Abstract

We present a new method for the form measurement of optical surfaces using the spatial coherence function, which enables a shearing interferometer in combination with an LED multispot illumination to function as a measurement device. A new evaluation approach connects the measured data with the surface form by inverse raytracing. First measurement results with the inverse evaluation procedures are shown. We present the whole measurement in combination with the evaluation procedure. In addition, the convergence and stability of the implemented optimization task is investigated.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. Z. Malacara and M. Servin, Interferogram Analysis for Optical Testing (CRC, 2016, Vol 84).
  2. P. Zhou and J. H. Burge, “Fabrication error analysis and experimental demonstration for computer-generated holograms,” Appl. Opt. 46(5), 657–663 (2007).
    [Crossref] [PubMed]
  3. J. C. Wyant and V. P. Bennett, Using Computer Generated Holograms to Test Aspheric Wavefronts (Springer, 2014).
  4. K. Fuerschbach, K. P. Thompson, and J. P. Rolland, “Interferometric measurement of a concave, φ-polynomial, Zernike mirror,” Opt. Lett. 39(1), 18–21 (2014).
    [Crossref] [PubMed]
  5. P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
    [Crossref]
  6. E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33(24), 2973–2975 (2008).
    [Crossref] [PubMed]
  7. I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, W. Osten, and C. Elster, “Evaluation of absolute form measurements using a tilted-wave interferometer,” Opt. Express 24(4), 3393–3404 (2016).
    [Crossref] [PubMed]
  8. U. Schnars, C. Falldorf, J. Watson, and W. Jüptner, Digital Holography and Wavefront Sensing (Springer, 2015).
  9. C. Falldorf, A. Simic, G. Ehret, M. Schulz, C. von Kopylow, and R. B. Bergmann, “Precise optical metrology using Computational Shear Interferometry and an LCD monitor as light source,” in Proceedings of Fringe 2013, ed. (Springer, 2014), pp.729–734.
  10. J.-H. Hagemann, G. Ehret, R. B. Bergmann, and C. Falldorf, “Realization of a shearing interferometer with LED multisport illumination for form characterization of optics,” in Proceedings of DGaO (2016).
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Book Company, 1988).
  12. D. Malacara, Optical Shop Testing, 3rd ed. (Wiley-VCH, 2007).
  13. C. Falldorf, J.-H. Hagemann, G. Ehret, and R. B. Bergmann, “Sparse light fields in coherent optical metrology [Invited],” Appl. Opt. 56(13), F14–F19 (2017).
    [Crossref] [PubMed]
  14. M. Schulz, I. Fortmeier, D. Sommer, and G. Blobel, “Concept of metrological reference surfaces for asphere and freeform metrology,” in Proceedings of the 17th International Conference of the European Society for Precision Engineering and Nanotechnology2017, pp. 365 −366.
  15. “Fringe Processor,” http://www.bias.de/wp-content/themes/bias/assets/pdf/OMOS_Flyer_v4_WEB.pdf

2017 (1)

2016 (1)

2014 (1)

2008 (1)

2007 (1)

2003 (1)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

Bergmann, R. B.

C. Falldorf, J.-H. Hagemann, G. Ehret, and R. B. Bergmann, “Sparse light fields in coherent optical metrology [Invited],” Appl. Opt. 56(13), F14–F19 (2017).
[Crossref] [PubMed]

J.-H. Hagemann, G. Ehret, R. B. Bergmann, and C. Falldorf, “Realization of a shearing interferometer with LED multisport illumination for form characterization of optics,” in Proceedings of DGaO (2016).

Blobel, G.

M. Schulz, I. Fortmeier, D. Sommer, and G. Blobel, “Concept of metrological reference surfaces for asphere and freeform metrology,” in Proceedings of the 17th International Conference of the European Society for Precision Engineering and Nanotechnology2017, pp. 365 −366.

Burge, J. H.

Dumas, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

Ehret, G.

C. Falldorf, J.-H. Hagemann, G. Ehret, and R. B. Bergmann, “Sparse light fields in coherent optical metrology [Invited],” Appl. Opt. 56(13), F14–F19 (2017).
[Crossref] [PubMed]

J.-H. Hagemann, G. Ehret, R. B. Bergmann, and C. Falldorf, “Realization of a shearing interferometer with LED multisport illumination for form characterization of optics,” in Proceedings of DGaO (2016).

Elster, C.

Falldorf, C.

C. Falldorf, J.-H. Hagemann, G. Ehret, and R. B. Bergmann, “Sparse light fields in coherent optical metrology [Invited],” Appl. Opt. 56(13), F14–F19 (2017).
[Crossref] [PubMed]

J.-H. Hagemann, G. Ehret, R. B. Bergmann, and C. Falldorf, “Realization of a shearing interferometer with LED multisport illumination for form characterization of optics,” in Proceedings of DGaO (2016).

Fleig, J.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 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. Photonics News 14(5), 38–43 (2003).
[Crossref]

Fortmeier, I.

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, W. Osten, and C. Elster, “Evaluation of absolute form measurements using a tilted-wave interferometer,” Opt. Express 24(4), 3393–3404 (2016).
[Crossref] [PubMed]

M. Schulz, I. Fortmeier, D. Sommer, and G. Blobel, “Concept of metrological reference surfaces for asphere and freeform metrology,” in Proceedings of the 17th International Conference of the European Society for Precision Engineering and Nanotechnology2017, pp. 365 −366.

Fuerschbach, K.

Garbusi, E.

Hagemann, J.-H.

C. Falldorf, J.-H. Hagemann, G. Ehret, and R. B. Bergmann, “Sparse light fields in coherent optical metrology [Invited],” Appl. Opt. 56(13), F14–F19 (2017).
[Crossref] [PubMed]

J.-H. Hagemann, G. Ehret, R. B. Bergmann, and C. Falldorf, “Realization of a shearing interferometer with LED multisport illumination for form characterization of optics,” in Proceedings of DGaO (2016).

Murphy, P.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

Osten, W.

Pruss, C.

Rolland, J. P.

Schulz, M.

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, W. Osten, and C. Elster, “Evaluation of absolute form measurements using a tilted-wave interferometer,” Opt. Express 24(4), 3393–3404 (2016).
[Crossref] [PubMed]

M. Schulz, I. Fortmeier, D. Sommer, and G. Blobel, “Concept of metrological reference surfaces for asphere and freeform metrology,” in Proceedings of the 17th International Conference of the European Society for Precision Engineering and Nanotechnology2017, pp. 365 −366.

Sommer, D.

M. Schulz, I. Fortmeier, D. Sommer, and G. Blobel, “Concept of metrological reference surfaces for asphere and freeform metrology,” in Proceedings of the 17th International Conference of the European Society for Precision Engineering and Nanotechnology2017, pp. 365 −366.

Stavridis, M.

Thompson, K. P.

Tricard, M.

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

Wiegmann, A.

Zhou, P.

Appl. Opt. (2)

Opt. Express (1)

Opt. Lett. (2)

Opt. Photonics News (1)

P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: A flexible solution for surface metrology,” Opt. Photonics News 14(5), 38–43 (2003).
[Crossref]

Other (9)

J. C. Wyant and V. P. Bennett, Using Computer Generated Holograms to Test Aspheric Wavefronts (Springer, 2014).

U. Schnars, C. Falldorf, J. Watson, and W. Jüptner, Digital Holography and Wavefront Sensing (Springer, 2015).

C. Falldorf, A. Simic, G. Ehret, M. Schulz, C. von Kopylow, and R. B. Bergmann, “Precise optical metrology using Computational Shear Interferometry and an LCD monitor as light source,” in Proceedings of Fringe 2013, ed. (Springer, 2014), pp.729–734.

J.-H. Hagemann, G. Ehret, R. B. Bergmann, and C. Falldorf, “Realization of a shearing interferometer with LED multisport illumination for form characterization of optics,” in Proceedings of DGaO (2016).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Book Company, 1988).

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

Z. Malacara and M. Servin, Interferogram Analysis for Optical Testing (CRC, 2016, Vol 84).

M. Schulz, I. Fortmeier, D. Sommer, and G. Blobel, “Concept of metrological reference surfaces for asphere and freeform metrology,” in Proceedings of the 17th International Conference of the European Society for Precision Engineering and Nanotechnology2017, pp. 365 −366.

“Fringe Processor,” http://www.bias.de/wp-content/themes/bias/assets/pdf/OMOS_Flyer_v4_WEB.pdf

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Figures (9)

Fig. 1
Fig. 1 Sketch of the measurement setup [9,10]. The surface under test (SUT) is illuminated by multiple LEDs and imaged to the image sensor plane by two lenses in a 4f-configuration. In the focal plane between the lenses, the spatial light modulator (SLM) is located which provides the shear. Two polarization filters are used to exploit the birefringent properties of the SLM.
Fig. 2
Fig. 2 Photo of the laboratory setup of the shearing interferometer with the multispot illumination (1). The light array consists of seven LED coupled fiber tips. In this example an aspheric lens with a focal length of 50 mm is placed in the measurement plane (2). Each light source yields an interferogram patch on the CCD sensor (5), which results in an overall resolvable interferogram shown on the screen. The interferometer is based on a spatial light modulator (4) as shearing element in the Fourier plane of a 4-f-configuration provided by two lenses (3).
Fig. 3
Fig. 3 Example of a measured mutual coherence function across the central region of a spherical lens (f = 50 mm): Amplitude (a) and phase (b) measured by the shear interferometer with the shear set to x = 240 µm and y = 160 µm. Each light source creates one sub-aperture in the sensor domain. The yellow rectangles show examples of three overlapping sub-apertures. Depending on the exact position of the light sources, we obtain dark or bright areas across the overlapping regions, showing the destructive and constructive superposition of the contributions from the individual sub-apertures in the complex mutual coherence function.
Fig. 4
Fig. 4 a) Sketch of the determination of a light source position. Rays with the direction of the klightsource-vectors, which are measured without the specimen, are propagated from the measurement plane to their crossing points; b) A kincident-vector is determined for every position of a krefracted-vector due to the knowledge of the corresponding light source position; c) Sketch of the form determination of the surface under test (SUT) by inverse raytracing. The solid line behind the measurement plane represents the rays in the direction of the krefracted-vectors, which are measured with the specimen, given by the spatial coherence function at its observation point. The dashed lines represent calculated rays refracted by the two shown exemplary surface forms of the assumed specimen. The form which provides the best match between measured and calculated rays corresponds to the surface under test.
Fig. 5
Fig. 5 Flow chart of the form determination by means of an optimization method in step two. The estimated surface form is changed by picking a set of random parameters from a given interval.
Fig. 6
Fig. 6 Photo of the measured spherical plan-convex lens with 50mm focal length and 25.4 mm diameter.
Fig. 7
Fig. 7 a) Calculated incident rays (red) originating from the light source positions which are determined by the first measurement of the coherence function. The black circles indicate the light source positions; b) Combined ray information of the measurements with (blue) and without (red) the specimen. Refraction at the measurement plane at z = 0 becomes visible; c) Result of the evaluation process. The known incident rays (red) are refracted at the surface determined (blue surface, only periphery visible) and also at the known reverse surface (yellow surface). We aim for the best match between the calculated rays (blue) and the measured exiting rays (blue Fig. 7(a)).
Fig. 8
Fig. 8 Reconstructed surface form of the plan-convex lens with 50 mm focal length.
Fig. 9
Fig. 9 Progress of the radius with ongoing iteration steps (measurement 3 of Table 1). The red line represents the measurement result from the PTB radius measurement bench.

Tables (2)

Tables Icon

Table 1 Multiple evaluation of the same data set for different starting parameters considering the radius of the best fit sphere and tilts. Every evaluation results in the same radius of curvature although the starting parameter is significantly changed. All values are in mm.

Tables Icon

Table 2 Multiple evaluations of the same data set considering a Zernike polynomial to describe the surface form. All values are in mm. Start values in all evaluations for parameters set to 0.

Equations (7)

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G ( x 1 , x 2 ) = U * ( x 1 , t ) U ( x 2 , t ) t = lim T 1 T t = T / 2 T / 2 U * ( x 1 , t ) U ( x 2 , t )   d t ,
U ( x , t ) = n U n ( x , t )
G ( x 1 , x 2 ) = n U n * ( x 1 ) U n ( x 2 ) .
I S ( x ) = | U ( x , t ) + U ( x + s , t ) | 2 T
I S ( x ) = I ( x ) + I ( x + s ) + 2 { G ( x , x + s ) } .
G ( x 0 , x 0 + s ) = n U n * ( x 0 ) U n ( x 0 + s ) = ψ ( s ; x 0 ) .
Ψ ( k x , k y ; x o ) = ψ ( s ; x 0 ) e i ( k x s x + k y s y ) d s x d s y ,

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