Abstract

Building on earlier work on the nodal aberration theory of third-order aberrations and a subset of fifth-order terms, this paper presents the multinodal field dependence of the family of aberrations describing the shape of the medial focal surface (the focal surface upon which the minimum RMS wavefront error is measured) and the astigmatic aberrations with respect to this surface through the fifth order. Specifically, the multinodal field dependence for W420M and W422 (the field-quartic medial surface and field-quartic astigmatism) are derived and presented as well as their influence on the magnitude and nodal field dependence of the companion lower-order terms, W220M and W222. This paper provides the first derivations of field-quartic aberrations presented by the author in the refereed literature.

© 2011 Optical Society of America

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References

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  1. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, 1976).
  2. R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).
  3. H. H. Hopkins, The Wave Theory of Aberrations (Clarendon, 1950).
  4. 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]
  5. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26, 1090–1100 (2009).
    [CrossRef]
  6. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27, 1490–1504 (2010).
    [CrossRef]
  7. K. P. Thompson, “Aberrations fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, 1980).
  8. K. P. Thompson, K. Fuerschbach, and J. P. Rolland, “An analytic expression for the field dependence of FRINGE Zernike coefficients in optical systems that are not rotationally symmetric,” Proc. SPIE 7849, 784906 (2010).
    [CrossRef]
  9. M. J. Crowe, A History of Vector Analysis (Notre Dame University, 1967), pp. 6–8.
  10. T. Needham, Visual Complex Analysis (Claredon, 1997), p. 3.
  11. K. P. Thompson, “Errata,” J. Opt. Soc. Am. A 26, 699 (2009).
    [CrossRef]
  12. 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 26, 1503–1517 (2009).
    [CrossRef]
  13. G. H. Spencer and M. V. R. K. Murty, “General ray-tracing procedure,” J. Opt. Soc. Am. 52, 672–678 (1962).
    [CrossRef]
  14. E. S. Barr, “Men and milestones in optics I. George Gabriel Stokes,” Appl. Opt. 1, 69–73 (1962).
    [CrossRef]
  15. G. B. Airy, “On the spherical aberration of eyepieces of telescopes,” Trans. Cambridge Philos. Soc. 3, 1–65 (1830).
    [CrossRef]
  16. Translation by R. Zehnder on behalf of J. Sasian, available from jose.sasian@optics.arizona.edu.
  17. A. E. Conrady, “The five aberrations of lens systems,” Mon. Not. R. Astron. Soc. 79, 60–66 (1918).
  18. M. C. Rimmer, “Optical aberration coefficients,” M.S. thesis (University of Rochester, 1963).
  19. J. Sasian, “The theory of sixth-order wave aberrations,” Appl. Opt. 49, D69–D95 (2010).
    [CrossRef] [PubMed]
  20. T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignment using Nodal Aberration Theory (NAT),” Opt. Express 18 (16), 17433–17447 (2010).
    [CrossRef] [PubMed]

2010 (4)

2009 (3)

2005 (1)

1997 (1)

T. Needham, Visual Complex Analysis (Claredon, 1997), p. 3.

1980 (2)

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

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

1976 (1)

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

1967 (1)

M. J. Crowe, A History of Vector Analysis (Notre Dame University, 1967), pp. 6–8.

1963 (1)

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

1962 (2)

1950 (1)

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

1918 (1)

A. E. Conrady, “The five aberrations of lens systems,” Mon. Not. R. Astron. Soc. 79, 60–66 (1918).

1830 (1)

G. B. Airy, “On the spherical aberration of eyepieces of telescopes,” Trans. Cambridge Philos. Soc. 3, 1–65 (1830).
[CrossRef]

Airy, G. B.

G. B. Airy, “On the spherical aberration of eyepieces of telescopes,” Trans. Cambridge Philos. Soc. 3, 1–65 (1830).
[CrossRef]

Barr, E. S.

Buchroeder, R. A.

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

Cakmakci, O.

Conrady, A. E.

A. E. Conrady, “The five aberrations of lens systems,” Mon. Not. R. Astron. Soc. 79, 60–66 (1918).

Crowe, M. J.

M. J. Crowe, A History of Vector Analysis (Notre Dame University, 1967), pp. 6–8.

Fuerschbach, K.

K. P. Thompson, K. Fuerschbach, and J. P. Rolland, “An analytic expression for the field dependence of FRINGE Zernike coefficients in optical systems that are not rotationally symmetric,” Proc. SPIE 7849, 784906 (2010).
[CrossRef]

Hopkins, H. H.

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

Murty, M. V. R. K.

Needham, T.

T. Needham, Visual Complex Analysis (Claredon, 1997), p. 3.

Rakich, A.

Rimmer, M. C.

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

Rolland, J. P.

Sasian, J.

Schmid, T.

Shack, R. V.

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

Spencer, G. H.

Thompson, K.

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

Thompson, K. P.

K. P. Thompson, K. Fuerschbach, and J. P. Rolland, “An analytic expression for the field dependence of FRINGE Zernike coefficients in optical systems that are not rotationally symmetric,” Proc. SPIE 7849, 784906 (2010).
[CrossRef]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27, 1490–1504 (2010).
[CrossRef]

T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignment using Nodal Aberration Theory (NAT),” Opt. Express 18 (16), 17433–17447 (2010).
[CrossRef] [PubMed]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26, 1090–1100 (2009).
[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 26, 1503–1517 (2009).
[CrossRef]

K. P. Thompson, “Errata,” J. Opt. Soc. Am. A 26, 699 (2009).
[CrossRef]

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

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Mon. Not. R. Astron. Soc. (1)

A. E. Conrady, “The five aberrations of lens systems,” Mon. Not. R. Astron. Soc. 79, 60–66 (1918).

Opt. Express (1)

Proc. SPIE (2)

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

K. P. Thompson, K. Fuerschbach, and J. P. Rolland, “An analytic expression for the field dependence of FRINGE Zernike coefficients in optical systems that are not rotationally symmetric,” Proc. SPIE 7849, 784906 (2010).
[CrossRef]

Trans. Cambridge Philos. Soc. (1)

G. B. Airy, “On the spherical aberration of eyepieces of telescopes,” Trans. Cambridge Philos. Soc. 3, 1–65 (1830).
[CrossRef]

Other (7)

Translation by R. Zehnder on behalf of J. Sasian, available from jose.sasian@optics.arizona.edu.

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

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

M. J. Crowe, A History of Vector Analysis (Notre Dame University, 1967), pp. 6–8.

T. Needham, Visual Complex Analysis (Claredon, 1997), p. 3.

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

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

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Figures (6)

Fig. 1
Fig. 1

In a rotationally nonsymmetric optical system, the fifth-order contribution to the medial focal surface (the surface of minimum RMS WFE) develops three collinear zeros in the field. The central zero is degenerate, and the field dependence is quadratic rather than linear from this node. The field dependence is linear from the remaining two nodes, which are symmetrically located. The magnitude of the focal shift for any particular point in the field is then the product of the distances to the individual nodes.

Fig. 2
Fig. 2

Response of the balance zone for the third-order plus fifth-order medial focal surface in a rotationally nonsymmetric system. (a) Ring balance zone for the medial focal surface in an aligned system. (b) Perturbations can result in the center of rotationally symmetric field dependence for the third-order term, which is field quadratic, displacing off the axis, and similarly for the fifth-order term, which has quartic field dependence, and may displace to a different center resulting in a distortion of the previously circular balance zone. (c) Additionally, the fifth-order term can develop additional binodal symmetry, illustrated in Fig. 1, which will result in further distortions to the balance zone. (d) Third-order node (blue) and fifth-order nodes (red) that created (c).

Fig. 3
Fig. 3

In a rotationally nonsymmetric optical system, field-quartic, aperture-quadratic astigmatism, W 422 (a fifth-order aberration), develops four nodes in the field, i.e., quadranodal behavior. The vectors locating the nodal positions a 422 , x ¯ 422 , and x ˜ 422 are developed in Appendix C and described in Eqs. (25, 26, 27, 28).

Fig. 4
Fig. 4

Intrinsic nodal properties of the medial focal surface through the fifth order. The single node (blue) is associated with the third-order aberration and is a degenerate field-squared node. The three nodes (red), the center one of which is a degenerate field-squared node, are associated with the fifth-order terms defining the focal surface that contains the minimum RMS WFE. The location of the third-order node and the magnitude of the third-order aberration are affected by the fifth-order term. See Section 3 and/or Appendix B for details.

Fig. 5
Fig. 5

Intrinsic astigmatic nodal properties through the fifth order. Two nodes (blue) are associated with the third-order aberration (binodal), and four nodes (red) are associated with the fifth-order aberration (quadranodal). The location of the third-order nodes and the magnitude of the third-order aberration are affected by the fifth-order term. See Section 4 and/or Appendix C for details.

Fig. 6
Fig. 6

Aperture and field vectors used in the generalized vector formulation of the wave aberration expansion of Hopkins as developed by Shack.

Equations (69)

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W = Δ W 20 ( ρ · ρ ) + Δ W 11 ( H · ρ ) + W 040 ( ρ · ρ ) 2 + W 131 E [ ( H a 131 E ) · ρ ] ( ρ · ρ ) + W 220 M [ ( H a 220 M ) · ( H a 220 M ) + b 220 M ] ( ρ · ρ ) + 1 2 W 222 [ ( H a 222 ) 2 + b 222 2 ] · ρ 2 + j W 311 j [ ( H σ j ) · ( H σ j ) ] [ ( H σ j ) · ρ ] + W 060 ( ρ · ρ ) 3 + W 151 [ ( H a 151 ) · ρ ] ( ρ · ρ ) 2 + W 240 M [ ( H a 240 M ) · ( H a 240 M ) + b 240 M ] ( ρ · ρ ) 2 + 1 2 W 242 { [ ( H a 242 ) 2 + b 242 2 ] · ρ 2 } ( ρ · ρ ) + W 331 M { [ ( H a 331 M ) 2 + b 331 M 2 ] ( H a 331 M ) * · ρ } ( ρ · ρ ) + 1 4 W 333 [ ( H a 333 ) 3 + 3 ( H a 333 ) b 333 2 c 333 3 ] · ρ 3 + j W 420 S 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 .
W = Δ W 20 ( ρ · ρ ) + W 220 M [ ( H a 220 M ) · ( H a 220 M ) + b 220 M ] ( ρ · ρ ) + 1 2 W 222 [ ( H a 222 ) 2 + b 222 2 ] · ρ 2 + j W 420 S 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 .
2 cos 2 ( θ ϕ ) = 1 + cos 2 ( θ ϕ )
W 420 M = W 420 S + 1 2 W 422 ,
W Min RMS Focal Surface + Astigmatism = Δ W 20 ( ρ · ρ ) + W 220 M [ ( H a 220 M ) · ( H a 220 M ) + b 220 M ] ( ρ · ρ ) + 1 2 W 222 [ ( H a 222 ) 2 + b 222 2 ] · ρ 2 + j W 420 M j [ ( H σ j ) · ( H σ j ) ] [ ( H σ j ) · ( H σ j ) ] ( ρ · ρ ) + 1 2 j W 422 j [ ( H σ j ) · ( H σ j ) ] [ ( H σ j ) 2 · ρ 2 ] .
W Min RMS Focal Surface = W 220 M [ ( H a 220 M ) · ( H a 220 M ) + b 220 M ] ( ρ · ρ ) + j W 420 M j [ ( H σ j ) · ( H σ j ) ] [ ( H σ j ) · ( H σ j ) ] ( ρ · ρ ) .
W Unnormalized Fifth Medial Surface = [ W 420 M ( H · H ) ( H · H ) 4 ( H · H ) ( H · A 420 M ) + 4 B 420 M ( H · H ) + 2 ( H 2 · B 420 M 2 ) 4 ( H · C 420 M ) + D 420 M ] ( ρ · ρ ) ,
W Third, Asymmetric, Medial = W 220 M E [ ( H a 220 M E ) 2 + b 220 M E ] ( ρ · ρ ) .
W 220 M E = W 220 M + 4 W 420 M b 420 M ,
a 220 M E = A 220 M + W 420 M ( 2 c 420 M 2 b 420 M 2 a 420 M * ) W 220 M E ,
B 220 M E = B 220 M + W 420 M ( d 420 M 2 a 420 M 2 · b 420 M 2 ) ,
b 220 M E = B 220 M E / W 220 M E a 220 M E · a 220 M E ,
W = W 420 M { [ ( H a 420 M ) 2 + 2 b 420 M 2 ] · ( H a 420 M ) 2 } ( ρ · ρ ) , W Normalized, Nonsymmetric Fifth, Medial = W 420 M [ ( H 420 M 2 + 2 b 420 M 2 ) · H 420 M 2 ] ( ρ · ρ ) ,
W Third and Fifth Astigmatism = 1 2 W 222 [ ( H a 222 ) 2 + b 222 2 ] · ρ 2 + 1 2 j W 422 j [ ( H σ j ) · ( H σ j ) ] [ ( H σ j ) 2 · ρ 2 ] .
W Third and Fifth Astigmatism = 1 2 W 222 [ ( H a 222 ) 2 + b 222 2 ] · ρ 2 + 1 2 j W 422 j [ ( H σ j ) · ( H σ j ) ] [ ( H σ j ) 2 · ρ 2 ] .
W Unnormalized Fifth Astigmatism = 1 2 W 422 [ ( H · H ) H 2 2 ( H · H ) H A 422 + 3 ( H · H ) B 422 2 2 ( H · A 422 ) H 2 C 422 3 H * + 3 B 422 H 2 3 H C 422 + D 422 2 ] · ρ 2 ,
W 422 : W = 1 2 W 422 [ H 422 3 H 422 * + 3 H 422 b 422 2 H 422 * c 422 3 H 422 * + 3 b 422 H 2 3 H c 422 + d 422 2 + 3 H * a 422 b 422 2 + 3 H a 422 * b 422 2 3 a 422 a 422 * b 422 2 c 422 3 a 422 * ] · ρ 2 .
W Nonsymmetric, Third Astigmatism = 1 2 W 222 E [ ( H a 222 E ) 2 + b 222 E 2 ] · ρ 2 ,
W 222 E W 222 + 3 W 422 b 422 ,
a 222 E A 222 + 3 2 W 422 ( c 422 b 422 2 a 422 * ) W 222 E ,
B 222 E 2 B 222 2 + W 422 ( d 422 2 c 422 3 a 422 * ) ,
b 222 E 2 B 222 E 2 / W 222 E a 222 E 2 .
W = 1 2 W 422 { [ ( H a 422 ) 3 + 3 ( H a 422 ) b 422 2 ( c 422 3 3 a 422 b 422 2 ) ] ( H a 422 ) * } · ρ 2 , = 1 2 W 422 { [ ( H a 422 ) 3 + 3 ( H a 422 ) b 422 2 ( c 422 3 ) ] ( H a 422 ) * } · ρ 2 ;
W Normalized, Nonsymmetric Fifth Astigmatism = 1 2 W 422 { [ H 422 3 + 3 H 422 b 422 2 ( c 422 3 ) ] H 422 * } · ρ 2 ,
( c 422 3 ) c 422 3 3 b 422 2 a 422 ,
x ¯ 422 R 422 + S 422 2 ,
x ˜ 422 R 422 S 422 2 ,
R 422 { ( c 422 3 ) 2 + { [ ( c 422 3 ) ] 2 4 + ( b 422 2 ) 3 27 } 1 2 } 1 3 , S 422 { ( c 422 3 ) 2 { [ ( c 422 3 ) ] 2 4 + ( b 422 2 ) 3 27 } 1 2 } 1 3 .
2 x ¯ 422 ,
x ¯ 422 + i 3 x ˜ 422 ,
x ¯ 422 i 3 x ˜ 422 .
W = Δ W 20 ( ρ · ρ ) + Δ W 11 ( H · ρ ) + W 040 ( ρ · ρ ) 2 + W 131 E ( H 131 E · ρ ) ( ρ · ρ ) + W 220 M E [ ( H 220 M E · H 220 M E ) + b 220 M E ] ( ρ · ρ ) + 1 2 W 222 E ( H 222 E 2 + b 222 E 2 ) · ρ 2 + j W 311 j [ ( H σ j ) · ( H σ j ) ] [ ( H σ j ) · ρ ] + W 060 ( ρ · ρ ) 3 + W 151 ( H 151 · ρ ) ( ρ · ρ ) 2 + W 240 M [ ( H 240 M · H 240 M ) + b 240 M ] ( ρ · ρ ) 2 + 1 2 W 242 [ ( H 242 2 + b 242 2 ) · ρ 2 ] ( ρ · ρ ) + W 331 M [ ( H 331 M 2 + b 331 M 2 ) H 331 M * · ρ ] ( ρ · ρ ) + 1 4 W 333 ( H 333 3 + 3 H 333 b 333 2 c 333 3 ) · ρ 3 + W 420 M [ ( H 420 M 2 + 2 b 420 M 2 ) · H 420 M 2 ] ( ρ · ρ ) + 1 2 W 422 { [ H 422 3 + 3 H 422 b 422 2 ( c 422 3 ) ] H 422 * } · 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 ,
W 420 M : W = j W 420 M j [ ( H σ j ) · ( H σ j ) ] 2 ( ρ · ρ ) , = j W 420 M j [ ( H · H 2 H · σ j + σ j · σ j ) ( H · H 2 H · σ j + σ j · σ j ) ] ( ρ · ρ ) , = [ j W 420 M j ( H · H ) ( H · H ) 4 ( H · H ) ( H · ( j W 420 M j σ j ) ) + 2 ( H · H ) ( j W 420 M j ( σ j · σ j ) + 4 j W 420 M j ( H · σ j ) ( H · σ j ) 4 H · ( j W 420 M j ( σ j · σ j ) σ j ) + j W 420 M j ( σ j · σ j ) ( σ j · σ j ) ] ( ρ · ρ ) .
4 j W 420 M j ( H · σ j ) ( H · σ j ) = 2 ( H · H ) ( j W 420 M j ( σ j · σ j ) ) + 2 H 2 · ( j W 420 M j σ j 2 ) ,
2 ( A · B ) ( A · C ) = ( A · A ) ( B · C ) + ( A 2 · B C ) ,
W Unnormalized, Fifth Medial Surface = [ W 420 M ( H · H ) ( H · H ) 4 ( H · H ) ( H · A 420 M ) + 4 B 420 M ( H · H ) + 2 ( H 2 · B 420 M 2 ) 4 ( H · C 420 M ) + D 420 M ] ( ρ · ρ ) .
W 420 M : = [ W 420 M ( H 420 M · H 420 M ) ( H 420 M · H 420 M ) + 4 B 420 M ( H · H ) 4 W 420 M ( a 420 M · a 420 M ) ( H · H ) + 2 ( H 2 · B 420 M 2 ) 2 W 420 M ( H 2 · a 420 M 2 ) 4 ( H · C 420 M ) + 4 W 420 M ( H · a 420 M ) ( a 420 M · a 420 M ) + D 420 M W 420 M ( a 420 M · a 420 M ) ( a 420 M · a 420 M ) ] ( ρ · ρ ) .
( H · H ) ( H · H ) = ( H 2 · H 2 ) .
W 420 M : = W 420 M [ ( H 420 M 2 · H 420 M 2 ) + 2 ( H 2 · b 420 M 2 ) + 4 b 420 M ( H · H ) 4 ( H · c 420 M ) + d 420 M ] ( ρ · ρ ) = W 420 M [ ( H 420 M 2 · H 420 M 2 ) + 2 ( H 420 M 2 · b 420 M 2 ) + 4 b 420 M ( H · H ) 4 ( H · c 420 M ) + d 420 M + 4 ( H a 420 M · b 420 M 2 ) 2 a 420 M 2 · b 420 M 2 ] ( ρ · ρ ) W = W 420 M [ ( H 420 M 2 + 2 b 420 M 2 ) · H 420 M 2 ] ( ρ · ρ ) + W 420 M [ 4 b 420 M ( H · H ) 2 H · ( 2 c 420 M 2 b 420 M 2 a 420 M * ) + ( d 420 M 2 a 420 M 2 · b 420 M 2 ) ] ( ρ · ρ ) ,
W 220 M + W 420 M : W = W 220 M [ ( H 220 M · H 220 M ) + b 220 M ] ( ρ · ρ ) + W 420 M [ ( H 420 M 2 + 2 b 420 M 2 ) ( H 420 M · H 420 M ) ] ( ρ · ρ ) + W 420 M [ 4 b 420 M ( H · H ) 2 H · ( 2 c 420 M 2 b 420 M 2 a 420 M * ) + ( d 420 M 2 a 420 M 2 · b 420 M 2 ) ] ( ρ · ρ ) .
W 220 M E W 220 M + 4 W 420 M b 420 M ,
a 220 M E = A 220 M + W 420 M ( 2 c 420 M 2 b 420 M 2 a 420 M * ) W 220 M E ,
B 220 M E B 220 M + W 420 M ( d 420 M 2 a 420 M 2 · b 420 M 2 ) ,
b 220 M E B 220 M E / W 220 M E a 220 M E · a 220 M E ,
W Normalized, Nonsymmetric Third+Fifth Medial = W 220 M E [ ( H 220 M E · H 220 M E ) + b 220 M E ] ( ρ · ρ ) + W 420 M [ ( H 420 M 2 + 2 b 420 M 2 ) · H 420 M 2 ] ( ρ · ρ ) .
W = Δ W 20 ρ 2 + W 220 S H 2 ρ 2 + W 222 H 2 ρ 2 cos 2 ϕ + W 420 S H 4 ρ 2 + W 422 H 4 ρ 2 cos 2 ϕ .
W = Δ W 20 ρ 2 + W 220 S H 2 ρ 2 + W 222 H 2 ρ 2 cos 2 ( θ ϕ ) + W 420 S H 4 ρ 2 + W 422 H 4 ρ 2 cos 2 ( θ ϕ ) .
W = Δ W 20 ( ρ · ρ ) + W 220 S ( H · H ) ( ρ · ρ ) + W 222 ( H · ρ ) 2 + W 420 S ( H · H ) 2 ( ρ · ρ ) + W 422 ( H · H ) ( H · ρ ) 2 .
2 cos 2 ( θ ϕ ) = 1 + cos [ 2 ( θ ϕ ) ] ,
W 220 M W 220 S + 1 2 W 222 , W 420 M W 420 S + 1 2 W 422 .
W = Δ W 20 ρ 2 + W 220 M H 2 ρ 2 + 1 2 W 222 H 2 ρ 2 cos [ 2 ( θ ϕ ) ] + W 420 M H 4 ρ 2 + 1 2 W 422 H 4 ρ 2 cos [ 2 ( θ ϕ ) ] ,
W = Δ W 20 ( ρ · ρ ) + W 220 M ( H · H ) ( ρ · ρ ) + 1 2 W 222 ( H 2 · ρ 2 ) + W 420 M ( H · H ) 2 ( ρ · ρ ) + 1 2 W 422 ( H · H ) ( H 2 · ρ 2 ) .
W = Δ W 20 ( ρ · ρ ) + j W 220 M j [ ( H σ j ) · ( H σ j ) ] ( ρ · ρ ) + 1 2 j W 222 j [ ( H σ j ) 2 · ρ 2 ] + j W 420 M j [ ( H σ j ) · ( H σ j ) ] 2 ( ρ · ρ ) + 1 2 j W 422 j [ ( H σ j ) · ( H σ j ) ] ( H σ j ) 2 · ρ 2 ,
W 422 : W = 1 2 j W 422 j [ ( H σ j ) · ( H σ j ) ] ( H σ j ) 2 · ρ 2 , = 1 2 j W 422 j [ ( H · H 2 ( H · σ j ) + σ j · σ j ) ( H 2 2 ( H σ j ) + σ j 2 ) ] · ρ 2 , = 1 2 { j W 422 j ( H · H ) H 2 · ρ 2 2 ( H · H ) [ H ( j W 422 j σ j ) ] · ρ 2 + ( H · H ) ( j W 422 j σ j 2 ) · ρ 2 2 [ H · ( j W 422 j σ j ) ( c ) ] H 2 · ρ 2 + 4 j W 422 j ( H · σ j ) H σ j · ρ 2 ( d ) + ( b ) 2 j W 422 j ( H · σ j ) σ j 2 · ρ 2 + [ j W 422 j ( σ j · σ j ) ] H 2 · ρ 2 2 H [ j W 422 j ( σ j · σ j ) σ j ] · ρ 2 + [ j W 422 j ( σ j · σ j ) σ j 2 ] · ρ 2 } .
( c ) 4 j W 422 j ( H · σ j ) ( H σ j · ρ 2 ) = 2 [ j W 422 j ( σ j · σ j ) ] ( H 2 · ρ 2 ) + 2 ( H · H ) ( j W 422 j σ j 2 ) · ρ 2 ,
( d ) 2 j W 422 j ( H · σ j ) ( σ j 2 · ρ 2 ) = H [ j W 422 j ( σ j · σ j ) σ j ] · ρ 2 ( j W 422 j σ j 3 ) · H ρ 2 ,
( b ) ( j W 422 j σ j 3 ) · H ρ 2 = ( j W 422 j σ j 3 ) H * · ρ 2 .
W Unnormalized Fifth Astigmatism = 1 2 W 422 [ ( H · H ) H 2 2 ( H · H ) H A 422 + 3 ( H · H ) B 422 2 2 ( H · A 422 ) H 2 C 422 3 H * + 3 B 422 H 2 3 H C 422 + D 422 2 ] · ρ 2 .
W 422 : W = 1 2 [ W 422 ( H 422 · H 422 ) H 422 2 + 3 ( H · H ) B 422 2 3 W 422 ( H · H ) a 422 2 C 422 3 H * + W 422 a 422 3 H * + 3 B 422 H 2 3 W 422 ( a 422 · a 422 ) H 2 3 H C 422 + 3 W 422 H a 422 ( a 422 · a 422 ) + D 422 2 W 422 ( a 422 · a 422 ) a 422 2 ] · ρ 2 .
W 422 : W = 1 2 W 422 [ H 422 3 H 422 * + 3 H 422 b 422 2 H 422 * c 422 3 H 422 * + 3 b 422 H 2 3 H c 422 + d 422 2 + 3 H * a 422 b 422 2 + 3 H a 422 * b 422 2 3 a 422 a 422 * b 422 2 c 422 3 a 422 * ] · ρ 2 ,
W 422 : W = 1 2 W 422 { [ H 422 3 + 3 H 422 b 422 2 ( c 422 3 3 a 422 b 422 2 ) ] H 422 * } · ρ 2 + 1 2 W 422 [ 3 b 422 H 2 3 H ( c 422 a 422 * b 422 2 ) + ( d 422 2 c 422 3 a 422 * ) ] · ρ 2 .
W 222 + W 422 : W = 1 2 W 222 ( H 222 2 + b 222 2 ) · ρ 2 + 1 2 W 422 { H 422 3 + 3 H 422 b 422 2 ( c 422 3 3 b 422 2 a 422 ) ] H 422 * } · ρ 2 + 1 2 W 422 [ 3 b 422 H 2 2 H ( 3 2 c 422 3 2 b 422 2 a 422 * ) + ( d 422 2 c 422 3 a 422 * ) ] · ρ 2 ,
( c 422 3 ) c 422 3 3 b 422 2 a 422 ,
W 222 E W 222 + 3 W 422 b 422 ,
a 222 E A 222 + 3 2 W 422 ( c 422 b 422 2 a 422 * ) W 222 E ,
B 222 E 2 B 222 2 + W 422 ( d 422 2 c 422 3 a 422 * ) ,
b 222 E 2 B 222 E 2 / W 222 E a 222 E 2 .
W Normalized, Nonsymmetric Third and Fifth Astigmatism = 1 2 W 222 E ( H 222 E 2 + b 222 E 2 ) · ρ 2 + 1 2 W 422 { [ H 422 3 + 3 H 422 b 422 2 ( c 422 3 ) ] H 422 * } · ρ 2 ,
( c 422 3 ) c 422 3 3 b 422 2 a 422 .

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