Abstract

The theoretical study of the fiber optic Faraday effect based on the mode theory is conducted for the first time, to my knowledge. An analytical formula that represents the Faraday effect in a single-mode fiber under the appropriate approximation is given as an explicit function of the refractive indices and Verdet constants of the core and cladding, the υ-value of the fiber, the magnetic field, and the fiber length. The difference between the fiber optic and bulk-optic Faraday effects is made clear. The effect of the modal field spreading in the core and cladding on the fiber optic Faraday effect is examined both analytically and numerically.

© 2005 Optical Society of America

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References

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  1. J.Dakin and B.Culshaw, eds., Optical Fiber Sensors (Artech House, 1989).
  2. T. Yoshino, M. Yokota, and T. Kenmochi, "High-speed all-fibre polarization controller," Electron. Lett. 39, 1800-1802 (2003).
    [CrossRef]
  3. M. Yokota, Y. Sato, I. Yamaguchi, T. Kenmochi, and T. Yoshino, "A compact polarimetric glucose sensor using a high-performance fiber-optic Faraday rotator," Meas. Sci. Technol. 15, 143-147 (2004).
    [CrossRef]
  4. T. Yoshino and S. Tanaka, "Longitudinal magneto-optic effect in ferromagnetic thin films. I," Jpn. J. Appl. Phys. 5, 989-993 (1966).
    [CrossRef]
  5. D. Marcuse, Light Transmission Optics (Van Nostrand Rheinhold, 1972), p. 13.
  6. D. Gloge, "Weakly guiding fibers," Appl. Opt. 10, 2252-2258 (1972).
    [CrossRef]
  7. L. B. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, 1983), pp. 22-23.

2004 (1)

M. Yokota, Y. Sato, I. Yamaguchi, T. Kenmochi, and T. Yoshino, "A compact polarimetric glucose sensor using a high-performance fiber-optic Faraday rotator," Meas. Sci. Technol. 15, 143-147 (2004).
[CrossRef]

2003 (1)

T. Yoshino, M. Yokota, and T. Kenmochi, "High-speed all-fibre polarization controller," Electron. Lett. 39, 1800-1802 (2003).
[CrossRef]

1972 (1)

1966 (1)

T. Yoshino and S. Tanaka, "Longitudinal magneto-optic effect in ferromagnetic thin films. I," Jpn. J. Appl. Phys. 5, 989-993 (1966).
[CrossRef]

Gloge, D.

Jeunhomme, L. B.

L. B. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, 1983), pp. 22-23.

Kenmochi, T.

M. Yokota, Y. Sato, I. Yamaguchi, T. Kenmochi, and T. Yoshino, "A compact polarimetric glucose sensor using a high-performance fiber-optic Faraday rotator," Meas. Sci. Technol. 15, 143-147 (2004).
[CrossRef]

T. Yoshino, M. Yokota, and T. Kenmochi, "High-speed all-fibre polarization controller," Electron. Lett. 39, 1800-1802 (2003).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Rheinhold, 1972), p. 13.

Sato, Y.

M. Yokota, Y. Sato, I. Yamaguchi, T. Kenmochi, and T. Yoshino, "A compact polarimetric glucose sensor using a high-performance fiber-optic Faraday rotator," Meas. Sci. Technol. 15, 143-147 (2004).
[CrossRef]

Tanaka, S.

T. Yoshino and S. Tanaka, "Longitudinal magneto-optic effect in ferromagnetic thin films. I," Jpn. J. Appl. Phys. 5, 989-993 (1966).
[CrossRef]

Yamaguchi, I.

M. Yokota, Y. Sato, I. Yamaguchi, T. Kenmochi, and T. Yoshino, "A compact polarimetric glucose sensor using a high-performance fiber-optic Faraday rotator," Meas. Sci. Technol. 15, 143-147 (2004).
[CrossRef]

Yokota, M.

M. Yokota, Y. Sato, I. Yamaguchi, T. Kenmochi, and T. Yoshino, "A compact polarimetric glucose sensor using a high-performance fiber-optic Faraday rotator," Meas. Sci. Technol. 15, 143-147 (2004).
[CrossRef]

T. Yoshino, M. Yokota, and T. Kenmochi, "High-speed all-fibre polarization controller," Electron. Lett. 39, 1800-1802 (2003).
[CrossRef]

Yoshino, T.

M. Yokota, Y. Sato, I. Yamaguchi, T. Kenmochi, and T. Yoshino, "A compact polarimetric glucose sensor using a high-performance fiber-optic Faraday rotator," Meas. Sci. Technol. 15, 143-147 (2004).
[CrossRef]

T. Yoshino, M. Yokota, and T. Kenmochi, "High-speed all-fibre polarization controller," Electron. Lett. 39, 1800-1802 (2003).
[CrossRef]

T. Yoshino and S. Tanaka, "Longitudinal magneto-optic effect in ferromagnetic thin films. I," Jpn. J. Appl. Phys. 5, 989-993 (1966).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

T. Yoshino, M. Yokota, and T. Kenmochi, "High-speed all-fibre polarization controller," Electron. Lett. 39, 1800-1802 (2003).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Yoshino and S. Tanaka, "Longitudinal magneto-optic effect in ferromagnetic thin films. I," Jpn. J. Appl. Phys. 5, 989-993 (1966).
[CrossRef]

Meas. Sci. Technol. (1)

M. Yokota, Y. Sato, I. Yamaguchi, T. Kenmochi, and T. Yoshino, "A compact polarimetric glucose sensor using a high-performance fiber-optic Faraday rotator," Meas. Sci. Technol. 15, 143-147 (2004).
[CrossRef]

Other (3)

D. Marcuse, Light Transmission Optics (Van Nostrand Rheinhold, 1972), p. 13.

L. B. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, 1983), pp. 22-23.

J.Dakin and B.Culshaw, eds., Optical Fiber Sensors (Artech House, 1989).

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Figures (2)

Fig. 1
Fig. 1

Schematic cross section of gyroelectric circular fiber and coordinate system.

Fig. 2
Fig. 2

Weighting functions α c and α cl for Faraday effect in a single-mode fiber calculated as a function of υ value.

Equations (67)

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[ ϵ ] = [ ϵ 0 ϵ 1 0 ϵ 1 ϵ 0 0 0 0 ϵ 0 ] ,
rot E = μ 0 H t ,
rot H = D t ,
D = [ ϵ ] E ,
div D = 0 ,
[ E x , E y , E z ] = [ A ( x , y ) , B ( x , y ) , C ( x , y ) ] exp [ i ( ω t β z ) ] . ( i 2 = 1 ) .
div D = [ ϵ 0 ( A x + B y + C z ) + ϵ 1 ( B x A y ) ] × exp [ i ( ω t β z ) ] = 0 ,
div E = ( ϵ 1 ϵ 0 ) ( B x A y ) .
2 A x 2 + 2 A y 2 β 2 A + ( ϵ 1 ϵ 0 ) ( 2 B x 2 2 A x y ) + ω 2 μ 0 ϵ 0 A + ω 2 μ 0 ϵ 1 B = 0 ,
2 B x 2 + 2 B y 2 β 2 B + ( ϵ 1 ϵ 0 ) ( 2 A y 2 + 2 B x y ) + ω 2 μ 0 ϵ 0 B ω 2 μ 0 ϵ 1 A = 0 ,
C = [ ( ϵ 1 ϵ 0 ) ( B x A y ) + ( A x + B y ) ] ( i β ) ,
ω 2 μ 0 ϵ 1 X ( ϵ 1 ϵ 0 ) 2 Y x y , ( ϵ 1 ϵ 0 ) 2 Y x 2 , ( ϵ 1 ϵ 0 ) 2 Y y 2 ( X , Y A or B ) .
2 A x 2 + 2 A y 2 β 2 A + ω 2 μ 0 ϵ 0 A + ω 2 μ 0 ϵ 1 B = 0 ,
2 B x 2 + 2 B y 2 β 2 B + ω 2 μ 0 ϵ 0 B ω 2 μ 0 ϵ 1 A = 0 .
B = p A ,
2 A x 2 + 2 A y 2 + [ ( β 2 + ω 2 μ 0 ϵ 0 ) + ω 2 μ 0 ϵ 1 p ] A = 0 ,
p ( 2 A x 2 + 2 A y 2 ) + [ ( β 2 + ω 2 μ 0 ϵ 0 ) p ω 2 μ 0 ϵ 1 ] A = 0 .
1 p = [ ( β 2 + ω 2 μ 0 ϵ 0 ) + ω 2 μ 0 ϵ 1 p ] [ ( β 2 + ω 2 μ 0 ϵ 0 ) p ω 2 μ 0 ϵ 1 ] ,
p = ± i ,
2 A ± x 2 + 2 A ± y 2 β 2 A ± + ω 2 μ 0 ( ϵ 0 ± i ϵ 1 ) A ± = 0 .
n 2 ± = ϵ 0 ± i ϵ 1 ,
ϵ 1 = i Q ϵ 0 ,
n ± = n 0 ( 1 ± Q 2 ) ( Q 1 ) ,
n c , ± = n c , 0 ( 1 ± Q c 2 ) ,
n cl , ± = n cl , 0 ( 1 ± Q cl 2 ) ,
H = [ C y + i β B , i β A C x , B x A y ] × [ i ( ω μ 0 ) ] exp [ i ( ω t β z ) ] ,
H = [ i β B , i β A , B x A y ] [ i ( ω μ 0 ) ] exp [ i ( ω t β z ) ] ,
H = [ β A , i β A , ± i A x A y ] [ i ( ω μ 0 ) ] exp [ i ( ω t β z ) ] ,
[ E x , E y , E z ] = [ A ± , ± i A ± , ( i A ± x ± A ± y ) × ( 1 ± Q ) β ] exp [ i ( ω t β ± z ) ] ,
[ H x , H y , H z ] = [ β ± A ± , i β ± A ± , ± i A ± x A ± y ] × [ i ( ω μ 0 ) exp ] [ i ( ω t β ± z ) ] ,
2 A ± x 2 + 2 A ± y 2 β ± 2 A ± + ω 2 μ 0 { n 2 c , ± n 2 cl , ± } A ± = 0 { in core in cladding } ,
υ 2 = ( k a ) 2 ( n c , 0 2 n cl , 0 2 ) ,
β 0 2 = k 2 n cl , 0 2 + ( 1.1428 υ 0.9960 ) 2 a 2 ,
β ± 2 = k 2 n cl , ± 2 + ( 1.1428 υ ± 0.9960 ) 2 a 2 ,
υ 2 ± = ( k a ) 2 ( n c , ± 2 n cl , ± 2 ) ,
Δ n c = n c , + n c , = n c , 0 Q c ,
Δ n cl = n cl , + n cl , = n cl , 0 Q cl .
n c , + + n c , = 2 n c , 0 ,
n cl , + + n cl , = 2 n cl , 0 ,
υ + υ = ( υ + 2 υ 2 ) ( 2 υ ) ,
β + β = ( β + 2 β 2 ) ( 2 β 0 ) .
Δ β = β + β .
β + 2 β 2 = k 2 ( n cl , + 2 n cl , 2 ) + [ ( 1.1428 υ + 0.9960 ) 2
[ ( 1.1428 υ + 0.9960 ) 2 ] a 2 [ Eq. ( 29 ) ]
= k 2 ( n cl , + 2 n cl , 2 ) + [ 1.1428 2 ( υ + 2 υ 2 )
[ 2 × 1.1428 × 0.9960 ( υ + υ ) ] a 2
= k 2 ( n cl , + 2 n cl , 2 ) + ( υ + 2 υ 2 ) ( 1.1428 2
( 1.1428 × 0.9960 υ ) a 2 [ Eq. ( 32 c ) ]
= k 2 ( n cl , + 2 n cl , 2 ) + k 2 [ ( n c , + 2 n c , 2 )
[ ( n cl , + 2 n cl , 2 ) ] ( 1.3060 1.1382 υ ) [ Eq. ( 30 ) ]
= 2 k 2 n cl , 0 Δ n cl + 2 k 2 ( n c , 0 Δ n c n cl , 0 Δ n cl ) ( 1.3060
( 1.1382 υ ) [ Eqs. ( 31 a ) ( 32 b ) ]
= 2 k 2 [ ( 1.3060 1.1382 υ ) n c , 0 Δ n c + ( 0.3060
( + 1.1382 υ ) [ n cl , 0 Δ n cl ] ,
Δ β = k 2 ( α c n c , 0 Δ n c + α cl n cl , 0 Δ n cl ) β 0 .
α c = 1.306 1.138 υ ,
α cl = 0.306 + 1.138 υ ,
α c + α cl = 1 .
β 0 = k ( γ c n c , 0 2 + γ cl n cl , 0 2 ) 1 2 ,
γ c = 1.306 ( 2.277 υ 0.992 ) υ 2 ,
γ cl = 0.306 + ( 2.277 υ 0.992 ) υ 2 ,
γ c + γ cl = 1 .
ϕ = ( Δ β 2 ) L ,
ϕ = ( k 2 ) ( α c n c , 0 Δ n c + α cl n cl , 0 Δ n cl ) L ( γ c n c , 0 2 + γ cl n cl , 0 2 ) 1 2 .
k Δ n c = 2 V c H ,
k Δ n cl = 2 V cl H .
ϕ = ( α c n c , 0 V c + α cl n cl , 0 V cl ) H L ( γ c n c , 0 2 + γ cl n cl , 0 2 ) 1 2 .

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