Abstract

Polynomial fit of interferograms is analyzed quantitatively. The errors from polynomial fit, such as fit error, digitization error, roundoff error, and finite sampling error, are explained. The advantage of using orthonormal polynomials are presented. The best reference wave front and the relative reference wave front are defined, and their characteristics are compared. The possibility of using nonorthonormal polynomials for analysis of noncircular aperture interferograms is discussed. A simulation using Zernike polynomials for an annular aperture interferogram is shown. Finally a method of obtaining the surface figure error information from several smaller subaperture interferograms is introduced, and a simulation of testing a large flat is shown.

© 1982 Optical Society of America

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References

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  1. M. P. Rimmer, C. M. King, D. G. Fox, Appl. Opt. 11, 2790 (1972).
    [Crossref] [PubMed]
  2. J. S. Loomis, FRINGE User’s Manual (Optical Sciences Center, U. Arizona, Tucson, 1976).
  3. J. Y. Wang, D. E. Silva, Appl. Opt. 19, 1510 (1980).
    [Crossref] [PubMed]
  4. W. H. Swantner, W. H. Lowrey, Appl. Opt. 19, 161 (1980).
    [Crossref] [PubMed]
  5. C.-J. Kim, “Polynomial Fit of Interferograms,” Ph.D. Dissertation, U. Arizona, Tucson (1982).
  6. C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  7. K.-L. Shu, R. E. Parks, R. R. Shannon, J. Opt. Soc. Am. 71, 1587 (1981).

1981 (1)

K.-L. Shu, R. E. Parks, R. R. Shannon, J. Opt. Soc. Am. 71, 1587 (1981).

1980 (2)

1972 (1)

Fox, D. G.

Hanson, R. J.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Kim, C.-J.

C.-J. Kim, “Polynomial Fit of Interferograms,” Ph.D. Dissertation, U. Arizona, Tucson (1982).

King, C. M.

Lawson, C. L.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Loomis, J. S.

J. S. Loomis, FRINGE User’s Manual (Optical Sciences Center, U. Arizona, Tucson, 1976).

Lowrey, W. H.

Parks, R. E.

K.-L. Shu, R. E. Parks, R. R. Shannon, J. Opt. Soc. Am. 71, 1587 (1981).

Rimmer, M. P.

Shannon, R. R.

K.-L. Shu, R. E. Parks, R. R. Shannon, J. Opt. Soc. Am. 71, 1587 (1981).

Shu, K.-L.

K.-L. Shu, R. E. Parks, R. R. Shannon, J. Opt. Soc. Am. 71, 1587 (1981).

Silva, D. E.

Swantner, W. H.

Wang, J. Y.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

K.-L. Shu, R. E. Parks, R. R. Shannon, J. Opt. Soc. Am. 71, 1587 (1981).

Other (3)

J. S. Loomis, FRINGE User’s Manual (Optical Sciences Center, U. Arizona, Tucson, 1976).

C.-J. Kim, “Polynomial Fit of Interferograms,” Ph.D. Dissertation, U. Arizona, Tucson (1982).

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

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Figures (8)

Fig. 1
Fig. 1

Least squares criterion in vector representation.

Fig. 2
Fig. 2

Advantages of using orthonormal polynomials.

Fig. 3
Fig. 3

Best reference and relative reference. S and R are the spaces formed by the surface polynomials and the reference polynomials.

Fig. 4
Fig. 4

Severely correlated polynomial. (SZ)i − 1 is the space formed by the polynomials from Z1 to Zi − 1.

Fig. 5
Fig. 5

Sources of correlation in the subaperture interferogram analysis.

Fig. 6
Fig. 6

Geometry of 2/5-radius subaperture test.

Fig. 7
Fig. 7

Simulations of 2/5-subaperture interferograms.

Fig. 8
Fig. 8

Error from 2/5 subaperture test (rms error is 0.053 wave).

Tables (2)

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Table I Statistics of the Interferogram and the Roundoff Error

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Table II Errors in the Coefficients of Circular Symmetric Terms a

Equations (8)

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i = 1 N [ W ( X i , Y i ) j = 1 M A j Z j ( X i , Y i ) ] 2 = min ,
W = 1 N [ W ( X 1 , Y 1 ) a ^ 1 + W ( X 2 , Y 2 ) a ^ 2 + + W ( X N , Y N ) a ^ N ] .
Z j = 1 N [ Z j ( X 1 , Y 1 ) a ^ 1 + Z j ( X 2 , Y 2 ) a ^ 2 + ... + Z j ( X N , Y N ) a ^ N ] .
|  U  | = [ 1 N i = 1 N U ( X i , Y i ) 2 ] 1 / 2 .
| W Z | = | W j = 1 M A j Z j | = | F | = min ,
| Δ W | = [ 1 N i = 1 N Δ W ( X i , Y i ) 2 ] 1 / 2 = σ N
| Δ Z | = M N | Δ W | = M N σ N .
1 ( W 1 i = 1 M r B 1 i Z 1 i j = M r + 1 N A j Z j ) 2 + 2 ( W 2 i = 1 M r B 2 i Z 2 i j = M r + 1 N A j Z j ) 2 + + K ( W K i = 1 M r B K i Z K i j = M r + 1 N A j Z j ) 2 = min ,

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