Abstract

We propose a novel method of quadrature demodulation with synchronous difference for suppressing noise in interferometric fiber-optic gyroscopes (IFOGs). For an IFOG with sine wave phase modulation, an in-phase result and a quadrature result are obtained simultaneously by coherent detection. Eigenfrequency modulation is used and a phase shift of 45° is set between the modulation signal and the reference signal, so that two results have the same expectation of amplitude but with opposite signs. A synchronous difference procedure is carried out for output, in which signals are added up and common noise between two results is eliminated. Theoretical analysis and experimental results show that both short term noise and long term instability of the IFOG are reduced by this method. In experimental comparison with the traditional demodulation method on the same IFOG with a 1982 m fiber coil, this method reduces the bias drift from 0.040°/h to 0.004°/h.

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References

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  1. E. J. Post, “Sagnac effect,” Rev. Mod. Phys.39, 475–493 (1967).
    [CrossRef]
  2. H. J. Arditty and H. C. Lefèvre, “Sagnac effect in fiber gyroscopes,” Opt. Lett.6, 401–403 (1981).
    [CrossRef] [PubMed]
  3. R. A. Bergh, H. C. Lefèvre, and H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol.2, 91–107 (1984).
    [CrossRef]
  4. H. C. Lefèvre, The Fiber-Optic Gyroscope (Artech House, 1993).
  5. G. B. Malykin, “On the ultimate sensitivity of fiber-optic gyroscopes,” Tech. Phys.54, 415–418 (2009).
    [CrossRef]
  6. I. A. Andronova and G. B. Malykin, “Physical problems of fiber gyroscopy based on the Sagnac effect,” Phys. Usp.45, 793–817 (2002).
    [CrossRef]
  7. P. Y. Chien and C. L. Pan, “Triangular phase-modulation approach to an open-loop fiber-optic gyroscope,” Opt. Lett.16, 1701–1703 (1991).
    [CrossRef] [PubMed]
  8. D. A. Jackson, A. D. Kersey, and A. C. Lewin, “Fibre gyroscope with passive quadrature detection,” Electron. Lett.20, 399–401 (1984).
    [CrossRef]
  9. J. Blake and I. S. Kim, “Distribution of relative intensity noise in signal and quadrature channels of a fiber-optic gyroscope,” Opt. Lett.19, 1648–1650 (1994)
    [CrossRef] [PubMed]
  10. X. Wang, C. He, and Z. Wang, “Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes,” Opt. Lett.36, 1191–1193 (2011).
    [CrossRef] [PubMed]
  11. W. K. Burns, R. P. Moeller, and A. Dandridge, “Excess noise in fiber gyroscope sources,” IEEE Photonic. Tech. Lett.2, 606–608 (1990).
    [CrossRef]
  12. R. P. Moeller and W. K. Burns, “1.06-ptm all-fiber gyroscope with noise subtraction,” Opt. Lett.16, 1902–1904 (1991).
    [CrossRef] [PubMed]
  13. R. C. Rabelo, R. T. Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol.18, 2146–2150 (2000).
    [CrossRef]
  14. P. Polynkin, J. Arruda, and J. Blake, “All-optical noise-subtraction scheme for a fiber-optic gyroscope,” Opt. Lett.25, 147–149 (2000).
    [CrossRef]
  15. R. Ulrich, “Fiber-optic rotation sensing with low drift,” Opt. Lett.5, 173–175 (1980).
    [CrossRef] [PubMed]
  16. K. Bohm, P. Marten, K. Petermann, E. Weidel, and R. Ulrich, “Low-drift fiber gyro using a superluminescent diode,” Electron. Lett.17, 352–353 (1981).
    [CrossRef]
  17. E. Jones and J. W. Parker, “Bias reduction by polarisation dispersion in the fibre-optic gyroscope,” Electron. Lett.22, 54–56 (1986).
    [CrossRef]
  18. S. L. A. Carrara, B. Y. Kim, and H. J. Shaw, “Bias drift reduction in polarization-maintaining fiber gyroscope,” Opt. Lett.12, 214–216 (1987).
    [CrossRef] [PubMed]
  19. O. Çelikel and F. Sametoǧlu, “Assessment of magneto-optic Faraday effect-based drift on interferometric single-mode fiber optic gyroscope (IFOG) as a function of variable degree of polarization (DOP),” Meas. Sci. Technol.23, 025104 (2012).
    [CrossRef]
  20. A. Lompado, J. C. Reinhardt, L. C. Heaton, J. L. Williams, and P. B. Ruffin, “Full Stokes polarimeter for characterization of fiber optic gyroscope coils,” Opt. Express17, 8370–8381 (2009).
    [CrossRef] [PubMed]
  21. Y. Yang, Z. Wang, and Z. Li, “Optically compensated dual-polarization interferometric fiber-optic gyroscope,” Opt. Lett.37, 2841–2843 (2012).
    [CrossRef] [PubMed]
  22. D. Kim and J. Kang, “Sagnac loop interferometer based on polarization maintaining photonic crystal fiber with reduced temperature sensitivity,” Opt. Express12, 4490–4495 (2004).
    [CrossRef] [PubMed]
  23. Y. Zhao, Y. Zheng, Y. Lin, and B. Li, “Step by step improvement of measurement methods for earth’s rotary rate using fiber optic gyro,” Measurement44, 1177–1182 (2011).
    [CrossRef]
  24. F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE74, 162–168 (1986).
    [CrossRef]
  25. O. Çelikel and S. E. San, “Design details and characterization of all digital closed-loop interferometric fiber optic gyroscope with superluminescent light emitting diode,” Opt. Rev.16, 35–43 (2009).
    [CrossRef]

2012 (2)

O. Çelikel and F. Sametoǧlu, “Assessment of magneto-optic Faraday effect-based drift on interferometric single-mode fiber optic gyroscope (IFOG) as a function of variable degree of polarization (DOP),” Meas. Sci. Technol.23, 025104 (2012).
[CrossRef]

Y. Yang, Z. Wang, and Z. Li, “Optically compensated dual-polarization interferometric fiber-optic gyroscope,” Opt. Lett.37, 2841–2843 (2012).
[CrossRef] [PubMed]

2011 (2)

X. Wang, C. He, and Z. Wang, “Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes,” Opt. Lett.36, 1191–1193 (2011).
[CrossRef] [PubMed]

Y. Zhao, Y. Zheng, Y. Lin, and B. Li, “Step by step improvement of measurement methods for earth’s rotary rate using fiber optic gyro,” Measurement44, 1177–1182 (2011).
[CrossRef]

2009 (3)

G. B. Malykin, “On the ultimate sensitivity of fiber-optic gyroscopes,” Tech. Phys.54, 415–418 (2009).
[CrossRef]

O. Çelikel and S. E. San, “Design details and characterization of all digital closed-loop interferometric fiber optic gyroscope with superluminescent light emitting diode,” Opt. Rev.16, 35–43 (2009).
[CrossRef]

A. Lompado, J. C. Reinhardt, L. C. Heaton, J. L. Williams, and P. B. Ruffin, “Full Stokes polarimeter for characterization of fiber optic gyroscope coils,” Opt. Express17, 8370–8381 (2009).
[CrossRef] [PubMed]

2004 (1)

2002 (1)

I. A. Andronova and G. B. Malykin, “Physical problems of fiber gyroscopy based on the Sagnac effect,” Phys. Usp.45, 793–817 (2002).
[CrossRef]

2000 (2)

1994 (1)

1991 (2)

1990 (1)

W. K. Burns, R. P. Moeller, and A. Dandridge, “Excess noise in fiber gyroscope sources,” IEEE Photonic. Tech. Lett.2, 606–608 (1990).
[CrossRef]

1987 (1)

1986 (2)

E. Jones and J. W. Parker, “Bias reduction by polarisation dispersion in the fibre-optic gyroscope,” Electron. Lett.22, 54–56 (1986).
[CrossRef]

F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE74, 162–168 (1986).
[CrossRef]

1984 (2)

R. A. Bergh, H. C. Lefèvre, and H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol.2, 91–107 (1984).
[CrossRef]

D. A. Jackson, A. D. Kersey, and A. C. Lewin, “Fibre gyroscope with passive quadrature detection,” Electron. Lett.20, 399–401 (1984).
[CrossRef]

1981 (2)

K. Bohm, P. Marten, K. Petermann, E. Weidel, and R. Ulrich, “Low-drift fiber gyro using a superluminescent diode,” Electron. Lett.17, 352–353 (1981).
[CrossRef]

H. J. Arditty and H. C. Lefèvre, “Sagnac effect in fiber gyroscopes,” Opt. Lett.6, 401–403 (1981).
[CrossRef] [PubMed]

1980 (1)

1967 (1)

E. J. Post, “Sagnac effect,” Rev. Mod. Phys.39, 475–493 (1967).
[CrossRef]

Allan, D. W.

F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE74, 162–168 (1986).
[CrossRef]

Andronova, I. A.

I. A. Andronova and G. B. Malykin, “Physical problems of fiber gyroscopy based on the Sagnac effect,” Phys. Usp.45, 793–817 (2002).
[CrossRef]

Arditty, H. J.

Arruda, J.

Bergh, R. A.

R. A. Bergh, H. C. Lefèvre, and H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol.2, 91–107 (1984).
[CrossRef]

Blake, J.

Bohm, K.

K. Bohm, P. Marten, K. Petermann, E. Weidel, and R. Ulrich, “Low-drift fiber gyro using a superluminescent diode,” Electron. Lett.17, 352–353 (1981).
[CrossRef]

Burns, W. K.

R. P. Moeller and W. K. Burns, “1.06-ptm all-fiber gyroscope with noise subtraction,” Opt. Lett.16, 1902–1904 (1991).
[CrossRef] [PubMed]

W. K. Burns, R. P. Moeller, and A. Dandridge, “Excess noise in fiber gyroscope sources,” IEEE Photonic. Tech. Lett.2, 606–608 (1990).
[CrossRef]

Carrara, S. L. A.

Carvalho, R. T.

Çelikel, O.

O. Çelikel and F. Sametoǧlu, “Assessment of magneto-optic Faraday effect-based drift on interferometric single-mode fiber optic gyroscope (IFOG) as a function of variable degree of polarization (DOP),” Meas. Sci. Technol.23, 025104 (2012).
[CrossRef]

O. Çelikel and S. E. San, “Design details and characterization of all digital closed-loop interferometric fiber optic gyroscope with superluminescent light emitting diode,” Opt. Rev.16, 35–43 (2009).
[CrossRef]

Chien, P. Y.

Dandridge, A.

W. K. Burns, R. P. Moeller, and A. Dandridge, “Excess noise in fiber gyroscope sources,” IEEE Photonic. Tech. Lett.2, 606–608 (1990).
[CrossRef]

He, C.

Heaton, L. C.

Jackson, D. A.

D. A. Jackson, A. D. Kersey, and A. C. Lewin, “Fibre gyroscope with passive quadrature detection,” Electron. Lett.20, 399–401 (1984).
[CrossRef]

Jones, E.

E. Jones and J. W. Parker, “Bias reduction by polarisation dispersion in the fibre-optic gyroscope,” Electron. Lett.22, 54–56 (1986).
[CrossRef]

Kang, J.

Kersey, A. D.

D. A. Jackson, A. D. Kersey, and A. C. Lewin, “Fibre gyroscope with passive quadrature detection,” Electron. Lett.20, 399–401 (1984).
[CrossRef]

Kim, B. Y.

Kim, D.

Kim, I. S.

Lefèvre, H. C.

R. A. Bergh, H. C. Lefèvre, and H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol.2, 91–107 (1984).
[CrossRef]

H. J. Arditty and H. C. Lefèvre, “Sagnac effect in fiber gyroscopes,” Opt. Lett.6, 401–403 (1981).
[CrossRef] [PubMed]

H. C. Lefèvre, The Fiber-Optic Gyroscope (Artech House, 1993).

Lewin, A. C.

D. A. Jackson, A. D. Kersey, and A. C. Lewin, “Fibre gyroscope with passive quadrature detection,” Electron. Lett.20, 399–401 (1984).
[CrossRef]

Li, B.

Y. Zhao, Y. Zheng, Y. Lin, and B. Li, “Step by step improvement of measurement methods for earth’s rotary rate using fiber optic gyro,” Measurement44, 1177–1182 (2011).
[CrossRef]

Li, Z.

Lin, Y.

Y. Zhao, Y. Zheng, Y. Lin, and B. Li, “Step by step improvement of measurement methods for earth’s rotary rate using fiber optic gyro,” Measurement44, 1177–1182 (2011).
[CrossRef]

Lompado, A.

Malykin, G. B.

G. B. Malykin, “On the ultimate sensitivity of fiber-optic gyroscopes,” Tech. Phys.54, 415–418 (2009).
[CrossRef]

I. A. Andronova and G. B. Malykin, “Physical problems of fiber gyroscopy based on the Sagnac effect,” Phys. Usp.45, 793–817 (2002).
[CrossRef]

Marten, P.

K. Bohm, P. Marten, K. Petermann, E. Weidel, and R. Ulrich, “Low-drift fiber gyro using a superluminescent diode,” Electron. Lett.17, 352–353 (1981).
[CrossRef]

Moeller, R. P.

R. P. Moeller and W. K. Burns, “1.06-ptm all-fiber gyroscope with noise subtraction,” Opt. Lett.16, 1902–1904 (1991).
[CrossRef] [PubMed]

W. K. Burns, R. P. Moeller, and A. Dandridge, “Excess noise in fiber gyroscope sources,” IEEE Photonic. Tech. Lett.2, 606–608 (1990).
[CrossRef]

Pan, C. L.

Parker, J. W.

E. Jones and J. W. Parker, “Bias reduction by polarisation dispersion in the fibre-optic gyroscope,” Electron. Lett.22, 54–56 (1986).
[CrossRef]

Petermann, K.

K. Bohm, P. Marten, K. Petermann, E. Weidel, and R. Ulrich, “Low-drift fiber gyro using a superluminescent diode,” Electron. Lett.17, 352–353 (1981).
[CrossRef]

Polynkin, P.

Post, E. J.

E. J. Post, “Sagnac effect,” Rev. Mod. Phys.39, 475–493 (1967).
[CrossRef]

Rabelo, R. C.

Reinhardt, J. C.

Ruffin, P. B.

Sametog?lu, F.

O. Çelikel and F. Sametoǧlu, “Assessment of magneto-optic Faraday effect-based drift on interferometric single-mode fiber optic gyroscope (IFOG) as a function of variable degree of polarization (DOP),” Meas. Sci. Technol.23, 025104 (2012).
[CrossRef]

San, S. E.

O. Çelikel and S. E. San, “Design details and characterization of all digital closed-loop interferometric fiber optic gyroscope with superluminescent light emitting diode,” Opt. Rev.16, 35–43 (2009).
[CrossRef]

Shaw, H. J.

S. L. A. Carrara, B. Y. Kim, and H. J. Shaw, “Bias drift reduction in polarization-maintaining fiber gyroscope,” Opt. Lett.12, 214–216 (1987).
[CrossRef] [PubMed]

R. A. Bergh, H. C. Lefèvre, and H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol.2, 91–107 (1984).
[CrossRef]

Ulrich, R.

K. Bohm, P. Marten, K. Petermann, E. Weidel, and R. Ulrich, “Low-drift fiber gyro using a superluminescent diode,” Electron. Lett.17, 352–353 (1981).
[CrossRef]

R. Ulrich, “Fiber-optic rotation sensing with low drift,” Opt. Lett.5, 173–175 (1980).
[CrossRef] [PubMed]

Walls, F. L.

F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE74, 162–168 (1986).
[CrossRef]

Wang, X.

Wang, Z.

Weidel, E.

K. Bohm, P. Marten, K. Petermann, E. Weidel, and R. Ulrich, “Low-drift fiber gyro using a superluminescent diode,” Electron. Lett.17, 352–353 (1981).
[CrossRef]

Williams, J. L.

Yang, Y.

Zhao, Y.

Y. Zhao, Y. Zheng, Y. Lin, and B. Li, “Step by step improvement of measurement methods for earth’s rotary rate using fiber optic gyro,” Measurement44, 1177–1182 (2011).
[CrossRef]

Zheng, Y.

Y. Zhao, Y. Zheng, Y. Lin, and B. Li, “Step by step improvement of measurement methods for earth’s rotary rate using fiber optic gyro,” Measurement44, 1177–1182 (2011).
[CrossRef]

Electron. Lett. (3)

K. Bohm, P. Marten, K. Petermann, E. Weidel, and R. Ulrich, “Low-drift fiber gyro using a superluminescent diode,” Electron. Lett.17, 352–353 (1981).
[CrossRef]

E. Jones and J. W. Parker, “Bias reduction by polarisation dispersion in the fibre-optic gyroscope,” Electron. Lett.22, 54–56 (1986).
[CrossRef]

D. A. Jackson, A. D. Kersey, and A. C. Lewin, “Fibre gyroscope with passive quadrature detection,” Electron. Lett.20, 399–401 (1984).
[CrossRef]

IEEE Photonic. Tech. Lett. (1)

W. K. Burns, R. P. Moeller, and A. Dandridge, “Excess noise in fiber gyroscope sources,” IEEE Photonic. Tech. Lett.2, 606–608 (1990).
[CrossRef]

J. Lightwave Technol. (2)

R. A. Bergh, H. C. Lefèvre, and H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol.2, 91–107 (1984).
[CrossRef]

R. C. Rabelo, R. T. Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol.18, 2146–2150 (2000).
[CrossRef]

Meas. Sci. Technol. (1)

O. Çelikel and F. Sametoǧlu, “Assessment of magneto-optic Faraday effect-based drift on interferometric single-mode fiber optic gyroscope (IFOG) as a function of variable degree of polarization (DOP),” Meas. Sci. Technol.23, 025104 (2012).
[CrossRef]

Measurement (1)

Y. Zhao, Y. Zheng, Y. Lin, and B. Li, “Step by step improvement of measurement methods for earth’s rotary rate using fiber optic gyro,” Measurement44, 1177–1182 (2011).
[CrossRef]

Opt. Express (2)

Opt. Lett. (9)

Opt. Rev. (1)

O. Çelikel and S. E. San, “Design details and characterization of all digital closed-loop interferometric fiber optic gyroscope with superluminescent light emitting diode,” Opt. Rev.16, 35–43 (2009).
[CrossRef]

Phys. Usp. (1)

I. A. Andronova and G. B. Malykin, “Physical problems of fiber gyroscopy based on the Sagnac effect,” Phys. Usp.45, 793–817 (2002).
[CrossRef]

Proc. IEEE (1)

F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE74, 162–168 (1986).
[CrossRef]

Rev. Mod. Phys. (1)

E. J. Post, “Sagnac effect,” Rev. Mod. Phys.39, 475–493 (1967).
[CrossRef]

Tech. Phys. (1)

G. B. Malykin, “On the ultimate sensitivity of fiber-optic gyroscopes,” Tech. Phys.54, 415–418 (2009).
[CrossRef]

Other (1)

H. C. Lefèvre, The Fiber-Optic Gyroscope (Artech House, 1993).

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Figures (6)

Fig. 1
Fig. 1

Quadrature demodulation with synchronous difference for an open-loop polarization maintaining IFOG. The experiment setup includes a C-Band ASE light source with a spectrum of 40 nm and output power of 13 dBmW, a circulator, a photodetector (PD) with a response of 10 μA/mW and load resistant of 100 KΩ, a Y-junction waveguide with polarizer and phase modulation function, a 1982 m long coil of polarization maintaining fiber (PMF), a dual-channel signal generator, a dual-channel digitizer with flexible resolution of 16–24 bits, and a computer.

Fig. 2
Fig. 2

Numerical simulation for IFOG outputs with optical intensity noise. The signal-to-noise ratio (SNR) of the source intensity is 35 dB. Theoretical value of the rotation rate is 9.667°/h in accordance with the following experiment. 5000 samples are given, where the sampling time is 0.15 s. Single channel results ΩI and ΩQ, differential result Ωout, and the traditional demodulation result ΩT are put forward for comparison.

Fig. 3
Fig. 3

(a) Numerical simulation for output noise amplitude versus SNR of source intensity. Noise amplitudes are quantified as standard deviations for ΩI, ΩQ, Ωout, and ΩT. (b) Numerical simulation for output bias versus SNR of source intensity. Biases are calculated by mean values of ΩI, ΩQ, Ωout, and ΩT.

Fig. 4
Fig. 4

Experimental results for IFOG outputs. The test was during a period of 50 min, where the sample time was 0.148 s. ΩI, ΩQ, Ωout, and ΩT are put forward for comparison. The test was carried out in a laboratory on the 4th floor of a building, where human activities might introduce additional noise due to acoustic vibration of light paths. This additional noise did not affect our comparison.

Fig. 5
Fig. 5

Distribution of ΩI and ΩQ detected in experiment. The solid curves represent probability density functions (PDFs) of ΩI, ΩQ, and Ωout. ΩI has a mean value of −6.80°/h and standard deviation of 1.048°/h. ΩQ has a mean value of 6.43°/h and standard deviation of 1.045°/h. Ωout has a mean value of 6.61°/h and standard deviation of 0.732°/h. They all have negative biases, as their absolute mean values are lower than the theoretical value of 9.666°/h. The stable bias is not included in IFOG noise for it can be calibrated.

Fig. 6
Fig. 6

Allan variance analysis of the IFOG. The bias drift of the traditional demodulated result is 0.040°/h, and the value is reduced to 0.004°/h by the quadrature demodulation with synchronous difference.

Equations (28)

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ϕ m ( t ) = ϕ 0 [ sin ( ω m t ) cos ( ω m t ) ] = 2 ϕ 0 sin ( ω m t π 4 ) ,
Δ ϕ ( t ) = ϕ m ( t ) ϕ m ( t τ ) = 2 2 ϕ 0 cos ( ω m t π 4 ) ,
I 1 H ( t ) = I 0 η J 1 ( ϕ b ) sin ( ϕ s ) sin ( ω m t + 3 π 4 ) = 2 2 I 0 η J 1 ( ϕ b ) sin ( ϕ s ) sin ( ω m t ) + 2 2 I 0 η J 1 ( ϕ b ) sin ( ϕ s ) cos ( ω m t ) ,
1 V f I ( t ) V I ( t ) = 2 2 I 0 η J 1 ( ϕ b ) sin ( ϕ s ) + h . c . ,
1 V f I ( t ) V Q ( t ) = 2 2 I 0 η J 1 ( ϕ b ) sin ( ϕ s ) + h . c . .
I I = 2 2 I 0 η J 1 ( ϕ b ) sin ( ϕ s ) ,
I Q = 2 2 I 0 η J 1 ( ϕ b ) sin ( ϕ s ) .
ϕ I = arctan 2 I I J 2 ( ϕ b ) I 2 H J 1 ( ϕ b ) = ϕ s ,
ϕ Q = arctan 2 I Q J 2 ( ϕ b ) I 2 H J 1 ( ϕ b ) = ϕ s ,
Ω I ( t ) = Ω + N + ( t ) N ( t ) + N 1 ( t ) ,
Ω Q ( t ) = Ω + N + ( t ) + N ( t ) + N 2 ( t ) ,
Ω out ( t ) = Ω Q ( t ) Ω I ( t ) 2 = Ω + N ( t ) + N 2 ( t ) N 1 ( t ) 2 .
< i N 2 > = < i T 2 > + < i S 2 > + < i I 2 > = ( 4 k T R L + 2 e < i > + < i > 2 Δ ν ) B ,
n P D ( t ) = η n S ( t ) g Q ( t ) ,
I P D = I D + n P D ( t ) = η ( I 0 + n S ( t ) ) g Q ( t ) ,
< [ n I ( t ) ] 2 > < [ n Q ( t ) ] 2 > 1 2 < [ n 1 H ( t ) ] 2 > ,
Ω I ( t ) = Ω + B I ( t ) = Ω + B + ( t ) B ( t ) + B 1 ( t ) ,
Ω Q ( t ) = Ω + B Q ( t ) = Ω + B + ( t ) + B ( t ) + B 2 ( t ) ,
Ω out ( t ) = Ω + B ( t ) + B 2 ( t ) B 1 ( t ) 2 .
R d B = 20 log [ min ( σ I , σ Q ) σ out ] ,
ϕ s = 2 π L D λ c Ω ,
I D = η I 0 g ( t ) = 1 2 η I 0 { 1 + cos [ ϕ s + Δ ϕ ( t ) ] }
Δ ϕ ( t ) = ϕ m ( t ) ϕ m ( t τ ) = 2 ϕ 0 sin ( ω m τ / 2 ) cos ( ω m t ) ,
g ( t ) = 1 2 { 1 + J 0 ( ϕ b ) cos ( ϕ s ) + 2 n = 1 ( 1 ) n J 2 n ( ϕ b ) cos [ 2 n ω m ( t τ 2 ) ] cos ( ϕ s ) + 2 n = 1 ( 1 ) n J 2 n 1 ( ϕ b ) cos [ ( 2 n 1 ) ω m ( t τ 2 ) ] sin ( ϕ s ) }
ϕ s = arctan I 1 H J 2 ( ϕ b ) I 2 H J 1 ( ϕ b ) .
I 1 H ( t ) = I 0 η J 1 ( ϕ b ) sin ( ϕ s ) cos ( ω m t π 2 ) = I 0 η J 1 ( ϕ b ) sin ( ϕ s ) sin ( ω m t + π ) ,
ϕ s = arcsin [ I 1 H I 0 η J 1 ( ϕ b ) ] .
I 1 H = I I 2 + I Q 2 ,

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