Abstract

With the application of multiwavelength digital holography in rough environments, such as where machine tools are used, we cannot rely on the complete absence of vibrations. The evaluation of temporal phase shifting in multiple interferograms regions allows one to determine and take into account random subwavelength tilt changes during image acquisition of the sensor with respect to a work piece. In this regard, experimental data inside a controlled laboratory setup, as well as data acquired within a five-axis machine tool suffering from random vibrations, are evaluated and affirmed by a simulation model.

© 2019 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Multiwavelength digital holography with autocalibration of phase shifts and artificial wavelengths

Daniel Carl, Markus Fratz, Marcel Pfeifer, Dominik M. Giel, and Heinrich Höfler
Appl. Opt. 48(34) H1-H8 (2009)

Multiwavelength-selective phase-shifting digital holography without mechanical scanning

Tatsuki Tahara and Yutaka Endo
Appl. Opt. 58(34) G218-G225 (2019)

Multiwavelength digital holography with wavelength-multiplexed holograms and arbitrary symmetric phase shifts

Tatsuki Tahara, Reo Otani, Kaito Omae, Takuya Gotohda, Yasuhiko Arai, and Yasuhiro Takaki
Opt. Express 25(10) 11157-11172 (2017)

References

  • View by:
  • |
  • |
  • |

  1. D. Carl, M. Fratz, and H. Höfler, “Digitale Mehrwellenlängen-Holografie für makroskopische Topografien in mikroskopischer Genauigkeit,” Tech. Mess. 77, 462–466 (2010).
    [Crossref]
  2. X. Luo and Y. Qin, eds., Hybrid Machining. Theory, Methods, and Case Studies (Elsevier, 2018).
  3. T. Seyler, M. Fratz, T. Beckmann, A. Bertz, and D. Carl, “Miniaturized multiwavelength digital holography sensor for extensive in-machine tool measurement,” Proc. SPIE 10329, 103290F (2017).
    [Crossref]
  4. T. Seyler, M. Fratz, T. Beckmann, A. Bertz, D. Carl, V. Grün, R. Börret, F. Ströer, and J. Seewig, “Extensive microstructural quality control inside a machine tool using multiwavelength digital holography,” Proc. SPIE 10834, 108342B (2018).
    [Crossref]
  5. T. Seyler and J. Engler, “HoloPort: 3D-sensor for machine tools,” in 20. GMA/ITG-Fachtagung Sensoren und Messsysteme (AMA, 2019), pp. 43–49.
  6. M. Fratz, T. Beckmann, A. Schiller, T. Seyler, A. Bertz, D. Carl, and K. Buse, “Digital holography: evolution from a research topic to a versatile tool for the inline 100% 3D quality control in industry,” in SENSOR 2017 and IRS2 2017 (2017), pp. 286–289.
  7. X. Huang, Z. Liu, and H. Xie, “Recent progress in residual stress measurement techniques,” Acta Mech. Solida Sin. 26, 570–583 (2013).
    [Crossref]
  8. T. M. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, 2005).
  9. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29, 183–185 (2004).
    [Crossref]
  10. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
    [Crossref]
  11. C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
    [Crossref]
  12. J. C. Dainty, A. E. Ennos, M. Françon, J. W. Goodman, T. S. McKechnie, and G. Parry, eds., Laser Speckle and Related Phenomena (Springer, 1975).
  13. J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
    [Crossref]
  14. R. D. Fiete, Modeling the Imaging Chain of Digital Cameras (SPIE, 2010).
  15. M. Weck and C. Brecher, Werkzeugmaschinen 3: mechatronische Systeme, Vorschubantriebe, Prozessdiagnose (Springer, 2006).
  16. A. Schiller, T. Beckmann, M. Fratz, D. Belzer, A. Bertz, D. Carl, and K. Buse, “Digital holography on moving objects: interference contrast as a function of velocity and aperture width,” Appl. Opt. 56, 4622–4628 (2017).
    [Crossref]
  17. A. Schiller, T. Beckmann, M. Fratz, A. Bertz, D. Carl, and K. Buse, “Motion compensation for interferometric off-center measurements of rotating objects with varying radii,” APL Photon. 4, 071301 (2019).
    [Crossref]
  18. T. Seyler, “HoloPort—3D-Inline-Messtechnik für die Werkzeugmaschine,” Jahrgang 109, 319–320 (2019).

2019 (2)

A. Schiller, T. Beckmann, M. Fratz, A. Bertz, D. Carl, and K. Buse, “Motion compensation for interferometric off-center measurements of rotating objects with varying radii,” APL Photon. 4, 071301 (2019).
[Crossref]

T. Seyler, “HoloPort—3D-Inline-Messtechnik für die Werkzeugmaschine,” Jahrgang 109, 319–320 (2019).

2018 (1)

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, D. Carl, V. Grün, R. Börret, F. Ströer, and J. Seewig, “Extensive microstructural quality control inside a machine tool using multiwavelength digital holography,” Proc. SPIE 10834, 108342B (2018).
[Crossref]

2017 (2)

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, and D. Carl, “Miniaturized multiwavelength digital holography sensor for extensive in-machine tool measurement,” Proc. SPIE 10329, 103290F (2017).
[Crossref]

A. Schiller, T. Beckmann, M. Fratz, D. Belzer, A. Bertz, D. Carl, and K. Buse, “Digital holography on moving objects: interference contrast as a function of velocity and aperture width,” Appl. Opt. 56, 4622–4628 (2017).
[Crossref]

2013 (1)

X. Huang, Z. Liu, and H. Xie, “Recent progress in residual stress measurement techniques,” Acta Mech. Solida Sin. 26, 570–583 (2013).
[Crossref]

2010 (1)

D. Carl, M. Fratz, and H. Höfler, “Digitale Mehrwellenlängen-Holografie für makroskopische Topografien in mikroskopischer Genauigkeit,” Tech. Mess. 77, 462–466 (2010).
[Crossref]

2004 (1)

2000 (1)

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[Crossref]

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[Crossref]

1976 (1)

Beckmann, T.

A. Schiller, T. Beckmann, M. Fratz, A. Bertz, D. Carl, and K. Buse, “Motion compensation for interferometric off-center measurements of rotating objects with varying radii,” APL Photon. 4, 071301 (2019).
[Crossref]

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, D. Carl, V. Grün, R. Börret, F. Ströer, and J. Seewig, “Extensive microstructural quality control inside a machine tool using multiwavelength digital holography,” Proc. SPIE 10834, 108342B (2018).
[Crossref]

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, and D. Carl, “Miniaturized multiwavelength digital holography sensor for extensive in-machine tool measurement,” Proc. SPIE 10329, 103290F (2017).
[Crossref]

A. Schiller, T. Beckmann, M. Fratz, D. Belzer, A. Bertz, D. Carl, and K. Buse, “Digital holography on moving objects: interference contrast as a function of velocity and aperture width,” Appl. Opt. 56, 4622–4628 (2017).
[Crossref]

M. Fratz, T. Beckmann, A. Schiller, T. Seyler, A. Bertz, D. Carl, and K. Buse, “Digital holography: evolution from a research topic to a versatile tool for the inline 100% 3D quality control in industry,” in SENSOR 2017 and IRS2 2017 (2017), pp. 286–289.

Belzer, D.

Bertz, A.

A. Schiller, T. Beckmann, M. Fratz, A. Bertz, D. Carl, and K. Buse, “Motion compensation for interferometric off-center measurements of rotating objects with varying radii,” APL Photon. 4, 071301 (2019).
[Crossref]

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, D. Carl, V. Grün, R. Börret, F. Ströer, and J. Seewig, “Extensive microstructural quality control inside a machine tool using multiwavelength digital holography,” Proc. SPIE 10834, 108342B (2018).
[Crossref]

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, and D. Carl, “Miniaturized multiwavelength digital holography sensor for extensive in-machine tool measurement,” Proc. SPIE 10329, 103290F (2017).
[Crossref]

A. Schiller, T. Beckmann, M. Fratz, D. Belzer, A. Bertz, D. Carl, and K. Buse, “Digital holography on moving objects: interference contrast as a function of velocity and aperture width,” Appl. Opt. 56, 4622–4628 (2017).
[Crossref]

M. Fratz, T. Beckmann, A. Schiller, T. Seyler, A. Bertz, D. Carl, and K. Buse, “Digital holography: evolution from a research topic to a versatile tool for the inline 100% 3D quality control in industry,” in SENSOR 2017 and IRS2 2017 (2017), pp. 286–289.

Börret, R.

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, D. Carl, V. Grün, R. Börret, F. Ströer, and J. Seewig, “Extensive microstructural quality control inside a machine tool using multiwavelength digital holography,” Proc. SPIE 10834, 108342B (2018).
[Crossref]

Brecher, C.

M. Weck and C. Brecher, Werkzeugmaschinen 3: mechatronische Systeme, Vorschubantriebe, Prozessdiagnose (Springer, 2006).

Buse, K.

A. Schiller, T. Beckmann, M. Fratz, A. Bertz, D. Carl, and K. Buse, “Motion compensation for interferometric off-center measurements of rotating objects with varying radii,” APL Photon. 4, 071301 (2019).
[Crossref]

A. Schiller, T. Beckmann, M. Fratz, D. Belzer, A. Bertz, D. Carl, and K. Buse, “Digital holography on moving objects: interference contrast as a function of velocity and aperture width,” Appl. Opt. 56, 4622–4628 (2017).
[Crossref]

M. Fratz, T. Beckmann, A. Schiller, T. Seyler, A. Bertz, D. Carl, and K. Buse, “Digital holography: evolution from a research topic to a versatile tool for the inline 100% 3D quality control in industry,” in SENSOR 2017 and IRS2 2017 (2017), pp. 286–289.

Cai, L. Z.

Carl, D.

A. Schiller, T. Beckmann, M. Fratz, A. Bertz, D. Carl, and K. Buse, “Motion compensation for interferometric off-center measurements of rotating objects with varying radii,” APL Photon. 4, 071301 (2019).
[Crossref]

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, D. Carl, V. Grün, R. Börret, F. Ströer, and J. Seewig, “Extensive microstructural quality control inside a machine tool using multiwavelength digital holography,” Proc. SPIE 10834, 108342B (2018).
[Crossref]

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, and D. Carl, “Miniaturized multiwavelength digital holography sensor for extensive in-machine tool measurement,” Proc. SPIE 10329, 103290F (2017).
[Crossref]

A. Schiller, T. Beckmann, M. Fratz, D. Belzer, A. Bertz, D. Carl, and K. Buse, “Digital holography on moving objects: interference contrast as a function of velocity and aperture width,” Appl. Opt. 56, 4622–4628 (2017).
[Crossref]

D. Carl, M. Fratz, and H. Höfler, “Digitale Mehrwellenlängen-Holografie für makroskopische Topografien in mikroskopischer Genauigkeit,” Tech. Mess. 77, 462–466 (2010).
[Crossref]

M. Fratz, T. Beckmann, A. Schiller, T. Seyler, A. Bertz, D. Carl, and K. Buse, “Digital holography: evolution from a research topic to a versatile tool for the inline 100% 3D quality control in industry,” in SENSOR 2017 and IRS2 2017 (2017), pp. 286–289.

Engler, J.

T. Seyler and J. Engler, “HoloPort: 3D-sensor for machine tools,” in 20. GMA/ITG-Fachtagung Sensoren und Messsysteme (AMA, 2019), pp. 43–49.

Fiete, R. D.

R. D. Fiete, Modeling the Imaging Chain of Digital Cameras (SPIE, 2010).

Fratz, M.

A. Schiller, T. Beckmann, M. Fratz, A. Bertz, D. Carl, and K. Buse, “Motion compensation for interferometric off-center measurements of rotating objects with varying radii,” APL Photon. 4, 071301 (2019).
[Crossref]

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, D. Carl, V. Grün, R. Börret, F. Ströer, and J. Seewig, “Extensive microstructural quality control inside a machine tool using multiwavelength digital holography,” Proc. SPIE 10834, 108342B (2018).
[Crossref]

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, and D. Carl, “Miniaturized multiwavelength digital holography sensor for extensive in-machine tool measurement,” Proc. SPIE 10329, 103290F (2017).
[Crossref]

A. Schiller, T. Beckmann, M. Fratz, D. Belzer, A. Bertz, D. Carl, and K. Buse, “Digital holography on moving objects: interference contrast as a function of velocity and aperture width,” Appl. Opt. 56, 4622–4628 (2017).
[Crossref]

D. Carl, M. Fratz, and H. Höfler, “Digitale Mehrwellenlängen-Holografie für makroskopische Topografien in mikroskopischer Genauigkeit,” Tech. Mess. 77, 462–466 (2010).
[Crossref]

M. Fratz, T. Beckmann, A. Schiller, T. Seyler, A. Bertz, D. Carl, and K. Buse, “Digital holography: evolution from a research topic to a versatile tool for the inline 100% 3D quality control in industry,” in SENSOR 2017 and IRS2 2017 (2017), pp. 286–289.

Goodman, J. W.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[Crossref]

Grün, V.

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, D. Carl, V. Grün, R. Börret, F. Ströer, and J. Seewig, “Extensive microstructural quality control inside a machine tool using multiwavelength digital holography,” Proc. SPIE 10834, 108342B (2018).
[Crossref]

Höfler, H.

D. Carl, M. Fratz, and H. Höfler, “Digitale Mehrwellenlängen-Holografie für makroskopische Topografien in mikroskopischer Genauigkeit,” Tech. Mess. 77, 462–466 (2010).
[Crossref]

Huang, X.

X. Huang, Z. Liu, and H. Xie, “Recent progress in residual stress measurement techniques,” Acta Mech. Solida Sin. 26, 570–583 (2013).
[Crossref]

Kreis, T. M.

T. M. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, 2005).

Liu, Q.

Liu, Z.

X. Huang, Z. Liu, and H. Xie, “Recent progress in residual stress measurement techniques,” Acta Mech. Solida Sin. 26, 570–583 (2013).
[Crossref]

Osten, W.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[Crossref]

Schiller, A.

A. Schiller, T. Beckmann, M. Fratz, A. Bertz, D. Carl, and K. Buse, “Motion compensation for interferometric off-center measurements of rotating objects with varying radii,” APL Photon. 4, 071301 (2019).
[Crossref]

A. Schiller, T. Beckmann, M. Fratz, D. Belzer, A. Bertz, D. Carl, and K. Buse, “Digital holography on moving objects: interference contrast as a function of velocity and aperture width,” Appl. Opt. 56, 4622–4628 (2017).
[Crossref]

M. Fratz, T. Beckmann, A. Schiller, T. Seyler, A. Bertz, D. Carl, and K. Buse, “Digital holography: evolution from a research topic to a versatile tool for the inline 100% 3D quality control in industry,” in SENSOR 2017 and IRS2 2017 (2017), pp. 286–289.

Seebacher, S.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[Crossref]

Seewig, J.

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, D. Carl, V. Grün, R. Börret, F. Ströer, and J. Seewig, “Extensive microstructural quality control inside a machine tool using multiwavelength digital holography,” Proc. SPIE 10834, 108342B (2018).
[Crossref]

Seyler, T.

T. Seyler, “HoloPort—3D-Inline-Messtechnik für die Werkzeugmaschine,” Jahrgang 109, 319–320 (2019).

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, D. Carl, V. Grün, R. Börret, F. Ströer, and J. Seewig, “Extensive microstructural quality control inside a machine tool using multiwavelength digital holography,” Proc. SPIE 10834, 108342B (2018).
[Crossref]

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, and D. Carl, “Miniaturized multiwavelength digital holography sensor for extensive in-machine tool measurement,” Proc. SPIE 10329, 103290F (2017).
[Crossref]

T. Seyler and J. Engler, “HoloPort: 3D-sensor for machine tools,” in 20. GMA/ITG-Fachtagung Sensoren und Messsysteme (AMA, 2019), pp. 43–49.

M. Fratz, T. Beckmann, A. Schiller, T. Seyler, A. Bertz, D. Carl, and K. Buse, “Digital holography: evolution from a research topic to a versatile tool for the inline 100% 3D quality control in industry,” in SENSOR 2017 and IRS2 2017 (2017), pp. 286–289.

Ströer, F.

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, D. Carl, V. Grün, R. Börret, F. Ströer, and J. Seewig, “Extensive microstructural quality control inside a machine tool using multiwavelength digital holography,” Proc. SPIE 10834, 108342B (2018).
[Crossref]

Wagner, C.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[Crossref]

Weck, M.

M. Weck and C. Brecher, Werkzeugmaschinen 3: mechatronische Systeme, Vorschubantriebe, Prozessdiagnose (Springer, 2006).

Xie, H.

X. Huang, Z. Liu, and H. Xie, “Recent progress in residual stress measurement techniques,” Acta Mech. Solida Sin. 26, 570–583 (2013).
[Crossref]

Yang, X. L.

Acta Mech. Solida Sin. (1)

X. Huang, Z. Liu, and H. Xie, “Recent progress in residual stress measurement techniques,” Acta Mech. Solida Sin. 26, 570–583 (2013).
[Crossref]

APL Photon. (1)

A. Schiller, T. Beckmann, M. Fratz, A. Bertz, D. Carl, and K. Buse, “Motion compensation for interferometric off-center measurements of rotating objects with varying radii,” APL Photon. 4, 071301 (2019).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Jahrgang (1)

T. Seyler, “HoloPort—3D-Inline-Messtechnik für die Werkzeugmaschine,” Jahrgang 109, 319–320 (2019).

Opt. Eng. (2)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[Crossref]

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (2)

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, and D. Carl, “Miniaturized multiwavelength digital holography sensor for extensive in-machine tool measurement,” Proc. SPIE 10329, 103290F (2017).
[Crossref]

T. Seyler, M. Fratz, T. Beckmann, A. Bertz, D. Carl, V. Grün, R. Börret, F. Ströer, and J. Seewig, “Extensive microstructural quality control inside a machine tool using multiwavelength digital holography,” Proc. SPIE 10834, 108342B (2018).
[Crossref]

Tech. Mess. (1)

D. Carl, M. Fratz, and H. Höfler, “Digitale Mehrwellenlängen-Holografie für makroskopische Topografien in mikroskopischer Genauigkeit,” Tech. Mess. 77, 462–466 (2010).
[Crossref]

Other (7)

X. Luo and Y. Qin, eds., Hybrid Machining. Theory, Methods, and Case Studies (Elsevier, 2018).

T. M. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, 2005).

T. Seyler and J. Engler, “HoloPort: 3D-sensor for machine tools,” in 20. GMA/ITG-Fachtagung Sensoren und Messsysteme (AMA, 2019), pp. 43–49.

M. Fratz, T. Beckmann, A. Schiller, T. Seyler, A. Bertz, D. Carl, and K. Buse, “Digital holography: evolution from a research topic to a versatile tool for the inline 100% 3D quality control in industry,” in SENSOR 2017 and IRS2 2017 (2017), pp. 286–289.

J. C. Dainty, A. E. Ennos, M. Françon, J. W. Goodman, T. S. McKechnie, and G. Parry, eds., Laser Speckle and Related Phenomena (Springer, 1975).

R. D. Fiete, Modeling the Imaging Chain of Digital Cameras (SPIE, 2010).

M. Weck and C. Brecher, Werkzeugmaschinen 3: mechatronische Systeme, Vorschubantriebe, Prozessdiagnose (Springer, 2006).

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 (11)

Fig. 1.
Fig. 1. Left, schematic integration of HoloPort measuring a work piece inside a Hermle C32U five-axis machine tool with critical axial movement visualized by orange arrow. Right, HoloPort optical design with two-dimensional beam arrangement for center of gravity mounting.
Fig. 2.
Fig. 2. Left, one interferogram $I( {x,y} )$ resulting from the simulation. Right, reconstructed height image of the simulated data. Contrast is enhanced in both color maps for illustration purposes. Lateral axes are given in pixels (Px), using a pixel size of 7 µm.
Fig. 3.
Fig. 3. Experimental setup including (a) a sketch and (b) a photograph showing the HoloPort sensor system and the piezo at the sample holder. The optical design of HoloPort is described in detail in [3,5]. (c) shows a height measurement of the milling sample at constant piezo voltage with enhanced contrast. Lateral axes are given in pixels (Px), featuring a camera pixel size of 3.45 µm, resulting in an object-sided pixel size of 7 µm.
Fig. 4.
Fig. 4. Reconstructed maps of the first phase shift ${\tilde \alpha _1}( {x,y} )$ for different slopes (left, $\beta \approx 17\frac{{pm}}{{Px}}$ , right: $\beta \approx 50\frac{{pm}}{{Px}}$ ) using ${25} \times {25}$ regions at a synthetic wavelength of 67.76 µm. Left, ${\tilde \alpha _1}( {x,y} )$ corresponding to a vibration of 0.05 µm. Right, ${\tilde \alpha _1}( {x,y} )$ corresponding to a vibration of 0.15 µm. The $c$ value of each section is shown by the color and used as a weight in the fit.
Fig. 5.
Fig. 5. Simulated standard deviation calculated from a composed stack of 10 measurements with vibrations of 0.05 µm amplitude. Left, using a constant phase shift map. Right, using the laterally resolved reconstructed phase-shift map.
Fig. 6.
Fig. 6. Reconstructed phase-shift map of composed stack with different piezo voltages corresponding to piezo voltage of 1 V (left) and 3 V (right). The $c$ value of each section is shown by the color and used as a weight in the fit.
Fig. 7.
Fig. 7. Slope of the plane fit in the phase-shift map dependent on the voltage applied to the piezo. A linear increase in the $x$ direction can be identified.
Fig. 8.
Fig. 8. Height image reconstructed from composed stack, evaluated using a global phase shift (left) and a laterally resolved phase-shift map (right). The mean micrometer of the standard deviation $\sigma $ over a part of the image shows a decrease using the new method, indicating a successful compensation of vibration.
Fig. 9.
Fig. 9. Left, experimental setup with HoloPort sensor integrated into a Hermle C32U milling machine to measure axial vibrations on a mirror test sample. Right, vibrations in the range of 50 nm at a frequency of  $\sim\!90\,\,{\rm Hz}$ can be observed.
Fig. 10.
Fig. 10. Two individual measurements on the same sample, taken with disabled (left) and enabled (right) spindle control. Left, reconstructed height (c) from smooth phase map (a) without any artifacts. Right, reconstructed height (d) with irregular phase map (b) close to $\pi$ and strong artifacts.
Fig. 11.
Fig. 11. Height maps of a single measurement inside the machine tool, reconstructed with a single phase shift (left) and laterally resolved phase shifts (right).

Equations (8)

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

α ~ n ( x , y ) = α n + ε n ( x , y ) .
I n ( x , y ) = a 0 ( x , y ) + a 1 ( x , y ) cos ( α ~ n ( x , y ) ) + a 2 ( x , y ) sin ( α ~ n ( x , y ) ) .
φ ( x , y ) = tan 1 ( a 2 ( x , y ) a 1 ( x , y ) ) .
I λ , n ( x , y ) = A r 2 ( x , y ) + A 0 2 ( x , y ) + 2 A r A 0 ( x , y ) × cos [ φ n ( x , y ) α ~ n ( x , y ) ] .
φ λ , n ( x , y ) = ( η ( x , y ) + 4 π h ( x , y ) λ n ) m o d 2 π .
ε λ , n ( x , y ) = 4 π β λ x .
I ( x , y ) = μ log ( μ ϑ ( x , y ) ) ,
σ I ( x , y ) = 1 F W C I F W C .

Metrics