Abstract

A method is proposed for exact discrete reconstruction of a two-dimensional wave front from four suitably designed lateral shearing experiments. The method reconstructs any wave front at evaluation points of a circular aperture exactly up to an arbitrary constant for noiseless data, and it shows excellent stability properties in the case of noisy data. Application of large shears is allowed, and high resolution of the reconstructed wave front can be achieved. Results of numerical experiments are presented that demonstrate the capability of the method.

© 2000 Optical Society of America

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References

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  1. V. Ronchi, “Le frangi die combazione nello studio delle superficie e dei sistemi ottici,” Riv. Ottica Mecc. Precis. 2, 9–35 (1923).
  2. W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc. 59, 940–952 (1947).
    [CrossRef]
  3. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–450 (1964).
    [CrossRef]
  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. 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]
  6. 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]
  7. 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]
  8. M. Servin, D. Malacara, J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt. 35, 4343–4348 (1996).
    [CrossRef] [PubMed]
  9. M. P. Rimmer, “Method for evaluating lateral shearing interferometer,” Appl. Opt. 13, 623–629 (1974).
    [CrossRef] [PubMed]
  10. 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]
  11. R. H. Hudgin, “Wavefront reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  12. R. J. Noll, “Phase estimates from slope-type wavefront sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  13. 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]
  14. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  15. W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  16. K. Freischlad, “Sensitivity of heterodyne shearing interferometers,” Appl. Opt. 26, 4053–4054 (1987).
    [CrossRef] [PubMed]
  17. H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  18. 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]
  19. F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
    [CrossRef] [PubMed]
  20. 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]
  21. G. W. R. Leibbrandt, G. Harbers, P. J. Kunst, “Wavefront analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt. 35, 6151–6161 (1996).
    [CrossRef] [PubMed]
  22. H. von Brug, “Zernike polynomials as a basis for wavefront fitting in lateral shearing interferometry,” Appl. Opt. 36, 2788–2790 (1997).
    [CrossRef]
  23. S. Loheide, I. Weingärtner, “New procedure for wavefront reconstruction,” Optik 108, 53–62 (1998).
  24. C. Elster, I. Weingärtner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. 16, 2281–2285 (1999).
    [CrossRef]
  25. C. Elster, I. Weingärtner, “Solution to the shearing problem,” Appl. Opt. 38, 5024–5031 (1999).
    [CrossRef]
  26. C. Elster, “Recovering wavefronts from difference measurements in lateral shearing interferometry,” J. Comp. Appl. Math. 110, 177–180 (1999).
    [CrossRef]
  27. C. Elster, “Evaluation of lateral shearing interferograms,” in Advanced Mathematical Tools in Metrology IV, P. Ciarlini, ed. (World Scientific, Singapore, 2000), pp. 76–87.
    [CrossRef]
  28. P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).
  29. IMSL Math/library (Visual Numerics Inc., Houston, Tex., 1994).

1999 (3)

C. Elster, I. Weingärtner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. 16, 2281–2285 (1999).
[CrossRef]

C. Elster, “Recovering wavefronts from difference measurements in lateral shearing interferometry,” J. Comp. Appl. Math. 110, 177–180 (1999).
[CrossRef]

C. Elster, I. Weingärtner, “Solution to the shearing problem,” Appl. Opt. 38, 5024–5031 (1999).
[CrossRef]

1998 (1)

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

1997 (2)

1996 (3)

1995 (1)

1991 (1)

1989 (1)

1988 (1)

1987 (1)

1986 (1)

1980 (1)

1979 (2)

1978 (1)

1977 (2)

1975 (1)

1974 (1)

1964 (1)

1947 (1)

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

1923 (1)

V. Ronchi, “Le frangi die combazione nello studio delle superficie e dei sistemi ottici,” Riv. Ottica Mecc. Precis. 2, 9–35 (1923).

Bates, W. J.

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

Baxter, B. S.

Elster, C.

C. Elster, I. Weingärtner, “Solution to the shearing problem,” Appl. Opt. 38, 5024–5031 (1999).
[CrossRef]

C. Elster, I. Weingärtner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. 16, 2281–2285 (1999).
[CrossRef]

C. Elster, “Recovering wavefronts from difference measurements in lateral shearing interferometry,” J. Comp. Appl. Math. 110, 177–180 (1999).
[CrossRef]

C. Elster, “Evaluation of lateral shearing interferograms,” in Advanced Mathematical Tools in Metrology IV, P. Ciarlini, ed. (World Scientific, Singapore, 2000), pp. 76–87.
[CrossRef]

Freischlad, K.

Freischlad, K. R.

Fried, D. L.

Frost, R. L.

Ghiglia, D. C.

Gill, P. E.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Harbers, G.

Hudgin, R. H.

Hunt, B. R.

Itoh, M.

Koliopoulos, C. L.

Kunst, P. J.

Leibbrandt, G. W. R.

Loheide, S.

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

Malacara, D.

Marroquin, J. L.

Murray, W.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Noll, R. J.

Rimmer, M. P.

Roddier, C.

Roddier, F.

Romero, L. A.

Ronchi, V.

V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–450 (1964).
[CrossRef]

V. Ronchi, “Le frangi die combazione nello studio delle superficie e dei sistemi ottici,” Riv. Ottica Mecc. Precis. 2, 9–35 (1923).

Rushforth, C. K.

Schreiber, H.

Schwider, J.

Servin, M.

Southwell, W. H.

Takahashi, T.

Takajo, H.

Tian, X.

von Brug, H.

Weingärtner, I.

C. Elster, I. Weingärtner, “Solution to the shearing problem,” Appl. Opt. 38, 5024–5031 (1999).
[CrossRef]

C. Elster, I. Weingärtner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. 16, 2281–2285 (1999).
[CrossRef]

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

Wright, M. H.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Wyant, J. C.

Yatagai, T.

Appl. Opt. (13)

V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–450 (1964).
[CrossRef]

M. P. Rimmer, “Method for evaluating lateral shearing interferometer,” 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 wavefront 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]

C. Elster, I. Weingärtner, “Solution to the shearing problem,” Appl. Opt. 38, 5024–5031 (1999).
[CrossRef]

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, “Wavefront 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]

K. Freischlad, “Sensitivity of heterodyne shearing interferometers,” Appl. Opt. 26, 4053–4054 (1987).
[CrossRef] [PubMed]

J. Comp. Appl. Math. (1)

C. Elster, “Recovering wavefronts from difference measurements in lateral shearing interferometry,” J. Comp. Appl. Math. 110, 177–180 (1999).
[CrossRef]

J. Opt. Soc. Am. (6)

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

Opt. Lett. (1)

Optik (1)

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

Proc. Phys. Soc. (1)

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

Riv. Ottica Mecc. Precis. (1)

V. Ronchi, “Le frangi die combazione nello studio delle superficie e dei sistemi ottici,” Riv. Ottica Mecc. Precis. 2, 9–35 (1923).

Other (3)

C. Elster, “Evaluation of lateral shearing interferograms,” in Advanced Mathematical Tools in Metrology IV, P. Ciarlini, ed. (World Scientific, Singapore, 2000), pp. 76–87.
[CrossRef]

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

IMSL Math/library (Visual Numerics Inc., Houston, Tex., 1994).

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Figures (1)

Fig. 1
Fig. 1

Flowchart description of algorithm M1.

Tables (1)

Tables Icon

Table 1 Results of Numerical Experimentsa

Equations (38)

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K :=ζ, η|ζ2+η2R2
Δsjfx, y=fx+sxj, y+syj-fx, y
x, yΩsj,  j=1,,J,
Ωsj:=ζ, η|ζ, ηK,  ζ+sxj, η+syjK;
fαβ=fxα, yβ,  α, βI0
Δsjfαβ=Δsjfxα, yβ=fxα+sxj, yβ+syj-fxα, yβ,α, βIj,  j=1,,J;
Q=ζ, η|-p/2ζ, ηp/2
p :=2R
δ := p/r1r2,
s1=r1δ, 0,  s2=r2δ, 0,s3=0, r1δ,  s4=0, r2δ.
xα, yβ=-p/2+δα, -p/2+δβ,α, βI0Z×Z,
I0:=α, β|-p/2+αδ, -p/2+βδK,
fαβ=fxα, yβ,  α, βI0
Δsjfαβ=fxα+sxj, yβ+syj-fxα, yβ,α, βIj,  j=1,,4,
Ij=α, β|xα, yβ=-p/2+αδ, -p/2+βδΩsj,j=1,,4,
xα, yβ,  withα, βI0 :=α, βI0|-p/2xα, yβ<p/2.
minfαβ,α,βI0 χ2,
χ2=χ02+f00-c02,
χ02=j=14α,βIjΔsjfαβ-fαβα,β,sj-fαβ2;
χ2fαβ10-6 for all α, βI0
c=1pα,βI0fαβ-fˆαβ
δ1:=maxα,βI0 |fαβ-fˆαβ|,δ2:=1pα,βI0fαβ-fˆαβ21/2.
fαβηαβ,  α, βI0.
ΔsjfαβΔsjfαβ+σαβ,α, βIj, j=1,,4,
fαβ,
α, βI0=α, βI0|-p/2xα, yβ<p/2,
xα, yβ=-p/2+αδ, -p/2+βδ,α, β=0,,N-1,
gαβx,1:=Δs1fαβ for α=0,,n1-1,β=0,,N-1,gαβx,2:=Δs2fαβ for α=0,,n2-1,β=0,,N-1,gαβy,1:=Δs3fαβ for α=0,,N-1,β=0,,n1-1,gαβy,2:=Δs4fαβ for α=0,,N-1,β=0,,n2-1,
n1:=r2-1r1,  n2:=r1-1r2.
gαβx,j=fα+rjβ-fαβ,  α=0,,nj-1,β=0,,N-1,j=1, 2,gαβy,j=fαβ+rj-fαβ,  α=0,,N-1,β=0,,nj-1,j=1, 2,
fαβ=n,m=0N-1 f˜nm exp2πinαNexp2πimβN,α, β=0,,N-1,f˜nm=1N2α,β=0N-1 fαβ exp-2πinαNexp-2πimβN,n, m=0,,N-1.
v˜nmx,j=f˜nmexp2πinrjN-1,n, m=0,,N-1,  j=1, 2,
v˜nmy,j=f˜nmexp2πimrjN-1,n, m=0,,N-1,  j=1, 2,
vαβx,j:=gαβx,jα=0,,nj-1,β=0,,N-1-l=1N/rj-1 gα-lrjβx,jα=nj,,N-1,β=0,,N-1,vαβy,j:=gαβy,jα=0,,N-1,β=0,,nj-1-l=1N/rj-1 gαβ-lrjy,jα=0,,N-1,β=nj,,N-1.
f˜ˆnm:= wnmx,1f˜ˆnmx,1+wnmx,2f˜ˆnmx,2+wnmy,1f˜ˆnmy,1+wnmy,2f˜ˆnmy,2,
f˜ˆnmx,j:=v˜nmx,jexp2πinrjN-1nrjN  Z0otherwise,f˜ˆnmy,j:=v˜nmy,jexp2πimrjN-1mrjN  Z0otherwise,
wnmx,j=wnjwn1+wn2+wm1+wm2,  j=1, 2,wnmy, j=wmjwn1+wn2+wm1+wm2,  j=1, 2,
wkj=1rjN3l=1N/rj-1sin2lπkrj/Nsin2πkrj/NkrjN  Z0otherwise,k=0,,N-1,  j=1, 2.

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