Gas modulation refractometry (GAMOR) is a methodology for assessment of gas refractivity, molar density, and pressure that, by a rapid gas modulation, exhibits a reduced susceptibility to various types of disturbances. Although previously demonstrated experimentally, no detailed analysis of its ability to reduce the pickup of drifts has yet been given. This work provides an explication of to what extent modulated refractometry in general, and GAMOR in particular, can reduce drifts, predominantly those of the cavity lengths, gas leakages, and outgassing. It is indicated that the methodology is insensitive to the linear parts of so-called campaign-persistent drifts and that it has a significantly reduced susceptibility to others. This makes the methodology suitable for high-accuracy assessments and out-of-laboratory applications

Gas modulation refractometry is a technique for assessment of gas refractivity, density, and pressure that, by a rapid modulation of the gas, provides a means to significantly reduce the pickup of fluctuations. Although its unique feature has previously been demonstrated, no detailed explication or analysis of this ability has yet been given. This work provides a theoretical explanation, in terms of the length of the modulation cycle, of the extent to which gas modulation can reduce the pickup of fluctuations. It is indicated that a rapid modulation can significantly reduce the influence of fluctuations with Fourier frequencies lower than the inverse of the modulation cycle length, which often are those that dominate. The predictions are confirmed experimentally

A technique for high-precision and high-accuracy assessment of both gas molar (and number) density and pressure, Gas Modulation Refractometry (GAMOR), is presented. The technique achieves its properties by assessing refractivity as a shift of a directly measurable beat frequency by use of Fabry-Perot cavity (FPC) based refractometry utilizing the Pound-Drever-Hall laser locking technique. Conventional FPC-based refractometry is, however, often limited by fluctuations and drifts of the FPC. GAMOR remedies this by an additional utilization of a gas modulation methodology, built upon a repeated filling and evacuation of the measurement cavity together with an interpolation of the empty cavity responses. The procedure has demonstrated an ability to reduce the influence of drifts in a non-temperature stabilized dual-FPC (DFPC)-based refractometry system, when assessing pressure, by more than three orders of magnitude. When applied to a DFPC system with active temperature stabilization, it has demonstrated, for assessment of pressure of N2 at 4304 Pa at room temperature, which corresponds to a gas molar density of 1.7 × 10−6 mol/cm3, a sub-0.1 ppm precision (i.e. a resolution of 0.34 mPa). It is claimed that the ability to assess gas molar density is at least as good as so far has been demonstrated for pressure (i.e. for the molar density addressed, a resolution of at least 1.2 × 10−13 mol/cm3). It has recently been argued that the methodology should be capable of providing an accuracy that is in the low ppm range. These levels of precision and accuracy are unprecedented among laser-based techniques for detection of atomic and molecular species. Since the molar polarizability of He can be calculated by ab initio quantum mechanical calculations with sub-ppm accuracy, it can also be used as a primary or semi-primary standard of both gas molar (and number) density and pressure. © 2021 The Author(s)

By measuring the refractivity and the temperature of a gas, its pressure can be assessed from fundamental principles. The highest performing instruments are based on Fabry-Perot cavities where a laser is used to probe the frequency of a cavity mode, which is shifted in relation to the refractivity of the gas in the cavity. Recent activities have indicated that such systems can demonstrate an extended uncertainty in the 10 ppm (parts-per-million or 10-6) range. As a means to reduce the influence of various types of disturbances (primarily drifts and fluctuations) a methodology based on modulation, denoted gas modulation refractometry (GAMOR), has recently been developed. Systems based on this methodology are in general high-performance, e.g. they have demonstrated precision in the sub-ppm range, and they are sturdy. They can also be made autonomous, allowing for automated and unattended operation for virtually infinite periods of time. To a large degree, the development of such instruments depends on the access to modern photonic components, e.g. narrow line-width lasers, electro-and acousto-optic components, and various types of fiber components. This work highlights the role of such modern devices in GAMOR-based instrumentation and provides a review on the recent development of such instruments in Sweden that has been carried out in a close collaboration between a research institute and the Academy. It is shown that the use of state-of-the-art photonic devices allows sturdy, automated and miniaturized instrumentation that, for the benefit of industry, can serve as standards for pressure and provide fast, unattended, and calibration-free pressure assessments at a fraction of the present cost. © 2022 The Author(s).

Assessments of refractivity in a Fabry-Perot (FP) cavity by refractometry often encompass a step in which the penetration depth of the light into the mirrors is estimated to correct for the fraction of the cavity length into which no gas can penetrate. However, as it is currently carried out, this procedure is not always coherently performed. Here, we discuss a common pitfall that can be a reason for this and provide a recipe on how to perform FP-cavity-based refractometry without any influence of mirror penetration depth. © 2021 Author(s).

Fabry–Pérot-based refractometry has demonstrated the ability to assess gas pressure with high accuracy and has been prophesized to be able to realize the SI unit for pressure, the pascal, based on quantum calculations of the molar polarizabilities of gases. So far, the technology has mostly been limited to well-controlled laboratories. However, recently, an easy-to-use transportable refractometer has been constructed. Although its performance has previously been assessed under well-controlled laboratory conditions, to assess its ability to serve as an actually transportable system, a ring-type comparison addressing various well-characterized pressure balances in the 10–90 kPa range at several European national metrology institutes is presented in this work. It was found that the transportable refractometer is capable of being transported and swiftly set up to be operational with retained performance in a variety of environments. The system could also verify that the pressure balances used within the ring-type comparison agree with each other. These results constitute an important step toward broadening the application areas of FP-based refractometry technology and bringing it within reach of various types of stakeholders, not least within industry.

Refractometry is a powerful technique for pressure assessments that, due to the recent redefinition of the SI system, also offers a new route to realizing the SI unit of pressure, the Pascal. Gas modulation refractometry (GAMOR) is a methodology that has demonstrated an outstanding ability to mitigate the influences of drifts and fluctuations, leading to long-term precision in the 10−7 region. However, its short-term performance, which is of importance for a variety of applications, has not yet been scrutinized. To assess this, we investigated the short-term performance (in terms of precision) of two similar, but independent, dual Fabry–Perot cavity refractometers utilizing the GAMOR methodology. Both systems assessed the same pressure produced by a dead weight piston gauge. That way, their short-term responses were assessed without being compromised by any pressure fluctuations produced by the piston gauge or the gas delivery system. We found that the two refractometer systems have a significantly higher degree of concordance (in the 10−8 range at 1 s) than what either of them has with the piston gauge. This shows that the refractometry systems under scrutiny are capable of assessing rapidly varying pressures (with bandwidths up to 2 Hz) with precision in the 10−8 range. © 2021 by the authors.

By measuring the refractivity and the temperature of a gas, its pressure can be assessed from fundamental principles. The highest performing instruments are based on Fabry-Perot cavities (FPC). Gas modulation refractometry (GAMOR) is a methodology that has the ability to reduce the influence of disturbances to such an extent that high-precision (sub-parts-per-million) assessments of pressure can be made by the use of FPCs of Invar. To allow for high accuracy assessments, it is of importance to assess the uncertainty contribution from the thermodynamic effects that are associated with the gas filling and emptying of the cavity (pV-work). This paper presents a detailed scrutiny of the influence of the gas exchange process on the assessment of gas temperature on an Invar-based dual-FPC (DFPC) instrumentation. It is shown that by virtue of a combination of a number of carefully selected design entities (a small cavity volume with a bore radius of 3 mm, a spacer material with high heat capacitance, large thermal conductivity, and no regions that are connected with low thermal conductance, i.e. no heat islands, and a continuous assessment of temperature of the cavity spacer) the system is not significantly affected by pV-work. Simulations show that 10 s after the filling all temperature gradients in the system are well into the sub-mK range. Experiments support that refractivity assessments initiated after 40 s are not significantly affected by the pV-work. The analysis given in this work indicates that an upper limit for the influence of pV-work on the Invar-based DFPC system using 100 s long gas modulation cycles is 0.5 mK/100 kPa (or 1.8 ppm/100 kPa). Consequently, thermodynamic effects will not be a limiting factor when the Invar-based DFPC GAMOR system is used for assessments of pressure or as a primary pressure standard up to atmospheric pressures. 

An updated version of an Invar-based dual Fabry-Perot cavity refractometer utilizing the gas modulation methodology has been characterized with regard to its ability to assess gas pressure in the low pressure regime, defined as the regime in which the instrumentation is mainly limited by the constant term a in the [ ( a ) 2 + ( b × P ) 2 ] 1 / 2 expression for the uncertainty. It is first concluded that this ability is predominantly limited by three entities, viz., the empty cavity repeatability, the residual gas pressures in the evacuated (measurement) cavity, and the contamination of the gas residing in the measurement cavity that originates from leaks and outgassing. We then present and utilize methods to separately estimate the uncertainty of the updated refractometer from these entities. It was found that, when utilizing gas modulation cycles of 100 s and when addressing nitrogen, the system can assess pressure in the low pressure regime with an expanded uncertainty ( k = 2 ) of 0.75 mPa, mainly limited by the empty cavity repeatability and outgassing of hydrogen. This is more than 1 order of magnitude below the previously assessed low pressure performance of the instrumentation.

An Invar-based Fabry-Perot cavity refractometer equipped with an automated, miniaturized galliumfixed-point cell for assessment of pressure is presented. The use of an Invar cavity spacer has previously demonstrated pressure assessments with sub-0.1 ppm precision. The fixed-point cell, whose design and implementation are presented here, provides a reference for temperature assessment of the gas inside the cavity with an uncertainty of 4 ppm. This opens up for a self-contained system for realization of the Pascal with an accuracy in the low ppm range. This is an important step towards disseminating the Pascal through fundamental principles. 

Gas modulation refractometry (GAMOR) is a methodology that can mitigate fluctuations and drifts in refractometry. This can open up for the use of non-conventional cavity spacer materials. In this paper, we report a dual-cavity system based on Invar that shows better precision for assessment of pressure than a similar system based on Zerodur. This refractometer shows for empty cavity measurements, up to 104 s, a white noise response (for N2) of 3 mPa s1=2. At 4303 Pa, the system has a minimum Allan deviation of 0.34 mPa (0.08 ppm) and a long-term stability (24 h) of 0.7 mPa. This shows that the GAMOR methodology allows for the use of alternative cavity materials.

By measuring the refractivity and the temperature of a gas, its pressure can be calculated from fundamental principles. The most sensitive instruments are currently based on Fabry-Perot cavities where a laser is used to probe the frequency of a cavity mode. However, for best accuracy, the realization of such systems requires exceptional mechanical stability. Gas modulation refractometry (GAMOR) has previously demonstrated an impressive ability to mitigate the influence of fluctuations and drifts whereby it can provide high-precision (sub-ppm, i.e., sub-parts-per-million or sub-10−6) assessment of gas refractivity and pressure. In this work, two independent GAMOR-based refractometers are individually characterized, compared to each other, and finally compared to a calibrated dead weight piston gauge with respect to their abilities to assess pressure in the 4-25 kPa range. The first system, referred to as the stationary optical pascal (SOP), uses a miniature fixed point gallium cell to measure the temperature. The second system, denoted the transportable optical pascal (TOP), relies on calibrated Pt-100 sensors. The expanded uncertainty for assessment of pressure (k=2) was estimated to, for the SOP and TOP, [(10mPa)2+(10×10−6P)2]1/2 and [(16mPa)2+(28×10−6P)2]1/2, respectively. While the uncertainty of the SOP is mainly limited by the uncertainty in the molar polarizability of nitrogen (8 ppm), the uncertainty of the TOP is dominated by the temperature assessment (26 ppm). To verify the long-term stability, the systems were compared to each other over a period of 5 months. It was found that all measurements fell within the estimated expanded uncertainty (k=2) for comparative measurements (27 ppm). This verified that the estimated error budget for the uncorrelated errors holds over this extensive period of time. © 2021 Author(s).

Based on a recent experimental determination of the static polarizability and a first-principle calculation of the frequency-dependent dipole polarizability of argon, this work presents, by using a Fabry–Perot refractometer operated at 1550 nm, a realization of the SI unit of pressure, the pascal, for pressures up to 100 kPa, with an uncertainty of [(1.0 mPa)2 + (5.8 × 10−6 P)2 + (26 × 10−12P2)2]1/2. The work also presents a value of the molar polarizability of N2 at 1550 nm and 302.9146 K of 4.396572(26) × 10−6 m3/mol, which agrees well with previously determined ones. 

A procedure is presented for in situ determination of the frequency penetration depth of coated mirrors in Fabry-Perot (FP) based refractometers and its influence on the assessment of refractivity and pressure. It is based on assessments of the absolute frequency of the laser and the free spectral range of the cavity. The procedure is demonstrated on an Invar-based FP cavity system with high-reflection mirrors working at 1.55 µm. The influence was assessed with such a low uncertainty that it does not significantly contribute to the uncertainties (k = 2) in the assessment of refractivity (<8 × 10−13) or pressure of nitrogen (<0.3 mPa).

A procedure for automated low uncertainty assessment of empty cavity mode frequencies in Fabry-Pérot cavity based refractometry that does not require access to laser frequency measuring instrumentation is presented. It requires a previously well-characterized system regarding mirror phase shifts, Gouy phase, and mode number, and is based on the fact that the assessed refractivity should not change when mode jumps take place. It is demonstrated that the procedure is capable of assessing mode frequencies with an uncertainty of 30 MHz, which, when assessing pressure of nitrogen, corresponds to an uncertainty of 0.3 mPa. 

A novel procedure for a robust assessment of cavity deformation in Fabry-Pérot (FP) refractometers is presented. It is based on scrutinizing the difference between two pressures: one assessed by the uncharacterized refractometer and the other provided by an external pressure reference system, at a series of set pressures for two gases with dissimilar refractivity (here, He and N 2). By fitting linear functions to these responses and extracting their slopes, it is possible to construct two physical entities of importance: one representing the cavity deformation and the other comprising a combination of the systematic errors of a multitude of physical entities, viz., those of the assessed temperature, the assessed or estimated penetration depth of the mirror, the molar polarizabilities, and the set pressure. This provides a robust assessment of cavity deformation with small amounts of uncertainties. A thorough mathematical description of the procedure is presented that serves as a basis for the evaluation of the basic properties and features of the procedure. The analysis indicates that the cavity deformation assessments are independent of systematic errors in both the reference pressure and the assessment of gas temperature and when the gas modulation refractometry methodology is used that they are insensitive to gas leakages and outgassing into the system. It also shows that when a high-precision (sub-ppm) refractometer is characterized according to the procedure, when high purity gases are used, the uncertainty in the deformation contributes to the uncertainty in the assessment of pressure of N 2 with solely a fraction (13%) of the uncertainty of its molar polarizability, presently to a level of a few ppm. This implies, in practice, that cavity deformation is no longer a limiting factor in FP-based refractometer assessments of pressure of N 2. © 2020 Author(s).

This work models and experimentally assesses the influence of absorption of laser light in mirrors in Fabry-Pérot based refractometers used for realization of pressure. Model parameters are assessed by experimental characterizations. Characterizations of two refractometers agree well with the predictions of the model. It is shown that, when pressures are assessed in the viscous region, the absorption of laser light in mirrors will give rise to a small alteration in the proportional response and a pressure-independent offset, where the latter is significant for He but considerably smaller for Ar and N2

Gas modulation refractometry (GAMOR) is a methodology that can mitigate the influence off luctuations and drifts in Fabry-Pérot cavity–based refractometry. This paper presents a thorough description of its principles, what it enables, and its applicability. An overview of the latest results is presented, including the realization of a system based upon a cavity spacer made of Invar that allows for detection of N2 with sub-ppm precision, and a characterization procedure that allows for assessment of N2 with an accuracy at low-ppm levels.