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.

Due to the introduction of double patterning schemes in the production processes of the semiconductor industry and the corresponding stringent specifications of the placement errors of the optical images of mask features (1.2 nm (3σ)) in the lithographic projection process, significant improvements of the registration measurements on photomasks are required. The Nanometer Comparator of the PTB, a one-dimensional vacuum-interference length comparator, has already demonstrated measurement reproducibility at this level on line scales and linear encoders, and therefore this comparator should also be applied for placement metrology on photomasks. Because a photomask is a relatively thin plate which is supported almost at its ends, the bending influence is much larger than for line scales and precautions had to be taken to reduce the influence of the sample support on the measurements results. A new sample mount was built, which exhibits mechanical alignment aids. In addition the position deviations caused by the mask bending were corrected by means of finite element calculations. Measurements performed with the Nanometer Comparator on a high-quality mask showed a maximum deviation of measured feature positions from the common mean of below 0.5 nm and a reproducibility of 0.2 nm. These results suggest that the use of a one-dimensional calibration of mask measuring machines allows us to perform traceable registration measurements with sufficient accuracy.