Abstract

We show that the various techniques commonly used to measure the linewidth enhancement factor can lead to different values when applied to quantum dot semiconductor lasers. Such behaviour is a direct consequence of the intrinsic capture/escape dynamics of quantum dot materials and of the free carrier plasma effects. This provides an explanation for the wide range of values experimentally measured and the linewidth re-broadening recently measured.

© 2006 Optical Society of America

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    [CrossRef]
  3. M. Kuntz, G. Fiol, M. Lämmlin, C. Schubert, A. R. Kovsh, A. Jacob, A. Umbach, and D. Bimberg, "10Gbit/s data modulation using 1.3 μm InGaAs quantum dot lasers," Electron. Lett. 41, 244-245 (2005).
    [CrossRef]
  4. D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
    [CrossRef]
  5. C. H. Henry, "Theory of the linewidth of semiconductor lasers," IEEE J. Quantum Electron. 18, 259-264 (1982).
    [CrossRef]
  6. 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]
  7. C. Harder, K. Vahala, and A. Yariv, "Measurement of the linewidth enhancement factor α of semiconductor lasers," Appl. Phys. Lett. 42, 328-330 (1983).
    [CrossRef]
  8. R. Jin, D. Boggavarapu, G. Khitrova, H. M. Gibbs, Y. Z. Hu, S. W. Koch, and N. Peyghambarian, "Linewidth broadening factor of a microcavity semiconductor laser," Appl. Phys. Lett. 61, 1883-1885 (1992).
    [CrossRef]
  9. Z. Toffano, A. Destrez, C. Birocheau, and L. Hassine, "New linewidth enhancement determination method in semiconductor lasers based on spectrum analysis above and below threshold," Electron. Lett. 28, 9-11 (1992).
    [CrossRef]
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    [CrossRef]
  12. J. Muszalski, J. Houlihan, G. Huyet, and B. Corbett, "Measurement of linewidth enhancement factor in self assembled quantum dot semiconductor lasers emitting at 1310nm," Electron. Lett. 40, 428-430 (2004).
    [CrossRef]
  13. A. A. Ukhanov, A. Stintz, P. G. Eliseev, and K. J. Malloy, "Comparison of the carrier induced refractive index, gain, and linewidth enhancement factor in quantum dot and quantum well lasers," Appl. Phys. Lett. 84, 1058-1060 (2004).
    [CrossRef]
  14. A. Markus, J. X. Chen, O. Gauthier-Lafaye, J.-G. Provost, C. Paranthoën and A. Fiore, "Impact of intraband relaxation on the performance of a quantum-dot laser," IEEE J. Sel. Top. Quantum Electron. 9, 1308-1314 (2003). Q1
    [CrossRef]
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    [CrossRef]
  16. A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, 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, 272-274 (2004).
    [CrossRef]
  17. J. Oksanen and J. Tulkki, "Linewidth enhancement factor and chirp in quantum dot lasers," J. Appl. Phys. 94, 1983-1989 (2003).
    [CrossRef]
  18. M. Sugawara, K. Mukai, and H. Shoji, "Effect of phonon bottleneck on quantum-dot laser performance," Appl. Phys. Lett. 71, 2791-2793 (1997).
    [CrossRef]
  19. A. V. Uskov, Y. Boucher, J. Le Bihan, and J. McInerney, "Theory of a self-assembled quantum-dot semiconductor laser with Auger carrier capture: quantumefficiency and nonlinear gain," Appl. Phys. Lett. 71, 1499-1501 (1998).
    [CrossRef]
  20. S. P. Hegarty, B. Corbett, J. G. McInerney and G. Huyet, "Free-carrier effect on index change in 1.3 μm quantumdot lasers," Electron. Lett. 41, 416-418 (2005).
    [CrossRef]
  21. H. C. Schneider, W. W. Chow and S. W. Koch, "Anomalous carrier-induced dispersion in quantum-dot active media," Phys. Rev. B 66, 041310 (2002).
    [CrossRef]
  22. H. C. Wong, G. B. Ren, J. M. Rorison. "Mode amplification in inhomogeneous QD semiconductor optical amplifiers," Opt. Quantum Electron. (to be published).
  23. H. Su, L. Zhang, R. Wang, T. C. Newell, A. L. Gray and L. F. Lester, "Linewidth study of InAs-InGaAs quantum dot distributed feedback lasers," IEEE Photon. Technol. Lett. 16, 2206-2208 (2004).
    [CrossRef]

2005 (4)

Z. Mi, P. Bhattacharya, and S. Fathpour, "High-speed 1.3 μm tunnel injection quantum-dot lasers," Appl. Phys. Lett. 86, 153109 (2005).
[CrossRef]

M. Kuntz, G. Fiol, M. Lämmlin, C. Schubert, A. R. Kovsh, A. Jacob, A. Umbach, and D. Bimberg, "10Gbit/s data modulation using 1.3 μm InGaAs quantum dot lasers," Electron. Lett. 41, 244-245 (2005).
[CrossRef]

B. Dagens, A. Markus, J. X. Chen, J.-G. Provost, D. Make, O. Le 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]

S. P. Hegarty, B. Corbett, J. G. McInerney and G. Huyet, "Free-carrier effect on index change in 1.3 μm quantumdot lasers," Electron. Lett. 41, 416-418 (2005).
[CrossRef]

2004 (5)

H. Su, L. Zhang, R. Wang, T. C. Newell, A. L. Gray and L. F. Lester, "Linewidth study of InAs-InGaAs quantum dot distributed feedback lasers," IEEE Photon. Technol. Lett. 16, 2206-2208 (2004).
[CrossRef]

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, 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, 272-274 (2004).
[CrossRef]

S. Schneider, P. Borri,W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, "Linewidth enhancement factor in InGaAs quantum-dot lasers," IEEE J. Quantum Electron. 40, 1423-1429 (2004).
[CrossRef]

J. Muszalski, J. Houlihan, G. Huyet, and B. Corbett, "Measurement of linewidth enhancement factor in self assembled quantum dot semiconductor lasers emitting at 1310nm," Electron. Lett. 40, 428-430 (2004).
[CrossRef]

A. A. Ukhanov, A. Stintz, P. G. Eliseev, and K. J. Malloy, "Comparison of the carrier induced refractive index, gain, and linewidth enhancement factor in quantum dot and quantum well lasers," Appl. Phys. Lett. 84, 1058-1060 (2004).
[CrossRef]

2003 (3)

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J.-G. Provost, C. Paranthoën and A. Fiore, "Impact of intraband relaxation on the performance of a quantum-dot laser," IEEE J. Sel. Top. Quantum Electron. 9, 1308-1314 (2003). Q1
[CrossRef]

J. Oksanen and J. Tulkki, "Linewidth enhancement factor and chirp in quantum dot lasers," J. Appl. Phys. 94, 1983-1989 (2003).
[CrossRef]

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

2002 (1)

H. C. Schneider, W. W. Chow and S. W. Koch, "Anomalous carrier-induced dispersion in quantum-dot active media," Phys. Rev. B 66, 041310 (2002).
[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]

1998 (1)

A. V. Uskov, Y. Boucher, J. Le Bihan, and J. McInerney, "Theory of a self-assembled quantum-dot semiconductor laser with Auger carrier capture: quantumefficiency and nonlinear gain," Appl. Phys. Lett. 71, 1499-1501 (1998).
[CrossRef]

1997 (1)

M. Sugawara, K. Mukai, and H. Shoji, "Effect of phonon bottleneck on quantum-dot laser performance," Appl. Phys. Lett. 71, 2791-2793 (1997).
[CrossRef]

1992 (2)

R. Jin, D. Boggavarapu, G. Khitrova, H. M. Gibbs, Y. Z. Hu, S. W. Koch, and N. Peyghambarian, "Linewidth broadening factor of a microcavity semiconductor laser," Appl. Phys. Lett. 61, 1883-1885 (1992).
[CrossRef]

Z. Toffano, A. Destrez, C. Birocheau, and L. Hassine, "New linewidth enhancement determination method in semiconductor lasers based on spectrum analysis above and below threshold," Electron. Lett. 28, 9-11 (1992).
[CrossRef]

1983 (1)

C. Harder, K. Vahala, and A. Yariv, "Measurement of the linewidth enhancement factor α of semiconductor lasers," Appl. Phys. Lett. 42, 328-330 (1983).
[CrossRef]

1982 (1)

C. H. Henry, "Theory of the linewidth of semiconductor lasers," IEEE J. Quantum Electron. 18, 259-264 (1982).
[CrossRef]

Bhattacharya, P.

Z. Mi, P. Bhattacharya, and S. Fathpour, "High-speed 1.3 μm tunnel injection quantum-dot lasers," Appl. Phys. Lett. 86, 153109 (2005).
[CrossRef]

Bimberg, D.

M. Kuntz, G. Fiol, M. Lämmlin, C. Schubert, A. R. Kovsh, A. Jacob, A. Umbach, and D. Bimberg, "10Gbit/s data modulation using 1.3 μm InGaAs quantum dot lasers," Electron. Lett. 41, 244-245 (2005).
[CrossRef]

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, 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, 272-274 (2004).
[CrossRef]

S. Schneider, P. Borri,W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, "Linewidth enhancement factor in InGaAs quantum-dot lasers," IEEE J. Quantum Electron. 40, 1423-1429 (2004).
[CrossRef]

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

Birocheau, C.

Z. Toffano, A. Destrez, C. Birocheau, and L. Hassine, "New linewidth enhancement determination method in semiconductor lasers based on spectrum analysis above and below threshold," Electron. Lett. 28, 9-11 (1992).
[CrossRef]

Boggavarapu, D.

R. Jin, D. Boggavarapu, G. Khitrova, H. M. Gibbs, Y. Z. Hu, S. W. Koch, and N. Peyghambarian, "Linewidth broadening factor of a microcavity semiconductor laser," Appl. Phys. Lett. 61, 1883-1885 (1992).
[CrossRef]

Borri, P.

S. Schneider, P. Borri,W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, "Linewidth enhancement factor in InGaAs quantum-dot lasers," IEEE J. Quantum Electron. 40, 1423-1429 (2004).
[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]

Boucher, Y.

A. V. Uskov, Y. Boucher, J. Le Bihan, and J. McInerney, "Theory of a self-assembled quantum-dot semiconductor laser with Auger carrier capture: quantumefficiency and nonlinear gain," Appl. Phys. Lett. 71, 1499-1501 (1998).
[CrossRef]

Chen, J. X.

B. Dagens, A. Markus, J. X. Chen, J.-G. Provost, D. Make, O. Le 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]

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J.-G. Provost, C. Paranthoën and A. Fiore, "Impact of intraband relaxation on the performance of a quantum-dot laser," IEEE J. Sel. Top. Quantum Electron. 9, 1308-1314 (2003). Q1
[CrossRef]

Chow, W. W.

H. C. Schneider, W. W. Chow and S. W. Koch, "Anomalous carrier-induced dispersion in quantum-dot active media," Phys. Rev. B 66, 041310 (2002).
[CrossRef]

Corbett, B.

S. P. Hegarty, B. Corbett, J. G. McInerney and G. Huyet, "Free-carrier effect on index change in 1.3 μm quantumdot lasers," Electron. Lett. 41, 416-418 (2005).
[CrossRef]

J. Muszalski, J. Houlihan, G. Huyet, and B. Corbett, "Measurement of linewidth enhancement factor in self assembled quantum dot semiconductor lasers emitting at 1310nm," Electron. Lett. 40, 428-430 (2004).
[CrossRef]

Dagens, B.

B. Dagens, A. Markus, J. X. Chen, J.-G. Provost, D. Make, O. Le 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]

Destrez, A.

Z. Toffano, A. Destrez, C. Birocheau, and L. Hassine, "New linewidth enhancement determination method in semiconductor lasers based on spectrum analysis above and below threshold," Electron. Lett. 28, 9-11 (1992).
[CrossRef]

Eliseev, P. G.

A. A. Ukhanov, A. Stintz, P. G. Eliseev, and K. J. Malloy, "Comparison of the carrier induced refractive index, gain, and linewidth enhancement factor in quantum dot and quantum well lasers," Appl. Phys. Lett. 84, 1058-1060 (2004).
[CrossRef]

Fathpour, S.

Z. Mi, P. Bhattacharya, and S. Fathpour, "High-speed 1.3 μm tunnel injection quantum-dot lasers," Appl. Phys. Lett. 86, 153109 (2005).
[CrossRef]

Fiol, G.

M. Kuntz, G. Fiol, M. Lämmlin, C. Schubert, A. R. Kovsh, A. Jacob, A. Umbach, and D. Bimberg, "10Gbit/s data modulation using 1.3 μm InGaAs quantum dot lasers," Electron. Lett. 41, 244-245 (2005).
[CrossRef]

Fiore, A.

B. Dagens, A. Markus, J. X. Chen, J.-G. Provost, D. Make, O. Le 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]

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J.-G. Provost, C. Paranthoën and A. Fiore, "Impact of intraband relaxation on the performance of a quantum-dot laser," IEEE J. Sel. Top. Quantum Electron. 9, 1308-1314 (2003). Q1
[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]

Gauthier-Lafaye, O.

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J.-G. Provost, C. Paranthoën and A. Fiore, "Impact of intraband relaxation on the performance of a quantum-dot laser," IEEE J. Sel. Top. Quantum Electron. 9, 1308-1314 (2003). Q1
[CrossRef]

Gibbs, H. M.

R. Jin, D. Boggavarapu, G. Khitrova, H. M. Gibbs, Y. Z. Hu, S. W. Koch, and N. Peyghambarian, "Linewidth broadening factor of a microcavity semiconductor laser," Appl. Phys. Lett. 61, 1883-1885 (1992).
[CrossRef]

Gray, A. L.

H. Su, L. Zhang, R. Wang, T. C. Newell, A. L. Gray and L. F. Lester, "Linewidth study of InAs-InGaAs quantum dot distributed feedback lasers," IEEE Photon. Technol. Lett. 16, 2206-2208 (2004).
[CrossRef]

Harder, C.

C. Harder, K. Vahala, and A. Yariv, "Measurement of the linewidth enhancement factor α of semiconductor lasers," Appl. Phys. Lett. 42, 328-330 (1983).
[CrossRef]

Hassine, L.

Z. Toffano, A. Destrez, C. Birocheau, and L. Hassine, "New linewidth enhancement determination method in semiconductor lasers based on spectrum analysis above and below threshold," Electron. Lett. 28, 9-11 (1992).
[CrossRef]

Hegarty, S. P.

S. P. Hegarty, B. Corbett, J. G. McInerney and G. Huyet, "Free-carrier effect on index change in 1.3 μm quantumdot lasers," Electron. Lett. 41, 416-418 (2005).
[CrossRef]

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

Henry, C. H.

C. H. Henry, "Theory of the linewidth of semiconductor lasers," IEEE J. Quantum Electron. 18, 259-264 (1982).
[CrossRef]

Houlihan, J.

J. Muszalski, J. Houlihan, G. Huyet, and B. Corbett, "Measurement of linewidth enhancement factor in self assembled quantum dot semiconductor lasers emitting at 1310nm," Electron. Lett. 40, 428-430 (2004).
[CrossRef]

Hu, Y. Z.

R. Jin, D. Boggavarapu, G. Khitrova, H. M. Gibbs, Y. Z. Hu, S. W. Koch, and N. Peyghambarian, "Linewidth broadening factor of a microcavity semiconductor laser," Appl. Phys. Lett. 61, 1883-1885 (1992).
[CrossRef]

Huyet, G.

S. P. Hegarty, B. Corbett, J. G. McInerney and G. Huyet, "Free-carrier effect on index change in 1.3 μm quantumdot lasers," Electron. Lett. 41, 416-418 (2005).
[CrossRef]

J. Muszalski, J. Houlihan, G. Huyet, and B. Corbett, "Measurement of linewidth enhancement factor in self assembled quantum dot semiconductor lasers emitting at 1310nm," Electron. Lett. 40, 428-430 (2004).
[CrossRef]

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, 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, 272-274 (2004).
[CrossRef]

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

Jacob, A.

M. Kuntz, G. Fiol, M. Lämmlin, C. Schubert, A. R. Kovsh, A. Jacob, A. Umbach, and D. Bimberg, "10Gbit/s data modulation using 1.3 μm InGaAs quantum dot lasers," Electron. Lett. 41, 244-245 (2005).
[CrossRef]

Jin, R.

R. Jin, D. Boggavarapu, G. Khitrova, H. M. Gibbs, Y. Z. Hu, S. W. Koch, and N. Peyghambarian, "Linewidth broadening factor of a microcavity semiconductor laser," Appl. Phys. Lett. 61, 1883-1885 (1992).
[CrossRef]

Kettler, T.

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

Khitrova, G.

R. Jin, D. Boggavarapu, G. Khitrova, H. M. Gibbs, Y. Z. Hu, S. W. Koch, and N. Peyghambarian, "Linewidth broadening factor of a microcavity semiconductor laser," Appl. Phys. Lett. 61, 1883-1885 (1992).
[CrossRef]

Koch, S. W.

H. C. Schneider, W. W. Chow and S. W. Koch, "Anomalous carrier-induced dispersion in quantum-dot active media," Phys. Rev. B 66, 041310 (2002).
[CrossRef]

R. Jin, D. Boggavarapu, G. Khitrova, H. M. Gibbs, Y. Z. Hu, S. W. Koch, and N. Peyghambarian, "Linewidth broadening factor of a microcavity semiconductor laser," Appl. Phys. Lett. 61, 1883-1885 (1992).
[CrossRef]

Kovsh, A. R.

M. Kuntz, G. Fiol, M. Lämmlin, C. Schubert, A. R. Kovsh, A. Jacob, A. Umbach, and D. Bimberg, "10Gbit/s data modulation using 1.3 μm InGaAs quantum dot lasers," Electron. Lett. 41, 244-245 (2005).
[CrossRef]

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

Kuntz, M.

M. Kuntz, G. Fiol, M. Lämmlin, C. Schubert, A. R. Kovsh, A. Jacob, A. Umbach, and D. Bimberg, "10Gbit/s data modulation using 1.3 μm InGaAs quantum dot lasers," Electron. Lett. 41, 244-245 (2005).
[CrossRef]

Laemmlin, M.

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

Lämmlin, M.

M. Kuntz, G. Fiol, M. Lämmlin, C. Schubert, A. R. Kovsh, A. Jacob, A. Umbach, and D. Bimberg, "10Gbit/s data modulation using 1.3 μm InGaAs quantum dot lasers," Electron. Lett. 41, 244-245 (2005).
[CrossRef]

Landreau, J.

B. Dagens, A. Markus, J. X. Chen, J.-G. Provost, D. Make, O. Le 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]

Langbein, W.

S. Schneider, P. Borri,W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, "Linewidth enhancement factor in InGaAs quantum-dot lasers," IEEE J. Quantum Electron. 40, 1423-1429 (2004).
[CrossRef]

Le Bihan, J.

A. V. Uskov, Y. Boucher, J. Le Bihan, and J. McInerney, "Theory of a self-assembled quantum-dot semiconductor laser with Auger carrier capture: quantumefficiency and nonlinear gain," Appl. Phys. Lett. 71, 1499-1501 (1998).
[CrossRef]

Le Gouezigou, O.

B. Dagens, A. Markus, J. X. Chen, J.-G. Provost, D. Make, O. Le 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]

Ledentsov, N. N.

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, 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, 272-274 (2004).
[CrossRef]

Lester, L. F.

H. Su, L. Zhang, R. Wang, T. C. Newell, A. L. Gray and L. F. Lester, "Linewidth study of InAs-InGaAs quantum dot distributed feedback lasers," IEEE Photon. Technol. Lett. 16, 2206-2208 (2004).
[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]

Make, D.

B. Dagens, A. Markus, J. X. Chen, J.-G. Provost, D. Make, O. Le 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.

A. A. Ukhanov, A. Stintz, P. G. Eliseev, and K. J. Malloy, "Comparison of the carrier induced refractive index, gain, and linewidth enhancement factor in quantum dot and quantum well lasers," Appl. Phys. Lett. 84, 1058-1060 (2004).
[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]

Markus, A.

B. Dagens, A. Markus, J. X. Chen, J.-G. Provost, D. Make, O. Le 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]

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J.-G. Provost, C. Paranthoën and A. Fiore, "Impact of intraband relaxation on the performance of a quantum-dot laser," IEEE J. Sel. Top. Quantum Electron. 9, 1308-1314 (2003). Q1
[CrossRef]

McInerney, J.

A. V. Uskov, Y. Boucher, J. Le Bihan, and J. McInerney, "Theory of a self-assembled quantum-dot semiconductor laser with Auger carrier capture: quantumefficiency and nonlinear gain," Appl. Phys. Lett. 71, 1499-1501 (1998).
[CrossRef]

McInerney, J. G.

S. P. Hegarty, B. Corbett, J. G. McInerney and G. Huyet, "Free-carrier effect on index change in 1.3 μm quantumdot lasers," Electron. Lett. 41, 416-418 (2005).
[CrossRef]

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

McPeake, D.

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, 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, 272-274 (2004).
[CrossRef]

Mi, Z.

Z. Mi, P. Bhattacharya, and S. Fathpour, "High-speed 1.3 μm tunnel injection quantum-dot lasers," Appl. Phys. Lett. 86, 153109 (2005).
[CrossRef]

Mikhrin, S. S.

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

Mukai, K.

M. Sugawara, K. Mukai, and H. Shoji, "Effect of phonon bottleneck on quantum-dot laser performance," Appl. Phys. Lett. 71, 2791-2793 (1997).
[CrossRef]

Muszalski, J.

J. Muszalski, J. Houlihan, G. Huyet, and B. Corbett, "Measurement of linewidth enhancement factor in self assembled quantum dot semiconductor lasers emitting at 1310nm," Electron. Lett. 40, 428-430 (2004).
[CrossRef]

Newell, T. C.

H. Su, L. Zhang, R. Wang, T. C. Newell, A. L. Gray and L. F. Lester, "Linewidth study of InAs-InGaAs quantum dot distributed feedback lasers," IEEE Photon. Technol. Lett. 16, 2206-2208 (2004).
[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]

O’Brien, D.

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

O’Reilly, E. P.

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, 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, 272-274 (2004).
[CrossRef]

Oksanen, J.

J. Oksanen and J. Tulkki, "Linewidth enhancement factor and chirp in quantum dot lasers," J. Appl. Phys. 94, 1983-1989 (2003).
[CrossRef]

Ouyang, D.

S. Schneider, P. Borri,W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, "Linewidth enhancement factor in InGaAs quantum-dot lasers," IEEE J. Quantum Electron. 40, 1423-1429 (2004).
[CrossRef]

Paranthoën, C.

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J.-G. Provost, C. Paranthoën and A. Fiore, "Impact of intraband relaxation on the performance of a quantum-dot laser," IEEE J. Sel. Top. Quantum Electron. 9, 1308-1314 (2003). Q1
[CrossRef]

Peyghambarian, N.

R. Jin, D. Boggavarapu, G. Khitrova, H. M. Gibbs, Y. Z. Hu, S. W. Koch, and N. Peyghambarian, "Linewidth broadening factor of a microcavity semiconductor laser," Appl. Phys. Lett. 61, 1883-1885 (1992).
[CrossRef]

Provost, J.-G.

B. Dagens, A. Markus, J. X. Chen, J.-G. Provost, D. Make, O. Le 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]

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J.-G. Provost, C. Paranthoën and A. Fiore, "Impact of intraband relaxation on the performance of a quantum-dot laser," IEEE J. Sel. Top. Quantum Electron. 9, 1308-1314 (2003). Q1
[CrossRef]

Ren, G. B.

H. C. Wong, G. B. Ren, J. M. Rorison. "Mode amplification in inhomogeneous QD semiconductor optical amplifiers," Opt. Quantum Electron. (to be published).

Rorison, J. M.

H. C. Wong, G. B. Ren, J. M. Rorison. "Mode amplification in inhomogeneous QD semiconductor optical amplifiers," Opt. Quantum Electron. (to be published).

Schneider, H. C.

H. C. Schneider, W. W. Chow and S. W. Koch, "Anomalous carrier-induced dispersion in quantum-dot active media," Phys. Rev. B 66, 041310 (2002).
[CrossRef]

Schneider, S.

S. Schneider, P. Borri,W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, "Linewidth enhancement factor in InGaAs quantum-dot lasers," IEEE J. Quantum Electron. 40, 1423-1429 (2004).
[CrossRef]

Schubert, C.

M. Kuntz, G. Fiol, M. Lämmlin, C. Schubert, A. R. Kovsh, A. Jacob, A. Umbach, and D. Bimberg, "10Gbit/s data modulation using 1.3 μm InGaAs quantum dot lasers," Electron. Lett. 41, 244-245 (2005).
[CrossRef]

Sellin, R. L.

S. Schneider, P. Borri,W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, "Linewidth enhancement factor in InGaAs quantum-dot lasers," IEEE J. Quantum Electron. 40, 1423-1429 (2004).
[CrossRef]

Shoji, H.

M. Sugawara, K. Mukai, and H. Shoji, "Effect of phonon bottleneck on quantum-dot laser performance," Appl. Phys. Lett. 71, 2791-2793 (1997).
[CrossRef]

Stintz, A.

A. A. Ukhanov, A. Stintz, P. G. Eliseev, and K. J. Malloy, "Comparison of the carrier induced refractive index, gain, and linewidth enhancement factor in quantum dot and quantum well lasers," Appl. Phys. Lett. 84, 1058-1060 (2004).
[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]

Su, H.

H. Su, L. Zhang, R. Wang, T. C. Newell, A. L. Gray and L. F. Lester, "Linewidth study of InAs-InGaAs quantum dot distributed feedback lasers," IEEE Photon. Technol. Lett. 16, 2206-2208 (2004).
[CrossRef]

Sugawara, M.

M. Sugawara, K. Mukai, and H. Shoji, "Effect of phonon bottleneck on quantum-dot laser performance," Appl. Phys. Lett. 71, 2791-2793 (1997).
[CrossRef]

Thedrez, B.

B. Dagens, A. Markus, J. X. Chen, J.-G. Provost, D. Make, O. Le 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]

Toffano, Z.

Z. Toffano, A. Destrez, C. Birocheau, and L. Hassine, "New linewidth enhancement determination method in semiconductor lasers based on spectrum analysis above and below threshold," Electron. Lett. 28, 9-11 (1992).
[CrossRef]

Tulkki, J.

J. Oksanen and J. Tulkki, "Linewidth enhancement factor and chirp in quantum dot lasers," J. Appl. Phys. 94, 1983-1989 (2003).
[CrossRef]

Ukhanov, A. A.

A. A. Ukhanov, A. Stintz, P. G. Eliseev, and K. J. Malloy, "Comparison of the carrier induced refractive index, gain, and linewidth enhancement factor in quantum dot and quantum well lasers," Appl. Phys. Lett. 84, 1058-1060 (2004).
[CrossRef]

Umbach, A.

M. Kuntz, G. Fiol, M. Lämmlin, C. Schubert, A. R. Kovsh, A. Jacob, A. Umbach, and D. Bimberg, "10Gbit/s data modulation using 1.3 μm InGaAs quantum dot lasers," Electron. Lett. 41, 244-245 (2005).
[CrossRef]

Uskov, A. V.

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, 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, 272-274 (2004).
[CrossRef]

A. V. Uskov, Y. Boucher, J. Le Bihan, and J. McInerney, "Theory of a self-assembled quantum-dot semiconductor laser with Auger carrier capture: quantumefficiency and nonlinear gain," Appl. Phys. Lett. 71, 1499-1501 (1998).
[CrossRef]

Ustinov, V. M.

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

Vahala, K.

C. Harder, K. Vahala, and A. Yariv, "Measurement of the linewidth enhancement factor α of semiconductor lasers," Appl. Phys. Lett. 42, 328-330 (1983).
[CrossRef]

Wang, R.

H. Su, L. Zhang, R. Wang, T. C. Newell, A. L. Gray and L. F. Lester, "Linewidth study of InAs-InGaAs quantum dot distributed feedback lasers," IEEE Photon. Technol. Lett. 16, 2206-2208 (2004).
[CrossRef]

Woggon, U.

S. Schneider, P. Borri,W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, "Linewidth enhancement factor in InGaAs quantum-dot lasers," IEEE J. Quantum Electron. 40, 1423-1429 (2004).
[CrossRef]

Wong, H. C.

H. C. Wong, G. B. Ren, J. M. Rorison. "Mode amplification in inhomogeneous QD semiconductor optical amplifiers," Opt. Quantum Electron. (to be published).

Yariv, A.

C. Harder, K. Vahala, and A. Yariv, "Measurement of the linewidth enhancement factor α of semiconductor lasers," Appl. Phys. Lett. 42, 328-330 (1983).
[CrossRef]

Zhang, L.

H. Su, L. Zhang, R. Wang, T. C. Newell, A. L. Gray and L. F. Lester, "Linewidth study of InAs-InGaAs quantum dot distributed feedback lasers," IEEE Photon. Technol. Lett. 16, 2206-2208 (2004).
[CrossRef]

Zhukov, A. E.

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

Appl. Phys. Lett. (7)

Z. Mi, P. Bhattacharya, and S. Fathpour, "High-speed 1.3 μm tunnel injection quantum-dot lasers," Appl. Phys. Lett. 86, 153109 (2005).
[CrossRef]

C. Harder, K. Vahala, and A. Yariv, "Measurement of the linewidth enhancement factor α of semiconductor lasers," Appl. Phys. Lett. 42, 328-330 (1983).
[CrossRef]

R. Jin, D. Boggavarapu, G. Khitrova, H. M. Gibbs, Y. Z. Hu, S. W. Koch, and N. Peyghambarian, "Linewidth broadening factor of a microcavity semiconductor laser," Appl. Phys. Lett. 61, 1883-1885 (1992).
[CrossRef]

A. A. Ukhanov, A. Stintz, P. G. Eliseev, and K. J. Malloy, "Comparison of the carrier induced refractive index, gain, and linewidth enhancement factor in quantum dot and quantum well lasers," Appl. Phys. Lett. 84, 1058-1060 (2004).
[CrossRef]

A. V. Uskov, E. P. O’Reilly, D. McPeake, N. N. Ledentsov, D. Bimberg, 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, 272-274 (2004).
[CrossRef]

M. Sugawara, K. Mukai, and H. Shoji, "Effect of phonon bottleneck on quantum-dot laser performance," Appl. Phys. Lett. 71, 2791-2793 (1997).
[CrossRef]

A. V. Uskov, Y. Boucher, J. Le Bihan, and J. McInerney, "Theory of a self-assembled quantum-dot semiconductor laser with Auger carrier capture: quantumefficiency and nonlinear gain," Appl. Phys. Lett. 71, 1499-1501 (1998).
[CrossRef]

Electron. Lett. (6)

S. P. Hegarty, B. Corbett, J. G. McInerney and G. Huyet, "Free-carrier effect on index change in 1.3 μm quantumdot lasers," Electron. Lett. 41, 416-418 (2005).
[CrossRef]

J. Muszalski, J. Houlihan, G. Huyet, and B. Corbett, "Measurement of linewidth enhancement factor in self assembled quantum dot semiconductor lasers emitting at 1310nm," Electron. Lett. 40, 428-430 (2004).
[CrossRef]

B. Dagens, A. Markus, J. X. Chen, J.-G. Provost, D. Make, O. Le 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]

Z. Toffano, A. Destrez, C. Birocheau, and L. Hassine, "New linewidth enhancement determination method in semiconductor lasers based on spectrum analysis above and below threshold," Electron. Lett. 28, 9-11 (1992).
[CrossRef]

M. Kuntz, G. Fiol, M. Lämmlin, C. Schubert, A. R. Kovsh, A. Jacob, A. Umbach, and D. Bimberg, "10Gbit/s data modulation using 1.3 μm InGaAs quantum dot lasers," Electron. Lett. 41, 244-245 (2005).
[CrossRef]

D. O’Brien, S. P. Hegarty, G. Huyet, J. G. McInerney, T. Kettler, M. Laemmlin, D. Bimberg, V. M. Ustinov, A. E. Zhukov, S. S. Mikhrin, A. R. Kovsh, "Feedback sensitivity of 1.3 μm InAs/GaAs quantum dot lasers," Electron. Lett. 39, 1819-1820 (2003).
[CrossRef]

IEEE J. Quantum Electron. (2)

C. H. Henry, "Theory of the linewidth of semiconductor lasers," IEEE J. Quantum Electron. 18, 259-264 (1982).
[CrossRef]

S. Schneider, P. Borri,W. Langbein, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, "Linewidth enhancement factor in InGaAs quantum-dot lasers," IEEE J. Quantum Electron. 40, 1423-1429 (2004).
[CrossRef]

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

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J.-G. Provost, C. Paranthoën and A. Fiore, "Impact of intraband relaxation on the performance of a quantum-dot laser," IEEE J. Sel. Top. Quantum Electron. 9, 1308-1314 (2003). Q1
[CrossRef]

IEEE Photon. Technol. Lett. (2)

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]

H. Su, L. Zhang, R. Wang, T. C. Newell, A. L. Gray and L. F. Lester, "Linewidth study of InAs-InGaAs quantum dot distributed feedback lasers," IEEE Photon. Technol. Lett. 16, 2206-2208 (2004).
[CrossRef]

J. Appl. Phys. (1)

J. Oksanen and J. Tulkki, "Linewidth enhancement factor and chirp in quantum dot lasers," J. Appl. Phys. 94, 1983-1989 (2003).
[CrossRef]

Opt. Quantum Electron. (1)

H. C. Wong, G. B. Ren, J. M. Rorison. "Mode amplification in inhomogeneous QD semiconductor optical amplifiers," Opt. Quantum Electron. (to be published).

Phys. Rev. B (1)

H. C. Schneider, W. W. Chow and S. W. Koch, "Anomalous carrier-induced dispersion in quantum-dot active media," Phys. Rev. B 66, 041310 (2002).
[CrossRef]

Other (2)

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

K. Petermann, Laser Diode Modulation and Noise (Kluwer, Dordrecht, 1991).

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Figures (2)

Fig. 1.
Fig. 1.

Values of the α-factor calculated by the following techniques: solid line is the ASE method, dashed line is the FM/AM method, dotted line is the linewidth method. (a) Laser operates for non-saturated dots (ρth = 0.6). (b) Laser operates close to full inversion (ρth = 0.9). For the non-saturated operating point (a) the α-factor does not show strong dependence on the injection current. This is not the case for operation close to full inversion (b).

Fig. 2.
Fig. 2.

(a) FM/AM response under small signal current modulation as defined by Eq. (6). The minimum of this function corresponds to αFM /AM as defined in Eq. (7). Dotted line is the value of α calculated just below threshold by the ASE method. Solid line is the FM/AM response just above threshold. Dashed line is the FM/AM response at twice threshold. Here ρth = 0.6. (b) Linewidth due to αlw . Solid line is for non-zero values of αd and αnr . Dashed line is for αnr = 0. Here the current ranges from threshold to 8 times threshold and αth = 0.75.

Equations (8)

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

α = 4 π λ dn dN dg dN ,
N ˙ = γ n N + J q 2 CN ( 1 ρ ) ,
ρ ˙ = γ n N + CN ( 1 ρ ) v g σ ( 2 ρ 1 ) E 2 ,
E ˙ = 1 2 γ s E + 1 2 v g g 0 ( 2 ρ 1 ) ( 1 + i α d ) E + i α nr v g g 0 NE ,
α = α d + α nr δN δρ .
F ( ω ) = 2 S dc δφ δS .
α FM AM F ( ω min ) = α d + α nr γ d C ( 1 ρ th ) 2 ρ th 1 + ρ th J dc J th 2 ρ th 1
α lw = α d + α nr 2 C N ss γ n + 2 C ( 1 ρ ss ) .

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