Abstract

Lateral shearing interferometry is a promising reference-free measurement technique for optical wave-front reconstruction. The wave front under study is coherently superposed by a laterally sheared copy of itself, and from the interferogram difference measurements of the wave front are obtained. From these difference measurements the wave front is then reconstructed. Recently, several new and efficient algorithms for evaluating lateral shearing interferograms have been suggested. So far, however, all evaluation methods are somewhat restricted, e.g., assume a priori knowledge of the wave front under study, or assume small shears, and so on. Here a new, to our knowledge, approach for the evaluation of lateral shearing interferograms is presented, which is based on an extension of the difference measurements. This so-called natural extension allows for reconstruction of that part of the underlying wave front whose information is contained in the given difference measurements. The method is not restricted to small shears and allows for high lateral resolution to be achieved. Since the method uses discrete Fourier analysis, the reconstructions can be efficiently calculated. Furthermore, it is shown that, by application of the method to the analysis of two shearing interferograms with suitably chosen shears, exact reconstruction of the underlying wave front at all evaluation points is obtained up to an arbitrary constant. The influence of noise on the results obtained by this reconstruction procedure is investigated in detail, and its stability is shown. Finally, applications to simulated measurements are presented. The results demonstrate high-quality reconstructions for single shearing interferograms and exact reconstructions for two shearing interferograms.

© 1999 Optical Society of America

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References

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  1. V. Ronchi, “Le frangie di combinazione 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. London 59, 940–952 (1947).
    [CrossRef]
  3. G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).
  5. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–450 (1964).
    [CrossRef]
  6. S. Bäumer, “Quantitative Mikro-Messtechnik mit einem Lateral Shearing Interferometer,” Ph.D. dissertation (Optics Institute of Berlin Technical University, Berlin, 1995).
  7. H. von Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. 36, 2788–2790 (1997).
    [CrossRef]
  8. 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]
  9. 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]
  10. 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]
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  13. 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]
  14. R. H. Hudgin, “Wavefront reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (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. 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]
  17. S. Loheide, I. Weingärtner, “New procedure for wavefront reconstruction,” Optik 108, 53–62 (1998).
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    [CrossRef]
  19. M. P. Rimmer, “Method for evaluating lateral shearing interferometer,” Appl. Opt. 13, 623–629 (1974).
    [CrossRef] [PubMed]
  20. 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]
  21. F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
    [CrossRef] [PubMed]
  22. H. Schreiber, J. Schwider, “Lateral shearing interferometer based on two Ronchi gratings in series,” Appl. Opt. 36, 5321–5324 (1997).
    [CrossRef] [PubMed]
  23. M. Servin, D. Malacara, J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt. 35, 4343–4348 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  27. K. Freischlad, “Sensitivity of heterodyne shearing interferometers,” Appl. Opt. 26, 4053–4054 (1987).
    [CrossRef] [PubMed]
  28. S. Loheide, I. Weingärtner, “Verfahren zur Ermittlung einer optischen, mechanischen, elektrischen oder anderen Messgrösse,” German patent197 05 609.1 (15May1997).
  29. I. Weingärtner, H. Stenger, “A simple shear-tilt interferometer for the measurement of wavefront aberration,” Optik 70, 124–126 (1985).
  30. J. B. Saunders, “Measurement of wavefronts without a reference standard. Part 1. The wavefront shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65B, 239–244 (1961).
    [CrossRef]
  31. A. Fricke, I. Weingärtner, “Verfahren zur Ermittlung einer Winkelteilung aus der Winkelteilungsdifferenz, bestimmt aus der Differenz zweier Winkelpositionen mit jeweils konstantem Winkelabstand,” German patent197 20 968.8 (patent pending).
  32. C. Elster, S. Loheide, I. Weingärtner, “Verfahren zur Ermittlung einer optischen, mechanischen, elektrischen oder anderen Messgrösse,” German patent198 33 269.6 (patent pending).
  33. C. Elster, I. Weingärtner, “Method for exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A (in press).

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)

1985 (1)

I. Weingärtner, H. Stenger, “A simple shear-tilt interferometer for the measurement of wavefront aberration,” Optik 70, 124–126 (1985).

1980 (1)

1979 (2)

1978 (1)

1977 (2)

1975 (1)

1974 (1)

1967 (1)

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

1964 (1)

1961 (1)

J. B. Saunders, “Measurement of wavefronts without a reference standard. Part 1. The wavefront shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65B, 239–244 (1961).
[CrossRef]

1947 (1)

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

1923 (1)

V. Ronchi, “Le frangie di combinazione 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. London 59, 940–952 (1947).
[CrossRef]

Bäumer, S.

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

Baxter, B. S.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).

Elster, C.

C. Elster, S. Loheide, I. Weingärtner, “Verfahren zur Ermittlung einer optischen, mechanischen, elektrischen oder anderen Messgrösse,” German patent198 33 269.6 (patent pending).

C. Elster, I. Weingärtner, “Method for exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A (in press).

Freischlad, K.

Freischlad, K. R.

Fricke, A.

A. Fricke, I. Weingärtner, “Verfahren zur Ermittlung einer Winkelteilung aus der Winkelteilungsdifferenz, bestimmt aus der Differenz zweier Winkelpositionen mit jeweils konstantem Winkelabstand,” German patent197 20 968.8 (patent pending).

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. dissertation (Optics Institute of Berlin Technical University, Berlin, 1996).

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).

S. Loheide, I. Weingärtner, “Verfahren zur Ermittlung einer optischen, mechanischen, elektrischen oder anderen Messgrösse,” German patent197 05 609.1 (15May1997).

C. Elster, S. Loheide, I. Weingärtner, “Verfahren zur Ermittlung einer optischen, mechanischen, elektrischen oder anderen Messgrösse,” German patent198 33 269.6 (patent pending).

Malacara, D.

Marroquin, J. L.

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 frangie di combinazione nello studio delle superficie e dei sistemi ottici,” Riv. Ottica Mecc. Precis. 2, 9–35 (1923).

Rushforth, C. K.

Saunders, J. B.

J. B. Saunders, “Measurement of wavefronts without a reference standard. Part 1. The wavefront shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65B, 239–244 (1961).
[CrossRef]

Schreiber, H.

Schulz, G.

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

Schwider, J.

Servin, M.

Southwell, W. H.

Stenger, H.

I. Weingärtner, H. Stenger, “A simple shear-tilt interferometer for the measurement of wavefront aberration,” Optik 70, 124–126 (1985).

Takahashi, T.

Takajo, H.

Tian, X.

von Brug, H.

Weingärtner, I.

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

I. Weingärtner, H. Stenger, “A simple shear-tilt interferometer for the measurement of wavefront aberration,” Optik 70, 124–126 (1985).

A. Fricke, I. Weingärtner, “Verfahren zur Ermittlung einer Winkelteilung aus der Winkelteilungsdifferenz, bestimmt aus der Differenz zweier Winkelpositionen mit jeweils konstantem Winkelabstand,” German patent197 20 968.8 (patent pending).

C. Elster, I. Weingärtner, “Method for exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A (in press).

S. Loheide, I. Weingärtner, “Verfahren zur Ermittlung einer optischen, mechanischen, elektrischen oder anderen Messgrösse,” German patent197 05 609.1 (15May1997).

C. Elster, S. Loheide, I. Weingärtner, “Verfahren zur Ermittlung einer optischen, mechanischen, elektrischen oder anderen Messgrösse,” German patent198 33 269.6 (patent pending).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).

Wyant, J. C.

Yatagai, T.

Appl. Opt. (12)

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 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]

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

J. Opt. Soc. Am. (5)

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

J. Res. Natl. Bur. Stand. Sect. B (1)

J. B. Saunders, “Measurement of wavefronts without a reference standard. Part 1. The wavefront shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65B, 239–244 (1961).
[CrossRef]

Opt. Acta (1)

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

Opt. Lett. (1)

Optik (2)

I. Weingärtner, H. Stenger, “A simple shear-tilt interferometer for the measurement of wavefront aberration,” Optik 70, 124–126 (1985).

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

Proc. Phys. Soc. London (1)

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

Riv. Ottica Mecc. Precis. (1)

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

Other (7)

S. Loheide, I. Weingärtner, “Verfahren zur Ermittlung einer optischen, mechanischen, elektrischen oder anderen Messgrösse,” German patent197 05 609.1 (15May1997).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1989).

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

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

A. Fricke, I. Weingärtner, “Verfahren zur Ermittlung einer Winkelteilung aus der Winkelteilungsdifferenz, bestimmt aus der Differenz zweier Winkelpositionen mit jeweils konstantem Winkelabstand,” German patent197 20 968.8 (patent pending).

C. Elster, S. Loheide, I. Weingärtner, “Verfahren zur Ermittlung einer optischen, mechanischen, elektrischen oder anderen Messgrösse,” German patent198 33 269.6 (patent pending).

C. Elster, I. Weingärtner, “Method for exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A (in press).

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Figures (4)

Fig. 1
Fig. 1

Test function (1) used for the reconstruction problem and data function (2) with a shear s = 1/(17, 5). Normal random numbers with zero mean and standard deviation 1 × 10-3 were added to the data function.

Fig. 2
Fig. 2

Test function (dotted curve) and reconstruction from data function (Fig. 1). The maximum difference between the test wave front and the approximated reconstruction obtained is approximately 8 × 10-3, and the mean reconstruction error is approximately 4 × 10-3.

Fig. 3
Fig. 3

Test function (1) used for the reconstruction problem and data functions (2 and 3) with shears s = 1/15 and s = 1/16, respectively. Normal random numbers with zero mean and standard deviation 1 × 10-3 were added to the data functions.

Fig. 4
Fig. 4

Test function and reconstruction from data functions (Fig. 3). The maximum difference between the test wave front and the approximated reconstruction obtained is approximately 2 × 10-3, and the mean reconstruction error is approximately 1 × 10-3.

Tables (1)

Tables Icon

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

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

Δfx=fx+s/2-fx-s/2,
Ffx+s/2ν=- fx+s/2exp-2πiνxdx=- fxexp-2πiνxexpiπνsdx,Ffx-s/2ν=- fx-s/2exp-2πiνxdx=- fxexp-2πiνxexp-iπνsdx,
Ffxν=TνFΔfxν,  Tν=1expiπνs-exp-iπνs=12i sinπνs.
fx=- exp2πiνxT˜νFΔfxνdν,
T˜ν=Tν,νks-1,kZ0,otherwise.
fx=k=- fkψkx,
ψkx=exp2πikx/pp,fk=ψk, f,
u, v:=0p u*xvxdx,
Δfx=k=-expiπksp-exp-iπkspfkψkx=k=- 2i sinπks/pfkψkx.
2i sinπks/pfk=ψk, Δf,
Δfx=fx+s/2-fx-s/2 over the interval s/2, p-s/2,
Δfpx=fpx+s/2-fpx-s/2,
Δfpx=Δfx, s/2xp-s/2-l=1p/s-1 Δfx+ls,  0x<s/2-l=1p/s-1 Δfx-ls,p-s/2<xp
ψkx+l=1p/s-1 ψkx±ls=ψkx1-exp±2πik1-exp±2πiksp=0 for all k with sinπks/p0,
ψkx=-l=1p/s-1 ψkx±ls for all k with sinπks/p0
fk=ψk, Δfp/2i sinπks/p,
Δfsmoothx=Δfx,  s/2xp-s/22Δfp-s/2-Δf{p-s/2-x-p-s/2},p-s/2<xp-s/2.
fsmoothx+s/2-fsmoothx-s/2=Δfx
Δf2x=Δfsmoothx-cs
fsmoothx=f2x+cx
c=Δfsmooths/2+Δfsmooths/2+s++Δfsmooths/2+p/s-1s/p.
ψk, Δfp=0p Δfpxψk*xdx=-0s/2l=1p/s-1 Δfx+lsψk*xdx+s/2p-s/2 Δfxψk*xdx-p-s/2pl=1p/s-1 Δfx-lsψk*xdx=s/2p-s/2 Δfxψk*x1-expiϕxdx,
yαj:=Δfζαj=fζαj+sj/2-fζαj-sj/2,  α=0, , nsj-1,  j=1, 2
ζαj=sj/2+α,  α=0, , nsj-1,  j=1, 2,
vαj:=       yαj,α=0, , nsj-1-l=1N/sj-1 yα-lsjj,α=nsj, , N-1,  j=1, 2,
v˜kj:=1Nl=0N-1 vlj exp-2πiklN,  k=0, , N-1,  j=1, 2.
v˜kj=f˜kexp2πiksjN-1,  k=0, , N-1,  j=1, 2,
yαj,σ=yαj+ηαj,  α=0, , nsj-1,  j=1, 2,
Eηαj=0,  Eηαj2=σ2.
k=1N-1 E|f˜ˆk-f˜k|2=δ2σ2,
f˜ˆk=ws1kf˜k1,σ+ws2kf˜k2,σws1k+ws2k,  k=1, , N-1,
f˜kj,σ=0,if sinπksj/N=0v˜kj,σexp2πiksj/N-1,otherwise,  k=1, , N-1,  j=1, 2,

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