Anomalous behavior of nonlinear refractive indexes of CO<sub>2</sub> and Xe in supercritical states

Evgenii Mareev; Victor Aleshkevich; Fedor Potemkin; Victor Bagratashvili; Nikita Minaev; Vyacheslav Gordienko

doi:10.1364/OE.26.013229

1. Introduction

Nowadays supercritical fluids (SCF) are the subject of major research in both academic science and industry [1–3] Fluid at supercritical state displays intermediate properties between a gas and a liquid, producing a unique medium which unites many of the advantages of either states of matter [4,5]. Supercritical fluids (SCF) are characterized by large density fluctuations and cluster formation [6], which lead to the emergence of anomalous features in their physical properties [7]. A SCF can change its density and therefore optical properties in a wide range of pressures and temperatures without a liquid-gas phase transition [8,9]. However, its chemical and physical properties strongly depend on large-scale density fluctuations [7,10] which maximum on the so-called ridge matches the location on the Widom line [11–15]. Such ridge divides supercritical region into two zones - gas-like supercritical fluid and liquid-like supercritical fluid [6,11–13]. The gas-like SCF approaching the ridge is filled with clusters, and the liquid-like SCF contains different pores (free volumes). The average density of these structures becomes identical at the ridge line [4,6]. The rigid line corresponds to the optimal condition for cluster formation [11,12]. Moreover, it is reasonable to expect that the nonlinear refractive index of the medium will also demonstrate a noticeable change in that range. However, to our knowledge no such investigation was ever performed. The change in nonlinear optical properties can be a sensitive tool for studying the modification state of SCF in the vicinity of the ridge. The simplest way that is commonly applied to estimate modifications of optical properties under different pressures involves the definition of proportional dependence between linear or nonlinear coefficients (n and n₂) and fluid density [16,17]. In case of low pressures such approximation is correct. However, we prove further in this paper that it becomes incorrect under the transition to the supercritical state.

The key point of the current research is to reveal the behavior of nonlinear properties of high pressure gases and supercritical fluids by performing spectral analysis after nonlinear propagation of femtosecond pulses in a weakly nonlinear regime. It is important to note that the optical properties of the supercritical fluid due to the combination of its high nonlinear properties (comparable or even higher than in condensed matter) and the possibility of their control, provide an opportunity for the elaboration of an adjustable supercontinuum laser source, which would allow to model complex processes such as filamentation [8].

2. Experimental setup

The experiments were carried out with a Cr:forsterite based laser system operating at λ = 1240 nm, at 10 Hz repetition rate, giving laser pulses of 200 fs and pulse energy up to 2mJ. The laser pulse energy was varied by a rotating half-wave plate and a Glan prism. The incoming energy was monitored by a Ge photodetector (Thorlabs PDA50B); it was attenuated in such a way that filamentation did not occurred; the start of filamentation process was determined by the change of intensity profile on the CCD camera (the laser beam radius is reduced and a bright conical emission is observed, when the filament is generated). The collimated laser beam passed through the supercritical cell with explored fluid. The supercritical cell (optical path 11.6 cm, pressures up to 200 bar and temperatures up to 80°C) is made of stainless steel with optical windows (sapphire) 5 mm thick. The temperature and pressure in the cell were measured with an iron-constantan thermocouple and a strain respectively. The precision of the temperature and pressure measurements was 0.1°C and 0.1 bar, respectively. The beam splitter was used to divide the input beam into two channels (30:70). The first channel pulse was transmitted through the cell and its spectrum was measured by the IR spectrometer (Solar SDH-IV). The frequency of the second pulse was doubled in the BBO crystal and was mixed with the first pulse in KDP crystal, giving the sum frequency (SF) signal at 413 nm, which was measured with the PMT (Hamamatsu). The optical path between the channels (the rise of fluid pressure increases the optical way in the first channel) was changed with the pair of motorized mirrors. As a result, the maximum of the second frequency signal as a function of time delay provides the value of the optical path (see Fig. 1).

Fig. 1 Schematic of the experimental setup.

Download Full Size | PDF

3. Theoretical background

Fluid at the critical point becomes opaque due to critical opalescence [18] and demonstrates extremely high scattering. However, over a critical point the transparency of the medium does not denote the absence of heterogeneities. This phenomenon was discovered by Nishikawa, using small-angle X-ray scattering [12], and the local area of maximum fluctuations was called “ridge”. The experiment also showed that the array ridges correspond to the extrema of isothermal compressibility. Later it was discovered that the line divides supercritical fluids into two regions corresponding to the liquid-like and the gas-like SCF [6,19]. This peak line now is called as the Widom line [6]. The ridge or the Widom line looks like the extension of the saturation curve in the supercritical zone. We carried out experiments in a relatively narrow region over the critical point in a comparatively small range of pressures and temperatures (up to 1.06 T_c and 1.6 p_c correspondingly). Following [9, 12, 15, 20] a number of thermodynamic parameters (isothermal compressibility, partial molar volume, sound velocity, thermal conductivity, etc.) have an extremum in the vicinity of the ridge. In addition, the trajectories of particle motion in liquid-like and gas-like SCF have a qualitatively different character: they reveal vibrational hopping transport in a liquid-like fluid, and ballistic collision dominated regime in a gas-like fluid [4]. The transition from one mode to another is accompanied by cluster formation with the maximum at the ridge. The cluster formation leads to the change of nonlinear properties of the medium, due to the change in a polarizability of matter structure units, which changes from molecules and atoms to clusters.

The refractive index of the media can be calculated using the modified polarization Langevin theory for dielectrics in which the dipole moment of molecule is induced by light field. In that case refractive index can be written in the form:

n = 1 + \frac{N β}{3} + \frac{4 π N β^{2}}{45 k T} E^{2} = n_{0} + n_{2} E^{2} .

n_{2} = \frac{4 π N β^{2}}{45 k_{B} T} ~ N β^{2} .

where, β is the polarizability of molecule, N is the number of molecules in a unit volume V, k_B is the Boltzmann constant, T is the thermodynamic temperature. The dipole moment of molecule induced by light field is p = βE, and in our experiment pE/k_BT = βE²/k_BT~10¹⁰. It means that molecules have time to orient in such strong light field. SCF structure can be approximated as a binary mixture of single particles (molecules or atoms) and clusters:

n_{2} ~ (N_{1} β^{2} + N_{2} β_{2}^{2}),

where N₁ is the number of single particles and N₂ is the number of clusters, β₂ is the cluster polarizability. The cluster polarizability in the first approximation can be written as:

β_{2} = β < s >^{γ},

where <s> is the mean number of molecules in a cluster, γ is the parameter which depends on the shape. In the case of elongated cluster type, when the directions of the largest polarizability of molecules are oriented mainly along the same course, the parameter γ = 1. While for a randomly oriented molecules, including the spherically symmetric cluster, γ = 0.5.

Taking into account that N = N₁ + N₂<s>, n₂ can be given in the form:

n_{2} ~ N β^{2} [1 + \frac{N_{2}}{N} (< s >^{2 γ} - < s >)] .

The cluster fraction at the ridge is in the range of 5-10 percent [21], and the mean cluster can contain 7-10 molecules [22–24]. It should be stressed that the main contribution to the nonlinear refractive index adds mostly elongated clusters mainly, while at γ = 0.5 the clusters aggregation does not increase the n₂ value. Using (5) one can expect the increase of n₂ at the ridge in several times, when the clustering is maximal [24,25]. The fraction of clusters in supercritical CO₂ [9,23] is much larger than that the fraction in supercritical Xe [22], the average number of molecules in the cluster is also larger in the molecular gas. Thus, the rise of n₂ in the ridge is more significant for CO₂ than for Xe.

After crossing the Widom line (with pressure increase), the structure of fluid changes from the array of chaotically moving particles and different clusters to the liquid-like structures, where molecules forms large structures (~10000 molecules) and pores (free volume), as shown in Fig. 2 [23,26,27]. Such structural transition is accompanied by sharp changes in various parameters of SCF. In particular, as mentioned above, the isothermal compressibility

Fig. 2 The schematic phase diagram. The painted region refers to supercritical state of fluid. The region is divided by the Widom line into gas like SCF (under the Widom line) and liquid-like (over the Widom line). The maximum cluster formation is observed in the close vicinity of critical point [21]. The shaded region shows the area one the phase diagram, where strong scattering complicates the measurements.

Download Full Size | PDF

k_{T} = - \frac{1}{V} {(\frac{\partial V}{\partial p})}_{T} = \frac{1}{ρ} {(\frac{\partial ρ}{\partial p})}_{T},

reaches maximum in the vicinity of the ridge.

The fluctuation of the number of molecules N_v in a volume V is proportional to the isothermal compressibility k_T [28]:

\frac{〈 {(Δ N_{V})}^{2} 〉}{{〈 N_{V} 〉}^{2}} = \frac{1}{V} k_{T} k_{B} T ~ \frac{1}{ρ} {(\frac{\partial ρ}{\partial p})}_{T},

where ΔN_V = N_V-<N_V>, <N_v> is the average value; term

{(\frac{\partial ρ}{\partial p})}_{T}

can be calculated with the help of data from NIST database [29], and has a maximum at the ridge (see Fig. 3).

Fig. 3 Pressure dependence of intensity fluctuations (RMS) for (a) CO₂ and (b) Xe at different temperatures. The dotted line is a result of calculation using Eq. (8). The shaded region shows pressure interval where determination of n₂ is hampered by critical opalescence. The ridge corresponds to the second maximum. Blue vertical line shows critical pressure.

Download Full Size | PDF

Fluid parameters in the vicinity of critical point fluctuate (from vapour to liquid and supercritical state) due to uneven conditions over SCF cell. It gives rise to high density fluctuations and critical opalescence [6, 18]. Such process leads to a huge increase of light scattering [30], and also to the suppression of filamentation and supercontinuum generation effects [8].

Thus, the development of clustering process in the vicinity of the ridge can lead to an increase in nonlinear refractive index of supercritical fluid. The value of n₂(z) in the presence of density fluctuations becomes a random function of propagation coordinate z, and can be given in the form:

n_{2} (z) = {\bar{n}}_{2} + δ n_{2},

where

{\bar{n}}_{2}

is the average value, δn₂ is the random fluctuation.

The nonlinear phase shift will therefore also be random; it has a maximal value for the center of laser pulse (with the intensity I₀) and is equal to

φ_{\max} = - L \frac{ω_{0}}{c} \frac{{\bar{n}}_{2}}{n_{0}} I_{0} - \frac{ω_{0}}{c} \frac{I_{0}}{n_{0}} \int_{0}^{L} δ n_{2} (z) d z = {\bar{φ}}_{\max} + δ φ_{\max},

where

{\bar{φ}}_{\max}

is the average value of nonlinear phase shift, δ

φ_{\max}

is the random fluctuation, ω₀ is the fundamental laser frequency, c is the speed of light, L is the length of the media.

Considering the correlation function,

γ (z_{1}, z_{2}) = < δ n_{2} (z_{1}) δ n_{2} (z_{2}) > = σ_{n}^{2} \exp (- \frac{{(z_{1} - z_{2})}^{2}}{z_{0}^{2}}),

where

z_{0}

is the typical size of heterogeneities, σ²_n is a fluctuation dispersion we have:

< δ φ_{\max}^{2} > = {(\frac{ω_{0} I_{0}}{c n_{0}})}^{2} \iint < δ n_{2} (z_{1}) δ n_{2} (z_{2}) > d z_{1} d z_{2} = \frac{\sqrt{π} σ_{n}^{2} z_{0}}{{\bar{n}}_{2}^{2} L} {\bar{φ}}_{\max}^{2} .

Because:

\frac{σ_{n}^{2}}{{\bar{n}}_{2}^{2}} = \frac{< {(δ ρ)}^{2} >}{ρ^{2}} = \frac{k_{T} k_{B} T}{z_{0} s},

where

z_{0} s

is the volume of fluid irradiated by laser pulse (s is sectional area), we can finally receive:

< δ φ^{2}_{\max} > = \frac{\sqrt{π} k_{T} k_{B} T}{s L} {\bar{φ}}_{\max}^{2} .

Therefore, in the close vicinity of critical point the fluctuations of the nonlinear phase are extremely large (the isothermal compressibility k_T increases dramatically).

In addition, at critical pressure, the intensity of the transmitted laser radiation decreases exponentially due to the strong scattering at the critical opalescence. The intensity decrease is larger when the temperature is closer to the critical state.

4. Measuring of nonlinear refractive index

In order to measure the nonlinear refractive index (n₂) the following method was employed. After passing nonlinear media with a known Kerr response the pulse spectrum was broadened by a self-phase modulation (SPM). The spectral broadening factor for a FTL Gaussian pulse according to [31] is given by:

\frac{Δ ω_{o u t}}{Δ ω_{i n}} = {(1 + \frac{4}{3 \sqrt{3}} {\bar{φ}}_{\max}^{2})}^{2} .

where Δω_out and Δω are the RMS spectral width of the output and input pulses;

{\bar{φ}}_{\max}

is the phase shift of the pulse after propagation and is given by (9). Therefore, by measuring the reference spectrum and the laser pulse spectrum after propagation through a nonlinear medium in a weakly nonlinear regime at known laser intensity one can determine the nonlinear refractive index n₂. However, as it was discussed in the Section 3 in the vicinity of critical pressure the intensity fades away and strongly fluctuates due to critical opalescence, which does not allow to obtain reliable data.

5. Results and discussion

Spectra obtained for CO₂ and Xe are given at the insets of Fig. 3. Laser pulse spectra for low pressures (1-20 bar) have Gaussian distributions. A modulation of spectra is increased with pressure, which supports the fact that the phase change becomes significant and is close to π. We calculated the spectral width as the RMS spectral width for complex spectral shapes [31]. Within the possible measurement error it coincides with FWHW value and shapes of the spectra are still close to the Gaussian. We measured n₂ value along two isotherms in CO₂ (33.5, 50 °C) and Xe (24, 34 °C) by changing pressure. To verify the accuracy of these n₂ measurements, we measured the n₂ value for the optical window from fused silica with known n₂ = 2.8 ± 0.3*10⁻²⁰m²/W [32] and a length of 5mm. Such verification revealed sufficient accuracy оf our n₂ measurements. The laser pulse spectrum was averaged over 20 laser pulses.

5.1 Intensity fluctuations

In the scope of experiments related to the determination of nonlinear refractive index, our primary aim was to find the typical values of intensity fluctuations in supercritical state. The laser intensity fluctuations were determined as RMS of intensity over 20 points for full-investigated pressure range in Xe and CO₂. In all cases there was observed a peak that corresponded to the transition to supercritical state (31.04 °C, 72.8 bar for CO₂ and 16.6°C, 57.6 bar for Xe). However, the second peak in intensity fluctuations was observed for all cases except for CO₂ at 33.5°C (see Fig. 3(a)). This peak, in our opinion, is located at the Widom line and shifts toward the high-pressure region with increase of the temperature (see Fig. 3) [6,12]. As it was discussed in Section 3 there are two regions with large intensity fluctuations: in the vicinity of critical point and in the vicinity of ridge. In the first region such fluctuations result in the critical opalescence. In the second region, the fluctuations are the result of the structure transition from gas-like SCF to liquid-like SCF which is fully consistent with our experimental data.

The absence of the second peak in case of CO₂ at 33.5°C is the result of coinciding at this temperature between the critical point and the ridge. This leads to overlapping of two peaks as it can be seen from a close look at the Fig. 3(a).

5.2 Linear refractive index

Special experiments were performed to determine temporal characteristics of laser pulse transmitted through the SCF cell. In our experiments, the fundamental radiation propagated through the fluid was mixed with its second harmonic in KDP crystal, thus giving auto-correlation function. By varying the time delay, we were able to determine the optical path of the laser beam. From these data we can calculate the refractive index of fluid in the cell. The determined value of n over all pressure regions for CO₂ and Xe are given in Fig. 4.

Fig. 4 Pressure dependence of linear refractive index (n) for CO₂ (a) and for Xe (b) at different temperatures. Dotted lines show pressure dependence of linear refractive index calculated under assumption that n is proportional to the density. The insets show the pressure dependence of optical path δL. Blue vertical line shows critical pressure.

Download Full Size | PDF

Taking into account that the linear refractive index is proportional to the density (dotted lines at Fig. 4) we can conclude (only with the exception of the critical pressure) that the experimental data can be fitted as n~ρ(p). In the critical point, where a critical opalescence takes place [18], phase transition occurs (not instantly due to unevenness of cell heat), therefore, the optical way of the laser beam changes (decreases) in the region where critical opalescence occurs [33]. That leads to a deepening in the pressure dependence of n (see Fig. 4). However, the passing of the ridge does not lead to a change in the refractive index, because there is no change in fluid density, which is uniform over the cell. In addition, as it was mentioned in section 3 cluster formation in the vicinity of the ridge is not accompanied by a change in refractive index n.

5.3 Nonlinear refractive index

The main purpose of our experiments was to determine the nonlinear refractive index (n₂) in different states of fluid. Following Eq. (4) we expected the increase of the n₂ in the vicinity of the ridge and also a sufficient broadening of laser pulse spectrum, which is fully confirmed by the experimental data (see Fig. 5). In the regions, where the clustering does not play a significant role and the density fluctuations are not substantial, the proportional dependence can be still applied for rough estimation of nonlinear properties. In the regions, where the structure of fluid is stable and clustering of fluid is not sufficient, the fraction of clusters N₂/N in Eq. (5) tends to zero, and n₂~Nβ²~ρ. Our experiment shows that the assumption can be applied for two pressure regions in CO₂: 1-60 bar, where n₂~(0.9 ± 0.2)*ρ (m⁵/(W*kg)) and 90-150 bar, where n₂~(0.8 ± 0.2)*ρ (m⁵/(W*kg)). For Xe these regions are 0-45 bar, where n₂~(0.7 ± 0.2)*ρ (m⁵/(W*kg)) and 70-90 bar, where n₂~(0.4 ± 0.1)*ρ (m⁵/(W*kg)).

Fig. 5 The pressure dependence of nonlinear refractive index (n₂) for (a) CO₂ and (b) Xe at different temperatures. The shaded zone shows the region where n₂ measurements are not allowed to obtain reliable data. The vertical blue line shows critical pressure. Dotted lines show ridge locations. The insets show the measured spectrum of 150-fs laser pulse passed through the cell at different pressures: 1 bar (close to initial spectrum) and 82 bar for CO₂ and 63 bar for Xe. First bar corresponds to the n₂ = 3.4 ± 0.6*10⁻²² m²/W in CO₂ and n₂ = 6.2 ± 1.6*10⁻²² m²/W in Xe at atmospheric pressure.

Download Full Size | PDF

Laser intensity decreases exponentially in the vicinity of the critical pressure due to the strong light scattering at the critical point, as was discussed in sections 3 and 4. As a result, the calculated nonlinear refractive index is lower than the actual one. To correct this estimation, the length L in (9) should be replaced by the length of transparency 1/α, where α is the attenuation coefficient in the Beer- Lambert-Bouguer-law. The region, where such effect takes place was marked at Fig. 6 with shaded zone. In addition, as it was mentioned in the Section 3, SCF clustering in the vicinity of the ridge (following Eq. (4)) leads to a rapid increase of the nonlinear refractive index n₂ (see Fig. 5). The increase of n₂ is larger for the temperatures closer to the critical point, because clustering of fluid is maximal, when the ridge coincides with the critical pressure, therefore N₂/N and <s> in Eq. (4) have maximum value. When pressure increases, the peak in the pressure dependence of n₂ becomes wider, and its amplitude falls, since the difference between gas-like and liquid-like SCF becomes smother and numbers of clusters decrease [4,6,24]. As the mean size of clusters and its number is larger in molecular gases (as it was discussed in Section 3), the maximum in n₂ pressure dependence is more pronounced (see Fig. 5). The broadening of the pulse spectrum in CO₂ in the vicinity of the ridge is not uniform. The Stokes wing is wider, which could be related to the increase of the rotational Raman part of the nonlinear refractive index [34]. It can also be confirmed by the fact that the laser pulse spectrum for atomic Xe even in the vicinity of ridge is more symmetric than that for CO₂. It is also worth mentioning that the rapid jump of the n₂ value in the vicinity of ridge is larger for molecular CO₂ than for atomic Xe. This correlates with the fact that the clustering is stronger in molecular gases.

6. Conclusion

Based on the modified Langevin polarization theory it has been suggested that the development of the clustering process in the vicinity of the ridge should lead to an anomalous increase in nonlinear refractive index (n₂) of the supercritical fluid. We have predicted theoretically and confirmed experimentally that the nonlinear refractive index reaches its maximum value in the vicinity of the ridge. It is important to note, that the predominant contribution to the rise of n₂ value is provided predominantly by elongated clusters.

The nonlinear refractive index of CO₂ and Xe in a wide range of pressures covering different states of fluid has been determined for the first time. The clustering of SCF, which manifests itself in the vicinity of the ridge, leads to the deviation of n₂ density dependence from linear one. In turn, it results in a rapid increase of n₂ (up to 4.8*10⁻²⁰m2/W in CO₂ and 3.5*10⁻²⁰m²/W in Xe) in the vicinity of the ridge. At the same time, the linear refractive index of fluid does not change at the ridge, because it depends on fluid density, rather than on its structure. The anomalous broadening of the laser pulse spectrum and maximum of its intensity fluctuations could serve as an indicator of passing the ridge in a pulse method.

We believe that analysis of the behavior of the nonlinear refractive index can provide better understanding of molecular dynamics of supercritical fluids.

Funding

Russian Science Foundation (14-33-00017); Federal Agency of Scientific Organizations (Agreement No 007-GZ/C3363/26)

Acknowledgments

The Authors wish to thank Anna Fursenko for the help with the preparation of the paper. The work of E. Mareev was supported by the Foundation for the advancement of theoretical physics and mathematics “BASIS”

References and links

1. Y. Arai, T. Sako, and Y. Takebayashi, SupercriticalFluids: Molecular Interactions, Physical Properties, and New Applications, (Springer, 2002).

2. M. McHugh and V. Krukonis, Supercritical Fluid Extraction: Principles and Practice, (Butterworth-Heinemann, 2013).

3. C. Domingo and P. Subra-Paternault, Supercritical Fluid Nanotechnology : Advances and Applications in Composites and Hybrid Nanomaterials, (CRC Press, 2016).

4. V. V. Brazhkin, A. G. Lyapin, V. N. Ryzhov, K. Trachenko, Y. D. Fomin, and E. N. Tsiok, “Where is the supercritical fluid on the phase diagram?” Phys. Uspekhi 55(11), 1061–1079 (2012). [CrossRef]

5. T. Morita, K. Nishikawa, M. Takematsu, H. Iida, and S. Furutaka, “Structure study of supercritical CO2 near high-order phase transition line by X-ray diffraction,” J. Phys. Chem. B 101(36), 7158–7162 (1997). [CrossRef]

6. B. Sedunov, “The Analysis of the Equilibrium Cluster Structure in Supercritical Carbon Dioxide,” Am. J. Anal. Chem. 3(12), 899–904 (2012). [CrossRef]

7. T. Morita, K. Kusano, H. Ochiai, K. Saitow, and K. Nishikawa, “Study of inhomogeneity of supercritical water by small-angle x-ray scattering,” J. Chem. Phys. 112(9), 4203–4211 (2000). [CrossRef]

8. E. Mareev, V. Bagratashvili, N. Minaev, F. Potemkin, and V. Gordienko, “Generation of an adjustable multi-octave supercontinuum under near-IR filamentation in gaseous, supercritical, and liquid carbon dioxide,” Opt. Lett. 41(24), 5760–5763 (2016). [CrossRef] [PubMed]

9. T. Sato, M. Sugiyama, K. Itoh, K. Mori, T. Fukunaga, M. Misawa, T. Otomo, and S. Takata, “Structural difference between liquidlike and gaslike phases in supercritical fluid,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(5), 051503 (2008). [CrossRef] [PubMed]

10. S. C. Tucker, “Solvent Density Inhomogeneities in Supercritical Fluids,” Chem. Rev. 99(2), 391–418 (1999). [CrossRef] [PubMed]

11. V. V. Brazhkin, A. G. Lyapin, V. N. Ryzhov, K. Trachenko, Y. D. Fomin, and E. N. Tsiok, “The Frenkel Line and Supercritical Technologies,” Russ. J. Phys. Chem. B 8(8), 1087–1094 (2014). [CrossRef]

12. K. Nishikawa, I. Tanaka, and Y. Amemiya, “Small-Angle X-ray Scattering Study of Supercritical Carbon Dioxide,” J. Phys. Chem. 100(1), 418–421 (1996). [CrossRef]

13. H. Nakayama, K. Saitow, M. Sakashita, K. Ishii, and K. Nishikawa, “Raman spectral changes of neat CO₂ across the ridge of density fluctuation in supercritical region,” Chem. Phys. Lett. 320(3-4), 323–327 (2000). [CrossRef]

14. D. Fomin, V. N. Ryzhov, E. N. Tsiok, V. V. Brazhkin, and K. Trachenko, “Thermodynamics and Widom lines in supercritical carbon dioxide,” Phys. Rev. E 91, 022111 (2014). [CrossRef]

15. L. Wang, C. Yang, M. T. Dove, Y. D. Fomin, V. V. Brazhkin, and K. Trachenko, “Direct links between dynamical, thermodynamic, and structural properties of liquids: Modeling results,” Phys. Rev. E 95(3), 032116 (2017). [CrossRef] [PubMed]

16. M. Azhar, N. Y. Joly, J. C. Travers, and P. St. J. Russell, “Nonlinear optics in Xe-filled hollow-core PCF in high pressure and supercritical regimes,” Appl. Phys. B. 112(4), 457–460 (2013). [CrossRef]

17. P. St. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8(4), 278–286 (2014). [CrossRef]

18. J. A. White and B. S. Maccabee, “Temperature Sependence of Critical Opalescence in Carbon Dioxide,” Phys. Rev. Lett. 26(24), 1468–1471 (1971). [CrossRef]

19. Y. D. Fomin, V. N. Ryzhov, E. N. Tsiok, and V. V. Brazhkin, “Thermodynamic properties of supercritical carbon dioxide: Widom and Frenkel lines,” Phys. Rev. E - Stat. Nonlinear. Soft Matter Phys. 91(2), 1–5 (2015).

20. D. Bolmatov, M. Zhernenkov, D. Zav’yalov, S. N. Tkachev, A. Cunsolo, and Y. Q. Cai, “The Frenkel Line: a direct experimental evidence for the new thermodynamic boundary,” Sci. Rep. 5(1), 15850 (2015). [CrossRef] [PubMed]

21. D. Bolmatov, D. Zav’yalov, M. Gao, and M. Zhernenkov, “Structural evolution of supercritical CO₂ across the Frenkel line,” J. Phys. Chem. Lett. 5(16), 2785–2790 (2014). [CrossRef] [PubMed]

22. M. Aoshima, T. Suzuki, and K. Kaneko, “Molecular Association-Mediated Micropore Filling of Supercritical Xe in a Graphite Slit Pore by Grand Canonical Monte Carlo Simulation,” Chem. Phys. Lett. 310(1–2), 1–7 (1999). [CrossRef]

23. H. J. Magnier, R. A. Curtis, and L. V. Woodcock, “Nature of the Supercritical Mesophase,” Nat. Sci. 6, 797–807 (2014).

24. T. Sato, M. Sugiyama, M. Misawa, S. Takata, T. Otomo, K. Itoh, K. Mori, and T. Fukunaga, “A new analysing approach for the structure of density fluctuation of supercritical fluid,” J. Phys. Condens. Matter 20(10), 104203 (2008). [CrossRef]

25. J. Baldyga, M. Henczka, and B. Y. Shekunov, “Fluid Dynamics, Mass Transfer, and Particle Formation in Supercritical Fluids” in Supercritical Fluid Technology for Drug Product Development, Eds. P. York, U. B. Kompella, B. Y. Shekunov, (CRC Press, 2004).

26. R. E. Ryltsev and N. M. Chtchelkatchev, “Multistage structural evolution in simple monatomic supercritical fluids: Superstable tetrahedral local order,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(5), 052101 (2013). [CrossRef] [PubMed]

27. T. J. Yoon, M. Y. Ha, W. B. Lee, and Y.-W. Lee, “Monte Carlo simulations on the local density inhomogeneities of sub- and supercritical carbon dioxide: Statistical analysis based on the Voronoi tessellation,” J. Supercrit. Fluids 119, 36–43 (2017). [CrossRef]

28. K. Nishikawa and T. Morita, “Inhomogeneity of molecular distribution in supercritical fluids Inhomogeneity of molecular distribution in supercritical fluids,” Chem. Phys. Lett. 316(3-4), 238–242 (2000). [CrossRef]

29. “NIST database.” [Online]. Available: http://webbook.nist.gov/.

30. V. N. Bagratashvili, K. P. Bestemyanov, V. M. Gordienko, A. N. Konovalov, V. K. Popov, and S. I. Tsypina, “Optical properties of CO2 in the vicinity of critical point,” Proc. SPIE 4705, 129–136 (2002). [CrossRef]

31. D. Wang, Y. Leng, and Z. Xu, “Measurement of nonlinear refractive index coefficient of inert gases with hollow-core fiber,” Appl. Phys. B Lasers Opt. 111(3), 447–452 (2013). [CrossRef]

32. D. Milam, “Review and assessment of measured values of the nonlinear refractive-index coefficient of fused silica,” Appl. Opt. 37(3), 546–550 (1998). [CrossRef] [PubMed]

33. H. Ushifusa, K. Inaba, K. Sugasawa, K. Takahashi, and K. Kishimoto, “Measurement and visualization of supercritical CO₂ in dynamic phase transition,” EPJ Web Conf., 92, 2103 (2015).

34. S. M. Howdle and V. N. Bagratashvili, “The effects of fluid density on the rotational Raman-spectrum of hydrogen dissolved in supercritical carbon dioxide,” Chem. Phys. Lett. 214(2), 215–219 (1993). [CrossRef]

Anomalous behavior of nonlinear refractive indexes of CO₂ and Xe in supercritical states

Abstract

1. Introduction

2. Experimental setup

3. Theoretical background

4. Measuring of nonlinear refractive index

5. Results and discussion

5.1 Intensity fluctuations

5.2 Linear refractive index

5.3 Nonlinear refractive index

6. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (5)

Equations (14)

Optics Express