Abstract

Peak–valley accuracy of λ/20 over a range of 2λ is not unusual in an interferometric null test. For the larger dynamic ranges of a nonnull test, however, the fringe-imaging optics degrades the accuracy. We classify the errors introduced and analyze them in the context of both general and third-order aberration theory. We can predict the measurement error from known interferometer parameters, and we illustrate this for a single mirror. The errors are tabulated for the specific case of a fourth-order asphere with 100 µm of sag. We show that the third-order approximation is comparable with exact ray-trace results for this case.

© 2000 Optical Society of America

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References

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  1. A. Offner, D. Malacara, “Null tests using compensators,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 427–454.
  2. 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]
  3. J. E. Greivenkamp, A. E. Lowman, R. J. Palum, “Sub-Nyquist interferometry: implementation and measurement capability,” Opt. Eng. 35, 2962–2969 (1996).
    [CrossRef]
  4. P. E. Murphy, T. G. Brown, D. T. Moore, “Optical Vernier interferometry for apsheric metrology,” in Emerging Lithographic Technologies III, Y. Vladimirsky, ed., Proc. SPIE3676, 643–652 (1999).
    [CrossRef]
  5. J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5258 (1987).
    [CrossRef] [PubMed]
  6. J. E. Greivenkamp, A. E. Lowman, “Modulation transfer function measurement of sparse-array sensors using a self-calibrating fringe pattern,” Appl. Opt. 33, 5029–5036 (1994).
    [CrossRef] [PubMed]
  7. Y.-Y. Cheng, J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. 23, 4539–4543 (1984).
    [CrossRef] [PubMed]
  8. R. Jozwicki, “Influence of aberrations of Fizeau interferometer elements on measurement errors,” Appl. Opt. 30, 3126–3132 (1991).
    [CrossRef]
  9. A. E. Lowman, J. E. Grivenkamp, “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]
  10. A. E. Lowman, J. E. Grievenkamp, “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]
  11. C. J. Evans, “Software based improvements in the accuracy of measurement of aspherics using a Fizeau inteferometer,” in Optical Fabrication and Testing, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 259–262.
  12. C. J. Evans, “Compensation for errors introduced by non-zero fringe densities in phase-measuring interferometers,” CIRP Annals 42/1, 577–580 (1993).
    [CrossRef]
  13. 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]
  14. 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).
  15. C. Huang, “Propagation errors in precision Fizeau interferometry,” Appl. Opt. 32, 7016–7021 (1993).
    [CrossRef] [PubMed]
  16. D. Malacara, “Twyman-Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 51–94.
  17. W. T. Welford, Aberrations of Optical Systems (A. Hilger, Boston, 1986).

1996

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

1994

1993

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

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

1991

1987

1984

Brown, T. G.

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

Cheng, Y.-Y.

DeShazer, L. G.

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).

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).
[CrossRef]

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]

C. J. Evans, “Software based improvements in the accuracy of measurement of aspherics using a Fizeau inteferometer,” in Optical Fabrication and Testing, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 259–262.

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, A. E. Lowman, “Modulation transfer function measurement of sparse-array sensors using a self-calibrating fringe pattern,” Appl. Opt. 33, 5029–5036 (1994).
[CrossRef] [PubMed]

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]

Grievenkamp, J. E.

A. E. Lowman, J. E. Grievenkamp, “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]

Grivenkamp, J. E.

A. E. Lowman, J. E. Grivenkamp, “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.

Jozwicki, R.

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).

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]

J. E. Greivenkamp, A. E. Lowman, “Modulation transfer function measurement of sparse-array sensors using a self-calibrating fringe pattern,” Appl. Opt. 33, 5029–5036 (1994).
[CrossRef] [PubMed]

A. E. Lowman, J. E. Grivenkamp, “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. Grievenkamp, “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]

Malacara, D.

D. Malacara, “Twyman-Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 51–94.

A. Offner, D. Malacara, “Null tests using compensators,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 427–454.

Moore, D. T.

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

Offner, A.

A. Offner, D. Malacara, “Null tests using compensators,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 427–454.

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 (A. 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).

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).
[CrossRef]

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]

Other

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

A. Offner, D. Malacara, “Null tests using compensators,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 427–454.

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. Grivenkamp, “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. Grievenkamp, “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 inteferometer,” in Optical Fabrication and Testing, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 259–262.

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).

D. Malacara, “Twyman-Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), pp. 51–94.

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

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Figures (13)

Fig. 1
Fig. 1

(a) Phase error with quadratic dependence on part position. (b) Mapping error induced by 20% third-order barrel distortion.

Fig. 2
Fig. 2

Rays traced through a conventional imaging lens.

Fig. 3
Fig. 3

(a) Simple interferometer setup. (b) Unfolded setup; lens should image the interferometer aperture stop onto the CCD.

Fig. 4
Fig. 4

Defining the chief ray of the lens (marginal and chief rays swap roles in pupil imaging).

Fig. 5
Fig. 5

When imaging coherent wavefronts, only two rays emerge from each object point.

Fig. 6
Fig. 6

Transverse ray aberrations of an imaging singlet (under incoherent illumination).

Fig. 7
Fig. 7

Interference imaging only uses two rays from the ray fan of each object point.

Fig. 8
Fig. 8

Test ray interfering with a reference ray from a different field point in the presence of transverse ray aberration.

Fig. 9
Fig. 9

Wave-front aberration of a general imaging system for an on-axis object point.

Fig. 10
Fig. 10

OPD and ε from an off-axis object point. M is chosen such that PO = ≅ PM, so ΔPOI is isosceles. Therefore ∠ MOH = ρθ a + b - α/2 and ∠ MHO = 90° - α/2. Length MH gives the path difference introduced from the exit pupil to the image plane.

Fig. 11
Fig. 11

Interferometer arm with pupil-centered spherical imaging mirror.

Fig. 12
Fig. 12

Phase error computed with aberrations compared with a real ray trace (dashed line). The test surface is a fourth-order asphere with 100 µm of sag, on an R/5, 250-mm base radius sphere.

Fig. 13
Fig. 13

Mapping error computed with aberrations compared with a real ray trace (dashed curve).

Tables (1)

Tables Icon

Table 1 Coefficients of the Phase Error Equationa

Equations (39)

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εyρ,h||ρ|1,-1h1.
Δyh=εyρtesth, h-εyρref, h
htesth=mh+εyρtesth, h,  test array,hrefh=mh+εy0, h,  reference ray.
hhtesthtest=hmhtest+εy[ρtesthtest, htest),
εyρ, h||ρ|1,-1h1=0,
ρtestρref,
Δyh=εyρtesth, h-εyρref, h0.
wρtesth, h-wρref, h0,
ρtesth=2 1tanθaddhsagh,
htesthtest=mhtest+εy[ρtesthtest, htest) Mapping,
ΔΨh=Ψoiρtesthtest-Ψoi(0, h Phase.
Ψoiρtesthtest, htest-Ψoi0, htest Nominal phase error,
Ψoi0, htest-Ψoi0, h Reference ray correction.
wρ, h=-Ψopρ, h-Ψop0, h.
Ψopρtesthtest, htest-Ψop0, htest+Ψpiρtesthtest, htest-Ψpi0, htest,
-wρtesthtest, htest+ΔΨpiρtesthtest, htest.
ερ; h= R tanαcosρθa+hθb+tanαsinρθa+hθbR tanαcosρθa+hθb,
ΔΨpiρ; h=R tanα×sinρθa+hθb-tan12 αcosρθa+hθbcosρθa+hθb+tanαsinρθa+hθbR tanαtanρθa+hθb,
α=arctan1addρwρ, h,
Ψoi0, h=Ψop0, 0-w0, h+Ψpi0, h,
w0, h-w0, htest+Ψpi0, htest-Ψpi0, h,
hrefh=mh+εy0, h.
wρ, h=w020ρ2+w111ρyh+w200h2 First order,
wρ, h=w040ρ4+w131ρyρ2h+w222ρy2h2+w220ρ2h2+w311ρyh3+w400h4 Third order,
εy=R1addρywρ, h,  εx=R1addρxwρ, h.
εyρ, h=Ra4w040ρ3+3w131ρ2h+2w222+w220ρh2+w311h3+2w020ρ+w111h.
ΔΨpiρ; hR1a wρθa+hθb=ρ+h θbθaw,
ΔΨoiρ; h=-wρ, h+ρ+h θbθaw.
w0, h-w0, hw400h4-h4+w200(h2-h2).
Ψpi0, h-Ψpi0, h=R2+y21/2-R2+y+εy21/2-yR εy=-hθbwθa
Ψpi0, h = R + R2 + y21/2
ΔΨoiρ; h=-wρ, h+ρ+h θbθaw+-hθbwθa+w400h4-h4+w200h2-h2.
ΔΨoi=-wρ, h+ρw+w400h4-h4+w200h2-h2.
ΔΨoi=κ04ρ4+κ13ρ3h+κ22ρ2h2+κ02ρ2+w400h4-h4+w200h2-h2.
ρtesth=2 1tanθaddhsagh,
do=Rt+Ri,  di=Rt+RiRi2Rt+Ri,m=-dido=-Ri2Rt+Ri,
w040=ua4Rt+Ri2Rt24Ri3,  w220=ua2ub2Rt4m,w311=-uaub3Ri2m.
ΔΨoih=31125 mm ua4+-3.125 mm ua2=3.54 μmh12-3.2 μmh8.
εyh=41125 mm ua3uaua+2-3.125 mm uauauah2+0.125 mmuauah3,εyh=29.5 μmh9-40 μmh5+25 μmh3.

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