Abstract

Resonance characteristics of Rayleigh backscattering in an optical passive ring-resonator gyro (OPRG) are theoretically formulated taking the temporal coherence of the optical source into account. This resonance has two peaks with separation equal to the Sagnac phase shift when rotation is induced. This phenomenon degrades the gyro’s linearity in a configuration to obtain the frequency output. Rayleigh backscattering also can induce an enhanced noise at specific input rotation rates. The methods of solving these problems are discussed. The theoretical limit of rotation sensing, given by the detector shot noise, is also computed taking into account the optical source coherence. A spectrum width narrower than several tens of kHz is required to realize the OPRG for navigation use.

© 1984 Optical Society of America

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  1. T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical Fiber Sensor Technology,” IEEE Trans. Microwave Theory Tech. MTT-30, 472 (1982).
    [CrossRef]
  2. K. Hotate, Y. Yoshida, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Fiber Laser Gyro with Easily Introduced Phase-Difference Bias,” Electron. Lett. 16, 941 (1980).
    [CrossRef]
  3. K. Hotate, Y. Yoshida, M. Higashiguchi, N. Niwa, “Fiber-Optic Laser Gyro with Easily Introduced Phase-Difference Bias,” Appl. Opt. 20, 4314 (1981).
    [CrossRef]
  4. K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Fiber Laser Gyro: Homodyne and Heterodyne Detections,” at International Conference on Fiberoptic Rotation Sensors and Related Technologies, MIT, Cambridge, Mass. (1981).
  5. 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]
  6. K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Heterodyne Fiber Gyro with Frequency Output,” at Symposium on Gyro Technology, Stuttgart, Germany (1982).
  7. G. A. Sanders, M. G. Prentiss, S. Ezekiel, “Passive Ring Resonator Method for Sensitive Inertial Rotation Measurements in Geophysics and Relativity,” Opt. Lett. 6, 569 (1981).
    [CrossRef] [PubMed]
  8. 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]
  9. L. F. Stokes, M. Chodorow, H. J. Show, “All Single-Mode Fiber Resonator,” Opt. Lett. 7, 288 (1982).
    [CrossRef] [PubMed]
  10. L. F. Stokes, M. Chodorow, H. J. Show, “Sensitive All-Single-Mode-Fiber Resonator Ring Interferometer,” IEEE OSAJ. Lightwave Technol. LT-1, 110 (1983).
    [CrossRef]
  11. R. G. Walker, C. D. W. Wilker, “Integrated Optical Ring Resonators Made by Silver Ion-Exchange in Glass,” Appl. Opt. 22, 1029 (1983).
    [CrossRef] [PubMed]
  12. B. Lamouroux, B. Prade, A. Orszag, “Polarization Effect in Optical-Fiber Ring Resonators,” Opt. Lett. 7, 8, 391 (1982).
    [CrossRef] [PubMed]
  13. Y. Ohstuka, “Optical Coherence Effects on a Fiber-Sensing Fabry-Perot Interferometer,” Appl. Opt. 21, 4316 (1982).
    [CrossRef]
  14. M. Nakazawa, “Rayleigh Backscattering Theory for Single-Mode Optical Fibers,” J. Opt. Soc. Am. 73, 1175 (1983).
    [CrossRef]
  15. E. Brinkmeyer, “Analysis of the Backscattering Method for Single-Mode Optical Fibers,” J. Opt. Soc. Am. 70, 1010 (1980).
    [CrossRef]
  16. Equations (25) and (26) are identical in this case. To apply these Eqs. to the more complicated case discussed in Sec. IV.B, such notations have been used.
  17. C. J. Nielsen, J. H. Osmundsen, “New Approach towards Frequency Stabilization of Linewidth-Narrowed Semiconductor Lasers,” Electron. Lett. 19, 644 (1983).
    [CrossRef]

1983

L. F. Stokes, M. Chodorow, H. J. Show, “Sensitive All-Single-Mode-Fiber Resonator Ring Interferometer,” IEEE OSAJ. Lightwave Technol. LT-1, 110 (1983).
[CrossRef]

C. J. Nielsen, J. H. Osmundsen, “New Approach towards Frequency Stabilization of Linewidth-Narrowed Semiconductor Lasers,” Electron. Lett. 19, 644 (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]

R. G. Walker, C. D. W. Wilker, “Integrated Optical Ring Resonators Made by Silver Ion-Exchange in Glass,” Appl. Opt. 22, 1029 (1983).
[CrossRef] [PubMed]

M. Nakazawa, “Rayleigh Backscattering Theory for Single-Mode Optical Fibers,” J. Opt. Soc. Am. 73, 1175 (1983).
[CrossRef]

1982

1981

G. A. Sanders, M. G. Prentiss, S. Ezekiel, “Passive Ring Resonator Method for Sensitive Inertial Rotation Measurements in Geophysics and Relativity,” Opt. Lett. 6, 569 (1981).
[CrossRef] [PubMed]

K. Hotate, Y. Yoshida, M. Higashiguchi, N. Niwa, “Fiber-Optic Laser Gyro with Easily Introduced Phase-Difference Bias,” Appl. Opt. 20, 4314 (1981).
[CrossRef]

1980

E. Brinkmeyer, “Analysis of the Backscattering Method for Single-Mode Optical Fibers,” J. Opt. Soc. Am. 70, 1010 (1980).
[CrossRef]

K. Hotate, Y. Yoshida, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Fiber Laser Gyro with Easily Introduced Phase-Difference Bias,” Electron. Lett. 16, 941 (1980).
[CrossRef]

Brinkmeyer, E.

Bucaro, J. A.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical Fiber Sensor Technology,” IEEE Trans. Microwave Theory Tech. MTT-30, 472 (1982).
[CrossRef]

Chodorow, M.

L. F. Stokes, M. Chodorow, H. J. Show, “Sensitive All-Single-Mode-Fiber Resonator Ring Interferometer,” IEEE OSAJ. Lightwave Technol. LT-1, 110 (1983).
[CrossRef]

L. F. Stokes, M. Chodorow, H. J. Show, “All Single-Mode Fiber Resonator,” Opt. Lett. 7, 288 (1982).
[CrossRef] [PubMed]

Cole, J. H.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical Fiber Sensor Technology,” IEEE Trans. Microwave Theory Tech. MTT-30, 472 (1982).
[CrossRef]

Dandridge, A.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical Fiber Sensor Technology,” IEEE Trans. Microwave Theory Tech. MTT-30, 472 (1982).
[CrossRef]

Ezekiel, S.

Giallorenzi, T. G.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical Fiber Sensor Technology,” IEEE Trans. Microwave Theory Tech. MTT-30, 472 (1982).
[CrossRef]

Higashiguchi, M.

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, Y. Yoshida, M. Higashiguchi, N. Niwa, “Fiber-Optic Laser Gyro with Easily Introduced Phase-Difference Bias,” Appl. Opt. 20, 4314 (1981).
[CrossRef]

K. Hotate, Y. Yoshida, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Fiber Laser Gyro with Easily Introduced Phase-Difference Bias,” Electron. Lett. 16, 941 (1980).
[CrossRef]

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

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Fiber Laser Gyro: Homodyne and Heterodyne Detections,” at International Conference on Fiberoptic Rotation Sensors and Related Technologies, MIT, Cambridge, Mass. (1981).

Hotate, K.

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, Y. Yoshida, M. Higashiguchi, N. Niwa, “Fiber-Optic Laser Gyro with Easily Introduced Phase-Difference Bias,” Appl. Opt. 20, 4314 (1981).
[CrossRef]

K. Hotate, Y. Yoshida, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Fiber Laser Gyro with Easily Introduced Phase-Difference Bias,” Electron. Lett. 16, 941 (1980).
[CrossRef]

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

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Fiber Laser Gyro: Homodyne and Heterodyne Detections,” at International Conference on Fiberoptic Rotation Sensors and Related Technologies, MIT, Cambridge, Mass. (1981).

Lamouroux, B.

Meyer, R. E.

Nakazawa, M.

Nielsen, C. J.

C. J. Nielsen, J. H. Osmundsen, “New Approach towards Frequency Stabilization of Linewidth-Narrowed Semiconductor Lasers,” Electron. Lett. 19, 644 (1983).
[CrossRef]

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] [PubMed]

K. Hotate, Y. Yoshida, M. Higashiguchi, N. Niwa, “Fiber-Optic Laser Gyro with Easily Introduced Phase-Difference Bias,” Appl. Opt. 20, 4314 (1981).
[CrossRef]

K. Hotate, Y. Yoshida, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Fiber Laser Gyro with Easily Introduced Phase-Difference Bias,” Electron. Lett. 16, 941 (1980).
[CrossRef]

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

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Fiber Laser Gyro: Homodyne and Heterodyne Detections,” at International Conference on Fiberoptic Rotation Sensors and Related Technologies, MIT, Cambridge, Mass. (1981).

Ohstuka, Y.

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] [PubMed]

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

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Fiber Laser Gyro: Homodyne and Heterodyne Detections,” at International Conference on Fiberoptic Rotation Sensors and Related Technologies, MIT, Cambridge, Mass. (1981).

Orszag, A.

Osmundsen, J. H.

C. J. Nielsen, J. H. Osmundsen, “New Approach towards Frequency Stabilization of Linewidth-Narrowed Semiconductor Lasers,” Electron. Lett. 19, 644 (1983).
[CrossRef]

Prade, B.

Prentiss, M. G.

Priest, R. G.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical Fiber Sensor Technology,” IEEE Trans. Microwave Theory Tech. MTT-30, 472 (1982).
[CrossRef]

Rashleigh, S. C.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical Fiber Sensor Technology,” IEEE Trans. Microwave Theory Tech. MTT-30, 472 (1982).
[CrossRef]

Sanders, G. A.

Show, H. J.

L. F. Stokes, M. Chodorow, H. J. Show, “Sensitive All-Single-Mode-Fiber Resonator Ring Interferometer,” IEEE OSAJ. Lightwave Technol. LT-1, 110 (1983).
[CrossRef]

L. F. Stokes, M. Chodorow, H. J. Show, “All Single-Mode Fiber Resonator,” Opt. Lett. 7, 288 (1982).
[CrossRef] [PubMed]

Sigel, G. H.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical Fiber Sensor Technology,” IEEE Trans. Microwave Theory Tech. MTT-30, 472 (1982).
[CrossRef]

Stokes, L. F.

L. F. Stokes, M. Chodorow, H. J. Show, “Sensitive All-Single-Mode-Fiber Resonator Ring Interferometer,” IEEE OSAJ. Lightwave Technol. LT-1, 110 (1983).
[CrossRef]

L. F. Stokes, M. Chodorow, H. J. Show, “All Single-Mode Fiber Resonator,” Opt. Lett. 7, 288 (1982).
[CrossRef] [PubMed]

Stowe, D. W.

Tekippe, V. J.

Walker, R. G.

Wilker, C. D. W.

Yoshida, Y.

K. Hotate, Y. Yoshida, M. Higashiguchi, N. Niwa, “Fiber-Optic Laser Gyro with Easily Introduced Phase-Difference Bias,” Appl. Opt. 20, 4314 (1981).
[CrossRef]

K. Hotate, Y. Yoshida, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Fiber Laser Gyro with Easily Introduced Phase-Difference Bias,” Electron. Lett. 16, 941 (1980).
[CrossRef]

Appl. Opt.

Electron. Lett.

K. Hotate, Y. Yoshida, M. Higashiguchi, N. Niwa, “Rotation Detection by Optical Fiber Laser Gyro with Easily Introduced Phase-Difference Bias,” Electron. Lett. 16, 941 (1980).
[CrossRef]

C. J. Nielsen, J. H. Osmundsen, “New Approach towards Frequency Stabilization of Linewidth-Narrowed Semiconductor Lasers,” Electron. Lett. 19, 644 (1983).
[CrossRef]

IEEE OSAJ. Lightwave Technol.

L. F. Stokes, M. Chodorow, H. J. Show, “Sensitive All-Single-Mode-Fiber Resonator Ring Interferometer,” IEEE OSAJ. Lightwave Technol. LT-1, 110 (1983).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

T. G. Giallorenzi, J. A. Bucaro, A. Dandridge, G. H. Sigel, J. H. Cole, S. C. Rashleigh, R. G. Priest, “Optical Fiber Sensor Technology,” IEEE Trans. Microwave Theory Tech. MTT-30, 472 (1982).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Other

K. Hotate, N. Okuma, M. Higashiguchi, N. Niwa, “Optical Fiber Laser Gyro: Homodyne and Heterodyne Detections,” at International Conference on Fiberoptic Rotation Sensors and Related Technologies, MIT, Cambridge, Mass. (1981).

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

Equations (25) and (26) are identical in this case. To apply these Eqs. to the more complicated case discussed in Sec. IV.B, such notations have been used.

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Figures (12)

Fig. 1
Fig. 1

Conceptual diagram of OPRG using the reflective characteristics of an optical ring resonator.

Fig. 2
Fig. 2

Resonance characteristic of the signal beam and the Rayleigh backscattering at rest: ω = 375 THz (λ = 0.8 μm); α t = 4 dB/km; αR = 3 dB/km; r′ = 0.95; α 0 = α 0 = 0.1 dB ; Fiber loop diam = 0.1 m; fiber loop turn = 424.

Fig. 3
Fig. 3

Numerical example of the Rayleigh backscattering at the resonance point: ω = 375 THz (λ = 0.8 μm), α t = 4 dB/km, αR = 3 dB/km, r′ = 0.95, fiber loop diam = 0.1 m, α 0 = α 0 : (a) interfered intensity σ of the signal beam and the Rayleigh backscattering; (b) Rayleigh backscattering intensity 〈I3〉.

Fig. 4
Fig. 4

Splitting of the resonance peaks of the Rayleigh backscattering 〈I3〉 caused by the Sagnac effect: ω = 375 THz (λ = 0.8 μm); r′ = 0.95; Δf = 5 kHz; α t = 4 dB/km; αR = 3 dB/km; α 0 = α 0 = 0.1 dB ; fiber loop diam = 0.1 m; fiber loop turn = 318; (a) θ s = 0.25 rad; (b) θ s = 0.5 rad.

Fig. 5
Fig. 5

Direct current and ac schemes to set the operation point.

Fig. 6
Fig. 6

Conceptual diagram of basic configuration of OPRG with phase-nulling and frequency output.

Fig. 7
Fig. 7

Specific ranges of input rotation rate affected by Rayleigh backscattering σ.

Fig. 8
Fig. 8

Conceptual drawing of the degradation of the linearity between Sagnac phase shift and frequency output.

Fig. 9
Fig. 9

Relation between the degradation of the linearity and the Sagnac phase shift. Two counterpropagating signal intensities are equal to each other: ω = 375 THz; α t = 4 dB/km; αR = 3 dB/km; α 0 = α 0 = 0.1 dB ; r′ = 0.95; Δf = 5 kHz; L = 31.4 m.

Fig. 10
Fig. 10

One method to eliminate noise caused by Rayleigh backscattering.

Fig. 11
Fig. 11

Limit of the rotation sensing due to the shot noise as a function of the fiber length: ω = 375 THz (λ = 0.8 μm); α t = 4 dB/km; αR = 3 dB/km; r′ = 0.95; η = 0.8; I0 = 1 mW; B s = 1 Hz; fiber loop diam = 0.1 m; α 0 = α 0 , when the spectrum width of the optical source is greater or smaller than the resonance width of the ring resonator itself is indicated by ● or ○, respectively.

Fig. 12
Fig. 12

Relation between the laser spectrum width and the limit of rotation sensing when the optimum fiber length r′ = 0.95; α 0 = α 0 = 0.1 dB ; B s = 1 Hz; η = 0.8; I0 = 1 mW; fiber loop diam = 0.1m. λ = 0.8 μm: α t = 4 dB/km, αR = 3 dB/km; λ = 1.3 μm: α t = 0.6 dB/km, αR = 0.4 dB/km; λ = 1.5 μm: α t = 0.2 dB/km, αR = 0.1 dB/km.

Equations (61)

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E ( t ) = E 0 { exp [ ω t + ϕ ( t ) ] } e ^ ,
exp j [ ϕ ( t + τ ) - ϕ ( t ) ] ¯ = exp ( - 2 π Δ f τ ) ,
E s ( t ) = E 0 exp ( j ω t ) { - r exp [ j ϕ ( t ) ] + T t f exp ( - j ω τ 0 ) m = 1 [ r t f exp ( - j ω τ 0 ) ] m - 1 × exp [ j ϕ ( t - m τ 0 ) } e ^
t f = exp [ - ( α c + α t L ) / 2 ] ,
T = ( 1 - r 2 ) exp ( - α c ) .
I 1 ( ψ ) = [ 1 - ρ L ( ψ ) ] I 0 ,
L ( ψ ) = ( 1 - R f ) 2 ( 1 - R f ) 2 + 4 R f sin 2 ( ψ / 2 ) ,
ψ = ω τ 0 ,
ρ = ( 1 - R ) ( 1 - R f 2 ) [ ( 1 - t f 2 ) + ( 1 - e - α c ) t f 2 ( 1 - R ) ] ( 1 - R f ) 2 ( 1 - R 2 ) , = T ( 1 - R f 2 ) ( 1 - t f 2 ) ( 1 - R f 2 ) 2 ( 1 - R 2 ) ,
R = r t f ,
R f = R exp ( - 2 π Δ f τ 0 ) ,
R = r 2
F = π R f 1 - R f ;
Δ ν 1 / 2 = 1 / ( 2 τ 0 F ) .
E i ( x , y , z , t ) = t a n = 0 ( r t f ) n exp ( - j n ω τ 0 ) exp [ - j ϕ ( t - n τ 0 - β z / ω ) ] × E 0 exp [ - ( 1 / 2 ) α t z ] exp [ j ( ω t - β z ) ] e ^ ( x , y ) ,
d E s ( z s , t ) = - j d a z exp [ ( - ½ ) α t z s ] × t a n = 0 ( r t f ) n exp ( - j n ω τ 0 ) exp [ j ϕ ( t - n τ 0 ) ] × exp [ j ( ω t - 2 β z s ) ] e ^ ,
d a z = E 0 ω d z s π W 0 2 c o exp [ - ( ½ ) α t z s ] × Δ n ( x s , y s , z s ) exp [ - 2 ( x s 2 + y s 2 ) W 0 2 ] d x s d y s ,
d E s D ( z s , t ) = t a n = 0 ( r t f ) n d E s ( z s , t - n τ 0 ) .
E s D ( t ) = z z = 0 z s = L d E s D ( z s , t ) = E 0 e ^ ( x , y ) exp ( j ω t ) ( - 2 j ω π W 0 2 c 0 ) × I ν T n = 0 R h n ( t - n τ 0 ) exp ( - j n ω τ 0 ) ,
h ( t ) m = 0 R m exp ( - j m ω τ 0 ) exp [ j ϕ ( t - m τ 0 ) ] ,
I ν = Δ n ( x s , y s , z s ) exp ( - α t z s ) × exp [ - 2 ( x 2 + y 2 ) / W 0 2 ] exp ( - j 2 β z s ) d V s .
I I 1 + I 2 + I 3 E S ( t ) ¯ 2 + 2 Re [ E S ( t ) E S D * ( t ) ¯ ] + E S D ( t ) ¯ 2 ,
I 2 = 0 ,
I 3 = α R S L { [ ρ 1 - t f 2 L ( ψ ) ] 2 + 2 T ( R 2 - R f 2 ) ( 1 - R 2 ) 2 ( 1 - R f 2 ) ρ 1 - t f 2 L ( ψ ) } I 0
σ 2 = ( I - I ) 2 = [ I 2 + ( I 3 - I 3 ) ] 2 I 2 2 ,
I 2 = 2 Re { A exp [ j ( ξ + ξ 0 ) ] } ,
σ 2 = [ 2 A cos ( ξ + ξ 0 ) ] 2 = 4 cos 2 ( ξ R + ξ 0 ) A A * ,
σ = 2 α R S L ( C r 2 + C i 2 ) 1 / 2 T I 0 cos ( ξ R + ξ 0 )
C r + j C i = - r ( 1 - r R ) 2 + T t f R { ( 1 - r R ) + [ 1 - R f exp ( j ω τ 0 ) ] } ( 1 - r R ) 2 [ 1 - R f exp ( j ω τ 0 ) ] 2 + T t f exp ( - j ω τ 0 ) exp ( - 2 π Δ f τ 0 ) ( 1 - r R ) 2 [ 1 - r exp ( - 2 π Δ f τ 0 ) exp ( - j ω τ 0 ) ] .
I est = I 1 + σ + I 3 ,
I 3 A = α R S L f ( 2 θ s ) { ( ρ 1 - t f 2 ) 2 ( 1 - 2 R f cos θ s + R f 2 ) ( 1 + 2 R f cos θ s + R f 2 ) ( 1 - R f 2 ) 2 × L ( ψ A ) L ( ψ A - 2 θ s ) + T ( R 2 - R f 2 ) ( 1 - R f 2 ) ( 1 - R 2 ) 2 ρ 1 - t f 2 [ L ( ψ A ) + L ( ψ A - 2 θ s ) ] } I 0 ,
I 3 B = α R S L f ( 2 θ s ) { ( ρ 1 - t f 2 ) 2 ( 1 - 2 R f cos θ s + R f 2 ) ( 1 + 2 R f cos θ s + R f 2 ) ( 1 - R f 2 ) 2 × L ( ψ B ) L ( ψ B + 2 θ s ) + T ( R 2 - R f 2 ) ( 1 - R f 2 ) ( 1 - R 2 ) 2 ρ 1 - t f 2 [ L ( ψ B ) + L ( ψ B + 2 θ s ) ] } I 0 ,
ψ A = ω τ 0 + θ s ,
ψ B = ω τ 0 - θ s ,
f ( x ) = ( 1 - R 2 ) 2 ( 1 - R 2 ) + 4 R 2 sin 2 ( x / 2 ) .
Δ ω s = m · p ,
Δ ω s = 2 τ 0 ( 1 - ) θ s ,
Δ i r = 2 χ Δ ν 1 / 2 ( 1 + χ 2 ) 3 / 2 ρ i o Δ f α ,
i ¯ i = [ 1 - ρ / ( 1 - χ 2 ) 1 / 2 ] i o ,
Ω S N = 3 3 4 π N c 0 λ 0 F ρ L 2 [ ( 1 - 2 / 3 ρ ) ω η I 0 B s ] 1 / 2 ,
I = I 1 + I 2 F + I 3 F ,
I 2 F = 2 r F T c r 2 + c i 2 I 0 cos ( 2 β z s + ξ 0 ) ,
I 3 F = r F 2 { [ ρ 1 - t f 2 L ( ψ ) ] 2 + 2 T ( R 2 - R f 2 ) ( 1 - R 2 ) 2 ( 1 - R f 2 ) ρ 1 - t f 2 L ( ψ ) } I 0 .
I A = u R I 1 A + u L I 3 A ,
I B = u L I 1 B + u R I 3 B ,
I 1 A = [ 1 - ρ L ( ψ B ) ] I 0 ,
I 1 B = [ 1 - ρ L ( ψ A ) ] I 0 ,
ψ A = ω τ 0 - θ f + θ s ,
ψ B = ω τ 0 + Δ ω s τ 0 - θ f - θ s ,
- u R ρ L ( ψ B ) + u L A [ L ( ψ A ) L ( ψ A - 2 θ s ) + L ( ψ A ) L ( ψ A - 2 θ s ) ] + u L B [ L ( ψ A ) + L ( ψ A - 2 θ s ) ] = 0 ,
- u L ρ L ( ψ A ) + u R A [ L ( ψ B ) L ( ψ B + 2 θ s ) + L ( ψ B ) L ( ψ B + 2 θ s ) ] + u R B [ L ( ψ B ) + L ( ψ B + 2 θ s ) ] = 0 ,
A = α R S L f ( 2 θ s ) ( ρ 1 - t f 2 ) ( 1 - 2 R f cos θ s + R f 2 ) ( 1 + 2 R f cos θ s + R f 2 ) ( 1 - R f 2 ) 2 ,
B = α R S L f ( 2 θ s ) T ( R 2 - R f 2 ) ( 1 - R f 2 ) ( 1 - R 2 ) 2 ρ 1 - t f 2 ,
L ( x ) 1 ,
L ( x ) = - L 2 ( x ) 2 R f ( 1 - R f ) 2 sin ( x ) - 2 R f ( 1 - R f ) 2 x ,
ψ A - ψ B = 2 ( A + B ) [ ( u L 2 + u R 2 ) ρ - 4 u L u R ( A + B ) ] u L u R [ ρ 2 - 4 ( A + B ) 2 ] θ s ,
ψ A - ψ B = 2 θ s - Δ ω s τ 0 .
Δ ω s = 1 τ 0 [ 2 θ s - ( ψ A - ψ B ) ] = 2 τ 0 ( 1 - ) θ s ,
= ( A + B ) [ ( u R 2 + u L 2 ) ρ - 4 u R u L ( A + B ) ] u L u R [ ρ 2 - 4 ( A + B ) 2 ] .
- ρ L ( - Δ ψ A ) + A [ L ( Δ ψ A ) L ( Δ ψ A - 2 θ s ) + L ( Δ ψ A ) L ( Δ ψ A - 2 θ s ) ] + B [ L ( Δ ψ A ) + L ( Δ ψ A - 2 θ s ) ] = 0
= ( Δ ψ A ) / θ s ,

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