Abstract

We report the complete determination of the polarization changes caused in linearly polarized incident light due to propagation in a magneto-optically active terbium gallium garnet (TGG) single crystal, at temperatures ranging from 6.3 to 300 K. A 28-fold increase in the Verdet constant of the TGG crystal is seen as its temperature decreases to 6.3 K. In contrast with polarimetry of light emerging from a Faraday material at room temperature, polarimetry at cryogenic temperatures cannot be carried out using the conventional fixed polarizer-analyzer technique because the assumption that ellipticity is negligible becomes increasingly invalid as temperature is lowered. It is shown that complete determination of light polarization in such a case requires the determination of its Stokes parameters, otherwise inaccurate measurements will result with negative implications for practical devices.

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  1. A. D. Buckingham and P. J. Stephens, “Magnetic Optical Activity,” Annu. Rev. Phys. Chem. 17, 399–432 (1966).
    [Crossref]
  2. E. Hecht, “The Faraday Effect,” in Optics, 4th ed. (Pearson Education Inc., San Francisco, 2002).
  3. C. A. Bennett, “Faraday Rotation,” in Principles of Physical Optics (John Wiley & Sons Inc., 2008).
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    [Crossref]
  6. R. W. Cooper and J. L. Page, “Magneto-optic light modulators,” Radio Electron. Eng. 39, 302–304 (1970).
    [Crossref]
  7. P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for Temperature-Dependence of the Faraday Effect by Optical Activity Temperature Shift,” IEEE Sens. J. 13, 832–837 (2013).
    [Crossref]
  8. T. Yoshino, S.-i. Torihata, M. Yokota, and N. Tsukada, “Faraday-Effect Optical Current Sensor with a Garnet Film /Ring Core in a Transverse Configuration,” Appl. Opt. 42, 1769–1772 (2003).
    [Crossref] [PubMed]
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  18. H. Hauser, R. Chabicovsky, and K. Riedling, “Magneto-Optical Methods,” in Handbook of Thin Films: Deposition and Processing, H. S. Nalwa, ed. (Academic Press, 2001).
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    [Crossref] [PubMed]
  21. E. Hecht, “Malus’s Law,” in Optics, 4th Edition ed. (Pearson Education Inc., San Francisco, 2002), pp. 332–333.
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    [Crossref]

2013 (1)

P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for Temperature-Dependence of the Faraday Effect by Optical Activity Temperature Shift,” IEEE Sens. J. 13, 832–837 (2013).
[Crossref]

2012 (1)

2011 (3)

A. V. Starobor, D. S. Zheleznov, O. V. Palashov, and E. A. Khazanov, “Magnetoactive media for cryogenic Faraday isolators,” J. Opt. Soc. Am. B 28, 1409–1415 (2011).
[Crossref]

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. S. J. Russell, “Complex Faraday Rotation in Microstructured Magneto-optical Fiber Waveguides,” Adv. Mater. 23, 2681–2688 (2011).
[Crossref] [PubMed]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

2008 (1)

S. Polisetty, J. Scheffler, S. Sahoo, Y. Wang, T. Mukherjee, X. He, and C. Binek, “Optimization of magneto-optical Kerr setup: Analyzing experimental assemblies using Jones matrix formalism,” Rev. Sci. Instrum. 79, 055107–055106 (2008).
[Crossref] [PubMed]

2007 (1)

2003 (1)

1995 (1)

J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7, 1468–1470 (1995).
[Crossref]

1992 (1)

1991 (1)

Y. Fujii, “High-isolation polarization-independent optical circulator coupled with single-mode fibers,” J. Light-wave Technol. 9, 456–460 (1991).
[Crossref]

1990 (1)

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Fundamental magneto-optics,” J. Appl. Phys. 68, 4203–4207 (1990).
[Crossref]

1987 (1)

G. Fischer, “The Faraday Optical Isolator,” J. Opt. Commun. 8, 18–21 (1987).

1984 (1)

1977 (1)

1970 (1)

R. W. Cooper and J. L. Page, “Magneto-optic light modulators,” Radio Electron. Eng. 39, 302–304 (1970).
[Crossref]

1967 (1)

R. P. Hunt, “Magneto-Optic Scattering from Thin Solid Films,” J. Appl. Phys. 38, 1652–1671 (1967).
[Crossref]

1966 (1)

A. D. Buckingham and P. J. Stephens, “Magnetic Optical Activity,” Annu. Rev. Phys. Chem. 17, 399–432 (1966).
[Crossref]

1964 (1)

1934 (1)

J. Van Vleck and M. Hebb, “On the Paramagnetic Rotation of Tysonite,” Phys. Rev. 46, 17–32 (1934).
[Crossref]

Bader, S. D.

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Fundamental magneto-optics,” J. Appl. Phys. 68, 4203–4207 (1990).
[Crossref]

Barnes, N. P.

Bennett, C. A.

C. A. Bennett, “Faraday Rotation,” in Principles of Physical Optics (John Wiley & Sons Inc., 2008).

Berry, H. G.

Binek, C.

S. Polisetty, J. Scheffler, S. Sahoo, Y. Wang, T. Mukherjee, X. He, and C. Binek, “Optimization of magneto-optical Kerr setup: Analyzing experimental assemblies using Jones matrix formalism,” Rev. Sci. Instrum. 79, 055107–055106 (2008).
[Crossref] [PubMed]

Bostwick, A.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Buckingham, A. D.

A. D. Buckingham and P. J. Stephens, “Magnetic Optical Activity,” Annu. Rev. Phys. Chem. 17, 399–432 (1966).
[Crossref]

Bunch, R. M.

Chabicovsky, R.

H. Hauser, R. Chabicovsky, and K. Riedling, “Magneto-Optical Methods,” in Handbook of Thin Films: Deposition and Processing, H. S. Nalwa, ed. (Academic Press, 2001).

Clarke, D.

D. Clarke and J.F. Grainger, “The description of polarized light,” in Polarized light and Optical Measurement, First ed. (Pergamon Press Ltd., Germany, 1971).

Cooper, R. W.

R. W. Cooper and J. L. Page, “Magneto-optic light modulators,” Radio Electron. Eng. 39, 302–304 (1970).
[Crossref]

Crassee, I.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Da, N.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. S. J. Russell, “Complex Faraday Rotation in Microstructured Magneto-optical Fiber Waveguides,” Adv. Mater. 23, 2681–2688 (2011).
[Crossref] [PubMed]

Davis, J. A.

Dawson, J. W.

J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7, 1468–1470 (1995).
[Crossref]

Fischer, G.

G. Fischer, “The Faraday Optical Isolator,” J. Opt. Commun. 8, 18–21 (1987).

Fujii, Y.

Y. Fujii, “High-isolation polarization-independent optical circulator coupled with single-mode fibers,” J. Light-wave Technol. 9, 456–460 (1991).
[Crossref]

Fujimoto, Y.

Gabrielse, G.

Grainger, J.F.

D. Clarke and J.F. Grainger, “The description of polarized light,” in Polarized light and Optical Measurement, First ed. (Pergamon Press Ltd., Germany, 1971).

Granzow, N.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. S. J. Russell, “Complex Faraday Rotation in Microstructured Magneto-optical Fiber Waveguides,” Adv. Mater. 23, 2681–2688 (2011).
[Crossref] [PubMed]

Hauser, H.

H. Hauser, R. Chabicovsky, and K. Riedling, “Magneto-Optical Methods,” in Handbook of Thin Films: Deposition and Processing, H. S. Nalwa, ed. (Academic Press, 2001).

He, X.

S. Polisetty, J. Scheffler, S. Sahoo, Y. Wang, T. Mukherjee, X. He, and C. Binek, “Optimization of magneto-optical Kerr setup: Analyzing experimental assemblies using Jones matrix formalism,” Rev. Sci. Instrum. 79, 055107–055106 (2008).
[Crossref] [PubMed]

Hebb, M.

J. Van Vleck and M. Hebb, “On the Paramagnetic Rotation of Tysonite,” Phys. Rev. 46, 17–32 (1934).
[Crossref]

Hecht, E.

E. Hecht, “The Faraday Effect,” in Optics, 4th ed. (Pearson Education Inc., San Francisco, 2002).

E. Hecht, “Malus’s Law,” in Optics, 4th Edition ed. (Pearson Education Inc., San Francisco, 2002), pp. 332–333.

Hernandez, E.

J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7, 1468–1470 (1995).
[Crossref]

Hunt, R. P.

R. P. Hunt, “Magneto-Optic Scattering from Thin Solid Films,” J. Appl. Phys. 38, 1652–1671 (1967).
[Crossref]

Jalil, M. B. A.

Kan, H.

Kawanaka, J.

Kawashima, T.

Khazanov, E. A.

Kotov, V. A.

A. K. Zvezdin and V. A. Kotov, “Magnetooptical effects,” in Modern Magnetooptics and Magnetooptical Materials (Taylor & Francis Group, New York, 1997).

Kuzmenko, A. B.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Lee, H. W.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. S. J. Russell, “Complex Faraday Rotation in Microstructured Magneto-optical Fiber Waveguides,” Adv. Mater. 23, 2681–2688 (2011).
[Crossref] [PubMed]

Levallois, J.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Liu, C.

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Fundamental magneto-optics,” J. Appl. Phys. 68, 4203–4207 (1990).
[Crossref]

Livingston, A. E.

MacDougall, T. W.

J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7, 1468–1470 (1995).
[Crossref]

Marel, D.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Martinez, J. C.

Mihailovic, P. M.

P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for Temperature-Dependence of the Faraday Effect by Optical Activity Temperature Shift,” IEEE Sens. J. 13, 832–837 (2013).
[Crossref]

Moog, E. R.

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Fundamental magneto-optics,” J. Appl. Phys. 68, 4203–4207 (1990).
[Crossref]

Mukherjee, T.

S. Polisetty, J. Scheffler, S. Sahoo, Y. Wang, T. Mukherjee, X. He, and C. Binek, “Optimization of magneto-optical Kerr setup: Analyzing experimental assemblies using Jones matrix formalism,” Rev. Sci. Instrum. 79, 055107–055106 (2008).
[Crossref] [PubMed]

Nakatsuka, M.

Nozawa, H.

Ostler, M.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Page, J. L.

R. W. Cooper and J. L. Page, “Magneto-optic light modulators,” Radio Electron. Eng. 39, 302–304 (1970).
[Crossref]

Palashov, O. V.

Petricevic, S. J.

P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for Temperature-Dependence of the Faraday Effect by Optical Activity Temperature Shift,” IEEE Sens. J. 13, 832–837 (2013).
[Crossref]

Petway, L. B.

Polisetty, S.

S. Polisetty, J. Scheffler, S. Sahoo, Y. Wang, T. Mukherjee, X. He, and C. Binek, “Optimization of magneto-optical Kerr setup: Analyzing experimental assemblies using Jones matrix formalism,” Rev. Sci. Instrum. 79, 055107–055106 (2008).
[Crossref] [PubMed]

Radunovic, J. B.

P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for Temperature-Dependence of the Faraday Effect by Optical Activity Temperature Shift,” IEEE Sens. J. 13, 832–837 (2013).
[Crossref]

Riedling, K.

H. Hauser, R. Chabicovsky, and K. Riedling, “Magneto-Optical Methods,” in Handbook of Thin Films: Deposition and Processing, H. S. Nalwa, ed. (Academic Press, 2001).

Robinson, C. C.

Rotenberg, E.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Russell, P. S. J.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. S. J. Russell, “Complex Faraday Rotation in Microstructured Magneto-optical Fiber Waveguides,” Adv. Mater. 23, 2681–2688 (2011).
[Crossref] [PubMed]

Sahoo, S.

S. Polisetty, J. Scheffler, S. Sahoo, Y. Wang, T. Mukherjee, X. He, and C. Binek, “Optimization of magneto-optical Kerr setup: Analyzing experimental assemblies using Jones matrix formalism,” Rev. Sci. Instrum. 79, 055107–055106 (2008).
[Crossref] [PubMed]

Scheffler, J.

S. Polisetty, J. Scheffler, S. Sahoo, Y. Wang, T. Mukherjee, X. He, and C. Binek, “Optimization of magneto-optical Kerr setup: Analyzing experimental assemblies using Jones matrix formalism,” Rev. Sci. Instrum. 79, 055107–055106 (2008).
[Crossref] [PubMed]

Schmidt, M. A.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. S. J. Russell, “Complex Faraday Rotation in Microstructured Magneto-optical Fiber Waveguides,” Adv. Mater. 23, 2681–2688 (2011).
[Crossref] [PubMed]

Seyller, T.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Starobor, A. V.

Stephens, P. J.

A. D. Buckingham and P. J. Stephens, “Magnetic Optical Activity,” Annu. Rev. Phys. Chem. 17, 399–432 (1966).
[Crossref]

Tan, S. G.

Tokita, S.

Torihata, S.-i.

Tsukada, N.

Van Vleck, J.

J. Van Vleck and M. Hebb, “On the Paramagnetic Rotation of Tysonite,” Phys. Rev. 46, 17–32 (1934).
[Crossref]

Walter, A. L.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Wang, Y.

S. Polisetty, J. Scheffler, S. Sahoo, Y. Wang, T. Mukherjee, X. He, and C. Binek, “Optimization of magneto-optical Kerr setup: Analyzing experimental assemblies using Jones matrix formalism,” Rev. Sci. Instrum. 79, 055107–055106 (2008).
[Crossref] [PubMed]

Wondraczek, L.

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. S. J. Russell, “Complex Faraday Rotation in Microstructured Magneto-optical Fiber Waveguides,” Adv. Mater. 23, 2681–2688 (2011).
[Crossref] [PubMed]

Yagi, H.

Yanagitani, T.

Yasuhara, R.

Yokota, M.

Yoshida, H.

Yoshino, T.

Zak, J.

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Fundamental magneto-optics,” J. Appl. Phys. 68, 4203–4207 (1990).
[Crossref]

Zheleznov, D. S.

Zvezdin, A. K.

A. K. Zvezdin and V. A. Kotov, “Magnetooptical effects,” in Modern Magnetooptics and Magnetooptical Materials (Taylor & Francis Group, New York, 1997).

Adv. Mater. (1)

M. A. Schmidt, L. Wondraczek, H. W. Lee, N. Granzow, N. Da, and P. S. J. Russell, “Complex Faraday Rotation in Microstructured Magneto-optical Fiber Waveguides,” Adv. Mater. 23, 2681–2688 (2011).
[Crossref] [PubMed]

Annu. Rev. Phys. Chem. (1)

A. D. Buckingham and P. J. Stephens, “Magnetic Optical Activity,” Annu. Rev. Phys. Chem. 17, 399–432 (1966).
[Crossref]

Appl. Opt. (3)

IEEE Photonics Technol. Lett. (1)

J. W. Dawson, T. W. MacDougall, and E. Hernandez, “Verdet constant limited temperature response of a fiber-optic current sensor,” IEEE Photonics Technol. Lett. 7, 1468–1470 (1995).
[Crossref]

IEEE Sens. J. (1)

P. M. Mihailovic, S. J. Petricevic, and J. B. Radunovic, “Compensation for Temperature-Dependence of the Faraday Effect by Optical Activity Temperature Shift,” IEEE Sens. J. 13, 832–837 (2013).
[Crossref]

J. Appl. Phys. (2)

R. P. Hunt, “Magneto-Optic Scattering from Thin Solid Films,” J. Appl. Phys. 38, 1652–1671 (1967).
[Crossref]

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Fundamental magneto-optics,” J. Appl. Phys. 68, 4203–4207 (1990).
[Crossref]

J. Light-wave Technol. (1)

Y. Fujii, “High-isolation polarization-independent optical circulator coupled with single-mode fibers,” J. Light-wave Technol. 9, 456–460 (1991).
[Crossref]

J. Opt. Commun. (1)

G. Fischer, “The Faraday Optical Isolator,” J. Opt. Commun. 8, 18–21 (1987).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Nat. Phys. (1)

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Marel, and A. B. Kuzmenko, “Giant Faraday rotation in single- and multilayer graphene,” Nat. Phys. 7, 48–51 (2011).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (1)

J. Van Vleck and M. Hebb, “On the Paramagnetic Rotation of Tysonite,” Phys. Rev. 46, 17–32 (1934).
[Crossref]

Radio Electron. Eng. (1)

R. W. Cooper and J. L. Page, “Magneto-optic light modulators,” Radio Electron. Eng. 39, 302–304 (1970).
[Crossref]

Rev. Sci. Instrum. (1)

S. Polisetty, J. Scheffler, S. Sahoo, Y. Wang, T. Mukherjee, X. He, and C. Binek, “Optimization of magneto-optical Kerr setup: Analyzing experimental assemblies using Jones matrix formalism,” Rev. Sci. Instrum. 79, 055107–055106 (2008).
[Crossref] [PubMed]

Other (6)

E. Hecht, “Malus’s Law,” in Optics, 4th Edition ed. (Pearson Education Inc., San Francisco, 2002), pp. 332–333.

E. Hecht, “The Faraday Effect,” in Optics, 4th ed. (Pearson Education Inc., San Francisco, 2002).

C. A. Bennett, “Faraday Rotation,” in Principles of Physical Optics (John Wiley & Sons Inc., 2008).

D. Clarke and J.F. Grainger, “The description of polarized light,” in Polarized light and Optical Measurement, First ed. (Pergamon Press Ltd., Germany, 1971).

H. Hauser, R. Chabicovsky, and K. Riedling, “Magneto-Optical Methods,” in Handbook of Thin Films: Deposition and Processing, H. S. Nalwa, ed. (Academic Press, 2001).

A. K. Zvezdin and V. A. Kotov, “Magnetooptical effects,” in Modern Magnetooptics and Magnetooptical Materials (Taylor & Francis Group, New York, 1997).

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Figures (7)

Fig. 1
Fig. 1

Faraday effect: The rotation of the plane of polarization of linearly polarized light on passing through a magneto-optically active material.

Fig. 2
Fig. 2

In polarimetry using the fixed polarizer-analyzer setup the transmission axes of polarizer P and analyzer A are oriented at 45° to each other, allowing one to determine the rotation.

Fig. 3
Fig. 3

(a) The convention used for determining the rotation θ, ellipticity |ψ| and handedness (LHCP or RHCP) of elliptically polarized light. (b) Configuration of the retarder W and analyzer A used to measure Stokes parameters.

Fig. 4
Fig. 4

The experimental arrangement including P = Polarizer, W = Retarder, A = Analyzer, PD = Photodetector.

Fig. 5
Fig. 5

The angles θ and ψ as a function of applied field B for the 6.3–300 K temperature range. For visual clarity the data for the 6.3–20 K and 40–300 K temperature ranges have been shown in separate plots. For the same reason the data for the 40–300 K temperature range are shown in 40 K increments, even though the raw data in this range were taken at 20 K increments.

Fig. 6
Fig. 6

The Verdet constant V plotted against temperature. The inset shows a plot of V against 1/T on logarithmic axes, highlighting a linear relationship between them.

Fig. 7
Fig. 7

Data obtained using the fixed polarizer-analyzer configuration in the temperature ranges of (a) 6.3–20 K (b) 40–300 K

Equations (10)

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θ = V B d
ε = ( ε x x i Q 0 i Q ε x x 0 0 0 ε z z )
θ = ω 2 c n x x Q d
ψ = ω 2 c n x x Q d
I = I 0 cos 2 ( θ π 4 ) + I min
I = | E x | 2 + | E y | 2 M = | E x | 2 | E y | 2 C = 2 | E x | | E y | cos ( δ y δ x ) , and S = 2 | E x | | E y | sin ( δ y δ x ) .
θ = 1 2 arctan ( C M ) , and | ψ | = arctan ( η ) .
2 η 1 + η 2 = | S | I p .
I T ( α , β , δ r ) = 1 2 [ I + ( M 2 cos 2 α + C 2 sin 2 α ) ( 1 + cos δ r ) ] + S 2 sin δ r sin ( 2 α 2 β ) + 1 4 [ ( M cos 2 α C sin 2 α ) cos 4 β + ( M sin 2 α + C cos 2 α ) sin 4 β ] ( 1 cos δ r )
I = C 0 1 + cos δ r 1 cos δ r [ C 4 cos ( 4 α + 4 β 0 ) + S 4 sin ( 4 α + 4 β 0 ) ] , M = 2 1 cos δ r [ C 4 cos ( 2 α + 4 β 0 ) + S 4 sin ( 2 α + 4 β 0 ) ] , C = 2 1 cos δ r [ S 4 cos ( 2 α + 4 β 0 ) C 4 sin ( 2 α + 4 β 0 ) ] , and S = S 2 sin δ r cos ( 2 α + 4 β 0 )

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