Abstract

A new method is presented for the reconstruction of a one-dimensional wave front on the basis of difference measurements from two shearing interferograms. The proposed algorithm reconstructs any wave front exactly up to an arbitrary constant. The method is not restricted to small shears. However, the shearing parameters have to be chosen such that certain constraints are satisfied. A procedure for determining such shearing parameters is given. In addition, it is shown that the procedure is stable with respect to noise introduced into the differences.

© 1999 Optical Society of America

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References

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  1. S. Bäumer, “Quantitative Mikro-Messtechnik mit einem Lateral Shearing Interferometer,” Ph.D. thesis (Optics Institute, Berlin Technical University, Berlin, 1995).
  2. 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]
  3. G. Harbers, P. J. Kunst, G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt. 35, 6162–6172 (1996).
    [CrossRef] [PubMed]
  4. H. Schreiber, J. Schwider, “Lateral shearing interferometer based on two Ronchi gratings in series,” Appl. Opt. 36, 5321–5324 (1997).
    [CrossRef] [PubMed]
  5. M. Servin, D. Malacara, J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt. 35, 4343–4348 (1996).
    [CrossRef] [PubMed]
  6. F. Roddier, C. Roddier, “Phase closure with rotational shear interferometers,” Opt. Commun. 60, 350–352 (1986).
    [CrossRef]
  7. E. Ribak, “Phase closure with rotational shear interferometer,” Appl. Opt. 26, 197–199 (1987).
    [CrossRef] [PubMed]
  8. J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
    [CrossRef]
  9. S. N. Srivastava, M. S. Tomar, R. S. Kasana, “Determination of the linear thermal expansion coefficient of long metallic bars by a Murty shearing interferometer,” Opt. Laser Technol. 22, 283–286 (1990).
    [CrossRef]
  10. H. von Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. 36, 2788–2790 (1997).
    [CrossRef]
  11. 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]
  12. 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]
  13. D. C. Ghiglia, L. A. Romero, “Direct phase estimation from phase differences using elliptic partial differential equation solvers,” Opt. Lett. 14, 1107–1109 (1989).
    [CrossRef] [PubMed]
  14. F. Guse, “Auswertung von Messungen mit Optimierungsverfahren demonstriert an Interferometrie und Ellipsometrie,” Ph.D. thesis (Optics Institute, Berlin Technical University, Berlin, 1996).
  15. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  16. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  17. G. W. R. Leibbrandt, G. Harbers, 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]
  18. S. Loheide, I. Weingärtner, “New procedure for wave front reconstruction,” Optik (Stuttgart) 108, 53–62 (1998).
  19. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  20. M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” Appl. Opt. 13, 623–629 (1974).
    [CrossRef] [PubMed]
  21. M. P. Rimmer, J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14, 142–150 (1975).
    [CrossRef] [PubMed]
  22. F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
    [CrossRef] [PubMed]
  23. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  24. H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  25. X. Tian, M. Itoh, T. Yatagai, “Simple algorithm for large-grid phase reconstruction of lateral-shearing interferometry,” Appl. Opt. 34, 7213–7220 (1995).
    [CrossRef] [PubMed]

1998 (1)

S. Loheide, I. Weingärtner, “New procedure for wave front reconstruction,” Optik (Stuttgart) 108, 53–62 (1998).

1997 (2)

1996 (3)

1995 (1)

1991 (1)

1990 (1)

S. N. Srivastava, M. S. Tomar, R. S. Kasana, “Determination of the linear thermal expansion coefficient of long metallic bars by a Murty shearing interferometer,” Opt. Laser Technol. 22, 283–286 (1990).
[CrossRef]

1989 (1)

1988 (1)

1987 (1)

1986 (2)

1980 (1)

1979 (2)

1978 (1)

1977 (3)

1975 (1)

1974 (1)

Bäumer, S.

S. Bäumer, “Quantitative Mikro-Messtechnik mit einem Lateral Shearing Interferometer,” Ph.D. thesis (Optics Institute, Berlin Technical University, Berlin, 1995).

Baxter, B. S.

Freischlad, K. R.

Fried, D. L.

Frost, R. L.

Ghiglia, D. C.

Guse, F.

F. Guse, “Auswertung von Messungen mit Optimierungsverfahren demonstriert an Interferometrie und Ellipsometrie,” Ph.D. thesis (Optics Institute, Berlin Technical University, Berlin, 1996).

Harbers, G.

Hardy, J. W.

Hudgin, R. H.

Hunt, B. R.

Itoh, M.

Kasana, R. S.

S. N. Srivastava, M. S. Tomar, R. S. Kasana, “Determination of the linear thermal expansion coefficient of long metallic bars by a Murty shearing interferometer,” Opt. Laser Technol. 22, 283–286 (1990).
[CrossRef]

Koliopoulos, C. L.

Kunst, P. J.

Lefebvre, J. E.

Leibbrandt, G. W. R.

Loheide, S.

S. Loheide, I. Weingärtner, “New procedure for wave front reconstruction,” Optik (Stuttgart) 108, 53–62 (1998).

Malacara, D.

Marroquin, J. L.

Noll, R. J.

Ribak, E.

Rimmer, M. P.

Roddier, C.

F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
[CrossRef] [PubMed]

F. Roddier, C. Roddier, “Phase closure with rotational shear interferometers,” Opt. Commun. 60, 350–352 (1986).
[CrossRef]

Roddier, F.

F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
[CrossRef] [PubMed]

F. Roddier, C. Roddier, “Phase closure with rotational shear interferometers,” Opt. Commun. 60, 350–352 (1986).
[CrossRef]

Romero, L. A.

Rushforth, C. K.

Schreiber, H.

Schwider, J.

Servin, M.

Southwell, W. H.

Srivastava, S. N.

S. N. Srivastava, M. S. Tomar, R. S. Kasana, “Determination of the linear thermal expansion coefficient of long metallic bars by a Murty shearing interferometer,” Opt. Laser Technol. 22, 283–286 (1990).
[CrossRef]

Takahashi, T.

Takajo, H.

Tian, X.

Tomar, M. S.

S. N. Srivastava, M. S. Tomar, R. S. Kasana, “Determination of the linear thermal expansion coefficient of long metallic bars by a Murty shearing interferometer,” Opt. Laser Technol. 22, 283–286 (1990).
[CrossRef]

von Brug, H.

Weingärtner, I.

S. Loheide, I. Weingärtner, “New procedure for wave front reconstruction,” Optik (Stuttgart) 108, 53–62 (1998).

Wyant, J. C.

Yatagai, T.

Appl. Opt. (11)

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

M. P. Rimmer, J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14, 142–150 (1975).
[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]

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

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

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

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

G. W. R. Leibbrandt, G. Harbers, 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]

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

F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
[CrossRef] [PubMed]

E. Ribak, “Phase closure with rotational shear interferometer,” Appl. Opt. 26, 197–199 (1987).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (6)

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

Opt. Commun. (1)

F. Roddier, C. Roddier, “Phase closure with rotational shear interferometers,” Opt. Commun. 60, 350–352 (1986).
[CrossRef]

Opt. Laser Technol. (1)

S. N. Srivastava, M. S. Tomar, R. S. Kasana, “Determination of the linear thermal expansion coefficient of long metallic bars by a Murty shearing interferometer,” Opt. Laser Technol. 22, 283–286 (1990).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

S. Loheide, I. Weingärtner, “New procedure for wave front reconstruction,” Optik (Stuttgart) 108, 53–62 (1998).

Other (2)

S. Bäumer, “Quantitative Mikro-Messtechnik mit einem Lateral Shearing Interferometer,” Ph.D. thesis (Optics Institute, Berlin Technical University, Berlin, 1995).

F. Guse, “Auswertung von Messungen mit Optimierungsverfahren demonstriert an Interferometrie und Ellipsometrie,” Ph.D. thesis (Optics Institute, Berlin Technical University, Berlin, 1996).

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Tables (1)

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Table 1 Values of Noise Amplification δ for Some Pairs of Selected Shears s1, s2

Equations (42)

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N  rν,Δ  p/N,
s1  νΔ,ns1  (r-1)ν,
s2  rΔ,ns2  (ν-1)r,
f(p)=y(p-sj/2)+f(p-sj)forj=1orj=2.
yαj=y(ζαj)=f(ζαj+sj/2)-f(ζαj-sj/2),
α=0,,nsj-1,j=1, 2
fl=f(xl),l=0,, N-1
xl=l,l=0,, N-1;
yαj=fα+sj-fα,α=0,, nsj-1,j=1, 2.
fl=k=0N-1f˜k exp2πiklN,l=0,, N-1,
f˜k  1N l=0N-1fl exp-2πiklN,k=0,, N-1,
f˜k,k=0,, N-1.
vαj
yαjα=0,, nsj-1-l=1(N/sj)-1yα-lsjjα=nsj,, N-1,j=1, 2,
v˜kj  1N l=0N-1vlj exp-2πiklN,
k=0,,N-1,j=1, 2.
nsj-(N/sj-1)sj=0,N-1-sj=nsj-1
v˜kj=f˜kexp2πiksjN-1,
k=0,, N-1,j=1, 2
Nv˜kj=α=0nsj-1yαj exp-2πikαN-α=nsjN-1l=1(N/sj)-1yα-lsjj exp-2πikαN=α=0nsj-1yαj exp-2πikαN-l=1(N/sj)-1α=nsj-lsjN-1-lsjyαj exp-2πik(α+lsj)N=l=1(N/sj)-1α=nsj-lsjN-1-lsjyαj exp-2πikαN-l=1(N/sj)-1α=nsj-lsjN-1-lsjyαj exp-2πik(α+lsj)N=l=1(N/sj)-11-exp-2πiklsjN×α=nsj-lsjN-1-lsjyαj exp-2πikαN,k=0,, N-1.
yαj,σ=yαj+ηαj,α=0,, nsj-1,j=1, 2,
E(ηαj)=0,E(ηαj)2=σ2
f˜kj,σ=0ifsinπ ksjN=0v˜kj,σexp2πiksjN-1otherwise,
k=1,, N-1,j=1, 2,
E(|f˜kj,σ-E(f˜kj,σ)|2)
=E1N 1exp2πi ksjN-1l=1(N/sj)-11-exp-2πiklsjN×α=nsj-lsjN-1-lsjηαj exp-2πikαN2=σ2 sjN2 l=1(N/sj)-1 sin2πl ksjNsin2π ksjN,
k=1,, N-1,j=1, 2.
wsj(k)=0ifsinπ ksjN=01/sjl=1(N/sj)-1 sin2πl ksjNsin2π ksjNotherwise,
k=1,, N-1,j=1, 2,
f˜^k=ws1(k)f˜k1,σ+ws2(k)f˜k2,σws1(k)+ws2(k),k=1,, N-1.
k=1N-1E(|f˜^k-f˜k|2)=δ2σ2,
δ2=1N2 k=1N-1 1ws1(k)+ws2(k).
1N α=0N-1 exp2πi(k-k)αN=δkk=1,k=k0,otherwise
fl  k=0N-1f˜k exp2πiklNk=0N-11N l=0N-1fl exp-2πiklNexp2πiklN.
v˜k=1N α=0N-1vα exp-2πikαN=1N α=0ns-1vα exp-2πikαN+α=nsN-1vα exp-2πikαN=1N α=0ns-1yα exp-2πikαN+α=nsN-1(-)l=1(N/s)-1yα-ls exp-2πikαN
=1N α=0ns-1[fα+s-fα]exp-2πikαN-α=nsN-1l=1(N/s)-1[fα-(l-1)s-fα-ls]exp-2πikαN
=1N α=0ns-1[fα+s-fα]exp-2πikαN-α=nsN-1l=0(N/s)-1[fα-(l-1)s-fα-ls]exp-2πikαN+1N α=nsN-1[fα+s-fα]exp-2πikαN
=1N α=0N-1[fα+s-fα]exp-2πikαN-α=nsN-1l=0(N/s)-1[fα-(l-1)s-fα-ls]exp-2πikαN
=1N α=0N-1k=0N-1 f˜kexp2πik(α+s)N-f˜k exp2πikαNexp-2πikαN-1Nα=nsN-1l=0(N/s)-1k=0N-1 f˜kexp2πik[α-(l-1)s]N-f˜kexp2πik(α-ls)Nexp-2πikαN
=1N α=0N-1k=0N-1f˜k exp2πikαNexp2πiksN-1×exp-2πikαN-1N α=nsN-1l=0(N/s)-1k=0N-1f˜k exp2πi(k-k)αN×exp-2πiksNlexp2πiksN-1
=k=0N-1f˜kexp2πiksN-1 1N α=0N-1 exp2πi(k-k)αN-1N α=nsN-1k=0N-1f˜k exp2πi(k-k)αNl=0(N/s)-1exp-2πiksNlexp2πiksN-1
=k=0N-1f˜kexp2πiksN-1δkk-1N α=nsN-1k=0N-1f˜k exp2πi(k-k)αN1-exp-2πiks(N/s)N1-exp-2πiksNexp2πiksN-1=f˜kexp2πiksN-1-1N α=nsN-1k=0N-1f˜k exp2πi(k-k)αN0=f˜kexp2πiksN-1.

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