**4. Methodology and Results**

In order to perform the measurements presented in this work, the GAMOR methodology has been used. Although this methodology has previously been described in the literature [**???** ], the following two sections, Sections **??** and **??**, give a brief overview of its principles. The latter one emphasizes how the methodology can be used to obtain pressure assessments in seconds. Section **??** provides a characterization of the two GAMOR-based

DFPC refractometry systems used, and Section **??** gives an example of an assessment. Finally, Section **??** presents the results of a series of assessments.

#### *4.1. Conventional Realization of the GAMOR Methodology*

As has been indicated previously [**???** ], the GAMOR methodology is based on two cornerstones: viz., (i) frequent referencing of filled measurement cavity beat frequencies to evacuated cavity beat frequencies, and (ii) an assessment of the evacuated measurement cavity beat frequency at the time of the assessment of the filled measurement cavity beat frequency by use of an interpolation between two evacuated measurement cavity beat frequency assessments, one performed before and one after the filled cavity assessments. The principles for the methodology when campaign-persistent drifts take place are schematically illustrated in Figure **??**.

Figure **??**a illustrates the pressure in the measurement cavity, which, according to cornerstone (i), is alternately evacuated and filled with gas (upper red curve), and the reference cavity is held at a constant pressure (lower blue curve). Campaign-persistent drifts will affect the frequencies of both the measurement and the reference lasers (although possibly to dissimilar extent, as shown in Figure **??**b) and thereby both the assessed beat frequency, *f*(*t*), and its unwrapped counterpart, *fUW*(*t*) (the latter displayed by the upper black curve in Figure **??**c). These curves indicate that the influence of drifts can be reduced by shortening the modulation cycle period; for a given drift rate, the shorter the gas modulation period, the less the assessed beat frequency will be affected by drifts [**?** ].

Furthermore, according to cornerstone (ii), the unwrapped evacuated measurement cavity beat frequency is, for each modulation cycle, not assessed by a single measurement. It is instead estimated by the use of a linear interpolation between two evacuated (unwrapped) measurement cavity beat frequency assessments performed in rapid succession—one taken directly prior to when the measurement cavity is filled with gas (for cycle *n*, at a time *tn*, denoted *f* (0) *UW*(*tn*)), and another directly after it has been evacuated (at a time *tn*+1, denoted *f* (0) *UW*(*tn*+1)), both marked by crosses in Figure **??**c. By this, the unwrapped evacuated measurement cavity beat frequency, ˜ *f* (0) *UW*(*tn*, *t*, *tn*+1), can be estimated at all times *t* during a modulation cycle. For cycle *n*, for which *tn* ≤ *t* ≤ *tn*+1, it is estimated as

$$
\tilde{f}\_{\rm LW}^{(0)}(t\_{n\prime}t\_{\prime}t\_{n+1}) = f\_{\rm LW}^{(0)}(t\_{n}) + \frac{f\_{\rm LW}^{(0)}(t\_{n+1}) - f\_{\rm LW}^{(0)}(t\_{n})}{t\_{n+1} - t\_{n}}(t - t\_{n}).\tag{6}
$$

For the case with campaign-persistent drifts, this interpolated value is represented by the green line in Figure **??**c.

By subtracting the estimated (interpolated) unwrapped evacuated measurement cavity beat frequency ( ¯ *f* (0) *UW*(*tn*, *t*, *tn*+1), the green line) from the measured (drift-influenced) unwrapped beat frequency during gas filling (*fUW*(*t*), the black curve), both in Figure **??**c, a campaign-persistent, drift-corrected net beat frequency, represented by the black curve in Figure **??**d, can be obtained. The average value of this curve a short time period just before the cavity is evacuated, at a time denoted *tg*, represents the Δ*fUW* to be used in the Equation (2) when GAMOR is performed. This shows that it is feasible to interpret GAMOR as "interpolated gas modulated refractometry."

**Figure 4.** A schematic illustration of the principles of GAMOR implemented on a system exposed to campaign-persistent drifts. Panel (**a**) shows, as functions of time, by the the upper red curve, *Pm*(*t*), the pressure in the measurement cavity, and by the lower blue curve, the pressure in the reference cavity, *Pr*(*t*). Panel (**b**) depicts the corresponding frequencies of the measurement and reference lasers, *νm*(*t*) (the lower red curve) and *νr*(*t*) (the upper blue curve), respectively, for display purposes, both offset by a common frequency. Panel (**c**) displays, by the upper black curve, the corresponding unwrapped beat frequency in the presence of gas, *fUW*(*t*), and the lower green line depicts the estimated evacuated measurement cavity beat frequency, ˜ *f* (0) *UW*(*tn*, *t*, *tn*+1). Panel (**d**) displays the drift-corrected shift in unwrapped beat frequency, Δ*fUW*(*t*). While the data that are used in ordinary GAMOR constitute the data points in the last part (10–20%) of section *I* (the time period between *tn* and slightly after *tg*) in panel (**d**), which are averaged to a single data value, in this work where cycle resolved assessments are performed, a significant part (ca. 80%) of the data in section *I* is used in an unaveraged manner. Note that the drifts have been greatly exaggerated for clarity.

#### *4.2. Use of GAMOR to Assess Short-Term Pressure Fluctuations*

GAMOR refractometry has so far been used to assess static pressures through the use of (and averaging over) a series of gas modulation cycles. It has been shown, for example, that, for the case with an Invar-based DFPC system, a minimum deviation could be achieved when averaging was performed over ten modulation cycles (i.e., over 103 s) [**?** ]. Such a mode of operation is suitable when static pressures (or slowly varying pressures, those that change slowly over time intervals corresponding to several gas modulation periods) are to be assessed. In such a case, the methodology first calculates a single pressure value for each individual gas modulation cycle (as schematically described in Figure **??**) and then takes the average over *n* such cycles. For the case when the instrumentation is mainly affected by white noise, this process will improve on the precision (decrease the influence of noise) by a factor of *n*<sup>−</sup>1/2.

The GAMOR methodology can though also be used for assessing short-term fluctuations of pressure. In this case, the assessment of the pressure, *P*(*t*), is continuously carried out from the shift in the incessantly assessed unwrapped beat frequency, Δ*fUW*(*t*), during individual modulation cycles. A calculation of the cycle resolved pressure began by assessing, for each refractometer, from the beat frequency, *f*(*t*), and the shifts in cavity mode numbers Δ*q*1(*t*) and Δ*q*2(*t*), the unwrapped (i.e., the mode-jump-corrected) beat frequency, *fUW*(*t*). The beat frequency was continuously sampled by a frequency counter with a readout rate of 4 Hz. The shift in cavity mode number Δ*qi*(*t*) was calculated as the nearest integer to [(*Vi*(*t*) − *V*0,*i*)/*VFSR*,*<sup>i</sup>* + *q*0,*i*(*Pcav*,*i*(*t*)/*P*0)(*n*<sup>0</sup> − 1)], where *Vi*(*t*) is the voltage sent to the tuning control of laser *i* (acting on its piezo stretcher), *V*0,*<sup>i</sup>* is the voltage for an empty cavity locked to mode *q*0,*i*, *VFSR*,*<sup>i</sup>* is the voltage required to tune the laser FSR, *Pcav*,*i*(*t*) is the pressure in the cavity *i*, and *n*<sup>0</sup> − 1 is the refractivity for the gas addressed that corresponds to the pressure *P*<sup>0</sup> (here taken as 10<sup>5</sup> Pa). The pressure *Pcav*,*i*(*t*) is assessed by the use of pressure gauge *GS*/*T*.1. Hence, by continuously measuring the voltages sent to the lasers, all information needed to calculate the changes of the cavity mode numbers, Δ*q*1(*t*) and Δ*q*2(*t*), is available at all times. Using this information and Equation (1) it is possible to calculate the unwrapped beat frequency, *fUW*(*t*), at all times during a modulation cycle.

### *4.3. System Characterizations*

Prior to the measurements, the two refractometers were first individually characterized by assessing the cavity deformations by the use of the methodology presented in [**?** ]. The results are presented in detail in [**?** ]. It was found that the cavity deformation parameters, *εm*, for the SOP and TOP when assessing nitrogen, were 0.001972(1) and 0.001927(1). Since (*n* − 1) ∝ (1 − *εm*), the measurement uncertainty in the cavity deformations will solely contribute to the total expanded uncertainty in pressure (k = 2) with 1 ppm. Furthermore, using a thorough evaluation, the two refractometers were attributed expanded uncertainties (k = 2) for assessment of pressure of nitrogen, of ((10 mPa)2 + (10 × <sup>10</sup>−6*P*)2) 1/2 for the SOP and ((16 mPa)2 + (28 × <sup>10</sup>−6*P*)2)1/2 for the TOP [**?** ]. It was found that while the SOP is predominantly limited by the uncertainty in the molar polarizability of nitrogen (8 ppm), the accuracy of the TOP is limited by the uncertainty of the temperature probes used for the temperature assessment (26 ppm). It should be noticed though, that both refractometers had smaller evaluated uncertainties than that of the DWPG, which was assessed to be ((60 mPa)2 + (41 × <sup>10</sup>−6*P*)2) 1/2.
