**1. Introduction**

Refractometry is a powerful technique for assessing gas pressure. It is based upon measuring, by optical means, the change in refractive index in a measurement compartment as gas is let into it. From the change in refractive index, under the condition that the molar polarizability (and higher order refractive virial coefficients) of the gas is known, the change in gas density can be calculated. From this, provided that the gas temperature is known, the pressure can be assessed by utilizing an equation of state. Moreover, since the Boltzmann constant was given a fixed value (i.e., without uncertainty) in the 2019 revision of the SI system of units [**?** ], refractometry also offers a new and independent route to realizing the SI unit of pressure, i.e., the Pascal [**?** ]. These exciting prospects have spurred a significant increase of interest within the field of refractometry. Work to explore and utilize the potential of optical methods for assessing the molar density and pressure of gas presently takes place at several national metrology institutes and universities [**??????? ?** ].

Besides being a potential primary method for measuring the Pascal, the technology also has several other highly interesting properties and advantages. As optical measurements do not utilize any mechanical actuators, the highest pressures that can be measured tend to be limited by the gas handling system used. The lowest pressure shifts that can be resolved are in turn limited by the laser locking. In practical terms, the dynamic range can be as high as eight orders of magnitude, typically covering the range from 1 mPa to 100 kPa. As optical measurements are performed by measuring changes in frequency, which can

**Citation:** Forssén, C.; Silander, I.; Zakrisson, J.; Axner, O.; Zelan, M. Short-Term Performances of Two Independent Gas Modulated Refractometers for Pressure Assessments. *Sensors* **2021**, *21*, 6272. https://doi.org/10.3390/s21186272

Academic Editors: Krzysztof M. Abramski and Piotr Jaworski

Received: 28 July 2021 Accepted: 13 September 2021 Published: 18 September 2021

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be measured swiftly and continuously with high accuracy, these systems can also be designed to give measurements of pressure with high time resolution. In practice, while the long-term performance is given by the stability of the cavity spacer, the time resolution and short-term performance are given by the acquisition rate and stability of the frequency counter. This combination of an extraordinary large dynamic range and a fast response facilitate accurate assessments of large pressure shifts with short settling times. Such measurements can be used to characterize pressure sensors, resolve the differences between sensor responses and actual pressure changes, study rapidly changing pressures, and investigate processes giving rise to such.

The best performing refractometers are based on Fabry–Perot (FP) cavities, where a laser is used to probe the frequency of a longitudinal cavity mode [**???????? ???** ]. By measuring the change in frequency between an empty (evacuated) and a gas-filled cavity, the refractivity can be assessed, from which the molar density and the pressure can be calculated. However, since such measurements cannot distinguish changes in refractivity of a gas from drifts in the physical length of the cavity, the most sophisticated FP refractometers utilize a dual FP cavity (DFPC) design in which one cavity acts as the measurement cavity and the other as a reference. This eliminates common-mode drifts in the cavity spacer caused by aging, temperature drifts, mechanical stress, etc. However, since the two cavities in a DFPC can drift dissimilarly, extraordinarily stable conditions are still required to achieve optimal performance. For a 15 cm long cavity, a drift in length of 1 pm gives rise to a shift in the assessed pressure of nitrogen of 2.5 mPa. As a means to remedy this, we developed a measurement methodology denoted gas modulation refractometry (GAMOR) [**?????** ].

The GAMOR methodology is based on repeated measurements performed on a relatively short timescale (typically using gas filling and evacuating cycles of 100 s) combined with an interpolation procedure in which the empty measurement cavity response is taken as the interpolated value of two such measurements—one taken just before and one directly after the filled measurement cavity measurement. That way, the influences of both long-term drifts and various types of fluctuations can be strongly suppressed [**???** ]. Furthermore, the influences of leaks and outgassing in the reference cavity can automatically be corrected for.

In order to perform high accuracy refractometry based on the GAMOR principle, care needs to be taken regarding the construction of the refractometers. To enable repeated filling and emptying of gas on relatively short time scales without introducing excessive amounts of PV work, cavities with small volumes have been implemented (<5 cm3) [**? ?** ]. The cavity spacers in these works were made from Invar which has both higher thermal conductivity and a larger volumetric heat capacity than commonly used glass materials (Zerodur and ULE glass).

This does not only eliminate any possible heat islands in the system; it also facilitates the assessment of gas temperature, which is performed by measurement of the temperature of the cavity spacer by the use of temperature probes placed in drilled holes in, or in direct contact with, the spacer. The use of Invar also eliminates effects of gas permeation which have been reported for ULE glass [**? ?** ]. Furthermore, to allow for fully automatic operation with sturdy laser locking and automated mode jumps, systems based on rugged narrow-banded fiber lasers working in the near IR (NIR) communication region (around 1.55 μm) have been used [**?** ].

This has lead to instrumentation that is capable of providing measurements with precision in the sub-ppm (sub-parts-per-million or sub-10<sup>−</sup>6) range [**????** ]. By then also using well-calibrated temperature sensors and accurately assessed molecular parameters (molar polarizabilties and virial coefficients), the systems can demonstrate good accuracy. Such a system, denoted the stationary optical pascal (SOP), was recently characterized in terms of its ability to realize the Pascal [**?** ]. It was found that its uncertainty was [(10 mPa)<sup>2</sup> + (10 × <sup>10</sup>−6*P*)2] 1/2, mainly limited by the uncertainty in the molar polarizability of nitrogen (8 ppm) [**?** ].

To assess the ability to realize a transportable refractometer, a similar system, denoted the transportable optical pascal (TOP), was recently developed and characterized. It was found that its uncertainty was [(16 mPa)<sup>2</sup> + (28 × <sup>10</sup>−6*P*)2] 1/2, mainly limited by the uncertainty of the temperature probes used for assessment of the temperature (26 ppm) [**?** ].

As was alluded to above, to make viable assessments of large pressure shifts with short settling times, which is needed for a number of applications, it is of importance that the system has a fast response. Although several types of refractometers have been scrutinized over the years [**???????????????????** ], virtually none of them has yet been assessed with respect to its short-term behavior. Access to two GAMOR-based refractometer systems allows for scrutiny of the short-term behavior of GAMOR-based refractometry in more detail. By comparing two fully independent GAMOR-based refractometer systems (the aforementioned SOP and TOP systems) connected to the same gas system, whose pressure was set by a dead weight piston gauge (DWPG), their short-term performances could be scrutinized in some detail. As the refractometers were completely independent, it could be concluded that deviations that are common to both systems are not inherent to one or the other refractometer, but rather the DWPG and/or the gas handling system. Thereby, we could ascertain the precision of the refractometers without any influence from the DWPG or gas handling system. Indeed, we assessed the short-term performances of two independent gas modulated refractometers regarding their ability to assess pressure. It was found that the refractometers can provide short-term precision on the 1 s time scale of 3 × <sup>10</sup><sup>−</sup>8, which is one order of magnitude better than the corresponding stability of the pressure provided by the DWPG. This opens up a number of novel applications for refractometry.

Although the SOP refractometer previously has been well described [**? ?** ], the TOP system has not. This system, including its construction and various components, is therefore described in some detail here. In addition, the theoretical model used for the evaluation of the data gathered is provided.
