Abstract

We derive a simplified rate equation model for the four-wave mixing (FWM) analysis on quantum dot (QD) semiconductor lasers subject to optical injection. The regenerative and the amplitude modulation spectra of the FWM signals with different intrinsic laser parameters and external injection conditions are investigated. By curve fitting the regenerative and the amplitude modulation spectra obtained experimentally, the intrinsic parameters of a commercial single-mode QD laser under different injection conditions are extracted. The linewidth enhancement factor α at different injection levels and detunings are shown, where a reduction of up to 39% from its free-running value is demonstrated. By increasing the injection strength, the α can be further reduced to minimized the chirp in optical communications.

© 2013 OSA

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  1. C. H. Lin, H. H. Lin, and F. Y. Lin, “Four-wave mixing analysis of quantum dot semiconductor lasers for linewidth enhancement factor extraction,” Opt. Express20, 101–110 (2012).
    [CrossRef] [PubMed]
  2. B. Lingnau, K. Lüdge, W. W. Chow, and E. Schöll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E86, 065201 (2012).
    [CrossRef]
  3. S. Melnik, G. Huyet, and A. Uskov, “The linewidth enhancement factor α of quantum dot semiconductor lasers,” Opt. Express14, 2950–2955 (2006).
    [CrossRef] [PubMed]
  4. S. K. Hwang and J. M. Liu, “Dynamical characteristics of an optically injected semiconductor laser,” Opt. Commun.183, 195–205 (2000).
    [CrossRef]
  5. Y. Okajima, S. K. Hwang, and J. M. Liu, “Experimental observation of chirp reduction in bandwidth-enhanced semiconductor lasers subject to strong optical injection,” Opt. Commun.219, 357–364 (2003).
    [CrossRef]
  6. B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett.41, 323–324 (2005).
    [CrossRef]
  7. T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, and L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photon. Technol. Lett.11, 1527–1529 (1999).
    [CrossRef]
  8. K. Kechaou, F. Grillot, J. G. Provost, B. Thedrez, and D. Erasme, “Self-injected semiconductor distributed feedback lasers for frequency chirp stabilization,” Opt. Express20, 26062–26074 (2012).
    [CrossRef] [PubMed]
  9. F. Grillot, B. Dagens, J. G. Provost, H. Su, and L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-μ m InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron.44, 946–951 (2008).
    [CrossRef]
  10. J. G. Provost and F. Grillot, “Measuring the chirp and the linewidth enhancement factor of optoelectronic devices with a Mach-Zehnder interferometer,” IEEE Photon. J.3, 476–488 (2011).
    [CrossRef]
  11. S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
    [CrossRef]
  12. T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron.43, 6–15 (2007).
    [CrossRef]
  13. K. Iiyama, K. Hayashi, and Y. Ida, “Simple method for measuring the linewidth enhancement factor of semiconductor lasers by optical injection locking,” Opt. Lett.17, 1128–1130 (1992).
    [CrossRef] [PubMed]
  14. R. Hui, A. Mecozzi, A. D’ottavi, and P. Spano, “Novel measurement technique of alpha factor in DFB semiconductor lasers by injection locking,” Electron. Lett.26, 997–998 (1990).
    [CrossRef]
  15. I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron.24, 148–154 (1988).
    [CrossRef]
  16. J. M. Liu and T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron.30, 957–965 (1994).
    [CrossRef]
  17. D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
    [CrossRef] [PubMed]
  18. D. O’Brien, S. P. Hegarty, G. Huyet, and A. V. Uskov, “Sensitivity of quantum-dot semiconductor lasers to optical feedback,” Opt. Lett.29, 1072–1074 (2004).
    [CrossRef]
  19. T. Erneux, E. A. Viktorov, B. Kelleher, D. Goulding, S. P. Hegarty, and G. Huyet, “Optically injected quantum-dot lasers,” Opt. Lett.35, 937–939 (2010).
    [CrossRef] [PubMed]
  20. B. Kelleher, D. Goulding, S. P. Hegarty, G. Huyet, E. A. Viktorov, and T. Erneux, “Optically injected single-mode quantum dot lasers,” in Quantum Dot Devices,Zhiming M. Wang, eds. (Springer, 2012), pp. 1–22.
    [CrossRef]
  21. M. Sugawara, N. Hatori, H. Ebe, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Modeling room-temperature lasing spectra of 1.3-μ m self-assembled InAs/GaAs quantum-dot lasers: Homogeneous broadening of optical gain under current injection,” J. Appl. Phys.97, 043523 (2005).
    [CrossRef]
  22. M. Gioannini, A. Sevega, and I. Montrosset, “Simulations of differential gain and linewidth enhancement factor of quantum dot semiconductor lasers,” Opt. Quantum Electron.38, 381–394 (2006).
    [CrossRef]
  23. L. F. Lester, F. Grillot, N. A. Naderi, and V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE8619, 861907, (2013).
    [CrossRef]
  24. N. A. Naderi, F. Grillot, V. Kovanis, and L. F. Lester, “Simultaneous low linewidth enhancement factor and high bandwidth quantum-dash injection-locked laser,” in Proceedings of IEEE Photonics Conference (IEEE, 2011), pp. 115–116.

2013 (1)

L. F. Lester, F. Grillot, N. A. Naderi, and V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE8619, 861907, (2013).
[CrossRef]

2012 (3)

2011 (1)

J. G. Provost and F. Grillot, “Measuring the chirp and the linewidth enhancement factor of optoelectronic devices with a Mach-Zehnder interferometer,” IEEE Photon. J.3, 476–488 (2011).
[CrossRef]

2010 (1)

2008 (2)

S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
[CrossRef]

F. Grillot, B. Dagens, J. G. Provost, H. Su, and L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-μ m InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron.44, 946–951 (2008).
[CrossRef]

2007 (2)

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron.43, 6–15 (2007).
[CrossRef]

2006 (2)

M. Gioannini, A. Sevega, and I. Montrosset, “Simulations of differential gain and linewidth enhancement factor of quantum dot semiconductor lasers,” Opt. Quantum Electron.38, 381–394 (2006).
[CrossRef]

S. Melnik, G. Huyet, and A. Uskov, “The linewidth enhancement factor α of quantum dot semiconductor lasers,” Opt. Express14, 2950–2955 (2006).
[CrossRef] [PubMed]

2005 (2)

M. Sugawara, N. Hatori, H. Ebe, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Modeling room-temperature lasing spectra of 1.3-μ m self-assembled InAs/GaAs quantum-dot lasers: Homogeneous broadening of optical gain under current injection,” J. Appl. Phys.97, 043523 (2005).
[CrossRef]

B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett.41, 323–324 (2005).
[CrossRef]

2004 (1)

2003 (1)

Y. Okajima, S. K. Hwang, and J. M. Liu, “Experimental observation of chirp reduction in bandwidth-enhanced semiconductor lasers subject to strong optical injection,” Opt. Commun.219, 357–364 (2003).
[CrossRef]

2000 (1)

S. K. Hwang and J. M. Liu, “Dynamical characteristics of an optically injected semiconductor laser,” Opt. Commun.183, 195–205 (2000).
[CrossRef]

1999 (1)

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, and L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photon. Technol. Lett.11, 1527–1529 (1999).
[CrossRef]

1994 (1)

J. M. Liu and T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron.30, 957–965 (1994).
[CrossRef]

1992 (1)

1990 (1)

R. Hui, A. Mecozzi, A. D’ottavi, and P. Spano, “Novel measurement technique of alpha factor in DFB semiconductor lasers by injection locking,” Electron. Lett.26, 997–998 (1990).
[CrossRef]

1988 (1)

I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron.24, 148–154 (1988).
[CrossRef]

Akiyama, T.

M. Sugawara, N. Hatori, H. Ebe, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Modeling room-temperature lasing spectra of 1.3-μ m self-assembled InAs/GaAs quantum-dot lasers: Homogeneous broadening of optical gain under current injection,” J. Appl. Phys.97, 043523 (2005).
[CrossRef]

Arakawa, Y.

M. Sugawara, N. Hatori, H. Ebe, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Modeling room-temperature lasing spectra of 1.3-μ m self-assembled InAs/GaAs quantum-dot lasers: Homogeneous broadening of optical gain under current injection,” J. Appl. Phys.97, 043523 (2005).
[CrossRef]

Bossert, D. J.

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, and L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photon. Technol. Lett.11, 1527–1529 (1999).
[CrossRef]

Chabran, C.

I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron.24, 148–154 (1988).
[CrossRef]

Chen, J. X.

B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett.41, 323–324 (2005).
[CrossRef]

Chow, W. W.

B. Lingnau, K. Lüdge, W. W. Chow, and E. Schöll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E86, 065201 (2012).
[CrossRef]

D’ottavi, A.

R. Hui, A. Mecozzi, A. D’ottavi, and P. Spano, “Novel measurement technique of alpha factor in DFB semiconductor lasers by injection locking,” Electron. Lett.26, 997–998 (1990).
[CrossRef]

Dagens, B.

F. Grillot, B. Dagens, J. G. Provost, H. Su, and L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-μ m InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron.44, 946–951 (2008).
[CrossRef]

B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett.41, 323–324 (2005).
[CrossRef]

Debarge, G.

I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron.24, 148–154 (1988).
[CrossRef]

Ebe, H.

M. Sugawara, N. Hatori, H. Ebe, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Modeling room-temperature lasing spectra of 1.3-μ m self-assembled InAs/GaAs quantum-dot lasers: Homogeneous broadening of optical gain under current injection,” J. Appl. Phys.97, 043523 (2005).
[CrossRef]

Erasme, D.

Erneux, T.

T. Erneux, E. A. Viktorov, B. Kelleher, D. Goulding, S. P. Hegarty, and G. Huyet, “Optically injected quantum-dot lasers,” Opt. Lett.35, 937–939 (2010).
[CrossRef] [PubMed]

B. Kelleher, D. Goulding, S. P. Hegarty, G. Huyet, E. A. Viktorov, and T. Erneux, “Optically injected single-mode quantum dot lasers,” in Quantum Dot Devices,Zhiming M. Wang, eds. (Springer, 2012), pp. 1–22.
[CrossRef]

Fiore, A.

B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett.41, 323–324 (2005).
[CrossRef]

Fischer, M.

S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
[CrossRef]

Forchel, A.

S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
[CrossRef]

Fordell, T.

T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron.43, 6–15 (2007).
[CrossRef]

Fuchs, B.

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, and L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photon. Technol. Lett.11, 1527–1529 (1999).
[CrossRef]

Gallion, P.

I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron.24, 148–154 (1988).
[CrossRef]

Gerhard, S.

S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
[CrossRef]

Gerschutz, F.

S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
[CrossRef]

Gioannini, M.

M. Gioannini, A. Sevega, and I. Montrosset, “Simulations of differential gain and linewidth enhancement factor of quantum dot semiconductor lasers,” Opt. Quantum Electron.38, 381–394 (2006).
[CrossRef]

Gouezigou, O. L.

B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett.41, 323–324 (2005).
[CrossRef]

Goulding, D.

T. Erneux, E. A. Viktorov, B. Kelleher, D. Goulding, S. P. Hegarty, and G. Huyet, “Optically injected quantum-dot lasers,” Opt. Lett.35, 937–939 (2010).
[CrossRef] [PubMed]

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

B. Kelleher, D. Goulding, S. P. Hegarty, G. Huyet, E. A. Viktorov, and T. Erneux, “Optically injected single-mode quantum dot lasers,” in Quantum Dot Devices,Zhiming M. Wang, eds. (Springer, 2012), pp. 1–22.
[CrossRef]

Greene, G.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Grillot, F.

L. F. Lester, F. Grillot, N. A. Naderi, and V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE8619, 861907, (2013).
[CrossRef]

K. Kechaou, F. Grillot, J. G. Provost, B. Thedrez, and D. Erasme, “Self-injected semiconductor distributed feedback lasers for frequency chirp stabilization,” Opt. Express20, 26062–26074 (2012).
[CrossRef] [PubMed]

J. G. Provost and F. Grillot, “Measuring the chirp and the linewidth enhancement factor of optoelectronic devices with a Mach-Zehnder interferometer,” IEEE Photon. J.3, 476–488 (2011).
[CrossRef]

F. Grillot, B. Dagens, J. G. Provost, H. Su, and L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-μ m InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron.44, 946–951 (2008).
[CrossRef]

N. A. Naderi, F. Grillot, V. Kovanis, and L. F. Lester, “Simultaneous low linewidth enhancement factor and high bandwidth quantum-dash injection-locked laser,” in Proceedings of IEEE Photonics Conference (IEEE, 2011), pp. 115–116.

Hartnett, M.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Hatori, N.

M. Sugawara, N. Hatori, H. Ebe, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Modeling room-temperature lasing spectra of 1.3-μ m self-assembled InAs/GaAs quantum-dot lasers: Homogeneous broadening of optical gain under current injection,” J. Appl. Phys.97, 043523 (2005).
[CrossRef]

Hayashi, K.

Hegarty, S. P.

T. Erneux, E. A. Viktorov, B. Kelleher, D. Goulding, S. P. Hegarty, and G. Huyet, “Optically injected quantum-dot lasers,” Opt. Lett.35, 937–939 (2010).
[CrossRef] [PubMed]

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

D. O’Brien, S. P. Hegarty, G. Huyet, and A. V. Uskov, “Sensitivity of quantum-dot semiconductor lasers to optical feedback,” Opt. Lett.29, 1072–1074 (2004).
[CrossRef]

B. Kelleher, D. Goulding, S. P. Hegarty, G. Huyet, E. A. Viktorov, and T. Erneux, “Optically injected single-mode quantum dot lasers,” in Quantum Dot Devices,Zhiming M. Wang, eds. (Springer, 2012), pp. 1–22.
[CrossRef]

Hofling, S.

S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
[CrossRef]

Hui, R.

R. Hui, A. Mecozzi, A. D’ottavi, and P. Spano, “Novel measurement technique of alpha factor in DFB semiconductor lasers by injection locking,” Electron. Lett.26, 997–998 (1990).
[CrossRef]

Huyet, G.

T. Erneux, E. A. Viktorov, B. Kelleher, D. Goulding, S. P. Hegarty, and G. Huyet, “Optically injected quantum-dot lasers,” Opt. Lett.35, 937–939 (2010).
[CrossRef] [PubMed]

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

S. Melnik, G. Huyet, and A. Uskov, “The linewidth enhancement factor α of quantum dot semiconductor lasers,” Opt. Express14, 2950–2955 (2006).
[CrossRef] [PubMed]

D. O’Brien, S. P. Hegarty, G. Huyet, and A. V. Uskov, “Sensitivity of quantum-dot semiconductor lasers to optical feedback,” Opt. Lett.29, 1072–1074 (2004).
[CrossRef]

B. Kelleher, D. Goulding, S. P. Hegarty, G. Huyet, E. A. Viktorov, and T. Erneux, “Optically injected single-mode quantum dot lasers,” in Quantum Dot Devices,Zhiming M. Wang, eds. (Springer, 2012), pp. 1–22.
[CrossRef]

Hwang, S. K.

Y. Okajima, S. K. Hwang, and J. M. Liu, “Experimental observation of chirp reduction in bandwidth-enhanced semiconductor lasers subject to strong optical injection,” Opt. Commun.219, 357–364 (2003).
[CrossRef]

S. K. Hwang and J. M. Liu, “Dynamical characteristics of an optically injected semiconductor laser,” Opt. Commun.183, 195–205 (2000).
[CrossRef]

Ida, Y.

Iiyama, K.

Ishida, M.

M. Sugawara, N. Hatori, H. Ebe, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Modeling room-temperature lasing spectra of 1.3-μ m self-assembled InAs/GaAs quantum-dot lasers: Homogeneous broadening of optical gain under current injection,” J. Appl. Phys.97, 043523 (2005).
[CrossRef]

Kamp, M.

S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
[CrossRef]

Kechaou, K.

Kelleher, B.

T. Erneux, E. A. Viktorov, B. Kelleher, D. Goulding, S. P. Hegarty, and G. Huyet, “Optically injected quantum-dot lasers,” Opt. Lett.35, 937–939 (2010).
[CrossRef] [PubMed]

B. Kelleher, D. Goulding, S. P. Hegarty, G. Huyet, E. A. Viktorov, and T. Erneux, “Optically injected single-mode quantum dot lasers,” in Quantum Dot Devices,Zhiming M. Wang, eds. (Springer, 2012), pp. 1–22.
[CrossRef]

Koeth, J.

S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
[CrossRef]

Kovanis, V.

L. F. Lester, F. Grillot, N. A. Naderi, and V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE8619, 861907, (2013).
[CrossRef]

N. A. Naderi, F. Grillot, V. Kovanis, and L. F. Lester, “Simultaneous low linewidth enhancement factor and high bandwidth quantum-dash injection-locked laser,” in Proceedings of IEEE Photonics Conference (IEEE, 2011), pp. 115–116.

Kovsh, A.

S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
[CrossRef]

Krestnikov, I.

S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
[CrossRef]

Landreau, J.

B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett.41, 323–324 (2005).
[CrossRef]

Lester, L. F.

L. F. Lester, F. Grillot, N. A. Naderi, and V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE8619, 861907, (2013).
[CrossRef]

F. Grillot, B. Dagens, J. G. Provost, H. Su, and L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-μ m InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron.44, 946–951 (2008).
[CrossRef]

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, and L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photon. Technol. Lett.11, 1527–1529 (1999).
[CrossRef]

N. A. Naderi, F. Grillot, V. Kovanis, and L. F. Lester, “Simultaneous low linewidth enhancement factor and high bandwidth quantum-dash injection-locked laser,” in Proceedings of IEEE Photonics Conference (IEEE, 2011), pp. 115–116.

Lin, C. H.

Lin, F. Y.

Lin, H. H.

Lindberg, A. M.

T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron.43, 6–15 (2007).
[CrossRef]

Lingnau, B.

B. Lingnau, K. Lüdge, W. W. Chow, and E. Schöll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E86, 065201 (2012).
[CrossRef]

Liu, J. M.

Y. Okajima, S. K. Hwang, and J. M. Liu, “Experimental observation of chirp reduction in bandwidth-enhanced semiconductor lasers subject to strong optical injection,” Opt. Commun.219, 357–364 (2003).
[CrossRef]

S. K. Hwang and J. M. Liu, “Dynamical characteristics of an optically injected semiconductor laser,” Opt. Commun.183, 195–205 (2000).
[CrossRef]

J. M. Liu and T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron.30, 957–965 (1994).
[CrossRef]

Lüdge, K.

B. Lingnau, K. Lüdge, W. W. Chow, and E. Schöll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E86, 065201 (2012).
[CrossRef]

Make, D.

B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett.41, 323–324 (2005).
[CrossRef]

Malloy, K. J.

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, and L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photon. Technol. Lett.11, 1527–1529 (1999).
[CrossRef]

Markus, A.

B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett.41, 323–324 (2005).
[CrossRef]

McInerney, J. G.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Mecozzi, A.

R. Hui, A. Mecozzi, A. D’ottavi, and P. Spano, “Novel measurement technique of alpha factor in DFB semiconductor lasers by injection locking,” Electron. Lett.26, 997–998 (1990).
[CrossRef]

Melnik, S.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

S. Melnik, G. Huyet, and A. Uskov, “The linewidth enhancement factor α of quantum dot semiconductor lasers,” Opt. Express14, 2950–2955 (2006).
[CrossRef] [PubMed]

Montrosset, I.

M. Gioannini, A. Sevega, and I. Montrosset, “Simulations of differential gain and linewidth enhancement factor of quantum dot semiconductor lasers,” Opt. Quantum Electron.38, 381–394 (2006).
[CrossRef]

Naderi, N. A.

L. F. Lester, F. Grillot, N. A. Naderi, and V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE8619, 861907, (2013).
[CrossRef]

N. A. Naderi, F. Grillot, V. Kovanis, and L. F. Lester, “Simultaneous low linewidth enhancement factor and high bandwidth quantum-dash injection-locked laser,” in Proceedings of IEEE Photonics Conference (IEEE, 2011), pp. 115–116.

Nakata, Y.

M. Sugawara, N. Hatori, H. Ebe, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Modeling room-temperature lasing spectra of 1.3-μ m self-assembled InAs/GaAs quantum-dot lasers: Homogeneous broadening of optical gain under current injection,” J. Appl. Phys.97, 043523 (2005).
[CrossRef]

Newell, T. C.

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, and L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photon. Technol. Lett.11, 1527–1529 (1999).
[CrossRef]

O’Brien, D.

Okajima, Y.

Y. Okajima, S. K. Hwang, and J. M. Liu, “Experimental observation of chirp reduction in bandwidth-enhanced semiconductor lasers subject to strong optical injection,” Opt. Commun.219, 357–364 (2003).
[CrossRef]

Otsubo, K.

M. Sugawara, N. Hatori, H. Ebe, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Modeling room-temperature lasing spectra of 1.3-μ m self-assembled InAs/GaAs quantum-dot lasers: Homogeneous broadening of optical gain under current injection,” J. Appl. Phys.97, 043523 (2005).
[CrossRef]

Petitbon, I.

I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron.24, 148–154 (1988).
[CrossRef]

Provost, J. G.

K. Kechaou, F. Grillot, J. G. Provost, B. Thedrez, and D. Erasme, “Self-injected semiconductor distributed feedback lasers for frequency chirp stabilization,” Opt. Express20, 26062–26074 (2012).
[CrossRef] [PubMed]

J. G. Provost and F. Grillot, “Measuring the chirp and the linewidth enhancement factor of optoelectronic devices with a Mach-Zehnder interferometer,” IEEE Photon. J.3, 476–488 (2011).
[CrossRef]

F. Grillot, B. Dagens, J. G. Provost, H. Su, and L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-μ m InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron.44, 946–951 (2008).
[CrossRef]

B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett.41, 323–324 (2005).
[CrossRef]

Rachinskii, D.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Rasskazov, O.

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Schilling, C.

S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
[CrossRef]

Schöll, E.

B. Lingnau, K. Lüdge, W. W. Chow, and E. Schöll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E86, 065201 (2012).
[CrossRef]

Sevega, A.

M. Gioannini, A. Sevega, and I. Montrosset, “Simulations of differential gain and linewidth enhancement factor of quantum dot semiconductor lasers,” Opt. Quantum Electron.38, 381–394 (2006).
[CrossRef]

Simpson, T. B.

J. M. Liu and T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron.30, 957–965 (1994).
[CrossRef]

Spano, P.

R. Hui, A. Mecozzi, A. D’ottavi, and P. Spano, “Novel measurement technique of alpha factor in DFB semiconductor lasers by injection locking,” Electron. Lett.26, 997–998 (1990).
[CrossRef]

Stintz, A.

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, and L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photon. Technol. Lett.11, 1527–1529 (1999).
[CrossRef]

Su, H.

F. Grillot, B. Dagens, J. G. Provost, H. Su, and L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-μ m InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron.44, 946–951 (2008).
[CrossRef]

Sugawara, M.

M. Sugawara, N. Hatori, H. Ebe, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Modeling room-temperature lasing spectra of 1.3-μ m self-assembled InAs/GaAs quantum-dot lasers: Homogeneous broadening of optical gain under current injection,” J. Appl. Phys.97, 043523 (2005).
[CrossRef]

Thedrez, B.

K. Kechaou, F. Grillot, J. G. Provost, B. Thedrez, and D. Erasme, “Self-injected semiconductor distributed feedback lasers for frequency chirp stabilization,” Opt. Express20, 26062–26074 (2012).
[CrossRef] [PubMed]

B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett.41, 323–324 (2005).
[CrossRef]

Uskov, A.

Uskov, A. V.

Viktorov, E. A.

T. Erneux, E. A. Viktorov, B. Kelleher, D. Goulding, S. P. Hegarty, and G. Huyet, “Optically injected quantum-dot lasers,” Opt. Lett.35, 937–939 (2010).
[CrossRef] [PubMed]

B. Kelleher, D. Goulding, S. P. Hegarty, G. Huyet, E. A. Viktorov, and T. Erneux, “Optically injected single-mode quantum dot lasers,” in Quantum Dot Devices,Zhiming M. Wang, eds. (Springer, 2012), pp. 1–22.
[CrossRef]

Electron. Lett. (2)

B. Dagens, A. Markus, J. X. Chen, J. G. Provost, D. Make, O. L. Gouezigou, J. Landreau, A. Fiore, and B. Thedrez, “Giant linewidth enhancement factor and purely frequency modulated emission from quantum dot laser,” Electron. Lett.41, 323–324 (2005).
[CrossRef]

R. Hui, A. Mecozzi, A. D’ottavi, and P. Spano, “Novel measurement technique of alpha factor in DFB semiconductor lasers by injection locking,” Electron. Lett.26, 997–998 (1990).
[CrossRef]

IEEE J. Quantum Electron. (4)

I. Petitbon, P. Gallion, G. Debarge, and C. Chabran, “Locking bandwidth and relaxation oscillations of an injection-locked semiconductor laser,” IEEE J. Quantum Electron.24, 148–154 (1988).
[CrossRef]

J. M. Liu and T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron.30, 957–965 (1994).
[CrossRef]

T. Fordell and A. M. Lindberg, “Experiments on the linewidth-enhancement factor of a vertical-cavity surface-emitting laser,” IEEE J. Quantum Electron.43, 6–15 (2007).
[CrossRef]

F. Grillot, B. Dagens, J. G. Provost, H. Su, and L. F. Lester, “Gain compression and above-threshold linewidth enhancement factor in 1.3-μ m InAs-GaAs quantum-dot lasers,” IEEE J. Quantum Electron.44, 946–951 (2008).
[CrossRef]

IEEE Photon. J. (1)

J. G. Provost and F. Grillot, “Measuring the chirp and the linewidth enhancement factor of optoelectronic devices with a Mach-Zehnder interferometer,” IEEE Photon. J.3, 476–488 (2011).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

S. Gerhard, C. Schilling, F. Gerschutz, M. Fischer, J. Koeth, I. Krestnikov, A. Kovsh, M. Kamp, S. Hofling, and A. Forchel, “Frequency-dependent linewidth enhancement factor of quantum-dot lasers,” IEEE Photon. Technol. Lett.20, 1736–1738 (2008).
[CrossRef]

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, and L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photon. Technol. Lett.11, 1527–1529 (1999).
[CrossRef]

J. Appl. Phys. (1)

M. Sugawara, N. Hatori, H. Ebe, M. Ishida, Y. Arakawa, T. Akiyama, K. Otsubo, and Y. Nakata, “Modeling room-temperature lasing spectra of 1.3-μ m self-assembled InAs/GaAs quantum-dot lasers: Homogeneous broadening of optical gain under current injection,” J. Appl. Phys.97, 043523 (2005).
[CrossRef]

Opt. Commun. (2)

S. K. Hwang and J. M. Liu, “Dynamical characteristics of an optically injected semiconductor laser,” Opt. Commun.183, 195–205 (2000).
[CrossRef]

Y. Okajima, S. K. Hwang, and J. M. Liu, “Experimental observation of chirp reduction in bandwidth-enhanced semiconductor lasers subject to strong optical injection,” Opt. Commun.219, 357–364 (2003).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

M. Gioannini, A. Sevega, and I. Montrosset, “Simulations of differential gain and linewidth enhancement factor of quantum dot semiconductor lasers,” Opt. Quantum Electron.38, 381–394 (2006).
[CrossRef]

Phys. Rev. E (1)

B. Lingnau, K. Lüdge, W. W. Chow, and E. Schöll, “Failure of the α factor in describing dynamical instabilities and chaos in quantum-dot lasers,” Phys. Rev. E86, 065201 (2012).
[CrossRef]

Phys. Rev. Lett. (1)

D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, and G. Huyet, “Excitability in a quantum dot semiconductor laser with optical injection,” Phys. Rev. Lett.98, 153903 (2007).
[CrossRef] [PubMed]

Proc. SPIE (1)

L. F. Lester, F. Grillot, N. A. Naderi, and V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE8619, 861907, (2013).
[CrossRef]

Other (2)

N. A. Naderi, F. Grillot, V. Kovanis, and L. F. Lester, “Simultaneous low linewidth enhancement factor and high bandwidth quantum-dash injection-locked laser,” in Proceedings of IEEE Photonics Conference (IEEE, 2011), pp. 115–116.

B. Kelleher, D. Goulding, S. P. Hegarty, G. Huyet, E. A. Viktorov, and T. Erneux, “Optically injected single-mode quantum dot lasers,” in Quantum Dot Devices,Zhiming M. Wang, eds. (Springer, 2012), pp. 1–22.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(a)–(c) FWM spectra of regenerative, FWM, and amplitude modulation signals without optical injection, respectively. (d)–(f) FWM spectra of regenerative, FWM, and amplitude modulation signals with optical injection, respectively. The green dots are from the numerical simulations and the blue curves are from the analytical solutions.

Fig. 2
Fig. 2

Calculated spectra of the (a)–(d) regenerative and (e)–(h) amplitude modulation signals of the QD laser with different α, γs, γd, and g, respectively.

Fig. 3
Fig. 3

Calculated spectra of the (a)–(b) regenerative (c)–(d) and amplitude modulation signals of the QD laser with different kinj and Δinj, respectively.

Fig. 4
Fig. 4

Schematic setup of the FWM analysis. TL: tunable laser; LD: QD laser diode; FR: Faraday rotator; HW: half-wave plate; PBS: polarizing beamsplitter; PD: photodiode; SA: spectrum analyzer; ATT: variable attenuator; AOM: acousto-optic modulator; PC: polarization controller; FC: 50/50 fiber coupler.

Fig. 5
Fig. 5

Experimentally obtained spectra of (a)–(d) regenerative and (e)–(h) amplitude modulation signals of the injection-locked QD laser with different injection powers (red dots). Blue curves are the least square fitting of the experimental data.

Fig. 6
Fig. 6

Experimentally obtained spectra of (a)–(d) regenerative and (e)–(h) amplitude modulation signals of the injection-locked QD laser with different detuning frequencies (red dots). Blue curves are the least square fitting of the experimental data.

Fig. 7
Fig. 7

Extracted linewidth enhancement factor α of the injection-locked QD laser with (a) different injection powers at Δinj = 0 and (b) different injection condition within the locking boundaries.

Tables (1)

Tables Icon

Table 1 The extracted parameters of the QD laser for different injection powers Pinj and detuning frequencies Δinj.

Equations (13)

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

d E d t = 1 2 υ g g 0 ( 2 ρ 1 1 + ε | E | 2 γ s υ g g 0 ) ( 1 i α ) E + γ s E p e i Δ t
d ρ d t = γ d ρ + C N W ( 1 ρ ) υ g ς ( 2 ρ 1 1 + ε | E | 2 ) | E | 2
d N W d t = γ N N W + J q 2 C N W ( 1 ρ ) ,
d E d t = γ s { 1 2 ( 1 i α ) [ g ( 2 ρ 1 ) 1 ] E + E inj + E p e i Δ t } + i Δ inj E
d ρ d t = γ d [ ρ + J ˜ ( 2 ρ 1 ) | E | 2 ] ,
E ( t ) = E 0 + E r e i Δ t + E f e i Δ t ,
ρ ( t ) = ρ 0 + ρ 1 e i Δ t + ρ 1 * e i Δ t ,
| E | 2 | E 0 | 2 ( 1 + σ e i Δ t + σ * e i Δ t )
E r E 0 = γ s [ ( 1 i α ) g ρ 1 + E p / E 0 ] i ( Δ inj + Δ ) + ( 1 i α ) G
E f E 0 = γ s ( 1 i α ) g ρ 1 * i ( Δ inj Δ ) + ( 1 i α ) G
σ = E r E 0 + ( E f E 0 ) * ,
ρ 1 = E p / E 0 Z + W , G = γ s 2 [ g ( 2 ρ 0 1 ) 1 ] , W = 2 g [ α Δ inj i Δ ( 1 + α 2 ) G ] i ( Δ inj Δ ) ( 1 + i α ) G
Z = [ 2 γ d ( J ˜ ρ 0 ) + ( γ d i Δ ) ( 2 ρ 0 1 ) ] [ i ( Δ inj + Δ ) + ( 1 i α ) G ] γ s γ d ( 2 ρ 0 1 ) ( J ˜ ρ 0 )

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