Interaction of pupil offset and fifth-order nodal aberration field properties in rotationally symmetric telescopes

Haili Hu; Jianjun Liu; Zhigang Fan

doi:10.1364/OE.21.017986

Optics Express
Vol. 21,
Issue 15,
pp. 17986-17998
(2013)
•https://doi.org/10.1364/OE.21.017986

Interaction of pupil offset and fifth-order nodal aberration field properties in rotationally symmetric telescopes

Haili Hu, Jianjun Liu, and Zhigang Fan

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Haili Hu, Jianjun Liu, and Zhigang Fan, "Interaction of pupil offset and fifth-order nodal aberration field properties in rotationally symmetric telescopes," Opt. Express 21, 17986-17998 (2013)

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About this Article
History
- Original Manuscript: April 19, 2013
- Revised Manuscript: June 23, 2013
- Manuscript Accepted: July 16, 2013
- Published: July 19, 2013

Abstract

In this paper we succeeded in deriving changes in the nodal positions of aberrations that belong to the fifth-order class in pupil dependence by applying a system level pupil decentration vector. Our treatment is specifically for rotationally symmetric multi-mirror optical designs that simply use an offset pupil as a means of creating an unobscured optical design. When the pupil is offset, only the vectors to determine the node locations are modified by the pupil decentration vector, while the nodal properties originally developed for titled/decentered optical systems are retained. In general, the modifications to the nodal vectors for any particular aberration type are contributed only by terms of higher order pupil dependence.

1. Introduction

Optical systems without rotational symmetry have been and will continue to be one of the hot topics in optical designs [1–7]. Among these optical systems, the pupil decentered configuration is proposed because the obstruction of primary aperture in telescopes can be avoided [5–7]. Aberration analysis is crucial for the optical design and fabrication, and significant research progresses have been made for optical systems without rotational symmetry [2–6, 8, 9]. The third-order aberration of the titled and decentered optical systems could be determined ether by ray-tracing or vector methods [5, 6]. The fifth-order aberration plays same role as third-order one in the optical design [9, 10]. Based on all elements’ parameters of optical systems, the multinodal property of the fifth-order aberrations was investigated by Thompson in [9]. Further, for some convenience applications, the pupil decentration vector was introduced to analyze the third-order aberration fields in pupil-decentered optical system [6]. However, this approach didn’t continue to apply the pupil decentration vector to analyze the fifth-order aberrations field property.

In this paper, we employed the pupil decentration vector to examine the impact of the pupil offset on the fifth-order nodal aberration field in pupil decentered optical systems. With the help of pupil decentration vector, the optical elements are considered as a whole but the pupil, and the fifth-order aberration nodal fields are independent of the parameters of individual lens. It is found that when the pupil is offset, the vectors to determine the node locations are modified by the pupil decentration vector and aberration coefficients.

2. Vector-form of aberration expansion in pupil decentered optical systems

2.1 Vector-form of wavefront aberration expansion

For centered rotationally symmetric optical system, wavefront aberrations are described by the scalar form of wave aberration expansion, which was proposed by Hopkins [11]. The wave aberration of surface j in an optical system with rotational symmetry is given by:

\begin{matrix} W_{j} (H, ρ, ρ \cos θ) = \sum_{p}^{\infty} \sum_{n}^{\infty} \sum_{m}^{\infty} {(W_{klm})}_{j} {(H^{2})}^{p} {(ρ^{2})}^{n} {(H ρ \cos θ)}^{m} \\ = \sum_{p}^{\infty} \sum_{n}^{\infty} \sum_{m}^{\infty} {(W_{klm})}_{j} H^{2 p + m} ρ^{2 n + m} \cos^{m} θ, \end{matrix}

where

\vec{ρ}

is the pupil vector with ρ_x and ρ_y along X and Y axis in the entrance/exit pupil,

\vec{H}

represents a field vector in the image plane with H_x and H_y along X and Y in the field coordinate system. The m, n, and p are positive integers. Terms k = 2p + m and l = 2n + m are exponents of field vector and pupil vector, respectively.

The pupil and field vector are depicted in Fig. 1. A left-hand coordinate system is employed, which follows the tradition in most commercial optical design software.

Fig. 1 Definitions of pupil vector $\vec{ρ}$ and field vector $\vec{H}$ .

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To analyze optical path differences between the ideal and arbitrary rays, the pupil and field vectors are expressed in forms of:

\vec{ρ} = | \vec{ρ} | \exp (i ϕ) = ρ_{x} x + ρ_{y} y,

\vec{H} = | \vec{H} | \exp (i φ) = H_{x} x + H_{y} y,

where

ϕ

and φ are angles measured in the pupil and field from the positive Y axis (in XOY plane), respectively.

To express the wave aberration in vector form, the field and the pupil position coordinate systems are converted to vectors and wave aberration expansion in vector form of the surface j can be written as [10–12]:

W = W_{j} [(\vec{H} \cdot \vec{H}), (\vec{ρ} \cdot \vec{ρ}), (\vec{H} \cdot \vec{ρ})] = \sum_{j} \sum_{p}^{\infty} \sum_{n}^{\infty} \sum_{m}^{\infty} {(W_{klm})}_{j} {(\vec{H} \cdot \vec{H})}^{p} {(\vec{ρ} \cdot \vec{ρ})}^{n} {(\vec{H} \cdot \vec{ρ})}^{m} .

2.2 Vector multiplication

The concept of the vector multiplication was named by Shack, and described in detail by Thompson in [12]. The algorithm of the vector multiplication is that: multiplication result of two vectors is a new vector coplanar with original vectors, and its magnitude (angle) is the product (sum) of those of two vectors. We take vectors $\vec{A}$ and $\vec{B}$ as examples to figure out the vector multiplication. Vectors $\vec{A}$ and $\vec{B}$ are defined as:

\vec{A} = a \exp (i α) = a_{x} \vec{i} + a_{y} \vec{j}, a_{x} = a \sin α, a_{y} = a \cos α,

\vec{B} = a \exp (i α) = a_{x} \vec{i} + a_{y} \vec{j}, a_{x} = a \sin α, a_{y} = a \cos α .

As shown in Fig. 2, the vector multiplication $\vec{A} \vec{B}$ is expressed by

Fig. 2 Operation algorithm for the vector multiplication of $\vec{A}$ and $\vec{B}$ . $\vec{i}$ and $\vec{j}$ are the normalized vectors in XOY coordinate system.

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\begin{matrix} \vec{A} \vec{B} = a b \exp [i (α + β)] = (a_{x} a_{x} + a_{x} a_{x}) \vec{i} + (a_{x} a_{x} + a_{x} a_{x}) \vec{i} \\ = a b \sin (α + β) \vec{i} + a b \sin (α + β) \vec{i} . \end{matrix}

Specifically, when $\vec{A} = \vec{B}$ , the vector multiplication $\vec{A} \vec{B}$ becomes:

\vec{A} \overset{2}{} = a^{2} \exp (i 2 α) = {(\vec{A} \overset{2}{})}_{x} \vec{i} + {(\vec{A} \overset{2}{})}_{y} \vec{j},

where

{(\vec{A} \overset{2}{})}_{x} = a^{2} \sin 2 α = 2 a_{x} a_{y}

and

{(\vec{A} \overset{2}{})}_{y} = a^{2} \cos 2 α = a_{y}^{2} - a_{x}^{2}

There are some operation algorithms of vector multiplication which will be used in the following analysis

{(\vec{A} - \vec{B})}^{3} = \vec{A} \overset{3}{} - 3 \vec{A} \overset{2}{} \vec{B} + 3 \vec{A} \vec{B} \overset{2}{} - \vec{B} \overset{3}{},

2 (\vec{A} \cdot \vec{B}) (\vec{A} \cdot \vec{C}) = (\vec{A} \cdot \vec{A}) (\vec{B} \cdot \vec{C}) + \vec{A} \overset{2}{} \cdot \vec{B} \vec{C} .

2.3 Pupil decentration vector

Unobscured all-reflective systems that are comprised of rotationally symmetric components on a common optical axis have two classes, offset pupil stop and field biased and often as a combination of these two. The issue discussed in this paper is the pupil-decentered optical system in which only the pupil is decentered while all the other lenses are laying on a common axis. Because the movement of principle ray along axis won’t cause changes of image center, we take the original axis as the reference coordinate of the decentered pupil. Then, pupil decentration vector representing the offset distance and direction from the new pupil is employed in following analysis.

As plotted in Fig. 3, the relation between the new pupil vector $\vec{ρ'}$ and the original one $\vec{ρ}$ is [6]:

\vec{ρ'} = \vec{ρ} + \vec{S},

where

\vec{S}

is the pupil decentration vector normalized to the original pupil.

Fig. 3 Pupil vector conversion during the pupil offset.

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3. Impact of pupil offset on fifth-order nodal aberration fields

Generally, the fifth-order terms of wave aberration expansion for a symmetric optical system are composed by nine parts [9], i.e., the fifth-order spherical W₀₆₀, the fifth-order field-linear coma W₁₅₁, a component of the fifth-order oblique spherical W₂₄₀, the fifth-order oblique spherical W₂₄₂, a field-cubed dependent form of third-order coma W₃₃₁, a field-cubed dependent form of third-order elliptical coma W₃₃₃, a component of the fifth-order field curvature W₄₂₀, the fifth-order astigmatism W₄₂₂, and the fifth-order distortion W₅₁₁. The vector form aberration expression including all the fifth-order aberrations is described by following:

\begin{matrix} W_{Fifth} = \sum_{j} W_{060 j} {(\vec{ρ} \cdot \vec{ρ})}^{3} + \sum_{j} W_{151 j} (\vec{H} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + \sum_{j} W_{240 j} (\vec{H} \cdot \vec{H}) {(\vec{ρ} \cdot \vec{ρ})}^{2} \\ + \sum_{j} W_{242 j} {(\vec{H} \cdot \vec{ρ})}^{2} (\vec{ρ} \cdot \vec{ρ}) + \sum_{j} W_{331 j} (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) + \sum_{j} W_{333 j} {(\vec{H} \cdot \vec{ρ})}^{3} \\ + \sum_{j} W_{420 j} {(\vec{H} \cdot \vec{H})}^{2} (\vec{ρ} \cdot \vec{ρ}) + \sum_{j} W_{422 j} (\vec{H} \cdot \vec{H}) {(\vec{H} \cdot \vec{ρ})}^{2} + \sum_{j} W_{511 j} {(\vec{H} \cdot \vec{H})}^{2} (\vec{H} \cdot \vec{ρ}) . \end{matrix}

With the help of pupil decentration vector, the vector-form expression of the fifth-order aberrations in pupil-decentered optical system is expanded into

\begin{matrix} W_{Fifth}^{S} = \sum_{j} W_{060 j} {[(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})]}^{3} + \sum_{j} W_{151 j} [\vec{H} \cdot (\vec{ρ} + \vec{S})] {[(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})]}^{2} \\ + \sum_{j} W_{240 M j} (\vec{H} \cdot \vec{H}) {[(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})]}^{2} + \sum_{j} W_{242 j} {[\vec{H} \cdot (\vec{ρ} + \vec{S})]}^{2} [(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})] \\ + \sum_{j} W_{331 M j} (\vec{H} \cdot \vec{H}) [\vec{H} \cdot (\vec{ρ} + \vec{S})] [(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})] + \sum_{j} W_{420 M j} {(\vec{H} \cdot \vec{H})}^{2} [(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})] \\ + \sum_{j} W_{333 j} {[\vec{H} \cdot (\vec{ρ} + \vec{S})]}^{3} + \sum_{j} W_{422 j} {(\vec{H} \cdot \vec{H})}^{} {[\vec{H} \cdot (\vec{ρ} + \vec{S})]}^{2} + \sum_{j} W_{511 j} {(\vec{H} \cdot \vec{H})}^{2} [\vec{H} \cdot (\vec{ρ} + \vec{S})], \end{matrix}

in which W_240M, W_331M, and W_420M are the medial components [2]:

W_{240 M} = W_{240} + \frac{1}{2} W_{242},

W_{331 M} = W_{331} + \frac{3}{4} W_{333},

W_{420 M} = W_{420} + \frac{1}{2} W_{422},

where

W_{klm} = \sum_{j} {(W_{klm})}_{j}

is the aberration coefficient of the rotationally symmetric optical system. By using polynomial expansion, we expand every part of the fifth-order aberrations in Eq. (13). These expanded expressions are complex and the detail expressions are shown in the Appendix A.

Pupil decentration vector was found its convenience when dealing with the third-order aberrations of pupil-decentered optical systems [6]. However, due to long and complex formulas it is impractical to use the expanded expressions for computing aberration coefficients directly. Inspired by the division method proposed in [13], our strategy is to categorize and group the expanded expressions by same exponents of the pupil vector (l = 2n + m). As a result, all fifth-order aberration expansions in the pupil-decentered system are divided into two categories: the first part with l≤3, acting as the “third-order” aberrations; the other part with l>3, performing the property of the fifth-order aberrations.

In this paper, we focused on the fifth-order pupil dependence to derive the changes in the field nodal positions, while the third-order aberrations of the pupil-decentered system have been discussed in [6]. We discussed the fifth-order aberrations part with l>3 in expanded expressions of the Appendix A, including three different terms:

(1). the aberration term l = 4 W_l _{= 4}: including terms {·} $\vec{(ρ} \cdot {\vec{ρ)}}^{2}$ and {·} $\vec{ρ} \overset{2}{} (\vec{ρ} \cdot \vec{ρ)}$ ;
(2). the aberration term l = 5 W_l _{= 5}: {·} $\cdot \vec{ρ} {(\vec{ρ} \cdot \vec{ρ})}^{2}$ ;
(3). the aberration term l = 6 W_l _{= 6}: {·} ${(\vec{ρ} \cdot \vec{ρ})}^{3}$ .

3.1 Fifth-order aberration term when l = 4 {·} $\vec{(ρ} \cdot {\vec{ρ)}}^{2}$ and {·} $\vec{ρ} \overset{2}{} (\vec{ρ} \cdot \vec{ρ)}$

A. The term $W_{l = 4}^{(1)}$ = {·} $\vec{(ρ} \cdot {\vec{ρ)}}^{2}$

The aberration term $W_{l = 4}^{(1)}$ = {·} $\vec{(ρ} \cdot {\vec{ρ)}}^{2}$ includes all components of ‘ $\vec{(ρ} \cdot {\vec{ρ)}}^{2}$ ’ in the Appendix A, and it is expressed by:

\begin{matrix} W_{l = 4}^{(1)} = W_{240 M} (\vec{H} \cdot \vec{H}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + 3 W_{151} (\vec{H} \cdot \vec{S}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + 9 W_{060} (\vec{S} \cdot \vec{S}) {(\vec{ρ} \cdot \vec{ρ})}^{2} \\ = [W_{240 M} (\vec{H} \cdot \vec{H}) + 3 W_{151} (\vec{H} \cdot \vec{S}) + 9 W_{060} (\vec{S} \cdot \vec{S})] {(\vec{ρ} \cdot \vec{ρ})}^{2}, \end{matrix}

where the first summation

W_{240 M} (\vec{H} \cdot \vec{H}) = \sum_{j} {(W_{240 M})}_{j} (\vec{H} \cdot \vec{H})

is the oblique spherical contribution from optical system with rotational symmetric.

When $W_{240 M} \neq 0$ , we defined following vectors

\vec{A_{240 M_{S}}} = - 1.5 W_{151} \vec{S},

\vec{a_{240 M_{S}}} = \vec{A_{240 M_{S}}} / W_{240},

and scalars

B_{240 M_{S}} = 9 W_{060} (\vec{S} \cdot \vec{S}),

b_{240 M_{S}} = B_{{240M}_{S}} / W_{240 M} - \vec{a_{240 M_{S}}} \cdot \vec{a_{240 M_{S}}} .

Then, $W_{l = 4}^{(1)}$ can be expressed as:

W_{l = 4}^{(1)} = W_{240 M} [(\vec{H} - \vec{a_{240 M_{S}}}) \cdot (\vec{H} - \vec{a_{{240M}_{S}}}) + b_{2 {40M}_{S}}] {(\vec{ρ} \cdot \vec{ρ})}^{2} .

The term $W_{l = 4}^{(1)}$ = {·} ${(\vec{ρ} \cdot \vec{ρ})}^{2}$ performs the multinodal property, which retain the property of the aberration W_240M analyzed in [9]. The Eq. (22) also means that the nodal properties of the W_240M term are not affected by pupil offset fundamentally. The aberration field nodal positions are determined by $\vec{a_{240}}$ and $b_{240}$ directly. According to the Eqs. (18-21), the terms of fifth-order pupil dependence have the impact on the nodal positions of the aberration term $W_{l = 4}^{(1)}$ = {·} ${(\vec{ρ} \cdot \vec{ρ})}^{2}$ . The nodal aberration field property of $W_{l = 4}^{(1)}$ = {·} ${(\vec{ρ} \cdot \vec{ρ})}^{2}$ also confirm that the case of $\vec{S}$ = 0 represents the rotational symmetric aberration theory, i.e., the multinodal positions degenerate to overlap at the symmetric center of a system with rotational symmetry.

In general, by concerning tilted and decentered elements in pupil-offset optical systems, we can modify the standard nodal vectors in the Appendix C of [2] by introducing modification terms $\vec{A_{240 M_{S}}}$ and $B_{240 M_{S}}$ :

\vec{A_{240M}} = \vec{A_{240 M}} + \vec{A_{240 M_{S}}} = \vec{A_{240 M}} - 1.5 W_{151} \vec{S},

B_{240 M} = B_{240 M} + B_{240 M_{S}} = B_{240 M} + 9 W_{060} (\vec{S} \cdot \vec{S}),

where

\vec{A_{240M}}

and

B_{240 M}

are the unnormalized standard vectors [2], as defined in the Appendix B.

B . The term = {·}

The part of the fifth-order aberration $W_{l = 4}^{(2)}$ = {·} $\vec{ρ} \overset{2}{} (\vec{ρ} \cdot \vec{ρ)}$ includes all components of ‘ $\vec{ρ} \overset{2}{} (\vec{ρ} \cdot \vec{ρ)}$ ’ in the Appendix A:

\begin{matrix} W_{l = 4}^{(2)} = \frac{1}{2} W_{242} (\vec{H} \overset{2}{} \cdot \vec{ρ} \overset{2}{}) (\vec{ρ} \cdot \vec{ρ}) + 2 W_{151} (\vec{H} \vec{S} \cdot \vec{ρ} \overset{2}{}) (\vec{ρ} \cdot \vec{ρ}) + 6 W_{060} (\vec{S} \overset{2}{} \cdot \vec{ρ} \overset{2}{}) (\vec{ρ} \cdot \vec{ρ}) \\ = \frac{1}{2} (W_{242} \vec{H} \overset{2}{} + 4 W_{151} \vec{H} \vec{S} + 12 W_{060} \vec{S} \overset{2}{}) \cdot \vec{ρ} \overset{2}{} (\vec{ρ} \cdot \vec{ρ}) . \end{matrix}

By defining following vectors and scalars (when W₂₄₂≠0),

\vec{A_{242_{S}}} = - 2 W_{151} \vec{S},

{\vec{B}}_{242_{S}}^{2} = 12 W_{060} \vec{S} \overset{2}{},

\vec{a_{242_{S}}} = \vec{A_{242_{S}}} / W_{242},

{\vec{b}}_{242_{S}}^{2} = {\vec{B}}_{242_{S}}^{2} / W_{242} - {\vec{a}}_{242_{S}}^{2},

the term

W_{l = 4}^{(2)}

can be expressed by

\begin{matrix} W_{l = 4}^{(2)} = \frac{1}{2} (W_{242} \vec{H} \overset{2}{} - 2 \vec{H} \vec{A_{242_{S}}} + {\vec{B_{242_{S}}}}^{2}) \cdot \vec{ρ} \overset{2}{} (\vec{ρ} \cdot \vec{ρ}) \\ = \frac{1}{2} W_{242} [{(\vec{H} - \vec{a_{242_{S}}})}^{2} + {\vec{b_{242_{S}}}}^{2}] \cdot \vec{ρ} \overset{2}{} (\vec{ρ} \cdot \vec{ρ}) . \end{matrix}

As indicated in Fig. 4, the term $W_{l = 4}^{(2)}$ = {·} $\cdot {\vec{ρ}}^{2} (\vec{ρ} \cdot \vec{ρ)}$ performs the two-nodal property, acting as the fifth-order oblique spherical W₂₄₂ discussed in [9]. Two nodal points locate at $\vec{H} = \vec{a_{242 S}} + i \vec{b_{242 S}}$ and $\vec{H} = \vec{a_{242 S}} - i \vec{b_{242 S}}$ , which are determined by pupil decentration vector directly. This confirms that the nodal properties of the W₂₄₂ term are not affected by pupil offset but field nodal positions are changed. The vector $\vec{a_{242 S}}$ denotes the midpoint between the two field nodes and the vector $\vec{b_{242 S}}$ is related to the distance between the field nodes, as released by Fig. 4. For applications, the aberration term $W_{l = 4}^{(2)}$ is important for optical design, because the performance of wide field-of-view unobscured three- and four-mirror optical systems is often limited by oblique spherical aberration.

Fig. 4 Two nodal points of fifth-order aberration term $W_{l = 4}^{(2)}$ . Nodal positions locate at $\vec{H} = \vec{a_{242 S}} + i \vec{b_{242 S}}$ and $\vec{H} = \vec{a_{242 S}} - i \vec{b_{242 S}}$ , respectively.

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In general, by including tilted and decentered elements, the modification for standard nodal vectors of W₂₄₂ is to add the pupil-decentered terms to the Appendix C of [2], and the modifications are expressed by

\vec{A_{242}} = \vec{A_{242}} + \vec{A_{242_{S}}} = \vec{A_{242}} - 2 W_{151} \vec{S},

{\vec{B_{242}}}^{2} = {\vec{B_{242}}}^{2} + {\vec{B_{242_{S}}}}^{2} = {\vec{B_{242}}}^{2} + 9 W_{060} (\vec{S} \cdot \vec{S}) .

3.2 Fifth-order aberration when l = 5 Wl = 5 = {•} $\cdot \vec{ρ} {(\vec{ρ} \cdot \vec{ρ})}^{2}$

The part of the fifth-order aberration $W_{l = 5}^{}$ = {·} $\cdot \vec{ρ} {(\vec{ρ} \cdot \vec{ρ})}^{2}$ includes all components of ‘ $\vec{ρ} {(\vec{ρ} \cdot \vec{ρ})}^{2}$ ’ in the Appendix A, and it is expressed by:

\begin{matrix} W_{l = 5} = W_{151} (\vec{H} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + 6 W_{060} (\vec{S} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2} \\ = (W_{151} \vec{H} - \vec{A_{151_{S}}}) \cdot \vec{ρ} {(\vec{ρ} \cdot \vec{ρ})}^{2} \\ = W_{151} (\vec{H} - \vec{a_{151_{S}}}) \cdot \vec{ρ} {(\vec{ρ} \cdot \vec{ρ})}^{2}, \end{matrix}

where the term

W_{151} \vec{H} = \sum_{j} {(W_{151})}_{j} \vec{H}

indicates the contribution from aberrations of the system with rotational symmetric; the term

\vec{A_{151_{S}}} = - 6 W_{060} \vec{S}

is the contribution from pupil offset in the pupil plane weighted by the fifth-order aberration coefficient

W_{060}

, and the vector

\vec{a_{151_{S}}} = \vec{A_{151_{S}}} / W_{151}

(when W₁₅₁≠0) is the normalized vector of

\vec{A_{151_{S}}}

The part $W_{l = 5}^{}$ = {·} $\cdot \vec{ρ} {(\vec{ρ} \cdot \vec{ρ})}^{2}$ performed the one nodal act, as released in Fig. 5 and its field dependence property is same as the fifth-order field-linear coma. The nodal position of the term $W_{l = 5}^{}$ in the image field is only modified by the vector $\vec{a_{151_{S}}}$ , which depends linearly on field-of-view. This result demonstrates that field-linear coma field of the optical system without rotational symmetry is unchanged but decentered from the center of the image field, as demonstrated by Fig. 5.

Fig. 5 The aberration field property of term W_l _{= 5} for a pupil-decentered optical system. Fifth-order aberration term W_l _{= 5}is displaced in the image plane to the point located by $\vec{H} = \vec{a_{151_{S}}}$ .

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In general, by including tilted and decentered elements the modification for the field nodal position vectors of W₁₅₁ is to add the pupil-decentered terms to Eq. (C-35) in [2]. Consequently, Eq. (C-35) in [2] is modified by:

\vec{A_{151}} = \vec{A_{151}} + \vec{A_{151_{S}}} = \vec{A_{151}} +6 W_{060} \vec{S} .

3.3 Fifth-order aberration when l = 6 W_{l = 6} = {·} ${(\vec{ρ} \cdot \vec{ρ})}^{3}$

The term $W_{l = 6} = W_{060} \vec{(ρ} \cdot \vec{ρ})^{3} = \sum_{j} {(W_{060})}_{j} \vec{(ρ} \cdot \vec{ρ})^{3}$ is just the fifth-order spherical aberration of rotationally symmetric optical system, which is not affected by the pupil decentration vector. Similar with the case for the third-order aberrations, the fifth-order spherical aberration is also independent of the field vector. It is confirmed that the asymmetric wave aberration term for the fifth-order spherical aberration is the rotationally symmetric term. The result is also consistent with the analysis in [9, 12] that the fifth-order spherical aberration is unaffected when the rotational symmetry is broken.

4. Conclusions

To conclude, by applying the pupil decentration vector to the fifth-order aberration, we have analyzed the interaction of the pupil offset and fifth-order nodal aberration fields in the pupil-decentered optical system. In expanded terms of fifth-order aberrations, aberration terms W_l _{= 4}, W_l _{= 5}, and W_l _{= 6} also retain the multinodal property, but the vectors that determine the node locations themselves are modified by the pupil decentration vector. Concerning tilted and decentered elements in pupil-offset systems, the nodal position vectors described in [2] are modified by aberrations of higher order in pupil and these modifications are extended here beyond those presented in [2]. Analytical results provide additional data for the aberration analysis in optical design.

APPENDIX A: Detail expansions of the fifth-order aberrations expanded by pupil decentration vector

(1). Fifth-order oblique spherical: $\begin{matrix} W_{242} = \sum_{j} W_{242 j} {(\vec{H} \cdot \vec{ρ})}^{2} (\vec{ρ} \cdot \vec{ρ}) = \sum_{j} W_{242 j} {[\vec{H} \cdot (\vec{ρ} + \vec{S})]}^{2} (\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S}) \\ = W_{242} [\begin{array}{l} {(\vec{H} \cdot \vec{ρ})}^{2} (\vec{ρ} \cdot \vec{ρ}) + {(\vec{H} \cdot \vec{S})}^{2} (\vec{ρ} \cdot \vec{ρ}) + 4 (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{S}) \\ + 2 (\vec{H} \cdot \vec{ρ}) (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) + {(\vec{H} \cdot \vec{ρ})}^{2} (\vec{S} \cdot \vec{S}) + 2 {(\vec{H} \cdot \vec{ρ})}^{2} (\vec{ρ} \cdot \vec{S}) \\ + 2 {(\vec{H} \cdot \vec{S})}^{2} (\vec{ρ} \cdot \vec{S}) + {(\vec{H} \cdot \vec{S})}^{2} (\vec{S} \cdot \vec{S}) + 2 (\vec{H} \cdot \vec{ρ}) (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) \end{array}] . \end{matrix}$
(2). Field-cubed coma: $\begin{matrix} W_{331 M} = \sum_{j} W_{331 M j} (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) = \sum_{j} W_{331 M j} (\vec{H} \cdot \vec{H}) [\vec{H} \cdot (\vec{ρ} + \vec{S})] [(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})] \\ = W_{331 M} [\begin{array}{l} (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) + 2 (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) + (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{S} \cdot \vec{S}) \\ + (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) + (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 2 (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{S}) \end{array}] . \end{matrix}$
(3). Fifth-order elliptical coma: $\begin{matrix} W_{333} = \sum_{j} W_{333 j} {(\vec{H} \cdot \vec{ρ})}^{3} = \sum_{j} W_{333 j} {[\vec{H} \cdot (\vec{ρ} + \vec{S})]}^{3} \\ = W_{333} [{(\vec{H} \cdot \vec{ρ})}^{3} + {(\vec{H} \cdot \vec{S})}^{3} + 3 {(\vec{H} \cdot \vec{ρ})}^{2} (\vec{H} \cdot \vec{S}) + 3 (\vec{H} \cdot \vec{ρ}) {(\vec{H} \cdot \vec{S})}^{2}] . \end{matrix}$
(4). A component of fifth-order field curvature: $\begin{matrix} W_{420 M} = \sum_{j} W_{420 M j} {(\vec{H} \cdot \vec{H})}^{2} (\vec{ρ} \cdot \vec{ρ}) = \sum_{j} W_{240 M j} {(\vec{H} \cdot \vec{H})}^{2} [(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})] \\ = W_{420 M} [{(\vec{H} \cdot \vec{H})}^{2} (\vec{ρ} \cdot \vec{ρ}) + 2 {(\vec{H} \cdot \vec{H})}^{2} (\vec{ρ} \cdot \vec{S}) + {(\vec{H} \cdot \vec{H})}^{2} (\vec{S} \cdot \vec{S})] . \end{matrix}$
(5). Fifth-order astigmatism: $\begin{matrix} W_{422} = \sum_{j} W_{422 j} (\vec{H} \cdot \vec{H}) {(\vec{H} \cdot \vec{ρ})}^{2} = \sum_{j} W_{240 j} (\vec{H} \cdot \vec{H}) {[\vec{H} \cdot (\vec{ρ} + \vec{S})]}^{2} \\ = W_{422} [(\vec{H} \cdot \vec{H}) {(\vec{H} \cdot \vec{ρ})}^{2} + 2 (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{H} \cdot \vec{S}) + (\vec{H} \cdot \vec{H}) {(\vec{H} \cdot \vec{S})}^{2}] . \end{matrix}$
(6). Fifth-order spherical: $\begin{matrix} W_{060} = \sum_{j} W_{060 j} {(\vec{ρ} \cdot \vec{ρ})}^{3} = \sum_{j} W_{060 j} {[(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})]}^{3} \\ = W_{060} [\begin{array}{l} {(\vec{ρ} \cdot \vec{ρ})}^{3} + {(\vec{S} \cdot \vec{S})}^{3} + 6 (\vec{ρ} \cdot \vec{S}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + 18 (\vec{S} \cdot \vec{S}) (\vec{ρ} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) \\ + 2 (\vec{ρ} \overset{3}{} \cdot \vec{S} \overset{3}{}) + 9 (\vec{S} \cdot \vec{S}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + 9 (\vec{ρ} \cdot \vec{ρ}) {(\vec{S} \cdot \vec{S})}^{2} \\ + 6 {(\vec{S} \cdot \vec{S})}^{2} (\vec{ρ} \cdot \vec{S}) + 6 (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \overset{2}{} \cdot \vec{S} \overset{2}{}) + 6 (\vec{S} \cdot \vec{S}) (\vec{ρ} \overset{2}{} \cdot \vec{S} \overset{2}{}) \end{array}] . \end{matrix}$
(7). Fifth-order field-linear coma: $\begin{matrix} W_{151} = \sum_{j} W_{151 j} (\vec{H} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2} = \sum_{j} W_{151 j} {[(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})]}^{2} [\vec{H} \cdot (\vec{ρ} + \vec{S})] \\ = W_{151} [\begin{array}{l} (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) + (\vec{H} \cdot \vec{ρ}) (\vec{S} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 4 (\vec{S} \cdot \vec{S}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ + 4 (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 4 (\vec{S} \cdot \vec{S}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) + (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ + (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 4 (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{ρ}) + 2 (\vec{H} \cdot \vec{S}) (\vec{ρ} \overset{2}{} \cdot \vec{S} \overset{2}{}) \\ + 4 (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) + 4 (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) + 2 (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \overset{2}{} \cdot \vec{S} \overset{2}{}) \end{array}] . \end{matrix}$
Based on Eq. (10), there is
$4 (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{ρ}) = 2 (\vec{ρ} \cdot \vec{ρ}) [(\vec{ρ} \cdot \vec{ρ}) (\vec{H} \cdot \vec{S}) + \vec{ρ} \overset{2}{} \cdot \vec{H} \vec{S}] .$
By substituting Eq. (41) into Eq. (40), we have
$\begin{matrix} W_{151} = \sum_{j} W_{151 j} (\vec{H} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2} = \sum_{j} W_{151 j} {[(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})]}^{2} [\vec{H} \cdot (\vec{ρ} + \vec{S})] \\ = W_{151} [\begin{array}{l} (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) + (\vec{H} \cdot \vec{ρ}) (\vec{S} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 4 (\vec{S} \cdot \vec{S}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ + 4 (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 4 (\vec{S} \cdot \vec{S}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) + 3 (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ + (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 2 (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \overset{2}{} \cdot \vec{H} \vec{S}) + 2 (\vec{H} \cdot \vec{S}) (\vec{S} \overset{2}{} \cdot \vec{ρ} \overset{2}{}) \\ + 4 (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) + 4 (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) + 2 (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \overset{2}{} \cdot \vec{S} \overset{2}{}) \end{array}] . \end{matrix}$
(8). Fifth-order distortion: $\begin{matrix} W_{511} = \sum_{j} W_{511 j} {(\vec{H} \cdot \vec{H})}^{2} (\vec{H} \cdot \vec{ρ}) = \sum_{j} W_{511 j} {(\vec{H} \cdot \vec{H})}^{2} [\vec{H} \cdot (\vec{ρ} + \vec{S})] \\ = W_{511} [{(\vec{H} \cdot \vec{H})}^{2} (\vec{H} \cdot \vec{ρ}) + {(\vec{H} \cdot \vec{H})}^{2} (\vec{H} \cdot \vec{S})] . \end{matrix}$
(9). A component of fifth-order oblique spherical: $\begin{matrix} W_{240 M} = \sum_{j} W_{240 M j} (\vec{H} \cdot \vec{H}) {(\vec{ρ} \cdot \vec{ρ})}^{2} = \sum_{j} W_{240 M j} (\vec{H} \cdot \vec{H}) {((\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S}))}^{2} \\ = W_{240 M} [\begin{array}{l} (\vec{H} \cdot \vec{H}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + 4 (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{ρ}) (\vec{S} \cdot \vec{S}) + 2 (\vec{H} \cdot \vec{H}) (\vec{ρ} \overset{2}{} \cdot \vec{S} \overset{2}{}) \\ + (\vec{H} \cdot \vec{H}) {(\vec{S} \cdot \vec{S})}^{2} + 4 (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) + 4 (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) \end{array}] . \end{matrix}$

APPENDIX B: Definitions for unnormalized standard nodal vectors.

\vec{A_{klm}} = \sum_{j} W_{klmj},

B_{klm} = \sum_{j} W_{klmj} (\vec{σ_{j}} \cdot \vec{σ_{j}}),

{\vec{B}}_{klm}^{2} = \sum_{j} W_{klmj} {\vec{σ_{j}}}^{2},

where the subscript j is surface number; k is power of field dependence; l is power of pupil dependence; m is power of cosine pupil dependence; the vector $\vec{σ_{j}}$ points to the center of the aberration field of a specific spherical surface, projected into the image plane.

Acknowledgments

Authors acknowledge Mr. Xiaoyi Zhao at University of Toronto for the helpful suggestions. Authors also thank Mr. Jian Wang from ChangChun Institute of Optics for detail explanations of the pupil decentration vector. The authors are also grateful to the reviewer and editors for their helpful and valuable comments.

References and links

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

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

3. J. R. Rogers, W. H. Taylor and D. T. Moore, eds., “Vector aberration theory and the design of off-axis systems,” in International Lens Design Conference, W. H. Taylor and D. T. Moore, eds. Proc. SPIE 554, 76–81 (1985).

4. J. W. Figoski, W. H. Taylor and D. T. Moore, eds., “Aberration characteristics of nonsymmetric systems,” in International Optical Design Conference, W.H. Taylor and D.T. Moore, eds. Proc. SPIE 554, 104–111 (1985).

5. S. H. Chang and A. Prata Jr., “Geometrical theory of aberrations near the axis in classical off-axis reflecting telescopes,” J. Opt. Soc. Am. A 22(11), 2454–2464 (2005). [CrossRef] [PubMed]

6. J. Wang, B. H. Guo, Q. Sun, and Z. W. Lu, “Third-order aberration fields of pupil decentered optical systems,” Opt. Express 20(11), 11652–11658 (2012). [CrossRef] [PubMed]

7. C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol. 2(1), 97–109 (2013).

8. J. L. F. de Meijere and C. H. F. Velzel, “Dependence of third- and fifth-order aberration coefficients on the definition of pupil coordinates,” J. Opt. Soc. Am. A 6(10), 1609–1617 (1989). [CrossRef]

9. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009). [CrossRef] [PubMed]

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

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

12. 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]

13. 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]

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Fig. 1 Definitions of pupil vector

\vec{ρ}

and field vector

\vec{H}

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Fig. 2 Operation algorithm for the vector multiplication of

\vec{A}

and

\vec{B}

\vec{i}

and

\vec{j}

are the normalized vectors in XOY coordinate system.

View in Article | Download Full Size | PDF

Fig. 3 Pupil vector conversion during the pupil offset.

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Fig. 4 Two nodal points of fifth-order aberration term

W_{l = 4}^{(2)}

. Nodal positions locate at

\vec{H} = \vec{a_{242 S}} + i \vec{b_{242 S}}

and

\vec{H} = \vec{a_{242 S}} - i \vec{b_{242 S}}

, respectively.

View in Article | Download Full Size | PDF

Fig. 5 The aberration field property of term W_l _{= 5} for a pupil-decentered optical system. Fifth-order aberration term W_l _{= 5}is displaced in the image plane to the point located by

\vec{H} = \vec{a_{151_{S}}}

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Equations (48)

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\begin{matrix} W_{j} (H, ρ, ρ \cos θ) = \sum_{p}^{\infty} \sum_{n}^{\infty} \sum_{m}^{\infty} {(W_{klm})}_{j} {(H^{2})}^{p} {(ρ^{2})}^{n} {(H ρ \cos θ)}^{m} \\ = \sum_{p}^{\infty} \sum_{n}^{\infty} \sum_{m}^{\infty} {(W_{klm})}_{j} H^{2 p + m} ρ^{2 n + m} \cos^{m} θ, \end{matrix}

\vec{ρ} = | \vec{ρ} | \exp (i ϕ) = ρ_{x} x + ρ_{y} y,

\vec{H} = | \vec{H} | \exp (i φ) = H_{x} x + H_{y} y,

W = W_{j} [(\vec{H} \cdot \vec{H}), (\vec{ρ} \cdot \vec{ρ}), (\vec{H} \cdot \vec{ρ})] = \sum_{j} \sum_{p}^{\infty} \sum_{n}^{\infty} \sum_{m}^{\infty} {(W_{klm})}_{j} {(\vec{H} \cdot \vec{H})}^{p} {(\vec{ρ} \cdot \vec{ρ})}^{n} {(\vec{H} \cdot \vec{ρ})}^{m} .

\vec{A} = a \exp (i α) = a_{x} \vec{i} + a_{y} \vec{j}, a_{x} = a \sin α, a_{y} = a \cos α,

\vec{B} = a \exp (i α) = a_{x} \vec{i} + a_{y} \vec{j}, a_{x} = a \sin α, a_{y} = a \cos α .

\begin{matrix} \vec{A} \vec{B} = a b \exp [i (α + β)] = (a_{x} a_{x} + a_{x} a_{x}) \vec{i} + (a_{x} a_{x} + a_{x} a_{x}) \vec{i} \\ = a b \sin (α + β) \vec{i} + a b \sin (α + β) \vec{i} . \end{matrix}

\vec{A} \overset{2}{} = a^{2} \exp (i 2 α) = {(\vec{A} \overset{2}{})}_{x} \vec{i} + {(\vec{A} \overset{2}{})}_{y} \vec{j},

{(\vec{A} - \vec{B})}^{3} = \vec{A} \overset{3}{} - 3 \vec{A} \overset{2}{} \vec{B} + 3 \vec{A} \vec{B} \overset{2}{} - \vec{B} \overset{3}{},

2 (\vec{A} \cdot \vec{B}) (\vec{A} \cdot \vec{C}) = (\vec{A} \cdot \vec{A}) (\vec{B} \cdot \vec{C}) + \vec{A} \overset{2}{} \cdot \vec{B} \vec{C} .

\vec{ρ'} = \vec{ρ} + \vec{S},

\begin{matrix} W_{Fifth} = \sum_{j} W_{060 j} {(\vec{ρ} \cdot \vec{ρ})}^{3} + \sum_{j} W_{151 j} (\vec{H} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + \sum_{j} W_{240 j} (\vec{H} \cdot \vec{H}) {(\vec{ρ} \cdot \vec{ρ})}^{2} \\ + \sum_{j} W_{242 j} {(\vec{H} \cdot \vec{ρ})}^{2} (\vec{ρ} \cdot \vec{ρ}) + \sum_{j} W_{331 j} (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) + \sum_{j} W_{333 j} {(\vec{H} \cdot \vec{ρ})}^{3} \\ + \sum_{j} W_{420 j} {(\vec{H} \cdot \vec{H})}^{2} (\vec{ρ} \cdot \vec{ρ}) + \sum_{j} W_{422 j} (\vec{H} \cdot \vec{H}) {(\vec{H} \cdot \vec{ρ})}^{2} + \sum_{j} W_{511 j} {(\vec{H} \cdot \vec{H})}^{2} (\vec{H} \cdot \vec{ρ}) . \end{matrix}

\begin{matrix} W_{Fifth}^{S} = \sum_{j} W_{060 j} {[(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})]}^{3} + \sum_{j} W_{151 j} [\vec{H} \cdot (\vec{ρ} + \vec{S})] {[(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})]}^{2} \\ + \sum_{j} W_{240 M j} (\vec{H} \cdot \vec{H}) {[(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})]}^{2} + \sum_{j} W_{242 j} {[\vec{H} \cdot (\vec{ρ} + \vec{S})]}^{2} [(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})] \\ + \sum_{j} W_{331 M j} (\vec{H} \cdot \vec{H}) [\vec{H} \cdot (\vec{ρ} + \vec{S})] [(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})] + \sum_{j} W_{420 M j} {(\vec{H} \cdot \vec{H})}^{2} [(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})] \\ + \sum_{j} W_{333 j} {[\vec{H} \cdot (\vec{ρ} + \vec{S})]}^{3} + \sum_{j} W_{422 j} {(\vec{H} \cdot \vec{H})}^{} {[\vec{H} \cdot (\vec{ρ} + \vec{S})]}^{2} + \sum_{j} W_{511 j} {(\vec{H} \cdot \vec{H})}^{2} [\vec{H} \cdot (\vec{ρ} + \vec{S})], \end{matrix}

W_{240 M} = W_{240} + \frac{1}{2} W_{242},

W_{331 M} = W_{331} + \frac{3}{4} W_{333},

W_{420 M} = W_{420} + \frac{1}{2} W_{422},

\begin{matrix} W_{l = 4}^{(1)} = W_{240 M} (\vec{H} \cdot \vec{H}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + 3 W_{151} (\vec{H} \cdot \vec{S}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + 9 W_{060} (\vec{S} \cdot \vec{S}) {(\vec{ρ} \cdot \vec{ρ})}^{2} \\ = [W_{240 M} (\vec{H} \cdot \vec{H}) + 3 W_{151} (\vec{H} \cdot \vec{S}) + 9 W_{060} (\vec{S} \cdot \vec{S})] {(\vec{ρ} \cdot \vec{ρ})}^{2}, \end{matrix}

\vec{A_{240 M_{S}}} = - 1.5 W_{151} \vec{S},

\vec{a_{240 M_{S}}} = \vec{A_{240 M_{S}}} / W_{240},

B_{240 M_{S}} = 9 W_{060} (\vec{S} \cdot \vec{S}),

b_{240 M_{S}} = B_{{240M}_{S}} / W_{240 M} - \vec{a_{240 M_{S}}} \cdot \vec{a_{240 M_{S}}} .

W_{l = 4}^{(1)} = W_{240 M} [(\vec{H} - \vec{a_{240 M_{S}}}) \cdot (\vec{H} - \vec{a_{{240M}_{S}}}) + b_{2 {40M}_{S}}] {(\vec{ρ} \cdot \vec{ρ})}^{2} .

\vec{A_{240M}} = \vec{A_{240 M}} + \vec{A_{240 M_{S}}} = \vec{A_{240 M}} - 1.5 W_{151} \vec{S},

B_{240 M} = B_{240 M} + B_{240 M_{S}} = B_{240 M} + 9 W_{060} (\vec{S} \cdot \vec{S}),

\begin{matrix} W_{l = 4}^{(2)} = \frac{1}{2} W_{242} (\vec{H} \overset{2}{} \cdot \vec{ρ} \overset{2}{}) (\vec{ρ} \cdot \vec{ρ}) + 2 W_{151} (\vec{H} \vec{S} \cdot \vec{ρ} \overset{2}{}) (\vec{ρ} \cdot \vec{ρ}) + 6 W_{060} (\vec{S} \overset{2}{} \cdot \vec{ρ} \overset{2}{}) (\vec{ρ} \cdot \vec{ρ}) \\ = \frac{1}{2} (W_{242} \vec{H} \overset{2}{} + 4 W_{151} \vec{H} \vec{S} + 12 W_{060} \vec{S} \overset{2}{}) \cdot \vec{ρ} \overset{2}{} (\vec{ρ} \cdot \vec{ρ}) . \end{matrix}

\vec{A_{242_{S}}} = - 2 W_{151} \vec{S},

{\vec{B}}_{242_{S}}^{2} = 12 W_{060} \vec{S} \overset{2}{},

\vec{a_{242_{S}}} = \vec{A_{242_{S}}} / W_{242},

{\vec{b}}_{242_{S}}^{2} = {\vec{B}}_{242_{S}}^{2} / W_{242} - {\vec{a}}_{242_{S}}^{2},

\begin{matrix} W_{l = 4}^{(2)} = \frac{1}{2} (W_{242} \vec{H} \overset{2}{} - 2 \vec{H} \vec{A_{242_{S}}} + {\vec{B_{242_{S}}}}^{2}) \cdot \vec{ρ} \overset{2}{} (\vec{ρ} \cdot \vec{ρ}) \\ = \frac{1}{2} W_{242} [{(\vec{H} - \vec{a_{242_{S}}})}^{2} + {\vec{b_{242_{S}}}}^{2}] \cdot \vec{ρ} \overset{2}{} (\vec{ρ} \cdot \vec{ρ}) . \end{matrix}

\vec{A_{242}} = \vec{A_{242}} + \vec{A_{242_{S}}} = \vec{A_{242}} - 2 W_{151} \vec{S},

{\vec{B_{242}}}^{2} = {\vec{B_{242}}}^{2} + {\vec{B_{242_{S}}}}^{2} = {\vec{B_{242}}}^{2} + 9 W_{060} (\vec{S} \cdot \vec{S}) .

\begin{matrix} W_{l = 5} = W_{151} (\vec{H} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + 6 W_{060} (\vec{S} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2} \\ = (W_{151} \vec{H} - \vec{A_{151_{S}}}) \cdot \vec{ρ} {(\vec{ρ} \cdot \vec{ρ})}^{2} \\ = W_{151} (\vec{H} - \vec{a_{151_{S}}}) \cdot \vec{ρ} {(\vec{ρ} \cdot \vec{ρ})}^{2}, \end{matrix}

\vec{A_{151}} = \vec{A_{151}} + \vec{A_{151_{S}}} = \vec{A_{151}} +6 W_{060} \vec{S} .

\begin{matrix} W_{242} = \sum_{j} W_{242 j} {(\vec{H} \cdot \vec{ρ})}^{2} (\vec{ρ} \cdot \vec{ρ}) = \sum_{j} W_{242 j} {[\vec{H} \cdot (\vec{ρ} + \vec{S})]}^{2} (\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S}) \\ = W_{242} [\begin{array}{l} {(\vec{H} \cdot \vec{ρ})}^{2} (\vec{ρ} \cdot \vec{ρ}) + {(\vec{H} \cdot \vec{S})}^{2} (\vec{ρ} \cdot \vec{ρ}) + 4 (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{S}) \\ + 2 (\vec{H} \cdot \vec{ρ}) (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) + {(\vec{H} \cdot \vec{ρ})}^{2} (\vec{S} \cdot \vec{S}) + 2 {(\vec{H} \cdot \vec{ρ})}^{2} (\vec{ρ} \cdot \vec{S}) \\ + 2 {(\vec{H} \cdot \vec{S})}^{2} (\vec{ρ} \cdot \vec{S}) + {(\vec{H} \cdot \vec{S})}^{2} (\vec{S} \cdot \vec{S}) + 2 (\vec{H} \cdot \vec{ρ}) (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) \end{array}] . \end{matrix}

\begin{matrix} W_{331 M} = \sum_{j} W_{331 M j} (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) = \sum_{j} W_{331 M j} (\vec{H} \cdot \vec{H}) [\vec{H} \cdot (\vec{ρ} + \vec{S})] [(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})] \\ = W_{331 M} [\begin{array}{l} (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) + 2 (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) + (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{S} \cdot \vec{S}) \\ + (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) + (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 2 (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{S}) \end{array}] . \end{matrix}

\begin{matrix} W_{333} = \sum_{j} W_{333 j} {(\vec{H} \cdot \vec{ρ})}^{3} = \sum_{j} W_{333 j} {[\vec{H} \cdot (\vec{ρ} + \vec{S})]}^{3} \\ = W_{333} [{(\vec{H} \cdot \vec{ρ})}^{3} + {(\vec{H} \cdot \vec{S})}^{3} + 3 {(\vec{H} \cdot \vec{ρ})}^{2} (\vec{H} \cdot \vec{S}) + 3 (\vec{H} \cdot \vec{ρ}) {(\vec{H} \cdot \vec{S})}^{2}] . \end{matrix}

\begin{matrix} W_{420 M} = \sum_{j} W_{420 M j} {(\vec{H} \cdot \vec{H})}^{2} (\vec{ρ} \cdot \vec{ρ}) = \sum_{j} W_{240 M j} {(\vec{H} \cdot \vec{H})}^{2} [(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})] \\ = W_{420 M} [{(\vec{H} \cdot \vec{H})}^{2} (\vec{ρ} \cdot \vec{ρ}) + 2 {(\vec{H} \cdot \vec{H})}^{2} (\vec{ρ} \cdot \vec{S}) + {(\vec{H} \cdot \vec{H})}^{2} (\vec{S} \cdot \vec{S})] . \end{matrix}

\begin{matrix} W_{422} = \sum_{j} W_{422 j} (\vec{H} \cdot \vec{H}) {(\vec{H} \cdot \vec{ρ})}^{2} = \sum_{j} W_{240 j} (\vec{H} \cdot \vec{H}) {[\vec{H} \cdot (\vec{ρ} + \vec{S})]}^{2} \\ = W_{422} [(\vec{H} \cdot \vec{H}) {(\vec{H} \cdot \vec{ρ})}^{2} + 2 (\vec{H} \cdot \vec{H}) (\vec{H} \cdot \vec{ρ}) (\vec{H} \cdot \vec{S}) + (\vec{H} \cdot \vec{H}) {(\vec{H} \cdot \vec{S})}^{2}] . \end{matrix}

\begin{matrix} W_{060} = \sum_{j} W_{060 j} {(\vec{ρ} \cdot \vec{ρ})}^{3} = \sum_{j} W_{060 j} {[(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})]}^{3} \\ = W_{060} [\begin{array}{l} {(\vec{ρ} \cdot \vec{ρ})}^{3} + {(\vec{S} \cdot \vec{S})}^{3} + 6 (\vec{ρ} \cdot \vec{S}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + 18 (\vec{S} \cdot \vec{S}) (\vec{ρ} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) \\ + 2 (\vec{ρ} \overset{3}{} \cdot \vec{S} \overset{3}{}) + 9 (\vec{S} \cdot \vec{S}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + 9 (\vec{ρ} \cdot \vec{ρ}) {(\vec{S} \cdot \vec{S})}^{2} \\ + 6 {(\vec{S} \cdot \vec{S})}^{2} (\vec{ρ} \cdot \vec{S}) + 6 (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \overset{2}{} \cdot \vec{S} \overset{2}{}) + 6 (\vec{S} \cdot \vec{S}) (\vec{ρ} \overset{2}{} \cdot \vec{S} \overset{2}{}) \end{array}] . \end{matrix}

\begin{matrix} W_{151} = \sum_{j} W_{151 j} (\vec{H} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2} = \sum_{j} W_{151 j} {[(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})]}^{2} [\vec{H} \cdot (\vec{ρ} + \vec{S})] \\ = W_{151} [\begin{array}{l} (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) + (\vec{H} \cdot \vec{ρ}) (\vec{S} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 4 (\vec{S} \cdot \vec{S}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ + 4 (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 4 (\vec{S} \cdot \vec{S}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) + (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ + (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 4 (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{ρ}) + 2 (\vec{H} \cdot \vec{S}) (\vec{ρ} \overset{2}{} \cdot \vec{S} \overset{2}{}) \\ + 4 (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) + 4 (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) + 2 (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \overset{2}{} \cdot \vec{S} \overset{2}{}) \end{array}] . \end{matrix}

4 (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{ρ}) = 2 (\vec{ρ} \cdot \vec{ρ}) [(\vec{ρ} \cdot \vec{ρ}) (\vec{H} \cdot \vec{S}) + \vec{ρ} \overset{2}{} \cdot \vec{H} \vec{S}] .

\begin{matrix} W_{151} = \sum_{j} W_{151 j} (\vec{H} \cdot \vec{ρ}) {(\vec{ρ} \cdot \vec{ρ})}^{2} = \sum_{j} W_{151 j} {[(\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S})]}^{2} [\vec{H} \cdot (\vec{ρ} + \vec{S})] \\ = W_{151} [\begin{array}{l} (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) + (\vec{H} \cdot \vec{ρ}) (\vec{S} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 4 (\vec{S} \cdot \vec{S}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ + 4 (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 4 (\vec{S} \cdot \vec{S}) (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) + 3 (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{ρ}) \\ + (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) + 2 (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \overset{2}{} \cdot \vec{H} \vec{S}) + 2 (\vec{H} \cdot \vec{S}) (\vec{S} \overset{2}{} \cdot \vec{ρ} \overset{2}{}) \\ + 4 (\vec{H} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) + 4 (\vec{ρ} \cdot \vec{S}) (\vec{H} \cdot \vec{S}) (\vec{ρ} \cdot \vec{ρ}) + 2 (\vec{H} \cdot \vec{ρ}) (\vec{ρ} \overset{2}{} \cdot \vec{S} \overset{2}{}) \end{array}] . \end{matrix}

\begin{matrix} W_{511} = \sum_{j} W_{511 j} {(\vec{H} \cdot \vec{H})}^{2} (\vec{H} \cdot \vec{ρ}) = \sum_{j} W_{511 j} {(\vec{H} \cdot \vec{H})}^{2} [\vec{H} \cdot (\vec{ρ} + \vec{S})] \\ = W_{511} [{(\vec{H} \cdot \vec{H})}^{2} (\vec{H} \cdot \vec{ρ}) + {(\vec{H} \cdot \vec{H})}^{2} (\vec{H} \cdot \vec{S})] . \end{matrix}

\begin{matrix} W_{240 M} = \sum_{j} W_{240 M j} (\vec{H} \cdot \vec{H}) {(\vec{ρ} \cdot \vec{ρ})}^{2} = \sum_{j} W_{240 M j} (\vec{H} \cdot \vec{H}) {((\vec{ρ} + \vec{S}) \cdot (\vec{ρ} + \vec{S}))}^{2} \\ = W_{240 M} [\begin{array}{l} (\vec{H} \cdot \vec{H}) {(\vec{ρ} \cdot \vec{ρ})}^{2} + 4 (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{ρ}) (\vec{S} \cdot \vec{S}) + 2 (\vec{H} \cdot \vec{H}) (\vec{ρ} \overset{2}{} \cdot \vec{S} \overset{2}{}) \\ + (\vec{H} \cdot \vec{H}) {(\vec{S} \cdot \vec{S})}^{2} + 4 (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{ρ}) (\vec{ρ} \cdot \vec{S}) + 4 (\vec{H} \cdot \vec{H}) (\vec{ρ} \cdot \vec{S}) (\vec{S} \cdot \vec{S}) \end{array}] . \end{matrix}

\vec{A_{klm}} = \sum_{j} W_{klmj},

B_{klm} = \sum_{j} W_{klmj} (\vec{σ_{j}} \cdot \vec{σ_{j}}),

{\vec{B}}_{klm}^{2} = \sum_{j} W_{klmj} {\vec{σ_{j}}}^{2},

Abstract

1. Introduction

2. Vector-form of aberration expansion in pupil decentered optical systems

2.1 Vector-form of wavefront aberration expansion

2.2 Vector multiplication

2.3 Pupil decentration vector

3. Impact of pupil offset on fifth-order nodal aberration fields

3.1 Fifth-order aberration term when l = 4 {·}(ρ→⋅ρ)→2 and {·}ρ→2(ρ→⋅ρ)→

A. The term Wl=4(1) = {·}(ρ→⋅ρ)→2

B . The term = {·}

3.2 Fifth-order aberration when l = 5 Wl = 5 = {•}⋅ρ→(ρ→⋅ρ→)2

3.3 Fifth-order aberration when l = 6 Wl = 6 = {·}(ρ→⋅ρ→)3

4. Conclusions

APPENDIX A: Detail expansions of the fifth-order aberrations expanded by pupil decentration vector

APPENDIX B: Definitions for unnormalized standard nodal vectors.

Acknowledgments

References and links

Cited By

Figures (5)

Equations (48)

Optics Express

3.1 Fifth-order aberration term when l = 4 {·} $\vec{(ρ} \cdot {\vec{ρ)}}^{2}$ and {·} $\vec{ρ} \overset{2}{} (\vec{ρ} \cdot \vec{ρ)}$

A. The term $W_{l = 4}^{(1)}$ = {·} $\vec{(ρ} \cdot {\vec{ρ)}}^{2}$

3.2 Fifth-order aberration when l = 5 Wl = 5 = {•} $\cdot \vec{ρ} {(\vec{ρ} \cdot \vec{ρ})}^{2}$

3.3 Fifth-order aberration when l = 6 W_{l = 6} = {·} ${(\vec{ρ} \cdot \vec{ρ})}^{3}$