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Refractive index measurement of compressed nitrogen using an infrared frequency-domain interferometer
Optik 164 (2018) 1–4
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Refractive index measurement of compressed nitrogen using an infrared frequency-domain interferometer Jiangtao Li, Lei Liu, Jun Tang, Guojun Li, Heli Ma, Yunjun Gu, Shenggang Liu, Jidong Weng, Qi Feng Chen ∗ National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, CAEP, Mianyang, Sichuan, 621900, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 20 January 2018 Accepted 20 February 2018 Keywords: Refractive index Broadband interferometer Frequency-domain interferometer Polarizability
a b s t r a c t Frequency-domain interferometry is used to obtain the refractive index of compressed nitrogen in the vicinity of 1550 nm as the pressure of nitrogen increases from 0 to 40 MPa at room temperature. By referring to the equation of state of nitrogen, the relation between the refractive index and the nitrogen density is obtained and the electric polarizability of a nitrogen molecule is thereby determined to be (1.82 ± 0.15) × 10−30 m−3 according to the Lorentz-Lorentz relation. © 2018 Elsevier GmbH. All rights reserved.
1. Introduction Refractive index of nitrogen is an important parameter for astrophysicists to infer the distance and speed of stones falling onto the earth from outer space. For such a falling stone in the atmosphere, a pressure gradient exists in the front, which is associated with a gradient of refractive index. In order to predict accurately when and where the stone will hit the earth, the relation between the refractive index of nitrogen and its density needs to be determined for the working wavelength of interferometer. Doppler Pin System (DPS) is one of the most popular velocity interferometer systems and is widely used in the velocity diagnostics of dynamic processes [1–7]. The working wavelength is 1550 nm, which is in the infrared range. In the visible range, refractive index of nitrogen has been determined by interferometric measurements [8–11]. However, in the infrared range, especially in the vicinity of 1550 nm, the data are absent. Frequency-domain interferometer has found a growing number of applications in displacement or deformation measurements [12–21]. If the distance between two reflecting interfaces along the optical path are known, frequency-domain interferometer can be inversely applied to determine the refractive index of the material between the two reflecting interfaces. In this work, the refractive index of nitrogen in the vicinity of 1550 nm is obtained using a frequency-domain interferometer when the pressure of nitrogen increases from 0 to 40 MPa at room temperature. The refractive index measurement are detailed in the experimental section. In the results and discussion section, the relation between the refractive index of nitrogen and its pressure (density) will be given and the mean polarizability of nitrogen molecule in the vicinity of 1550 nm will be determined.
∗ Corresponding author. E-mail address: [email protected] (Q.F. Chen). https://doi.org/10.1016/j.ijleo.2018.02.056 0030-4026/© 2018 Elsevier GmbH. All rights reserved.
2
J. Li et al. / Optik 164 (2018) 1–4
Fig. 1. A schematic diagram of the experimental setup for the measurement of the refractive index of nitrogen.
Fig. 2. (a) The interference in the frequency-domain shown from 1545 nm to 1555 nm. (b) The three most notable peaks obtained from a transformation of (a). The horizontal axis of (b) is a product of refractive index and distance.
2. Experimental A schematic diagram of the experimental setup is shown in Fig. 1. The setup includes a gas source of nitrogen (purity > 99.999%), a gas booster and pump system, a gas cell which is enclosed by a stainless steel container and a sapphire window, and a frequency-domain interferometer which is composed of a broadband infrared laser, a circulator and an infrared spectrometer. The components of the interferometer are connected by optical fibers. The frequency-domain interferometer uses a broadband infrared laser to provide a coherent light beam. The beam goes through the circulator in the anti-clockwise direction, and arrives at the front surface of the sapphire window, which serves as the first reflector. The second reflector is the back surface of the sapphire window, and the third one is the front surface of the mirror inside the gas cell. The distance between the front surface of the mirror and the back surface of the sapphire window is maintained by a drilled ring spacer which is not shown in Fig. 1. The reflected light beams from the three reflectors will re-enter the circulator and will finally get into the infrared spectrometer. The fringe pattern in the frequency domain will be recorded by the CCD of the spectrometer. Since we are only interested in the spectral range around 1550 nm, the spectra from 1545 nm to 1555 nm were recorded when we increased the nitrogen pressure in the gas cell from 0 to 40 MPa. The gas pressure was given by a calibrated pressure gauge with an uncertainty of 0.5%, the ambient temperature was provided by a calibrated temperature gauge with an uncertainty of 0.5 K.
3. Results and discussion Fig. 2(a) shows the interference recorded by the frequency-domain interferometer from 1545 nm to 1555 nm. A constant offset is applied for each spectrum to show the periodic structure along the wavelength direction. The periodicity originates
J. Li et al. / Optik 164 (2018) 1–4
3
Table 1 The refractive index of nitrogen (n) and its uncertainty with respect to the gas pressure and the ambient temperature. Pressure MPa
Temperature K
Densitya g/cm3
n
sigma(n)
4.00E-06 5.18 10.29 19.92 29.86 39.39
296.2 296.2 296.0 296.0 296.0 296.0
4.80E-08 0.0594 0.1171 0.2168 0.3013 0.3649
1.0000 1.0128 1.0284 1.0508 1.0803 1.1003
0.0022 0.0031 0.0027 0.0026 0.0026 0.0024
a
The density is calculated using the equation of state of nitrogen [22].
Fig. 3. (a) The relation between the refractive index of nitrogen and its pressure at aboutT = 296 K. (b) The relation between the refractive index of nitrogen and its density at about T = 296 K. A linear fit of the data is also shown.
from the phase differences of the light beams reflected from the three reflectors as mentioned in the experimental section. The phase difference can written as: () = 2 ·
n · 2d .
(1)
here, n is the refractive index and d is the distance between the reflectors. If the optical media between the two reflectors has a layered structure, the term in the numerator in Eq. (1) needs to be replaced by a sum of optical paths. It can be seen from Eq. (1) that, the phase difference varies periodically if 1/ changes. Therefore, a Fourier transformation can be applied, which changes the variable from to nd, as shown by Fig. 2(b). Fig. 2(b) shows three peaks on the axis of nd. Peak 1 is associated with the interference of light beams coming from the first and second reflectors, which is exactly the thickness of the sapphire window multiplied by the refractive index of the sapphire. That also explain why location of peak 1 is invariant as the gas pressure increases. Peak 2 is associated with the interference of light beams coming from the second and the third reflectors. The shift of the peak 2 reflects the increase of refractive index of the gas when the nitrogen pressure increases. Peak 3 is associated with the interference of light beams coming from the first and third reflectors. That is why the amount of shift for peak 2 and 3 are the same for all the measurements. Based on the analysis above, the refractive index of nitrogen can be obtained by peak 2 location in Fig. 2(b) using the following formula: n(p) =
n(p) · d , n(p = 0) · d
(2)
here, the term on the right side of Eq. (2) is obtained by a Gaussian peak fitting procedure with the uncertainty determined. We also assume that the refractive index is 1 when p = 0. The results are given in Table 1. Fig. 3(a) shows the relation between the refractive index of nitrogen and its pressure, as measured directly from experiment. Using the equation of state of nitrogen [22], the relation between the refractive index of nitrogen and its density can also be obtained, as shown in Fig. 3(b). According to the linear fit, the relation between the refractive index of nitrogen and its density in the vicinity of 1550 nm can be written as: n = 1 + 0.264(g/cm3 )
(3)
4
J. Li et al. / Optik 164 (2018) 1–4
The electric polarizability of a nitrogen molecule can be obtained according to the Lorentz-Lorentz relation [8,23]: n2 − 1 4 = N˛, 3 n2 + 2
(4)
here, N is the number density of nitrogen molecules, ˛ is the electric polarizability of a nitrogen molecule, ε0 is the electric permittivity of free space. The value of the electric polarizability of a nitrogen molecule according to Eq. (4) is determined to be (1.82 ± 0.15) × 10−30 m−3 , which is very close to the value of 1.75 × 10−30 m−3 , which is extrapolated from the visible regime using the semi-empirical equation given by Kerl et al. [8]. 4. Conclusions and outlook The refractive index of nitrogen in the vicinity of 1550 nm is measured using a frequency-domain interferometer when the gas pressure increases from 0 to 40 MPa at room temperature. By referring to the equation of state of nitrogen, the relation between the refractive index of nitrogen and its density is obtained, which can be approximated by a linear formula: n = 1 + 0.264(g/cm3 ). Furthermore, the electric polarizability of a nitrogen molecule is determined to be (1.82 ± 0.15) × 10−30 m−3 , which is consistent with the extrapolated result of a semi-empirical equation from literature. Acknowledgements We would like to thank Tianjiong Tao for helpful discussion. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11604315, 11504352, 11674292, and 11604313), the Foundation of National Key Laboratory of Shock Wave and Detonation Physics, CAEP (Grant Nos. 6142A03010203 and 9140C670103150C67289), the Science Challenge Project (Grant No. TZ2016001), the President Foundation of China Academy of Engineering Physics (201501032), and the Science and Technology Development Foundation of China Academy of Engineering Physics (2015B0102001). References [1] J. Zheng, Q.F. Chen, Y.J. Gu, J.T. Li, Z.G. Li, C.J. Li, Z.Y. Chen, Shock-adiabatic to quasi-isentropic compression of warm dense helium up to 150 GPa, Phys. Rev. B 95 (2017) 224104. [2] J. Zheng, Q. Chen, Y. Gu, Z. Li, Z. Shen, Multishock compression properties of warm dense argon, Sci. Rep. 5 (2015) 16041. [3] M.D. Knudson, M.P. Desjarlais, A. Becker, R.W. Lemke, K.R. Cochrane, M.E. Savage, D.E. Bliss, T.R. Mattsson, R. Redmer, Direct observation of an abrupt insulator-to-metal transition in dense liquid deuterium, Science 348 (2015) 1455–1460. [4] M.D. Knudson, M.P. Desjarlais, Shock compression of quartz to 1.6 TPa: redefining a pressure standard, Phys. Rev. Lett. 102 (2009) 225501. [5] M.D. Knudson, D.L. Hanson, J.E. Bailey, C.A. Hall, J.R. Asay, C. Deeney, Principal Hugoniot, reverberating wave, and mechanical reshock measurements of liquid deuterium to 400 GPa using plate impact techniques, Phys. Rev. B 69 (2004) 144209. [6] J. Zheng, Q. Chen, Y. Gu, Z. Chen, Hugoniot measurements of double-shocked precompressed denxe xenon plasmas, Phys. Rev. E 86 (2012) 066406. [7] K. Falk, S.P. Regan, J. Vorberger, B.J.B. Crowley, S.H. Glenzer, S.X. Hu, C.D. Murphy, P.B. Radha, A.P. Jephcoat, J.S. Wark, D.O. Gericke, G. Gregori, Comparison between X-ray scattering and velocity-interferometry measurements from shocked liquid deuterium, Phys. Rev. E 87 (2013) 043112. [8] K. Kerl, U. Hohm, H. Varchmin, Polarizability of a(w,T,rho) of small molecules in the gas phase, Ber. Bunsenges. Phys. Chem. 96 (1992) 728–733. [9] U. Hohm, K. Kerl, Temperature dependence of mean molecular polarizability of gas molecules, Mol. Phys. 58 (1986) 541–550. [10] R.C. Burns, C. Graham, A.R.M. Weller, Direct measurement and calculation of the second refractivity virial coefficients of gases, Mol. Phys. 59 (1986) 41–64. [11] D. Reolon, M. Jacquot, I. Verrier, G. Brun, C. Veillas, High resolution group refractive index measurement by broadband supercontinuum interferometry and wavelet-transform analysis, Opt. Express 14 (2006) 12744–12750. [12] Y.K. Milyukov, V.N. Rudenko, B.S. Klyachko, A.M. Kart, A.V. Myasnikov, Broadband laser interferometer for monitoring deformations of the earth’s body, in: Bulletin of the Russian Academy of Sciences. Physics, 1999, pp. 948–951. [13] D. Reolon, M. Jacquot, I. Verrier, G. Brun, C. Veillas, Broadband supercontinuum interferometer for high-resolution profilometry, Opt. Express 14 (2006) 128–137. [14] A. Araya, Laser-interferometric broadband seismometer for ocean borehole observations, in: Symposium on Underwater Technology and Workshop on Scientific Use of Submarine Cables and Related Technologies, 2007, pp. 245–248. [15] S.G. Liu, Z.R. Li, Q.M. Jing, Y. Zhang, H.L. Ma, T.J. Tao, X. Wang, Y. Bi, J.D. Weng, J.A. Xu, Note: a novel method to measure the deformation of diamond anvils under high pressure, Rev. Sci. Instrum. 85 (2014) 046113. [16] J.D. Weng, T.J. Tao, S.G. Liu, H.L. Ma, X. Wang, C.L. Liu, H. Tan, Optical-fiber frequency domain interferometer with nanometer resolution and centimeter measuring range, Rev. Sci. Instrum. 84 (2013) 113103. [17] K.N. Joo, S.W. Kim, Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser, Opt. Express 14 (2006) 5954–5960. [18] P. Salieres, L.L. Deroff, T. Auguste, P. Monot, P. d’Oliveira, D. Campo, J.F. Hergott, H. Merdji, B. Carre, Frequency-domain interferometry in the XUV with high-order harmonics, Phys. Rev. Lett. 83 (1999) 5483–5486. [19] E. Kudeki, G.R. Stitt, Frequency domain interferometry: a high resolution radar technique for studies of atmosphere turbulence, Geophys. Res. Lett. 14 (1987) 198–201. [20] J.P. Geindre, P. Audebert, A. Rousse, F. Fallies, J.C. Gauthier, Frequency-domain interferometer for measuring the phase and amplitude of a femtosecond pulse probing a laser-produced plasma, Opt. Lett. 19 (1994) 1997–1999. [21] Pd. Groot, L. Deck, Surface profiling by analysis of white-light interferograms in the spatial frequency domain, J. Mod. Opt. 42 (1995) 389–401. [22] R. Span, E.W. Lemmon, R.T. Jacobsen, W. Wagner, A. Yokozeki, A reference equation of state for the thermodynamic properties of nitrogen for temperatures from 63.151 to 1000 K and pressures to 2200 MPa, J. Phys. Chem. Ref. Data 29 (2000) 1361–1433. [23] E. Hecht, Optics, 4th ed., Addison Wesley, San Francisco, Boston, New York, Capetown, Hong Kong, London, Madrid, Mexico City, Montreal, Munich, Paris, Singapore, Sydney, Tokyo, Toronto, 2002.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Refractive index measurement of compressed nitrogen using an infrared frequency-domain interferometer Jiangtao Li, Lei Liu, Jun Tang, Guojun Li, Heli Ma, Yunjun Gu, Shenggang Liu, Jidong Weng, Qi Feng Chen ∗ National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, CAEP, Mianyang, Sichuan, 621900, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 20 January 2018 Accepted 20 February 2018 Keywords: Refractive index Broadband interferometer Frequency-domain interferometer Polarizability
a b s t r a c t Frequency-domain interferometry is used to obtain the refractive index of compressed nitrogen in the vicinity of 1550 nm as the pressure of nitrogen increases from 0 to 40 MPa at room temperature. By referring to the equation of state of nitrogen, the relation between the refractive index and the nitrogen density is obtained and the electric polarizability of a nitrogen molecule is thereby determined to be (1.82 ± 0.15) × 10−30 m−3 according to the Lorentz-Lorentz relation. © 2018 Elsevier GmbH. All rights reserved.
1. Introduction Refractive index of nitrogen is an important parameter for astrophysicists to infer the distance and speed of stones falling onto the earth from outer space. For such a falling stone in the atmosphere, a pressure gradient exists in the front, which is associated with a gradient of refractive index. In order to predict accurately when and where the stone will hit the earth, the relation between the refractive index of nitrogen and its density needs to be determined for the working wavelength of interferometer. Doppler Pin System (DPS) is one of the most popular velocity interferometer systems and is widely used in the velocity diagnostics of dynamic processes [1–7]. The working wavelength is 1550 nm, which is in the infrared range. In the visible range, refractive index of nitrogen has been determined by interferometric measurements [8–11]. However, in the infrared range, especially in the vicinity of 1550 nm, the data are absent. Frequency-domain interferometer has found a growing number of applications in displacement or deformation measurements [12–21]. If the distance between two reflecting interfaces along the optical path are known, frequency-domain interferometer can be inversely applied to determine the refractive index of the material between the two reflecting interfaces. In this work, the refractive index of nitrogen in the vicinity of 1550 nm is obtained using a frequency-domain interferometer when the pressure of nitrogen increases from 0 to 40 MPa at room temperature. The refractive index measurement are detailed in the experimental section. In the results and discussion section, the relation between the refractive index of nitrogen and its pressure (density) will be given and the mean polarizability of nitrogen molecule in the vicinity of 1550 nm will be determined.
∗ Corresponding author. E-mail address: [email protected] (Q.F. Chen). https://doi.org/10.1016/j.ijleo.2018.02.056 0030-4026/© 2018 Elsevier GmbH. All rights reserved.
2
J. Li et al. / Optik 164 (2018) 1–4
Fig. 1. A schematic diagram of the experimental setup for the measurement of the refractive index of nitrogen.
Fig. 2. (a) The interference in the frequency-domain shown from 1545 nm to 1555 nm. (b) The three most notable peaks obtained from a transformation of (a). The horizontal axis of (b) is a product of refractive index and distance.
2. Experimental A schematic diagram of the experimental setup is shown in Fig. 1. The setup includes a gas source of nitrogen (purity > 99.999%), a gas booster and pump system, a gas cell which is enclosed by a stainless steel container and a sapphire window, and a frequency-domain interferometer which is composed of a broadband infrared laser, a circulator and an infrared spectrometer. The components of the interferometer are connected by optical fibers. The frequency-domain interferometer uses a broadband infrared laser to provide a coherent light beam. The beam goes through the circulator in the anti-clockwise direction, and arrives at the front surface of the sapphire window, which serves as the first reflector. The second reflector is the back surface of the sapphire window, and the third one is the front surface of the mirror inside the gas cell. The distance between the front surface of the mirror and the back surface of the sapphire window is maintained by a drilled ring spacer which is not shown in Fig. 1. The reflected light beams from the three reflectors will re-enter the circulator and will finally get into the infrared spectrometer. The fringe pattern in the frequency domain will be recorded by the CCD of the spectrometer. Since we are only interested in the spectral range around 1550 nm, the spectra from 1545 nm to 1555 nm were recorded when we increased the nitrogen pressure in the gas cell from 0 to 40 MPa. The gas pressure was given by a calibrated pressure gauge with an uncertainty of 0.5%, the ambient temperature was provided by a calibrated temperature gauge with an uncertainty of 0.5 K.
3. Results and discussion Fig. 2(a) shows the interference recorded by the frequency-domain interferometer from 1545 nm to 1555 nm. A constant offset is applied for each spectrum to show the periodic structure along the wavelength direction. The periodicity originates
J. Li et al. / Optik 164 (2018) 1–4
3
Table 1 The refractive index of nitrogen (n) and its uncertainty with respect to the gas pressure and the ambient temperature. Pressure MPa
Temperature K
Densitya g/cm3
n
sigma(n)
4.00E-06 5.18 10.29 19.92 29.86 39.39
296.2 296.2 296.0 296.0 296.0 296.0
4.80E-08 0.0594 0.1171 0.2168 0.3013 0.3649
1.0000 1.0128 1.0284 1.0508 1.0803 1.1003
0.0022 0.0031 0.0027 0.0026 0.0026 0.0024
a
The density is calculated using the equation of state of nitrogen [22].
Fig. 3. (a) The relation between the refractive index of nitrogen and its pressure at aboutT = 296 K. (b) The relation between the refractive index of nitrogen and its density at about T = 296 K. A linear fit of the data is also shown.
from the phase differences of the light beams reflected from the three reflectors as mentioned in the experimental section. The phase difference can written as: () = 2 ·
n · 2d .
(1)
here, n is the refractive index and d is the distance between the reflectors. If the optical media between the two reflectors has a layered structure, the term in the numerator in Eq. (1) needs to be replaced by a sum of optical paths. It can be seen from Eq. (1) that, the phase difference varies periodically if 1/ changes. Therefore, a Fourier transformation can be applied, which changes the variable from to nd, as shown by Fig. 2(b). Fig. 2(b) shows three peaks on the axis of nd. Peak 1 is associated with the interference of light beams coming from the first and second reflectors, which is exactly the thickness of the sapphire window multiplied by the refractive index of the sapphire. That also explain why location of peak 1 is invariant as the gas pressure increases. Peak 2 is associated with the interference of light beams coming from the second and the third reflectors. The shift of the peak 2 reflects the increase of refractive index of the gas when the nitrogen pressure increases. Peak 3 is associated with the interference of light beams coming from the first and third reflectors. That is why the amount of shift for peak 2 and 3 are the same for all the measurements. Based on the analysis above, the refractive index of nitrogen can be obtained by peak 2 location in Fig. 2(b) using the following formula: n(p) =
n(p) · d , n(p = 0) · d
(2)
here, the term on the right side of Eq. (2) is obtained by a Gaussian peak fitting procedure with the uncertainty determined. We also assume that the refractive index is 1 when p = 0. The results are given in Table 1. Fig. 3(a) shows the relation between the refractive index of nitrogen and its pressure, as measured directly from experiment. Using the equation of state of nitrogen [22], the relation between the refractive index of nitrogen and its density can also be obtained, as shown in Fig. 3(b). According to the linear fit, the relation between the refractive index of nitrogen and its density in the vicinity of 1550 nm can be written as: n = 1 + 0.264(g/cm3 )
(3)
4
J. Li et al. / Optik 164 (2018) 1–4
The electric polarizability of a nitrogen molecule can be obtained according to the Lorentz-Lorentz relation [8,23]: n2 − 1 4 = N˛, 3 n2 + 2
(4)
here, N is the number density of nitrogen molecules, ˛ is the electric polarizability of a nitrogen molecule, ε0 is the electric permittivity of free space. The value of the electric polarizability of a nitrogen molecule according to Eq. (4) is determined to be (1.82 ± 0.15) × 10−30 m−3 , which is very close to the value of 1.75 × 10−30 m−3 , which is extrapolated from the visible regime using the semi-empirical equation given by Kerl et al. [8]. 4. Conclusions and outlook The refractive index of nitrogen in the vicinity of 1550 nm is measured using a frequency-domain interferometer when the gas pressure increases from 0 to 40 MPa at room temperature. By referring to the equation of state of nitrogen, the relation between the refractive index of nitrogen and its density is obtained, which can be approximated by a linear formula: n = 1 + 0.264(g/cm3 ). Furthermore, the electric polarizability of a nitrogen molecule is determined to be (1.82 ± 0.15) × 10−30 m−3 , which is consistent with the extrapolated result of a semi-empirical equation from literature. Acknowledgements We would like to thank Tianjiong Tao for helpful discussion. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11604315, 11504352, 11674292, and 11604313), the Foundation of National Key Laboratory of Shock Wave and Detonation Physics, CAEP (Grant Nos. 6142A03010203 and 9140C670103150C67289), the Science Challenge Project (Grant No. TZ2016001), the President Foundation of China Academy of Engineering Physics (201501032), and the Science and Technology Development Foundation of China Academy of Engineering Physics (2015B0102001). References [1] J. Zheng, Q.F. Chen, Y.J. Gu, J.T. Li, Z.G. Li, C.J. Li, Z.Y. Chen, Shock-adiabatic to quasi-isentropic compression of warm dense helium up to 150 GPa, Phys. Rev. B 95 (2017) 224104. [2] J. Zheng, Q. Chen, Y. Gu, Z. Li, Z. Shen, Multishock compression properties of warm dense argon, Sci. Rep. 5 (2015) 16041. [3] M.D. Knudson, M.P. Desjarlais, A. Becker, R.W. Lemke, K.R. Cochrane, M.E. Savage, D.E. Bliss, T.R. Mattsson, R. Redmer, Direct observation of an abrupt insulator-to-metal transition in dense liquid deuterium, Science 348 (2015) 1455–1460. [4] M.D. Knudson, M.P. Desjarlais, Shock compression of quartz to 1.6 TPa: redefining a pressure standard, Phys. Rev. Lett. 102 (2009) 225501. [5] M.D. Knudson, D.L. Hanson, J.E. Bailey, C.A. Hall, J.R. Asay, C. Deeney, Principal Hugoniot, reverberating wave, and mechanical reshock measurements of liquid deuterium to 400 GPa using plate impact techniques, Phys. Rev. B 69 (2004) 144209. [6] J. Zheng, Q. Chen, Y. Gu, Z. Chen, Hugoniot measurements of double-shocked precompressed denxe xenon plasmas, Phys. Rev. E 86 (2012) 066406. [7] K. Falk, S.P. Regan, J. Vorberger, B.J.B. Crowley, S.H. Glenzer, S.X. Hu, C.D. Murphy, P.B. Radha, A.P. Jephcoat, J.S. Wark, D.O. Gericke, G. Gregori, Comparison between X-ray scattering and velocity-interferometry measurements from shocked liquid deuterium, Phys. Rev. E 87 (2013) 043112. [8] K. Kerl, U. Hohm, H. Varchmin, Polarizability of a(w,T,rho) of small molecules in the gas phase, Ber. Bunsenges. Phys. Chem. 96 (1992) 728–733. [9] U. Hohm, K. Kerl, Temperature dependence of mean molecular polarizability of gas molecules, Mol. Phys. 58 (1986) 541–550. [10] R.C. Burns, C. Graham, A.R.M. Weller, Direct measurement and calculation of the second refractivity virial coefficients of gases, Mol. Phys. 59 (1986) 41–64. [11] D. Reolon, M. Jacquot, I. Verrier, G. Brun, C. Veillas, High resolution group refractive index measurement by broadband supercontinuum interferometry and wavelet-transform analysis, Opt. Express 14 (2006) 12744–12750. [12] Y.K. Milyukov, V.N. Rudenko, B.S. Klyachko, A.M. Kart, A.V. Myasnikov, Broadband laser interferometer for monitoring deformations of the earth’s body, in: Bulletin of the Russian Academy of Sciences. Physics, 1999, pp. 948–951. [13] D. Reolon, M. Jacquot, I. Verrier, G. Brun, C. Veillas, Broadband supercontinuum interferometer for high-resolution profilometry, Opt. Express 14 (2006) 128–137. [14] A. Araya, Laser-interferometric broadband seismometer for ocean borehole observations, in: Symposium on Underwater Technology and Workshop on Scientific Use of Submarine Cables and Related Technologies, 2007, pp. 245–248. [15] S.G. Liu, Z.R. Li, Q.M. Jing, Y. Zhang, H.L. Ma, T.J. Tao, X. Wang, Y. Bi, J.D. Weng, J.A. Xu, Note: a novel method to measure the deformation of diamond anvils under high pressure, Rev. Sci. Instrum. 85 (2014) 046113. [16] J.D. Weng, T.J. Tao, S.G. Liu, H.L. Ma, X. Wang, C.L. Liu, H. Tan, Optical-fiber frequency domain interferometer with nanometer resolution and centimeter measuring range, Rev. Sci. Instrum. 84 (2013) 113103. [17] K.N. Joo, S.W. Kim, Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser, Opt. Express 14 (2006) 5954–5960. [18] P. Salieres, L.L. Deroff, T. Auguste, P. Monot, P. d’Oliveira, D. Campo, J.F. Hergott, H. Merdji, B. Carre, Frequency-domain interferometry in the XUV with high-order harmonics, Phys. Rev. Lett. 83 (1999) 5483–5486. [19] E. Kudeki, G.R. Stitt, Frequency domain interferometry: a high resolution radar technique for studies of atmosphere turbulence, Geophys. Res. Lett. 14 (1987) 198–201. [20] J.P. Geindre, P. Audebert, A. Rousse, F. Fallies, J.C. Gauthier, Frequency-domain interferometer for measuring the phase and amplitude of a femtosecond pulse probing a laser-produced plasma, Opt. Lett. 19 (1994) 1997–1999. [21] Pd. Groot, L. Deck, Surface profiling by analysis of white-light interferograms in the spatial frequency domain, J. Mod. Opt. 42 (1995) 389–401. [22] R. Span, E.W. Lemmon, R.T. Jacobsen, W. Wagner, A. Yokozeki, A reference equation of state for the thermodynamic properties of nitrogen for temperatures from 63.151 to 1000 K and pressures to 2200 MPa, J. Phys. Chem. Ref. Data 29 (2000) 1361–1433. [23] E. Hecht, Optics, 4th ed., Addison Wesley, San Francisco, Boston, New York, Capetown, Hong Kong, London, Madrid, Mexico City, Montreal, Munich, Paris, Singapore, Sydney, Tokyo, Toronto, 2002.