*2.1. Refractivity*

As has previously been outlined [**?** ], each DFPC refractometer addresses the empty cavity mode *q*<sup>01</sup> or *q*<sup>02</sup> with light of frequency *ν*<sup>01</sup> or *ν*02, respectively. The beat frequency between the two lasers, *f* , which is the measured entity, is given by the difference between the two laser frequencies, defined as | *ν*<sup>1</sup> − *ν*<sup>2</sup> |. Since the lasers have a limited tuning range, automatic mode jumps will take place when the change in pressure becomes large. This implies that *f* is a non-monotonic (i.e., a wrapped) function. It is therefore convenient to define an unwrapped beat frequency as

$$f\_{\rm LW} = \pm f - \left(\frac{\Delta q\_1}{q\_{01}} \nu\_{01} - \frac{\Delta q\_2}{q\_{02}} \nu\_{02}\right),\tag{1}$$

where Δ*q*<sup>1</sup> and Δ*q*2, counted from *q*<sup>01</sup> and *q*02, are the mode jumps and where the ± sign refers to the cases when *ν*<sup>1</sup> > *ν*<sup>2</sup> and *ν*<sup>1</sup> < *ν*2.

The refractivity can then be expressed as a function of the shift of the unwrapped beat frequency when gas is let out of (or into) the measurement cavity, Δ*fUW*. As has been shown recently [**?** ], while denoting the measurement cavity as *m*, the refractivity can be expressed as a function of the unwrapped beat frequency when GAMOR is used as

$$n - 1 = \frac{|\,\Delta f\_{lIN}\,\,|\,\,\upsilon\_{0m}$$

$$1 - |\,\,\Delta f\_{lIN}\,\,|\,\,\upsilon\_{0m} + \Delta \eta\_{m} \,/\eta\_{0m} + \varepsilon\_{m}\,'\,}\,\tag{2}$$

where *ε<sup>m</sup>* is a deformation parameter comprising the refractivity-normalized relative difference in lengths of the two cavities due to pressurization, given by [(*δL*/*L*0)*<sup>m</sup>* − (*δL*/*L*0)*r*]/(*n* − 1), where (*δL*/*L*0)*<sup>m</sup>* and (*δL*/*L*0)*<sup>r</sup>* are the relative changes in length of the measurement and reference cavities when the measurement cavity is pressurized [**???** ]. It is worth noting that *ε<sup>m</sup>* can be assessed with high accuracy by a methodology developed by Zakrisson et al. [**?** ].

In Equation (2) the influences of the mirror dispersion and the finite penetration depth of the mirrors have been neglected. The former since the systems in this work use light in the communication band (around 1.55 μm), for which there are mirrors with a minimum of (linear) dispersion. The latter since the effect is smaller than the uncertainty of the molar polarizability of the gas [**? ?** ].
