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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 15 — Jul. 29, 2013
  • pp: 17986–17998
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Interaction of pupil offset and fifth-order nodal aberration field properties in rotationally symmetric telescopes

Haili Hu, Jianjun Liu, and Zhigang Fan  »View Author Affiliations


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


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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.

© 2013 OSA

1. Introduction

Optical systems without rotational symmetry have been and will continue to be one of the hot topics in optical designs [1

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

7

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).

]. Among these optical systems, the pupil decentered configuration is proposed because the obstruction of primary aperture in telescopes can be avoided [5

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]

7

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).

]. Aberration analysis is crucial for the optical design and fabrication, and significant research progresses have been made for optical systems without rotational symmetry [2

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

6

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]

, 8

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

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]

]. The third-order aberration of the titled and decentered optical systems could be determined ether by ray-tracing or vector methods [5

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

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]

]. The fifth-order aberration plays same role as third-order one in the optical design [9

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

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

]. Based on all elements’ parameters of optical systems, the multinodal property of the fifth-order aberrations was investigated by Thompson in [9

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]

]. Further, for some convenience applications, the pupil decentration vector was introduced to analyze the third-order aberration fields in pupil-decentered optical system [6

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]

]. 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

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

]. The wave aberration of surface j in an optical system with rotational symmetry is given by:
Wj(H,ρ,ρcosθ)=pnm(Wklm)j(H2)p(ρ2)n(Hρcosθ)m=pnm(Wklm)jH2p+mρ2n+mcosmθ,
(1)
where ρ is the pupil vector with ρx and ρy along X and Y axis in the entrance/exit pupil, H represents a field vector in the image plane with Hx and Hy 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
Fig. 1 Definitions of pupil vector ρ and field vector H.
. A left-hand coordinate system is employed, which follows the tradition in most commercial optical design software.

To analyze optical path differences between the ideal and arbitrary rays, the pupil and field vectors are expressed in forms of:
ρ=|ρ|exp(iϕ)=ρxx+ρyy,
(2)
H=|H|exp(iφ)=Hxx+Hyy,
(3)
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

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

12

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]

]:

W=Wj[(HH),(ρρ),(Hρ)]=jpnm(Wklm)j(HH)p(ρρ)n(Hρ)m.
(4)

2.2 Vector multiplication

The concept of the vector multiplication was named by Shack, and described in detail by Thompson in [12

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]

]. 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 A and B as examples to figure out the vector multiplication. Vectors A and B are defined as:

A=aexp(iα)=axi+ayj,ax=asinα,ay=acosα,
(5)
B=aexp(iα)=axi+ayj,ax=asinα,ay=acosα.
(6)

As shown in Fig. 2
Fig. 2 Operation algorithm for the vector multiplication of A and B. i and j are the normalized vectors in XOY coordinate system.
, the vector multiplication AB is expressed by

AB=abexp[i(α+β)]=(axax+axax)i+(axax+axax)i=absin(α+β)i+absin(α+β)i.
(7)

Specifically, when A=B, the vector multiplication AB becomes:
A2=a2exp(i2α)=(A2)xi+(A2)yj,
(8)
where (A2)x=a2sin2α=2axay and (A2)y=a2cos2α=ay2ax2.

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

(AB)3=A33A2B+3AB2B3,
(9)
2(AB)(AC)=(AA)(BC)+A2BC.
(10)

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
Fig. 3 Pupil vector conversion during the pupil offset.
, the relation between the new pupil vector ρ' and the original one ρ is [6

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]

]:
ρ'=ρ+S,
(11)
where S is the pupil decentration vector normalized to the original pupil.

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

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]

], 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:

WFifth=jW060j(ρρ)3+jW151j(Hρ)(ρρ)2+jW240j(HH)(ρρ)2+jW242j(Hρ)2(ρρ)+jW331j(HH)(Hρ)(ρρ)+jW333j(Hρ)3+jW420j(HH)2(ρρ)+jW422j(HH)(Hρ)2+jW511j(HH)2(Hρ).
(12)

With the help of pupil decentration vector, the vector-form expression of the fifth-order aberrations in pupil-decentered optical system is expanded into
WFifthS=jW060j[(ρ+S)(ρ+S)]3+jW151j[H(ρ+S)][(ρ+S)(ρ+S)]2+jW240Mj(HH)[(ρ+S)(ρ+S)]2+jW242j[H(ρ+S)]2[(ρ+S)(ρ+S)]+jW331Mj(HH)[H(ρ+S)][(ρ+S)(ρ+S)]+jW420Mj(HH)2[(ρ+S)(ρ+S)]+jW333j[H(ρ+S)]3+jW422j(HH)[H(ρ+S)]2+jW511j(HH)2[H(ρ+S)],
(13)
in which W240M, W331M, and W420M are the medial components [2

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

]:
W240M=W240+12W242,
(14)
W331M=W331+34W333,
(15)
W420M=W420+12W422,
(16)
where Wklm=j(Wklm)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

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]

]. 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

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]

], 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

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]

]. 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 Wl = 4: including terms {·}(ρρ)2 and {·}ρ2(ρρ);
  • (2). the aberration term l = 5 Wl = 5: {·}ρ(ρρ)2;
  • (3). the aberration term l = 6 Wl = 6: {·}(ρρ)3.

3.1 Fifth-order aberration term when l = 4 {·}(ρρ)2 and {·}ρ2(ρρ)

A. The term Wl=4(1) = {·}(ρρ)2

The aberration term Wl=4(1) = {·}(ρρ)2 includes all components of ‘(ρρ)2’ in the Appendix A, and it is expressed by:
Wl=4(1)=W240M(HH)(ρρ)2+3W151(HS)(ρρ)2+9W060(SS)(ρρ)2=[W240M(HH)+3W151(HS)+9W060(SS)](ρρ)2,
(17)
where the first summation W240M(HH)=j(W240M)j(HH) is the oblique spherical contribution from optical system with rotational symmetric.

When W240M0, we defined following vectors
A240MS=1.5W151S,
(18)
a240MS=A240MS/W240,
(19)
and scalars

B240MS=9W060(SS),
(20)
b240MS=B240MS/W240Ma240MSa240MS.
(21)

Then, Wl=4(1) can be expressed as:

Wl=4(1)=W240M[(Ha240MS)(Ha240MS)+b240MS](ρρ)2.
(22)

The term Wl=4(1) = {·}(ρρ)2 performs the multinodal property, which retain the property of the aberration W240M analyzed in [9

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]

]. The Eq. (22) also means that the nodal properties of the W240M term are not affected by pupil offset fundamentally. The aberration field nodal positions are determined by a240 and b240 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 Wl=4(1) = {·}(ρρ)2. The nodal aberration field property of Wl=4(1) = {·}(ρρ)2 also confirm that the case of 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

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

] by introducing modification terms A240MS and B240MS:
A240M=A240M+A240MS=A240M1.5W151S,
(23)
B240M=B240M+B240MS=B240M+9W060(SS),
(24)
where A240M and B240M are the unnormalized standard vectors [2

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

], as defined in the Appendix B.

B . The term = {·}

The part of the fifth-order aberration Wl=4(2) = {·}ρ2(ρρ) includes all components of ‘ρ2(ρρ)’ in the Appendix A:

Wl=4(2)=12W242(H2ρ2)(ρρ)+2W151(HSρ2)(ρρ)+6W060(S2ρ2)(ρρ)=12(W242H2+4W151HS+12W060S2)ρ2(ρρ).
(25)

By defining following vectors and scalars (when W242≠0),
A242S=2W151S,
(26)
B242S2=12W060S2,
(27)
a242S=A242S/W242,
(28)
b242S2=B242S2/W242a242S2,
(29)
the term Wl=4(2) can be expressed by

Wl=4(2)=12(W242H22HA242S+B242S2)ρ2(ρρ)=12W242[(Ha242S)2+b242S2]ρ2(ρρ).
(30)

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

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

], and the modifications are expressed by

A242=A242+A242S=A2422W151S,
(31)
B2422=B2422+B242S2=B2422+9W060(SS).
(32)

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

The part of the fifth-order aberration Wl=5 = {·}ρ(ρρ)2 includes all components of ‘ρ(ρρ)2’ in the Appendix A, and it is expressed by:
Wl=5=W151(Hρ)(ρρ)2+6W060(Sρ)(ρρ)2=(W151HA151S)ρ(ρρ)2=W151(Ha151S)ρ(ρρ)2,
(33)
where the term W151H=j(W151)jH indicates the contribution from aberrations of the system with rotational symmetric; the term A151S=6W060S is the contribution from pupil offset in the pupil plane weighted by the fifth-order aberration coefficient W060, and the vector a151S=A151S/W151(when W151≠0) is the normalized vector of A151S.

The part Wl=5 = {·}ρ(ρρ)2 performed the one nodal act, as released in Fig. 5
Fig. 5 The aberration field property of term Wl = 5 for a pupil-decentered optical system. Fifth-order aberration term Wl = 5is displaced in the image plane to the point located by H=a151S.
and its field dependence property is same as the fifth-order field-linear coma. The nodal position of the term Wl=5 in the image field is only modified by the vector a151S, 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.

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

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

]. Consequently, Eq. (C-35) in [2

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

] is modified by:

A151=A151+A151S=A151+6W060S.
(34)

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

The term Wl=6=W060(ρρ)3=j(W060)j(ρρ)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

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]

, 12

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]

] 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 Wl = 4, Wl = 5, and Wl = 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

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

] are modified by aberrations of higher order in pupil and these modifications are extended here beyond those presented in [2

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

]. 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:
    W242=jW242j(Hρ)2(ρρ)=jW242j[H(ρ+S)]2(ρ+S)(ρ+S)=W242[(Hρ)2(ρρ)+(HS)2(ρρ)+4(Hρ)(ρS)(HS)+2(Hρ)(HS)(ρρ)+(Hρ)2(SS)+2(Hρ)2(ρS)+2(HS)2(ρS)+(HS)2(SS)+2(Hρ)(HS)(SS)].
    (35)
  • (2). Field-cubed coma:
    W331M=jW331Mj(HH)(Hρ)(ρρ)=jW331Mj(HH)[H(ρ+S)][(ρ+S)(ρ+S)]=W331M[(HH)(Hρ)(ρρ)+2(HH)(Hρ)(ρS)+(HH)(Hρ)(SS)+(HH)(HS)(ρρ)+(HH)(HS)(SS)+2(HH)(HS)(ρS)].
    (36)
  • (3). Fifth-order elliptical coma:
    W333=jW333j(Hρ)3=jW333j[H(ρ+S)]3=W333[(Hρ)3+(HS)3+3(Hρ)2(HS)+3(Hρ)(HS)2].
    (37)
  • (4). A component of fifth-order field curvature:
    W420M=jW420Mj(HH)2(ρρ)=jW240Mj(HH)2[(ρ+S)(ρ+S)]=W420M[(HH)2(ρρ)+2(HH)2(ρS)+(HH)2(SS)].
    (38)
  • (5). Fifth-order astigmatism:
    W422=jW422j(HH)(Hρ)2=jW240j(HH)[H(ρ+S)]2=W422[(HH)(Hρ)2+2(HH)(Hρ)(HS)+(HH)(HS)2].
    (39)
  • (6). Fifth-order spherical:
    W060=jW060j(ρρ)3=jW060j[(ρ+S)(ρ+S)]3=W060[(ρρ)3+(SS)3+6(ρS)(ρρ)2+18(SS)(ρS)(ρρ)+2(ρ3S3)+9(SS)(ρρ)2+9(ρρ)(SS)2+6(SS)2(ρS)+6(ρρ)(ρ2S2)+6(SS)(ρ2S2)].
    (40)
  • (7). Fifth-order field-linear coma:
    W151=jW151j(Hρ)(ρρ)2=jW151j[(ρ+S)(ρ+S)]2[H(ρ+S)]=W151[(Hρ)(ρρ)(ρρ)+(Hρ)(SS)(SS)+4(SS)(Hρ)(ρρ)+4(ρS)(HS)(SS)+4(SS)(Hρ)(ρS)+(HS)(ρρ)(ρρ)+(HS)(SS)(SS)+4(ρρ)(ρS)(Hρ)+2(HS)(ρ2S2)+4(HS)(SS)(ρρ)+4(ρS)(HS)(ρρ)+2(Hρ)(ρ2S2)].
    (41)

    Based on Eq. (10), there is
    4(ρρ)(ρS)(Hρ)=2(ρρ)[(ρρ)(HS)+ρ2HS].
    (42)

    By substituting Eq. (41) into Eq. (40), we have
    W151=jW151j(Hρ)(ρρ)2=jW151j[(ρ+S)(ρ+S)]2[H(ρ+S)]=W151[(Hρ)(ρρ)(ρρ)+(Hρ)(SS)(SS)+4(SS)(Hρ)(ρρ)+4(ρS)(HS)(SS)+4(SS)(Hρ)(ρS)+3(HS)(ρρ)(ρρ)+(HS)(SS)(SS)+2(ρρ)(ρ2HS)+2(HS)(S2ρ2)+4(HS)(SS)(ρρ)+4(ρS)(HS)(ρρ)+2(Hρ)(ρ2S2)].
    (43)

  • (8). Fifth-order distortion:
    W511=jW511j(HH)2(Hρ)=jW511j(HH)2[H(ρ+S)]=W511[(HH)2(Hρ)+(HH)2(HS)].
    (44)
  • (9). A component of fifth-order oblique spherical:
    W240M=jW240Mj(HH)(ρρ)2=jW240Mj(HH)((ρ+S)(ρ+S))2=W240M[(HH)(ρρ)2+4(HH)(ρρ)(SS)+2(HH)(ρ2S2)+(HH)(SS)2+4(HH)(ρρ)(ρS)+4(HH)(ρS)(SS)].
    (45)

APPENDIX B: Definitions for unnormalized standard nodal vectors.

Aklm=jWklmj,
(46)
Bklm=jWklmj(σjσj),
(47)
Bklm2=jWklmjσj2,
(48)

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 σj 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]

OCIS Codes
(080.2740) Geometric optics : Geometric optical design
(080.1005) Geometric optics : Aberration expansions

ToC Category:
Geometric Optics

History
Original Manuscript: April 19, 2013
Revised Manuscript: June 23, 2013
Manuscript Accepted: July 16, 2013
Published: July 19, 2013

Citation
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)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-15-17986


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References

  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., “Geometrical theory of aberrations near the axis in classical off-axis reflecting telescopes,” J. Opt. Soc. Am. A22(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. Express20(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. A6(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. A26(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. A22(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. Express16(20), 15655–15670 (2008). [CrossRef] [PubMed]

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