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Using helium as a standard of refractive index: Correcting errors in a gas refractometer
Metrologia
Abstract
The refractive index of helium at atmospheric pressure can be determined from ab initio calculations in combination with careful pressure and temperature measurements. Therefore, helium can serve as a theory-based standard of refractive index; it might be used as a medium of known refractive index for high-accuracy interferometric length measurements or it can be used to characterize and correct errors in a gas refractometer. We have used helium to correct for pressure-induced distortions of two refractometers built by us, where both refractometers basically consist of a laser locked to the transmission maximum of a simple Fabry–Perot cavity. As a proof-of-principle of the helium-correction technique, we have used our device to measure the molar refractivity of nitrogen and we find reasonable agreement with previous measurements. When our two refractometers simultaneously measure the refractive index of a common nitrogen sample, we find that the two systems agree with each other within a few parts in 109.
... To calculate the density ρ m , the m is calculated first at NTP since V is known. The molar refractivity A R for helium gas is calculated using the following expression in [56] A R = 0.51725407 + 1197.5410 ...
... where λ is the wavelength of light used. The refractivity virial coefficient B R is calculated using the expression suggested by [56]: ...
... Please note that the above expression for B R was developed at a wavelength of 633 nm [56]. However, we used this value of B R at 1550 nm, since B R has only a small effect on the refractive index (modify the result less than 2 × 10 −10 ) [56]. ...
Neutron and gamma irradiation is known to compact silica, resulting in macroscopic changes in refractive index (RI) and geometric structure. The change in RI and linear compaction in a radiation environment is caused by three well-known mechanisms: (i) radiation-induced attenuation (RIA), (ii) radiation-induced compaction (RIC), and (iii) radiation-induced emission (RIE). These macroscopic changes induce errors in monitoring physical parameters such as temperature, pressure, and strain in optical fiber-based sensors, which limit their application in radiation environments. We present a cascaded Fabry–Perot interferometer (FPI) technique to measure macroscopic properties, such as radiation-induced change in RI and length compaction in real time to actively account for sensor drift. The proposed cascaded FPI consists of two cavities: the first cavity is an air cavity, and the second is a silica cavity. The length compaction from the air cavity is used to deduce the RI change within the silica cavity. We utilize fast Fourier transform (FFT) algorithm and two bandpass filters for the signal extraction of each cavity. Inclusion of such a simple cascaded FPI structure will enable accurate determination of physical parameters under the test
... However, precise measurements of the frequency shift in the gas mode are not so simple [70]. For this reason, it is unclear if there will be a definite improvement with respect to the classical experiments-in particular, with respect to Piccard-Stahel and Joos. ...
... where ρ is the molar density and A R = (4/3)πN A α is the product of the Avogadro number N A and the molecular polarizability α (see, e.g., [70]). The coefficient B R takes into account two-body interactions, which, for air and helium at atmospheric pressure, can be ignored. ...
... As an example, for helium and a wavelength of λ = 633 nm, where α ∼ 0.52 mol −1 · cm 3 [70], this gives ∼ 3.5 · 10 −5 . Thus, in this simple approximation, where the temperature dependence of is ...
The dominant CMB dipole anisotropy is a Doppler effect due to a particular motion of the solar system with a velocity of 370 km/s. Since this derives from peculiar motions and local inhomogeneities, one could meaningfully consider a fundamental frame of rest Σ associated with the Universe as a whole. From the group properties of Lorentz transformations, two observers, individually moving within Σ, would still be connected by the relativistic composition rules. However, the ultimate implications could be substantial. Physical interpretation is thus traditionally demanded in order to correlate some of the dragging of light observed in the laboratory with the direct CMB observations. Today, the small residuals—from those of Michelson–Morley to present experiments with optical resonators—are just considered instrumental artifacts. However, if the velocity of light in the interferometers is not the same parameter “c” of Lorentz transformations, nothing would prevent a non-zero dragging. Furthermore, the observable effects would be much smaller than what is classically expected and would most likely be of an irregular nature. We review an alternative reading of experiments that leads to remarkable correlations with the CMB observations. Notably, we explain the irregular 10−15 fractional frequency shift presently measured with optical resonators operating in vacuum and solid dielectrics. For integration times of about 1 s and a typical Central European latitude, we also predict daily variations of the Allan variance in the range (5÷12)·10−16.
... Other means to alleviate the limitations are to construct the DFPC of low thermal expansion glass, e.g., ultralow expansion glass (ULE) [1,24] or Zerodur [2,[6][7][8][9][10][11]13,[26][27][28][29], place it in a highly temperature stabilized environment (a combined gas and vacuum chamber) [1], and let the system relax and equilibrate for long time periods after each gas filling or emptying process [1]. However, several of these actions are cumbersome to pursue and increase the complexity of the systems. ...
... Similarly, based on Eqs. (22), (27), and (28) [and by use of Eqs. (SM-18) and (SM-26) in Supplement 1], the error (or uncertainty) in the assessment of pressure for MNI refractometry, δ P MNI , can be expressed as ...
... When UMI is performed, since the drifts in the lengths of the cavities are of Type I and the leakages and the outgassing into the measurement cavity are of Type IIb, the error (or uncertainty) in the assessment of pressure, δ P UMI , can, based on Eqs. (25), (27), and (28) [and by use of Eqs. (SM-19) and (SM-26) in Supplement 1], be written as ...
Gas modulation refractometry (GAMOR) is a methodology for assessment of gas refractivity, molar density, and pressure that, by a rapid gas modulation, exhibits a reduced susceptibility to various types of disturbances. Although previously demonstrated experimentally, no detailed analysis of its ability to reduce the pickup of drifts has yet been given. This work provides an explication of to what extent modulated refractometry in general, and GAMOR in particular, can reduce drifts, predominantly those of the cavity lengths, gas leakages, and outgassing. It is indicated that the methodology is insensitive to the linear parts of so-called campaign-persistent drifts and that it has a significantly reduced susceptibility to others. This makes the methodology suitable for high-accuracy assessments and out-of-laboratory applications.
... The mirror substrate is fabricated using an optically transparent material such as ULE®, fused silica, or another optical glass [2][3][4][5][6][7][8][9][10][11][12]. When using FP cavities made of different materials, the physical properties of the mirror materials, as well as the physical properties of the spacer material, must be considered [13,14]. In a previous study, Stone et al. used two FP cavities consisting of spacers made of ZERODUR® and fused silica mirrors [13,14]. ...
... When using FP cavities made of different materials, the physical properties of the mirror materials, as well as the physical properties of the spacer material, must be considered [13,14]. In a previous study, Stone et al. used two FP cavities consisting of spacers made of ZERODUR® and fused silica mirrors [13,14]. The lengths of the FP cavities were 452 mm and 94 mm. ...
... The lengths of the FP cavities were 452 mm and 94 mm. The authors expected that both FP cavities would experience the same resonant frequency shifts in response to changes in the pressure around the FP cavities, but in fact, differences were observed [13,14]. The cause of this was thought to be mirror deformation arising from the mismatch between the compressibility of the mirror material and that of the spacer material [13,14]. ...
An optical pressure measurement system using Fabry–Pérot (FP) cavities are being developed in several national metrology institutes as a future pressure standard. One of the challenges in these research endeavors is the deformation of the FP cavity. The purpose of this study is to evaluate mirror deformation in FP cavities constructed using different materials for the spacer and mirrors, both by simulation and experiment. Experimental and finite element method (FEM) simulation results are in broad agreement. The mirror deformation amount can be determined using the dual cavity method without using the nominal values of material and without depending on conventional pressure standards. Our results suggest that the dual cavity method as well as FEM simulation should be adopted for situations in which the mirror deformation is large and/or high accuracy is required. The results of this study contribute to the development of the primary pressure standard using FP cavities.
... This is much smaller than many effects which must preliminarily be subtracted. For instance, by changing from vacuum to the gas case under pressure, and for a typical cavity length of 10 cm, the effect of cavity deformations is about 10 MHz [57]. Theoretically, this should not depend on the gas used but only on the solid parts of the apparatus. ...
... Theoretically, this should not depend on the gas used but only on the solid parts of the apparatus. Yet, experimental measurements at atmospheric pressure show that there is a difference between nitrogen and helium of about 0.6 MHz [57]. Therefore, one should lower the pressure to reduce this spurious effect. ...
... The starting point is the Lorentz-Lorentz equation (see, e.g., [57]) ...
The possibility to correlate ether-drift measurements in laboratory and direct CMB observations with satellites in space would definitely confirm the existence of a fundamental preferred frame for relativity. Today, the small residuals observed so far (from Michelson-Morley onward) are just considered typical instrumental effects in experiments with better and better sensitivity. Though, if the velocity of light propagating in the various interferometers is not exactly the same parameter c of Lorentz transformations, nothing would really prevent to observe an ether drift. Thus, for the Earth cosmic velocity v = 370 km/s, we argue that a fundamental 10-15 light anisotropy, as presently observed in vacuum and in solid dielectrics, is revealing a 10-9 difference in the vacuum effective refractivity between an apparatus in an ideal freely falling frame and an apparatus on the Earth surface. In this perspective, the stochastic nature of the physical vacuum could also explain the irregular character of the signal and the observed substantial reduction from its instantaneous 10-15 value to its statistical average 10-18 (or smaller). For the same v = 370 km/s the different refractivities, respectively, O(10-4) and O(10-5) for air or helium at atmospheric pressure, could also explain the observed light anisotropy, respectively O(10-10) and O(10-11) . However, for consistency, one should also understand the physical mechanism which enhances the signal in weakly bound gaseous matter but remains ineffective in solid dielectrics where the refractivity is O(1) . This mechanism is naturally identified in a non-local, tiny temperature gradient of a fraction of millikelvin which is found in all classical experiments and might ultimately be related to the CMB temperature dipole of ±3 mK or reflect the fundamental energy flow associated with a Lorentz-non-invariant vacuum state. The importance of the issue would deserve more stringent tests with dedicated experiments and significant improvements in the data analysis.-1
... As helium exhibits a nonlinear refractive index n 2 which is two orders of magnitude lower than that of air [31,32] it is commonly used in mode-locked oscillators to mitigate nonlinear effects [33,34]. Thanks to its 9-times lower thermal dispersion [35][36][37] it was also used in cw thin-disk oscillators [38,39] and TDMPAs [21,22] to reduce thermal lensing and the effects of the air wedge, which are both consequences of the heated ambient gas [40]. ...
... For the degradation of the beam quality of the pulses with the higher seed pulse energies (100 µJ and 50 µJ), possible intrinsic or thermally introduced deformations of the optics can be excluded since the pulses with the low seeded pulse energy of 10 µJ pass the same optics as the pulses of the other configurations, but show excellent beam quality although they were amplified to even higher average output powers. Using the same reasoning, it can also be ruled out that the higher thermal dispersion of air amounting to dn/dT ≈ − 9 ·10 −7 [35,37] degrades the beam quality; helium has a dn/dT ≈ − 1 ·10 −7 [36] at ambient pressure for a wavelength around 1 µm. We hence conclude that the nonlinearities, i.e. the spatial Kerr-effect lead to an increase of the M 2 -value. ...
We present an experimental investigation on the benefits of helium as an atmospheric gas in CPA-free thin-disk multipass amplifiers (TDMPAs) for the amplification to average powers exceeding 1 kW and pulse peak powers reaching 5 GW. Both the performance of the amplifier and the properties of the amplified sub-400 fs laser pulses centred at a wavelength of 1030 nm are compared for different helium concentrations in air, outlining and quantifying the benefits of a helium-rich atmosphere. The amplification of 100 µJ pulses in an atmosphere with 60% helium instead of air led to a maximum increase in efficiency from 24% to 29%. This translated into an increase of average output power and pulse energy of 34 W (i.e +19%) and 0.34 mJ (i.e. +19%) respectively. At the same time an improvement of the beam quality from M² = 1.18 to M² = 1.14 was achieved. For the amplification of 10 µJ pulses to over 1 kW of average power an atmosphere with 33% helium led to an improved beam pointing stability by a factor of 2. Moreover, the beam propagation factor M² improved by 0.1, and the power stability improved by approximately 10%.
... The frequency of a given mode in an FP cavity when pressure induced cavity deformation, the phase shift of light from the mirrors, and the Gouy phase, are taken into account, can be obtained by the use of a round-trip resonance condition for the phase of the light. As is shown in part 1 of the Supplement 1, such a condition can, for the m th TEM 00 mode of an FP cavity with DBR mirrors, be written as [13,20] 2k in (L 0 + δL) ...
... Once the numbers of the modes addressed, i.e. the m 0i , had been uniquely assessed, it was possible, by use of Eq. (20), to obtain a more accurate (i.e. the actual) value of the γ ′ s,i of each cavity. These entities were assessed, for each cavity separately, by the use of four consecutive assessments of the FSR. ...
A procedure is presented for in situ determination of the frequency penetration depth of coated mirrors in Fabry-Perot (FP) based refractometers and its influence on the assessment of refractivity and pressure. It is based on assessments of the absolute frequency of the laser and the free spectral range of the cavity. The procedure is demonstrated on an Invar-based FP cavity system with high-reflection mirrors working at 1.55 μm. The influence was assessed with such a low uncertainty that it does not significantly contribute to the uncertainties (k = 2) in the assessment of refractivity (<8 × 10⁻¹³) or pressure of nitrogen (<0.3 mPa).
... Since the CTEs of the spacer and the mirrors are different, the mirrors are deformed when temperature changes. The amount of mirror deformation can be estimated by associating the deformation due to temperature and the deformation due to pressure [16]. The difference in the length changes due to pressure change between the spacer and the mirror in the 25.4 mm area where the mirrors are joined to the spacer is 91.4 pm/kPa. ...
... Fig. 7 shows the cavity length in a vacuum measured intermittently. The aging rate was -12.7 pm/day (120.7 kHz/day) against 50 mm, which was equivalent to the literature value [16]. The chamber was not always evacuated, and any pressure was sealed during these intermittent experiments. ...
Optical pressure standards (OPS) using Fabry-Perot (FP) cavities can measure a wide pressure range with high accuracy. National metrology institute of Japan (NMIJ) is developing an OPS for the pressure range of 1 Pa–100 kPa. In this study, the target uncertainty of the OPS, the challenges for measuring below 100 Pa, and its current specifications in the range from 10 Pa to 120 kPa are reported. In addition, the methods for solving the challenges to achieve the target uncertainty is considered.
... The best performing refractometers are based on Fabry-Perot (FP) cavities, where a laser is used to probe the frequency of a longitudinal cavity mode [5][6][7][8][9][10][11][12][13][14][15]. By measuring the change in frequency between an empty (evacuated) and a gas-filled cavity, the refractivity can be assessed, from which the molar density and the pressure can be calculated. ...
... As was alluded to above, to make viable assessments of large pressure shifts with short settling times, which is needed for a number of applications, it is of importance that the system has a fast response. Although several types of refractometers have been scrutinized over the years [3][4][5][6][7][8][10][11][12][13][14][15][16][17][18][19][24][25][26], virtually none of them has yet been assessed with respect to its short-term behavior. Access to two GAMOR-based refractometer systems allows for scrutiny of the short-term behavior of GAMOR-based refractometry in more detail. ...
Refractometry is a powerful technique for pressure assessments that, due to the recent redefinition of the SI system, also offers a new route to realizing the SI unit of pressure, the Pascal. Gas modulation refractometry (GAMOR) is a methodology that has demonstrated an outstanding ability to mitigate the influences of drifts and fluctuations, leading to long-term precision in the 10−7 region. However, its short-term performance, which is of importance for a variety of applications, has not yet been scrutinized. To assess this, we investigated the short-term performance (in terms of precision) of two similar, but independent, dual Fabry–Perot cavity refractometers utilizing the GAMOR methodology. Both systems assessed the same pressure produced by a dead weight piston gauge. That way, their short-term responses were assessed without being compromised by any pressure fluctuations produced by the piston gauge or the gas delivery system. We found that the two refractometer systems have a significantly higher degree of concordance (in the 10−8 range at 1 s) than what either of them has with the piston gauge. This shows that the refractometry systems under scrutiny are capable of assessing rapidly varying pressures (with bandwidths up to 2 Hz) with precision in the 10−8 range.
... The most sensitive refractometers are based on Fabry-Pérot (FP) cavities in which a laser is used to probe the frequency of a longitudinal mode. [6][7][8][9][10][11][12][13] Although different realizations of such refractometers have shown promising results, a crucial limiting issue that needs to be addressed is the fact that the cavities are subjected to pressure-induced deformation that will compress (or elongate) the cavities when they are exposed to gas pressure. Without taking this effect into consideration properly, pressure assessments can be adversely affected on the permille to percent range. ...
... As is shown in Sec. V C of the supplementary material, 19 it is also possible to extract, from the two expressions for k i s given in Eq. (8), this time by elimination of κ, an expression for an experimentally quantifiable part of ψ 0 , denoted ψ 0 0 , that can be expressed as ...
A novel procedure for a robust assessment of cavity deformation in Fabry–Pérot (FP) refractometers is presented. It is based on scrutinizing the difference between two pressures: one assessed by the uncharacterized refractometer and the other provided by an external pressure reference system, at a series of set pressures for two gases with dissimilar refractivity (here, He and N 2). By fitting linear functions to these responses and extracting their slopes, it is possible to construct two physical entities of importance: one representing the cavity deformation and the other comprising a combination of the systematic errors of a multitude of physical entities, viz., those of the assessed temperature, the assessed or estimated penetration depth of the mirror, the molar polarizabilities, and the set pressure. This provides a robust assessment of cavity deformation with small amounts of uncertainties. A thorough mathematical description of the procedure is presented that serves as a basis for the evaluation of the basic properties and features of the procedure. The analysis indicates that the cavity deformation assessments are independent of systematic errors in both the reference pressure and the assessment of gas temperature and when the gas modulation refractometry methodology is used that they are insensitive to gas leakages and outgassing into the system. It also shows that when a high-precision (sub-ppm) refractometer is characterized according to the procedure, when high purity gases are used, the uncertainty in the deformation contributes to the uncertainty in the assessment of pressure of N 2 with solely a fraction (13%) of the uncertainty of its molar polarizability, presently to a level of a few ppm. This implies, in practice, that cavity deformation is no longer a limiting factor in FP-based refractometer assessments of pressure of N 2.
... Although it is simple in theory to realize FP-based instrumentation for refractometry, it is not trivial in practice. One reason is that such cavities are influenced by various types of drifts, predominantly those from changes of the length of the cavity (which, in turn, mainly originate from alterations in the temperature of the cavity spacer or from material creeping), gas leaks, and outgassing [16,[27][28][29][30]. For example, a drift in length of 1 pm of a 30 cm long cavity during a measurement corresponds to an error in the assessment of refractivity and pressure of 3 × 10 -12 and (for N2) 1 mPa, respectively. ...
... , will not solely originate from the change in the refractivity of the gas in the measurement cavity, as indicated by Eq. (12); it will also be influenced by the drifts in the system that take place between the instances when the measurement cavity is filled with gas (i.e. when (0, ) g f is assessed) and when it is evacuated (i.e. when (0,0) f is assessed). Such drifts comprise predominantly alterations in the length of the cavities (originating from drifts of the temperature of the spacer material) [16,[27][28][29][30]. For the DFPCbased techniques, they can also be caused by gas leaks or outgassing into the reference cavity, which, to reduce the risk for unintentional fluctuations in its gas density, often is evacuated solely once in the beginning of a measurement series. ...
Gas modulation refractometry (GAMOR) is a technique based on a dual-Fabry-Perot (FP) cavity (DFPC) for assessment of gas refractivity, density, and pressure that can alleviate significant limitations of conventional refractometry systems, predominantly those related to drifts. Repeated assessments of the beat frequency when the measurement cavity is evacuated provide conditions under which the methodology is immune to the linear parts of the drifts in the system, both those from length changes of the cavities and those from gas leaks and outgassing. This implies that the technique is solely influenced by the non-linear parts of the drifts. This work provides a description of the principle behind the GAMOR methodology and explicates the background to its unique property. Based on simple models of the drifts of the temperature in the cavity spacer and the residual gas in the reference cavity, this work predicts that a GAMOR system, when used for assessment of refractivity, can sustain significant temperature drifts and leakage rates without being affected by noticeable errors or uncertainties. The cavity spacer can be exposed to temperature fluctuations of 100 mK over 103 s, and the reference cavity can have a leakage that fills it up with gas on a timescale of days, without providing errors or uncertainties in the assessment of refractivity that are 3 x 10^(-12), which, for N2, corresponds to 0.01 ppm (parts per million) of the value under atmospheric pressure conditions, and thereby 1 mPa. Since well-designed systems often have temperature fluctuations and leakage rates that are smaller than these, it is concluded that there will, in practice, not be any appreciable influence from cavity length drifts, gas leaks, and outgassing in the GAMOR methodology.
... In a second step, we used the temperature distribution and combined it with the gas thermo-optic coefficient [33][34][35] to calculate the n(x,y,z)-1 distribution in front of the disk, see Fig. 5 for 1-bar air and 1-bar He. Using the three-dimensional Helmholtz equation in the paraxial approximation, we subsequently studied the evolution of an incoming Gaussian laser beam while impinging orthogonally on the disk and propagating through the gas. ...
... Thermal and optical properties of vacuum, helium, nitrogen, and air, at 25°C and 1030 nm[33][34][35]. ...
We unveil a gas-lens effect in kW-class thin-disk lasers, which accounts in our experiments for 33% of the overall disk thermal lensing. By operating the laser in vacuum, the gas lens vanishes. This leads to a lower overall thermal lensing and hence to a significantly extended power range of optimal beam quality. In our high-power continuous-wave (cw) thin-disk laser, we obtain single-transverse-mode operation, i.e. M² < 1.1, in a helium or vacuum environment over an output-power range from 300 W to 800 W, which is 70% broader than in an air environment. In order to predict the magnitude of the gas-lens effect in different thin-disk laser systems and gain a deeper understanding of the effect of the heated gas in front of the disk, we develop a new numerical model. It takes into account the heat transfer between the thin disk and the surrounding gas and calculates the lensing effect of the heated gas. Using this model, we accurately reproduce our experimental results and additionally predict, for the first time by means of a theoretical tool, the existence of the known gas-wedge effect due to gas convection. The gas-lens and gas-wedge effects are relevant to all high-power thin-disk systems, both oscillators and amplifiers, operating in cw as well as pulsed mode. Specifically, canceling the gas-lens effect becomes crucial for kW power scaling of thin-disk oscillators because of the larger mode area on the disk and the resulting higher sensitivity to the disk thermal lens.
... In general, the main limiting factors when realizing pressure using FP refractometers are the knowledge of the molar polarizability of the gas and the pressure-induced deformation of the cavity. While He offers the lowest uncertainty of the polarizability (known from first-principle calculations) [3], the cavity deformation, which is normally assessed by the two-gas method and requires knowledge of the molar polarizability of an additional gas, cannot yet be assessed down to a level that, for He, provides a similarly low uncertainty [5]. ...
Based on a recent experimental determination of the static polarizability and a first-principle calculation of the frequency-dependent dipole polarizability of argon, this work presents, by using a Fabry–Perot refractometer operated at 1550 nm, a realization of the SI unit of pressure, the pascal, for pressures up to 100 kPa, with an uncertainty of [(1.0 mPa)² + (5.8 × 10⁻⁶ P)² + (26 × 10⁻¹²P²)²]1/2. The work also presents a value of the molar polarizability of N2 at 1550 nm and 302.9146 K of 4.396572(26) × 10⁻⁶ m³/mol, which agrees well with previously determined ones.
... On the other hand, with the latest revision of the International System of Units in 2019 [5], an alternative way is offered to realize the pascal unit without any mechanical actuator but rather depends on the thermodynamic aspect of a gas. By calculating the refractivity of a gas inside a refractometer using frequency measurements of a laser beam, its density can be derived from the Lorentz-Lorenz equation [6]. In the case where air is used, semi empirical equations such as the revised formulae of Elden's and Ciddor's [7]- [9] can then be used to deduce the pressure inside the refractometer. ...
... While RI variation due to pressure and temperature is rather well documented for air and nitrogen gas, the straightforward relationship between temperature and pressurized helium gas has yet to be researched. As a result, we followed the ideal gas law and Lorentz-Lorenz relation to observe the pressure induced RI, expressible as [23]: ...
We present an experimental comparison of gas pressure sensors based on differently structured extrinsic Fabry-Pérot interferometers (FPIs). Different fabrication techniques were adopted to foster the practical realization of these sensors. A novel photonic crystal fiber (PCF) based FPI with an angle cleaved end face that enables gas to easily flow in and out of the cavity while also avoiding the mixed signal from the cascaded structure is also presented. We measured helium gas pressure from 0.1 MPa (atmospheric pressure) to 5 MPa at room temperature and obtained a gas pressure sensitivity coefficient of 0.5±0.01 nm/MPa. Furthermore, the gas pressure sensitivities are numerically analyzed to validate the experimental results.
... Nonetheless, different uncertainties may change the GMR wavelength in some practical applications, for instance, sensors working at the resonance wavelengths of gases. To name a few, fabrication variations, thermal expansion of the layers, and stress/strain after deposition of layers (which may change the RI of the materials), are some of these uncertainties [27]. Hence, these devices must be calibrated or tuned after fabrication. ...
The high quality factor of guided mode resonance (GMR) structures and their sensitivity to the adjacent media’s refractive index (RI) make them excellent candidates for sensing applications. In this study, we derive analytical sensitivity equations for the GMR structures for the first time, which can be utilized to better understand and optimize their behavior. The simulated structures (based on the finite element method) show excellent agreement with the analytical sensitivities for both the transverse electric and magnetic (TE and TM) modes. The effects of waveguide height, working wavelength, and material RIs on the obtained sensitivities were surveyed, which helped achieve a design approach based on the analytical equations. The substrate RI should be as low as possible for any application. For analytes, the RI of the waveguide thin film should be low, whereas this trend is different for gas sensors; there is an optimum RI for the waveguide in the TE mode, and in the TM mode, the waveguide should have a higher possible RI. Moreover, the TM-mode gas sensors exhibit optimum sensitivity for a specified waveguide height. The design approach was used for a gas sensor operating at methane resonance wavelength (1653.7 nm), which can be tuned using a double-layer graphene sheet. A linear tunability of up to 5 nm was achieved using 1 eV chemical potential adjustment, whereas the device sensitivity remains between 291.9 to 295.5 nm/RIU. Moreover, higher sensitivities up to 950 nm/RIU can be obtained for analyte sensors at 1550 nm.
... The application of optical sensors continues to increase for the detection of different chemicals and materials in a wide range of refractive indices (RIs), including diluted and condensed gases, as well as aqua-based analytes, biomolecules, and other types of materials [1][2][3][4][5][6]. Advantages such as an immunity against electromagnetic interferences [7], cost-effectiveness due to the low cost of telecommunication instruments, and progress in the integration of optical and opto-electronic devices [8] has led to the rapid growth of optical sensors. ...
This work performs a quantitative comparison of different dielectric-based guided-mode resonance (D-GMR) sensors. To this end, diverse D-GMR structures are classified into three different classes, and their sensitivity (S) is compared to each other. For one of these classes in various schemes, the sensitivity is investigated for the TE and TM modes. Moreover, grating height effects are studied for different cases in this category, and analytical sensitivity equations are used as benchmarks. Then, the three classes are compared and, based on the numerical results and analytical equations, various applications are proposed for different structures in the refractive index (RI) of interest. Comparing our results to other recent works, we prove that the proposed classification leads to great sensing performances and the predictions are reliable. A comparison has been performed for methane as a gas sample (with RI of 1.0003) and a hemoglobin solution and toluene as two different analytes (with RIs of 1.33 and 1.4778, respectively). The results show a sensitivity of ${\rm S} = {1427.3}\;{\rm nm/refractive}$ S = 1427.3 n m / r e f r a c t i v e index unit (nm/RIU) for methane with a detection precision of one to a few volume percentages in the air, which can also be calibrated to illuminate the fabrication variation errors. For hemoglobin, a sensitivity of 1073.4 nm/RIU is obtained, with a limit of detection of 116.15 mg/lit for 65-87 g/lit of hemoglobin in water; for the toluene sensor, ${\rm S} = {1019.7}\;{\rm nm/RIU}$ S = 1019.7 n m / R I U is calculated. As a general result, a high figure of merit/sensitivity can be achieved over a wide range of applications, from gases to high RI analytes, using our proposed classifications.
... Since the CTEs of the spacer and the mirrors are different, the mirrors are deformed when temperature changes. The amount of mirror deformation can be estimated by associating the deformation due to temperature and the deformation due to pressure [16]. The difference in [13]. ...
Optical pressure standards (OPS) using Fabry-Perot (FP) cavities can measure a wide pressure range with high accuracy. National metrology institute of Japan (NMIJ) is developing an OPS for a pressure range of 1 Pa–100 kPa. In this study, the target uncertainty of the OPS, its current specifications in the range from 40 kPa to 120 kPa, and the challenges for measuring below 100 Pa are reported. In addition, the methods for solving the challenges to achieve the target uncertainty is considered.
... In Eq In practice one could employ helium back-filling or simply measure the cavity mode frequency in a known pressure of helium. The refractive index of helium as a function of frequency and temperature is presently known theoretically with an uncertainty of approximately ±3 x10 -9 [16]. At near infrared wavelengths the value is approximately an order of magnitude smaller than that of air (I-n = 2.5 x10 -5 at 10 5 Pa). ...
... It is known that they remain constant even at high pressures when n differs from unity [Born 1999]. Experimental measurement of nitrogen molar refractivity is described in [Stone 2004] giving a value of 4.445 × 10 6 m 3 /mol. With this value and the one calculated for hydrogen (no experimental value was found), the density profiles are reconstructed in (see Figure 3.31). ...
In this joint thesis, performed between the French Institute CENBG (Bordeaux) and the Canadian Institute INRS (Varennes), laser driven ion acceleration and an application of the beams are studied. The first part, carried out at CENBG and on the PICO2000 laser facility of the LULI laboratory, studies both experimentally and using numerical particle-in-cell (PIC) simulations, the interaction of a high power infrared laser with a high density gas target. The second part, performed at ALLS laser facility of the EMT-INRS institute, investigates the utilization of laser generated beams for elementary analysis of various materials and artifacts. In this work, firstly the characteristics of the two lasers, the experimental configurations, and the different employed particle diagnostics (Thomson parabolas, radiochromic films, etc.) employed are introduced.In the first part, a detailed study of the supersonic high density gas jets which have been used as targets at LULI is presented, from their conceptual design using fluid dynamics simulations, up to the characterization of their density profiles using Mach-Zehnder interferometry. Other optical methods such as strioscopy have been implemented to control the dynamics of the gas jet and thus define the optimal instant to perform the laser shot. The spectra obtained in different interaction conditions are presented, showing maximum energies of up to 6 MeV for protons and 16 MeV for Helium ions in the laser direction. Numerical simulations carried out with the PIC code PICLS are presented and used to discuss the different structures seen in the spectra and the underlying acceleration mechanisms.The second part presents an experiment using laser based sources generated by the ALLS laser to perform a material analysis by the Particle-induced X-ray emission (PIXE) and X-ray fluorescence (XRF) techniques. Proton and X-ray beams produced by the interaction of the laser with Aluminum, Copper and Gold targets were used to make these analyzes. The relative importance of XRF or PIXE is studied depending on the nature of the particle production target. Several spectra obtained for different materials are presented and discussed. The dual contribution of both processes is analyzed and indicates that a combination improves the retrieval of constituents in materials and allows for volumetric analysis up to tens of microns on cm^2 large areas, up to a detection threshold of ppms.
... 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][28][29][30]. For lower pressures, these fluctuations in cavity length will limit the sensitivity of the instrument. ...
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.
... The most sensitive refractometers are based on Fabry-Perot (FP) cavities where a laser is used to probe the frequency of a longitudinal mode [12][13][14][15][16][17][18][19]. Although such refractometers have the potential to provide highly accurate measurements, their cavities, usually bored in low thermal expansion glass, e.g., Zerodur or ultra-low expansion glass (ULE), are sensitive to mechanical disturbances and long-term length drifts [10,20,21]. ...
Gas modulation refractometry (GAMOR) is a methodology that can mitigate fluctuations and drifts in refractometry. This can open up for the use of non-conventional cavity spacer materials. In this paper, we report a dual-cavity system based on Invar that shows better precision for assessment of pressure than a similar system based on Zerodur. This refractometer shows for empty cavity measurements, up to ${10^4}$ 10 4 s, a white noise response (for ${{\rm N}_2}$ N 2 ) of 3 mPa ${{\rm s}^{1/2}}$ s 1 / 2 . At 4303 Pa, the system has a minimum Allan deviation of 0.34 mPa (0.08 ppm) and a long-term stability (24 h) of 0.7 mPa. This shows that the GAMOR methodology allows for the use of alternative cavity materials.
... The influence of the humidity on the refractive index of air (at a temperature of 20 °C and a pressure of 1 atm) was found to be minor. Finally, we have performed similar considerations also for thin-disk lasers operated in a helium atmosphere using the information published in [18] to calculate the refractive index. Both the tilt induced to the laser beam as well as the dioptric power of the thermally induced spherical contribution to the wavefront distortion were found to be about one order of magnitude smaller than the values obtained in ambient air. ...
In this paper, we present a FEM-model that can be used to investigate the effects of thermally induced natural convection at the thin-disk laser crystal. Based on this simulation, we calculated the distribution of the refractive index of the ambient gas for the case of air and helium. By evaluating the optical path difference of a beam at normal incidence, the angular tilt (gas wedge) in the plane of the direction of convection as well as the spherical contribution (gas lens) was calculated for a set of different pump spot geometries and temperatures of the pumped area on the surface of the laser disk. Equations were derived that allow to simply calculate the tilt angle and the focal length of the gas lens for different temperatures of the disk and pump spot diameters for air as ambient medium.
... For microwave [8] and optical [9,10] methods, the dependence of α d on frequency ω is relevant, but for helium the frequency dependent part of α d (ω) is small [18] for experimentally useful frequencies [19], and does not have to be known with high relative accuracy. ...
The QED contribution to the dipole polarizability of the He4 atom was computed, including the effect of finite nuclear mass. The computationally most challenging contribution of the second electric-field derivative of the Bethe logarithm was obtained using two different methods: the integral representation method of Schwartz and the sum-over-states approach of Goldman and Drake. The results of both calculations are consistent, although the former method turned out to be much more accurate. The obtained value of the electric-field derivative of the Bethe logarithm, equal to 0.048 557 2(14) in atomic units, confirms the small magnitude of this quantity found in the only previous calculation [G. Łach, B. Jeziorski, and K. Szalewicz, Phys. Rev. Lett. 92, 233001 (2004)], but differs from it by about 5%. The origin of this difference is explained. The total QED correction of the order of α3 in the fine-structure constant α amounts to 30.6671(1)×10−6, including the 0.1822×10−6 contribution from the electric-field derivative of the Bethe logarithm and the 0.01112(1)×10−6 correction for the finite nuclear mass, with all values in atomic units. The resulting theoretical value of the molar polarizability of helium-4 is 0.51725408(5)cm3/mol with the error estimate dominated by the uncertainty of the QED corrections of order α4 and higher. Our value is in agreement with but an order of magnitude more accurate than the result 0.5172544(10)cm3/mol of the most recent experimental determination [C. Gaiser and B. Fellmuth, Phys. Rev. Lett. 120, 123203 (2018)].
... For microwave [8] and optical [9,10] methods, the dependence of α d on frequency ω is relevant, but for helium the frequency dependent part of α d (ω) is small [18] for experimentally useful frequencies [19], and does not have to be known with high relative accuracy. ...
The QED contribution to the dipole polarizability of the $^4$He atom was computed, including the effect of finite nuclear mass. The computationally most challenging contribution of the second electric-field derivative of Bethe logarithm was obtained using two different methods: the integral representation method of Schwartz and the sum-over-states approach of Goldman and Drake. The results of both calculations are consistent, although the former method turned out to be much more accurate. The obtained value of the electric-field derivative of Bethe logarithm, equal to $0.0485572(14)$ in atomic units, confirms the small magnitude of this quantity found in the only previous calculation [Phys. Rev. Lett. 92, 233001 (2004)], but differs from it by about 5%. The origin of this difference is explained. The total QED correction of the order of $\alpha^3$ in the fine-structure constant $\alpha$, amounts to $30.6671(1)\cdot10^{-6}$, including the $0.1822\cdot10^{-6}$ contribution from the electric field derivative of Bethe logarithm and the $0.01112(1)\cdot10^{-6}$ correction for the finite nuclear mass, all values in atomic units. The resulting theoretical value of the molar polarizability of helium-4 is $0.517\,254\,08(5)$ cm$^3$/mol with the error estimate dominated by the uncertainty of the QED corrections of order $\alpha^4$ and higher. Our value is in agreement but is an order of magnitude more accurate than the result $0.517\,254\,4(10)$ cm$^3$/mol of the most recent experimental determination [Phys. Rev. Lett. 120, 123203 (2018)].
... One method to measure the refractive index is to use a Fabry-Perot cavity Ref. [16]. Fabry-Perot cavities consist of two mirrors facing each other. ...
Optical pressure measurement systems can precisely measure pressure from vacuum to high pressure with a single apparatus. However, the continuously measurable range without interrupting measurement is limited to less than 1 kPa. In this study, an optical pressure measurement system with a wide continuous measurable range was developed. The optical system can continuously measure a pressure range of 18 kPa when using nitrogen gas by using a Littrow external cavity diode laser as its light source. First, discrete pressure points from vacuum to atmospheric pressure were measured. The repeatability and non-linearity of the differences between the measured values had standard deviations of 0.1 Pa and 0.5 Pa, respectively. Next, a pressure range from 91 kPa to 109 kPa was continuously measured. The results showed that the linearity characteristics of gauges as a function of pressure can be evaluated precisely using our optical pressure measurement system.
... It is known that they remain constant even at high pressures when n differs from unity 27 . An experimental measurement of nitrogen molar refractivity is described in ref. 29 giving a value of 4.445x10 −6 m 3 /mol. We have used this value to reconstruct the density profiles shown in figure 4. Due to the opacity of the gas jet at high densities, reconstruction of the density is only possible in a moderate density region (10 19 -10 20 cm −3 ). ...
Computational fluid dynamics simulations are performed to design gas nozzles, associated with a 1000 bars backing pressure system, capable of generating supersonic gas jet targets with densities close to the critical density for 1053 nm laser radiation (10²¹ cm⁻³). Such targets should be suitable for laser-driven ion acceleration at a high repetition rate. The simulation results are compared to the density profiles measured by interferometry, and characterization of the gas jet dynamics is performed using strioscopy. Proton beams with maximum energies up to 2 MeV have been produced from diatomic hydrogen gas jet targets in a first experiment.
... The first type of Fabry-Perot cavity based refractometry systems utilized a single measurement cavity drilled in a low thermal expansion material. 4,[7][8][9][14][15][16][17][18][19][20][21][22][23] In these, the reference laser was often stabilized with respect to an external frequency reference, e.g., an iodine transition or a frequency comb. However, the drifts in these were significant and were often the liming factor for their performance. ...
Gas modulation refractometry (GAMOR) is a methodology that, by performing repeated reference assessments with the measurement cavity being evacuated while the reference cavity is held at a constant pressure, can mitigate drifts in dual Fabry-Perot cavity based refractometry. A novel realization of GAMOR, referred to as gas equilibration GAMOR, that outperforms the original realization of GAMOR, here referred to as single cavity modulated GAMOR (SCM-GAMOR), is presented. In this, the reference measurements are carried out by equalizing the pressures in the two cavities, whereby the time it takes to reach adequate conditions for the reference measurements has been reduced. This implies that a larger fraction of the measurement cycle can be devoted to data acquisition, which reduces white noise and improves on its short-term characteristics. The presented realization also encompasses a new cavity design with improved temperature stabilization and assessment. This has contributed to improved long-term characteristics of the GAMOR methodology. The system was characterized with respect to a dead weight pressure balance. It was found that the system shows a significantly improved precision with respect to SCM-GAMOR for all integration times. For a pressure of 4303 Pa, it can provide a response for short integration times (up to 10 min) of 1.5 mPa (cycle)1/2, while for longer integration times (up to 18 h), it shows an integration time-independent Allan deviation of 1 mPa (corresponding to a precision, defined as twice the Allan deviation, of 0.5 ppm), exceeding the original SCM-GAMOR system by a factor of 2 and 8, respectively. When used for low pressures, it can provide a precision in the sub-mPa region; for the case with an evacuated measurement cavity, the system provided, for up to 40 measurement cycles (ca. 1.5 h), a white noise of 0.7 mPa (cycle)1/2, and a minimum Allan deviation of 0.15 mPa. It shows a purely linear response in the 2.8–10.1 kPa range. This implies that the system can be used for the transfer of calibration over large pressure ranges with exceptional low uncertainty.
... A Fabry-Perot cavity (optical resonator) can be used to measure the refractive index of gases with very high resolution [1]. It's very promising to establish a so-called optical pressure standard based on refractive index measurements, which allows for a new realization of the Pascal as suggested [2][3][4][5]. ...
The National Institute of Metrology (NIM) has constructed a dual optical cavity for the optical pressure standard project. The elastic deformation induced by pressure is investigated by finite element analysis (FEA) simulation. The distortion coefficient, fractional length change per unit pressure, is calculated to be (9.936 ± 0.014)×10⁻¹²/Pa for the measurement channel and (1.49 ~ 1.54)×10⁻¹¹/Pa for the reference channel. The gravity-induced deformation is also modelled to evaluate the sensitivity to the low-frequency vibration. The cavity is modelled to be supported symmetrically on four points. With optimized support position, the sensitivity to vertical acceleration reduces to 19 kHz/(m/s²) at 633 nm.
... For what concern the measurement of fringes u i , it will be necessary improve the signal to noise ratio and reduce the effects of non-linearities of the interferometer as well as an implementation of a more complex compensation algorithm [22]. After that, helium could be used, because its refraction index can be obtained by first principles with high accuracy and its molar refractivity may be determined with lower uncertainty [23,24,25]; this fact would allow to compare the results obtained by ''first principles calculations" with the measurement performed by the optical pressure standard to estimate the pressure dependence of thermomechanical deformations of the double mirror multiplication set-up. ...
A novel pressure standard based on optical interferometry has been realized. The paper presents a preliminary achievement of the system, in which the standard pressure between 1 kPa and 120 kPa is obtained through the measurement of the refractive index of a gas. The response of the optical system has been initially evaluated measuring its sensitivity for nitrogen and air, in terms of the ratio between a pressure variation and the number of detected interferometric fringes. Afterwards, the interferometer has been geometrical characterized in order to perform an absolute measurement of the refractive index of the gas, allowing an optical measurement of the gas pressure. The following step has been to compare the results of a series of measurements obtained with the optical system with a reference pressure standard at 120 kPa. Finally, the standard has been compared with two capacitance diaphragm gauges in the range between 1 kPa and 120 kPa. The device has a relative standard uncertainty equal to 170 ppm at atmospheric pressure.
... The first type of Fabry-Perot cavity based refractometry systems utilized a single measurement cavity drilled in a low thermal expansion material. 4,[7][8][9][14][15][16][17][18][19][20][21][22][23] In these, the reference laser was often stabilized with respect to an external frequency reference, e.g., an iodine transition or a frequency comb. However, the drifts in these were significant and were often the liming factor for their performance. ...
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 instrumentation is based on a dual Fabry-Perot cavity refractometer in which the beat frequency between the light fields locked to two different cavities, one measurement and one reference cavity, is measured. The GAMOR methodology comprises a process in which the measurement cavity sequentially is filled and evacuated while the reference cavity is constantly evacuated. By performing beat frequency measurements both before and after the finite-pressure measurement, zero point references are periodically created. This opens up for high precision refractometry under nontemperature-stabilized conditions. A first version of an instrumentation based on the GAMOR methodology has been realized and its basic performance has been scrutinized. The refractometer consists of a Zerodur cavity-block and tunable narrow linewidth fiber lasers operating within the C34 communication channel (i.e., around 1.55 μm) at which there are a multitude of fiber coupled off-the-shelf optical, electro-optic, and acousto-optic components. The system is fully computer controlled, which implies it can perform unattended gas assessments over any foreseeable length of time. When applied to a system with no active temperature stabilization, the GAMOR methodology has demonstrated a 3 orders of magnitude improvement of the precision with respect to conventional static detection. When referenced to a dead weight pressure scale the instrumentation has demonstrated assessment of pressures in the kilo-Pascal range (4303 and 7226 Pa) limited by white noise with standard deviations in the 3.2N−1/2–3.5N−1/2 mPa range, where N is the number of measurement cycles (each being 100 s long). For short measurement times (up to around 10³ s), the system exhibits a (1σ) total relative precision of 0.7 (0.5) ppm for assessment of pressures in the 4 kPa region and 0.5 (0.4) ppm for pressures around 7 kPa, where the numbers in parentheses represent the part of the total noise that has been attributed to the refractometer. As long as the measurement procedure is performed over short time scales, the inherent properties of the GAMOR methodology allow for high precision assessments by the use of instrumentation that is not actively temperature stabilized or systems that are affected by outgassing or leaks. They also open up for a variety of applications within metrology; e.g., transfer of calibration and characterization of pressure gauges, including piston gauges.
Ce mémoire traite du développement et de la caractérisation d’un réfractomètre de type Fabry-Perot à simple cavité fonctionnant à 532 nm, conçu pour la mesure de la pression de gaz purs entre 100 Pa à 100 kPa. L’objectif de ce nouvel instrument est d’améliorer les meilleures capacités actuelles de mesurage réalisées en France par des méthodes conventionnelles basées sur la mesure d’une force appliquée sur une surface. Dans un premier temps, les principes fondamentaux de la réfractométrie sont rappelés, en détaillant les relations entre la réfractivité d’un gaz pur, sa densité molaire et sa pression thermodynamique. Les valeurs des coefficients du viriel de densité et de réfractivité du gaz ainsi que leurs incertitudes ont été extrapolées à la longueur d’onde et température de travail du réfractomètre développé au LNE-Cnam. Après une description détaillée de la cavité Fabry-Perot, les mesures de ses paramètres intrinsèques et leurs incertitudes (coefficient de dilatation thermique, coefficient de déformation mécanique, dérive à long terme et intervalle spectral libre) sont présentées. Elles permettent de corriger les erreurs sur la mesure de pression liées aux déformations subies par la cavité pendant la procédure de mesure. Les méthodes de contrôle de la température au millikelvin, de la stabilisation de la pression du gaz ainsi que la procédure permettant de limiter les effets thermiques liés à la détente-compression du gaz sont exposées. Une caractérisation métrologique de l’instrument a permis d’établir un budget d’incertitude complet et d’identifier les améliorations à apporter pour atteindre le niveau d’incertitude des références nationales autour de la pression atmosphérique et les surpasser pour des pressions plus basses. La reproductibilité des mesures a été évaluée à 2,7 ppm. Une comparaison avec une balance de pression de référence PG7607 entre 30 kPa à 100 kPa montre un écart relatif inférieur à 8 ppm, qui est plus faible que les meilleures incertitudes actuelles. Enfin, cette thèse explore une application innovante de la réfractométrie pour la mesure de pression acoustique dans la gamme infrasonore, permettant ainsi d’établir une continuité entre les mesures de pression statique et acoustique, qui fait défaut à l’heure actuelle.
After the redefinition of the International System of Units (SI), it has been realized that the pascal, the SI unit of pressure, can be calculated more accurately because, according to ideal gas law, the pressure depends on temperature, gas number density, and few fundamental constants like Boltzmann’s constant, the universal gas constant, and Avogadro’s number. Among those, the temperature is redefined by evaluating the thermal energy. The fundamental constants are already calculated using quantum electrodynamics (QED) with the utmost accuracy. Therefore, the precise measurement of pascal depends majorly on gas number density, which can be computed using optical techniques. Quantum calculations involve calculating pressure using optical methods, also known as the quantum realization of pascal. This chapter reviewed the optical techniques for measuring pressure using refractometry proposed by various national metrology institutes.
Ring lasers are commonly used as gyroscopes for aircraft navigation and attitude control. The largest ring lasers are sensitive enough that they can be used for high resolution inertial rotation sensing of the Earth in order to detect tiny perturbations to the Earth's rotation caused by earthquakes or global mass transport. This book describes the latest advances in the development of large ring lasers for applications in geodesy and geophysics using the most sensitive and stable devices available. Chapters cover our current knowledge of the physics of the laser gyroscope, how to acquire and analyse data from ring lasers, and what the potential applications are in the geosciences. It is a valuable reference for those working with ring lasers or using the data for applications in geodesy and geophysics; as well as researchers in laser physics, photonics and navigation.
Helium (He) is a noble gas with unique properties. It is the smallest atom, completely inert, stable, and whose characteristics are unmatched by other noble gases, elements, and compounds. Helium is used in several industries such as welding, cryogenics, medical, military, aerospace, as an atmosphere for growing silicon and germanium chips, and of course party balloons. There are two isotopes of Helium: ³He and ⁴He, the former is very rare and the latter is very common. Helium is generally found by accident with natural gas and occasionally carbon dioxide with percentages varying from as little as 0.04% to rarely 10% in many natural gas deposits. There are five field facilities that produce over 80% of the helium in the world that are located in Algiers, Qatar, Russia, and USA (two fields). The use of helium is growing at the rate of at least 6% per year and helium is considered by the USA as a critical mineral.
A procedure is presented which calibrates a wavelength/refractive-index tracker, so that it can compensate for the absolute refractive index of air within 3 × 10⁻⁸ ⋅ n. The procedure employs ultrahigh-purity helium and argon as reference gases of known n(p, T) to deduce the two unknown parameters in the working equation of the tracker: gas pathlength and pressure-induced distortion error. The performance of the gas calibration procedure is evaluated by comparing the corrected tracker against a master refractometer based on a Fabry–Perot cavity in nitrogen, a third reference gas of known n(p, T). In nitrogen, the calibrated trackers demonstrate accuracy at the level of 4 × 10⁻⁹ ⋅ n. Testing in a fourth reference gas—water vapor—reveals that the working equation of the trackers must include a third unknown parameter: an end-effect caused by a moisture-dependence of the reflection phase-shift. Correcting for this moisture-related error represents the largest contribution to measurement uncertainty, and explains why performance of the calibrated trackers is an order-of-magnitude worse in moist air than in pure gas. In air, the Fabry–Perot cavity-based refractometer performs within 5 × 10⁻⁹ ⋅ n, but is not a commercially-available device.
By measuring the refractivity and the temperature of a gas, its pressure can be assessed from fundamental principles. The highest performing instruments are based on Fabry-Perot cavities where a laser is used to probe the frequency of a cavity mode, which is shifted in relation to the refractivity of the gas in the cavity. Recent activities have indicated that such systems can demonstrate an extended uncertainty in the 10 ppm (parts-per-million or 10-6) range. As a means to reduce the influence of various types of disturbances (primarily drifts and fluctuations) a methodology based on modulation, denoted Gas Modulation Refractometry (GAMOR), has recently been developed. Systems based on this methodology are in general high-performance, e.g. they have demonstrated precision in the sub-ppm range, and they are sturdy. They can also be made autonomous, allowing for automated and unattended operation for virtually infinite periods of time. To a large degree, the development of such instruments depends on the access to modern photonic components, e.g. narrow line-width lasers, electro- and acousto-optic components, and various types of fiber components. This work highlights the role of such modern devices in GAMOR-based instrumentation and provide a review on the recent development of such instruments in Sweden that has been carried out in a close collaboration between a research institute and the Academy. It is shown that the use of state-of-the-art photonic devices allows sturdy, automated and miniaturised instrumentation that, for the benefit of industry, can serve as standards for pressure and provide fast, unattended, and calibration-free pressure assessments at a fraction of the present cost.
By measuring the refractivity and the temperature of a gas, its pressure can be calculated from fundamental principles. The most sensitive instruments are currently based on Fabry–Perot cavities where a laser is used to probe the frequency of a cavity mode. However, for best accuracy, the realization of such systems requires exceptional mechanical stability. Gas modulation refractometry (GAMOR) has previously demonstrated an impressive ability to mitigate the influence of fluctuations and drifts whereby it can provide high-precision (sub-ppm, i.e., sub-parts-per-million or sub- 10 − 6) assessment of gas refractivity and pressure. In this work, two independent GAMOR-based refractometers are individually characterized, compared to each other, and finally compared to a calibrated dead weight piston gauge with respect to their abilities to assess pressure in the 4–25 kPa range. The first system, referred to as the stationary optical pascal (SOP), uses a miniature fixed point gallium cell to measure the temperature. The second system, denoted the transportable optical pascal (TOP), relies on calibrated Pt-100 sensors. The expanded uncertainty for assessment of pressure ( k = 2) was estimated to, for the SOP and TOP, [ ( 10 mPa ) 2 + ( 10 × 10 − 6 P ) 2 ] 1 / 2 and [ ( 16 mPa ) 2 + ( 28 × 10 − 6 P ) 2 ] 1 / 2, respectively. While the uncertainty of the SOP is mainly limited by the uncertainty in the molar polarizability of nitrogen (8 ppm), the uncertainty of the TOP is dominated by the temperature assessment (26 ppm). To verify the long-term stability, the systems were compared to each other over a period of 5 months. It was found that all measurements fell within the estimated expanded uncertainty ( k = 2) for comparative measurements (27 ppm). This verified that the estimated error budget for the uncorrelated errors holds over this extensive period of time.
A technique for high-precision and high-accuracy assessment of both gas molar (and number) density and pressure, Gas Modulation Refractometry (GAMOR), is presented. The technique achieves its properties by assessing refractivity as a shift of a directly measurable beat frequency by use of Fabry-Perot cavity (FPC) based refractometry utilizing the Pound-Drever-Hall laser locking technique. Conventional FPC-based refractometry is, however, often limited by fluctuations and drifts of the FPC. GAMOR remedies this by an additional utilization of a gas modulation methodology, built upon a repeated filling and evacuation of the measurement cavity together with an interpolation of the empty cavity responses. The procedure has demonstrated an ability to reduce the influence of drifts in a non-temperature stabilized dual-FPC (DFPC)-based refractometry system, when assessing pressure, by more than three orders of magnitude. When applied to a DFPC system with active temperature stabilization, it has demonstrated, for assessment of pressure of N2 at 4304 Pa at room temperature, which corresponds to a gas molar density of 1.7 × 10⁻⁶ mol/cm³, a sub-0.1 ppm precision (i.e. a resolution of 0.34 mPa). It is claimed that the ability to assess gas molar density is at least as good as so far has been demonstrated for pressure (i.e. for the molar density addressed, a resolution of at least 1.2 × 10⁻¹³ mol/cm³). It has recently been argued that the methodology should be capable of providing an accuracy that is in the low ppm range. These levels of precision and accuracy are unprecedented among laser-based techniques for detection of atomic and molecular species. Since the molar polarizability of He can be calculated by ab initio quantum mechanical calculations with sub-ppm accuracy, it can also be used as a primary or semi-primary standard of both gas molar (and number) density and pressure.
The Boltzmann constant k is a scaling factor between macroscopic (thermodynamic temperature) and microscopic (thermal energy) physics.
Gas sensing based on bulk refractive index (RI) changes, has been a challenging task for localized surface plasmon resonance (LSPR) spectroscopy, presenting only a limited number of reports in this field. In this work, it is demonstrated that a plasmonic thin film composed of Au nanoparticles embedded in a CuO matrix can be used to detect small changes (as low as 6×10-5 RIU) in bulk RI of gases at room temperature, using a High-Resolution LSPR spectroscopy system. To optimize the film’s surface, a simple Ar plasma treatment revealed to be enough to remove the top layers of the film and to partially expose the embedded nanoparticles, and thus enhance the film’s gas sensing capabilities. The treated sample exhibits high sensitivity to inert gases (Ar, N2), presenting a refractive index sensitivity (RIS) to bulk RI changes of 425 nm/RIU. Furthermore, a 2-fold signal increase is observed for O2, showing that the film is clearly more sensitive to this gas due to its oxidizing nature. The results showed that the Au:CuO thin film system is a RI sensitive platform able to detect inert gases, which can be more sensitive to detect non-inert gases as O2 or even other reactive species.
Laser refractometers are approaching accuracy levels where gas pressures in the range 1Pa<p<1MPa inferred by measurements of gas refractivity at a known temperature will be competitive with the best existing pressure standards and sensors. Here, the authors develop the relationship between pressure and refractivity p=c1⋅(n−1)+c2⋅(n−1)2+c3⋅(n−1)3+⋯, via measurement at T=293.1529(13)K and λ=632.9908(2)nm for p≤500kPa. The authors give values of the coefficients c1,c2,c3 for six gases: Ne, Ar, Xe, N2, CO2, and N2O. For each gas, the resulting molar polarizability AR≡2RT3c1 has a standard uncertainty within 16×10−6⋅AR. In these experiments, pressure was realized via measurements of helium refractivity at a known temperature: for He, the relationship between pressure and refractivity is known through calculation much more accurately than it can presently be measured. This feature allowed them to calibrate a pressure transducer in situ with helium and subsequently use the transducer to accurately gage the relationship between pressure and refractivity on an isotherm for other gases of interest.
A novel realization of Gas Modulation Refractometry (GAMOR) that outperforms the original realization [Single Cavity Modulated GAMOR (SCM-GAMOR)], is presented. The reference measurements are carried out by equalizing the pressures in the two cavities. By this, the time it takes to reach adequate conditions for the reference measurements has been reduced. This implies that a larger fraction of the measurement cycle can be devoted to data acquisition, which reduces white noise and improves on short-term characteristics. The presented realization also encompasses a new cavity design with improved temperature stabilization and assessment. This has contributed to an improved long-term characteristics. The system was characterized with respect to a dead weight piston gauge. For short integration times (up to 10 min) it can provide a response that exceeds that of the original SCM-GAMOR system by a factor of two. For longer integration times, and up to 18 hours, the system shows, at 4303 Pa, an integration time independent Allan deviation of 1 mPa (corresponding to a precision defined as twice the Allan deviation, of 0.5 ppm). This implies that the novel system shows a significant improvement with respect to the original realization for all integration times (by a factor of 8 for an integration time of 18 hours). When used for low pressures, it can provide a precision in the sub-mPa region; for the case with an evacuated measurement cavity, the system provided, up to ca. 40 measurement cycles (ca. 1.5 hours), a white-noise limited noise of 0.7 mPa sqrt(cycle), and minimum Allan deviation of 0.15 mPa. Furthermore, over the pressure range investigated, i.e. in the 2.8 - 10.1 kPa range, it shows, with respect to a dead weight piston gauge, a purely linear response. This implies that the system can be used for transfer of calibration over large pressure ranges with exceptional low uncertainty.
Accurately quantifying mixture composition in supersonic flows is vital to the experimental development of fuel–air mixing enhancement strategies for scramjet propulsion applications. Mixture composition can be experimentally quantified by using either intrusive probe-based techniques or nonintrusive laser-based optical diagnostic techniques, with the different methods each offering unique advantages and disadvantages. To this end, the work presented herein provides a comparison of mixture composition measurements made with an intrusive gas-sampling probe to measurements of the same quantity made with the nonintrusive filtered Rayleigh scattering technique. The measurements are made downstream of a strut injector, which sonically injects helium parallel to a Mach 2.5 airflow, creating a two-dimensional planar shear layer composed of a binary mixture of helium and air. The experimental results show that the helium mole fraction measurements obtained using these two different techniques compare quite favorably. Specifically, at the primary downstream station of interest, all of the points surveyed with the gas-sampling probe lie within the 95% uncertainty bands of the filtered Rayleigh scattering measurements, with an average absolute difference in helium mole fraction of |ΔχHe|=0.0195 between the two techniques.
Recently, the resolution of length measurements based on optical interferometry reached sub-nanometer order. However, the accuracy of the optical interferometry is limited by the knowledge of absolute air refractive index. In this work, we have proposed a method to measure the absolute air refractive index by measuring the free-spectral-range and the resonance frequency of a Fabry-Perot cavity, whose inside is filled by vacuum and by air. During the air refractive index measurement, the geometrical length of the Fabry-Perot cavity must be constant in vacuum and in air. In this paper, a compensation of the Fabry-Perot cavity length deformation owing to a large pressure difference between air and vacuum is proposed.
Refractometry of air is a central problem for interferometer-based dimensional measurements. Refractometry at the 10⁻⁹ level is only valid if air temperature gradients are controlled at the millikelvin (mK) level. Very precise tests of second-generation National Institute of Standards and Technology (NIST) refractometers involve comparing two instruments (two optical cavities made from ultralow expansion glass) that are located in nominally the same environment; temperature gradients must be kept below a few millikelvin to achieve satisfactory precision of these tests. This paper describes a thermal stabilization scheme that maintains < 1 mK thermal gradients over 100 h in a 0.5 m × 0.15 m × 0.15 m volume. The approach uses passive (aluminum envelopes and foam insulation) and active (thermistors, foil heaters, and proportional–integral–derivative control) temperature stabilization. Thermal gradients are sensed with thermocouples and a nanovoltmeter and switch; the reference junctions of the thermocouples being in thermal contact with a thermistor temperature standard. Indirect evidence shows that performance is better than the < 1 mK uncertainty of measuring temperature gradients, the uncertainty of which is due to limitations of the nanovoltmeter and switch.
We present a simple method of simultaneous measuring of the refractive index and dispersion for transparent materials over a wide wavelength region from 430 nm to 660 nm. It is mainly composed of the broadband illuminating source of stable incident light intensity and spectrometer with CCD detector. In order to verify a reliable and appropriate performance of measurements of the present method, sapphire and K9 glass were used as the samples to test the method. The measuring results show that present method is in agreement with the minimum deviation method. It validates the feasibility of present technique.
Recent advancements in the field of Fabry-Perot cavity based optical refractometry (OR), FPC-OR, comprising the use of proposed novel methodologies based on drift-free Dual FPC-OR (DF-DFPC-OR), have alleviated much of earlier problems with FPC-OR, in particular drifts of the cavity spacer, and therefore open up for new levels of accurate characterization of gases. However, to be able to use these techniques for highly accurate assessments, general and explicit expressions are needed for how the gas density, ${\rho_n}$, is related to the refractivity of the gas, $n - 1$, and how the latter depends on the change in frequency of laser light that follows an evacuation of the cavity, $\Delta {\nu_l}$. This paper presents such relations that properly acknowledge the influence of all conceivable phenomena and higher order (non-linear) contributions to the assessment of these entities under drift-free conditions. The relative contributions of each of these phenomena to measurements of gas refractivity and density are individually assessed. It is shown, among other things, that, the influence of cavity deformation is larger for an open cavity (placed in a compartment in which gas is introduced) than a closed one; if not accounted for appropriately, under standard pressure and temperature conditions it can contribute to the accuracy of the technique with up to a few times 10$^{-3}$ if an open cavity is used, while it can be an order of magnitude smaller, i.e. in the order of 10$^{-4}$, for a well-designed closed cavity. It is also shown that the non-linearities from dispersion are smaller for the cases when relocking takes place than when it is not used. The expressions derived can serve as a basis for assessment of refractivity or gas density, and changes in such, in future realizations of OR in such a way that they can fully benefit from the extraordinary power of DF-DFPC-OR.
We consider the feasibility of basing a pressure standard on measurements of the dielectric constant ϵ and the thermodynamic temperature T of helium near 0 °C. The pressure p of the helium would be calculated from fundamental constants, quantum mechanics, and statistical mechanics. At present, the relative standard uncertainty of the pressure ur(p) would exceed 20 × 10⁻⁶, the relative uncertainty of the value of the molar polarizability of helium Aϵ calculated ab initio. If the relativistic corrections to Aϵ were calculated as accurately as the classical value is now known, a capacitance-based pressure standard might attain ur(p) < 6 × 10⁻⁶ for pressures near 1 MPa, a result of considerable interest for pressure metrology. One obtains p by eliminating the density from the virial expansions for p and ϵ − 1. If ϵ − 1 were measured with a very stable, 0.5 pF toroidal cross capacitor, the small capacitance and the small values of ϵ − 1 would require state-of-the-art capacitance measurements to achieve a useful pressure standard.
Recent quantum mechanical calculations of the interaction energy of pairs of helium atoms are accurate and some include reliable estimates of their uncertainty. We combined these ab initio results with earlier published results to obtain a helium-helium interatomic potential that includes relativistic retardation effects over all ranges of interaction. From this potential, we calculated the thermophysical properties of helium, i.e., the second virial coefficients, the dilute-gas viscosities, and the dilute-gas thermal conductivities of 3He, 4He, and their equimolar mixture from 1 K to 104 K. We also calculated the diffusion and thermal diffusion coefficients of mixtures of 3He and 4He. For the pure fluids, the uncertainties of the calculated values are dominated by the uncertainties of the potential; for the mixtures, the uncertainties of the transport properties also include contributions from approximations in the transport theory. In all cases, the uncertainties are smaller than the corresponding experimental uncertainties; therefore, we recommend the ab initio results be used as standards for calibrating instruments relying on these thermophysical properties. We present the calculated thermophysical properties in easy-to-use tabular form.
We introduce two new approaches for near-real-time, high-precision tracking of the refractive index of the ambient atmosphere. The methods can be realized at low cost and are expected to have important practical application in those accurate dimensional metrology applications employing interferometry in air. A valuable potential application is the control of step-and-repeat mask positioning for integrated circuit production in which temporal stability time scales over days can be crucial. Extension of the methods to absolute index measurement is discussed.
A new system, consisting of a double-channel Fabry-Perot etalon and laser diodes emitting around 780 nm, is described and proposed for use for measuring air-refractive index. The principle of this refractometer is based on frequency measurements between optical laser sources. It permits quasi-instantaneous measurement with a resolution of better than 10(-9) and uncertainty in the 10(-8) range. Some preliminary results on the stability of this system and the measurements of the refractive index of air with this apparatus are presented. The first measurements of the index of air at 780 nm are, within an experimental uncertainty of the order of 2 x 10(-8), in agreement with the predicted values by the so-called revised Edlén equations. This result is, to the best of our knowledge, the first to extend to the near IR the validity of the revised Edlén equation derived for the wavelength range of 350-650 nm.
The refractive index of nitrogen at 25 degree C has been measured accurately between 40 and 400 bar, and the refractivity R has been calculated. Particular attention is given to the determination of the density, which is the limitation of accuracy on R. The refractivity virial coefficients A//R, B//R, and C//R, have been calculated, the last one being given for the first time. The results are compared with the published data, the dielectric virial coefficients, and the induced absorption. The Buckingham expression of B//R is calculated for N//2 and compared with the present values.
Es wird ein Laserinterferometer vorgestellt, mit dem sich der Druck in Gasen kontinuierlich über den gemessenen Gangunterschied bestimmen läßt. Als Meßgas wird handelsüblicher Stickstoff verwendet. Die Meßgröße, der Gangunterschied, wird vorzeichenrichtig gezählt und digital angezeigt. Das sogenannte Druckmeßinterferometer wurde mit einem Präzisions- Kolbenmanometer bei der PTB verglichen. Die Unsicherheit der Druckmessung liegt bei 0,01%.
We have characterized the accuracy of atmospheric wavelength tracking based on a laser servolocked to a simple Fabry-Perot cavity. The motivations are (1) to explore a method for air refractive index measurement and (2) to determine the stability and accuracy of these cavities when employed as a length reference, with potential application to absolute distance interferometry, air-wavelength stabilized lasers, or similar applications. The Fabry-Perot cavity consists of mirrors optically contacted to an ultra-low-expansion spacer with the interior of the cavity open along its length to the surrounding air. Changes in laser frequency are monitored to determine changes in the refractive index of the gas in the cavity. We have studied limitations of this technique that arise from humidity effects, thermal distortion, and (for absolute refractive index measurements where the cavity must be evacuated) pressure-induced distortions. Comparing results from two cavities with very different lengths gives us a very sensitive probe of errors associated with end effects, and pressure-induced distortions can be measured by filling the cavity with helium, whose index of refraction is believed to be well known from ab initio calculations. The uncertainty of refractive index measurements can be greatly reduced when these sources of error are measured and corrected.
The increasing use of laser interferometers for ultraprecise length measurements in the free atmosphere has produced a corresponding requirement for the accurate determination of the refractive index of ambient air. Consequently the National Physical Laboratory (NPL) (Teddington, UK) has developed an automatic interference air refractometer sensors for this purpose. The measurement of refractive-index variations of gases at different wavelengths, temperatures, and pressures also has high importance, for example, by enhancing the understanding of the molecular structures involved. Therefore the NPL's air refractometer was also used for determining refractive indices and other associated refractometric parameters of a range of atmospheric gases from which a study was made on the validity of the Lorentz-Lorentz equation when applied to air. The gases that were studied at a wavelength of 633 nm were nitrogen, oxygen, argon, carbon dioxide, neon, helium, nitrous oxide (N{sub 2}O), krypton, and dry air.
We briefly describe a heterodyne refractometer developed at the BIPM in collaboration with the BNM/INM conservatory. The heart of the refractometer, a double Fabry-Perot interferometer, is placed inside the balance case of a very sensitive 1 kg mass comparator, the FB2 balance. Comparisons between methods using refractometry and the NPL revised Edlén formulas, carried out for a period of nine months, yielded a difference in air index of refraction of 4×10-8 with a standard deviation of 1×10-8. The variation of air index of refraction was about 1.5×10-5 during the study. Precise determinations of the short-term and long-term stability of the Fabry-Perot cavity, made of Zerodur, were also achieved. For monitoring air density, results obtained with the refractometry method were compared with those deduced from two other methods: the CIPM formula for the density of moist air and the use of buoyancy artifacts. The response characteristics for the three determinations were comparable and the agreement among the air density determinations was within 1×10-5 kg m-3.
This paper calculates the optical phase shifts on reflection from dielectric coated mirrors. The technique consists of measuring the transmission spectrum of the mirrors. These data are fitted with a theoretical expression based on an equivalent, quarter-wave stack. This expression then gives the phase-shift corrections for the mirrors. A fast algorithm, based on a series representation for Chebyshev polynomials, calculates the reflectivity and phase shift for a single wavelength in 3 sec in BASIC on an inexpensive home computer. An application is to measure absolute wavelengths with equipment commonly present in laser laboratories, namely, scanning Fabry-Perot interferometers with dielectric coated mirrors. The method is that of exact fractions, which requires one primary wavelength standard, with a less accurate, secondary standard (or wavemeter). The accuracy of the technique is equal to that of the primary standard. The precision is that of reading the interferometer and can be many times that of the secondary standard.
This paper presents the principle of an air wavelength standard for high accuracy length metrology. According to the definition of the Mètre, nanometric accuracy by interferometric measurement techniques can be reached only for measurements made under vacuum or by taking into account the fluctuations of the refractive index of air. We have developed a new type of laser source whose wavelength is insensitive to slow fluctuations of the refractive index of air. The sensitivity of our air wavelength standard to some characteristics (such as temperature, pressure, ageing behaviour...) has been measured. Results show that the relative uncertainty level of the wavelength of our source is below 10^{-8}.
The refractivity of helium computed from experimental measurements made at 15 different wavelengths by use of a corner-reflector Michelson interferometer, and computed from recent theoretical values of oscillator strength at 26 different wavelengths, is reported. The experimental measurements range from 4801 to 20586 Å. The absolute refractivity at 760 torr, 0°C, 5462.258 Å is 3.4950×10-5. A dispersion formula describing the data is (n-1)×105 = 1470.091/(423.98-σ2).Other dispersion formulas are discussed. The experimental and theoretical computations are compared with the experimental and theoretical work of various authors. The comparison includes computations of effective F values, absorption wavelength, dielectric constant, molecular polarizability, and Verdet constant.
Highly accurate and precise meaurements of the refractive index function F and refractive index measurements n(P,T) up to pressures of 35 MPa have been measured for a gaseous CH4, C2H4, C2H6, CO2, SF6, H2, N2, He, and Ar. Overflow experiments to determine F were extended to high pressure values. The third refractivity virial coefficient (RVC) C(R) becomes significant for all the gases examined, except for He, when the measurements approach approximately the value 0.3 of the reduced density rho/rho(c). The fourth RVC D(R) becomes significant for C2H4, C2H6, and SF6 when rho/rho(c) approaches approximately one. The agreement between the classically calculated values of second RFC B(R) and the present experiment B(R) values is very good for CH4, C2H5, C2H6, SF6, and Ar, but poor for CO2, H2, N2, and He. For He, the quantum mechanical calculation of B(R) obtained by including both the long-range and short-range effects on polarizability is in remarkably good agreement with the present experimental value.
We have obtained very accurately calculated nonrelativistic values of the index of refraction and the Verdet constant of helium gas. We have used the Breit-Pauli operator to obtain corrections of order α2 to the same optical quantities. This required us to analyze the scattering of light by helium atoms correct up to third order: second order in the electromagnetic field and first order in the Breit-Pauli operator. We compare these results with experimental values and find some significant discrepancies.
Relativistic and leading QED corrections to the static electric dipole polarizability of helium are calculated. The resulting theoretical uncertainty is estimated to be under 2 ppm.
We describe a new simple, compact refractometer for air refractive index measurements. It consists of a double plane–plane Fabry Perot interferometer. Both interferometers consisting of Zerodur spacers of thickness of 1 and 100 mm are illuminated independently by the same single mode laser diode. The shorter cavity allows unambiguous identification of the transmission peak of the longer one to which the laser frequency is servo-locked. The refractive index of air is obtained via a heterodyne comparison with a second laser locked to a hyperfine component of the rubidium D2 line. We obtain a resolution of order 10−10 and accuracy of a few times 10−8. The metrological characteristics of the interferometer in vacuum are presented. Initial results for refractive index measurements agree with values calculated using the revised Edlen formulas. We also describe how this refractometer is used to measure variations of the density of air and their correlation with changes of refractive index of air. The density of air is used to make buoyancy corrections when comparing mass standards of different volume. Our preliminary results indicate that the values of air density determined by refractometry agree with those calculated using the Comité International des Poids et Mesures formula, which is based on measurements of temperature, pressure, moisture content, and CO2 concentration. © 1999 American Institute of Physics.
Precise electromagnetic measurements of the polarizability (dielectricity, refractivity) of a macroscopic sample of helium gas are re-examined in view of recently published ab initio calculations of the helium atomic polarizability, including relativistic effects. The collation yields a new estimate of the Boltzmann constant as well as a general assessment of the reliability of optical and electrical measurements of the polarization properties of helium gas.
A spherical Fabry-Perot interferometer is used to study compressed gases to a few hundred bar. First problem is the determination of the true length of gas. It requires an accurate measurement of the optical length of the interferometer under vacuum, and different corrections: phase variation at reflection from multidielectric coatings, which varies with wavelength, compressibility of the spacer, etc.
Air-buoyancy corrections are often a dominant source of error in the transfer of the unit of mass through weighing of nominally equal weights of different densities. The possibilities of monitoring air density through measurement of the refractive index of air are considered. A critical discussion is given of the present knowledge of the refractivities of air and its constituent gases. A simple laser refractometer capable of resolving air-density variations of 0.15 × 10-3 kg m-3, corresponding to a change in air buoyancy of 9 μg, is demonstrated in weighings of the Swedish prototype kilogram with stainless-steel standards.
A four-ring, toroidal cross capacitor was used to measure accurately the relative dielectric permittivity (p,T) of He, Ar, N2, O2, CH4, C2H6, C3H8, and CO2. ( is often called the dielectric constant.) The data are in the range from 0 to 50C and, in many cases, extend up to 7 MPa. The accurate measurement of (p,T) required a good understanding of the deformation of the gas-filled capacitors with applied pressure. This understanding was tested in two ways. First, the experimental values of (p,T) for helium were compared with theoretical values. The average difference was within the noise,
expt–
theory=(–0.050.21)10–6, demonstrating that the four-ring cross capacitor deformed as predicted. Second, (p,T) of argon was measured simultaneously on three isotherms using two capacitors: the four-ring capacitor, and a 16-rod cross capacitor made using different materials and a different geometry. The results for the two capacitors are completely consistent, within the specifications of the capacitance bridge. There was a small inconsistency that was equivalent to 110–6 of the measured capacitances, or, for argon, 310–5
A
, where A
is the zero-density limit of the molar polarizability (–1)/[(+2)].
The dielectric, refractivity, Kerr, and hyperpolarizability second virial coefficients for the helium and argon gases are evaluated for a wide range of temperatures using a semiclassical approach and the high quality frequency-dependent interaction induced electric polarizabilities and second hyperpolarizabilities of the previous paper. For helium and argon we obtain satisfactory agreement with most of the experimental data for the dielectric and the refractivity second virial coefficients. Our results confirm that the helium gas second Kerr virial coefficient is very small at temperatures beyond 70 K. For argon we obtain a very good agreement with a recent experimental determination at 632.8 nm, whereas we suggest that previous experimental results for 458 nm might be inaccurate. The ESHG results indicate a possible disagreement between a recent experimental determination and the semiclassical ansatz for the second hyperpolarizability virial coefficients. (C) 1999 American Institute of Physics. [S0021-9606(99)31046-1].
An interference refractometer for absolute measurement of the refractive index of air has been developed. It is essentially based on components of a commercial Zeeman type laser interferometer system and uses a differential interferometer and a vacuum chamber with four measurement cells. The resolution and accuracy of interference phase interpolation is improved by an electronic phase meter and a detection technique which compensates for interferometer non-linearity. The system has been designed for an accuracy of a few parts in 108. Experimental investigations of individual components and comparison measurements with pure gases, with Edlén's formula and with the BCR interference refractometer confirm this goal.
The use of a long, thermally stable Fabry-Perot etalon as a refractometer is considered in detail in this study of the refractive index of air. The etalon consists of two flat plates of fused silica 60 mm in diameter, with a cylindrical spacer made of Zerodur (a polycrystalline glass ceramic of extremely low thermal expansion) 200 mm long. The interferogram of light from a frequency-stabilized He-Ne laser is imaged with large-diameter mirror optics. The principal result is a demonstration of the effects of changes in atmospheric pressure on the etalon. The measured refractive-index values deviate by 2 parts in 10(7) from calculated values. Possible causes of error are considered in detail.
Large Gaussian-type geminal wave function expansions and direct perturbation theory (DPT) of relativistic effects have been applied to calculate the relativistic contribution to the static dipole polarizability of the helium atom. It has been demonstrated that DPT is superior for this purpose to traditional Breit-Pauli calculations. The resulting value of the molar polarizability of 4He is 0.517254(1) cm3 x mol(-1), including a literature estimate of QED effects. As a by-product, a very accurate value of the nonrelativistic helium second hyperpolarizability, gamma = 43.104227(1) atomic units (without the mass-polarization correction), has been obtained.
Measurements of the dispersion of the refractive index using a nonlinear interferometer are described. A sample of the optical material to be measured is interposed between two optically nonlinear crystals, and a moderately intense laser is passed through the combination. By observing the interference between the second harmonics produced in the two nonlinear crystals, the difference between the refractive indices of the sample at the laser frequency and its second harmonic frequency can be determined very precisely. We have used the interferometer to measure the dispersion in several gases between 1064 and 532 nm. We have also used it to determine the dispersion of two transparent solids. The method can measure the index difference to better than 0.1%, which for gases such as helium represents an absolute accuracy of ~10-9.
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