Abstract

Geiger mode single-photon detectors have been used in pulse ranging system and three-dimensional imaging systems due to high sensitivity and easy integration. The ranging accuracy and precision is influenced by many factors. Based on statistical theory, five main factors are discussed in this article, namely, echo signal intensity, pulse width, detector quantum efficiency, target position, and background noise. An analytical relationship among the ranging accuracy, precision, and these factors is obtained for a Q-switched laser pulse. Through this relationship, it is shown that the echo signal intensity and pulse width are more important than other factors and higher echo signal intensity and narrower pulse width can result in better accuracy and precision.

© 2010 Optical Society of America

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  1. 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, 7671–7678 (2002).
    [CrossRef]
  2. R. M. Marino and W. R. Davis, Jr., “Jigsaw: a foliage-penetrating 3D imaging laser radar system,” Lincoln Lab. J. 15, 23–36(2005).
  3. M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (2002).
  4. S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
    [CrossRef]
  5. F. Zappa, S. Tisa, A. Tosi, and S. Cova, “Principles and features of single-photon avalanche diode arrays,” Sens. Actuators A, Phys. 140, 103–112 (2007).
    [CrossRef]
  6. D. R. Gerwe, P. S. Idell, and J. Vaughn, “Cramer-Rao bound analysis of target characterization accuracy limits for imaging systems,” Proc. SPIE 4490, 245–255 (2001).
    [CrossRef]
  7. S. Johnson and S. Cain, “Bound on range precision for shot-noise limited ladar systems,” Appl. Opt. 47, 5147–5154 (2008).
    [CrossRef] [PubMed]
  8. O. Steinvall and T. Chevalier, “Range accuracy and resolution for laser radars,” Proc. SPIE 5988, 598808 (2005).
    [CrossRef]
  9. D. G. Fouche, “Detection and false-alarm probabilities for laser radars that use Geiger-mode detectors,” Appl. Opt. 42, 5388–5398 (2003).
    [CrossRef] [PubMed]
  10. P. Gatt, S. Johnson, and T. Nichols, “Geiger-mode avalanche photodiode ladar receiver performance characteristics and detection statistics,” Appl. Opt. 48, 3261–3276 (2009).
    [CrossRef] [PubMed]
  11. G. R. Osche, Optical Detection Theory for Laser Applications (Wiley-Interscience, 2002).

2009

2008

2007

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

F. Zappa, S. Tisa, A. Tosi, and S. Cova, “Principles and features of single-photon avalanche diode arrays,” Sens. Actuators A, Phys. 140, 103–112 (2007).
[CrossRef]

2005

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

O. Steinvall and T. Chevalier, “Range accuracy and resolution for laser radars,” Proc. SPIE 5988, 598808 (2005).
[CrossRef]

2003

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, 7671–7678 (2002).
[CrossRef]

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (2002).

2001

D. R. Gerwe, P. S. Idell, and J. Vaughn, “Cramer-Rao bound analysis of target characterization accuracy limits for imaging systems,” Proc. SPIE 4490, 245–255 (2001).
[CrossRef]

Albota, M. A.

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (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, 7671–7678 (2002).
[CrossRef]

Aull, B. F.

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, 7671–7678 (2002).
[CrossRef]

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (2002).

Cain, S.

Campisi, A.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Carlson, R. R.

Chevalier, T.

O. Steinvall and T. Chevalier, “Range accuracy and resolution for laser radars,” Proc. SPIE 5988, 598808 (2005).
[CrossRef]

Cosentino, L.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Cova, S.

F. Zappa, S. Tisa, A. Tosi, and S. Cova, “Principles and features of single-photon avalanche diode arrays,” Sens. Actuators A, Phys. 140, 103–112 (2007).
[CrossRef]

Davis, W. R.

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

Fallica, G.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Finocchiaro, P.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Fouche, D. G.

Gatt, P.

Gerwe, D. R.

D. R. Gerwe, P. S. Idell, and J. Vaughn, “Cramer-Rao bound analysis of target characterization accuracy limits for imaging systems,” Proc. SPIE 4490, 245–255 (2001).
[CrossRef]

Heinrichs, R. M.

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (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, 7671–7678 (2002).
[CrossRef]

Idell, P. S.

D. R. Gerwe, P. S. Idell, and J. Vaughn, “Cramer-Rao bound analysis of target characterization accuracy limits for imaging systems,” Proc. SPIE 4490, 245–255 (2001).
[CrossRef]

Johnson, S.

Kocher, D. G.

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (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, 7671–7678 (2002).
[CrossRef]

Lanzan, L.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Lombardo, S.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Marino, R. M.

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

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (2002).

Mazzillo, M.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Mooney, J.

Mooney, J. G.

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (2002).

Musumeci, F.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Newbury, N. R.

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (2002).

Nichols, T.

O’Brien, M. E.

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (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, 7671–7678 (2002).
[CrossRef]

Osche, G. R.

G. R. Osche, Optical Detection Theory for Laser Applications (Wiley-Interscience, 2002).

Player, B. E.

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, 7671–7678 (2002).
[CrossRef]

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (2002).

Privitera, S.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Sanfilippo, D.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Sciacca, E.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Scordino, A.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Steinvall, O.

O. Steinvall and T. Chevalier, “Range accuracy and resolution for laser radars,” Proc. SPIE 5988, 598808 (2005).
[CrossRef]

Tisa, S.

F. Zappa, S. Tisa, A. Tosi, and S. Cova, “Principles and features of single-photon avalanche diode arrays,” Sens. Actuators A, Phys. 140, 103–112 (2007).
[CrossRef]

Tosi, A.

F. Zappa, S. Tisa, A. Tosi, and S. Cova, “Principles and features of single-photon avalanche diode arrays,” Sens. Actuators A, Phys. 140, 103–112 (2007).
[CrossRef]

Tudisco, S.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Valvo, G.

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

Vaughn, J.

D. R. Gerwe, P. S. Idell, and J. Vaughn, “Cramer-Rao bound analysis of target characterization accuracy limits for imaging systems,” Proc. SPIE 4490, 245–255 (2001).
[CrossRef]

Willard, B. C.

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (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, 7671–7678 (2002).
[CrossRef]

Zappa, F.

F. Zappa, S. Tisa, A. Tosi, and S. Cova, “Principles and features of single-photon avalanche diode arrays,” Sens. Actuators A, Phys. 140, 103–112 (2007).
[CrossRef]

Zayhowski, J. J.

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, 7671–7678 (2002).
[CrossRef]

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (2002).

Appl. Opt.

Lincoln Lab. J.

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

M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with Geiger-mode avalanche photodiode arrays,” Lincoln Lab. J. 13, 351–370 (2002).

Proc. SPIE

S. Tudisco, S. Privitera, F. Musumeci, L. Lanzan, A. Scordino, A. Campisi, L. Cosentino, P. Finocchiaro, G. Fallica, S. Lombardo, M. Mazzillo, D. Sanfilippo, E. Sciacca, and G. Valvo, “A new generation of SPAD: single photon avalanche diodes,” Proc. SPIE 6619, 66193N (2007).
[CrossRef]

D. R. Gerwe, P. S. Idell, and J. Vaughn, “Cramer-Rao bound analysis of target characterization accuracy limits for imaging systems,” Proc. SPIE 4490, 245–255 (2001).
[CrossRef]

O. Steinvall and T. Chevalier, “Range accuracy and resolution for laser radars,” Proc. SPIE 5988, 598808 (2005).
[CrossRef]

Sens. Actuators A, Phys.

F. Zappa, S. Tisa, A. Tosi, and S. Cova, “Principles and features of single-photon avalanche diode arrays,” Sens. Actuators A, Phys. 140, 103–112 (2007).
[CrossRef]

Other

G. R. Osche, Optical Detection Theory for Laser Applications (Wiley-Interscience, 2002).

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Figures (8)

Fig. 1
Fig. 1

Typical pulse ranging system structure.

Fig. 2
Fig. 2

Laser pulse shape.

Fig. 3
Fig. 3

Relationship among the gate signal starting time t s , the gate width g w , and the target position t d .

Fig. 4
Fig. 4

Accuracy and precision versus intensity and pulse width with g w = 100 ns , t d = 50 ns , n b = 50 kHz , and η = 0.5 .

Fig. 5
Fig. 5

Accuracy and precision versus intensity and the product of quantum efficiency and detection probability of the detector with g w = 100 ns , t d = 50 ns , n b = 50 kHz , and p w = 10 ns .

Fig. 6
Fig. 6

Accuracy and precision versus intensity and target position with g w = 100 ns , p w = 10 ns , n b = 50 kHz , and η = 0.5 .

Fig. 7
Fig. 7

Accuracy and precision versus intensity and background noise: (a) g w = 100 ns , p w = 10 ns , t d = 20 ns , and η = 0.5 (b) g w = 100 ns , p w = 10 ns , t d = 80 ns , and η = 0.5 .

Fig. 8
Fig. 8

Simulation results of accuracy and precision versus intensity and pulse with g w = 100 ns , t d = 50 ns , n b = 50 kHz , and η = 0.5 .

Equations (18)

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

s ( t ) = t τ exp [ ( t / τ ) n ] .
p w = 3.5 τ .
0 s ( t ) d t = 0 t τ exp ( ( t / τ ) ) d t = τ .
f ( t ) = 1 τ s ( t ) = 1 τ t τ exp ( ( t / τ ) ) ,
0 f ( t ) d t = 1.
N b = η K b + N d = η ( K b + N d / η ) = η n b ,
I R ( t ) = N 0 · f ( t ) = { N 0 τ t t d τ e t t d τ ( t > t d ) 0 ( t t d ) .
N ( t ) ¯ = 0 t [ I R ( τ ) ] d τ + n b t = 0 t t d τ N 0 x e x d x + n b t = N 0 ( 1 e t t d τ t t d τ e t t d τ ) + n b t .
N e ¯ ( t ) = η × N ( t ) ¯ ,
P ( k ) = e N e ( t ) ¯ N e ( t ) ¯ k k ! ,
P ( k > 0 ) = 1 P ( k = 0 ) = 1 e N e ( t ) ¯ .
P ( g w ) = 1 e N e ( g w ) ¯ 1 e η N 0 η n b g w .
P ( t ) = P ( k > 0 ) 1 e N e ( g w ) ¯ 1 e N e ( t ) ¯ 1 e η N 0 η n b g w .
f p ( t ) = P ( t ) t = { η n b e η n b t 1 e N 0 n b g w ( 0 < t t d ) [ η N 0 1 τ ( t t d ) τ e ( t t d ) τ + η n b ] · e N e ¯ ( t ) 1 e η N 0 η n b g w ( t d < t < g w ) .
t ¯ = 0 t · f p ( t ) d t = 0 t d t · η n b e η n b t 1 e η N 0 η n b g w d t + t d g w t · [ η N 0 1 τ ( t t d ) τ e ( t t d ) τ + η n b ] · e N e ¯ ( t ) 1 e η N 0 η n b g w d t
σ 2 = 0 t 2 · f p ( t ) d t t ¯ 2 = 0 t d t 2 · η n b e η n b t 1 e η N 0 η n b g w d t + t d g w t 2 · [ η N 0 1 τ ( t t d ) τ e ( t t d ) τ + η n b ] · e 1 τ ( τ e t t d τ τ + e t t d τ t d e t t d τ t ) n b t 1 e η N 0 η n b g w d t t ¯ 2
R a = t ¯ t d .
R p = σ .

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