Abstract

Direct measurement of pressure dependent nonlinear refractive index of CO2 and Xe in subcritical and supercritical states are reported. In the vicinity of the ridge (or the Widom line), corresponding to the maximum density fluctuations, the nonlinear refractive index reaches a maximum value (up to 4.8*10−20m2/W in CO2 and 3.5*10−20m2/W in Xe). Anomalous behavior of the nonlinear refractive index in the vicinity of a ridge is caused by the cluster formation. That corresponds to the results of our theoretical assumption based on the modified Langevin theory.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (2)

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]

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]

2016 (1)

2015 (2)

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).

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]

2014 (5)

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]

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]

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

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]

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]

2013 (3)

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]

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]

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]

2012 (2)

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]

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

2008 (2)

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]

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]

2002 (1)

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]

2000 (3)

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]

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]

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]

1999 (2)

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

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]

1998 (1)

1997 (1)

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]

1996 (1)

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]

1993 (1)

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]

1971 (1)

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

Abdolvand, A.

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]

Amemiya, Y.

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]

Aoshima, M.

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]

Azhar, M.

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]

Bagratashvili, V.

Bagratashvili, V. N.

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]

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]

Bestemyanov, K. P.

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]

Bolmatov, D.

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]

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]

Brazhkin, V. V.

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]

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).

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]

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]

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]

Cai, Y. Q.

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]

Chang, W.

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]

Chtchelkatchev, N. M.

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]

Cunsolo, A.

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]

Curtis, R. A.

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

Dove, M. T.

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]

Fomin, D.

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]

Fomin, Y. D.

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]

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).

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]

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]

Fukunaga, T.

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]

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]

Furutaka, S.

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]

Gao, M.

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]

Gordienko, V.

Gordienko, V. M.

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]

Ha, M. Y.

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]

Hölzer, P.

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]

Howdle, S. M.

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]

Iida, H.

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]

Ishii, K.

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]

Itoh, K.

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]

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]

Joly, N. Y.

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]

Kaneko, K.

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]

Konovalov, A. N.

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]

Kusano, K.

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]

Lee, W. B.

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]

Lee, Y.-W.

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]

Leng, Y.

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]

Lyapin, A. G.

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]

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]

Maccabee, B. S.

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

Magnier, H. J.

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

Mareev, E.

Milam, D.

Minaev, N.

Misawa, M.

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]

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]

Mori, K.

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]

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]

Morita, T.

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]

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]

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]

Nakayama, H.

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]

Nishikawa, K.

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]

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]

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]

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]

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]

Ochiai, H.

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]

Otomo, T.

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]

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]

Popov, V. K.

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]

Potemkin, F.

Russell, P. St. J.

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]

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]

Ryltsev, R. E.

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]

Ryzhov, V. N.

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).

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]

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]

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]

Saitow, K.

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]

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]

Sakashita, M.

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]

Sato, T.

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]

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]

Sedunov, B.

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

Sugiyama, M.

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]

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]

Suzuki, T.

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]

Takata, S.

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]

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]

Takematsu, M.

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]

Tanaka, I.

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]

Tkachev, S. N.

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]

Trachenko, K.

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]

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]

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]

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]

Travers, J. C.

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]

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]

Tsiok, E. N.

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).

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]

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]

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]

Tsypina, S. I.

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]

Tucker, S. C.

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

Wang, D.

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]

Wang, L.

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]

White, J. A.

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

Woodcock, L. V.

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

Xu, Z.

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]

Yang, C.

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]

Yoon, T. J.

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]

Zav’yalov, D.

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]

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]

Zhernenkov, M.

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]

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]

Am. J. Anal. Chem. (1)

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

Appl. Opt. (1)

Appl. Phys. B Lasers Opt. (1)

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]

Appl. Phys. B. (1)

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]

Chem. Phys. Lett. (4)

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]

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]

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]

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]

Chem. Rev. (1)

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

J. Chem. Phys. (1)

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]

J. Phys. Chem. (1)

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]

J. Phys. Chem. B (1)

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]

J. Phys. Chem. Lett. (1)

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]

J. Phys. Condens. Matter (1)

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]

J. Supercrit. Fluids (1)

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]

Nat. Photonics (1)

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]

Nat. Sci. (1)

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

Opt. Lett. (1)

Phys. Rev. E (2)

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]

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]

Phys. Rev. E - Stat. Nonlinear. Soft Matter Phys. (1)

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).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

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]

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]

Phys. Rev. Lett. (1)

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

Phys. Uspekhi (1)

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]

Proc. SPIE (1)

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]

Russ. J. Phys. Chem. B (1)

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]

Sci. Rep. (1)

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

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).

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Figures (5)

Fig. 1
Fig. 1 Schematic of the experimental setup.
Fig. 2
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.
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.
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.
Fig. 5
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.

Equations (14)

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n=1+ Nβ 3 + 4πN β 2 45kT E 2 = n 0 + n 2 E 2 .
n 2 = 4πN β 2 45 k B T ~N β 2 .
n 2 ~( N 1 β 2 + N 2 β 2 2 ),
β 2 =β<s > γ ,
n 2 ~N β 2 [ 1+ N 2 N ( <s > 2γ <s> ) ].
k T = 1 V ( V p ) T = 1 ρ ( ρ p ) T ,
( Δ N V ) 2 N V 2 = 1 V k T k B T~ 1 ρ ( ρ p ) T ,
n 2 (z)= n ¯ 2 +δ n 2 ,
φ max =L ω 0 c n ¯ 2 n 0 I 0 ω 0 c I 0 n 0 0 L δ n 2 (z) dz= φ ¯ max +δ φ max ,
γ( z 1 , z 2 )=<δ n 2 ( z 1 )δ n 2 ( z 2 )>= σ n 2 exp( ( z 1 z 2 ) 2 z 0 2 ),
<δ φ max 2 >= ( ω 0 I 0 c n 0 ) 2 <δ n 2 ( z 1 ) δ n 2 ( z 2 )>d z 1 d z 2 = π σ n 2 z 0 n ¯ 2 2 L φ ¯ max 2 .
σ n 2 n ¯ 2 2 = < (δρ) 2 > ρ 2 = k T k B T z 0 s ,
<δ φ 2 max >= π k T k B T sL φ ¯ max 2 .
Δ ω out Δ ω in = ( 1+ 4 3 3 φ ¯ max 2 ) 2 .

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