Measurement of the refractive index of air in a low-pressure regime and the applicability of traditional empirical formulae

The refractive index of air is a major limiting factor in length measurements by interferometry, which are mostly performed under atmospheric conditions. Therefore, especially in the last century, measurement and description of the air refractive index was a key point in order to achieve accuracy in the realisation of the length by interferometry. Nevertheless, interferometric length measurements performed in vacuum are much more accurate since the wavelength of the light is not affected by the air refractive index. However, compared with thermal conditions in air, in high vacuum heat conduction is missing. In such a situation, dependent on the radiative thermal equilibrium, a temperature distribution can be very inhomogeneous. Using a so-called contact gas instead of high vacuum is a very effective way to enable heat conduction on nearly the same level as under atmospheric pressure conditions whereby keeping the effect of the air refractive index on a small level. As physics predicts, and as we have demonstrated previously, helium seems like the optimal contact gas because of its large heat conduction and its refractive index that can be calculated from precisely known parameters. On the other hand, helium gas situated in a vacuum chamber could easily be contaminated, e.g. by air leakage from outside. Above the boiling point of oxygen (-183 degrees C) it is therefore beneficial to use dry air as a contact gas. In such an approach, the air refractive index could be calculated based on measured quantities for pressure and temperature. However, existing formulas for the air refractive index are not valid in the low-pressure regime. Although it seems reasonable that the refractivity (n-1) of dry air simply downscales with the pressure, to our knowledge there is no experimental evidence for the applicability of any empirical formula. This evidence is given in the present paper which reports on highly accurate measurements of the air refractive index n for the wavelengths 532 nm, 633 nm and 780 nm in the low-pressure regime from 0 Pa to 1300 Pa. In our approach, using a vacuum cell, n-1 is obtained from the comparison of optical path lengths in vacuum and air along the same path by imaging interferometry. These measured values are compared with the ones obtained from Bonsch's formula. An agreement of +/- 10(-9) is found in the low-pressure regime. Accordingly, this formula could be applied for the accurate determination of the refractive index of dry air even at low pressures, provided that the pressure is measured with high accuracy.