Abstract

An algorithm is proposed to reconstruct two-dimensional wave-front from phase differences measured by lateral shearing interferometer. Two one-dimensional phase profiles of object wave-front are computed using Fourier transform from phase differences, and then the two-dimensional wave-front distribution is retrieved by use of least-square fitting. The algorithm allows large shear amount and works fast based on fast Fourier transform. Investigations into reconstruction accuracy and reliability are carried out by numerical experiments, in which effects of different shear amounts and noises on reconstruction accuracy are evaluated. Optical measurement is made in a lateral shearing interferometer based on double-grating.

© 2006 Optical Society of America

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References

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  1. W. J. Bates, "A wavefront shearing interferometer," Proc. Phys. Soc. 59, 940-952 (1947).
    [CrossRef]
  2. V. Ronchi, "Forty years of history of a grating interferometers," Appl. Opt. 3, 427-450 (1964).
    [CrossRef]
  3. M. V. R. K. Murty, "The use of a single plane parallel plate as a lateral shearing interferometer with a visible gas laser source," Appl. Opt. 3, 531-534 (1964).
    [CrossRef]
  4. M. P. Rimmer, "Method for evaluating lateral shearing interferograms," App. Opt. 13, 623-629 (1974).
    [CrossRef]
  5. M. P. Rimmer and J. C. Wyant, "Evaluation of large aberration using a lateral-shear interferometer having variable shear," App. Opt. 14, 142-150 (1975).
  6. R. R. Gruenzel, "Application of lateral shearing interferometry to stochastic inputs," J. Opt. Soc. Am. 66, 1341-1347 (1976).
    [CrossRef]
  7. R. S. Kasana and K. J. Rosenbruch, "Determination of the refractive index of a lens using the Murty shearing interferometer," Appl. Opt. 22, 3526-3531 (1983).
    [CrossRef] [PubMed]
  8. T. Yatagai and T. Kanou, "Aspherical surface testing with shearing interferometer using fringe scanning detection method," Opt. Eng. 23, 357-360 (1984).
  9. T. Nomura, K. Kamiya, S. Okuda, H. Miyashiro, K. Yoshikawa, and H. Tashiro, "Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running," Precis. Eng. 22, 185-189 (1998).
    [CrossRef]
  10. L. Erdman and R. Kowarschik, "Testing of refractive silicon microlenses by use of a lateral shearing interferometer in transmission," Appl. Opt. 37, 676-682 (1998).
    [CrossRef]
  11. F. Quercioli, B. Tiribilli, and M. Vassalli, "Wavefront-division lateral shearing autocorrelator for ultrafast laser microscopy," Opt. Express 12, 4303-4312 (2004).
    [CrossRef] [PubMed]
  12. D. Mehta, S. Dubey, M. Hossain, and C. Shakher, "Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry," Appl. Opt. 44, 7515-7521 (2005).
    [CrossRef] [PubMed]
  13. R. H. Hudgin, "Wave-front reconstruction for compensated imaging," J. Opt. Soc. Am. 67, 375-378 (1977).
    [CrossRef]
  14. D. L. Fried, "Least-squares fitting a wavefront distortion estimate to an array of phase-difference measurements," J. Opt. Soc. Am. 67, 370-375 (1977).
    [CrossRef]
  15. B. R. Hunt, "Matrix formulation of the reconstruction of phase values from phase differences," J. Opt. Soc. Am. 69, 393-399 (1979).
    [CrossRef]
  16. W. H. Southwell, "Wavefront estimation from wavefront slope measurements," J. Opt. Soc. Am. 70, 998-1006 (1980).
    [CrossRef]
  17. K. R. Freischlad and C. L. Koliopoulos, "Modal estimation of wave front from difference measurements using the discrete Fourier transform," J. Opt. Soc. Am. A 3, 1852-1861 (1986).
    [CrossRef]
  18. G. Harbers, P. J. Kunst and G. W. R. Leibbrandt, "Analysis of lateral shearing interferograms by use of Zernike polynomials," Appl. Opt. 35, 6162-6172 (1996).
    [CrossRef] [PubMed]
  19. X. Tian and T. Yatagai, "Simple algorithm for large-grid phase reconstruction of lateral-shearing interferometry," Appl. Opt. 34, 7213-7220 (1995).
    [CrossRef] [PubMed]
  20. H. von Brug, "Zernike polynomials as basis for wave-front fitting in lateral shearing interferometry," Appl. Opt. 36, 2788-2790 (1997).
    [CrossRef]
  21. M. Servin, D. Malacara, and J. L. Marroquin, "Wave-front recovery from two orthogonal sheared interferograms," Appl. Opt. 35, 4343-4348 (1996).
    [CrossRef] [PubMed]
  22. C. Elster and I. Weingartner, "Exact wave-front reconstruction from two lateral shearing interferograms," J. Opt. Soc. Am. A. 16, 2281-2285 (1999).
    [CrossRef]
  23. C. Elster and I. Weingartner, "Solution to the shearing problem," Appl. Opt. 38, 5024-5031 ( 1999).
    [CrossRef]
  24. C. Elster, "Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears," Appl. Opt. 39, 5353-5359 (2000).
    [CrossRef]
  25. A. Dubra, C. Paterson, and C. Dainty, "Wavefront reconstruction from shear phase maps using the discrete Fourier transform," Appl. Opt. 43, 1108-1113 (2004).
    [CrossRef] [PubMed]
  26. S. Okuda, T. Nomura, K. Kazuhide, H. Miyashiro, H. Hatsuzo, and K. Yoshikawa, "High precision analysis of lateral shearing interferogram using the integration method and polynomials," Appl. Opt. 39, 5179-5186 (2000).
    [CrossRef]
  27. P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26, 2504-2505 (1987).
    [CrossRef] [PubMed]
  28. M. H. Takeda, H. Ina, and S. Kobayashi, "Fourier transforms method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, 156-160 (1982).
    [CrossRef]
  29. T. J. Flynn, "Two-dimensional phase unwrapping with minimum weighted discontinuity," J. Opt. Soc. Am. A. 14, 2692-2701 (1997).
    [CrossRef]
  30. J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057-2060 (1973).
    [CrossRef] [PubMed]
  31. K. Patorski, "Grating shearing interferometer with variable shear and fringe orientation," Appl. Opt. 25, 4192-4198 (1986)
    [CrossRef] [PubMed]
  32. G. W. R. Leibbrandt, G. Harbers, and P. J. Kunst, "Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer," Appl. Opt. 35, 6151-6161 (1996)
    [CrossRef] [PubMed]
  33. H. Schreiber and J. Schwider, "Lateral shearing interferometer based on two Ronchi phase gratings in series," Appl. Opt. 36, 5321-5324 (1997).
    [CrossRef] [PubMed]

App. Opt.

M. P. Rimmer, "Method for evaluating lateral shearing interferograms," App. Opt. 13, 623-629 (1974).
[CrossRef]

M. P. Rimmer and J. C. Wyant, "Evaluation of large aberration using a lateral-shear interferometer having variable shear," App. Opt. 14, 142-150 (1975).

Appl. Opt.

V. Ronchi, "Forty years of history of a grating interferometers," Appl. Opt. 3, 427-450 (1964).
[CrossRef]

M. V. R. K. Murty, "The use of a single plane parallel plate as a lateral shearing interferometer with a visible gas laser source," Appl. Opt. 3, 531-534 (1964).
[CrossRef]

R. S. Kasana and K. J. Rosenbruch, "Determination of the refractive index of a lens using the Murty shearing interferometer," Appl. Opt. 22, 3526-3531 (1983).
[CrossRef] [PubMed]

L. Erdman and R. Kowarschik, "Testing of refractive silicon microlenses by use of a lateral shearing interferometer in transmission," Appl. Opt. 37, 676-682 (1998).
[CrossRef]

D. Mehta, S. Dubey, M. Hossain, and C. Shakher, "Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry," Appl. Opt. 44, 7515-7521 (2005).
[CrossRef] [PubMed]

G. Harbers, P. J. Kunst and G. W. R. Leibbrandt, "Analysis of lateral shearing interferograms by use of Zernike polynomials," Appl. Opt. 35, 6162-6172 (1996).
[CrossRef] [PubMed]

X. Tian and T. Yatagai, "Simple algorithm for large-grid phase reconstruction of lateral-shearing interferometry," Appl. Opt. 34, 7213-7220 (1995).
[CrossRef] [PubMed]

H. von Brug, "Zernike polynomials as basis for wave-front fitting in lateral shearing interferometry," Appl. Opt. 36, 2788-2790 (1997).
[CrossRef]

M. Servin, D. Malacara, and J. L. Marroquin, "Wave-front recovery from two orthogonal sheared interferograms," Appl. Opt. 35, 4343-4348 (1996).
[CrossRef] [PubMed]

C. Elster and I. Weingartner, "Solution to the shearing problem," Appl. Opt. 38, 5024-5031 ( 1999).
[CrossRef]

C. Elster, "Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears," Appl. Opt. 39, 5353-5359 (2000).
[CrossRef]

A. Dubra, C. Paterson, and C. Dainty, "Wavefront reconstruction from shear phase maps using the discrete Fourier transform," Appl. Opt. 43, 1108-1113 (2004).
[CrossRef] [PubMed]

S. Okuda, T. Nomura, K. Kazuhide, H. Miyashiro, H. Hatsuzo, and K. Yoshikawa, "High precision analysis of lateral shearing interferogram using the integration method and polynomials," Appl. Opt. 39, 5179-5186 (2000).
[CrossRef]

P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm," Appl. Opt. 26, 2504-2505 (1987).
[CrossRef] [PubMed]

J. C. Wyant, "Double frequency grating lateral shear interferometer," Appl. Opt. 12, 2057-2060 (1973).
[CrossRef] [PubMed]

K. Patorski, "Grating shearing interferometer with variable shear and fringe orientation," Appl. Opt. 25, 4192-4198 (1986)
[CrossRef] [PubMed]

G. W. R. Leibbrandt, G. Harbers, and P. J. Kunst, "Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer," Appl. Opt. 35, 6151-6161 (1996)
[CrossRef] [PubMed]

H. Schreiber and J. Schwider, "Lateral shearing interferometer based on two Ronchi phase gratings in series," Appl. Opt. 36, 5321-5324 (1997).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

M. H. Takeda, H. Ina, and S. Kobayashi, "Fourier transforms method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, 156-160 (1982).
[CrossRef]

R. H. Hudgin, "Wave-front reconstruction for compensated imaging," J. Opt. Soc. Am. 67, 375-378 (1977).
[CrossRef]

D. L. Fried, "Least-squares fitting a wavefront distortion estimate to an array of phase-difference measurements," J. Opt. Soc. Am. 67, 370-375 (1977).
[CrossRef]

B. R. Hunt, "Matrix formulation of the reconstruction of phase values from phase differences," J. Opt. Soc. Am. 69, 393-399 (1979).
[CrossRef]

W. H. Southwell, "Wavefront estimation from wavefront slope measurements," J. Opt. Soc. Am. 70, 998-1006 (1980).
[CrossRef]

R. R. Gruenzel, "Application of lateral shearing interferometry to stochastic inputs," J. Opt. Soc. Am. 66, 1341-1347 (1976).
[CrossRef]

J. Opt. Soc. Am. A

K. R. Freischlad and C. L. Koliopoulos, "Modal estimation of wave front from difference measurements using the discrete Fourier transform," J. Opt. Soc. Am. A 3, 1852-1861 (1986).
[CrossRef]

J. Opt. Soc. Am. A.

C. Elster and I. Weingartner, "Exact wave-front reconstruction from two lateral shearing interferograms," J. Opt. Soc. Am. A. 16, 2281-2285 (1999).
[CrossRef]

T. J. Flynn, "Two-dimensional phase unwrapping with minimum weighted discontinuity," J. Opt. Soc. Am. A. 14, 2692-2701 (1997).
[CrossRef]

Opt. Eng.

T. Yatagai and T. Kanou, "Aspherical surface testing with shearing interferometer using fringe scanning detection method," Opt. Eng. 23, 357-360 (1984).

Opt. Express

F. Quercioli, B. Tiribilli, and M. Vassalli, "Wavefront-division lateral shearing autocorrelator for ultrafast laser microscopy," Opt. Express 12, 4303-4312 (2004).
[CrossRef] [PubMed]

Precis. Eng.

T. Nomura, K. Kamiya, S. Okuda, H. Miyashiro, K. Yoshikawa, and H. Tashiro, "Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running," Precis. Eng. 22, 185-189 (1998).
[CrossRef]

Proc. Phys. Soc., 1947

W. J. Bates, "A wavefront shearing interferometer," Proc. Phys. Soc. 59, 940-952 (1947).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Computer simulation of wave-front reconstruction from two phase differences in orthogonal directions: (a) is the original phase function, (b) and (c) are the x- and y-directional phase difference distributions, respectively, and (d) is the reconstructed phase distribution.

Fig. 2.
Fig. 2.

Phase difference imposed by uniformly random noise with a level of 30%, where noise level is defined as the ratio between the average of noise absolute value and the one of the phase difference. (a) and (b) are the x- and y-directional phase difference distributions, respectively. (c) is the reconstructed phase distribution.

Fig. 3.
Fig. 3.

Deviation of the reconstructed to the original phase vs. noise level, where noise level is defined as the ratio between the average of noise absolute value and the one of the phase difference

Fig. 4.
Fig. 4.

Double-grating lateral shearing interferometer: Rotating ground glass is used to lower spatial coherence of the illumination light for reducing speckle; Two Ronchi gratings produce both shear and phase shift.

Fig. 5.
Fig. 5.

Result of the optic surface testing with the double-grating lateral shearing interferometer. (a) The x-directional lateral-shearing interferogram. (b) The y-directional lateral-shearing interferogram. (c) The reconstructed phase distribution.

Tables (1)

Tables Icon

Table 1. Deviation of the reconstructed phase to the original phase vs. the shear amount s. (λ=632.8nm)

Equations (17)

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D x m n = φ m n φ m s n m , n = 0,1,2 , . N 1 .
D y m n = φ m n φ ( m , n s ) . m , n = 0,12 , N 1 .
FT x { φ m n } = FT x { D x m n } 1 exp ( i 2 π v x s N ) .
FT y { φ m n } = FT y { D y m n } 1 exp ( i 2 π v y s N ) .
f x m n = FT x 1 { FT x { D x m n } 1 exp ( i 2 π v x s N ) } .
f y m n = FT y 1 { FT y { D y m n } 1 exp ( i 2 π v y s N ) } .
f x m n + c n = φ m n + N x m n .
f x m n + d m = φ m n + N y m n .
c n = 1 N n = 0 N 1 [ f x N 1 n f y N 1 n ] 1 N m = 0 N 1 [ f x m n f n m n ] + 1 N 2 m = 0 N 1 n = 0 N 1 [ f x m n f n m n ] . n = 0,1,2 , , N 1 .
d m = 1 N n = 0 N 1 [ f x m n f y m n ] 1 N m = 0 N 1 [ f x N 1 n f y N 1 n ] . m = 0,1,2 , N 1 .
d N 1 = 0 .
φ m n = [ f x m n + c n ] + [ f y m n + d m ] 2 .
p = 0 ( N s ) 1 D x ( m + ps , n ) = 0 . m = 0,1,2 , s 1 ; n = 0,1,2 , , N 1 .
q = 0 ( N s ) 1 D y ( m , n + qs ) = 0 . n = 0,1,2 , s 1 ; m = 0,1,2 , , N 1 .
D x m n = p = 1 N s 1 D x ( m + ps , n ) . m = 0,1,2 , , s 1 ; n = 0,1,2 , , N 1
D y m n = q = 1 N s 1 D y ( m , n + qs ) . n = 0,1,2 , , s 1 ; m = 0,1,2 , , N 1
φ x y = 2 π × [ 0.1665 × ( x 2 + y 2 2 ) ( x 2 + y 2 + 1 ) 0.8325 × ( x 2 y 2 ) 0.6660 × ( x 2 + y 2 ) ( x 2 y 2 ) ] .

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