Abstract

The theory of four-wave mixing (FWM) in the quantum dot (QD) semiconductor optical amplifiers (SOAs) is discussed by combining the QD rate equations system, the quantum-mechanical density-matrix theory, and the pulse propagation in QD SOAs including the three region of QD structure ground state (GS), excited state (ES), and wetting layer. Also, relations for differential gain, gain integral, and nonlinear susceptibility of both pump, probe, and signal pulses were discussed. Gain and differential gain have been calculated for QD structure. FWM efficiency and its components [spectral hole burning (SHB), carrier heating, and carrier density pulsation] are calculated. It is found that inclusion of ES in the formulas and in the calculations is essential since it works as a carrier reservoir for GS. It is found that QD SOA with enough capture time from ES to GS will reduce the SHB component, and so it is suitable for telecommunication applications that require symmetric conversion and independent detuning.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. W. Berg, J. Mørk, and J. M. Hvam, “Gain dynamics and saturation in semiconductor quantum dot amplifiers,” New J. Phys. 6, 178 (2004).
    [CrossRef]
  2. J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Static gain saturation model of quantum-dot semiconductor optical amplifiers,” IEEE J. Quantum Electron. 44, 658–666 (2008).
    [CrossRef]
  3. J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Numerical simulation of temporal and spectral variation of gain and phase recovery in quantum-dot semiconductor optical amplifiers,” IEEE J. Quantum Electron. 46, 405–413 (2010).
    [CrossRef]
  4. D. Nielsen and S. L. Chuang, “Four-wave mixing and wavelength conversion in quantum dots,” Phys. Rev. B 81, 035305 (2010).
    [CrossRef]
  5. C. Wang, F. Grillot, and J. Even, “Impacts of wetting layer and excited state on the modulation response of quantum-dot lasers”, IEEE J. Quantum Electron. 48, 1144–1150(2012).
    [CrossRef]
  6. O. Qasaimeh, “Characteristics of cross-gain (XG) wavelength conversion in quantum dot semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 16, 542–544 (2004).
    [CrossRef]
  7. Y. Ben-Ezra, M. Haridim, and B. I. Lembrikov, “Theoretical analysis of gain-recovery time and chirp in QD-SOA,” IEEE Photon. Technol. Lett. 17, 1803–1805 (2005).
    [CrossRef]
  8. Y. Ben-Ezra, B. I. Lembrikov, and M. Haridim, “Acceleration of gain recovery and dynamics of electrons in QD-SOA,” IEEE J. Quantum Electron. 41, 1268–1273 (2005).
    [CrossRef]
  9. T. W. Berg and J. Mørk, “Saturation and noise properties of quantum-dot optical amplifiers,” IEEE J. Quantum Electron. 40, 1527–1539 (2004).
    [CrossRef]
  10. S. Grosse, J. H. Sandman, G. von Plessen, J. Feldman, H. Lipsanen, M. Sopanen, J. Tulkki, and J. Ahopelto, “Carrier relaxation dynamics in quantum dots: scattering mechanisms and state-filing effects,” Phys. Rev. B 55, 4473–4476(1997).
    [CrossRef]
  11. 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. B 69, 235332 (2004).
    [CrossRef]
  12. T. W. Berg, S. Bischoff, I. Magnusdottir, and J. Mørk, “Ultrafast gain recovery and modulation limitations in self-assembled quantum-dot devices,” IEEE Photon. Technol. Lett. 13, 541–543 (2001).
    [CrossRef]
  13. F. Grillot and M. Gioannini, “Spectral analysis of 1.55 μm InAs–InP(113)B quantum-dot lasers based on a multipopulation rate equations mode,” IEEE J. Quantum Electron. 45, 872–878 (2009).
    [CrossRef]
  14. M. Gioannini and I. Montrosset, “Numerical analysis of the frequency chirp in quantum-dot semiconductor lasers,” IEEE J. Quantum Electron. 43, 941–949 (2007).
    [CrossRef]
  15. D. Bimberg, N. Kistaedter, N. N. Ledentsov, Z. Alferov, P. Kopev, and V. Ustinov, “InGaAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 3, 196–205 (1997).
    [CrossRef]
  16. R. Maram, Q. H. Baghban, H. Rasooli, S. R. Ghorbani, and A. Rostami, “Equivalent circuit model of quantum dot semiconductor optical amplifiers: dynamic behavior and saturation properties,” J. Opt. A 11, 105205 (2009).
    [CrossRef]
  17. A. Uskov, J. Mork, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769–1781 (1994).
    [CrossRef]
  18. M. Shtaif and G. Eisenstein, “Analytical solution of wave mixing between short optical pulses in a semiconductor optical amplifier,” Appl. Phys. Lett. 66, 1458 (1995).
    [CrossRef]
  19. S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
    [CrossRef]
  20. T. Akiyama, H. Kuwatsuka, N. Hatori, Y. Nakata, H. Ebe, and M. Sugawara, “Symmetric highly efficient (∼0  dB) wavelength conversion based on four-wave mixing in quantum dot optical amplifiers,” IEEE Photon. Technol. Lett. 14, 1139 (2002).
    [CrossRef]
  21. M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, K. Otsubo, and H. Ebe, “Quantum-dot semiconductor optical amplifiers,” Proc. SPIE 4905, 259–275 (2002).
    [CrossRef]

2012 (1)

C. Wang, F. Grillot, and J. Even, “Impacts of wetting layer and excited state on the modulation response of quantum-dot lasers”, IEEE J. Quantum Electron. 48, 1144–1150(2012).
[CrossRef]

2010 (2)

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Numerical simulation of temporal and spectral variation of gain and phase recovery in quantum-dot semiconductor optical amplifiers,” IEEE J. Quantum Electron. 46, 405–413 (2010).
[CrossRef]

D. Nielsen and S. L. Chuang, “Four-wave mixing and wavelength conversion in quantum dots,” Phys. Rev. B 81, 035305 (2010).
[CrossRef]

2009 (2)

F. Grillot and M. Gioannini, “Spectral analysis of 1.55 μm InAs–InP(113)B quantum-dot lasers based on a multipopulation rate equations mode,” IEEE J. Quantum Electron. 45, 872–878 (2009).
[CrossRef]

R. Maram, Q. H. Baghban, H. Rasooli, S. R. Ghorbani, and A. Rostami, “Equivalent circuit model of quantum dot semiconductor optical amplifiers: dynamic behavior and saturation properties,” J. Opt. A 11, 105205 (2009).
[CrossRef]

2008 (1)

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

2007 (1)

M. Gioannini and I. Montrosset, “Numerical analysis of the frequency chirp in quantum-dot semiconductor lasers,” IEEE J. Quantum Electron. 43, 941–949 (2007).
[CrossRef]

2005 (3)

Y. Ben-Ezra, M. Haridim, and B. I. Lembrikov, “Theoretical analysis of gain-recovery time and chirp in QD-SOA,” IEEE Photon. Technol. Lett. 17, 1803–1805 (2005).
[CrossRef]

Y. Ben-Ezra, B. I. Lembrikov, and M. Haridim, “Acceleration of gain recovery and dynamics of electrons in QD-SOA,” IEEE J. Quantum Electron. 41, 1268–1273 (2005).
[CrossRef]

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[CrossRef]

2004 (4)

T. W. Berg and J. Mørk, “Saturation and noise properties of quantum-dot optical amplifiers,” IEEE J. Quantum Electron. 40, 1527–1539 (2004).
[CrossRef]

T. W. Berg, J. Mørk, and J. M. Hvam, “Gain dynamics and saturation in semiconductor quantum dot amplifiers,” New J. Phys. 6, 178 (2004).
[CrossRef]

O. Qasaimeh, “Characteristics of cross-gain (XG) wavelength conversion in quantum dot semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 16, 542–544 (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. B 69, 235332 (2004).
[CrossRef]

2002 (2)

T. Akiyama, H. Kuwatsuka, N. Hatori, Y. Nakata, H. Ebe, and M. Sugawara, “Symmetric highly efficient (∼0  dB) wavelength conversion based on four-wave mixing in quantum dot optical amplifiers,” IEEE Photon. Technol. Lett. 14, 1139 (2002).
[CrossRef]

M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, K. Otsubo, and H. Ebe, “Quantum-dot semiconductor optical amplifiers,” Proc. SPIE 4905, 259–275 (2002).
[CrossRef]

2001 (1)

T. W. Berg, S. Bischoff, I. Magnusdottir, and J. Mørk, “Ultrafast gain recovery and modulation limitations in self-assembled quantum-dot devices,” IEEE Photon. Technol. Lett. 13, 541–543 (2001).
[CrossRef]

1997 (2)

D. Bimberg, N. Kistaedter, N. N. Ledentsov, Z. Alferov, P. Kopev, and V. Ustinov, “InGaAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 3, 196–205 (1997).
[CrossRef]

S. Grosse, J. H. Sandman, G. von Plessen, J. Feldman, H. Lipsanen, M. Sopanen, J. Tulkki, and J. Ahopelto, “Carrier relaxation dynamics in quantum dots: scattering mechanisms and state-filing effects,” Phys. Rev. B 55, 4473–4476(1997).
[CrossRef]

1995 (1)

M. Shtaif and G. Eisenstein, “Analytical solution of wave mixing between short optical pulses in a semiconductor optical amplifier,” Appl. Phys. Lett. 66, 1458 (1995).
[CrossRef]

1994 (1)

A. Uskov, J. Mork, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769–1781 (1994).
[CrossRef]

Ahopelto, J.

S. Grosse, J. H. Sandman, G. von Plessen, J. Feldman, H. Lipsanen, M. Sopanen, J. Tulkki, and J. Ahopelto, “Carrier relaxation dynamics in quantum dots: scattering mechanisms and state-filing effects,” Phys. Rev. B 55, 4473–4476(1997).
[CrossRef]

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. B 69, 235332 (2004).
[CrossRef]

T. Akiyama, H. Kuwatsuka, N. Hatori, Y. Nakata, H. Ebe, and M. Sugawara, “Symmetric highly efficient (∼0  dB) wavelength conversion based on four-wave mixing in quantum dot optical amplifiers,” IEEE Photon. Technol. Lett. 14, 1139 (2002).
[CrossRef]

M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, K. Otsubo, and H. Ebe, “Quantum-dot semiconductor optical amplifiers,” Proc. SPIE 4905, 259–275 (2002).
[CrossRef]

Alferov, Z.

D. Bimberg, N. Kistaedter, N. N. Ledentsov, Z. Alferov, P. Kopev, and V. Ustinov, “InGaAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 3, 196–205 (1997).
[CrossRef]

Alferov, Z. I.

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[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. B 69, 235332 (2004).
[CrossRef]

Baghban, Q. H.

R. Maram, Q. H. Baghban, H. Rasooli, S. R. Ghorbani, and A. Rostami, “Equivalent circuit model of quantum dot semiconductor optical amplifiers: dynamic behavior and saturation properties,” J. Opt. A 11, 105205 (2009).
[CrossRef]

Ben-Ezra, Y.

Y. Ben-Ezra, M. Haridim, and B. I. Lembrikov, “Theoretical analysis of gain-recovery time and chirp in QD-SOA,” IEEE Photon. Technol. Lett. 17, 1803–1805 (2005).
[CrossRef]

Y. Ben-Ezra, B. I. Lembrikov, and M. Haridim, “Acceleration of gain recovery and dynamics of electrons in QD-SOA,” IEEE J. Quantum Electron. 41, 1268–1273 (2005).
[CrossRef]

Berg, T. W.

T. W. Berg and J. Mørk, “Saturation and noise properties of quantum-dot optical amplifiers,” IEEE J. Quantum Electron. 40, 1527–1539 (2004).
[CrossRef]

T. W. Berg, J. Mørk, and J. M. Hvam, “Gain dynamics and saturation in semiconductor quantum dot amplifiers,” New J. Phys. 6, 178 (2004).
[CrossRef]

T. W. Berg, S. Bischoff, I. Magnusdottir, and J. Mørk, “Ultrafast gain recovery and modulation limitations in self-assembled quantum-dot devices,” IEEE Photon. Technol. Lett. 13, 541–543 (2001).
[CrossRef]

Bimberg, D.

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Numerical simulation of temporal and spectral variation of gain and phase recovery in quantum-dot semiconductor optical amplifiers,” IEEE J. Quantum Electron. 46, 405–413 (2010).
[CrossRef]

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

D. Bimberg, N. Kistaedter, N. N. Ledentsov, Z. Alferov, P. Kopev, and V. Ustinov, “InGaAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 3, 196–205 (1997).
[CrossRef]

Bischoff, S.

T. W. Berg, S. Bischoff, I. Magnusdottir, and J. Mørk, “Ultrafast gain recovery and modulation limitations in self-assembled quantum-dot devices,” IEEE Photon. Technol. Lett. 13, 541–543 (2001).
[CrossRef]

Chuang, S. L.

D. Nielsen and S. L. Chuang, “Four-wave mixing and wavelength conversion in quantum dots,” Phys. Rev. B 81, 035305 (2010).
[CrossRef]

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. B 69, 235332 (2004).
[CrossRef]

M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, K. Otsubo, and H. Ebe, “Quantum-dot semiconductor optical amplifiers,” Proc. SPIE 4905, 259–275 (2002).
[CrossRef]

T. Akiyama, H. Kuwatsuka, N. Hatori, Y. Nakata, H. Ebe, and M. Sugawara, “Symmetric highly efficient (∼0  dB) wavelength conversion based on four-wave mixing in quantum dot optical amplifiers,” IEEE Photon. Technol. Lett. 14, 1139 (2002).
[CrossRef]

Eisenstein, G.

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Numerical simulation of temporal and spectral variation of gain and phase recovery in quantum-dot semiconductor optical amplifiers,” IEEE J. Quantum Electron. 46, 405–413 (2010).
[CrossRef]

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

M. Shtaif and G. Eisenstein, “Analytical solution of wave mixing between short optical pulses in a semiconductor optical amplifier,” Appl. Phys. Lett. 66, 1458 (1995).
[CrossRef]

Even, J.

C. Wang, F. Grillot, and J. Even, “Impacts of wetting layer and excited state on the modulation response of quantum-dot lasers”, IEEE J. Quantum Electron. 48, 1144–1150(2012).
[CrossRef]

Feldman, J.

S. Grosse, J. H. Sandman, G. von Plessen, J. Feldman, H. Lipsanen, M. Sopanen, J. Tulkki, and J. Ahopelto, “Carrier relaxation dynamics in quantum dots: scattering mechanisms and state-filing effects,” Phys. Rev. B 55, 4473–4476(1997).
[CrossRef]

Ghorbani, S. R.

R. Maram, Q. H. Baghban, H. Rasooli, S. R. Ghorbani, and A. Rostami, “Equivalent circuit model of quantum dot semiconductor optical amplifiers: dynamic behavior and saturation properties,” J. Opt. A 11, 105205 (2009).
[CrossRef]

Gioannini, M.

F. Grillot and M. Gioannini, “Spectral analysis of 1.55 μm InAs–InP(113)B quantum-dot lasers based on a multipopulation rate equations mode,” IEEE J. Quantum Electron. 45, 872–878 (2009).
[CrossRef]

M. Gioannini and I. Montrosset, “Numerical analysis of the frequency chirp in quantum-dot semiconductor lasers,” IEEE J. Quantum Electron. 43, 941–949 (2007).
[CrossRef]

Grillot, F.

C. Wang, F. Grillot, and J. Even, “Impacts of wetting layer and excited state on the modulation response of quantum-dot lasers”, IEEE J. Quantum Electron. 48, 1144–1150(2012).
[CrossRef]

F. Grillot and M. Gioannini, “Spectral analysis of 1.55 μm InAs–InP(113)B quantum-dot lasers based on a multipopulation rate equations mode,” IEEE J. Quantum Electron. 45, 872–878 (2009).
[CrossRef]

Grosse, S.

S. Grosse, J. H. Sandman, G. von Plessen, J. Feldman, H. Lipsanen, M. Sopanen, J. Tulkki, and J. Ahopelto, “Carrier relaxation dynamics in quantum dots: scattering mechanisms and state-filing effects,” Phys. Rev. B 55, 4473–4476(1997).
[CrossRef]

Haridim, M.

Y. Ben-Ezra, B. I. Lembrikov, and M. Haridim, “Acceleration of gain recovery and dynamics of electrons in QD-SOA,” IEEE J. Quantum Electron. 41, 1268–1273 (2005).
[CrossRef]

Y. Ben-Ezra, M. Haridim, and B. I. Lembrikov, “Theoretical analysis of gain-recovery time and chirp in QD-SOA,” IEEE Photon. Technol. Lett. 17, 1803–1805 (2005).
[CrossRef]

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. B 69, 235332 (2004).
[CrossRef]

M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, K. Otsubo, and H. Ebe, “Quantum-dot semiconductor optical amplifiers,” Proc. SPIE 4905, 259–275 (2002).
[CrossRef]

T. Akiyama, H. Kuwatsuka, N. Hatori, Y. Nakata, H. Ebe, and M. Sugawara, “Symmetric highly efficient (∼0  dB) wavelength conversion based on four-wave mixing in quantum dot optical amplifiers,” IEEE Photon. Technol. Lett. 14, 1139 (2002).
[CrossRef]

Hvam, J. M.

T. W. Berg, J. Mørk, and J. M. Hvam, “Gain dynamics and saturation in semiconductor quantum dot amplifiers,” New J. Phys. 6, 178 (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. B 69, 235332 (2004).
[CrossRef]

Kim, J.

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Numerical simulation of temporal and spectral variation of gain and phase recovery in quantum-dot semiconductor optical amplifiers,” IEEE J. Quantum Electron. 46, 405–413 (2010).
[CrossRef]

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

Kistaedter, N.

D. Bimberg, N. Kistaedter, N. N. Ledentsov, Z. Alferov, P. Kopev, and V. Ustinov, “InGaAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 3, 196–205 (1997).
[CrossRef]

Kopev, P.

D. Bimberg, N. Kistaedter, N. N. Ledentsov, Z. Alferov, P. Kopev, and V. Ustinov, “InGaAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 3, 196–205 (1997).
[CrossRef]

Kovsh, A. R.

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[CrossRef]

Kozhukhov, A. V.

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[CrossRef]

Krestnikov, I. L.

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[CrossRef]

Kuwatsuka, H.

T. Akiyama, H. Kuwatsuka, N. Hatori, Y. Nakata, H. Ebe, and M. Sugawara, “Symmetric highly efficient (∼0  dB) wavelength conversion based on four-wave mixing in quantum dot optical amplifiers,” IEEE Photon. Technol. Lett. 14, 1139 (2002).
[CrossRef]

Laemmlin, M.

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

Ledentsov, N. N.

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[CrossRef]

D. Bimberg, N. Kistaedter, N. N. Ledentsov, Z. Alferov, P. Kopev, and V. Ustinov, “InGaAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 3, 196–205 (1997).
[CrossRef]

Lembrikov, B. I.

Y. Ben-Ezra, M. Haridim, and B. I. Lembrikov, “Theoretical analysis of gain-recovery time and chirp in QD-SOA,” IEEE Photon. Technol. Lett. 17, 1803–1805 (2005).
[CrossRef]

Y. Ben-Ezra, B. I. Lembrikov, and M. Haridim, “Acceleration of gain recovery and dynamics of electrons in QD-SOA,” IEEE J. Quantum Electron. 41, 1268–1273 (2005).
[CrossRef]

Lipsanen, H.

S. Grosse, J. H. Sandman, G. von Plessen, J. Feldman, H. Lipsanen, M. Sopanen, J. Tulkki, and J. Ahopelto, “Carrier relaxation dynamics in quantum dots: scattering mechanisms and state-filing effects,” Phys. Rev. B 55, 4473–4476(1997).
[CrossRef]

Livshits, D. A.

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[CrossRef]

Magnusdottir, I.

T. W. Berg, S. Bischoff, I. Magnusdottir, and J. Mørk, “Ultrafast gain recovery and modulation limitations in self-assembled quantum-dot devices,” IEEE Photon. Technol. Lett. 13, 541–543 (2001).
[CrossRef]

Maram, R.

R. Maram, Q. H. Baghban, H. Rasooli, S. R. Ghorbani, and A. Rostami, “Equivalent circuit model of quantum dot semiconductor optical amplifiers: dynamic behavior and saturation properties,” J. Opt. A 11, 105205 (2009).
[CrossRef]

Mark, J.

A. Uskov, J. Mork, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769–1781 (1994).
[CrossRef]

Maximov, M. V.

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[CrossRef]

Meuer, C.

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Numerical simulation of temporal and spectral variation of gain and phase recovery in quantum-dot semiconductor optical amplifiers,” IEEE J. Quantum Electron. 46, 405–413 (2010).
[CrossRef]

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

Mikhrin, S. S.

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[CrossRef]

Montrosset, I.

M. Gioannini and I. Montrosset, “Numerical analysis of the frequency chirp in quantum-dot semiconductor lasers,” IEEE J. Quantum Electron. 43, 941–949 (2007).
[CrossRef]

Mork, J.

A. Uskov, J. Mork, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769–1781 (1994).
[CrossRef]

Mørk, J.

T. W. Berg, J. Mørk, and J. M. Hvam, “Gain dynamics and saturation in semiconductor quantum dot amplifiers,” New J. Phys. 6, 178 (2004).
[CrossRef]

T. W. Berg and J. Mørk, “Saturation and noise properties of quantum-dot optical amplifiers,” IEEE J. Quantum Electron. 40, 1527–1539 (2004).
[CrossRef]

T. W. Berg, S. Bischoff, I. Magnusdottir, and J. Mørk, “Ultrafast gain recovery and modulation limitations in self-assembled quantum-dot devices,” IEEE Photon. Technol. Lett. 13, 541–543 (2001).
[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. B 69, 235332 (2004).
[CrossRef]

T. Akiyama, H. Kuwatsuka, N. Hatori, Y. Nakata, H. Ebe, and M. Sugawara, “Symmetric highly efficient (∼0  dB) wavelength conversion based on four-wave mixing in quantum dot optical amplifiers,” IEEE Photon. Technol. Lett. 14, 1139 (2002).
[CrossRef]

M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, K. Otsubo, and H. Ebe, “Quantum-dot semiconductor optical amplifiers,” Proc. SPIE 4905, 259–275 (2002).
[CrossRef]

Nielsen, D.

D. Nielsen and S. L. Chuang, “Four-wave mixing and wavelength conversion in quantum dots,” Phys. Rev. B 81, 035305 (2010).
[CrossRef]

Novikov, I. I.

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[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. B 69, 235332 (2004).
[CrossRef]

M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, K. Otsubo, and H. Ebe, “Quantum-dot semiconductor optical amplifiers,” Proc. SPIE 4905, 259–275 (2002).
[CrossRef]

Qasaimeh, O.

O. Qasaimeh, “Characteristics of cross-gain (XG) wavelength conversion in quantum dot semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 16, 542–544 (2004).
[CrossRef]

Rasooli, H.

R. Maram, Q. H. Baghban, H. Rasooli, S. R. Ghorbani, and A. Rostami, “Equivalent circuit model of quantum dot semiconductor optical amplifiers: dynamic behavior and saturation properties,” J. Opt. A 11, 105205 (2009).
[CrossRef]

Rostami, A.

R. Maram, Q. H. Baghban, H. Rasooli, S. R. Ghorbani, and A. Rostami, “Equivalent circuit model of quantum dot semiconductor optical amplifiers: dynamic behavior and saturation properties,” J. Opt. A 11, 105205 (2009).
[CrossRef]

Sandman, J. H.

S. Grosse, J. H. Sandman, G. von Plessen, J. Feldman, H. Lipsanen, M. Sopanen, J. Tulkki, and J. Ahopelto, “Carrier relaxation dynamics in quantum dots: scattering mechanisms and state-filing effects,” Phys. Rev. B 55, 4473–4476(1997).
[CrossRef]

Shernyakov, Y. M.

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[CrossRef]

Shtaif, M.

M. Shtaif and G. Eisenstein, “Analytical solution of wave mixing between short optical pulses in a semiconductor optical amplifier,” Appl. Phys. Lett. 66, 1458 (1995).
[CrossRef]

Sopanen, M.

S. Grosse, J. H. Sandman, G. von Plessen, J. Feldman, H. Lipsanen, M. Sopanen, J. Tulkki, and J. Ahopelto, “Carrier relaxation dynamics in quantum dots: scattering mechanisms and state-filing effects,” Phys. Rev. B 55, 4473–4476(1997).
[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. B 69, 235332 (2004).
[CrossRef]

T. Akiyama, H. Kuwatsuka, N. Hatori, Y. Nakata, H. Ebe, and M. Sugawara, “Symmetric highly efficient (∼0  dB) wavelength conversion based on four-wave mixing in quantum dot optical amplifiers,” IEEE Photon. Technol. Lett. 14, 1139 (2002).
[CrossRef]

M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, K. Otsubo, and H. Ebe, “Quantum-dot semiconductor optical amplifiers,” Proc. SPIE 4905, 259–275 (2002).
[CrossRef]

Tulkki, J.

S. Grosse, J. H. Sandman, G. von Plessen, J. Feldman, H. Lipsanen, M. Sopanen, J. Tulkki, and J. Ahopelto, “Carrier relaxation dynamics in quantum dots: scattering mechanisms and state-filing effects,” Phys. Rev. B 55, 4473–4476(1997).
[CrossRef]

Uskov, A.

A. Uskov, J. Mork, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769–1781 (1994).
[CrossRef]

Ustinov, V.

D. Bimberg, N. Kistaedter, N. N. Ledentsov, Z. Alferov, P. Kopev, and V. Ustinov, “InGaAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 3, 196–205 (1997).
[CrossRef]

Ustinov, V. M.

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[CrossRef]

von Plessen, G.

S. Grosse, J. H. Sandman, G. von Plessen, J. Feldman, H. Lipsanen, M. Sopanen, J. Tulkki, and J. Ahopelto, “Carrier relaxation dynamics in quantum dots: scattering mechanisms and state-filing effects,” Phys. Rev. B 55, 4473–4476(1997).
[CrossRef]

Wang, C.

C. Wang, F. Grillot, and J. Even, “Impacts of wetting layer and excited state on the modulation response of quantum-dot lasers”, IEEE J. Quantum Electron. 48, 1144–1150(2012).
[CrossRef]

Appl. Phys. Lett. (1)

M. Shtaif and G. Eisenstein, “Analytical solution of wave mixing between short optical pulses in a semiconductor optical amplifier,” Appl. Phys. Lett. 66, 1458 (1995).
[CrossRef]

IEEE J. Quantum Electron. (9)

F. Grillot and M. Gioannini, “Spectral analysis of 1.55 μm InAs–InP(113)B quantum-dot lasers based on a multipopulation rate equations mode,” IEEE J. Quantum Electron. 45, 872–878 (2009).
[CrossRef]

M. Gioannini and I. Montrosset, “Numerical analysis of the frequency chirp in quantum-dot semiconductor lasers,” IEEE J. Quantum Electron. 43, 941–949 (2007).
[CrossRef]

D. Bimberg, N. Kistaedter, N. N. Ledentsov, Z. Alferov, P. Kopev, and V. Ustinov, “InGaAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron. 3, 196–205 (1997).
[CrossRef]

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

J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, “Numerical simulation of temporal and spectral variation of gain and phase recovery in quantum-dot semiconductor optical amplifiers,” IEEE J. Quantum Electron. 46, 405–413 (2010).
[CrossRef]

C. Wang, F. Grillot, and J. Even, “Impacts of wetting layer and excited state on the modulation response of quantum-dot lasers”, IEEE J. Quantum Electron. 48, 1144–1150(2012).
[CrossRef]

Y. Ben-Ezra, B. I. Lembrikov, and M. Haridim, “Acceleration of gain recovery and dynamics of electrons in QD-SOA,” IEEE J. Quantum Electron. 41, 1268–1273 (2005).
[CrossRef]

T. W. Berg and J. Mørk, “Saturation and noise properties of quantum-dot optical amplifiers,” IEEE J. Quantum Electron. 40, 1527–1539 (2004).
[CrossRef]

A. Uskov, J. Mork, and J. Mark, “Wave mixing in semiconductor laser amplifiers due to carrier heating and spectral-hole burning,” IEEE J. Quantum Electron. 30, 1769–1781 (1994).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

T. Akiyama, H. Kuwatsuka, N. Hatori, Y. Nakata, H. Ebe, and M. Sugawara, “Symmetric highly efficient (∼0  dB) wavelength conversion based on four-wave mixing in quantum dot optical amplifiers,” IEEE Photon. Technol. Lett. 14, 1139 (2002).
[CrossRef]

O. Qasaimeh, “Characteristics of cross-gain (XG) wavelength conversion in quantum dot semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. 16, 542–544 (2004).
[CrossRef]

Y. Ben-Ezra, M. Haridim, and B. I. Lembrikov, “Theoretical analysis of gain-recovery time and chirp in QD-SOA,” IEEE Photon. Technol. Lett. 17, 1803–1805 (2005).
[CrossRef]

T. W. Berg, S. Bischoff, I. Magnusdottir, and J. Mørk, “Ultrafast gain recovery and modulation limitations in self-assembled quantum-dot devices,” IEEE Photon. Technol. Lett. 13, 541–543 (2001).
[CrossRef]

J. Opt. A (1)

R. Maram, Q. H. Baghban, H. Rasooli, S. R. Ghorbani, and A. Rostami, “Equivalent circuit model of quantum dot semiconductor optical amplifiers: dynamic behavior and saturation properties,” J. Opt. A 11, 105205 (2009).
[CrossRef]

New J. Phys. (1)

T. W. Berg, J. Mørk, and J. M. Hvam, “Gain dynamics and saturation in semiconductor quantum dot amplifiers,” New J. Phys. 6, 178 (2004).
[CrossRef]

Phys. Rev. B (3)

D. Nielsen and S. L. Chuang, “Four-wave mixing and wavelength conversion in quantum dots,” Phys. Rev. B 81, 035305 (2010).
[CrossRef]

S. Grosse, J. H. Sandman, G. von Plessen, J. Feldman, H. Lipsanen, M. Sopanen, J. Tulkki, and J. Ahopelto, “Carrier relaxation dynamics in quantum dots: scattering mechanisms and state-filing effects,” Phys. Rev. B 55, 4473–4476(1997).
[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. B 69, 235332 (2004).
[CrossRef]

Proc. SPIE (1)

M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, K. Otsubo, and H. Ebe, “Quantum-dot semiconductor optical amplifiers,” Proc. SPIE 4905, 259–275 (2002).
[CrossRef]

Semicond. Sci. Technol. (1)

S. S. Mikhrin, A. R. Kovsh, I. L. Krestnikov, A. V. Kozhukhov, D. A. Livshits, N. N. Ledentsov, Y. M. Shernyakov, I. I. Novikov, M. V. Maximov, V. M. Ustinov, and Z. I. Alferov, “High power temperature-insensitive 1.3 μm InAs/InGaAs/GaAs quantum dot lasers,” Semicond. Sci. Technol. 20, 340–342 (2005).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1.
Fig. 1.

Diagram of quantum-dot band-structure and carrier relaxation processes for three level system.

Fig. 2.
Fig. 2.

Physical picture of FWM frequencies.

Fig. 3.
Fig. 3.

InAs QD material gain spectrum at carrier density Nw=3×1023m3.

Fig. 4.
Fig. 4.

Differential gain spectrum of InAs QD.

Fig. 5.
Fig. 5.

FWM efficiency in QD SOA with its components: CDP, carrier density pulsation; SHB, spectral hole burning; CH, carrier heating. Both curves of positive and negative branches of FWM efficiency are shown.

Fig. 6.
Fig. 6.

Positive and negative branches of FWM efficiency in QD SOA.

Fig. 7.
Fig. 7.

Comparison between the theoretical calculations from this model with the experimental data from [20].

Fig. 8.
Fig. 8.

Nonlinear gain coefficient for SHB, κSHB, is plotted versus (a) the escape time τ12 and (b) the capture time τ21.

Fig. 9.
Fig. 9.

FWM efficiency at three escape times τ12. For each escape time, positive and negative detuning curves are plotted. Capture time is τ21=0.16ps. Arrows refers to curves of τ12=1.1ps and τ12=2.1ps which are coincides.

Fig. 10.
Fig. 10.

FWM efficiency is calculated at different capture times τ21. For each capture time, positive and negative detuning curves are plotted. Escape time is τ12=1.2ps.

Tables (1)

Tables Icon

Table 1. Parameters Used in the Calculation [4718]

Equations (59)

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

dNwdt=JeLwNw(1h)τw2+Nwhτ2wNwτwr,
dhdt=NwLw(1h)NQτw2+f(1h)τ12NwLwhNQτ2wh(1f)τ21,
dfdt=h(1f)τ21f(1h)τ12f2τ1r+Ct,
τw2=1AW+CWNw,
τ21=1AE+CENw,
τ12=τ21μGsμEse(EEsEGs)κβT,
τ2W=τW2μGsNQNμEsρWe(EWLEEs)κβT,
ρW=mWLκβTπ2.
Ct=iVDjμcv,j(ρcv,jρvc,j)E(t),
E(z,t)=E1eiω1t+E0eiω0t+E2eiω2t+c.c,
Ct=C¯t+C˜teiδt+C˜t*eiδt.
C¯t=i2DVj=i,k|μcv,j|2χj(ω)(ρ¯c,j+ρ¯v,j1)Eo2,
C˜t=jiμ22(ρ¯cv,j+ρ¯vc,j1)×{[χj(ω1)χj*(ωo)]E1E0*+[χj(ωo)χj*(ω2)]E0E2*}jiμ22(ρ˜c+ρ˜v)[χj(ω1)χj*(ω2)]|E0|2,
χj(ω)=1(ωωj)+iγ
g(ω)=(i2)(ωε0nc)(1V)|μcv|2D(ω)(ρ¯cd,i+ρ¯vd,i1)[χ(ω)χ*(ω)]dω,
D(ω)=D2πσ2e((ωω0))22σ2,
(ρ¯cd,i+ρ¯vd,i1)=2ρ¯h(τRτ21)1,
τR=[1τ1r+1τ12+ρ¯h(1τ211τ12)]1,
X(N)=n¯cω(α+i)a(NN0),
dX(N)dN=n¯cω(α+i)dg(N)dN
α=Re(X(N))Im(X(N)).
dg¯dNw(α+i)=(ωcnε0V)|μcv|2D(ω)χ(E)[dρ¯cd,idNw+dρ¯vd,idNw]dω.
[dρ¯cd,idNw+dρ¯vd,idNw]=(4τw2τ211τR2JeLwN¯w2)(τR1ρ¯h(1τ211τ12)).
dg¯dNw(α+i)=(4τw2τ211τR2JeLwN¯w2)(τR1ρ¯h(1τ211τ12))(ωcnε0)1V|μcv|2D(ω)χ(ω)dω.
P(t)=P1eiω1t+P0eiω0t+P2eiω2t+c.c.
P(t)=1Vj=i,k|μcv,j|(ρcv,j+ρvc,j).
ρcv,j=σj,1eiω1t+σj,0eiω0t+σj,2eiω2t,
ρc,j=ρ¯c,j+ρ˜c,jeiδt+ρ˜c,j*eiδt,
Nj=N¯j+N˜jeiδt+N˜j*eiδt,
σcv,0=μχi(ωo)[(ρ¯cd,i+ρ¯vd,i1)Eo+(ρ˜cd,i+ρ˜vd,i)E2+(ρ˜cd,i*+ρ˜vd,i*)E1]σcv,0μχi(ωo)(ρ¯cd,i+ρ¯vd,i1)Eo,
σcv,1=μχi(ω1)[(ρ¯cd,i+ρ¯vd,i1)E1+(ρ˜cd,i+ρ˜cv,i)E0],
σcv,2=μχi(ω2)[(ρ¯cd,i+ρ¯vd,i1)E2+(ρ˜vd,i*+ρ˜vd,i*)E0],
P(t)=1Viμ2χi(ωo)[(ρ¯cd,i+ρ¯vd,i1)Eo]eiω0t+1Viμ2χi(ω1)[(ρ¯cd,i+ρ¯vd,i1)E1+(ρ˜cd,i+ρ˜vd,i)E0]eiω1t+1Viμ2χi(ω2)[(ρ¯cd,i+ρ¯vd,i1)E2+(ρ˜cd,i*+ρ˜vd,i*)E0]eiω2t.
P0=1Viμ2χi(ωo)[(ρ¯cd,i+ρ¯vd,i1)Eo],
P0=ε0XL(ωo)E0,
P1=1Viμ2χi(ω1)[(ρ¯cd,i+ρ¯vd,i1)E1+(ρ˜cd,i+ρ˜vd,i)E0],
P2=1Viμ2χi(ω2)[(ρ¯cd,i+ρ¯vd,i1)E2+(ρ˜cd,i*+ρ˜vd,i*)E0].
(ρ˜cd,i+ρ˜vd,i)=[2τ211(ρ¯cd,i+ρ¯vd,i)(1τ211τ12)]τRρ˜h,i+2τRC˜(t)(iδτR+1).
ρ˜h,i=Re(τw2N¯w(JeLwN¯wiδ)N˜w)=Jτw2N˜weLwN¯w2,
(ρ˜cd,i+ρ˜vd,i)=[2τ2112ρ¯h,iτ211τR1(1τ211τ12)]Jτw2τReLwN¯w2N˜w+2τRC˜(t)(iδτR+1).
(ρ˜cd,i+ρ˜vd,i)=12[τR1ρ¯h,i(1τ211τ12)]4Jτw2τ211τR2eLwN¯w2N˜w
(ρ˜cd,i+ρ˜vd,i)=12[dρ¯cd,idNw+dρ¯vd,idNw]N˜w.
2iδρ˜hiδ(ρ˜c,j+ρ˜v,j)=2N˜wDτw22(N˜wρ¯h+N¯wρ˜h)Dτw22(N˜wρ¯h+N¯wρ˜h)Dτ2w(ρ˜c,j+ρ˜v,j)τ1r+2C˜t.
N˜w=τ1riV2|μcv,i|2D(ω)(ρ¯cd,i+ρ¯vd,i1)[χi(ω)χi*(ω)](E1E0*+E0E2*)dωζ+iτ1r|E0|22V2|μcv,i|2D(ω)(dρ¯cd,idNw+dρ¯vd,idNw)(χi(ω0)χi*(ω0))dω,
ζ=[(iδ+N¯wD(1τw2+1τ2w))(2Jτw2τ1reLwNw2)][2D(1τw2ρ¯h,i(1τw2+1τ2w)τ1r)](iδτ1r+1).
N˜w=τ1r|E0|24cnε0ω0g(ω0)[E1E0*+E0E2*]2ζ+|E0|2Esat2,
Esat2=ω12cnε0τ1rdg(Nw)dNw.
(ρ˜cd,i+ρ˜vd,i)=([dρ¯cd,idNw+dρ¯vd,idNw])(τ1r|E0|24cnε0ω0g(ω0)2ξ+|E0|2Esat2[E1E0*+E0E2*])+2τRC˜(t)(iδτR+1).
P1=1VD(ω)|μcv,i|2χi(ω1)(ρ¯cd,i+ρ¯vd,i1)E1dω+1Dτdτc1(1iδτd)1VD(ω)|μcv,i|2χi(ω1)E0N˜w[1(ρ¯cd,i+ρ¯vd,i1)]dω+(2iτd)(1iδτd)D(ω)1V|μcv,i|43χi(ω1)(ρ¯cd,i+ρ¯vd,i1)[χi(ω1)χi*(ωo)]E1|E0|2dω+(2iτd)(1iδτd)D(ω)1V|μcv,i|43χi(ω1)(ρ¯cd,i+ρ¯vd,i1)[χi(ωo)χi*(ω2)]E02E2*dω.
XCDP(ω1;ω0;ω1)=ε04(cn)2ω0ω1dgdNW(α(ω1)+i)(iδ10τR+1)(τ1rg(ω0)|E0|22ξ+|E0|2Esat2),
XiSHB(ω1;ω0;ω1)=D(ωl)iVε0|μ|43χi(ω1)τR(iδ10τR+1)|E0|2×(ρ¯cv,i+ρ¯vc,i1)[χi(ω1)χi*(ω0)],
E0(L)=eG(τ)/2(1iα)[1+F(δ)|E1|2]E0,
E1(L)=eG(τ)/2(1iα)[1+F+(δ)|E0|2]E1,
E2(L)=eG(τ)/2(1iα)F(δ)E02E1*.
F3LevelQD(±δ)=CeG3L(τ)12[1iα2ξ±+|E0|2Esat2+x=SHB,CH(1iαx)κx1±iδτx].
dG3Ldτ=0L[(Γdg(Nw)dNw{JeLwNw0(1τw2+1τwrh(τ)[1τ2w+1τw2])})g(Nw)(1τw2+1τwrh(τ)(1τ2w+1τw2))]dz.
κSHB(1iαSHB)=(ω0τR(cnε0τ1rdgdNw))×|μcv|43D(ω)χi(ω)(ρ¯cv,i+ρ¯vc,i1)[χi(ω0)χi*(ω)]dω|μcv|22D(ω)(ρ¯cd,i+ρ¯vd,i1)(χi(ω)χi*(ω))dω,
κCH=3τCHNwΔE2πτ1r(kβT)3m*hc,
hc=πkβ2Tm*32.

Metrics