# Abstract

Electronic autocollimators are utilised versatilely for non-contact angle measurements in applications like straightness measurements and profilometry. Yet, no calibration of the angle measurement of an autocollimator has been available when both its measurement axes are engaged. Additionally, autocollimators have been calibrated at fixed distances to the reflector, although its distance may vary during the use of an autocollimator. To extend the calibration capabilities of the Physikalisch-Technische Bundesanstalt (PTB) regarding spatial angles and variable distances, a novel calibration device has been set up: the spatial angle autocollimator calibrator (SAAC). In this paper, its concept and its mechanical realisation will be presented. The focus will be on the system’s mathematical modelling and its application in spatial angle calibrations. The model considers the misalignments of the SAAC’s components, including the non-orthogonalities of the measurement axes of the autocollimators and of the rotational axes of the tilting unit. It allows us to derive specific measurement procedures to determine the misalignments *in situ* and, in turn, to correct the measurements of the autocollimators. Finally, the realisation and the results of a traceable spatial angle calibration of an autocollimator will be presented. This is the first calibration of this type worldwide.

# Acknowledgments

Part of this research was undertaken within the European Metrology Research Programme (EMRP) Joint Research Project (JRP) SIB 58 Angle Metrology. The EMRP is jointly funded by the EMRP participating countries within the European Association of National Metrology Institutes (EURAMET) and the European Union.

## Appendix

**Method for the determination of the non-orthogonalities of the rotational axes of the tilting unit and the measurement axes of the autocollimator**

The non-orthogonalities in this illustration are strongly exaggerated. The autocollimator’s horizontal and vertical measurement directions are regarded as seen from the autocollimator standing upright (roll angle adjustment: 0°). They are co-rotated with the autocollimator. For the presentation of this method, the roll angle adjustments of the autocollimator are assumed to be ideal. See the note below regarding this constraint.

**First roll angle adjustment: 0°**

The autocollimator is adjusted as depicted in Figure 11: The reticle of the vertical measurement axis (V-Rtc, red line) is aligned parallel to the horizontal deflection direction of the reflected beam (H-Dfl, dashed line). For a horizontal tilting of the mirror, the autocollimator measures the deflection angle of the beam (along the H-Dfl line) with its horizontal CCD (H-CCD, thin black line), while the vertical measurement value is constantly zero. When the mirror is tilted vertically, not only the vertical deflection angle (along the V-Dfl line) is measured by the autocollimator’s vertical CCD (V-CCD) but also the horizontal measurement value is =0 due to the non-orthogonalities of the reticles (*v*) and the rotational axes of the tilting unit (*α*).

### Figure 11:

For this roll angle adjustment, the angle *θ*_{1} is calculated by evaluating the measurement values of the autocollimator (using the approximation tan *θ*_{1}≈sin *θ*_{1}≈*θ*_{1} for small angles *θ*_{1}):

where *H* is the horizontal, and *V* is the vertical measurement value of the autocollimator.

A note on the experimental application of the method: A residual misalignment (non-parallelism) between the reticle of the autocollimator’s vertical measurement axis (V-Rtc) and the horizontal deflection direction of the tilting system (H-Dfl) can be evaluated by the use of autocollimator measurements performed during the horizontal deflection of the tilting system. The resulting angle between V-Rtc and H-Dfl can then be used for correcting the angle *θ*_{1} by adding or subtracting it.

**Second roll angle adjustment: 90°**

For the second roll angle adjustment of the autocollimator, it is rotated by approximately 90° (Figure 12) with respect to its optical axis to align the reticle of its horizontal measurement axis (H-Rtc) parallel to the horizontal deflection direction of the reflected beam (H-Dfl). For a horizontal tilting of the mirror, the autocollimator measures the deflection angle of the beam (along the H-Dfl line) with its vertical CCD (V-CCD), while the horizontal measurement value is constantly zero. The mirror is tilted vertically again. The horizontal CCD (H-CCD) measures the vertical deflection (along the V-Dfl line). The vertical measurement value is ≠0, due to the non-orthogonalities.

### Figure 12:

For the second roll angle adjustment, the angle *θ*_{2} is given by:

With equation (A.1) and (A.2) one obtains

To be compliant with the parameter definitions of the SAAC, *α*=*r*_{yα} and *v*=*V*_{0v} are valid.

### References

[1] H. Lammert, F. Senf and M. Berger, Proc. SPIE 3152, 168–179 (1997). Search in Google Scholar

[2] R. D. Geckeler and A. Just, Proc. SPIE 6704, 670407-1 (2007). Search in Google Scholar

[3] R. D. Geckeler, A. Just, M. Krause and V. V. Yashchuk Nucl. Instrum. Methods Phys. Res. Sect. A 616, 140 (2010). Search in Google Scholar

[4] F. Siewert, H. Lammert and T. Zeschke, in ‘Modern Developments in X-Ray and Neutron Optics’, Eds. By A. Erko, M. Idir, T. Krist and A.G. Michette (Springer, Berlin, 2008) pp. 193–200. Search in Google Scholar

[5] V. V. Yashchuk, S. Barbera, E. E. Domninga, J. L. Kirschmana, G. Y. Morrisona, et al., Nucl. Instrum. Methods Phys. Res. Sect. A 616, 212 (2010). Search in Google Scholar

[6] G. Ehret, M. Schulz, M. Stavridis and C. Elster, Meas. Sci. Technol. 23, 094007 (2012). Search in Google Scholar

[7] F. Siewert, J. Buchheim, S. Boutet, G. J. Williams, P. A. Montanez, et al., Opt. Express 20, 4 (2012). Search in Google Scholar

[8] A. Just, M. Krause, R. Probst and R. Wittekopf, Metrologia 40, 288 (2003). Search in Google Scholar

[9] R. D. Geckeler and A. Just, Proc. SPIE 7077, 70770B (2008). Search in Google Scholar

[10] R. Probst, G. Fuetterer, J. Illemann, J. Mokros, P.K. Lui, et al. Proc. 7th Int. Conf. Euspen 2, 121 (2007). Search in Google Scholar

[11] R. Probst, R. Wittekopf, M. Krause, H. Dangschat and A. Ernst, Meas. Sci. Technol. 9, 1059 (1998). Search in Google Scholar

[12] R. D. Geckeler, O. Kranz, A. Just and M. Krause, Adv. Opt. Technol. 1, 427 (2012). Search in Google Scholar

[13] http://www.moeller-wedel-optical.com/en/produkte/electronic-autocollimators/p-elcomat-3000.html (accessed 2014-11-13). Search in Google Scholar

[14] R. D. Geckeler and A. Just, Meas. Sci. Technol. 25, 105009 (2014). Search in Google Scholar

[15] O. Kranz, R. D. Geckeler, A. Just and M. Krause, Proc. SPIE 8789, 87890D (2013). Search in Google Scholar

[16] R. D. Geckeler, Meas. Sci. Technol. 18, 115 (2007). Search in Google Scholar

[17] S. K. Barber, R. D. Geckeler, V. V. Yashchuk, M. V. Gubarev, J. Buchheim, et al., Opt. Eng. 50, 073602 (2011). Search in Google Scholar

[18] P. J. Sim, in ‘Modern Techniques in Metrology’, Ed. By P. Hewitt (World Scientific, Singapore, 1984) pp. 102–112. Search in Google Scholar

[19] International vocabulary of metrology – Basic and general concepts and associated terms (VIM) JCGM 200:2012 http://www.bipm.org/en/publications/guides/vim.html (accessed 2014-11-13). Search in Google Scholar

**Received:**2015-2-23

**Accepted:**2015-4-23

**Published Online:**2015-5-21

**Published in Print:**2015-10-1

©2015 THOSS Media & De Gruyter