, and coordinate frame attached to the image plane, respectively. It is also shown that the entrance and exit pupils, image plane are located at (0, 0, z), (0, 0, ze), and (0, 0, zn), respectively, defined by (xyz)0.

The errors of the transferred ray aberrations from equations (1) or (2) are not quantified analytically, as stated in the Abstract of [13].

However, wavefront aberrations can also be determined in terms of the object space parameters without using the Gaussian imaging equation [14–17]. For example, the unit directional vector ${\overline{\mathcal{l}}}_0$ originating from a point object ${\overline{P}}_0=(0,h_0,P_{0z})$ is expressed in terms of the intercepted in-plane point $\overline{P}=(x,y)$ of the entrance pupil (Fig. 1). The Taylor series expansion can then be employed to obtain the wavefront aberration function as W(h₀, x, y), where (xyz) are coordinate frame in the object space attached to the entrance pupil. Nonetheless, transforming the wavefront aberrations into ray aberrations based on a coordinate frame in the object space is still challenging. To address this problem, this study converts equation (2) from the image space to the object space using the chain rule. The numerical results show that the ray aberrations obtained by this converted equation are in very close agreement with the Zemax results.

2 Relationship between wavefront and ray aberrations in object space

This work distinguishes between a “ray” and a “ray path.” Referring to Figure 2, a ray ${\overline{R}}_i=({\overline{P}}_i,{\overline{\mathcal{l}}}_i)$ includes the incidence point ${\overline{P}}_i$ and its unit directional vector ${\overline{\mathcal{l}}}_i$. When a source ray ${\overline{R}}_0=({\overline{P}}_0,{\overline{\mathcal{l}}}_0)$ travels through an axially symmetrical optical system, multiple rays (${\overline{R}}_i$, i = 0 to n) are distributed along the ray path $\overline{H}({\overline{P}}_0,{\overline{\mathcal{l}}}_0)$ (Fig. 2).

Fig. 2 Axially symmetrical system with nine boundaries [18]. The ray path $\overline{H}({\overline{P}}_0,{\overline{\mathcal{l}}}_0)$ is the path between points ${\overline{P}}_0$ and ${\overline{P}}_n$.

Equation (2) must be mathematically converted in order to operate on the polar coordinate frame in object space. One of the most useful coordinate frames in object space is (xyz), attached to the entrance pupil. Generally, the relationship of the intercepted points of a general ray path on the entrance pupil and exit pupil, $\overline{P}=(\rho ,\varphi )$ and ${\overline{P}}_e=({\rho }_e,{\varphi }_e)$, are given by ρ_e = ρ_e (ρ, ϕ) and ϕ_e = ϕ_e (ρ, ϕ). Thus, equation (2) can be converted using the following chain rule:$$\left[\begin{array}{c}\Delta P_{\mathrm{nx}}\\ \Delta P_{\mathrm{ny}}\end{array}\right]\approx \frac{R_s}{{\xi }_s}\left\{\left[\begin{array}{c}\sin {\varphi }_e\\ \cos {\varphi }_e\end{array}\right]\left(\frac{\partial W}{\partial \rho }\frac{\partial \rho }{\partial {\rho }_e}+\frac{\partial W}{\partial \varphi }\frac{\partial \varphi }{\partial {\rho }_e}\right)+\frac{1}{{\rho }_e}\left[\begin{array}{c}\cos {\varphi }_e\\ -\sin {\varphi }_e\end{array}\right]\left(\frac{\partial W}{\partial \rho }\frac{\partial \rho }{\partial {\varphi }_e}+\frac{\partial W}{\partial \varphi }\frac{\partial \varphi }{\partial {\varphi }_e}\right)\right\}.$$(3)

From the numerical simulation results, it is found that ∂ϕ/∂ϕ_e ≈ 1, ∂ϕ/∂ρ_e ≈ 0, and ∂ρ/∂ϕ_e ≈ 0 hold over the entire entrance pupil domain. Thus, equation (3) can be further simplified as$$\left[\begin{array}{c}\Delta P_{\mathrm{nx}}\\ \Delta P_{\mathrm{ny}}\end{array}\right]\approx \frac{R_s}{{\xi }_s}\left\{\left[\begin{array}{c}\sin {\varphi }_e\\ \cos {\varphi }_e\end{array}\right]\frac{\partial W}{\partial \rho }\frac{\partial \rho }{\partial {\rho }_e}+\frac{1}{{\rho }_e}\left[\begin{array}{c}\cos {\varphi }_e\\ -\sin {\varphi }_e\end{array}\right]\frac{\partial W}{\partial \varphi }\right\}.$$(4)

Since equation (4) is an approximation equation, it can be further simplified without degrading its effectiveness by setting cosϕ_e = cosϕ, sinϕ_e = sinϕ, and ρ/ρ_e = ∂ρ/∂ρ_e to obtain$$\left[\begin{array}{c}\Delta P_{\mathrm{nx}}\\ \Delta P_{\mathrm{ny}}\end{array}\right]\approx \frac{R_s}{{\xi }_s}\frac{\partial \rho }{\partial {\rho }_e}\left\{\left[\begin{array}{c}\sin \varphi \\ \cos \varphi \end{array}\right]\frac{\partial W}{\partial \rho }+\frac{1}{\rho }\left[\begin{array}{c}\cos \varphi \\ -\sin \varphi \end{array}\right]\frac{\partial W}{\partial \varphi }\right\}.$$(5)

The numerical value of ∂ρ/∂ρ_e depends on ρ, the radial coordinate of the entrance pupil intercepted by the ray path $\overline{H}({\overline{P}}_0,{\overline{\mathcal{l}}}_0)$. To simplify equation (5), ∂ρ/∂ρ_e can be approximated either by$$\frac{\partial \rho }{\partial {\rho }_e}=\frac{{\rho }_{\max }}{{\rho }_{e/\max }},$$(6)where ρ_max and ρ_e/max are the maximum entrance and exit pupil radii, respectively, or$$\frac{\partial \rho }{\partial {\rho }_e}={\left(\frac{\partial \rho }{\partial {\rho }_e}\right)}_{\mathrm{axis}},$$(7)where (∂ρ/∂ρ_e)_axis is the pupil magnification evaluated by paraxial optics. The numerical results show that, compared with equation (7), equation (6) may introduce additional errors into equation (5). It is thus discarded here. By substituting equation (7) into equation (5), the following relationship between the wavefront and ray aberrations applicable in the object space is obtained:$$\left[\begin{array}{c}\Delta P_{\mathrm{nx}}\\ \Delta P_{\mathrm{ny}}\end{array}\right]\approx \frac{R_s}{{\xi }_s}{\left(\frac{\partial \rho }{\partial {\rho }_e}\right)}_{\mathrm{axis}}\left\{\left[\begin{array}{c}\sin \varphi \\ \cos \varphi \end{array}\right]\frac{\partial W}{\partial \rho }+\left[\begin{array}{c}\cos \varphi \\ -\sin \varphi \end{array}\right]\frac{1}{\rho }\frac{\partial W}{\partial \varphi }\right\}.$$(8)

Consider the illustrative optical system with a Gaussian image plane shown in Figure 2. Assume that the point object is located at ${\overline{P}}_0=(0,h_0,P_{0z})=(\mathrm{0,17},-200)$. The lengths of this paper are specified in millimeters. The parameters of the optical system are presented in Table 1. The primary wavefront aberrations W(h₀, ρ, ϕ) and the ray aberrations (ΔP_nx, ΔP_ny), based on the polar coordinate frame built at the entrance pupil, are given respectively by (Eq. (1) of [15] and Eqs. (2a) and (2b) of [14]) as follows:$$\mathrm{W}(h_0,\rho ,\varphi )=C_{040}{\rho }^4+C_{131}{h}_0{\rho }^3\cos \varphi +C_{222}{h}_0^2{\rho }^2(\cos \varphi {)}^2+C_{220}{h}_0^2{\rho }^2+C_{311}{h}_0^3\rho \cos \varphi ,$$(9) $$\left[\begin{array}{c}\Delta P_{\mathrm{nx}}\\ \Delta P_{\mathrm{ny}}\end{array}\right]=B_1{\rho }^3\left[\begin{array}{c}\sin \varphi \\ \cos \varphi \end{array}\right]+B_2{h}_0{\rho }^2\left[\begin{array}{c}\sin \left(2\varphi \right)\\ 2+\cos \left(2\varphi \right)\end{array}\right]+B_3{h}_0^2\rho \left[\begin{array}{c}\sin \varphi \\ 3\cos \varphi \end{array}\right]+B_4{h}_0^2\rho \left[\begin{array}{c}\sin \varphi \\ \cos \varphi \end{array}\right]+B_5{h}_0^3\left[\begin{array}{c}0\\ 1\end{array}\right].$$(10)

It is noted that the second-order wavefront aberration (i.e., defocus wavefront aberration C₀₂₀ ρ ², magnification wavefront aberration C₁₁₁ h₀ ρcosϕ) and first-order defocus ray aberration (i.e., A₁ ρ(sinϕ, cosϕ)) are not included in equations (9) and (10), respectively, since their coefficients, C₀₂₀, C₁₁₁, and A₁, are zero-valued when the image plane is Gaussian. By substituting equations (9) and (10) into equation (8), the following transformed ray aberration coefficients are obtained:$$B_1=4C_{040}\frac{R_s}{{\xi }_s}{\left(\frac{\partial \rho }{\partial {\rho }_e}\right)}_{\mathrm{axis}},$$(11) $$B_2=C_{131}\frac{R_s}{{\xi }_s}{\left(\frac{\partial \rho }{\partial {\rho }_e}\right)}_{\mathrm{axis}},$$(12) $$B_3=C_{222}\frac{R_s}{{\xi }_s}{\left(\frac{\partial \rho }{\partial {\rho }_e}\right)}_{\mathrm{axis}},$$(13) $$B_4=\left(2{\mathrm{C}}_{220}-{\mathrm{C}}_{222}\right)\frac{R_s}{{\xi }_s}{\left(\frac{\partial \rho }{\partial {\rho }_e}\right)}_{\mathrm{axis}},$$(14) $$B_5=C_{311}\frac{R_s}{{\xi }_s}{\left(\frac{\partial \rho }{\partial {\rho }_e}\right)}_{\mathrm{axis}},$$(15)where (∂ρ/∂ρ_e)_axis = 1.11521263, and the radius and refractive index of the reference sphere are R_s = 131.564842 and ξ_s = 1, respectively.

The numerical values of the transformed ray aberration coefficients obtained from equations (11) to (15) are given in column No. 3 of Table 2. The ray aberration coefficients from Zemax simulation are listed in No. 4 (see Appendix). The percentage errors of these transformed ray aberration coefficients relative to the values in columns No. 2 and No. 4 are listed in columns No. 5 and No. 6, respectively. The results presented in columns No. 5 and No. 6 show that equations (11)–(15) transfer the wavefront aberration coefficients to ray aberration coefficients with significant accuracy. The peak values of the transferred ray aberrations (i.e., $B_1{\rho }_{\max }^3$, $B_2{\rho }_{\max }^2$, ${\mathrm{B}}_3{h}_0^2{\rho }_{\max }$, ${\mathrm{B}}_4{h}_0^2{\rho }_{\max }$, and $B_5{h}_0^3$, where object height h₀ = 17 mm and maximum opening radius of the entrance pupil ρ_max = 21 mm) and those obtained from [14] and Zemax simulation are listed in columns No. 7, No. 8, and No. 9, respectively. The percentage errors of No. 7 with respect to No. 8 and No. 9 are shown in columns No. 10 and No. 11, respectively.

It should be noted that the numerical values of the ray aberration coefficients based on (xyz) are different from those based on (xyz)_e due to the non-linearities of ρ_e = ρ_e (ρ, ϕ) and ϕ_e = ϕ_e(ρ, ϕ). The ray aberration coefficients computed by Zemax based on (xyz) and (xyz)_e are given in columns No. 4 and No. 12 of Table 2, respectively (see Appendix). It can be seen from column No. 13, which shows the percentage difference between them, that the spherical aberration coefficient B₁ and coma coefficient B₂ have percentage differences of −29.777% and −20.995% between frames (xyz) and (xyz)_e, respectively. The percentage differences of the astigmatism coefficient B₃ and field curvature coefficient B₄ in the two coordinate frames are both −11.116%. The distortion coefficients in (xyz) and (xyz)_e are identical since the distortion is a function of the object height h₀ only.

Equations (8)–(15) are also valid for systems with an object at infinity, provided that the object height h₀ is replaced by the field angle β₀. Table 3 lists the numerical results for the system in Figure 2 with an object at infinity and β₀ = 4.377°. The percentage errors compared with the values listed in column No. 2 and those obtained from Zemax (No. 4), respectively, are very small. Thus, the results confirm that equation (8) can transform wavefront aberrations into ray aberrations with high precision when the image plane is Gaussian.

Similarly, equation (1) can be revised to transform the ray aberration coefficients based on the Cartesian coordinate frame (xyz) attached to the entrance pupil by using the expressions x_e = x_e(x, y) and y_e = y_e(x, y), to obtain$$\left[\begin{array}{c}\Delta P_{\mathrm{nx}}\\ \Delta P_{\mathrm{ny}}\end{array}\right]\approx \frac{R_s}{{\xi }_s}{\left(\frac{\partial y}{\partial y_e}\right)}_{\mathrm{axis}}\left[\begin{array}{c}\partial W(h_0,x,y)/\partial x\\ \partial W(h_0,x,y)/\partial y\end{array}\right].$$(16)

Equation (16) is obtained by assuming ∂x/∂y_e ≈ 0 and ∂y/∂x_e ≈ 0, and setting ∂y/∂y_e = ∂x/∂x_e to be (∂y/∂y_e)_axis, which is evaluated along the optical axis of the axially symmetrical system. Again, equation (16) is also valid for a system with an object at infinity, provided that the object height h₀ is replaced by the field angle β₀.

3 Conclusions

Many wavefront aberrations are determined using the Gaussian imaging equation. In other words, they are expressed in terms of the object height h₀ and coordinates in the image space as W(h₀, x_e, y_e), where (xyz)_e is attached to the exit pupil. Thus, the classic relationship between the wavefront aberrations W(h₀, x_e, y_e) and ray aberrations (ΔP_nx, ΔP_ny) is confined to coordinate frames in the image space. Alternatively, the primary wavefront aberrations can be determined without using the Gaussian imaging equation by using the Taylor series expansion of the optical path length to obtain the expression W(h₀, x, y), where (xyz) is the coordinate frame in the object space attached to the entrance pupil. To estimate the ray aberrations (ΔP_nx, ΔP_ny) from the wavefront aberrations W(h₀, x, y) of a system with a Gaussian image plane, this study has converted the classic relationship equation between the ray and wavefront aberrations [6] using the chain rule based on ρ_e = ρ_e (ρ, ϕ) and ϕ_e = ϕ_e (ρ, ϕ). It has been shown that the percentage errors of the transformed ray aberrations relative to those derived in [14] and obtained by Zemax are negligible for an object placed at ${\overline{P}}_0=(\mathrm{0,17},-200)$ or at infinity with β₀ = 4.377° in the illustrative axially symmetrical system. Thus, the validity of the transformation equation is confirmed.

Funding

National Science and Technology Council (113-2221-E-006-103-).

Conflicts of interest

The author declares no conflicts of interest.

Data availability statement

The author confirms that the data supporting the findings of this study are either available within the article and its supplementary materials or could be obtained from the authors upon reasonable request.

References

Appendix

Zemax simulation can provide the peak values of ray aberrations (i.e., spherical aberration D1, coma D2, astigmatism D3, field curvature D4, and distortion D5) and wavefront aberrations (denoted as spherical aberration W040, coma W131, astigmatism W222, field curvature W220S, and distortion W311). Their values are identical in object and image spaces, leading that we have the following equations:$$D1=B_1{\rho }_{\max }^3=B_{e/1}{\rho }_{e/\max }^3,$$(A1) $$D2={\mathrm{B}}_2{h}_0{\rho }_{\max }^2={\mathrm{B}}_{e/2}{h}_0{\rho }_{e/\max }^2,$$(A2) $$D3={\mathrm{B}}_3{h}_0^2{\rho }_{\max }={\mathrm{B}}_{\mathrm{e}/3}{h}_0^2{\rho }_{e/\max },$$(A3) $$D4={\mathrm{B}}_4{h}_0^2{\rho }_{\max }={\mathrm{B}}_{e/4}{h}_0^2{\rho }_{e/\max },$$(A4) $$D5=B_5{h}_0^3=B_{e/5}{h}_0^3,$$(A5) $$W040=C_{040}{\rho }_{\max }^4/{\upsilon }_0=C_{e/040}{\rho }_{e/\max }^4/{\upsilon }_0,$$(A6) $$W131=C_{131}{h}_0{\rho }_{\max }^3/{\upsilon }_0=C_{e/131}{h}_0{\rho }_{e/\max }^3/{\upsilon }_0,$$(A7) $$W222=C_{222}{h}_0^2{\rho }_{\max }^2/{\upsilon }_0=C_{e/222}{h}_0^2{\rho }_{e/\max }^2/{\upsilon }_0,$$(A8) $$W220S=C_{220}{h}_0^2{\rho }_{\max }^2/{\upsilon }_0=C_{e/220}{h}_0^2{\rho }_{e/\max }^2/{\upsilon }_0,$$(A9) $$W311=C_{311}{h}_0^3{\rho }_{\max }/{\upsilon }_0=C_{e/311}{h}_0^3{\rho }_{e/\max }/{\upsilon }_0,$$(A10)where B_j and B_e/j (j = 1–5) are ray aberration coefficients in object and image spaces. ρ_max and ρ_e/max are respectively the maximum radii of the entrance and exit pupils. C₀₄₀, C₁₃₁, C₂₂₂, C₂₂₀, C₃₁₁ (C coefficients) are wavefront aberration coefficients in object space, and C_e/040, C_e/131, C_e/222, C_e/220, C_e/311 (C_e coefficients) are wavefront aberration coefficients in image space. Thus one can obtain the aberration coefficients, B_j (listed in No. 4), B_e/j (given in No. 12), C, and C_e from the Zemax’s peak values of wavefront and ray aberrations.

All Tables

All Figures

Fig. 1 Four coordinate frames are frequently used in geometrical optics: (xyz)0, (xyz), (xyz)e, and (xyz)n. These frames are referred to as the world coordinate frame, coordinate frame attached to the entrance pupil, coordinate frame attached to the exit pupil, and coordinate frame attached to the image plane, respectively. It is also shown that the entrance and exit pupils, image plane are located at (0, 0, z), (0, 0, ze), and (0, 0, zn), respectively, defined by (xyz)0.

In the text

	Fig. 2 Axially symmetrical system with nine boundaries [18]. The ray path $\overline{H}({\overline{P}}_0,{\overline{\mathcal{l}}}_0)$ is the path between points ${\overline{P}}_0$ and ${\overline{P}}_n$.
In the text