Abstract

Novel theory is developed for the avalanche multiplication process in avalanche photodiodes (APDs) under time-varying reverse-biasing conditions. Integral equations are derived characterizing the statistics of the multiplication factor and the impulse-response function of APDs, as well as their breakdown probability, all under the assumption that the electric field driving the avalanche process is time varying and spatially nonuniform. Numerical calculations generated by the model predict that by using a bit-synchronous sinusoidal biasing scheme to operate the APD in an optical receiver, the pulse-integrated gain-bandwidth product can be improved by a factor of 5 compared to the same APD operating under the conventional static biasing. The bit-synchronized periodic modulation of the electric field in the multiplication region serves to (1) produce large avalanche multiplication factors with suppressed avalanche durations for photons arriving in the early phase of each optical pulse; and (2) generate low avalanche gains and very short avalanche durations for photons arriving in the latter part of each optical pulse. These two factors can work together to reduce intersymbol interference in optical receivers without sacrificing sensitivity.

© 2012 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010 (1)

J. Zhang, P. Eraerds, N. Walenta, C. Barreiro, R. Thew, and H. Zbinden, “2.23 GHz gating InGaAs/InP single-photon avalanche diode for quantum key distribution,” Proc. SPIE 7681, 76810Z1–76810Z8 (2010).

2009 (4)

2008 (1)

L. J. J. Tan, J. S. Ng, C. H. Tan, and J. P. R. David, “Avalanche noise characteristics in submicron InP diodes,” IEEE J. Quantum Electron. 44(4), 378–382 (2008).
[CrossRef]

2006 (1)

2005 (1)

K. Makita, T. Nakata, K. Shiba, and T. Takeuchi, “40Gbps waveguide photodiodes,” NEC J. Adv. Tech. 2, 234–240 (2005).

2004 (1)

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

2003 (2)

C. H. Tan, P. J. Hambleton, J. P. R. David, R. C. Tozer, and G. J. Rees, “Calculation of APD impulse response using a space- and time-dependent ionization probability distribution function,” J. Lightwave Technol. 21(1), 155–159 (2003).
[CrossRef]

J. S. Ng, C. H. Tan, J. P. David, G. Hill, and G. J. Rees, “Field dependence of impact ionization coefficients in In0.53Ga0.47As,” IEEE Trans. Electron. Dev. 50(4), 901–905 (2003).
[CrossRef]

2002 (2)

M. M. Hayat, O.-H. Kwon, Y. Pan, P. Sotirelis, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Gain-bandwidth characteristics of thin avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(5), 770–781 (2002).
[CrossRef]

M. M. Hayat, O.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Boundary effects on multiplication noise in thin heterostructure avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(12), 2114–2123 (2002).
[CrossRef]

2001 (1)

D. C. Herbert and E. T. R. Chidley, “Very low noise avalanche detection,” IEEE Trans. Electron. Dev. 48(7), 1475–1477 (2001).
[CrossRef]

2000 (2)

T. Nakata, I. Watanabe, K. Makita, and T. Torikai, “InAlAs avalanche photodiodes with very thin multiplication layer of 0.1 μm for high-speed and low-voltage-operation optical receiver,” Electron. Lett. 36(21), 1807–1808 (2000).
[CrossRef]

M. A. Saleh, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Dead-space-based theory correctly predicts excess noise factor for thin GaAs and AlGaAs avalanche photodiodes,” IEEE Trans. Electron. Dev. 47(3), 625–633 (2000).
[CrossRef]

1992 (3)

M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Effect of dead space on gain and noise of double-carrier-multiplication avalanche photodiodes,” IEEE Trans. Electron. Dev. 39(3), 546–552 (1992).
[CrossRef]

M. M. Hayat and B. E. A. Saleh, “Statistical properties of the impulse response function of double carrier multiplication avalanche photodiodes including the effect of dead space,” J. Lightwave Technol. 10(10), 1415–1425 (1992).
[CrossRef]

M. M. Hayat, W. L. Sargeant, and B. E. A. Saleh, “Effect of dead space on gain and noise in Si and GaAs avalanche photodiodes,” IEEE J. Quantum Electron. 28(5), 1360–1365 (1992).
[CrossRef]

1987 (1)

B. L. Kasper and J. C. Campbell, “Multigigabit-per-second avalanche photodiode lightwave receivers,” J. Lightwave Technol. 5(10), 1351–1364 (1987).
[CrossRef]

1984 (1)

F. Osaka, T. Mikawa, and T. Kaneda, “Electron and hole ionization coefficients in (100) oriented Ga0.33In0.67As0.70P0.30,” Appl. Phys. Lett. 45(3), 292–293 (1984).
[CrossRef]

1966 (1)

R. J. McIntyre, “Multiplication noise in uniform avalanche diodes,” IEEE Trans. Electron. Dev. 13(1), 164–168 (1966).
[CrossRef]

Adachi, S.

Barreiro, C.

J. Zhang, P. Eraerds, N. Walenta, C. Barreiro, R. Thew, and H. Zbinden, “2.23 GHz gating InGaAs/InP single-photon avalanche diode for quantum key distribution,” Proc. SPIE 7681, 76810Z1–76810Z8 (2010).

Beck, A.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

Beck, J. D.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

Beling, A.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

Bowers, J. E.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

W. S. Zaoui, H.-W. Chen, J. E. Bowers, Y. Kang, M. Morse, M. J. Paniccia, A. Pauchard, and J. C. Campbell, “Frequency response and bandwidth enhancement in Ge/Si avalanche photodiodes with over 840 GHz gain-bandwidth-product,” Opt. Express 17(15), 12641–12649 (2009).
[CrossRef] [PubMed]

Campbell, J. C.

W. S. Zaoui, H.-W. Chen, J. E. Bowers, Y. Kang, M. Morse, M. J. Paniccia, A. Pauchard, and J. C. Campbell, “Frequency response and bandwidth enhancement in Ge/Si avalanche photodiodes with over 840 GHz gain-bandwidth-product,” Opt. Express 17(15), 12641–12649 (2009).
[CrossRef] [PubMed]

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

M. M. Hayat, O.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Boundary effects on multiplication noise in thin heterostructure avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(12), 2114–2123 (2002).
[CrossRef]

M. M. Hayat, O.-H. Kwon, Y. Pan, P. Sotirelis, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Gain-bandwidth characteristics of thin avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(5), 770–781 (2002).
[CrossRef]

B. L. Kasper and J. C. Campbell, “Multigigabit-per-second avalanche photodiode lightwave receivers,” J. Lightwave Technol. 5(10), 1351–1364 (1987).
[CrossRef]

Chen, H. W.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

Chen, H.-W.

Chidley, E. T. R.

D. C. Herbert and E. T. R. Chidley, “Very low noise avalanche detection,” IEEE Trans. Electron. Dev. 48(7), 1475–1477 (2001).
[CrossRef]

Coldren, L. A.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

David, J. P.

J. S. Ng, C. H. Tan, J. P. David, G. Hill, and G. J. Rees, “Field dependence of impact ionization coefficients in In0.53Ga0.47As,” IEEE Trans. Electron. Dev. 50(4), 901–905 (2003).
[CrossRef]

David, J. P. R.

Decobert, J.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

Demiguel, S.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

Eraerds, P.

J. Zhang, P. Eraerds, N. Walenta, C. Barreiro, R. Thew, and H. Zbinden, “2.23 GHz gating InGaAs/InP single-photon avalanche diode for quantum key distribution,” Proc. SPIE 7681, 76810Z1–76810Z8 (2010).

Guo, X.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

Hambleton, P. J.

Hayat, M. M.

D. S. G. Ong, J. S. Ng, M. M. Hayat, P. Sun, and J. P. R. David, “Optimization of InP APDs for high-speed lightwave systems,” J. Lightwave Technol. 27(15), 3294–3302 (2009).
[CrossRef]

P. Sun, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Statistical correlation of gain and buildup time in APDs and its effects on receiver performance,” J. Lightwave Technol. 24(2), 755–768 (2006).
[CrossRef]

M. M. Hayat, O.-H. Kwon, Y. Pan, P. Sotirelis, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Gain-bandwidth characteristics of thin avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(5), 770–781 (2002).
[CrossRef]

M. M. Hayat, O.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Boundary effects on multiplication noise in thin heterostructure avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(12), 2114–2123 (2002).
[CrossRef]

M. A. Saleh, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Dead-space-based theory correctly predicts excess noise factor for thin GaAs and AlGaAs avalanche photodiodes,” IEEE Trans. Electron. Dev. 47(3), 625–633 (2000).
[CrossRef]

M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Effect of dead space on gain and noise of double-carrier-multiplication avalanche photodiodes,” IEEE Trans. Electron. Dev. 39(3), 546–552 (1992).
[CrossRef]

M. M. Hayat and B. E. A. Saleh, “Statistical properties of the impulse response function of double carrier multiplication avalanche photodiodes including the effect of dead space,” J. Lightwave Technol. 10(10), 1415–1425 (1992).
[CrossRef]

M. M. Hayat, W. L. Sargeant, and B. E. A. Saleh, “Effect of dead space on gain and noise in Si and GaAs avalanche photodiodes,” IEEE J. Quantum Electron. 28(5), 1360–1365 (1992).
[CrossRef]

Herbert, D. C.

D. C. Herbert and E. T. R. Chidley, “Very low noise avalanche detection,” IEEE Trans. Electron. Dev. 48(7), 1475–1477 (2001).
[CrossRef]

Hill, G.

J. S. Ng, C. H. Tan, J. P. David, G. Hill, and G. J. Rees, “Field dependence of impact ionization coefficients in In0.53Ga0.47As,” IEEE Trans. Electron. Dev. 50(4), 901–905 (2003).
[CrossRef]

Huntington, A.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

Inoue, S.

Kaneda, T.

F. Osaka, T. Mikawa, and T. Kaneda, “Electron and hole ionization coefficients in (100) oriented Ga0.33In0.67As0.70P0.30,” Appl. Phys. Lett. 45(3), 292–293 (1984).
[CrossRef]

Kang, Y.

Kang, Y. M.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

Kasper, B. L.

B. L. Kasper and J. C. Campbell, “Multigigabit-per-second avalanche photodiode lightwave receivers,” J. Lightwave Technol. 5(10), 1351–1364 (1987).
[CrossRef]

Kinch, M. A.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

Kuo, Y. H.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

Kwon, O.-H.

M. M. Hayat, O.-H. Kwon, Y. Pan, P. Sotirelis, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Gain-bandwidth characteristics of thin avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(5), 770–781 (2002).
[CrossRef]

M. M. Hayat, O.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Boundary effects on multiplication noise in thin heterostructure avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(12), 2114–2123 (2002).
[CrossRef]

Li, X.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

Litski, S.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

Liu, H. D.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

Ma, F.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

Makita, K.

K. Makita, T. Nakata, K. Shiba, and T. Takeuchi, “40Gbps waveguide photodiodes,” NEC J. Adv. Tech. 2, 234–240 (2005).

T. Nakata, I. Watanabe, K. Makita, and T. Torikai, “InAlAs avalanche photodiodes with very thin multiplication layer of 0.1 μm for high-speed and low-voltage-operation optical receiver,” Electron. Lett. 36(21), 1807–1808 (2000).
[CrossRef]

McIntosh, D. C.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

McIntyre, R. J.

R. J. McIntyre, “Multiplication noise in uniform avalanche diodes,” IEEE Trans. Electron. Dev. 13(1), 164–168 (1966).
[CrossRef]

Mikawa, T.

F. Osaka, T. Mikawa, and T. Kaneda, “Electron and hole ionization coefficients in (100) oriented Ga0.33In0.67As0.70P0.30,” Appl. Phys. Lett. 45(3), 292–293 (1984).
[CrossRef]

Morse, M.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

W. S. Zaoui, H.-W. Chen, J. E. Bowers, Y. Kang, M. Morse, M. J. Paniccia, A. Pauchard, and J. C. Campbell, “Frequency response and bandwidth enhancement in Ge/Si avalanche photodiodes with over 840 GHz gain-bandwidth-product,” Opt. Express 17(15), 12641–12649 (2009).
[CrossRef] [PubMed]

Nakata, T.

K. Makita, T. Nakata, K. Shiba, and T. Takeuchi, “40Gbps waveguide photodiodes,” NEC J. Adv. Tech. 2, 234–240 (2005).

T. Nakata, I. Watanabe, K. Makita, and T. Torikai, “InAlAs avalanche photodiodes with very thin multiplication layer of 0.1 μm for high-speed and low-voltage-operation optical receiver,” Electron. Lett. 36(21), 1807–1808 (2000).
[CrossRef]

Namekata, N.

Ng, J. S.

D. S. G. Ong, J. S. Ng, M. M. Hayat, P. Sun, and J. P. R. David, “Optimization of InP APDs for high-speed lightwave systems,” J. Lightwave Technol. 27(15), 3294–3302 (2009).
[CrossRef]

L. J. J. Tan, J. S. Ng, C. H. Tan, and J. P. R. David, “Avalanche noise characteristics in submicron InP diodes,” IEEE J. Quantum Electron. 44(4), 378–382 (2008).
[CrossRef]

J. S. Ng, C. H. Tan, J. P. David, G. Hill, and G. J. Rees, “Field dependence of impact ionization coefficients in In0.53Ga0.47As,” IEEE Trans. Electron. Dev. 50(4), 901–905 (2003).
[CrossRef]

Ong, D. S. G.

Osaka, F.

F. Osaka, T. Mikawa, and T. Kaneda, “Electron and hole ionization coefficients in (100) oriented Ga0.33In0.67As0.70P0.30,” Appl. Phys. Lett. 45(3), 292–293 (1984).
[CrossRef]

Pan, Y.

M. M. Hayat, O.-H. Kwon, Y. Pan, P. Sotirelis, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Gain-bandwidth characteristics of thin avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(5), 770–781 (2002).
[CrossRef]

Paniccia, M. J.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

W. S. Zaoui, H.-W. Chen, J. E. Bowers, Y. Kang, M. Morse, M. J. Paniccia, A. Pauchard, and J. C. Campbell, “Frequency response and bandwidth enhancement in Ge/Si avalanche photodiodes with over 840 GHz gain-bandwidth-product,” Opt. Express 17(15), 12641–12649 (2009).
[CrossRef] [PubMed]

Pauchard, A.

W. S. Zaoui, H.-W. Chen, J. E. Bowers, Y. Kang, M. Morse, M. J. Paniccia, A. Pauchard, and J. C. Campbell, “Frequency response and bandwidth enhancement in Ge/Si avalanche photodiodes with over 840 GHz gain-bandwidth-product,” Opt. Express 17(15), 12641–12649 (2009).
[CrossRef] [PubMed]

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

Rees, G. J.

J. S. Ng, C. H. Tan, J. P. David, G. Hill, and G. J. Rees, “Field dependence of impact ionization coefficients in In0.53Ga0.47As,” IEEE Trans. Electron. Dev. 50(4), 901–905 (2003).
[CrossRef]

C. H. Tan, P. J. Hambleton, J. P. R. David, R. C. Tozer, and G. J. Rees, “Calculation of APD impulse response using a space- and time-dependent ionization probability distribution function,” J. Lightwave Technol. 21(1), 155–159 (2003).
[CrossRef]

Saleh, B. E. A.

P. Sun, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Statistical correlation of gain and buildup time in APDs and its effects on receiver performance,” J. Lightwave Technol. 24(2), 755–768 (2006).
[CrossRef]

M. M. Hayat, O.-H. Kwon, Y. Pan, P. Sotirelis, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Gain-bandwidth characteristics of thin avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(5), 770–781 (2002).
[CrossRef]

M. M. Hayat, O.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Boundary effects on multiplication noise in thin heterostructure avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(12), 2114–2123 (2002).
[CrossRef]

M. A. Saleh, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Dead-space-based theory correctly predicts excess noise factor for thin GaAs and AlGaAs avalanche photodiodes,” IEEE Trans. Electron. Dev. 47(3), 625–633 (2000).
[CrossRef]

M. M. Hayat and B. E. A. Saleh, “Statistical properties of the impulse response function of double carrier multiplication avalanche photodiodes including the effect of dead space,” J. Lightwave Technol. 10(10), 1415–1425 (1992).
[CrossRef]

M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Effect of dead space on gain and noise of double-carrier-multiplication avalanche photodiodes,” IEEE Trans. Electron. Dev. 39(3), 546–552 (1992).
[CrossRef]

M. M. Hayat, W. L. Sargeant, and B. E. A. Saleh, “Effect of dead space on gain and noise in Si and GaAs avalanche photodiodes,” IEEE J. Quantum Electron. 28(5), 1360–1365 (1992).
[CrossRef]

Saleh, M. A.

M. A. Saleh, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Dead-space-based theory correctly predicts excess noise factor for thin GaAs and AlGaAs avalanche photodiodes,” IEEE Trans. Electron. Dev. 47(3), 625–633 (2000).
[CrossRef]

Sargeant, W. L.

M. M. Hayat, W. L. Sargeant, and B. E. A. Saleh, “Effect of dead space on gain and noise in Si and GaAs avalanche photodiodes,” IEEE J. Quantum Electron. 28(5), 1360–1365 (1992).
[CrossRef]

Sarid, G.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

Shiba, K.

K. Makita, T. Nakata, K. Shiba, and T. Takeuchi, “40Gbps waveguide photodiodes,” NEC J. Adv. Tech. 2, 234–240 (2005).

Sotirelis, P.

M. M. Hayat, O.-H. Kwon, Y. Pan, P. Sotirelis, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Gain-bandwidth characteristics of thin avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(5), 770–781 (2002).
[CrossRef]

Sun, P.

Takeuchi, T.

K. Makita, T. Nakata, K. Shiba, and T. Takeuchi, “40Gbps waveguide photodiodes,” NEC J. Adv. Tech. 2, 234–240 (2005).

Tan, C. H.

L. J. J. Tan, J. S. Ng, C. H. Tan, and J. P. R. David, “Avalanche noise characteristics in submicron InP diodes,” IEEE J. Quantum Electron. 44(4), 378–382 (2008).
[CrossRef]

J. S. Ng, C. H. Tan, J. P. David, G. Hill, and G. J. Rees, “Field dependence of impact ionization coefficients in In0.53Ga0.47As,” IEEE Trans. Electron. Dev. 50(4), 901–905 (2003).
[CrossRef]

C. H. Tan, P. J. Hambleton, J. P. R. David, R. C. Tozer, and G. J. Rees, “Calculation of APD impulse response using a space- and time-dependent ionization probability distribution function,” J. Lightwave Technol. 21(1), 155–159 (2003).
[CrossRef]

Tan, L. J. J.

L. J. J. Tan, J. S. Ng, C. H. Tan, and J. P. R. David, “Avalanche noise characteristics in submicron InP diodes,” IEEE J. Quantum Electron. 44(4), 378–382 (2008).
[CrossRef]

Teich, M. C.

P. Sun, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Statistical correlation of gain and buildup time in APDs and its effects on receiver performance,” J. Lightwave Technol. 24(2), 755–768 (2006).
[CrossRef]

M. M. Hayat, O.-H. Kwon, Y. Pan, P. Sotirelis, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Gain-bandwidth characteristics of thin avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(5), 770–781 (2002).
[CrossRef]

M. M. Hayat, O.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Boundary effects on multiplication noise in thin heterostructure avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(12), 2114–2123 (2002).
[CrossRef]

M. A. Saleh, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Dead-space-based theory correctly predicts excess noise factor for thin GaAs and AlGaAs avalanche photodiodes,” IEEE Trans. Electron. Dev. 47(3), 625–633 (2000).
[CrossRef]

M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Effect of dead space on gain and noise of double-carrier-multiplication avalanche photodiodes,” IEEE Trans. Electron. Dev. 39(3), 546–552 (1992).
[CrossRef]

Thew, R.

J. Zhang, P. Eraerds, N. Walenta, C. Barreiro, R. Thew, and H. Zbinden, “2.23 GHz gating InGaAs/InP single-photon avalanche diode for quantum key distribution,” Proc. SPIE 7681, 76810Z1–76810Z8 (2010).

Torikai, T.

T. Nakata, I. Watanabe, K. Makita, and T. Torikai, “InAlAs avalanche photodiodes with very thin multiplication layer of 0.1 μm for high-speed and low-voltage-operation optical receiver,” Electron. Lett. 36(21), 1807–1808 (2000).
[CrossRef]

Tozer, R. C.

Tscherptner, N.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

Walenta, N.

J. Zhang, P. Eraerds, N. Walenta, C. Barreiro, R. Thew, and H. Zbinden, “2.23 GHz gating InGaAs/InP single-photon avalanche diode for quantum key distribution,” Proc. SPIE 7681, 76810Z1–76810Z8 (2010).

Wang, S.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

M. M. Hayat, O.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Boundary effects on multiplication noise in thin heterostructure avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(12), 2114–2123 (2002).
[CrossRef]

Watanabe, I.

T. Nakata, I. Watanabe, K. Makita, and T. Torikai, “InAlAs avalanche photodiodes with very thin multiplication layer of 0.1 μm for high-speed and low-voltage-operation optical receiver,” Electron. Lett. 36(21), 1807–1808 (2000).
[CrossRef]

Zadka, M.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

Zaoui, W. S.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

W. S. Zaoui, H.-W. Chen, J. E. Bowers, Y. Kang, M. Morse, M. J. Paniccia, A. Pauchard, and J. C. Campbell, “Frequency response and bandwidth enhancement in Ge/Si avalanche photodiodes with over 840 GHz gain-bandwidth-product,” Opt. Express 17(15), 12641–12649 (2009).
[CrossRef] [PubMed]

Zbinden, H.

J. Zhang, P. Eraerds, N. Walenta, C. Barreiro, R. Thew, and H. Zbinden, “2.23 GHz gating InGaAs/InP single-photon avalanche diode for quantum key distribution,” Proc. SPIE 7681, 76810Z1–76810Z8 (2010).

Zhang, J.

J. Zhang, P. Eraerds, N. Walenta, C. Barreiro, R. Thew, and H. Zbinden, “2.23 GHz gating InGaAs/InP single-photon avalanche diode for quantum key distribution,” Proc. SPIE 7681, 76810Z1–76810Z8 (2010).

Zheng, X.

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

Zheng, X. G.

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

Appl. Phys. Lett. (1)

F. Osaka, T. Mikawa, and T. Kaneda, “Electron and hole ionization coefficients in (100) oriented Ga0.33In0.67As0.70P0.30,” Appl. Phys. Lett. 45(3), 292–293 (1984).
[CrossRef]

Electron. Lett. (1)

T. Nakata, I. Watanabe, K. Makita, and T. Torikai, “InAlAs avalanche photodiodes with very thin multiplication layer of 0.1 μm for high-speed and low-voltage-operation optical receiver,” Electron. Lett. 36(21), 1807–1808 (2000).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. M. Hayat, W. L. Sargeant, and B. E. A. Saleh, “Effect of dead space on gain and noise in Si and GaAs avalanche photodiodes,” IEEE J. Quantum Electron. 28(5), 1360–1365 (1992).
[CrossRef]

L. J. J. Tan, J. S. Ng, C. H. Tan, and J. P. R. David, “Avalanche noise characteristics in submicron InP diodes,” IEEE J. Quantum Electron. 44(4), 378–382 (2008).
[CrossRef]

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

J. C. Campbell, S. Demiguel, F. Ma, A. Beck, X. Guo, S. Wang, X. Zheng, X. Li, J. D. Beck, M. A. Kinch, A. Huntington, L. A. Coldren, J. Decobert, and N. Tscherptner, “Recent advances in avalanche photodiodes,” IEEE J. Sel. Top. Quantum Electron. 10(4), 777–787 (2004).
[CrossRef]

IEEE Trans. Electron. Dev. (7)

D. C. Herbert and E. T. R. Chidley, “Very low noise avalanche detection,” IEEE Trans. Electron. Dev. 48(7), 1475–1477 (2001).
[CrossRef]

R. J. McIntyre, “Multiplication noise in uniform avalanche diodes,” IEEE Trans. Electron. Dev. 13(1), 164–168 (1966).
[CrossRef]

M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Effect of dead space on gain and noise of double-carrier-multiplication avalanche photodiodes,” IEEE Trans. Electron. Dev. 39(3), 546–552 (1992).
[CrossRef]

M. M. Hayat, O.-H. Kwon, Y. Pan, P. Sotirelis, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Gain-bandwidth characteristics of thin avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(5), 770–781 (2002).
[CrossRef]

M. M. Hayat, O.-H. Kwon, S. Wang, J. C. Campbell, B. E. A. Saleh, and M. C. Teich, “Boundary effects on multiplication noise in thin heterostructure avalanche photodiodes,” IEEE Trans. Electron. Dev. 49(12), 2114–2123 (2002).
[CrossRef]

M. A. Saleh, M. M. Hayat, B. E. A. Saleh, and M. C. Teich, “Dead-space-based theory correctly predicts excess noise factor for thin GaAs and AlGaAs avalanche photodiodes,” IEEE Trans. Electron. Dev. 47(3), 625–633 (2000).
[CrossRef]

J. S. Ng, C. H. Tan, J. P. David, G. Hill, and G. J. Rees, “Field dependence of impact ionization coefficients in In0.53Ga0.47As,” IEEE Trans. Electron. Dev. 50(4), 901–905 (2003).
[CrossRef]

J. Lightwave Technol. (5)

Nat. Photonics (1)

Y. M. Kang, H. D. Liu, M. Morse, M. J. Paniccia, M. Zadka, S. Litski, G. Sarid, A. Pauchard, Y. H. Kuo, H. W. Chen, W. S. Zaoui, J. E. Bowers, A. Beling, D. C. McIntosh, X. G. Zheng, and J. C. Campbell, “Monolithic germanium/silicon avalanche photodiodes with 340 GHz gain-bandwidth product,” Nat. Photonics 3(1), 59–63 (2009).
[CrossRef]

NEC J. Adv. Tech. (1)

K. Makita, T. Nakata, K. Shiba, and T. Takeuchi, “40Gbps waveguide photodiodes,” NEC J. Adv. Tech. 2, 234–240 (2005).

Opt. Express (2)

Proc. SPIE (1)

J. Zhang, P. Eraerds, N. Walenta, C. Barreiro, R. Thew, and H. Zbinden, “2.23 GHz gating InGaAs/InP single-photon avalanche diode for quantum key distribution,” Proc. SPIE 7681, 76810Z1–76810Z8 (2010).

Other (6)

Y. Kang, Z. Huang, Y. Saado, J. Campbell, A. Pauchard, J. Bowers, and M. J. Paniccia, “High performance Ge/Si avalanche photodiodes development in Intel,” Opt. Fiber Comm. Conf. & Expo. (OFC/NFOEC), 1–3 (2011).

G. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002).

N. Yasuoka, H. Kuwatsuka, M. Makiuchi, T. Uchida, and A. Yasaki, “Large multiplication-bandwidth products in APDs with a thin InP multiplication layer,” Proc. IEEE Laser & Electro Opt. Soc. Ann. Meeting LEOS' 2003, 999–1000 (2003).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley-Interscience, 2007).

P. Bhattacharya, Semiconductor Optoelectronic Devices (Prentice Hall, 1996).

R. G. Smith and S. D. Personick, “Receiver Design for Optical Fiber Communication Systems,” Semiconductor Devices for Optical Communication, H. Kressel ed. (Springer-Verlag, 1980).

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Figures (6)

Fig. 1
Fig. 1

Schematic of the proposed dynamic biasing approach (blue curve) repeated periodically over optical-pulse intervals. The green straight line represents the traditional constant bias. The periodic change in the reverse bias from the first to the second half of the optical-pulse period causes (1) photons that arrive early in the pulse window to trigger high gains but pulse-limited durations, and (2) late photons to trigger avalanches with low gains and almost transit-time limited avalanche durations. Such modulation of impact ionization results in a much higher average GBP compared to the conventional biasing scheme.

Fig. 2
Fig. 2

Schematic of a separate-absorption-multiplication (SAM) APD.

Fig. 3
Fig. 3

Mean impulse-response functions triggered by a hole under a constant electric field of VB = 14.30 V as a function of the integer multiples of the transit time, which is simply v/w = 2.985 ps. Different curves correspond to different ages (in transit times) of the initiating hole: red: age s = 0; blue: age s = 0.75; black: age s = 1.25; magenta: age s = 1.9; and cyan: age s = 2.6. The mean gain is calculated as 28 for each case. As expected these curves are simply shifted versions of one another.

Fig. 4
Fig. 4

Calculated age-dependent impulse response function under a sinusoidal dynamic bias. Different curves correspond to different ages (in transit times) of the initiating hole: red: age s = 0; black: age s = 1.4; magenta: age s = 2.6; and green: age s = 4. The dynamic-biasing parameters used are: B = 13 V, C = 6 V and ϕ = 0.

Fig. 5
Fig. 5

Calculated age-dependent impulse response function under dynamic biasing using the following values for the dynamic-bias parameters: B = 13 V, C = 6 V and ϕ = π/3.

Fig. 6
Fig. 6

Calculated pulse response due to a 8.3-ps rectangular optical pulse with the shown sinusoidal-dynamic bias function. The pulse response corresponding to a conventional static bias is also shown for comparison. A five-fold enhancement in the pulse-integrated gain-bandwidth-product is predicted.

Tables (1)

Tables Icon

Table 1 Ionization Parameters for InP [28]

Equations (32)

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

α ( x , t ) = A e exp ( [ E c , e / E ( x ; t ) ] m e ) ,
β ( x , t ) = A h exp ( [ E c , h / E ( x ; t ) ] m h ) ,
h e ( ξ ; x , s ) = { α ( ξ , s + ( ξ x ) / v e ) exp { x + d e ( x , s ) ξ α ( σ , s + ( σ x ) / v e ) d σ } , ξ x + d e ( x , s ) 0 , otherwise
h h ( ξ ; x , s ) = { β ( ξ , s + ( x ξ ) / v h ) exp { ξ x d h ( x , s ) β ( σ , s + ( x σ ) / v h ) d σ } , ξ x d h ( x , s ) 0 , otherwise ,
q x x + d E ( y , s + ( y x ) / v e ) d y = E t h , e ( x + d ) ,
q x d x E ( y , s + ( x y ) / v h ) d y = E t h , h ( x d ) ,
M ( x , s ) = 0. 5 [ Z ( x , s ) + Y ( x , s ) ] .
G ( s ) = M ( w , s ) = 0.5 [ 1 + Y ( w , s ) ] .
E [ Z ( x , s ) ] = E [ E [ Z ( x , s ) | X e ] ]                               = E [ Z 1 ( X e , s + ( X e x ) / v e ) + Z 2 ( X e , s + ( X e x ) / v e ) + Y ( X e , s + ( X e x ) / v e ) ] ,
z ( x , s ) = E [ 2 z ( X e , s + ( X e x ) / v e ) + y ( X e , s + ( X e x ) / v e ) ] .
z ( x , s ) = w h e ( ξ ; x , s ) d ξ + x w [ 2 z ( ξ , s + ( ξ x ) / v e ) + y ( ξ , s + ( ξ x ) / v e ) ] h e ( ξ ; x , s ) d ξ 0 x w , s 0.
y ( x , s ) = x h h ( ξ ; x , s ) d ξ + 0 x [ 2 y ( ξ , s + ( x ξ ) / v h ) + z ( ξ , s + ( x ξ ) / v h ) ] h h ( ξ ; x , s ) d ξ 0 x w , s 0.
E [ I e ( t , x , s ) | X e ] = 2 i e ( t , X e , s + ( X e x ) / v e ) + i h ( t , X e , s + ( X e x ) / v e ) ,
E [ I e ( t , x , s ) | X e > w ] = ( q v e / w ) { u ( t ) u ( t ( w x ) / v e ) } ,
i e ( t , x , s ) = ( q v e / w ) [ u ( t ) u ( t ( w x ) / v e ) ] w h e ( ξ ; x , s ) d ξ + x min ( w , x + v e t ) [ 2 i e ( t ( ξ x ) / v e , ξ , s + ( ξ x ) / v e ) + i h ( t ( ξ x ) / v e , ξ , s + ( ξ x ) / v e ) ] h e ( ξ ; x , s ) d ξ
i h ( t , x , s ) = ( q v h / w ) [ u ( t ) u ( t x / v h ) ] x h h ( ξ ; x , s ) d ξ + max ( 0 , x v h t ) x [ 2 i h ( t ( x ξ ) / v h , ξ , s + ( x ξ ) / v h ) + i e ( t ( x ξ ) / v h , ξ , s + ( x ξ ) / v h ) ] h h ( ξ ; x , s ) d ξ
i p ( t ) = 0 T i ( t , s ) p p h ( s ) d s ,
g ¯ p = T 1 0 T g a ( s ) d s
G B P p = g p B p .
V B D ( t ) =   B + C sin ( 2 π f c t + ϕ ) ,
F a ( s ) = y 2 ( w , s ) + 2 y ( w , s ) + 1 [ y ( w , s ) + 1 ] 2 .
E [ Z ( x , s ) 2 ] = E [ E [ Z ( x , s ) 2 | X e ] ]                                       = E [ { Z 1 ( X e , s + ( X e x ) / v e ) + Z 2 ( X e , s + ( X e x ) / v e ) + Y ( X e , s + ( X e x ) / v e ) } 2 ]                                       = E [ Z 1 ( X e , s + ( X e x ) / v e ) 2 + Z 2 ( X e , s + ( X e x ) / v e ) 2 + Y ( X e , s + ( X e x ) / v e ) 2 ]                                         + 2E [ Z 1 ( X e , s + ( X e x ) / v e ) Z 2 ( X e , s + ( X e x ) / v e ) + 2 Z 1 ( X e , s + ( X e x ) / v e ) Y ( X e , s + ( X e x ) / v e ) ]                                         + 2E [ Z 2 ( X e , s + ( X e x ) / v e ) Y ( X e , s + ( X e x ) / v e ) ] ,
z 2 ( x , s ) = E [ 2 z 2 ( X e , s + ( X e x ) / v e ) + y 2 ( X e , s + ( X e x ) / v e ) + 2 z ( X e , s + ( X e x ) / v e ) 2 + 4 z ( X e , s + ( X e x ) / v e ) y ( X e , s + ( X e x ) / v e ) ] .
z 2 ( x , s ) = w h e ( ξ ; x , s ) d ξ + x w [ 2 z 2 ( ξ , s + ( ξ x ) / v e ) + y 2 ( ξ , s + ( ξ x ) / v e ) + 4 z ( ξ , s + ( ξ x ) / v e ) y ( ξ , s + ( ξ x ) / v e ) + 2 z ( ξ , s + ( ξ x ) / v e ) 2 ] h e ( ξ ; x , s ) d ξ , 0 x w , s 0.
y 2 ( x , s ) = x h h ( ξ ; x , s ) d ξ + 0 x [ 2 y 2 ( ξ , s + ( x ξ ) / v h ) + z 2 ( ξ , s + ( x ξ ) / v h ) + 4 z ( ξ , s + ( x ξ ) / v h ) y ( ξ , s + ( x ξ ) / v h ) + 4 y ( ξ , s + ( x ξ ) / v h ) 2 ] h h ( ξ ; x , s ) d ξ , 0 x w , s 0.
P Z ( x , s ) = w h e ( ξ ; x , s ) d ξ + x w P Z ( ξ , s + ( ξ x ) / v e ) 2 P Y ( ξ , s + ( ξ x ) / v e ) h e ( ξ ; x , s ) d ξ 0 x w , s 0.
P Y ( x , s ) = x h h ( ξ ; x , s ) d ξ + 0 x P Y ( ξ , s + ( x ξ ) / v h ) 2 P Z ( ξ , s + ( x ξ ) / v h ) h h ( ξ ; x , s ) d ξ 0 x w , s 0.
P B ( s ) = 1 P Z ( w , s ) .
f Z ( x , s , m ) = δ m 1 w h e ( ξ ; x , s ) d ξ + x w f Z ( ξ , s + ( ξ x ) / v e , m ) * f Z ( ξ , s + ( ξ x ) / v e , m ) * f Y ( ξ , s + ( ξ x ) / v e , m ) × h e ( ξ ; x , s ) d ξ , 0 x w , s 0 , m = 1 , 2 , 3 ,
f Y ( x , s , m ) = δ m 1 x h h ( ξ ; x , s ) d ξ + 0 x f Y ( ξ , s + ( x ξ ) / v h , m ) * f Y ( ξ , s + ( x ξ ) / v h , m ) * f Z ( ξ , s + ( x ξ ) / v h , m ) × h h ( ξ ; x , s ) d ξ , 0 x w , s 0 , m = 1 , 2 , 3 ,
F Z ( x , s , ω ) = m = 0 f Z ( x , s , m ) e j ω m , F Y ( x , s , ω ) = m = 0 f Y ( x , s , m ) e j ω m , π < ω π , and obtain F Z ( x , s , ω ) = e j ω w h e ( ξ ; x , s ) d ξ + x w F Z ( ξ , s + ( ξ x ) / v e , ω ) 2 F Y ( ξ , s + ( ξ x ) / v e , ω ) × h e ( ξ ; x , s ) d ξ , 0 x w , s 0 , π < ω π
F Y ( x , s , ω ) = e j ω x h h ( ξ ; x , s ) d ξ + o x F Z ( ξ , s + ( x ξ ) / v h , m ) 2 F Z ( ξ , s + ( x ξ ) / v h , m ) × h h ( ξ ; x , s ) d ξ , 0 x w , s 0 , π < ω π

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