Abstract

Compared with typical diode lasers (DLs), transistor lasers (TLs) support not only current-controlled but also voltage-controlled modulation. In this work, we theoretically investigate the small-signal voltage modulation of TLs based on the Franz-Keldysh (F-K) absorption and related optoelectronic feedback. In addition to the conventional rate equations relevant to DLs, our model physically includes various F-K effects. An optically induced current due to the F-K absorption may dramatically alter the voltage response of TLs. A model composed of the intrinsic optical response and an electrical transfer function which is fed back by this optical response is proposed to explain the true behaviors of voltage modulation in TLs.

© 2016 Optical Society of America

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    [Crossref]

2016 (1)

M. Feng, J. Qiu, C. Y. Wang, and N. Holonyak., “Intra-cavity photon-assisted tunneling collector-base voltage-mediated electron-hole spontaneous-stimulated recombination transistor laser,” J. Appl. Phys. 119(8), 084502 (2016).
[Crossref]

2015 (2)

Y.-W. Chern, C.-H. Chang, and C.-H. W. Wu, “The effect of voltage-dependent charge-removing mechanism on the optical modulation bandwidths of light-emitting transistors,” IEEE Trans. Electron Dev. 62(12), 4076–4081 (2015).
[Crossref]

H.-L. Wang, Y.-H. Huang, G.-S. Cheng, S. W. Chang, and C.-H. Wu, “Analysis of tunable internal loss caused by Franz–Keldysh absorption in transistor lasers,” IEEE J. Sel. Top. Quantum Electron. 21(6), 270–276 (2015).
[Crossref]

2014 (2)

2013 (2)

H.-L. Wang, Y.-J. Huang, and C.-H. Wu, “Optical frequency response analysis of light-emitting transistors under different microwave configurations,” Appl. Phys. Lett. 103(5), 051110 (2013).
[Crossref]

H.-L. Wang, P.-H. Chou, and C.-H. Wu, “Microwave determination of quantum-well capture and escape time in light-emitting transistors,” IEEE Trans. Electron Dev. 60(3), 1088–1091 (2013).
[Crossref]

2012 (1)

2011 (1)

F. Tan, R. Bambery, M. Feng, and N. Holonyak., “Transistor laser with simultaneous electrical and optical output at 20 and 40 Gb/s data rate modulation,” Appl. Phys. Lett. 99(6), 061105 (2011).
[Crossref]

2010 (2)

2009 (3)

G. Walter, C. H. Wu, H. W. Then, M. Feng, and N. Holonyak., “4.3 GHz optical bandwidth light emitting transistor,” Appl. Phys. Lett. 94(24), 241101 (2009).
[Crossref]

M. Feng, N. Holonyak, H. W. Then, C. H. Wu, and G. Walter, “Tunnel junction transistor laser,” Appl. Phys. Lett. 94(4), 041118 (2009).
[Crossref]

N. H. Zhu, H. G. Zhang, J. W. Man, H. L. Zhu, J. H. Ke, Y. Liu, X. Wang, H. Q. Yuan, L. Xie, and W. Wang, “Microwave generation in an electro-absorption modulator integrated with a DFB laser subject to optical injection,” Opt. Express 17(24), 22114–22123 (2009).
[Crossref] [PubMed]

2007 (2)

A. James, G. Walter, M. Feng, and N. Holonyak., “Photon-assisted breakdown, negative resistance, and switching in a quantum-well transistor laser,” Appl. Phys. Lett. 90(15), 152109 (2007).
[Crossref]

A. James, N. Holonyak, M. Feng, and G. Walter, “Franz–Keldysh photon-assisted voltage-operated switching of a transistor laser,” IEEE Photonics Technol. Lett. 19(9), 680–682 (2007).
[Crossref]

2004 (3)

G. Walter, N. Holonyak, M. Feng, and R. Chan, “Laser operation of a heterojunction bipolar light-emitting transistor,” Appl. Phys. Lett. 85(20), 4768 (2004).
[Crossref]

M. Feng, N. Holonyak, and W. Hafez, “Light-emitting transistor: Light emission from InGaP/GaAs heterojunction bipolar transistors,” Appl. Phys. Lett. 84(1), 151–153 (2004).
[Crossref]

M. Feng, N. Holonyak, and R. Chan, “Quantum-well-base heterojunction bipolar light-emitting transistor,” Appl. Phys. Lett. 84(11), 1952–1954 (2004).
[Crossref]

2000 (1)

K. Kudo, K. Yashiki, T. Sasaki, Y. Yokoyama, K. Hamamoto, T. Morimoto, and M. Yamaguchi, “1.55-μm wavelength-selectable microarray DFB-LD’s with monolithically integrated MMI combiner, SOA, and EA-Modulator,” IEEE Photonics Technol. Lett. 12(3), 242–244 (2000).
[Crossref]

1999 (1)

C. H. Chen, M. Hargis, J. M. Woodall, M. R. Melloch, J. S. Reynolds, E. Yablonovitch, and W. Wang, “GHz bandwidth GaAs light-emitting diodes,” Appl. Phys. Lett. 74(21), 3140–3142 (1999).
[Crossref]

1976 (1)

J. Heinen, W. Hurber, and W. Harth, “Light-emitting diodes with a modulation bandwidth of more than 1 GHz,” Electron. Lett. 12(21), 553–554 (1976).
[Crossref]

1963 (1)

K. Tharmalingam, “Optical absorption in the presence of a uniform field,” Phys. Rev. 130(6), 2204–2206 (1963).
[Crossref]

Arai, S.

Bambery, R.

F. Tan, R. Bambery, M. Feng, and N. Holonyak., “Transistor laser with simultaneous electrical and optical output at 20 and 40 Gb/s data rate modulation,” Appl. Phys. Lett. 99(6), 061105 (2011).
[Crossref]

Baureis, P.

P. Baureis, “Compact modeling of electrical, thermal and optical LED behavior,” in Proceedings of 35th European IEEE Solid-State Device Research Conference (2005).
[Crossref]

Chai, G.

Chan, R.

M. Feng, N. Holonyak, and R. Chan, “Quantum-well-base heterojunction bipolar light-emitting transistor,” Appl. Phys. Lett. 84(11), 1952–1954 (2004).
[Crossref]

G. Walter, N. Holonyak, M. Feng, and R. Chan, “Laser operation of a heterojunction bipolar light-emitting transistor,” Appl. Phys. Lett. 85(20), 4768 (2004).
[Crossref]

Chang, C.-H.

Y.-W. Chern, C.-H. Chang, and C.-H. W. Wu, “The effect of voltage-dependent charge-removing mechanism on the optical modulation bandwidths of light-emitting transistors,” IEEE Trans. Electron Dev. 62(12), 4076–4081 (2015).
[Crossref]

Chang, S. W.

H.-L. Wang, Y.-H. Huang, G.-S. Cheng, S. W. Chang, and C.-H. Wu, “Analysis of tunable internal loss caused by Franz–Keldysh absorption in transistor lasers,” IEEE J. Sel. Top. Quantum Electron. 21(6), 270–276 (2015).
[Crossref]

Chen, C. H.

C. H. Chen, M. Hargis, J. M. Woodall, M. R. Melloch, J. S. Reynolds, E. Yablonovitch, and W. Wang, “GHz bandwidth GaAs light-emitting diodes,” Appl. Phys. Lett. 74(21), 3140–3142 (1999).
[Crossref]

Cheng, G.-S.

H.-L. Wang, Y.-H. Huang, G.-S. Cheng, S. W. Chang, and C.-H. Wu, “Analysis of tunable internal loss caused by Franz–Keldysh absorption in transistor lasers,” IEEE J. Sel. Top. Quantum Electron. 21(6), 270–276 (2015).
[Crossref]

Chern, Y.-W.

Y.-W. Chern, C.-H. Chang, and C.-H. W. Wu, “The effect of voltage-dependent charge-removing mechanism on the optical modulation bandwidths of light-emitting transistors,” IEEE Trans. Electron Dev. 62(12), 4076–4081 (2015).
[Crossref]

Choi, W.-Y.

Chou, P.-H.

H.-L. Wang, P.-H. Chou, and C.-H. Wu, “Microwave determination of quantum-well capture and escape time in light-emitting transistors,” IEEE Trans. Electron Dev. 60(3), 1088–1091 (2013).
[Crossref]

Chrostowski, L.

Duan, Z.

Feng, M.

M. Feng, J. Qiu, C. Y. Wang, and N. Holonyak., “Intra-cavity photon-assisted tunneling collector-base voltage-mediated electron-hole spontaneous-stimulated recombination transistor laser,” J. Appl. Phys. 119(8), 084502 (2016).
[Crossref]

F. Tan, R. Bambery, M. Feng, and N. Holonyak., “Transistor laser with simultaneous electrical and optical output at 20 and 40 Gb/s data rate modulation,” Appl. Phys. Lett. 99(6), 061105 (2011).
[Crossref]

H. W. Then, M. Feng, and N. Holonyak., “Microwave circuit model of the three-port transistor laser,” J. Appl. Phys. 107(9), 094509 (2010).
[Crossref]

G. Walter, C. H. Wu, H. W. Then, M. Feng, and N. Holonyak., “4.3 GHz optical bandwidth light emitting transistor,” Appl. Phys. Lett. 94(24), 241101 (2009).
[Crossref]

M. Feng, N. Holonyak, H. W. Then, C. H. Wu, and G. Walter, “Tunnel junction transistor laser,” Appl. Phys. Lett. 94(4), 041118 (2009).
[Crossref]

A. James, N. Holonyak, M. Feng, and G. Walter, “Franz–Keldysh photon-assisted voltage-operated switching of a transistor laser,” IEEE Photonics Technol. Lett. 19(9), 680–682 (2007).
[Crossref]

A. James, G. Walter, M. Feng, and N. Holonyak., “Photon-assisted breakdown, negative resistance, and switching in a quantum-well transistor laser,” Appl. Phys. Lett. 90(15), 152109 (2007).
[Crossref]

M. Feng, N. Holonyak, and R. Chan, “Quantum-well-base heterojunction bipolar light-emitting transistor,” Appl. Phys. Lett. 84(11), 1952–1954 (2004).
[Crossref]

M. Feng, N. Holonyak, and W. Hafez, “Light-emitting transistor: Light emission from InGaP/GaAs heterojunction bipolar transistors,” Appl. Phys. Lett. 84(1), 151–153 (2004).
[Crossref]

G. Walter, N. Holonyak, M. Feng, and R. Chan, “Laser operation of a heterojunction bipolar light-emitting transistor,” Appl. Phys. Lett. 85(20), 4768 (2004).
[Crossref]

Hafez, W.

M. Feng, N. Holonyak, and W. Hafez, “Light-emitting transistor: Light emission from InGaP/GaAs heterojunction bipolar transistors,” Appl. Phys. Lett. 84(1), 151–153 (2004).
[Crossref]

Hamamoto, K.

K. Kudo, K. Yashiki, T. Sasaki, Y. Yokoyama, K. Hamamoto, T. Morimoto, and M. Yamaguchi, “1.55-μm wavelength-selectable microarray DFB-LD’s with monolithically integrated MMI combiner, SOA, and EA-Modulator,” IEEE Photonics Technol. Lett. 12(3), 242–244 (2000).
[Crossref]

Han, L.

Hargis, M.

C. H. Chen, M. Hargis, J. M. Woodall, M. R. Melloch, J. S. Reynolds, E. Yablonovitch, and W. Wang, “GHz bandwidth GaAs light-emitting diodes,” Appl. Phys. Lett. 74(21), 3140–3142 (1999).
[Crossref]

Harth, W.

J. Heinen, W. Hurber, and W. Harth, “Light-emitting diodes with a modulation bandwidth of more than 1 GHz,” Electron. Lett. 12(21), 553–554 (1976).
[Crossref]

Heinen, J.

J. Heinen, W. Hurber, and W. Harth, “Light-emitting diodes with a modulation bandwidth of more than 1 GHz,” Electron. Lett. 12(21), 553–554 (1976).
[Crossref]

Holonyak, N.

M. Feng, J. Qiu, C. Y. Wang, and N. Holonyak., “Intra-cavity photon-assisted tunneling collector-base voltage-mediated electron-hole spontaneous-stimulated recombination transistor laser,” J. Appl. Phys. 119(8), 084502 (2016).
[Crossref]

F. Tan, R. Bambery, M. Feng, and N. Holonyak., “Transistor laser with simultaneous electrical and optical output at 20 and 40 Gb/s data rate modulation,” Appl. Phys. Lett. 99(6), 061105 (2011).
[Crossref]

H. W. Then, M. Feng, and N. Holonyak., “Microwave circuit model of the three-port transistor laser,” J. Appl. Phys. 107(9), 094509 (2010).
[Crossref]

G. Walter, C. H. Wu, H. W. Then, M. Feng, and N. Holonyak., “4.3 GHz optical bandwidth light emitting transistor,” Appl. Phys. Lett. 94(24), 241101 (2009).
[Crossref]

M. Feng, N. Holonyak, H. W. Then, C. H. Wu, and G. Walter, “Tunnel junction transistor laser,” Appl. Phys. Lett. 94(4), 041118 (2009).
[Crossref]

A. James, N. Holonyak, M. Feng, and G. Walter, “Franz–Keldysh photon-assisted voltage-operated switching of a transistor laser,” IEEE Photonics Technol. Lett. 19(9), 680–682 (2007).
[Crossref]

A. James, G. Walter, M. Feng, and N. Holonyak., “Photon-assisted breakdown, negative resistance, and switching in a quantum-well transistor laser,” Appl. Phys. Lett. 90(15), 152109 (2007).
[Crossref]

M. Feng, N. Holonyak, and R. Chan, “Quantum-well-base heterojunction bipolar light-emitting transistor,” Appl. Phys. Lett. 84(11), 1952–1954 (2004).
[Crossref]

M. Feng, N. Holonyak, and W. Hafez, “Light-emitting transistor: Light emission from InGaP/GaAs heterojunction bipolar transistors,” Appl. Phys. Lett. 84(1), 151–153 (2004).
[Crossref]

G. Walter, N. Holonyak, M. Feng, and R. Chan, “Laser operation of a heterojunction bipolar light-emitting transistor,” Appl. Phys. Lett. 85(20), 4768 (2004).
[Crossref]

Huang, X.

Huang, Y.-H.

H.-L. Wang, Y.-H. Huang, G.-S. Cheng, S. W. Chang, and C.-H. Wu, “Analysis of tunable internal loss caused by Franz–Keldysh absorption in transistor lasers,” IEEE J. Sel. Top. Quantum Electron. 21(6), 270–276 (2015).
[Crossref]

Huang, Y.-J.

H.-L. Wang, Y.-J. Huang, and C.-H. Wu, “Optical frequency response analysis of light-emitting transistors under different microwave configurations,” Appl. Phys. Lett. 103(5), 051110 (2013).
[Crossref]

Huo, W.

Hurber, W.

J. Heinen, W. Hurber, and W. Harth, “Light-emitting diodes with a modulation bandwidth of more than 1 GHz,” Electron. Lett. 12(21), 553–554 (1976).
[Crossref]

James, A.

A. James, N. Holonyak, M. Feng, and G. Walter, “Franz–Keldysh photon-assisted voltage-operated switching of a transistor laser,” IEEE Photonics Technol. Lett. 19(9), 680–682 (2007).
[Crossref]

A. James, G. Walter, M. Feng, and N. Holonyak., “Photon-assisted breakdown, negative resistance, and switching in a quantum-well transistor laser,” Appl. Phys. Lett. 90(15), 152109 (2007).
[Crossref]

Ke, J. H.

Kudo, K.

K. Kudo, K. Yashiki, T. Sasaki, Y. Yokoyama, K. Hamamoto, T. Morimoto, and M. Yamaguchi, “1.55-μm wavelength-selectable microarray DFB-LD’s with monolithically integrated MMI combiner, SOA, and EA-Modulator,” IEEE Photonics Technol. Lett. 12(3), 242–244 (2000).
[Crossref]

Lee, M.-J.

Liang, S.

Liu, Y.

Man, J. W.

Melloch, M. R.

C. H. Chen, M. Hargis, J. M. Woodall, M. R. Melloch, J. S. Reynolds, E. Yablonovitch, and W. Wang, “GHz bandwidth GaAs light-emitting diodes,” Appl. Phys. Lett. 74(21), 3140–3142 (1999).
[Crossref]

Morimoto, T.

K. Kudo, K. Yashiki, T. Sasaki, Y. Yokoyama, K. Hamamoto, T. Morimoto, and M. Yamaguchi, “1.55-μm wavelength-selectable microarray DFB-LD’s with monolithically integrated MMI combiner, SOA, and EA-Modulator,” IEEE Photonics Technol. Lett. 12(3), 242–244 (2000).
[Crossref]

Nishiyama, N.

Park, K.-Y.

Qiu, J.

M. Feng, J. Qiu, C. Y. Wang, and N. Holonyak., “Intra-cavity photon-assisted tunneling collector-base voltage-mediated electron-hole spontaneous-stimulated recombination transistor laser,” J. Appl. Phys. 119(8), 084502 (2016).
[Crossref]

Reynolds, J. S.

C. H. Chen, M. Hargis, J. M. Woodall, M. R. Melloch, J. S. Reynolds, E. Yablonovitch, and W. Wang, “GHz bandwidth GaAs light-emitting diodes,” Appl. Phys. Lett. 74(21), 3140–3142 (1999).
[Crossref]

Rücker, H.

Sasaki, T.

K. Kudo, K. Yashiki, T. Sasaki, Y. Yokoyama, K. Hamamoto, T. Morimoto, and M. Yamaguchi, “1.55-μm wavelength-selectable microarray DFB-LD’s with monolithically integrated MMI combiner, SOA, and EA-Modulator,” IEEE Photonics Technol. Lett. 12(3), 242–244 (2000).
[Crossref]

Sato, N.

Sato, T.

Shi, W.

Shirao, M.

Tan, F.

F. Tan, R. Bambery, M. Feng, and N. Holonyak., “Transistor laser with simultaneous electrical and optical output at 20 and 40 Gb/s data rate modulation,” Appl. Phys. Lett. 99(6), 061105 (2011).
[Crossref]

Tan, S.

Tharmalingam, K.

K. Tharmalingam, “Optical absorption in the presence of a uniform field,” Phys. Rev. 130(6), 2204–2206 (1963).
[Crossref]

Then, H. W.

H. W. Then, M. Feng, and N. Holonyak., “Microwave circuit model of the three-port transistor laser,” J. Appl. Phys. 107(9), 094509 (2010).
[Crossref]

G. Walter, C. H. Wu, H. W. Then, M. Feng, and N. Holonyak., “4.3 GHz optical bandwidth light emitting transistor,” Appl. Phys. Lett. 94(24), 241101 (2009).
[Crossref]

M. Feng, N. Holonyak, H. W. Then, C. H. Wu, and G. Walter, “Tunnel junction transistor laser,” Appl. Phys. Lett. 94(4), 041118 (2009).
[Crossref]

Walter, G.

G. Walter, C. H. Wu, H. W. Then, M. Feng, and N. Holonyak., “4.3 GHz optical bandwidth light emitting transistor,” Appl. Phys. Lett. 94(24), 241101 (2009).
[Crossref]

M. Feng, N. Holonyak, H. W. Then, C. H. Wu, and G. Walter, “Tunnel junction transistor laser,” Appl. Phys. Lett. 94(4), 041118 (2009).
[Crossref]

A. James, G. Walter, M. Feng, and N. Holonyak., “Photon-assisted breakdown, negative resistance, and switching in a quantum-well transistor laser,” Appl. Phys. Lett. 90(15), 152109 (2007).
[Crossref]

A. James, N. Holonyak, M. Feng, and G. Walter, “Franz–Keldysh photon-assisted voltage-operated switching of a transistor laser,” IEEE Photonics Technol. Lett. 19(9), 680–682 (2007).
[Crossref]

G. Walter, N. Holonyak, M. Feng, and R. Chan, “Laser operation of a heterojunction bipolar light-emitting transistor,” Appl. Phys. Lett. 85(20), 4768 (2004).
[Crossref]

Wang, C. Y.

M. Feng, J. Qiu, C. Y. Wang, and N. Holonyak., “Intra-cavity photon-assisted tunneling collector-base voltage-mediated electron-hole spontaneous-stimulated recombination transistor laser,” J. Appl. Phys. 119(8), 084502 (2016).
[Crossref]

Wang, H.-L.

H.-L. Wang, Y.-H. Huang, G.-S. Cheng, S. W. Chang, and C.-H. Wu, “Analysis of tunable internal loss caused by Franz–Keldysh absorption in transistor lasers,” IEEE J. Sel. Top. Quantum Electron. 21(6), 270–276 (2015).
[Crossref]

H.-L. Wang, P.-H. Chou, and C.-H. Wu, “Microwave determination of quantum-well capture and escape time in light-emitting transistors,” IEEE Trans. Electron Dev. 60(3), 1088–1091 (2013).
[Crossref]

H.-L. Wang, Y.-J. Huang, and C.-H. Wu, “Optical frequency response analysis of light-emitting transistors under different microwave configurations,” Appl. Phys. Lett. 103(5), 051110 (2013).
[Crossref]

Wang, W.

Wang, X.

Woodall, J. M.

C. H. Chen, M. Hargis, J. M. Woodall, M. R. Melloch, J. S. Reynolds, E. Yablonovitch, and W. Wang, “GHz bandwidth GaAs light-emitting diodes,” Appl. Phys. Lett. 74(21), 3140–3142 (1999).
[Crossref]

Wu, C. H.

M. Feng, N. Holonyak, H. W. Then, C. H. Wu, and G. Walter, “Tunnel junction transistor laser,” Appl. Phys. Lett. 94(4), 041118 (2009).
[Crossref]

G. Walter, C. H. Wu, H. W. Then, M. Feng, and N. Holonyak., “4.3 GHz optical bandwidth light emitting transistor,” Appl. Phys. Lett. 94(24), 241101 (2009).
[Crossref]

Wu, C.-H.

H.-L. Wang, Y.-H. Huang, G.-S. Cheng, S. W. Chang, and C.-H. Wu, “Analysis of tunable internal loss caused by Franz–Keldysh absorption in transistor lasers,” IEEE J. Sel. Top. Quantum Electron. 21(6), 270–276 (2015).
[Crossref]

H.-L. Wang, Y.-J. Huang, and C.-H. Wu, “Optical frequency response analysis of light-emitting transistors under different microwave configurations,” Appl. Phys. Lett. 103(5), 051110 (2013).
[Crossref]

H.-L. Wang, P.-H. Chou, and C.-H. Wu, “Microwave determination of quantum-well capture and escape time in light-emitting transistors,” IEEE Trans. Electron Dev. 60(3), 1088–1091 (2013).
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Figures (6)

Fig. 1
Fig. 1

The layer structure and band diagram of the TL. A QW is incorporated in the p-type base for photon emissions. The additional holes generated by the F-K absorption flow back into the base (active) region and participate in the carrier-photon interaction.

Fig. 2
Fig. 2

(a) The LIV family curves of the TL. The more significant F-K absorption at the larger V CB makes the light output turns smaller. (b) The carrier density versus i B and V CB . .At a fixed i B , if V CB is larger than a critical value, the device would not lase anymore and turn into a spontaneous-emission source.

Fig. 3
Fig. 3

(a) The magnitude of the intrinsic optical response in the logarithmic scale under different i B (0) from 60 to 90 mA at V CB (0) = 1.6 V. The modulation bandwidth increases with i B (0) since the photon density is raised in the cavity. (b) The counterparts at different V CB (0) from 1.2 to 2.4 V ( i B (0) fixed at 70 mA). The modulation bandwidth decreases with V CB (0) due to the less photon density in the cavity.

Fig. 4
Fig. 4

The small-signal circuit of the TL. An additional current source which is resulted from the F-K absorption in the BC junction is incorporated in this electrical model. The admittance corresponding to this additional current can be deduced from rate equations.

Fig. 5
Fig. 5

The comparison among various responses related to the voltage-controlled modulations of the TL at a V CB (0) of (a) 1.8 V, (b) 2 V, and (c) 2.2 V. The magnitude of the electrical transfer function is minimal as the magnitude of the intrinsic optical response is at its peak.

Fig. 6
Fig. 6

(a) The electrical transfer function T e (ω) and (b) overall response T overall (ω) at different collector resistances R C . In the TL, the response T overall (ω) is sensitive to the collector resistance. As the collector resistance increases, its normalized magnitude decreases but becomes relatively flat.

Tables (1)

Tables Icon

Table 1 List of parameters for TLs.

Equations (26)

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dN dt = η i i B e V a R v g g N p + η BC Γ J BC Γ a v g α N p ,
d N p dt =( Γ a v g g 1 τ p ) N p + Γ a β sp R sp Γ J BC v g α N p ,
α θ F { η Ai 2 (η)+ [ Ai ' (η)] 2 },
η= E g ω p θ F , θ F = ( eF 2μ ) 2/3 ,
dΔN dt =Δ R sp v g Δg N p (0) + v g g (0) Δ N p + η BC Γ J BC Γ a ( v g Δα N p (0) + v g α (0) Δ N p ),
dΔ N p dt = Γ a v g Δg N p (0) + Γ a v g g (0) Δ N p Δ N p τ p Γ J BC v g Δα N p (0) Γ J BC v g α (0) Δ N p + Γ a Δ( β sp R sp ),
Δg=aΔN a p Δ N p ,
a= g N | N= N (0) , N p = N p (0) = a 0 1+ε N p (0) ,
a p = g N p | N= N (0) , N p = N p (0) = ε g (0) 1+ε N p (0) .
Δα=KΔF, K= α (0) 3 F (0) { η (0) Ai 2 ( η (0) )+ [ Ai ' ( η (0) )] 2 η (0) Ai 2 ( η (0) )+ [ Ai ' ( η (0) )] 2 }.
d dt ( ΔN Δ N p )=( γ NN γ NP γ PN γ PP )( ΔN Δ N p )+( γ Nα Δα γ Pα Δα ),
γ NN 1 τ sp + v g a N p (0) , γ NP v g g (0) v g a p N p (0) η BC Γ J BC Γ a v g α (0) , γ Nα η BC Γ J BC Γ a v g N p (0) ,
γ PN v g a Γ a N p (0) , γ PP v g a p Γ a N p (0) Γ a v g g (0) + 1 τ p + Γ J BC v g α (0) , γ Pα Γ J BC v g N p (0) .
( γ NN +jω γ NP γ PN γ PP +jω )( N 1 N p,1 )=( γ Nα α 1 γ Pα α 1 ).
N 1 = H(ω)[ ( γ PP +jω) γ Nα + γ NP γ Pα ] ω R 2 α 1 ,
N p,1 = H(ω)[ ( γ NN +jω) γ Pα + γ Nα γ PN ] ω R 2 α 1 ,
H(ω) ω R 2 Δ = ω R 2 ω R 2 ω 2 +jωγ , Δ= γ NN γ PP + γ NP γ PN +jω( γ NN + γ PP ) ω 2 ,
ω R 2 = γ NP γ PN + γ NN γ PP =( v g g (0) v g a p N p (0) η BC Γ J BC Γ a v g α (0) ) v g a Γ a N p (0)
+( 1 τ sp + v g a N p (0) )( v g a p Γ a N p (0) Γ a v g g (0) + 1 τ p + Γ J BC v g α (0) ),
γ= γ NN + γ PP = 1 τ sp + v g a N p (0) + v g a p Γ a N p (0) Γ a v g g (0) + 1 τ p + Γ J BC v g α (0) .
T p (ω) N p,1 V 1 = [ v g a Γ a N p (0) η BC Γ J BC Γ a v g N p (0) ( jω+ 1 τ sp + v g a N p (0) ) 1 Γ J BC v g N p (0) ]K [ jω v g a Γ a N p (0) ( v g a p N p (0) v g g (0) + η BC Γ J BC Γ a v g α (0) ) ( jω+ v g a N p (0) + 1 τ sp ) 1 + v g a p Γ a N p (0) Γ a v g g (0) + 1 τ p + Γ J BC v g α (0) ] .
U(ω)=10log | T p (ω) T p (0) | 2 .
T overall (ω) N p,1 V CB,1 = N p,1 V C ' B,1 × V C ' B,1 V CB,1 = T p (ω)× T e (ω),
i add =e V a η ' Γ J BC Γ a v g α N p ,
i add,1 =Y(ω) V C ' B,1 ,
Y(ω)=e V a η ' Γ J BC Γ a v g ( α V C ' B N p (0) + α (0) N p,1 V C ' B,1 ) V C ' B V C ' B (0) , i B = i B (0) =e V a η ' Γ J BC Γ a v g [ K N p (0) + α (0) T p (ω) ].

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