Abstract

The noise suppressions in the chaos lidar (CLIDAR) and the synchronized chaos lidar (S-CLIDAR) systems with the optoelectronic feedback (OEF) and optical feedback (OF) schemes are studied numerically. Compared with the CLIDAR system, the S-CLIDAR system with the OEF scheme has better correlation coefficients in the large noise regime for SNR < 15 dB. For the S-CLIDAR system with the OF scheme, better detections are also achieved in wide ranges depending on the levels of the phase noise presented in the channel. To have the best synchronization and detection quality, the optimized conditions for the coupling and feedback strengths in the S-CLIDAR system are also discussed.

© 2010 Optical Society of America

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  1. F. Y. Lin, and J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10, 991–997 (2004).
    [CrossRef]
  2. T. Mukai, and K. Otsuka, “New route to optical chaos: Successive-subharmonic-oscil- lation cascade in a semiconductor laser coupled to an external cavity,” Phys. Rev. Lett. 55, 1711–1714 (1985).
    [CrossRef] [PubMed]
  3. B. T. J. Mork, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28, 93–108 (1992).
    [CrossRef]
  4. F. Y. Lin, and J. M. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron. 39, 562–568 (2003).
    [CrossRef]
  5. S. Tang, and J. Liu, “Chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron. 37, 329–336 (2001).
    [CrossRef]
  6. F. Y. Lin, and J. M. Liu, “Harmonic frequency locking in a semiconductor laser with delayed negative optoelectronic feedback,” Appl. Phys. Lett. 81, 3128–3130 (2002).
    [CrossRef]
  7. F. Y. Lin, and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221, 173–180 (2003).
    [CrossRef]
  8. V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, “Dynamic behavior and locking of a semiconductor laser subjected to external injection,” IEEE J. Quantum Electron. 34, 2350–2357 (1998).
    [CrossRef]
  9. S. Wieczorek, B. Krauskopf, and D. Lenstra, “A unifying view of bifurcations in a semiconductor laser subject to optical injection,” Opt. Commun. 172, 279–295 (1999).
    [CrossRef]
  10. N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation cw lidar,” Appl. Opt. 22, 1382–1386 (1983).
    [CrossRef] [PubMed]
  11. C. Nagasawa, M. Abo, H. Yamamoto, and O. Uchino, “Random modulation cw lidar using new random sequence,” Appl. Opt. 29, 1466–1470 (1990).
    [CrossRef] [PubMed]
  12. Y. Emery, and C. Flesia, “Use of the a1- and the a2-sequences to modulate continuous-wave pseudorandom noise lidar,” Appl. Opt. 37, 2238–2241 (1998).
    [CrossRef]
  13. V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996).
    [CrossRef]
  14. T. B. Simpson, and J. M. Liu, “Phase and amplitude characteristics of nearly degenerate four-wave mixing in fabry–perot semiconductor lasers,” J. Appl. Phys. 73, 2587–2589 (1993).
    [CrossRef]
  15. A. B. Utkin, A. Lavrov, and R. Vilar, “Laser rangefinder architecture as a cost-effective platform for lidar fire surveillance,” Opt. Laser Technol. 41, 862–870 (2009).
    [CrossRef]
  16. T. Cossio, K. Slatton, W. Carter, K. Shrestha, and D. Harding, “Predicting topographic and bathymetric measurement performance for low-snr airborne lidar,” IEEE Trans. Geosci. Rem. Sens. 47, 2298–2315 (2009).
    [CrossRef]

2009 (2)

A. B. Utkin, A. Lavrov, and R. Vilar, “Laser rangefinder architecture as a cost-effective platform for lidar fire surveillance,” Opt. Laser Technol. 41, 862–870 (2009).
[CrossRef]

T. Cossio, K. Slatton, W. Carter, K. Shrestha, and D. Harding, “Predicting topographic and bathymetric measurement performance for low-snr airborne lidar,” IEEE Trans. Geosci. Rem. Sens. 47, 2298–2315 (2009).
[CrossRef]

2004 (1)

F. Y. Lin, and J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10, 991–997 (2004).
[CrossRef]

2003 (2)

F. Y. Lin, and J. M. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron. 39, 562–568 (2003).
[CrossRef]

F. Y. Lin, and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221, 173–180 (2003).
[CrossRef]

2002 (1)

F. Y. Lin, and J. M. Liu, “Harmonic frequency locking in a semiconductor laser with delayed negative optoelectronic feedback,” Appl. Phys. Lett. 81, 3128–3130 (2002).
[CrossRef]

2001 (1)

S. Tang, and J. Liu, “Chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron. 37, 329–336 (2001).
[CrossRef]

1999 (1)

S. Wieczorek, B. Krauskopf, and D. Lenstra, “A unifying view of bifurcations in a semiconductor laser subject to optical injection,” Opt. Commun. 172, 279–295 (1999).
[CrossRef]

1998 (2)

Y. Emery, and C. Flesia, “Use of the a1- and the a2-sequences to modulate continuous-wave pseudorandom noise lidar,” Appl. Opt. 37, 2238–2241 (1998).
[CrossRef]

V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, “Dynamic behavior and locking of a semiconductor laser subjected to external injection,” IEEE J. Quantum Electron. 34, 2350–2357 (1998).
[CrossRef]

1996 (1)

V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996).
[CrossRef]

1993 (1)

T. B. Simpson, and J. M. Liu, “Phase and amplitude characteristics of nearly degenerate four-wave mixing in fabry–perot semiconductor lasers,” J. Appl. Phys. 73, 2587–2589 (1993).
[CrossRef]

1992 (1)

B. T. J. Mork, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28, 93–108 (1992).
[CrossRef]

1990 (1)

C. Nagasawa, M. Abo, H. Yamamoto, and O. Uchino, “Random modulation cw lidar using new random sequence,” Appl. Opt. 29, 1466–1470 (1990).
[CrossRef] [PubMed]

1985 (1)

T. Mukai, and K. Otsuka, “New route to optical chaos: Successive-subharmonic-oscil- lation cascade in a semiconductor laser coupled to an external cavity,” Phys. Rev. Lett. 55, 1711–1714 (1985).
[CrossRef] [PubMed]

1983 (1)

N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation cw lidar,” Appl. Opt. 22, 1382–1386 (1983).
[CrossRef] [PubMed]

Abo, M.

C. Nagasawa, M. Abo, H. Yamamoto, and O. Uchino, “Random modulation cw lidar using new random sequence,” Appl. Opt. 29, 1466–1470 (1990).
[CrossRef] [PubMed]

Annovazzi-Lodi, V.

V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, “Dynamic behavior and locking of a semiconductor laser subjected to external injection,” IEEE J. Quantum Electron. 34, 2350–2357 (1998).
[CrossRef]

V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996).
[CrossRef]

Baba, H.

N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation cw lidar,” Appl. Opt. 22, 1382–1386 (1983).
[CrossRef] [PubMed]

Carter, W.

T. Cossio, K. Slatton, W. Carter, K. Shrestha, and D. Harding, “Predicting topographic and bathymetric measurement performance for low-snr airborne lidar,” IEEE Trans. Geosci. Rem. Sens. 47, 2298–2315 (2009).
[CrossRef]

Cossio, T.

T. Cossio, K. Slatton, W. Carter, K. Shrestha, and D. Harding, “Predicting topographic and bathymetric measurement performance for low-snr airborne lidar,” IEEE Trans. Geosci. Rem. Sens. 47, 2298–2315 (2009).
[CrossRef]

Donati, S.

V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, “Dynamic behavior and locking of a semiconductor laser subjected to external injection,” IEEE J. Quantum Electron. 34, 2350–2357 (1998).
[CrossRef]

Donati,, S.

V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996).
[CrossRef]

Emery, Y.

Y. Emery, and C. Flesia, “Use of the a1- and the a2-sequences to modulate continuous-wave pseudorandom noise lidar,” Appl. Opt. 37, 2238–2241 (1998).
[CrossRef]

Flesia, C.

Y. Emery, and C. Flesia, “Use of the a1- and the a2-sequences to modulate continuous-wave pseudorandom noise lidar,” Appl. Opt. 37, 2238–2241 (1998).
[CrossRef]

Harding, D.

T. Cossio, K. Slatton, W. Carter, K. Shrestha, and D. Harding, “Predicting topographic and bathymetric measurement performance for low-snr airborne lidar,” IEEE Trans. Geosci. Rem. Sens. 47, 2298–2315 (2009).
[CrossRef]

Krauskopf, B.

S. Wieczorek, B. Krauskopf, and D. Lenstra, “A unifying view of bifurcations in a semiconductor laser subject to optical injection,” Opt. Commun. 172, 279–295 (1999).
[CrossRef]

Lavrov, A.

A. B. Utkin, A. Lavrov, and R. Vilar, “Laser rangefinder architecture as a cost-effective platform for lidar fire surveillance,” Opt. Laser Technol. 41, 862–870 (2009).
[CrossRef]

Lenstra, D.

S. Wieczorek, B. Krauskopf, and D. Lenstra, “A unifying view of bifurcations in a semiconductor laser subject to optical injection,” Opt. Commun. 172, 279–295 (1999).
[CrossRef]

Lin, F. Y.

F. Y. Lin, and J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10, 991–997 (2004).
[CrossRef]

F. Y. Lin, and J. M. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron. 39, 562–568 (2003).
[CrossRef]

F. Y. Lin, and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221, 173–180 (2003).
[CrossRef]

F. Y. Lin, and J. M. Liu, “Harmonic frequency locking in a semiconductor laser with delayed negative optoelectronic feedback,” Appl. Phys. Lett. 81, 3128–3130 (2002).
[CrossRef]

Liu, J.

S. Tang, and J. Liu, “Chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron. 37, 329–336 (2001).
[CrossRef]

Liu, J. M.

F. Y. Lin, and J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10, 991–997 (2004).
[CrossRef]

F. Y. Lin, and J. M. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron. 39, 562–568 (2003).
[CrossRef]

F. Y. Lin, and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221, 173–180 (2003).
[CrossRef]

F. Y. Lin, and J. M. Liu, “Harmonic frequency locking in a semiconductor laser with delayed negative optoelectronic feedback,” Appl. Phys. Lett. 81, 3128–3130 (2002).
[CrossRef]

T. B. Simpson, and J. M. Liu, “Phase and amplitude characteristics of nearly degenerate four-wave mixing in fabry–perot semiconductor lasers,” J. Appl. Phys. 73, 2587–2589 (1993).
[CrossRef]

Mark, J.

B. T. J. Mork, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28, 93–108 (1992).
[CrossRef]

Mork, B. T. J.

B. T. J. Mork, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28, 93–108 (1992).
[CrossRef]

Mukai, T.

T. Mukai, and K. Otsuka, “New route to optical chaos: Successive-subharmonic-oscil- lation cascade in a semiconductor laser coupled to an external cavity,” Phys. Rev. Lett. 55, 1711–1714 (1985).
[CrossRef] [PubMed]

Nagasawa, C.

C. Nagasawa, M. Abo, H. Yamamoto, and O. Uchino, “Random modulation cw lidar using new random sequence,” Appl. Opt. 29, 1466–1470 (1990).
[CrossRef] [PubMed]

Otsuka, K.

T. Mukai, and K. Otsuka, “New route to optical chaos: Successive-subharmonic-oscil- lation cascade in a semiconductor laser coupled to an external cavity,” Phys. Rev. Lett. 55, 1711–1714 (1985).
[CrossRef] [PubMed]

Sakurai, K.

N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation cw lidar,” Appl. Opt. 22, 1382–1386 (1983).
[CrossRef] [PubMed]

Scire, A.

V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, “Dynamic behavior and locking of a semiconductor laser subjected to external injection,” IEEE J. Quantum Electron. 34, 2350–2357 (1998).
[CrossRef]

V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996).
[CrossRef]

Shrestha, K.

T. Cossio, K. Slatton, W. Carter, K. Shrestha, and D. Harding, “Predicting topographic and bathymetric measurement performance for low-snr airborne lidar,” IEEE Trans. Geosci. Rem. Sens. 47, 2298–2315 (2009).
[CrossRef]

Simpson, T. B.

T. B. Simpson, and J. M. Liu, “Phase and amplitude characteristics of nearly degenerate four-wave mixing in fabry–perot semiconductor lasers,” J. Appl. Phys. 73, 2587–2589 (1993).
[CrossRef]

Slatton, K.

T. Cossio, K. Slatton, W. Carter, K. Shrestha, and D. Harding, “Predicting topographic and bathymetric measurement performance for low-snr airborne lidar,” IEEE Trans. Geosci. Rem. Sens. 47, 2298–2315 (2009).
[CrossRef]

Sorel, M.

V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, “Dynamic behavior and locking of a semiconductor laser subjected to external injection,” IEEE J. Quantum Electron. 34, 2350–2357 (1998).
[CrossRef]

Sugimoto, N.

N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation cw lidar,” Appl. Opt. 22, 1382–1386 (1983).
[CrossRef] [PubMed]

Takeuchi, N.

N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation cw lidar,” Appl. Opt. 22, 1382–1386 (1983).
[CrossRef] [PubMed]

Tang, S.

S. Tang, and J. Liu, “Chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron. 37, 329–336 (2001).
[CrossRef]

Uchino, O.

C. Nagasawa, M. Abo, H. Yamamoto, and O. Uchino, “Random modulation cw lidar using new random sequence,” Appl. Opt. 29, 1466–1470 (1990).
[CrossRef] [PubMed]

Utkin, A. B.

A. B. Utkin, A. Lavrov, and R. Vilar, “Laser rangefinder architecture as a cost-effective platform for lidar fire surveillance,” Opt. Laser Technol. 41, 862–870 (2009).
[CrossRef]

Vilar, R.

A. B. Utkin, A. Lavrov, and R. Vilar, “Laser rangefinder architecture as a cost-effective platform for lidar fire surveillance,” Opt. Laser Technol. 41, 862–870 (2009).
[CrossRef]

Wieczorek, S.

S. Wieczorek, B. Krauskopf, and D. Lenstra, “A unifying view of bifurcations in a semiconductor laser subject to optical injection,” Opt. Commun. 172, 279–295 (1999).
[CrossRef]

Yamamoto, H.

C. Nagasawa, M. Abo, H. Yamamoto, and O. Uchino, “Random modulation cw lidar using new random sequence,” Appl. Opt. 29, 1466–1470 (1990).
[CrossRef] [PubMed]

Appl. Opt. (3)

N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation cw lidar,” Appl. Opt. 22, 1382–1386 (1983).
[CrossRef] [PubMed]

C. Nagasawa, M. Abo, H. Yamamoto, and O. Uchino, “Random modulation cw lidar using new random sequence,” Appl. Opt. 29, 1466–1470 (1990).
[CrossRef] [PubMed]

Y. Emery, and C. Flesia, “Use of the a1- and the a2-sequences to modulate continuous-wave pseudorandom noise lidar,” Appl. Opt. 37, 2238–2241 (1998).
[CrossRef]

Appl. Phys. Lett. (1)

F. Y. Lin, and J. M. Liu, “Harmonic frequency locking in a semiconductor laser with delayed negative optoelectronic feedback,” Appl. Phys. Lett. 81, 3128–3130 (2002).
[CrossRef]

IEEE J. Quantum Electron. (5)

V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, “Dynamic behavior and locking of a semiconductor laser subjected to external injection,” IEEE J. Quantum Electron. 34, 2350–2357 (1998).
[CrossRef]

B. T. J. Mork, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28, 93–108 (1992).
[CrossRef]

F. Y. Lin, and J. M. Liu, “Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback,” IEEE J. Quantum Electron. 39, 562–568 (2003).
[CrossRef]

S. Tang, and J. Liu, “Chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron. 37, 329–336 (2001).
[CrossRef]

V. Annovazzi-Lodi, S. Donati, and A. Scire, “Synchronization of chaotic injected-laser systems and its application to optical cryptography,” IEEE J. Quantum Electron. 32, 953–959 (1996).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

F. Y. Lin, and J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10, 991–997 (2004).
[CrossRef]

IEEE Trans. Geosci. Rem. Sens. (1)

T. Cossio, K. Slatton, W. Carter, K. Shrestha, and D. Harding, “Predicting topographic and bathymetric measurement performance for low-snr airborne lidar,” IEEE Trans. Geosci. Rem. Sens. 47, 2298–2315 (2009).
[CrossRef]

J. Appl. Phys. (1)

T. B. Simpson, and J. M. Liu, “Phase and amplitude characteristics of nearly degenerate four-wave mixing in fabry–perot semiconductor lasers,” J. Appl. Phys. 73, 2587–2589 (1993).
[CrossRef]

Opt. Commun. (2)

S. Wieczorek, B. Krauskopf, and D. Lenstra, “A unifying view of bifurcations in a semiconductor laser subject to optical injection,” Opt. Commun. 172, 279–295 (1999).
[CrossRef]

F. Y. Lin, and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221, 173–180 (2003).
[CrossRef]

Opt. Laser Technol. (1)

A. B. Utkin, A. Lavrov, and R. Vilar, “Laser rangefinder architecture as a cost-effective platform for lidar fire surveillance,” Opt. Laser Technol. 41, 862–870 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

T. Mukai, and K. Otsuka, “New route to optical chaos: Successive-subharmonic-oscil- lation cascade in a semiconductor laser coupled to an external cavity,” Phys. Rev. Lett. 55, 1711–1714 (1985).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

(Color online) Schematic setups of the CLIDAR and the S-CLIDAR systems with the OEF and the OF schemes. Tx and Rx are the transmitter and the receiver lasers. ηT and ηR are the feedback strengths of the Tx and Rx and ηC is the coupling strength from the channel to the Rx, respectively. τT and τR are the feedback delay times of the Tx and Rx and τC is the target range delay in the channel, respectively.

Fig. 2
Fig. 2

(a) Time series and (b) autocorrelation of the transmitted waveform from the Tx of the OEF scheme with a delay time τT = 9.5 and a feedback strength ηOEF,T = 0.123. (c)–(f) The detected waveforms in the receivers and their corresponding correlations to the transmitted waveform for the CLIDAR and the S-CLIDAR systems, respectively.

Fig. 6
Fig. 6

(Color online) The optimized coupling strengths of ηOEF,C and ηOF,C for the S-CLIDAR system with (a) the OEF and (b) the OF schemes under different levels of noise

Fig. 3
Fig. 3

(a) Time series and (b) autocorrelation of the transmitted waveform from the Tx of the OF scheme with a delay time τT = 9.5 ns and a feedback strength ηOEF,T = 0.2. (c)–(j) The detected waveforms in the receivers and their corresponding correlations to the transmitted waveform for the CLIDAR and the S-CLIDAR systems with the phase noise levels of m = 0, 0.5, and 0.75, respectively.

Fig. 4
Fig. 4

(Color online) Correlation coefficients of the CLIDAR and the S-CLIDAR systems with (a) the OEF and (b) the OF schemes for different levels of noise.

Fig. 5
Fig. 5

(Color online) The differences of the correlation coefficients Δρ between the S-CLIDAR and the CLIDAR systems for different levels of noise

Equations (11)

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

da T dt = 1 2 [ γ c γ n γ s J ˜ T n ˜ T γ p ( 2 a T + a T 2 ) ] ( 1 + a T ) + η OF , T ( 1 + a T ( t τ T ) ) cos ( ϕ T ( t τ T ) ϕ T ( t ) + ω T τ T )
d ϕ T d t = b 2 [ γ c γ n γ s J ˜ T n ˜ T γ p ( 2 a T + a T 2 ) ] + η OF , T ( 1 + a T ( t τ T ) ) 1 + a T sin ( ϕ T ( t τ T ) ϕ T ( t ) + ω T τ T )
d n ˜ T dt = γ s n ˜ T γ n ( 1 + a T ) 2 n ˜ T γ s J ˜ T ( 2 a T + a T 2 ) + γ s γ p γ c J ˜ T ( 2 a T + a T 2 ( 1 + a T ) 2 + η OEF , T γ s ( J ˜ T + 1 ) ( 1 + 2 a T ( t τ T ) ) + a T ( t τ T ) 2
d a R dt = 1 2 [ γ c γ n γ s J ˜ R n ˜ R γ p ( 2 a R + a R 2 ) ] ( 1 + a R ) + η OF , R ( 1 + a R ( t τ R ) ) cos ( ϕ R ( t τ R ) ϕ R ( t ) + ω R τ R ) + η OF , C ( 1 + a C ( t τ C ) ) cos ( ϕ C ( t τ C ) ϕ R ( t ) + ω T τ C Δ ω t )
d ϕ R dt = b 2 [ γ c γ n γ s J ˜ R n ˜ R γ p ( 2 a R + a R 2 ) ] + η OF , R ( 1 + a R ( t τ R ) ) 1 + a R sin ( ϕ R ( t τ R ) ϕ R ( t ) + ω R τ R ) + η OF , C ( 1 + a C ( t τ C ) ) 1 + a R sin ( ϕ C ( t τ C ) ϕ R ( t ) + ω T τ C Δ ω t )
d n ˜ R dt = γ s n ˜ R γ n ( 1 + a R ) 2 n ˜ R γ s J ˜ R ( 2 a R + a R 2 ) + γ s γ p γ c J ˜ R ( 2 a R + a R 2 ( 1 + a R ) 2 + η OEF , R γ s ( J ˜ R + 1 ) ( 1 + 2 a R ( t τ R ) ) + a R ( t τ R ) 2 + η OEF , C γ s ( J ˜ R + 1 ) ( 1 + 2 a C ( t τ C ) ) + a C ( t τ C ) 2 ,
a C ( t ) = a T ( t ) + RN 1 ( t )
ϕ C ( t ) = ϕ T ( t ) + mRN 2 ( t ) , 0 m 1
SNR = 10 log P s σ RN1 2 ,
ρ ( Δ τ ) = [ S T ( t ) S T ( t ) ] [ S R ( t + Δ τ ) S R ( t ) ] | S T ( t ) S T ( t ) | 2 1 2 | S R ( t ) S R ( t ) | 2 1 2 ,
Δ ρ = ρ S CLIDAR ρ CLIDAR .

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