Abstract

Aspheric optical surfaces are often tested using computer-generated holograms (CGHs). For precise measurement, the wavefront errors caused by the CGH must be known and characterized. A parametric model relating the wavefront errors to the CGH fabrication errors is introduced. Methods are discussed for measuring the fabrication errors in the CGH substrate, duty cycle, etching depth, and effect of surface roughness. An example analysis of the wavefront errors from fabrication nonuniformities for a phase CGH is given. The calibration of these effects for a CGH null corrector is demonstrated to cause measurement error less than 1 nm.

© 2007 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).
  2. S. M. Arnold, "How to test an asphere with a computer generated hologram," in Holographic Optics,Proc. SPIE 1052, 191-197 (1989).
  3. H. J. Tiziani, S. Reichelt, C. Pruss, M. Rocktaeschel, and U. Hofbauer, "Testing of aspheric surfaces," in Lithographic and Micromachining Techniques for Optical Component Fabrication,Proc. SPIE 4440, 109-119 (2001).
    [CrossRef]
  4. R. Schreiner, T. Herrmann, J. Röder, S. Müller-Pfeiffer, and O. Falkenstörfer, "Design considerations for computer generated holograms as supplement to Fizeau interferometers," in Optical Fabrication, Testing, and Methodology II,Proc. SPIE 5965, 59650K (2005).
    [CrossRef]
  5. E. Curatu and M. Wang, "Tolerancing and testing of CGH aspheric nulls," in Optical Manufacturing and Testing III,Proc. SPIE 3782, 581-599 (1999).
  6. Y. C. Chang and J. H. Burge, "Errors analysis for CGH optical testing," in Optical Manufacturing and Testing III,Proc. SPIE 3782, 358-366 (1999).
    [CrossRef]
  7. 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]
  8. P. Zhou and J. H. Burge are preparing a paper to be called "Coupling of surface roughness on the wavefront performance for diffraction grating."
  9. A. F. Fercher, "Computer-generated holograms for testing optical elements: error analysis and error compensation," Opt. Acta 23, 347-365 (1976).
    [CrossRef]
  10. J. C. Wyant, P. K. O'Neill, and A. J. MacGovern, "Interferometric method of measuring plotter distortion," Appl. Opt. 13, 1549-1551 (1974).
    [CrossRef] [PubMed]
  11. A. Ono and J. C. Wyant, "Plotting errors measurement of CGH using an improved interferometric method," Appl. Opt. 23, 3905-3910 (1984).
    [CrossRef] [PubMed]
  12. J. Burge, "A null test for null correctors: error analysis," in Quality and Reliability for Optical Systems,Proc. SPIE 1993, 86-97 (1993).
    [CrossRef]
  13. P. Zhou and J. H. Burge are preparing a paper to be called "Optimization design of computer generated holograms to improve the performance."

2006

2005

R. Schreiner, T. Herrmann, J. Röder, S. Müller-Pfeiffer, and O. Falkenstörfer, "Design considerations for computer generated holograms as supplement to Fizeau interferometers," in Optical Fabrication, Testing, and Methodology II,Proc. SPIE 5965, 59650K (2005).
[CrossRef]

2001

H. J. Tiziani, S. Reichelt, C. Pruss, M. Rocktaeschel, and U. Hofbauer, "Testing of aspheric surfaces," in Lithographic and Micromachining Techniques for Optical Component Fabrication,Proc. SPIE 4440, 109-119 (2001).
[CrossRef]

1999

E. Curatu and M. Wang, "Tolerancing and testing of CGH aspheric nulls," in Optical Manufacturing and Testing III,Proc. SPIE 3782, 581-599 (1999).

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

1993

J. Burge, "A null test for null correctors: error analysis," in Quality and Reliability for Optical Systems,Proc. SPIE 1993, 86-97 (1993).
[CrossRef]

1989

S. M. Arnold, "How to test an asphere with a computer generated hologram," in Holographic Optics,Proc. SPIE 1052, 191-197 (1989).

1984

1976

A. F. Fercher, "Computer-generated holograms for testing optical elements: error analysis and error compensation," Opt. Acta 23, 347-365 (1976).
[CrossRef]

1974

Arnold, S. M.

S. M. Arnold, "How to test an asphere with a computer generated hologram," in Holographic Optics,Proc. SPIE 1052, 191-197 (1989).

Burge, J.

J. Burge, "A null test for null correctors: error analysis," in Quality and Reliability for Optical Systems,Proc. SPIE 1993, 86-97 (1993).
[CrossRef]

Burge, J. H.

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," in Optical Manufacturing and Testing III,Proc. SPIE 3782, 358-366 (1999).
[CrossRef]

Chang, Y. C.

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," in Optical Manufacturing and Testing III,Proc. SPIE 3782, 358-366 (1999).
[CrossRef]

Curatu, E.

E. Curatu and M. Wang, "Tolerancing and testing of CGH aspheric nulls," in Optical Manufacturing and Testing III,Proc. SPIE 3782, 581-599 (1999).

Falkenstörfer, O.

R. Schreiner, T. Herrmann, J. Röder, S. Müller-Pfeiffer, and O. Falkenstörfer, "Design considerations for computer generated holograms as supplement to Fizeau interferometers," in Optical Fabrication, Testing, and Methodology II,Proc. SPIE 5965, 59650K (2005).
[CrossRef]

Fercher, A. F.

A. F. Fercher, "Computer-generated holograms for testing optical elements: error analysis and error compensation," Opt. Acta 23, 347-365 (1976).
[CrossRef]

Herrmann, T.

R. Schreiner, T. Herrmann, J. Röder, S. Müller-Pfeiffer, and O. Falkenstörfer, "Design considerations for computer generated holograms as supplement to Fizeau interferometers," in Optical Fabrication, Testing, and Methodology II,Proc. SPIE 5965, 59650K (2005).
[CrossRef]

Hofbauer, U.

H. J. Tiziani, S. Reichelt, C. Pruss, M. Rocktaeschel, and U. Hofbauer, "Testing of aspheric surfaces," in Lithographic and Micromachining Techniques for Optical Component Fabrication,Proc. SPIE 4440, 109-119 (2001).
[CrossRef]

MacGovern, A. J.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

Müller-Pfeiffer, S.

R. Schreiner, T. Herrmann, J. Röder, S. Müller-Pfeiffer, and O. Falkenstörfer, "Design considerations for computer generated holograms as supplement to Fizeau interferometers," in Optical Fabrication, Testing, and Methodology II,Proc. SPIE 5965, 59650K (2005).
[CrossRef]

O'Neill, P. K.

Ono, A.

Pruss, C.

H. J. Tiziani, S. Reichelt, C. Pruss, M. Rocktaeschel, and U. Hofbauer, "Testing of aspheric surfaces," in Lithographic and Micromachining Techniques for Optical Component Fabrication,Proc. SPIE 4440, 109-119 (2001).
[CrossRef]

Reichelt, S.

H. J. Tiziani, S. Reichelt, C. Pruss, M. Rocktaeschel, and U. Hofbauer, "Testing of aspheric surfaces," in Lithographic and Micromachining Techniques for Optical Component Fabrication,Proc. SPIE 4440, 109-119 (2001).
[CrossRef]

Rocktaeschel, M.

H. J. Tiziani, S. Reichelt, C. Pruss, M. Rocktaeschel, and U. Hofbauer, "Testing of aspheric surfaces," in Lithographic and Micromachining Techniques for Optical Component Fabrication,Proc. SPIE 4440, 109-119 (2001).
[CrossRef]

Röder, J.

R. Schreiner, T. Herrmann, J. Röder, S. Müller-Pfeiffer, and O. Falkenstörfer, "Design considerations for computer generated holograms as supplement to Fizeau interferometers," in Optical Fabrication, Testing, and Methodology II,Proc. SPIE 5965, 59650K (2005).
[CrossRef]

Schreiner, R.

R. Schreiner, T. Herrmann, J. Röder, S. Müller-Pfeiffer, and O. Falkenstörfer, "Design considerations for computer generated holograms as supplement to Fizeau interferometers," in Optical Fabrication, Testing, and Methodology II,Proc. SPIE 5965, 59650K (2005).
[CrossRef]

Tiziani, H. J.

H. J. Tiziani, S. Reichelt, C. Pruss, M. Rocktaeschel, and U. Hofbauer, "Testing of aspheric surfaces," in Lithographic and Micromachining Techniques for Optical Component Fabrication,Proc. SPIE 4440, 109-119 (2001).
[CrossRef]

Wang, M.

E. Curatu and M. Wang, "Tolerancing and testing of CGH aspheric nulls," in Optical Manufacturing and Testing III,Proc. SPIE 3782, 581-599 (1999).

Wyant, J. C.

Zhou, P.

Appl. Opt.

Opt. Acta

A. F. Fercher, "Computer-generated holograms for testing optical elements: error analysis and error compensation," Opt. Acta 23, 347-365 (1976).
[CrossRef]

Proc. SPIE

S. M. Arnold, "How to test an asphere with a computer generated hologram," in Holographic Optics,Proc. SPIE 1052, 191-197 (1989).

H. J. Tiziani, S. Reichelt, C. Pruss, M. Rocktaeschel, and U. Hofbauer, "Testing of aspheric surfaces," in Lithographic and Micromachining Techniques for Optical Component Fabrication,Proc. SPIE 4440, 109-119 (2001).
[CrossRef]

R. Schreiner, T. Herrmann, J. Röder, S. Müller-Pfeiffer, and O. Falkenstörfer, "Design considerations for computer generated holograms as supplement to Fizeau interferometers," in Optical Fabrication, Testing, and Methodology II,Proc. SPIE 5965, 59650K (2005).
[CrossRef]

E. Curatu and M. Wang, "Tolerancing and testing of CGH aspheric nulls," in Optical Manufacturing and Testing III,Proc. SPIE 3782, 581-599 (1999).

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

J. Burge, "A null test for null correctors: error analysis," in Quality and Reliability for Optical Systems,Proc. SPIE 1993, 86-97 (1993).
[CrossRef]

Other

P. Zhou and J. H. Burge are preparing a paper to be called "Optimization design of computer generated holograms to improve the performance."

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

P. Zhou and J. H. Burge are preparing a paper to be called "Coupling of surface roughness on the wavefront performance for diffraction grating."

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Figures (6)

Fig. 1
Fig. 1

Binary, linear grating profile.

Fig. 2
Fig. 2

Setup of CGH substrate measurement.

Fig. 3
Fig. 3

Monte Carlo simulation on (left) etching depth and (right) duty cycle.

Fig. 4
Fig. 4

(Color online) Sensitivity of diffraction efficiencies to (left) etching depth and (right) duty cycle.

Fig. 5
Fig. 5

(Color online) Transmitted wavefront of the CGH substrate.

Fig. 6
Fig. 6

Setup of the CGH substrate calibration.

Tables (6)

Tables Icon

Table 1 Summary of Equations for Parametric Model Analysis

Tables Icon

Table 2 Duty Cycle and Etching Depth of the Five Positions

Tables Icon

Table 3 Surface Roughness Measurement

Tables Icon

Table 4 Wavefront Errors from CGH Fabrication Nonuniformities without Substrate Calibration

Tables Icon

Table 5 Wavefront Errors from CGH Fabrication Errors after Subtracting the Zeroth-Order Measurement

Tables Icon

Table 6 Substrate Calibration

Equations (97)

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D = b / S
A 0
A 1
2 π ( n 1 ) t / λ
Ψ / D
Ψ / ϕ
Δ D
Δ ϕ
Ψ / ( A 0 / A 1 )
Δ ( A 0 / A 1 )
A 0 / A 1
Δ W D = 1 2 π Ψ D Δ D = 1 2 π Ψ D ( Δ D D ) D ,
Δ W ϕ = Ψ ϕ Δ ϕ = Ψ ϕ ( Δ ϕ ϕ ) ϕ ,
Δ W A 0 / A 1 = 1 2 π Ψ ( A 0 / A 1 ) Δ ( A 0 A 1 ) ,
Δ D
Δ W D
Δ ϕ
Δ W e
Δ ( A 0 / A 1 )
A 0
A 1
Δ W A 0 / A 1
Δ W ( x , y ) = m λ ε S ,
δ s
2 δ s
( n 1 ) δ s
0.33 λ
A 0
A 1
± 1 %
± 3 %
± 5 %
δ t
δ D
δ t
η / ϕ
η / D
0.33 λ
δ t
± 1 %
δ t
δ D
0.0005 λ
I s c a t ( 2 π σ ) 2 = [ 2 π ( n 1 ) R q / λ ] 2 ,
σ 2
R q
I s c a t = ( 4 π R q / λ ) 2
I = 1 I s c a t .
632.8   nm
2 ± 0.5   nm
0.01 % ± 0.005 %
0.33 λ
0.00086   nm
( 1   in. = 2.54   cm )
0.33 λ
0.33 λ
13.6   nm
1   mm
20   ms
13.6   nm
0.3   nm
0.1 μ m
20 μ m
632.8   nm
3.2   nm
0.35 λ
632.8   nm
15   mm
15   nm
1   nm
1   nm
A 0 2 ( 1 D ) 2 + A 1 2 D 2 + 2 A 0 A 1 D ( 1 D ) cos ( ϕ )
[ A 0 2 + A 1 2 2 A 0 A 1  cos ( ϕ ) ] D 2  sin c 2 ( m D )
tan ( Ψ )
A 1 D   sin ( ϕ ) A 0 ( 1 D ) + A 1 D   cos ( ϕ )
A 1   sin ( ϕ ) sinc ( m D ) [ A 0 + A 1   cos ( ϕ ) ] sinc ( m D )
η D
2 A 0 2 ( 1 D ) + 2 A 1 2 D + 2 A 0 A 1 ( 1 2 D ) cos   ϕ
2 [ A 0 2 + A 1 2 2 A 0 A 1   cos ( ϕ ) ] D   sinc ( 2 m D )
η ϕ
2 A 0 A 1 D ( 1 D ) sin   ϕ
2 A 0 A 1   sin   ϕ D 2  sin c 2 ( m D )
Ψ D
A 0 A 1   sin   ϕ A 1 2 D 2 + A 0 2 ( 1 D ) 2 + 2 A 0 A 1 D ( 1 D ) cos   ϕ
{ , 0 , for   sinc ( m D ) = 0 otherwise
Ψ ϕ
A 1 2 D 2 + A 0 A 1 D ( 1 D ) cos   ϕ A 1 2 D 2 + A 0 2 ( 1 D ) 2 + 2 A 0 A 1 D ( 1 D ) cos   ϕ
A 1 2 A 0 A 1  cos   ϕ A 1 2 + A 0 2 2 A 0 A 1   cos   ϕ
Ψ ( A 0 / A 1 )
D ( 1 D ) sin   ϕ ( A 0 / A 1 ) 2 ( 1 D ) 2 + D 2 + 2 D ( 1 D ) ( A 0 / A 1 ) cos   ϕ
sin   ϕ ( A 0 / A 1 ) 2 + 1 2 ( A 0 / A 1 ) cos   ϕ
| A 0 / A 1 |
δ Ψ m = 1 δ ϕ
δ Ψ m = 1 δ ( A 0 / A 1 )
Ψ m = 1 ϕ Ψ m = 0 ϕ
Ψ m = 0 D
Ψ m = 1 ( A 0 / A 1 ) Ψ m = 0 ( A 0 / A 1 )

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