Abstract

Building off an earlier work on multinodal third-order aberrations [J. Opt. Soc. Am. A 22, 1389 (2005) ], this is the first in a series of papers that derives and illustrates the characteristic multinodal geometry for each of the fifth-order aberrations. Part I (as this paper will be referred to) will present the spherical aberration family: specifically, W060, W240M and W242, and W080 (fifth-order spherical, oblique spherical, and seventh-order spherical). Nodal aberration theory is proving to be very effective as both an optical design tool for fully unobscured off-axis telescopes and as an analysis method, particularly in the context of the response of any imaging optical systems to misalignment. It is important to recognize that this multinodal approach to aberration theory is not restricted to small perturbations. The remaining papers in this series will result in a complete presentation of the intrinsic characteristic multinodal properties of each of the fifth-order aberrations. As such, this series provides a definitive theory of the optical aberrations of (nonanamorphic) imaging systems with a circular aperture stop.

© 2009 Optical Society of America

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References

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  1. K. P. Thompson, “Aberrations fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, 1980).
  2. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389-1401 (2005).
    [CrossRef]
  3. L. Seidel, “About the third order expansion that describes the path of a light beam outside the plane of the axis through an optical system of refracting elements,” translated from German by R. Zehnder on behalf of J. Sasian, available from jose.sasian@optics.arizona.edu (2007). Originally published in Astronomische Nachrichten, 1027, 1028, 1029 (1865).
  4. H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon, 1950).
  5. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, 1976).
  6. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “A real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A (to be published).
  7. J. Figoski, “Aberration characteristics of nonsymmetric systems,” Proc. SPIE 554, 104-109 (1985).
  8. K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16, 20345-20353 (2008).
    [CrossRef] [PubMed]
  9. H. A. Buchdahl, Optical Aberration Coefficients (Oxford U. Press, 1954).
  10. M. C. Rimmer “Optical aberration coefficients,” M.S. thesis (University of Rochester, 1963).
  11. R. V. Shack, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721. Phone, 520-621-1356. (Personal communication, 1977).
  12. B. A. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Jpn. 108, 217-219 (1996).
    [CrossRef]

2008 (1)

2005 (1)

1996 (1)

B. A. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Jpn. 108, 217-219 (1996).
[CrossRef]

1985 (1)

J. Figoski, “Aberration characteristics of nonsymmetric systems,” Proc. SPIE 554, 104-109 (1985).

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Oxford U. Press, 1954).

Buchroeder, R. A.

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

Cakmakci, O.

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “A real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A (to be published).

Figoski, J.

J. Figoski, “Aberration characteristics of nonsymmetric systems,” Proc. SPIE 554, 104-109 (1985).

Hopkins, H. H.

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

McLeod, B. A.

B. A. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Jpn. 108, 217-219 (1996).
[CrossRef]

Rimmer, M. C.

M. C. Rimmer “Optical aberration coefficients,” M.S. thesis (University of Rochester, 1963).

Rolland, J. P.

K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16, 20345-20353 (2008).
[CrossRef] [PubMed]

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “A real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A (to be published).

Schmid, T.

K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16, 20345-20353 (2008).
[CrossRef] [PubMed]

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “A real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A (to be published).

Seidel, L.

L. Seidel, “About the third order expansion that describes the path of a light beam outside the plane of the axis through an optical system of refracting elements,” translated from German by R. Zehnder on behalf of J. Sasian, available from jose.sasian@optics.arizona.edu (2007). Originally published in Astronomische Nachrichten, 1027, 1028, 1029 (1865).

Shack, R. V.

R. V. Shack, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721. Phone, 520-621-1356. (Personal communication, 1977).

Thompson, K. P.

K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16, 20345-20353 (2008).
[CrossRef] [PubMed]

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

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “A real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A (to be published).

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

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Proc. SPIE (1)

J. Figoski, “Aberration characteristics of nonsymmetric systems,” Proc. SPIE 554, 104-109 (1985).

Publ. Astron. Soc. Jpn. (1)

B. A. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Jpn. 108, 217-219 (1996).
[CrossRef]

Other (8)

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

H. A. Buchdahl, Optical Aberration Coefficients (Oxford U. Press, 1954).

M. C. Rimmer “Optical aberration coefficients,” M.S. thesis (University of Rochester, 1963).

R. V. Shack, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721. Phone, 520-621-1356. (Personal communication, 1977).

L. Seidel, “About the third order expansion that describes the path of a light beam outside the plane of the axis through an optical system of refracting elements,” translated from German by R. Zehnder on behalf of J. Sasian, available from jose.sasian@optics.arizona.edu (2007). Originally published in Astronomische Nachrichten, 1027, 1028, 1029 (1865).

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

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

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “A real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A (to be published).

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Figures (5)

Fig. 1
Fig. 1

Conventions for the pupil vector ρ and the field vector H .

Fig. 2
Fig. 2

Representation of the location of the center of rotational symmetry for the aberration field components associated with surface j and the effective field height vector, H A j .

Fig. 3
Fig. 3

(a) The geometric shape of the 0.7 aperture zone (projected 90° onto the plane of the paper) can be used to uniquely identify a focal surface for oblique spherical aberration. (b) The curvature of the quadratic focal surface associated with oblique spherical aberration. ( W 242 , W 240 M > 0 , W 240 M > W 242 ) . (c) The thru-focus on-axis caustic for spherical aberration. For oblique spherical, the size of the caustic varies as the square of the field as an addition, or subtraction, to the third-order contribution, which is field independent. The minimum spot size occurs where the 0.7 aperture rays reach focus across the field, and this defines the medial focal surface.

Fig. 4
Fig. 4

In a perturbed system, the tangential oblique spherical aberration (TOBSA) with respect to the medial surface for OBSA is zero at two points (nodes) in the field.

Fig. 5
Fig. 5

The characteristic field behavior of the medial focal surface for oblique spherical aberration in an optical system without symmetry is for the vertex of the point of quadratic departure to decenter to a point located by the vector a 240 M and the surface to be defocused by an amount proportional to b 240 M .

Equations (46)

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W = j p n m ( W k l m ) j H k ρ l cos m ϕ ,
k = 2 p + m , l = 2 n + m .
W = j p n m ( W k l m ) j ( H H ) p ( ρ ρ ) n ( H ρ ) m ,
k = 2 p + m , l = 2 n + m .
ρ = ρ exp ( i ϕ ) = ρ x x ̂ + ρ y y ̂ ,
H = H exp ( i θ ) = H x x ̂ + H y y ̂ ,
H A j = H σ j ,
W = j p n m ( W k l m ) j ( H A j H A j ) p ( ρ ρ ) n ( H A j ρ ) m = j p n m ( W k l m ) j [ ( H σ j ) ( H σ j ) ] p ( ρ ρ ) n [ ( ( H σ j ) ρ ) ] m , k = 2 p + m , l = 2 n + m .
W = Δ W 020 ( ρ ρ ) + Δ W 111 ( H ρ ) + j W 040 j ( ρ ρ ) 2 + j W 131 j [ ( H σ j ) ρ ] ( ρ ρ ) + j W 220 j [ ( H σ j ) ( H σ j ) ] ( ρ ρ ) + j W 222 j [ ( H σ j ) ρ ] 2 + j W 311 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] + j W 060 j ( ρ ρ ) 3 + j W 151 j [ ( H σ j ) ρ ] ( ρ ρ ) 2 + j W 240 j [ ( H σ j ) ( H σ j ) ] ( ρ ρ ) 2 + j W 242 j [ ( H σ j ) ρ ] 2 ( ρ ρ ) + j W 331 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] ( ρ ρ ) + j W 333 j [ ( H σ j ) ρ ] 3 + j W 420 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ( H σ j ) ] ( ρ ρ ) + j W 422 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] 2 + j W 511 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] + j W 080 j ( ρ ρ ) 4 ,
W = Δ W 020 ( ρ ρ ) + j W 040 j ( ρ ρ ) 2 + j W 060 j ( ρ ρ ) 3 + j W 240 j [ ( H σ j ) ( H σ j ) ] ( ρ ρ ) 2 + j W 242 j [ ( H σ j ) ρ ] 2 ( ρ ρ ) + j W 080 j ( ρ ρ ) 4 .
W = j W 060 j ( ρ ρ ) 3 = W 060 ( ρ ρ ) 3
W = Δ W 20 ( ρ ρ ) + j W 240 j ( H H ) ( ρ ρ ) 2 + j W 242 j ( H ρ ) 2 ( ρ ρ ) .
W 240 M W 240 + 1 2 W 242
W = Δ W 20 ρ 2 + W 240 H 2 ρ 4 + W 242 H 2 ρ 4 ( 1 2 + 1 2 cos 2 ϕ ) = Δ W 20 ρ 2 + W 240 M H 2 ρ 4 + 1 2 W 242 H 2 ρ 4 cos 2 ϕ .
W = Δ W 20 ( ρ ρ ) + W 240 M ( H H ) ( ρ ρ ) 2 + 1 2 W 242 ( H 2 ρ 2 ) ( ρ ρ ) .
W = 1 2 j W 242 j [ ( H σ j ) 2 ρ 2 ] ( ρ ρ ) = 1 2 { [ j W 242 j H 2 2 H ( j W 242 j σ j ) + j W 242 j σ j 2 ] ρ 2 } ( ρ ρ ) .
A 242 j W 242 j σ j ,
B 242 2 j W 242 j σ j 2 .
a 242 A 242 W 242 ,
b 242 2 B 242 2 W 242 a 242 2 ,
W = 1 2 W 242 ( [ ( H a 242 ) 2 + b 242 2 ] ρ 2 ) ( ρ ρ ) = 1 2 W 242 ( [ H 242 2 + b 242 2 ] ρ 2 ) ( ρ ρ ) ,
0 = ( H a 242 ) 2 + b 242 2
H = a 242 + ( b 242 2 ) 1 2 .
H = a 242 ± i b 242 ,
b 242 2 = b 242 2 exp ( i 2 β ) ,
b 242 = b 242 2 1 2 exp ( i β ) = b 242 exp ( i β ) ,
± i b 242 = b 242 exp [ i ( β ± 90 ) ] .
Δ W 020 = j W 240 M j [ ( H σ j ) ( H σ j ) ] = j W 240 M j ( H H ) 2 H ( j W 240 M j σ j ) + j W 240 M j ( σ j σ j ) .
A 240 M j W 240 M j σ j ,
B 240 M j W 240 M j ( σ j σ j ) ( a scalar ) ,
a 240 M A 240 M W 240 M ,
b 240 M B 240 M W 240 M a 240 M a 240 M ,
Δ W 020 = W 240 M [ ( H a 240 M ) ( H a 240 M ) + b 240 M ] = W 240 M [ ( H 240 M H 240 M ) + b 240 M ] ,
δ z 240 M = 8 ( f # ) 2 W 240 M b 240 M .
W = j W 080 j ( ρ ρ ) 4 = W 080 ( ρ ρ ) 4 .
W = Δ W 020 ( ρ ρ ) + j W 040 j ( ρ ρ ) 2 + j W 060 j ( ρ ρ ) 3 + j W 240 j [ ( H σ j ) ( H σ j ) ] ( ρ ρ ) 2 + j W 242 j [ ( H σ j ) ρ ] 2 ( ρ ρ ) + j W 080 j ( ρ ρ ) 4 ,
W = Δ W 020 ( ρ ρ ) + W 040 ( ρ ρ ) 2 + W 060 ( ρ ρ ) 3 + W 240 M [ ( H a 240 M ) ( H a 240 M ) + b 240 M ] + 1 2 W 242 ( [ ( H a 242 ) 2 + b 242 2 ] ρ 2 ) ( ρ ρ ) + W 080 ( ρ ρ ) 4 .
W = Δ W 020 ( ρ ρ ) + Δ W 111 ( H ρ ) + W 040 ( ρ ρ ) 2 + W 131 [ ( H a 131 ) ρ ] ( ρ ρ ) + W 220 M [ ( H a 220 M ) ( H a 220 M ) + b 220 M ] + 1 2 W 222 ( [ ( H a 222 ) 2 + b 222 2 ] ρ 2 ) ( ρ ρ ) + j W 311 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] + W 060 ( ρ ρ ) 3 + j W 151 j [ ( H σ j ) ρ ] ( ρ ρ ) 2 + W 240 M [ ( H a 240 M ) ( H a 240 M ) + b 240 M ] + 1 2 W 242 ( [ ( H a 242 ) 2 + b 242 2 ] ρ 2 ) ( ρ ρ ) + j W 331 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] ( ρ ρ ) + j W 333 j [ ( H σ j ) ρ ] 3 + j W 420 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ( H σ j ) ] ( ρ ρ ) + j W 422 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] 2 + j W 511 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] + W 080 ( ρ ρ ) 4 .
A 2 = A 2 exp ( i 2 α ) ,
( A 2 ) 1 2 = A exp ( i α ) .
0 = ( H a 242 ) 2 + b 242 2 ,
( H a 242 ) 2 = b 242 2 ,
[ ( H a 242 ) 2 ] 1 2 = [ b 242 2 ] 1 2 ,
( H a 242 ) = ± i b 242 ,
H = a 242 ± i b 242
A B = ( see Ref. 2 ) = A B exp [ i ( α + β ) ] .

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