Abstract

A mathematical model is presented to characterize the performance of a lanthanum-modified lead zirconate titanate electro-optical cylindrical structure with an arbitrary number N of lateral electrodes switched with a series of voltages V 0 and V 0 + ΔV. The method yields a solution that can be written as a sum of numerical and analytical components, and the contribution of numerical components becomes increasingly smaller as the electrode wrapping angle is decreasing below π/3N. The effects of the electrode wrapping angle and the switching voltages on the induced refractive-index distribution and electrode capacitance are analyzed. It is shown that the design offers a simple and compact structure that performs dynamic light diverging with a focal length that can be controlled by the applied voltage. The possibility of light steering is considered for a two-electrode structure.

© 1998 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  9. T. Feng, W. Yi-Zun, P.-D. Ye, “Improved coupled mode theory for anisotropic waveguide and modulators,” IEEE J. Quantum Electron. 24, 531–536 (1988).
    [CrossRef]
  10. N. Kuleshov, G. Beilin, “Optimization of electrode pattern for multichannel spatial light modulators on the basis of PLZT ceramics with quadratic electrooptic effect,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 345–355 (1995).
    [CrossRef]
  11. I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Secs. 1.441.1 and 1.441.4.
  12. L. Rade, Beta Mathematics Handbook, 2nd ed. (CRC, Boca Raton, Fla., 1997), p. 232.
  13. G. H. Haertling, “PLZT electro-optic materials and applications—a review,” Ferroelectrics 75, 25–55 (1987).
    [CrossRef]
  14. A. Ghatak, K. Thyagarajan, “Graded index optical waveguides: review,” Prog. Opt. 18, 1–126 (1980).
    [CrossRef]
  15. L. G. Atkinson, D. T. Moore, N. J. Sullo, “Imaging capabilities of a long gradient-index rod,” Appl. Opt. 21, 1004–1007 (1982).
    [CrossRef] [PubMed]

1996

1991

T. Tatebayashi, T. Yamamoto, H. Sato, “Electro-optic variable focal-length lens using PLZT ceramic,” Appl. Opt. 34, 5049–5055 (1991).
[CrossRef]

1990

1989

D. Marcuse, “Electrostatic field of coplanar lines computed with the point matching method,” IEEE J. Quantum Electron. 25, 939–947 (1989).
[CrossRef]

1988

T. Feng, W. Yi-Zun, P.-D. Ye, “Improved coupled mode theory for anisotropic waveguide and modulators,” IEEE J. Quantum Electron. 24, 531–536 (1988).
[CrossRef]

1987

G. H. Haertling, “PLZT electro-optic materials and applications—a review,” Ferroelectrics 75, 25–55 (1987).
[CrossRef]

1986

A. Sawicki, K. Sachse, “Lower and upper bound calculations on the capacitance of multiconductor printed transmission line using the spectral-domain approach and variational method,” IEEE Trans. Microwave Theory Tech. MTT-34, 236–244 (1986).
[CrossRef]

1984

1982

1980

A. Ghatak, K. Thyagarajan, “Graded index optical waveguides: review,” Prog. Opt. 18, 1–126 (1980).
[CrossRef]

1968

H. E. Stinehelfer, “An accurate calculation of uniform microstrip transmission lines,” IEEE Trans. Microwave Theory Tech. MTT-16, 439–444 (1968).
[CrossRef]

Atkinson, L. G.

Beilin, G.

N. Kuleshov, G. Beilin, “Optimization of electrode pattern for multichannel spatial light modulators on the basis of PLZT ceramics with quadratic electrooptic effect,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 345–355 (1995).
[CrossRef]

Bussjager, R.

Cleverly, D. S.

Feng, T.

T. Feng, W. Yi-Zun, P.-D. Ye, “Improved coupled mode theory for anisotropic waveguide and modulators,” IEEE J. Quantum Electron. 24, 531–536 (1988).
[CrossRef]

Ghatak, A.

A. Ghatak, K. Thyagarajan, “Graded index optical waveguides: review,” Prog. Opt. 18, 1–126 (1980).
[CrossRef]

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Secs. 1.441.1 and 1.441.4.

Haertling, G. H.

G. H. Haertling, “PLZT electro-optic materials and applications—a review,” Ferroelectrics 75, 25–55 (1987).
[CrossRef]

Kornreich, P. G.

Kowel, S. T.

Kuleshov, N.

N. Kuleshov, G. Beilin, “Optimization of electrode pattern for multichannel spatial light modulators on the basis of PLZT ceramics with quadratic electrooptic effect,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 345–355 (1995).
[CrossRef]

Lee, S. H.

Marcuse, D.

D. Marcuse, “Electrostatic field of coplanar lines computed with the point matching method,” IEEE J. Quantum Electron. 25, 939–947 (1989).
[CrossRef]

Moore, D. T.

Osman, J.

Rade, L.

L. Rade, Beta Mathematics Handbook, 2nd ed. (CRC, Boca Raton, Fla., 1997), p. 232.

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Secs. 1.441.1 and 1.441.4.

Sachse, K.

A. Sawicki, K. Sachse, “Lower and upper bound calculations on the capacitance of multiconductor printed transmission line using the spectral-domain approach and variational method,” IEEE Trans. Microwave Theory Tech. MTT-34, 236–244 (1986).
[CrossRef]

Sato, H.

T. Tatebayashi, T. Yamamoto, H. Sato, “Electro-optic variable focal-length lens using PLZT ceramic,” Appl. Opt. 34, 5049–5055 (1991).
[CrossRef]

Sawicki, A.

A. Sawicki, K. Sachse, “Lower and upper bound calculations on the capacitance of multiconductor printed transmission line using the spectral-domain approach and variational method,” IEEE Trans. Microwave Theory Tech. MTT-34, 236–244 (1986).
[CrossRef]

Song, Q. W.

Stinehelfer, H. E.

H. E. Stinehelfer, “An accurate calculation of uniform microstrip transmission lines,” IEEE Trans. Microwave Theory Tech. MTT-16, 439–444 (1968).
[CrossRef]

Sullo, N. J.

Tatebayashi, T.

T. Tatebayashi, T. Yamamoto, H. Sato, “Electro-optic variable focal-length lens using PLZT ceramic,” Appl. Opt. 34, 5049–5055 (1991).
[CrossRef]

Thyagarajan, K.

A. Ghatak, K. Thyagarajan, “Graded index optical waveguides: review,” Prog. Opt. 18, 1–126 (1980).
[CrossRef]

Title, M. A.

Wang, X.-M.

Yamamoto, T.

T. Tatebayashi, T. Yamamoto, H. Sato, “Electro-optic variable focal-length lens using PLZT ceramic,” Appl. Opt. 34, 5049–5055 (1991).
[CrossRef]

Ye, P.-D.

T. Feng, W. Yi-Zun, P.-D. Ye, “Improved coupled mode theory for anisotropic waveguide and modulators,” IEEE J. Quantum Electron. 24, 531–536 (1988).
[CrossRef]

Yeh, Y.

Yi-Zun, W.

T. Feng, W. Yi-Zun, P.-D. Ye, “Improved coupled mode theory for anisotropic waveguide and modulators,” IEEE J. Quantum Electron. 24, 531–536 (1988).
[CrossRef]

Zeng, Q.

Appl. Opt.

Ferroelectrics

G. H. Haertling, “PLZT electro-optic materials and applications—a review,” Ferroelectrics 75, 25–55 (1987).
[CrossRef]

IEEE J. Quantum Electron.

D. Marcuse, “Electrostatic field of coplanar lines computed with the point matching method,” IEEE J. Quantum Electron. 25, 939–947 (1989).
[CrossRef]

T. Feng, W. Yi-Zun, P.-D. Ye, “Improved coupled mode theory for anisotropic waveguide and modulators,” IEEE J. Quantum Electron. 24, 531–536 (1988).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

A. Sawicki, K. Sachse, “Lower and upper bound calculations on the capacitance of multiconductor printed transmission line using the spectral-domain approach and variational method,” IEEE Trans. Microwave Theory Tech. MTT-34, 236–244 (1986).
[CrossRef]

H. E. Stinehelfer, “An accurate calculation of uniform microstrip transmission lines,” IEEE Trans. Microwave Theory Tech. MTT-16, 439–444 (1968).
[CrossRef]

Opt. Lett.

Prog. Opt.

A. Ghatak, K. Thyagarajan, “Graded index optical waveguides: review,” Prog. Opt. 18, 1–126 (1980).
[CrossRef]

Other

N. Kuleshov, G. Beilin, “Optimization of electrode pattern for multichannel spatial light modulators on the basis of PLZT ceramics with quadratic electrooptic effect,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 345–355 (1995).
[CrossRef]

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Secs. 1.441.1 and 1.441.4.

L. Rade, Beta Mathematics Handbook, 2nd ed. (CRC, Boca Raton, Fla., 1997), p. 232.

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Figures (10)

Fig. 1
Fig. 1

General definition of the problem to be studied: N lateral electrodes are placed upon a PLZT cylinder and are switched by the voltage sequence V 0, V 0 + ΔV.

Fig. 2
Fig. 2

Self-capacitance C 11 (solid curve) and mutual capacitance C 12 (dotted curve) of the two-electrode system (N = 2) versus electrode wrapping angle ϑ0 for ε1 = 1, ε2 = 5000, and r 0 = 2 mm.

Fig. 3
Fig. 3

Pseudocapacitance of an electrode as a function of electrode number N for ε1 = 1, ε2 = 5000, and r 0 = 2 mm.

Fig. 4
Fig. 4

Examples of the normalized electric potential distribution φ(r, ϑ)/V 0 inside the PLZT cylinder: (a) four electrodes (N = 4), ΔV = -2V 0, and electrode wrapping angle ϑ0 = 60°; (b) eight electrodes (N = 8), ΔV = -0.5V 0, and ϑ0 = 20°.

Fig. 5
Fig. 5

Refractive-index distribution inside the PLZT cylinder with eight lateral electrodes (ϑ0 = 22.5°) for x-polarized light. The data were calculated for the following parameter set: ε1 = 1, ε2 = 5000, r 0 = 2 mm, n 0 = 2.5, V 0 = 100 V, ΔV = -2V 0, ϑ0 = 22.5°, N = 8, R 11 = 2.42 × 10-16 m2/V2, and R 12 = 1.94 × 10-16 m2/V2.

Fig. 6
Fig. 6

Cross section of the refractive-index three-dimensional distribution (ϑ = 0°) (a) for three values of the general voltage: V 0 = 300 V (solid curve), V 0 = 200 V (short-dashed curve), and V 0 = 100 V (long-dashed curve) and (b) for three values of the bias voltage: ΔV = -2V 0 (solid curve), ΔV = -V 0 (short-dashed curve), and ΔV = -0.5V 0 (long-dashed curve). The curve data were calculated for the following parameter set: ε1 = 1, ε2 = 5000, r 0 = 2 mm, n 0 = 2.5, ϑ0 = 30°, N = 6, R 11 = 2.42 × 10-16 m2/V2, and R 12 = 1.94 × 10-16 m2/V2.

Fig. 7
Fig. 7

Four meridional cross section of the refractive-index-three dimensional surface for ϑ = 0° (solid curves), ϑ = 30° (dotted curve), ϑ = 60° (dashed curves), and ϑ = 90° (dotted–dashed curve): (a) for N = 6 and ϑ0 = 30° (the curves for ϑ = 0° and ϑ = 90° and for ϑ = 30° and ϑ = 60° practically coincide) and (b) for N = 8 and ϑ0 = 22.5°. The data were calculated for the following parameter set: ε1 = 1, ε2 = 5000, r 0 = 2 mm, n 0 = 2.5, V 0 = 400 V, ΔV = -2V 0, R 11 = 2.42 × 10-16 m2/V2, and R 12 = 1.94 × 10-16 m2/V2.

Fig. 8
Fig. 8

Meridional cross section of the refractive-index three-dimensional surface for three values of electrode duty ratio: ϑ0 = 20° (solid curve), ϑ0 = 30° (short-dashed curve), and ϑ0 = 40° (long-dashed curve). The data were calculated for the following parameter set: ε1 = 1, ε2 = 5000, r 0 = 1 mm, n 0 = 2.5, V 0 = 400 V, ΔV = -2V 0, N = 6, R 11 = 2.42 × 10-16 m2/V2, and R 12 = 1.94 × 10-16 m2/V2.

Fig. 9
Fig. 9

Three-dimensional refractive-index distribution for the two-electrode structure. The data were calculated for the following parameter set: ϑ0 = 30°, ε1 = 1, ε2 = 5000, r 0 = 1 mm, n 0 = 2.5, V 0 = 300 V, ΔV = -2V 0, N = 2, R 11 = 2.42 × 10-16 m2/V2, and R 12 = 1.94 × 10-16 m2/V2.

Fig. 10
Fig. 10

Three-dimensional refractive-index distribution for the two-electrode structure immersed in a dielectric medium with permittivity close to the permittivity of the electro-optic material, i.e., ε1 = 3000. The data were calculated for the following parameter set: ϑ0 = 30°, ε2 = 5000, r 0 = 1 mm, n 0 = 2.5, V 0 = 300 V, ΔV = -V 0, N = 2, R 11 = 2.42 × 10-16 m2/V2, and R 12 = 1.94 × 10-16 m2/V2.

Equations (46)

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2 φ r ,   ϑ r 2 + 1 r φ r ,   ϑ r + 1 r 2 2 φ r ,   ϑ ϑ 2 = 0 .
φ r ,   ϑ = V 0 B 0 + n = 1   B n r r 0 Nn / 2 cos Nn ϑ   2 , 0 r < r 0 ,
φ * r ,   ϑ = V 0 C 0 - C 0   ln r r 0 + n = 1   C n r 0 r Nn / 2 cos Nn 2   ϑ , r 0 r < ,
φ r 0 ,   ϑ = φ * r 0 ,   ϑ ,     0 ϑ < 2 π ,
φ r 0 ,   ϑ = V 0 ,     - ϑ 0 2 ± 2 π N   m ϑ + ϑ 0 2 ± 2 π N   m ,
φ r 0 ,   ϑ = V 0 + Δ V ,     2 π N - ϑ 0 2 ± 2 π N   m ϑ 2 π N + ϑ 0 2 ± 2 π N   m ,
ε 1 φ * r 0 ,   ϑ r = ε 2 φ r 0 ,   ϑ r ,     ϑ 0 2 ± 2 π N   m < ϑ < 2 π N - ϑ 0 2 ± 2 π N   m ,
n = 1   B n   cos Nn 2   ϑ = 1 - B 0 , 0 ϑ + ϑ 0 2 ,
n = 1   B n   cos Nn 2   ϑ = 1 + Δ V V 0 - B 0 , 2 π N - ϑ 0 2 ϑ 2 π N ,
B 0 ε 1 ε 1 + ε 2 + n = 1 Nn 2   B n   cos Nn 2   ϑ = 0 , ϑ 0 2 ϑ 2 π N - ϑ 0 2 .
ρ ϑ = V 0 r 0 B 0 ε 1 ε 1 + ε 2 + n = 1 Nn 2   B n   cos nN ϑ 2 = ρ 1 ϑ , 0 ϑ ϑ 0 2 0 , ϑ 0 2 < ϑ < 2 π N - ϑ 0 2 ρ 2 ϑ , 2 π N - ϑ 0 2 ϑ 2 π N .
B 0 ε 1 ε 1 + ε 2 = N 2 π 0 ϑ 0 / 2   ρ 1 ξ d ξ + N 2 π 0 ϑ 0 / 2   ρ 2 2 π / N - ξ d ξ ,
nN 2   B n = N π 0 ϑ 0 / 2   ρ 1 ξ cos Nn ξ 2 d ξ + N - 1 n π 0 ϑ 0 / 2   ρ 2 2 π N - ξ cos Nn ξ 2 d ξ .
n = 1 - 1 n cos Nn ϑ 2 cos Nn ξ 2 n = - 1 2 ln 2 cos N ϑ 2 + cos N ξ 2 ,
n = 1 cos Nn ϑ 2 cos Nn ξ 2 n = - 1 2 ln 2 cos N ϑ 2 - cos N ξ 2 .
- N 2 π 0 ϑ 0 / 2 ln 2 cos N ϑ 2 - cos N ξ 2 ρ 1 ξ d ξ - N 2 π 0 ϑ 0 / 2 ln 2 cos N ϑ 2 + cos N ξ 2 ρ 2 2 π N - ξ d ξ = 1 - B 0 ,     0 ϑ ϑ 0 2 ,
- N 2 π 0 ϑ 0 / 2 ln 2 cos N ϑ 2 - cos N ξ 2 ρ 1 ξ d ξ - N 2 π 0 ϑ 0 / 2 ln 2 cos N ϑ 2 + cos N ξ 2 ρ 2 2 π N - ξ d ξ = 1 + Δ V V 0 - B 0 ,     2 π N - ϑ 0 2 ϑ 2 π N .
cos N ϑ / 2 = α + β   cos N ϑ 1 / 2 , cos N ξ / 2 = α + β   cos N ξ 1 / 2 ,    
cos N ϑ / 2 = - α - β   cos N ϑ 1 / 2 , cos N ξ / 2 = α + β   cos N ξ 1 / 2 ,
- N 2 π ln β 0 2 π / N   ρ 1 ξ 1 d ξ d ξ 1 d ξ 1 + N π 0 2 π / N m = 1 1 m cos Nm ϑ 1 2 cos Nm ξ 1 2 ρ 1 ξ 1 d ξ d ξ 1 d ξ 1 - N 2 π ln 4 α 0 2 π / N   ρ 2 ξ 1 d ξ d ξ 1 d ξ 1 + N 2 π 0 2 π / N m = 1 - 1 m m α 4 β m cos N ϑ 1 / 2 + cos N ξ 1 / 2 m ρ 2 ξ 1 d ξ d ξ 1 d ξ 1 = 1 - B 0 ,
- N 2 π ln 4 α 0 2 π / N   ρ 1 ξ 1 d ξ d ξ 1 d ξ 1 - N 2 π ln β 0 2 π / N   ρ 2 ξ 1 d ξ d ξ 1 d ξ 1 + N 2 π 0 2 π / N m = 1 - 1 m m β 4 α m cos N ϑ 1 / 2 + cos N ξ 1 / 2 m ρ 1 ξ 1 d ξ d ξ 1 d ξ 1 + N π 0 2 π / N m = 1 1 m cos Nm ϑ 1 2 cos Nm ξ 1 2   ρ 2 ξ 1 d ξ d ξ 1 d ξ 1 = 1 + Δ V V 0 - B 0 .
ρ 1 ξ 1 d ξ d ξ 1 = n = 0   a n 1 cos Nn ξ 1 2 , ρ 2 ξ 1 d ξ d ξ 1 = p = 0   a p 2 cos Nn ξ 1 2 .
- a 0 1 ln β - a 0 2 ln 4 α + G 0 a 0 2 2 + 1 4 n = 1   G n a n 2 = 1 - ε 1 + ε 2 ε 2 a 0 1 + a 0 2 ,
- a 0 1 ln 4 α - a 0 2 ln β + G 0 a 0 1 2 + 1 4 n = 1   G n a 0 1 = 1 + Δ V V 0 - ε 1 + ε 2 ε 2 a 0 1 + a 0 2 ;
n = 1   G ms a m 2 + a s 1 s = - G s a 0 2 , n = 1   G ms a m 1 + a s 1 s = - G s a 0 1 ,
G ns = 1 2 m = 1 - 1 m α ns m m 1 2 tan N ϑ 0 8 m , G n = m = 1 - 1 m β n m m 1 2 tan N ϑ 0 8 m , G 0 = m = 1 - 1 m β 0 m m 1 2 tan N ϑ 0 8 m , α ns m = N 2 π 2 0 2 π / N 0 2 π / N cos Nn ξ 1 / 2 cos Ns ϑ 1 / 2 cos N ξ 1 / 2 + cos N ϑ 1 / 2 m d ξ 1 d ϑ 1 = 2 - m + 2 m !   1 = 0 m j = 0 1 / 2 δ | s - 1 + 2 j | 0 1 - j ! j ! × i = 0 m - 1 / 2 δ | n - m + 1 + 2 i | 0 m - 1 - i ! i ! ,
ρ 1 ϑ = V 0 r 0 d ϑ 1 d ϑ m = 0   a m 1 cos Nm ϑ 1 ϑ 2 = V 0 r 0 cos N ϑ 4 cos 2 N ϑ 4 - cos 2 N ϑ 0 8 1 / 2 × m = 0 - 1 m a m 1 T 2 m sin N ϑ / 4 sin N ϑ 0 / 8 ,
ρ 2 ϑ = V 0 r 0 d ϑ 1 d ϑ m = 0   a m 2 cos Nm ϑ 1 ϑ 2 = V 0 r 0 cos N 4 π - ϑ cos 2 N 4 π - ϑ - cos 2 N ϑ 0 8 1 / 2 × m = 0 - 1 m a m 2 T 2 m sin N π - ϑ / 4 sin N ϑ 0 / 8 .
Q 1 Q 2 Q N = C 11 C 12 C 1 N C 21 C 22 C 2 N C N 1 C N 2 C NN V 0 V 0   +   Δ V V 0   +   Δ V ,
nB n = 2 N 2 b 0 2 n x 0 + m = 1 2 n   b m 2 n x m 2 b 0 2 n - 1 y 0 + m = 1 2 n - 1   b m 2 n - 1 y m ,
cos Nn ξ ξ 1 2 = p = 0 n   b p n cos Np ξ 1 2 .
b j n = 2 - 1 n + j k = 0 n i = 0 2 n - k   g k 2 n sin N ϑ 0 8 2 n - k × 2 n - k ! δ | j + n - k - i | 0 i ! 2 n - 2 k - i ! ,
T 2 n x = k = 0 n   g k 2 n x 2 n - 2 k ,
T n + 1 x = 2 xT n x - T n - 1 x .
E r = - φ r = 2 V 0 r x 0 n = 1   b 0 2 n ρ Nn cos nN ϑ + y 0 n = 1   b 0 2 n - 1 ρ 2 n - 1 N / 2 cos N 2 n - 1 ϑ / 2 ,
E ϑ = - 1 r φ ϑ = 2 V 0 r x 0 n = 1   b 0 2 n ρ Nn sin Nn ϑ + y 0 n = 1   b 0 2 n - 1 ρ N 2 n - 1 / 2 sin N 2 n - 1 ϑ / 2 ,
b 0 n = N 2 π 0 ϑ 0 / 2 2 cos N ξ / 4 cos Nn ξ / 2 d ξ cos N ξ / 2 - cos N ϑ 0 / 4 1 / 2 = N 4 P n cos N ϑ 0 / 4 + P n - 1 cos N ϑ 0 / 4 .
n = 1   ρ 2 n cos 2 nx = ρ 2 cos 2 x - ρ 2 1 + ρ 4 - 2 ρ 2 cos   2 x , n = 1   ρ 2 n - 1 cos 2 n - 1 x ) = ρ   cos x 1 - ρ 2 1 + ρ 4 - 2 ρ 2 cos   2 x ,
n = 1   ρ 2 n sin 2 nx = ρ 2 sin 2 x 1 + ρ 4 - 2 ρ 2 cos   2 x , n = 1   ρ 2 n - 1 sin 2 n - 1 x ) = ρ   sin x 1 + ρ 2 1 + ρ 4 - 2 ρ 2 cos   2 x .
sin γ = sin N ξ / 4 sin N ϑ 0 / 8 .
Δ   1 n 2 i , j , k , l = 1 2 k l   R i , j , k , l E k E l       k ,   l = x ,   y ,   z ,
R i , j = R 11 R 12 R 12 0 0 0 R 12 R 11 R 12 0 0 0 R 12 R 12 R 11 0 0 0 0 0 0 R 66 0 0 0 0 0 0 R 66 0 0 0 0 0 0 R 66 .
n x = n 0 1 - n 0 2 2 R 11 E x 2 + R 12 E y 2 , n y = n 0 1 - n 0 2 2 R 12 E x 2 + R 11 E y 2 , n z = n 0 1 - n 0 2 2   R 12 E x 2 + E y 2 ,
E x = E r   cos ϑ - E ϑ   sin ϑ , E y = E r   sin ϑ + E ϑ   cos ϑ .
n x , y r ,   ϑ n 0 1 - A ϑ 0 ,   V 0 2   r 2 + B ϑ 0 ,   V 0 r 4 + .
z = 2 π / A .

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