Abstract

A simple zonal approach is proposed for estimating phase distribution on large grids. The estimation is based on phase differences that are precisely measured in two orthogonal directions by a lateral-shearing interferometer. It requires only O(N 2) operations for reconstructing a phase distribution on an N × N grid. Computer simulation and experimental results are demonstrated to show the effectiveness of the new algorithm.

© 1995 Optical Society of America

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References

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  1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  2. M. P. Rimmer, “Methods for evaluating lateral shearing interferograms,” Appl. Opt. 13, 623–629 (1974).
    [CrossRef] [PubMed]
  3. D. L. Fried, “Least-square fitting of a wave-front distortion estimation to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  4. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  5. B. R. Hunt, “Matrix formulation of the reconstruction of phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  6. R. L. Frost, C. K. Rushforth, B. S. Baxter, “Fast FFT-based algorithm for phase estimation in speckle imaging,” Appl. Opt. 18, 2056–2061 (1979).
    [CrossRef] [PubMed]
  7. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  8. A. Menikoff, “Wave-front reconstruction with a square aperture,” J. Opt. Soc. Am. A 6, 1027–1030 (1989).
    [CrossRef]
  9. 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]
  10. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  11. K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
    [CrossRef]
  12. D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
    [CrossRef]
  13. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  14. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  15. Phase shift interferometry is well known as an effective technique for phase measurement. With an improved phase-calculation algorithm, one can get a peak-to-valley measuring accuracy of λ/100 without difficulty. See, for example, Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3048–3052 (1985); C. Ai, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shift interferometry,” Appl. Opt. 26, 1112–1116 (1987).
    [CrossRef] [PubMed]

1990 (1)

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

1989 (2)

1986 (1)

1985 (1)

Phase shift interferometry is well known as an effective technique for phase measurement. With an improved phase-calculation algorithm, one can get a peak-to-valley measuring accuracy of λ/100 without difficulty. See, for example, Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3048–3052 (1985); C. Ai, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shift interferometry,” Appl. Opt. 26, 1112–1116 (1987).
[CrossRef] [PubMed]

1980 (2)

1979 (3)

1978 (1)

1977 (2)

1974 (2)

Baxter, B. S.

Brangaccio, D. J.

Bruning, J. H.

Carpio-Valadez, J. M.

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

Cheng, Y.-Y.

Phase shift interferometry is well known as an effective technique for phase measurement. With an improved phase-calculation algorithm, one can get a peak-to-valley measuring accuracy of λ/100 without difficulty. See, for example, Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3048–3052 (1985); C. Ai, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shift interferometry,” Appl. Opt. 26, 1112–1116 (1987).
[CrossRef] [PubMed]

Cubalchini, R.

Freischlad, K. R.

Fried, D. L.

Frost, R. L.

Gallagher, J. E.

Ghiglia, D. C.

Herriott, D. R.

Herrmann, J.

Hudgin, R. H.

Hunt, B. R.

Koliopoulos, C. L.

Malacara, D.

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

Menikoff, A.

Noll, R. J.

Rimmer, M. P.

Romero, L. A.

Rosenfeld, D. P.

Rushforth, C. K.

Sanchez-Mondragon, J. J.

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

Southwell, W. H.

White, A. D.

Wyant, J. C.

Phase shift interferometry is well known as an effective technique for phase measurement. With an improved phase-calculation algorithm, one can get a peak-to-valley measuring accuracy of λ/100 without difficulty. See, for example, Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3048–3052 (1985); C. Ai, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shift interferometry,” Appl. Opt. 26, 1112–1116 (1987).
[CrossRef] [PubMed]

Appl. Opt. (4)

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

M. P. Rimmer, “Methods for evaluating lateral shearing interferograms,” Appl. Opt. 13, 623–629 (1974).
[CrossRef] [PubMed]

R. L. Frost, C. K. Rushforth, B. S. Baxter, “Fast FFT-based algorithm for phase estimation in speckle imaging,” Appl. Opt. 18, 2056–2061 (1979).
[CrossRef] [PubMed]

Phase shift interferometry is well known as an effective technique for phase measurement. With an improved phase-calculation algorithm, one can get a peak-to-valley measuring accuracy of λ/100 without difficulty. See, for example, Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3048–3052 (1985); C. Ai, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shift interferometry,” Appl. Opt. 26, 1112–1116 (1987).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

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

Opt. Lett. (1)

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Figures (4)

Fig. 1
Fig. 1

M × N grid on which phase reconstruction is performed. D x ′(i, j) and D y ′(i, j) are directly measured slopes in the x and the y directions, respectively. f x ′(i, j) and f y ′(i, j) are integral values of Eqs. (1) and (2) at a point (i, j). f x ′(i + 1, j) is the integral value in the x direction at point (i + 1, j). f y ′(i, j + 1) is the integral value in the y direction at point (i, j + 1).

Fig. 2
Fig. 2

Noise propagator of the proposed algorithm versus size of grid N.

Fig. 3
Fig. 3

Results of a computer simulation with noisy data: (a) a plot of the original phase function, (b) the phase differences in the x direction (with noise added), (c) the phase differences in the y direction (with noise added), (d) a plot of the reconstructed phase distribution.

Fig. 4
Fig. 4

Results of measuring the surface topography of a film recorded with a computer-generated hologram by the use of a differential interference contrast microscope and the new phase-estimation algorithm: (a) the lateral-shearing interferogram when the direction of the shear is in the x direction, (b) the lateral-shearing interferogram when the direction of the shear is in the y direction, (c) a plot of the reconstructed surface topography of the test sample.

Equations (32)

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f x ( i , j ) = { - m = i M / 2 - 1 D x ( m , j ) , i < M / 2 0 , i = M / 2 m = M / 2 i - 1 D x ( m , j ) , i > M / 2 ,
f y ( i , j ) = { - n = j N / 2 - 1 D y ( i , n ) , j < N / 2 0 , j = N / 2 n = N / 2 j - 1 D y ( i , n ) , j > N / 2 ,
f x ( i , j ) + c j = ϕ ( i , j ) + N x ( x , y )             ( j = 1 , 2 , , N ) ,
f y ( i , j ) + d i = ϕ ( i , j ) + N y ( x , y )             ( i = 1 , 2 , , M ) ,
S = i = 1 M j = 1 N [ N x 2 ( i , j ) + N y 2 ( i , j ) ] = i = 1 M j = 1 N { [ f x ( i , j ) + c j - ϕ ( i , j ) ] 2 + [ f y ( i , j ) + d i - ϕ ( i , j ) ] 2 } .
S ϕ ( i , j ) = 0             ( i = 1 , 2 , , M ; j = 1 , 2 , , N ) ,
S c j = 0             ( j = 1 , 2 , , N ) ,
S d j = 0             ( i = 1 , 2 , , M ) .
ϕ ( i , j ) = [ f x ( i , j ) + c j ] + [ f y ( i , j ) + d i ] 2             ( i = 1 , 2 , , M ; j = 1 , 2 , , N ) .
i = 1 M [ f x ( i , j ) + c j - f y ( i , j ) - d i ] = 0             ( j = 1 , 2 , , N ) ,
j = 1 N [ - f x ( i , j ) - c j + f y ( i , j ) + d i ] = 0             ( i = 1 , 2 , , M ) .
AX = B ,
A = M 0 0 0 M 0 0 0 M - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1             - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 N 0 0 0 N 0 0 0 N ,
X = c 1 c 2 c N d 1 d 2 d M - 1 ,
B = i = 1 M [ f y ( i , 1 ) - f x ( i , 1 ) ] i = 1 M [ f y ( i , 2 ) - f x ( i , 2 ) ] i = 1 M [ f y ( i , N ) - f x ( i , N ) ] j = 1 N [ f x ( 1 , j ) - f y ( 1 , j ) ] j = 1 N [ f x ( 2 , j ) - f y ( 2 , j ) ] j = 1 N [ f x ( M - 1 , j ) - f y ( M - 1 , j ) ] .
A - 1 = V Q P P 1 N 1 N 1 N P Q P 1 N 1 N 1 N P P Q 1 N 1 N 1 N 1 N 1 N 1 N 2 N 1 N 1 N 1 N 1 N 1 N 1 N 2 N 1 N 1 N 1 N 1 N 1 N 1 N 2 N ,
X = A - 1 B .
c j = U + 1 M i = 1 M [ f y ( i , j ) - f x ( i , j ) ]             ( j = 1 , 2 , , N ) ,
d i = V + 1 N j = 1 N [ f x ( i , j ) - f y ( i , j ) ]             ( i = 1 , 2 , , M ) ,
V = - 1 N j = 1 N [ f x ( M , j ) - f y ( M , j ) ] ,
U = V - 1 M N j = 1 N i = 1 M [ f y ( i , j ) - f x ( i , j ) ] .
ϕ ( i , j ) = ½ [ f x ( i , j ) + c j + f y ( i , j ) + d i ] ,
ϕ ˜ ( i , j ) = ½ [ f ˜ x ( i , j ) + c ˜ j + f ˜ y ( i , j ) + d ˜ i ] .
e ( i , j ) = ϕ ˜ ( i , j ) - ϕ ( i , j ) = ½ { [ f x ( i , j ) - f x ( i , j ) ] + ( c ˜ j - c j ) + [ f ˜ y ( i , j ) - f y ( i , j ) ] + ( d ˜ i - d y ) } .
e ( i , j ) = 1 2 m = 1 M - 1 n = 1 N - 1 [ α i j m n n y ( m , n ) + β i j m n n x ( m , n ) ] ,
n u ( i , j ) n v ( i , j ) = σ n 2 δ ( u , v ) δ ( i , i ) δ ( j , j ) ,
σ ϕ 2 ( i , j ) = e ( i , j ) e * ( i , j ) .
σ ϕ 2 ( i , j ) = 1 4 σ n 2 m = 1 M - 1 n = 1 N - 1 ( α i j m n 2 + β i j m n 2 ) .
σ ϕ 2 ¯ = 1 N M i = 1 M j = 1 N σ ϕ 2 ( i , j ) ,
σ ϕ 2 ¯ = σ n 2 4 M N i = 1 M j = 1 N [ m = 1 M - 1 n = 1 N - 1 ( α i j m n 2 + β i j m n 2 ) ] .
σ ϕ 2 ¯ / σ n 2 = 0.0652 + 0.8305 ln ( N ) .
W ( x , y ) = 2 π 0.6328 [ 1.45 ( x 2 + y 2 ) 2 + 7.0 y ( x 2 + y 2 ) + 0.3 ( x 2 + 3 y 2 ) + ( x 2 + y 2 ) ] , ( x , y ) { - 1.0 x 1.0 , - 1.0 y 1.0 } ,

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