## Abstract

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.

## 1. Introduction

The theory about aberration fields of nonsymmetric optical systems was developed by Shack [1] and Thompson [2,3] in 1970s, based on the work of Buchroeder [4]. In a nonsymmetric optical system, third-order aberrations are still the sum of the individual surface contributions just like in an axially symmetric system [4]. Each surface has its shifted center of the aberration field from the perturbed Gaussian image center, which is often denoted by a vector$σ→j$. The total field of each aberration type is a function of the vector $σ→j$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 surface$j$in a vector form can be written as [3]:

$Wj=∑p∞∑n∞∑m∞(Wklm)j(H→⋅H→)p(ρ→⋅ρ→)n(H→⋅ρ→)m ,k=2p+m, l=2n+m ,$
where $H→$is the normalized field vector and $ρ→$is the normalized aperture vector. The indices $p$, $m$, $n$ are integer numbers, $k$ and $l$ are the power of $H→$and $ρ→$, respectively.

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

$ρ→′=ρ→+s→ ,$
where $s→$is the normalized pupil decentration vector. For simplicity but without loss of generality, we make $s→$in the y axis direction. The vector-form aberration expansion for a pupil decentered system can be modified as:

Fig. 1 Pupil vector relation before and after pupil decentration.

$Wj=∑p∞∑n∞∑m∞(Wklm)j(H→⋅H→)p[(ρ→+s→)⋅(ρ→+s→)]n[H→⋅(ρ→+s→)]m .$

The total aberration field is simply the sum of individual surfaces:

$W=∑jWj .$

According to Eq. (3) and Eq. (4), the aberration function of a pupil decentered system through third order is given as:

$W=∑jW020j(ρ→+s→)⋅(ρ→+s→)+∑jW111jH→⋅(ρ→+s→)+∑jW040j[(ρ→+s→)⋅(ρ→+s→)]2 +∑jW131j[H→⋅(ρ→+s→)][(ρ→+s→)⋅(ρ→+s→)]+12∑jW222j[H→2⋅(ρ→+s→)2] +∑jW220Mj(H→⋅H→)[(ρ→+s→)⋅(ρ→+s→)]+∑jW311j(H→⋅H→)[H→⋅(ρ→+s→)] ,$
in which $W220M$, the medial astigmatic component proposed by Hopkins [5], is used:

$W220M=W220+12W222 .$

Because the vector $s→$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.

Expand Eq. (5) by using the vector multiplication operation [3], and the aberration function changes to:

$W=W020(ρ→⋅ρ→)+2W020(ρ→⋅s→)+W020s2 +W111(H→⋅ρ→)+W111(H→⋅s→) +[W040(ρ→⋅ρ→)2+W040s4+2W040s→2⋅ρ→2+4W040s2(ρ→⋅ρ→) +4W040(s→⋅ρ→)(ρ→⋅ρ→)+4W040s2(s→⋅ρ→)] +[W131(H→⋅ρ→)(ρ→⋅ρ→)+2W131s2(H→⋅ρ→)+W131(s→H→⋅ρ→2)+2W131(s→⋅H→)(ρ→⋅ρ→)+W131s2(s→⋅H→)+W131(s→2H→∗⋅ρ→)] +12W222(H→2⋅ρ→2)+12W222(H→2⋅s→2)+W222(H→2s→∗⋅ρ→) +W220M(H→⋅H→)(ρ→⋅ρ→)+W220Ms2(H→⋅H→)+2W220M(H→⋅H→)(s→⋅ρ→) +W311(H→⋅H→)(H→⋅ρ→)+W311(H→⋅H→)(H→⋅s→) ,$
where $s2=s→⋅s→$ is a constant scalar caused by pupil decentration, $s→2=s→s→$ is a square vector manipulated by vector multiplication operation, $Wklm=∑j(Wklm)j$ is the aberration coefficient of the optical system with axial symmetry. Each aberration group listed in 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 [6]. In this way, the aberration expansion in Eq. (7) for a pupil decentered system can be grouped as:

$W=W040(ρ→⋅ρ→)2 +(4W040s→+W131H→)⋅ρ→(ρ→⋅ρ→) +(12W222H→2+W131s→H→+2W040s→2)⋅ρ→2 +[W220M(H→⋅H→)+2W131(s→⋅H→)+(W020+4W040s2 ) ](ρ→⋅ρ→) +[2W020s→ +W111H→+4W040s2s→+2W131s2H→ +W131s→2H→∗ +W222H→2s→∗+2W220M(H→⋅H→)s→+W311(H→⋅H→)H→]⋅ρ→ + [W020s2+W111(H→⋅s→)+W040s4 +W131s2(s→⋅H→)+12W222(H→2⋅s→2 )+W220Ms2(H→⋅H→)+W311(H→⋅H→)(H→⋅s→)] .$

A similar result was also given by Shack in his course notes [1]. 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 [3].

#### 3.2 Coma

The second group is third-order coma.

$W=(4W040s→+W131H→)⋅ρ→(ρ→⋅ρ→) .$

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 $W040$nor $W131$equals 0, Eq. (9) can be described as:

$W=W131(H→+4W040W131s→)⋅ρ→(ρ→⋅ρ→) .$

Define a vector

$a→c≡−4W040W131s→ ,$
and third-order coma can be shown as

$W=W131(H→−a→c)⋅ρ→(ρ→⋅ρ→) .$

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 $s→$. 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 .

Fig. 2 The nodal characteristic of coma field for a pupil decentered optical system: the node lies on the line along the vector direction of pupil decentration.

Fig. 3 Full field display of a pupil decentered system for third-order coma.

#### 3.3 Astigmatism

The third group stands for astigmatism.

$W=(12W222H→2+W131s→H→+2W040s→2)⋅ρ→2 .$

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$W222≠0$, Eq. (13) leads to:

$W =12W222(H→2+2W131s→W222H→+4W040s→2W222)⋅ρ→2 =12W222[(H→+W131s→W222)2−(W1312−4W040W222)s→2W2222]⋅ρ→2 .$

Define

$a→a≡−W131s→W222 ,$
and
$b→a≡W1312−4W040W222W2222s→ ,$
then third-order astigmatism in Eq. (14) is written as:

$W= 12W222[(H→−a→a)2−b→a2]⋅ρ→2 .$

Equation (17) reveals there are two nodal points for third-order astigmatism field

$H→=a→a±b→a .$

From Eq. (16) and Eq. (18), the locations of the two nodal points depend on the sign of the item$(W1312−4W040W222)$. They may be located on the line along the direction of vector$s→$when $(W1312−4W040W222)>0$, or at two points that are symmetric about the line when $(W1312−4W040W222)<0$,as shown in Fig. 4 . The corresponding full field maps which illustrate these two conditions are shown in Fig. 5 .

Fig. 4 Locations of the two nodal points of astigmatism field in a pupil decentered optical system. (a) On the line along the direction of pupil decentration vector. (b) Symmetric about the pupil vector direction line.

Fig. 5 Full field maps of two pupil decentered systems which correspond to the two different nodal positions as shown in Fig. 4 (a) and (b).

#### 3.4 Defocus and field curvature

The fourth group stands for defocus and field curvature.

$W=[W220M(H→⋅H→)+2W131(s→⋅H→)+(W020+4W040s2 ) ](ρ→⋅ρ→) =W220M[(H→⋅H→)+2W131W220M(s→⋅H→)+(W020+4W040s2 )W220M ](ρ→⋅ρ→) =W220M[(H→+W131W220Ms→)⋅(H→+W131W220Ms→)+W020W220M+4W040W220Ms2−W1312s2W220M2 ](ρ→⋅ρ→) .$

This also brings the same conclusion in [3], i.e. there is a longitudinal focal shift apart from the decenter of the vertex of the quadratic medial surface.

#### 3.5 Distortion

The fifth group is the distortion item.

$W=[2W020s→ +W011H→+4W040s2s→+2W131s2H→ +W131s→2H→∗ +W222H→2s→∗+2W220M(H→⋅H→)s→+W311(H→⋅H→)H→]⋅ρ→ .$

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.

## 4. Conclusions

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.

## Acknowledgments

The work was supported by the Changjitu Special Cooperation Foundation of Jilin Province and Chinese Academy of Sciences (grants 2011CJT0004).

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).

3. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005). [CrossRef]   [PubMed]

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).

6. L. B. Moore, A. M. Hvisc, and J. Sasian, “Aberration fields of a combination of plane symmetric systems,” Opt. Express 16(20), 15655–15670 (2008). [CrossRef]   [PubMed]

### References

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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).
3. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005).
[Crossref] [PubMed]
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).
6. L. B. Moore, A. M. Hvisc, and J. Sasian, “Aberration fields of a combination of plane symmetric systems,” Opt. Express 16(20), 15655–15670 (2008).
[Crossref] [PubMed]

#### Other (4)

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).

H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon Press, 1950).

R. V. Shack, “Aberration theory, OPTI 514 course notes,” College of Optical Sciences, University of Arizona, Tucson, Arizona.

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).

### Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

### Figures (5)

Fig. 1 Pupil vector relation before and after pupil decentration.
Fig. 2 The nodal characteristic of coma field for a pupil decentered optical system: the node lies on the line along the vector direction of pupil decentration.
Fig. 3 Full field display of a pupil decentered system for third-order coma.
Fig. 4 Locations of the two nodal points of astigmatism field in a pupil decentered optical system. (a) On the line along the direction of pupil decentration vector. (b) Symmetric about the pupil vector direction line.
Fig. 5 Full field maps of two pupil decentered systems which correspond to the two different nodal positions as shown in Fig. 4 (a) and (b).

### Equations (20)

$W j = ∑ p ∞ ∑ n ∞ ∑ m ∞ ( W klm ) j ( H → ⋅ H → ) p ( ρ → ⋅ ρ → ) n ( H → ⋅ ρ → ) m , k=2p+m, l=2n+m ,$
$ρ → ′ = ρ → + s → ,$
$W j = ∑ p ∞ ∑ n ∞ ∑ m ∞ ( W klm ) j ( H → ⋅ H → ) p [ ( ρ → + s → )⋅( ρ → + s → ) ] n [ H → ⋅( ρ → + s → ) ] m .$
$W= ∑ j W j .$
$W= ∑ j W 020j ( ρ → + s → )⋅( ρ → + s → )+ ∑ j W 111j H → ⋅( ρ → + s → )+ ∑ j W 040j [ ( ρ → + s → )⋅( ρ → + s → ) ] 2 + ∑ j W 131j [ H → ⋅( ρ → + s → ) ] [ ( ρ → + s → )⋅( ρ → + s → ) ]+ 1 2 ∑ j W 222j [ H → 2 ⋅ ( ρ → + s → ) 2 ] + ∑ j W 220 M j ( H → ⋅ H → ) [ ( ρ → + s → )⋅( ρ → + s → ) ]+ ∑ j W 311j ( H → ⋅ H → )[ H → ⋅( ρ → + s → ) ] ,$
$W 220 M = W 220 + 1 2 W 222 .$
$W= W 020 ( ρ → ⋅ ρ → )+2 W 020 ( ρ → ⋅ s → )+ W 020 s 2 + W 111 ( H → ⋅ ρ → )+ W 111 ( H → ⋅ s → ) +[ W 040 ( ρ → ⋅ ρ → ) 2 + W 040 s 4 +2 W 040 s → 2 ⋅ ρ → 2 +4 W 040 s 2 ( ρ → ⋅ ρ → ) +4 W 040 ( s → ⋅ ρ → )( ρ → ⋅ ρ → )+4 W 040 s 2 ( s → ⋅ ρ → ) ] +[ W 131 ( H → ⋅ ρ → )( ρ → ⋅ ρ → )+2 W 131 s 2 ( H → ⋅ ρ → )+ W 131 ( s → H → ⋅ ρ → 2 ) +2 W 131 ( s → ⋅ H → )( ρ → ⋅ ρ → )+ W 131 s 2 ( s → ⋅ H → )+ W 131 ( s → 2 H → ∗ ⋅ ρ → ) ] + 1 2 W 222 ( H → 2 ⋅ ρ → 2 )+ 1 2 W 222 ( H → 2 ⋅ s → 2 )+ W 222 ( H → 2 s → ∗ ⋅ ρ → ) + W 220 M ( H → ⋅ H → )( ρ → ⋅ ρ → )+ W 220 M s 2 ( H → ⋅ H → )+2 W 220 M ( H → ⋅ H → )( s → ⋅ ρ → ) + W 311 ( H → ⋅ H → )( H → ⋅ ρ → )+ W 311 ( H → ⋅ H → )( H → ⋅ s → ) ,$
$W= W 040 ( ρ → ⋅ ρ → ) 2 +( 4 W 040 s → + W 131 H → )⋅ ρ → ( ρ → ⋅ ρ → ) +( 1 2 W 222 H → 2 + W 131 s → H → +2 W 040 s → 2 )⋅ ρ → 2 +[ W 220 M ( H → ⋅ H → )+2 W 131 ( s → ⋅ H → )+( W 020 +4 W 040 s 2 ) ]( ρ → ⋅ ρ → ) +[ 2 W 020 s → + W 111 H → +4 W 040 s 2 s → +2 W 131 s 2 H → + W 131 s → 2 H → ∗ + W 222 H → 2 s → ∗ +2 W 220 M ( H → ⋅ H → ) s → + W 311 ( H → ⋅ H → ) H → ]⋅ ρ → + [ W 020 s 2 + W 111 ( H → ⋅ s → )+ W 040 s 4 + W 131 s 2 ( s → ⋅ H → ) + 1 2 W 222 ( H → 2 ⋅ s → 2 )+ W 220 M s 2 ( H → ⋅ H → )+ W 311 ( H → ⋅ H → )( H → ⋅ s → ) ] .$
$W=( 4 W 040 s → + W 131 H → )⋅ ρ → ( ρ → ⋅ ρ → ) .$
$W= W 131 ( H → + 4 W 040 W 131 s → )⋅ ρ → ( ρ → ⋅ ρ → ) .$
$a → c ≡− 4 W 040 W 131 s → ,$
$W= W 131 ( H → − a → c )⋅ ρ → ( ρ → ⋅ ρ → ) .$
$W=( 1 2 W 222 H → 2 + W 131 s → H → +2 W 040 s → 2 )⋅ ρ → 2 .$
$W = 1 2 W 222 ( H → 2 + 2 W 131 s → W 222 H → + 4 W 040 s → 2 W 222 )⋅ ρ → 2 = 1 2 W 222 [ ( H → + W 131 s → W 222 ) 2 − ( W 131 2 −4 W 040 W 222 ) s → 2 W 222 2 ]⋅ ρ → 2 .$
$a → a ≡− W 131 s → W 222 ,$
$b → a ≡ W 131 2 −4 W 040 W 222 W 222 2 s → ,$
$W= 1 2 W 222 [ ( H → − a → a ) 2 − b → a 2 ]⋅ ρ → 2 .$
$H → = a → a ± b → a .$
$W=[ W 220 M ( H → ⋅ H → )+2 W 131 ( s → ⋅ H → )+( W 020 +4 W 040 s 2 ) ]( ρ → ⋅ ρ → ) = W 220 M [ ( H → ⋅ H → )+ 2 W 131 W 220 M ( s → ⋅ H → )+ ( W 020 +4 W 040 s 2 ) W 220 M ]( ρ → ⋅ ρ → ) = W 220 M [ ( H → + W 131 W 220 M s → )⋅( H → + W 131 W 220 M s → )+ W 020 W 220 M +4 W 040 W 220 M s 2 − W 131 2 s 2 W 220 M 2 ]( ρ → ⋅ ρ → ) .$
$W=[ 2 W 020 s → + W 011 H → +4 W 040 s 2 s → +2 W 131 s 2 H → + W 131 s → 2 H → ∗ + W 222 H → 2 s → ∗ +2 W 220 M ( H → ⋅ H → ) s → + W 311 ( H → ⋅ H → ) H → ]⋅ ρ → .$