Abstract

Four modal methods of reconstructing a wavefront from its difference fronts based on Zernike polynomials in lateral shearing interferometry are currently available, namely the Rimmer–Wyant method, elliptical orthogonal transformation, numerical orthogonal transformation, and difference Zernike polynomial fitting. The present study compared these four methods by theoretical analysis and numerical experiments. The results show that the difference Zernike polynomial fitting method is superior to the three other methods due to its high accuracy, easy implementation, easy extension to any high order, and applicability to the reconstruction of a wavefront on an aperture of arbitrary shape. Thus, this method is recommended for use in lateral shearing interferometry for wavefront reconstruction.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” Appl. Opt. 13, 623–629 (1974).
    [CrossRef]
  5. 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]
  6. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  7. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399(1979).
    [CrossRef]
  8. J. Herrmann, “Least-squares wave front errors with minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  9. 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]
  10. 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]
  11. J. Herrmann, “Cross coupling and aliasing in modal wavefront estimation,” J. Opt. Soc. Am. 71, 989–992 (1981).
    [CrossRef]
  12. 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]
  13. 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).
    [CrossRef]
  14. D. Korwan, “Lateral shearing interferogram analysis,” Proc. SPIE 429, 194–198 (1983).
    [CrossRef]
  15. W. Shen, M. Chang, and D. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36, 905–913 (1997).
    [CrossRef]
  16. 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]
  17. 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]
  18. 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]
  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. 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]
  21. H. von Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. 36, 2788–2790 (1997).
    [CrossRef]
  22. 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]
  23. G.-M. Dai, “Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A 13, 1218–1225 (1996).
    [CrossRef]
  24. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. A 70, 998–1006 (1980).
    [CrossRef]
  25. J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. 11 of Applied Optics and Optical Engineering Series (Academic, 1992), p. 28.
  26. G.-M. Dai and V. N. Mahajan, “Nonrecursive orthonormal polynomials with matrix formulation,” Opt. Lett. 32, 74–76 (2007).
    [CrossRef]
  27. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33, 8121–8124 (1994).
    [CrossRef]
  28. Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
    [CrossRef]
  29. A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer, 2007).

2012 (1)

2009 (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]

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)

2006 (1)

2005 (1)

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)

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

2001 (1)

2000 (1)

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)

1986 (1)

1983 (1)

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

1981 (1)

1980 (2)

J. Herrmann, “Least-squares wave front errors with minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
[CrossRef]

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. A 70, 998–1006 (1980).
[CrossRef]

1979 (1)

1977 (2)

1975 (1)

1974 (1)

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. 11 of Applied Optics and Optical Engineering Series (Academic, 1992), p. 28.

Dai, F.

Dai, G.-M.

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]

Ding, J.

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]

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]

Hara, S.

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[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]

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[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]

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[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.

Ishi, M.

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

Jin, Z.

Kakuchi, O.

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

Kamiya, K.

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]

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

Kazuhide, K.

Koliopoulos, C. L.

Kondo, H.

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

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]

Mahajan, V. N.

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]

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

Nomura, T.

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]

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.

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]

Oshino, T.

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

Ota, K.

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

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]

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[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]

Quarteroni, A.

A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer, 2007).

Rimmer, M. P.

Sacco, R.

A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer, 2007).

Saito, J.

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

Saleri, F.

A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer, 2007).

Sasaki, O.

Sekine, Y.

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[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.

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. A 70, 998–1006 (1980).
[CrossRef]

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]

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

Suzuki, A.

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

Takeuchi, S.

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

Tang, F.

Tashiro, H.

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.

Watanabe, Y.

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

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).
[CrossRef]

J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. 11 of Applied Optics and Optical Engineering Series (Academic, 1992), p. 28.

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.

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]

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]

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

Appl. Opt. (9)

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

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

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]

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]

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]

V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33, 8121–8124 (1994).
[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. (5)

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys. (1)

Y. Zhu, K. Sugisaki, K. Murakami, K. Ota, H. Kondo, M. Ishi, J. Kawakami, T. Oshino, J. Saito, A. Suzuki, M. Hasegawa, Y. Sekine, S. Takeuchi, C. Ouchi, O. Kakuchi, Y. Watanabe, T. Hasegawa, and S. Hara, “Shearing interferometry for at wavelength wavefront measurement of extreme-ultraviolet lithography projection optics,” Jpn. J. Appl. Phys. 42, 5844–5847 (2003).
[CrossRef]

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 (3)

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]

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]

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

Other (2)

A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer, 2007).

J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. 11 of Applied Optics and Optical Engineering Series (Academic, 1992), p. 28.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1.

Methods of wavefront reconstruction for lateral shearing interferometry.

Fig. 2.
Fig. 2.

Simulation conditions: (a) input Zernike coefficients; (b) original wavefront. Reconstruction errors of (c) the R–W method; (d) EOT; (e) NOT; and (f) DZF.

Fig. 3.
Fig. 3.

Cross-coupling matrices of (a) the R–W method; (b) EOT; (c) NOT; and (d) DZF. Cross-coupling matrix differences (e) between EOT and DZF and (f) between NOT and DZF.

Fig. 4.
Fig. 4.

RMS reconstruction errors of the four methods under different shear ratios.

Fig. 5.
Fig. 5.

Data regions in the x direction used to reconstruct the wavefront: (a) the overlap region of two interfering circular beams; (b) the largest elliptical region that lies in the overlap regions.

Fig. 6.
Fig. 6.

RMS reconstruction errors of EOT and DZF under different shear ratios.

Fig. 7.
Fig. 7.

Simulated measurement noises in (a) the random noise and (b) the modal noise. Reconstruction errors under (c) the random noise and (d) the modal noise.

Fig. 8.
Fig. 8.

RMS reconstruction errors of the three methods when the coefficient error was added to the difference fronts successively.

Fig. 9.
Fig. 9.

Characteristics of the different shear ratios: (a) condition numbers of the four methods; (b) RMS reconstruction errors of the four methods under 1% modal noise; (c) noise levels of the difference fronts in the x and y directions under a certain absolute noise; and (d) RMS reconstruction errors of the four methods under the same noise as that in (c).

Tables (3)

Tables Icon

Table 1. Zernike and Difference Zernike Polynomials

Tables Icon

Table 2. RMS Reconstruction Errors of the Four Methods

Tables Icon

Table 3. Comparative Results for the Four Methods

Equations (32)

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

W(x,y)=j=2JajZj(x,y),
Ax=Sxa,
Ay=Sya,
A=Sa,
A=(AxAy)andS=(SxSy).
a^=SA^,
ΔWx(x,y;s)=W(x+s,y)W(xs,y)=j=1JAx,jRZj(x,y),
Ex,j(x,y)=Zj(x/(1s),y/1s).
Sx,j,iE=NjΣxEj(x,y)ΔZx,i(x,y;s)dxdy,
Fx=ZxMxT,
SxN=(MxT)1SxR.
ΔWx=ΔZxa,
ΔWy=ΔZya.
ΔW=ΔZa,
ΔW=(ΔWxΔWy)andΔZ=(ΔZxΔZy).
a^=ΔZΔW,
ΔWμ=(GμfGμr)(AμfAμr)=GμrAμr+GμrAμr,
A^μf=Gμf(ΔWμ+eμ),
A^μf=Aμf+CμAμr+Gμfeμ,
(AμfAμr)=(Sμf1Sμf2Sμr1Sμr2)(afar),
Aμf=Sμf1af+Sμf2ar,
Aμr=Sμr1af+Sμr2ar.
a^f=Sf1A^f=Sf1(A^xfA^yf)withSf1=(Sxf1Syf1),
a^f=Sf1(Axf+CxAxr+GxfexAyf+CyAyr+Gyfey).
a^f=Sf1(Sxf1af+Sxf2ar+Cx(Sxr1af+Sxr2ar)+GxfexSyf1af+Syf2ar+Cy(Syr1af+Syr2ar)+Gyfey)=af+Sf1CSr1af+Sf1(Sf2+CSr2)ar+Sf1Gfe,
C=(Cx00Cy),Sr1=(Sxr1Syr1),Sf2=(Sxf2Syf2)andSr2=(Sxr2Syr2),Gf=(Gxf00Gyf),e=(exey).
a^f=af+Sf1(Sf2+CSr2)ar+Sf1Gfe=af+TSar+HSe,
ΔWμ=(ΔZμfΔZμr)(afar)=ΔZμfaf+ΔZμrar,
a^=ΔZf(ΔW+e)=ΔZf(ΔWxΔWy)+ΔZfe.
a^=ΔZf(ΔZxfΔZyf)af+ΔZf(ΔZxrΔZyr)ar+ΔZfe=af+TDar+HDe,
TD=ΔZfΔZr,ΔZr=(ΔZxrΔZyr)andHD=ΔZf.
C(Q)=σmax(Q)/σmin(Q),

Metrics