## Absolute flatness measurement using oblique incidence setup and an iterative algorithm. A demonstration on synthetic data |

Optics Express, Vol. 22, Issue 3, pp. 3538-3546 (2014)

http://dx.doi.org/10.1364/OE.22.003538

Acrobat PDF (2887 KB)

### Abstract

A method to provide absolute planarity measurements through an interferometric oblique incidence setup and an iterative algorithm is presented. With only three measurements, the calibration of absolute planarity is achieved in a fast and effective manner. Demonstration with synthetic data is provided, and the possible application to very long flat mirrors is pointed out.

© 2014 Optical Society of America

## 1. Introduction

## 2. Oblique incidence setup

*K*) and a reference flat (here named

*M*). The flats

*K*and

*M*are used to define a measurement cavity where the test surface (here named

*L*) is inserted at a certain incidence reflecting angle

*α*[Fig. 1(a)]. The setup is on the horizontal plane, putting

*L*,

*K*and

*M*surfaces along the vertical, to minimize deformations due to gravity. After the measurement with

*L*in place, the cavity itself is measured removing

*L*and realigning

*K*and

*M*in front of each other [Fig. 1(b)]. The latter measurement is then repeated rotating the flat

*M*on its plane through an angle

*β*[Fig. 1(c)], so that information on the azimuthal error of the cavity is also acquired.

*K*(

*x,y*),

*M*(

*x,y*) and

*L*(

*x,y*), with (

*x*,

*y*) the local coordinates of each surface. Referring to the sequence (a), (b), (c) in Fig. 1, we also indicate the maps of optical path difference (

*OPD*) measured with the interferometer as

*KLM*(

*x,y*),

*KM*(

*x,y*),

*KM*(

_{β}*x,y*), respectively. To represent the measuring configurations and the posture of the flats in each step, we introduce a set of operators acting on the data maps or the surface coordinates [26]; in particular we use the operators listed in Table 1.

*M*through an angle

*β*as in Fig. 1(c), the result is represented with

*L*, it represents the fact that such surface seen from the beam is stretched along the

*x-*axis; also, the height is multiplied times a scaling factor depending on the angle

*α*. As to the latter factor, it includes two different effects, as shown in Fig. 2 where a height step Δ

*z*is drawn: the first effect is that the

*OPD*changes from 2Δ

*z*to 2 cosα Δ

*z*; the second effect is a further factor of two due to double pass, so that the overall scaling factor with respect to the case of normal incidence is 2 cosα. In detail, being

*α*≤ 60 degrees, the scaling factor is equal to or greater than unity, so the sensitivity of the setup is the same or even higher compared to normal incidence.

*M*in Eq. (1) is because the beam impinging on

*M*is mirrored by the surface

*L*about the y-axis, then causing a reversal of the

*x*-coordinates on the

*OPD*introduced by

*M*. Naturally, in Eq. (2) relating to the case of empty cavity such a reversal is not occurring, and

*β*in Eq. (3), its value should be selected avoiding integer sub-multiples of 360 degrees [26]; the optimum value suggested in the literature and widely used in practice is

*β*= 54 degrees [27].

*K*,

*L*and

*M*using a random selection of low spatial frequency Zernike polynomials. To better account for real experimental conditions, we also added a Gaussian distribution of white noise. We then computed the interference patterns

*KLM*(

*x,y*),

*KM*(

*x,y*),

*KM*(

_{β}*x,y*) that would be produced in the measuring sequence of Fig. 1. As an example, here we report the results we obtained using a selection of low spatial frequency terms with 100 nm Peak-to-Valley (P-V) and 10 nm root-mean-square (rms), and a white noise with 25 nm P-V and 2.5 nm rms. We set

*α*= 45 degrees and

*β*= 54 degrees. In Fig. 3 we present the expected measurement results according to Eqs. (1)-(3).

*KLM*appears laterally compressed. In fact, if the dimensions of all the three flats are the same, the surfaces

*K*and

*M*are vignetted by

*L*. If the flat

*L*has an elliptical footprint, the tilt angle

*α*could be adjusted to minimize the compression effect, then improving the spatial resolution and the sensitivity. It is understood that in real measurements the tilt angle needs be optimized depending on the size and shape of the test surface

*L*. For that reason, final results cannot be easily retrieved with standard data processing. To have a procedure reliable and easy to be optimized, we here present a data processing approach based on an iterative algorithm [28,29]. The latter has been already used with standard three-flat setup, both in horizontal and in vertical configurations, showing clear advantages as to versatility and effectiveness [30,31]; it was also used in the area of applied research, where it allowed to observe the viscoelastic behavior of fused silica glass at room temperature [32,33]. In the present case, the iterative algorithm is used to retrieve the absolute figure of the test surface

*L*with pixel resolution and sub-nanometric rms accuracy.

## 3. Data processing with Iterative Algorithm

*KLM*and

*KM*:The figure map of

*M*can be divided in two separate contributions: a rotationally invariant part, as spherical aberration, and a rotation dependent part, as astigmatism or cylinder. The rotationally invariant part is also mirror invariant, so it is automatically cancelled in Eq. (5) by the difference

*KM*and

*KM*as inputs. To this purpose, the following steps are implemented:

_{β}- 1. Generate an initial set of trial surfaces
*K*and*M*(they can be created randomly or also initially taken as a matrix of zeroes). - 5. Update the maps replacing
*K*and*M*with*K*,_{new}*M*and go back to step 2. The cycle exits if the rms values of the difference Δ(_{new}*KM*) and Δ(*KM*) computed in step 3 reach a minimum or become smaller than a given threshold. We can also set the cycle to be repeated for a fixed number of times._{β}

*K*and

*M*shown in Fig. 4. Computing the differences between the retrieved maps and the starting ones, we can clearly see that the angular part is fully reconstructed while the rotationally invariant part is not: however, since we are going to use Eq. (5) where the latter is automatically discarded, this missing reconstruction has no consequences.

*M*, we now calculate the angular part

*L*, in principle we could define and use an inverse operator

*KLM*, we stretched

*L*and so we lost some lateral resolution. This naturally occurs also in real measurements, looking at

*L*with an oblique incidence angle. So, to evaluate the algorithm in itself we here do not consider lateral resolution, limiting ourselves to undo the scaling only. The result is shown in Fig. 4, where the retrieved surfaces

*K*,

*L*(stretched),

*M*are presented, along with the pertaining error maps. Remarkably, while significant (basically, rotation invariant) errors on

*K*and

*M*are still present, the oblique map of

*L*is retrieved with sub-nanometric rms accuracy.

## 4. Application to mirrors with elliptical footprint

*X*and

*Y*, with

*X*>

*Y*: if the diameter of the laser beam out of the interferometer is

*D*, with

## 5. Other error sources

*KM*and

*KM*measurements: the flats are not displaced but only rotated, and a mechanical reference can be used to avoid mismatches between the two measurements. The measurement

_{β}*KLM*is instead more critical: the tilted surface is cropping the beam, so that we have fewer fiducial references, and small misalignments are more likely to occur. To quantify the influence of these effects, we made some simulations.

*α-*value used to generate the

*KLM*synthetic map; after that, we applied the iterative algorithm, assuming that we still had a tilt angle of exactly 45 degrees. The error computed on the difference map between the retrieved

*L-*map and the original one so becomes 0.57 nm rms, slightly larger than previously.

*KLM*is made. During all the procedure, the flat

*K*is remaining attached to the Fizeau interferometer, so we can assume that its correspondence to the pixels matrix is fixed. The flat

*L*is used in this measurement for the first time, so repositioning errors are not occurring. The position of the flat

*M*is more critical because it is realigned to fit the tilt angle setup: it is possible to have some mismatch between the actual pixel map and the one considered in the other measurements, in particular if

*L*is small and most of the beam is vignetted. Such a mismatch can be reduced to the pixel level using proper alignment aids, for example introducing fiducial masks to identify the center of each surface. Based on these considerations, we made a simulation imparting to the flat

*M*a misalignment of ± 1 pixel along the

*x*and

*y*axes. Working out the map of residuals between the retrieved flat

*L*and the original one, typical values of 2.6 nm rms error are found. This relatively large error is mostly originated from the distribution of white noise considered in this example, creating high spatial frequency residuals that cannot be retrieved with even a small mismatch. This effect is generally present in every three-flat data-retrieving algorithm. If we repeat this simulation using maps with a ten times lower noise level, we find a maximum value of 0.64 nm rms on the

*L*-map residual error.

## 6. Conclusions

## Acknowledgments

## References and links

1. | M. Zeuner and S. Kiontke, “Ion beam figuring technology in optics manufacturing,” Optik & Photonik |

2. | M. Weiser, “Ion beam figuring for lithography optics,” Nucl. Instrum. Methods Phys. Res. B |

3. | J. Arkwright, J. Burke, and M. Gross, “A deterministic optical figure correction technique that preserves precision-polished surface quality,” Opt. Express |

4. | A. Schutze, J. Y. Jeong, S. E. Babayan, and J. Park, “The atmospheric-pressure plasma jet: a review and comparison to other plasma sources,” IEEE Trans. Plasma Sci. |

5. | T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, and A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Meth. A |

6. | Y. Mori, K. Yamauchi, and K. Endo, “Elastic emission machining,” Precis. Eng. |

7. | S. Matsuyama, T. Wakioka, N. Kidani, T. Kimura, H. Mimura, Y. Sano, Y. Nishino, M. Yabashi, K. Tamasaku, T. Ishikawa, and K. Yamauchi, “One-dimensional Wolter optics with a sub-50 nm spatial resolution,” Opt. Lett. |

8. | W. Gao, P. S. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. |

9. | M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. |

10. | R. D. Geckeler, “Optimal use of pentaprism in highly accurate deflectometric scanning,” Meas. Sci. Technol. |

11. | P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2m flat using a pentaprism scanning system,” Opt. Eng. |

12. | F. Siewert, J. Buchheim, S. Boutet, G. J. Williams, P. A. Montanez, J. Krzywinski, and R. Signorato, “Ultra-precise characterization of LCLS hard X-ray focusing mirrors by high resolution slope measuring deflectometry,” Opt. Express |

13. | J. Yellowhair and J. H. Burge, “Analysis of a scanning pentaprism system for measurements of large flat mirrors,” Appl. Opt. |

14. | J. Ojeda-Castañeda, “Foucault, wire and phase modulation tests,” in |

15. | L. Rayleigh, “Interference bands and their applications,” Nature |

16. | B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. |

17. | R. E. Parks, L.-Z. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. |

18. | M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Stuttg.) |

19. | U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. |

20. | V. Greco and G. Molesini, “Micro-temperature effects on absolute flatness test plates,” Pure Appl. Opt. |

21. | V. Greco, R. Tronconi, C. Del Vecchio, M. Trivi, and G. Molesini, “Absolute measurement of planarity with Fritz’s method: uncertainty evaluation,” Appl. Opt. |

22. | L. Zhang, B. Xuan, and J. Xie, “Combination of skip-flat test with Ritchey-Common test for the large rectangular flat,” Proc. SPIE |

23. | P. Hariharan, “Interferometric testing of optical surfaces: absolute measurements of flatness,” Opt. Eng. |

24. | D. Malacara, “Twyman-Green interferometer,” in |

25. | Z. Han, L. Chen, T. Wulan, and R. Zhu, “The absolute flatness measurements of two aluminum coated mirrors based on the skip flat test,” Optik (Stuttg.) |

26. | M. Vannoni, A. Sordini, and G. Molesini, “Calibration of absolute planarity flats: generalized iterative approach,” Opt. Eng. |

27. | V. B. Gubin and V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. |

28. | M. Vannoni and G. Molesini, “Iterative algorithm for three flat test,” Opt. Express |

29. | M. Vannoni and G. Molesini, “Absolute planarity with three-flat test: an iterative approach with Zernike polynomials,” Opt. Express |

30. | M. Vannoni and G. Molesini, “Three-flat test with plates in horizontal posture,” Appl. Opt. |

31. | C. Morin and S. Bouillet, “Absolute calibration of three reference flats based on an iterative algorithm: study and implementation,” Proc. Soc. Photo Opt. Instrum. Eng. |

32. | M. Vannoni, A. Sordini, and G. Molesini, “Long-term deformation at room temperature observed in fused silica,” Opt. Express |

33. | M. Vannoni, A. Sordini, and G. Molesini, “Relaxation time and viscosity of fused silica glass at room temperature,” Eur Phys J E Soft Matter |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.4800) Instrumentation, measurement, and metrology : Optical standards and testing

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: November 6, 2013

Revised Manuscript: January 9, 2014

Manuscript Accepted: January 10, 2014

Published: February 6, 2014

**Citation**

Maurizio Vannoni, "Absolute flatness measurement using oblique incidence setup and an iterative algorithm. A demonstration on synthetic data," Opt. Express **22**, 3538-3546 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3538

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### References

- M. Zeuner, S. Kiontke, “Ion beam figuring technology in optics manufacturing,” Optik & Photonik 7(2), 56–58 (2012).
- M. Weiser, “Ion beam figuring for lithography optics,” Nucl. Instrum. Methods Phys. Res. B 267(8-9), 1390–1393 (2009).
- J. Arkwright, J. Burke, M. Gross, “A deterministic optical figure correction technique that preserves precision-polished surface quality,” Opt. Express 16(18), 13901–13907 (2008).
- A. Schutze, J. Y. Jeong, S. E. Babayan, J. Park, “The atmospheric-pressure plasma jet: a review and comparison to other plasma sources,” IEEE Trans. Plasma Sci. 26(6), 1685–1694 (1998).
- T. Arnold, G. Böhm, R. Fechner, J. Meister, A. Nickel, F. Frost, T. Hänsel, A. Schindler, “Ultra-precision surface finishing by ion beam and plasma jet techniques—status and outlook,” Nucl. Instrum. Meth. A 616(2-3), 147–156 (2010).
- Y. Mori, K. Yamauchi, K. Endo, “Elastic emission machining,” Precis. Eng. 9(3), 123–128 (1987).
- S. Matsuyama, T. Wakioka, N. Kidani, T. Kimura, H. Mimura, Y. Sano, Y. Nishino, M. Yabashi, K. Tamasaku, T. Ishikawa, K. Yamauchi, “One-dimensional Wolter optics with a sub-50 nm spatial resolution,” Opt. Lett. 35(21), 3583–3585 (2010).
- W. Gao, P. S. Huang, T. Yamada, S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26(4), 396–404 (2002).
- M. Schulz, C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. 45(6), 060503 (2006).
- R. D. Geckeler, “Optimal use of pentaprism in highly accurate deflectometric scanning,” Meas. Sci. Technol. 18(1), 115–125 (2007).
- P. C. V. Mallik, C. Zhao, J. H. Burge, “Measurement of a 2m flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).
- F. Siewert, J. Buchheim, S. Boutet, G. J. Williams, P. A. Montanez, J. Krzywinski, R. Signorato, “Ultra-precise characterization of LCLS hard X-ray focusing mirrors by high resolution slope measuring deflectometry,” Opt. Express 20(4), 4525–4536 (2012).
- J. Yellowhair, J. H. Burge, “Analysis of a scanning pentaprism system for measurements of large flat mirrors,” Appl. Opt. 46(35), 8466–8474 (2007).
- J. Ojeda-Castañeda, “Foucault, wire and phase modulation tests,” in Optical Shop Testing, third edn., D. Malacara ed., (Wiley and Sons, Hoboken 2007), pp. 310–312.
- L. Rayleigh, “Interference bands and their applications,” Nature 48(1235), 212–214 (1893).
- B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23(4), 234379 (1984).
- R. E. Parks, L.-Z. Shao, C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37(25), 5951–5956 (1998).
- M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Stuttg.) 112(9), 381–391 (2001).
- U. Griesmann, Q. Wang, J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46(9), 093601 (2007).
- V. Greco, G. Molesini, “Micro-temperature effects on absolute flatness test plates,” Pure Appl. Opt. 7(6), 1341–1346 (1998).
- V. Greco, R. Tronconi, C. Del Vecchio, M. Trivi, G. Molesini, “Absolute measurement of planarity with Fritz’s method: uncertainty evaluation,” Appl. Opt. 38(10), 2018–2027 (1999).
- L. Zhang, B. Xuan, J. Xie, “Combination of skip-flat test with Ritchey-Common test for the large rectangular flat,” Proc. SPIE 7656, 76564W (2010).
- P. Hariharan, “Interferometric testing of optical surfaces: absolute measurements of flatness,” Opt. Eng. 36(9), 2478–2481 (1997).
- D. Malacara, “Twyman-Green interferometer,” in Optical Shop Testing, third edn., D. Malacara ed., (Wiley and Sons, Hoboken 2007), pp. 78–79.
- Z. Han, L. Chen, T. Wulan, R. Zhu, “The absolute flatness measurements of two aluminum coated mirrors based on the skip flat test,” Optik (Stuttg.) 124(19), 3781–3785 (2013).
- M. Vannoni, A. Sordini, G. Molesini, “Calibration of absolute planarity flats: generalized iterative approach,” Opt. Eng. 51(8), 081510 (2012).
- V. B. Gubin, V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147–148 (1990).
- M. Vannoni, G. Molesini, “Iterative algorithm for three flat test,” Opt. Express 15(11), 6809–6816 (2007).
- M. Vannoni, G. Molesini, “Absolute planarity with three-flat test: an iterative approach with Zernike polynomials,” Opt. Express 16(1), 340–354 (2008).
- M. Vannoni, G. Molesini, “Three-flat test with plates in horizontal posture,” Appl. Opt. 47(12), 2133–2145 (2008).
- C. Morin, S. Bouillet, “Absolute calibration of three reference flats based on an iterative algorithm: study and implementation,” Proc. Soc. Photo Opt. Instrum. Eng. 8169, 816915 (2011).
- M. Vannoni, A. Sordini, G. Molesini, “Long-term deformation at room temperature observed in fused silica,” Opt. Express 18(5), 5114–5123 (2010).
- M. Vannoni, A. Sordini, G. Molesini, “Relaxation time and viscosity of fused silica glass at room temperature,” Eur Phys J E Soft Matter 34(9), 92 (2011).

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