Wavefront aberration of non-round ²⁸Si-spheres

For the determination of the volume to realize the kilogram according to the 2019 redefinition, silicon spheres with the smallest possible form deviation (PV < 100 nm) were used. PV stands for peak-to-valley and describes the maximum value from the highest peak to the lowest valley of the sphere.

To achieve the required measurement uncertainty of the volume of less than 1x10^-8, the largest contributions in the measurement uncertainty budget were investigated in detail. The dominant contribution to date was the uncertainty of the wavefront, which became accurately describable using optical simulations and could therefore be drastically reduced. Since then, a large number of simulations with spheres of different form deviation have been performed and the simulated data of a disturbed wavefront were compared with the corresponding reference data set. Thus, the contribution to the uncertainty budget could be calculated, but not yet experimentally verified. One result of the investigations is that the influence of the wavefront of very round spheres with a few 1x10^-10 is so small that it cannot be resolved by measurement. Nevertheless, the disturbance of the wavefront has a non-negligible contribution in the measurement uncertainty budget.

If the topography of the sphere is considered, the uncertainty can also increase significantly locally. Especially at the peak-valley transition, the reflected wavefront deviates more from the initial wavefront than the one hitting the extrema. If the deviation from the ideal spherical form increases, the influence due to the disturbed wavefront may even exceed the total uncertainty of a typical sphere. For spheres with large topographic gradient (PV > 350 nm), the effect due to wavefront aberration reaches the range of measurable resolution.

The reason why the wavefront effect could only be determined with optical simulations so far is due to the missing prerequisite that for the experimental investigation of the wavefront aberration spheres from the same crystal as well as the same surface properties, but very different roundness deviation are needed. However, since this year two ²⁸Si-spheres are available which make such an investigation possible. In addition to a sphere with typical PV values between 20 and 30 nm, which is of the usual high quality, it has now been possible to produce a sphere with the same surface properties (polishing time and fineness), which can be expected to have a significantly larger roundness deviation. Initial measurements in the sphere interferometer showed a PV value of about 540 nm (Fig. 1).

Based on the present two different spheres from the same crystal, it is possible to directly compare their density (measurements of working group 1.13). In a second measurement the masses of the spheres can be determined (measurements of working group 1.11). By means of an XRF/XPS measurement (by working group 1.11 and 7.22), the surfaces of the spheres, here in particular the water layer, a remaining hydrocarbon layer and the oxide layer on the silicon, can be determined and used to correct the three measured quantities density, mass and volume. If the measured values are then inserted into the physical equation ρ=m/V, there should be a difference for a sphere with significant deviation from roundness, but the magnitude of this difference should correspond to the result of the simulation. An initial estimate using optical simulations shows a vanishingly small wavefront aberration of 2.5 pm for the very round sphere (sphere 19-72 with PV = 29 nm). For the non-round sphere (19-73 with PV = 540 nm), the wavefront aberration increases to 0.7 nm. In terms of volume, the expected wavefront aberration is 4x10^-8 ·V. Therefore, the demands on all measurement quantities are high and require best possible results in order to estimate this uncertainty contribution in the volume measurement more accurately. Especially for spheres with larger roundness deviations a distinct reduction of measurement uncertainty is expected.

   
Fig. 1: Topography of the non-round ²⁸Si-sphere 19-73 with PV = 540 nm.