Abstract

We analyze the linearity and modulation depth of ac magnetic-field sensors or current sensors, using a ferrimagnetic or ferromagnetic film as the Faraday rotator and employing the detection of only the zeroth-order optical diffraction component from the rotator. It is theoretically shown that for this class of sensor the condition of a constant modulation depth and that of a constant ratio error give an identical series of curves for the relationship between Faraday rotation angle Θ and polarizer/analyzer relative angle Φ. We give some numerical examples to demonstrate the usefulness of the result with reference to a rare-earth iron garnet film as the rotator.

© 1996 Optical Society of America

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References

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  1. G. W. Day, “Recent advances in Faraday effect sensors,” in Optical Fiber Sensors, Vol. 44 of Springer Proceedings in Physics (Springer-Verlag, Berlin, 1989), pp. 250–254.
  2. E. Aikawa, U. Ueda, M. Watanabe, H. Takahashi, M. Imataki, “Fiber-optic magnetic field sensor using (Cd1–xMnx)Te single crystals,” in Technical Digest of the Ninth Sensor Symposium (Tokosha, Tokyo, 1990), pp. 55–58.
  3. K. Machida, Y. Asahara, H. Ishikawa, K. Nakajima, Y. Fujii, “Magneto-optical properties of Bi-substituted epitaxial rare earth iron garnet films,” J. Appl. Phys. 61, 3256–3258 (1987).
  4. Y. Asahara, N. Nakamura, “The rare-earth iron garnet film with small temperature dependence of sensitivity used in magnetic field sensors,” in Proceedings of the Sixth International Conference on Ferrites (Japanese Society of Powder and Powder Metallurgy, Tokyo, 1990), pp. 1617–1620.
  5. T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Magneto-optical hysteresis loops of multidomain materials—calculation for fixed analyzer method,” J. Magn. Soc. Jpn. 14, 642–647 (1990) (in Japanese).
  6. T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Nonlinearity of Faraday loops,” IEEE Trans. Magn. 26, 1358–1360 (1990).
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1966), Chap. 8, p. 392.
  8. B. Kuhlow, M. Lambeck, “Light diffraction by magnetic domains,” Physica B 80, 374–380 (1975).
  9. O. Kamada, Y. Tsujimoto, Y. Hayashi, “Fiber-optic current sensors using mixed rare earth iron garnet crystals,” in Proceedings of the Third Sensor Symposium (Tokosha, Tokyo, 1983), pp. 167–169.

1990

T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Magneto-optical hysteresis loops of multidomain materials—calculation for fixed analyzer method,” J. Magn. Soc. Jpn. 14, 642–647 (1990) (in Japanese).

T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Nonlinearity of Faraday loops,” IEEE Trans. Magn. 26, 1358–1360 (1990).

1987

K. Machida, Y. Asahara, H. Ishikawa, K. Nakajima, Y. Fujii, “Magneto-optical properties of Bi-substituted epitaxial rare earth iron garnet films,” J. Appl. Phys. 61, 3256–3258 (1987).

1975

B. Kuhlow, M. Lambeck, “Light diffraction by magnetic domains,” Physica B 80, 374–380 (1975).

Aikawa, E.

E. Aikawa, U. Ueda, M. Watanabe, H. Takahashi, M. Imataki, “Fiber-optic magnetic field sensor using (Cd1–xMnx)Te single crystals,” in Technical Digest of the Ninth Sensor Symposium (Tokosha, Tokyo, 1990), pp. 55–58.

Asahara, Y.

K. Machida, Y. Asahara, H. Ishikawa, K. Nakajima, Y. Fujii, “Magneto-optical properties of Bi-substituted epitaxial rare earth iron garnet films,” J. Appl. Phys. 61, 3256–3258 (1987).

Y. Asahara, N. Nakamura, “The rare-earth iron garnet film with small temperature dependence of sensitivity used in magnetic field sensors,” in Proceedings of the Sixth International Conference on Ferrites (Japanese Society of Powder and Powder Metallurgy, Tokyo, 1990), pp. 1617–1620.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1966), Chap. 8, p. 392.

Day, G. W.

G. W. Day, “Recent advances in Faraday effect sensors,” in Optical Fiber Sensors, Vol. 44 of Springer Proceedings in Physics (Springer-Verlag, Berlin, 1989), pp. 250–254.

Fujii, Y.

K. Machida, Y. Asahara, H. Ishikawa, K. Nakajima, Y. Fujii, “Magneto-optical properties of Bi-substituted epitaxial rare earth iron garnet films,” J. Appl. Phys. 61, 3256–3258 (1987).

Hayashi, Y.

O. Kamada, Y. Tsujimoto, Y. Hayashi, “Fiber-optic current sensors using mixed rare earth iron garnet crystals,” in Proceedings of the Third Sensor Symposium (Tokosha, Tokyo, 1983), pp. 167–169.

Imataki, M.

E. Aikawa, U. Ueda, M. Watanabe, H. Takahashi, M. Imataki, “Fiber-optic magnetic field sensor using (Cd1–xMnx)Te single crystals,” in Technical Digest of the Ninth Sensor Symposium (Tokosha, Tokyo, 1990), pp. 55–58.

Inokuti, S.

T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Magneto-optical hysteresis loops of multidomain materials—calculation for fixed analyzer method,” J. Magn. Soc. Jpn. 14, 642–647 (1990) (in Japanese).

T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Nonlinearity of Faraday loops,” IEEE Trans. Magn. 26, 1358–1360 (1990).

Ishikawa, H.

K. Machida, Y. Asahara, H. Ishikawa, K. Nakajima, Y. Fujii, “Magneto-optical properties of Bi-substituted epitaxial rare earth iron garnet films,” J. Appl. Phys. 61, 3256–3258 (1987).

Kamada, O.

O. Kamada, Y. Tsujimoto, Y. Hayashi, “Fiber-optic current sensors using mixed rare earth iron garnet crystals,” in Proceedings of the Third Sensor Symposium (Tokosha, Tokyo, 1983), pp. 167–169.

Kuhlow, B.

B. Kuhlow, M. Lambeck, “Light diffraction by magnetic domains,” Physica B 80, 374–380 (1975).

Lambeck, M.

B. Kuhlow, M. Lambeck, “Light diffraction by magnetic domains,” Physica B 80, 374–380 (1975).

Machida, K.

K. Machida, Y. Asahara, H. Ishikawa, K. Nakajima, Y. Fujii, “Magneto-optical properties of Bi-substituted epitaxial rare earth iron garnet films,” J. Appl. Phys. 61, 3256–3258 (1987).

Nakajima, K.

K. Machida, Y. Asahara, H. Ishikawa, K. Nakajima, Y. Fujii, “Magneto-optical properties of Bi-substituted epitaxial rare earth iron garnet films,” J. Appl. Phys. 61, 3256–3258 (1987).

Nakamura, N.

Y. Asahara, N. Nakamura, “The rare-earth iron garnet film with small temperature dependence of sensitivity used in magnetic field sensors,” in Proceedings of the Sixth International Conference on Ferrites (Japanese Society of Powder and Powder Metallurgy, Tokyo, 1990), pp. 1617–1620.

Numata, T.

T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Magneto-optical hysteresis loops of multidomain materials—calculation for fixed analyzer method,” J. Magn. Soc. Jpn. 14, 642–647 (1990) (in Japanese).

T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Nonlinearity of Faraday loops,” IEEE Trans. Magn. 26, 1358–1360 (1990).

Sakurai, Y.

T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Magneto-optical hysteresis loops of multidomain materials—calculation for fixed analyzer method,” J. Magn. Soc. Jpn. 14, 642–647 (1990) (in Japanese).

T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Nonlinearity of Faraday loops,” IEEE Trans. Magn. 26, 1358–1360 (1990).

Takahashi, H.

E. Aikawa, U. Ueda, M. Watanabe, H. Takahashi, M. Imataki, “Fiber-optic magnetic field sensor using (Cd1–xMnx)Te single crystals,” in Technical Digest of the Ninth Sensor Symposium (Tokosha, Tokyo, 1990), pp. 55–58.

Tanaike, H.

T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Nonlinearity of Faraday loops,” IEEE Trans. Magn. 26, 1358–1360 (1990).

T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Magneto-optical hysteresis loops of multidomain materials—calculation for fixed analyzer method,” J. Magn. Soc. Jpn. 14, 642–647 (1990) (in Japanese).

Tsujimoto, Y.

O. Kamada, Y. Tsujimoto, Y. Hayashi, “Fiber-optic current sensors using mixed rare earth iron garnet crystals,” in Proceedings of the Third Sensor Symposium (Tokosha, Tokyo, 1983), pp. 167–169.

Ueda, U.

E. Aikawa, U. Ueda, M. Watanabe, H. Takahashi, M. Imataki, “Fiber-optic magnetic field sensor using (Cd1–xMnx)Te single crystals,” in Technical Digest of the Ninth Sensor Symposium (Tokosha, Tokyo, 1990), pp. 55–58.

Watanabe, M.

E. Aikawa, U. Ueda, M. Watanabe, H. Takahashi, M. Imataki, “Fiber-optic magnetic field sensor using (Cd1–xMnx)Te single crystals,” in Technical Digest of the Ninth Sensor Symposium (Tokosha, Tokyo, 1990), pp. 55–58.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1966), Chap. 8, p. 392.

IEEE Trans. Magn.

T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Nonlinearity of Faraday loops,” IEEE Trans. Magn. 26, 1358–1360 (1990).

J. Appl. Phys.

K. Machida, Y. Asahara, H. Ishikawa, K. Nakajima, Y. Fujii, “Magneto-optical properties of Bi-substituted epitaxial rare earth iron garnet films,” J. Appl. Phys. 61, 3256–3258 (1987).

J. Magn. Soc. Jpn.

T. Numata, H. Tanaike, S. Inokuti, Y. Sakurai, “Magneto-optical hysteresis loops of multidomain materials—calculation for fixed analyzer method,” J. Magn. Soc. Jpn. 14, 642–647 (1990) (in Japanese).

Physica B

B. Kuhlow, M. Lambeck, “Light diffraction by magnetic domains,” Physica B 80, 374–380 (1975).

Other

O. Kamada, Y. Tsujimoto, Y. Hayashi, “Fiber-optic current sensors using mixed rare earth iron garnet crystals,” in Proceedings of the Third Sensor Symposium (Tokosha, Tokyo, 1983), pp. 167–169.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1966), Chap. 8, p. 392.

Y. Asahara, N. Nakamura, “The rare-earth iron garnet film with small temperature dependence of sensitivity used in magnetic field sensors,” in Proceedings of the Sixth International Conference on Ferrites (Japanese Society of Powder and Powder Metallurgy, Tokyo, 1990), pp. 1617–1620.

G. W. Day, “Recent advances in Faraday effect sensors,” in Optical Fiber Sensors, Vol. 44 of Springer Proceedings in Physics (Springer-Verlag, Berlin, 1989), pp. 250–254.

E. Aikawa, U. Ueda, M. Watanabe, H. Takahashi, M. Imataki, “Fiber-optic magnetic field sensor using (Cd1–xMnx)Te single crystals,” in Technical Digest of the Ninth Sensor Symposium (Tokosha, Tokyo, 1990), pp. 55–58.

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Figures (6)

Fig. 1
Fig. 1

Optical configuration of the magnetic-field sensor.

Fig. 2
Fig. 2

Schematic of the diffraction of the incident beam by stripe magnetic domains in the Faraday rotator.

Fig. 3
Fig. 3

Definitions of the coordinate systems for the rotator with a parallel stripe domain structure.

Fig. 4
Fig. 4

Plot of the Faraday rotation angle Θ and polarizer/analyzer relative angle Φ for fixed values of αh 1 taken as the parameter, where h 1 is the normalized field amplitude and α is a parameter determined by the modulation depth (see text).

Fig. 5
Fig. 5

Schematic of the modulation depth versus magnetic field. The continuous curve is the modulation depth, and the dash–dot line connects the origin and point (H 0, C 0)(see text).

Fig. 6
Fig. 6

Plot of η and the ratio error K against t(=y/h 0 x) for various values of r(=h 1/h 0).

Equations (24)

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E x ( x , y ) = cos Θ , E y ( x , y ) = sin Θ , j = { 2 [ ( x j p ) / w ] 1 } ,
M = M up M down = ( 2 w p ) / p M s ,
E y ( 0 ) = w / p  sin  Θ + ( p w ) / p ( sin Θ ) = M / M s sin Θ ,
[ E x ( 0 ) E y ( 0 ) ] = ( cos Θ H / H s sin Θ ) .
E x ( 1 ) = E x ( 2 ) = = 0 , E y ( n ) = 2 / n π sin [ n π / 2 ( 1 + M / M s ) ] sin Θ ( n = ± 1 , ± 2 , ) ,
E ( 0 ) = ( 1 0 0 0 ) ( cos Φ sin Φ sin Φ cos Φ ) ( cos Θ H / H s sin Θ ) = ( cos Φ cos Θ + H / H s sin Φ sin Θ 0 ) .
I out ( 0 ) = ( cos Φ cos Θ + H / H s  sin Φ sin Θ ) 2 .
V d c = V 0 ( cos 2 Φ cos 2 Θ + 1 2 H 1 2 H s 2 sin 2 Φ sin 2 Θ ) ,
V ac = V 0 ( 1 8 H 1 2 H s 2 sin 2 2 Φ sin 2 2 Θ + 1 8 H 1 4 H s 4 cos 4 Φ cos 4 Θ ) 1 / 2 ,
C = V ac / V dc .
x = sin Φ sin Θ , y = cos Φ cos Θ
V dc = y 2 + 1 2 h 1 2 x 2 ,
V ac = ( 2 h 1 2 x 2 y 2 + 1 8 h 1 4 x 4 ) 1 / 2 .
A 2 w 2 + ( A 2 2 ) w 2 + ( 1 4 A 2 1 8 ) = 0.
w = ( y / h 1 x ) 2 = [ 2 A 2 + ( 4 7 2 A 2 ) 1 / 2 ] / 2 A 2 .
Θ = arccot ( α h 1 tan Φ ) .
K = C 0 H 1 / C 1 H 0 1
η = ( y 2 + 1 2 h 1 2 x 2 ) ( 2 x 2 y 2 + 1 8 h 0 2 x 4 ) 1 / 2 / ( y 2 + 1 2 h 0 2 x 2 ) ( 2 x 2 y 2 + 1 8 h 1 2 x 4 ) 1 / 2 .
η = ( 2 t 2 + r 2 ) ( 16 t 2 + 1 ) 1 / 2 / ( 2 t 2 + 1 ) ( 16 t 2 + r 2 ) 1 / 2 .
64 ( η 2 1 ) u 3 + [ 64 ( η 2 r 2 ) + 4 ( η 2 r 2 1 ) ] u 2 + [ 16 ( η 2 r 4 ) + 4 r 2 ( η 2 1 ) ] u + r 2 ( η 2 r 2 ) = 0.
Θ = arccot ( β h 0 tan Φ ) .
E x ( 0 ) = cos Φ cos Θ + H / H s sin Φ sin Θ , E x ( n ) = 2 / n π sin Φ sin Θ sin [ n π / 2 ( 1 + M / M s ) ] ( n = ± 1 , ± 2 , ) ,
I tot = n = E ( n ) 2 = E x ( 0 ) 2 + 2 n = 1 E ( n ) 2 + n m m = E ( m ) E ( n ) .
I tot = 1 2 ( 1 + cos 2 Φ cos 2 Θ + H / H s sin 2 Φ sin 2 Θ ) .

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