Abstract

Aspheric surfaces are often measured using interferometers with null correctors, either refractive or diffractive. The use of null correctors allows high accuracy in the measurement, but also introduces imaging aberrations, such as mapping distortion and field curvature. These imaging aberrations couple with diffraction effects and limit the accuracy of the measurements, causing high frequency features in the surface under test to be filtered out and creating artifacts near boundaries, especially at edges. We provide a concise methodology for analyzing these effects using the astigmatic field curves to define the aberration, and showing how this couples with diffraction as represented by the Talbot effect and Fresnel edge diffraction. The resulting relationships are validated with both computer simulations and direct measurements from an interferometer with CGH null corrector.

© 2012 OSA

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References

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  1. D. Malacara, Optical Shop Testing, 3rd ed. (Wiley 2007).
  2. J. M. Sasian, “Design of null lens correctors for the testing of astronomical optics,” Opt. Eng. 27, 1051 (1988).
  3. J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” Proc. SPIE 2576, 258–269 (1995).
    [CrossRef]
  4. C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L(2007).
    [CrossRef]
  5. M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313, 706313-8 (2008).
    [CrossRef]
  6. P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49(28), 5351–5359 (2010).
    [CrossRef] [PubMed]
  7. P. Zhou and J. H. Burge, “Diffraction effects in interferometry,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OMA3.
  8. P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790, 77900L (2010).
  9. J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010).
    [CrossRef]
  10. E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high spatial frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997).
    [CrossRef]
  11. L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE 1400, 24–32 (1990).
  12. P. E. Murphy, T. G. Brown, and D. T. Moore, “Measurement and calibration of interferometric imaging aberrations,” Appl. Opt. 39(34), 6421–6429 (2000).
    [CrossRef] [PubMed]
  13. J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005), pp. 88–91.
  14. C. Zhao and J. H. Burge, “Generalization of the Coddington equations to include hybrid diffractive surfaces,” Proc. SPIE 7652, 76522U (2010).
    [CrossRef]
  15. Zemax, “Design tools,” http://www.zemax.com/ .

2010 (4)

P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790, 77900L (2010).

J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010).
[CrossRef]

C. Zhao and J. H. Burge, “Generalization of the Coddington equations to include hybrid diffractive surfaces,” Proc. SPIE 7652, 76522U (2010).
[CrossRef]

P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49(28), 5351–5359 (2010).
[CrossRef] [PubMed]

2008 (1)

M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313, 706313-8 (2008).
[CrossRef]

2007 (1)

C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L(2007).
[CrossRef]

2000 (1)

1997 (1)

E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high spatial frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997).
[CrossRef]

1995 (1)

J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” Proc. SPIE 2576, 258–269 (1995).
[CrossRef]

1990 (1)

L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE 1400, 24–32 (1990).

1988 (1)

J. M. Sasian, “Design of null lens correctors for the testing of astronomical optics,” Opt. Eng. 27, 1051 (1988).

Ai, C.

E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high spatial frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997).
[CrossRef]

Brown, T. G.

Burge, J. H.

J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010).
[CrossRef]

C. Zhao and J. H. Burge, “Generalization of the Coddington equations to include hybrid diffractive surfaces,” Proc. SPIE 7652, 76522U (2010).
[CrossRef]

P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49(28), 5351–5359 (2010).
[CrossRef] [PubMed]

P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790, 77900L (2010).

M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313, 706313-8 (2008).
[CrossRef]

C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L(2007).
[CrossRef]

J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” Proc. SPIE 2576, 258–269 (1995).
[CrossRef]

Moore, D. T.

Murphy, P. E.

Novak, E.

E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high spatial frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997).
[CrossRef]

Novak, M.

M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313, 706313-8 (2008).
[CrossRef]

Sasian, J. M.

J. M. Sasian, “Design of null lens correctors for the testing of astronomical optics,” Opt. Eng. 27, 1051 (1988).

Selberg, L. A.

L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE 1400, 24–32 (1990).

Wyant, J. C.

E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high spatial frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997).
[CrossRef]

Zhao, C.

C. Zhao and J. H. Burge, “Generalization of the Coddington equations to include hybrid diffractive surfaces,” Proc. SPIE 7652, 76522U (2010).
[CrossRef]

P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790, 77900L (2010).

J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010).
[CrossRef]

M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313, 706313-8 (2008).
[CrossRef]

C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L(2007).
[CrossRef]

Zhou, P.

P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790, 77900L (2010).

P. Zhou and J. H. Burge, “Analysis of wavefront propagation using the Talbot effect,” Appl. Opt. 49(28), 5351–5359 (2010).
[CrossRef] [PubMed]

J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010).
[CrossRef]

Appl. Opt. (2)

Opt. Eng. (1)

J. M. Sasian, “Design of null lens correctors for the testing of astronomical optics,” Opt. Eng. 27, 1051 (1988).

Proc. SPIE (8)

J. H. Burge, “Applications of computer-generated holograms for interferometric measurement of large aspheric optics,” Proc. SPIE 2576, 258–269 (1995).
[CrossRef]

C. Zhao and J. H. Burge, “Imaging aberrations from null correctors,” Proc. SPIE 6723, 67230L(2007).
[CrossRef]

M. Novak, C. Zhao, and J. H. Burge, “Distortion mapping correction in aspheric null testing,” Proc. SPIE 7063, 706313, 706313-8 (2008).
[CrossRef]

P. Zhou, J. H. Burge, and C. Zhao, “Imaging issues for interferometric measurement of aspheric surfaces using CGH null correctors,” Proc. SPIE 7790, 77900L (2010).

J. H. Burge, C. Zhao, and P. Zhou, “Imaging issues for interferometry with CGH null correctors,” Proc. SPIE 7739, 77390T (2010).
[CrossRef]

E. Novak, C. Ai, and J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high spatial frequency phase measurements,” Proc. SPIE 3134, 114–121 (1997).
[CrossRef]

L. A. Selberg, “Interferometer accuracy and precision,” Proc. SPIE 1400, 24–32 (1990).

C. Zhao and J. H. Burge, “Generalization of the Coddington equations to include hybrid diffractive surfaces,” Proc. SPIE 7652, 76522U (2010).
[CrossRef]

Other (4)

Zemax, “Design tools,” http://www.zemax.com/ .

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley 2007).

J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005), pp. 88–91.

P. Zhou and J. H. Burge, “Diffraction effects in interferometry,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OMA3.

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Figures (17)

Fig. 1
Fig. 1

The refractive a) and diffractive b) null correctors are often used to test the aspheric surfaces. The null corrector is designed for wavefront performance, but not for good imaging. Therefore, the system suffers imaging aberration which will limit the measurement accuracy.

Fig. 2
Fig. 2

Propagation in a converging space is converted to equivalent propagation in a collimated space. The period and amplitude of wavefront ripples at R1 will change due to diffraction as they propagate to R2. The change in the period scales according to geometry. The change in magnitude can be evaluated using Eq. (4), which was also derived and validated in Ref. 6.

Fig. 3
Fig. 3

The amplitude and phase variation calculated from the Fresnel integrals for the case of diffraction of collimated light from a knife edge.

Fig. 4
Fig. 4

An ideal lens can be used to convert the converging/diverging wavefront to a collimated one, and then we can use the Talbot effect to analyze the phase smoothing due to defocus error. (a) diffractive null optics; (b) refractive null optics. Solid line: wavefront model showing that rays leave the test surface at the normal direction and come to an ideal focus after the null optics; Dotted line: imaging model showing a point on the test surface is imaged through the null optics and the interferometer stop limits the illuminated area on the null optics and the spatial frequencies that passed by the system.

Fig. 5
Fig. 5

The edge diffraction from outer/inner edge. It is shown that the edge diffraction gets better by focusing at the inner edge in this particular case. (a) Mapping distortion from object to image; (b) Focus at inner edge; (c) Focus at outer edge.

Fig. 6
Fig. 6

The field curves for an axisymmetric system. The short solid lines in (b) and (c) represent the magnitude and orientation of the two principal axes.

Fig. 7
Fig. 7

The Zemax simulation of a coherent imaging system allows the geometric analysis of the field curves and complete diffraction propagation analysis as the incident phase ripple is rotated away from the principal axes.

Fig. 8
Fig. 8

Comparison of the computer simulation of a coherent imaging system with the theoretical calculation based on Talbot effects. The phase ripple has a spatial period of 0.5 mm and the wavelength of this test is 1µm. The imaging astigmatism creates two principal focus positions, represented by field curves zt and zs, which are at −190 mm and 138 mm respectively. The analytical transfer function can be calculated using Eqs. (1) and (7). The Zemax diffraction simulation uses the model shown in Fig. 7. The simulation matches the analytical prediction. The Zemax simulation has no data points near where the transfer functions are close to zero because of the low signal-noise-ratio.

Fig. 9
Fig. 9

Simulation of a 20 nm P-V phase ripple with a period of 0.5 mm passing through an imaging system with a transfer function shown in Fig. 8: a) original phase ripple; b) phase ripple after passing the aberrated imaging system. (Note: although the images have the appearance of an interferogram, the gray scale is used here to depict phase, not intensity.)

Fig. 10
Fig. 10

Use a 4f system to show the effects of the quadratic imaging aberration on the phase. The three dots superimposed on the three phase plates are three diffraction orders (1, 0 and −1 orders).

Fig. 11
Fig. 11

The mapping distortion caused by the three quadratic phase terms. (a) 2 λ P-V Cylinder x; (b) 2 λ P-V Cylinder y; (c) 2 λ P-V Astigmatism 45°

Fig. 12
Fig. 12

Experiment setup: a hologram is used to convert the spherical wavefront from the interferometer to a cylindrical wavefront and match the cylinder mirror under test.

Figure 13
Figure 13

The two field curves are almost parallel because the hologram has power in one direction and no power in the other direction. A paraxial lens with the focal length of 516.8 mm is used to convert the beam to a collimated one.

Fig. 14
Fig. 14

Images of a binary mask with the interferometer focused at different planes for the CGH test of the cylinder. a) at vertical focus, showing sharp contrast; b) at intermediate focus where the contrast is predicted to go to zero; c) at horizontal focus, where the transfer function predicts high contrast, but with a phase reversal. A rectangular opaque mask (shown as red rectangle in the image) was added in front the amplitude mask to help visualize the reverse contrast at the horizontal focus.

Fig. 15
Fig. 15

Images of circular aperture stop at different focal plane. Red arrows indicate the positions where diffraction occurs strongly. a) at horizontal focus; b) at intermediate focus; c) at vertical focus.

Fig. 16
Fig. 16

A close-up phase map due to diffraction from a vertical knife edge around x = 0 mm.

Fig. 17
Fig. 17

A line profile of edge diffraction from vertical knife edge around x = 0 mm. The experiment is in a good agreement with theoretical calculation.

Equations (8)

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TF= W' W =cos( 2π L z T )=cos( πλL p 2 ),
p 1 a 1 = p 2 a 2 and a 1 R 1 = a 2 R 2 .
Z 2 ' Z 1 ' = f 2 ( 1 R 2 1 R 1 ),
TF= W' W =cos( πλ( Z 2 ' Z 1 ' ) ( p 1 ' ) 2 )=cos( πλ R 1 ( R 1 R 2 ) R 2 p 1 2 )=cos( πλ L e ν n 2 4 a 1 2 ),
U= 1 2 [ C(u)C()+i( S(u)S() ) ],
T F t (x,y)=cos[ πλ ν n 2 4 a 2 m t (x,y) 2 z t (x,y) ] T F s (x,y)=cos[ πλ ν n 2 4 a 2 m s (x,y) 2 z s (x,y) ],
z (x,y) θ = z s (x,y) cos 2 (θ)+ z t (x,y) sin 2 (θ) = 1 2 [ z s (x,y)+ z t (x,y) ]+ 1 2 cos(2θ)[ z s (x,y) z t (x,y) ],
m (x,y) θ = m s 2 (x,y) cos 2 (θ)+ m t 2 (x,y) sin 2 (θ) = 1 2 [ m s 2 (x,y)+ m t 2 (x,y) ]+ 1 2 cos(2θ)[ m s 2 (x,y) m t 2 (x,y) ] ,

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