By introducing a transformed pupil vector into the aberration expansions of an axially symmetric optical system, the aberration coefficients through third order of a pupil decentered off-axis optical system are obtained. Nodal aberration characteristics are revealed only by means of the pupil decentration vector and the aberration coefficients of the axially symmetric system, which shows great convenience since parameters of individual surface such as radius of curvature, decenter as well as the shifted center of the aberration field are not used in the analysis.
©2012 Optical Society of America
The theory about aberration fields of nonsymmetric optical systems was developed by Shack  and Thompson [2,3] in 1970s, based on the work of Buchroeder . In a nonsymmetric optical system, third-order aberrations are still the sum of the individual surface contributions just like in an axially symmetric system . Each surface has its shifted center of the aberration field from the perturbed Gaussian image center, which is often denoted by a vector. The total field of each aberration type is a function of the vector and the aberration coefficient of individual surfaces. This theory is applicable to all types of off-axis optical systems.
In reflective optical systems, a principal application of off-axis formation is to avoid the obstruction of the primary aperture by the secondary optical element. One of the off axis methods is to decenter the pupil. Although aberrations of such systems can undoubtedly be analyzed with the available vectorial aberration theory, it relies on the detailed parameters of all the optical elements. In this paper we describe the third-order aberration fields of pupil decentered optical systems by considering these optical elements as a whole excluding the pupil (equally the aperture stop). The aberration fields are thus independent of the parameters of individual elements.
2. Vector-form aberration expansion of pupil decentered systems
The wave aberration expansion of surfacein a vector form can be written as :
When it comes to the problem in this paper, only the pupil is decentered while the other elements hold their common axis. As shown in Fig. 1 , the relation between the new pupil coordinate and the old one is
The total aberration field is simply the sum of individual surfaces:5], is used:
Because the vector is independent of surface indices, the aberration coefficients expansion in Eq. (5) can be further arranged. The aberrations of individual surfaces can be added together directly, which is the same state as in an axially symmetric system.Eq. (7) is induced by one type of optical aberration, in sequence as defocus, tilt, spherical, coma, astigmatism, field curvature and distortion.
3. Aberration function and field characteristics
The aberration coefficients can be grouped according to their dependence on the power of the aperture vector as in . In this way, the aberration expansion in Eq. (7) for a pupil decentered system can be grouped as:
A similar result was also given by Shack in his course notes . Because piston is not a true optical aberration and it’s often neglected in aberration analysis, attention will be paid to the five monochromatic optical aberrations, especially to coma and astigmatism.
3.1 Spherical aberration
The first item in Eq. (8) is third-order spherical aberration. It can be seen when an optical system becomes off axis by decentering the pupil, spherical aberration doesn’t change. This can also be concluded from the available theory by Thompson .
The second group is third-order coma.
From Eq. (9) we can find an interesting property about third-order coma. When a symmetric system does not have spherical aberration, coma will not change with the decentration of the pupil. This is just like the coma property of a spherical-aberration-free system when the aperture stop is shifted axially. When the symmetric system is coma-free but has residual spherical aberration, the system with pupil decentration will demonstrate a constant coma, and the coma is linearly dependent on the pupil decentration magnitude. When neither nor equals 0, Eq. (9) can be described as:
Define a vector
Equation (12) gives the usual characteristic field behavior of coma in a pupil decentered optical system. A node exists away from the center of the Gaussian image, and the displacement relates to the ratio of spherical aberration and coma of the original symmetric system, as well as the pupil decentration magnitude. According to Eq. (11), the coma node lies on the line along the vector direction of . The nodal characteristic of coma field for a pupil decentered system is shown in Fig. 2 , and the corresponding full field map is shown in Fig. 3 .
The third group stands for astigmatism.
It is shown that astigmatism, coma and spherical aberration in the rotationally symmetric system induce second-order, first-order and constant items of astigmatism in the new nonsymmetric system, respectively. A special condition is considered at first. When the rotationally symmetric system is astigmatism-free, then after decentering the pupil, linear or constant astigmatism appears in the new system. The concrete astigmatism field form depends on the magnitude of spherical aberration, coma and pupil decentration.
When, Eq. (13) leads to:
DefineEq. (14) is written as:
Equation (17) reveals there are two nodal points for third-order astigmatism field
From Eq. (16) and Eq. (18), the locations of the two nodal points depend on the sign of the item. They may be located on the line along the direction of vectorwhen , or at two points that are symmetric about the line when ,as shown in Fig. 4 . The corresponding full field maps which illustrate these two conditions are shown in Fig. 5 .
3.4 Defocus and field curvature
The fourth group stands for defocus and field curvature.
This also brings the same conclusion in , i.e. there is a longitudinal focal shift apart from the decenter of the vertex of the quadratic medial surface.
The fifth group is the distortion item.
Since distortion doesn’t affect the imaging quality and can often be corrected via image procession, it is not discussed here. But from Eq. (20), it can inferred that distortion field has a three-node characteristic.
We demonstrate the aberration fields of pupil decentered optical systems through third order in this paper. By describing the off axis factor with a pupil decentration vector in the aberrations expansion, the optical system excluding the pupil is manipulated as a whole. The aberration coefficients we get do not contain parameters of individual surfaces, and they are only functions of the pupil decentration vector and the system aberration coefficients of the rotationally symmetric system, so the aberration field of a pupil decentered optical system can be inferred from the original rotationally symmetric form. This work and its follow-up into higher-order optical aberrations are applicable to the design and analysis of off-axis systems that are formed by pupil decentration.
The work was supported by the Changjitu Special Cooperation Foundation of Jilin Province and Chinese Academy of Sciences (grants 2011CJT0004).
References and links
1. R. V. Shack, “Aberration theory, OPTI 514 course notes,” College of Optical Sciences, University of Arizona, Tucson, Arizona.
2. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).
4. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).
5. H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon Press, 1950).