Abstract

Resonance characteristics of an optical fiber ring resonator are derived taking the effect of birefringence into account. We introduce the concept of the eigenstate of polarization to discuss the resonance characteristics resulting from the polarization fluctuation. Using this concept, the polarization problem in an optical passive ring-resonator gyro (OPRG) is discussed to reveal that this causes gyro drift. A way to reduce the drift is to use not polarizers but two polarization controllers. The precision required for polarization control is estimated; it is found difficult to do in practice. We discuss another configuration to solve the polarization problem in an OPRG without polarization controllers.

© 1986 Optical Society of America

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References

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  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]
  2. K. Hotate, “Fiber Optic Gyroscope,” in JARECT, Optical Devices and Fibers, Vol. 17, Y. Suematsu, Ed (OHM and North-Holland, Tokyo, 1985), p. 333.
  3. K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Heterodyne Fiber Gyro with Frequency Output,” Opt. Lett. 7, 331 (1982).
    [CrossRef] [PubMed]
  4. N. Furestenau, V. Wetzing, E. Leubeck, G. Siol, W. Auch, M. Koch, “Drift and Scale Factor Tests on SEL Fiber Gyro in Controlled Environment,” presented at Symposium on Gyro Technology, Stuttgart (1984), pp. 3.0–3.30.
  5. B. Y. Kim, H. J. Shaw, “Phase-Reading All-Fiber-Optic Gyroscope,” Opt. Lett. 9, 378 (1984).
    [CrossRef] [PubMed]
  6. H. C. Lefevre, P. 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.
  7. K. Iwatsuki, K. Hotate, M. Higashiguchi, “Effect of Rayleigh Backscattering in an Optical Passive Ring-Resonator Gyro,” Appl. Opt. 23, 3916 (1984).
    [CrossRef] [PubMed]
  8. K. Iwatsuki, K. Hotate, M. Higashiguchi, “Rayleigh Backscattering in Optical Passive Ring-Resonator Gyro,” presented at Second International Conference on Optical Fiber Sensors, Stuttgart (1984), pp. 313–316.
  9. R. E. Meyer, S. Ezekiel, D. W. Stowe, V. J. Tekippe, “Passive Fiber-Optic Ring Resonator for Rotation Sensing,” Opt. Lett. 8, 644 (1983).
    [CrossRef] [PubMed]
  10. K. Iwatsuki, K. Hotate, M. Higashiguchi, “Kerr Effect in Optical Passive Ring-Resonator Gyros,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), paper THGG13.
  11. B. F. Lamouroux, B. S. Prade, A. G. Orszag, “Polarization Effect in Optical-Fiber Ring Resonators,” Opt. Lett. 7, 391 (1982).
    [CrossRef] [PubMed]
  12. R. J. Mears, M. P. Varnham, D. N. Payne, A. J. Barlow, “Polarization Control in Resonant-Ring Fiber Gyroscope,” presented at Second International Conference on Optical Fiber Sensors, Stuttgart (1984), pp. 309–312.
  13. R. Ulrich, M. Johnson, “Fiber-Ring Interferometer: Polarization Analysis,” Opt. Lett. 4, 152 (1979).
    [CrossRef] [PubMed]
  14. C. Chen, W. K. Burns, “Polarization Characteristics of Single-Mode Fiber Coupler,” IEEE J. Quantum Electron. QE-18, 1589 (1982).
    [CrossRef]
  15. M. P. Varnham, D. N. Payne, A. J. Barlow, E. J. Tarbox, “Coiled-Birefringent-Fiber Polarizers,” Opt. Lett. 9, 306 (1984).
    [CrossRef] [PubMed]
  16. S. Ezekiel, H. J. Arditty, Eds., Fiber-Optic Rotation Sensors and Related Technologies (Springer-Verlag, New York, 1982), pp. 52–77.
  17. J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “Single-Polarization Fiber,” IEEE/OSA J. Lightware Technol. LT-1, 370 (1983).
    [CrossRef]
  18. I. Yokohama, K. Okamoto, J. Noda, “Fiber-Optic Polarising Beam Splitter Employing Birefringent-Fiber Coupler,” Electron. Lett. 21, 415 (1985).
    [CrossRef]

1985 (1)

I. Yokohama, K. Okamoto, J. Noda, “Fiber-Optic Polarising Beam Splitter Employing Birefringent-Fiber Coupler,” Electron. Lett. 21, 415 (1985).
[CrossRef]

1984 (4)

1983 (2)

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “Single-Polarization Fiber,” IEEE/OSA J. Lightware Technol. LT-1, 370 (1983).
[CrossRef]

R. E. Meyer, S. Ezekiel, D. W. Stowe, V. J. Tekippe, “Passive Fiber-Optic Ring Resonator for Rotation Sensing,” Opt. Lett. 8, 644 (1983).
[CrossRef] [PubMed]

1982 (3)

1979 (1)

Arditty, H. J.

H. C. Lefevre, P. 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.

Auch, W.

N. Furestenau, V. Wetzing, E. Leubeck, G. Siol, W. Auch, M. Koch, “Drift and Scale Factor Tests on SEL Fiber Gyro in Controlled Environment,” presented at Symposium on Gyro Technology, Stuttgart (1984), pp. 3.0–3.30.

Barlow, A. J.

M. P. Varnham, D. N. Payne, A. J. Barlow, E. J. Tarbox, “Coiled-Birefringent-Fiber Polarizers,” Opt. Lett. 9, 306 (1984).
[CrossRef] [PubMed]

R. J. Mears, M. P. Varnham, D. N. Payne, A. J. Barlow, “Polarization Control in Resonant-Ring Fiber Gyroscope,” presented at Second International Conference on Optical Fiber Sensors, Stuttgart (1984), pp. 309–312.

Bergh, R. A.

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

Burns, W. K.

C. Chen, W. K. Burns, “Polarization Characteristics of Single-Mode Fiber Coupler,” IEEE J. Quantum Electron. QE-18, 1589 (1982).
[CrossRef]

Chen, C.

C. Chen, W. K. Burns, “Polarization Characteristics of Single-Mode Fiber Coupler,” IEEE J. Quantum Electron. QE-18, 1589 (1982).
[CrossRef]

Ezekiel, S.

Furestenau, N.

N. Furestenau, V. Wetzing, E. Leubeck, G. Siol, W. Auch, M. Koch, “Drift and Scale Factor Tests on SEL Fiber Gyro in Controlled Environment,” presented at Symposium on Gyro Technology, Stuttgart (1984), pp. 3.0–3.30.

Graindorge, P.

H. C. Lefevre, P. 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] [PubMed]

K. Iwatsuki, K. Hotate, M. Higashiguchi, “Rayleigh Backscattering in Optical Passive Ring-Resonator Gyro,” presented at Second International Conference on Optical Fiber Sensors, Stuttgart (1984), pp. 313–316.

K. Iwatsuki, K. Hotate, M. Higashiguchi, “Kerr Effect in Optical Passive Ring-Resonator Gyros,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), paper THGG13.

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] [PubMed]

K. Hotate, “Fiber Optic Gyroscope,” in JARECT, Optical Devices and Fibers, Vol. 17, Y. Suematsu, Ed (OHM and North-Holland, Tokyo, 1985), p. 333.

K. Iwatsuki, K. Hotate, M. Higashiguchi, “Rayleigh Backscattering in Optical Passive Ring-Resonator Gyro,” presented at Second International Conference on Optical Fiber Sensors, Stuttgart (1984), pp. 313–316.

K. Iwatsuki, K. Hotate, M. Higashiguchi, “Kerr Effect in Optical Passive Ring-Resonator Gyros,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), paper THGG13.

Howard, R. E.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “Single-Polarization Fiber,” IEEE/OSA J. Lightware Technol. LT-1, 370 (1983).
[CrossRef]

Iwatsuki, 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. Iwatsuki, K. Hotate, M. Higashiguchi, “Rayleigh Backscattering in Optical Passive Ring-Resonator Gyro,” presented at Second International Conference on Optical Fiber Sensors, Stuttgart (1984), pp. 313–316.

K. Iwatsuki, K. Hotate, M. Higashiguchi, “Kerr Effect in Optical Passive Ring-Resonator Gyros,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), paper THGG13.

Johnson, M.

Kim, B. Y.

Koch, M.

N. Furestenau, V. Wetzing, E. Leubeck, G. Siol, W. Auch, M. Koch, “Drift and Scale Factor Tests on SEL Fiber Gyro in Controlled Environment,” presented at Symposium on Gyro Technology, Stuttgart (1984), pp. 3.0–3.30.

Lamouroux, B. F.

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]

H. C. Lefevre, P. 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.

Leubeck, E.

N. Furestenau, V. Wetzing, E. Leubeck, G. Siol, W. Auch, M. Koch, “Drift and Scale Factor Tests on SEL Fiber Gyro in Controlled Environment,” presented at Symposium on Gyro Technology, Stuttgart (1984), pp. 3.0–3.30.

MacChesney, J. B.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “Single-Polarization Fiber,” IEEE/OSA J. Lightware Technol. LT-1, 370 (1983).
[CrossRef]

Mears, R. J.

R. J. Mears, M. P. Varnham, D. N. Payne, A. J. Barlow, “Polarization Control in Resonant-Ring Fiber Gyroscope,” presented at Second International Conference on Optical Fiber Sensors, Stuttgart (1984), pp. 309–312.

Meyer, R. E.

Niwa, N.

Noda, J.

I. Yokohama, K. Okamoto, J. Noda, “Fiber-Optic Polarising Beam Splitter Employing Birefringent-Fiber Coupler,” Electron. Lett. 21, 415 (1985).
[CrossRef]

Okamoto, K.

I. Yokohama, K. Okamoto, J. Noda, “Fiber-Optic Polarising Beam Splitter Employing Birefringent-Fiber Coupler,” Electron. Lett. 21, 415 (1985).
[CrossRef]

Okuma, N.

Orszag, A. G.

Papuchon, M.

H. C. Lefevre, P. 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.

Payne, D. N.

M. P. Varnham, D. N. Payne, A. J. Barlow, E. J. Tarbox, “Coiled-Birefringent-Fiber Polarizers,” Opt. Lett. 9, 306 (1984).
[CrossRef] [PubMed]

R. J. Mears, M. P. Varnham, D. N. Payne, A. J. Barlow, “Polarization Control in Resonant-Ring Fiber Gyroscope,” presented at Second International Conference on Optical Fiber Sensors, Stuttgart (1984), pp. 309–312.

Pleibel, W.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “Single-Polarization Fiber,” IEEE/OSA J. Lightware Technol. LT-1, 370 (1983).
[CrossRef]

Prade, B. S.

Sears, F. M.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “Single-Polarization Fiber,” IEEE/OSA J. Lightware Technol. LT-1, 370 (1983).
[CrossRef]

Shaw, H. J.

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

B. Y. Kim, H. J. Shaw, “Phase-Reading All-Fiber-Optic Gyroscope,” Opt. Lett. 9, 378 (1984).
[CrossRef] [PubMed]

Simpson, J. R.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “Single-Polarization Fiber,” IEEE/OSA J. Lightware Technol. LT-1, 370 (1983).
[CrossRef]

Siol, G.

N. Furestenau, V. Wetzing, E. Leubeck, G. Siol, W. Auch, M. Koch, “Drift and Scale Factor Tests on SEL Fiber Gyro in Controlled Environment,” presented at Symposium on Gyro Technology, Stuttgart (1984), pp. 3.0–3.30.

Stolen, R. H.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “Single-Polarization Fiber,” IEEE/OSA J. Lightware Technol. LT-1, 370 (1983).
[CrossRef]

Stowe, D. W.

Tarbox, E. J.

Tekippe, V. J.

Ulrich, R.

Varnham, M. P.

M. P. Varnham, D. N. Payne, A. J. Barlow, E. J. Tarbox, “Coiled-Birefringent-Fiber Polarizers,” Opt. Lett. 9, 306 (1984).
[CrossRef] [PubMed]

R. J. Mears, M. P. Varnham, D. N. Payne, A. J. Barlow, “Polarization Control in Resonant-Ring Fiber Gyroscope,” presented at Second International Conference on Optical Fiber Sensors, Stuttgart (1984), pp. 309–312.

Vatoux, S.

H. C. Lefevre, P. 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.

Wetzing, V.

N. Furestenau, V. Wetzing, E. Leubeck, G. Siol, W. Auch, M. Koch, “Drift and Scale Factor Tests on SEL Fiber Gyro in Controlled Environment,” presented at Symposium on Gyro Technology, Stuttgart (1984), pp. 3.0–3.30.

Yokohama, I.

I. Yokohama, K. Okamoto, J. Noda, “Fiber-Optic Polarising Beam Splitter Employing Birefringent-Fiber Coupler,” Electron. Lett. 21, 415 (1985).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

I. Yokohama, K. Okamoto, J. Noda, “Fiber-Optic Polarising Beam Splitter Employing Birefringent-Fiber Coupler,” Electron. Lett. 21, 415 (1985).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. Chen, W. K. Burns, “Polarization Characteristics of Single-Mode Fiber Coupler,” IEEE J. Quantum Electron. QE-18, 1589 (1982).
[CrossRef]

IEEE/OSA J. Lightware Technol. (1)

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. MacChesney, R. E. Howard, “Single-Polarization Fiber,” IEEE/OSA J. Lightware Technol. LT-1, 370 (1983).
[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 (7)

R. J. Mears, M. P. Varnham, D. N. Payne, A. J. Barlow, “Polarization Control in Resonant-Ring Fiber Gyroscope,” presented at Second International Conference on Optical Fiber Sensors, Stuttgart (1984), pp. 309–312.

K. Iwatsuki, K. Hotate, M. Higashiguchi, “Kerr Effect in Optical Passive Ring-Resonator Gyros,” in Technical Digest, Third International Conference on Optical Fiber Sensors (Optical Society of America, Washington, DC, 1985), paper THGG13.

S. Ezekiel, H. J. Arditty, Eds., Fiber-Optic Rotation Sensors and Related Technologies (Springer-Verlag, New York, 1982), pp. 52–77.

H. C. Lefevre, P. 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.

K. Iwatsuki, K. Hotate, M. Higashiguchi, “Rayleigh Backscattering in Optical Passive Ring-Resonator Gyro,” presented at Second International Conference on Optical Fiber Sensors, Stuttgart (1984), pp. 313–316.

N. Furestenau, V. Wetzing, E. Leubeck, G. Siol, W. Auch, M. Koch, “Drift and Scale Factor Tests on SEL Fiber Gyro in Controlled Environment,” presented at Symposium on Gyro Technology, Stuttgart (1984), pp. 3.0–3.30.

K. Hotate, “Fiber Optic Gyroscope,” in JARECT, Optical Devices and Fibers, Vol. 17, Y. Suematsu, Ed (OHM and North-Holland, Tokyo, 1985), p. 333.

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Figures (7)

Fig. 1
Fig. 1

Configuration of optical fiber ring resonator.

Fig. 2
Fig. 2

Resonance characteristics generated by two ESOPs in the optical fiber ring resonator.

Fig. 3
Fig. 3

Basic optical setup of an OPRG.

Fig. 4
Fig. 4

Basic configuration of an OPRG to solve the polarization problem using two polarization controllers.

Fig. 5
Fig. 5

Configuration of the OPRG to achieve phase nulling, frequency output, and elimination of Rayleigh backscattering noise. (The two polarization controllers are omitted for simplicity).

Fig. 6
Fig. 6

Drift due to mismatch between the input SOP into the ring resonator and one of the ESOPs as a function of resonance peak separation generated by two ESOPs: Δζ = 0.01 rad, Δξ = 0.0 rad.

Fig. 7
Fig. 7

Gyro drift due to mismatch between the input SOP into the ring resonator and one of the ESOPs as a function of the fiber-loop length: Δζ = 10 rad; Δξ = 0.0 rad; λ = 0.8 μm; a = 0.05 m; R = 0.98; α c = 0.005 dB; α t = 4dB; I0 = 1mW; quantum efficiency = 0.8; integration time = 1 s.

Equations (43)

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[ E 3 E 4 ] = exp ( - α c / 2 ) [ C r C t C t C r ] [ E 1 E 2 ] ,
E 4 = exp ( - α c / 2 ) n = 0 [ exp ( - α c / 2 ) C r F b ] n C t E 1 ,
E 2 = F b E 4 = exp ( - α c / 2 ) F b n = 0 [ exp ( - α c / 2 ) C r F b ] n C t E 1 ,
exp ( - α c ) [ E 1 2 + E 2 2 ] = E 3 2 + E 4 2 .
I D = E 3 2 = exp ( - α c ) E 1 { I - C t S [ I - exp ( - α c ) F b F b ] SC t } E 1 ,
S = n = 0 ( S ¯ ) n = n = 0 [ exp ( - α c / 2 ) C r F b ] n ,
C r C r = C r C r = r 2 I ,
C t C t = C t C t = t 2 I ,
F b F b = F b F b = exp ( - α t L ) I ,
I D = exp ( - α c ) { e 1 2 [ 1 - ρ Γ ( β 1 L ) ] + e 2 2 [ 1 - ρ Γ ( β 2 L ) ] } ,
e i = v i t C t E 1 / t ,             ( i = 1 , 2 )
t f = exp [ - ( α c + α t L ) / 2 ] ,
ρ = t 2 ( 1 - t f 2 ) ( 1 - r t f ) 2 ,
Γ ( x ) = ( 1 - r t f ) 2 ( 1 - r t f ) 2 - 4 r t f sin 2 ( x / 2 ) .
C r = [ r x 0 0 r y ] ,
C t = [ t x 0 0 t y ] ,
F b = exp ( - α t L / 2 ) exp ( - j β L ) I ,
I D = exp ( - α c ) { [ 1 - ( 1 - t f 2 ) t x 2 1 - r x t f exp ( - j β L ) 2 ] E 1 x 2 + [ 1 - ( 1 - t f 2 ) t y 2 1 - r y t f exp ( - j β L ) 2 ] E 1 y 2 } .
I D = exp ( - α c ) [ I 0 - 1 - t f 2 + r y x / r x 2 1 - r x t f exp ( - j β L ) 2 g 2 ] ,
g = ( 1 0 ) t C t E 1 ,
I 0 = E 1 2 ,
C r = [ r x r x y r y x r y ] .
I D = exp ( - α c ) [ I 0 - ( 1 - t f 2 ) ( V - 1 C t E 1 ) Λ 2 ( V - 1 C t E 1 ) - ( 1 - t f 2 ) ( V - 1 C t E 1 ) Λ J Λ ( V - 1 C t E 1 ) ] ,
V = [ v 1 , v 2 ] ,
J = V ¯ V - I .
E 4 R = n = 0 ( C r F b ) n C t E 1 = SC t E 1 .
E 4 L = [ n = 0 ( C r F b ) n ] t F b t C t t E 3 = S t F b t C t t E 3 .
I R = E 4 R 2 = E 1 C t S SC t E 1 ,
I L = E 4 L 2 = E 3 C t * F b * S * S t F b t C t t E 3 ,
I R = E 1 C t V [ λ 1 2 0 0 λ 2 2 ] V C t E 1 ,
I L = E 3 C t * F b * V * [ λ 1 2 0 0 λ 2 2 ] ( V * ) F b t C t t E 3 ,
I A = { ( 1 - K A ) [ 1 - ρ Γ ( x ) ] + K A [ 1 - ρ Γ ( x - Δ β L ) ] } u A exp ( - α c ) I 0 ,
I B = { ( 1 - K B ) [ 1 - ρ Γ ( y ) ] + K B [ 1 - ρ Γ ( y - Δ β L ) ] } u B exp ( - α c ) I 0 ,
Δ β = β 1 - β 2 ,
x = θ s + θ f + Δ ω τ 0 ,
y = θ s - θ f ,
( 1 - Δ K ) Γ ( x ) + Δ K Γ ( x - Δ β L ) = 0 ,
Γ ( y ) = 0.
θ p = x - y ,
Δ K = 1 - v 1 · v R 2 = Δ ζ 2 - Δ ξ 2 cos 4 ζ .
θ p m 2 Δ K F = 2 F ( Δ ζ 2 - Δ ξ 2 cos 4 ζ ) ,
Ω p = c λ 4 π L a θ p m = c λ 4 π L a 2 ( 1 - R f ) π R f ( Δ ζ 2 - Δ ξ 2 cos 4 ζ ) ,
R f = R t f exp ( - 2 π Δ f τ 0 ) ,

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