Abstract

Many practical modulator materials include combinations of electrooptically induced birefringence, optical activity, and/or Faraday rotation. Thus, there is a need for a procedure to design and analyze devices fabricated with materials exhibiting any or all of these effects. In this paper a simple procedure employing an extension of the general Jacobi method is introduced for determining the properties of the two allowed elliptical eigenpolarizations for an arbitrary direction of propagation and for the principal indices and axes of a general lossless, electrooptic, and gyrotropic medium. The procedure uses an iterative application of unitary transformations to diagonalize the Hermitian impermeability tensor. A complex polarization variable is defined from elements of the unitary transformation matrix to determine the ellipticity, azimuth angle, relative amplitude and phase, and handedness of the two orthogonal elliptical polarizations. The phase velocity indices of refraction are readily calculated with simple derived expressions. The procedure is numerically stable and accurate for any crystal class, external field direction, and direction of propagation.

© 1989 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. A. Yariv, J. F. Lotspeich, “Coupled-Mode Analysis of Light Propagation in Optically Active Crystals,” J. Opt. Soc. Am. 72, 273–277 (1982).
    [CrossRef]
  4. P. Van Den Keybus, W. Grevendonk, “Comparison of Optical Activity and Faraday Rotation in Crystalline SiO2,” Phys. Status Solidi B 136, 651–659 (1986).
    [CrossRef]
  5. M. P. Silverman, “Effects of Circular Birefringence on Light Propagation and Reflection,” Am. J. Phys. 54, 69–76 (1986).
    [CrossRef]
  6. S. Mallick, D. Rouede, A. G. Apostolidis, “Efficiency and Polarization Characteristics of Photorefractive Diffraction in a Bi12SiO20 Crystal,” J. Opt. Soc. Am. B 4, 1247–1259 (1987).
    [CrossRef]
  7. A. Marrakchi, R. V. Johnson, A. R. Tanguay, “Polarization Properties of Photorefractive Diffraction in Electrooptic and Optically Active Sillenite Crystals (Bragg Regime),” J. Opt. Soc. Am. B 3, 321–336 (1986).
    [CrossRef]
  8. O. S. Eritsyan, “Optical Problems in the Electrodynamics of Gyrotropic Media,” Sov. Phys. Usp. 25, 919–935 (1983).
    [CrossRef]
  9. E. U. Condon, “Theories of Optical Rotatory Power,” Rev. Mod. Phys. 9, 432–457 (1937).
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    [CrossRef]
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    [CrossRef]
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  13. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983).
  14. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960).
  15. J. F. Nye, Physical Properties of Crystals (Oxford U.P., London, 1957).
  16. V. M. Agranovich, V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer-Verlag, New York, 1984).
  17. A. V. Shubnikov, Principles of Optical Crystallography (Consultants Bureau, New York, 1960).
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  19. B. V. Bokut, A. N. Serdyukov, “On the Phenomenological Theory of Natural Optical Activity,” Sov. Phys. JETP 34, 962–964 (1972).
  20. A. Yariv, Optical Electronics (Holt, Rinehart, & Winston, New York, 1976).
  21. O. G. Vlokh, “Electrogyration Properties of Crystals,” Ferroelectrics 75, 119–137 (1987).
    [CrossRef]
  22. J. Kobayashi, T. Asahi, S. Takahashi, “Simultaneous Measurements of Electrogyration and Electrooptic Effects of α-quartz,” Ferroelectrics 75, 139–152 (1987).
    [CrossRef]
  23. F. Vachss, L. Hesselink, “Measurement of the Electrogyratory and Electro-Optic Effects in BSO and BGO,” Opt. Commun. 62, 159–165 (1987).
    [CrossRef]
  24. F. A. Hopf, G. I. Stegeman, Applied Classical Electrodynamics, Vol. 1: Linear Optics (Wiley, New York, 1985).
  25. M. J. Freiser, “A Survey of Magnetooptic Effects,” IEEE Trans. Magn. MAG-4, 152–161 (1968).
    [CrossRef]
  26. P. S. Pershan, “Magneto-Optical Effects,” J. Appl. Phys. 38, 1482–1490 (1967).
    [CrossRef]
  27. J. F. Dillon, “Magnetooptics and Its Uses,” J. Magn. and Magn. Mater. 31–34, 1–9 (1983).
    [CrossRef]
  28. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U.P., London, 1965).
  29. P. Hlawiczka, Gyrotropic Waveguides (Academic, New York, 1981).
  30. IMSL Library Reference Manual (International Mathematical & Statistical Libraries, Inc., Houston, TX, 1980), ed. 9.2.
  31. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science, New York, 1987).

1988 (2)

1987 (4)

O. G. Vlokh, “Electrogyration Properties of Crystals,” Ferroelectrics 75, 119–137 (1987).
[CrossRef]

J. Kobayashi, T. Asahi, S. Takahashi, “Simultaneous Measurements of Electrogyration and Electrooptic Effects of α-quartz,” Ferroelectrics 75, 139–152 (1987).
[CrossRef]

F. Vachss, L. Hesselink, “Measurement of the Electrogyratory and Electro-Optic Effects in BSO and BGO,” Opt. Commun. 62, 159–165 (1987).
[CrossRef]

S. Mallick, D. Rouede, A. G. Apostolidis, “Efficiency and Polarization Characteristics of Photorefractive Diffraction in a Bi12SiO20 Crystal,” J. Opt. Soc. Am. B 4, 1247–1259 (1987).
[CrossRef]

1986 (3)

A. Marrakchi, R. V. Johnson, A. R. Tanguay, “Polarization Properties of Photorefractive Diffraction in Electrooptic and Optically Active Sillenite Crystals (Bragg Regime),” J. Opt. Soc. Am. B 3, 321–336 (1986).
[CrossRef]

P. Van Den Keybus, W. Grevendonk, “Comparison of Optical Activity and Faraday Rotation in Crystalline SiO2,” Phys. Status Solidi B 136, 651–659 (1986).
[CrossRef]

M. P. Silverman, “Effects of Circular Birefringence on Light Propagation and Reflection,” Am. J. Phys. 54, 69–76 (1986).
[CrossRef]

1983 (2)

O. S. Eritsyan, “Optical Problems in the Electrodynamics of Gyrotropic Media,” Sov. Phys. Usp. 25, 919–935 (1983).
[CrossRef]

J. F. Dillon, “Magnetooptics and Its Uses,” J. Magn. and Magn. Mater. 31–34, 1–9 (1983).
[CrossRef]

1982 (1)

1973 (2)

A. Yariv, “Coupled-Mode Theory for Guided-Wave Optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

V. M. Agranovich, V. L. Ginzburg, “Phenomenological Electrodynamics of Gyrotropic Media,” Sov. Phys. JETP 36, 440–443 (1973).

1972 (1)

B. V. Bokut, A. N. Serdyukov, “On the Phenomenological Theory of Natural Optical Activity,” Sov. Phys. JETP 34, 962–964 (1972).

1968 (2)

R. M. Hornreich, S. Shtrikman, “Theory of Gyrotropic Birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

M. J. Freiser, “A Survey of Magnetooptic Effects,” IEEE Trans. Magn. MAG-4, 152–161 (1968).
[CrossRef]

1967 (1)

P. S. Pershan, “Magneto-Optical Effects,” J. Appl. Phys. 38, 1482–1490 (1967).
[CrossRef]

1937 (1)

E. U. Condon, “Theories of Optical Rotatory Power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich, V. L. Ginzburg, “Phenomenological Electrodynamics of Gyrotropic Media,” Sov. Phys. JETP 36, 440–443 (1973).

V. M. Agranovich, V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer-Verlag, New York, 1984).

Apostolidis, A. G.

Asahi, T.

J. Kobayashi, T. Asahi, S. Takahashi, “Simultaneous Measurements of Electrogyration and Electrooptic Effects of α-quartz,” Ferroelectrics 75, 139–152 (1987).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science, New York, 1987).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science, New York, 1987).

Bokut, B. V.

B. V. Bokut, A. N. Serdyukov, “On the Phenomenological Theory of Natural Optical Activity,” Sov. Phys. JETP 34, 962–964 (1972).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Condon, E. U.

E. U. Condon, “Theories of Optical Rotatory Power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Dillon, J. F.

J. F. Dillon, “Magnetooptics and Its Uses,” J. Magn. and Magn. Mater. 31–34, 1–9 (1983).
[CrossRef]

Eimerl, D.

Eritsyan, O. S.

O. S. Eritsyan, “Optical Problems in the Electrodynamics of Gyrotropic Media,” Sov. Phys. Usp. 25, 919–935 (1983).
[CrossRef]

Freiser, M. J.

M. J. Freiser, “A Survey of Magnetooptic Effects,” IEEE Trans. Magn. MAG-4, 152–161 (1968).
[CrossRef]

Gaylord, T. K.

Ginzburg, V. L.

V. M. Agranovich, V. L. Ginzburg, “Phenomenological Electrodynamics of Gyrotropic Media,” Sov. Phys. JETP 36, 440–443 (1973).

V. M. Agranovich, V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer-Verlag, New York, 1984).

Grevendonk, W.

P. Van Den Keybus, W. Grevendonk, “Comparison of Optical Activity and Faraday Rotation in Crystalline SiO2,” Phys. Status Solidi B 136, 651–659 (1986).
[CrossRef]

Hesselink, L.

F. Vachss, L. Hesselink, “Measurement of the Electrogyratory and Electro-Optic Effects in BSO and BGO,” Opt. Commun. 62, 159–165 (1987).
[CrossRef]

Hlawiczka, P.

P. Hlawiczka, Gyrotropic Waveguides (Academic, New York, 1981).

Hopf, F. A.

F. A. Hopf, G. I. Stegeman, Applied Classical Electrodynamics, Vol. 1: Linear Optics (Wiley, New York, 1985).

Hornreich, R. M.

R. M. Hornreich, S. Shtrikman, “Theory of Gyrotropic Birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

Johnson, R. V.

Kobayashi, J.

J. Kobayashi, T. Asahi, S. Takahashi, “Simultaneous Measurements of Electrogyration and Electrooptic Effects of α-quartz,” Ferroelectrics 75, 139–152 (1987).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960).

Lotspeich, J. F.

Maldonado, T. A.

Mallick, S.

Marrakchi, A.

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford U.P., London, 1957).

Pershan, P. S.

P. S. Pershan, “Magneto-Optical Effects,” J. Appl. Phys. 38, 1482–1490 (1967).
[CrossRef]

Rouede, D.

Serdyukov, A. N.

B. V. Bokut, A. N. Serdyukov, “On the Phenomenological Theory of Natural Optical Activity,” Sov. Phys. JETP 34, 962–964 (1972).

Shtrikman, S.

R. M. Hornreich, S. Shtrikman, “Theory of Gyrotropic Birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

Shubnikov, A. V.

A. V. Shubnikov, Principles of Optical Crystallography (Consultants Bureau, New York, 1960).

Silverman, M. P.

M. P. Silverman, “Effects of Circular Birefringence on Light Propagation and Reflection,” Am. J. Phys. 54, 69–76 (1986).
[CrossRef]

Stegeman, G. I.

F. A. Hopf, G. I. Stegeman, Applied Classical Electrodynamics, Vol. 1: Linear Optics (Wiley, New York, 1985).

Takahashi, S.

J. Kobayashi, T. Asahi, S. Takahashi, “Simultaneous Measurements of Electrogyration and Electrooptic Effects of α-quartz,” Ferroelectrics 75, 139–152 (1987).
[CrossRef]

Tanguay, A. R.

Vachss, F.

F. Vachss, L. Hesselink, “Measurement of the Electrogyratory and Electro-Optic Effects in BSO and BGO,” Opt. Commun. 62, 159–165 (1987).
[CrossRef]

Van Den Keybus, P.

P. Van Den Keybus, W. Grevendonk, “Comparison of Optical Activity and Faraday Rotation in Crystalline SiO2,” Phys. Status Solidi B 136, 651–659 (1986).
[CrossRef]

Vlokh, O. G.

O. G. Vlokh, “Electrogyration Properties of Crystals,” Ferroelectrics 75, 119–137 (1987).
[CrossRef]

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U.P., London, 1965).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Yariv, A.

A. Yariv, J. F. Lotspeich, “Coupled-Mode Analysis of Light Propagation in Optically Active Crystals,” J. Opt. Soc. Am. 72, 273–277 (1982).
[CrossRef]

A. Yariv, “Coupled-Mode Theory for Guided-Wave Optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983).

A. Yariv, Optical Electronics (Holt, Rinehart, & Winston, New York, 1976).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983).

Am. J. Phys. (1)

M. P. Silverman, “Effects of Circular Birefringence on Light Propagation and Reflection,” Am. J. Phys. 54, 69–76 (1986).
[CrossRef]

Appl. Opt. (1)

Ferroelectrics (2)

O. G. Vlokh, “Electrogyration Properties of Crystals,” Ferroelectrics 75, 119–137 (1987).
[CrossRef]

J. Kobayashi, T. Asahi, S. Takahashi, “Simultaneous Measurements of Electrogyration and Electrooptic Effects of α-quartz,” Ferroelectrics 75, 139–152 (1987).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Yariv, “Coupled-Mode Theory for Guided-Wave Optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

IEEE Trans. Magn. (1)

M. J. Freiser, “A Survey of Magnetooptic Effects,” IEEE Trans. Magn. MAG-4, 152–161 (1968).
[CrossRef]

J. Appl. Phys. (1)

P. S. Pershan, “Magneto-Optical Effects,” J. Appl. Phys. 38, 1482–1490 (1967).
[CrossRef]

J. Magn. and Magn. Mater. (1)

J. F. Dillon, “Magnetooptics and Its Uses,” J. Magn. and Magn. Mater. 31–34, 1–9 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

F. Vachss, L. Hesselink, “Measurement of the Electrogyratory and Electro-Optic Effects in BSO and BGO,” Opt. Commun. 62, 159–165 (1987).
[CrossRef]

Phys. Rev. (1)

R. M. Hornreich, S. Shtrikman, “Theory of Gyrotropic Birefringence,” Phys. Rev. 171, 1065–1074 (1968).
[CrossRef]

Phys. Status Solidi B (1)

P. Van Den Keybus, W. Grevendonk, “Comparison of Optical Activity and Faraday Rotation in Crystalline SiO2,” Phys. Status Solidi B 136, 651–659 (1986).
[CrossRef]

Rev. Mod. Phys. (1)

E. U. Condon, “Theories of Optical Rotatory Power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Sov. Phys. JETP (2)

V. M. Agranovich, V. L. Ginzburg, “Phenomenological Electrodynamics of Gyrotropic Media,” Sov. Phys. JETP 36, 440–443 (1973).

B. V. Bokut, A. N. Serdyukov, “On the Phenomenological Theory of Natural Optical Activity,” Sov. Phys. JETP 34, 962–964 (1972).

Sov. Phys. Usp. (1)

O. S. Eritsyan, “Optical Problems in the Electrodynamics of Gyrotropic Media,” Sov. Phys. Usp. 25, 919–935 (1983).
[CrossRef]

Other (12)

A. Yariv, Optical Electronics (Holt, Rinehart, & Winston, New York, 1976).

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U.P., London, 1965).

P. Hlawiczka, Gyrotropic Waveguides (Academic, New York, 1981).

IMSL Library Reference Manual (International Mathematical & Statistical Libraries, Inc., Houston, TX, 1980), ed. 9.2.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (Elsevier Science, New York, 1987).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983).

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960).

J. F. Nye, Physical Properties of Crystals (Oxford U.P., London, 1957).

V. M. Agranovich, V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer-Verlag, New York, 1984).

A. V. Shubnikov, Principles of Optical Crystallography (Consultants Bureau, New York, 1960).

F. A. Hopf, G. I. Stegeman, Applied Classical Electrodynamics, Vol. 1: Linear Optics (Wiley, New York, 1985).

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Figures (10)

Fig. 1
Fig. 1

Sense of optical rotation relative to the direction of propagation k for (a) natural optical activity and (b) Faraday rotation.

Fig. 2
Fig. 2

Index ellipsoid cross section (crosshatched) that is normal to the wavevector k and passes through the origin. The principal axes of the crosshatched ellipse represent the directions of the allowed linear polarizations D1 and D2. D1, D2, and k form an orthogonal triad.

Fig. 3
Fig. 3

Gyration surface for right-handed quartz (class 32). The white surface depicts right-handed optical rotation with the maximum rotation occurring for propagation along the optic axis. The dark surface depicts left-handed rotation with maximum rotation along a direction perpendicular to the optic axis. There is no optical rotation for propagation ~56° from the optic axis.

Fig. 4
Fig. 4

Gyration surface for Faraday active crystals. Maximum optical rotation occurs for propagation parallel and antiparallel to B. The white surface depicts rotation of one sense while the dark surface depicts rotation of the opposite sense.

Fig. 5
Fig. 5

Flattened helical contour of an elliptically polarized propagating wave at an instant of time. The radial vectors to the contour represent the displacement vector D (⊥k).

Fig. 6
Fig. 6

Orthogonal transformation of the (x,y,z) dielectric axes to the (x″,y″,z″) coordinate system of the wavevector k (z″ ∥ k) represented in polar coordinates (ϕk,θk).

Fig. 7
Fig. 7

Cross-sectional ellipse of the index ellipsoid in the (x″,y″) plane. The x‴ and y‴ axes represent the major axes of the eigenpolarizations oriented relative to x″ and y″. The two eigenstates have orthogonal major axes, opposite handedness, and the same ellipticity. The wavevector k and the z″ axis are normal to the plane of the figure.

Fig. 8
Fig. 8

Cartesian complex plane of polarization. Each point in the plane represents a polarization state. The basis states are the horizontal linear polarization at the origin and the vertical linear polarization at infinity. The dashed circle represents the unit circle (unit relative amplitude). The radial line represents a contour of constant relative phase of π/4.

Fig. 9
Fig. 9

Principal transverse crystal orientation of Bi12SiO20 (BSO). The external electric field is applied in the [ 1 ¯ 1 ¯ 0 ] direction, and the direction of propagation is along [ 1 ¯ 1 0 ]. The (x,y,z) coordinate system represents the unperturbed dielectric axes. The new coordinate system resulting from the electrooptic effect is represented by (x′,y′,z′). The coordinate system of the wavevector k is given by (x″,y″,z″) with z″ ∥ k. Finally, the (x‴,y‴,z‴) coordinate system with z‴ || z″ || k represents the principal axis coordinate system of the eigenstates for the given k.

Fig. 10
Fig. 10

Gyration surface for BSO in the (x,z) or (y,z) principal plane. The dashed circle represents the surface projection with no applied electric field, and therefore, the optical rotation is invariant with the wavevector k direction. The solid heart-shape contour depicts the surface when a field is applied in the [ 1 ¯ 1 ¯ 0 ] direction. (This contour depicts actual calculations using Eq. (36) with g11 = 2, ζ 41 = 1 / 2, and E = 1.) The magnitude of radial vectors from the origin to the surface is a measure of the optical rotation per unit length which depends on the direction of k. Note that optical rotation is not affected by the electric field for propagation in the (x,y) plane.

Tables (1)

Tables Icon

Table I Gyration Tensors gij for All Crystal Classes Exhibiting Natural Optical Activity13

Equations (52)

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[ D x D y D z ] = [ x 0 0 0 y 0 0 0 z ] [ E x E y E z ] ,
D = [ ] E + i 0 G × E = [ ] E ,
D = { [ ] + i 0 [ G ] } E = [ ] E .
E = [ ] D = [ η ] D = ( 1 / 0 ) { [ η ] i [ η ] [ G ] [ η ] } D .
Im [ η ] = [ η ] [ G ] [ η ] = [ 0 η x η y G z η x η z G y η x η y G z 0 η y η z G x η x η z G y η y η z G x 0 ] .
D i = i j E j + γ i j l ( E j / x l ) = i j E j ,
i j = i j ( ω , k ) = i j ( ω ) + i γ i j l ( ω ) k l ,
G x = G sin θ k cos ϕ k , G y = G sin θ k sin ϕ k , G z = G cos θ k , | G | = G = g 11 sin 2 θ k cos 2 ϕ k + g 22 sin 2 θ k sin 2 ϕ k + g 33 cos 2 θ k + 2 g 12 sin 2 θ k sin ϕ k cos ϕ k + 2 g 13 sin θ k cos θ k cos ϕ k + 2 g 23 sin θ k cos θ k sin ϕ k ,
Δ ( 1 / n 2 ) i = j r i j E j i = 1 , , 6 , j = x , y , z = 1 , 2 , 3 ,
[ Δ ( 1 / n 2 ) 1 Δ ( 1 / n 2 ) 2 Δ ( 1 / n 2 ) 3 Δ ( 1 / n 2 ) 4 Δ ( 1 / n 2 ) 5 Δ ( 1 / n 2 ) 6 ] = [ r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 r 41 r 42 r 43 r 51 r 52 r 53 r 61 r 62 r 63 ] [ E x E y E z ] .
[ 1 / n 2 ] = [ 1 / n 1 2 + Δ ( 1 / n 2 ) 1 Δ ( 1 / n 2 ) 6 Δ ( 1 / n 2 ) 5 Δ ( 1 / n 2 ) 6 1 / n 2 2 + Δ ( 1 / n 2 ) 2 Δ ( 1 / n 2 ) 4 Δ ( 1 / n 2 ) 5 Δ ( 1 / n 2 ) 4 1 / n 3 2 + Δ ( 1 / n 2 ) 3 ] .
g i j = g i j + ζ i j k E k ,
D = [ ] E + i 0 ψ B × E ,
G x = ψ B sin θ B cos ϕ B , G y = ψ B sin θ B sin ϕ B , G z = ψ B cos θ B .
[ 1 / n 2 ] = [ η ] = [ η x x η x y η x z η x y * η y y η y z η x z * η y z * η z z ] ,
( x 2 / n x 2 ) + ( y 2 / n y 2 ) + ( z 2 / n z 2 ) = 1 ,
G = g i j k i k j ,
[ a ] = [ cos Φ exp ( i B ) sin Φ exp ( i B ) sin Φ cos Φ ] .
x = x cos θ k cos ϕ k + y cos θ k sin ϕ k z sin θ k , y = x sin ϕ k + y cos ϕ k , z = x sin θ k cos ϕ k + y sin θ k sin ϕ k + z cos θ k .
[ η ] = [ η x x η x y η x y * η z z ] ,
η x x = ( η x x cos 2 ϕ k + η y y sin 2 ϕ k ) cos 2 θ k + η z z sin 2 θ k + 2 η x y r cos 2 θ k cos ϕ k sin ϕ k 2 cos θ k sin θ k ( η x z r cos ϕ k + η y z r sin ϕ k ) , η y y = η x x sin 2 ϕ k + η y y cos 2 ϕ k 2 η x y r cos ϕ k sin ϕ k , η x y = ( η y y η x x ) cos θ k cos ϕ k sin ϕ k + cos θ k ( η x y cos 2 ϕ k η x y * sin 2 ϕ k ) + sin θ k ( η x z * sin ϕ k η y z * cos ϕ k ) = η y x * ,
Φ = 1 2 tan 1 { 2 sgn ( η x y r ) | η x y | / ( η x x η y y ) } ,
n 1 = { η x x cos 2 Φ + η y y sin 2 Φ + 2 sgn ( η x y r ) | η x y | cos Φ sin Φ } 1 / 2 , n 2 = { η x x sin 2 Φ + η y y cos 2 Φ 2 sgn ( η x y r ) | η x y | cos Φ sin Φ } 1 / 2 .
D 1 = [ cos Φ exp ( i B ) sin Φ ] H , D 2 = [ exp ( i B ) sin Φ cos Φ ] H ,
D 1 = [ cos Φ exp ( i B ) sin Φ ] , D 2 = [ exp ( i B ) sin Φ cos Φ ] .
η x x x 2 + η y y y 2 + 2 η x y r x y = 1 ,
β 1 = 1 2 tan 1 [ 2 η x y r / ( η x x η y y ) ] .
D i = [ D x D y ] i = [ | D x | exp ( i δ x ) | D y | exp ( i δ y ) ] i .
χ 1 = exp ( i B ) tan Φ ,
χ 2 = 1 / χ 1 * = exp ( i B ) cot Φ .
tan 2 β 1 = 2 Re ( χ 1 ) / ( 1 | χ 1 | 2 ) , sin 2 ξ 1 = 2 Im ( χ 1 ) / ( 1 + | χ 1 | 2 ) .
ξ 1 = 1 2 sin 1 ( sin B sin 2 Φ ) .
β 2 = β 1 + π / 2 , ξ 2 = ξ 1 , r 2 = cot Φ , Δ δ 2 = π + Δδ 1 .
[ 1 / n 2 ] = [ 1 / n 0 2 0 0 0 1 / n 0 2 0 0 0 1 / n 0 2 ] ,
[ g ] = [ g 11 0 ζ 41 E y 0 g 11 ζ 41 E x ζ 41 E y ζ 41 E x g 11 ] ,
G = g 11 2 ζ 41 E sin θ k cos θ k ( sin ϕ k + cos ϕ k ) ,
[ η ] = [ 1 / n 0 2 0 1 / 2 { r 41 E i g 11 / n 0 4 } 0 1 / n 0 2 1 / 2 { r 41 E i g 11 / n 0 4 } 1 / 2 { r 41 E + i g 11 / n 0 4 } 1 / 2 { r 41 E i g 11 / n 0 4 } 1 / n 0 2 ] , c
n x = 2 . 52996 x = [ 1 / 2 1 / 2 1 / 2 ] T n y = 2 . 53 y = [ 1 / 2 1 / 2 0 ] T ( y k ) . n z = 2 . 53004 z = [ 1 / 2 1 / 2 1 / 2 ] T
[ η ] = [ 1 / n 0 2 ( r 41 E + i g 11 / n 0 4 ) ( r 41 E i g 11 / n 0 4 ) 1 / n 0 2 ] ,
1 / n 0 2 ( x 2 + y 2 ) 2 r 41 E x y = 1 .
Φ = 1 2 tan 1 { 2 sgn ( η x y r ) | η x y | / ( 1 / n 0 2 1 / n 0 2 ) } = 45 ° ,
n 1 = { 1 / n 0 2 + | η x y | } 1 / 2 = 2 . 52785 ( fast wave ) , n 2 = { 1 / n 0 2 | η x y | } 1 / 2 = 2 . 53216 ( slow wave ) .
D 1 = [ 1 / 2 ( 0 . 01656 + i 0 . 99986 ) 1 / 2 ] , D 2 = [ ( 0 . 01656 i 0 . 99986 ) 1 / 2 1 / 2 ] ,
ϕ = 1 2 tan 1 [ 2 | H 12 | / ( H 11 H 22 ) ] .
[ a ] ϕ = [ cos ϕ exp ( i A ) sin ϕ 0 exp ( i A ) sin ϕ cos ϕ 0 0 0 1 ] ,
H ϕ 11 = H 11 cos 2 ϕ + H 22 sin 2 ϕ + 2 | H 12 | cos ϕ sin ϕ , H ϕ 22 = H 11 sin 2 ϕ + H 22 cos 2 ϕ 2 | H 12 | cos ϕ sin ϕ , H ϕ 33 = H 33 , H ϕ 12 = ( H 22 H 11 ) ( H 12 / | H 12 | ) cos ϕ sin ϕ + H 12 ( cos 2 ϕ sin 2 ϕ ) = H 21 * = 0 , H ϕ 13 = H 13 cos ϕ + H 23 ( H 12 / | H 12 | ) sin ϕ = H 31 * , H ϕ 23 = H 13 ( H 12 * / | H 12 | ) sin ϕ + H 23 cos ϕ = H 32 * . }
θ = 1 2 tan 1 [ 2 | H 13 | / ( H 11 H 33 ) ] .
[ a ] θ = [ cos θ 0 exp ( i B ) sin θ 0 1 0 exp ( i B ) sin θ 0 cos θ ] ,
H θ 11 = H 11 cos 2 θ + H 33 sin 2 θ + 2 | H 13 | cos θ sin θ , H θ 22 = H 22 , H θ 33 = H 11 sin 2 θ + H 33 cos 2 θ 2 | H 13 | cos θ sin θ , H θ 12 = H 12 cos θ + H 23 * ( H 13 / | H 13 | ) sin θ = H 21 * , H θ 13 = ( H 33 H 11 ) ( H 13 / | H 13 | ) cos θ sin θ + H 13 ( cos 2 θ sin 2 θ ) = H 31 * = 0 , H θ 23 = H 12 * ( H 13 / | H 13 | ) sin θ + H 23 cos θ = H 32 * . }
ψ = 1 2 tan 1 [ 2 | H 23 | / ( H 22 H 33 ) ] .
[ a ] ψ = [ 1 0 0 0 cos ψ exp ( i C ) sin ψ 0 exp ( i C ) sin ψ cos ψ ] ,
H ψ 11 = H 11 , H ψ 22 = H 22 cos 2 ψ + H 33 sin 2 ψ + 2 | H 23 | cos ψ sin ψ , H ψ 33 = H 22 sin 2 ψ + H 33 cos 2 ψ 2 | H 23 | cos ψ sin ψ , H ψ 12 = H 12 cos ψ + H 13 ( H 23 * / | H 23 | ) sin ψ = H 21 * , H ψ 13 = H 12 ( H 23 / | H 23 | ) sin ψ + H 13 cos ψ = H 31 * , H ψ 23 = ( H 33 H 22 ) ( H 23 / | H 23 | ) cos ψ sin ψ + H 23 ( cos 2 ψ sin 2 ψ ) = H 32 * = 0 . }

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