Abstract

A new interferometric scheme for measuring mode conversion distributed locally along a polarization-maintaining fiber is presented. Using this technique the power coupling coefficient, varying with magnitude and angle of external pressure transversely applied to a fiber, was evaluated both theoretically and experimentally. The coupling point location is determined with ±1.5-cm accuracy and resolution of better than 10 cm for a 220-m long fiber having modal birefringence of 4.4 × 10−4. The coupling coefficient was proportional to the external force in the range from 5 × 10−3 to 0.1 kg/mm. The relationships determined experimentally reflected those predicted by theory.

© 1988 Optical Society of America

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References

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  1. I. P. Kaminow, “Polarization in Optical Fibers,” IEEE J. Quantum Electron. QE-17, 15 (1981).
    [CrossRef]
  2. J. Sakai, T. Kimura, “Birefringence and Polarization Characteristics of Single-Mode Optical Fibers Under Elastic Deformations,” IEEE J. Quantum Electron. QE-17, 1041 (1981).
    [CrossRef]
  3. J. Sakai, T. Kimura, “Polarization Behavior in Multiply Perturbed Single-Mode Fibers,” IEEE J. Quantum Electron QE-18, 59 (1982).
    [CrossRef]
  4. M. Nakazawa, N. Shibata, M. Tokuda, Y. Negishi, “Measurements of Polarization Mode Couplings Along Polarization-Maintaining Single-Mode Optical Fibers,” J. Opt. Soc. Am. A 1, 285 (1984).
    [CrossRef]
  5. N. Shibata, M. Tateda, S. Seikai, N. Uchida, “Spatial Technique for Measuring Modal Delay Differences in a Dual-Mode Optical Fiber,” Appl. Opt. 19, 1489 (1980).
    [CrossRef] [PubMed]
  6. N. Shibata, M. Tsubokawa, S. Seikai, “Measurement of Polarization Mode Dispersion by Optical Heterodyne Detection,” Electron. Lett. 20, 1055 (1984).
    [CrossRef]
  7. M. Tsubokawa, N. Shibata, S. Seikai, “Evaluation of Polarization Mode Coupling Coefficient from Measurement of Polarization Mode Dispersion,” IEEE/OSA J. Lightwave Technol. LT-3, 850 (1985).
    [CrossRef]
  8. M. Tsubokawa, T. Higashi, S. Seikai, “Fiber-Optic Sensor for Measuring External Force Distributed Along a Fiber,” Jpn. J. Appl. Phys. Lett. 26, 587 (1987).
    [CrossRef]
  9. A. R. Nelson, D. H. McMahon, R. L. Gravel, “Passive Multiplexing System for Fiber-Optic Sensors,” Appl. Opt. 19, 2917 (1980).
    [CrossRef] [PubMed]
  10. J. L. Brooks, R. C. Youngquist, R. H. Wentworth, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence Multiplexing of Fiber-Optic Interferometric Sensors,” IEEE/OSA J. Lightwave Technol. LT-3, 1062 (1985).
    [CrossRef]
  11. S. P. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970).
  12. A. Simon, R. Ulrich, “Evolution of Polarization Along a Single-Mode Fiber,” Appl. Phys. Lett. 31, 517 (1977).
    [CrossRef]
  13. J. Noda, K. Okamoto, Y. Sasaki, “Polarization-Maintaining Fibers and Their Applications,” IEEE/OSA J. Lightwave Technol. LT-4, 1071 (1986).
    [CrossRef]
  14. N. Shibata, K. Okamoto, M. Tateda, S. Seikai, Y. Sasaki, “Modal Birefringence and Polarization Mode Dispersion in Stress-Inducing Anisotropy,” IEEE J. Quantum Electron. QE-19, 1110 (1983).
    [CrossRef]
  15. K. Okamoto, T. Edahiro, N. Shibata, “Polarization Properties of Single-Polarization Fibers,” Opt. Lett. 7, 569 (1982).
    [CrossRef] [PubMed]
  16. N. Shibata, K. Okamoto, K. Suzuki, Y. Ishida, “Polarization-Mode Properties of Elliptical-Core Fibers and Stress-Induced Birefringent Fibers,” J. Opt. Soc. Am. 73, 1792 (1983).
    [CrossRef]

1987 (1)

M. Tsubokawa, T. Higashi, S. Seikai, “Fiber-Optic Sensor for Measuring External Force Distributed Along a Fiber,” Jpn. J. Appl. Phys. Lett. 26, 587 (1987).
[CrossRef]

1986 (1)

J. Noda, K. Okamoto, Y. Sasaki, “Polarization-Maintaining Fibers and Their Applications,” IEEE/OSA J. Lightwave Technol. LT-4, 1071 (1986).
[CrossRef]

1985 (2)

J. L. Brooks, R. C. Youngquist, R. H. Wentworth, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence Multiplexing of Fiber-Optic Interferometric Sensors,” IEEE/OSA J. Lightwave Technol. LT-3, 1062 (1985).
[CrossRef]

M. Tsubokawa, N. Shibata, S. Seikai, “Evaluation of Polarization Mode Coupling Coefficient from Measurement of Polarization Mode Dispersion,” IEEE/OSA J. Lightwave Technol. LT-3, 850 (1985).
[CrossRef]

1984 (2)

N. Shibata, M. Tsubokawa, S. Seikai, “Measurement of Polarization Mode Dispersion by Optical Heterodyne Detection,” Electron. Lett. 20, 1055 (1984).
[CrossRef]

M. Nakazawa, N. Shibata, M. Tokuda, Y. Negishi, “Measurements of Polarization Mode Couplings Along Polarization-Maintaining Single-Mode Optical Fibers,” J. Opt. Soc. Am. A 1, 285 (1984).
[CrossRef]

1983 (2)

N. Shibata, K. Okamoto, K. Suzuki, Y. Ishida, “Polarization-Mode Properties of Elliptical-Core Fibers and Stress-Induced Birefringent Fibers,” J. Opt. Soc. Am. 73, 1792 (1983).
[CrossRef]

N. Shibata, K. Okamoto, M. Tateda, S. Seikai, Y. Sasaki, “Modal Birefringence and Polarization Mode Dispersion in Stress-Inducing Anisotropy,” IEEE J. Quantum Electron. QE-19, 1110 (1983).
[CrossRef]

1982 (2)

K. Okamoto, T. Edahiro, N. Shibata, “Polarization Properties of Single-Polarization Fibers,” Opt. Lett. 7, 569 (1982).
[CrossRef] [PubMed]

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

1981 (2)

I. P. Kaminow, “Polarization in Optical Fibers,” IEEE J. Quantum Electron. QE-17, 15 (1981).
[CrossRef]

J. Sakai, T. Kimura, “Birefringence and Polarization Characteristics of Single-Mode Optical Fibers Under Elastic Deformations,” IEEE J. Quantum Electron. QE-17, 1041 (1981).
[CrossRef]

1980 (2)

1977 (1)

A. Simon, R. Ulrich, “Evolution of Polarization Along a Single-Mode Fiber,” Appl. Phys. Lett. 31, 517 (1977).
[CrossRef]

Brooks, J. L.

J. L. Brooks, R. C. Youngquist, R. H. Wentworth, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence Multiplexing of Fiber-Optic Interferometric Sensors,” IEEE/OSA J. Lightwave Technol. LT-3, 1062 (1985).
[CrossRef]

Edahiro, T.

Goodier, J. N.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970).

Gravel, R. L.

Higashi, T.

M. Tsubokawa, T. Higashi, S. Seikai, “Fiber-Optic Sensor for Measuring External Force Distributed Along a Fiber,” Jpn. J. Appl. Phys. Lett. 26, 587 (1987).
[CrossRef]

Ishida, Y.

Kaminow, I. P.

I. P. Kaminow, “Polarization in Optical Fibers,” IEEE J. Quantum Electron. QE-17, 15 (1981).
[CrossRef]

Kim, B. Y.

J. L. Brooks, R. C. Youngquist, R. H. Wentworth, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence Multiplexing of Fiber-Optic Interferometric Sensors,” IEEE/OSA J. Lightwave Technol. LT-3, 1062 (1985).
[CrossRef]

Kimura, T.

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

J. Sakai, T. Kimura, “Birefringence and Polarization Characteristics of Single-Mode Optical Fibers Under Elastic Deformations,” IEEE J. Quantum Electron. QE-17, 1041 (1981).
[CrossRef]

McMahon, D. H.

Nakazawa, M.

Negishi, Y.

Nelson, A. R.

Noda, J.

J. Noda, K. Okamoto, Y. Sasaki, “Polarization-Maintaining Fibers and Their Applications,” IEEE/OSA J. Lightwave Technol. LT-4, 1071 (1986).
[CrossRef]

Okamoto, K.

J. Noda, K. Okamoto, Y. Sasaki, “Polarization-Maintaining Fibers and Their Applications,” IEEE/OSA J. Lightwave Technol. LT-4, 1071 (1986).
[CrossRef]

N. Shibata, K. Okamoto, M. Tateda, S. Seikai, Y. Sasaki, “Modal Birefringence and Polarization Mode Dispersion in Stress-Inducing Anisotropy,” IEEE J. Quantum Electron. QE-19, 1110 (1983).
[CrossRef]

N. Shibata, K. Okamoto, K. Suzuki, Y. Ishida, “Polarization-Mode Properties of Elliptical-Core Fibers and Stress-Induced Birefringent Fibers,” J. Opt. Soc. Am. 73, 1792 (1983).
[CrossRef]

K. Okamoto, T. Edahiro, N. Shibata, “Polarization Properties of Single-Polarization Fibers,” Opt. Lett. 7, 569 (1982).
[CrossRef] [PubMed]

Sakai, J.

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

J. Sakai, T. Kimura, “Birefringence and Polarization Characteristics of Single-Mode Optical Fibers Under Elastic Deformations,” IEEE J. Quantum Electron. QE-17, 1041 (1981).
[CrossRef]

Sasaki, Y.

J. Noda, K. Okamoto, Y. Sasaki, “Polarization-Maintaining Fibers and Their Applications,” IEEE/OSA J. Lightwave Technol. LT-4, 1071 (1986).
[CrossRef]

N. Shibata, K. Okamoto, M. Tateda, S. Seikai, Y. Sasaki, “Modal Birefringence and Polarization Mode Dispersion in Stress-Inducing Anisotropy,” IEEE J. Quantum Electron. QE-19, 1110 (1983).
[CrossRef]

Seikai, S.

M. Tsubokawa, T. Higashi, S. Seikai, “Fiber-Optic Sensor for Measuring External Force Distributed Along a Fiber,” Jpn. J. Appl. Phys. Lett. 26, 587 (1987).
[CrossRef]

M. Tsubokawa, N. Shibata, S. Seikai, “Evaluation of Polarization Mode Coupling Coefficient from Measurement of Polarization Mode Dispersion,” IEEE/OSA J. Lightwave Technol. LT-3, 850 (1985).
[CrossRef]

N. Shibata, M. Tsubokawa, S. Seikai, “Measurement of Polarization Mode Dispersion by Optical Heterodyne Detection,” Electron. Lett. 20, 1055 (1984).
[CrossRef]

N. Shibata, K. Okamoto, M. Tateda, S. Seikai, Y. Sasaki, “Modal Birefringence and Polarization Mode Dispersion in Stress-Inducing Anisotropy,” IEEE J. Quantum Electron. QE-19, 1110 (1983).
[CrossRef]

N. Shibata, M. Tateda, S. Seikai, N. Uchida, “Spatial Technique for Measuring Modal Delay Differences in a Dual-Mode Optical Fiber,” Appl. Opt. 19, 1489 (1980).
[CrossRef] [PubMed]

Shaw, H. J.

J. L. Brooks, R. C. Youngquist, R. H. Wentworth, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence Multiplexing of Fiber-Optic Interferometric Sensors,” IEEE/OSA J. Lightwave Technol. LT-3, 1062 (1985).
[CrossRef]

Shibata, N.

M. Tsubokawa, N. Shibata, S. Seikai, “Evaluation of Polarization Mode Coupling Coefficient from Measurement of Polarization Mode Dispersion,” IEEE/OSA J. Lightwave Technol. LT-3, 850 (1985).
[CrossRef]

N. Shibata, M. Tsubokawa, S. Seikai, “Measurement of Polarization Mode Dispersion by Optical Heterodyne Detection,” Electron. Lett. 20, 1055 (1984).
[CrossRef]

M. Nakazawa, N. Shibata, M. Tokuda, Y. Negishi, “Measurements of Polarization Mode Couplings Along Polarization-Maintaining Single-Mode Optical Fibers,” J. Opt. Soc. Am. A 1, 285 (1984).
[CrossRef]

N. Shibata, K. Okamoto, M. Tateda, S. Seikai, Y. Sasaki, “Modal Birefringence and Polarization Mode Dispersion in Stress-Inducing Anisotropy,” IEEE J. Quantum Electron. QE-19, 1110 (1983).
[CrossRef]

N. Shibata, K. Okamoto, K. Suzuki, Y. Ishida, “Polarization-Mode Properties of Elliptical-Core Fibers and Stress-Induced Birefringent Fibers,” J. Opt. Soc. Am. 73, 1792 (1983).
[CrossRef]

K. Okamoto, T. Edahiro, N. Shibata, “Polarization Properties of Single-Polarization Fibers,” Opt. Lett. 7, 569 (1982).
[CrossRef] [PubMed]

N. Shibata, M. Tateda, S. Seikai, N. Uchida, “Spatial Technique for Measuring Modal Delay Differences in a Dual-Mode Optical Fiber,” Appl. Opt. 19, 1489 (1980).
[CrossRef] [PubMed]

Simon, A.

A. Simon, R. Ulrich, “Evolution of Polarization Along a Single-Mode Fiber,” Appl. Phys. Lett. 31, 517 (1977).
[CrossRef]

Suzuki, K.

Tateda, M.

N. Shibata, K. Okamoto, M. Tateda, S. Seikai, Y. Sasaki, “Modal Birefringence and Polarization Mode Dispersion in Stress-Inducing Anisotropy,” IEEE J. Quantum Electron. QE-19, 1110 (1983).
[CrossRef]

N. Shibata, M. Tateda, S. Seikai, N. Uchida, “Spatial Technique for Measuring Modal Delay Differences in a Dual-Mode Optical Fiber,” Appl. Opt. 19, 1489 (1980).
[CrossRef] [PubMed]

Timoshenko, S. P.

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970).

Tokuda, M.

Tsubokawa, M.

M. Tsubokawa, T. Higashi, S. Seikai, “Fiber-Optic Sensor for Measuring External Force Distributed Along a Fiber,” Jpn. J. Appl. Phys. Lett. 26, 587 (1987).
[CrossRef]

M. Tsubokawa, N. Shibata, S. Seikai, “Evaluation of Polarization Mode Coupling Coefficient from Measurement of Polarization Mode Dispersion,” IEEE/OSA J. Lightwave Technol. LT-3, 850 (1985).
[CrossRef]

N. Shibata, M. Tsubokawa, S. Seikai, “Measurement of Polarization Mode Dispersion by Optical Heterodyne Detection,” Electron. Lett. 20, 1055 (1984).
[CrossRef]

Tur, M.

J. L. Brooks, R. C. Youngquist, R. H. Wentworth, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence Multiplexing of Fiber-Optic Interferometric Sensors,” IEEE/OSA J. Lightwave Technol. LT-3, 1062 (1985).
[CrossRef]

Uchida, N.

Ulrich, R.

A. Simon, R. Ulrich, “Evolution of Polarization Along a Single-Mode Fiber,” Appl. Phys. Lett. 31, 517 (1977).
[CrossRef]

Wentworth, R. H.

J. L. Brooks, R. C. Youngquist, R. H. Wentworth, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence Multiplexing of Fiber-Optic Interferometric Sensors,” IEEE/OSA J. Lightwave Technol. LT-3, 1062 (1985).
[CrossRef]

Youngquist, R. C.

J. L. Brooks, R. C. Youngquist, R. H. Wentworth, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence Multiplexing of Fiber-Optic Interferometric Sensors,” IEEE/OSA J. Lightwave Technol. LT-3, 1062 (1985).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

A. Simon, R. Ulrich, “Evolution of Polarization Along a Single-Mode Fiber,” Appl. Phys. Lett. 31, 517 (1977).
[CrossRef]

Electron. Lett. (1)

N. Shibata, M. Tsubokawa, S. Seikai, “Measurement of Polarization Mode Dispersion by Optical Heterodyne Detection,” Electron. Lett. 20, 1055 (1984).
[CrossRef]

IEEE J. Quantum Electron (1)

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

IEEE J. Quantum Electron. (3)

I. P. Kaminow, “Polarization in Optical Fibers,” IEEE J. Quantum Electron. QE-17, 15 (1981).
[CrossRef]

J. Sakai, T. Kimura, “Birefringence and Polarization Characteristics of Single-Mode Optical Fibers Under Elastic Deformations,” IEEE J. Quantum Electron. QE-17, 1041 (1981).
[CrossRef]

N. Shibata, K. Okamoto, M. Tateda, S. Seikai, Y. Sasaki, “Modal Birefringence and Polarization Mode Dispersion in Stress-Inducing Anisotropy,” IEEE J. Quantum Electron. QE-19, 1110 (1983).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (3)

J. Noda, K. Okamoto, Y. Sasaki, “Polarization-Maintaining Fibers and Their Applications,” IEEE/OSA J. Lightwave Technol. LT-4, 1071 (1986).
[CrossRef]

J. L. Brooks, R. C. Youngquist, R. H. Wentworth, M. Tur, B. Y. Kim, H. J. Shaw, “Coherence Multiplexing of Fiber-Optic Interferometric Sensors,” IEEE/OSA J. Lightwave Technol. LT-3, 1062 (1985).
[CrossRef]

M. Tsubokawa, N. Shibata, S. Seikai, “Evaluation of Polarization Mode Coupling Coefficient from Measurement of Polarization Mode Dispersion,” IEEE/OSA J. Lightwave Technol. LT-3, 850 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. Lett. (1)

M. Tsubokawa, T. Higashi, S. Seikai, “Fiber-Optic Sensor for Measuring External Force Distributed Along a Fiber,” Jpn. J. Appl. Phys. Lett. 26, 587 (1987).
[CrossRef]

Opt. Lett. (1)

Other (1)

S. P. Timoshenko, J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970).

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Figures (13)

Fig. 1
Fig. 1

Schematic diagram of an optical interferometer for evaluating model conversion.

Fig. 2
Fig. 2

Dependence of resolvable length Δz on the polarization dispersion T p and the ratio Δλ/λ0 of light emitted from the source.

Fig. 3
Fig. 3

Coordinate systems with respect to the fiber’s principal axes and the direction of transversely applied pressure.

Fig. 4
Fig. 4

Experimental arrangement for measuring the locations and magnitudes of mode couplings due to external forces.

Fig. 5
Fig. 5

Microscopic photograph of the cross-sectional area of the test PANDA fiber13; SIR denotes stress-inducing rod.

Fig. 6
Fig. 6

(a) Power spectral distribution of light emitted from the LD. (b) Beat amplitude vs the optical path difference ΔL for interference between identically polarized modes.

Fig. 7
Fig. 7

Beat amplitude as a function of ΔL: (a) no external pressures act on the fiber; (b),(c) external pressures of 0.025 kg/mm transversely applied across the fiber at z = 68, 69, and 70 m with θ p = 45°. The horizontal axis in (c) is magnified 40 times more than in (b).

Fig. 8
Fig. 8

Beat amplitudes observed when pressures of (a) F = 0.005 kg/mm, (b) F = 0.055 kg/mm, and (c) F = 0.505 kg/mm are imposed on the fiber at z = 70 m.

Fig. 9
Fig. 9

Coupling coefficient h dependence on the force magnitude F. Symbols indicate the measured data and lines indicate the calculated results.

Fig. 10
Fig. 10

Beat amplitudes observed for (a) θ p = 11.5°, (b) θ p = 22.5°, and (c) θ p = 45°.

Fig. 11
Fig. 11

Angle θ p dependence of the coupling coefficient h. Symbols denote the measured data and the curve denotes the calculated h.

Fig. 12
Fig. 12

Relationship between h and intrinsic modal birefringence B in a fiber.

Fig. 13
Fig. 13

Dependence of h on fiber length d of the pressure-applied region; d B on the horizontal axis indicates the beat length of a fiber at λ = 1.27 μm.

Tables (1)

Tables Icon

Table I Waveguide and Transmission Parameters of the Test Polarization-Maintaining Fiber

Equations (41)

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

S ˜ = r Re [ n = 1 N E x y n * · E y y ] ,
E x y n = h n / ( 1 h n ) p = 1 N ( 1 h p ) 0 Q ( ω ) × exp i [ ( β x β y ) z n + β y L ω t ] d ω ,
E y y = p = 1 N 1 h p 0 Q ( ω ) exp i [ β y L ω ( t Δ L / c ) ] d ω .
P ( ω ) = exp [ ( ω ω 0 ) 2 / 2 Δ ω 2 ] ,
S = r n = 1 N h n / ( 1 h n ) p = 1 N ( 1 h p ) exp [ ( Δ ω Z n ) 2 / 2 c 2 ] ,
Z n = ( β x o β y o ) c z n Δ L ,
S r = r p = 1 N 1 h p .
S / S r = n = 1 N h n / ( 1 h n ) exp [ ( Δ ω Z n ) 2 / 2 c 2 ] ,
Δ z = 2 / [ ( β x o β y o ) Δ ω ] = L T c / T .
S n / S r = h n / ( 1 h n ) .
h n = ( S n / S r ) 2 / [ 1 ( S n / S r ) 2 ] ,
= ( S n / S r ) 2 for h 1 .
E x = [ cos φ cos ( δ β d / 2 ) i cos ( φ + α ) sin ( δ β d / 2 ) ] × exp [ i ( N x x + N y y ) d / 2 ] ,
E y = [ sin φ cos ( δ β d / 2 ) + i sin ( φ + α ) sin ( δ β d / 2 ) ] × exp [ i ( N x x + N y y ) d / 2 ] ,
tan α = [ 2 N x y / ( N x x N y y ) ] .
δ β = ( N x x N y y ) 2 + | 2 N x y | 2 .
[ δ ε e ] ξζ z = [ δ ε e ξ 0 0 0 δ ε e ζ 0 0 0 δ ε e z ] ,
δ ε e ξ = 2 n 0 [ C 1 σ e ξ + C 2 ( σ e ζ + σ e z ) ] , δ ε e ζ = 2 n 0 [ C 1 σ e ζ + C 2 ( σ e ξ + σ e z ) ] , δ ε e z = 2 n 0 [ C 1 σ e z + C 2 ( σ e ξ + σ e ζ ) ] ,
σ e ξ = 3 F / π b ,
σ e ζ = F / π b ;
[ δ ε e ] x y z = T 1 [ δ ε e ] ξ ζ z T = [ δ ε e x δ ε e x y 0 δ ε e x y δ ε e y 0 0 0 0 ] ,
δ ε e x = δ ε e ξ cos 2 θ p + δ ε e ζ sin 2 θ p , δ ε e y = δ ε e ξ sin 2 θ p + δ ε e ζ cos 2 θ p , δ ε e x y = ( 1 / 2 ) ( δ ε e ξ δ ε e ζ ) sin ( 2 θ p ) .
T = [ cos θ p sin θ p 0 sin θ p cos θ p 0 0 0 1 ] .
N x x N y y = ( 2 π / λ ) [ B C ( 4 F / π d ) cos ( 2 θ p ) ] ,
N x y = N y x = ( π / λ ) ( 4 F / π d ) C sin ( 2 θ p ) ,
S = ( r / 2 ) 0 P ( ω ) [ sin ( 2 α ) sin 2 ( δ β d / 2 ) ] d ω .
S / S 0 = 0 P ( ω ) sin ( 2 α ) sin 2 ( δ β d / 2 ) d ω / 2 0 P ( ω ) d ω ,
S / S 0 = ( 1 / 4 ) sin ( 2 α ) { 1 cos ( 2 π d K / λ 0 ) exp [ ( π d Δ λ K / λ 0 2 ) 2 / 2 ] } ,
K = [ B 2 + C 2 ( 4 F / π b ) 2 2 C B ( 4 F / π b ) cos ( 2 θ p ) ] 1 / 2 ,
h = 1 2 1 2 ( 1 1 4 sin 2 ( 2 α ) { 1 cos ( 2 π d K λ 0 ) × exp [ 1 2 ( π d Δ λ K λ 0 ) 2 ] } 2 ) 1 / 2 ,
h = ( 1 / 16 ) sin 2 ( 2 α ) { 1 cos ( 2 π d K / λ 0 ) exp [ ( π Δ λ d K / λ 0 2 ) 2 / 2 ] } 2 .
η = h L .
E ( z ) = 0 Q ( ω ) [ A ( z ) e x + B ( z ) e y ] exp ( i ω t ) d ω ,
d d z [ A ( z ) B ( z ) ] = i [ N x x N x y N y x N y y ] [ A ( z 0 ) B ( z 0 ) ] ,
N μ μ = β μ + ω ε 0 0 2 π 0 E μ * [ δ ε e ] E μ r d r d θ , ( μ = x , y ) ,
N x y = N y x * = ω ε 0 0 2 π 0 E x * [ δ ε e ] E y r d r d θ .
N x x N y y = β x β y + ω ε 0 0 2 π 0 [ δ ε e x δ ε e y ] | E | 2 r d r d θ ,
N x x N y y = ( 2 π / λ ) C [ ( σ i x σ i y ) + ( σ e ξ σ e ζ ) 0 cos ( 2 θ p ) ] H ( V ) = ( 2 π / λ ) [ B + C ( σ e ξ σ e ζ ) 0 H ( V ) cos ( 2 θ p ) ,
H ( V ) = 0 2 π 0 ( σ e ξ σ e ζ ) | E | 2 r d r d θ ( σ e ξ σ e ζ ) 0 0 2 π 0 | E | 2 r d r d θ ,
V = ( 2 π / λ ) n 0 a 2 Δ i ,
N x y = ( 1 / 2 ) 0 2 π 0 ω ε 0 ( δ ε e x δ ε e y ) | E | 2 sin ( 2 θ p ) r d r d θ = ( π / λ ) C ( σ e ξ σ e ζ ) 0 H ( V ) sin ( 2 θ p ) .

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