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Form and Wavefront Metrology

Working Group 4.21

Form measurement of curved optical surfaces

Sub-aperture interferometry

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

Fig. 1: Aspheric synchrotron optic at Helmholtz-Zentrum Berlin für Materialien und Energie GmbH

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. Interferometric techniques offer a high sensitivity and the advantage to be directly traced back to the wavelength of the used Laser. While full aperture interferometers can be successfully employed for the highly accurate testing of plane and spherical surfaces, they are no longer applicable for more complex surfaces.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 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. In order to reconstruct the whole surface, the small interferometer has to be moved over the specimen, thereby conducting many sub-surfaces at different positions. These sub-surfaces are combined to a reconstruction of 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 magnitudes larger than the systematic error of the small interferometer when the number of sub-surfaces is large (cf. Fig. 2, left).

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 the measurements of the sub-surfaces to be carried out according to a suitable design of experiment, a model-based analysis of the data then yields profiles of the surface [1-3] (uniquely up to straight lines), while simultaneously scanning stage errors and systematic interferometer errors are accounted for (cf. Fig. 2, right).

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

Since the interferometer needs no longer be calibrated prior to the measurement, this measurement principle also allows for the calibration of interferometers without using a (known) reference surface [4] (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.

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

Virtual experiments: 3D Simulation environment

Design and assessment of new optical surface measurement devices is significantly eased by using virtual experiments. In a virtual experiment, the measurement process is modelled mathematically and carried out on a computer, thereby allowing for 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. [5]. We developed a flexible 3D simulation environment which is used to test and assess different possible designs. The simulation environment accounts in particular for the interaction of all axis movements and the different measurement devices (Fig. 4).

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 [6] which utilizes additional lateral positioning measurements. The virtual experiments can also be used to optimize the sensor spacing for improving the lateral resolution [7].

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.