Six European National Measurement Institutes (NMIs) have joined forces within the European Metrology Research Programme funded project NANOTRACE to develop the next generation of optical interferometers having a target uncertainty of 10 pm. These are needed for NMIs to provide improved traceable dimensional metrology that can be disseminated to the wider nanotechnology community, thereby supporting the growth in nanotechnology. Several approaches were followed in order to develop the interferometers. This paper briefly describes the different interferometers developed by the various partners and presents the results of a comparison of performance of the optical interferometers using an x-ray interferometer to generate traceable reference displacements.

The PTB developed a new optical heterodyne interferometer in the context of the European joint research project ‘Nanotrace’. A new optical concept using plane-parallel plates and spatially separated input beams to minimize the periodic nonlinearities was realized. Furthermore, the interferometer has the resolution of a double-path interferometer, compensates for possible angle variations between the mirrors and the interferometer optics and offers a minimal path difference between the reference and the measurement arm. Additionally, a new heterodyne phase evaluation based on an analogue to digital converter board with embedded field programmable gate arrays was developed, providing a high-resolving capability in the single-digit picometre range. The nonlinearities were characterized by a comparison with an x-ray interferometer, over a measurement range of 2.2 periods of the optical interferometer. Assuming an error-free x-ray interferometer, the nonlinearities are considered to be the deviation of the measured displacement from a best-fit line. For the proposed interferometer, nonlinearities smaller than ±10 pm were observed without any quadrature fringe correction.

In order to be able to resolve displacements of a picometer with widely used commercially available heterodyne interferometers, an advanced phase meter was developed at PTB. Key to this level of accuracy is the use of a state-of-the-art analogue-to-digital converter (ADC) board enabling the implementation of a phase-evaluation method by using embedded field programmable gate arrays. Experimental results obtained with commercially available heterodyne laser interferometer components prove that the proposed phase-evaluation procedure is capable of interpolating an optical fringe down into the picometer regime. The phase evaluation was moreover extended to track simultaneously two heterodyne beat frequencies with only two photodetectors and ADCs. Potential limitations of the long-term stability of heterodyne interferometers are discussed. The phase meter was tested, has been readily applied, can be easily adapted and is therefore to be used in a wide field of applications.

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We revisited the treatment of the influence of the support conditions and the resulting bending of a scale on the positions of its graduation lines. In contrast to earlier publications, we did not calculate the position deviation with respect to the case without any bending. Due to the production processes used today, this is inappropriate and leads to an overestimation of the related uncertainty contribution. We also extended the treatment from two lines to all lines of the scale and included a finite starting point of the graduation. We verified our analytical model by means of FEM calculations. In addition, we showed that a first order Taylor expansion yields sufficiently accurate results for the position deviations and leads to simple equations for their size. Because line scale measurements are related to the zero line the constant term of the Taylor expansion cancels and the remaining coefficient is identical to the sensitivity coefficient required in the determination of the standard uncertainty contribution. If it is sufficient to suppress the position dependence and to consider only the case where the sample is supported at symmetric positions, then a new, simple equation is obtained for the resulting uncertainty contribution. Finally we showed that due to the position dependence of the deviations the scale coefficient of the scale, which is obtained by a linear fit to the deviations of the line positions of the scale from their nominal values, is also influenced, which has apparently not been noticed up to now. If the line standard used to disseminate the unit of length has not been designed carefully then the resulting change in the scale coefficient exposes a practical limit to the related achievable length-dependent uncertainty contribution.