Abstract

The equation for evolution of the four-component Stokes vector in weakly anisotropic and smoothly inhomogeneous media is derived on the basis of a quasi-isotropic approximation of the geometrical optics method, which provides the consequent asymptotic solution of Maxwell’s equations. Our equation generalizes previous results obtained for the normal propagation of electromagnetic waves in stratified media. It is valid for curvilinear rays with torsion and is capable of describing normal mode conversion in inhomogeneous media. Remarkably, evolution of the four-component Stokes vector is described by the Bargmann–Michel–Telegdi equation for relativistic spin precession, whereas the equation for the three-component Stokes vector resembles the Landau–Lifshitz equation describing spin precession in ferromagnetic systems. The general theory is applied for analysis of polarization evolution in a magnetized plasma. We also emphasize fundamental features of the non-Abelian polarization evolution in anisotropic inhomogeneous media and illustrate them by simple examples.

© 2007 Optical Society of America

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  1. K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, 1961).
  2. V. I. Ginzburg, Propagation of Electromagnetic Waves in Plasma (Gordon & Breach, 1970).
  3. D. B. Melrose and R. C. McPhedran, Electromagnetic Processes in Dispersive Media (Cambridge U. Press, 1991).
    [CrossRef]
  4. V. V. Zheleznyakov, V. V. Kocharovski, and Vl. V. Kocharovski, "Linear interaction of electromagnetic waves in inhomogeneous weakly anisotropic media," Usp. Fiz. Nauk 141, 257-282 (1983) V. V. Zheleznyakov, V. V. Kocharovski, and Vl. V. Kocharovski,[Sov. Phys. Usp. 26, 877-902 (1983)].
    [CrossRef]
  5. Yu. A. Kravtsov, "Quasi-isotropic geometrical optics approximation," Sov. Phys. Dokl. 13, 1125-1127 (1969).
  6. Yu. A. Kravtsov, O. N. Naida, and A. A. Fuki, "Waves in weakly anisotropic 3D inhomogeneous media: quasi-isotropic approximation of geometrical optics," Phys. Usp. 39, 129-134 (1996).
    [CrossRef]
  7. A. A. Fuki, Yu. A. Kravtsov, and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, 1997).
  8. Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
    [CrossRef]
  9. Yu. A. Kravtsov, Geometrical Optics in Engineering Physics (Alpha Science, 2005).
  10. G. N. Ramachandran and S. Ramaseshan, Crystal Optics, Vol. 25 of Encyclopedia of Physics (Springer-Verlag, 1961).
  11. R. M. A. Azzam, "Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4×4 matrix calculus," J. Opt. Soc. Am. A 68, 1756-1767 (1978).
    [CrossRef]
  12. C. Brosseau, "Evolution of the Stokes parameters in optically anisotropic media," Opt. Lett. 20, 1221-1223 (1995).
    [CrossRef] [PubMed]
  13. C. S. Brown and A. E. Bak, "Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber," Opt. Eng. 34, 1625-1635 (1995).
    [CrossRef]
  14. S. E. Segre, "On the use of polarization modulation in combined interferometry and polarimetry," Plasma Phys. Controlled Fusion 40, 153-161 (1998).
    [CrossRef]
  15. S. E. Segre, "A review of plasma polarimetry - theory and methods," Plasma Phys. Controlled Fusion 41, R57-R100 (1999).
    [CrossRef]
  16. S. E. Segre, "Evolution of the polarization state for radiation propagating in a nonuniform, birefringent, optically active and dichroic medium: the case of magnetized plasma," J. Opt. Soc. Am. A 17, 95-100 (2000).
    [CrossRef]
  17. S. E. Segre, "Effect of ray refraction in evolution of the polarization state of radiation propagating in a nonuniform, birefringent, optically active and dichroic medium," J. Opt. Soc. Am. A 17, 1682-1683 (2000).
    [CrossRef]
  18. S. E. Segre, "New formalism for the analysis of polarization evolution for radiation in a weakly nonuniform, fully anisotropic medium: a magnetized plasma," J. Opt. Soc. Am. A 18, 2601-2606 (2001).
    [CrossRef]
  19. S. E. Segre, "Comparison between two alternative approaches for the analysis of polarization evolution of EM waves in a nonuniform, fully anisotropic medium: a magnetized plasma," Preprint RT/ERG/FUS/2001/13 (ENEA, 2001).
  20. S. E. Segre, "Polarization evolution of radiation propagating in weakly non-uniform magnetized plasma with dissipation," J. Phys. D 36, 2806-2810 (2003).
    [CrossRef]
  21. S. E. Segre and V. Zanza, "Derivation of the pure Faraday and Cotton-Mouton effects when polarimetric effects in a tokamak are large," Plasma Phys. Controlled Fusion 48, 339-351 (2006).
    [CrossRef]
  22. K. Yu. Bliokh, D. Yu. Frolov, and Yu. A. Kravtsov, "Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium," Phys. Rev. A 75, 053821 (2007).
    [CrossRef]
  23. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  24. M. M. Popov, "Eigen-oscillations of multi-mirrors resonators," Vestn. Leningr. Univ., Ser. 4: Fiz., Khim. 22, 44-54 (1969) (in Russian). Derivation of the Popov's orthogonal coordinate system is also reproduced in Chap. 9.
  25. V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Springer-Verlag, 1990) [original Russian edition, V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wavelength Diffraction Problems: The Model Problem Method (Nauka, 1972)].
  26. V. Cervený, Seismic Ray Theory (Cambridge U. Press, 2001).
    [CrossRef]
  27. P. Berczynski, K. Yu. Bliokh, Yu. A. Kravtsov, and A. Stateczny, "Diffraction of Gaussian beam in 3D smoothly inhomogeneous media: eikonal-based complex geometrical optics approach," J. Opt. Soc. Am. A 23, 1442-1451 (2006).
    [CrossRef]
  28. For a review, see S. I. Vinitskii, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii, "Topological phases in quantum mechanics and polarization optics," Usp. Fiz. Nauk 160(6), 1-49 (1990) S. I. Vinitsky, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii,[Sov. Phys. Usp. 33, 403-450 (1990)].
    [CrossRef]
  29. R. Barakat, "Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory," Opt. Commun. 38, 159-161 (1981).
    [CrossRef]
  30. R. Simon, "The connection between Mueller and Jones matrices of polarization optics," Opt. Commun. 42, 293-297 (1982).
    [CrossRef]
  31. S. R. Cloude, "Group theory and polarization algebra," Optik (Stuttgart) 75, 26-36 (1986).
  32. K. Kim, L. Mandel, and E. Wolf, "Relationship between Jones and Mueller matrices for random media," J. Opt. Soc. Am. A 4, 433-437 (1987).
    [CrossRef]
  33. D. Han, Y. S. Kim, and M. E. Noz, "Polarization optics and bilinear representation of the Lorentz group," Phys. Lett. A 219, 26-32 (1996).
    [CrossRef]
  34. V. B. Berestetskii, E. M. Lifshits, and L. P. Pitaevskii, Quantum Electrodynamics (Pergamon, 1982).
  35. V. Bargmann, L. Michel, and V. L. Telegdi, "Precession of the polarization of particles moving in a homogeneous electromagnetic field," Phys. Rev. Lett. 2, 435-436 (1959).
    [CrossRef]
  36. J. Bolte and S. Keppeler, "A semiclassical approach to the Dirac equation," Appl. Phys. (N.Y.) 274, 125-162 (1999).
    [CrossRef]
  37. H. Spohn, "Semiclassical limit of the Dirac equation and spin precession," Appl. Phys. (N.Y.) 282, 420-431 (2000).
    [CrossRef]
  38. A. I. Akhiezer, V. G. Baryakhtar, and S. V. Peletminskii, Spin Waves (North-Holland, 1968).
  39. C. P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, 1989).
  40. Yu. A. Kravtsov and O. N. Naida, "Linear transformation of electromagnetic waves in three-dimensional inhomogeneous magneto-active plasma," Sov. Phys. JETP 44, 122-126 (1976).
  41. Yu. A. Kravtsov and O. N. Naida, "Theory for Cotton-Mouton diagnostics of magnetized plasma," J. Tech. Phys. 41, 155-160 (2000).
  42. Yu. A. Kravtsov and O. N. Naida, "Manifestation of the Cotton-Mouton effect in the ionosphere plasma," Adv. Space Res. 27, 1233-1237 (2001).
    [CrossRef]
  43. Z. H. Czyz, B. Bieg, and Yu. A. Kravtsov, "Complex polarization angle: relation to traditional polarization parameters and application to microwave plasma polarimetry," Phys. Lett. A 368, 101-107 (2007).
    [CrossRef]
  44. H. Kuratsuji and S. Kakigi, "Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters," Phys. Rev. Lett. 80, 1888-1891 (1998).
    [CrossRef]
  45. R. Botet, H. Kuratsuji, and R. Seto, "Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media," Prog. Theor. Phys. 116, 285-294 (2006).
    [CrossRef]

2007 (2)

K. Yu. Bliokh, D. Yu. Frolov, and Yu. A. Kravtsov, "Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium," Phys. Rev. A 75, 053821 (2007).
[CrossRef]

Z. H. Czyz, B. Bieg, and Yu. A. Kravtsov, "Complex polarization angle: relation to traditional polarization parameters and application to microwave plasma polarimetry," Phys. Lett. A 368, 101-107 (2007).
[CrossRef]

2006 (3)

R. Botet, H. Kuratsuji, and R. Seto, "Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media," Prog. Theor. Phys. 116, 285-294 (2006).
[CrossRef]

S. E. Segre and V. Zanza, "Derivation of the pure Faraday and Cotton-Mouton effects when polarimetric effects in a tokamak are large," Plasma Phys. Controlled Fusion 48, 339-351 (2006).
[CrossRef]

P. Berczynski, K. Yu. Bliokh, Yu. A. Kravtsov, and A. Stateczny, "Diffraction of Gaussian beam in 3D smoothly inhomogeneous media: eikonal-based complex geometrical optics approach," J. Opt. Soc. Am. A 23, 1442-1451 (2006).
[CrossRef]

2003 (1)

S. E. Segre, "Polarization evolution of radiation propagating in weakly non-uniform magnetized plasma with dissipation," J. Phys. D 36, 2806-2810 (2003).
[CrossRef]

2001 (2)

2000 (4)

1999 (2)

S. E. Segre, "A review of plasma polarimetry - theory and methods," Plasma Phys. Controlled Fusion 41, R57-R100 (1999).
[CrossRef]

J. Bolte and S. Keppeler, "A semiclassical approach to the Dirac equation," Appl. Phys. (N.Y.) 274, 125-162 (1999).
[CrossRef]

1998 (2)

H. Kuratsuji and S. Kakigi, "Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters," Phys. Rev. Lett. 80, 1888-1891 (1998).
[CrossRef]

S. E. Segre, "On the use of polarization modulation in combined interferometry and polarimetry," Plasma Phys. Controlled Fusion 40, 153-161 (1998).
[CrossRef]

1996 (2)

Yu. A. Kravtsov, O. N. Naida, and A. A. Fuki, "Waves in weakly anisotropic 3D inhomogeneous media: quasi-isotropic approximation of geometrical optics," Phys. Usp. 39, 129-134 (1996).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, "Polarization optics and bilinear representation of the Lorentz group," Phys. Lett. A 219, 26-32 (1996).
[CrossRef]

1995 (2)

C. Brosseau, "Evolution of the Stokes parameters in optically anisotropic media," Opt. Lett. 20, 1221-1223 (1995).
[CrossRef] [PubMed]

C. S. Brown and A. E. Bak, "Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber," Opt. Eng. 34, 1625-1635 (1995).
[CrossRef]

1990 (1)

For a review, see S. I. Vinitskii, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii, "Topological phases in quantum mechanics and polarization optics," Usp. Fiz. Nauk 160(6), 1-49 (1990) S. I. Vinitsky, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii,[Sov. Phys. Usp. 33, 403-450 (1990)].
[CrossRef]

1987 (1)

1986 (1)

S. R. Cloude, "Group theory and polarization algebra," Optik (Stuttgart) 75, 26-36 (1986).

1983 (1)

V. V. Zheleznyakov, V. V. Kocharovski, and Vl. V. Kocharovski, "Linear interaction of electromagnetic waves in inhomogeneous weakly anisotropic media," Usp. Fiz. Nauk 141, 257-282 (1983) V. V. Zheleznyakov, V. V. Kocharovski, and Vl. V. Kocharovski,[Sov. Phys. Usp. 26, 877-902 (1983)].
[CrossRef]

1982 (1)

R. Simon, "The connection between Mueller and Jones matrices of polarization optics," Opt. Commun. 42, 293-297 (1982).
[CrossRef]

1981 (1)

R. Barakat, "Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory," Opt. Commun. 38, 159-161 (1981).
[CrossRef]

1978 (1)

R. M. A. Azzam, "Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4×4 matrix calculus," J. Opt. Soc. Am. A 68, 1756-1767 (1978).
[CrossRef]

1976 (1)

Yu. A. Kravtsov and O. N. Naida, "Linear transformation of electromagnetic waves in three-dimensional inhomogeneous magneto-active plasma," Sov. Phys. JETP 44, 122-126 (1976).

1969 (2)

Yu. A. Kravtsov, "Quasi-isotropic geometrical optics approximation," Sov. Phys. Dokl. 13, 1125-1127 (1969).

M. M. Popov, "Eigen-oscillations of multi-mirrors resonators," Vestn. Leningr. Univ., Ser. 4: Fiz., Khim. 22, 44-54 (1969) (in Russian). Derivation of the Popov's orthogonal coordinate system is also reproduced in Chap. 9.

1959 (1)

V. Bargmann, L. Michel, and V. L. Telegdi, "Precession of the polarization of particles moving in a homogeneous electromagnetic field," Phys. Rev. Lett. 2, 435-436 (1959).
[CrossRef]

Akhiezer, A. I.

A. I. Akhiezer, V. G. Baryakhtar, and S. V. Peletminskii, Spin Waves (North-Holland, 1968).

Azzam, R. M. A.

R. M. A. Azzam, "Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4×4 matrix calculus," J. Opt. Soc. Am. A 68, 1756-1767 (1978).
[CrossRef]

Babich, V. M.

V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Springer-Verlag, 1990) [original Russian edition, V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wavelength Diffraction Problems: The Model Problem Method (Nauka, 1972)].

Bak, A. E.

C. S. Brown and A. E. Bak, "Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber," Opt. Eng. 34, 1625-1635 (1995).
[CrossRef]

Barakat, R.

R. Barakat, "Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory," Opt. Commun. 38, 159-161 (1981).
[CrossRef]

Bargmann, V.

V. Bargmann, L. Michel, and V. L. Telegdi, "Precession of the polarization of particles moving in a homogeneous electromagnetic field," Phys. Rev. Lett. 2, 435-436 (1959).
[CrossRef]

Baryakhtar, V. G.

A. I. Akhiezer, V. G. Baryakhtar, and S. V. Peletminskii, Spin Waves (North-Holland, 1968).

Berczynski, P.

Berestetskii, V. B.

V. B. Berestetskii, E. M. Lifshits, and L. P. Pitaevskii, Quantum Electrodynamics (Pergamon, 1982).

Bieg, B.

Z. H. Czyz, B. Bieg, and Yu. A. Kravtsov, "Complex polarization angle: relation to traditional polarization parameters and application to microwave plasma polarimetry," Phys. Lett. A 368, 101-107 (2007).
[CrossRef]

Bliokh, K. Yu.

K. Yu. Bliokh, D. Yu. Frolov, and Yu. A. Kravtsov, "Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium," Phys. Rev. A 75, 053821 (2007).
[CrossRef]

P. Berczynski, K. Yu. Bliokh, Yu. A. Kravtsov, and A. Stateczny, "Diffraction of Gaussian beam in 3D smoothly inhomogeneous media: eikonal-based complex geometrical optics approach," J. Opt. Soc. Am. A 23, 1442-1451 (2006).
[CrossRef]

Bolte, J.

J. Bolte and S. Keppeler, "A semiclassical approach to the Dirac equation," Appl. Phys. (N.Y.) 274, 125-162 (1999).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Botet, R.

R. Botet, H. Kuratsuji, and R. Seto, "Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media," Prog. Theor. Phys. 116, 285-294 (2006).
[CrossRef]

Brosseau, C.

Brown, C. S.

C. S. Brown and A. E. Bak, "Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber," Opt. Eng. 34, 1625-1635 (1995).
[CrossRef]

Budden, K. G.

K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, 1961).

Buldyrev, V. S.

V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Springer-Verlag, 1990) [original Russian edition, V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wavelength Diffraction Problems: The Model Problem Method (Nauka, 1972)].

Cervený, V.

V. Cervený, Seismic Ray Theory (Cambridge U. Press, 2001).
[CrossRef]

Cloude, S. R.

S. R. Cloude, "Group theory and polarization algebra," Optik (Stuttgart) 75, 26-36 (1986).

Czyz, Z. H.

Z. H. Czyz, B. Bieg, and Yu. A. Kravtsov, "Complex polarization angle: relation to traditional polarization parameters and application to microwave plasma polarimetry," Phys. Lett. A 368, 101-107 (2007).
[CrossRef]

Debrov, V. L.

For a review, see S. I. Vinitskii, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii, "Topological phases in quantum mechanics and polarization optics," Usp. Fiz. Nauk 160(6), 1-49 (1990) S. I. Vinitsky, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii,[Sov. Phys. Usp. 33, 403-450 (1990)].
[CrossRef]

Dubovik, V. M.

For a review, see S. I. Vinitskii, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii, "Topological phases in quantum mechanics and polarization optics," Usp. Fiz. Nauk 160(6), 1-49 (1990) S. I. Vinitsky, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii,[Sov. Phys. Usp. 33, 403-450 (1990)].
[CrossRef]

Frolov, D. Yu.

K. Yu. Bliokh, D. Yu. Frolov, and Yu. A. Kravtsov, "Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium," Phys. Rev. A 75, 053821 (2007).
[CrossRef]

Fuki, A. A.

Yu. A. Kravtsov, O. N. Naida, and A. A. Fuki, "Waves in weakly anisotropic 3D inhomogeneous media: quasi-isotropic approximation of geometrical optics," Phys. Usp. 39, 129-134 (1996).
[CrossRef]

A. A. Fuki, Yu. A. Kravtsov, and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, 1997).

Ginzburg, V. I.

V. I. Ginzburg, Propagation of Electromagnetic Waves in Plasma (Gordon & Breach, 1970).

Han, D.

D. Han, Y. S. Kim, and M. E. Noz, "Polarization optics and bilinear representation of the Lorentz group," Phys. Lett. A 219, 26-32 (1996).
[CrossRef]

Kakigi, S.

H. Kuratsuji and S. Kakigi, "Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters," Phys. Rev. Lett. 80, 1888-1891 (1998).
[CrossRef]

Keppeler, S.

J. Bolte and S. Keppeler, "A semiclassical approach to the Dirac equation," Appl. Phys. (N.Y.) 274, 125-162 (1999).
[CrossRef]

Kim, K.

Kim, Y. S.

D. Han, Y. S. Kim, and M. E. Noz, "Polarization optics and bilinear representation of the Lorentz group," Phys. Lett. A 219, 26-32 (1996).
[CrossRef]

Kocharovski, V. V.

V. V. Zheleznyakov, V. V. Kocharovski, and Vl. V. Kocharovski, "Linear interaction of electromagnetic waves in inhomogeneous weakly anisotropic media," Usp. Fiz. Nauk 141, 257-282 (1983) V. V. Zheleznyakov, V. V. Kocharovski, and Vl. V. Kocharovski,[Sov. Phys. Usp. 26, 877-902 (1983)].
[CrossRef]

Kocharovski, Vl. V.

V. V. Zheleznyakov, V. V. Kocharovski, and Vl. V. Kocharovski, "Linear interaction of electromagnetic waves in inhomogeneous weakly anisotropic media," Usp. Fiz. Nauk 141, 257-282 (1983) V. V. Zheleznyakov, V. V. Kocharovski, and Vl. V. Kocharovski,[Sov. Phys. Usp. 26, 877-902 (1983)].
[CrossRef]

Kravtsov, Yu. A.

K. Yu. Bliokh, D. Yu. Frolov, and Yu. A. Kravtsov, "Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium," Phys. Rev. A 75, 053821 (2007).
[CrossRef]

Z. H. Czyz, B. Bieg, and Yu. A. Kravtsov, "Complex polarization angle: relation to traditional polarization parameters and application to microwave plasma polarimetry," Phys. Lett. A 368, 101-107 (2007).
[CrossRef]

P. Berczynski, K. Yu. Bliokh, Yu. A. Kravtsov, and A. Stateczny, "Diffraction of Gaussian beam in 3D smoothly inhomogeneous media: eikonal-based complex geometrical optics approach," J. Opt. Soc. Am. A 23, 1442-1451 (2006).
[CrossRef]

Yu. A. Kravtsov and O. N. Naida, "Manifestation of the Cotton-Mouton effect in the ionosphere plasma," Adv. Space Res. 27, 1233-1237 (2001).
[CrossRef]

Yu. A. Kravtsov and O. N. Naida, "Theory for Cotton-Mouton diagnostics of magnetized plasma," J. Tech. Phys. 41, 155-160 (2000).

Yu. A. Kravtsov, O. N. Naida, and A. A. Fuki, "Waves in weakly anisotropic 3D inhomogeneous media: quasi-isotropic approximation of geometrical optics," Phys. Usp. 39, 129-134 (1996).
[CrossRef]

Yu. A. Kravtsov and O. N. Naida, "Linear transformation of electromagnetic waves in three-dimensional inhomogeneous magneto-active plasma," Sov. Phys. JETP 44, 122-126 (1976).

Yu. A. Kravtsov, "Quasi-isotropic geometrical optics approximation," Sov. Phys. Dokl. 13, 1125-1127 (1969).

A. A. Fuki, Yu. A. Kravtsov, and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, 1997).

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[CrossRef]

Yu. A. Kravtsov, Geometrical Optics in Engineering Physics (Alpha Science, 2005).

Kuratsuji, H.

R. Botet, H. Kuratsuji, and R. Seto, "Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media," Prog. Theor. Phys. 116, 285-294 (2006).
[CrossRef]

H. Kuratsuji and S. Kakigi, "Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters," Phys. Rev. Lett. 80, 1888-1891 (1998).
[CrossRef]

Lifshits, E. M.

V. B. Berestetskii, E. M. Lifshits, and L. P. Pitaevskii, Quantum Electrodynamics (Pergamon, 1982).

Mandel, L.

Markovski, B. L.

For a review, see S. I. Vinitskii, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii, "Topological phases in quantum mechanics and polarization optics," Usp. Fiz. Nauk 160(6), 1-49 (1990) S. I. Vinitsky, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii,[Sov. Phys. Usp. 33, 403-450 (1990)].
[CrossRef]

McPhedran, R. C.

D. B. Melrose and R. C. McPhedran, Electromagnetic Processes in Dispersive Media (Cambridge U. Press, 1991).
[CrossRef]

Melrose, D. B.

D. B. Melrose and R. C. McPhedran, Electromagnetic Processes in Dispersive Media (Cambridge U. Press, 1991).
[CrossRef]

Michel, L.

V. Bargmann, L. Michel, and V. L. Telegdi, "Precession of the polarization of particles moving in a homogeneous electromagnetic field," Phys. Rev. Lett. 2, 435-436 (1959).
[CrossRef]

Naida, O. N.

Yu. A. Kravtsov and O. N. Naida, "Manifestation of the Cotton-Mouton effect in the ionosphere plasma," Adv. Space Res. 27, 1233-1237 (2001).
[CrossRef]

Yu. A. Kravtsov and O. N. Naida, "Theory for Cotton-Mouton diagnostics of magnetized plasma," J. Tech. Phys. 41, 155-160 (2000).

Yu. A. Kravtsov, O. N. Naida, and A. A. Fuki, "Waves in weakly anisotropic 3D inhomogeneous media: quasi-isotropic approximation of geometrical optics," Phys. Usp. 39, 129-134 (1996).
[CrossRef]

Yu. A. Kravtsov and O. N. Naida, "Linear transformation of electromagnetic waves in three-dimensional inhomogeneous magneto-active plasma," Sov. Phys. JETP 44, 122-126 (1976).

A. A. Fuki, Yu. A. Kravtsov, and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, 1997).

Noz, M. E.

D. Han, Y. S. Kim, and M. E. Noz, "Polarization optics and bilinear representation of the Lorentz group," Phys. Lett. A 219, 26-32 (1996).
[CrossRef]

Orlov, Yu. I.

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[CrossRef]

Peletminskii, S. V.

A. I. Akhiezer, V. G. Baryakhtar, and S. V. Peletminskii, Spin Waves (North-Holland, 1968).

Pitaevskii, L. P.

V. B. Berestetskii, E. M. Lifshits, and L. P. Pitaevskii, Quantum Electrodynamics (Pergamon, 1982).

Popov, M. M.

M. M. Popov, "Eigen-oscillations of multi-mirrors resonators," Vestn. Leningr. Univ., Ser. 4: Fiz., Khim. 22, 44-54 (1969) (in Russian). Derivation of the Popov's orthogonal coordinate system is also reproduced in Chap. 9.

Ramachandran, G. N.

G. N. Ramachandran and S. Ramaseshan, Crystal Optics, Vol. 25 of Encyclopedia of Physics (Springer-Verlag, 1961).

Ramaseshan, S.

G. N. Ramachandran and S. Ramaseshan, Crystal Optics, Vol. 25 of Encyclopedia of Physics (Springer-Verlag, 1961).

Segre, S. E.

S. E. Segre and V. Zanza, "Derivation of the pure Faraday and Cotton-Mouton effects when polarimetric effects in a tokamak are large," Plasma Phys. Controlled Fusion 48, 339-351 (2006).
[CrossRef]

S. E. Segre, "Polarization evolution of radiation propagating in weakly non-uniform magnetized plasma with dissipation," J. Phys. D 36, 2806-2810 (2003).
[CrossRef]

S. E. Segre, "New formalism for the analysis of polarization evolution for radiation in a weakly nonuniform, fully anisotropic medium: a magnetized plasma," J. Opt. Soc. Am. A 18, 2601-2606 (2001).
[CrossRef]

S. E. Segre, "Effect of ray refraction in evolution of the polarization state of radiation propagating in a nonuniform, birefringent, optically active and dichroic medium," J. Opt. Soc. Am. A 17, 1682-1683 (2000).
[CrossRef]

S. E. Segre, "Evolution of the polarization state for radiation propagating in a nonuniform, birefringent, optically active and dichroic medium: the case of magnetized plasma," J. Opt. Soc. Am. A 17, 95-100 (2000).
[CrossRef]

S. E. Segre, "A review of plasma polarimetry - theory and methods," Plasma Phys. Controlled Fusion 41, R57-R100 (1999).
[CrossRef]

S. E. Segre, "On the use of polarization modulation in combined interferometry and polarimetry," Plasma Phys. Controlled Fusion 40, 153-161 (1998).
[CrossRef]

S. E. Segre, "Comparison between two alternative approaches for the analysis of polarization evolution of EM waves in a nonuniform, fully anisotropic medium: a magnetized plasma," Preprint RT/ERG/FUS/2001/13 (ENEA, 2001).

Seto, R.

R. Botet, H. Kuratsuji, and R. Seto, "Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media," Prog. Theor. Phys. 116, 285-294 (2006).
[CrossRef]

Simon, R.

R. Simon, "The connection between Mueller and Jones matrices of polarization optics," Opt. Commun. 42, 293-297 (1982).
[CrossRef]

Slichter, C. P.

C. P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, 1989).

Spohn, H.

H. Spohn, "Semiclassical limit of the Dirac equation and spin precession," Appl. Phys. (N.Y.) 282, 420-431 (2000).
[CrossRef]

Stateczny, A.

Stepanovskii, Yu. P.

For a review, see S. I. Vinitskii, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii, "Topological phases in quantum mechanics and polarization optics," Usp. Fiz. Nauk 160(6), 1-49 (1990) S. I. Vinitsky, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii,[Sov. Phys. Usp. 33, 403-450 (1990)].
[CrossRef]

Telegdi, V. L.

V. Bargmann, L. Michel, and V. L. Telegdi, "Precession of the polarization of particles moving in a homogeneous electromagnetic field," Phys. Rev. Lett. 2, 435-436 (1959).
[CrossRef]

Vinitskii, S. I.

For a review, see S. I. Vinitskii, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii, "Topological phases in quantum mechanics and polarization optics," Usp. Fiz. Nauk 160(6), 1-49 (1990) S. I. Vinitsky, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii,[Sov. Phys. Usp. 33, 403-450 (1990)].
[CrossRef]

Wolf, E.

Zanza, V.

S. E. Segre and V. Zanza, "Derivation of the pure Faraday and Cotton-Mouton effects when polarimetric effects in a tokamak are large," Plasma Phys. Controlled Fusion 48, 339-351 (2006).
[CrossRef]

Zheleznyakov, V. V.

V. V. Zheleznyakov, V. V. Kocharovski, and Vl. V. Kocharovski, "Linear interaction of electromagnetic waves in inhomogeneous weakly anisotropic media," Usp. Fiz. Nauk 141, 257-282 (1983) V. V. Zheleznyakov, V. V. Kocharovski, and Vl. V. Kocharovski,[Sov. Phys. Usp. 26, 877-902 (1983)].
[CrossRef]

Adv. Space Res. (1)

Yu. A. Kravtsov and O. N. Naida, "Manifestation of the Cotton-Mouton effect in the ionosphere plasma," Adv. Space Res. 27, 1233-1237 (2001).
[CrossRef]

Appl. Phys. (N.Y.) (2)

J. Bolte and S. Keppeler, "A semiclassical approach to the Dirac equation," Appl. Phys. (N.Y.) 274, 125-162 (1999).
[CrossRef]

H. Spohn, "Semiclassical limit of the Dirac equation and spin precession," Appl. Phys. (N.Y.) 282, 420-431 (2000).
[CrossRef]

J. Opt. Soc. Am. A (6)

J. Phys. D (1)

S. E. Segre, "Polarization evolution of radiation propagating in weakly non-uniform magnetized plasma with dissipation," J. Phys. D 36, 2806-2810 (2003).
[CrossRef]

J. Tech. Phys. (1)

Yu. A. Kravtsov and O. N. Naida, "Theory for Cotton-Mouton diagnostics of magnetized plasma," J. Tech. Phys. 41, 155-160 (2000).

Opt. Commun. (2)

R. Barakat, "Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory," Opt. Commun. 38, 159-161 (1981).
[CrossRef]

R. Simon, "The connection between Mueller and Jones matrices of polarization optics," Opt. Commun. 42, 293-297 (1982).
[CrossRef]

Opt. Eng. (1)

C. S. Brown and A. E. Bak, "Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber," Opt. Eng. 34, 1625-1635 (1995).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

S. R. Cloude, "Group theory and polarization algebra," Optik (Stuttgart) 75, 26-36 (1986).

Phys. Lett. A (2)

D. Han, Y. S. Kim, and M. E. Noz, "Polarization optics and bilinear representation of the Lorentz group," Phys. Lett. A 219, 26-32 (1996).
[CrossRef]

Z. H. Czyz, B. Bieg, and Yu. A. Kravtsov, "Complex polarization angle: relation to traditional polarization parameters and application to microwave plasma polarimetry," Phys. Lett. A 368, 101-107 (2007).
[CrossRef]

Phys. Rev. A (1)

K. Yu. Bliokh, D. Yu. Frolov, and Yu. A. Kravtsov, "Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium," Phys. Rev. A 75, 053821 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

H. Kuratsuji and S. Kakigi, "Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters," Phys. Rev. Lett. 80, 1888-1891 (1998).
[CrossRef]

V. Bargmann, L. Michel, and V. L. Telegdi, "Precession of the polarization of particles moving in a homogeneous electromagnetic field," Phys. Rev. Lett. 2, 435-436 (1959).
[CrossRef]

Phys. Usp. (1)

Yu. A. Kravtsov, O. N. Naida, and A. A. Fuki, "Waves in weakly anisotropic 3D inhomogeneous media: quasi-isotropic approximation of geometrical optics," Phys. Usp. 39, 129-134 (1996).
[CrossRef]

Plasma Phys. Controlled Fusion (3)

S. E. Segre, "On the use of polarization modulation in combined interferometry and polarimetry," Plasma Phys. Controlled Fusion 40, 153-161 (1998).
[CrossRef]

S. E. Segre, "A review of plasma polarimetry - theory and methods," Plasma Phys. Controlled Fusion 41, R57-R100 (1999).
[CrossRef]

S. E. Segre and V. Zanza, "Derivation of the pure Faraday and Cotton-Mouton effects when polarimetric effects in a tokamak are large," Plasma Phys. Controlled Fusion 48, 339-351 (2006).
[CrossRef]

Prog. Theor. Phys. (1)

R. Botet, H. Kuratsuji, and R. Seto, "Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media," Prog. Theor. Phys. 116, 285-294 (2006).
[CrossRef]

Sov. Phys. Dokl. (1)

Yu. A. Kravtsov, "Quasi-isotropic geometrical optics approximation," Sov. Phys. Dokl. 13, 1125-1127 (1969).

Sov. Phys. JETP (1)

Yu. A. Kravtsov and O. N. Naida, "Linear transformation of electromagnetic waves in three-dimensional inhomogeneous magneto-active plasma," Sov. Phys. JETP 44, 122-126 (1976).

Usp. Fiz. Nauk (2)

V. V. Zheleznyakov, V. V. Kocharovski, and Vl. V. Kocharovski, "Linear interaction of electromagnetic waves in inhomogeneous weakly anisotropic media," Usp. Fiz. Nauk 141, 257-282 (1983) V. V. Zheleznyakov, V. V. Kocharovski, and Vl. V. Kocharovski,[Sov. Phys. Usp. 26, 877-902 (1983)].
[CrossRef]

For a review, see S. I. Vinitskii, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii, "Topological phases in quantum mechanics and polarization optics," Usp. Fiz. Nauk 160(6), 1-49 (1990) S. I. Vinitsky, V. L. Debrov, V. M. Dubovik, B. L. Markovski, and Yu. P. Stepanovskii,[Sov. Phys. Usp. 33, 403-450 (1990)].
[CrossRef]

Vestn. Leningr. Univ., Ser. 4: Fiz., Khim. (1)

M. M. Popov, "Eigen-oscillations of multi-mirrors resonators," Vestn. Leningr. Univ., Ser. 4: Fiz., Khim. 22, 44-54 (1969) (in Russian). Derivation of the Popov's orthogonal coordinate system is also reproduced in Chap. 9.

Other (14)

V. M. Babich and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Springer-Verlag, 1990) [original Russian edition, V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wavelength Diffraction Problems: The Model Problem Method (Nauka, 1972)].

V. Cervený, Seismic Ray Theory (Cambridge U. Press, 2001).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

V. B. Berestetskii, E. M. Lifshits, and L. P. Pitaevskii, Quantum Electrodynamics (Pergamon, 1982).

A. I. Akhiezer, V. G. Baryakhtar, and S. V. Peletminskii, Spin Waves (North-Holland, 1968).

C. P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, 1989).

K. G. Budden, Radio Waves in the Ionosphere (Cambridge U. Press, 1961).

V. I. Ginzburg, Propagation of Electromagnetic Waves in Plasma (Gordon & Breach, 1970).

D. B. Melrose and R. C. McPhedran, Electromagnetic Processes in Dispersive Media (Cambridge U. Press, 1991).
[CrossRef]

A. A. Fuki, Yu. A. Kravtsov, and O. N. Naida, Geometrical Optics of Weakly Anisotropic Media (Gordon & Breach, 1997).

Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
[CrossRef]

Yu. A. Kravtsov, Geometrical Optics in Engineering Physics (Alpha Science, 2005).

G. N. Ramachandran and S. Ramaseshan, Crystal Optics, Vol. 25 of Encyclopedia of Physics (Springer-Verlag, 1961).

S. E. Segre, "Comparison between two alternative approaches for the analysis of polarization evolution of EM waves in a nonuniform, fully anisotropic medium: a magnetized plasma," Preprint RT/ERG/FUS/2001/13 (ENEA, 2001).

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Figures (3)

Fig. 1
Fig. 1

Coordinate frames locally attached to the ray: Popov’s (parallel transport) basis ( e 1 , e 2 , l ) versus Frenet (natural trihedral) basis ( n , b , l ) . The ray segment is shown by the dotted curve.

Fig. 2
Fig. 2

External magnetic field B 0 in the Popov’s coordinate frame. Longitudinal and transverse (with respect to the ray) components are given by B 0 = l ( B 0 l ) and B 0 = B 0 l ( B 0 l ) .

Fig. 3
Fig. 3

Schematic illustration of the peculiarities of the polarization evolution in weakly anisotropic inhomogeneous media: (a) noncommutativity, (b) conversion of modes, (c) nonreversibility.

Equations (80)

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μ G = 1 k 0 L = λ 0 2 π L 1 ,
ε ̂ = ε 0 I ̂ 3 + ν ̂ ,
μ A = max ν i j ε 0 1
curl curl E = k 0 2 ε ̂ E ,
μ = max ( μ G , μ A ) .
E = A Γ exp ( i k 0 Ψ i ω t ) .
( Ψ ) 2 = ε 0 ,
2 A Ψ + A Δ Ψ + k 0 ε 0 A = 0 .
r ̇ = l , l ̇ = ln n 0 ,
Γ = Γ 1 e 1 + Γ 2 e 2 ,
e 1 = n cos φ + b sin φ ,
e 2 = n sin φ + b cos φ .
Γ ̇ = i 2 J ̂ Γ , with J ̂ = k 0 n 0 [ ν 11 ν 12 i h ν 21 + i h ν 22 ] .
l ̇ = K n , n ̇ = K l + κ b , b ̇ = κ n .
φ = 0 σ κ d σ
e ̇ i = ( e i ln n 0 ) l ,
J ̂ = k 0 n 0 [ ν 11 ν 12 ν 21 ν 22 ] .
Γ Γ ̃ = V ̂ Γ , J ̂ J ̃ ̂ = V ̂ J ̂ V ̂ , with
V ̂ = 1 2 [ 1 1 i i ]
J ̃ ̂ = G α σ ̂ α ,
σ ̂ 0 = I ̂ 2 , σ ̂ 1 = [ 0 1 1 0 ] ,
σ ̂ 2 = [ 0 i i 0 ] , σ ̂ 3 = [ 1 0 0 1 ] ,
Γ ̃ ̇ = i 2 G α σ ̂ α Γ ̃ ,
G 0 = k 0 2 n 0 ( ν 11 + ν 22 ) , G 1 = k 0 2 n 0 ( ν 11 ν 22 ) ,
G 2 = k 0 2 n 0 ( ν 12 + ν 21 ) , G 3 = i k 0 2 n 0 ( ν 12 ν 21 ) .
S 0 = Γ 1 2 + Γ 2 2 , S 1 = Γ 1 2 Γ 2 2 , S 2 = 2 Re ( Γ 1 * Γ 2 ) ,
S 3 = 2 Im ( Γ 1 * Γ 2 ) .
S α = Γ ̃ σ ̂ α Γ ̃ .
S ̇ α = i 2 Γ ̃ ( G β σ ̂ α σ ̂ β G β * σ ̂ β σ ̂ α ) Γ ̃ = i 2 Γ ̃ ( [ σ ̂ α , σ ̂ β ] Re G β + i { σ ̂ α , σ ̂ β } Im G β ) Γ ̃ ,
S ̇ 0 = Im G β S β = Im G 0 S 0 + Im G S ,
S ̇ = Re G × S + Im G 0 S + Im G S 0 ,
S ̇ α = M ̂ α β S β ,
M ̂ = [ Im G 0 Im G 1 Im G 2 Im G 3 Im G 1 Im G 0 Re G 3 Re G 2 Im G 2 Re G 3 Im G 0 Re G 1 Im G 3 Re G 2 Re G 1 Im G 0 ] ,
M ̂ = M ̂ a + M ̂ d + M ̂ b .
M ̂ a = Im G 0 I ̂ 4 ,
M ̂ d = [ 0 Im G 1 Im G 2 Im G 3 Im G 1 0 0 0 Im G 2 0 0 0 Im G 3 0 0 0 ]
M ̂ b = [ 0 0 0 0 0 0 Re G 3 Re G 2 0 Re G 3 0 Re G 1 0 Re G 2 Re G 1 0 ]
S ̇ α = F α β S β + Im G 0 S α ,
F α β = [ 0 Im G 1 Im G 2 Im G 3 Im G 1 0 Re G 3 Re G 2 Im G 2 Re G 3 0 Re G 1 Im G 3 Re G 2 Re G 1 0 ] ( Im G , Re G )
s ̇ = ( Re G + s × Im G ) × s ,
s ̇ = Ω × s ,
s ( σ ) = R ̂ Ω ( Ω σ ) s ( 0 ) ,
[ R ̂ Ω ( Ω σ ) ] i j = cos ( Ω σ ) δ i j sin ( Ω σ ) ϵ i j k Ω k Ω + [ 1 cos ( Ω σ ) ] Ω i Ω j Ω 2
ε ̂ = [ ε x x ε x y 0 ε y x ε y y 0 0 0 ε z z ] .
ε x x = ε y y = 1 v ( 1 + i w ) ( 1 + i w ) 2 u , ε z z = 1 v 1 + i w ,
ε x y = ε y x = i v u ( 1 + i w ) 2 u ,
u = ( ω c ω ) 2 = ( e B 0 m c ω ) 2 , v = ( ω p ω ) 2 = 4 π e 2 N e m ω 2 , w = ν e f f ω
w 1 ,
v 1 or u 1 .
ε x x = ε y y 1 v 1 u + i v ( 1 + u ) ( 1 u ) 2 w = 1 V + i P w ,
ε z z 1 v + i v w ,
ε x y = ε y x i v u 1 u + 2 v u ( 1 u ) 2 w = i V u + R w ,
ε z z ε x x u v 1 u i u v ( 3 u ) ( 1 u ) 2 w = u V i Q w ,
P = v ( 1 + u ) ( 1 u ) 2 , Q = u v ( 3 u ) ( 1 u ) 2 , R = 2 v u ( 1 u ) 2 , V = v 1 u .
ε ̂ A ̂ ε ̂ A ̂ , A ̂ = [ cos θ cos ϕ cos θ sin ϕ sin θ sin ϕ cos ϕ 0 sin θ cos ϕ sin θ sin ϕ cos θ ] .
ε 11 = ε x x + ( ε z z ε x x ) sin 2 θ cos 2 ϕ ,
ε 12 = ε x y cos θ + ( ε z z ε x x ) sin 2 θ sin ϕ cos ϕ ,
ε 21 = ε x y cos θ + ( ε z z ε x x ) sin 2 θ sin ϕ cos ϕ ,
ε 22 = ε x x + ( ε z z ε x x ) sin 2 θ sin 2 ϕ .
ν 11 = u V sin 2 θ cos 2 ϕ + i ( P Q sin 2 θ cos 2 ϕ ) w ,
ν 12 = ( i u V + R w ) cos θ + ( u V i Q w ) sin 2 θ sin ϕ cos ϕ ,
ν 21 = ( i u V + R w ) cos θ + ( u V i Q w ) sin 2 θ sin ϕ cos ϕ ,
ν 22 = u V sin 2 θ sin 2 ϕ + i ( P Q sin 2 θ sin 2 ϕ ) w .
G 0 = k 0 2 n 0 [ u V sin 2 θ + i ( 2 P Q sin 2 θ ) w ] ,
G 1 = k 0 2 n 0 ( u V i Q w ) sin 2 θ cos 2 ϕ ,
G 2 = k 0 2 n 0 ( u V i Q w ) sin 2 θ sin 2 ϕ ,
G 3 = k 0 n 0 ( u V i R w ) cos θ ,
M ̂ a = k 0 ( 2 P Q sin 2 θ ) w 2 1 V I ̂ 4 ,
M ̂ d = k 0 w 2 1 V [ 0 Q sin 2 θ cos 2 ϕ Q sin 2 θ sin 2 ϕ 2 R cos θ Q sin 2 θ cos 2 ϕ 0 0 0 Q sin 2 θ sin 2 ϕ 0 0 0 2 R cos θ 0 0 0 ] ,
M ̂ b = k 0 V 2 1 V [ 0 0 0 0 0 0 2 u cos θ u sin 2 θ sin 2 ϕ 0 2 u cos θ 0 u sin 2 θ cos 2 ϕ 0 u sin 2 θ sin 2 ϕ u sin 2 θ cos 2 ϕ 0 ] .
Ω = k 0 V 2 1 V [ u sin 2 θ cos 2 ϕ u sin 2 θ sin 2 ϕ 2 u cos θ ] .
s ± = ± Ω Ω ,
T = 1 s out s in 2 , T [ 0 , 1 ] .
s out = R ̂ Ω II ( Ω II σ II ) R ̂ Ω I ( Ω I σ I ) s in .
s out = R ̂ Ω I ( Ω I σ I ) R ̂ Ω II ( Ω II σ II ) s in .
s out = R ̂ Ω I ( Ω I σ I ) R ̂ Ω II ( Ω II σ II ) R ̂ Ω I ( Ω I σ I ) s in .
Ω = k 0 V u 2 1 V [ u sin 2 θ 0 2 cos θ ] .
Ω I k 0 V u 2 1 V [ 0 0 2 cos θ ] , Ω II = k 0 V u 2 1 V [ u 0 0 ] .
T = 1 cos ( k 0 V u σ II 2 1 V ) 2 .
Ω I = k 0 V u 2 1 V [ u sin 2 θ 0 2 cos θ ] , Ω II = k 0 V u 2 1 V [ u sin 2 θ 0 2 cos θ ] ,

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