Abstract

Computer-generated holograms (CGHs), such as those used in optical testing, are created by etching patterns into an optical substrate. Imperfections in the etching can cause small scale surface roughness that varies across the pattern. The variation in this roughness affects the phase and amplitude of the wavefronts in the various diffraction orders. A simplified model is developed and validated that treats the scattering loss from the roughness as an amplitude effect. We demonstrate the use of this model for engineering analysis and provide a graphical method for understanding the application. Furthermore, we investigate the magnitude of this effect for the application of optical testing and show that the effect on measurement accuracy is limited to 1nm for typical CGHs.

© 2007 Optical Society of America

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References

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  1. T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
    [CrossRef]
  2. P. Zhou and J. H. Burge, "Fabrication error analysis and experimental demonstration for computer-generated holograms," Appl. Opt. 46, (2007).
    [CrossRef] [PubMed]
  3. Y. C. Chang and J. H. Burge, "Errors analysis for CGH optical testing," Proc. SPIE 3782, 358-366 (1999).
    [CrossRef]
  4. Y. C. Chang, P. Zhou, and J. H. Burge, "Analysis of phase sensitivity for binary computer generated holograms," Appl. Opt. 45, 4223-4234 (2006).
    [CrossRef] [PubMed]
  5. D. W. Ricks and L. V. Chizek, "Light scattering from binary optics," Proc. SPIE 1211, 24-37 (1990).
    [CrossRef]
  6. C. Saxer and K. Freischlad, "Interference microscope for sub-Angstrom surface roughness measurement," Proc. SPIE 5144, 37-45 (2003).
    [CrossRef]

2007 (1)

P. Zhou and J. H. Burge, "Fabrication error analysis and experimental demonstration for computer-generated holograms," Appl. Opt. 46, (2007).
[CrossRef] [PubMed]

2006 (1)

2003 (1)

C. Saxer and K. Freischlad, "Interference microscope for sub-Angstrom surface roughness measurement," Proc. SPIE 5144, 37-45 (2003).
[CrossRef]

1999 (1)

Y. C. Chang and J. H. Burge, "Errors analysis for CGH optical testing," Proc. SPIE 3782, 358-366 (1999).
[CrossRef]

1990 (1)

D. W. Ricks and L. V. Chizek, "Light scattering from binary optics," Proc. SPIE 1211, 24-37 (1990).
[CrossRef]

1985 (1)

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[CrossRef]

Burge, J. H.

P. Zhou and J. H. Burge, "Fabrication error analysis and experimental demonstration for computer-generated holograms," Appl. Opt. 46, (2007).
[CrossRef] [PubMed]

Y. C. Chang, P. Zhou, and J. H. Burge, "Analysis of phase sensitivity for binary computer generated holograms," Appl. Opt. 45, 4223-4234 (2006).
[CrossRef] [PubMed]

Y. C. Chang and J. H. Burge, "Errors analysis for CGH optical testing," Proc. SPIE 3782, 358-366 (1999).
[CrossRef]

Chang, Y. C.

Chizek, L. V.

D. W. Ricks and L. V. Chizek, "Light scattering from binary optics," Proc. SPIE 1211, 24-37 (1990).
[CrossRef]

Freischlad, K.

C. Saxer and K. Freischlad, "Interference microscope for sub-Angstrom surface roughness measurement," Proc. SPIE 5144, 37-45 (2003).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[CrossRef]

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[CrossRef]

Ricks, D. W.

D. W. Ricks and L. V. Chizek, "Light scattering from binary optics," Proc. SPIE 1211, 24-37 (1990).
[CrossRef]

Saxer, C.

C. Saxer and K. Freischlad, "Interference microscope for sub-Angstrom surface roughness measurement," Proc. SPIE 5144, 37-45 (2003).
[CrossRef]

Zhou, P.

P. Zhou and J. H. Burge, "Fabrication error analysis and experimental demonstration for computer-generated holograms," Appl. Opt. 46, (2007).
[CrossRef] [PubMed]

Y. C. Chang, P. Zhou, and J. H. Burge, "Analysis of phase sensitivity for binary computer generated holograms," Appl. Opt. 45, 4223-4234 (2006).
[CrossRef] [PubMed]

Appl. Opt. (2)

P. Zhou and J. H. Burge, "Fabrication error analysis and experimental demonstration for computer-generated holograms," Appl. Opt. 46, (2007).
[CrossRef] [PubMed]

Y. C. Chang, P. Zhou, and J. H. Burge, "Analysis of phase sensitivity for binary computer generated holograms," Appl. Opt. 45, 4223-4234 (2006).
[CrossRef] [PubMed]

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[CrossRef]

Proc. SPIE (3)

D. W. Ricks and L. V. Chizek, "Light scattering from binary optics," Proc. SPIE 1211, 24-37 (1990).
[CrossRef]

C. Saxer and K. Freischlad, "Interference microscope for sub-Angstrom surface roughness measurement," Proc. SPIE 5144, 37-45 (2003).
[CrossRef]

Y. C. Chang and J. H. Burge, "Errors analysis for CGH optical testing," Proc. SPIE 3782, 358-366 (1999).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Illustrates an ideal binary, linear grating.

Fig. 2
Fig. 2

Actual profile of a binary, linear grating.

Fig. 3
Fig. 3

Rough grating and its equivalent model.

Fig. 4
Fig. 4

(Color online) Zero-order diffraction efficiency and wavefront phase across the grating (50% duty cycle and 0.35 λ etching depth).

Fig. 5
Fig. 5

(Color online) First-order diffraction efficiency and wavefront phase across the grating (50% duty cycle and 0.35 λ etching depth).

Fig. 6
Fig. 6

(Color online) Diffraction efficiency sensitivity for the (left) zero order and the (right) first order.

Fig. 7
Fig. 7

(Color online) Wavefront phase sensitivity for the (left) zero order and the (right) first order.

Fig. 8
Fig. 8

Graphical representation for (left) zero- and (right) first-order diffraction.

Tables (3)

Tables Icon

Table 1 Summary of Diffraction Efficiencies and Wavefront Phases

Tables Icon

Table 2 Sensitivity Functions for Diffraction Efficiencies and Wavefront Phases

Tables Icon

Table 3 Diffraction Efficiency Variation and Wavefront Phase Variation From Surface Roughness (50% Duty Cycle and 0.35λ Etching Depth)

Equations (7)

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u ( x ) = A 0 + ( A 1 e i ϕ A 0 ) rect ( x b ) 1 S   comb ( x S ) .
U ( ξ ) = { { A 0 + [ A 1   cos ( ϕ ) A 0 ] D } + i { A 1   sin ( ϕ ) D } m = 0 { [ A 1   cos ( ϕ ) A 0 ] D   sinc ( m D ) } + i { A 1   sin ( ϕ ) D   sinc ( m D ) } m = ±1 , ±2 ,  … ,
TIS r = I scat I ref = ( 2 π σ ) 2 = ( 4 π λ R q ) 2 .
TIS t = I scat I trans = ( 2 π σ ) 2 = [ 2 π ( n 1 ) R q / λ ] 2 ,
A = A Fresnel 1 TIS ,
Δ η A 1 = η A 1 Δ A 1 ,
Δ W A 1 = 1 2 π Ψ A 1 Δ A 1 ,

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