Abstract

A theoretical and experimental analysis of optical pulse coding techniques applied to distributed optical fiber temperature sensors based on spontaneous Brillouin scattering using the Landau-Placzek ratio (LPR) scheme is presented, quantifying in particular the impact of Simplex coding on stimulated Brillouin and Raman power thresholds. The signal-to-noise ratio (SNR) enhancement and temperature resolution improvement provided by coding are also characterized. Experimental results confirm that, differently from Raman-based sensors, pulse coding affects the stimulated Brillouin threshold, resulting in lower optimal input power levels; these features allow one to achieve high sensing performance avoiding the use of high peak power pulses.

© 2008 Optical Society of America

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References

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  1. "Optical-fibre Sensors," Tech. Focus Nature Photon. 2, 143-158 (2008).
  2. H. H. Kee, G. P. Lees, and T. P. Newson, "1.65 ?m Raman-based distributed temperature sensor," Electron. Lett. 35, 1869-1871 (1999).
    [CrossRef]
  3. M. Niklès, L. Thévenaz, and P. A. Robert, "Simple distributed fiber sensor based on Brillouin gain spectrum analysis," Opt. Lett. 21, 758-760 (1996).
    [CrossRef] [PubMed]
  4. X. Bao, L. Zou, Q. Yu, and L. Chen, "Development and applications of the distributed temperature and strain sensors based on Brillouin scattering," in Proceeding of IEEE Sensors Conf.2004, vol 3, pp. 1210 - 1213.
  5. Y. T. Cho, M. Alahbabi, M. J. Gunning, and T. P. Newson, "50-km single-ended spontaneous-Brillouin-based distributed-temperature sensor exploiting pulsed Raman amplification," Opt. Lett. 28, pp. 1651-1653 (2003).
    [CrossRef] [PubMed]
  6. X. Bao, D. J. Webb, and D. A. Jackson, "Combined distributed temperature and strain sensor based on Brillouin loss in an optical fiber," Opt. Lett. 19, 141-143 (1994).
    [CrossRef] [PubMed]
  7. K. Hotate and M. Tanaka, "Distributed fiber Brillouin strain sensing with 1-cm spatial resolution by correlation-cased continuous-wave technique," IEEE Photon. Technol. Lett. 14, 197-199 (2002).
    [CrossRef]
  8. A. Minardo et al., "A reconstruction technique for long-range stimulated Brillouin scattering distributed fibre-optic sensors: experimental results," Meas. Sci. Technol. 16, 900-908 (2005).
    [CrossRef]
  9. M. A. Soto, P. K. Sahu, G. Bolognini, and F. Di Pasquale, "Brillouin-based distributed temperature sensor employing pulse coding," IEEE Sens. J. 8, 225-226 (2008).
    [CrossRef]
  10. P. C. Wait and T. P. Newson, "Landau Placzek ratio applied to distributed fiber sensing," Opt. Commun. 122, 141-146 (1996).
    [CrossRef]
  11. K. De. Souza et al, "Improvement of signal-to-noise capabilities of a distributed temperature sensor using optical preamplification," Meas. Sci. Technol. 12, 952- 957 (2001).
  12. P. C. Wait, K. De Souza, and T. P. Newson, "A theoretical comparison of spontaneous Raman and Brillouin based fibre optic distributed temperature sensors," Opt. Commun. 144, 17-23 (1997).
    [CrossRef]
  13. K. De Souza and T. P. Newson, "Brillouin-based fiber-optic distributed temperature sensor with optical preamplification," Opt. Lett. 25, 1331-1333 (2000).
    [CrossRef]
  14. Y. T. Cho, M. N. Alahbabi, M. J. Gunning, and T. P. Newson, "Enhanced performance of long range Brillouin intensity based temperature sensors using remote Raman amplification," Meas. Sci. Technol. 15, 1548-1552 (2004).
    [CrossRef]
  15. Y. T. Cho, M. N. Alahbabi, G. Brambilla, and T. P. Newson, "Distributed Raman Amplification Combined With a Remotely Pumped EDFA Utilized to Enhance the Performance of Spontaneous Brillouin-Based Distributed Temperature Sensors," IEEE Photon. Technol. Lett. 17, 1256-1258 (2005).
    [CrossRef]
  16. J. Park et al, "Raman-based distributed temperature sensor with Simplex coding and link optimization," IEEE Photon. Technol. Lett. 18, 1879-1881 (2006).
    [CrossRef]
  17. M. Nazarathy et al., "Real-time long-range complementary correlation optical time-domain reflectometer," J. Lightwave Technol. 7, 24-38 (1989).
    [CrossRef]
  18. M. D. Jones, "Using Simplex codes to improve OTDR Sensitivity," IEEE Photon. Technol. Lett. 15, 822-824 (1993).
    [CrossRef]
  19. D. Lee et al, "Analysis and Experimental Demonstration of Simplex Coding Technique for SNR Enhancement of OTDR," In Proceeding of IEEE LTIMC, (New York, USA, 2004), pp. 118-122,
  20. M. Harwit and N. J. A. Sloane, Hadamard Transform Optics (New York: Academic, 1979).
  21. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (New York: Academic, 1995).
  22. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, (Wiley New York, 1962).
  23. Y. Aoki, K. Tajima, and I. Mito, "Input Power Limits of Single-Mode Optical Fibers due to Stimulated Brillouin Scattering in Optical Communication Systems," J. Lightwave Technol. 6, 710-719 (1988).
    [CrossRef]
  24. K. De Souza, "Significance of coherent Rayleigh noise in fibre-optic distributed temperature sensing based on spontaneous Brillouin scattering," Meas. Sci. Technol. 17, 1065-1069 (2006).
    [CrossRef]

2008

"Optical-fibre Sensors," Tech. Focus Nature Photon. 2, 143-158 (2008).

M. A. Soto, P. K. Sahu, G. Bolognini, and F. Di Pasquale, "Brillouin-based distributed temperature sensor employing pulse coding," IEEE Sens. J. 8, 225-226 (2008).
[CrossRef]

2006

J. Park et al, "Raman-based distributed temperature sensor with Simplex coding and link optimization," IEEE Photon. Technol. Lett. 18, 1879-1881 (2006).
[CrossRef]

K. De Souza, "Significance of coherent Rayleigh noise in fibre-optic distributed temperature sensing based on spontaneous Brillouin scattering," Meas. Sci. Technol. 17, 1065-1069 (2006).
[CrossRef]

2005

Y. T. Cho, M. N. Alahbabi, G. Brambilla, and T. P. Newson, "Distributed Raman Amplification Combined With a Remotely Pumped EDFA Utilized to Enhance the Performance of Spontaneous Brillouin-Based Distributed Temperature Sensors," IEEE Photon. Technol. Lett. 17, 1256-1258 (2005).
[CrossRef]

A. Minardo et al., "A reconstruction technique for long-range stimulated Brillouin scattering distributed fibre-optic sensors: experimental results," Meas. Sci. Technol. 16, 900-908 (2005).
[CrossRef]

2004

Y. T. Cho, M. N. Alahbabi, M. J. Gunning, and T. P. Newson, "Enhanced performance of long range Brillouin intensity based temperature sensors using remote Raman amplification," Meas. Sci. Technol. 15, 1548-1552 (2004).
[CrossRef]

2003

2002

K. Hotate and M. Tanaka, "Distributed fiber Brillouin strain sensing with 1-cm spatial resolution by correlation-cased continuous-wave technique," IEEE Photon. Technol. Lett. 14, 197-199 (2002).
[CrossRef]

2001

K. De. Souza et al, "Improvement of signal-to-noise capabilities of a distributed temperature sensor using optical preamplification," Meas. Sci. Technol. 12, 952- 957 (2001).

2000

1999

H. H. Kee, G. P. Lees, and T. P. Newson, "1.65 ?m Raman-based distributed temperature sensor," Electron. Lett. 35, 1869-1871 (1999).
[CrossRef]

1997

P. C. Wait, K. De Souza, and T. P. Newson, "A theoretical comparison of spontaneous Raman and Brillouin based fibre optic distributed temperature sensors," Opt. Commun. 144, 17-23 (1997).
[CrossRef]

1996

P. C. Wait and T. P. Newson, "Landau Placzek ratio applied to distributed fiber sensing," Opt. Commun. 122, 141-146 (1996).
[CrossRef]

M. Niklès, L. Thévenaz, and P. A. Robert, "Simple distributed fiber sensor based on Brillouin gain spectrum analysis," Opt. Lett. 21, 758-760 (1996).
[CrossRef] [PubMed]

1994

1993

M. D. Jones, "Using Simplex codes to improve OTDR Sensitivity," IEEE Photon. Technol. Lett. 15, 822-824 (1993).
[CrossRef]

1989

M. Nazarathy et al., "Real-time long-range complementary correlation optical time-domain reflectometer," J. Lightwave Technol. 7, 24-38 (1989).
[CrossRef]

1988

Y. Aoki, K. Tajima, and I. Mito, "Input Power Limits of Single-Mode Optical Fibers due to Stimulated Brillouin Scattering in Optical Communication Systems," J. Lightwave Technol. 6, 710-719 (1988).
[CrossRef]

Alahbabi, M.

Alahbabi, M. N.

Y. T. Cho, M. N. Alahbabi, G. Brambilla, and T. P. Newson, "Distributed Raman Amplification Combined With a Remotely Pumped EDFA Utilized to Enhance the Performance of Spontaneous Brillouin-Based Distributed Temperature Sensors," IEEE Photon. Technol. Lett. 17, 1256-1258 (2005).
[CrossRef]

Y. T. Cho, M. N. Alahbabi, M. J. Gunning, and T. P. Newson, "Enhanced performance of long range Brillouin intensity based temperature sensors using remote Raman amplification," Meas. Sci. Technol. 15, 1548-1552 (2004).
[CrossRef]

Aoki, Y.

Y. Aoki, K. Tajima, and I. Mito, "Input Power Limits of Single-Mode Optical Fibers due to Stimulated Brillouin Scattering in Optical Communication Systems," J. Lightwave Technol. 6, 710-719 (1988).
[CrossRef]

Bao, X.

Bolognini, G.

M. A. Soto, P. K. Sahu, G. Bolognini, and F. Di Pasquale, "Brillouin-based distributed temperature sensor employing pulse coding," IEEE Sens. J. 8, 225-226 (2008).
[CrossRef]

Brambilla, G.

Y. T. Cho, M. N. Alahbabi, G. Brambilla, and T. P. Newson, "Distributed Raman Amplification Combined With a Remotely Pumped EDFA Utilized to Enhance the Performance of Spontaneous Brillouin-Based Distributed Temperature Sensors," IEEE Photon. Technol. Lett. 17, 1256-1258 (2005).
[CrossRef]

Cho, Y. T.

Y. T. Cho, M. N. Alahbabi, G. Brambilla, and T. P. Newson, "Distributed Raman Amplification Combined With a Remotely Pumped EDFA Utilized to Enhance the Performance of Spontaneous Brillouin-Based Distributed Temperature Sensors," IEEE Photon. Technol. Lett. 17, 1256-1258 (2005).
[CrossRef]

Y. T. Cho, M. N. Alahbabi, M. J. Gunning, and T. P. Newson, "Enhanced performance of long range Brillouin intensity based temperature sensors using remote Raman amplification," Meas. Sci. Technol. 15, 1548-1552 (2004).
[CrossRef]

Y. T. Cho, M. Alahbabi, M. J. Gunning, and T. P. Newson, "50-km single-ended spontaneous-Brillouin-based distributed-temperature sensor exploiting pulsed Raman amplification," Opt. Lett. 28, pp. 1651-1653 (2003).
[CrossRef] [PubMed]

De Souza, K.

K. De Souza, "Significance of coherent Rayleigh noise in fibre-optic distributed temperature sensing based on spontaneous Brillouin scattering," Meas. Sci. Technol. 17, 1065-1069 (2006).
[CrossRef]

K. De Souza and T. P. Newson, "Brillouin-based fiber-optic distributed temperature sensor with optical preamplification," Opt. Lett. 25, 1331-1333 (2000).
[CrossRef]

P. C. Wait, K. De Souza, and T. P. Newson, "A theoretical comparison of spontaneous Raman and Brillouin based fibre optic distributed temperature sensors," Opt. Commun. 144, 17-23 (1997).
[CrossRef]

De. Souza, K.

K. De. Souza et al, "Improvement of signal-to-noise capabilities of a distributed temperature sensor using optical preamplification," Meas. Sci. Technol. 12, 952- 957 (2001).

Di Pasquale, F.

M. A. Soto, P. K. Sahu, G. Bolognini, and F. Di Pasquale, "Brillouin-based distributed temperature sensor employing pulse coding," IEEE Sens. J. 8, 225-226 (2008).
[CrossRef]

Gunning, M. J.

Y. T. Cho, M. N. Alahbabi, M. J. Gunning, and T. P. Newson, "Enhanced performance of long range Brillouin intensity based temperature sensors using remote Raman amplification," Meas. Sci. Technol. 15, 1548-1552 (2004).
[CrossRef]

Y. T. Cho, M. Alahbabi, M. J. Gunning, and T. P. Newson, "50-km single-ended spontaneous-Brillouin-based distributed-temperature sensor exploiting pulsed Raman amplification," Opt. Lett. 28, pp. 1651-1653 (2003).
[CrossRef] [PubMed]

Hotate, K.

K. Hotate and M. Tanaka, "Distributed fiber Brillouin strain sensing with 1-cm spatial resolution by correlation-cased continuous-wave technique," IEEE Photon. Technol. Lett. 14, 197-199 (2002).
[CrossRef]

Jackson, D. A.

Jones, M. D.

M. D. Jones, "Using Simplex codes to improve OTDR Sensitivity," IEEE Photon. Technol. Lett. 15, 822-824 (1993).
[CrossRef]

Kee, H. H.

H. H. Kee, G. P. Lees, and T. P. Newson, "1.65 ?m Raman-based distributed temperature sensor," Electron. Lett. 35, 1869-1871 (1999).
[CrossRef]

Lees, G. P.

H. H. Kee, G. P. Lees, and T. P. Newson, "1.65 ?m Raman-based distributed temperature sensor," Electron. Lett. 35, 1869-1871 (1999).
[CrossRef]

Minardo, A.

A. Minardo et al., "A reconstruction technique for long-range stimulated Brillouin scattering distributed fibre-optic sensors: experimental results," Meas. Sci. Technol. 16, 900-908 (2005).
[CrossRef]

Mito, I.

Y. Aoki, K. Tajima, and I. Mito, "Input Power Limits of Single-Mode Optical Fibers due to Stimulated Brillouin Scattering in Optical Communication Systems," J. Lightwave Technol. 6, 710-719 (1988).
[CrossRef]

Nazarathy, M.

M. Nazarathy et al., "Real-time long-range complementary correlation optical time-domain reflectometer," J. Lightwave Technol. 7, 24-38 (1989).
[CrossRef]

Newson, T. P.

Y. T. Cho, M. N. Alahbabi, G. Brambilla, and T. P. Newson, "Distributed Raman Amplification Combined With a Remotely Pumped EDFA Utilized to Enhance the Performance of Spontaneous Brillouin-Based Distributed Temperature Sensors," IEEE Photon. Technol. Lett. 17, 1256-1258 (2005).
[CrossRef]

Y. T. Cho, M. N. Alahbabi, M. J. Gunning, and T. P. Newson, "Enhanced performance of long range Brillouin intensity based temperature sensors using remote Raman amplification," Meas. Sci. Technol. 15, 1548-1552 (2004).
[CrossRef]

Y. T. Cho, M. Alahbabi, M. J. Gunning, and T. P. Newson, "50-km single-ended spontaneous-Brillouin-based distributed-temperature sensor exploiting pulsed Raman amplification," Opt. Lett. 28, pp. 1651-1653 (2003).
[CrossRef] [PubMed]

K. De Souza and T. P. Newson, "Brillouin-based fiber-optic distributed temperature sensor with optical preamplification," Opt. Lett. 25, 1331-1333 (2000).
[CrossRef]

H. H. Kee, G. P. Lees, and T. P. Newson, "1.65 ?m Raman-based distributed temperature sensor," Electron. Lett. 35, 1869-1871 (1999).
[CrossRef]

P. C. Wait, K. De Souza, and T. P. Newson, "A theoretical comparison of spontaneous Raman and Brillouin based fibre optic distributed temperature sensors," Opt. Commun. 144, 17-23 (1997).
[CrossRef]

P. C. Wait and T. P. Newson, "Landau Placzek ratio applied to distributed fiber sensing," Opt. Commun. 122, 141-146 (1996).
[CrossRef]

Niklès, M.

Park, J.

J. Park et al, "Raman-based distributed temperature sensor with Simplex coding and link optimization," IEEE Photon. Technol. Lett. 18, 1879-1881 (2006).
[CrossRef]

Robert, P. A.

Sahu, P. K.

M. A. Soto, P. K. Sahu, G. Bolognini, and F. Di Pasquale, "Brillouin-based distributed temperature sensor employing pulse coding," IEEE Sens. J. 8, 225-226 (2008).
[CrossRef]

Soto, M. A.

M. A. Soto, P. K. Sahu, G. Bolognini, and F. Di Pasquale, "Brillouin-based distributed temperature sensor employing pulse coding," IEEE Sens. J. 8, 225-226 (2008).
[CrossRef]

Tajima, K.

Y. Aoki, K. Tajima, and I. Mito, "Input Power Limits of Single-Mode Optical Fibers due to Stimulated Brillouin Scattering in Optical Communication Systems," J. Lightwave Technol. 6, 710-719 (1988).
[CrossRef]

Tanaka, M.

K. Hotate and M. Tanaka, "Distributed fiber Brillouin strain sensing with 1-cm spatial resolution by correlation-cased continuous-wave technique," IEEE Photon. Technol. Lett. 14, 197-199 (2002).
[CrossRef]

Thévenaz, L.

Wait, P. C.

P. C. Wait, K. De Souza, and T. P. Newson, "A theoretical comparison of spontaneous Raman and Brillouin based fibre optic distributed temperature sensors," Opt. Commun. 144, 17-23 (1997).
[CrossRef]

P. C. Wait and T. P. Newson, "Landau Placzek ratio applied to distributed fiber sensing," Opt. Commun. 122, 141-146 (1996).
[CrossRef]

Webb, D. J.

Electron. Lett.

H. H. Kee, G. P. Lees, and T. P. Newson, "1.65 ?m Raman-based distributed temperature sensor," Electron. Lett. 35, 1869-1871 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

K. Hotate and M. Tanaka, "Distributed fiber Brillouin strain sensing with 1-cm spatial resolution by correlation-cased continuous-wave technique," IEEE Photon. Technol. Lett. 14, 197-199 (2002).
[CrossRef]

Y. T. Cho, M. N. Alahbabi, G. Brambilla, and T. P. Newson, "Distributed Raman Amplification Combined With a Remotely Pumped EDFA Utilized to Enhance the Performance of Spontaneous Brillouin-Based Distributed Temperature Sensors," IEEE Photon. Technol. Lett. 17, 1256-1258 (2005).
[CrossRef]

J. Park et al, "Raman-based distributed temperature sensor with Simplex coding and link optimization," IEEE Photon. Technol. Lett. 18, 1879-1881 (2006).
[CrossRef]

M. D. Jones, "Using Simplex codes to improve OTDR Sensitivity," IEEE Photon. Technol. Lett. 15, 822-824 (1993).
[CrossRef]

IEEE Sens. J.

M. A. Soto, P. K. Sahu, G. Bolognini, and F. Di Pasquale, "Brillouin-based distributed temperature sensor employing pulse coding," IEEE Sens. J. 8, 225-226 (2008).
[CrossRef]

J. Lightwave Technol.

M. Nazarathy et al., "Real-time long-range complementary correlation optical time-domain reflectometer," J. Lightwave Technol. 7, 24-38 (1989).
[CrossRef]

Y. Aoki, K. Tajima, and I. Mito, "Input Power Limits of Single-Mode Optical Fibers due to Stimulated Brillouin Scattering in Optical Communication Systems," J. Lightwave Technol. 6, 710-719 (1988).
[CrossRef]

Meas. Sci. Technol.

K. De Souza, "Significance of coherent Rayleigh noise in fibre-optic distributed temperature sensing based on spontaneous Brillouin scattering," Meas. Sci. Technol. 17, 1065-1069 (2006).
[CrossRef]

Y. T. Cho, M. N. Alahbabi, M. J. Gunning, and T. P. Newson, "Enhanced performance of long range Brillouin intensity based temperature sensors using remote Raman amplification," Meas. Sci. Technol. 15, 1548-1552 (2004).
[CrossRef]

K. De. Souza et al, "Improvement of signal-to-noise capabilities of a distributed temperature sensor using optical preamplification," Meas. Sci. Technol. 12, 952- 957 (2001).

A. Minardo et al., "A reconstruction technique for long-range stimulated Brillouin scattering distributed fibre-optic sensors: experimental results," Meas. Sci. Technol. 16, 900-908 (2005).
[CrossRef]

Opt. Commun.

P. C. Wait, K. De Souza, and T. P. Newson, "A theoretical comparison of spontaneous Raman and Brillouin based fibre optic distributed temperature sensors," Opt. Commun. 144, 17-23 (1997).
[CrossRef]

P. C. Wait and T. P. Newson, "Landau Placzek ratio applied to distributed fiber sensing," Opt. Commun. 122, 141-146 (1996).
[CrossRef]

Opt. Lett.

Tech. Focus Nature Photon.

"Optical-fibre Sensors," Tech. Focus Nature Photon. 2, 143-158 (2008).

Other

D. Lee et al, "Analysis and Experimental Demonstration of Simplex Coding Technique for SNR Enhancement of OTDR," In Proceeding of IEEE LTIMC, (New York, USA, 2004), pp. 118-122,

M. Harwit and N. J. A. Sloane, Hadamard Transform Optics (New York: Academic, 1979).

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (New York: Academic, 1995).

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, (Wiley New York, 1962).

X. Bao, L. Zou, Q. Yu, and L. Chen, "Development and applications of the distributed temperature and strain sensors based on Brillouin scattering," in Proceeding of IEEE Sensors Conf.2004, vol 3, pp. 1210 - 1213.

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Figures (9)

Fig. 1.
Fig. 1.

Fig. 1. Characteristic t-z curves defined by Eqs. (8)(9) employed in the solution of the system of Eqs. (5)(6). (a) Single-pulse case. (b) Coded-pulses case.

Fig. 2.
Fig. 2.

Threshold power for both SRS (dotted line) and SBS (solid line) versus number of bits within a codeword, for different spatial resolution settings.

Fig. 3.
Fig. 3.

Experimental set-up of the implemented sensor system.

Fig. 4.
Fig. 4.

Transmission and reflection spectra of the narrowband FBG.

Fig. 5.
Fig. 5.

Temperature distribution along the fiber (TCC at 343 K) for single pulses BDTS (gray line) and 127-bit Simplex-coded BDTS (black line). Inset, spatial resolution achieved by Simplex-coded and single-pulse BDTS.

Fig. 6.
Fig. 6.

Temperature resolution as function of fiber length, for conventional-BDTS, and 127-bit Simplex coded BDTS with and without λ-averaging.

Fig. 7.
Fig. 7.

Coherent Rayleigh noise fluctuations versus distance obtained with (black line) and without (gray line) λ-averaging.

Fig. 8.
Fig. 8.

Measured temperature and error range (standard deviation) at 4 km distance versus input peak power for 127-bit Simplex-coded BDTS (squares) and conventional-BDTS (circles).

Fig. 9.
Fig. 9.

Temperature distribution along the sensing fiber, with input Ppeak=16 dBm, for conventional (gray line) and Simplex-coded (black line) BDTS.

Tables (1)

Tables Icon

Table 1. Maximum temperature error in measurements at 4 km distance

Equations (18)

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

I RS I SpBS = T f T ( β T ρ 0 ν s 2 1 )
T = 1 K T ( 1 LPR ( T ) LPR ( T R ) ) + T R
P th = A eff C R , B g R , B L eff
G cod = L c + 1 2 L c
I P z + n c I P t = g B I P I B α I P
I B z n c I B t = g B I P I B + σ I B
I P ( z = 0 , t ) = I Po ( t ) , I B [ z = z L ( t ) , t ] = I Bo ( t )
Γ 1 : χ P = t n c z
Γ 2 : χ B = t + n c z .
I P ( z , t ) = I Po ( t n c z ) exp ( α z )
I B ( z = 0 , t ) = I Bo ( t 2 ) exp [ α ( c 2 n t ) + g B z i ( t ) z L ( t ) I P ( ξ , t n c ξ ) d ξ ]
z i ( t ) = c 2 n ( t T ) , z L ( t ) = c 2 n t
z i = 0 z L I P ( ξ , t MAX n c ξ ) d ξ = I Po z i = 0 z L exp ( α ξ ) d ξ = I Po α [ 1 exp ( α z L ) ]
z L = c 2 n T B ( L c + 1 ) 2
I B ( z = 0 , t MAX ) = I Bo ( t MAX / 2 ) exp ( α z L ' + g B I Po L eff )
L eff = 1 α [ 1 exp ( α L p ( L c + 1 ) 4 ) ]
P th = A eff C R , B g R , B L eff ( Δ ν B + Δ ν P Δ ν B )
f CRN = V g 4 Δ z Δ f

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