Abstract

We present a new technique for using the information of two orthogonal lateral-shear interferograms to estimate an aspheric wave front. The wave-front estimation from sheared inteferometric data may be considered an ill-posed problem in the sense of Hadamard. We apply Thikonov regularization theory to estimate the wave front that has produced the lateral sheared interferograms as the minimizer of a positive definite-quadratic cost functional. The introduction of the regularization term permits one to find a well-defined and stable solution to the inverse shearing problem over the wave-front aperture as well as to reduce wave-front noise as desired.

© 1996 Optical Society of America

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References

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  1. D. Malacara, “Twyman-Green interferometer,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 51–94.
  2. M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 1–49.
  3. A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 321–365.
  4. D.-S. Wan, D.-T. Lin, “Ronchi test and a new phase reduction algorithm,” Appl. Opt. 29, 3255–3265 (1988).
    [CrossRef]
  5. K. Omura, T. Yatagai, “Phase measuring Ronchi test,” Appl. Opt. 27, 523–528 (1988).
    [CrossRef] [PubMed]
  6. M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 123–172.
  7. 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]
  8. D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  9. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  10. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  11. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  12. H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  13. D. C. Ghiglia, L. A. Romero, “Direct phase estimation from phase differences using fast elliptic partial differential equation solvers,” Opt. Lett. 14, 1107–1109 (1989).
    [CrossRef] [PubMed]
  14. J. Hadamard, “Sur les problems aux derives partielles et leur signification physique,” Princeton University Bulletin 13 (Princeton University, Princeton, N.J., 1902).
  15. A. N. Thikonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).
  16. D. Malcara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 455–500.
  17. J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” CVGIP Graphic. Models Image Process. 55, 408–417 (1993).
    [CrossRef]
  18. G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).
  19. J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393–2400 (1995).
    [CrossRef]

1995

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393–2400 (1995).
[CrossRef]

1993

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” CVGIP Graphic. Models Image Process. 55, 408–417 (1993).
[CrossRef]

1989

1988

1979

1978

1977

1975

1963

A. N. Thikonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

Cornejo-Rodriguez, A.

A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 321–365.

DeVore, S. L.

D. Malcara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 455–500.

Fried, D. L.

Ghiglia, D. C.

Gollub, G. H.

G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).

Hadamard, J.

J. Hadamard, “Sur les problems aux derives partielles et leur signification physique,” Princeton University Bulletin 13 (Princeton University, Princeton, N.J., 1902).

Hudgin, R. H.

Hunt, B. R.

Lin, D.-T.

Malacara, D.

D. Malacara, “Twyman-Green interferometer,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 51–94.

Malcara, D.

D. Malcara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 455–500.

Mantravadi, M. V.

M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 1–49.

M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 123–172.

Marroquin, J. L.

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393–2400 (1995).
[CrossRef]

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” CVGIP Graphic. Models Image Process. 55, 408–417 (1993).
[CrossRef]

Noll, R. J.

Omura, K.

Rimmer, M. P.

Rivera, M.

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393–2400 (1995).
[CrossRef]

Romero, L. A.

Takahashi, T.

Takajo, H.

Thikonov, A. N.

A. N. Thikonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

Van Loan, C. F.

G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).

Wan, D.-S.

Wyant, J. C.

Yatagai, T.

Appl. Opt.

CVGIP Graphic. Models Image Process.

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” CVGIP Graphic. Models Image Process. 55, 408–417 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Sov. Math. Dokl.

A. N. Thikonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

Other

D. Malcara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 455–500.

G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).

D. Malacara, “Twyman-Green interferometer,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 51–94.

M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 1–49.

A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 321–365.

M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), pp. 123–172.

J. Hadamard, “Sur les problems aux derives partielles et leur signification physique,” Princeton University Bulletin 13 (Princeton University, Princeton, N.J., 1902).

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Figures (6)

Fig. 1
Fig. 1

A sinusoidally shaped wave front that gives no inteference fringes for that particular amount of lateral shear.

Fig. 2
Fig. 2

Frequency response of the interferometer [Eq. (3)] plus wave-front recovery system; the solid trace shows the response for the proposed inverse filter [Eq. (13) with λ = 0.01] and the dashed line shows the response for the Fried–Hudgin technique of least-squares integration [Eq. (17)].

Fig. 3
Fig. 3

Shearing interferograms along the x and y directions calculated according to Eqs. (19) and (20).

Fig. 4
Fig. 4

Estimated wave front over the full wave-front aperture P(x, y) for the proposed technique. (a) Estimated wave front. (b) Error between the analytical wave front [Eq. (19)] and its estimation. The peak error was 0.29 rad, and its standard deviation was 0.0007 rad.

Fig. 5
Fig. 5

Estimated wave front over the aperture’s domain P(x, y) for the least-squares integration technique proposed by Freid8 and Hudgin.9 (a) Estimated wave front. (b) Error between the analytical wave front [Eq. (19)] and its estimation. The peak error was 5.9 rad, and its standard deviation was 0.016 rad. As is shown here, the estimation error over regions of no overlap and the poor frequency response of this system have increased the detection error.

Fig. 6
Fig. 6

(a) Central line of the actual wave front that modulates the shearograms shown in Fig. 3. (b) Central line of the estimated wave front according to the proposed technique. (c) Central line of the estimated wave front for the Fried–Hudgin technique of least-squares integration.

Equations (20)

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I x ( x , y ) = a ( x , y ) + b ( x , y ) cos × { 2 π λ [ 2 D x + W ( x δ x , y ) W ( x + δ x , y ) ] } × P ( x δ x , y ) P ( x + δ x , y ) ,
Δ x ( x , y ) = [ W ( x δ x , y ) W ( x + δ x , y ) ] × P ( x δ x , y ) P ( x + δ x , y ) ,
H SI ( ω x , ω y ) = F [ Δ x ( x , y ) ] F [ W ( x , y ) ] = 2 sin ( δ x ω x ) × ( π ω x π ) ,
δ x ω x = n π, n = 0 , 1 , 2 , 3 , ….
U = ( x , y ) L U x ( x , y ) 2 + U y ( x , y ) 2 , ( x , y ) L ,
U x ( x , y ) = [ W ˆ ( x + δ x , y ) W ˆ ( x δ x , y ) Δ x ( x , y ) ] × P ( x δ x , y ) P ( x + δ x , y ) , U y ( x , y ) = [ W ˆ ( x , y + δ y ) W ˆ ( x , y δ y ) Δ y ( x , y ) ] × P ( x , y δ y ) P ( x , y + δ y ) .
Δ x ( x , y ) = [ W ( x δ x , y ) W ( x + δ x , y ) ] × P ( x δ x , y ) P ( x + δ x , y ) , Δ y ( x , y ) = [ W ( x , y δ y ) W ( x , y + δ y ) ] | × P ( x , y δ y ) P ( x , y + δ y ) .
R x ( x , y ) = { W ˆ ( x 1 , y ) 2 W ˆ ( x , y ) + W ˆ ( x + 1 , y ) } × P ( x 1 , y ) P ( x + 1 , y ) , R y ( x , y ) = { W ˆ ( x , y 1 ) 2 W ˆ ( x , y ) + W ˆ ( x , y + 1 ) } × P ( x , y 1 ) P ( x , y + 1 ) .
U = ( x , y ) L U x ( x , y ) 2 + U y ( x , y ) 2 + λ × [ R x ( x , y ) 2 + R y ( x , y ) 2 ] , ( x , y ) L .
U W ˆ ( x , y ) = U x ( x + δ x , y ) + U x ( x δ x , y ) U y ( x , y + δ y ) + U y ( x , y δ y ) + λ [ R x ( x + 1 , y ) 2 R x ( x , y ) + R x ( x 1 , y ) ] + λ [ R y ( x , y + 1 ) 2 R y ( x , y ) + R y ( x , y 1 ) ] = 0 .
W ˆ ( x , y ) k + 1 = W ˆ ( x , y ) k η U W ˆ ( x , y ) ,
Δ x ( x , y ) = δ x δ x Δ x ( x , y ) , Δ y ( x , y ) = δ y δ y Δ x ( x , y ) .
| H ( ω x , λ ) | = | F [ W ˆ ( x , y ) ] F [ Δ x ( x , y ) ] | = | 2 sin ( δ x ω x ) 2 2 cos ( 2 δ x ω x ) + λ [ 6 8 cos ( ω x ) + 2 cos ( 2 ω x ) ] | ,
U = ( x , y ) L U x ( x , y ) 2 + U y ( x , y ) 2 ( x , y ) L ,
U x ( x , y ) = [ W ˆ ( x + 1 , y ) W ˆ ( x , y ) Δ x ( x , y ) 2 δ x ] × P ( x + 1 , y ) P ( x , y ) , U y ( x , y ) = [ W ˆ ( x , y + 1 ) W ˆ ( x , y ) Δ y ( x , y ) 2 δ x ] × P ( x , y + 1 ) P ( x , y ) .
W ˆ ( x , y ) k + 1 = W ˆ ( x , y ) k η [ U x ( x 1 , y ) U x ( x , y ) + U y ( x , y 1 ) U y ( x , y ) ] ,
| H FH ( ω x ) | = | F [ W ˆ ( x , y ) ] F [ Δ x ( x , y ) ] | = | { [ 1 cos ( ω x ) ] 2 + sin ( ω x ) 2 } 1 / 2 2 δ x [ 2 2 cos ( 2 ω x ) ] | .
| H ˆ ( ω x ) FH | = | H FH ( ω x ) | | H SI ( ω x ) | , | H ˆ ( ω x ) proposed | = | H ( ω x , λ ) | | H SI ( ω x ) | .
W ( x , y ) = 10 λ [ ( x 2 + 1 . 4 y 2 ) 2 1 . 5 ( x 2 + y 2 ) + 0 . 008 cos ( 70 x 2 + y 2 ) ] P ( x , y ) .
I x = 127 + 100 cos × { 2 π λ [ y + W ( x 0 . 08 , y ) W ( x + 0 . 08 , y ) ] } × P ( x 0 . 08 , y ) P ( x + 0 . 08 , y ) , ( 1 x , y 1 ) , I y = 127 + 100 cos × { 2 π λ [ x + W ( x , y 0 . 08 ) W ( x , y + 0 . 08 ) ] } × P ( x , y 0 . 08 ) P ( x , y + 0 . 08 ) , ( 1 x , y 1 ) .

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