1. Introduction

In 1856, Philip Ludwig von Seidel extended the Gaussian theory of rotationally-symmetric lens systems to include monochromatic ray aberrations up to the third order. In his study, an object ${\bar{\textrm P}_0}$ located in the meridional plane of a rotationally-symmetric lens system was described as ${\bar{\textrm P}_0} = {[{0\;\;{\textrm{h}_0}\;\;{\textrm{P}_{0\textrm{z}}}\;\;{1}} ]^\textrm{T}}$, where ${\textrm{h}_0}$ is the object height perpendicular to optical axis at ${\textrm{P}_{0\textrm{z}}}$ as shown in Fig. 1. For a ray, as depicted in Fig. 2, originating from object ${\bar{\textrm P}_0}$ and passing through the entrance pupil at a point having polar coordinates $\rho$ and $\phi$, the intersection point of the ray on the image plane is given by

 

Fig. 1. Ray originating from an object, passing through entrance pupil, and then intersecting the image plane. The ${\textrm{x}_0}\;{\textrm{and}\ }{\textrm{y}_0}$ axes are shown displaced from the object plane for clarity.

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Fig. 2. Entrance pupil with Cartesian coordinates $({\textrm{x}_\textrm{a}},{\textrm{y}_\textrm{a}})$ and polar coordinates $(\rho ,\phi )$.

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Advances in the design and development of optical systems have been made possible by the continued development of the theoretical understanding of optical aberrations that spans almost four centuries [2]. Many approaches for deriving equations to compute the third-order or Seidel primary aberration coefficients have been proposed [3–12,15–20]. One of the most widely used methods was developed by Buchdahl which can compute third-order aberrations as well as in principle any higher-order term [3]. In his approach, the marginal and principal paraxial rays are traced and the results are then used to compute the unconverted third-order Buchdahl aberration coefficients (denoted as ${\sigma _\textrm{j}},\;\textrm{j} = 1 - 5$) which correspond to the B coefficients in general meaning. These sigma coefficients can then easily be converted into transverse, longitudinal, and wave aberration values [13]. Buchdahl’s approach has been implemented in most commercial optical design and analysis programs. Furthermore, these σ aberration coefficients are computed for the maximum entrance pupil radius ${\rho_{\max }}$ and maximum object height ${\textrm{h}_{0/\max }}$. These aberration coefficients are specific for the lens system configuration being analyzed where the entrance pupil radius and object height are each considered to be normalized, i.e., $\tilde{\rho}\;{\textrm{and}}\;{\tilde{\textrm{h}}_0}$ each range from 0 to 1. The primary ray aberrations for any ray within the pupil are then determined from these coefficients using the appropriate fractional scaling factors of pupil radius ${\tilde{\rho}}$ and object height ${\tilde{\textrm{h}}_0}$. For example, the primary tangential spherical aberration and tangential sagittal coma at height ${\textrm{h}_{0/\max }}/2$ and pupil radius of ${{{{\rho}_{\max }}} \mathord{\left/ {\vphantom {{{\rho_{\max }}} {\sqrt 2 }}} \right.} {\sqrt 2 }}$ are given by $\textrm{TSPH} = {\sigma _1}{\tilde{\rho }^3} = {\sigma _1}{\left( {{1 \mathord{\left/ {\vphantom {1 {\sqrt 2 }}} \right. } {\sqrt 2 }}} \right)^3}$ and $\textrm{TSCO} = {\sigma _2}{\tilde{\rho }^2}{\tilde{\textrm{h}}_0} = {\sigma _2}{\left( {{1 \mathord{\left/ {\vphantom {1 {\sqrt 2 }}} \right. } {\sqrt 2 }}} \right)^2}({0.5} )$, respectively.

A paraxial ray is by definition a ray propagating through the lens system having infinitesimal heights and angles. The imagery formed is stigmatic since there are no aberrations. However, it is possible to trace paraxial rays at finite heights and angles if the all of the angles (nu) are interpreted as $n\tan (u)$ where n is the index of refraction [14]. These “finite” paraxial rays are aberration free. Buchdahl and others used the finite paraxial ray tracing principle in determining the optical aberrations and first-order layout of lens systems [13]. In contrast, the path of real or trigonometrically-traced rays are computed using finite heights and actual angles and are aberrated with all orders of aberrations. The new method presented in this paper utilizes the tracing of real “pseudoparaxial” skew rays. These real rays are propagated in very close proximity to the optical axis with tiny heights and angles and are aberrated like trigonometrically-traced rays having finite heights and angles.

This paper describes an alternative method for determining the A and B coefficients of a rotationally-symmetric lens system based on the relationship between the Seidel primary ray aberrations and the third-order derivative matrix of a skew ray passing through the system. This alternative new method can be considered to be based upon a form of differential skew ray tracing proximate to the optical axis. In particular, the values of ${\textrm{A}_1}$ and ${\textrm{A}_2}$ are determined using a ray Jacobian matrix approach [21], while the Seidel aberration B coefficients are computed using finite-difference (FD) methods based on the ray Hessian matrix [22]. It is shown in Section 3.2 that aberrations determined using the coefficient values obtained from conventional methods and the B coefficients obtained in the present investigation are equivalent. As will be seen, the A and B coefficients determined using the new method are also constants for a given system that are normalized for unit entrance pupil radius and unit object height. For example, if the maximum entrance pupil radius is 12 and the object height is 8, then the primary tangential spherical aberration and tangential sagittal coma are given by $\textrm{TSPH} = {\textrm{B}_1}{({12} )^3}$ and $\textrm{TSCO} = {\textrm{B}_2}{({12} )^2}(8 )$, respectively.

2. Ray aberration polynomial

2.1 Ray aberration concept

Figure 3 shows the rotationally-symmetric optical system considered in this investigation. This optical system is a Tessar type configuration comprising a cemented doublet, two singlets, and a stop. The system prescription is contained in Table 1. The source ray ${\bar{\textrm{R}}_0} = {[{{{\bar{\textrm P}}_0}\;\;{{\bar{\ell }}_0}} ]^\textrm{T}}$ originates from an object ${\bar{\textrm P}_0}$ with directional vector $\bar{\boldsymbol{\ell }}{}_0$ located in the object plane denoted as surface “0”. Since the system is rotationally-symmetric about its optical axis, the object point ${\bar{\textrm P}_0}$ can be positioned with no loss of generality as lying in the meridional plane with a height ${\textrm{h}_0}$ , i.e.,

 

Fig. 3. Example rotationally-symmetric optical system having a Tessar type configuration.

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Table 1. Prescription for example rotationally-symmetric optical system shown in Fig. 3. The entrance pupil is located 22.103275 from the vertex of surface 1.

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Seidel explored the aberrations of a ray originating from an object at height ${\textrm{h}_0}$ with the directional vector $\boldsymbol{\bar{\ell}}_0$. The source ray vector is given by

2.2 Alternative ray aberration formulation

The intercept point on the image plane, denoted as ${\bar{\textrm P}_\textrm{n}}({{{\bar{\textrm{X}}}_{\textrm{aber}}}} )$, can be approximated by its Taylor series expansion with respect to ${\bar{\textrm{X}}_{\textrm{aber}}}$ as

Using FD Method #1 and Eq. (15) to approximate a third-order derivative involves two ray-tracing operations. The first operation uses ${\bar{\textrm{X}}_{\textrm{aber}}} = {[{0\;\;0\;\;0} ]^\textrm{T}}$ to produce ${({{{{\partial^2}{\textrm{P}_{\textrm{nx}}}} \mathord{\left/ {\vphantom {{{\partial^2}{\textrm{P}_{\textrm{nx}}}} {\partial {\alpha_0}\partial {\beta_0}}}} \right.} {\partial {\alpha_0}\partial {\beta_0}}} } )_1}$ and the second operation uses the same parameters with object height increased by a miniscule amount $\Delta {\textrm{h}_0}$, e.g., $\Delta {\textrm{h}_0} = {10^{ - 6}}$, to determine ${({{{{\partial^2}{\textrm{P}_{\textrm{nx}}}} \mathord{\left/ {\vphantom {{{\partial^2}{\textrm{P}_{\textrm{nx}}}} {\partial {\alpha_0}\partial {\beta_0}}}} \right.} {\partial {\alpha_0}\partial {\beta_0}}}} )_2}$. Likewise, FD Method #2 and FD Method #3 are used to compute, respectively,

Table 2. Third-order derivatives computed by three Finite-Difference (FD) methods.

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The estimation accuracy of the three FD methods to determine the third-order derivatives was investigated for three different values of $\Delta {\textrm{h}_0}$, $\Delta {\alpha _0}$, and $\Delta {\beta _0}$ with each having the same value, viz., ${10^{ - 4}},\,{1}{{0}^{ - 2}},\,\,{\textrm{and}\,\, 1}{{0}^{ - 1}}$. Table 3 lists the corresponding percentage errors relative to the values of Table 2. For the example lens system, Table 3 indicates that if $\Delta {\textrm{h}_0} \le {10^{ - 2}}$ mm, $\Delta {\alpha _0} < {10^{ - 4}}$ rad, and $\Delta {\beta _0} < {10^{ - 4}}$ rad, then the computational errors are negligible.

Table 3. The percentage errors, relative to the values in Table 2, of the third-order derivatives estimated by different FD methods.

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3. Primary ray aberration coefficients

3.1 Aberration coefficients formulation

Consider the circular entrance pupil shown in Fig. 2 having ${({\textrm{xyz}} )_\textrm{a}}$ as its coordinate frame. If $\rho $ is the pupil radius measured along the ${\textrm{y}_\textrm{a}}$ axis from the origin of ${({\textrm{xyz}} )_\textrm{a}}$, the Cartesian coordinates of the entrance pupil can be obtained simply by rotating the generating line ${[{0\;\;\rho \;\;{0\;\; 1}} ]^\textrm{T}}$ about the $- {\textrm{z}_\textrm{a}}$ axis through an angle $\phi $ $({0 \le \phi < 2\pi } )$ as shown by Eq. (18).

The notation ${({\,} )_{\textrm{prin}}}$ shown in Eqs. (25a)–(27) indicates the third-order differentials inside the parentheses are evaluated using the principal ray. Equations (22)–(28) can determine the transverse aberrations not only for meridional and sagittal rays, but also for any skew rays defined within the entire domain of ${\bar{\textrm{X}}_{\textrm{aber}}} = {[{{\textrm{h}_0}\;\;{\alpha_0}\;\;{\beta_0}} ]^\textrm{T}}$. Also, two expressions are available to determine ${\textrm{A}_1},\;{\textrm{B}_1},\, \textrm{and}\,\, {\textrm{B}_2}$. Numerical evaluations show that the values of ${\textrm{A}_1}$ from Eq. (22), ${\textrm{B}_1}$ from Eqs. (24a) and (24b), and ${\textrm{B}_2}$ from Eqs. (25a) and (25b), respectively, evaluated using a ray defined by ${\bar{\textrm{X}}_{\textrm{aber/optical}\;\;{\textrm{axis}}}} = {[{0\;\;0\;\;0} ]^\textrm{T}}$ are essentially identical. It is noted that the partial derivatives for the B coefficients should be computed in the Gaussian image plane while the partial derivatives for the A coefficients should be computed in the defocused image plane.

3.2 Aberration coefficients for example lens

The Buchdahl coefficients are determined for a normalized entrance pupil radius and normalized field angle while the new aberration coefficients are normalized for unit entrance pupil radius and unit object height. In this subsection, the A and B ray aberration coefficients are determined for the rotationally-symmetric lens system shown in Fig. 3 and compared to the Buchdahl coefficients. The object is positioned at ${\textrm{P}_{0\textrm{z}}} = - 200$ with respect to the vertex of surface 1, the image is located at the Gaussian image plane, the maximum radius of the entrance pupil is ${\rho_{\max }} = 21$, and the object height is 17. Equations (22)–(28) are evaluated by using the ray defined as ${\bar{\textrm{X}}_{\textrm{aber/optical}\,{\textrm{axis}}}} = {[{0\;\;0\;\;{0}} ]^\textrm{T}}$ [Eq. (8)] that requires the terms ${{{\alpha _0}} \mathord{\left/ {\vphantom {{{\alpha_0}} {{\textrm{x}_\textrm{a}}}}}\right. } {{\textrm{x}_\textrm{a}}}}$, ${{{\beta _0}} \mathord{\left/ {\vphantom {{{\beta_0}} {{\textrm{y}_\textrm{a}}}}} \right.} {{\textrm{y}_\textrm{a}}}}$, ${{{\beta _{0/\textrm{prin}}}} \mathord{\left/ {\vphantom {{{\beta_{0/\textrm{prin}}}} {{\textrm{y}_\textrm{a}}}}} \right.} {{\textrm{y}_\textrm{a}}}}$, ${{{\beta _{0/\textrm{prin}}}} \mathord{\left/ {\vphantom {{{\beta_{0/\textrm{prin}}}} {{\textrm{h}_0}}}} \right.} {{\textrm{h}_0}}}$, and ${{{\alpha _0}} \mathord{\left/ {\vphantom {{{\alpha_0}} {{\textrm{h}_0}}}} \right.} {{\textrm{h}_0}}}$ which also have to be determined for this specific ray. A ray intercepts the entrance pupil at $({{\textrm{x}_\textrm{a}},{\textrm{y}_\textrm{a}}} )$ that originates from ${\bar{\textrm P}_0} = {[{0\;\;{\textrm{h}_0}\;\;{-\ 200\ 1}} ]^\textrm{T}}$ with direction ${\bar{\ell }_0} = {[{\textrm{S}{\alpha_0}\textrm{C}{\beta_0}\;\;{\textrm{S}}{\beta_0}\;\;\textrm{C}{\alpha_0}\textrm{S}{\beta_0}\;\;{0}} ]^\textrm{T}}$. Since the entrance pupil is located at ${\textrm{d}_{\textrm{entrance}}} = 22.103275$, measured from the vertex of surface 1, the intercept point $({{\textrm{x}_\textrm{a}},{\textrm{y}_\textrm{a}}} )$ is given by [23],

3.2.1 Defocus aberration coefficient and transverse magnification

When the object is positioned at ${\textrm{P}_{0\textrm{z}}} = - 200$ and the image is located at the Gaussian image plane $({{\textrm{d}_{8/\textrm{Gaussian}}} = 92.088474} )$, the values of the defocus aberration coefficient ${\textrm{A}_1}$ from Eq. (22) and the transverse magnification ${\textrm{A}_2}$ from Eq. (23) are ${\textrm{A}_1} = 0$ and ${\textrm{A}_2} = - 0.66041336$. Numerical evaluation also shows that when the image plane is defocused from the Gaussian image plane by for example, ${\textrm{d}_8} = 85.088474$, then ${\textrm{A}_1} = - 4.7722950 \times {10^{ - 2}}$ and ${\textrm{A}_2} = - 0.62517748$.

3.2.2 Spherical aberration coefficient

Primary spherical aberration is the only third-order term that (i) does not depend on ${\textrm{h}_0}$, (ii) is an axial aberration, and (iii) has a constant value over the entire field-of-view. ${\textrm{B}_1}$ can be determined using Eq. (24b) by considering only meridional rays and by setting ${\alpha _0} = {\textrm{x}_\textrm{a}} = 0$ to give

3.2.3 Coma coefficient

Coma is defined as a variation in magnification from one zone to another zone in the entrance pupil. The new method coma coefficient ${\textrm{B}_2}$ can be determined by using Eqs. (25a) or (25b) and real pseudoparaxial principal rays. The two ${\textrm{B}_2}$ formulations each utilize two principal rays, viz., one ray travelling along optical axis and the other principal ray having $\Delta {\textrm{h}_0} = {10^{ - 6}},$ or other very small value. Now, the needed third-order partial derivatives are calculated to be

3.2.4 Astigmatism coefficients

The new coefficients ${\textrm{B}_3} + {\textrm{B}_4}$ [Eq. (26)] and $3{\textrm{B}_3} + {\textrm{B}_4}$ [Eq. (27)] are related to the (SAST) sagittal and (TAST) tangential astigmatism [Eq. (2a) and Eq. (2b)], respectively. Evaluation of Eqs. (26) and (27) requires Eqs. (29a), (29c), (30b), (31c), (31d), (31f) and terms in Table 4. The values of ${{{\beta _0}} \mathord{\left/ {\vphantom {{{\beta_0}} {{\textrm{h}_0}}}} \right.} {{\textrm{h}_0}}}$ and ${{{\beta _{0/\textrm{prin}}}} \mathord{\left/ {\vphantom {{{\beta_{0/\textrm{prin}}}} {{\textrm{h}_0}}}} \right.} {{\textrm{h}_0}}}$ in Table 4 can be determined by taking the partial derivative with respect to ${\textrm{h}_0}$ of Eqs. (29a) and (29c), respectively, and then evaluate them by the ray defined by ${\bar{\textrm{X}}_{\textrm{aber/optical}\;\;{\textrm{axis}}}} = {[{0\;\;0\;\;{0}} ]^\textrm{T}}$. It follows that ${{\partial {\beta _0}} \mathord{\left/ {\vphantom {{\partial {\beta_0}} {\partial {\textrm{h}_0} = }}} \right.} {\partial {\textrm{h}_0} = }}{{{\beta _0}} \mathord{\left/ {\vphantom {{{\beta_0}} {{\textrm{h}_0}}}} \right.} {{\textrm{h}_0}}}$ and ${{\partial {\beta _{0/\textrm{prin}}}} \mathord{\left/ {\vphantom {{\partial {\beta_{0/\textrm{prin}}}} {\partial {\textrm{h}_0} = }}} \right.} {\partial {\textrm{h}_0} = }}{{{\beta _{0/\textrm{prin}}}} \mathord{\left/ {\vphantom {{{\beta_{0/\textrm{prin}}}} {{\textrm{h}_0}}}} \right.} {{\textrm{h}_0}}}$ in the neighborhood of the optical axis. The evaluation results of Eqs. (26) and (27) are

Table 4. Terms necessary to compute B₃ and B₄.

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Solving for ${\textrm{B}_3}$ yields that ${\textrm{B}_3} = 1.34525828 \times {10^{ - 6}}$. The unconverted Buchdahl transverse astigmatism coefficient is ${\sigma _3} = 2.337794 \times {10^{ - 3}}$. The contribution to the sagittal astigmatism due to ${\textrm{B}_3}$ and ${\sigma _3}$ are ${\textrm{B}_3}\textrm{h}_0^2{\rho _{\textrm{a}/\max }} = 8.164373 \times {10^{ - 3}}$ and ${{{\sigma _3}} \mathord{\left/ {\vphantom {{{\sigma_3}} {({ - 2{n_8}{u_8}} )= 8.164434 \times {{10}^{ - 3}}}}} \right.} {({ - 2{{n}_8}{{u}_8}} )= 8.164434 \times {{10}^{ - 3}}}}$. It is evident that the new method ${\textrm{B}_3}$ and the Buchdahl method yield the same result recognizing that, in general, ${\textrm{B}_3} = {{{\sigma _3}} \mathord{\left/ {\vphantom {{{\sigma_3}} {({ - 2h_0^2{\rho_{\textrm{a}/\max }}{n_{\textrm{final}}}{u_{\textrm{final}}}} )}}} \right.} {({ - 2{\textrm{h}}_0^2{\rho_{\textrm{a}/\max }}{n_{\textrm{final}}}{u_{\textrm{final}}}} )}}$. An alternative direct solution for ${\textrm{B}_3}$ is given by

3.2.5 Field or Petzval curvature coefficient

The Petzval or field curvature coefficient is determined from Eqs. (33) and (34) by solving for ${\textrm{B}_4}$ thereby yielding that ${\textrm{B}_4} = - 5.53941436 \times {10^{ - 6}}$. The unconverted Buchdahl Petzval coefficient is ${\sigma _4} = - 9.626366 \times {10^{ - 3}}$. The contribution to the sagittal and tangential astigmatism (SAST and TAST, respectively) due to ${\textrm{B}_4}$ and ${\sigma _4}$ are ${\textrm{B}_4}\textrm{h}_0^2{\rho_{\textrm{a}/\max }} = - 3.361871 \times {10^{ - 2}}$ and ${{{\sigma _4}} \mathord{\left/ {\vphantom {{{\sigma_4}} {({ - 2{{n}_8}{{u}_8}} )= - 3.361880 \times {{10}^{ - 2}}}}} \right.} {({ - 2{{n}_8}{{u}_8}} )= - 3.361880 \times {{10}^{ - 2}}}}$. It is evident that the new method ${\textrm{B}_4}$ and the Buchdahl method produce the same results recognizing that in general ${\textrm{B}_4} = {{{\sigma _4}} \mathord{\left/ {\vphantom {{{\sigma_4}} {({ - 2h_0^2{\rho_{\textrm{a}/\max }}{n_{\textrm{final}}}{u_{\textrm{final}}}} )}}} \right.} {({ - 2{\textrm{h}}_0^2{\rho_{\textrm{a}/\max }}{n_{\textrm{final}}}{u_{\textrm{final}}}} )}}$. An alternative solution for ${\textrm{B}_4}$ is given by

3.2.6 Distortion coefficient

Determination of the distortion coefficient ${\textrm{B}_5}$ requires the following third-order derivatives of the ray defined by ${\bar{\textrm{X}}_{\textrm{aber/optical}\,{\textrm{axis}}}} = {[{0\;\;0\;\;{0}} ]^\textrm{T}}$.

4. Conclusion

Since Seidel published his method to compute the five third-order aberrations over 150 years ago, a number of others, such as Conrady [28], Herzberger [29], Welford [4], Cox [30], Brouwer [31], Luneburg [32], Slyusarev [33], and Buchdahl [3], have created alternative approaches to compute these aberrations. Some are based upon polynomial expansion of the wave front, Characteristic Function, or ray trace intercepts. Buchdahl took a distinctly different path and created the aberration terms by expansion of the quasi-invariant he invented following the Lagrange invariant. From this, he formulated a process to generate aberration coefficients of any order although ninth order was the practical limit due to the enormous algebraic complexity. Buchdahl generated unconverted aberration coefficients that can be easily converted into transverse, longitudinal, or wave aberrations. His coefficients were computed surface by surface and then summed to obtain the system aberrations. In addition, like most others, he normalized the entrance pupil radius and the object height so that the coefficients themselves indicated the aberration at full aperture and object height. Some of the other researchers approaches yielded only the system aberrations.

The new method to compute the Seidel aberrations presented in this paper is arguably unique since the aberrations are determined by tracing real pseudoparaxial skew rays and computing the Jacobian, Hessian, and third-order derivative matrices numerically. The aberration coefficients determined by the new method have been shown to be associated with a unit entrance pupil radius and unit object height, and the relationship between these coefficients and the Buchdahl coefficients were presented. Unlike the Buchdahl method that determines the aberration coefficients surface by surface and then sums to obtain the system aberration coefficients, the new method provides only the resultant system aberration coefficients. While Buchdahl’s approach requires evaluation of a large number of complex algebraic equations that grows rapidly as the aberration order increases, the new method requires the direct determination of a number of derivatives by differential ray tracing and numerical computations. The accuracy of the derivative matrices is a function of the number of bits comprising a ray trace datum (word size) and the size of parametric differentials (e.g., $\Delta {\textrm{h}_0}$) used in computing the derivatives. As demonstrated in this paper, the accuracy of the Seidel aberration coefficients using the new method is equivalent to the Buchdahl method when proper computation of the derivative matrices is performed. Once the derivative matrices are generated, determination of the aberration coefficients is straightforward, and not computationally intensive, by evaluation of Eqs. (22)–(28).

Several of the aforementioned researchers determined the fifth-order aberration coefficients that required significant algebraic effort in their development. The evaluation of these aberration coefficients is also substantial. The nature of the new method to use numerically-determined derivatives to generate the information needed to compute the Seidel aberration coefficients appears to offer the opportunity to compute higher-order aberration coefficients by generating higher-order partial derivative matrices. It is anticipated that the new method will provide a more rapid evaluation of the system aberrations in rotationally-symmetric optical systems. Future plans include the development of the equations for the fifth-order aberration coefficients.