Abstract

In a recent paper [J. Opt. Soc. Am. A 29, 2038 (2012)], we proposed a generalized high spatial resolution zonal wavefront reconstruction method for lateral shearing interferometry. The test wavefront can be reconstructed with high spatial resolution by using linear interpolation on a subgrid for initial values estimation. In the current paper, we utilize the difference between the Zernike polynomial fitting method and linear interpolation in determining the subgrid initial values. The validity of the proposed method is investigated through comparison with the previous high spatial resolution zonal method. Simulation results show that the proposed method is more accurate and more stable to shear ratios compared with the previous method. A comprehensive comparison of the properties of the proposed method, the previous high spatial resolution zonal method, and the modal method is performed.

© 2013 Optical Society of America

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  1. 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]
  2. P. Hegeman, X. Christmann, M. Visser, and J. Braat, “Experimental study of a shearing interferometer concept for at-wavelength characterization of extreme-ultraviolet optics,” Appl. Opt. 40, 4526–4533 (2001).
    [CrossRef]
  3. J. C. Chanteloup, “Multiple-wave lateral shearing interferometry for wave-front sensing,” Appl. Opt. 44, 1559–1571 (2005).
    [CrossRef]
  4. K. R. Freischlad and 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]
  5. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]
  6. M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14, 142–150 (1975).
  7. J. Herrmann, “Cross coupling and aliasing in modal wavefront estimation,” J. Opt. Soc. Am. 71, 989–992 (1981).
    [CrossRef]
  8. D. Korwan, “Lateral shearing interferogram analysis,” Proc. SPIE 429, 194–198 (1983).
    [CrossRef]
  9. G.-M. Dai, “Modified Hartmann–Shack wavefront sensing and iterative wavefront reconstruction,” Proc. SPIE 2201, 562–573 (1994).
    [CrossRef]
  10. 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]
  11. G.-M. Dai, “Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A 13, 1218–1225 (1996).
    [CrossRef]
  12. G. Leibbrandt, G. Harbers, and P. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt. 35, 6151–6161 (1996).
    [CrossRef]
  13. W. Shen, M. Chang, and D. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
    [CrossRef]
  14. H. von Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. 36, 2788–2790 (1997).
    [CrossRef]
  15. 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]
  16. S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, S. Grilli, I. Gurov, and V. Chugunov, “Wave front aberration analysis in two beam reversal shearing interferometry by elliptical Zernike polynomials,” Proc. SPIE 5481, 27–36 (2004).
    [CrossRef]
  17. G.-M. Dai, “Wavefront reconstruction from slope data within pupils of arbitrary shapes using iterative Fourier transform,” Open Optics J. 1, 1–3 (2007).
  18. K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
    [CrossRef]
  19. F. Dai, F. Tang, X. Wang, P. Feng, and O. Sasaki, “Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms,” Opt. Express 20, 1530–1544 (2012).
    [CrossRef]
  20. F. Dai, F. Tang, X. Wang, O. Sasaki, and P. Feng, “Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: comparisons of existing algorithms,” Appl. Opt. 51, 5028–5037 (2012).
    [CrossRef]
  21. J. B. Saunders, “Measurement of wave fronts without a reference standard. Part 1. The wave-front-shearing interferometer,” J. Res. Natl. Bur. Stand. Sect. B 65, 239–244 (1961).
    [CrossRef]
  22. M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” Appl. Opt. 13, 623–629 (1974).
    [CrossRef]
  23. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  24. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  25. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  26. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  27. J. Herrmann, “Least-square wave-front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  28. D. C. Ghiglia and L. A. Romero, “Direct phase estimation from phase differences using fast elliptic partial differential equation solvers,” Opt. Lett. 14, 1107–1109 (1989).
    [CrossRef]
  29. X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A 11, 045702 (2009).
    [CrossRef]
  30. H. Takajo and T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  31. H. Takajo and T. Takahashi, “Noniterative method for obtaining the exact solution for the normal equation in least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988).
    [CrossRef]
  32. P. Liang, J. Ding, Z. Jin, C. S. Guo, and H. T. Wang, “Two-dimensional wave-front reconstruction from lateral shearing interferograms,” Opt. Express 14, 625–634 (2006).
    [CrossRef]
  33. W. Zou and Z. Zhang, “Generalized wave-front reconstruction algorithm applied in a Shack–Hartmann test,” Appl. Opt. 39, 250–268 (2000).
    [CrossRef]
  34. W. Zou and J. P. Rolland, “Iterative zonal wave-front estimation algorithm for optical testing with general shaped pupils,” J. Opt. Soc. Am. A 22, 938–951 (2005).
    [CrossRef]
  35. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  36. W. Zou and J. P. Rolland, “Quantifications of error propagation in slope-based wavefront estimations,” J. Opt. Soc. Am. A 23, 2629–2638 (2006).
    [CrossRef]
  37. C. Elster and I. Weingärtner, “Solution to the shearing problem,” Appl. Opt. 38, 5024–5031 (1999).
    [CrossRef]
  38. C. Elster and I. Weingärtner, “Exact wave-front reconstruction from two lateral shearing interferograms,” J. Opt. Soc. Am. A 16, 2281–2285 (1999).
    [CrossRef]
  39. C. Elster, “Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears,” Appl. Opt. 39, 5353–5359 (2000).
    [CrossRef]
  40. Z. Yin, “Exact wavefront recovery with tilt from lateral shear interferograms,” Appl. Opt. 48, 2760–2766 (2009).
    [CrossRef]
  41. T. Nomura, S. Okuda, K. Kamiya, H. Tashiro, and K. Yoshikawa, “Improved Saunders method for the analysis of lateral shearing interferograms,” Appl. Opt. 41, 1954–1961 (2002).
    [CrossRef]
  42. F. Dai, F. Tang, X. Wang, and O. Sasaki, “Generalized zonal wavefront reconstruction for high spatial resolution in lateral shearing interferometry,” J. Opt. Soc. Am. A 29, 2038–2047 (2012).
    [CrossRef]
  43. J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. XI of Applied Optics and Optical Engineering Series (Academic, 1992), p. 28.
  44. B. H. Dean and C. W. Bowers, “Diversity selection for phase-diverse phase retrieval,” J. Opt. Soc. Am. A 20, 1490–1504 (2003).
    [CrossRef]
  45. D. Malacara, Optical Shop Testing, 3rd ed. (CRC Press, 2007).

2012 (3)

2009 (2)

Z. Yin, “Exact wavefront recovery with tilt from lateral shear interferograms,” Appl. Opt. 48, 2760–2766 (2009).
[CrossRef]

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A 11, 045702 (2009).
[CrossRef]

2008 (1)

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

2007 (1)

G.-M. Dai, “Wavefront reconstruction from slope data within pupils of arbitrary shapes using iterative Fourier transform,” Open Optics J. 1, 1–3 (2007).

2006 (2)

2005 (2)

2004 (1)

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, S. Grilli, I. Gurov, and V. Chugunov, “Wave front aberration analysis in two beam reversal shearing interferometry by elliptical Zernike polynomials,” Proc. SPIE 5481, 27–36 (2004).
[CrossRef]

2003 (1)

2002 (1)

2001 (1)

2000 (3)

1999 (2)

1998 (1)

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]

1997 (2)

W. Shen, M. Chang, and D. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
[CrossRef]

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

1996 (3)

1994 (1)

G.-M. Dai, “Modified Hartmann–Shack wavefront sensing and iterative wavefront reconstruction,” Proc. SPIE 2201, 562–573 (1994).
[CrossRef]

1989 (1)

1988 (2)

1986 (1)

1983 (1)

D. Korwan, “Lateral shearing interferogram analysis,” Proc. SPIE 429, 194–198 (1983).
[CrossRef]

1981 (1)

1980 (2)

1979 (2)

1978 (1)

1977 (2)

1975 (1)

1974 (1)

1961 (1)

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

Bowers, C. W.

Braat, J.

Chang, M.

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A 11, 045702 (2009).
[CrossRef]

W. Shen, M. Chang, and D. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
[CrossRef]

Chanteloup, J. C.

Christmann, X.

Chugunov, V.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, S. Grilli, I. Gurov, and V. Chugunov, “Wave front aberration analysis in two beam reversal shearing interferometry by elliptical Zernike polynomials,” Proc. SPIE 5481, 27–36 (2004).
[CrossRef]

Creath, K.

J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. XI of Applied Optics and Optical Engineering Series (Academic, 1992), p. 28.

Cubalchini, R.

Dai, F.

Dai, G.-M.

G.-M. Dai, “Wavefront reconstruction from slope data within pupils of arbitrary shapes using iterative Fourier transform,” Open Optics J. 1, 1–3 (2007).

G.-M. Dai, “Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A 13, 1218–1225 (1996).
[CrossRef]

G.-M. Dai, “Modified Hartmann–Shack wavefront sensing and iterative wavefront reconstruction,” Proc. SPIE 2201, 562–573 (1994).
[CrossRef]

De Nicola, S.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, S. Grilli, I. Gurov, and V. Chugunov, “Wave front aberration analysis in two beam reversal shearing interferometry by elliptical Zernike polynomials,” Proc. SPIE 5481, 27–36 (2004).
[CrossRef]

Dean, B. H.

Ding, J.

Elster, C.

Feng, P.

Ferraro, P.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, S. Grilli, I. Gurov, and V. Chugunov, “Wave front aberration analysis in two beam reversal shearing interferometry by elliptical Zernike polynomials,” Proc. SPIE 5481, 27–36 (2004).
[CrossRef]

Finizio, A.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, S. Grilli, I. Gurov, and V. Chugunov, “Wave front aberration analysis in two beam reversal shearing interferometry by elliptical Zernike polynomials,” Proc. SPIE 5481, 27–36 (2004).
[CrossRef]

Freischlad, K. R.

Fried, D. L.

Gao, Y.

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A 11, 045702 (2009).
[CrossRef]

Ghiglia, D. C.

Grilli, S.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, S. Grilli, I. Gurov, and V. Chugunov, “Wave front aberration analysis in two beam reversal shearing interferometry by elliptical Zernike polynomials,” Proc. SPIE 5481, 27–36 (2004).
[CrossRef]

Guo, C. S.

Gurov, I.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, S. Grilli, I. Gurov, and V. Chugunov, “Wave front aberration analysis in two beam reversal shearing interferometry by elliptical Zernike polynomials,” Proc. SPIE 5481, 27–36 (2004).
[CrossRef]

Harbers, G.

Hasegawa, M.

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

Hasegawa, T.

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

Hatsuzo, H.

Hegeman, P.

Herrmann, J.

Honda, T.

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

Hudgin, R. H.

Hunt, B. R.

Jin, Z.

Kamiya, K.

T. Nomura, S. Okuda, K. Kamiya, H. Tashiro, and K. Yoshikawa, “Improved Saunders method for the analysis of lateral shearing interferograms,” Appl. Opt. 41, 1954–1961 (2002).
[CrossRef]

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]

Kato, S.

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

Kawakami, J.

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

Kazuhide, K.

Koliopoulos, C. L.

Korwan, D.

D. Korwan, “Lateral shearing interferogram analysis,” Proc. SPIE 429, 194–198 (1983).
[CrossRef]

Kunst, P.

Kunst, P. J.

Leibbrandt, G.

Leibbrandt, G. W. R.

Liang, P.

Liu, X.

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A 11, 045702 (2009).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing, 3rd ed. (CRC Press, 2007).

Miyashiro, H.

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]

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]

Murakami, K.

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

Noll, R. J.

Nomura, T.

Okada, M.

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

Okuda, S.

Otaki, K.

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

Ouchi, C.

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

Pierattini, G.

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, S. Grilli, I. Gurov, and V. Chugunov, “Wave front aberration analysis in two beam reversal shearing interferometry by elliptical Zernike polynomials,” Proc. SPIE 5481, 27–36 (2004).
[CrossRef]

Rimmer, M. P.

Rolland, J. P.

Romero, L. A.

Sasaki, O.

Saunders, J. B.

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

Shen, W.

W. Shen, M. Chang, and D. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
[CrossRef]

Southwell, W. H.

Sugisaki, K.

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

Takahashi, T.

Takajo, H.

Tang, F.

Tashiro, H.

T. Nomura, S. Okuda, K. Kamiya, H. Tashiro, and K. Yoshikawa, “Improved Saunders method for the analysis of lateral shearing interferograms,” Appl. Opt. 41, 1954–1961 (2002).
[CrossRef]

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]

Visser, M.

von Brug, H.

Wan, D.

W. Shen, M. Chang, and D. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
[CrossRef]

Wang, H. T.

Wang, X.

Weingärtner, I.

Wyant, J. C.

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

J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. XI of Applied Optics and Optical Engineering Series (Academic, 1992), p. 28.

Yin, Z.

Yokota, H.

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

Yoshikawa, K.

Zhang, Z.

Zhu, Y.

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

Zou, W.

Appl. Opt. (14)

P. Hegeman, X. Christmann, M. Visser, and J. Braat, “Experimental study of a shearing interferometer concept for at-wavelength characterization of extreme-ultraviolet optics,” Appl. Opt. 40, 4526–4533 (2001).
[CrossRef]

J. C. Chanteloup, “Multiple-wave lateral shearing interferometry for wave-front sensing,” Appl. Opt. 44, 1559–1571 (2005).
[CrossRef]

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

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]

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

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

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]

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

F. Dai, F. Tang, X. Wang, O. Sasaki, and P. Feng, “Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: comparisons of existing algorithms,” Appl. Opt. 51, 5028–5037 (2012).
[CrossRef]

W. Zou and Z. Zhang, “Generalized wave-front reconstruction algorithm applied in a Shack–Hartmann test,” Appl. Opt. 39, 250–268 (2000).
[CrossRef]

C. Elster and I. Weingärtner, “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]

Z. Yin, “Exact wavefront recovery with tilt from lateral shear interferograms,” Appl. Opt. 48, 2760–2766 (2009).
[CrossRef]

T. Nomura, S. Okuda, K. Kamiya, H. Tashiro, and K. Yoshikawa, “Improved Saunders method for the analysis of lateral shearing interferograms,” Appl. Opt. 41, 1954–1961 (2002).
[CrossRef]

J. Opt. A (1)

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A 11, 045702 (2009).
[CrossRef]

J. Opt. Soc. Am. (8)

J. Opt. Soc. Am. A (9)

K. R. Freischlad and 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]

G.-M. Dai, “Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A 13, 1218–1225 (1996).
[CrossRef]

W. Zou and J. P. Rolland, “Quantifications of error propagation in slope-based wavefront estimations,” J. Opt. Soc. Am. A 23, 2629–2638 (2006).
[CrossRef]

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

W. Zou and J. P. Rolland, “Iterative zonal wave-front estimation algorithm for optical testing with general shaped pupils,” J. Opt. Soc. Am. A 22, 938–951 (2005).
[CrossRef]

H. Takajo and T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
[CrossRef]

H. Takajo and T. Takahashi, “Noniterative method for obtaining the exact solution for the normal equation in least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988).
[CrossRef]

F. Dai, F. Tang, X. Wang, and O. Sasaki, “Generalized zonal wavefront reconstruction for high spatial resolution in lateral shearing interferometry,” J. Opt. Soc. Am. A 29, 2038–2047 (2012).
[CrossRef]

B. H. Dean and C. W. Bowers, “Diversity selection for phase-diverse phase retrieval,” J. Opt. Soc. Am. A 20, 1490–1504 (2003).
[CrossRef]

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

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

Open Optics J. (1)

G.-M. Dai, “Wavefront reconstruction from slope data within pupils of arbitrary shapes using iterative Fourier transform,” Open Optics J. 1, 1–3 (2007).

Opt. Eng. (1)

W. Shen, M. Chang, and D. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Precis. Eng. (1)

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. SPIE (4)

D. Korwan, “Lateral shearing interferogram analysis,” Proc. SPIE 429, 194–198 (1983).
[CrossRef]

G.-M. Dai, “Modified Hartmann–Shack wavefront sensing and iterative wavefront reconstruction,” Proc. SPIE 2201, 562–573 (1994).
[CrossRef]

K. Sugisaki, M. Okada, K. Otaki, Y. Zhu, J. Kawakami, K. Murakami, C. Ouchi, M. Hasegawa, S. Kato, T. Hasegawa, H. Yokota, and T. Honda, “EUV wavefront measurement of six-mirror optics using EWMS,” Proc. SPIE 6921, 69212U (2008).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, S. Grilli, I. Gurov, and V. Chugunov, “Wave front aberration analysis in two beam reversal shearing interferometry by elliptical Zernike polynomials,” Proc. SPIE 5481, 27–36 (2004).
[CrossRef]

Other (2)

D. Malacara, Optical Shop Testing, 3rd ed. (CRC Press, 2007).

J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. XI of Applied Optics and Optical Engineering Series (Academic, 1992), p. 28.

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Figures (11)

Fig. 1.
Fig. 1.

Schematics of (a) the discretization of test wavefront, (b) and (c) lateral shearing and discretization of the difference wavefront in the x direction for modal reconstruction.

Fig. 2.
Fig. 2.

Schematics of (a) the lateral shearing and (b) discretization of the difference wavefront in the x direction for zonal reconstruction.

Fig. 3.
Fig. 3.

Schematic of the subgrid at which the values of the test wavefront are estimated as initial values.

Fig. 4.
Fig. 4.

Simulation conditions of (a) the test wavefront and (b) Zernike coefficients of the test wavefront.

Fig. 5.
Fig. 5.

Reconstructed wavefronts of GZR, DZF, and CZMR under the shear amounts (a)–(c) Sx=Sy=5, (d)–(f) Sx=Sy=15, and (g)–(i) Sx=Sy=25.

Fig. 6.
Fig. 6.

Reconstruction errors of GZR, DZF, and CZMR under the shear amounts (a)–(c) Sx=Sy=5, (d)–(f) Sx=Sy=15, and (g)–(i) Sx=Sy=25.

Fig. 7.
Fig. 7.

Reconstruction errors of GZR, DZF, and CZMR in RMS and PV versus shear ratios under different initial point of the subgrid.

Fig. 8.
Fig. 8.

Reconstruction errors of DZF and CZMR in RMS and PV versus reconstruction terms under the shear amounts of (a), (b) Sx=Sy=5; (c), (d) Sx=Sy=15; and (e), (f) Sx=Sy=25.

Fig. 9.
Fig. 9.

First 37 Zernike coefficients error of GZR, DZF, and CZMR under the shear amount Sx=Sy=15.

Fig. 10.
Fig. 10.

Zernike coefficient errors of the terms Z8, Z11, Z17, Z20, Z27, and Z35 of GZR, DZF, and CZMR versus shear ratios.

Fig. 11.
Fig. 11.

RMS reconstruction errors of GZR, DZF, and CZMR versus different random noise levels under the shear amounts (a) Sx=Sy=5, (b) Sx=Sy=15, and (c) Sx=Sy=25.

Tables (1)

Tables Icon

Table 1. Computation Time of the Three Methods Under Different Sample Size and Different Shear Amount, in Which S=Sx=Sy

Equations (15)

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W(x,y)=j=2JajZj(x,y),
ΔWx(x,y;sx)=W(x+2sx,y)W(x,y)=j=2Jaj[Zj(x+2sx,y)Zj(x,y)]=j=2JajΔZx,j(x,y;sx),
ΔWxm=ΔZxa,
ΔWym=ΔZya,
ΔWm=ΔZawithΔWm=(ΔWxmΔWym)andΔZ=(ΔZxΔZy).
a^=(ΔZTΔZ)1ΔZTΔWm.
W^m=Za^,
MxW=ΔWxz,
MyW=ΔWyz,
MW=ΔWzwithM=[MxMy]andΔWz=[ΔWxzΔWyz],
W˜[r+i,c+j]=ΔWyz[r,c]Sy×i+ΔWxz[r,c]Sx×j,
HW=W˜s,
MeW=ΔWewithMe=[MH]andΔWe=[ΔWzW˜s].
W^z=(MeTMe)1MeTΔWe.
W˜[r+i,c+j]=W^m[r+i,c+j],

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