Abstract

The occurrence of linear birefringence is inevitable when dealing with fiber optics. Intrinsic birefringence can be minimized, but deploying the fiber on an experiment will introduce stress birefringence due to bending and pressure. We have studied the effects of this extraneous linear birefringence on the measurement of current-induced circular birefringence in a fiber which also has a strong twist circular birefringence bias. Orienting the analyzing polarizer by a proscribed procedure gives minimum error. Quantitative error limits on the current for given fiber coil radii and winding tensile stress are calculated. Additional restrictions on the fiber lead-in and lead-out sections are discussed.

© 1988 Optical Society of America

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References

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  1. G. I. Chandler, P. R. Forman, F. C. Jahoda, K. A. Kare, “Fiber-Optic Heterodyne Phase-Shift Measurements of Plasma Current,” Appl. Opt. 25, 1770 (1986).
    [CrossRef] [PubMed]
  2. 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 (1969).
    [CrossRef]
  3. R. Ulrich, A. Simon, “Polarization Optics of Twisted Single-Moded Fibers,” Appl. Opt. 18, 2241 (1979).
    [CrossRef] [PubMed]
  4. e.g., H. S. Lassing, W. J. Mastop, A. F. G. van der Meer, A. A. M. Oomens, “Plasma Current Measurements by Faraday Rotation in a Single-Mode Fiber,” Appl. Opt. 26, 2456 (1987).
    [CrossRef] [PubMed]
  5. S. C. Rashleigh, “Origins and Control of Polarization Effects in Single-Mode Fibers,” IEEE/OSA J. Lightwave Technol. LT-1, 312 (1983).
    [CrossRef]
  6. S. C. Rashleigh, R. Ulrich, “High-Birefringence in Tension Coiled Single-Mode Fibers,” Opt. Lett. 5, 345 (1980).
    [CrossRef]
  7. H. S. Lassing, A. A. M. Oomens, R. Woltjer, P. C. T. van der Laan, G. G. Wolzak, “Development of a Magneto-Optic Current Sensor for High, Pulsed Currents,” Rev. Sci. Instrum. 57, 851 (1986).
    [CrossRef]
  8. H. Hurwitz, R. Clark Jones, “A New Calculus for the Treatment of Optical Systems, II,” J. Opt. Soc. Am. 31, 493 (1941).
    [CrossRef]

1987 (1)

1986 (2)

G. I. Chandler, P. R. Forman, F. C. Jahoda, K. A. Kare, “Fiber-Optic Heterodyne Phase-Shift Measurements of Plasma Current,” Appl. Opt. 25, 1770 (1986).
[CrossRef] [PubMed]

H. S. Lassing, A. A. M. Oomens, R. Woltjer, P. C. T. van der Laan, G. G. Wolzak, “Development of a Magneto-Optic Current Sensor for High, Pulsed Currents,” Rev. Sci. Instrum. 57, 851 (1986).
[CrossRef]

1983 (1)

S. C. Rashleigh, “Origins and Control of Polarization Effects in Single-Mode Fibers,” IEEE/OSA J. Lightwave Technol. LT-1, 312 (1983).
[CrossRef]

1980 (1)

1979 (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 (1969).
[CrossRef]

1941 (1)

Chandler, G. I.

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 (1969).
[CrossRef]

Clark Jones, R.

Forman, P. R.

Hurwitz, H.

Jahoda, F. C.

Kare, K. A.

Lassing, H. S.

e.g., H. S. Lassing, W. J. Mastop, A. F. G. van der Meer, A. A. M. Oomens, “Plasma Current Measurements by Faraday Rotation in a Single-Mode Fiber,” Appl. Opt. 26, 2456 (1987).
[CrossRef] [PubMed]

H. S. Lassing, A. A. M. Oomens, R. Woltjer, P. C. T. van der Laan, G. G. Wolzak, “Development of a Magneto-Optic Current Sensor for High, Pulsed Currents,” Rev. Sci. Instrum. 57, 851 (1986).
[CrossRef]

Mastop, W. J.

Oomens, A. A. M.

e.g., H. S. Lassing, W. J. Mastop, A. F. G. van der Meer, A. A. M. Oomens, “Plasma Current Measurements by Faraday Rotation in a Single-Mode Fiber,” Appl. Opt. 26, 2456 (1987).
[CrossRef] [PubMed]

H. S. Lassing, A. A. M. Oomens, R. Woltjer, P. C. T. van der Laan, G. G. Wolzak, “Development of a Magneto-Optic Current Sensor for High, Pulsed Currents,” Rev. Sci. Instrum. 57, 851 (1986).
[CrossRef]

Rashleigh, S. C.

S. C. Rashleigh, “Origins and Control of Polarization Effects in Single-Mode Fibers,” IEEE/OSA J. Lightwave Technol. LT-1, 312 (1983).
[CrossRef]

S. C. Rashleigh, R. Ulrich, “High-Birefringence in Tension Coiled Single-Mode Fibers,” Opt. Lett. 5, 345 (1980).
[CrossRef]

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 (1969).
[CrossRef]

Ulrich, R.

van der Laan, P. C. T.

H. S. Lassing, A. A. M. Oomens, R. Woltjer, P. C. T. van der Laan, G. G. Wolzak, “Development of a Magneto-Optic Current Sensor for High, Pulsed Currents,” Rev. Sci. Instrum. 57, 851 (1986).
[CrossRef]

van der Meer, A. F. G.

Woltjer, R.

H. S. Lassing, A. A. M. Oomens, R. Woltjer, P. C. T. van der Laan, G. G. Wolzak, “Development of a Magneto-Optic Current Sensor for High, Pulsed Currents,” Rev. Sci. Instrum. 57, 851 (1986).
[CrossRef]

Wolzak, G. G.

H. S. Lassing, A. A. M. Oomens, R. Woltjer, P. C. T. van der Laan, G. G. Wolzak, “Development of a Magneto-Optic Current Sensor for High, Pulsed Currents,” Rev. Sci. Instrum. 57, 851 (1986).
[CrossRef]

Appl. Opt. (3)

IEEE/OSA J. Lightwave Technol. (1)

S. C. Rashleigh, “Origins and Control of Polarization Effects in Single-Mode Fibers,” IEEE/OSA J. Lightwave Technol. LT-1, 312 (1983).
[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 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

H. S. Lassing, A. A. M. Oomens, R. Woltjer, P. C. T. van der Laan, G. G. Wolzak, “Development of a Magneto-Optic Current Sensor for High, Pulsed Currents,” Rev. Sci. Instrum. 57, 851 (1986).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Plot of Γ − Γ0, the apparent change in phase, vs 2FL, the actual change in phase due to Faraday rotation. Two orientations of the polarizer are presented, 45 and 0°, and three values of δ/2T are assumed for each graph; 0.0,0.4,0.8.

Fig. 2
Fig. 2

Plot of ratio of Faraday rotation integral with offset current distribution to the same integral but with the current centered. The assumed values of δ, F, and T are appropriate for our experimental data.

Fig. 3
Fig. 3

Schematic drawing of experimental setup used to measure current.

Fig. 4
Fig. 4

Observed intensity of transmitted light vs polarizer orientation when circularly polarized light is injected into the fiber.

Fig. 5
Fig. 5

Fiber sensor current measurement plotted along with conventional Rogowski coil current traces for data which are close to the extremes in polarizer orientation. The dashed line is the fiber sensor measurement. Above each graph is the corresponding Lissajous figure along with a least-squares fit to a circle to that data.

Fig. 6
Fig. 6

Normalized sum of the squares of the deviations of fits of the Lissajous figures to a centered circle vs polarizer orientation.

Fig. 7
Fig. 7

Best fit of the data to theoretical prediction. To achieve this fit an artificial shift of 15° in the angle θ was required. This shift in the same direction as the shift observed in the previous figure. Two values of δ yield equally good fits, 0.0112 and 0.0214.

Fig. 8
Fig. 8

Schematic of exclusive or phase detector circuit which yields direct analog signal corresponding to the phase, Γ − Γ0.

Equations (29)

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( E x E y ) out = ( A - B B A * ) ( E x E y ) in ,
A = cos ( ϕ / 2 ) + i sin ( ϕ / 2 ) cos χ ,
B = sin ( ϕ / 2 ) sin χ ,
ϕ = [ ( δ ) 2 + ( 2 F + 2 T ) 2 ] 1 / 2 L ,
tan χ = ( 2 F + 2 T ) / δ .
( E x E y ) out = ( cos 2 θ sin θ cos θ sin θ cos θ sin 2 θ ) ( A - B B A * ) ( cos ω t / 2 sin ω t / 2 ) in ,
E 2 = [ 1 + cos ( ω t + Γ ) ] / 2 ,
tan Γ = sin 2 θ sin γ + cos 2 θ sin ( Ω + π / 2 ) sin 2 θ cos γ + cos 2 θ cos ( Ω + π / 2 ) ,
tan γ = sin ϕ sin χ cos ϕ
tan Ω = sin ϕ sin χ cos 2 ϕ / 2 + sin 2 ϕ / 2 cos 2 χ ,
ϕ ( ν ) = 0 2 π R { ( δ ) 2 + [ 2 F ( s ) · e ^ + 2 T ] 2 } 1 / 2 d s ,
δ b = 0.25 k n 3 ( p 11 - p 12 ) ( 1 + ν P ) ρ 2 R 2 ,
δ t c = k ( n 3 / 2 ) ( p 11 - p 12 ) ( 1 + ν P ) ( 2 - 3 ν P ) ( 1 - ν P ) ρ R z ,
( E x E y ) = ( cos 2 θ sin θ cos θ sin θ cos θ sin 2 θ ) ( exp ( - τ ) 0 0 exp ( - i τ ) ) ( 1 i ) ,
E 2 = 1 + sin 2 θ sin 2 τ .
E max 2 - E max 2 E max 2 + E min 2 = sin 2 τ = 0.575.
n = 1.46 , ν P = 0.17 , ( p 11 - p 12 ) = - 0.15 , k = 2 π / λ vac , and λ vac = 0.633 μ m .
δ t c = - 4.87 × 10 4 ρ R z rad / cm .
M in = exp ( i μ ) ( cos α - sin α sin α cos α ) ( exp ( i τ ) 0 0 exp ( - i τ ) ) ( cos - sin sin cos ) ,
[ cos α exp ( i 2 τ ) cos ( ω t / 2 ) - sin α sin ( ω t / 2 ) sin α exp ( i 2 τ ) cos ( ω t / 2 ) + cos α sin ( ω t / 2 ) ] ,
E E * = ½ [ 1 + X sin ( ω t + ) + Y cos ( ω t + ) ] ,
X = [ cos χ sin ϕ sin 2 θ + sin 2 χ sin 2 ( ϕ / 2 ) cos 2 θ ] sin 2 τ + ( cos 2 θ { sin 2 α [ cos 2 ( ϕ / 2 ) + sin 2 ( ϕ / 2 ) cos 2 χ ] - cos 2 α sin χ sin ϕ } - sin 2 θ [ sin 2 α sin χ sin ϕ + cos 2 α cos ϕ ] ) cos 2 τ , Y = cos 2 θ { cos 2 α [ cos 2 ( ϕ / 2 ) + sin 2 ( ϕ / 2 ) cos 2 χ ] + sin 2 α sin χ sin ϕ } + sin 2 θ [ sin 2 α cos ϕ - cos 2 α sin χ sin ϕ ] .
{ cos [ ( ϕ 2 / 2 ) d l ] + i sin [ ( ϕ 2 / 2 ) d l ] cos χ 2 - sin [ ( ϕ 2 / 2 ) d l ] sin χ 2 sin [ ( ϕ 2 / 2 ) d l ] sin χ 2 cos [ ( ϕ 2 / 2 ) d l ] - i sin [ ( ϕ 2 / 2 ) d l ] cos χ 2 } × { cos [ ( ϕ 1 / 2 ) d l ] + i sin [ ( ϕ 1 / 2 ) d l ] cos χ 1 - sin [ ( ϕ 1 / 2 ) d l ] sin χ 1 sin [ ( ϕ 1 / 2 ) d l ] sin χ 1 cos [ ( ϕ 1 / 2 ) d l ] - i sin [ ( ϕ 1 / 2 ) d l ] cos χ 1 } = ( A - B B A * ) ,
A = cos [ ( ϕ 1 / 2 ) d l ] cos [ ( ϕ 2 / 2 ) d l ] - sin [ ( ϕ 1 / 2 ) d l ] sin [ ( ϕ 2 / 2 ) d l ] cos ( χ 2 - χ 1 ) + i [ cos [ ( ϕ 2 / 2 ) d l ] sin [ ( ϕ 1 / 2 ) d l ] cos χ 1 + cos [ ( ϕ 1 / 2 ) d l ] sin [ ( ϕ 2 ) d l ] cos χ 2 } , B = sin χ 1 sin [ ( ϕ 1 / 2 ) d l ] cos [ ( ϕ 2 / 2 ) d l ] + sin χ 2 sin [ ( ϕ 2 / 2 ) d l ] cos [ ( ϕ 1 / 2 ) d l ] - i sin [ ( ϕ 1 / 2 ) d l ] sin [ ( ϕ 2 / 2 ) d l ] sin ( χ 2 - χ 1 ) .
( A - B B A * ) = ( cos ϕ / 2 + i sin ϕ / 2 cos χ - sin ϕ / 2 sin χ sin ϕ / 2 sin χ cos ϕ / 2 - i sin ϕ / 2 cos χ ) ,
[ cos ( ϕ e / 2 ) + i sin ( ϕ e / 2 ) cos χ e - sin ( ϕ e / 2 ) sin χ e sin ( ϕ e / 2 ) sin χ e cos ( ϕ e / 2 ) - i sin ( ϕ e / 2 ) cos χ e ] ,
cos τ = cos 2 ( ϕ e / 2 ) + sin 2 ( ϕ e / 2 ) sin 2 χ e .
E E * = 1 / 2 ( 1 + G cos ω t + H sin ω t ) ,
G = [ cos 2 ( ϕ / 2 ) + sin 2 ( ϕ / 2 ) cos 2 χ ] × ( cos 2 α 2 cos 2 θ - sin 2 α 2 sin 2 θ cos 2 τ 2 ) + sin 2 ( ϕ / 2 ) sin 2 χ sin 2 θ sin 2 τ 2 - sin ϕ sin χ ( sin 2 α 2 cos 2 θ + cos 2 α 2 sin 2 θ cos 2 τ 2 ) , H = - sin ϕ sin χ ( cos 2 α 2 cos 2 θ - sin 2 α 2 sin 2 θ cos 2 τ 2 ) + sin ϕ cos χ sin 2 θ sin 2 τ 2 - cos ϕ ( sin 2 α 2 cos 2 θ + cos 2 α 2 sin 2 θ cos 2 τ 2 ) .

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