Abstract

One common way to measure asphere and freeform surfaces is the interferometric Null test, where a computer generated hologram (CGH) is placed in the object path of the interferometer. If undetected phase errors are present in the CGH, the measurement will show systematic errors. Therefore the absolute phase of this element has to be known. This phase is often calculated using scalar diffraction theory. In this paper we discuss the limitations of this theory for the prediction of the absolute phase generated by different implementations of CGH. Furthermore, for regions where scalar approximation is no longer valid, rigorous simulations are performed to identify phase sensitive structure parameters and evaluate fabrication tolerances for typical gratings.

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2012 (1)

2011 (2)

L. Li and Gérard Granet, “Field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings,” J. Opt. Soc. Am. A28738–746 (2011).
[CrossRef]

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “Systematic analysis of the measurement of cone angles using high line density computer-generated holograms,” Opt. Eng.50(2011).

2007 (2)

2006 (1)

2002 (2)

S. Reichelt, C. Pruss, and H. J. Tiziani, “Specification and characterization of CGHs for interferometrical optical testing,” Proc. SPIE4778 (2002).
[CrossRef]

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A19 (2002).
[CrossRef]

2001 (3)

M. Totzeck, “Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields,” Optik - International Journal for Light and Electron Optics112 (2001).
[CrossRef]

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, “Investigation of large null-CGH realization,” Proc. SPIE4440 (2001).
[CrossRef]

N. Bokor, R. Shechter, N. Davidson, A. A. Friesem, and Erez Hasman, “Achromatic phase retarder by slanted illumination of a dielectric grating with period comparable with the wavelength,” Appl. Opt.402076–2080 (2001).
[CrossRef]

2000 (1)

1997 (1)

1996 (1)

P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt.432063–2085 (1996).
[CrossRef]

1995 (1)

1989 (1)

S. M. Arnold, “How to test an asphere with a computer-generated hologram,” Proc. SPIE1052191–197 (1989).
[CrossRef]

1972 (1)

Arnold, S. M.

S. M. Arnold, “How to test an asphere with a computer-generated hologram,” Proc. SPIE1052191–197 (1989).
[CrossRef]

Bennett, V. P.

Bokor, N.

Burge, J. H.

Chang, Y.-C.

Davidson, N.

Drauschke, A.

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, “Investigation of large null-CGH realization,” Proc. SPIE4440 (2001).
[CrossRef]

Fainman, Yeshayahu

Feng, D.

Friesem, A. A.

Gao, Z.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “Systematic analysis of the measurement of cone angles using high line density computer-generated holograms,” Opt. Eng.50(2011).

Gaskill, J. D.

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

Gralak, B.

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A19 (2002).
[CrossRef]

Granet, Gérard

Häfner, M.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “Systematic analysis of the measurement of cone angles using high line density computer-generated holograms,” Opt. Eng.50(2011).

Hasman, Erez

Heitkamp, B.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “Systematic analysis of the measurement of cone angles using high line density computer-generated holograms,” Opt. Eng.50(2011).

Ichioka, Y.

Kley, E.-B.

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, “Investigation of large null-CGH realization,” Proc. SPIE4440 (2001).
[CrossRef]

Konishi, T.

Lalanne, P.

P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt.432063–2085 (1996).
[CrossRef]

Larochelle, S.

Lemercier-Lalanne, D.

P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt.432063–2085 (1996).
[CrossRef]

Li, L.

Ma, J.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “Systematic analysis of the measurement of cone angles using high line density computer-generated holograms,” Opt. Eng.50(2011).

Nevière, M.

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A19 (2002).
[CrossRef]

Osten, W.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “Systematic analysis of the measurement of cone angles using high line density computer-generated holograms,” Opt. Eng.50(2011).

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press, 1991).

Popov, E.

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A19 (2002).
[CrossRef]

Pruss, C.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “Systematic analysis of the measurement of cone angles using high line density computer-generated holograms,” Opt. Eng.50(2011).

S. Reichelt, C. Pruss, and H. J. Tiziani, “Specification and characterization of CGHs for interferometrical optical testing,” Proc. SPIE4778 (2002).
[CrossRef]

Reichelt, S.

S. Reichelt, C. Pruss, and H. J. Tiziani, “Specification and characterization of CGHs for interferometrical optical testing,” Proc. SPIE4778 (2002).
[CrossRef]

Richter, I.

Rockstroh, W.

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, “Investigation of large null-CGH realization,” Proc. SPIE4440 (2001).
[CrossRef]

Schmidt, H.

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, “Investigation of large null-CGH realization,” Proc. SPIE4440 (2001).
[CrossRef]

Shechter, R.

Sheng, Y.

Sun, P.-C.

Takahara, K.

Tayeb, G.

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A19 (2002).
[CrossRef]

Tiziani, H. J.

S. Reichelt, C. Pruss, and H. J. Tiziani, “Specification and characterization of CGHs for interferometrical optical testing,” Proc. SPIE4778 (2002).
[CrossRef]

Totzeck, M.

M. Totzeck, “Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields,” Optik - International Journal for Light and Electron Optics112 (2001).
[CrossRef]

Turunen, J.

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications, (Akademie Verlag, 1997).

Wittig, L.-C.

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, “Investigation of large null-CGH realization,” Proc. SPIE4440 (2001).
[CrossRef]

Wyant, J. C.

Wyrowski, F.

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, “Investigation of large null-CGH realization,” Proc. SPIE4440 (2001).
[CrossRef]

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications, (Akademie Verlag, 1997).

Xu, F.

Yotsuya, T.

Yu, W.

Yuan, C.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “Systematic analysis of the measurement of cone angles using high line density computer-generated holograms,” Opt. Eng.50(2011).

Zhou, P.

Zhu, R.

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “Systematic analysis of the measurement of cone angles using high line density computer-generated holograms,” Opt. Eng.50(2011).

Appl. Opt. (6)

J. Mod. Opt. (1)

P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt.432063–2085 (1996).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

J. Ma, C. Pruss, M. Häfner, B. Heitkamp, R. Zhu, Z. Gao, C. Yuan, and W. Osten, “Systematic analysis of the measurement of cone angles using high line density computer-generated holograms,” Opt. Eng.50(2011).

Opt. Express (1)

Optik - International Journal for Light and Electron Optics (1)

M. Totzeck, “Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields,” Optik - International Journal for Light and Electron Optics112 (2001).
[CrossRef]

Proc. SPIE (3)

S. Reichelt, C. Pruss, and H. J. Tiziani, “Specification and characterization of CGHs for interferometrical optical testing,” Proc. SPIE4778 (2002).
[CrossRef]

S. M. Arnold, “How to test an asphere with a computer-generated hologram,” Proc. SPIE1052191–197 (1989).
[CrossRef]

E.-B. Kley, W. Rockstroh, H. Schmidt, A. Drauschke, F. Wyrowski, and L.-C. Wittig, “Investigation of large null-CGH realization,” Proc. SPIE4440 (2001).
[CrossRef]

Other (4)

E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press, 1991).

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications, (Akademie Verlag, 1997).

W. Iff, S. Glaubrecht, N. Lindlein, and J. Schwider, “Untersuchung der Abweichungen zwischen skalarer und rigoroser Rechnung an CGHs in Twyman-Green-Interferometern zur Linsenprüfung,” DGaO-Proceedings http://www.dgao-proceedings.de (2010).

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

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Figures (18)

Fig. 1
Fig. 1

Null test for an asphere: A CGH is placed in the object path to generate a wavefront that matches the asphere. Fabrication errors of the asphere can be determined, if the effects of the interferometer itself and the CGH are known.

Fig. 2
Fig. 2

Comparison of scalar approximation and rigorous simulation for (a) binary grating with duty cycle of 0.5 and variable grating period and (b) variable duty cycle for different grating periods. Illumination wavelength is set to 633 nm.

Fig. 3
Fig. 3

Models of the simulated gratings for use in transmission and reflection.

Fig. 4
Fig. 4

Convergence check of the used binary gratings: Both types of gratins show a quick convergence, even for slanted illumination. The truncation order M was set to 50.

Fig. 5
Fig. 5

Changes in phase for line gratings used in transmission for linewidth, height and sidewall variations of 1 % using TE polarised light. Column 1 shows the +1st order, column 2 shows the 0th order and column 3 shows the −1st order.

Fig. 6
Fig. 6

Angular phase dependence of the 1st order for a structure with period of 2 μm: (a) output phase, (b) phase after unwrapping and (c) Intensity distribution for the given structure.

Fig. 7
Fig. 7

Simulation results for the phase change caused by 1% parameter variation for a binary grating with a Cr-layer used in reflection.

Fig. 8
Fig. 8

(a) Model of the blazed transmission grating used in the calculations (b) Comparison between scalar and rigorous simulation for the generated phase of a blazed grating. For small structure sizes the difference between scalar and rigorous are increasing rapidly.

Fig. 9
Fig. 9

Calculated phase changes for 1 % parameter change of a blazed grating with a height of 1.39 μm.

Fig. 10
Fig. 10

Binary blaze structure with a period of 1 μm and a misalignment Δx of 0 and ±75 nm.

Fig. 11
Fig. 11

Comparison of scalar and rigorous calculations of the phase for misaligned masks and perpendicular incident light.

Fig. 12
Fig. 12

Dependence of the generated phase over misalignment Δx for grating period Λ = 1 μm: It can be seen that even for a small misalignment the resulting change in phase can not be neglected. There is also a strong dependence of the phase on the angle of incidence.

Fig. 13
Fig. 13

Modelled structure with edge rounding. For the calculations 78 layers were used, resulting in a height of 17.79 nm for each layer.

Fig. 14
Fig. 14

Dependence of the generated phase over the number of used layers for RCWA simulation.

Fig. 15
Fig. 15

Change of induced phase change by small parameter variation over increasing edge rounding. The rigorous simulations were done for perpendicular incident light and 17° oblique incident light. For a comparison scalar results are also shown for perpendicular incident light.

Fig. 16
Fig. 16

Changes in phase for line gratings used in transmission for line width, height and side wall variations of 1 % using TM polarised light.

Fig. 17
Fig. 17

Calculated phase changes for 1 % parameter change of a blazed grating with pitch of 1 μm and height of 1.39 μm.

Fig. 18
Fig. 18

Simulation results for the phase change caused by 1% parameter variation for a binary grating with a Cr-layer used in reflection.

Equations (6)

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U = { A 0 + [ A 1 cos ( ξ ) A 0 ] D + i A 1 sin ( ξ ) D m = 0 [ A 1 cos ( ξ ) A 0 ] D sinc ( m D ) + i A 1 sin ( ξ ) D sinc ( m D ) m = ± 1 , ± 2
ξ = ( n 1 ) d λ 2 π
tan ϕ = { U } { U } = { A 1 sin ( ξ ) D A 0 + [ A 1 cos ( ξ ) A 0 ] D m = 0 A 1 sin ( ξ ) sinc ( m D ) [ A 1 cos ( ξ ) A 0 ] sinc + ( m D ) m = ± 1 , ± 2
W = ϕ 2 π
Δ W = Δ W rig Δ W scalar
g round ( x ) = g ( x ) * 1 2 π σ exp ( x 2 2 σ 2 )

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