Abstract

Aberration functions that are a complete, orthogonal, and normalized set over a weighted spherical pupil are developed. A general weighting is considered, for which special cases are applicable to systems satisfying the Abbe sine condition and the Herschel condition. Paraboloidal mirrors are also considered. This weighting can also be used to account empirically for Fresnel reflection losses in the optical system. The functions can be expressed in an analytic form. Expressions are given for 24 low-order aberrations.

© 2004 Optical Society of America

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Errata

Colin J. R. Sheppard, "Orthogonal aberration functions for high-aperture optical systems: erratum," J. Opt. Soc. Am. A 21, 2468-2469 (2004)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-21-12-2468

References

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  1. F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
    [CrossRef]
  2. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).
  3. D. Malacara, Optical Shop Testing (Wiley, New York, 1992).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. C. J. R. Sheppard, “Aberrations in high-aperture conventional and confocal imaging systems,” Appl. Opt. 27, 4782–4786 (1988).
    [CrossRef] [PubMed]
  8. C. J. R. Sheppard, “Comment: vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138, 262–264 (1997).
    [CrossRef]
  9. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  10. C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
    [CrossRef]
  11. C. J. R. Sheppard, K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).
    [CrossRef] [PubMed]
  12. P. Török, P. Varga, G. Németh, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
    [CrossRef]

1997

C. J. R. Sheppard, “Comment: vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138, 262–264 (1997).
[CrossRef]

1995

1993

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

1988

1987

1981

1976

1959

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1934

F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).

Day, R. D.

Gu, M.

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

Larkin, K. G.

Lawrence, G. N.

Mahajan, V. N.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1992).

Németh, G.

Noll, R.

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, “Comment: vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138, 262–264 (1997).
[CrossRef]

C. J. R. Sheppard, K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).
[CrossRef] [PubMed]

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

C. J. R. Sheppard, “Aberrations in high-aperture conventional and confocal imaging systems,” Appl. Opt. 27, 4782–4786 (1988).
[CrossRef] [PubMed]

Török, P.

Varga, P.

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).

Zernike, F.

F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

C. J. R. Sheppard, “Comment: vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138, 262–264 (1997).
[CrossRef]

Physica

F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[CrossRef]

Proc. R. Soc. London Ser. A

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).

D. Malacara, Optical Shop Testing (Wiley, New York, 1992).

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Figures (5)

Fig. 1
Fig. 1

Form of some of the lower orders of aberration for an aplanatic system ( p = 1 / 2 ) for α = 90 ° : (a) m = 0 , (b) m = 1 , (c) m = 2 .

Fig. 2
Fig. 2

Form of some of the lower orders of aberration for a uniform angular distribution ( p = 0 ) for α = 90 ° : (a) m = 0 , (b) m = 1 , (c) m = 2 .

Fig. 3
Fig. 3

Behavior of the aberrations for an aplanatic system for a numerical aperture of 0.95: (a) m = 0 , (b) m = 1 , (c) m = 2 .

Fig. 4
Fig. 4

Behavior of the aberrations for an aplanatic system for a numerical aperture of 0.5: (a) m = 0 , (b) m = 1 , (c) m = 2 .

Fig. 5
Fig. 5

Form of some of the lower orders of aberration for a paraboloid mirror for α = 90 ° : (a) m = 0 , (b) m = 1 , (c) m = 2 .

Tables (1)

Tables Icon

Table 1 Coefficients for the Aberration Terms for Some Particular Cases

Equations (63)

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E = 0 2 π 0 α a ( θ ) exp ( 2 π i Φ ) ( 1 + cos   θ ) sin   θ d θ d ϕ ,
E = 0 2 π 0 α cos p   θ ( 1 + cos   θ ) sin   θ d θ d ϕ ,
c = cos   θ ,
E = 2 π cos   α 1 c p ( 1 + c ) d c ,
f l ( p ,   cos   α ) = cos   α 1 w ( c ) d c = cos   α 1 c p + l ( 1 + c ) d c = ( 1 - cos p + l + 1   α ) p + l + 1 + ( 1 - cos p + l + 2   α ) p + l + 2 ,
E = 2 π f 0 .
cos n   θ   sin m   θ   sin   m ϕ cos n   θ   sin m   θ   cos   m ϕ
m 0 , n 0 .
S = 1 - 4 π 2 [ Φ 2 ¯ - ( Φ ¯ ) 2 ] .
Φ 00 = A 00 = 1 .
2 π cos   α 1 A 00 2 c p ( 1 + c ) d c = 2 π f 0 .
Φ 01 = A 01 sin   θ   sin   ϕ ,
A 01 2 = 2 f 0 f 0 - f 2 .
Φ 10 = A 10 ( f 0 cos   θ - f 1 ) ,
A 10 2 = 1 f 0 f 2 - f 1 2 .
Φ 11 = A 11 [ ( f 0 - f 2 ) cos   θ - ( f 1 - f 3 ) ] sin   θ   sin   ϕ ,
A 11 2 = 2 f 0 ( f 0 - f 2 ) [ ( f 0 - f 2 ) ( f 2 - f 4 ) - ( f 1 - f 3 ) 2 ] .
Φ 02 = A 02 sin 2   θ   sin   2 ϕ ,
A 02 2 = 2 f 0 f 0 - 2 f 2 + f 4 .
Φ 20 = A 20 [ ( f 0 f 2 - f 1 2 ) cos 2   θ - ( f 0 f 3 - f 1 f 2 ) cos   θ + ( f 1 f 3 - f 2 2 ) ] ,
A 20 2 = f 0 ( f 0 f 2 - f 1 2 ) [ ( f 0 f 2 - f 1 2 ) f 4 - f 0 f 3 2 + 2 f 1 f 2 f 3 - f 2 3 ] .
ϕ 0 m = A 0 m sin m   θ   sin   m ϕ ,
A 0 m 2 = 2 f 0 / cos   α 1 ( 1 - c 2 ) m w ( c ) d c
= 2 f 0 / f 0 - mf 2 + m ( m - 1 ) 2 !   f 4 - + ( - 1 ) m f 2 m ,
= 2 f 0 / F 0 ,
F p = f p - mf p + 2 + m ( m - 1 ) 2 !   f p + 4 - + ( - 1 ) m f p + 2 m .
A 03 2 = 2 f 0 f 0 - 3 f 2 + 3 f 4 - f 6 ( trefoil ) ,
A 04 2 = 2 f 0 f 0 - 4 f 2 + 6 f 4 - 4 f 6 + f 8 ( tetrafoil ) ,
A 05 2 = 2 f 0 f 0 - 5 f 2 + 10 f 4 - 10 f 6 + 5 f 8 - f 10
( pentafoil ) ,
Φ 1 m = A 1 m ( F 0 cos   θ - F 1 ) sin m   θ   sin   m ϕ ,
A 1 m 2 = 2 f 0 F 0 ( F 0 F 2 - F 1 2 ) .
Φ 12 = A 12 [ ( f 0 - 2 f 2 + f 4 ) cos   θ - ( f 1 - 2 f 3 + f 5 ) ] sin 2   θ   sin   2 ϕ ,
A 12 2 = 2 f 0 ( f 0 - 2 f 2 + f 4 ) [ ( f 0 - 2 f 2 + f 4 ) ( f 2 - 2 f 4 + f 6 ) - ( f 1 - 2 f 3 + f 5 ) 2 ] ,
Φ 13 = A 12 [ ( f 0 - 3 f 2 + 3 f 4 - f 6 ) cos   θ - ( f 1 - 3 f 3 + 3 f 5 - f 7 ) ] sin 3   θ   sin   3 ϕ ,
A 13 2 = 2 f 0 ( f 0 - 3 f 2 + 3 f 4 - f 6 ) [ ( f 0 - 3 f 2 + 3 f 4 - f 6 ) ( f 2 - 3 f 4 + 3 f 6 - f 8 ) - ( f 1 - 3 f 3 + 3 f 5 - f 7 ) 2 ] ,
Φ 2 m = A 2 m [ ( F 0 F 2 - F 1 2 ) cos 2   θ - ( F 0 F 3 - F 1 F 2 ) cos   θ + ( F 1 F 3 - F 2 2 ) ] sin m   θ   sin   m ϕ ,
A 2 m 2 = 2 f 0 ( F 0 F 2 - F 1 2 ) [ ( F 0 F 2 - F 1 2 ) F 4 - F 0 F 3 2 + 2 F 1 F 2 F 3 - F 2 3 ] ,
Φ 21 = A 22 { [ ( f 0 - f 2 ) ( f 2 - f 4 ) - ( f 1 - f 3 ) 2 ] cos 2   θ } - [ ( f 0 - f 2 ) ( f 3 - f 5 ) - ( f 1 - f 3 ) ( f 2 - f 4 ) ] cos   θ + [ ( f 1 - f 3 ) ( f 3 - f 5 ) - ( f 2 - f 4 ) 2 ] } sin   θ   sin   ϕ ,
A 21 2 = 2 f 0 [ ( f 0 - f 2 ) ( f 2 - f 4 ) - ( f 1 - f 3 ) 2 ] - 1 × { [ ( f 0 - f 2 ) ( f 2 - f 4 ) - ( f 1 - f 3 ) 2 ] ( f 4 - f 6 ) - ( f 0 - f 2 ) ( f 3 - f 5 ) 2 + 2 ( f 1 - 2 f 3 + f 5 )× ( f 2 - 2 f 4 + f 6 ) ( f 3 - 2 f 5 + f 7 ) - ( f 2 - 2 f 4 + f 6 ) 3 } - 1 ,
Φ 22 = A 22 { [ ( f 0 - 2 f 2 + f 4 ) ( f 2 - 2 f 4 + f 6 ) - ( f 1 - 2 f 3 + f 5 ) 2 ] cos 2   θ - [ ( f 0 - 2 f 2 + f 4 ) ( f 3 - 2 f 5 + f 7 ) - ( f 1 - 2 f 3 + f 5 ) ( f 2 - 2 f 4 + f 6 ) ] cos   θ + [ ( f 1 - 2 f 3 + f 5 ) ( f 3 - 2 f 5 + f 7 ) - ( f 2 - 2 f 4 + f 6 ) 2 ] } sin 2   θ   sin   2 ϕ ,
A 22 2 = 2 f 0 [ ( f 0 - 2 f 2 + f 4 ) ( f 2 - 2 f 4 + f 6 ) - ( f 1 - 2 f 3 + f 5 ) 2 ] - 1 × { [ ( f 0 - 2 f 2 + f 4 ) ( f 2 - 2 f 4 + f 6 ) - ( f 1 - 2 f 3 + f 5 ) 2 ] ( f 4 - 2 f 6 + f 8 ) - ( f 0 - 2 f 2 + f 4 ) ( f 3 - 2 f 5 + f 7 ) 2 + 2 ( f 1 - 2 f 3 + f 5 ) ( f 2 - 2 f 4 + f 6 ) ( f 3 - 2 f 5 + f 7 ) - ( f 2 - 2 f 4 + f 6 ) 3 } - 1 .
Φ nm = A nm ( b n c n + b n - 1 c n - 1 + + b 0 ) ( 1 - c 2 ) m sin   m ϕ ,
F 0 F 1 F 2 F 3 0 G 1 G 2 G 3 0 0 H 2 H 3 0 0 0 b n b n - 1 b 0 = 0 ,
G 1 = F 0 F 1 F 1 F 2 , G 2 = F 0 F 1 F 2 F 3 ,
H 2 = F 0 F 1 F 2 F 1 F 2 F 3 F 2 F 3 F 4 , H 3 = F 0 F 1 F 2 F 1 F 2 F 3 F 3 F 4 F 5 ,
A nm 2 = f 0 b n 2 F 2 n + 2 b n b n - 1 F 2 n - 1 + + b 0 2 F 0 ,
= 1 , m = 0 ,
= 2 , m 0 .
Φ 3 m = A 3 m H 2   cos 3   θ - H 3   cos 2   θ + 1 G 1   G 2 G 3 H 2 H 3 cos   θ - 1 F 0 G 1   F 1 F 2 F 3 G 1 G 2 G 3 0 H 2 H 3 sin m   θ   sin   m ϕ
= A 3 m { H 2 cos 3   θ - H 3 cos 2   θ + [ G 2 F 5 - F 4 ( F 0 F 4 - F 2 2 ) + F 3 ( F 1 F 4 - F 2 F 3 ) ] cos   θ - [ F 5 ( F 1 F 3 - F 2 2 ) - F 1 F 4 2 + 2 F 2 F 3 F 4 - F 3 3 ] } sin m   θ   sin   m ϕ .
Φ 30 = A 3 m { h 2 cos 3   θ - h 3 cos 2   θ + [ g 2 f 5 - f 4 ( f 0 f 4 - f 2 2 ) + f 3 ( f 1 f 4 - f 2 f 3 ) ] cos   θ - [ f 5 ( f 1 f 3 - f 2 2 ) - f 1 f 4 2 + 2 f 2 f 3 f 4 - f 3 3 ] } ,
s = 2   sin θ 2 .
Φ nm = A nm ( d n s 2 n + d n - 1 s 2 n - 2 + + d 0 ) sin m   θ   sin   m ϕ .
ρ = s 2   sin ( α / 2 ) .
Φ nm = A nm d n 2   sin α 2 2 n ρ 2 n + d n - 1 2   sin α 2 2 n - 2 ρ 2 n - 2 + + d 0 × ρ m 2   sin   α 2 m 1 - ρ 2   sin 2   α 2 m / 2 sin   m ϕ
= A nm ( ρ 2 n + m + e n - 1 ρ 2 n + m - 2 + e 0 ρ m )× 1 - ρ 2   sin 2   α 2 m / 2 sin   m ϕ ,
e k = d k d n 2   sin   α 2 - 2 ( n - k )
A nm = A nm d n 2   sin   α 2 2 n + m .
Φ nm = A nm ( ρ 2 n + m + e n - 1 ρ 2 n + m - 2 + + e 0 ρ m ) sin   m ϕ ,
ρ sin   θ sin   α .
a ( c ) = 2 1 + c ,
f l ( p ,   cos   α ) = cos   α 1 2 1 + c   c p + l ( 1 + c ) d c = 2 ( 1 - cos p + l + 1 α ) p + l + 1 .

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