Abstract

The measurement of aspheric and free-form surfaces in a non-null test configuration has the advantage that no compensation optics is required. However, if a surface is measured in a non-null test configuration, retrace errors are introduced to the measurement. We describe a method to calibrate the test space of an interferometer, enabling to compensate retrace errors. The method is effective even for strong deviations from null test configuration up to several 100 waves, enabling the fast and flexible measurement of aspheres and free-form surfaces. In this paper we present the application of the method to the calibration of the Tilted Wave Interferometer. Furthermore, the method can be generalized to the calibration of other setups.

© 2014 Optical Society of America

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References

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  1. B. Braunecker, R. Hentschel, and H. J. Tiziani, Advanced Optics Using Aspherical Elements (SPIE Press Monograph PM173, 2008).
    [Crossref]
  2. D. Malacara, K. Creath, J. Schmit, and C. Wyant, Optical Shop Testing (Wiley, 2007), chap. 12.12 Interferometers using synthetic holograms, 3rd ed.
    [Crossref]
  3. M. F. Kuechel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916 (2009).
    [Crossref]
  4. P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photon. News 14, 38–43 (2003).
    [Crossref]
  5. E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 2973–2975 (2008).
    [Crossref] [PubMed]
  6. J. W. Goodman, Introduction to Fourier Optics (MaGraw-Hill, 2005), 3rd ed.
  7. H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. AIEE 47, 617–644 (1928).
  8. C. J. Evans and J. B. Bryan, “Compensation for errors introduced by nonzero fringe sensities in phase-measuring interferometers,” CIRP Annals Manufacturing Technology 42(1) 577–580 (1993).
    [Crossref]
  9. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University, 1970).
  10. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” JOSA 66, 207–211 (1976).
    [Crossref]
  11. J. Liesener, “Zum einsatz räumlicher lichtmodulatoren in der interferometrischen wellenfrontmesstechnik,” Ph.D. thesis, University of Stuttgart (2006).
  12. E. Garbusi and W. Osten, “Perturbation methods in optics: Application to the interferometric measurement of surfaces,” J. Opt. Soc. A 26, 2538–2549 (2009).
    [Crossref]
  13. G. Baer, J. Schindler, J. Siepmann, C. Pruss, W. Osten, and M. Schulz, “Measurement of aspheres and free-form surfaces in a non-null test interferometer: reconstruction of high-frequency errors,” Proc. SPIE 8788, 878818 (2013).
    [Crossref]
  14. I. Widdershoven, M. Baas, and H. Spaan, “Ultra-precision 3D coordinate metrology results showing 10nm accuracy,” in Proceedings of the 11th international symposium of measurement technology and intelligent instruments, (2013), pp. 1–5.
  15. I. Widdershoven, M. Baas, and H. Spaan, “Tactile coordinate metrology fur ultra-precision measurement of optics: Results and intercomparison,” in Proceedings of 2014 ASPE Summer Topical Vol 48, (2014), pp. 92–97.
  16. G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” JEOS:RP8 (2013), 1–5.

2013 (1)

G. Baer, J. Schindler, J. Siepmann, C. Pruss, W. Osten, and M. Schulz, “Measurement of aspheres and free-form surfaces in a non-null test interferometer: reconstruction of high-frequency errors,” Proc. SPIE 8788, 878818 (2013).
[Crossref]

2009 (2)

E. Garbusi and W. Osten, “Perturbation methods in optics: Application to the interferometric measurement of surfaces,” J. Opt. Soc. A 26, 2538–2549 (2009).
[Crossref]

M. F. Kuechel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916 (2009).
[Crossref]

2008 (1)

2003 (1)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photon. News 14, 38–43 (2003).
[Crossref]

1993 (1)

C. J. Evans and J. B. Bryan, “Compensation for errors introduced by nonzero fringe sensities in phase-measuring interferometers,” CIRP Annals Manufacturing Technology 42(1) 577–580 (1993).
[Crossref]

1976 (1)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” JOSA 66, 207–211 (1976).
[Crossref]

1928 (1)

H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. AIEE 47, 617–644 (1928).

Baas, M.

I. Widdershoven, M. Baas, and H. Spaan, “Ultra-precision 3D coordinate metrology results showing 10nm accuracy,” in Proceedings of the 11th international symposium of measurement technology and intelligent instruments, (2013), pp. 1–5.

I. Widdershoven, M. Baas, and H. Spaan, “Tactile coordinate metrology fur ultra-precision measurement of optics: Results and intercomparison,” in Proceedings of 2014 ASPE Summer Topical Vol 48, (2014), pp. 92–97.

Baer, G.

G. Baer, J. Schindler, J. Siepmann, C. Pruss, W. Osten, and M. Schulz, “Measurement of aspheres and free-form surfaces in a non-null test interferometer: reconstruction of high-frequency errors,” Proc. SPIE 8788, 878818 (2013).
[Crossref]

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” JEOS:RP8 (2013), 1–5.

Braunecker, B.

B. Braunecker, R. Hentschel, and H. J. Tiziani, Advanced Optics Using Aspherical Elements (SPIE Press Monograph PM173, 2008).
[Crossref]

Bryan, J. B.

C. J. Evans and J. B. Bryan, “Compensation for errors introduced by nonzero fringe sensities in phase-measuring interferometers,” CIRP Annals Manufacturing Technology 42(1) 577–580 (1993).
[Crossref]

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University, 1970).

Creath, K.

D. Malacara, K. Creath, J. Schmit, and C. Wyant, Optical Shop Testing (Wiley, 2007), chap. 12.12 Interferometers using synthetic holograms, 3rd ed.
[Crossref]

Dumas, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photon. News 14, 38–43 (2003).
[Crossref]

Evans, C. J.

C. J. Evans and J. B. Bryan, “Compensation for errors introduced by nonzero fringe sensities in phase-measuring interferometers,” CIRP Annals Manufacturing Technology 42(1) 577–580 (1993).
[Crossref]

Fleig, J.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photon. News 14, 38–43 (2003).
[Crossref]

Forbes, G.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photon. News 14, 38–43 (2003).
[Crossref]

Garbusi, E.

E. Garbusi and W. Osten, “Perturbation methods in optics: Application to the interferometric measurement of surfaces,” J. Opt. Soc. A 26, 2538–2549 (2009).
[Crossref]

E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 2973–2975 (2008).
[Crossref] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (MaGraw-Hill, 2005), 3rd ed.

Hentschel, R.

B. Braunecker, R. Hentschel, and H. J. Tiziani, Advanced Optics Using Aspherical Elements (SPIE Press Monograph PM173, 2008).
[Crossref]

Kuechel, M. F.

M. F. Kuechel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916 (2009).
[Crossref]

Liesener, J.

J. Liesener, “Zum einsatz räumlicher lichtmodulatoren in der interferometrischen wellenfrontmesstechnik,” Ph.D. thesis, University of Stuttgart (2006).

Malacara, D.

D. Malacara, K. Creath, J. Schmit, and C. Wyant, Optical Shop Testing (Wiley, 2007), chap. 12.12 Interferometers using synthetic holograms, 3rd ed.
[Crossref]

Murphy, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photon. News 14, 38–43 (2003).
[Crossref]

Noll, R. J.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” JOSA 66, 207–211 (1976).
[Crossref]

Nyquist, H.

H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. AIEE 47, 617–644 (1928).

Osten, W.

G. Baer, J. Schindler, J. Siepmann, C. Pruss, W. Osten, and M. Schulz, “Measurement of aspheres and free-form surfaces in a non-null test interferometer: reconstruction of high-frequency errors,” Proc. SPIE 8788, 878818 (2013).
[Crossref]

E. Garbusi and W. Osten, “Perturbation methods in optics: Application to the interferometric measurement of surfaces,” J. Opt. Soc. A 26, 2538–2549 (2009).
[Crossref]

E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 2973–2975 (2008).
[Crossref] [PubMed]

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” JEOS:RP8 (2013), 1–5.

Pruss, C.

G. Baer, J. Schindler, J. Siepmann, C. Pruss, W. Osten, and M. Schulz, “Measurement of aspheres and free-form surfaces in a non-null test interferometer: reconstruction of high-frequency errors,” Proc. SPIE 8788, 878818 (2013).
[Crossref]

E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 2973–2975 (2008).
[Crossref] [PubMed]

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” JEOS:RP8 (2013), 1–5.

Schindler, J.

G. Baer, J. Schindler, J. Siepmann, C. Pruss, W. Osten, and M. Schulz, “Measurement of aspheres and free-form surfaces in a non-null test interferometer: reconstruction of high-frequency errors,” Proc. SPIE 8788, 878818 (2013).
[Crossref]

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” JEOS:RP8 (2013), 1–5.

Schmit, J.

D. Malacara, K. Creath, J. Schmit, and C. Wyant, Optical Shop Testing (Wiley, 2007), chap. 12.12 Interferometers using synthetic holograms, 3rd ed.
[Crossref]

Schulz, M.

G. Baer, J. Schindler, J. Siepmann, C. Pruss, W. Osten, and M. Schulz, “Measurement of aspheres and free-form surfaces in a non-null test interferometer: reconstruction of high-frequency errors,” Proc. SPIE 8788, 878818 (2013).
[Crossref]

Siepmann, J.

G. Baer, J. Schindler, J. Siepmann, C. Pruss, W. Osten, and M. Schulz, “Measurement of aspheres and free-form surfaces in a non-null test interferometer: reconstruction of high-frequency errors,” Proc. SPIE 8788, 878818 (2013).
[Crossref]

Spaan, H.

I. Widdershoven, M. Baas, and H. Spaan, “Ultra-precision 3D coordinate metrology results showing 10nm accuracy,” in Proceedings of the 11th international symposium of measurement technology and intelligent instruments, (2013), pp. 1–5.

I. Widdershoven, M. Baas, and H. Spaan, “Tactile coordinate metrology fur ultra-precision measurement of optics: Results and intercomparison,” in Proceedings of 2014 ASPE Summer Topical Vol 48, (2014), pp. 92–97.

Tiziani, H. J.

B. Braunecker, R. Hentschel, and H. J. Tiziani, Advanced Optics Using Aspherical Elements (SPIE Press Monograph PM173, 2008).
[Crossref]

Tricard, M.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photon. News 14, 38–43 (2003).
[Crossref]

Widdershoven, I.

I. Widdershoven, M. Baas, and H. Spaan, “Tactile coordinate metrology fur ultra-precision measurement of optics: Results and intercomparison,” in Proceedings of 2014 ASPE Summer Topical Vol 48, (2014), pp. 92–97.

I. Widdershoven, M. Baas, and H. Spaan, “Ultra-precision 3D coordinate metrology results showing 10nm accuracy,” in Proceedings of the 11th international symposium of measurement technology and intelligent instruments, (2013), pp. 1–5.

Wyant, C.

D. Malacara, K. Creath, J. Schmit, and C. Wyant, Optical Shop Testing (Wiley, 2007), chap. 12.12 Interferometers using synthetic holograms, 3rd ed.
[Crossref]

CIRP Annals Manufacturing Technology (1)

C. J. Evans and J. B. Bryan, “Compensation for errors introduced by nonzero fringe sensities in phase-measuring interferometers,” CIRP Annals Manufacturing Technology 42(1) 577–580 (1993).
[Crossref]

J. Opt. Soc. A (1)

E. Garbusi and W. Osten, “Perturbation methods in optics: Application to the interferometric measurement of surfaces,” J. Opt. Soc. A 26, 2538–2549 (2009).
[Crossref]

JOSA (1)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” JOSA 66, 207–211 (1976).
[Crossref]

Opt. Lett. (1)

Opt. Photon. News (1)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photon. News 14, 38–43 (2003).
[Crossref]

Proc. SPIE (2)

M. F. Kuechel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916 (2009).
[Crossref]

G. Baer, J. Schindler, J. Siepmann, C. Pruss, W. Osten, and M. Schulz, “Measurement of aspheres and free-form surfaces in a non-null test interferometer: reconstruction of high-frequency errors,” Proc. SPIE 8788, 878818 (2013).
[Crossref]

Trans. AIEE (1)

H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. AIEE 47, 617–644 (1928).

Other (8)

J. Liesener, “Zum einsatz räumlicher lichtmodulatoren in der interferometrischen wellenfrontmesstechnik,” Ph.D. thesis, University of Stuttgart (2006).

I. Widdershoven, M. Baas, and H. Spaan, “Ultra-precision 3D coordinate metrology results showing 10nm accuracy,” in Proceedings of the 11th international symposium of measurement technology and intelligent instruments, (2013), pp. 1–5.

I. Widdershoven, M. Baas, and H. Spaan, “Tactile coordinate metrology fur ultra-precision measurement of optics: Results and intercomparison,” in Proceedings of 2014 ASPE Summer Topical Vol 48, (2014), pp. 92–97.

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” JEOS:RP8 (2013), 1–5.

B. Braunecker, R. Hentschel, and H. J. Tiziani, Advanced Optics Using Aspherical Elements (SPIE Press Monograph PM173, 2008).
[Crossref]

D. Malacara, K. Creath, J. Schmit, and C. Wyant, Optical Shop Testing (Wiley, 2007), chap. 12.12 Interferometers using synthetic holograms, 3rd ed.
[Crossref]

J. W. Goodman, Introduction to Fourier Optics (MaGraw-Hill, 2005), 3rd ed.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University, 1970).

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Figures (9)

Fig. 1
Fig. 1 Schematic setup of the interferometer with the central source and one exemplary off-axis source indicated. L: Laser source; BS1, BS2: Beam splitter; C1, C2, C3, Lens; PSA: Point source array; AA: Aperture array; M: Mirror; O: Objective; SUT : Surfaces under test; A: Aperture; IO: Imaging optics; C: Camera;
Fig. 2
Fig. 2 Definition of the reference plane for the illumination part. The OPD of the example ray (red dotted line) from FQ to EQ can be calculated from WQ(M, N, X, Y). PSA: Point source array; BS1: Beam splitter; C1: Collimation lens; O: Objective; EQ: Reference plane;
Fig. 3
Fig. 3 Definition of the reference plane for the imaging part. The OPD of the example ray (red dotted line) from EP to FP can be calculated from WP(m, n, x, y). BS1: Beam splitter; C1: Colimation lens; O: Objective; Ep: Reference plane; A: Aperture; IO: Imaging optics; C: Camera;
Fig. 4
Fig. 4 Measured modulo 2π phase maps with the reference sphere in the null-test of the central source as well as in the null-test positions of the surrounding eight sources where the fringe density is minimal.
Fig. 5
Fig. 5 Measured modulo 2π phase map with the reference sphere placed in a defocused position, containing rays from four different sources in one measurement.
Fig. 6
Fig. 6 Visualization of the side conditions sc1 to sc6. The coordinate system is illustrated by the black arrows with the optical axis being the z axis. The small red arrows illustrate the side conditions.
Fig. 7
Fig. 7 Measurement results of a weak aspheric surface obtained with a Tilted Wave Interferometer (a) and a tactile Isara 400 coordinate measurement machine (b)
Fig. 8
Fig. 8 Measurement of a weak aspheric surface
Fig. 9
Fig. 9 Measurement of an steep aspheric surface with a slope deviation of up to 5° from the spherical form

Equations (15)

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W Q ( X , Y ) = i q i Z i ( X , Y )
q i = q i ( M , N ) = j Q i j Z j ( M , N )
W Q ( X , Y , M , N ) = i j Q i j Z j ( M , N ) Z i ( X , Y )
W P ( x , y , m , n ) = k l P k l Z l ( m , n ) Z k ( x , y )
OPD ( X , Y , x , y , c x , c y , c z , r ) = W Q ( X , Y , M , N ) + W P ( x , y , m , n ) + OPD geom ( c x , c y , c z , r )
b = OPD ( X , Y , x , y , c x , c y , c z , r )
A = ( A Q A P A c )
A Q = ( b 1 ( Q ε ( 1 , 1 ) ) b 1 ( Q ε ( i , j ) ) b n ( Q ε ( 1 , 1 ) ) b n ( Q ε ( i , j ) ) )
A P = ( b 1 ( P ε ( 1 , 1 ) ) b 1 ( P ε ( k , l ) ) b n ( P ε ( 1 , 1 ) ) b n ( P ε ( k , l ) ) )
A c = ( b 1 ( c x ε ( 1 ) ) b 1 ( c z ε ( t ) ) b n ( c x ε ( 1 ) ) b n ( c z ε ( t ) ) )
δ b = Ax
min x Ax δ b 2
c x ( 1 ) = 0 , c y ( 1 ) = 0 , c z ( 1 ) = 0
c x ( 2 ) = 0 , c y ( 2 ) = 0
c y ( 3 ) = 0

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