Abstract

The dielectric tensor of an anisotropic crystal with multiple perturbations is presented to include the effects of multiple perturbations. To study electromagnetic wave propagation in anisotropic crystals subject to various influences the perturbed dielectric tensor is substituted into Maxwell’s equation. Then, a 2 × 2 transmission matrix formalism, based on a normal-mode approach, is extended to anisotropic crystals possessing multiple birefringences to develop compensation schemes for ac optical sensors employing the crystal. It is shown that a new compensation method utilizing two analyzers can eliminate the effects of both unwanted linear birefringences and unwanted circular birefringences on the stability of the ac bulk polarimetric optical sensor. The conditions (here referred to as the quenching condition) in which the compensation method becomes important are also derived for both the voltage (or electric field) and current (or magnetic field) sensors.

© 1989 Optical Society of America

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References

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  1. G. A. Massey, D. C. Erickson, R. A. Kadlec, “Electromagnetic Field Components: Their Measurement Using Linear Electrooptic and Magnetooptic Effects,” Appl. Opt. 14, 2712–2719 (1975).
    [CrossRef] [PubMed]
  2. K. Kyuma, S. Tai, M. Nunoshita, N. Mikai, Y. Ida, “Fiber-Optic Current and Voltage Sensors Using a Bi12GeO20 Single Crystal,” IEEE/OSA J. Lightwave Technol. LT-1, 93–97 (1983).
    [CrossRef]
  3. K. Shibata, “A Fiber Optic Electric Field Sensor Using the Electro-Optic Effect of Bi4Ge3O12,” Fusi Electric Corporate Research and Development, Japan.
  4. Y. Kuroda, Y. Abe, H. Kuwahara, K. Yoshinaga, “Field Test of Fiber-Optic Voltage and Current Sensors Applied to Gas Insulated Substation,” Proc. Soc. Photo-Opt. Instrum. Eng. 586, 30–37 (1985).
  5. D. P. Bortfeld, H. Meier, “Refractive Indices and Electro-Optic Coefficients of the Eulities Bi4Ge3O12 and Bi4Si3O12,” J. Appl. Phys. 43, 5110–5111 (1972).
    [CrossRef]
  6. K. S. Lee, G. W. Day, “The Effect of the Temperature and Pressure Dependent Birefringence on a Voltage Sensor and, the Elimination of its Effect on a Voltage Sensor Using a Bulk-Type Bismuth Germanate: Experiment,” to be submitted for publication.
  7. W. J. Tabor, F. S. Chen, “Electromagnetic Propagation Through Materials Possessing Both Faraday Rotation and Birefringence: Experiments with Ytterbium Orthoferrite,” J. Appl. Phys. 40, 2760–2765 (1969).
    [CrossRef]
  8. P. S. Pershan, “Magneto-Optical Effects,” J. Appl. Phys. 38, 1482–1490 (1967).
    [CrossRef]
  9. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960), pp. 251–253.
  10. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  11. J. Sakai, T. Kimura, “Polarization Behavior in Multiply Perturbed Single-Mode Fibers,” IEEE J. Quantum Electron. QE-18, 59–65 (1982).
    [CrossRef]
  12. N. Imoto, N. Yoshizawa, J. Sakai, H. Tsuchiya, “Birefringence in Single-Mode Optical Fiber due to Elliptical Core Deformation and State Anisotropy,” IEEE J. Quantum Electron. QE-16, 1267–1271 (1980).
    [CrossRef]
  13. C. S. Namba, “Electro-Optical Effect of Zincblende,” J. Opt. Soc. Am. 51, 76–79 (1961).
    [CrossRef]
  14. T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, New York, 1981).
    [CrossRef]
  15. R. Ulrich, A. Simon, “Polarization Optics of Twisted Single-Mode Fibers,” Appl. Opt. 18, 2241–2251 (1979).
    [CrossRef] [PubMed]
  16. W. R. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962).
  17. K. S. Lee, D. Conrad, G. W. Day, P. D. Hale, “Measurement of Optical, Electro-Optical, and Photoelastic Properties of Crystalline Bi4Ge3O12,” Appl. Opt., submitted for publication.
  18. F. Bucholtz, K. P. Koo, A. D. Kersey, A. Dandridge, “Fiber-Optic Magnetic Sensor Development,” Proc. Soc. Photo-Opt. Instrum. Eng. 718, 56–65 (1986).
  19. A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, NJ, 1976).

1986

F. Bucholtz, K. P. Koo, A. D. Kersey, A. Dandridge, “Fiber-Optic Magnetic Sensor Development,” Proc. Soc. Photo-Opt. Instrum. Eng. 718, 56–65 (1986).

1985

Y. Kuroda, Y. Abe, H. Kuwahara, K. Yoshinaga, “Field Test of Fiber-Optic Voltage and Current Sensors Applied to Gas Insulated Substation,” Proc. Soc. Photo-Opt. Instrum. Eng. 586, 30–37 (1985).

1983

K. Kyuma, S. Tai, M. Nunoshita, N. Mikai, Y. Ida, “Fiber-Optic Current and Voltage Sensors Using a Bi12GeO20 Single Crystal,” IEEE/OSA J. Lightwave Technol. LT-1, 93–97 (1983).
[CrossRef]

1982

J. Sakai, T. Kimura, “Polarization Behavior in Multiply Perturbed Single-Mode Fibers,” IEEE J. Quantum Electron. QE-18, 59–65 (1982).
[CrossRef]

1980

N. Imoto, N. Yoshizawa, J. Sakai, H. Tsuchiya, “Birefringence in Single-Mode Optical Fiber due to Elliptical Core Deformation and State Anisotropy,” IEEE J. Quantum Electron. QE-16, 1267–1271 (1980).
[CrossRef]

1979

1975

1972

D. P. Bortfeld, H. Meier, “Refractive Indices and Electro-Optic Coefficients of the Eulities Bi4Ge3O12 and Bi4Si3O12,” J. Appl. Phys. 43, 5110–5111 (1972).
[CrossRef]

1969

W. J. Tabor, F. S. Chen, “Electromagnetic Propagation Through Materials Possessing Both Faraday Rotation and Birefringence: Experiments with Ytterbium Orthoferrite,” J. Appl. Phys. 40, 2760–2765 (1969).
[CrossRef]

1967

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

1961

Abe, Y.

Y. Kuroda, Y. Abe, H. Kuwahara, K. Yoshinaga, “Field Test of Fiber-Optic Voltage and Current Sensors Applied to Gas Insulated Substation,” Proc. Soc. Photo-Opt. Instrum. Eng. 586, 30–37 (1985).

Bortfeld, D. P.

D. P. Bortfeld, H. Meier, “Refractive Indices and Electro-Optic Coefficients of the Eulities Bi4Ge3O12 and Bi4Si3O12,” J. Appl. Phys. 43, 5110–5111 (1972).
[CrossRef]

Bucholtz, F.

F. Bucholtz, K. P. Koo, A. D. Kersey, A. Dandridge, “Fiber-Optic Magnetic Sensor Development,” Proc. Soc. Photo-Opt. Instrum. Eng. 718, 56–65 (1986).

Chen, F. S.

W. J. Tabor, F. S. Chen, “Electromagnetic Propagation Through Materials Possessing Both Faraday Rotation and Birefringence: Experiments with Ytterbium Orthoferrite,” J. Appl. Phys. 40, 2760–2765 (1969).
[CrossRef]

Conrad, D.

K. S. Lee, D. Conrad, G. W. Day, P. D. Hale, “Measurement of Optical, Electro-Optical, and Photoelastic Properties of Crystalline Bi4Ge3O12,” Appl. Opt., submitted for publication.

Dandridge, A.

F. Bucholtz, K. P. Koo, A. D. Kersey, A. Dandridge, “Fiber-Optic Magnetic Sensor Development,” Proc. Soc. Photo-Opt. Instrum. Eng. 718, 56–65 (1986).

Day, G. W.

K. S. Lee, D. Conrad, G. W. Day, P. D. Hale, “Measurement of Optical, Electro-Optical, and Photoelastic Properties of Crystalline Bi4Ge3O12,” Appl. Opt., submitted for publication.

K. S. Lee, G. W. Day, “The Effect of the Temperature and Pressure Dependent Birefringence on a Voltage Sensor and, the Elimination of its Effect on a Voltage Sensor Using a Bulk-Type Bismuth Germanate: Experiment,” to be submitted for publication.

Erickson, D. C.

Hale, P. D.

K. S. Lee, D. Conrad, G. W. Day, P. D. Hale, “Measurement of Optical, Electro-Optical, and Photoelastic Properties of Crystalline Bi4Ge3O12,” Appl. Opt., submitted for publication.

Ida, Y.

K. Kyuma, S. Tai, M. Nunoshita, N. Mikai, Y. Ida, “Fiber-Optic Current and Voltage Sensors Using a Bi12GeO20 Single Crystal,” IEEE/OSA J. Lightwave Technol. LT-1, 93–97 (1983).
[CrossRef]

Imoto, N.

N. Imoto, N. Yoshizawa, J. Sakai, H. Tsuchiya, “Birefringence in Single-Mode Optical Fiber due to Elliptical Core Deformation and State Anisotropy,” IEEE J. Quantum Electron. QE-16, 1267–1271 (1980).
[CrossRef]

Kadlec, R. A.

Kersey, A. D.

F. Bucholtz, K. P. Koo, A. D. Kersey, A. Dandridge, “Fiber-Optic Magnetic Sensor Development,” Proc. Soc. Photo-Opt. Instrum. Eng. 718, 56–65 (1986).

Kimura, T.

J. Sakai, T. Kimura, “Polarization Behavior in Multiply Perturbed Single-Mode Fibers,” IEEE J. Quantum Electron. QE-18, 59–65 (1982).
[CrossRef]

Koo, K. P.

F. Bucholtz, K. P. Koo, A. D. Kersey, A. Dandridge, “Fiber-Optic Magnetic Sensor Development,” Proc. Soc. Photo-Opt. Instrum. Eng. 718, 56–65 (1986).

Kuroda, Y.

Y. Kuroda, Y. Abe, H. Kuwahara, K. Yoshinaga, “Field Test of Fiber-Optic Voltage and Current Sensors Applied to Gas Insulated Substation,” Proc. Soc. Photo-Opt. Instrum. Eng. 586, 30–37 (1985).

Kuwahara, H.

Y. Kuroda, Y. Abe, H. Kuwahara, K. Yoshinaga, “Field Test of Fiber-Optic Voltage and Current Sensors Applied to Gas Insulated Substation,” Proc. Soc. Photo-Opt. Instrum. Eng. 586, 30–37 (1985).

Kyuma, K.

K. Kyuma, S. Tai, M. Nunoshita, N. Mikai, Y. Ida, “Fiber-Optic Current and Voltage Sensors Using a Bi12GeO20 Single Crystal,” IEEE/OSA J. Lightwave Technol. LT-1, 93–97 (1983).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960), pp. 251–253.

Lee, K. S.

K. S. Lee, D. Conrad, G. W. Day, P. D. Hale, “Measurement of Optical, Electro-Optical, and Photoelastic Properties of Crystalline Bi4Ge3O12,” Appl. Opt., submitted for publication.

K. S. Lee, G. W. Day, “The Effect of the Temperature and Pressure Dependent Birefringence on a Voltage Sensor and, the Elimination of its Effect on a Voltage Sensor Using a Bulk-Type Bismuth Germanate: Experiment,” to be submitted for publication.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960), pp. 251–253.

Massey, G. A.

Meier, H.

D. P. Bortfeld, H. Meier, “Refractive Indices and Electro-Optic Coefficients of the Eulities Bi4Ge3O12 and Bi4Si3O12,” J. Appl. Phys. 43, 5110–5111 (1972).
[CrossRef]

Mikai, N.

K. Kyuma, S. Tai, M. Nunoshita, N. Mikai, Y. Ida, “Fiber-Optic Current and Voltage Sensors Using a Bi12GeO20 Single Crystal,” IEEE/OSA J. Lightwave Technol. LT-1, 93–97 (1983).
[CrossRef]

Namba, C. S.

Narasimhamurty, T. S.

T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, New York, 1981).
[CrossRef]

Nunoshita, M.

K. Kyuma, S. Tai, M. Nunoshita, N. Mikai, Y. Ida, “Fiber-Optic Current and Voltage Sensors Using a Bi12GeO20 Single Crystal,” IEEE/OSA J. Lightwave Technol. LT-1, 93–97 (1983).
[CrossRef]

Nussbaum, A.

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, NJ, 1976).

Pershan, P. S.

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

Phillips, R. A.

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, NJ, 1976).

Sakai, J.

J. Sakai, T. Kimura, “Polarization Behavior in Multiply Perturbed Single-Mode Fibers,” IEEE J. Quantum Electron. QE-18, 59–65 (1982).
[CrossRef]

N. Imoto, N. Yoshizawa, J. Sakai, H. Tsuchiya, “Birefringence in Single-Mode Optical Fiber due to Elliptical Core Deformation and State Anisotropy,” IEEE J. Quantum Electron. QE-16, 1267–1271 (1980).
[CrossRef]

Shibata, K.

K. Shibata, “A Fiber Optic Electric Field Sensor Using the Electro-Optic Effect of Bi4Ge3O12,” Fusi Electric Corporate Research and Development, Japan.

Shurcliff, W. R.

W. R. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962).

Simon, A.

Tabor, W. J.

W. J. Tabor, F. S. Chen, “Electromagnetic Propagation Through Materials Possessing Both Faraday Rotation and Birefringence: Experiments with Ytterbium Orthoferrite,” J. Appl. Phys. 40, 2760–2765 (1969).
[CrossRef]

Tai, S.

K. Kyuma, S. Tai, M. Nunoshita, N. Mikai, Y. Ida, “Fiber-Optic Current and Voltage Sensors Using a Bi12GeO20 Single Crystal,” IEEE/OSA J. Lightwave Technol. LT-1, 93–97 (1983).
[CrossRef]

Tsuchiya, H.

N. Imoto, N. Yoshizawa, J. Sakai, H. Tsuchiya, “Birefringence in Single-Mode Optical Fiber due to Elliptical Core Deformation and State Anisotropy,” IEEE J. Quantum Electron. QE-16, 1267–1271 (1980).
[CrossRef]

Ulrich, R.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yoshinaga, K.

Y. Kuroda, Y. Abe, H. Kuwahara, K. Yoshinaga, “Field Test of Fiber-Optic Voltage and Current Sensors Applied to Gas Insulated Substation,” Proc. Soc. Photo-Opt. Instrum. Eng. 586, 30–37 (1985).

Yoshizawa, N.

N. Imoto, N. Yoshizawa, J. Sakai, H. Tsuchiya, “Birefringence in Single-Mode Optical Fiber due to Elliptical Core Deformation and State Anisotropy,” IEEE J. Quantum Electron. QE-16, 1267–1271 (1980).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

J. Sakai, T. Kimura, “Polarization Behavior in Multiply Perturbed Single-Mode Fibers,” IEEE J. Quantum Electron. QE-18, 59–65 (1982).
[CrossRef]

N. Imoto, N. Yoshizawa, J. Sakai, H. Tsuchiya, “Birefringence in Single-Mode Optical Fiber due to Elliptical Core Deformation and State Anisotropy,” IEEE J. Quantum Electron. QE-16, 1267–1271 (1980).
[CrossRef]

IEEE/OSA J. Lightwave Technol.

K. Kyuma, S. Tai, M. Nunoshita, N. Mikai, Y. Ida, “Fiber-Optic Current and Voltage Sensors Using a Bi12GeO20 Single Crystal,” IEEE/OSA J. Lightwave Technol. LT-1, 93–97 (1983).
[CrossRef]

J. Appl. Phys.

D. P. Bortfeld, H. Meier, “Refractive Indices and Electro-Optic Coefficients of the Eulities Bi4Ge3O12 and Bi4Si3O12,” J. Appl. Phys. 43, 5110–5111 (1972).
[CrossRef]

W. J. Tabor, F. S. Chen, “Electromagnetic Propagation Through Materials Possessing Both Faraday Rotation and Birefringence: Experiments with Ytterbium Orthoferrite,” J. Appl. Phys. 40, 2760–2765 (1969).
[CrossRef]

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

J. Opt. Soc. Am.

Proc. Soc. Photo-Opt. Instrum. Eng.

F. Bucholtz, K. P. Koo, A. D. Kersey, A. Dandridge, “Fiber-Optic Magnetic Sensor Development,” Proc. Soc. Photo-Opt. Instrum. Eng. 718, 56–65 (1986).

Y. Kuroda, Y. Abe, H. Kuwahara, K. Yoshinaga, “Field Test of Fiber-Optic Voltage and Current Sensors Applied to Gas Insulated Substation,” Proc. Soc. Photo-Opt. Instrum. Eng. 586, 30–37 (1985).

Other

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, NJ, 1976).

T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, New York, 1981).
[CrossRef]

W. R. Shurcliff, Polarized Light (Harvard U.P., Cambridge, 1962).

K. S. Lee, D. Conrad, G. W. Day, P. D. Hale, “Measurement of Optical, Electro-Optical, and Photoelastic Properties of Crystalline Bi4Ge3O12,” Appl. Opt., submitted for publication.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960), pp. 251–253.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

K. S. Lee, G. W. Day, “The Effect of the Temperature and Pressure Dependent Birefringence on a Voltage Sensor and, the Elimination of its Effect on a Voltage Sensor Using a Bulk-Type Bismuth Germanate: Experiment,” to be submitted for publication.

K. Shibata, “A Fiber Optic Electric Field Sensor Using the Electro-Optic Effect of Bi4Ge3O12,” Fusi Electric Corporate Research and Development, Japan.

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Figures (2)

Fig. 1
Fig. 1

Compensation scheme of the voltage sensor with electrooptic crystals possessing multiple birefringences.

Fig. 2
Fig. 2

Compensation scheme of the magnetic field sensor with magnetooptic materials possessing multiple birefringences.

Equations (79)

Equations on this page are rendered with MathJax. Learn more.

D = [ ] E ,
[ ] = [ ] + i 00 [ G ] .
i j = j i * ,
( x 0 0 0 y 0 0 0 z ) ,
[ 0 ] = A 0 ( x 0 0 0 y 0 0 0 z ) A 0 1 = ( x cos 2 Ө 0 + y sin 2 Ө 0 ( y x ) 2 sin 2 Ө 0 0 ( y x ) 2 sin 2 Ө 0 x sin 2 Ө 0 + y cos 2 Ө 0 0 0 0 z ) ,
A 0 = ( cos Ө 0 sin Ө 0 0 sin Ө 0 cos Ө 0 0 0 0 1 ) .
[ ] = [ 0 ] + n = 1 N [ Δ n l ] + m = 1 M [ Δ m c ] ,
[ Δ n l ] = A n ( Δ n l 2 0 0 0 Δ n l 2 0 0 0 0 ) A n 1 ,
A n = ( cos Ө n sin Ө n 0 sin Ө n cos Ө n 0 0 0 1 ) ,
[ Δ n l ] = ( Δ n l 2 cos 2 Ө n Δ n l 2 sin 2 Ө n 0 Δ n l 2 sin 2 Ө n Δ n l 2 cos 2 Ө n 0 0 0 0 ) ,
[ Δ m c ] = ( 0 Δ m c 0 Δ m c 0 0 0 0 0 ) ,
Δ n l = 2 δ n l 0 k 0 l n 0 ,
Δ m c = 2 i 0 Φ m k 0 n 0 l ,
δ n l = B n l k 0 l ,
Φ m = B m c k 0 l / 2 ,
Δ l = 2 0 n 0 2 r 41 E ,
( 0 ) = ( θ 0 0 0 0 0 0 0 0 ) ,
Δ l = 2 0 n e ( n o 3 r 13 n e 3 r 33 ) E n o ( n e + n o ) ,
δ l = k 0 [ q 11 1 2 ( q 12 + q 13 ) + q 44 ] n o 3 P l / 4 ,
Δ l = 1 2 0 n 0 2 P [ q 11 1 2 ( q 12 + q 13 ) + q 44 ] .
Φ = V d B l ,
Δ c = ± 2 i 0 V d B / k 0 n o ,
α = n o 2 p 44 τ .
Δ c = i 2 0 n o p 44 τ / k 0 ,
[ ] = ( x x x y 0 y x y y 0 0 0 z z ) ,
x x = x cos 2 Ө 0 + y sin 2 Ө 0 + 1 2 n = 1 N Δ n l cos 2 Ө n , y y = x sin 2 Ө 0 + y cos 2 Ө 0 1 2 n = 1 N Δ n l cos 2 Ө n , x y = ( y x ) cos Ө 0 sin Ө 0 1 2 n = 1 N Δ n l sin 2 Ө n + m = 1 M Δ m c = y x * ,
× × E + ω 2 μ [ ] E = 0 ,
( ω 2 μ x x k 2 ω 2 μ x y 0 ω 2 μ y x ω 2 μ y y k 2 0 0 0 ω 2 μ z z ) ( E x E y E z ) = 0 ,
k ± 2 = ω 2 μ 2 [ ( x x + y y ) ± ( x x y y ) 2 + 4 x y y x ] .
( E x E y E z ) = A ( 1 P + 0 ) exp [ i ( ω t k + Z ) ] ,
( E x E y E z ) = A ( 1 P 0 ) exp [ i ( ω t k Z ) ] ,
P ± = [ ( y y x x ) ( y y x x ) 2 + 4 x y y x ] / 2 x y ,
( E x E y ) z = l = ( A B C D ) ( E x E y ) z = 0 .
( E x ( z ) E y ( z ) ) = ( A B C D ) ( E x ( 0 ) E y ( 0 ) ) ,
A = cos ϕ i [ ( y y x x ) / ( y y x x ) 2 + 4 x y y x ] sin ϕ = D * , B = ( 2 i x y / ( y y x x ) 2 + 4 x y y x ) sin ϕ , C = ( 2 i y x / ( y y x x ) 2 + 4 x y y x ) sin ϕ , ϕ = ( k + k ) l / 2 ,
P P i = 1 2 [ 1 + 2 x y sin 2 ϕ ( y y x x ) 2 + 4 x y y x ] ,
P P i = 1 2 [ 1 2 x y sin 2 ϕ ( y y x x ) 2 + 4 x y y x ] ,
x x = 0 + 1 2 n = 1 N Δ n l cos 2 Ө n , y y = 0 1 2 n = 1 N Δ n l cos 2 Ө n , x y = y x * = Δ 0 l 2 1 2 n = 1 N Δ n l sin 2 Ө n ,
ϕ l 4 ω 2 μ 0 ( x x y y ) 2 + 4 x y y x .
P P i = 1 2 [ 1 ( δ 0 l + n = 1 N δ n l sin 2 Ө n ) ] ,
P P i = 1 2 [ 1 + ( δ 0 l + n = 1 N δ n l sin 2 Ө n ) ] ,
Γ m ( T , P ) = Γ m 1 n = 1 N δ n l ( T , P ) sin 2 Ө n ,
Γ m ( T , P ) = Γ m 1 + n = 1 N δ n l ( T , P ) sin 2 Ө n ,
Γ m = 2 π λ 0 n o 3 r 41 V m l d ( for cubic crystal class 4 ¯ 3 m ) .
Γ m ( T , P ) + Γ m ( T , P ) = 2 Γ m ( 1 + A 2 + A 4 + )
[ Γ m ( T , P ) + Γ m ( T , P ) ] / 2 Γ m
P ± 45 P i = 1 2 [ 1 ± 2 ( y y x x ) ( x y + y x ) sin 2 ϕ ( y y x x ) 2 + 4 x y y x ± i ( x y y x ) ( y y x x ) 2 + 4 x y y x sin 2 ϕ ] ,
P ± 45 P i = 1 2 [ 1 ± 1 2 ( n = 1 N δ n l cos 2 Ө n ) ( n = 1 N δ n l sin 2 Ө n ) ( 2 V d B l + 2 m = 1 M Φ m ) ] ,
Γ m + 45 ( T , P ) = Γ m 1 + 1 2 [ n = 1 N δ n l ( T , P ) cos 2 Ө n ] [ n = 1 N δ n l ( T , P ) sin 2 Ө n ] 2 m = 1 M Φ m ( T , P ) ,
Γ m 45 ( T , P ) = Γ m 1 1 2 [ n = 1 N δ n l ( T , P ) cos 2 Ө n ] [ n = 1 N δ n l ( T , P ) sin 2 Ө n ] + 2 m = 1 M Φ m ( T ) ,
Γ m = 2 V d B m l ,
[ Γ m + 45 ( T , P ) + Γ m 45 ( T , P ) ] / 2 Γ m
δ 0 l = n o 3 r 41 E k 0 l n = 1 N δ n l sin 2 Ө n .
1 2 ( m = 1 N δ n l cos 2 Ө n ) ( m = 1 N δ n l sin 2 Ө n ) 2 V d B l .
m = 1 M Φ m V d B l .
( exp ( i ϕ ) 0 0 exp ( i ϕ ) ) ,
δ n l = 2 ϕ , or = ( k + k ) l , or = ω μ ( x x y y ) ,
x x y y = x y ± 1 2 Δ n l ,
n x x y y n x y ( 1 ± Δ n l / 4 x y ) ,
δ n l = 2 ( n x x n y y ) π l / λ 0
δ n l = n o k 0 Δ n l l / 2 0 ,
δ n l = 2 π [ ( n e n o ) 1 2 ( n e 3 r 33 n o 3 r 13 ) E ] l / λ 0 ,
Δ l = 2 0 n e ( n o 3 r 13 n e 3 r 33 ) E n o ( n e + n o ) .
( x x Δ m c 0 Δ m c y y 0 0 0 z z ) ,
Φ m = ( n r n l ) π l / λ 0 ,
1 00 ( x x Δ m c 0 Δ m c x x 0 0 0 x x + n 2 ) ( E x E y E z ) = ( n 2 0 0 0 n 2 0 0 0 n 2 ) ( E x E y E z )
| x x n 2 00 Δ m c Δ m c x x n 2 00 | = 0 .
n 2 = x x 00 ± i Δ m c 00 ,
Φ m = i k 0 n 0 l 2 0 Δ m c
P / P i = | 1 / 2 ( 0 0 0 1 ) ( 1 i i 1 ) ( A B C D ) ( 0 1 ) | 2 = 1 / 2 | ( 0 i B + D ) | 2 = 1 2 [ DD * + BB * + i ( B * D BD * ) ] ,
P / P i = | 1 / 2 ( 1 0 0 0 ) ( 1 i i 1 ) ( A B C D ) ( 0 1 ) | 2 = 1 / 2 | ( B i D 0 ) | 2 = 1 2 [ DD * + BB * i ( B * D BD * ) ] ,
( A B C D )
1 / 2 ( 1 i i 1 )
( 0 0 0 1 )
P P i = 1 2 { 1 + [ ( x y * + x y ) sin 2 ϕ ( y y x x ) 2 + 4 x y y x + 2 i ( x y * x y ) ( y y x x ) sin 2 ϕ ( y y x x ) 2 + 4 x y y x ] } , P P i = 1 2 { 1 [ ( x y * + x y ) sin 2 ϕ ( y y x x ) 2 + 4 x y y x + 2 i ( x y * x y ) ( y y x x ) sin 2 ϕ ( y y x x ) 2 + 4 x y y x ] } ,
x x = 0 + 1 2 n = 1 N Δ n l cos 2 Ө n , y y = 0 1 2 n = 1 N Δ n l cos 2 Ө n , x y = y x * = Δ 0 l 2 1 2 n = 1 N Δ n l sin 2 Ө n + m = 1 M Δ m c .
P P i = 1 2 [ 1 ( δ 0 l + n = 1 N δ n l sin 2 Ө n + 2 n = 1 N δ n l cos 2 Ө n · m = 1 M Φ m ) ] P P i = 1 2 [ 1 + ( δ 0 l + n = 1 N δ n l sin 2 Ө n + 2 n = 1 N δ n l cos 2 Ө n · m = 1 M Φ m ) ]
Γ m = Γ m 1 ( n = 1 N δ n l sin 2 Ө n + 2 n = 1 N δ n l cos 2 Ө n · m = 1 M Φ m ) , Γ m = Γ m 1 + ( n = 1 N δ n l sin 2 Ө n + 2 n = 1 N δ n l cos 2 Ө n · m = 1 M Φ m ) ,
Γ m = 2 π λ 0 n o 3 r 41 V m l d .

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