Abstract

We present analytical derivations of aberration functions for annular sector apertures. We show that the Zernike functions for circular apertures can be generalized for any aperture shape. Interferogram reduction when Zernike functions were used as a basis set was performed on annular sectors. We have created a computer program to generate orthogonal aberration functions. Completely general aperture shapes and user-selected basis sets may be treated with a digital Gram–Schmidt orthonormalization approach.

© 1994 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 464.
  2. B. Tatian, “Aberration balancing in rotationally symmetric lenses,” J. Opt. Soc. Am. 64, 1083–1091 (1974).
    [CrossRef]
  3. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
    [CrossRef]
  4. R. Barakat, “Optimum balance wave-front aberrations for radially symmetric amplitude distributions: generalizations of Zernike polynomials,” J. Opt. Soc. Am. 70, 739–742 (1980).
    [CrossRef]
  5. W. H. Swantner, W. H. Lowrey, “Zernike–Tatian polynomials for interferogram reduction,” Appl. Opt. 19, 161–163 (1980).
    [CrossRef] [PubMed]
  6. C. R. Hayslett, W. H. Swantner, “Wave-front derivation by three computer programs,” Appl. Opt. 19, 3401–3406 (1980).
    [CrossRef] [PubMed]
  7. J. Loomis, “A computer program for analysis of interferometric data,” in Optical Interferograms—Reduction and Interpretation, A. H. Guenther, D. H. Liebenberg, eds. (American Society for Testing and Materials, Philadelphia, Pa., 1978), pp. 71–86.
    [CrossRef]
  8. D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
    [CrossRef]
  9. M. P. Rimmer, J. C. Wyant, “Evaluation of large aberrations using a lateral shear interferometer having variable shear,” Appl. Opt. 14, 142–150 (1975).
    [PubMed]
  10. W. Wolfe, Optical Sciences Center, University of Arizona, Tucson, Ariz. 85721 (personal communication, 1983).
  11. F. Scheid, Numerical Analysis (McGraw-Hill, New York, 1968), Chap. 5, p. 148.
  12. W. Swantner, “Wave fronts of axicon systems,” Opt. Eng. 21, 333–339 (1982).
  13. S. Steinberg of the University of New Mexico used the vaxima14 program [Version 10 (1983)] to derive the orthogonal aberration functions for the W = r4 + Dy problem for a sector aperture.
  14. vaxima/macsyma are copyrighted by the Mathlab Group, Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, Mass.

1990 (1)

D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

1982 (1)

W. Swantner, “Wave fronts of axicon systems,” Opt. Eng. 21, 333–339 (1982).

1981 (1)

1980 (3)

1975 (1)

1974 (1)

Barakat, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 464.

Carpio-Valadez, J. M.

D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Hayslett, C. R.

Loomis, J.

J. Loomis, “A computer program for analysis of interferometric data,” in Optical Interferograms—Reduction and Interpretation, A. H. Guenther, D. H. Liebenberg, eds. (American Society for Testing and Materials, Philadelphia, Pa., 1978), pp. 71–86.
[CrossRef]

Lowrey, W. H.

Mahajan, V. N.

Malacara, D.

D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Rimmer, M. P.

Sanchez-Mondragon, J. J.

D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Scheid, F.

F. Scheid, Numerical Analysis (McGraw-Hill, New York, 1968), Chap. 5, p. 148.

Swantner, W.

W. Swantner, “Wave fronts of axicon systems,” Opt. Eng. 21, 333–339 (1982).

Swantner, W. H.

Tatian, B.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 464.

Wolfe, W.

W. Wolfe, Optical Sciences Center, University of Arizona, Tucson, Ariz. 85721 (personal communication, 1983).

Wyant, J. C.

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

Opt. Eng. (2)

W. Swantner, “Wave fronts of axicon systems,” Opt. Eng. 21, 333–339 (1982).

D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[CrossRef]

Other (6)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 464.

W. Wolfe, Optical Sciences Center, University of Arizona, Tucson, Ariz. 85721 (personal communication, 1983).

F. Scheid, Numerical Analysis (McGraw-Hill, New York, 1968), Chap. 5, p. 148.

S. Steinberg of the University of New Mexico used the vaxima14 program [Version 10 (1983)] to derive the orthogonal aberration functions for the W = r4 + Dy problem for a sector aperture.

vaxima/macsyma are copyrighted by the Mathlab Group, Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, Mass.

J. Loomis, “A computer program for analysis of interferometric data,” in Optical Interferograms—Reduction and Interpretation, A. H. Guenther, D. H. Liebenberg, eds. (American Society for Testing and Materials, Philadelphia, Pa., 1978), pp. 71–86.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Simulated interferogram of a corner-cube retroreflector.

Fig. 2
Fig. 2

Simulated interferogram of a segment of an annular sector aperture.

Fig. 3
Fig. 3

Computer-generated contours of the wave front, W = 10r2 + 50y.

Fig. 4
Fig. 4

Computer-generated contours of the residual wave-front error, W = r2 − 1.217y + 0.2238.

Fig. 5
Fig. 5

Computer-generated contours of the residual wave-front error, W= r4 − 1.3623r2 + 0.453.

Tables (1)

Tables Icon

Table 1 Results from Test Cases Run with sector

Equations (25)

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W 2 = A 2 r 4 + 2 A r 6 + r 8 ,
W 2 = 0 2 π d θ 1 r ( A 2 r 4 + 2 A r 6 + r 8 ) d r 0 2 π d θ 1 r d r ,
W = 0 2 π d θ 1 d r r ( A r 2 + r 4 ) 0 2 π d θ 1 d r r .
σ 2 = A 2 12 ( 1 2 ) 2 + A 6 { 1 2 [ 1 + 2 ( 1 2 ) ] } + 4 45 2 45 2 4 14 6 45 + 4 8 45 .
A = ( 1 + 2 ) .
W = 1 6 ( 1 + 4 2 + 4 ) ,
σ 2 = 1 180 ( 1 2 ) 4 .
Z ( r , θ ) = r 4 ( 1 + 2 ) r 2 + 1 6 ( 1 + 4 2 + 4 ) .
W = r 2 + D r sin θ,
y = r sin θ .
W 2 = α α d θ 1 r ( r 2 + D r cos ϕ ) 2 d r α α d ϕ 1 r d r ,
W = α α d ϕ 1 r ( r 2 + D r cos ϕ ) d r α α d ϕ 1 r d r ,
σ 2 = 1 12 ( 1 2 ) 2 + D [ 4 sin α 5 α 1 5 1 2 2 sin α 9 α ( 1 + 2 ) ( 1 3 ) 1 2 ] + D 2 [ ( 1 4 + sin 2 α 8 α ) ( 1 + 2 ) ( 2 sin α α 1 3 1 2 ) 2 ] .
D = 4 sin α 5 α 1 5 1 2 + 2 sin α 9 α ( 1 + 2 ) ( 1 3 ) 1 2 ( 1 2 + sin 2 α 4 α ) ( 1 + 2 ) 2 ( 2 sin α 3 α 1 3 1 2 ) 2 .
W = α 2 ( 1 4 ) + 2 D 3 ( 1 3 ) sin α .
Z = r 2 1 . 217 y W = r 2 1 . 217 y + 0 . 2238 .
W = 10 r 2 + 50 y .
G 1 = Z 1 / 28 . 95 , G 3 = ( Z 3 0 . 8736 ) / 1 . 759 , G 4 = ( Z 4 4 . 288 G 3 19 . 62 G 1 ) / 3 . 209 ,
W = 1507 . 12584 G 1 + 109 . 37365 G 3 + 16 . 0470727 G 4 .
Z 1 = 1 , Z 3 = y , Z 4 = 2 r 2 1 ,
W = 5 . 0006 [ 2 r 2 1 2 . 4377 ( y 0 . 8736 ) 0 . 6777 ] + 62 . 1794 ( y 0 . 8736 ) + 52 . 0596 ,
W = 10 . 001 r 2 + 49 . 989 y + 0 . 0005 .
W = 43 . 2792 G 1 + 9 . 0417 G 3 + 0 . 5230 G 4 ,
σ 2 = 43 . 2792 2 + 9 . 0417 2 + 0 . 5230 2 = 1955 . 1166 .
S . D . = ( 9 . 0417 2 + 0 . 5230 2 ) 1 / 2 = 9 . 057 .

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