Abstract

The drift of an optical fiber gyroscope caused by the earth's magnetic field through the Faraday effect is studied theoretically in a simple model and also in a practical model. The mechanism of its reduction using polarization-maintaining optical fiber is clarified; it depends on the twist and the birefringence of the fiber. The presence of the twist component whose period is just equal to one turn of the sensing fiber loop causes the drift; ways of suppressing it are described. Influence of the practical parameters, such as imperfection of the polarizer, is also discussed.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. G. Giallorenzi et al., “Optical Fiber Sensor Technology,” IEEE J. Quantum Electron. QE-18, 626 (1982).
    [CrossRef]
  2. K. Hotate, “Fiber Optic Gyroscope,” JARECT, Optical Devices & Fibers, 17, P. 333, Ohmsha and North-Holland (1985).
  3. R. A. Bergh, H. C. Lefevre, H. J. Shaw, “Overview of Fiber-Optic Gyroscopes,” IEEE/OSA J. Lightwave Technol. LT-2, 91 (1984).
    [CrossRef]
  4. K. Iwatsuki, K. Hotate, M. Higashiguchi, “Effect of Rayleigh Backscattering in an Optical Passive Ring-Resonator Gyro,” Appl. Opt. 23, 3916 (1984).
    [CrossRef] [PubMed]
  5. R. A. Bergh, B. Culshaw, C. C. Cutler, H. C. Lefevre, H. J. Shaw, “Source Statistics and the Kerr Effect in Fiber-Optic Gyroscopes,” Opt. Lett. 7, 563 (1982).
    [CrossRef] [PubMed]
  6. H. C. Lefevre, R. A. Bergh, H. J. Shaw, “All-Fiber Gyroscope with Inertial-Navigation Short-Tem Sensitivity,” Opt. Lett. 7, 454 (1982).
    [CrossRef] [PubMed]
  7. K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Heterodyne Fiber Gyro with Frequency Output,” Opt. Lett. 7, 331 (1982).
    [CrossRef]
  8. K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Heterodyne Fiber Gyro with Frequency Output,” paper presented at Symposium Gyro Technology, Stuttgart (1982).
  9. K. Hotate, M. Higashiguchi, N. Ohgi, Y. Ueno, “Micro-Optics Optical-Heterodyne Fiber Gyro with Drift Compensation,” paper presented at Symposium Gyro Technology, Stuttgart (1984).
  10. B. Y. Kim, H. J. Shaw, “Phase-Reading, All-Fiber-Optic Gyroscope,” Opt. Lett. 9, 378 (1984).
    [CrossRef] [PubMed]
  11. W. Auch et al., “Fibre-Optic Gyro with Polarization-Preserving Fibre,” paper presented at Symposium on Gyro Technology, Stuttgart (1983).
  12. D. Eberhard, E. Voges, “Fiber Gyroscope with Phase-Modulated Single-Sideband Detection,” Opt. Lett. 9, 22 (1984).
    [CrossRef] [PubMed]
  13. H. C. Lefevre, Ph. Graindorge, H. J. Arditty, S. Vatoux, M. Papuchon, “Double Closed-Loop Hybrid Fiber Gyroscope Using Digital Phase Ramp,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), postdeadline paper PDS7.
  14. H. C. Lefevre, H. J. Arditty, P. Graindorge, “Compact All-Fiber Gyroscope Brassboard,” paper presented at International Conference on Optical Fiber Sensors, Stuttgart (1984).
  15. M. Boehm, “Fiber Gyroscopes in Europe,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), paper WBB1.
  16. Y. Nishiura, K. Ono, S. Hiraoka, “Development of Compact Phase-Modulated Fiber-Optic Gyroscope with Stabilized Scale Factor,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), postdeadline paper PDS-6.
  17. K. Boehm, K. Petermann, E. Weidel, “Sensitivity of a Fiber-Optic Gyroscope to Environmental Magnetic Fields,” Opt. Lett. 7, 180 (1982).
    [CrossRef]
  18. W. Auch, E. Schlemper, “Drift Behaviour of a Fiber-Optic Rotation Sensor Using Polarization-Preserving Fibre,” paper presented at International Conference on Optical Fiber Sensors, London (1983).
  19. T. Okoshi, K. Kikuchi, N. Shimosaka, R. Ishibashi, “Trial Manufacture and Characteristic Analysis of Optical Fiber Isolator,” Paper of Technical Group, IECEJapan, OQE 84-68 (1984), in Japanese;an English version will be published soon. Equation (1) was derived in this paper. The derivation can be performed using Refs. 20 and 21.
  20. T. Okoshi, K. Okamoto, K. Hotate, “Optical Fiber” (in Japanese), p. 172, Ohmsha (1983).
  21. A. Yariv, “Coupled-Mode Theory for Guided-Wave Optics,” IEEE J. Quantum Electron. QE-9, 919 (1973).
    [CrossRef]

1984 (4)

1982 (5)

1973 (1)

A. Yariv, “Coupled-Mode Theory for Guided-Wave Optics,” IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

Arditty, H. J.

H. C. Lefevre, Ph. Graindorge, H. J. Arditty, S. Vatoux, M. Papuchon, “Double Closed-Loop Hybrid Fiber Gyroscope Using Digital Phase Ramp,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), postdeadline paper PDS7.

H. C. Lefevre, H. J. Arditty, P. Graindorge, “Compact All-Fiber Gyroscope Brassboard,” paper presented at International Conference on Optical Fiber Sensors, Stuttgart (1984).

Auch, W.

W. Auch, E. Schlemper, “Drift Behaviour of a Fiber-Optic Rotation Sensor Using Polarization-Preserving Fibre,” paper presented at International Conference on Optical Fiber Sensors, London (1983).

W. Auch et al., “Fibre-Optic Gyro with Polarization-Preserving Fibre,” paper presented at Symposium on Gyro Technology, Stuttgart (1983).

Bergh, R. A.

Boehm, K.

Boehm, M.

M. Boehm, “Fiber Gyroscopes in Europe,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), paper WBB1.

Culshaw, B.

Cutler, C. C.

Eberhard, D.

Giallorenzi, T. G.

T. G. Giallorenzi et al., “Optical Fiber Sensor Technology,” IEEE J. Quantum Electron. QE-18, 626 (1982).
[CrossRef]

Graindorge, P.

H. C. Lefevre, H. J. Arditty, P. Graindorge, “Compact All-Fiber Gyroscope Brassboard,” paper presented at International Conference on Optical Fiber Sensors, Stuttgart (1984).

Graindorge, Ph.

H. C. Lefevre, Ph. Graindorge, H. J. Arditty, S. Vatoux, M. Papuchon, “Double Closed-Loop Hybrid Fiber Gyroscope Using Digital Phase Ramp,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), postdeadline paper PDS7.

Higashiguchi, M.

K. Iwatsuki, K. Hotate, M. Higashiguchi, “Effect of Rayleigh Backscattering in an Optical Passive Ring-Resonator Gyro,” Appl. Opt. 23, 3916 (1984).
[CrossRef] [PubMed]

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Heterodyne Fiber Gyro with Frequency Output,” Opt. Lett. 7, 331 (1982).
[CrossRef]

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Heterodyne Fiber Gyro with Frequency Output,” paper presented at Symposium Gyro Technology, Stuttgart (1982).

K. Hotate, M. Higashiguchi, N. Ohgi, Y. Ueno, “Micro-Optics Optical-Heterodyne Fiber Gyro with Drift Compensation,” paper presented at Symposium Gyro Technology, Stuttgart (1984).

Hiraoka, S.

Y. Nishiura, K. Ono, S. Hiraoka, “Development of Compact Phase-Modulated Fiber-Optic Gyroscope with Stabilized Scale Factor,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), postdeadline paper PDS-6.

Hotate, K.

K. Iwatsuki, K. Hotate, M. Higashiguchi, “Effect of Rayleigh Backscattering in an Optical Passive Ring-Resonator Gyro,” Appl. Opt. 23, 3916 (1984).
[CrossRef] [PubMed]

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Heterodyne Fiber Gyro with Frequency Output,” Opt. Lett. 7, 331 (1982).
[CrossRef]

K. Hotate, “Fiber Optic Gyroscope,” JARECT, Optical Devices & Fibers, 17, P. 333, Ohmsha and North-Holland (1985).

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Heterodyne Fiber Gyro with Frequency Output,” paper presented at Symposium Gyro Technology, Stuttgart (1982).

K. Hotate, M. Higashiguchi, N. Ohgi, Y. Ueno, “Micro-Optics Optical-Heterodyne Fiber Gyro with Drift Compensation,” paper presented at Symposium Gyro Technology, Stuttgart (1984).

T. Okoshi, K. Okamoto, K. Hotate, “Optical Fiber” (in Japanese), p. 172, Ohmsha (1983).

Ishibashi, R.

T. Okoshi, K. Kikuchi, N. Shimosaka, R. Ishibashi, “Trial Manufacture and Characteristic Analysis of Optical Fiber Isolator,” Paper of Technical Group, IECEJapan, OQE 84-68 (1984), in Japanese;an English version will be published soon. Equation (1) was derived in this paper. The derivation can be performed using Refs. 20 and 21.

Iwatsuki, K.

Kikuchi, K.

T. Okoshi, K. Kikuchi, N. Shimosaka, R. Ishibashi, “Trial Manufacture and Characteristic Analysis of Optical Fiber Isolator,” Paper of Technical Group, IECEJapan, OQE 84-68 (1984), in Japanese;an English version will be published soon. Equation (1) was derived in this paper. The derivation can be performed using Refs. 20 and 21.

Kim, B. Y.

Lefevre, H. C.

R. A. Bergh, H. C. Lefevre, H. J. Shaw, “Overview of Fiber-Optic Gyroscopes,” IEEE/OSA J. Lightwave Technol. LT-2, 91 (1984).
[CrossRef]

R. A. Bergh, B. Culshaw, C. C. Cutler, H. C. Lefevre, H. J. Shaw, “Source Statistics and the Kerr Effect in Fiber-Optic Gyroscopes,” Opt. Lett. 7, 563 (1982).
[CrossRef] [PubMed]

H. C. Lefevre, R. A. Bergh, H. J. Shaw, “All-Fiber Gyroscope with Inertial-Navigation Short-Tem Sensitivity,” Opt. Lett. 7, 454 (1982).
[CrossRef] [PubMed]

H. C. Lefevre, Ph. Graindorge, H. J. Arditty, S. Vatoux, M. Papuchon, “Double Closed-Loop Hybrid Fiber Gyroscope Using Digital Phase Ramp,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), postdeadline paper PDS7.

H. C. Lefevre, H. J. Arditty, P. Graindorge, “Compact All-Fiber Gyroscope Brassboard,” paper presented at International Conference on Optical Fiber Sensors, Stuttgart (1984).

Nishiura, Y.

Y. Nishiura, K. Ono, S. Hiraoka, “Development of Compact Phase-Modulated Fiber-Optic Gyroscope with Stabilized Scale Factor,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), postdeadline paper PDS-6.

Niwa, N.

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Heterodyne Fiber Gyro with Frequency Output,” Opt. Lett. 7, 331 (1982).
[CrossRef]

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Heterodyne Fiber Gyro with Frequency Output,” paper presented at Symposium Gyro Technology, Stuttgart (1982).

Ohgi, N.

K. Hotate, M. Higashiguchi, N. Ohgi, Y. Ueno, “Micro-Optics Optical-Heterodyne Fiber Gyro with Drift Compensation,” paper presented at Symposium Gyro Technology, Stuttgart (1984).

Okamoto, K.

T. Okoshi, K. Okamoto, K. Hotate, “Optical Fiber” (in Japanese), p. 172, Ohmsha (1983).

Okoshi, T.

T. Okoshi, K. Kikuchi, N. Shimosaka, R. Ishibashi, “Trial Manufacture and Characteristic Analysis of Optical Fiber Isolator,” Paper of Technical Group, IECEJapan, OQE 84-68 (1984), in Japanese;an English version will be published soon. Equation (1) was derived in this paper. The derivation can be performed using Refs. 20 and 21.

T. Okoshi, K. Okamoto, K. Hotate, “Optical Fiber” (in Japanese), p. 172, Ohmsha (1983).

Okuma, N.

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Heterodyne Fiber Gyro with Frequency Output,” Opt. Lett. 7, 331 (1982).
[CrossRef]

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Heterodyne Fiber Gyro with Frequency Output,” paper presented at Symposium Gyro Technology, Stuttgart (1982).

Ono, K.

Y. Nishiura, K. Ono, S. Hiraoka, “Development of Compact Phase-Modulated Fiber-Optic Gyroscope with Stabilized Scale Factor,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), postdeadline paper PDS-6.

Papuchon, M.

H. C. Lefevre, Ph. Graindorge, H. J. Arditty, S. Vatoux, M. Papuchon, “Double Closed-Loop Hybrid Fiber Gyroscope Using Digital Phase Ramp,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), postdeadline paper PDS7.

Petermann, K.

Schlemper, E.

W. Auch, E. Schlemper, “Drift Behaviour of a Fiber-Optic Rotation Sensor Using Polarization-Preserving Fibre,” paper presented at International Conference on Optical Fiber Sensors, London (1983).

Shaw, H. J.

Shimosaka, N.

T. Okoshi, K. Kikuchi, N. Shimosaka, R. Ishibashi, “Trial Manufacture and Characteristic Analysis of Optical Fiber Isolator,” Paper of Technical Group, IECEJapan, OQE 84-68 (1984), in Japanese;an English version will be published soon. Equation (1) was derived in this paper. The derivation can be performed using Refs. 20 and 21.

Ueno, Y.

K. Hotate, M. Higashiguchi, N. Ohgi, Y. Ueno, “Micro-Optics Optical-Heterodyne Fiber Gyro with Drift Compensation,” paper presented at Symposium Gyro Technology, Stuttgart (1984).

Vatoux, S.

H. C. Lefevre, Ph. Graindorge, H. J. Arditty, S. Vatoux, M. Papuchon, “Double Closed-Loop Hybrid Fiber Gyroscope Using Digital Phase Ramp,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), postdeadline paper PDS7.

Voges, E.

Weidel, E.

Yariv, A.

A. Yariv, “Coupled-Mode Theory for Guided-Wave Optics,” IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

T. G. Giallorenzi et al., “Optical Fiber Sensor Technology,” IEEE J. Quantum Electron. QE-18, 626 (1982).
[CrossRef]

A. Yariv, “Coupled-Mode Theory for Guided-Wave Optics,” IEEE J. Quantum Electron. QE-9, 919 (1973).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (1)

R. A. Bergh, H. C. Lefevre, H. J. Shaw, “Overview of Fiber-Optic Gyroscopes,” IEEE/OSA J. Lightwave Technol. LT-2, 91 (1984).
[CrossRef]

Opt. Lett. (6)

Other (11)

W. Auch, E. Schlemper, “Drift Behaviour of a Fiber-Optic Rotation Sensor Using Polarization-Preserving Fibre,” paper presented at International Conference on Optical Fiber Sensors, London (1983).

T. Okoshi, K. Kikuchi, N. Shimosaka, R. Ishibashi, “Trial Manufacture and Characteristic Analysis of Optical Fiber Isolator,” Paper of Technical Group, IECEJapan, OQE 84-68 (1984), in Japanese;an English version will be published soon. Equation (1) was derived in this paper. The derivation can be performed using Refs. 20 and 21.

T. Okoshi, K. Okamoto, K. Hotate, “Optical Fiber” (in Japanese), p. 172, Ohmsha (1983).

H. C. Lefevre, Ph. Graindorge, H. J. Arditty, S. Vatoux, M. Papuchon, “Double Closed-Loop Hybrid Fiber Gyroscope Using Digital Phase Ramp,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), postdeadline paper PDS7.

H. C. Lefevre, H. J. Arditty, P. Graindorge, “Compact All-Fiber Gyroscope Brassboard,” paper presented at International Conference on Optical Fiber Sensors, Stuttgart (1984).

M. Boehm, “Fiber Gyroscopes in Europe,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), paper WBB1.

Y. Nishiura, K. Ono, S. Hiraoka, “Development of Compact Phase-Modulated Fiber-Optic Gyroscope with Stabilized Scale Factor,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), postdeadline paper PDS-6.

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Heterodyne Fiber Gyro with Frequency Output,” paper presented at Symposium Gyro Technology, Stuttgart (1982).

K. Hotate, M. Higashiguchi, N. Ohgi, Y. Ueno, “Micro-Optics Optical-Heterodyne Fiber Gyro with Drift Compensation,” paper presented at Symposium Gyro Technology, Stuttgart (1984).

W. Auch et al., “Fibre-Optic Gyro with Polarization-Preserving Fibre,” paper presented at Symposium on Gyro Technology, Stuttgart (1983).

K. Hotate, “Fiber Optic Gyroscope,” JARECT, Optical Devices & Fibers, 17, P. 333, Ohmsha and North-Holland (1985).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Simple model of an optical fiber gyroscope with one-turn square loop: (a) portion of the sensing fiber loop in the magnetic field H; (b) twist of the fiber, AB = BC = CD = DE.

Fig. 2
Fig. 2

Light propagation in twisted fiber with the magnetic field.

Fig. 3
Fig. 3

Relationship between bias and fiber birefringence.

Fig. 4
Fig. 4

Relationship between bias and fiber length.

Fig. 5
Fig. 5

Relationship between bias and fiber twist.

Fig. 6
Fig. 6

Relationship between bias and magnetic field direction.

Fig. 7
Fig. 7

Optical fiber with random twist.

Fig. 8
Fig. 8

Practical model of the optical fiber gyroscope.

Fig. 9
Fig. 9

Relationship between drift and twist power-spectrum component whose period is just equal to one turn of the sensing loop.

Fig. 10
Fig. 10

Relationship between drift and fiber birefringence.

Fig. 11
Fig. 11

Relationship between the axis of the polarizer and that at the fiber endface.

Equations (61)

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

[ Ê x Ê y ] = exp ( j β a υ z ) C ( ϕ , ζ ) [ E x E y ] ,
C ( ϕ , ζ ) = [ C 11 C 12 C 21 C 22 ] ,
C 11 = C 22 * = cos η z j Δ β 2 η sin η z ,
C 12 = C 21 = ϕ ζ η sin η z ,
β a υ = β x + β y 2 ,
η = ( Δ β 2 ) 2 + ( ϕ ζ ) 2 ,
[ Ê xr Ê yr ] = P C ED C DC C CB C BA P [ E x E y ] ,
P = [ 1 0 0 0 ] ,
C BA = C [ ϕ , ζ cos ( θ + π 4 ) ] ,
C CB = C [ ϕ , ζ sin ( θ + π 4 ) ] ,
C DC = C [ ϕ , ζ cos ( θ + π 4 ) ] ,
C ED = C [ ϕ , ζ sin ( θ + π 4 ) ] .
Ê xr = f ( ϕ ) E x .
[ Ê xl Ê yl ] = P C AB C BC C CD C DE P [ E x E y ] ,
C AB = C [ ϕ , ζ cos ( θ + π 4 ) ] ,
C BC = C [ ϕ , ζ sin ( θ + π 4 ) ] ,
C CD = C [ ϕ , ζ cos ( θ + π 4 ) ] ,
C DE = C [ ϕ , ζ sin ( θ + π 4 ) ] ;
Ê xl = f ( ϕ ) E x .
Δ ψ = arg ( Ê xl ) arg ( Ê xr ) = arg { f ( ϕ ) } arg { f ( ϕ ) } .
C = F + G ,
F = [ exp ( j η z ) 0 0 exp ( j η z ) ] ,
G = [ g 1 1 1 g 1 * ] g 2 sin η z ,
g 1 = j ϕ ζ η + Δ β / 2 ,
g 2 = ϕ ζ η .
S = C n C n 1 C 1 = ( F n + G n ) ( F n 1 + G n 1 ) ( F 1 + G 1 ) .
η k z k = 2 π ; k = 1 , 2 , , n 1
z k = 2 π η k 2 2 π Δ β .
S = ( F n + G n ) [ exp { j 2 ( n 1 ) π } 0 0 exp { j 2 ( n 1 ) π } ] .
S r = ( F rn + G rn ) [ exp { j 2 ( n 1 ) π } 0 0 exp { j 2 ( n 1 ) π } ] ,
η rk = ( Δ β 2 ) 2 ( ϕ rk ζ rk ) 2 .
S l = ( F l 1 + G l 1 ) [ exp { j 2 ( n 1 ) π } 0 0 exp { j 2 ( n 1 ) π } ] ,
η lk = ( Δ β 2 ) 2 + ( ϕ lk + ζ lk ) 2 .
s r = exp { j η rn z rn j 2 ( n 1 ) π } ,
s l = exp { j η l 1 z l 1 j 2 ( n 1 ) π } .
2 ( n 1 ) π = k = 1 n 1 η rk z rk = k = 2 n η lk z lk ,
s r = exp ( j k = 1 n η rk z rk ) ,
s l = exp ( j k = 1 n η lk z lk ) .
Δ ψ = k = 1 n ( η lk z lk η rk z rk ) .
Δ ψ = 0 L ( η l η r ) dz = 0 L 4 ϕ ζ η l + η r dz ,
Δ ψ = 0 L 4 ϕ ζ Δ β dz .
ζ = ζ 0 sin ( z r θ 0 ) ,
Δ ψ = 4 ζ 0 Δ β 0 L ϕ ( z ) sin ( z r θ 0 ) dz .
Δ ψ = 4 ζ 0 r Δ β 0 2 m π ϕ ( θ ) sin ( θ θ 0 ) d θ ,
Δ ψ = 4 ζ 0 r Δ β ( a cos θ 0 b sin θ 0 ) ,
a = 0 2 m π ϕ ( θ ) sin θ d θ ,
b = 0 2 m π ϕ ( θ ) cos θ d θ .
Δ ψ = 4 ζ 0 r Δ β a 2 + b 2 cos ( θ 0 + ξ ) = 4 ζ 0 r Δ β | Φ | 2 cos ( θ 0 + ξ )
Φ = 0 2 m π ϕ ( θ ) exp ( j θ ) d θ .
W = | Φ | 2 2 m π .
Δ ψ = 4 ζ 0 r Δ β 2 m π W cos ( θ 0 + ξ ) .
δ = 8 ζ 0 r Δ β 2 m π W = 8 ζ 0 Δ β 2 AW ,
R ( α ) = [ cos α sin α sin α cos α ] .
S i = [ exp ( j ψ i ) 0 0 exp ( j ψ i ) ] ,
ψ i = 0 L η i dz ,
E r = P R ( α 2 ) S r R ( α 1 ) P E in ,
P = [ 1 0 0 ] .
I = I 0 { 1 + cos ( Δ ψ + ψ s ) } ,
I 0 = 2 ( | E x | 2 + ɛ 2 | E y | 2 ) × { cos 2 α 1 cos 2 α 2 + sin 2 α 1 sin 2 α 2 + 2 cos α 1 cos α 2 sin α 1 sin α 2 cos ( ψ r + ψ l ) } ,
Δ ψ = ψ l ψ r ,
I 0 = 2 ( | E x | 2 + ɛ 2 | E y | 2 ) ( cos 2 α 1 cos 2 α 2 + sin 2 α 1 sin 2 α 2 ) .

Metrics