Abstract

We have used terahertz time-domain spectroscopy to investigate the complex optical constants and birefringence of a widely used liquid crystal mixture E7 in both nematic and isotropic phases (26°C70°C). The extinction coefficient of E7 at room temperature is less than 0.035 and without sharp absorption features in the frequency range of 0.2–2.0 THz. The extraordinary (ne) and ordinary (no) indices of refraction at 26°C are 1.690–1.704 and 1.557–1.581, respectively, giving rise to a birefringence of 0.130–0.148 in this frequency range. The temperature-dependent (26°C70°C) order parameter extracted from the birefringence data agrees with that in the visible region quite well. Further, the temperature gradients of the terahertz optical constants of E7 are also determined. The optical constants of E7 in the terahertz or sub-millimeter wave range are found to deviate significantly from values predicated by the usual extended Cauchy equations used in the visible and near-infrared.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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  24. J. Li, S. Gauza, and S.-T. Wu, “Temperature effect on liquid crystal refractive indices,” J. Appl. Phys. 96, 19–24 (2004).
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    [CrossRef]
  28. F. Z. Yang and J. R. Sambles, “Determination of the microwave permittivities of nematic liquid crystals using a single-metallic slit technique,” Appl. Phys. Lett. 81, 2047–2049 (2002).
    [CrossRef]
  29. F. Z. Yang and J. R. Sambles, “Determination of the permittivity of nematic liquid crystals in the microwave region,” Liq. Cryst. 30, 599–602 (2003).
    [CrossRef]
  30. F. Z. Yang and J. R. Sambles, “Microwave liquid crystal wavelength selector,” Appl. Phys. Lett. 79, 3717–3719 (2001).
    [CrossRef]
  31. T. S. Perova, “Far-infrared and low-frequency Raman spectra of condensed media,” in Advances in Chemical Physics: Relaxation Phenomena in Condensed Matter, W.Coffey, eds. (Wiley, 1994), Vol. 87, pp. 427–480.
  32. G. J. Evans and M. Evans, “High and low frequency torsional absorptions in nematic K21,” J. Chem. Soc., Faraday Trans. 2 73, 285–292 (1977).
    [CrossRef]

2010 (1)

2009 (3)

R. Wilk, N. Vieweg, O. Kopschinski, and M. Koch, “Liquid crystal based electrically switchable Bragg structure for THz waves,” Opt. Express 17, 7377–7382 (2009).
[CrossRef] [PubMed]

X.-C.Zhang, R.Beigang, and K.Tanaka, eds., “Special issue on THz wave photonics,” J. Opt. Soc. Am. B 25, A1–A125 (2009).

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3, 148–151 (2009).
[CrossRef]

2008 (6)

I.-C. Ho, C.-L. Pan, C.-F. Hsieh, and R.-P. Pan, “Liquid-crystal-based terahertz tunable Solc filter,” Opt. Lett. 33, 1401–1403 (2008).
[CrossRef] [PubMed]

Y.J.Ding, Q.Hu, M.Kock, and C.E.Stutz, eds., “Special issue on THz materials, devices, and applications,” IEEE J. Sel. Top. Quantum Electron. 14, 257–259 (2008).
[CrossRef]

S. A. Jewell, E. Hendry, T. H. Issac, and J. R. Sambles, “Tuneable Fabry–Perot etalon for terahertz radiation,” New J. Phys. 10, 033012 (2008).
[CrossRef]

C.-F. Hsieh, Y.-C. Lai, R.-P. Pan, and C.-L. Pan, “Polarizing terahertz waves with nematic liquid crystals,” Opt. Lett. 33, 1174–1176 (2008).
[CrossRef] [PubMed]

C.-J. Lin, Y.-T. Li, C.-F. Hsieh, R.-P. Pan, and C.-L. Pan, “Manipulating terahertz wave by a magnetically tunable liquid crystal phase grating,” Opt. Express 16, 2995–3001 (2008).
[CrossRef] [PubMed]

R.-P. Pan, C.-F. Hsieh, C.-L. Pan, and C.-Y. Chen, “Temperature-dependent optical constants and birefringence of nematic liquid crystal 5CB in the terahertz frequency range,” J. Appl. Phys. 103, 093523 (2008).
[CrossRef]

2007 (2)

G. Abbate, V. Tkachenko, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, and L. De Stefano, “Optical characterization of liquid crystals by combined ellipsometry and half-leaky-guided-mode spectroscopy in the visible-near infrared range,” J. Appl. Phys. 101, 073105 (2007).
[CrossRef]

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[CrossRef]

2006 (2)

S.-T. Wu and D.-K. Yang, Fundamentals of Liquid Crystal Devices, Wiley Series in Display Technology (Wiley, 2006).

C.-Y. Chen, C.-L. Pan, C.-F. Hsieh, Y.-F. Lin, and R.-P. Pan, “Liquid-crystal-based terahertz tunable Lyot filter,” Appl. Phys. Lett. 88, 101107 (2006).
[CrossRef]

2005 (3)

C.-L. Pan, C.-F. Hsieh, R.-P. Pan, M. Tanaka, F. Miyamaru, M. Tani, and M. Hangyo, “Control of enhanced THz transmission through metallic hole arrays using nematic liquid crystal,” Opt. Express 13, 3921–3930 (2005).
[CrossRef] [PubMed]

J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive indices of liquid crystals for display applications,” J. Disp. Technol. 1, 51–61 (2005).
[CrossRef]

J. Li, S.-T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

2004 (4)

S. Brugioni and R. Meucci, “Liquid crystals in the mid-infrared region and their applications,” Infrared Phys. Technol. 46, 17–21 (2004).
[CrossRef]

C.-Y. Chen, C.-F. Hsieh, Y.-F. Lin, R.-P. Pan, and C.-L. Pan, “Magnetically tunable room-temperature 2π liquid crystal terahertz phase shifter,” Opt. Express 12, 2625–2630 (2004).
[CrossRef] [PubMed]

J. Li, S. Gauza, and S.-T. Wu, “Temperature effect on liquid crystal refractive indices,” J. Appl. Phys. 96, 19–24 (2004).
[CrossRef]

J. Li, S. Gauzia, and S.-T. Wu, “High temperature-gradient refractive index liquid crystal,” Opt. Express 12, 2002–2010 (2004).
[CrossRef] [PubMed]

2003 (1)

F. Z. Yang and J. R. Sambles, “Determination of the permittivity of nematic liquid crystals in the microwave region,” Liq. Cryst. 30, 599–602 (2003).
[CrossRef]

2002 (2)

F. Z. Yang and J. R. Sambles, “Determination of the microwave permittivities of nematic liquid crystals using a single-metallic slit technique,” Appl. Phys. Lett. 81, 2047–2049 (2002).
[CrossRef]

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nature Mater. 1, 26–33 (2002).
[CrossRef]

2001 (1)

F. Z. Yang and J. R. Sambles, “Microwave liquid crystal wavelength selector,” Appl. Phys. Lett. 79, 3717–3719 (2001).
[CrossRef]

2000 (1)

R. Kersting, G. Strasser, and K. Unterrainer, “Terahertz phase modulator,” Electron. Lett. 36, 1156–1158 (2000).
[CrossRef]

1994 (1)

T. S. Perova, “Far-infrared and low-frequency Raman spectra of condensed media,” in Advances in Chemical Physics: Relaxation Phenomena in Condensed Matter, W.Coffey, eds. (Wiley, 1994), Vol. 87, pp. 427–480.

1992 (2)

P. F. Goldsmith, “Quasi-optical techniques,” Proc. IEEE 80, 1729–1747 (1992).
[CrossRef]

B. Bahadur, Liquid Crystals—Applications and Uses (World Scientific, 1992).
[CrossRef]

1987 (1)

1986 (1)

S.-T. Wu, “Birefringence dispersions of liquid crystal,” Phys. Rev. A 33, 1270–1274 (1986).
[CrossRef] [PubMed]

1977 (1)

G. J. Evans and M. Evans, “High and low frequency torsional absorptions in nematic K21,” J. Chem. Soc., Faraday Trans. 2 73, 285–292 (1977).
[CrossRef]

Abbate, G.

G. Abbate, V. Tkachenko, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, and L. De Stefano, “Optical characterization of liquid crystals by combined ellipsometry and half-leaky-guided-mode spectroscopy in the visible-near infrared range,” J. Appl. Phys. 101, 073105 (2007).
[CrossRef]

Averitt, R. D.

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3, 148–151 (2009).
[CrossRef]

Azad, A. K.

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3, 148–151 (2009).
[CrossRef]

Bahadur, B.

B. Bahadur, Liquid Crystals—Applications and Uses (World Scientific, 1992).
[CrossRef]

Brugioni, S.

J. Li, S.-T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

S. Brugioni and R. Meucci, “Liquid crystals in the mid-infrared region and their applications,” Infrared Phys. Technol. 46, 17–21 (2004).
[CrossRef]

Chen, C. -Y.

R.-P. Pan, C.-F. Hsieh, C.-L. Pan, and C.-Y. Chen, “Temperature-dependent optical constants and birefringence of nematic liquid crystal 5CB in the terahertz frequency range,” J. Appl. Phys. 103, 093523 (2008).
[CrossRef]

C.-Y. Chen, C.-L. Pan, C.-F. Hsieh, Y.-F. Lin, and R.-P. Pan, “Liquid-crystal-based terahertz tunable Lyot filter,” Appl. Phys. Lett. 88, 101107 (2006).
[CrossRef]

C.-Y. Chen, C.-F. Hsieh, Y.-F. Lin, R.-P. Pan, and C.-L. Pan, “Magnetically tunable room-temperature 2π liquid crystal terahertz phase shifter,” Opt. Express 12, 2625–2630 (2004).
[CrossRef] [PubMed]

Chen, H. -T.

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3, 148–151 (2009).
[CrossRef]

Cich, M. J.

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3, 148–151 (2009).
[CrossRef]

De Stefano, L.

G. Abbate, V. Tkachenko, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, and L. De Stefano, “Optical characterization of liquid crystals by combined ellipsometry and half-leaky-guided-mode spectroscopy in the visible-near infrared range,” J. Appl. Phys. 101, 073105 (2007).
[CrossRef]

Evans, G. J.

G. J. Evans and M. Evans, “High and low frequency torsional absorptions in nematic K21,” J. Chem. Soc., Faraday Trans. 2 73, 285–292 (1977).
[CrossRef]

Evans, M.

G. J. Evans and M. Evans, “High and low frequency torsional absorptions in nematic K21,” J. Chem. Soc., Faraday Trans. 2 73, 285–292 (1977).
[CrossRef]

Faetti, S.

J. Li, S.-T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

Ferguson, B.

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nature Mater. 1, 26–33 (2002).
[CrossRef]

Gauza, S.

J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive indices of liquid crystals for display applications,” J. Disp. Technol. 1, 51–61 (2005).
[CrossRef]

J. Li, S. Gauza, and S.-T. Wu, “Temperature effect on liquid crystal refractive indices,” J. Appl. Phys. 96, 19–24 (2004).
[CrossRef]

Gauzia, S.

Giocondo, M.

G. Abbate, V. Tkachenko, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, and L. De Stefano, “Optical characterization of liquid crystals by combined ellipsometry and half-leaky-guided-mode spectroscopy in the visible-near infrared range,” J. Appl. Phys. 101, 073105 (2007).
[CrossRef]

Goldsmith, P. F.

P. F. Goldsmith, “Quasi-optical techniques,” Proc. IEEE 80, 1729–1747 (1992).
[CrossRef]

Hangyo, M.

Hendry, E.

S. A. Jewell, E. Hendry, T. H. Issac, and J. R. Sambles, “Tuneable Fabry–Perot etalon for terahertz radiation,” New J. Phys. 10, 033012 (2008).
[CrossRef]

Ho, I. -C.

Hsieh, C. -F.

Issac, T. H.

S. A. Jewell, E. Hendry, T. H. Issac, and J. R. Sambles, “Tuneable Fabry–Perot etalon for terahertz radiation,” New J. Phys. 10, 033012 (2008).
[CrossRef]

Jansen, C.

Jewell, S. A.

S. A. Jewell, E. Hendry, T. H. Issac, and J. R. Sambles, “Tuneable Fabry–Perot etalon for terahertz radiation,” New J. Phys. 10, 033012 (2008).
[CrossRef]

Kersting, R.

R. Kersting, G. Strasser, and K. Unterrainer, “Terahertz phase modulator,” Electron. Lett. 36, 1156–1158 (2000).
[CrossRef]

Koch, M.

Kopschinski, O.

Krumbholz, N.

Lai, Y. -C.

Li, J.

J. Li, S.-T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive indices of liquid crystals for display applications,” J. Disp. Technol. 1, 51–61 (2005).
[CrossRef]

J. Li, S. Gauzia, and S.-T. Wu, “High temperature-gradient refractive index liquid crystal,” Opt. Express 12, 2002–2010 (2004).
[CrossRef] [PubMed]

J. Li, S. Gauza, and S.-T. Wu, “Temperature effect on liquid crystal refractive indices,” J. Appl. Phys. 96, 19–24 (2004).
[CrossRef]

Li, Y. -T.

Lim, K. -C.

Lin, C. -J.

Lin, Y. -F.

C.-Y. Chen, C.-L. Pan, C.-F. Hsieh, Y.-F. Lin, and R.-P. Pan, “Liquid-crystal-based terahertz tunable Lyot filter,” Appl. Phys. Lett. 88, 101107 (2006).
[CrossRef]

C.-Y. Chen, C.-F. Hsieh, Y.-F. Lin, R.-P. Pan, and C.-L. Pan, “Magnetically tunable room-temperature 2π liquid crystal terahertz phase shifter,” Opt. Express 12, 2625–2630 (2004).
[CrossRef] [PubMed]

Lu, R.

J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive indices of liquid crystals for display applications,” J. Disp. Technol. 1, 51–61 (2005).
[CrossRef]

Marino, A.

G. Abbate, V. Tkachenko, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, and L. De Stefano, “Optical characterization of liquid crystals by combined ellipsometry and half-leaky-guided-mode spectroscopy in the visible-near infrared range,” J. Appl. Phys. 101, 073105 (2007).
[CrossRef]

Mazzulla, A.

G. Abbate, V. Tkachenko, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, and L. De Stefano, “Optical characterization of liquid crystals by combined ellipsometry and half-leaky-guided-mode spectroscopy in the visible-near infrared range,” J. Appl. Phys. 101, 073105 (2007).
[CrossRef]

Meucci, R.

J. Li, S.-T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

S. Brugioni and R. Meucci, “Liquid crystals in the mid-infrared region and their applications,” Infrared Phys. Technol. 46, 17–21 (2004).
[CrossRef]

Mikulics, M.

Miyamaru, F.

Padilla, W. J.

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3, 148–151 (2009).
[CrossRef]

Pan, C. -L.

Pan, R. -P.

Perova, T. S.

T. S. Perova, “Far-infrared and low-frequency Raman spectra of condensed media,” in Advances in Chemical Physics: Relaxation Phenomena in Condensed Matter, W.Coffey, eds. (Wiley, 1994), Vol. 87, pp. 427–480.

Sambles, J. R.

S. A. Jewell, E. Hendry, T. H. Issac, and J. R. Sambles, “Tuneable Fabry–Perot etalon for terahertz radiation,” New J. Phys. 10, 033012 (2008).
[CrossRef]

F. Z. Yang and J. R. Sambles, “Determination of the permittivity of nematic liquid crystals in the microwave region,” Liq. Cryst. 30, 599–602 (2003).
[CrossRef]

F. Z. Yang and J. R. Sambles, “Determination of the microwave permittivities of nematic liquid crystals using a single-metallic slit technique,” Appl. Phys. Lett. 81, 2047–2049 (2002).
[CrossRef]

F. Z. Yang and J. R. Sambles, “Microwave liquid crystal wavelength selector,” Appl. Phys. Lett. 79, 3717–3719 (2001).
[CrossRef]

Scheller, M.

Shakfa, M. K.

Strasser, G.

R. Kersting, G. Strasser, and K. Unterrainer, “Terahertz phase modulator,” Electron. Lett. 36, 1156–1158 (2000).
[CrossRef]

Tanaka, M.

Tani, M.

Taylor, A. J.

H.-T. Chen, W. J. Padilla, M. J. Cich, A. K. Azad, R. D. Averitt, and A. J. Taylor, “A metamaterial solid-state terahertz phase modulator,” Nat. Photonics 3, 148–151 (2009).
[CrossRef]

Tkachenko, V.

G. Abbate, V. Tkachenko, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, and L. De Stefano, “Optical characterization of liquid crystals by combined ellipsometry and half-leaky-guided-mode spectroscopy in the visible-near infrared range,” J. Appl. Phys. 101, 073105 (2007).
[CrossRef]

Tonouchi, M.

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007).
[CrossRef]

Unterrainer, K.

R. Kersting, G. Strasser, and K. Unterrainer, “Terahertz phase modulator,” Electron. Lett. 36, 1156–1158 (2000).
[CrossRef]

Vieweg, N.

Vita, F.

G. Abbate, V. Tkachenko, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, and L. De Stefano, “Optical characterization of liquid crystals by combined ellipsometry and half-leaky-guided-mode spectroscopy in the visible-near infrared range,” J. Appl. Phys. 101, 073105 (2007).
[CrossRef]

Wen, C. -H.

J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive indices of liquid crystals for display applications,” J. Disp. Technol. 1, 51–61 (2005).
[CrossRef]

Wilk, R.

Wu, S. -T.

S.-T. Wu and D.-K. Yang, Fundamentals of Liquid Crystal Devices, Wiley Series in Display Technology (Wiley, 2006).

J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive indices of liquid crystals for display applications,” J. Disp. Technol. 1, 51–61 (2005).
[CrossRef]

J. Li, S.-T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

J. Li, S. Gauzia, and S.-T. Wu, “High temperature-gradient refractive index liquid crystal,” Opt. Express 12, 2002–2010 (2004).
[CrossRef] [PubMed]

J. Li, S. Gauza, and S.-T. Wu, “Temperature effect on liquid crystal refractive indices,” J. Appl. Phys. 96, 19–24 (2004).
[CrossRef]

S.-T. Wu and K.-C. Lim, “Absorption and scattering measurements of nematic liquid crystals,” Appl. Opt. 26, 1722–1727 (1987).
[CrossRef] [PubMed]

S.-T. Wu, “Birefringence dispersions of liquid crystal,” Phys. Rev. A 33, 1270–1274 (1986).
[CrossRef] [PubMed]

Yang, D. -K.

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Figures (10)

Fig. 1
Fig. 1

Simplified sketches of the experimental configuration for transmission of ordinary ray through (a) the LC and (b) reference cells. The symbols are defined in the text. The blue ellipsoids in (a) represent schematically the LC molecules, which are homogeneously aligned in the cell.

Fig. 2
Fig. 2

The real refractive indices of E7 are plotted as functions of frequency. The blue circles and the red triangles are the extraordinary and ordinary refractive indices of E7, n e and n o , at 26 ° C , respectively. The black squares are the real refractive indices of E7 in the isotropic phase (at 60 ° C ). The error bars are comparable to or smaller than the symbols.

Fig. 3
Fig. 3

Frequency dependence of the birefringence of E7 measured at 26 ° C . The error bars are comparable to or smaller than the symbols.

Fig. 4
Fig. 4

The imaginary refractive indices of E7 are plotted as functions of frequency. The blue circles and the red triangles are κ e and κ o at 26 ° C , respectively. The black squares are the imaginary refractive indices of E7 in the isotropic phase (at 60 ° C ). The error bars are comparable to or smaller than the symbols.

Fig. 5
Fig. 5

The refractive indices of E7 reported in the literature from the near-infrared [4, 27], mid-infrared [4], to the millimeter [28, 29, 30] waves together with those of this work in the millimeter and sub-millimeter wave ranges.

Fig. 6
Fig. 6

The birefringence of E7 reported in the literature from the visible [5], near-infrared [4, 27], mid-infrared [4], to the millimeter [28, 29, 30] waves together with those of this work in the millimeter and sub-millimeter wave ranges.

Fig. 7
Fig. 7

Average refractive indices of E7 versus T - T c at frequencies of 0.34, 0.70, 0.98, and 1.40 THz. The open circles represent the experimental data and the solid lines are the results of fitting.

Fig. 8
Fig. 8

Birefringence of E7 versus T - T c at frequencies of 0.34, 0.70, 0.98, and 1.40 THz. The closed circles represent the experimental data and the solid curves are the results of fitting.

Fig. 9
Fig. 9

Extraordinary and ordinary refractive indices of E7 versus T - T c at frequencies of 0.34, 0.70, 0.98, and 1.40 THz. The open and closed circles represent n e and n o , respectively. The crosses represent the refractive indices of E7 in the isotropic phase. The solid lines are the fitting curves. The dashed lines are the average indices at temperatures below T c .

Fig. 10
Fig. 10

Temperature dependence of d n e / d T , d n o / d T , and d ( Δ n ) / d T of E7 at 0.34 THz. The solid and dashed curves represent d n e / d T and d n o / d T , respectively. The dotted curve is for d ( Δ n ) / d T .

Tables (2)

Tables Icon

Table 1 The Fitting Parameters of Eq. (8) for Frequencies from 0.34 to 1.40 THz

Tables Icon

Table 2 The Fitting Parameters of Eq. (9) for Frequencies from 0.34 to 1.40 THz

Equations (13)

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E ref ( ω ) = E 0 ( ω ) t ̃ a w t ̃ w a   exp [ i ( ω / c ) n ̃ w ( d 1 + d 2 ) ] exp [ i ( ω / c ) n ̃ a ( d a + Δ d ) ] ,
E LC ( ω ) = E 0 ( ω ) t ̃ a w t ̃ w LC t ̃ LC w t ̃ w a   exp [ i ( ω / c ) n ̃ w ( d 3 + d 4 ) ] F P ˜ LC ( ( ω , d LC ) ,
F P ˜ LC ( ω , d LC ) = m = 0 N ( r ̃ LC w 2 m   exp { i ( ω / c ) [ n ̃ LC d LC ( 2 m + 1 ) ] } ) ,
T ( ω ) = E LC ( ω ) / E ref ( ω ) = t ̃ w LC t ̃ LC w   exp { i ( ω / c ) [ Δ d ( n ̃ w n ̃ a ) + d LC ( n ̃ LC n ̃ a ) ] } m = 0 N r ̃ LC w 2 m   exp [ i 2 m ( ω / c ) n ̃ LC d LC ] .
n LC = n 0 + [ arg ( T ( ω ) ) arg ( t ̃ w LC t ̃ LC w   exp [ ω ( Δ d κ w + d LC κ LC ) / c ] m = 0 N r ̃ LC w 2 m exp ( i 2 m n ̃ LC d LC ω / c ) ) ] c / ( ω d LC ) Δ d ( n w n 0 ) / d LC ,
κ LC = κ w Δ d / d LC ln [ | T ( ω ) | / { | m = 0 N r ̃ LC w 2 m   exp [ i 2 m ( ω / c ) n ̃ LC d LC ] | | t ̃ w LC t ̃ LC w | } ] c / ( ω d LC ) .
n o , e = A o , e + B o , e λ 2 + C o , e λ 4 .
n ( T ) = A B T .
Δ n ( T ) = ( Δ n ) 0 S = ( Δ n ) 0 ( 1 T / T c ) β ,
n e ( T ) = A B T + ( 2 ( Δ n ) 0 / 3 ) ( 1 T / T c ) β ,
n o ( T ) = A B T ( ( Δ n ) 0 / 3 ) ( 1 T / T c ) β .
d n e / d T = B 2 β ( Δ n ) 0 / [ 3 T c ( 1 T / T c ) 1 β ] ,
d n o / d T = B + β ( Δ n ) 0 / [ 3 T c ( 1 T / T c ) 1 β ] .

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