Abstract

Tilted-wave interferometry is a promising measurement technique for the highly accurate measurement of aspheres and freeform surfaces. However, the interferometric fringe evaluation of the sub-apertures causes unknown patch offsets, which currently prevent this measurement technique from providing absolute measurements. Simple strategies, such as constructing differences of optical path length differences (OPDs) or ignoring the piston parameter, can diminish the accuracy resulting from the absolute form measurement. Additional information is needed instead; in this paper, the required accuracy of such information is explored in virtual experiments. Our simulation study reveals that, when one absolute OPD is known within a range of 500 nm, the accuracy of the final measurement result is significantly enhanced.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Analytical Jacobian and its application to tilted-wave interferometry

Ines Fortmeier, Manuel Stavridis, Axel Wiegmann, Michael Schulz, Wolfgang Osten, and Clemens Elster
Opt. Express 22(18) 21313-21325 (2014)

Form determination of optical surfaces by measuring the spatial coherence function using shearing interferometry

Jan-hendrik Hagemann, Claas Falldorf, Gerd Ehret, and Ralf B. Bergmann
Opt. Express 26(21) 27991-28001 (2018)

Calibration of a non-null test interferometer for the measurement of aspheres and free-form surfaces

Goran Baer, Johannes Schindler, Christof Pruss, Jens Siepmann, and Wolfgang Osten
Opt. Express 22(25) 31200-31211 (2014)

References

  • View by:
  • |
  • |
  • |

  1. G. Schulz, Aspheric Surfaces, in Progress in Optics, E. Wolf, ed. (North-Holland Physics Publishing, 1988), pp. 349–415.
    [Crossref]
  2. B. Braunecker, R. Hentschel, and H. J. Tiziani, eds., Advanced Optics Using Aspherical Elements (SPIE, 2008), pp. 292–307.
  3. A. E. Lowman, “Calibration of a non-null interferometer for aspheric testing,” PhD thesis, University of Arizona (1995).
  4. R. Henselmans, “Non-contact measurement machine for freeform optics,” PhD thesis, Technical University of Eindhoven (2009).
  5. E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33(24), 2973–2975 (2008).
    [Crossref] [PubMed]
  6. E. Garbusi and W. Osten, “Perturbation methods in optics: application to the interferometric measurement of surfaces,” J. Opt. Soc. Am. A 26(12), 2538–2549 (2009).
    [Crossref]
  7. W. Osten, B. Dörband, E. Garbusi, C. Pruss, and L. Seifert, “Testing aspheric lenses: New approaches,” Optoelectronics, Instrumentation and Data Processing 46, 329–339 (2010).
    [Crossref]
  8. I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, W. Osten, and C. Elster, “Analytical Jacobian and its application to tilted-wave interferometry,” Opt. Express 22(18), 21313–21325 (2014).
    [Crossref] [PubMed]
  9. D. Malacara, Optical Shop Testing (John Wiley & Sons, Inc., 2007).
    [Crossref]
  10. G. Baer, J. Schindler, C. Pruss, and W. Osten, “Calibration of a non-null test interferometer for the measurement of aspheres and free-form surfaces,” Opt. Express 22(25), 31200–31211 (2014).
    [Crossref]
  11. R. Draper and H. Smith, Applied Regression Analysis, 3rd edition (John Wiley & Sons, Inc., 1998).
  12. P. De Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39, 2658–2663 (2000).
    [Crossref]
  13. K. Meiners-Hagen, R. Schödel, and F. Pollinger, “Multi-wavelength interferometry for length measurements using diode lasers,” Measurement Science Review 9, 16–26 (2009).
    [Crossref]
  14. J. C. Wyant, “White light interferometry,” Proc. SPIE 4737, 98–107 (2002).
    [Crossref]
  15. G. Baer, C. Pruss, and W. Osten, “Verkippte Objektwellen nutzendes und ein Fizeau-Interferometerobjektiv aufweisendes Interferometer,” German patent application102015222366.3 (2015).

2014 (2)

2010 (1)

W. Osten, B. Dörband, E. Garbusi, C. Pruss, and L. Seifert, “Testing aspheric lenses: New approaches,” Optoelectronics, Instrumentation and Data Processing 46, 329–339 (2010).
[Crossref]

2009 (2)

K. Meiners-Hagen, R. Schödel, and F. Pollinger, “Multi-wavelength interferometry for length measurements using diode lasers,” Measurement Science Review 9, 16–26 (2009).
[Crossref]

E. Garbusi and W. Osten, “Perturbation methods in optics: application to the interferometric measurement of surfaces,” J. Opt. Soc. Am. A 26(12), 2538–2549 (2009).
[Crossref]

2008 (1)

2002 (1)

J. C. Wyant, “White light interferometry,” Proc. SPIE 4737, 98–107 (2002).
[Crossref]

2000 (1)

Baer, G.

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Calibration of a non-null test interferometer for the measurement of aspheres and free-form surfaces,” Opt. Express 22(25), 31200–31211 (2014).
[Crossref]

G. Baer, C. Pruss, and W. Osten, “Verkippte Objektwellen nutzendes und ein Fizeau-Interferometerobjektiv aufweisendes Interferometer,” German patent application102015222366.3 (2015).

De Groot, P.

Dörband, B.

W. Osten, B. Dörband, E. Garbusi, C. Pruss, and L. Seifert, “Testing aspheric lenses: New approaches,” Optoelectronics, Instrumentation and Data Processing 46, 329–339 (2010).
[Crossref]

Draper, R.

R. Draper and H. Smith, Applied Regression Analysis, 3rd edition (John Wiley & Sons, Inc., 1998).

Elster, C.

Fortmeier, I.

Garbusi, E.

Henselmans, R.

R. Henselmans, “Non-contact measurement machine for freeform optics,” PhD thesis, Technical University of Eindhoven (2009).

Lowman, A. E.

A. E. Lowman, “Calibration of a non-null interferometer for aspheric testing,” PhD thesis, University of Arizona (1995).

Malacara, D.

D. Malacara, Optical Shop Testing (John Wiley & Sons, Inc., 2007).
[Crossref]

Meiners-Hagen, K.

K. Meiners-Hagen, R. Schödel, and F. Pollinger, “Multi-wavelength interferometry for length measurements using diode lasers,” Measurement Science Review 9, 16–26 (2009).
[Crossref]

Osten, W.

Pollinger, F.

K. Meiners-Hagen, R. Schödel, and F. Pollinger, “Multi-wavelength interferometry for length measurements using diode lasers,” Measurement Science Review 9, 16–26 (2009).
[Crossref]

Pruss, C.

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Calibration of a non-null test interferometer for the measurement of aspheres and free-form surfaces,” Opt. Express 22(25), 31200–31211 (2014).
[Crossref]

W. Osten, B. Dörband, E. Garbusi, C. Pruss, and L. Seifert, “Testing aspheric lenses: New approaches,” Optoelectronics, Instrumentation and Data Processing 46, 329–339 (2010).
[Crossref]

E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33(24), 2973–2975 (2008).
[Crossref] [PubMed]

G. Baer, C. Pruss, and W. Osten, “Verkippte Objektwellen nutzendes und ein Fizeau-Interferometerobjektiv aufweisendes Interferometer,” German patent application102015222366.3 (2015).

Schindler, J.

Schödel, R.

K. Meiners-Hagen, R. Schödel, and F. Pollinger, “Multi-wavelength interferometry for length measurements using diode lasers,” Measurement Science Review 9, 16–26 (2009).
[Crossref]

Schulz, G.

G. Schulz, Aspheric Surfaces, in Progress in Optics, E. Wolf, ed. (North-Holland Physics Publishing, 1988), pp. 349–415.
[Crossref]

Schulz, M.

Seifert, L.

W. Osten, B. Dörband, E. Garbusi, C. Pruss, and L. Seifert, “Testing aspheric lenses: New approaches,” Optoelectronics, Instrumentation and Data Processing 46, 329–339 (2010).
[Crossref]

Smith, H.

R. Draper and H. Smith, Applied Regression Analysis, 3rd edition (John Wiley & Sons, Inc., 1998).

Stavridis, M.

Wiegmann, A.

Wyant, J. C.

J. C. Wyant, “White light interferometry,” Proc. SPIE 4737, 98–107 (2002).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Measurement Science Review (1)

K. Meiners-Hagen, R. Schödel, and F. Pollinger, “Multi-wavelength interferometry for length measurements using diode lasers,” Measurement Science Review 9, 16–26 (2009).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Optoelectronics, Instrumentation and Data Processing (1)

W. Osten, B. Dörband, E. Garbusi, C. Pruss, and L. Seifert, “Testing aspheric lenses: New approaches,” Optoelectronics, Instrumentation and Data Processing 46, 329–339 (2010).
[Crossref]

Proc. SPIE (1)

J. C. Wyant, “White light interferometry,” Proc. SPIE 4737, 98–107 (2002).
[Crossref]

Other (7)

G. Baer, C. Pruss, and W. Osten, “Verkippte Objektwellen nutzendes und ein Fizeau-Interferometerobjektiv aufweisendes Interferometer,” German patent application102015222366.3 (2015).

R. Draper and H. Smith, Applied Regression Analysis, 3rd edition (John Wiley & Sons, Inc., 1998).

D. Malacara, Optical Shop Testing (John Wiley & Sons, Inc., 2007).
[Crossref]

G. Schulz, Aspheric Surfaces, in Progress in Optics, E. Wolf, ed. (North-Holland Physics Publishing, 1988), pp. 349–415.
[Crossref]

B. Braunecker, R. Hentschel, and H. J. Tiziani, eds., Advanced Optics Using Aspherical Elements (SPIE, 2008), pp. 292–307.

A. E. Lowman, “Calibration of a non-null interferometer for aspheric testing,” PhD thesis, University of Arizona (1995).

R. Henselmans, “Non-contact measurement machine for freeform optics,” PhD thesis, Technical University of Eindhoven (2009).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Setup of the TWI with some example rays traced through the system.
Fig. 2
Fig. 2 Example of a comparison of the reconstruction results for the reconstruction of simulated data: (a) shows the deviation of the topography from its design. The difference between the reconstructed and the real topography is shown in (b) for referencing each OPD to a chosen OPD of each patch, in (c) for referencing the OPDs to a chosen OPD of each patch and not estimating the piston parameter, and in (d) with additional absolute data and assuming that the data are affected only by small measurement errors which can be modelled by white Gaussian noise (standard deviation 5 nm). (e) shows the Zernike coefficients τj corresponding to the residual topography TrecT of (b), and (f) shows the Zernike coefficients τj corresponding to the residual topography TrecT shown in (c). The topography reconstructions were realized by using a Zernike polynomial function of 153 Zernike coefficients.
Fig. 3
Fig. 3 Standard deviations used to draw random Zernike coefficients that represent the deviation of the topography from its design from a normal distribution.
Fig. 4
Fig. 4 Schematic sketch of the proposed measurement setup of the TWI for including an additional absolute measurement.
Fig. 5
Fig. 5 In (a) and (b) the mean RMS reconstruction errors in dependence on the chosen standard deviation of the added errors of the additional absolute patch offset measurement are shown for a topography with design 1 and for a topography with design 2, respectively (blue points). Additionally, in (c) and (d) the mean PV reconstruction errors in dependence on the chosen standard deviation of the added errors of the additional absolute patch offset measurement are shown for a topography with design 1 and for a topography with design 2, respectively (blue points). The dotted red line marks in each diagram the mean RMS- or PV-reconstruction error if no additional data were included and the OPDs were referenced to a chosen OPD of a patch. The dashed green line assigns the mean RMS- or PV-reconstruction error if no additional data were included, the OPDs were referenced, and additionally the piston parameter was not estimated.
Fig. 6
Fig. 6 RMS reconstruction error as a function of the position error of the topography along the optical axis for a topography with design 1 (a) and a topography with design 2 (b) for different reconstruction methods: Referencing the OPDs of each patch to a chosen OPD, with including additional absolute patch offset data having a measurement error of 100 nm, and referencing the OPDs of each patch to a chosen OPD as well as ignoring the piston parameter.

Tables (1)

Tables Icon

Table 1 Parameters of the design of the two aspheres used in the examples.

Equations (8)

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

T ( x , y ) = T 0 ( x , y ) + j c j Z j ( x , y ) ,
J 0 c = L L 0 .
L meas i k = L i b k + ε i ,
Δ L = ( Δ L T Δ L add T b L add T ) T .
J ext = [ L 1 c 1 L 1 c 2 L 1 c a 1 0 0 0 L 2 c 1 L 2 c 2 L 2 c a 1 0 0 0 L max c 1 L max c 2 L max c a 0 0 1 0 L 1 add c 1 L 1 add c 2 L 1 add c a 0 0 0 1 L max add c 1 L max add c 2 L max add c a 0 0 0 1 0 0 0 0 0 0 0 0 1 ] .
V i n = [ σ Δ L 1 2 0 0 0 0 0 0 σ Δ L 2 2 0 0 0 0 0 0 0 σ Δ L max 2 0 0 0 0 0 0 σ Δ L 1 add 2 0 0 0 0 0 0 0 σ Δ L max add 2 0 0 0 0 0 0 σ b L add 2 ] .
χ 2 = ( J ext θ Δ L ) T ( V i n ) 1 ( J ext θ Δ L )
θ ^ = ( J ext T ( V i n ) 1 J ext ) 1 J ext T ( V i n ) 1 Δ L

Metrics