Gas modulation refractometry for high-precision assessment of pressure under non-temperature-stabilized conditions

The authors report on the realization of a novel methodology for refractometry-GAs modulation refractometry (GAMOR)-that decreases the influence of drifts in Fabry Perot cavity refractometry. The i ...


I. INTRODUCTION
The prevailing types of instrumentation for accurate assessment of pressures in the 1 Pa to 100 kPa range are liquid column manometers and force-balanced piston gauges.The most sensitive of these is the ultrasonic interferometer manometer 1,2 and the force-balanced piston gauge, 3 which both have regularly demonstrated detection of pressures with uncertainties in the mPa region.
][6][7][8][9][10][11] In this technique, the frequency of laser light, which is locked to a mode of a Fabry-Perot (FP) cavity in which gas is let in, is monitored.As shown in Sec.II, changes in the density of the gas give rise to alterations in the refractivity, which, in turn, provide a shift of the frequency.When done correctly, this approach has the potential to outperform manometers for assessments of the presence of gas, both in terms of its density and pressure.
However, although prophesized to be extraordinarily sensitive, the measurements have to be performed under extremely well-controlled conditions in order to reach the highest performance.It has also been shown that the assessments is, in practice, often limited by processes in the cavity spacer material, e.g., drifts, distortions, and relaxations, 12 as well as absorption of gas into its walls and outgassing. 13This implies that a cavity spacer made of a material with a number of conflicting properties, e.g., both a low thermal expansion and a low gas absorption, often subjected to particular treatments, a) Electronic mail: martin.zelan@ri.seb) Electronic mail: ove.axner@umu.see.g., cured (so as to achieve a minimum of internal stress), is needed.This puts severe restrictions on its use for everyday assessments.
To improve on this, systems based on two FP cavities, here referred to as a dual Fabry-Perot cavity (DFPC) refractometer, were developed. 5,6,10,12,13,24,25In these, one cavity is used as the measurement cavity while the other serves as a reference cavity.The justification of this is to eliminate common-mode effects of thermal drifts and relaxations of the spacer on the assessments.Besides reducing these, the introduction of a reference cavity also provides a well-defined reference light field to which the measured change in frequency of the measurement cavity can be assessed, measured as a radio-frequency (RF) beat frequency.Recently, such a wellcharacterized system, utilizing cavities drilled in ultralow expansion (ULE) glass under exceptionally well-stabilized temperature conditions together with HeNe lasers, has reported an expanded uncertainty of [(2 mPa) 2 þ (8.8 Â 10 À6 Â p) 2 ] 0.5 , where p is given in mPa.Here, the pressure dependent uncertainty originates mainly from the uncertainty in the refractivity of nitrogen while the pressure independent term predominantly comes from outgassing. 13o realize a refractometry system that is less subjected to environmental restrictions, we present in this paper a measurement methodology termed GAs modulation refractometry (GAMOR).In this, the measurement cavity is repeatedly filled and evacuated with the gas whose properties are to be assessed.7][28][29][30][31][32] Measurements of the beat frequency between the light fields locked to the measurement and reference cavities are performed with the measurement cavity sequentially being filled and evacuated, while the reference cavity is constantly evacuated.The measurements are performed both before and after the finitepressure measurement is made and can hence be used as means to provide zero point references for the assessments of the refractivity of the gas let into the cavity.By linear interpolation between the two reference assessments, the linear drift in the length of the cavity that takes place during the measurement of the finite pressure can be corrected for.The resulting "cavity-drift-corrected" beat frequency is also not affected by the drifts of the cavity spacer that take place on timescales longer than the gas modulation cycle.
As one major contribution to cavity length drifts is thermal expansion, this novel gas modulation methodology does not only relax the requirements on temperature stability but also implies that the cavity does not have to provide both a low thermal expansion and a low gas absorption, which often are seen as conflicting requirements.This opens up for the use of a larger choice of materials, e.g., Zerodur instead of ULE glass, which provides improved gas absorption properties.It also allows for high-precision assessments (with subparts per million resolution) in systems that are not constructed for exceptionally low drifts.This methodology thus mitigates the need for specialized temperature stabilized cavities allowing refractometry systems to be used for a wider range of applications.
In addition, the use of tunable narrow linewidth fiber lasers both facilities locking of the laser frequencies to modes of the cavities by the sturdy Pound-Drever-Hall (PDH) locking technique down to sub-kilo-hertz ranges and provides tunability over a few free-spectral ranges (FSRs) of the cavity.This opens up for assessment of any pressure that corresponds to cavity mode shifts of a few FSRs (for gases such as N 2 and Ar and a cavity with a length of around 20 cm, a few kilo-Pascal).It also allows for implementation of an automatic mode jumping (and relocking) procedure during gas filling and evacuation that allows for assessments of pressures for which the cavity modes are shifted more than the piezoelectric transducer-induced tunability of the laser while not shifting the laser frequency, Dv, more than at most one FSR of the cavity, v FSR .While the former allows for assessment of a wide range of pressures, the latter implies that effects of dispersion (from both the mirrors and the gas) can be minimized.
Moreover, the construction is simplified by the fact that the lasers are operating at standard wavelengths within the C34 communication channel (i.e., around 1.55 lm), at which there are a multitude of fiber coupled off-the-shelf optical, electro-optic, and acousto-optic components.All this implied that the system rather easily could be made fully computer controlled for unattended gas assessments over foreseeable time.
We provide in this work a description and assessment of the GAMOR methodology.To demonstrate its powerfulness, it is applied to an instrumentation that is based on an untreated cavity spacer (i.e., not exposed to any curing) equipped with mirrors that are simply pressed against the spacer material (hence with neither any gluing, nor optical contacting, which easily can be cleaned), under nontemperature regulated conditions.
Before the experimental realization is described, the principles of the methodology are schematically depicted.The performance of the system is then assessed by a series of measurements, both when no gas was let into the cavity, so as to assess the inherent properties of the methodology, and when it was referenced with respect to a dead weight piston gauge, when finite amounts of gas was assessed (up to 7.2 kPa).The procedures for how the GAMOR methodology is carried out when the pressure is provided by a piston gauge are described in some detail.
To illustrate the advantage of the GAMOR mode of detection, assessments were performed utilizing the system with two different evaluation methodologies: one in which the assessments were made under conventional (henceforth referred to as static) conditions and one when the GAMOR methodology was used.By studying the gas filling and emptying processes and by use of Allan plots of the measured refractometry signal expressed in pressure units, suitable gas modulation and detection conditions were assessed.It is shown that the GAMOR methodology can improve considerably on the precision of DFPC refractometry using a conventional (static) mode of detection, by more than 3 orders of magnitude, yielding sub-parts per million resolution.The experimental results presented in the paper concerns mainly noise and short-term stability, contributing to the precision.Despite that the accuracy of the instrumentation could not be assessed, the precision demonstrated indicates that the methodology can be useful for transfer of calibration between pressure standards and gauges over a large range of pressures, including systems with no overlapping working ranges.

II. THEORY
Evaluation of gas parameters from the GAMOR signal can be done by the use of conventional expressions for Fabry-Perot refractometry.As the amount of gas in the cavity is altered, the change in refractivity, Dðn À 1Þ, can, for pressures up to 10 kPa, be related to the change in frequency of the laser, Dv, and the change in cavity mode number to which the laser is locked, Dq, as 33 Dðn À 1Þ where v 0 is the frequency of the laser, q 0 is the cavity mode to which the laser is locked, both assessed for an empty cavity, 1 and g are gas and wavelength specific correction coefficients that take mirror dispersion and gas dispersion into account, respectively, while e is a refractometer-specific parameter coefficient that takes the relative elongation of the cavity due to the presence of gas into account, defined as DL E =L 0 ¼ e Á ðn À 1Þ, where DL E is the elongation of the cavity due to the presence of the gas.For the cases with low dispersion mirrors, and for most types of gases, e.g., N 2 , as is used in this work, the terms containing 1 and g can be neglected, in particular, for the cases when mode jumping is used (whereby Dv=v 0 is limited to v FSR =v 0 ), simplifying the expression above.
For the pressure range considered here, the refractivity can be related to the (molar) density of the gas, q, as 33 where A R and Bq are the molecular polarizability and the first density virial coefficient, respectively.The latter is given by Àð1 þ 4B R =A 2 R Þ=6, where, in turn, B R is the first refractive virial coefficient in the Lorentz-Lorenz equation. 34imilarly, the pressure can be related to density as 12 where B p ðTÞ is the first pressure virial coefficient.The values for the coefficients and constants used in this work, which are retrieved from the literature, 12,13,35 and, whenever needed, recalculated to the laser wavelength using in this work, are presented in Table I.
The pressure produced by the piston gauge, P PG , can be expressed as 36 where g is the local gravity acceleration, m p and m w are the masses of the piston and the weight placed on the piston, respectively, A p is the effective area of the piston, a is the thermal expansion coefficient of the piston and cylinder, T p is the piston temperature, T 0 is the temperature at which the area of the piston was measured, and P hood is the hood pressure.For the pressures used in this work (<100 kPa), the contribution from elastic distortion has been neglected.This implies that a change in the pressure in a cavity can be estimated from the change in the frequency of the laser (in practice, measured as a change in the beat frequency) and temperature by the use of the Eqs.( 1)-(3) using data from Table I, while the pressure provided by the piston gauge is given by Eq. ( 4).

III. EXPERIMENT
The heart of the refractometer comprised a DFPC that consisted of one reference and one measurement cavity, drilled in a common block of Zerodur with a size of 100 Â 100 (T À 300) 2 .This implies that for a temperature of 296 K, it is given by À5.36(9) Â 10 À6 m 3 /mol.c The cavity deformation coefficient e has been estimated by the use of a simple finite element simulation not including surface roughness or the geometry of the contact surfaces between the cavity spacer and the mirrors.d The mirror dispersion coefficient, 1, is given by v FSR v 0 GDDðv 0 Þ=2, where GDDðv 0 Þ is the group delay dispersion at the laser frequency v 0 which, for low-dispersion mirrors, can be as low as a few tens of fs 2 (and even close to zero when specifically made for the laser frequency).e The dispersion for nitrogen is, at 1530 nm, in the order of 6 Â 10 À21 Hz À1 (Ref.35).Hence the gas dispersion coefficient, g, which is given by ðv 0 =n 0 Þð@n=@vÞ, takes a value of 1.2 Â 10 À6 .
03E105-3 Silander et al.: Gas modulation refractometry for high-precision assessment 03E105-3 Â 190 mm.As can be seen in Fig. 1, the DFPC was placed on a 450 Â 600 Â 12.7 mm thin breadboard, which in turn was placed on an air-supported optical table.The breadboard was, together with the optical, acousto optic, and electro optical components, placed in a 1200 Â 600 Â 250 mm Plexiglas box.
Although the temperature of the entire laboratory was regulated, no part of the refractometer system was actively temperature stabilized.
Figure 2 shows a drawing of the cavity assembly.As is shown, the instrumentation was based on closed cavities with a small diameter (6.2 mm, separated by a distance of 50 mm, center-to-center distance).This implies that the relative elongation of the cavity due to the presence of gas (given by e) is significantly smaller than when open cavities or closed cavities with large diameters are used, 33 as previously often has been the case.
The cavities were made up of 1/2 in.mirrors (Layertec, 106587) that are mounted in aluminum holders, which were pressed against the spacer with the help of an aluminum frame.In contrast to earlier designs 20,22 the mirrors were simply pressed against the Zerodur block with standard 1/2 in.O-rings providing the seal.This required neither any gluing, nor any optical contacting.This design has the advantage that it enables a replacement or a cleaning of the cavity mirrors with relative ease.During the course of this work, this was regularly achieved within 30 min, from laser lock to laser lock.The finesse was assessed to 10 4 .
As is illustrated in Fig. 3, each of the two arms of the refractometer was addressed by light from a narrow linewidth Er-doped fiber laser, NTK, Koheras Adjustik E15),  emitting light with a wavelength around 1.55 lm (i.e., within the C34 communication channel).Before impinging upon the cavity, the light from each laser was sent through a fibercoupled acousto-optic modulator (AOM, AA Opto-Electronic, MT110-IR25-3FIO).The first order output from this was split by a 90/10 polarization maintaining fiber splitter (Thorlabs, PMC1550-90B-FC), whose high transmission output was connected to an electro-optic modulator (EOM, General Photonics, LPM-001-15) that was modulated at 12.5 MHz for the purpose of PDH locking. 37The output of the EOM was then coupled into free space by an output coupler and sent through a mode-matching lens, a polarizing beam splitter (PBS), and a quarter-wave (k/4) plate, before it entered the cavity.The low power outputs of the 90/10 fiber splitters of the two arms were merged by a 50/50 fiber coupler (Thorlabs, PMC1550-50B-FC) whose output was sent to a beat note detecting detector (Thorlabs, PDA8GS).
For each of the cavities, for the locking, both the back reflected and the transmitted light were used.The reflected light was passed through the quarter-wave plate (k/4) and was deflected by the PBS before it was collected with a fast photo-detector (Thorlabs, PDA10CE-EC).The transmitted light was detected by a large area photo-detector (Thorlabs, PDA50B-EC).The outputs from the two photo detectors were connected to a commercial digital servo module based on a field programmable gate array (FPGA, Toptica, Digilock 110).The reflected signal was demodulated at 12.5 MHz to produce a PDH error signal that was routed to two proportional-integral-derivative servos.One constituted a slow servo with a bandwidth up to 100 Hz, while the other consisted of a fast servo with a bandwidth up to around 100 kHz.The slow servo controlled the piezoelectric transducer of the fiber laser, while the fast was connected to the voltage controlled oscillator that produced an RF frequency of around 110 MHz for the fast frequency tuning of the first order output of the AOM.The transmission signal was used to control the automatic relocking enabling controlled mode hops.The relocking process takes typically a few tenths of a second.The output of the FPGA that was routed to the laser was limited to a voltage corresponding to a change of the laser frequency of two FSR.When the feedback voltage reaches this limit, the automatic relocking routine of the module relocks the laser to an adjacent mode with a frequency in the center of its working range.
The output signal of the beat note detecting detector was sent to a frequency counter (Tektronix FCA3003).The beat frequency was acquired with a rate of 10 Hz.As the frequency counter is limited to 3 GHz, temperature tuning was used to tune the frequencies of the two lasers so that the frequency difference (i.e., the beat frequency) was in the center of that range.After this initial manual (coarse) tuning, the automatic relocking routine kept the beat signal within the 0.5-3 GHz range for all measurement conditions.
The temperature of the cavity spacer was monitored by a thin foil Pt-100 sensor (JUMO, 20 Â 30 mm) mounted on top of the cavity spacer that was connected to a benchtop multimeter (Agilent, 34410A).The distance between the sensor and the measurement cavity wall was thereby approximately 50 mm.
In addition to the DFPC, the system consists of an automated vacuum system, shown in Fig. 4, for fast evacuation and filling of gas.The system comprises three solenoid valves (denoted 1, 2, and 3), four pressure gauges (marked by A, B, C, and D), and a piston gauge (RUSKA 2465A).In addition, there is a needle valve (unmarked) to regulate the flow of gas.The gas used was nitrogen, with a purity of 99.996% (suppled centrally at our facility).A computer was used to control the solenoid valves and to read the status of the pressure gauges, the temperatures of the piston in the piston gauge and the cavity spacer, as well as the beat frequency (from the frequency counter).
The wavelength of the laser was 1550.1325(3)nm, assessed by the use of a wavelength meter (Burleigh, WA-1500), which corresponds to a frequency of 193.3979(4)THz.The free spectral range (FSR) for the measurement cavity when being evacuated was assessed by the use of controlled mode jumps to 788.807(1) MHz.This implies that the cavity mode number addressed for an empty measurement cavity, q 0 , could be assessed to 245 178.
By a comparison of two piston gauges with each other, both set at 100 kPa, a short-term stability of this type of piston gauge has previously been assessed to 0.5 mPa. 38easurements of the long-term stability have revealed stabilities approaching 1 Â 10 À6 per decade. 36,38

IV. PRINCIPLES AND PROCEDURES A. Cavity-drift-corrected measurements based on gas modulation-Principles
The general principle for the GAMOR methodology is schematically illustrated in Fig. 5, which displays a single measurement cycle of the methodology in an idealized situation.As is shown in panel (a), starting from a state in which the measurement cavity is empty, it is, after a given time (in this case 15 s), filled with a certain pressure of gas.At a given time thereafter (50 s in this case), it is evacuated.The beat signal is measured during the entire measurement cycle.
However, as is shown by the dashed curve in panel (b), due to drifts of the cavity spacer, the measured beat signal will not be a complete replica of the gas pressure.To alleviate this, as is shown by the red line, the beat frequency from an evacuated cavity is estimated at all instances of the measurement cycle by use of a linear interpolation between two beat signals, measured with the measurement cavity being evacuated, viz., one measured before the filling of the gas and one measured after evacuation of it, respectively.A cavitydrift-corrected beat frequency corresponding to the gas in the cavity is calculated as the difference between the beat frequency measured (the dashed curve) and that interpolated from the two evacuated cavity measurements (the red line).
The corrected beat frequency is then used to calculate the pressure during the entire measurement cycle using the formalism given earlier.By this procedure, the effect of the linear drift of the cavities is efficiently eliminated from the assessment of gas density.For the time during which the beat frequency is stable (encircled by the black box), the latter is used for the assessment of the pressure in the connected system.However, to improve on the duty cycle, the cavity is, in practice, not fully evacuated between fillings; the emptying process is terminated when the pressure in the cavity has reached a given preset evacuation pressure.The pressure that remains in the cavity after the termination of the evacuation, which is measure by a pressure gauge, is referred to as the residual gas pressure.The shift of the beat frequency that corresponds to this residual pressure (relative to an empty cavity, i.e., vacuum) is estimated from the residual gas measurement by the use of the responsivity of the system obtained from earlier characterizations of the system.This beat frequency shift is used to calculate the actual beat frequency from a fully evacuated (i.e., emptied) cavity at low pressures.Two such fully evacuated cavity beat frequencies are then used for linear interpolation to obtain the cavity-drift-corrected beat frequency by the use of the idealized procedure described earlier.
The length of the evacuation period should be chosen wisely.The longer the evaluation period, the lower the residual pressure (i.e., the closer it will be to the ideal case of vacuum) and the higher the precision of the "fully evacuated cavity" measurements.However, too long evacuation times will increase the modulation period, which, in turn, will reduce the duty cycle and make the system more sensitive to nonlinear drifts.

B. Cavity-drift-corrected measurements-Experimental procedure
To assess the performance of the refractometer, it was connected to a dead weight piston gauge.As is described below, the practical realization of the GAMOR methodology differs in this case slightly from the general principle above.In practice, after a warm up time to allow for settling of transient effects in the system, a single measurement cycle is in these cases carried out by a series of computer controlled valve states, (i) -(iii), that result in a filling and an emptying of the measurement cavity in the refractometer, presented in Fig. 6, as follows: (i) The gas in the dead weight piston gauge that remains from the previous cycle is let into the refractometer by opening valve 2. This gives rise to an instantaneous pressure rise (to 3400 Pa at 18 s in Fig. 6).Valve 1 is simultaneously (or shortly thereafter) opened whereby the pressure in the combined volume of the piston gauge and the refractometer is increased at a slow rate (restricted by the needle valve placed before valve 1 in the system, unmarked in Fig. 4), shown as the steadily increasing pressure between 15 and 65 s in Fig. 6.This continues until the pressure measured by the pressure gauge B (P B ) exceeds a preset system pressure (P set system ), which has been chosen to be slightly above the set value of the piston gauge (P PG ).When this condition is reached, valve 1 is closed.The actual pressure to which the system was filled during this valve state is denoted as P 0 system .(ii) The pressure in the combined volume of piston gauge and refractometer is then let to stabilize, from P 0 system , to that regulated by the dead weight piston gauge, P PG .This takes place for a preset time that allows for both a proper settling process and a measuring time (illustrated by region ii in Fig. 6).A measurement of the beat frequency for a "filled" cavity is taken as the average over a part of the region over which the beat frequency is stable.After this, valve 2 is closed.(iii) The refractometer is then emptied by opening valve 3.
As was alluded to above, this continues until the pressure measured by pressure gauge C, P C , is smaller than a preset evacuation pressure, P set evac , i.e., until P C < P set evac (where the latter in this work is set to 0.1 Pa).A measurement of the beat frequency from an evacuated cavity is taken as the last data point before the next measurement cycle starts.The actual pressure in the system when the measurement of the beat frequency is performed in this valve state is called the residual pressure, P res .
The net beat frequency corresponding to the amount of gas in the cavity is evaluated from the beat frequency assessed with a filled cavity, corrected for the drift of the cavity by the cavity-drift-correction procedure described earlier.
The sharp "sparks" during the filling process in region i in Fig. 6 are artifacts from the relocking process during the automated (discrete) mode jumps (Dq ¼ À1) for which the measurement laser is unlocked for a short time.Since the evaluation procedure is not using data points within region i, they do not affect the assessments.
As a part of the characterization of the system, Fig. 7 displays zoomed parts of the measurement data illustrated in Fig. 6.The panels (a) and (b) show the pressure assessed during the final parts of the states ii and iii, respectively.The curve in panel (a) and the uppermost curve in panel (b) (both blue in color) display the pressure assessed by the refractometer.The lower curve in panel (b) (green in color) represents the pressure measured by pressure gauge C.
Panel (a) shows that in valve state ii, in which the pressure is regulated by the piston gauge, the pressure decreases exponentially, from the pressure the system obtained after closing valve 1, i.e., P 0 system , toward the pressure created by the piston gauge, P PG .This pressure equilibration takes place by a slow rise of the piston in the piston gauge.Based on this type of measurement, the length of valve state ii was chosen so it allowed for both a sufficiently long settling time and an adequate measuring time to assess the beat frequency that corresponds to P PG with sufficient precision.To allow for assessments of pressures in the entire pressure range addressed, this valve state was chosen to be 60 s of which the beat signal was averaged over 30 s.
The second panel in Fig. 7 shows the pressure in the system in valve state iii as measured by pressure gauge C and the refractometer (the red and the blue curves, respectively).The data show that during this valve state, the pressure decreases monotonically, although differently fast for the pressure gauge and the refractometer.This is due to the fact that the pressure gauge is mounted outside of the measurement cavity and hence will be pumped down faster than the refractometer.This shows that the pressure measured by pressure gauge C, i.e., P C , is not a fully correct representation of the residual pressure in the cavity, P evac .However, since the evacuation process is reproducible, a characterization of the process can be done.By using a model for gas conductivity, and a set of measurements, such as those displayed in panel (b), it was revealed that the actual pressure in the cavity, P evac , under the prevailing conditions, is 55% higher than P C , irrespective of the pressure provided by the piston gauge.This value was therefore used for the estimation of the residual pressure in the cavity from the pressure measured by pressure gauge C.

V. RESULT
To assess the advantages of the GAMOR methodology with respect to the conventional (static) mode of detection, measurements were carried out by the aforementioned gas modulation procedure (thus with the cavity being repeatedly filled and evacuated with gas) but evaluated in two different ways; i.e., both according to (1) the conventional (static) mode of evaluation, performed by evaluating the beat frequency between the two cavities solely from the periods when the measurement cavity is filled with gas (i.e., without periodically referencing it to the beat frequency of an evacuated measurement cavity), and (2) the GAMOR methodology, in which the net beat frequency is assessed as the difference between the beat frequency measured with a filled cavity and the linear fit between the beat signals that corresponds to empty cavities before the filling and after the evacuation of the cavity, by use of the cavity-drift-correction procedure described above.
By this procedure, it was guaranteed that the two sets of assessments, (1) and ( 2), representing the two modes of evaluation, static and GAMOR, respectively, were based on the same set of measurements.This implies that the differences in performance between the two detection methodologies can be considered conclusive.

A. Evaluation of the GAMOR instrumentation by the use of an evacuated measurement cavity
To make a thorough evaluation of the GAMOR methodology, a comparison of the two modes of evaluation (GAMOR and static) was first assessed for an evacuated measurement cavity.In this case, despite that no gas was filled into the measurement cavity, the GAMOR data were still evaluated by the methodology described in Fig. 5 as if it would be exposed to a periodic filling and emptying of gas in the measurement cavity over a hundred second cycle, i.e., by subtracting the beat frequency from a linear interpolation between two cavity evacuated measurements from the beat frequency measured at the time instance when the cavity normally contains gas. Figure 8 displays a 24 h long series of measurement when the measurement cavity was not exposed to any filling or emptying of gas.Panel (a) shows, by the red curve, the pressure assessed by the static mode of evaluation, i.e., (i) above, while the black (almost fully horizontal) curve, seemingly overlapping the x-axis, displays the corresponding entity when the GAMOR methodology was used, i.e., (ii), both with respect to the left axis.The various data points, which for each series merge into a continuouslooking curve, represent in both cases the pressure estimated from the measured beat frequency between the light fields locked to the measurement and reference cavities by the use of Eqs. ( 1)-(3) using data from Table I.
The blue curve (the right axis) represents the measured temperature variations in the cavity spacer.The cavity spacer temperature was taken as a rolling mean over 100 s, which was found to minimize both the noise and the drift of the refractometer assessments.It can be noticed that this period also corresponds to the time of an individual measurement cycle.As is shown by the blue curve, the temperature of the nontemperature stabilized cavity was found to drift around 300 mK over The red curve in panel (a) shows that the static mode of evaluation is significantly affected by drifts (in this case with a standard deviation of 6.4 Pa).As can be seen by a comparison with the (blue) temperature curve, despite that a DFPC was used (so that the changes in the measured beat frequency from an evacuated measurement cavity correspond to changes in the difference in length of the two cavities), the drifts of this mode of detection correlate with the temperature; the assessed pressure increases with decreasing temperature (and vice versa).This shows that the drifts of the static mode of detection originate largely from drifts in the length of the cavity spacer caused by drifts in the temperature.For comparison, it can be noted that a drift in the assessed pressure of 1 Pa corresponds to a change in the difference in length of the two cavities of solely 0.6 nm (representing a few atom layers).
The data taken with the GAMOR methodology (the black curves in the two panels), on the other hand, do not show the same behavior.They do not display any drift that is correlated to the temperature; there are no visible drifts of this signal over the 24 h measurement on the scale used in panel (a); their fluctuations, which are enlarged in panel (b), are mostly composed of high frequency noise.Consequently, the data taken with the GAMOR methodology have significantly lower standard deviations, for the data shown in Fig. 8, 1.5 and 3.5 mPa for the 1 and 24 h long series of measurements, respectively.This is more than 3 orders of magnitude smaller than for the static mode of detection (6.4 Pa).This demonstrates that the GAMOR methodology has an immense resistance (almost a complete immunity) to thermal expansion and creeping of the cavity spacer.

B. Evaluation of the GAMOR instrumentation for assessment of finite pressures
To assess the precision of the GAMOR mode of detection when finite pressures are assessed, measurements were performed at two pressure levels.The levels were selected among the ones the piston gauge could provide and to those the refractometer in its present configuration could handle (where the latter was limited to around 10 4 Pa because of the load on the turbo pump during evacuations).Figure 9 displays two series of measurements from $700 repeated fillings of the refractometer (taken over 20 h) with pressures of 4303 and 7226 Pa of nitrogen, respectively [provided under regulatory conditions by the piston gauge, given by Eq. ( 4)].In this case, the data represent, for each measurement cycle, the difference in pressure assessed by the refractometer and that produced by the piston gauge, where the former again was assessed from Eqs. ( 1)-( 3) by use of the net beat frequency, the number of mode jumps performed, and the measured temperature.The latter was estimated from Eq. ( 4) by the use of the weight placed on the piston, its temperature, and the measured hood pressure.The data displayed in panel (a) correspond to the static mode of evaluation (i.e., with no reference to an evacuated measurement cavity) while those in the panels (b) and (c) originate from the GAMOR mode of evaluation.The two sets of data displayed represent two series out of several sets of measurements that show significantly different amounts of drifts in the static mode of evaluation.For simplicity, all curves have been shifted so they provide a value of zero at the start of the measurement series.Panel (a) shows, by the blue curve, taken at 7226 Pa, that pressures assessed by refractometry by the use of the static mode of evaluation can deviate significantly from those produced by the piston gauge (in this case, 15 Pa for a measurement time of 20 h).A comparison of the two sets of data (blue and red curves) reveals though that this deviation can vary significantly between various measurement series; the standard deviations of the two series of measurements displayed in panel (a) are 0.4 and 4.3 Pa, respectively.The reason for the large spread of drifts among the various measurement series is that the drifts originate mainly from drifts of temperature, which can (and will) vary considerably over time.It can also be noticed that the drifts displayed by an evacuated cavity [the red curve in Fig. 8(a)] are of similar magnitude to the measurement series that shows the largest drifts at 7226 Pa (i.e., the blue curve in Fig. 9).
Panel (b) shows that when the GAMOR mode of detection is used for the evaluation of the same set of data, it again does not pick up more than a minute fraction of the drifts in the system.The deviations between the pressure assessed by the refractometer and that provided by the piston gauge are again significantly smaller than when the data are evaluated by the static mode of evaluation, this time almost 3 orders of magnitude smaller ($8 and $11 mPa vs $4.3 Pa).This shows that also when finite (kPa) pressures are assessed, the GAMOR methodology has a significant advantage over the static mode of detection; it produces a significantly lower level of fluctuations than the static mode of evaluation, especially from those that originates from drifts of the length of the cavity spacer.
Panel (c) displays the data from solely the first hour of the measurement series shown in panel (b).It was found that the standard deviations of these two sets of data were smaller than those for the 20 h measurement series (3.1 and 4.4 mPa vs 8 and 11 mPa, respectively).Other 1-h sections of the 20h long measurement series show standard deviations with similar magnitude as that of the first hour.The fact that the standard deviations of the shorter sets of data are smaller than those of the longer indicates that although the GAMOR mode of detection reduces drifts of the measured beat frequencies significantly with respect to the static mode of detection, the data still exhibit some minor (residual) degrees of drift.
The dashed horizontal lines in the panels (b) and (c) correspond to 6 2r measured over the length of the curves, which amount to 16 and 22 mPa for the 20 h measurement [displayed in panel (b)], and 6.2 and 8.8 mPa for the 1 h set of data (the lower panel).
Since the optimum detection conditions for the GAMOR methodology are expected to encompass those where the combined influence of white noise and drifts is minimized, it is of interest to assess the stability of the system for the GAMOR mode of evaluation over various time scales.This can most conveniently be done by the use of Allan plots.

C. Allan deviation analysis of the system
The Allan variance, r 2 A ðsÞ, first introduced by Allan in 1966 for characterization of frequency standards, 39 is a measure of the difference between consecutive points or groups of data, each averaged over a time s.Hence, it is conveniently displayed as a function of the averaging time the data are exposed to.This implies that it constitutes an assessment of the time stability of a measurement system.It is therefore an excellent tool for assessment of which types of noise a system is affected by and it can be utilized to identify suitable modes of operation.As long as the system is affected by white noise, as often is the case for short integration times, the Allan deviation, r A ðsÞ, which is the square root of the Allan variance, decreases with averaging time as s À1=2 .In a log-log plot of r A ðsÞ vs s, this corresponds to a slope of À1/2.However, for longer integration times, other types of noise usually show up, eventually making the Allan deviation to increase with s, indicating that the system is dominated by drifts on these time scales.For the case when the data show a white-noise behavior, the Allan deviation is identical to the standard deviation. 39he Allan deviations of the data shown in Figs. 8 and 9 are presented in Fig. 10.While the blue and the red curves display the deviations of the pressure assessments in the kPa region (representing the two sets of data shown in Fig. 9, thus corresponding to 4303 and 7226 Pa, respectively), the two black curves represent a measurement series with the entire system being evacuated and not referenced to the piston gauge.The solid black curve corresponds to the black curve in Fig. 8.To assess the combined influence of the noise and drifts of the residual pressure gauge (i.e., gauge C) and the residual pressure in the cavity on the assessment, the dashed-dotted curve represents the same data when no reference is made to this pressure gauge (i.e., assuming that P evac was truly zero).As means to guide the eye, the horizontal dashed lines represent Allan deviations of 1.5 and 4 mPa, respectively.
All four data sets show white noise behavior up to around ten measurement cycles, i.e., $10 3 s.When the refractometer is not referenced to the piston gauge and when no reference is made to the residual pressure gauge (the black dashed-dotted curve), the uncertainty of a GAMOR assessment of an evacuated measurement cavity shows a white noise dependence with an Allan deviation, r wn EC , of 1:4N À1=2 mPa where N is the number of measurement cycles, for all measurement times investigated (up to 100 measurement cycles, corresponding to 10 4 s).Since the Allan deviation is identical to the standard deviation when the data show a white-noise behavior, 39 this indicates that the system exhibits a white noise response with a standard deviation with the same value.This shows that, in the absence of the residual pressure gauge, a single measurement cycle has a (1r) uncertainty of 1.4 mPa while, when 100 cycles are averaged, it decreases to a value slightly above 0.1 mPa.
Since the difference between the solid and the dasheddotted black curves corresponds to the influences of the residual pressure and pressure gauge C, it can be concluded from a comparison of these two curves that for averaging up to 10 cycles (10 3 s) the GAMOR assessment is not significantly affected by noise or drifts of the residual pressure gauge.For longer averaging times, however, and with the instrumentation used in this work, a GAMOR assessment is affected by noise or drifts of the residual pressure gauge.
All this implies that when pressure measurements are made on an evacuated measurement cavity the GAMORmethodology-based system displays a (1r) uncertainty below 1 mPa when the data are averaged over 2-10 measurement cycles.Since the Allan deviation is not directly related to the standard deviation when the data are not white-noise limited, i.e., when it is affected by other types of noise (e.g., drifts), it is not possible to make a direct assessment of the precision provided by the data from the Allan deviation for averaging times above ten measurement cycles.
When the refractometer is referenced to the piston gauge, and when finite pressures are being assessed (the solid blue and red curves), the amount of noise increases.The upper two curves in Fig. 10 show that when kilo-Pascal pressures are assessed over short time scales (s < 5 cycles), the refractometer is still limited by white-noise, although this time with a standard deviation in the 3:0N À1=2 -3:5N À1=2 mPa range.

A. Precision
Since the Allan deviation is identical to the standard deviation when the data show a white-noise behavior, 39 it is possible to assess the precision of the instrumentation by use of the white noise dominated parts of the Allan deviation.Before assessing this, it should be concluded that while one part of the difference between the Allan deviation plots from the assessments with an evacuated measurement cavity and the ones addressing kPa pressures can be attributed to the refractometer, another part is considered to originate from the piston gauge.This implies that the noise levels shown by the blue and red curves in Fig. 10 can be interpreted as upper levels for the noise of the refractometer.

Assessment of sources of white noise from the refractometer
There are at least two quantifiable sources of white noise that can be attributed to the refractometer.
First, in the present realization of the GAMOR instrumentation, there is a certain amount of timing jitter between the assessments of the pressure provided by pressure gauge C and the beat frequency assessed by the frequency counter (originating from the data acquisition procedure).Since the gas evacuation process in valve state iii has not reached a complete equilibrium when the residual gas pressure data are taken, the lack of full synchronization between the acquisitions of the pressure and the beat frequency will appear as "steps" in the data [as can be seen in Fig. 7(b)].This will contribute to the uncertainty of the final assessment.For the present realization of the system, the standard deviation of the timing jitter was assessed to 40 ms.Since it was found that at the end of valve state iii (at which the pressure is around 0.1 Pa) the pressure is decreasing with a rate of around 10 mPa/s, this implies that this phenomenon can give rise to an uncertainty in the assessment of pressure, r wn sync , that can contribute to the precision with 0:4N À1=2 mPa.
Second, another quantifiable source of noise is the assessment of the temperature of the gas.The white noise from the measurement of the resistance of the Pt-100 probe, expressed as a relative uncertainty, i.e., as r wn T =T, was found to be 3 Â 10 À6 Hz À1/2 .Since, in this work, the cavity temperature was assessed by taking a rolling average over one measurement cycle, the noise of the temperature assessment amounts to 100 lK, which, for the cases with 4303 and 7226 Pa, correspond to white noise levels in the assessment of the pressure of 1:3N À1=2 and 2:2N À1=2 mPa, respectively.
Under the assumption that these sources of noise are uncorrelated, their contributions to the precision of the refractometer, expressed in terms of a standard deviation r wn P , (which thus is relevant for integration times up to around 10 cycles) can be estimated to be given by This indicates that, for the pressures assessed above, i.e., at 4303 and 7226 Pa, it can be estimated that the standard deviations of the white noise levels of the refractometer can contribute to the precision by at least 2:0N À1=2 and 2:6N À1=2 mPa, respectively.

Assessment of sources of white noise from the piston gauge
Since the piston gauge has not been fully characterized, its contribution to the precision of the measured data is presently not fully known.Despite this, however, it is possible to provide rough estimates of some of its possible contribution.
One contribution is the variation in the pressure of the piston gauge that originates from variations of the height of the piston, which, in turn, come from the fact that the relative effective area of the piston can have a dependence on the piston height.When this is the case, there will be cycle-to-cycle variations in the pressure that are caused by the variations of P 0 system during valve state i.As is shown in an endnote, it can be estimated that this process can contribute with white noise (and thereby the precision) on the order of 2N À1=2 mPa. 40nother contributor to the noise is the hood pressure gauge.The white noise contribution of this pressure gauge was assessed to 1 mPa Hz À1/2 , which, for an averaging period of 30 s (the time over which data were averaged) provides a contribution to the white noise of 0:2N À1=2 mPa.This indicates that, when finite pressures are assessed, the standard deviation of the white noise attributed to the piston gauge is dominated by the variations in the height of the piston, which in turn are caused by the variations of P 0 system during valve state i.

Other sources of white noise in the system
While the aforementioned sources of noise can account for the major part of the experimentally assessed white noise in the system, and thereby the precision of the instrumentation, it is possible that there are also contributions from other sources, which are less straightforward to estimate or assess.Two examples of such are fluctuations in the temperature between the temperature probe and the gas 41 and vibrations and mechanical noise. 42It is possible though to see the remaining parts of the assessed white noise as an upper limit for the combined contribution of these.However, since the system still is under development, it was found sufficient, at this stage, to identify the existence of the latter sources of noise; the contribution from each of these sources of noise will be properly assessed in the future.

B. Accuracy
Based on the properties of the GAMOR methodology, in particular its reduced susceptibility to drifts of the cavity, the main aim of this work has been to demonstrate the method's capability to provide high precision assessments of gas pressure in systems that are not necessarily well stabilized with respect to their drifts.To assess this, the methodology was applied to a nontemperature-stabilized cavity made out of Zerodur onto which the mirrors are simply pressed.Although this served the purpose of assessing the potential of the GAMOR methodology to perform high-precision assessments of gas in system exposed to non-negligible thermal drifts, this first realization of a GAMOR instrumentation, was not primarily constructed for high accuracy.
In general, an assessment of the accuracy of a system requires calibration of the system under a set of well-defined conditions, and that all physical entities that affect the response of the system (e.g., the temperature of the gas) can be accurately assessed during the period over which the system is used.Since no pressure standard has been available during the course of the development the GAMOR methodology, it was not possible to assess the accuracy of the instrumentation.
However, it was possible to perform a relative comparison between the estimated pressure set by the piston gauge [according to Eq. ( 4)] and the pressure estimated from the refractometer measurements [according to Eqs. ( 1)-( 3) and the estimated quantities given in Table I].Such a comparison indicated that the system presently provides a pressure that supersedes that predicted by the piston gauge by 0.2%.This deviation can have a number of causes, e.g., uncertainties in the temperature assessment (primarily due the placement of the probe but also, to a certain extent, a lack of calibration), the mechanical stability of the cavity when exposed to gas pressure, the estimate of the molecular coefficients and constants given in Table I, and to a smaller degree, the accuracy of the piston gauge pressure.Based on an improved version of the system (with improved temperature assessment) that is to be constructed at RISE Research Institutes of Sweden (where pressure standards are accessible), the accuracy of the GAMOR methodology will be addressed in more detail in a future work.At the current stage, it is only possible to discuss potential differences between the accuracy when pressure is assessed by use of the GAMOR methodology and when it is measured by use of the static mode of detection.Most of the known contributions to the uncertainty of DFPC refractometry are the same in the GAMOR methodology as when the static mode of detection is used.One distinct advantage with the GAMOR methodology though, is that it has a lower susceptibility than the static mode of detection to cavity length drifts; as is illustrated by Fig. 5, the GAMOR methodology is only affected by nonlinear drifts during a single measurement cycle, and it is hardly affected at all by drifts and relaxations between measurement cycles.This implies that the GAMOR methodology reduces the need for a spacer material with an exceptionally low heat expansion coefficient.It thereby provides a potential to realize well-constructed systems with accuracies that are higher than when the static mode of detection is used.However, a disadvantage is that the transfer of gas adds an uncertainty related to the gas transfer process, including the periodic heat transport with the gas in and out of the cavity, which can be significant at higher pressures (around atmospheric pressure).This implies that the accuracy of a given instrumentation based on the GAMOR methodology will largely depend on the system design (the spacer material, the degree of temperature stabilization, the heat capacity of the gas in the cavity etc.).
It is also worth to note that one particular interesting application of the GAMOR methodology is the transfer of calibration between two units-a "standard" and a unit to which the calibration is transferred, possibly at a different pressure.For this purpose it is sufficient to ensure that the two assessments are performed on such a time scale that the measurement is not significantly affected by drifts, and that the response is linear over the pressure range over which the calibration is transferred [or, alternatively, the coefficients of the response given by the Eqs.( 1)-( 3) are known with sufficient accuracy].In this case, the measurement on the standard calibrates the system for the duration of the calibration transfer process.This implies that the accuracy of the calibrated unit will be given by a combination of the accuracy of the standard (if the system is used in its nonlinear region, the accuracy of the coefficients describing the refractometer response) and the precision of the refractometer (the latter over the timescale of the calibration process).Therefore, if the transfer of calibration is performed on a restricted time scale (in this particular case around 10 3 s) the system will benefit from the high precision presented above.

VII. SUMMARY AND CONCLUSIONS
In this work, we have presented and demonstrated a new type of detection and evaluation methodology for assessment of gas pressure by refractometry, termed GAs modulation refractometry and denoted GAMOR.The instrumentation is based on a DFPC refractometer in which one cavity is used as the measurement cavity (in which the gas under scrutiny is let) while the other serves as the reference (which constantly is evacuated).Although ordinary refractometry has been shown to be sensitive, and thereby capable of detecting small changes in gas density, the technique is, in its standard configuration, unfortunately notoriously perceptive to changes (noise and drift) in the length of the cavity.This often sets the limit for its performance.
In the GAMOR methodology, the measurement cavity is repeatedly filled and evacuated with the gas whose properties are to be assessed.This implies that assessments of properties of a gas can be performed without being affected by the linear parts of the drifts of the cavity length that take place during the measurement and the drifts that take place on timescales longer than the gas modulation cycle that conventional refractometers often are limited by.A first version of such an instrumentation has been realized and its basic performance has been scrutinized.
It has been shown that the GAMOR methodology can significantly reduce the susceptibility of optical refractometry to cavity drifts.This implies that it allows for reduced requirements on the design of the cavity.A first instrumentation was therefore constructed around off-the-shelf components, including a nontemperature stabilized DFPC refractometer made of Zerodur onto which the mirrors are pressed toward the cavity spacer (which allows for easily servicing and cleaning).To allow for an easy implementation, the instrumentation is also based on tunable narrow linewidth fiber lasers, which facilitates locking of the laser frequencies to some modes of the cavities by the PDH technique and the implementation of automatic mode jumping (and relocking) features.The lasers are operating at standard wavelengths within the C34 communication channel (i.e., around 1.55 lm), at which there are a multitude of off-the-shelf optical, electro-optic, and acousto-optic components (of which many are fiber coupled).The system is fully computer controlled and can perform unattended gas assessments over any foreseeable length of time (unattended 24 h measurement series are repeatedly performed).Using this instrumentation, in comparison with a static mode of detection, the GAMOR methodology has demonstrated an immense resistance (almost a complete immunity) to thermal expansion and creeping of the cavity spacer, which has resulted in a 3 orders of magnitude improvement of the precision.
To assess the performance at low pressures, the system was scrutinized by the use of an evacuated measurement cavity.In this case, when the system was not referenced to any pressure gauge for assessment of the residual pressure in the cavity, an analysis based on Allan plots was performed.It showed that the system displays a precision with a white noise behavior with a standard deviation of 1:4N À1=2 , which thus corresponds to a (1r) uncertainty of slightly above 0.1 mPa for an integration time of 100 cycles (10 4 s).When the residual pressure was assessed by a pressure gauge, which is needed to implement cavity-drift-corrected assessments, it was found that the system expresses the same amount of white noise up to averaging times of ten measurement cycles.For averaging times above this, the performance of the refractometer is limited by the combined noise from the residual pressure gauge and the residual pressure.
To assess finite amounts of gas, the refractometer was referenced to a dead weight piston gauge (RUSKA 2465A) that acted both as a pressure reference and stabilizer.The system has so far been applied to pressures up to 10 kPa.It was found that for short measurements times (up to a few measurement cycles) the instrumentation could provide assessment of pressures in the kPa range (4303 and 7226 Pa) limited by white noise with standard deviations of 3:2N À1=2 and 3:5N À1=2 mPa, respectively.
Since the precision assessed in this work includes effects from noise from both the refractometer and the piston gauge, it is possible to conclude that the noise levels of the GAMOR system is presumably lower than those assessed experimentally.It was estimated that for assessment of 4303 and 7226 Pa pressures identified noise sources from the refractometer contribute to the white noise of the system with standard deviations of 2.0 and 2.6 mPa, respectively.This implies that the system exhibits a (1r) relative precision of 0.7 (0.5) ppm for assessment of pressures in the 4 kPa region and 0.5 (0.4) ppm around 7 kPa, where the numbers in parentheses represent the part that can be identified to originate from the refractometer.In the present realization, the system is limited by the noise in the assessments of the temperature of the gas and timing jitter between the assessment of the residual gas pressure in the cavity and the beat frequency.
Since this is the first realization of a GAMOR system, and further improvements of it is currently under way, it is assumed that, in the closest future, some of the sources of noise can be reduced and that the origin of the amount of noise experienced can be more precisely assessed.
It has previously been argued that conventional DFPC refractometers have the ability to assess pressure over a large pressure range, limited by the cavity deformation. 13We expect the GAMOR methodology to have the same property.The fact that an automatic mode jumping and relocking process has been demonstrated implies that the GAMOR methodology in principle can rapidly accommodate a variety of pressures.This will open up for the use of the instrumentation for efficient transfer of calibration between systems, including those with no overlapping working range.It can also be used for detailed characterization of pressure gauges, including piston gauges.Due to the periodic gas exchange, the methodology is particularly applicable to situations when the system cannot be actively temperature stabilized or when the system is affected by outgassing or leaks.
Since the GAMOR methodology is based on a modulation procedure, with all benefits such a process can bring, it can be applied to a variety of types of cavities; not only the one used in this work.This opens up for the realization of many types of systems, dedicated to a number of application areas.
Moreover, since density can be assessed without any reference to gas temperature, it is assumed that density assessments can be performed with even better precision than pressure.We therefore prophesize that the new GAMOR methodology can be used for high precision assessments of also density and refractivity under a wide range of conditions. 40The fluctuations in piston height, dh, can be estimated from the variation of the pressure to which the system was filled during valve state i, i.e., dP 0 system , as dh ¼ ðdP 0 system =P PG ÞðV=A p Þ, where V is the combined volume of the piston gauge and the refractometer and A P is the piston area.For the system used, the ratio of the two latter can be assessed to 1 m.Under the assumption that the relative effective area of the piston has a dependence on the piston height, a, that is comparable to those of other piston gauges, which has been assessed to 0.4 ppm/mm (Ref.38), the fluctuations in the pressure provided by the piston gauge, dP PG , due to variations of the height of the piston, can be estimated from dP PG ¼ a P PG dh.Since it was estimated that a typical fluctuation of the pressure to which the system was filled during valve state i (i.e., dP 0 system ) was 5 Pa, this shows that the fluctuations of the filling pressure due to cycle-to-cycle variations of P 0 system , i.e., dP PG , can be estimated to be 2 mPa. 41Due to the finite thermal conductivity of the cavity spacer material, and the present construction of the cavity, it is possible that not all fluctuations (noise or drifts) in the temperature of the gas in the cavity can be directly assessed by the temperature probe.In this case, it is possible that fast fluctuations in the gas temperature can cause a white noise uncertainty in the pressure assessed. 42Vibrations and mechanical noise can couple in to the system in a variety of ways, e.g., through the length of the cavity, through the capacitive gauge diaphragm, and by pick up through the pressure gauge (e.g., changes in g).

FIG. 1 .
FIG. 1. (Color online) Refractometer part of the GAMOR instrumentation.The refractometer is placed in 1200 Â 600 Â 250 mm Plexiglas box to reduce environmental fluctuations.The DFPC, some gas lines, and a number of free space optics are placed in the left part of the box, while the fiber optical components appear under the right box lid.

FIG. 2 .
FIG. 2. (Color online) Drawing of the DFPC assembly.Four 1/2 in.mirrors are mounted in mirror holders that are pressed onto the cavity spacer.The seal between the spacer and the mirror is achieved by the use of a standard 1/2 in.O-ring.To avoid stress passing from the frame to the cavity spacer the mirror holders are suspended on standard silicone lab tubing.The Zerodur spacer rests on a part of the aluminum frame, which is separated from the spacer by a 1 mm Teflon sheet.

FIG. 5 .
FIG.5.(Color online) Principle of the GAMOR methodology.Panel (a) displays a single measurement cycle in which the cavity, which originally is empty, first is filled with a certain pressure of gas before it is emptied again.Panel (b) shows, by the dashed curve, the corresponding beat frequency measured with the system (for simplicity in the absence of mode hop), starting from an arbitrary chosen beat frequency.The two crosses (red in color) represent two assessments of the beat frequency with the measurement cavity being evacuated.The line between the crosses (red in color) represents the evacuated cavity beat frequency at all times during the measurement cycle, estimated by interpolation between the two evacuated cavity assessments.The solid curve in the same panel shows the cavity-drift-corrected change in beat frequency due to the presence of the gas, calculated as the difference between the measured beat frequency (the dashed curve) and the estimated evacuated-cavity beat frequency (the line interpolated between the crosses, red in color).The box indicates a period of time over which the pressure is assessed from the cavity-drift-corrected change in beat frequency.

FIG. 7 .
FIG. 7. (Color online) Zoom in of the pressures assessed in the valve states ii [panel (a), during stabilization of the pressure] and iii [panel (b), emptying of the refractometer].The curve in panel (a) and the uppermost curve in panel (b) (both blue in color) display the pressure assessed by the refractometer, while the lower curve in panel (b) (green in color) displays the pressure assessed with the gauge C. The dashed lines represent the time instances for which there are changes in the valve states.
24 h.Panel (b) shows an enlargement of the first hour of the data evaluated by the GAMOR methodology [the black curve in panel (a)].

FIG. 8 .
FIG. 8. (Color online) Panel (a): A 24 h long series of measurement of an empty measurement cavity evaluated by two different means: the lowermost curve (red in color)-without gas modulation, referred to as a static mode of detection, and the almost fully horizontal curve (black in color)-by use of the GAMOR methodology (both left axis).The uppermost curve (blue in color and right axis) represents the temperature.Panel (b): a zoom in of the first hour section of the data taken with the GAMOR methodology.

FIG. 9 .
FIG. 9. (Color online) Difference in the pressure assessed by the refractometer and that provided by the piston gauge for pressures of nitrogen of 4303 Pa (lowermost and blue curves) and 7226 Pa (uppermost and red curves).Panel (a): Pressure evaluated by the static mode of detection, thus with no reference to an evacuated cavity.Panel (b): Pressure evaluated by the GAMOR methodology, thus with recurring reference to an evacuated cavity.Panel (c): The first hour data from the measurements displayed in panel (b).In the latter two panels, the dashed lines represent the levels for 6 twice the standard deviation of the data over the time displayed [16 and 22 mPa in panel (b), and 6.2 and 8.8 mPa in panel (c), for 4303 and 7226 Pa, respectively].

FIG. 10 .
FIG. 10. (Color online) Allan deviation of the measurements presented in Figs. 8 and 9.The two uppermost solid curves represent the Allan deviations of the refractometer referred to the piston gauge for pressures of 4303 Pa (the predominantly uppermost curve, blue in color) and 7226 Pa (the slightly lower curve, red in color).The two lowermost curves (black in color) represent the deviations of measurements of an evacuated cavity not referenced to the piston gauge.The solid and dashed-dotted black curves represent data taken with and without the use of pressure gauge C, respectively.The horizontal dashed lines represent Allan deviations of 1.5 and 4 mPa.

TABLE I .
Coefficients and constants used in this work.