Abstract

In this paper, we propose a combined system of heterodyne detection with laser pulse and photon counting based on Geiger-mode avalanche photodiode (GM-APD) that is designed to achieve the range of remote non-cooperative target. Based on the heterodyne principle and assuming that the creation of primary electrons in GM-APD is Poisson-distributed, the range accuracy model is established. The factors that influence the range accuracy, namely pulse width, echo intensity, local oscillator (LO) intensity, noise, echo position, and beat frequency, are discussed. The results show that these six factors have significant influence on the range accuracy when the echo intensity is extremely weak. In case that the primary electrons of the echo signal are beyond 4, the pulse width and echo intensity are the main influence factors. It is also shown that the stronger echo intensity, narrower pulse width, low noise, large echo position, and small beat frequency produce higher range accuracy in a pulsed photon heterodyne detection system based on GM-APD.

© 2013 OSA

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    [CrossRef]
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  5. D. Renker, “Geiger-mode avalanche photodiodes, history, properties and problems,” Nucl. Instrum. Methods Phys. Res., Sect. A.567, 48–56 (2006).
  6. S. Johnson, P. Gatt, and T. Nichols, “Analysis of Geiger-mode APD laser radars,” Proc. SPIE5086, 359–368 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  26. M. Henriksson, “Detection probabilities for photon-counting avalanche photodiodes applied to a laser radar system,” Appl. Opt.44(24), 5140–5147 (2005).
    [CrossRef] [PubMed]
  27. D. G. Fouche, “Detection and false-alarm probabilities for laser radars that use Geiger-mode detectors,” Appl. Opt.42(27), 5388–5398 (2003).
    [CrossRef] [PubMed]
  28. Y. M. Chi, B. Qi, W. Zhu, L. Qian, H. K. Lo, S. H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys.13(1), 1–18 (2011).
    [CrossRef]

2012

L. Liu, H. Zhang, J. Guo, S. Zhao, and T. Wang, “Photon time-interval statistics applied to the analysis of laser heterodyne signal with photon counter,” Opt. Commun.285(18), 3820–3826 (2012).
[CrossRef]

C. Niclass, K. Ito, M. Soga, H. Matsubara, I. Aoyagi, S. Kato, and M. Kagami, “Design and characterization of a 256 x 64-pixel single-photon imager in CMOS for a MEMS-based laser scanning time-of-flight sensor,” Opt. Express20(11), 11863–11881 (2012).
[CrossRef] [PubMed]

2011

Y. M. Chi, B. Qi, W. Zhu, L. Qian, H. K. Lo, S. H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys.13(1), 1–18 (2011).
[CrossRef]

2010

F. Wang, Y. Zhao, Y. Zhang, and X. Sun, “Range accuracy limitation of pulse ranging systems based on Geiger mode single-photon detectors,” Appl. Opt.49(29), 5561–5566 (2010).
[CrossRef] [PubMed]

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun.283(2), 304–308 (2010).
[CrossRef]

2008

L. A. Jiang and J. X. Luu, “Heterodyne detection with a weak local oscillator,” Appl. Opt.47(10), 1486–1503 (2008).
[CrossRef] [PubMed]

P. Mercier, J. Bénier, P. A. Frugier, G. Contencin, J. Veaux, S. L. Basseuil, and M. Debruyne, “Heterodyne velocimetry and detonics experiments,” Proc. SPIE7126, 71261O (2008).
[CrossRef]

2007

D. Stoppa, L. Pancheri, M. Scandiuzzo, L. Gonzo, G. F. D. Betta, and A. Simoni, “A CMOS 3-D imager based on single photon avalanche diode,” IEEE Trans. Circ. Syst.54(1), 4–12 (2007).
[CrossRef]

2006

J. X. Luu and L. A. Jiang, “Saturation effects in heterodyne detection with Geiger-mode InGaAs avalanche photodiode detector arrays,” Appl. Opt.45(16), 3798–3804 (2006).
[CrossRef] [PubMed]

D. Renker, “Geiger-mode avalanche photodiodes, history, properties and problems,” Nucl. Instrum. Methods Phys. Res., Sect. A.567, 48–56 (2006).

P. Andersson, “Long-range three-dimensional imaging using range-gated laser radar images,” Opt. Eng.45(3), 034301 (2006).
[CrossRef]

2005

O. Steinvall and T. Chevalier, “Range accuracy and resolution for laser radars,” Proc. SPIE5988, 598808 (2005).
[CrossRef]

M. Henriksson, “Detection probabilities for photon-counting avalanche photodiodes applied to a laser radar system,” Appl. Opt.44(24), 5140–5147 (2005).
[CrossRef] [PubMed]

R. M. Marino and W. R. D. Jr, “Jigsaw: a foliage-penetrating 3D imaging laser radar system,” Lincoln Lab. J.15, 23–36 (2005).

2004

M. Salem and A. Dogariu, “Optical heterodyne detection of random electromagnetic beams,” J. Mod. Opt.51, 2305–2313 (2004).

2003

D. G. Fouche, “Detection and false-alarm probabilities for laser radars that use Geiger-mode detectors,” Appl. Opt.42(27), 5388–5398 (2003).
[CrossRef] [PubMed]

S. Johnson, P. Gatt, and T. Nichols, “Analysis of Geiger-mode APD laser radars,” Proc. SPIE5086, 359–368 (2003).
[CrossRef]

2002

M. A. Albota, R. M. Heinrichs, D. G. Kocher, D. G. Fouche, B. E. Player, M. E. O’Brien, B. F. Aull, J. J. Zayhowski, J. Mooney, B. C. Willard, and R. R. Carlson, “Three-dimensional imaging laser radar with a photon-counting avalanche photodiode array and microchip laser,” Appl. Opt.41(36), 7671–7678 (2002).
[CrossRef] [PubMed]

B. F. Aull, A. H. Loomis, D. J. Young, R. M. Heinrichs, B. J. Felton, P. J. Daniels, and D. J. Landers, “Geiger-mode avalanche photodiodes for three-dimensional imaging,” Lincoln Lab. J.13, 335–350 (2002).

H. Ailisto, V. Heikkinen, R. Mitikka, R. Myllylä, J. Kostamovaara, A. Mäntyniemi, and M. Koskinen, “Scannerless imaging pulse-laser range finding,” J. Opt. A.4(6), S337–S346 (2002).
[CrossRef]

2001

M. C. Amann, T. B. M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng.40(1), 10–19 (2001).
[CrossRef]

1997

1984

1979

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” J. Mod. Opt.26, 627–644 (1979).

1976

1975

1965

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE53(11), 1688–1700 (1965).
[CrossRef]

Abbas, M. M.

Ailisto, H.

H. Ailisto, V. Heikkinen, R. Mitikka, R. Myllylä, J. Kostamovaara, A. Mäntyniemi, and M. Koskinen, “Scannerless imaging pulse-laser range finding,” J. Opt. A.4(6), S337–S346 (2002).
[CrossRef]

Albota, M. A.

Amann, M. C.

M. C. Amann, T. B. M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng.40(1), 10–19 (2001).
[CrossRef]

Andersson, P.

P. Andersson, “Long-range three-dimensional imaging using range-gated laser radar images,” Opt. Eng.45(3), 034301 (2006).
[CrossRef]

Aoyagi, I.

Aull, B. F.

Basseuil, S. L.

P. Mercier, J. Bénier, P. A. Frugier, G. Contencin, J. Veaux, S. L. Basseuil, and M. Debruyne, “Heterodyne velocimetry and detonics experiments,” Proc. SPIE7126, 71261O (2008).
[CrossRef]

Bénier, J.

P. Mercier, J. Bénier, P. A. Frugier, G. Contencin, J. Veaux, S. L. Basseuil, and M. Debruyne, “Heterodyne velocimetry and detonics experiments,” Proc. SPIE7126, 71261O (2008).
[CrossRef]

Betta, G. F. D.

D. Stoppa, L. Pancheri, M. Scandiuzzo, L. Gonzo, G. F. D. Betta, and A. Simoni, “A CMOS 3-D imager based on single photon avalanche diode,” IEEE Trans. Circ. Syst.54(1), 4–12 (2007).
[CrossRef]

Buhl, D.

Carlson, R. R.

Chellappa, R.

Chevalier, T.

O. Steinvall and T. Chevalier, “Range accuracy and resolution for laser radars,” Proc. SPIE5988, 598808 (2005).
[CrossRef]

Chi, Y. M.

Y. M. Chi, B. Qi, W. Zhu, L. Qian, H. K. Lo, S. H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys.13(1), 1–18 (2011).
[CrossRef]

Contencin, G.

P. Mercier, J. Bénier, P. A. Frugier, G. Contencin, J. Veaux, S. L. Basseuil, and M. Debruyne, “Heterodyne velocimetry and detonics experiments,” Proc. SPIE7126, 71261O (2008).
[CrossRef]

Daniels, P. J.

B. F. Aull, A. H. Loomis, D. J. Young, R. M. Heinrichs, B. J. Felton, P. J. Daniels, and D. J. Landers, “Geiger-mode avalanche photodiodes for three-dimensional imaging,” Lincoln Lab. J.13, 335–350 (2002).

Debruyne, M.

P. Mercier, J. Bénier, P. A. Frugier, G. Contencin, J. Veaux, S. L. Basseuil, and M. Debruyne, “Heterodyne velocimetry and detonics experiments,” Proc. SPIE7126, 71261O (2008).
[CrossRef]

Der, S.

Dogariu, A.

M. Salem and A. Dogariu, “Optical heterodyne detection of random electromagnetic beams,” J. Mod. Opt.51, 2305–2313 (2004).

Felton, B. J.

B. F. Aull, A. H. Loomis, D. J. Young, R. M. Heinrichs, B. J. Felton, P. J. Daniels, and D. J. Landers, “Geiger-mode avalanche photodiodes for three-dimensional imaging,” Lincoln Lab. J.13, 335–350 (2002).

Fink, D.

Fouche, D. G.

Frugier, P. A.

P. Mercier, J. Bénier, P. A. Frugier, G. Contencin, J. Veaux, S. L. Basseuil, and M. Debruyne, “Heterodyne velocimetry and detonics experiments,” Proc. SPIE7126, 71261O (2008).
[CrossRef]

Gatt, P.

S. Johnson, P. Gatt, and T. Nichols, “Analysis of Geiger-mode APD laser radars,” Proc. SPIE5086, 359–368 (2003).
[CrossRef]

Gonzo, L.

D. Stoppa, L. Pancheri, M. Scandiuzzo, L. Gonzo, G. F. D. Betta, and A. Simoni, “A CMOS 3-D imager based on single photon avalanche diode,” IEEE Trans. Circ. Syst.54(1), 4–12 (2007).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE53(11), 1688–1700 (1965).
[CrossRef]

Guo, J.

L. Liu, H. Zhang, J. Guo, S. Zhao, and T. Wang, “Photon time-interval statistics applied to the analysis of laser heterodyne signal with photon counter,” Opt. Commun.285(18), 3820–3826 (2012).
[CrossRef]

Heikkinen, V.

H. Ailisto, V. Heikkinen, R. Mitikka, R. Myllylä, J. Kostamovaara, A. Mäntyniemi, and M. Koskinen, “Scannerless imaging pulse-laser range finding,” J. Opt. A.4(6), S337–S346 (2002).
[CrossRef]

Heinrichs, R. M.

Henriksson, M.

Hong, K. H.

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun.283(2), 304–308 (2010).
[CrossRef]

Ito, K.

Jiang, L. A.

Johnson, S.

S. Johnson, P. Gatt, and T. Nichols, “Analysis of Geiger-mode APD laser radars,” Proc. SPIE5086, 359–368 (2003).
[CrossRef]

Jr, W. R. D.

R. M. Marino and W. R. D. Jr, “Jigsaw: a foliage-penetrating 3D imaging laser radar system,” Lincoln Lab. J.15, 23–36 (2005).

Kagami, M.

Kato, S.

Kim, B. W.

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun.283(2), 304–308 (2010).
[CrossRef]

Kim, T. H.

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun.283(2), 304–308 (2010).
[CrossRef]

Kocher, D. G.

Kong, H. J.

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun.283(2), 304–308 (2010).
[CrossRef]

Koskinen, M.

H. Ailisto, V. Heikkinen, R. Mitikka, R. Myllylä, J. Kostamovaara, A. Mäntyniemi, and M. Koskinen, “Scannerless imaging pulse-laser range finding,” J. Opt. A.4(6), S337–S346 (2002).
[CrossRef]

Kostamovaara, J.

H. Ailisto, V. Heikkinen, R. Mitikka, R. Myllylä, J. Kostamovaara, A. Mäntyniemi, and M. Koskinen, “Scannerless imaging pulse-laser range finding,” J. Opt. A.4(6), S337–S346 (2002).
[CrossRef]

Kostiuk, T.

Landers, D. J.

B. F. Aull, A. H. Loomis, D. J. Young, R. M. Heinrichs, B. J. Felton, P. J. Daniels, and D. J. Landers, “Geiger-mode avalanche photodiodes for three-dimensional imaging,” Lincoln Lab. J.13, 335–350 (2002).

Lescure, T. B. M.

M. C. Amann, T. B. M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng.40(1), 10–19 (2001).
[CrossRef]

Liu, L.

L. Liu, H. Zhang, J. Guo, S. Zhao, and T. Wang, “Photon time-interval statistics applied to the analysis of laser heterodyne signal with photon counter,” Opt. Commun.285(18), 3820–3826 (2012).
[CrossRef]

Lo, H. K.

Y. M. Chi, B. Qi, W. Zhu, L. Qian, H. K. Lo, S. H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys.13(1), 1–18 (2011).
[CrossRef]

Loomis, A. H.

B. F. Aull, A. H. Loomis, D. J. Young, R. M. Heinrichs, B. J. Felton, P. J. Daniels, and D. J. Landers, “Geiger-mode avalanche photodiodes for three-dimensional imaging,” Lincoln Lab. J.13, 335–350 (2002).

Luu, J. X.

Lvovsky, A. I.

Y. M. Chi, B. Qi, W. Zhu, L. Qian, H. K. Lo, S. H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys.13(1), 1–18 (2011).
[CrossRef]

Malota, F.

Mäntyniemi, A.

H. Ailisto, V. Heikkinen, R. Mitikka, R. Myllylä, J. Kostamovaara, A. Mäntyniemi, and M. Koskinen, “Scannerless imaging pulse-laser range finding,” J. Opt. A.4(6), S337–S346 (2002).
[CrossRef]

Marino, R. M.

R. M. Marino and W. R. D. Jr, “Jigsaw: a foliage-penetrating 3D imaging laser radar system,” Lincoln Lab. J.15, 23–36 (2005).

Matsubara, H.

Mercier, P.

P. Mercier, J. Bénier, P. A. Frugier, G. Contencin, J. Veaux, S. L. Basseuil, and M. Debruyne, “Heterodyne velocimetry and detonics experiments,” Proc. SPIE7126, 71261O (2008).
[CrossRef]

Mitikka, R.

H. Ailisto, V. Heikkinen, R. Mitikka, R. Myllylä, J. Kostamovaara, A. Mäntyniemi, and M. Koskinen, “Scannerless imaging pulse-laser range finding,” J. Opt. A.4(6), S337–S346 (2002).
[CrossRef]

Mooney, J.

Mumma, M. J.

Myllylä, R.

H. Ailisto, V. Heikkinen, R. Mitikka, R. Myllylä, J. Kostamovaara, A. Mäntyniemi, and M. Koskinen, “Scannerless imaging pulse-laser range finding,” J. Opt. A.4(6), S337–S346 (2002).
[CrossRef]

M. C. Amann, T. B. M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng.40(1), 10–19 (2001).
[CrossRef]

Nichols, T.

S. Johnson, P. Gatt, and T. Nichols, “Analysis of Geiger-mode APD laser radars,” Proc. SPIE5086, 359–368 (2003).
[CrossRef]

Niclass, C.

O’Brien, M. E.

Oh, M. S.

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun.283(2), 304–308 (2010).
[CrossRef]

Pancheri, L.

D. Stoppa, L. Pancheri, M. Scandiuzzo, L. Gonzo, G. F. D. Betta, and A. Simoni, “A CMOS 3-D imager based on single photon avalanche diode,” IEEE Trans. Circ. Syst.54(1), 4–12 (2007).
[CrossRef]

Player, B. E.

Qi, B.

Y. M. Chi, B. Qi, W. Zhu, L. Qian, H. K. Lo, S. H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys.13(1), 1–18 (2011).
[CrossRef]

Qian, L.

Y. M. Chi, B. Qi, W. Zhu, L. Qian, H. K. Lo, S. H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys.13(1), 1–18 (2011).
[CrossRef]

Redman, B.

Renker, D.

D. Renker, “Geiger-mode avalanche photodiodes, history, properties and problems,” Nucl. Instrum. Methods Phys. Res., Sect. A.567, 48–56 (2006).

Rioux, M.

M. C. Amann, T. B. M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng.40(1), 10–19 (2001).
[CrossRef]

Salem, M.

M. Salem and A. Dogariu, “Optical heterodyne detection of random electromagnetic beams,” J. Mod. Opt.51, 2305–2313 (2004).

Scandiuzzo, M.

D. Stoppa, L. Pancheri, M. Scandiuzzo, L. Gonzo, G. F. D. Betta, and A. Simoni, “A CMOS 3-D imager based on single photon avalanche diode,” IEEE Trans. Circ. Syst.54(1), 4–12 (2007).
[CrossRef]

Simoni, A.

D. Stoppa, L. Pancheri, M. Scandiuzzo, L. Gonzo, G. F. D. Betta, and A. Simoni, “A CMOS 3-D imager based on single photon avalanche diode,” IEEE Trans. Circ. Syst.54(1), 4–12 (2007).
[CrossRef]

Soga, M.

Steinvall, O.

O. Steinvall and T. Chevalier, “Range accuracy and resolution for laser radars,” Proc. SPIE5988, 598808 (2005).
[CrossRef]

Stoppa, D.

D. Stoppa, L. Pancheri, M. Scandiuzzo, L. Gonzo, G. F. D. Betta, and A. Simoni, “A CMOS 3-D imager based on single photon avalanche diode,” IEEE Trans. Circ. Syst.54(1), 4–12 (2007).
[CrossRef]

Sun, X.

Tian, L.

Y. M. Chi, B. Qi, W. Zhu, L. Qian, H. K. Lo, S. H. Youn, A. I. Lvovsky, and L. Tian, “A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution,” New J. Phys.13(1), 1–18 (2011).
[CrossRef]

Veaux, J.

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Figures (6)

Fig. 1
Fig. 1

Block diagram of photon heterodyne detection system.

Fig. 2
Fig. 2

Time sequence of pulse heterodyne detection ranging system.

Fig. 3
Fig. 3

Detection probability distribution with M e 2 =0.32 , f IF =100MHz , p w =50ns , τ d =50ns , T G =1μs , N PE =50kHz , and N s = N LO =3MHz .

Fig. 4
Fig. 4

Influence of pulse width on range accuracy with M e 2 =0.32 , f IF =100MHz , τ d =50ns , T G =1μs , N PE =50kHz , and N LO =3MHz .

Fig. 5
Fig. 5

(a) Influence of noise on range accuracy with M e 2 =0.32 , f IF =100MHz , p w =10ns , τ d =50ns , T G =1μs , and N LO =3MHz ; (b) Influence of echo position on range accuracy with M e 2 =0.32 , f IF =100MHz , p w =10ns , T G =1μs , N PE =50kHz , and N LO =3MHz .

Fig. 6
Fig. 6

(a) Influence of LO intensity on range accuracy with M e 2 =0.32 , f IF =100MHz , p w =10ns , τ d =50ns , T G =1μs , and N PE =50kHz ; (b) Influence of beat frequency on range accuracy with M e 2 =0.32 , p w =10ns , τ d =50ns , T G =1μs , N PE =50kHz and N LO =3MHz .

Equations (19)

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E l ( r ,t)= e l ^ A l ( r )exp[ j ω l t φ l ( r ) ]
E s ( r ,t)= e s ^ A s ( r ) S r ( t )exp[ j ω s t φ s ( r ) ]
S r ( t )= 1 τ p t τ p exp( t/ τ p ),( t0 )
i IF ( t )= η q e hυ { σ | A l ( r ) | 2 d 2 r + S r ( t t d ) σ | A l ( r ) | 2 d 2 r +2 S r ( t t d ) σ | A l ( r ) || A s ( r ) |cos[ ω IF ( t t d )+Δφ( r ) ] d 2 r }
R Beat ( t )= N LO + N s S r ( t τ d )+2 M e S r ( t τ d ) N LO N s cos[ ω IF ( t τ d )+Δφ ]
S PE ( t )={ N PE ,0<t τ d N PE + R Beat ( t ), τ d <t τ d + p w N PE , τ d + p w <t T G
N e ( t ) ¯ = 0 t S PE ( ξ )dξ
P( k, t 1 , t 2 )= [ t 1 t 2 S PE ( t )dt ] k k! exp( t 1 t 2 S PE ( t )dt )
P D ( T G )=1 e N e ( T G ) ¯
P D ( t )= P( k>0 ) 1 e N e ( T G ) ¯ = 1P( k=0 ) 1 e N e ( T G ) ¯ = 1 e N e ( t ) ¯ 1 e N e ( T G ) ¯
p( t )= P D ( t ) t
t ¯ = 0 tp( t )dt
σ 2 = 0 t 2 p( t )dt t ¯ 2
ΔR= ( t ¯ τ d )c 2
R σ = σc 2
N e ( t ) ¯ ={ N PE t,0t τ d N PE t+( N LO + N s )( t τ d )+ 2 M e N LO N s ω IF { sin[ ω IF ( t τ d )+Δφ ]sinΔφ } , τ d t τ d + p w N PE t+( N LO + N s ) p w + 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] , τ d + p w t T G
p( t )={ N PE e N PE t 1 e N PE T G ( N LO + N s ) p w 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] ,( t τ d ) 1 1 e N PE T G ( N LO + N s ) p w 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] e { N PE t+( N LO + N s )( t τ d )+ 2 M e N LO N s ω IF { sin[ ω IF ( t τ d )+Δφ ]sinΔφ } } ×{ N PE + N LO + N s +2 M e N LO N s cos[ ω IF ( t τ d )+Δφ ] },( τ d <t τ d + p w ) N PE 1 e N PE T G ( N LO + N s ) p w 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] e { N PE t+( N LO + N s ) p w + 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] } ,( τ d + p w <t T G )
t ¯ = 0 τ d t N PE e N PE t 1 e N PE T G ( N LO + N s ) p w 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] dt + τ d τ d + p w t× 1 1 e N PE T G ( N LO + N s ) p w 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] e { N PE t+( N LO + N s )( t τ d )+ 2 M e N LO N s ω IF { sin[ ω IF ( t τ d )+Δφ ]sinΔφ } } ×{ N PE + N LO + N s +2 M e N LO N s cos[ ω IF ( t τ d )+Δφ ] }dt+ τ d + p w T G t N PE 1 e N PE T G ( N LO + N s ) p w 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] e { N PE t+( N LO + N s ) p w + 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] } dt
σ 2 = 0 τ d t 2 N PE e N PE t 1 e N PE T G ( N LO + N s ) p w 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] dt + τ d τ d + p w t 2 × 1 1 e N PE T G ( N LO + N s ) p w 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] e { N PE t+( N LO + N s )( t τ d )+ 2 M e N LO N s ω IF { sin[ ω IF ( t τ d )+Δφ ]sinΔφ } } ×{ N PE + N LO + N s +2 M e N LO N s cos[ ω IF ( t τ d )+Δφ ] }dt+ τ d + p w T G t 2 × N PE 1 e N PE T G ( N LO + N s ) p w 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] e { N PE t+( N LO + N s ) p w + 2 M e N LO N s ω IF [ sin( ω IF p w +Δφ )sinΔφ ] } dt t ¯ 2

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