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Noncontact terahertz time-domain spectroscopy was employed to measure pressure-dependent refractive indices of gases such as helium (He), argon (Ar), krypton (Kr), oxygen (O2), nitrogen (N2), methane (CH4), and carbon dioxide (CO2). The refractive indices of these gases scaled linearly with pressure, for pressures in the 55–3,750 torr range. At the highest pressure, the refractive indices ((n-1) x 106) of He and CO2 were 170 and 2,390, respectively. The refractive index of CO2 was 14.1-fold higher than that of He, owing to the stronger polarizability of CO2. Although the studied gases differed in terms of their molecular structure, their refractive indices were strongly determined by polarizability. The measured refractive indices agreed well with the theoretical calculations.
© 2016 Optical Society of America
1. Introduction
In the work that introduced the first THz system in 1989, THz time-domain spectroscopy (THz-TDS) was performed on water vapor [1]. The THz pulse propagated through humid air, providing information on absorption and dispersion in the water vapor [2,3]. After developed the first THz-TDS, a high-quality N2O gas was investigated using a gas cell that was located in the middle of the THz beam’s path [4,5]. The THz pulse propagated through the gas cell filled with gas. This method has been commonly used to measure the THz characteristics of gases [4–6], along with the newly suggested THz resonance method that uses grooves or slits in parallel-plate waveguides [7,8]. Recently, significant attention has been paid to studying rotational motion and inter- and intra-vibrational motion of molecular gases such as carbon monoxide (CO) [9], ammonia (NH3) [10], and hydrochloric acid (HCl) [11] because for many molecular gases kinetic energy is in the THz region. Meanwhile, refractive index is an important parameter in applied interferometry, which is used in heat transfer studies [12], for efficiently sensing the resonance of the metasurface [13,14], and for detecting CH4 and CO2 gases distilled from shale oil [15].
Here, we measured the pressure dependence of the refractive indices of He, Ar, Kr, O2, N2, CH4, and CO2 gases, for frequencies in the 0.3–4.5 THz range. Among them, He, Ar, and Kr belong to the group VIII family in the periodic table and are monatomic. O2 and N2 are diatomic, with double and triple bonds, respectively. CH4 and CO2 are mixed gases, composed of two different elements. We selected these three groups to study the molar mass (He, Ar, and Kr), number of bonds (O2 and N2), and molecule structure (CH4 and CO2)-dependent refractive indices with different pressures. These gases have been fundamentally studied for refractive index resonances using the ultraviolet and visible light regions [16–22]. Because the resonance lines of these gases are in the ultraviolet and visible regions, there is a need to expand to the THz region for THz applications. Recently, the refractive indices of CO2 and CH4 were measured in the THz region, but only at atmospheric pressure and room temperature [15]. The key parameters that determine the refractive index of a gas are the gas polarizability, the gas temperature, and the gas pressure [23]. Among these, polarizability is determined by the gas itself. Temperature and pressure can be varied to determine their effect on the refractive index of a gas. Here, we measured the refractive indices of the above gases at different pressures at room temperature. The results of our measurements were in a good agreement with theoretical calculations.
2. Experimental setup
The experimental setup in the present study was based on a conventional THz-TDS system, schematically shown in Fig. 1. In this system, a 10-cm-diameter and 30-mm-length cylindrical gas cell was located between two parabolic mirrors. The cell was sealed at both ends by 3-mm-thick undoped silicon windows and O-rings. Two tube lines connected a precision pressure gauge to the input tube line and a drain valve to the output tube line. The THz-TDS system, including the gas cell and other parts such as the tube lines, valves, gauge, and gas control units was kept in an airtight box filled with dry air (humidity below 3%), to prevent humid air from penetrating into the gas cell. Only the gas container and a connection tube line with a vacuum pump were located outside of the airtight box. Because the studied gases had very high quality, a small amount of humid air in the gas cell can significantly alter experimental results. To prevent humid air from remaining into the gas cell, the gas cell was filled with dry air, following which the dry air was removed using the vacuum pump. Using these methods, we obtained a water vapor-free experimental system. We measured the transmitted THz pulses by scanning the cell at room temperature (20 °C) for different gas pressures, ranging from low vacuum (here, 55 torr) to a high pressure (3,750 torr). Owing to manual pressure setting, the pressures for the different gas samples were not exactly same. However, for each sample the pressure variation was ± 15 torr (2 KPa). The temporal resolution in our studies was 0.013 ps, which corresponds to a 2 µm step of the mechanical delay line, and the dynamic frequency range spanned 1.5 GHz after zeros were padded to the time domain data. All of the studied gas samples were at least 99.95% pure, and all were optically transparent.
3. Measurement and analysis
When the refractive index of a gas exhibits a resonance at a specific wavelength (or frequency), it can be represented using the Sellmeier formula [24]:
where n is the refractive index, λ is the wavelength, and Bi and Ci are the experimentally determined Sellmeier coefficients [16–22]. In our studies, the coefficients Ci, capturing the resonance wavelengths of the gas, were in the ultraviolet and visible regions. Therefore, it was assume that the refractive indices of the studied gases saturated in the THz region. Meanwhile, according to the Hauf–Grigull relation [12], the refractive index of a rare gas is:where r is the gas refractivity, M is the gas molar mass, P is the pressure, R is the ideal gas constant, and T is the temperature which is constant (20 °C) during measurement. For the studied gases, Table 1 lists the molar masses, densities (at 760 torr (1 atm) and 293.12 K), refractive indices (at 1 THz obtained from Eq. (1)), refractivity (calculated from Eq. (2)), and polarizability.
Table 1. Parameters for comparing experimental measurements to theoretical calculations (temperature: 293.12 K; pressure: 760 torr))
The reference THz pulse was obtained by scanning the gas cell in a low vacuum state (pressure 55 torr). Figure 2 shows the measured reference and sample pulses for the gas cell filled with He, O2, Kr, and CO2 gases for pressures increasing by 375 torr (50 KPa) to 3,750 torr (500 KPa). Because of the very stable laser (Mai Tai XF-1, Spectra-Physics) and THz-TDS system, each measured THz pulse is also highly stable, and the pulse shape is almost identical to that shown in Fig. 2 (a). The amplitude change per the increase in pressure is very small because the gases have very low power absorption. For example, the amplitude variation between the reference and the sample pulse at maximal pressure is only 8%, which indicates the power absorption at the maximum pressure is 0.0022 cm−1. For He, the maximal temporal delay between the reference pulse and the sample pulse for in the maximal pressure scenario is only 0.173 ps. For CO2, the maximal temporal delay is 2.368 ps, which is nearly 13.7 times longer than the delay for He. The longer temporal delay indicates the higher refractive index of this sample. The refractive index of a sample gas is n = c/(L/Δt), where c is the speed of light, L is the cell length, and Δt is the temporal delay. THz-TDS can directly measure the temporal delay. The refractive index of a sample gas can be also measured using the phase delay in the frequency domain, as n = 1 + ΔΦ/(2πfL/c), where ΔΦ is the phase difference and f is the frequency. Because the refractive index variations were almost constant in the THz region, the refractive indices measured in the time and frequency domains varied within ± 0.9% for the considered pressure.
Fig. 2 Measured THz reference pulse (transmitted through an empty gas cell at a 55 torr pressure (block)) and sample pulses for different gas pressures. The vertical dashed lines indicate the peak positions of the first and the last THz pulses. (a) The results for helium. (b) The results for oxygen. (c) The results for krypton. (d) The results for carbon dioxide.
Pressure dependence of the refractive indices of the different gases is summarized in Fig. 3. The measured frequency expended up to 4.5 THz because of the very stable and narrow THz pulse measurement. The refractive indices remained constant with increasing frequency but increased with increasing pressure. For He and CO2, the refractive indices ((n-1) 106) at the highest pressure were 170 and 2,390, respectively. The refractive index of CO2 was 14.1-fold higher than that of He. This ratio is very close to the ratio of He to CO2 pulse delays which is 13.7.
Fig. 3 Pressure dependence of refractive indices ((n-1) 106). (a) The results for helium. (b) The results for oxygen. (c) The results for krypton. (d) The results for carbon dioxide.
Figure 4 plots the refractive indices of the different studied gases as a function of the gas pressure. The dots and solid lines indicate the refractive indices obtained, respectively, using the phase delay measurement and theoretical calculations based on Eq. (2) and Table 1. For all gases except CO2, the measured and the calculated values agreed very well. The calculated refractive index for CO2 was obtained from the best fit at 1 THz using Eq. (1) and the Sellmeier coefficients, as explained in [18], Bideau-Mehu et al.. CO2 exhibits five resonance peaks, at 77.6, 112.1, 133.3, 147.4, and 4,135 nm; these peaks span from the near ultraviolet region to the near infrared region. Therefore, in the case of CO2 Eq. (1) is a fifth-degree polynomial. For these reasons, the best fit in the THz region may be inaccurate. However, we directly measured the refractive index of CO2 in the THz region, and the measured result agreed well with previously reported values [15]. Table 2 lists the refractive indices of CO2,obtained from calculations, time domain measurements, and frequency domain measurements, for different pressures. Because the calculated result for atmospheric pressure deviates by 10.8% from the phase measurement result, the deviation accumulates at higher pressures. However, for the remaining gas samples and nearly atmospheric pressures, the calculated results agree well with the corresponding experimental results, as shown in the inset of Fig. 4.
Fig. 4 Refractive indices ((n-1) 106) vs. pressure. The solid lines indicate theoretical calculations using the Sellmeier coefficients. The dashed line indicates theoretical calculation using the polarizability of CO2 obtained from the CRC handbook [25]. The inset shows an expanded view, for pressures from 340 torr to 800 torr.
Table 2. Refractive indices (n-1) 106 of CO2, at different pressures (temperature: 293.12 K).
For a constant temperature, the refractive index depends on the gas pressure and molar refractivity (Rn) as [26]:
where ρ is the density (in PM/RT). The molar refractivity importantly determines the gas refractive index. The molar refractivities of N2, Ar, and O2 were 4.390, 4.142, and 3.690, respectively. Because the molar refractivities of these three gases were similar, the refractive indices were also similar and followed the order of molar refractivities. For reasons mentioned above, the refractive indices of CO2, CH4, and Kr, whose molar refractivities were 6.963, 6.365, and 6.257, respectively, were similar, with that of CO2 being the largest.On the other hand, polarizability (α’) as well determines the gas refractive index. The relationship between the gas molar refractivity and polarizability can be written as [26]
where NA is the Avogadro constant. Larger polarizability yields higher refractive indices. The polarizability of CO2 obtained from calculations and from the CRC handbook [25] are 2.76210−24 cm3 and 2.91110−24 cm3, respectively. The difference is 0.14910−24 cm3 (5.4%). However, the experimental results for the other gas samples are in a good agreement with the calculated and CRC handbook data, as shown in Table 1. When we use the calculated polarizability of CO2, the measurement result and the result of theoretical calculations (solid line of CO2 in Fig. 4) are different at higher pressures. However, if we use the polarizability given in the CRC handbook, the measurement result and the result of theoretical calculations (dashed line in Fig. 4) are very similar.In addition, for a constant temperature, we can write Δ(n-1)/(n-1) = Δp/p for specific gases captured by Eq. (2). Therefore, a gas with higher polarizability is more sensitive to pressure variations than a gas with smaller polarizability. Figure 5 shows the dependence of gas polarizability on the refractive index, for different pressures. The solid lines are the fits to the pressure-dependence measurements. From Eqs. (3) and (4), we obtain . The gas polarizability is inversely proportional to the gas pressure. At low pressures, the measured data are not well aligned compared with high-pressure data, because it was impossible to precisely set the target pressure using the manual regulator of the gas container. Although the pressure variation is under 15 torr, relatively large variations are observed at lower pressures compared with high pressure. However, the data points are still located on the fitting lines. The fit slopes corresponding to the pressures of 375 torr and 3,750 torr are 7.74 /cm3 and 7.75 /cm3, respectively. Therefore, the refractive indices of gas samples with higher polarizability and higher pressure are more sensitive than those of gas samples with lower polarizability and lower pressure. Based on the above analysis, pressure dependence of refractive indices can be precisely determined using the THz-TDS method.
Fig. 5 Gas polarizability vs. the gas refractive index ((n-1) 106). The solid lines indicate the fits to the pressure-dependence measurements.
4. Summary and conclusions
Using the powerful THz-TDS technique, we have experimentally and theoretically studied the refractive indices of several gases. The studied gases were categorized into three groups: 1) a monatomic group that included He, Ar, and Kr; 2) a diatomic group that included O2 (double band structure) and N2 (triple bond structure); and 3) gases containing two different elements – this group included CH4 and CO2. Although the different studied gases had different molecular and atomic structures, their refractive indices linearly depended on the gas polarizability and pressure when the temperature was constant. The polarizability, determined by the gas molar refractivity, is gas-specific. However, gas pressure importantly affects the gas refractive index. Higher polarizability and higher pressure yield larger refractive indices. The refractive indices ((n-1) 106) of He and CO2 at the highest pressure were 170 and 2,390, respectively. The refractive index of CO2 was 14.1-fold larger and more sensitive than that of He. The measured and calculated results agreed very well, except for the CO2 sample, for which the results exhibited a small deviation owing to the ultraviolet and visible range resonances and a complicated polynomial expression for fitting the data in the THz range. However, if we use the polarizability given in the CRC handbook, the measurement result and the result of theoretical calculations are very similar. When measuring the refractive index of a known gas, the gas pressure can be determined using our results, or vice versa. In the present study, we demonstrated that THz-TDS is a powerful noninvasive method that can be used for calculating the refractive index of a gas at different gas pressures. Consequently, measurements of pressure-dependence of the gas refractive index using the THz-TDS method is likely to find use in interferometry, in studying heat transfer in gases, or in real-time pressure measurements in the natural gas industry.
Funding
This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1A2B4012523).
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