Abstract

We study errors that occur in geometry and phase reconstruction when using scalar diffraction theory in line gratings with periods below 10 μm. The application of those gratings in so-called computer-generated holograms in high-precision interferometric testing of aspheres and free-form surfaces imposes high demands on the generated phase, leading to error budgets in the range of λ/100. Using rigorous simulations as references, we identify the limits where scalar diffraction theory fails to accurately describe grating geometries and identify the significant error mechanisms.

© 2014 Optical Society of America

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References

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W. Cai, P. Zhou, C. Zhao, and J. H. Burge, Opt. Eng. 52, 124101 (2013).
[CrossRef]

S. Peterhänsel, C. Pruss, and W. Osten, Opt. Express 21, 11638 (2013).
[CrossRef]

2011 (1)

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, Opt. Eng. 50, 055801 (2011).
[CrossRef]

2009 (1)

2008 (1)

2006 (1)

2003 (1)

U. Levy, E. Marom, and D. Mendlovic, Opt. Commun. 299, 11 (2003).

2002 (1)

S. Reichelt, C. Pruss, and H. J. Tiziani, Proc. Soc. Photo-Opt. Instrum. Eng. 4778, 158 (2002).

2001 (1)

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, Proc. SPIE 4440, 135 (2001).

1996 (3)

1995 (1)

1989 (1)

S. M. Arnold, Proc. SPIE 1052, 191 (1989).

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S. T. Peng, T. Tamir, and H. L. Bertoni, IEEE Trans. Microwave Theor. Tech. 23, 123 (1975).
[CrossRef]

1972 (1)

Arnold, S. M.

S. M. Arnold, Proc. SPIE 1052, 191 (1989).

Bennett, V. P.

Bertoni, H. L.

S. T. Peng, T. Tamir, and H. L. Bertoni, IEEE Trans. Microwave Theor. Tech. 23, 123 (1975).
[CrossRef]

Burge, J. H.

W. Cai, P. Zhou, C. Zhao, and J. H. Burge, Opt. Eng. 52, 124101 (2013).
[CrossRef]

Y.-C. Chang, P. Zhou, and J. H. Burge, Appl. Opt. 45, 4223 (2006).
[CrossRef]

Cai, W.

W. Cai, P. Zhou, C. Zhao, and J. H. Burge, Opt. Eng. 52, 124101 (2013).
[CrossRef]

Chang, Y.-C.

Chavel, P.

Drauschke, A.

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, Proc. SPIE 4440, 135 (2001).

Fang, Z.

Gao, Z.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, Opt. Eng. 50, 055801 (2011).
[CrossRef]

Garnet, G.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics, (Wiley, 1978).

Gaylord, T. K.

Goudail, F.

Grann, E. B.

Guzial, B.

Häfner, M.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, Opt. Eng. 50, 055801 (2011).
[CrossRef]

Heitkamp, B.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, Opt. Eng. 50, 055801 (2011).
[CrossRef]

Kley, E.-B.

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, Proc. SPIE 4440, 135 (2001).

Kuang, D.

Lalanne, Ph.

Levy, U.

U. Levy, E. Marom, and D. Mendlovic, Opt. Commun. 299, 11 (2003).

Li, L.

Ma, J.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, Opt. Eng. 50, 055801 (2011).
[CrossRef]

Marom, E.

U. Levy, E. Marom, and D. Mendlovic, Opt. Commun. 299, 11 (2003).

Mendlovic, D.

U. Levy, E. Marom, and D. Mendlovic, Opt. Commun. 299, 11 (2003).

Moharam, M. G.

Morris, G. M.

Moulin, G.

Osten, W.

S. Peterhänsel, C. Pruss, and W. Osten, Opt. Express 21, 11638 (2013).
[CrossRef]

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, Opt. Eng. 50, 055801 (2011).
[CrossRef]

Peng, S. T.

S. T. Peng, T. Tamir, and H. L. Bertoni, IEEE Trans. Microwave Theor. Tech. 23, 123 (1975).
[CrossRef]

Peterhänsel, S.

Pommet, D. A.

Pruss, C.

S. Peterhänsel, C. Pruss, and W. Osten, Opt. Express 21, 11638 (2013).
[CrossRef]

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, Opt. Eng. 50, 055801 (2011).
[CrossRef]

S. Reichelt, C. Pruss, and H. J. Tiziani, Proc. Soc. Photo-Opt. Instrum. Eng. 4778, 158 (2002).

Reichelt, S.

S. Reichelt, C. Pruss, and H. J. Tiziani, Proc. Soc. Photo-Opt. Instrum. Eng. 4778, 158 (2002).

Rockstroh, W.

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, Proc. SPIE 4440, 135 (2001).

Schmidt, H.

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, Proc. SPIE 4440, 135 (2001).

Swanson, G. J.

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” technical report (Massachusetts Institute of Technology, 1991).

Tamir, T.

S. T. Peng, T. Tamir, and H. L. Bertoni, IEEE Trans. Microwave Theor. Tech. 23, 123 (1975).
[CrossRef]

Tiziani, H. J.

S. Reichelt, C. Pruss, and H. J. Tiziani, Proc. Soc. Photo-Opt. Instrum. Eng. 4778, 158 (2002).

Wang, H.

Wittig, L.-C.

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, Proc. SPIE 4440, 135 (2001).

Wyant, J. C.

Wyrowski, F.

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, Proc. SPIE 4440, 135 (2001).

Yuan, C.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, Opt. Eng. 50, 055801 (2011).
[CrossRef]

Zhao, C.

W. Cai, P. Zhou, C. Zhao, and J. H. Burge, Opt. Eng. 52, 124101 (2013).
[CrossRef]

Zhou, P.

W. Cai, P. Zhou, C. Zhao, and J. H. Burge, Opt. Eng. 52, 124101 (2013).
[CrossRef]

Y.-C. Chang, P. Zhou, and J. H. Burge, Appl. Opt. 45, 4223 (2006).
[CrossRef]

Zhu, R.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, Opt. Eng. 50, 055801 (2011).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Microwave Theor. Tech. (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, IEEE Trans. Microwave Theor. Tech. 23, 123 (1975).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Commun. (1)

U. Levy, E. Marom, and D. Mendlovic, Opt. Commun. 299, 11 (2003).

Opt. Eng. (2)

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, Opt. Eng. 50, 055801 (2011).
[CrossRef]

W. Cai, P. Zhou, C. Zhao, and J. H. Burge, Opt. Eng. 52, 124101 (2013).
[CrossRef]

Opt. Express (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

S. Reichelt, C. Pruss, and H. J. Tiziani, Proc. Soc. Photo-Opt. Instrum. Eng. 4778, 158 (2002).

Proc. SPIE (2)

S. M. Arnold, Proc. SPIE 1052, 191 (1989).

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, Proc. SPIE 4440, 135 (2001).

Other (3)

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” technical report (Massachusetts Institute of Technology, 1991).

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics, (Wiley, 1978).

W. Iff, S. Glaubrecht, N. Lindlein, and J. Schwider, in DGaO (Deutsche Gesellschaft für angewandte Optik) Proceedings http://www.dgao-proceedings.de (2010).

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Figures (5)

Fig. 1.
Fig. 1.

Cross section of the binary line grating model used. The period is given by p and the duty cycle d is defined as the ratio b/p. The grating height h defines, together with the refractive index n and the used wavelength λ, the phase step ϕ between the two levels. A0 and A1 are the corresponding amplitudes of the output wavefronts.

Fig. 2.
Fig. 2.

Dependence of reconstruction of parameters d and h (left) and φ (right) on period p. While the reconstruction of d approaches the correct value rather quickly, for height reconstruction the deviation drops below 5 nm only for p>8μm, resulting in phase differences larger than λ/100 even for periods of 10 μm.

Fig. 3.
Fig. 3.

Absolute difference between simulated and reconstructed values for gratings with a period of 6 μm; color bars indicate deviation in d and h [nm], respectively. The d reconstruction shows large deviations for shallow gratings as well as for d=0.5. For the reconstructed h values, the smallest deviations are found for height values around 300–350 nm; for smaller and larger values, the reconstruction is leading to an increase of error.

Fig. 4.
Fig. 4.

Comparison of the phase difference Δφ between simulated phase and calculated phase from the scalar reconstructed values for different grating periods: (a) 4 μm, (b) 6 μm, and (c) 8 μm.

Fig. 5.
Fig. 5.

Scalar diffraction theory predicts a symmetric behavior of efficiency over d. The shift in symmetry is increasing with increasing height and reduced if period is increased.

Equations (3)

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ηsca={(1d)2+d2+2d(1d)cos(ϕ)m=02d2(1cos(ϕ))sinc2(md)m0,
tan(φsca)={dsin(ϕ)(1d)+dcos(ϕ)m=0sin(ϕ)sinc(md)(cos(ϕ)1)sinc(md)m0.
min(η⃗measη⃗sca(d,h)).

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