Abstract

Interferometry is an optical testing technique that quantifies the optical path difference (OPD) between a reference wave front and a test wave front based on the interference of light. Fringes are formed when the OPD is an integral multiple of the illuminating wavelength. The resultant two-dimensional pattern is called an interferogram. The function of any interferogram analysis program is to extract this OPD and to produce a representation of the test wave front (or surface). This is accomplished through a three-step process of sampling, ordering, and fitting. We develop a generalized linear-algebra vector-notation model of the interferogram sampling and fitting process.

© 1993 Optical Society of America

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References

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  1. M. P. Rimmer, C. M. King, D. G. Fox, “Computer program for the analysis of interferometric test data,” Appl. Opt. 11, 2790–2796 (1972).
    [CrossRef] [PubMed]
  2. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  3. Cheol-Jung Kim, “Polynomial fit of interferograms,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1982).
  4. D. Malacara, J. M. Caprio-Valadez, J. J. Sanchez-Mon-dragon, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 24, 672–675 (1990).
    [CrossRef]
  5. J. L. Lewis, W. P. Kuhn, H. P. Stahl, “The evaluation of a random sampling error on the polynomial fit of subaperture test data,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 88–94 (1988).
  6. W. H. Swantner, W. H. Lowrey, “Zernike–Tatian polynomials for interferogram reduction,” Appl. Opt. 19, 161–163 (1980).
    [CrossRef] [PubMed]
  7. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
    [CrossRef]
  8. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annual pupils: errata,” J. Opt. Soc. Am. 71, 1408 (1981).
    [CrossRef]
  9. J. L. Rayces, “Least-squares fitting of orthogonal polynomials to the wave-aberration function,” Appl. Opt. 31, 2223–2228 (1992).
    [CrossRef] [PubMed]

1992 (1)

1990 (1)

D. Malacara, J. M. Caprio-Valadez, J. J. Sanchez-Mon-dragon, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 24, 672–675 (1990).
[CrossRef]

1981 (2)

1980 (2)

1972 (1)

Caprio-Valadez, J. M.

D. Malacara, J. M. Caprio-Valadez, J. J. Sanchez-Mon-dragon, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 24, 672–675 (1990).
[CrossRef]

Fox, D. G.

Kim, Cheol-Jung

Cheol-Jung Kim, “Polynomial fit of interferograms,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1982).

King, C. M.

Kuhn, W. P.

J. L. Lewis, W. P. Kuhn, H. P. Stahl, “The evaluation of a random sampling error on the polynomial fit of subaperture test data,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 88–94 (1988).

Lewis, J. L.

J. L. Lewis, W. P. Kuhn, H. P. Stahl, “The evaluation of a random sampling error on the polynomial fit of subaperture test data,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 88–94 (1988).

Lowrey, W. H.

Mahajan, V. N.

Malacara, D.

D. Malacara, J. M. Caprio-Valadez, J. J. Sanchez-Mon-dragon, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 24, 672–675 (1990).
[CrossRef]

Rayces, J. L.

Rimmer, M. P.

Sanchez-Mon-dragon, J. J.

D. Malacara, J. M. Caprio-Valadez, J. J. Sanchez-Mon-dragon, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 24, 672–675 (1990).
[CrossRef]

Silva, D. E.

Stahl, H. P.

J. L. Lewis, W. P. Kuhn, H. P. Stahl, “The evaluation of a random sampling error on the polynomial fit of subaperture test data,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 88–94 (1988).

Swantner, W. H.

Wang, J. Y.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

Opt. Eng. (1)

D. Malacara, J. M. Caprio-Valadez, J. J. Sanchez-Mon-dragon, “Wave-front fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 24, 672–675 (1990).
[CrossRef]

Other (2)

J. L. Lewis, W. P. Kuhn, H. P. Stahl, “The evaluation of a random sampling error on the polynomial fit of subaperture test data,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 88–94 (1988).

Cheol-Jung Kim, “Polynomial fit of interferograms,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Ariz., 1982).

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Figures (7)

Fig. 1
Fig. 1

When an interferogram is sampled by digitization, data points are usually placed along each fringe (either bright or dark).

Fig. 2
Fig. 2

When an interferogram is sampled by phase-measuring interferometry, data points are placed at each detector location. This usually results in a uniform high-density sampling of the pupil.

Fig. 3
Fig. 3

Top curve, One-dimensional slice of a parabolic wave front. Bottom curve, Resultant irradiance pattern formed when this slice interferes with a plane wave front. The plusses along the bottom of the graph represent data points sampled at irradiance minima.

Fig. 4
Fig. 4

When plotted, the ordered sample points clearly represent a parabolic shape.

Fig. 5
Fig. 5

Two arbitrary vectors in a two-dimensional space. The first vector is defined to be the first orthogonal vector of the space.

Fig. 6
Fig. 6

Two orthogonal vectors in a two-dimensional space. The function of the Gram–Schmidt orthogonalization process is to create these two vectors.

Fig. 7
Fig. 7

Gram–Schmidt orthogonalization process, which subtracts the projection of the second vector onto the proceeding vector to create an orthogonal vector.

Equations (28)

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Z ( x , y ) = j = 1 M F j ( x , y ) G j ,
Z r ( x r , y r ) = j = 1 M F j ( x r , y r ) G j ,
[ Z r ] = [ F 1 ( x r , y r ) , , F M ( x r , y r ) ] [ G 1 G M ] .
[ Z 1 Z N ] ,
[ Z 1 Z N ] = [ F 1 ( x 1 , y 1 ) , , F M ( x 1 , y 1 ) F 1 ( x N , y N ) , , F M ( x N , y N ) ] [ G 1 G M ] .
Z = [ Z 1 Z N ] , F j = [ F j ( x 1 , y 1 ) F j ( x N , y N ) ] , G = [ G 1 G M ] .
Z = [ F 1 , , F M ] G .
S | [ Z [ F 1 , , F M ] G ] | 2 ,
S r = 1 N [ Z r j = 1 M G j F j ( x r , y r ) ] 2 .
r = 1 N Z r F k ( x r , y r ) = r = 1 N j = 1 M G j F j ( x r , y r ) F k ( x r , y r ) .
[ r = 1 N Z r F 1 ( x r , y r ) r = 1 N Z r F M ( x r , y r ) ] = [ r = 1 N F 1 ( x r , y r ) F 1 ( x r , y r ) r = 1 N F M ( x r , y r ) F 1 ( x r , y r ) r = 1 N F 1 ( x r , y r ) F M ( x r , y r ) r = 1 N F M ( x r , y r ) F M ( x r , y r ) ] × [ G 1 G M ] .
[ Z · F 1 Z · F M ] = [ F 1 · F 1 F 1 · F M F M · F 1 F M · F M ] G ,
F i · F j = r = 1 N F i ( x r , y r ) F j ( x r , y r ) .
Z · F j = r = 1 N Z r F j ( x r , y r ) .
[ Z · F 1 Z · F M ] = [ F 1 · F 1 0 0 F M · F M ] G ,
G = [ Z · F 1 F 1 2 Z · F M F M 2 ] .
Φ j = F j s = 1 j 1 F j · Φ s Φ s · Φ s Φ s .
D j s = F s · Φ s Φ s · Φ s ,
Φ s = F j s = 1 j 1 D j s Φ s .
Z = [ Φ 1 , , Φ M ] Γ .
Γ = [ Z · Φ 1 Φ 1 2 Z · Φ M Φ M 2 ] .
[ Φ 1 T Φ M T ] = [ F 1 T F M T ] [ 0 0 D 21 0 0 D 31 D 32 0 0 D M 1 D M 2 D M M 1 0 ] [ Φ 1 T Φ M T ] .
[ Φ 1 T Φ M T ] = ( I + D ) 1 [ F 1 T F M T ] .
[ F 1 , , F M ] G = [ Φ 1 , , Φ M ] Γ .
[ Φ 1 , , Φ M ] T = ( I + D ) 1 [ F 1 , , F M ] T ,
[ Φ 1 , , Φ M ] = [ F 1 , , F M ] [ ( I + D ) 1 ] T .
[ F 1 , , F M ] G = [ F 1 , , F M ] [ ( I + D ) 1 ] T Γ ,
G = [ ( I + D ) 1 ] T Γ .

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