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Form measurement of curved optical surfaces

Working Group 8.42


Sub-aperture interferometry

Modern optical systems consist of increasingly complex optical surfaces, as required, e.g., in lithography, for synchrotron optics or even consumer camera and cell phone camera objectives.

Aspheric synchrotron optic at BESSY ( Berliner Elektronenspeicherring-Gesellschaft für Synchrotronstrahlung m.b.H.).

Fig. 1: Aspherical synchrotron optics at BESSY ( Berliner Elektronenspeicherring-Gesellschaft für Synchrotronstrahlung m.b.H.).


Testing the form of a complex optical surface within the required high accuracy is challenging and the development of corresponding measurement principles is a task of ongoing research. For many applications profile measurements are sufficient. Due to their contactless nature, optical measurement techniques are the method of choice for optical surfaces. The highest accuracy can be reached with interferometric techniques. While full aperture interferometers can be successfully employed for the highly accurate testing of planes and spherical surfaces, they are no longer applicable for more complex surfaces. The reason is that interferometric techniques are limited by the maximum relative tilt between the interferometer reference surface and the corresponding (part of the) surface of the specimen. Therefore, interferometers with small apertures and higher resolutions are employed for complex surfaces. These small interferometers measure only a (possibly small) part of the surface of the test specimen. In order to reconstruct the whole surface, the small interferometer has to be moved over the specimen, thereby recording many sub-surfaces at different positions. These sub-surfaces are combined to reconstruct the whole surface by stitching techniques. The difficulty that arises with the application of these techniques is that even small (unavoidable) systematic errors of the small interferometer cumulate, and the resulting maximum reconstruction error can even be orders of magnitude larger than the systematic error of the small interferometer when the number of sub-surfaces is large.

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Math-based topography reconstruction

The cumulation of the systematic interferometer errors during stitching can be avoided when the tilt of the interferometer is measured in addition. The combination of the sub-surfaces, the influence of the individual scanning stage errors as well as the systematic interferometer errors can be described in terms of linear, discrete models. Assuming that the measurements of the sub-surfaces are carried out according to a suitable design of the experiment, a model-based analysis of the data then yields profiles of the whole surface (uniquely up to straight lines), while simultaneously scanning stage errors and systematic interferometer errors are taken into account.

Comparison of common stitching techniques and TMS for simulated data.

Fig. 2: Comparison of common stitching techniques and TMS for simulated data.


Since the interferometer no longer needs to be calibrated prior to the measurement, this measurement principle also allows the calibration of interferometers without using a (known) reference surface (Fig. 3). We call this measurement system Traceable Multi-Sensor system (TMS), since the topography measurement can be traced back to angle and length measurements.

Systematic sensor error of an interferometer determined by TMS from 20 measurements.
Fig. 3 : Systematic sensor error of an interferometer determined by TMS from 20 measurements (blue lines). The orange line shows the average of the 20 curves. The reproducibility is better than 0.5 nm.
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Virtual experiments: 3D simulation environment

The design and assessment of new optical surface measurement devices is significantly eased by using virtual experiments. In a virtual experiment, the measurement process is modeled mathematically and simulated on a computer, thereby allowing a quantitative sensitivity analysis of the possible error sources such as interferometer measurement errors, positioning errors of the scanning stage, assumptions of mathematical reconstruction procedures, etc. PTB Working Group 8.42 has developed a flexible 3D simulation environment which is used to test and assess different possible geometries and measurement principles. The simulation environment accounts in particular for the interaction of all axis movements and the different measurement devices.

Virtual 3D optical surface measurements.

Fig. 4: Virtual 3D optical surface measurements.


Virtual experiments are also helpful for the development and improvement of reconstruction algorithms since they allow reconstruction errors to be assessed realistically. Figure 5 shows in comparison resulting reconstruction errors for the TMS procedure and a recently improved variant which utilizes additional lateral positioning measurements.

Reconstruction error for TMS and for the improved TMS algorithm.

Fig. 5: Reconstruction error for TMS (left) and for the improved TMS algorithm (right). For the simulations a sinusoidal topography with a PV (peak to valley) of 200 nm and wavelength λ was used; "pos. unc." characterizes the accuracy of the scanning stage.
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The development of new methods for the form measurement of curved optical surfaces is undertaken jointly by Working Group Opens internal link in current window4.24 (Asphere Metrology) and Opens internal link in current windowWorking Group 4.21 (Form and Wavefront Metrology).

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Publication single view


Title: Reconstructing surface profiles from curvature measurements
Author(s): C. Elster, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz;I. Weingärtner
Journal: Optik - International Journal for Light and Electron Optics
Year: 2002
Volume: 113
Issue: 4
Pages: 154 - 158
DOI: 10.1078/0030-4026-00138
ISSN: 0030-4026
Web URL: http://www.sciencedirect.com/science/article/pii/S0030402604701345
Keywords: Runge-Kutta method
Tags: 8.42, Form
Abstract: Summary Recently, the measurement of curvature has been suggested as a promising new technique for the highly accurate determination of large-area surface profiles on the nanometer scale. It was shown that the curvature can be measured with highest accuracy and high lateral resolution. However, the reconstruction of surface profiles from curvature data involves the numerical solution of an ordinary differential equation for which initial or boundary values must be specified. This paper investigates the accuracy with which surface profiles can be reconstructed from curvature data. The stability of the reconstructions is examined with respect to the presence of measurement noise and the accuracy of the initial values. The assessment of the reconstruction accuracy is based on an analytical solution (up to numerical integration) derived for the case when the measurement results are given in Cartesian coordinates, and on numerical results in the polar case. The results presented for the latter case allow, in particular, conclusions to be drawn regarding the minimum accuracy of data and initial values required for reconstructing aspheres from curvature measurements with nanometer accuracy.

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