## Abstract

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.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Refractometry constitutes an optical technique for assessment of refractivity, gas density, and pressure that has demonstrated high precision and accuracy [1–16]. It has been indicated that the technology can not only be used to assess pressure, but it also has the potential to replace current mechanical pressure standards, in particular in the 1–100 kPa range [17–23].

A common means to realize refractometry is to base it on a Fabry–Perot (FP) cavity (FPC) [1,3,24–26]. When gas whose refractivity (density or pressure) is to be assessed is let into such a cavity, the frequencies of its cavity modes will shift [3,14]. By locking a laser to one of its modes, the shift of the frequency of the mode will be transferred to a shift in the frequency of the laser light [3,14]. A common way to assess such a shift is to mix the frequency of the laser light down to an RF by use of another laser (a reference laser). This can be achieved by merging the two laser fields onto a photodiode and measuring the beat frequency with a frequency counter. By this, the shift in the frequency of the cavity mode addressed when the amount of gas in the cavity is altered is converted to a shift in a measured beat frequency.

Although it is simple in theory to realize FPC-based instrumentation, it is not trivial in practice to carry out high-precision refractivity assessments. One reason is that FPCs often are exposed to fluctuations and the assessments are influenced by various types of disturbances, not least from changes of the length of the cavity [16,27–30]. For lower pressures, these fluctuations in cavity length will limit the sensitivity of the instrument.

To alleviate some of these limitations, FPC-based refractometry is often based on a dual-Fabry–Perot cavity (DFPC) in which two cavities are simultaneously addressed by two laser fields [3,4,9,13–17,29,30]. An advantage of this is that any change in length of the cavity spacer (into which both cavities are bored) does not affect the assessment. However, since the lengths of two cavities also can fluctuate dissimilarly over time, DFPC-based refractometry will also pick up disturbances, although often to a lesser extent [29].

As a means to remove these shortcomings, gas modulation refractometry (GAMOR) was recently developed [31,32]. GAMOR is a methodology that utilizes a rapid gas modulation procedure for a repetitive referencing of the filled measurement cavity assessment to a reference assessment. It is built upon two cornerstones; (i) one is that the refractivity of the gas in the measurement cavity is assessed by the use of a rapid gas modulation procedure that allows for a repetitive referencing of the filled measurement cavity beat frequency assessments to empty cavity assessments, while (ii) the other is that the empty measurement cavity beat frequency is estimated based on an interpolation between two assessed empty measurement cavity beat frequency measurements. This methodology reduces the pickup of various types of fluctuations and drifts in refractometry systems, not only those from changes in length of the cavity (e.g., those caused by drifts in the temperature of the spacer material) but also those that have other origins [31].

It has previously been shown that the GAMOR methodology could significantly reduce fluctuations and drifts in a DFPC-based system with no active temperature stabilization, from the Pa to the mPa range [31]. When applied to a system with active temperature stabilization, it demonstrated a sub-mPa precision for low pressures and a sub-ppm stability at 4304 Pa [32]. Its extraordinary ability to reduce the pickup of fluctuations and drifts recently made it possible to realize high-precision (sub-ppm) refractometry using a cavity spacer made of Invar [33]. When assessing pressure at 4303 Pa, the system could provide (for measurement times around ${{10}^3}\;{\rm s}$) a minimum Allan deviation of 0.34 mPa, which corresponds to a relative deviation of 0.08 ppm. Despite these demonstrations, the fundamental cause for its unique ability has not yet been elucidated, nor has the influence of using a rapid gas modulation process on refractometry been subjected to any scrutiny.

To remedy this, we present, in this work, a mathematically based explication as to why a rapid sequence of referencing the gas measurement to a reference assessment (i.e., a short gas filling-and-emptying cycle), i.e., the first of the two cornerstones on which GAMOR relies, can reduce the influence of fluctuations [34]. To confirm the model, we then provide an experimental investigation of how the time between the empty and the filled measurement cavity assessments affects the pickup of fluctuations and disturbances in the assessment of pressure in DFPC-based refractometry. The data clearly confirm that the performance of DFPC-based refractometry can be significantly improved by utilizing gas modulation, where the amount of fluctuations picked up decreases with decreasing length of the gas modulation cycle.

By this combination of mathematical modeling and experimental confirmation, this work provides a conclusive elucidation of the underlying causes as to why the GAMOR methodology is capable of reducing the number of fluctuations in the detection process pickup.

## 2. THEORY

Although GAMOR can be performed by several different modes of operation [31,32], we will in this work restrict ourselves to analysis of its original realization, termed single-cavity-modulated GAMOR (SCM-GAMOR), in which the amount of gas in one of the cavities is regularly modulated while that in the other one is held constant [31]. This has most in common with the modes of operation of (conventional) unmodulated DFPC-based refractivity and allows therefore for both a direct and straightforward comparison of the two methodologies and an investigation of the dependence of the length of the gas modulation cycle on the ability to reduce the influence of fluctuations and disturbances on the assessed refractivity.

#### A. Assessment of Gas Refractivity from the Measured Change in Beat Frequency

An expression for DFPC-based refractivity that relates the change in the measured beat frequency to the change in refractivity of the gas in the measurement cavity when the amount of gas in it is changed has previously been derived in the literature. As is shown in Ref. [32], it is possible to express the refractivity of the gas in the measurement cavity, $n - 1$, in terms of the shift in beat frequency when the cavity is filled with gas (or emptied to vacuum), ${\Delta}\!f$, as

Moreover, in Eq. (1), $\overline {\Delta q} _m $ is given by $\Delta q_m/q_m^{(0)}$, where $\Delta q_m$ denotes the number of mode jumps the measurement laser performs when the cavity is filled with gas (or when it is evacuated). Finally, $\varepsilon _m$ represents a refractivity-normalized relative deformation coefficient of the measurement cavity, defined by $\varepsilon _m(n - 1) = {\delta}\! L_m/L_m^{(0)}$, in which, in turn, ${\delta}\! L_m$ is the elongation of the measurement cavity when it is filled with gas, while $L_m^{(0)}$ is the length of the measurement cavity when empty [35–37].

A compilation of the nomenclature used is given in the supplementary material of Ref. [32]. Note that Eq. (1) is valid for all types of refractometry in which the amount of gas in a single cavity is altered, including both unmodulated and modulated DFPC-based refractometry, the latter of which is a part of SCM-GAMOR.

#### B. Assessment of Gas Density and Pressure from the Gas Refractivity

The conversion of a given refractivity, $n - 1$, to gas density, $\rho$, is performed through the extended Lorentz–Lorenz equation, given by Eq. (SM-20) in the supplementary material of Ref. [32], which implies that the density can be assessed from the assessed refractivity by

where ${A_R}$ and ${\tilde B_\rho}$ are the molar dynamic polarizability and a refractivity virial coefficient, respectively, where the latter is given by ${-}(1 + 4{B_R}/A_R^2)/6$, where, in turn, ${B_R}$ is the refractivity virial coefficient in the Lorentz–Lorenz equation [14,38,39].The corresponding pressure, $P$, can thereafter be obtained from the density as

where ${k_B}$ is the Boltzmann constant, $T$ is the temperature of the gas, ${N_A}$ is the Avogadro’s number, and ${B_p}(T)$ is a density virial coefficient [20,29].## 3. MODEL FOR THE ROLE OF THE LENGTH OF THE GAS FILLING-AND-EMPTYING CYCLE ON THE PICKUP OF FLUCTUATIONS IN THE ASSESSMENT OF REFRACTIVITY IN DFPC-BASED REFRACTOMETRY TECHNIQUES

#### A. Schematic Illustration of Unmodulated and Modulated DFPC Refractometry

As was alluded to above, since refractivity assessments comprise measurements performed at different time instances, with at least one cavity having dissimilar amounts of gas, they are affected by omnipresent fluctuations of various physical entities and disturbances in the measurement process that take place between the individual measurement instances. To provide an intuitive understanding of how the length of the gas filling-and-emptying cycle can influence how much of a given fluctuation the detection process will pick up, Fig. 1 displays, by the panels (a) and (c), the gas filling-and-emptying process for unmodulated and modulated DFPC refractometry, respectively. The panels (b) and (d) show the developments of the associated beat frequencies (see below). The instances for the two beat frequency measurements in panel (a) are marked by vertical dashed lines (leftmost, the empty measurement cavity assessment, and rightmost, the filled measurement cavity assessment). Although there is one beat frequency assessment for each gas modulation cycle in the modulated case, the corresponding instances for the modulated situation are, in panel (c), for illustrative purposes, marked by vertical lines solely on the fifth one.

Each assessment of beat frequency will comprise an averaging over several data points. For simplicity, we will below assume that the averaging time is significantly shorter than both the length of the gas filling-and-emptying cycle in the unmodulated case and the gas modulation cycle in the modulated case, and, for simplicity, the same in all cases (taken as 10 s).

Although the figure depicts the modulated case with a gas modulation cycle that is solely 1 order of magnitude shorter than the length of the gas filling-and-emptying cycle in the unmodulated situation, in reality the relative difference between the two cases can be significantly larger; we will below assume that the length of the gas filling-and-emptying cycle in the unmodulated case [corresponding to panel (a)]) is ${{10}^5}\;{\rm s}$, while that in the modulated case [panel (c)] is ${{10}^2}\;{\rm s}$ (corresponding to what has been used in previous realizations [31,32]). This implies that the gas modulation period is ${{10}^3}$ times shorter than the gas filling-and-emptying cycle for the unmodulated case and that there will be ${{10}^3}$ gas modulation cycles within the time for an unmodulated gas filling-and-emptying cycle. Let us also assume that the entire series of assessments make up a measurement campaign that can comprise one or several unmodulated assessments and therefore one or several thousand gas modulation cycles.

#### B. Modeling of the Fluctuations in Terms of Fourier Components

To provide a comprehensible description of the extent to which the length of the gas filling-and-emptying cycle influences how much fluctuations the detection process will pick up, we will start by assuming that the system is affected by fluctuations, $\delta [f(t)]$, that are the same in both $f_{(0,g)}(t)$ and ${f_{(0,0)}}(t)$ [henceforth jointly denoted $f(t)$]. Let us then assume that the fluctuations can be described in terms of a set of Fourier components, e.g., as

Let us thereafter specifically consider an individual Fourier component of the fluctuations, $\delta {[f(t)]_{{{\tilde f}_D}}}$, that has a given Fourier frequency, ${\tilde f_D}$, a given amplitude ${a_{{{\tilde f}_D}}}$, and a given phase ${\varphi _{{{\tilde f}_D}}}$, viz.,

Such a Fourier component is schematically shown by the green (lower) curves [representing the empty measurement cavity beat frequency, ${f_{(0,0)}}(t)$] and as parts of (the envelope of) the black (upper) curves [corresponding to the filled measurement cavity beat frequencies, $f_{(0,g)}(t)$] in the panels (b) and (d) in Fig. 1.

#### C. Influence of a Given Fourier Component of the Fluctuations on the Assessment of Refractivity

Since the shift of the beat frequency, $\Delta\! f$, comprises the difference between a filled and an empty measurement cavity assessment, at $t = {t_{\rm mod}}$ and 0, respectively [represented by an open circle and a cross in the panels (b) and (d)], given by $f({t_{\rm mod}}) - f(0)$, the assessment will be affected by fluctuations with the Fourier frequency ${\tilde f_D}$ by an amount of $\delta {[f({t_{\rm mod}})]_{{{\tilde f}_D}}} - \delta {[f(0)]_{{{\tilde f}_D}}}$. This implies that the assessment of the shift of the beat frequency will be associated with a fluctuation, denoted $\delta {[\Delta\! f({t_{\rm mod}})]_{{{\tilde f}_D}}}$, given by

#### D. Influence of Averaging Time and the Length of the Gas Filling-and-Emptying Cycle on the Pickup of Various Frequency Components of the Fluctuations on the Assessment of Refractivity

### 1. Influence of Averaging on the Amount of Fluctuations Picked Up from a Given Fourier Component

As was alluded to above, to reduce the influence of high-frequency fluctuations, it is common to average the measured beat frequency over a certain time. When the averaging is performed over a period of ${t_{\rm avg}}$, the fluctuation picked up, $\delta {[\Delta\! f({t_{\rm mod}},{t_{\rm avg}})]_{{{\tilde f}_D}}}$, can be expressed as

Evaluating the integrals, this can be written as

This shows that the amount of a given fluctuation that is picked up is given by the relation between its Fourier frequency, ${\tilde f_D}$, and the two time scales in the system, i.e., the length of the gas filling-and-emptying cycle, ${t_{\rm mod}}$, and the averaging time, ${t_{\rm avg}}$ (and, to a certain extent, the phase ${\varphi _{{{\!\tilde f\!}_D}}}$).

### 2. Influence of Fluctuations Whose Frequencies Are Higher Than the
Inverse of ($\pi$* Times) the
Averaging Time (i.e., for ${\tilde f_D} \gt
1/\pi\! {t_{\rm avg}}$)*

For the components of the fluctuations whose frequencies are higher than the inverse of ($\pi$ times) the averaging time, i.e., for the case when ${\tilde f_D} \gt 1/\pi\! {t_{\rm avg}}$, henceforth referred to as “fast” fluctuations, it is possible to conclude from Eq. (11) that the maximum amount picked up will be inversely proportional to its Fourier frequency, viz., $4{a_{{{\!\tilde f\!}_D}}}/(2\pi\! {\tilde f_D}{t_{\rm avg}})$. The averaging will, though, give rise to certain resonance conditions, i.e., specific frequencies of the fluctuations (those for which the frequency is $n/{t_{\rm avg}}$ where $n$ is an integer) that will not be picked up at all. A depiction of the situation for the case when ${t_{\rm avg}}$ is 10 s is given [in terms of its normalized root-mean-square (rms) value ${\sigma _{\Delta\! f}}/{a_{{{\!\tilde f\!}_D}}}$] by the envelope of the rightmost part (${\tilde f_D}\; {\gt}\;{3} \times {{10}^{- 2}}\;{\rm Hz}$) of the red curve in Fig. 2(a), or, when disregarding the resonances, by the black sloping straight line.

### 3. Influence of Fluctuations Whose Frequencies Are Lower Than the
Inverse of ($\pi$* Times) the
Averaging Time (i.e., for ${\tilde f_D} \lt
1/\pi\! {t_{\rm avg}}$)*

The averaging process will, though, not significantly affect the components of the fluctuations whose frequencies are lower than the inverse of ($\pi$ times) the averaging time, i.e., for the case when ${\tilde f_D} \lt 1/\pi\! {t_{\rm avg}}$. For these components, it can be shown, by series expansion of Eq. (11), that the amount of the fluctuation picked up can be expressed as

Since ${\varphi _{{{\!\tilde f\!}_D}}}$ can vary between consecutive assessments, a series of assessments performed during a given measurement campaign will effectively be affected by the rms value of Eq. (12), denoted ${\sigma _{\Delta\! f}}$, which can be written

This implies that also the finite gas filling-and-emptying cycle will give rise to certain resonance conditions, this time for the Fourier components whose frequencies are integers of the inverse of the gas filling-and-emptying cycle period, i.e., $n/{t_{\rm mod}}$. This effect can be seen by the middle part of the red curve in Fig. 2(a) (i.e., for ${3} \times {{10}^{- 6}}\;{\rm Hz}\; \lt {\tilde f_D} {\lt}\;{3} \times {{10}^{- 2}}\;{\rm Hz}$).

The maximum amount of the Fourier component considered that can be picked up is given by the horizontal envelope of the response (the black horizontal line), for which ${\sigma _{\Delta\! f}}$ is $\sqrt 2 {a_{{{\!\tilde f\!}_D}}}$. It can be shown though that, on average, the system will, in this frequency interval (i.e., $1/\pi\! {t_{\rm mod}} \lt {\tilde f_D} \lt 1/\pi\! {t_{\rm avg}}$), pick up around the average of all ${a_{{{\!\tilde f\!}_D}}}$ of the Fourier components considered.

### 4. Influence of Fluctuations Whose Frequencies Are Lower Than the
Inverse of ($\pi$* Times) the
Gas Filling-and-Emptying Cycle Period (i.e., for ${\tilde f_D} \lt
1/\pi\! {t_{\rm mod}}$)*

More importantly though, for fluctuations whose frequencies also are lower than the inverse of ($\pi$ times) the gas filling-and-emptying cycle period [but still higher than the inverse of ($\pi$ times) the measurement campaign], i.e., for the case when $1/\pi\! {t_{\rm camp}} \lt {\tilde f_D} \lt 1/\pi\! {t_{\rm mod}}$, Eq. (14) takes a limiting value, given by

This shows that such fluctuations will be picked up to a significantly lesser extent, and, due to the ${\tilde f_D}$ dependence, the reduction in the pickup is more pronounced the lower the frequency of the fluctuation [indicated by the leftmost sloping line in Fig. 2(a)].

#### E. Effect of Reducing the Length of the Gas Filling-and-Emptying Cycle

Equation (15) shows that, for the lowest Fourier frequencies, i.e., those with ${\tilde f_D} \lt 1/\pi\! {t_{\rm mod}}$, the shorter ${t_{\rm mod}}$ is, the lesser amount of a given Fourier frequency the system will pick up. This implies that when the length of the gas filling-and-emptying cycle is reduced a given amount, the influence of such a Fourier component will be reduced by the same amount.

In Fig. 2, a gas filling-and-emptying cycle period (i.e., the gas modulation period) of ${{10}^5}\;{\rm s}$ [panel (a)] is compared with a period of ${{10}^2}\;{\rm s}$ [panel (b)], typically used in GAMOR. The slowest fluctuations, i.e., those with Fourier coefficients below ${3} \times {{10}^{- 6}}\;{\rm Hz}$ (the leftmost parts of the two panels in Fig. 2) will, in the modulated case, be attenuated ${{10}^3}$ times more than in the “unmodulated” system (with a gas cycling period of ${{10}^5}\;{\rm s}$). In addition, it will also pick up fewer of most of the fluctuations with Fourier coefficients between ${3} \times {{10}^{- 6}}$ and ${3} \times {{10}^{- 3}}\;{\rm Hz}$, given by Eq. (14).

Since the low-frequency components of the fluctuation often contain significantly (order of magnitude) more power than what the high-frequency components hold, as fluctuations of the $1/\!{f^a}$-type do, the rapid gas modulation process can be significantly more effective than the averaging process in reducing the number of fluctuations the system picks up. This is the advantage of the gas modulation methodology.

#### F. Case with Slow Fluctuations and Drifts: A Short Comment

Although not considered by the analysis above, but one that is scrutinized in some detail in a separate publication [41], it can be concluded that also the influences of the slowest fluctuations (those for which the Fourier frequency is lower than $1/\pi\!{t_{\rm camp}}$ for which the Fourier decomposition is given by the Eqs. (6) and (7) is not valid) and linear and nonlinear drifts (whose contribution to the assessment of the shift of the beat frequency can be expressed as a Taylor series in terms of $t$) are expected to be reduced by a shortening of the measurement cycle period. However, since such slow fluctuations and drifts also affect the accuracy (and not solely the precision of the assessment, which is considered in this work), and they are efficiently eliminated by the use of the interpolation procedure of GAMOR (which is not dealt with here), they will not be considered in the present work; they are treated in detail in a separate publication [41].

## 4. EXPERIMENTAL SETUP

The experimental setup used to confirm the predictions given above is basically the same as in Ref. [33]. The system comprises two main parts: a refractometer that is used to measure the difference in refractivity between the two cavities (schematically displayed in Fig. 3), and a gas handling/vacuum system that is used to fill and empty the cavities.

The refractometer setup is based upon a 150 mm long Invar-based DFPC and consists of two identical arms, each comprising a laser whose frequency is locked to a mode in an FPC. Each cavity of the DFPC (denoted 1 and 2) is made of two 12.7 mm high reflective concave mirrors with a radius of curvature of 500 mm. The mirrors are pressed onto the spacer, yielding FPCs with free spectral ranges (FSRs) of 1 GHz and finesse values of around $10^{4}$.

The light in each arm is produced by an Er-doped fiber laser (EDFL) that is coupled into an acousto-optic modulator (AOM). The first order output of the AOM is, in turn, coupled into a 90/10 fiber splitter, whose 90% output is sent to an electro-optic modulator (EOM) that is modulated at 12.5 MHz for Pound–Drever–Hall (PDH) locking [42]. The output of the EOM is sent through an optical circulator to a collimator whose output is mode-matched to the ${{\rm TEM}_{00}}$ mode of the FPC. The light transmitted through the cavity is monitored by a large area photodetector, while the backreflected light is picked up by the collimator and routed through the circulator onto the fast photodetector. The outputs from the photodetectors are connected to a field programmable gate array (FPGA).

In the FPGA, the signals from the reflection detectors are demodulated at 12.5 MHz to produce the PDH error signals, while the signals from the transmission detectors are used to control the feedback when a laser is making a mode hop. The slow components of the feedback ($\lt{100}\;{\rm Hz}$) are sent to the laser while the fast ones ($\gt{100}\;{\rm Hz}$) are routed to a voltage-controlled oscillator (VCO) that produces an RF voltage that drives the AOM at 110 MHz.

The remaining light from the fiber splitters (their 10% outputs) is sampled and combined with light from the second arm in a 50/50 fiber coupler. To detect the beat frequency between the two arms, the combined light is sent to a fiber-coupled photodetector (Beat. Detector) whose RF output in turn is routed to a frequency counter.

The lasers are initially (i.e., before the measurement series) tuned by temperature to suitable cavity modes so as to produce frequencies that provide a beat frequency that is in the center of the working range of the frequency counter. After this manual (coarse) setting, to allow for automated assessments, an automatic relocking routine is engaged, which keeps the beat frequency within the dynamic range of the frequency counter (between 2 and 6 GHz) under all measurement conditions.

To be able to assess the pressure from the density, by use of Eq. (5), it is important to accurately assess the temperature of the gas. In the experimental setup used in this work, the temperature of the cavity spacer is monitored by three calibrated Pt-100 sensors that are placed in holes drilled into the cavity spacer for monitoring of the temperature and its distribution in the spacer [43]. Both the (temporal) stability and the (spatial) gradients in temperature of the cavity were estimated to be below 5 mK.

The DFPC spacer is connected to a gas and vacuum system whose details are given elsewhere [33]. High-purity nitrogen is supplied from a central gas unit. The residual amount of gas in the system was monitored by a pressure gauge.

A measurement campaign starts by assessment of the frequencies of the two lasers when locked to evacuated cavities, i.e., $\nu _m^{(0)}$ and $\nu _r^{(0)}$, which can be assessed with a relative uncertainty of ${2} \times {{10}^{- 7}}$ by the use of a wavelength meter (Burleigh, WA-1500), and the mode number addressed for the empty cavity, i.e., $q_m^{(0)}$, which can be assessed uniquely (i.e., with no uncertainty) from the fact that $n - 1$, as given by Eq. (1), should be a continuous function when the measurement laser is making a mode hop.

Moreover, the cavity module has recently been preliminarily characterized with respect to its pressure-induced cavity deformation by use of a novel two-gas method, using helium and nitrogen. As is further discussed in detail in an upcoming work, it was found that the relative elongation of the cavity being used, $(\delta\! L_m/L_m^{(0)})/P$, is ${5.271}({5}) \times {{10}^{- 12}}\;{{\rm Pa}^{- 1}}$, which implies that, when ${{\rm N}_2}$ is being detected, ${\varepsilon _m}$ is ${1.968}({1}) \times {{10}^{- 3}}$ [44].

## 5. RESULTS: INVESTIGATION OF THE DEPENDENCE OF THE LENGTH OF THE GAS FILLING-AND-EMPTYING CYCLE ON THE PICKUP OF FLUCTUATIONS IN THE ASSESSMENT OF A REFRACTIVITY IN A DFPC-BASED REFRACTOMETER

To assess the influence of the length of the gas filling-and-emptying cycle on the pickup of the fluctuations and disturbances, the system is exposed to a study that was performed with a set of dissimilar times between the empty and the filled measurement cavity assessments (ranging from 100 s to ${3} \times {{10}^4}\;{\rm s}$) [45]. To incontestably assess this dependence, a series of measurements [a 50 h-long series of gas modulations, as indicated in Fig. 1(c), with 1800 modulation cycles (each being 100 s long)] that could be used for all assessments was taken. The data were evaluated by eight different means, so as to represent eight different gas filling-and-emptying cycle lengths.

The situation with the shortest cycle length (100 s), representing the situation in GAMOR, was retrieved by evaluating the measured beat frequencies (averaged over 10 s) on a cycle-to-cycle basis, as is indicated in the panels (c) and (d) in Fig. 1. The situations with longer gas filling-and-emptying cycles were obtained by referring the (averaged) beat frequency of a given cycle to an earlier (averaged) empty cavity measurement. To evaluate the situation with a gas modulation cycle with a length of $m \times 100$ s, every cycle $k$ (with $k$ ranging from $m$ to $N$ in steps of $m$, where $N$ is the number of actual gas filling-and-emptying cycles) was referred to the (averaged) empty cavity measurements of cycle $1 + k - m$. This procedure was performed for eight values of $m$ (1, 3, 5, 10, 30, 50, 100, and 300), yielding lengths of the gas filling-and-emptying cycle (the gas modulation cycle) ranging from 100 s to ${3} \times {{10}^4}\;{\rm s}$.

To not be affected by fluctuations and drifts of the temperature assessment, which affects the assessment of pressure from the density, so as to more clearly focus the study upon the influence of fluctuations of the cavity and disturbances of the detection process, the measurements were performed with an empty measurement cavity.

The resulting data sets for the eight gas modulation cycle lengths are shown by the eight overlaid curves in Fig. 4, expressed in units of pressure. Whereas the data set for the longest gas filling-and-emptying cycle considered (30,000 s, 8.3 h) solely consists of seven data points, the one with the shortest cycle (100 s) comprises 1800. It can be concluded from the figure that the data sets with the longest gas filling-and-emptying cycles are the ones that provide the largest deviations; the shorter the gas filling-and-emptying cycles, the less the data spread. The data set with the shortest cycle length (100 s) is, for clarity, redrawn with an enlarged scale in the inset. It is here possible to conclude that this data set displays fluctuations of only a few mPa, considerably less than what the data curves corresponding to the longer cycle lengths in the main window display.

To assess to which degree the various modes of detection (using various lengths of the gas filling-and-emptying cycle) pick up the fluctuations and disturbances that the system is exposed to, Fig. 5 displays the standard deviations, ${\sigma}_{\textit{P}}$, of the assessments shown in the main window of Fig. 4 as a function of the length of the gas filling-and-emptying cycle. The figure shows that the deviation decreases significantly with a decreased length of the gas filling-and-emptying cycle (gas modulation cycle period). Moreover, the standard deviation of the assessment using the shortest gas modulation cycle period (100 s) is significantly (in this case more than 50 times) lower than that for the longest length of the gas filling-and-emptying cycle (30,000 s). This confirms the predictions of the model given above [46].

To further confirm the alleged advantage of short gas modulation times, the Allan deviations of the data taken with the three shortest gas modulation cycle lengths [100 s (lowermost curve), 300 s, and 500 s (uppermost curve)] are shown in Fig. 6. In agreement with the data shown in Figs. 4 and 5, the Allan deviation of the signal that results when the shortest gas modulation cycle (100 s) is used is consistently smaller than those of the others; it displays a deviation of 0.9 mPa for a single modulation cycle (100 s) that decreases monotonically with averaging time (with an almost pure white-noise dependence) down to 6000 s, at which it shows an Allan deviation of 0.15 mPa. For longer averaging times, it starts to be affected by drifts. The reason why the data are not affected by drifts until such a considerable time as 6000 s is that the gas modulation procedure that is used in this work not only reduces the influence of fluctuations (as has been explicated in this work) but also various types of drifts (as is described elsewhere [41]). The Allan deviations from the data taken with longer gas modulation cycle lengths (300 and 500 s) show a higher level of noise and no evidence of drift for averaging times up to 6000 s.

## 6. SUMMARY AND CONCLUSIONS

It has previously been shown that GAMOR has the ability to reduce the influence of fluctuations and drifts on refractometry assessments [31,32]. In an early realization, it was demonstrated that the technique was capable of reducing drifts from a DFPC-based refractometry system with no active temperature control by up to 3 orders of magnitude, from the pascal to the millipascal range [31]. Recently, a GAMOR system has shown to provide, when assessing pressure at 4303 Pa, a minimum Allan deviation of 0.34 mPa, which corresponds to a relative deviation of 0.08 ppm [33]. This extraordinary ability has been attributed to a measurement procedure that is based upon two cornerstones: (i) one being that the refractivity of the gas in the measurement cavity is assessed by the use of a rapid gas modulation procedure that allows for a repetitive referencing of the filled measurement cavity beat frequency assessments to empty cavity assessments, while (ii) the other is that the empty measurement cavity beat frequency is estimated based on an interpolation between two assessed empty measurement cavity beat frequency measurements.

It has been alleged that the first of these two features contributes significantly to the technique’s ability to reduce the amount of fluctuations the methodology picks up in a series of refractivity assessments, which, in turn, improves its precision. However, hitherto the fundamental cause for this claim has not yet been elucidated, nor has the influence of using rapid modulation processes on refractometry been subjected to any study. This work has therefore scrutinized the role of the length of the time period between the pair-wise (empty and filled) measurement cavity assessments (for simplicity, here referred to as the length of the gas filling-and-emptying cycle or the length of the gas modulation cycle) on the pickup of fluctuations in the assessment of refractivity in FPC-based refractometry techniques.

A mathematical model of the data acquisition (DAQ) process in DFPC-based refractometry is presented that explicates the basis for why rapid gas modulation is capable of reducing the amount of fluctuations the detection system picks up. It indicates that a sequence of referencing the gas measurement to a reference assessment reduces the influence of fluctuations with Fourier frequencies lower than the inverse of the gas modulation cycle length. This implies that the methodology reduces the fluctuations the system picks up to a larger degree the shorter the length of the modulation cycle, both over a larger range of frequencies and, to a larger extent, over each frequency component. It also implies that it reduces the influence of other frequency components than what the conventionally used averaging processes do (which decreases the influence of fast fluctuations, i.e., the components of the fluctuations whose Fourier frequencies are higher than the inverse of the integration time). Hence, similar to what has been concluded about other types of modulation techniques, e.g., wavelength modulation spectrometry [47], the model indicates that rapid gas modulation has the ability to reduce the influence of the low-frequency fluctuations that often are the dominating ones in measurement systems (due to their anticipated $1/\!{f^a}$ dependence).

Measurements performed under a given set of conditions have confirmed that the length of the gas filling-and-emptying cycle indeed plays a significant role in eliminating the pickup of fluctuations and disturbances in the system. It was shown that, for a given set of data (taken over 50 h with a gas modulation cycle of 100 s), evaluated in eight different ways so as to mimic detection using eight different gas-and-filling cycle times, ranging from 100 s to ${3} \times {{10}^4}\;{\rm s}$, the utilization of a short gas modulation cycle (100 s) can provide detection conditions that provide standard deviations that are significantly smaller (more than 50 times) than those for a length of the gas filling-and-emptying cycle of 30,000 s (8.3 h) (0.9 mPa versus 50 mPa). This does not only verify the predictions of the mathematical model given above, but also the alleged assumption that a rapid gas modulation process, which is one of the cornerstones on which the GAMOR methodology relies, can be highly beneficial for refractometry.

## Funding

Vetenskapsrådet (621-2015-04374); Umeå University Industrial doctoral school (IDS); VINNOVA (2018-04570, 2019-05029); Kempestiftelserna (1823, U12).

## Disclosures

The authors declare no conflicts of interest.

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**34. **The effect of the second cornerstone on the ability to reduce the
influence of drifts is dealt with in an upcoming publication
[41].

**35. **In practice, it is not possible to evacuate the measurement cavity
to pure vacuum during a measurement cycle; it will therefore,
during the evacuation stage, always contain a residual amount of
gas. This implies that it is not possible to directly assess $f_{(0,0)}$; what in reality is measured is
the beat frequency that results when the measurement cavity
contains the residual amount of gas, denoted $f_{(0,{\rm
Res})}$. On the other hand, since the
amount of the residual gas can be assessed by the use of a
pressure gauge with good accuracy, and this can be directly
related to a corresponding refractivity, $n_{\rm
Res}-1$, it is possible to assess $f_{(0,0)}$ in terms of the measurable
entities $f_{(0,{\rm
Res})}$ and $n_{\rm
Res}-1$ as $f_{(0,0)}=
f_{(0,{\rm Res})}+(n_{\rm Res}-1) \nu_m^{(0)}$, allowing Eq. (1) to be used in its given
form also in the presence of a residual gas pressure in the
measurement cavity.

**36. **Note that in reality, $\delta
L_m$ is the difference in elongation
of the measurement and the empty cavity when the measurement
cavity is filled with gas. However, in most practical cases, the
elongation of the empty cavity is only a fraction of that of the
measurement cavity.

**37. **It is of importance to note that Eq. (1) shows that the refractivity (and thereby the
molar density and pressure) can be assessed without any explicit
assessment or monitoring of the length of the measurement cavity, i.e., $
L_m^{(0)}$. The reason for this is that the
change in cavity mode frequency when the measurement cavity is
evacuated (or filled with gas) is measured directly as a shift of
a cavity mode, $\Delta f$. There is neither any need to
explicitly assess the FSR. The reason for this is that $\overline {\Delta
q}_m$ is not expressed in terms of the
FSR but instead in the mode number of the mode addressed, $q_m^{(0)}$, which is an integer that can be
assessed uniquely (i.e., with no uncertainty) by ensuring that the
assessed refractivity is a continuous function when the
measurement laser is making a (controlled) mode jump. An
alternative means to assess $q_m^{(0)}$ is as the closest integer to the
ratio of the frequency of the laser addressing the evacuated
measurement cavity, $\nu_m^{(0)}$, and the FSR of the cavity. It
should be noted that the former of these means does not require
assessment of any physical entity.

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**39. **Note that the expression for $B\rho$ given in Ref. [19] is incorrect. The correct
expression, given here, is derived in Ref. [32].

**40. **Fluctuations and disturbances that give rise to Fourier components
that are not included in the description given by Eq. (6), i.e., predominantly
those with Fourier components whose frequencies are lower than the
inverse of the measurement campaign and those that provide linear
or nonlinear drifts, which can be described as a low-order
polynomial in $t$, will not only affect the
precision of an assessment, but also the accuracy. Their influence
on the assessment of refractivity is therefore not considered in
the present work; they are treated in detail in a separate
publication [41].

**41. **O. Axner, I. Silander, C. Forssén, J. Zakrisson, and M. Zelan are preparing a manuscript to be called
“Ability of gas modulation refractometry to reduce the
influence of drifts.”

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**43. **The holes were bored so that the distance between the sensors and
the measurement cavity wall is 5 mm. The sensors are connected to
two DAQ modules. The probes and the DAQ modules were calibrated at
RISE Research Institutes of Sweden, Borås, Sweden, to within a
combined uncertainty of 10 mK. To monitor drifts in the DAQ
modules, a $100{\Omega}$ standard reference resistor is
simultaneously monitored.

**44. **J. Zakrisson, I. Silander, C. Forssén, M. Zelan, and O. Axner are preparing a manuscript to be called
“Procedure for robust characterization of Fabry Perot
based refractometer.”

**45. **The length of the shortest gas modulation cycle was chosen to be
similar to those used in previous GAMOR assessments [31,32], viz., 100 s, since this cycle time
previously has been considered to be sufficiently long to allow
for a complete (equilibrated) filling of the measurement cavity
with gas and to achieve an equilibration between the temperatures
of the gas and the cavity walls (which, for the gas pressure
assessed in this work, both take place on a scale of seconds),
while still being as short as possible to reduce as much as
possible of the pickup of low-frequency
fluctuations.

**46. **It is not possible to perform a quantitative verification, since
this would require a knowledge of the amount of all Fourier
components of the fluctuations
and disturbances in the system, which is beyond the scope of this
work.

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