Abstract

We present a novel theoretical time-domain model for a quantum dot semiconductor optical amplifier, that allows to simulate subpicosecond pulse propagation including power-based and phase-based effects. Static results including amplified spontaneous emission spectra, continuous wave amplification, and four-wave mixing experiments in addition to dynamic pump-probe simulations are presented for different injection currents. The model uses digital filters to describe the frequency dependent gain and microscopically calculated carrier-carrier scattering rates for the interband carrier dynamics. It can be used to calculate the propagation of multiple signals with different wavelengths or one wideband signal with high bitrate.

© 2012 OSA

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  1. L. Jacak, P. Hawrylak, and A. Wójs, Quantum Dots (Springer, 1998).
  2. D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (John Willey & Sons, 1999).
  3. T. W. Berg, Quantum Dot Semiconductor Optical Amplifiers, Physics and Application (Research Center COM, 2004).
  4. W. W. Chow and S. W. Koch, “Theory of semiconductor quantum-dot laser dynamics,” IEEE J. Quantum Electron.41(4), 495–505 (2005).
    [CrossRef]
  5. P. Blood, “Gain and recombination in quantum dot lasers,” IEEE J. Sel. Top. Quantum Electron.15(3), 808–818 (2009).
    [CrossRef]
  6. J. Kim and S. L. Chuang, “Theoretical and experimental study of optical gain, refractive index change, and linewidth enhancement factor of p-doped quantum-dot lasers,” IEEE J. Quantum Electron.42(9), 942–952 (2006).
    [CrossRef]
  7. J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
    [CrossRef] [PubMed]
  8. J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Theoretical and experimental study of high-speed small-signal cross-gain modulation of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.45(3), 240–248 (2009).
    [CrossRef]
  9. J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Static gain saturation model of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.44(7), 658–666 (2008).
    [CrossRef]
  10. J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” App. Phys. Lett.94, 041112 (2009).
    [CrossRef]
  11. K. Lüdge, M. J. P. Bormann, E. Malić, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, “Turn-on dynamics and modulation response in semiconductor quantum dot lasers,” Phys. Rev. B78, 035316 (2008).
    [CrossRef]
  12. K. Lüdge and E. Schöll, “Quantum dot lasers - desynchronized nonlinear dynamics of electrons and holes,” IEEE J. Quantum Electron.45(11), 1396–1403 (2009).
    [CrossRef]
  13. M. Wegert, N. Majer, K. Lüdge, S. Dommers-Völkel, J. Gomis-Bresco, A. Knorr, U. Woggon, and E. Schöll, “Nonlinear gain dynamics of quantum dot optical amplifiers,” Semicond. Sci. Technol.25, 014008 (2010).
  14. N. Majer, K. Lüdge, and E. Schöll, “Cascading enables ultrafast gain recovery dynamics of quantum dot semiconductor optical amplifiers,” Phys. Rev. B82, 235301 (2010).
    [CrossRef]
  15. M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers,” Phys. Rev. B69, 235332 (2004).
    [CrossRef]
  16. G. Toptchiyski, S. Kindt, K. Petermann, E. Hilliger, S. Diez, and H. G. Weber, “Time-domain modeling of semiconductor optical amplifiers for OTDM applications,” J. Lightwave Technol.17, 2577–2583 (1999).
    [CrossRef]
  17. R. E. Bogner and A. G. Constantinides, Introduction to Digital Filtering (John Wiley & Sons, 1975).
  18. J. Kim, “Effect of Free-carrier Absorption on the Carrier Dynamics of Quantum-dot Semiconductor Optical Amplifiers,” J. Korean Phys. Soc.55(2), 512–516 (2009).
    [CrossRef]
  19. A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, and G. Huyet, “Carrier-induced refractive index in quantum dot structures due to transitions from discrete quantum dot levels to continuum states,” Appl. Phys. Lett.84(2), 272–274 (2004).
    [CrossRef]
  20. D. G. Deppe, H. Huang, and O. B. Shchekin, “Modulation characteristics of quantum-dot lasers: the influence of p-type doping and the electronic density of states on obtaining high speed,” IEEE J. Quantum Electron.38(12), 1587–1593 (2002).
    [CrossRef]
  21. T. Vallaitis, C. Koos, R. Bonk, W. Freude, M. Laemmlin, C. Meuer, D. Bimberg, and J. Leuthold, “Slow and fast dynamics of gain and phase in a quantum dot semiconductor optical amplifier,” Opt. Express16, 4177–4191 (2008).
    [CrossRef]
  22. V. Cesari, P. Borri, M. Rosetti, A. Fiore, and W. Langbein, “Refractive index dynamic and linewidth enhancement fator in p-doped InAs-GaAs quantum-dot amplifiers,” IEEE J. Quantum Electron.45(6), 579–585 (2009).
    [CrossRef]

2010 (2)

M. Wegert, N. Majer, K. Lüdge, S. Dommers-Völkel, J. Gomis-Bresco, A. Knorr, U. Woggon, and E. Schöll, “Nonlinear gain dynamics of quantum dot optical amplifiers,” Semicond. Sci. Technol.25, 014008 (2010).

N. Majer, K. Lüdge, and E. Schöll, “Cascading enables ultrafast gain recovery dynamics of quantum dot semiconductor optical amplifiers,” Phys. Rev. B82, 235301 (2010).
[CrossRef]

2009 (6)

J. Kim, “Effect of Free-carrier Absorption on the Carrier Dynamics of Quantum-dot Semiconductor Optical Amplifiers,” J. Korean Phys. Soc.55(2), 512–516 (2009).
[CrossRef]

V. Cesari, P. Borri, M. Rosetti, A. Fiore, and W. Langbein, “Refractive index dynamic and linewidth enhancement fator in p-doped InAs-GaAs quantum-dot amplifiers,” IEEE J. Quantum Electron.45(6), 579–585 (2009).
[CrossRef]

K. Lüdge and E. Schöll, “Quantum dot lasers - desynchronized nonlinear dynamics of electrons and holes,” IEEE J. Quantum Electron.45(11), 1396–1403 (2009).
[CrossRef]

P. Blood, “Gain and recombination in quantum dot lasers,” IEEE J. Sel. Top. Quantum Electron.15(3), 808–818 (2009).
[CrossRef]

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Theoretical and experimental study of high-speed small-signal cross-gain modulation of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.45(3), 240–248 (2009).
[CrossRef]

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” App. Phys. Lett.94, 041112 (2009).
[CrossRef]

2008 (4)

K. Lüdge, M. J. P. Bormann, E. Malić, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, “Turn-on dynamics and modulation response in semiconductor quantum dot lasers,” Phys. Rev. B78, 035316 (2008).
[CrossRef]

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Static gain saturation model of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.44(7), 658–666 (2008).
[CrossRef]

J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
[CrossRef] [PubMed]

T. Vallaitis, C. Koos, R. Bonk, W. Freude, M. Laemmlin, C. Meuer, D. Bimberg, and J. Leuthold, “Slow and fast dynamics of gain and phase in a quantum dot semiconductor optical amplifier,” Opt. Express16, 4177–4191 (2008).
[CrossRef]

2006 (1)

J. Kim and S. L. Chuang, “Theoretical and experimental study of optical gain, refractive index change, and linewidth enhancement factor of p-doped quantum-dot lasers,” IEEE J. Quantum Electron.42(9), 942–952 (2006).
[CrossRef]

2005 (1)

W. W. Chow and S. W. Koch, “Theory of semiconductor quantum-dot laser dynamics,” IEEE J. Quantum Electron.41(4), 495–505 (2005).
[CrossRef]

2004 (2)

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, and G. Huyet, “Carrier-induced refractive index in quantum dot structures due to transitions from discrete quantum dot levels to continuum states,” Appl. Phys. Lett.84(2), 272–274 (2004).
[CrossRef]

M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers,” Phys. Rev. B69, 235332 (2004).
[CrossRef]

2002 (1)

D. G. Deppe, H. Huang, and O. B. Shchekin, “Modulation characteristics of quantum-dot lasers: the influence of p-type doping and the electronic density of states on obtaining high speed,” IEEE J. Quantum Electron.38(12), 1587–1593 (2002).
[CrossRef]

1999 (1)

Akiyama, T.

M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers,” Phys. Rev. B69, 235332 (2004).
[CrossRef]

Arakawa, Y.

M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers,” Phys. Rev. B69, 235332 (2004).
[CrossRef]

Berg, T. W.

T. W. Berg, Quantum Dot Semiconductor Optical Amplifiers, Physics and Application (Research Center COM, 2004).

Bimberg, D.

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Theoretical and experimental study of high-speed small-signal cross-gain modulation of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.45(3), 240–248 (2009).
[CrossRef]

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” App. Phys. Lett.94, 041112 (2009).
[CrossRef]

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Static gain saturation model of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.44(7), 658–666 (2008).
[CrossRef]

T. Vallaitis, C. Koos, R. Bonk, W. Freude, M. Laemmlin, C. Meuer, D. Bimberg, and J. Leuthold, “Slow and fast dynamics of gain and phase in a quantum dot semiconductor optical amplifier,” Opt. Express16, 4177–4191 (2008).
[CrossRef]

J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
[CrossRef] [PubMed]

K. Lüdge, M. J. P. Bormann, E. Malić, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, “Turn-on dynamics and modulation response in semiconductor quantum dot lasers,” Phys. Rev. B78, 035316 (2008).
[CrossRef]

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, and G. Huyet, “Carrier-induced refractive index in quantum dot structures due to transitions from discrete quantum dot levels to continuum states,” Appl. Phys. Lett.84(2), 272–274 (2004).
[CrossRef]

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (John Willey & Sons, 1999).

Blood, P.

P. Blood, “Gain and recombination in quantum dot lasers,” IEEE J. Sel. Top. Quantum Electron.15(3), 808–818 (2009).
[CrossRef]

Bogner, R. E.

R. E. Bogner and A. G. Constantinides, Introduction to Digital Filtering (John Wiley & Sons, 1975).

Bonk, R.

Bormann, M. J. P.

K. Lüdge, M. J. P. Bormann, E. Malić, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, “Turn-on dynamics and modulation response in semiconductor quantum dot lasers,” Phys. Rev. B78, 035316 (2008).
[CrossRef]

Borri, P.

V. Cesari, P. Borri, M. Rosetti, A. Fiore, and W. Langbein, “Refractive index dynamic and linewidth enhancement fator in p-doped InAs-GaAs quantum-dot amplifiers,” IEEE J. Quantum Electron.45(6), 579–585 (2009).
[CrossRef]

Cesari, V.

V. Cesari, P. Borri, M. Rosetti, A. Fiore, and W. Langbein, “Refractive index dynamic and linewidth enhancement fator in p-doped InAs-GaAs quantum-dot amplifiers,” IEEE J. Quantum Electron.45(6), 579–585 (2009).
[CrossRef]

Chow, W. W.

W. W. Chow and S. W. Koch, “Theory of semiconductor quantum-dot laser dynamics,” IEEE J. Quantum Electron.41(4), 495–505 (2005).
[CrossRef]

Chuang, S. L.

J. Kim and S. L. Chuang, “Theoretical and experimental study of optical gain, refractive index change, and linewidth enhancement factor of p-doped quantum-dot lasers,” IEEE J. Quantum Electron.42(9), 942–952 (2006).
[CrossRef]

Constantinides, A. G.

R. E. Bogner and A. G. Constantinides, Introduction to Digital Filtering (John Wiley & Sons, 1975).

Deppe, D. G.

D. G. Deppe, H. Huang, and O. B. Shchekin, “Modulation characteristics of quantum-dot lasers: the influence of p-type doping and the electronic density of states on obtaining high speed,” IEEE J. Quantum Electron.38(12), 1587–1593 (2002).
[CrossRef]

Diez, S.

Dommers, S.

J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
[CrossRef] [PubMed]

Dommers-Völkel, S.

M. Wegert, N. Majer, K. Lüdge, S. Dommers-Völkel, J. Gomis-Bresco, A. Knorr, U. Woggon, and E. Schöll, “Nonlinear gain dynamics of quantum dot optical amplifiers,” Semicond. Sci. Technol.25, 014008 (2010).

Ebe, H.

M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers,” Phys. Rev. B69, 235332 (2004).
[CrossRef]

Eisenstein, G.

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” App. Phys. Lett.94, 041112 (2009).
[CrossRef]

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Theoretical and experimental study of high-speed small-signal cross-gain modulation of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.45(3), 240–248 (2009).
[CrossRef]

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Static gain saturation model of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.44(7), 658–666 (2008).
[CrossRef]

Fiore, A.

V. Cesari, P. Borri, M. Rosetti, A. Fiore, and W. Langbein, “Refractive index dynamic and linewidth enhancement fator in p-doped InAs-GaAs quantum-dot amplifiers,” IEEE J. Quantum Electron.45(6), 579–585 (2009).
[CrossRef]

Freude, W.

Gomis-Bresco, J.

M. Wegert, N. Majer, K. Lüdge, S. Dommers-Völkel, J. Gomis-Bresco, A. Knorr, U. Woggon, and E. Schöll, “Nonlinear gain dynamics of quantum dot optical amplifiers,” Semicond. Sci. Technol.25, 014008 (2010).

J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
[CrossRef] [PubMed]

Grundmann, M.

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (John Willey & Sons, 1999).

Hatori, N.

M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers,” Phys. Rev. B69, 235332 (2004).
[CrossRef]

Hawrylak, P.

L. Jacak, P. Hawrylak, and A. Wójs, Quantum Dots (Springer, 1998).

Hilliger, E.

Hövel, P.

K. Lüdge, M. J. P. Bormann, E. Malić, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, “Turn-on dynamics and modulation response in semiconductor quantum dot lasers,” Phys. Rev. B78, 035316 (2008).
[CrossRef]

Huang, H.

D. G. Deppe, H. Huang, and O. B. Shchekin, “Modulation characteristics of quantum-dot lasers: the influence of p-type doping and the electronic density of states on obtaining high speed,” IEEE J. Quantum Electron.38(12), 1587–1593 (2002).
[CrossRef]

Huyet, G.

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, and G. Huyet, “Carrier-induced refractive index in quantum dot structures due to transitions from discrete quantum dot levels to continuum states,” Appl. Phys. Lett.84(2), 272–274 (2004).
[CrossRef]

Ishida, M.

M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers,” Phys. Rev. B69, 235332 (2004).
[CrossRef]

Jacak, L.

L. Jacak, P. Hawrylak, and A. Wójs, Quantum Dots (Springer, 1998).

Kim, J.

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Theoretical and experimental study of high-speed small-signal cross-gain modulation of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.45(3), 240–248 (2009).
[CrossRef]

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” App. Phys. Lett.94, 041112 (2009).
[CrossRef]

J. Kim, “Effect of Free-carrier Absorption on the Carrier Dynamics of Quantum-dot Semiconductor Optical Amplifiers,” J. Korean Phys. Soc.55(2), 512–516 (2009).
[CrossRef]

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Static gain saturation model of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.44(7), 658–666 (2008).
[CrossRef]

J. Kim and S. L. Chuang, “Theoretical and experimental study of optical gain, refractive index change, and linewidth enhancement factor of p-doped quantum-dot lasers,” IEEE J. Quantum Electron.42(9), 942–952 (2006).
[CrossRef]

Kindt, S.

Knorr, A.

M. Wegert, N. Majer, K. Lüdge, S. Dommers-Völkel, J. Gomis-Bresco, A. Knorr, U. Woggon, and E. Schöll, “Nonlinear gain dynamics of quantum dot optical amplifiers,” Semicond. Sci. Technol.25, 014008 (2010).

K. Lüdge, M. J. P. Bormann, E. Malić, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, “Turn-on dynamics and modulation response in semiconductor quantum dot lasers,” Phys. Rev. B78, 035316 (2008).
[CrossRef]

J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
[CrossRef] [PubMed]

Koch, S. W.

W. W. Chow and S. W. Koch, “Theory of semiconductor quantum-dot laser dynamics,” IEEE J. Quantum Electron.41(4), 495–505 (2005).
[CrossRef]

Koos, C.

Kuntz, M.

K. Lüdge, M. J. P. Bormann, E. Malić, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, “Turn-on dynamics and modulation response in semiconductor quantum dot lasers,” Phys. Rev. B78, 035316 (2008).
[CrossRef]

Laemmlin, M.

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Theoretical and experimental study of high-speed small-signal cross-gain modulation of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.45(3), 240–248 (2009).
[CrossRef]

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Static gain saturation model of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.44(7), 658–666 (2008).
[CrossRef]

T. Vallaitis, C. Koos, R. Bonk, W. Freude, M. Laemmlin, C. Meuer, D. Bimberg, and J. Leuthold, “Slow and fast dynamics of gain and phase in a quantum dot semiconductor optical amplifier,” Opt. Express16, 4177–4191 (2008).
[CrossRef]

Lämmlin, M.

J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
[CrossRef] [PubMed]

Langbein, W.

V. Cesari, P. Borri, M. Rosetti, A. Fiore, and W. Langbein, “Refractive index dynamic and linewidth enhancement fator in p-doped InAs-GaAs quantum-dot amplifiers,” IEEE J. Quantum Electron.45(6), 579–585 (2009).
[CrossRef]

Ledentsov, N. N.

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, and G. Huyet, “Carrier-induced refractive index in quantum dot structures due to transitions from discrete quantum dot levels to continuum states,” Appl. Phys. Lett.84(2), 272–274 (2004).
[CrossRef]

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (John Willey & Sons, 1999).

Leuthold, J.

Lüdge, K.

M. Wegert, N. Majer, K. Lüdge, S. Dommers-Völkel, J. Gomis-Bresco, A. Knorr, U. Woggon, and E. Schöll, “Nonlinear gain dynamics of quantum dot optical amplifiers,” Semicond. Sci. Technol.25, 014008 (2010).

N. Majer, K. Lüdge, and E. Schöll, “Cascading enables ultrafast gain recovery dynamics of quantum dot semiconductor optical amplifiers,” Phys. Rev. B82, 235301 (2010).
[CrossRef]

K. Lüdge and E. Schöll, “Quantum dot lasers - desynchronized nonlinear dynamics of electrons and holes,” IEEE J. Quantum Electron.45(11), 1396–1403 (2009).
[CrossRef]

K. Lüdge, M. J. P. Bormann, E. Malić, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, “Turn-on dynamics and modulation response in semiconductor quantum dot lasers,” Phys. Rev. B78, 035316 (2008).
[CrossRef]

Majer, N.

M. Wegert, N. Majer, K. Lüdge, S. Dommers-Völkel, J. Gomis-Bresco, A. Knorr, U. Woggon, and E. Schöll, “Nonlinear gain dynamics of quantum dot optical amplifiers,” Semicond. Sci. Technol.25, 014008 (2010).

N. Majer, K. Lüdge, and E. Schöll, “Cascading enables ultrafast gain recovery dynamics of quantum dot semiconductor optical amplifiers,” Phys. Rev. B82, 235301 (2010).
[CrossRef]

Malic, E.

K. Lüdge, M. J. P. Bormann, E. Malić, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, “Turn-on dynamics and modulation response in semiconductor quantum dot lasers,” Phys. Rev. B78, 035316 (2008).
[CrossRef]

J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
[CrossRef] [PubMed]

McPeake, D.

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, and G. Huyet, “Carrier-induced refractive index in quantum dot structures due to transitions from discrete quantum dot levels to continuum states,” Appl. Phys. Lett.84(2), 272–274 (2004).
[CrossRef]

Meuer, C.

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Theoretical and experimental study of high-speed small-signal cross-gain modulation of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.45(3), 240–248 (2009).
[CrossRef]

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” App. Phys. Lett.94, 041112 (2009).
[CrossRef]

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Static gain saturation model of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.44(7), 658–666 (2008).
[CrossRef]

T. Vallaitis, C. Koos, R. Bonk, W. Freude, M. Laemmlin, C. Meuer, D. Bimberg, and J. Leuthold, “Slow and fast dynamics of gain and phase in a quantum dot semiconductor optical amplifier,” Opt. Express16, 4177–4191 (2008).
[CrossRef]

Nakata, Y.

M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers,” Phys. Rev. B69, 235332 (2004).
[CrossRef]

O’Reilly, E. P.

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, and G. Huyet, “Carrier-induced refractive index in quantum dot structures due to transitions from discrete quantum dot levels to continuum states,” Appl. Phys. Lett.84(2), 272–274 (2004).
[CrossRef]

Otsubo, K.

M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers,” Phys. Rev. B69, 235332 (2004).
[CrossRef]

Petermann, K.

Richter, M.

J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
[CrossRef] [PubMed]

Rosetti, M.

V. Cesari, P. Borri, M. Rosetti, A. Fiore, and W. Langbein, “Refractive index dynamic and linewidth enhancement fator in p-doped InAs-GaAs quantum-dot amplifiers,” IEEE J. Quantum Electron.45(6), 579–585 (2009).
[CrossRef]

Schöll, E.

M. Wegert, N. Majer, K. Lüdge, S. Dommers-Völkel, J. Gomis-Bresco, A. Knorr, U. Woggon, and E. Schöll, “Nonlinear gain dynamics of quantum dot optical amplifiers,” Semicond. Sci. Technol.25, 014008 (2010).

N. Majer, K. Lüdge, and E. Schöll, “Cascading enables ultrafast gain recovery dynamics of quantum dot semiconductor optical amplifiers,” Phys. Rev. B82, 235301 (2010).
[CrossRef]

K. Lüdge and E. Schöll, “Quantum dot lasers - desynchronized nonlinear dynamics of electrons and holes,” IEEE J. Quantum Electron.45(11), 1396–1403 (2009).
[CrossRef]

K. Lüdge, M. J. P. Bormann, E. Malić, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, “Turn-on dynamics and modulation response in semiconductor quantum dot lasers,” Phys. Rev. B78, 035316 (2008).
[CrossRef]

J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
[CrossRef] [PubMed]

Shchekin, O. B.

D. G. Deppe, H. Huang, and O. B. Shchekin, “Modulation characteristics of quantum-dot lasers: the influence of p-type doping and the electronic density of states on obtaining high speed,” IEEE J. Quantum Electron.38(12), 1587–1593 (2002).
[CrossRef]

Sugawara, M.

M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers,” Phys. Rev. B69, 235332 (2004).
[CrossRef]

Temnov, V. V.

J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
[CrossRef] [PubMed]

Toptchiyski, G.

Uskov, A. V.

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, and G. Huyet, “Carrier-induced refractive index in quantum dot structures due to transitions from discrete quantum dot levels to continuum states,” Appl. Phys. Lett.84(2), 272–274 (2004).
[CrossRef]

Vallaitis, T.

Weber, H. G.

Wegert, M.

M. Wegert, N. Majer, K. Lüdge, S. Dommers-Völkel, J. Gomis-Bresco, A. Knorr, U. Woggon, and E. Schöll, “Nonlinear gain dynamics of quantum dot optical amplifiers,” Semicond. Sci. Technol.25, 014008 (2010).

Woggon, U.

M. Wegert, N. Majer, K. Lüdge, S. Dommers-Völkel, J. Gomis-Bresco, A. Knorr, U. Woggon, and E. Schöll, “Nonlinear gain dynamics of quantum dot optical amplifiers,” Semicond. Sci. Technol.25, 014008 (2010).

J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
[CrossRef] [PubMed]

Wójs, A.

L. Jacak, P. Hawrylak, and A. Wójs, Quantum Dots (Springer, 1998).

App. Phys. Lett. (1)

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Role of carrier reservoirs on the slow phase recovery of quantum dot semiconductor optical amplifiers,” App. Phys. Lett.94, 041112 (2009).
[CrossRef]

Appl. Phys. Lett. (1)

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, and G. Huyet, “Carrier-induced refractive index in quantum dot structures due to transitions from discrete quantum dot levels to continuum states,” Appl. Phys. Lett.84(2), 272–274 (2004).
[CrossRef]

IEEE J. Quantum Electron. (7)

D. G. Deppe, H. Huang, and O. B. Shchekin, “Modulation characteristics of quantum-dot lasers: the influence of p-type doping and the electronic density of states on obtaining high speed,” IEEE J. Quantum Electron.38(12), 1587–1593 (2002).
[CrossRef]

V. Cesari, P. Borri, M. Rosetti, A. Fiore, and W. Langbein, “Refractive index dynamic and linewidth enhancement fator in p-doped InAs-GaAs quantum-dot amplifiers,” IEEE J. Quantum Electron.45(6), 579–585 (2009).
[CrossRef]

K. Lüdge and E. Schöll, “Quantum dot lasers - desynchronized nonlinear dynamics of electrons and holes,” IEEE J. Quantum Electron.45(11), 1396–1403 (2009).
[CrossRef]

W. W. Chow and S. W. Koch, “Theory of semiconductor quantum-dot laser dynamics,” IEEE J. Quantum Electron.41(4), 495–505 (2005).
[CrossRef]

J. Kim and S. L. Chuang, “Theoretical and experimental study of optical gain, refractive index change, and linewidth enhancement factor of p-doped quantum-dot lasers,” IEEE J. Quantum Electron.42(9), 942–952 (2006).
[CrossRef]

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Theoretical and experimental study of high-speed small-signal cross-gain modulation of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.45(3), 240–248 (2009).
[CrossRef]

J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Static gain saturation model of quantum-dot semiconductor optical amplifier,” IEEE J. Quantum Electron.44(7), 658–666 (2008).
[CrossRef]

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

P. Blood, “Gain and recombination in quantum dot lasers,” IEEE J. Sel. Top. Quantum Electron.15(3), 808–818 (2009).
[CrossRef]

J. Korean Phys. Soc. (1)

J. Kim, “Effect of Free-carrier Absorption on the Carrier Dynamics of Quantum-dot Semiconductor Optical Amplifiers,” J. Korean Phys. Soc.55(2), 512–516 (2009).
[CrossRef]

J. Lightwave Technol. (1)

Opt. Express (1)

Phys. Rev. B (3)

N. Majer, K. Lüdge, and E. Schöll, “Cascading enables ultrafast gain recovery dynamics of quantum dot semiconductor optical amplifiers,” Phys. Rev. B82, 235301 (2010).
[CrossRef]

M. Sugawara, H. Ebe, N. Hatori, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Theory of optical signal amplification and processing by quantum-dot semiconductor optical amplifiers,” Phys. Rev. B69, 235332 (2004).
[CrossRef]

K. Lüdge, M. J. P. Bormann, E. Malić, P. Hövel, M. Kuntz, D. Bimberg, A. Knorr, and E. Schöll, “Turn-on dynamics and modulation response in semiconductor quantum dot lasers,” Phys. Rev. B78, 035316 (2008).
[CrossRef]

Phys. Rev. Lett. (1)

J. Gomis-Bresco, S. Dommers, V. V. Temnov, U. Woggon, M. Lämmlin, D. Bimberg, E. Malić, M. Richter, E. Schöll, and A. Knorr, “Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots,” Phys. Rev. Lett.101, 256803 (2008).
[CrossRef] [PubMed]

Semicond. Sci. Technol. (1)

M. Wegert, N. Majer, K. Lüdge, S. Dommers-Völkel, J. Gomis-Bresco, A. Knorr, U. Woggon, and E. Schöll, “Nonlinear gain dynamics of quantum dot optical amplifiers,” Semicond. Sci. Technol.25, 014008 (2010).

Other (4)

R. E. Bogner and A. G. Constantinides, Introduction to Digital Filtering (John Wiley & Sons, 1975).

L. Jacak, P. Hawrylak, and A. Wójs, Quantum Dots (Springer, 1998).

D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (John Willey & Sons, 1999).

T. W. Berg, Quantum Dot Semiconductor Optical Amplifiers, Physics and Application (Research Center COM, 2004).

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Figures (16)

Fig. 1
Fig. 1

Optical power spectrum P(f) and material gain g(f) as functions of a frequency f for inhomogeneously distributed QDs for a convention model with frequency slots (a) and the presented one (b) for propagation of two CW signals with different central frequencies. Calculated gain profiles are presented above and signal spectra are below. In a usual model (a) both input CW signals are located in two different time domain models. In this case we can almost obtain correct frequency dependence of the gain, but absence of the phase based effect. The presented model (b) allows to have both correct gain and phase modulations.

Fig. 2
Fig. 2

Optical power spectrum P(f) and material gain g(f) as functions of a frequency f for inhomogeneously distributed QDs for a convention model with frequency slots (a) and the presented one (b) for a short pulse propagation with a broad spectrum. Calculated gain profiles are presented above and signals spectra are below. For a conventional model the whole signal is located in one time-domain model and the stimulated recombination is calculated as if all photons have the same frequency, equal to the central frequency of this time-domain model. This creates a significant spectral hole burning and the same gain for all spectral components. The presented model (b) avoids these defects. It can provide both correct gain saturation (stroke line represents unsaturated gain) and correct amplification for all spectral components.

Fig. 3
Fig. 3

The schematic diagram of the the bidirectional light propagation ℰF(R) through the SOA segments dz (above) and the calculated effects in each segment for one time step dt (below). Calculated carrier dynamics are based on the initial carrier distribution ( n Q W e ( h ) and f G S ( E S ) e ( h )) for this segment. Carrier dynamics and optical signal segments interact with each other through the material gain gGS(ES), stimulated recombination R G S ( E S ) stim and spontaneous recombination R G S ( E S ) spont. Final carrier distributions ( n Q W e ( h ) and f G S ( E S ) e ( h )) are used as initial for the same segment and next time step t + dt, final values of signals ℰF(R) are used as initial for the next z + dz (previous zdz) segment for the next time step.

Fig. 4
Fig. 4

Energy levels with carrier transitions for a single quantum dot (left) and inhomogeneous distribution of M QDs with a common QW (right). ρ(E) represents the total density of states for each level.

Fig. 5
Fig. 5

Measured and calculated amplified spontaneous emission spectra from GS and ES for two different injection currents.

Fig. 6
Fig. 6

The principle of filter-based signal amplification for one length and time step. Left part shows the signal processing schematically, the right part represents corresponding power spectrum: ℰ (t) - input signal for current length step and ℰm(t) - signal on the output of m-th filter, ℰSE(t) - spontaneous emission from QD groups, and ℰ(t +dt) - output signal for the current step and input signal for next length step and time moment. Dashed line in the right part of the figure shows spectrum before applying filters or spontaneous emission, respectively, for comparison.

Fig. 7
Fig. 7

Calculated magnitude (a) and phase (b) responses of gain filters for two QDs groups with transition frequencies of 219.4 THz meV and 225.4 THz (relative frequencies in the model are −17.7 THz and −11.7 THz) and gain 0.0004 and 0.0006 per length step.

Fig. 8
Fig. 8

Calculated magnitude and phase modulation after the amplification by a single filter for different calculation time steps (20 fs and 5 fs). For both time steps we obtain the same modulation around central frequency and visible distortions exist only for high detuning.

Fig. 9
Fig. 9

Calculated magnitude (a) and phase (b) responses of a single gain filter and multiple filters within inhomogeneous distribution. The total amplification is calculated as a summation over all filters.

Fig. 10
Fig. 10

Calculated magnitude (left) and phase(right) responses of SE filters with relative central frequencies −17.7 THz and −11.7 THz. The form of the magnitude responses is equal to the gain filters (Fig. 7), but is normalized.

Fig. 11
Fig. 11

Measured and calculated device gain saturations as a function of output power of the QD-SOA for CW input signals at the wavelength 1310 nm.

Fig. 12
Fig. 12

Measured and calculated pump-probe dynamics for 10 GHz pulses for injected current 250 mA (a) and 500 mA (b).

Fig. 13
Fig. 13

Carrier dynamics in the last device segment for 10 GHz pulses. (a) shows the occupation probability of the central dot in the inhomogeneous distribution, (b) shows 2D carrier density in QW and total carrier density (including QD levels) per layer.

Fig. 14
Fig. 14

Phase recovery for a 10 GHz pulse sequence propagation through the SOA. The net gain is a function of GS carrier density and TPA while the phase mainly depends on the QW total carrier density.

Fig. 15
Fig. 15

A calculated spectrum of the output signal with FWM products (detuning 500 GHz) (a) and calculated and experimental FWM up- and down- conversion efficiencies for different detuning between input signals (b). FWM products of the first order (f3 and f4) and the second order (f5 and f6) are visible in the spectrum. ASE was not included in this simulations to receive more “clean” results.

Fig. 16
Fig. 16

Filter structures for gain (a) and SE (b). A and B are filter coefficients and Z−1 is the delay element. Both filters have the same form and the only difference is a multiplication by material gain factor (G − 1) for calculation of the signal amplification but different coefficients.

Tables (1)

Tables Icon

Table 1 List of parameters and coefficients used in the simulations

Equations (25)

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D ( m ) = 1 σ G S , 0 2 2 exp ( ( m M / 2 ) 2 2 σ G S , 0 2 ) σ G S , 0 = η G S , 0 Δ E G S , 0 2 2 ln 2
E G S R S ( I Inj ) = E G S , 0 1 1 + k G S R S ( I Inj 250 m A ) Δ E G S R S ( I Inj ) = Δ E G S , 0 ( 1 + k Δ E R S ( I Inj 250 m A ) )
η Inj = 0.35 + 0.25 I Inj 250 m A 250 m A
P F ( R ) ( t ) = | F ( R ) ( t ) exp ( i 2 π f 0 t ) | 2 h ¯ ω 0
F z = [ 1 2 α int ( γ T P A 2 + i b 2 ) ( | F | 2 + | R | 2 ) h ¯ ω i ω c 0 Δ n F C A ] × × ( F + m = 1 M H m * F + S E )
H m = { ( G ( m ) 1 ) exp ( i 2 π ( f F ( m ) f 0 ) t t T 2 ) , t 0 0 , t < 0
Δ f F W H M = γ 0 h = 1 π T 2
G G S ( E S ) ( m ) = exp ( 1 2 g G S ( E S ) ( m ) d z ) g G S ( E S ) ( m ) = Γ a g D ( m ) ε G S ( E S ) D 2 D N lay × × ( f G S ( E S ) e ( m ) + f G S ( E S ) h ( m ) 1 )
F z = 1 2 α int F
F z = ( 1 2 γ T P A + i b 2 ) ( | F | 2 + | R | 2 ) h ¯ ω F
C Drude = e 2 2 η b g ε 0 m * ω 2
Δ η F C A = N e , h 3 D C Drude Δ φ F C A z = Δ η F C A ω η b g c 0
N G S ( E S ) S E ( m ) t = f G S ( E S ) e ( m ) f G S ( E S ) h ( m ) τ G S ( E S ) s p × × D ( m ) ε G S ( E S ) D 2 D W d z N lay
G S ( E S ) W N ( m ) = exp ( i θ ) β S E N G S , E S S E ( m ) k S E
S E ( t , z ) = m = 1 M G S Lor ( m , t , z ) + m = 1 M E S Lor ( m , t , z )
n Q W e ( h ) t = η Inj I inj e d z L B Q W n Q W e n Q W h m S Q W E S e ( h ) ( m ) ( 1 f E S e ( h ) ( m ) ) ε E S D ( m ) N Q D + m S E S Q W e ( h ) ( m ) f E S e ( h ) ( m ) ε E S D ( m ) D 2 D m S Q W G S e ( h ) ( m ) ( 1 f G S e ( h ) ( m ) ) ε G S D ( m ) D 2 D + m S G S Q W e ( h ) ( m ) f G S e ( h ) ( m ) ε G S D ( m ) D 2 D
f E S e ( h ) ( m ) t = S Q W E S e ( h ) ( m ) ( 1 f E S e ( h ) ( m ) ) S E S Q W e ( h ) ( m ) f E S e , h ( m ) S E S G S e ( h ) ( m ) f E S e ( h ) ( m ) ( 1 f G S e ( h ) ( m ) ) + S G S E S e ( h ) ( m ) f G S e ( h ) ( m ) ( 1 f E S e ( h ) ( m ) ) ε G S ε E S f E S e ( m ) f E S h ( m ) τ E S R E S stim ( m ) D 2 D ε G S ( E S ) D ( m ) W d z N lay
f G S e ( h ) ( m ) t = S Q W G S e ( h ) ( m ) ( 1 f G S e ( h ) ( m ) ) S G S Q W e ( h ) ( n ) f G S e , h ( n ) + S E S G S e ( h ) ( m ) f E S e ( h ) ( m ) ( 1 f G S e ( h ) ( m ) ) ε E S ε G S S G S E S e ( h ) ( m ) f G S e ( h ) ( m ) ( 1 f E S e ( h ) ( m ) ) f G S e ( m ) f G S h ( m ) τ G S R G S stim ( m ) D 2 D ε G S ( E S ) D ( m ) W d z N lay
R G S ( E S ) stim ( m ) = ( | F , G S ( E S ) , out ( m ) | 2 | F , G S ( E S ) , in ( m ) | 2 + + | R , G S ( E S ) , out ( m ) | 2 | R , G S ( E S ) , in ( m ) | 2 )
H ( Z ) = ( G 1 ) B 1 + A Z 1
out ( m , t ) = B ( G ( m , t ) 1 ) in ( m , t ) + Z G ( m , t d t ) Z G ( m , t ) = A ( m ) in ( m , t )
A ( m ) = exp ( i 2 π f F ( m ) d t d t T 2 )
B = 1 | A | = 1 exp ( d t T 2 )
H ( Z ) = B 1 + A Z 1
S E , out ( m , t ) = B S E , in ( m , t ) + Z S E ( m , t d t ) Z S E ( m , t ) = A ( m ) S E , in ( m , t )

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