Abstract

Phase-shifting interferometry is the standard method for testing figure error on optical surfaces. Instruments measuring spheres and flats are readily available, but the accurate measurement of aspheres requires null correction. One problem with the general (nonull) testing of aspheres is the loss of common path. Systematic errors are introduced into the measurement by the fringe imaging optics. The sources and types of error are reviewed, as well as their effect on a wave-front measurement. These nonnull errors are predicted generally, with third-order analytic expressions derived for a tilted or a defocused test surface. An interferometer is built to test the expressions. The imaging system is a single lens, nominally image telecentric. Measurements are performed on a test surface defocused from -5 to 5 mm. The resulting measurement bias is shown to be in good agreement with third-order aberration theory predictions.

© 2000 Optical Society of America

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References

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  1. Y.-Y. Cheng, J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. 23, 4539–4543 (1984).
    [CrossRef] [PubMed]
  2. J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5258 (1987).
    [CrossRef] [PubMed]
  3. R. J. Palum, J. E. Greivenkamp, “Sub-Nyquist interferometry: results and implementation issues,” in Laser Interferometry: Qualitative Analysis of Interferograms: Third in a Series, J. E. Wampler, ed., Proc. SPIE1162, 378–388 (1989).
    [CrossRef]
  4. J. E. Greivenkamp, A. E. Lowman, R. J. Palum, “Sub-Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35, 2962–2969 (1996).
    [CrossRef]
  5. P. E. Murphy, T. G. Brown, D. T. Moore, “Optical Vernier interferometry for aspheric metrology,” in Emerging Lithographic Technologies III, Y. Vladimirsky, ed., Proc. SPIE3676, 643–652 (1999).
    [CrossRef]
  6. A. E. Lowman, J. E. Greivenkamp, “Interferometer induced wavefront errors when testing in a non-null configuration,” in Interferometry VI: Applications, R. J. Pryptniewicz, G. M. Brown, W. E. Jeuptner, eds., Proc. SPIE2004, 173–181 (1993).
    [CrossRef]
  7. A. E. Lowman, J. E. Greivenkamp, “Modeling an interferometer for non-null testing of aspheres,” in Optical Manufacturing and Testing, V. J. Daugherty, H. Stabl, eds., Proc. SPIE2536, 139–147 (1995).
    [CrossRef]
  8. C. J. Evans, “Compensation for errors introduced by non-zero fringe densities in phase-measuring interferometers,” CIRP Annals 42/1, 577–580 (1993).
  9. C. J. Evans, “Software based improvements in the accuracy of measurement of aspherics using a Fizeau interferometer,” Optical Fabrication and Testing, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 259–267.
  10. R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, B. Loucks, “Measurements of the LIGO Pathfinder optics,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
    [CrossRef]
  11. H. Kurita, K. Saito, M. Kato, T. Yatagai, “Influence of system aberrations on interferometric aspheric surface testing,” in Laser and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. SPIE680, 47–52 (1987).
  12. C. Huang, “Propagation errors in precision Fizeau interferometry,” Appl. Opt. 32, 7016–7021 (1993).
    [CrossRef] [PubMed]
  13. P. E. Murphy, T. G. Brown, D. T. Moore, “Interference imaging for aspheric surface testing,” Appl. Opt. 39, 2122–2129 (2000).
    [CrossRef]
  14. W. T. Welford, Aberrations of Optical Systems (Hilger, Boston, 1986).
  15. H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
    [CrossRef]

2000

1999

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

1996

J. E. Greivenkamp, A. E. Lowman, R. J. Palum, “Sub-Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35, 2962–2969 (1996).
[CrossRef]

1993

C. J. Evans, “Compensation for errors introduced by non-zero fringe densities in phase-measuring interferometers,” CIRP Annals 42/1, 577–580 (1993).

C. Huang, “Propagation errors in precision Fizeau interferometry,” Appl. Opt. 32, 7016–7021 (1993).
[CrossRef] [PubMed]

1987

1984

Brown, T. G.

P. E. Murphy, T. G. Brown, D. T. Moore, “Interference imaging for aspheric surface testing,” Appl. Opt. 39, 2122–2129 (2000).
[CrossRef]

P. E. Murphy, T. G. Brown, D. T. Moore, “Optical Vernier interferometry for aspheric metrology,” in Emerging Lithographic Technologies III, Y. Vladimirsky, ed., Proc. SPIE3676, 643–652 (1999).
[CrossRef]

Burton, D. R.

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

Cheng, Y.-Y.

Evans, C. J.

C. J. Evans, “Compensation for errors introduced by non-zero fringe densities in phase-measuring interferometers,” CIRP Annals 42/1, 577–580 (1993).

C. J. Evans, “Software based improvements in the accuracy of measurement of aspherics using a Fizeau interferometer,” Optical Fabrication and Testing, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 259–267.

R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, B. Loucks, “Measurements of the LIGO Pathfinder optics,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp, A. E. Lowman, R. J. Palum, “Sub-Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35, 2962–2969 (1996).
[CrossRef]

J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5258 (1987).
[CrossRef] [PubMed]

R. J. Palum, J. E. Greivenkamp, “Sub-Nyquist interferometry: results and implementation issues,” in Laser Interferometry: Qualitative Analysis of Interferograms: Third in a Series, J. E. Wampler, ed., Proc. SPIE1162, 378–388 (1989).
[CrossRef]

A. E. Lowman, J. E. Greivenkamp, “Modeling an interferometer for non-null testing of aspheres,” in Optical Manufacturing and Testing, V. J. Daugherty, H. Stabl, eds., Proc. SPIE2536, 139–147 (1995).
[CrossRef]

A. E. Lowman, J. E. Greivenkamp, “Interferometer induced wavefront errors when testing in a non-null configuration,” in Interferometry VI: Applications, R. J. Pryptniewicz, G. M. Brown, W. E. Jeuptner, eds., Proc. SPIE2004, 173–181 (1993).
[CrossRef]

Huang, C.

Kato, M.

H. Kurita, K. Saito, M. Kato, T. Yatagai, “Influence of system aberrations on interferometric aspheric surface testing,” in Laser and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. SPIE680, 47–52 (1987).

Kurita, H.

H. Kurita, K. Saito, M. Kato, T. Yatagai, “Influence of system aberrations on interferometric aspheric surface testing,” in Laser and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. SPIE680, 47–52 (1987).

Lalor, M. J.

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

Loucks, B.

R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, B. Loucks, “Measurements of the LIGO Pathfinder optics,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
[CrossRef]

Lowman, A. E.

J. E. Greivenkamp, A. E. Lowman, R. J. Palum, “Sub-Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35, 2962–2969 (1996).
[CrossRef]

A. E. Lowman, J. E. Greivenkamp, “Modeling an interferometer for non-null testing of aspheres,” in Optical Manufacturing and Testing, V. J. Daugherty, H. Stabl, eds., Proc. SPIE2536, 139–147 (1995).
[CrossRef]

A. E. Lowman, J. E. Greivenkamp, “Interferometer induced wavefront errors when testing in a non-null configuration,” in Interferometry VI: Applications, R. J. Pryptniewicz, G. M. Brown, W. E. Jeuptner, eds., Proc. SPIE2004, 173–181 (1993).
[CrossRef]

Moore, D. T.

P. E. Murphy, T. G. Brown, D. T. Moore, “Interference imaging for aspheric surface testing,” Appl. Opt. 39, 2122–2129 (2000).
[CrossRef]

P. E. Murphy, T. G. Brown, D. T. Moore, “Optical Vernier interferometry for aspheric metrology,” in Emerging Lithographic Technologies III, Y. Vladimirsky, ed., Proc. SPIE3676, 643–652 (1999).
[CrossRef]

Murphy, P. E.

P. E. Murphy, T. G. Brown, D. T. Moore, “Interference imaging for aspheric surface testing,” Appl. Opt. 39, 2122–2129 (2000).
[CrossRef]

P. E. Murphy, T. G. Brown, D. T. Moore, “Optical Vernier interferometry for aspheric metrology,” in Emerging Lithographic Technologies III, Y. Vladimirsky, ed., Proc. SPIE3676, 643–652 (1999).
[CrossRef]

Palum, R. J.

J. E. Greivenkamp, A. E. Lowman, R. J. Palum, “Sub-Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35, 2962–2969 (1996).
[CrossRef]

R. J. Palum, J. E. Greivenkamp, “Sub-Nyquist interferometry: results and implementation issues,” in Laser Interferometry: Qualitative Analysis of Interferograms: Third in a Series, J. E. Wampler, ed., Proc. SPIE1162, 378–388 (1989).
[CrossRef]

Parks, R. E.

R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, B. Loucks, “Measurements of the LIGO Pathfinder optics,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
[CrossRef]

Saito, K.

H. Kurita, K. Saito, M. Kato, T. Yatagai, “Influence of system aberrations on interferometric aspheric surface testing,” in Laser and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. SPIE680, 47–52 (1987).

Shao, L. Z.

R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, B. Loucks, “Measurements of the LIGO Pathfinder optics,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
[CrossRef]

Sullivan, P. J.

R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, B. Loucks, “Measurements of the LIGO Pathfinder optics,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, Boston, 1986).

Wyant, J. C.

Yatagai, T.

H. Kurita, K. Saito, M. Kato, T. Yatagai, “Influence of system aberrations on interferometric aspheric surface testing,” in Laser and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. SPIE680, 47–52 (1987).

Zhang, H.

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

Appl. Opt.

CIRP Annals

C. J. Evans, “Compensation for errors introduced by non-zero fringe densities in phase-measuring interferometers,” CIRP Annals 42/1, 577–580 (1993).

Opt. Eng.

J. E. Greivenkamp, A. E. Lowman, R. J. Palum, “Sub-Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35, 2962–2969 (1996).
[CrossRef]

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

Other

W. T. Welford, Aberrations of Optical Systems (Hilger, Boston, 1986).

R. J. Palum, J. E. Greivenkamp, “Sub-Nyquist interferometry: results and implementation issues,” in Laser Interferometry: Qualitative Analysis of Interferograms: Third in a Series, J. E. Wampler, ed., Proc. SPIE1162, 378–388 (1989).
[CrossRef]

P. E. Murphy, T. G. Brown, D. T. Moore, “Optical Vernier interferometry for aspheric metrology,” in Emerging Lithographic Technologies III, Y. Vladimirsky, ed., Proc. SPIE3676, 643–652 (1999).
[CrossRef]

A. E. Lowman, J. E. Greivenkamp, “Interferometer induced wavefront errors when testing in a non-null configuration,” in Interferometry VI: Applications, R. J. Pryptniewicz, G. M. Brown, W. E. Jeuptner, eds., Proc. SPIE2004, 173–181 (1993).
[CrossRef]

A. E. Lowman, J. E. Greivenkamp, “Modeling an interferometer for non-null testing of aspheres,” in Optical Manufacturing and Testing, V. J. Daugherty, H. Stabl, eds., Proc. SPIE2536, 139–147 (1995).
[CrossRef]

C. J. Evans, “Software based improvements in the accuracy of measurement of aspherics using a Fizeau interferometer,” Optical Fabrication and Testing, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 259–267.

R. E. Parks, C. J. Evans, P. J. Sullivan, L. Z. Shao, B. Loucks, “Measurements of the LIGO Pathfinder optics,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 95–111 (1997).
[CrossRef]

H. Kurita, K. Saito, M. Kato, T. Yatagai, “Influence of system aberrations on interferometric aspheric surface testing,” in Laser and Nonlinear Optical Materials, L. G. DeShazer, ed., Proc. SPIE680, 47–52 (1987).

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Figures (7)

Fig. 1
Fig. 1

Illustration of mapping error; distortion introduces aspheric terms on a defocused sphere. The solid curve represents three waves of defocus. The dashed curve results after 10% barrel distortion. The dotted curve is the difference between the two, illustrating the error introduced when distortion is unaccounted for.

Fig. 2
Fig. 2

Schematic of experimental interferometer (imaging lens position may vary).

Fig. 3
Fig. 3

Raw phase data from a spherical test surface defocused 5 mm.

Fig. 4
Fig. 4

Reconstructed wave front of a surface defocused 5 mm.

Fig. 5
Fig. 5

Data slice of the 5-mm defocused wave front along the x axis.

Fig. 6
Fig. 6

Measured wave fronts after removal of the quadratic term: (a) wave front along the x axis, (b) wave front along the y axis.

Fig. 7
Fig. 7

Amount of fourth-order error for different amounts of defocus. Experimental data points for the x and the y axes given by × and +, respectively. Dashed curve, prediction with aberration theory; solid curve, best fit to the experimental data. Dotted curve, residual error between the prediction and this experimental fit.

Tables (3)

Tables Icon

Table 1 Measurements Along the x Axis of a Defocused Test Sphere

Tables Icon

Table 2 Component Specificationsa

Tables Icon

Table 3 Computed Third-Order Aberrations of the Imaging Lens

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

Mapping:  Δrhtest=yρhtest, htest=1uaρywρhtest, htest,  rΔθhtest=xρhtest, htest=1uaρxwρhtest, htest,
Phase:  ΔΨhtest=Ψoiρhtest, htest-Ψoi0, href
ρhtest21/uasaghtest.
Δz=-zimg·=-zr y+1rzθ x,
=1uawρ, h=yx.
mzimg=2sag.
zimg=ρhimgua.
Δz=-ρhimg·w.
ΔΨhtest-wρ, htest+ρw+w400href4-htest4+w200href2-htest2.
ΔΨcalhtest-wρ, htest+w400href4-htest4+w200href2-htest2,
wρ, h=w020ρ2+w040ρ4+w131ρyρ2h+w222ρy2h2+w220ρ2h2+w311ρyh3.
Ψh2 sagh+2 Figh+Nullh+Envh+ΔΨcalh.
Tilt:  zhtest=αy=αymaxhtest=zmaxhtest;  so ρhtest2uazmaxymax.
Defocus:  zhtest=χ2yR2=χymax22R2 htest2=zmaxhtest2;  so ρhtest4uazmaxymax htest.
ΨNullh2 Figh+Nullh+EnvNullh.
Ψh2 sagh+ΔΨcalh+Δ Envh.
Ψhtest-2zhtest+-wρhtest, htest.
-ΔΨh=2zmaxhy+w020ρ02+w040ρ04+w131ρ03hy+w222ρ02hy2+w220ρ02h2+w311ρ0h2hy,
-ΔΨh=2zmaxh2+w020ρmax2h2+w040ρmax4h4+w131ρmax3h4+w222ρmax2h4+w220ρmax2h4+w311ρmaxh4,
-ΔΨh=w020ρ02+w040ρ04+2zmax+w131ρ03hy+w222ρ02hy2+w220ρ02h2+w311ρ0h2hy.
-ΔΨh=2zmax+w020ρmax2h2+w040ρmax4+w131ρmax3+w222+w220ρmax2+w311ρmaxh4.
ΔΨpredictedh=0.0162ρmax2+-0.1633ρmaxh4.
ΔΨregressionh=0.0163ρmax2+-0.191ρmaxh4.
residualh=0.0001ρmax2+-0.028ρmaxh4.
ΔΨpistonhtestw400href4-htest4+w200href2-htest2.
href=htest+w/H.
ΔΨpistonhtest=4w400htest3wH+2w200htestwH.
wρ, h=2w020ρ+2w222+w220ρh2.
40.7986λ0.01678 mmδz0.003922h4+20.0159λh6ρmax=0.00302λ/mmδzh4+0.00402λh6ρmax.
ΔΨ=ψrtest, θtest-ψrref, θref=ψhyb+ρyya, ϕb+ρxya/yb-ψhyb, ϕb.
ΔΨhρyyaψrhyb+ρxyaybψθϕb,
ΔΨhρmaxh0.322 mmψr4.436 mmh.

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