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Optical measurement of the gas number density in a Fabry–Perot cavity
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2013 Meas. Sci. Technol. 24 105207
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IOP PUBLISHING
MEASUREMENT SCIENCE AND TECHNOLOGY
doi:10.1088/0957-0233/24/10/105207
Meas. Sci. Technol. 24 (2013) 105207 (5pp)
Optical measurement of the gas number
density in a Fabry–Perot cavity
Isak Silander 1 , Martin Zelan 2 , Ove Axner 1 , Fredrik Arrhén 2 ,
Leslie Pendrill 2 and Aleksandra Foltynowicz 1
1
2
Department of Physics, Umeå University, SE-901 87 Umeå, Sweden
SP Swedish Technical Research Institute, SE-50 115 Borås, Sweden
E-mail: [email protected] and [email protected]
Received 8 May 2013, in final form 12 August 2013
Published 17 September 2013
Online at stacks.iop.org/MST/24/105207
Abstract
An optical method for measuring the gas density by monitoring the refractive index inside a
high-finesse Fabry–Perot cavity is presented. The frequency of a narrow linewidth Er:fiber
laser, locked to a mode of the cavity, is measured with the help of an optical frequency comb
while the gas density inside the cavity changes. A resolution of 1.4 × 10−6 mol m−3 is
achieved in 3 s for nitrogen, which allows measurement of a relative gas density change of
3.4 × 10−8 at atmospheric pressure.
Keywords: laser refractometry, gas density measurement, metrology, optical frequency comb,
Fabry–Perot cavity
(Some figures may appear in colour only in the online journal)
1. Introduction
index [4], which, in turn, has spurred significant advances
in calculations of atomic properties such as refractivity and
polarizability [5]. Optical refractometry, and in particular
laser refractometry, has been successfully used to measure
the absolute refractive index of dry gas to very high accuracy
[6, 7]. It has also been applied to measurements of the density
of air in order to compensate for the buoyancy in precision
mass metrology [8, 9]. Interferometry with optical frequency
combs (OFCs) has been employed for measurements of the
refractive index of air and other gases [10–12].
We demonstrate a highly sensitive technique for the
measurement of the number density of a gas inside a highfinesse Fabry–Perot cavity, based on an earlier feasibility
study [13, 14]. In this method, a narrow linewidth fiber
laser is locked to a mode of the cavity and the change in
laser frequency caused by the shift of the cavity mode with
changing refractive index is monitored with the use of an
OFC. The technique provides fast time response where the
achieved resolution in density determination is limited by
the OFC reference and is independent of the absolute gas
pressure. This makes the technique particularly appealing in
higher density ranges, where the conventional techniques lack
accuracy.
Measurement of changes of gas density has several
applications, where the most straightforward are for gas flow
and leak detection and calibration. Conventional methods
for the determination of gas density variation involve a
measurement of changes of the pressure (under constant
volume) or the volume (under constant pressure) [1]. In both
these cases, the assessment relies on the use of the equation of
state of a gas, whereby it is limited by the intrinsic accuracy
of either the pressure gauge or accuracy of temperature
measurement and control. As a consequence, the accuracy of
conventional techniques is insufficient for applications aiming
to detect small leaks or flows under atmospheric (or higher)
pressure conditions [2].
By detecting instead the change of the refractive index,
e.g., by monitoring the refractivity or the permittivity of the
gas, an assessment of the gas density can be made independent
(to first order) of both the pressure and the temperature.
This enables single-parameter monitoring of gas density and
its variation with significantly increased accuracy [3]. As
a consequence, the last decade has seen the introduction
of a number of new and improved optical and dielectric
measurement techniques for measurements of the refractive
0957-0233/13/105207+05$33.00
1
© 2013 IOP Publishing Ltd
Printed in the UK & the USA
Meas. Sci. Technol. 24 (2013) 105207
I Silander et al
Figure 1. Experimental setup. A narrow linewidth fiber laser is locked to a mode of a Fabry–Perot cavity by the PDH technique. An
electro-optic modulator (EOM) is phase-modulating the laser light, and an error signal is derived from the light reflected from the cavity
monitored by a fast photodiode. The laser servo keeps the fiber laser locked to a cavity mode. The cavity is connected to a vacuum system in
which the gas density can be controlled and the pressure can be measured. To measure the frequency of the fiber laser, the output from the
laser is split and a beat note is created between the laser and an OFC.
2. Theory
can be used for the determination of absolute gas density,
with the accuracy to which the refractive index, molecular
polarizability and optical frequency are known. The main
experimental uncertainty originates from the measurement of
the optical frequencies, and it has two contributions. The first
comes from the finite resolution of the frequency measurement,
δν, determined by the linewidth of the continuous wave (cw)
laser or the frequency reference (whichever is larger). The
second comes from the shift of the cavity mode frequencies
due to the thermal drift of the cavity length, ∂L/∂T = αL,
where α is the thermal expansion coefficient of the cavity
spacer. Including the temperature and pressure dependence of
molar polarizability that originates from the deviation from
the ideal gas law, the accuracy of the determination of the gas
density is given by
2 2
2δν 2
ν ∂A
ν ∂A
2n
δT +
δp .
+ αδT +
δρ ≈
3A
ν0
Aν0 ∂T
Aν0 ∂ p
(6)
The gas number density, ρ, defined as N/V , where N is the
amount of gas (in mol) and V is the volume (in m3), is related
to the index of refraction (the optical refractivity), n, as
2
(n − 1),
(1)
ρ=
3A
where A is the molar polarizability (in m3 mol–1) [8, 15].
The refractive index can be assessed accurately by placing the
gas sample inside a Fabry–Perot resonator and measuring the
frequency of a given mode of the cavity. The frequency of
the qth cavity mode can be expressed as
cq
,
(2)
ν=
2Ln
where c is the speed of light in vacuum, L is the cavity length
and q is the mode number (under the assumption that the
reflection phase shift due to mirror coating dispersion can
be neglected, which is valid when operating near the design
wavelength of the cavity mirrors). Solving equation (2) for n
and differentiating with respect to ν shows that a change in the
refractive index can be related to a change in the frequency of
the cavity mode as
n
dn
=− .
(3)
dν
ν
This implies that the change of gas density can be assessed by
the measurement of the change in optical frequency of a cavity
mode according to
dρ dn
2 n
dρ
=
=−
.
(4)
dν
dn dν
3A ν
Integrating both sides of the equation yields
2n
2n ν
2n
,
(5)
ρ − ρ0 = −
ln(ν) +
ln(ν0 ) ≈ −
3A
3A
3A ν0
where ν0 is the optical frequency corresponding to the
reference density ρ0 , and the last step is valid for ν =
ν − ν0 < ν0 . By setting the reference density to zero
(corresponding to an evacuated cavity), the above equation
3. Experimental setup
The experimental setup is illustrated in figure 1. A narrow
linewidth Er:fiber laser (Koheras, Adjustik E15) operating
around 1531 nm is locked using the Pound–Drever–Hall
(PDH) [16] technique to a Fabry–Perot cavity with a finesse of
7400 and a free spectral range of 428 MHz. The cavity mirrors
are glued to a 35 cm spacer made of Zerodur placed directly
on an optical table, which, in turn, is placed in a temperaturestabilized room. The temperature of the room, optical table
and the Zerodur spacer is monitored by PT-100 sensors (Pico
Technology, PT-104). The cavity is connected to a vacuum
system, which allows changes of the amount of gas in the
chamber as well as continuous measurement of the pressure
inside the cavity by the use of a pressure gauge (Oerlikon
Leybold Vacuum, Ceravac CTR91-1Torr) with 1 s resolution.
2
I Silander et al
35
54
28
58
21
62
Frequency [MHz]
Pressure [Pa]
The sidebands for the PDH locking are created at
20 MHz with the use of a fiber-coupled LiNbO3 electrooptic modulator (EOM, General Photonics, LPM-001-15) and
a fixed frequency source. The PDH error signal is obtained by
phase sensitive detection of light intensity in cavity reflection
using a high-bandwidth photo-detector. The bandwidth of the
PDH servo loop is 10 kHz, limited by the resonance of the
PZT in the fiber stretcher (at 35 kHz) to which the correction
feedback is applied [17]. Since the linewidth of the fiber laser
is narrow (less than 1 kHz @ 1 ms), such a bandwidth is
sufficient to keep the laser locked to the center of the cavity
mode, whose linewidth is 29 kHz (HWHM).
The optical frequency of the cw laser is measured using
an Er:fiber OFC (MenloSystems, FC-1500-250-WG) as a
reference. A fraction of the cw light is split off with a
fiber-splitter and used to create a beat note with the output
of the frequency comb. The repetition rate and the carrier
envelope offset frequency of the comb are stabilized to a GPS
referenced rubidium oscillator (Symmetricom, TSC4410A),
and the values of the two parameters are chosen so that the
frequency of the beat note between the cw laser and the
nearest comb line is around 60 MHz. The number of the closest
comb line is determined by measuring the frequency of the cw
laser with a wavemeter (Burleigh, WA-1500), which provides
30 MHz resolution. The sign of the beat note is determined
by changing the repetition rate and carrier envelope offset
frequency in a controlled way and observing the sign of the
change of the beat note. The frequency of the optical beat note
is monitored using either a counter integrated in the frequency
comb system with 1 Hz sampling rate or with a fast external
counter (1 kHz sampling rate, Agilent 53230A).
14
7
70
66
65
60
70
55
0
0
0
0.8
Frequency [MHz]
Meas. Sci. Technol. 24 (2013) 105207
5
10
15
20
Pressure [Pa]
1.6
Time [min]
25
30
74
3.2
2.4
Figure 2. Time dependence of pressure (solid red lower curve, left
axis) and optical frequency (dashed blue upper curve, right axis) as
the number density of nitrogen in the cavity is varied. The curves are
offset for clarity. The inset shows the optical beat note frequency as
a function of pressure with a linear fit.
4
Allan deviation [Hz]
10
4. Results
3
10
To test the performance of the method, we repeatedly filled
the cavity with pure nitrogen to a given pressure and quickly
evacuated it, while simultaneously monitoring the intracavity
pressure and beat frequency between the cw laser and the
OFC. Figure 2 shows the result for one cycle of filling
and evacuating the cavity; the solid red curve displays the
pressure in the chamber, the dashed blue curve—the beat note
frequency. As expected, the two curves, offset for clarity, have
the same general form. A closer scrutiny of the data showed
that when the cavity was repeatedly filled and emptied, the
laser frequency corresponding to the evacuated chamber was
drifting slowly. This was caused by thermal expansion of the
cavity spacer and hysteresis in its compressibility [6]. To
correct for this drift in a series of repeated measurements,
we fitted a third-order polynomial to the vacuum frequencies,
ν0 , in between consecutive fillings and subtracted it from
the measured data. The resulting dependence between the
corrected frequency and pressure is shown by the inset in
figure 2, where the solid line represents a linear fit with a
slope of −519.26(7) kHz Pa–1. This clearly illustrates that the
measured beat frequency has a linear dependence on pressure,
within the accuracy of the pressure gauge.
To investigate the accuracy as well as the long-term
stability of the system, we measured the optical beat note
10
-1
10
0
1
10
Integration time [s]
2
10
3
10
Figure 3. The Allan deviation of the optical beat note between the
laser locked to the evacuated cavity and the OFC.
frequency for 2 h while keeping the cavity at vacuum. Under
these conditions, the experimental uncertainty in density,
equation (6), can be rewritten in terms of the measured
frequency as
2δν 2
δνbeatnote ≈ ν0
+ (αδT )2 .
(7)
ν0
Figure 3 shows the Allan deviation of the beat note
measurement. At integration times up to 3 s, the data follow
white noise behavior. At 1 s, the uncertainty is 1.4 kHz, which
corresponds to a relative frequency uncertainty, 2δν/ν0 , of
7 × 10−12. This implies that at short integration times, the
uncertainty is limited by the stability of the Rb oscillator
to which the frequency comb is stabilized (specified to be
5 × 10−12 at 1 s). The drift at longer integration times is caused
by thermal expansion of the cavity spacer. The temperature of
the cavity spacer drifted at a rate of 0.01 K h−1 during the
measurement. This change in spacer temperature results in
3
Meas. Sci. Technol. 24 (2013) 105207
I Silander et al
a frequency drift of 3.2 kHz in 100 s (assuming a thermal
expansion coefficient of Zerodur, α, of 6 × 10−8 K−1), in
agreement with the measured Allan deviation.
The highest resolution in the determination of the optical
beat note frequency is obtained at an integration time of
3 s and is equal to 900 Hz. Provided that the temperature
and pressure dependence of molar polarizability can be
neglected, the frequency uncertainty can be recalculated to
the minimum change of gas density using the first term in
equation (6). The value of the molar polarizability of nitrogen
at atmospheric pressure and 23 ◦ C is calculated from the
definition, A = 2 (n − 1) RT Z/3p (where R is the molar gas
constant, and Z is the compressibility), and equations (1),
(2) and (4) in [18] to be 4.40 × 10−6 m3 mol−1. Thus the
minimum change of gas density that can be resolved with
the optical technique is 1.4 × 10−6 mol m−3 in 3 s, which
corresponds to a relative accuracy in the density determination
of 3.4 × 10−8 at atmospheric pressure. This result is valid
provided that the error contributions from the temperature and
pressure dependence of molar polarizability, i.e. the third and
fourth terms in equation (6), are lower than the first term.
The two terms, (∂A/∂T )/A and (∂A/∂ p)/A, are calculated by
differentiating the definition of A to be −1 × 10−5 K−1 (at
atmospheric pressure) and 2.6 × 10−9 Pa−1, respectively. Thus
the variation of molar polarizability is negligible compared to
the frequency uncertainty at 3 s if the temperature is constant
to within 3 mK and the pressure to within 13 Pa.
It is noteworthy that the accuracy of the optical technique
is significantly less affected by changes in temperature and
pressure than the conventional technique of determining gas
density based on the equation of state of the gas. The accuracy
of the conventional technique is directly dependent on the
accuracy to which the temperature and pressure can be
controlled and measured. Since the resolution of a pressure
gauge is usually a fraction of its measurement range, typically
of the order of 10−4, the gas number density assessment cannot
be made with relative accuracy better than this fraction. Thus
the presented optical technique has the potential to supersede
the conventional technique by orders of magnitude.
lasers locked to the two cavities [8, 9]. In an even simpler
approach, one laser can be locked to both cavities with the aid
of an acousto-optical modulator.
The developed method can find application in several
areas where the gas density change is the main observable. The
most straightforward are the measurement of gas flows and the
calibration of gas flow meters by monitoring the increase (or
decrease) of gas density to/from a known volume. With the
possibility of using differently sized volumes, the method can
be tuned for maximal accuracy, with large volumes used for
large flows and small volumes for small flows. By measuring
the filling of an initially empty volume, the method can also
be used in applications where the absolute gas density is of
importance. Among them are currently employed methods
for calibration of reference gas leaks for vacuum applications
[19, 20], where one of the main uncertainties comes from the
initial absolute pressure measurement.
The method also gives a possibility of detecting small
density changes even at high pressures, which might enable the
detection of small critical leaks of unwanted (i.e. hazardous)
gas at higher pressure. Besides applications in gas flow
measurement and calibration, assessment of the gas number
density can also be a promising start of a future redefinition of
the current artifact-based pressure standard. A future standard
based on the equation of state of a gas, often reduced to the ideal
gas law, would allow a standard that is based on fundamental
physical constants. Such a method will also make it possible to
eliminate the use of hazardous mercury in the current worldwide spread methods for pressure standards.
Acknowledgments
This project was supported by the Faculty of Science and
Technology, Umeå University, and the Swedish Research
Council under the projects 621-2011-5123 and 621-20123650.
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4
Meas. Sci. Technol. 24 (2013) 105207
I Silander et al
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5