Abstract

The effects of birefringence and total internal reflection on the performance of current sensors that use the Faraday effect have been modeled numerically. The results show that, when these effects are present, the output electric field azimuth is affected by the position of the current-carrying conductor. In sensors that use highly circular birefringent fibers, however, the output azimuth is not affected.

© 1992 Optical Society of America

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References

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  1. B. Culshaw, J. Dakin, eds., Optical Fiber Sensors: Systems and Applications (Artech, Norwood, Mass., 1980), Vol. 2, p. 746.
  2. A. Papp, H. Harms, “Magnetooptical current transformer. 1: Principles,” Appl. Opt. 19, 3729–3743 (1980).
    [CrossRef] [PubMed]
  3. S. P. Bush, D. A. Jackson, “Dual channel Faraday effect current sensor capable of simultaneous measurement of two independent currents,” Opt. Lett. 16, 955–957 (1991).
    [CrossRef] [PubMed]
  4. G. I. Chandler, F. C. Jahoda, “Current measurements by Faraday rotation in single-mode optical fibers,” Rev. Sci. Instrum. 56, 852–854 (1985).
    [CrossRef]
  5. A. M. Smith, “Polarization and magnetooptic properties of single-mode optical fiber,” Appl. Opt. 17, 52–56 (1978).
    [CrossRef] [PubMed]
  6. P. R. Forman, F. C. Jahoda, “Linear birefringence effects on fiber-optic current sensors,” Appl. Opt. 27, 3088–3096 (1988).
    [CrossRef] [PubMed]
  7. R. I. Laming, D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” IEEE J. Lightwave Technol. 7, 2084–2094 (1989).
    [CrossRef]
  8. F. Maystre, A. Bertholds, “Magneto-optic current sensor using a helical fiber Fabry–Perot resonator,” Opt. Lett. 14, 587–589 (1989).
    [CrossRef] [PubMed]
  9. J. Stone, “Stress-optic effects, birefringence, and reduction of birefringence by annealing in fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. 6, 1245–1248 (1988).
    [CrossRef]
  10. D. Tang, G. W. Day, “Progress in the development of miniature optical fiber current sensors,” in Proceedings of the IEEE Lasers and Electro-Optics Society 1988 Annual MeetingInstitute of Electrical and Electronics Engineers, New York, 1988), pp. 306–307.
  11. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 38, 671–685 (1948).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).
  13. A. D. Kersey, D. A. Jackson, “Current sensing utilizing heterodyne detection of the Faraday effect in single-mode optical fiber,” IEEE J. Lightwave Technol. LT-4, 640–643 (1986).
    [CrossRef]
  14. R. Ulrich, S. C. Rashleigh, W. Eichoff, “Bending-induced birefringence in single-mode fibers,” Opt. Lett. 5, 273–275 (1980).
    [CrossRef] [PubMed]
  15. 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]

1991 (1)

1989 (2)

R. I. Laming, D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” IEEE J. Lightwave Technol. 7, 2084–2094 (1989).
[CrossRef]

F. Maystre, A. Bertholds, “Magneto-optic current sensor using a helical fiber Fabry–Perot resonator,” Opt. Lett. 14, 587–589 (1989).
[CrossRef] [PubMed]

1988 (2)

P. R. Forman, F. C. Jahoda, “Linear birefringence effects on fiber-optic current sensors,” Appl. Opt. 27, 3088–3096 (1988).
[CrossRef] [PubMed]

J. Stone, “Stress-optic effects, birefringence, and reduction of birefringence by annealing in fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. 6, 1245–1248 (1988).
[CrossRef]

1986 (1)

A. D. Kersey, D. A. Jackson, “Current sensing utilizing heterodyne detection of the Faraday effect in single-mode optical fiber,” IEEE J. Lightwave Technol. LT-4, 640–643 (1986).
[CrossRef]

1985 (1)

G. I. Chandler, F. C. Jahoda, “Current measurements by Faraday rotation in single-mode optical fibers,” Rev. Sci. Instrum. 56, 852–854 (1985).
[CrossRef]

1980 (2)

1978 (1)

1969 (1)

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]

1948 (1)

Bertholds, A.

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).

Bush, S. P.

Chandler, G. I.

G. I. Chandler, F. C. Jahoda, “Current measurements by Faraday rotation in single-mode optical fibers,” Rev. Sci. Instrum. 56, 852–854 (1985).
[CrossRef]

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]

Day, G. W.

D. Tang, G. W. Day, “Progress in the development of miniature optical fiber current sensors,” in Proceedings of the IEEE Lasers and Electro-Optics Society 1988 Annual MeetingInstitute of Electrical and Electronics Engineers, New York, 1988), pp. 306–307.

Eichoff, W.

Forman, P. R.

Harms, H.

Jackson, D. A.

S. P. Bush, D. A. Jackson, “Dual channel Faraday effect current sensor capable of simultaneous measurement of two independent currents,” Opt. Lett. 16, 955–957 (1991).
[CrossRef] [PubMed]

A. D. Kersey, D. A. Jackson, “Current sensing utilizing heterodyne detection of the Faraday effect in single-mode optical fiber,” IEEE J. Lightwave Technol. LT-4, 640–643 (1986).
[CrossRef]

Jahoda, F. C.

P. R. Forman, F. C. Jahoda, “Linear birefringence effects on fiber-optic current sensors,” Appl. Opt. 27, 3088–3096 (1988).
[CrossRef] [PubMed]

G. I. Chandler, F. C. Jahoda, “Current measurements by Faraday rotation in single-mode optical fibers,” Rev. Sci. Instrum. 56, 852–854 (1985).
[CrossRef]

Jones, R. C.

Kersey, A. D.

A. D. Kersey, D. A. Jackson, “Current sensing utilizing heterodyne detection of the Faraday effect in single-mode optical fiber,” IEEE J. Lightwave Technol. LT-4, 640–643 (1986).
[CrossRef]

Laming, R. I.

R. I. Laming, D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” IEEE J. Lightwave Technol. 7, 2084–2094 (1989).
[CrossRef]

Maystre, F.

Papp, A.

Payne, D. N.

R. I. Laming, D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” IEEE J. Lightwave Technol. 7, 2084–2094 (1989).
[CrossRef]

Rashleigh, S. C.

Smith, A. M.

Stone, J.

J. Stone, “Stress-optic effects, birefringence, and reduction of birefringence by annealing in fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. 6, 1245–1248 (1988).
[CrossRef]

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]

Tang, D.

D. Tang, G. W. Day, “Progress in the development of miniature optical fiber current sensors,” in Proceedings of the IEEE Lasers and Electro-Optics Society 1988 Annual MeetingInstitute of Electrical and Electronics Engineers, New York, 1988), pp. 306–307.

Ulrich, R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).

Appl. Opt. (3)

IEEE J. Lightwave Technol. (3)

R. I. Laming, D. N. Payne, “Electric current sensors employing spun highly birefringent optical fibers,” IEEE J. Lightwave Technol. 7, 2084–2094 (1989).
[CrossRef]

J. Stone, “Stress-optic effects, birefringence, and reduction of birefringence by annealing in fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. 6, 1245–1248 (1988).
[CrossRef]

A. D. Kersey, D. A. Jackson, “Current sensing utilizing heterodyne detection of the Faraday effect in single-mode optical fiber,” IEEE J. Lightwave Technol. LT-4, 640–643 (1986).
[CrossRef]

J. Appl. Phys. (1)

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]

J. Opt. Soc. Am. (1)

Opt. Lett. (3)

Rev. Sci. Instrum. (1)

G. I. Chandler, F. C. Jahoda, “Current measurements by Faraday rotation in single-mode optical fibers,” Rev. Sci. Instrum. 56, 852–854 (1985).
[CrossRef]

Other (3)

B. Culshaw, J. Dakin, eds., Optical Fiber Sensors: Systems and Applications (Artech, Norwood, Mass., 1980), Vol. 2, p. 746.

D. Tang, G. W. Day, “Progress in the development of miniature optical fiber current sensors,” in Proceedings of the IEEE Lasers and Electro-Optics Society 1988 Annual MeetingInstitute of Electrical and Electronics Engineers, New York, 1988), pp. 306–307.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).

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Figures (12)

Fig. 1
Fig. 1

Bulk glass sensor head: S,, source; I, current-carrying conductor; x, y, Jones matrices reference axes; 1,2,3,4, Faraday effect Jones matrices; 1,2,3, reflection Jones matrices; I′, external current. All currents flow into the page.

Fig. 2
Fig. 2

Geometry that was used to calculate the Faraday rotation along a straight path a perpendicular distance s away from a current-carrying conductor I.

Fig. 3
Fig. 3

Fiber loop sensor head: S, source; O, center of loop; (p, ξ), (q, σ), polar coordinates of point P and internal current position Q, respectively, I′, external current; dl, path element; Bt, tangential component of the magnetic field that is due to the conductor at Q.

Fig. 4
Fig. 4

Output electric field azimuth and ellipticity versus input azimuth for a square bulk glass sensor head of refractive index n = 1.5.

Fig. 5
Fig. 5

Deviation from linearity for a square bulk glass sensor head of refractive index n = 1.5.

Fig. 6
Fig. 6

Heterodyne intensity signal for an ac that produces a Faraday modulation of 20° at half of the carrier frequency.

Fig. 7
Fig. 7

Deviation of the output azimuth from that for a central current versus conductor placement for the square bulk glass sensor head. The square lies in the X, Y plane and the light enters the square at X = Y = 0 while it travels in the positive X direction as indicated by the arrows.

Fig. 8
Fig. 8

Deviation from zero of the output azimuth for an external current at position I′ in Fig. 1 for the square bulk glass sensor head.

Fig. 9
Fig. 9

Deviation from linearity for a fiber loop sensor head with a linear birefringence of 10°.

Fig. 10
Fig. 10

Heterodyne intensity signal for an ac that produces a Faraday modulation of 20° at half of the carrier frequency for a fiber loop with 40° of linear birefringence.

Fig. 11
Fig. 11

Deviation of the output azimuth from that for a central current versus conductor placement for the fiber loop sensor head. The loop lies in the X, Y plane and the light enters the loop at X = 1.0, Y = 0 while it travels in the positive Y direction as indicated by the arrows.

Fig. 12
Fig. 12

Deviation from zero of the output azimuth for an external current at position I′ in Fig. 3 for the fiber loop head as a function of linear birefringence.

Equations (30)

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E out = M s · E in ,
E out = [ a exp ( i η ) b ] ,
β = ( 1 / 2 ) tan - 1 ( tan 2 ζ cos η ) ,
e = tan χ ,
I = P · S ( Ω ) · E out 2 ,
P = ( 1 0 0 0 ) ,
S ( Ω ) = ( cos Ω - sin Ω sin Ω cos Ω ) .
I = I 0 cos 2 ( θ + Ω ) ,
I = I 0 cos 2 ( ω t + θ + Ω ) .
θ = V C B · d l ,
E out = M G · E in = F 4 · R 3 · F 3 · R 2 · F 2 · R 1 · F 1 · E in ,
R = ( ρ x 0 0 ρ y ) ,
ρ x = 1 + i α 1 - i α ,
ρ y = n 0 2 + i n 2 α n 0 2 - i n 2 α ,
α = ( n 2 sin 2 ψ - n 0 2 ) 1 / 2 n cos ψ ,
Δ θ k = Γ [ tan - 1 ( z 2 s ) - tan - 1 ( z 1 s ) ] ,
δ b = K r 2 / R 2             deg / m ,
δ T = 2 π R δ b = 2 π K r 2 / R             deg / turn .
F = ( A - B B A * ) ,
A = cos ( ϕ / 2 ) + i sin ( ϕ / 2 ) cos ( ɛ ) ,
B = sin ( ϕ / 2 ) sin ( ɛ ) ,
( ϕ / 2 ) 2 = ( δ / 2 ) 2 + θ 2 ,
tan ( ɛ ) = 2 θ δ ,
F ( θ 2 , δ 2 ) · F ( θ 1 , δ 1 ) = F ( θ 1 , + θ 2 , δ 1 + δ 2 ) ,
M F = S ( γ m ) · F m · S ( - γ m ) · S ( γ m - 1 ) · F m - 1 · S ( - γ m - 1 ) · S ( γ 2 ) · F 2 · S ( - γ 2 ) · S ( γ 1 ) · F 1 · S ( - γ 1 ) ,
E out = M F · E in .
B t = ( μ 0 I 2 π ) p - q cos ( σ - ξ ) p 2 + q 2 - 2 p q cos ( σ - ξ ) .
θ = V B · d l = Γ p B t d ξ .
I ( ξ , σ ) = Γ { ξ - σ 2 + tan - 1 [ p + q p - q tan ( ξ - σ 2 ) ] } .
Δ θ = I ( σ , ξ + Δ ξ ) - I ( σ , ξ ) .

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