Abstract

By optically injecting a quantum dash laser and simultaneously producing a significant lowering of the device threshold, a large enhancement in the differential gain is realized. This effect is observed by way of a dramatic reduction in the linewidth enhancement factor and a large increase in the 3-dB modulation bandwidth, especially as the injection wavelength is blue-shifted. Compared to its free-running value, a 50X improvement in the laser’s differential gain is found.

© 2014 Optical Society of America

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  1. G. T. Liu, A. Stintz, H. Li, K. J. Malloy, L. F. Lester, “Extremely low room-temperature threshold current density diode lasers using lnAs dots in In0.15Ga0.85As quantum well,” Electron. Lett. 35(14), 1163–1165 (1999).
    [CrossRef]
  2. O. B. Shchekin, J. Ahn, D. G. Deppe, “High temperature performance of self-organized quantum dot laser with stacked p-doped active region,” Electron. Lett. 38(14), 712–713 (2002).
    [CrossRef]
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  4. M. Asada, Y. Miyamoto, Y. Suematsu, “Gain and the threshold of three-dimensional quantum-box lasers,” IEEE J. Quantum Electron. 22(9), 1915–1921 (1986).
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  5. T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photonics Technol. Lett. 11(12), 1527–1529 (1999).
    [CrossRef]
  6. K. Y. Lau, S. Xin, W. I. Wang, N. Bar-Chaim, M. Mittelstein, “Enhancement of modulation bandwidth in InGaAs strained-layer single quantum well lasers,” Appl. Phys. Lett. 55(12), 1173–1175 (1989).
    [CrossRef]
  7. Y. Arakawa, A. Yariv, “Theory of gain, modulation response, and spectral linewidth in AlGaAs quantum well lasers,” IEEE J. Quantum Electron. 21(10), 1666–1674 (1985).
    [CrossRef]
  8. K. J. Vahala, C. E. Zah, “Effect of doping on the optical gain and the spontaneous noise enhancement factor in quantum well amplifiers and lasers studied by simple analytical expressions,” Appl. Phys. Lett. 52(23), 1945–1947 (1988).
    [CrossRef]
  9. Z. Mi, P. Bhattacharya, “DC and dynamic characteristics of P-doped and tunnel injection 1.65-μm InAs quantum-dash lasers grown on InP (001),” IEEE J. Quantum Electron. 42, 1224–1232 (2006).
    [CrossRef]
  10. A. Martinez, Y. Li, L. F. Lester, A. L. Gray, “Microwave frequency characterization of undoped and p-doped quantum dot lasers,” Appl. Phys. Lett. 90(25), 251101 (2007).
    [CrossRef]
  11. M. Bayer, A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65(4), 041308 (2002).
    [CrossRef]
  12. D. R. Matthews, H. D. Summers, P. M. Smowton, M. Hopkinson, “Experimental investigation of the effect of wetting-layer states on the gain-current characteristic of quantum-dot lasers,” Appl. Phys. Lett. 81(26), 4904–4906 (2002).
    [CrossRef]
  13. P. M. Smowton, E. J. Pearce, H. C. Schneider, W. W. Chow, M. Hopkinson, “Filamentation and linewidth enhancement factor in InGaAs quantum dot lasers,” Appl. Phys. Lett. 81(17), 3251–3253 (2002).
    [CrossRef]
  14. 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(2), 272–274 (2004).
    [CrossRef]
  15. J. Muszalski, J. Houlihan, G. Huyet, B. Corbett, “Measurement of linewidth enhancement factor in self-assembled quantum dot semiconductor lasers emitting at 1310nm,” Electron. Lett. 40(7), 428–430 (2004).
    [CrossRef]
  16. A. Markus, J. X. Chen, O. Gauthier-Lafaye, J. Provost, C. Paranthoen, A. Fiore, “Impact of intraband relaxation on the performance of a quantum-dot laser,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1308–1314 (2003).
    [CrossRef]
  17. T. B. Simpson, J. M. Liu, A. Gavrielides, “Small-signal analysis of modulation characteristics in a semiconductor laser subject to strong optical injection,” IEEE J. Quantum Electron. 32(8), 1456–1468 (1996).
    [CrossRef]
  18. E. K. Lau, H. K. Sung, M. C. Wu, “Frequency response enhancement of optical injection locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008).
    [CrossRef]
  19. B. Riou, N. Trenado, F. Grillot, F. Mallecot, V. Colson, M. F. Martineau, B. Thédrez, L. Silvestre, D. Meichenin, K. Merghem, and A. Ramdane, “High performance strained-layer InGaAsP/InP laser with low linewidth enhancement factor over 30 nm,” in Proceedings of IEEE European Conference on Optical Communication (ECOC) (2003), paper We4.P.85, Rimini, Italy.
  20. T. B. Simpson, J.-M. Liu, M. AlMulla, N. G. Usechak, V. Kovanis, “Tunable photonic microwave oscillator self-locked by polarization- rotated optical feedback,” in Proc. IEEE Int. Freq. Control Symp., May 2012, pp. 1–5.
    [CrossRef]
  21. S.-Z. Chan, “Analysis of an optically injected semiconductor laser for microwave generation,” IEEE J. Quantum Electron. 46(3), 421–428 (2010).
    [CrossRef]
  22. S. Chan, J.-M. Liu, “Tunable narrow-linewidth photonic microwave generation using semiconductor laser dynamics,” IEEE J. Sel. Top. Quantum Electron. 10(5), 1025–1032 (2004).
    [CrossRef]
  23. D. J. Bossert, D. Gallant, “Improved method for gain/index measurements of semiconductor lasers,” Electron. Lett. 32(4), 338–339 (1996).
    [CrossRef]
  24. C. Harder, K. Vahala, A. Yariv, “Measurement of the linewidth enhancement factor α of semiconductor lasers,” Appl. Phys. Lett. 42(4), 328–330 (1983).
    [CrossRef]
  25. N. A. Naderi, M. Pochet, F. Grillot, N. B. Terry, V. Kovanis, L. F. Lester, “Modeling the injection-locked behavior of a quantum dash semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 15(3), 563–571 (2009).
    [CrossRef]
  26. M. Pochet, N. A. Naderi, N. Terry, V. Kovanis, L. F. Lester, “Dynamic behavior of an injection-locked quantum-dash Fabry-Perot laser at zero-detuning,” Opt. Express 17(23), 20623–20630 (2009).
    [CrossRef] [PubMed]
  27. M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22(11), 763–765 (2010).
    [CrossRef]
  28. L. F. Lester, F. Grillot, N. A. Naderi, V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE 8619, 861907 (2013).
    [CrossRef]

2013 (1)

L. F. Lester, F. Grillot, N. A. Naderi, V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE 8619, 861907 (2013).
[CrossRef]

2010 (2)

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22(11), 763–765 (2010).
[CrossRef]

S.-Z. Chan, “Analysis of an optically injected semiconductor laser for microwave generation,” IEEE J. Quantum Electron. 46(3), 421–428 (2010).
[CrossRef]

2009 (2)

N. A. Naderi, M. Pochet, F. Grillot, N. B. Terry, V. Kovanis, L. F. Lester, “Modeling the injection-locked behavior of a quantum dash semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 15(3), 563–571 (2009).
[CrossRef]

M. Pochet, N. A. Naderi, N. Terry, V. Kovanis, L. F. Lester, “Dynamic behavior of an injection-locked quantum-dash Fabry-Perot laser at zero-detuning,” Opt. Express 17(23), 20623–20630 (2009).
[CrossRef] [PubMed]

2008 (1)

E. K. Lau, H. K. Sung, M. C. Wu, “Frequency response enhancement of optical injection locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008).
[CrossRef]

2007 (1)

A. Martinez, Y. Li, L. F. Lester, A. L. Gray, “Microwave frequency characterization of undoped and p-doped quantum dot lasers,” Appl. Phys. Lett. 90(25), 251101 (2007).
[CrossRef]

2006 (1)

Z. Mi, P. Bhattacharya, “DC and dynamic characteristics of P-doped and tunnel injection 1.65-μm InAs quantum-dash lasers grown on InP (001),” IEEE J. Quantum Electron. 42, 1224–1232 (2006).
[CrossRef]

2004 (3)

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(2), 272–274 (2004).
[CrossRef]

J. Muszalski, J. Houlihan, G. Huyet, B. Corbett, “Measurement of linewidth enhancement factor in self-assembled quantum dot semiconductor lasers emitting at 1310nm,” Electron. Lett. 40(7), 428–430 (2004).
[CrossRef]

S. Chan, J.-M. Liu, “Tunable narrow-linewidth photonic microwave generation using semiconductor laser dynamics,” IEEE J. Sel. Top. Quantum Electron. 10(5), 1025–1032 (2004).
[CrossRef]

2003 (1)

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J. Provost, C. Paranthoen, A. Fiore, “Impact of intraband relaxation on the performance of a quantum-dot laser,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1308–1314 (2003).
[CrossRef]

2002 (4)

M. Bayer, A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65(4), 041308 (2002).
[CrossRef]

D. R. Matthews, H. D. Summers, P. M. Smowton, M. Hopkinson, “Experimental investigation of the effect of wetting-layer states on the gain-current characteristic of quantum-dot lasers,” Appl. Phys. Lett. 81(26), 4904–4906 (2002).
[CrossRef]

P. M. Smowton, E. J. Pearce, H. C. Schneider, W. W. Chow, M. Hopkinson, “Filamentation and linewidth enhancement factor in InGaAs quantum dot lasers,” Appl. Phys. Lett. 81(17), 3251–3253 (2002).
[CrossRef]

O. B. Shchekin, J. Ahn, D. G. Deppe, “High temperature performance of self-organized quantum dot laser with stacked p-doped active region,” Electron. Lett. 38(14), 712–713 (2002).
[CrossRef]

1999 (2)

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photonics Technol. Lett. 11(12), 1527–1529 (1999).
[CrossRef]

G. T. Liu, A. Stintz, H. Li, K. J. Malloy, L. F. Lester, “Extremely low room-temperature threshold current density diode lasers using lnAs dots in In0.15Ga0.85As quantum well,” Electron. Lett. 35(14), 1163–1165 (1999).
[CrossRef]

1996 (2)

T. B. Simpson, J. M. Liu, A. Gavrielides, “Small-signal analysis of modulation characteristics in a semiconductor laser subject to strong optical injection,” IEEE J. Quantum Electron. 32(8), 1456–1468 (1996).
[CrossRef]

D. J. Bossert, D. Gallant, “Improved method for gain/index measurements of semiconductor lasers,” Electron. Lett. 32(4), 338–339 (1996).
[CrossRef]

1989 (1)

K. Y. Lau, S. Xin, W. I. Wang, N. Bar-Chaim, M. Mittelstein, “Enhancement of modulation bandwidth in InGaAs strained-layer single quantum well lasers,” Appl. Phys. Lett. 55(12), 1173–1175 (1989).
[CrossRef]

1988 (1)

K. J. Vahala, C. E. Zah, “Effect of doping on the optical gain and the spontaneous noise enhancement factor in quantum well amplifiers and lasers studied by simple analytical expressions,” Appl. Phys. Lett. 52(23), 1945–1947 (1988).
[CrossRef]

1986 (1)

M. Asada, Y. Miyamoto, Y. Suematsu, “Gain and the threshold of three-dimensional quantum-box lasers,” IEEE J. Quantum Electron. 22(9), 1915–1921 (1986).
[CrossRef]

1985 (1)

Y. Arakawa, A. Yariv, “Theory of gain, modulation response, and spectral linewidth in AlGaAs quantum well lasers,” IEEE J. Quantum Electron. 21(10), 1666–1674 (1985).
[CrossRef]

1983 (1)

C. Harder, K. Vahala, A. Yariv, “Measurement of the linewidth enhancement factor α of semiconductor lasers,” Appl. Phys. Lett. 42(4), 328–330 (1983).
[CrossRef]

Ahn, J.

O. B. Shchekin, J. Ahn, D. G. Deppe, “High temperature performance of self-organized quantum dot laser with stacked p-doped active region,” Electron. Lett. 38(14), 712–713 (2002).
[CrossRef]

AlMulla, M.

T. B. Simpson, J.-M. Liu, M. AlMulla, N. G. Usechak, V. Kovanis, “Tunable photonic microwave oscillator self-locked by polarization- rotated optical feedback,” in Proc. IEEE Int. Freq. Control Symp., May 2012, pp. 1–5.
[CrossRef]

Arakawa, Y.

Y. Arakawa, A. Yariv, “Theory of gain, modulation response, and spectral linewidth in AlGaAs quantum well lasers,” IEEE J. Quantum Electron. 21(10), 1666–1674 (1985).
[CrossRef]

Asada, M.

M. Asada, Y. Miyamoto, Y. Suematsu, “Gain and the threshold of three-dimensional quantum-box lasers,” IEEE J. Quantum Electron. 22(9), 1915–1921 (1986).
[CrossRef]

Bar-Chaim, N.

K. Y. Lau, S. Xin, W. I. Wang, N. Bar-Chaim, M. Mittelstein, “Enhancement of modulation bandwidth in InGaAs strained-layer single quantum well lasers,” Appl. Phys. Lett. 55(12), 1173–1175 (1989).
[CrossRef]

Bayer, M.

M. Bayer, A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65(4), 041308 (2002).
[CrossRef]

Bhattacharya, P.

Z. Mi, P. Bhattacharya, “DC and dynamic characteristics of P-doped and tunnel injection 1.65-μm InAs quantum-dash lasers grown on InP (001),” IEEE J. Quantum Electron. 42, 1224–1232 (2006).
[CrossRef]

Bimberg, 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(2), 272–274 (2004).
[CrossRef]

Bossert, D. J.

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photonics Technol. Lett. 11(12), 1527–1529 (1999).
[CrossRef]

D. J. Bossert, D. Gallant, “Improved method for gain/index measurements of semiconductor lasers,” Electron. Lett. 32(4), 338–339 (1996).
[CrossRef]

Chan, S.

S. Chan, J.-M. Liu, “Tunable narrow-linewidth photonic microwave generation using semiconductor laser dynamics,” IEEE J. Sel. Top. Quantum Electron. 10(5), 1025–1032 (2004).
[CrossRef]

Chan, S.-Z.

S.-Z. Chan, “Analysis of an optically injected semiconductor laser for microwave generation,” IEEE J. Quantum Electron. 46(3), 421–428 (2010).
[CrossRef]

Chen, J. X.

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J. Provost, C. Paranthoen, A. Fiore, “Impact of intraband relaxation on the performance of a quantum-dot laser,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1308–1314 (2003).
[CrossRef]

Chow, W. W.

P. M. Smowton, E. J. Pearce, H. C. Schneider, W. W. Chow, M. Hopkinson, “Filamentation and linewidth enhancement factor in InGaAs quantum dot lasers,” Appl. Phys. Lett. 81(17), 3251–3253 (2002).
[CrossRef]

Corbett, B.

J. Muszalski, J. Houlihan, G. Huyet, B. Corbett, “Measurement of linewidth enhancement factor in self-assembled quantum dot semiconductor lasers emitting at 1310nm,” Electron. Lett. 40(7), 428–430 (2004).
[CrossRef]

Deppe, D. G.

O. B. Shchekin, J. Ahn, D. G. Deppe, “High temperature performance of self-organized quantum dot laser with stacked p-doped active region,” Electron. Lett. 38(14), 712–713 (2002).
[CrossRef]

Fiore, A.

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J. Provost, C. Paranthoen, A. Fiore, “Impact of intraband relaxation on the performance of a quantum-dot laser,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1308–1314 (2003).
[CrossRef]

Forchel, A.

M. Bayer, A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65(4), 041308 (2002).
[CrossRef]

Fuchs, B.

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photonics Technol. Lett. 11(12), 1527–1529 (1999).
[CrossRef]

Gallant, D.

D. J. Bossert, D. Gallant, “Improved method for gain/index measurements of semiconductor lasers,” Electron. Lett. 32(4), 338–339 (1996).
[CrossRef]

Gauthier-Lafaye, O.

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J. Provost, C. Paranthoen, A. Fiore, “Impact of intraband relaxation on the performance of a quantum-dot laser,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1308–1314 (2003).
[CrossRef]

Gavrielides, A.

T. B. Simpson, J. M. Liu, A. Gavrielides, “Small-signal analysis of modulation characteristics in a semiconductor laser subject to strong optical injection,” IEEE J. Quantum Electron. 32(8), 1456–1468 (1996).
[CrossRef]

Gray, A. L.

A. Martinez, Y. Li, L. F. Lester, A. L. Gray, “Microwave frequency characterization of undoped and p-doped quantum dot lasers,” Appl. Phys. Lett. 90(25), 251101 (2007).
[CrossRef]

Grillot, F.

L. F. Lester, F. Grillot, N. A. Naderi, V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE 8619, 861907 (2013).
[CrossRef]

N. A. Naderi, M. Pochet, F. Grillot, N. B. Terry, V. Kovanis, L. F. Lester, “Modeling the injection-locked behavior of a quantum dash semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 15(3), 563–571 (2009).
[CrossRef]

Harder, C.

C. Harder, K. Vahala, A. Yariv, “Measurement of the linewidth enhancement factor α of semiconductor lasers,” Appl. Phys. Lett. 42(4), 328–330 (1983).
[CrossRef]

Hopkinson, M.

D. R. Matthews, H. D. Summers, P. M. Smowton, M. Hopkinson, “Experimental investigation of the effect of wetting-layer states on the gain-current characteristic of quantum-dot lasers,” Appl. Phys. Lett. 81(26), 4904–4906 (2002).
[CrossRef]

P. M. Smowton, E. J. Pearce, H. C. Schneider, W. W. Chow, M. Hopkinson, “Filamentation and linewidth enhancement factor in InGaAs quantum dot lasers,” Appl. Phys. Lett. 81(17), 3251–3253 (2002).
[CrossRef]

Houlihan, J.

J. Muszalski, J. Houlihan, G. Huyet, B. Corbett, “Measurement of linewidth enhancement factor in self-assembled quantum dot semiconductor lasers emitting at 1310nm,” Electron. Lett. 40(7), 428–430 (2004).
[CrossRef]

Huyet, G.

J. Muszalski, J. Houlihan, G. Huyet, B. Corbett, “Measurement of linewidth enhancement factor in self-assembled quantum dot semiconductor lasers emitting at 1310nm,” Electron. Lett. 40(7), 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(2), 272–274 (2004).
[CrossRef]

Kovanis, V.

L. F. Lester, F. Grillot, N. A. Naderi, V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE 8619, 861907 (2013).
[CrossRef]

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22(11), 763–765 (2010).
[CrossRef]

M. Pochet, N. A. Naderi, N. Terry, V. Kovanis, L. F. Lester, “Dynamic behavior of an injection-locked quantum-dash Fabry-Perot laser at zero-detuning,” Opt. Express 17(23), 20623–20630 (2009).
[CrossRef] [PubMed]

N. A. Naderi, M. Pochet, F. Grillot, N. B. Terry, V. Kovanis, L. F. Lester, “Modeling the injection-locked behavior of a quantum dash semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 15(3), 563–571 (2009).
[CrossRef]

T. B. Simpson, J.-M. Liu, M. AlMulla, N. G. Usechak, V. Kovanis, “Tunable photonic microwave oscillator self-locked by polarization- rotated optical feedback,” in Proc. IEEE Int. Freq. Control Symp., May 2012, pp. 1–5.
[CrossRef]

Lau, E. K.

E. K. Lau, H. K. Sung, M. C. Wu, “Frequency response enhancement of optical injection locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008).
[CrossRef]

Lau, K. Y.

K. Y. Lau, S. Xin, W. I. Wang, N. Bar-Chaim, M. Mittelstein, “Enhancement of modulation bandwidth in InGaAs strained-layer single quantum well lasers,” Appl. Phys. Lett. 55(12), 1173–1175 (1989).
[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(2), 272–274 (2004).
[CrossRef]

Lester, L. F.

L. F. Lester, F. Grillot, N. A. Naderi, V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE 8619, 861907 (2013).
[CrossRef]

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22(11), 763–765 (2010).
[CrossRef]

M. Pochet, N. A. Naderi, N. Terry, V. Kovanis, L. F. Lester, “Dynamic behavior of an injection-locked quantum-dash Fabry-Perot laser at zero-detuning,” Opt. Express 17(23), 20623–20630 (2009).
[CrossRef] [PubMed]

N. A. Naderi, M. Pochet, F. Grillot, N. B. Terry, V. Kovanis, L. F. Lester, “Modeling the injection-locked behavior of a quantum dash semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 15(3), 563–571 (2009).
[CrossRef]

A. Martinez, Y. Li, L. F. Lester, A. L. Gray, “Microwave frequency characterization of undoped and p-doped quantum dot lasers,” Appl. Phys. Lett. 90(25), 251101 (2007).
[CrossRef]

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photonics Technol. Lett. 11(12), 1527–1529 (1999).
[CrossRef]

G. T. Liu, A. Stintz, H. Li, K. J. Malloy, L. F. Lester, “Extremely low room-temperature threshold current density diode lasers using lnAs dots in In0.15Ga0.85As quantum well,” Electron. Lett. 35(14), 1163–1165 (1999).
[CrossRef]

Li, H.

G. T. Liu, A. Stintz, H. Li, K. J. Malloy, L. F. Lester, “Extremely low room-temperature threshold current density diode lasers using lnAs dots in In0.15Ga0.85As quantum well,” Electron. Lett. 35(14), 1163–1165 (1999).
[CrossRef]

Li, Y.

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22(11), 763–765 (2010).
[CrossRef]

A. Martinez, Y. Li, L. F. Lester, A. L. Gray, “Microwave frequency characterization of undoped and p-doped quantum dot lasers,” Appl. Phys. Lett. 90(25), 251101 (2007).
[CrossRef]

Liu, G. T.

G. T. Liu, A. Stintz, H. Li, K. J. Malloy, L. F. Lester, “Extremely low room-temperature threshold current density diode lasers using lnAs dots in In0.15Ga0.85As quantum well,” Electron. Lett. 35(14), 1163–1165 (1999).
[CrossRef]

Liu, J. M.

T. B. Simpson, J. M. Liu, A. Gavrielides, “Small-signal analysis of modulation characteristics in a semiconductor laser subject to strong optical injection,” IEEE J. Quantum Electron. 32(8), 1456–1468 (1996).
[CrossRef]

Liu, J.-M.

S. Chan, J.-M. Liu, “Tunable narrow-linewidth photonic microwave generation using semiconductor laser dynamics,” IEEE J. Sel. Top. Quantum Electron. 10(5), 1025–1032 (2004).
[CrossRef]

T. B. Simpson, J.-M. Liu, M. AlMulla, N. G. Usechak, V. Kovanis, “Tunable photonic microwave oscillator self-locked by polarization- rotated optical feedback,” in Proc. IEEE Int. Freq. Control Symp., May 2012, pp. 1–5.
[CrossRef]

Malloy, K. J.

G. T. Liu, A. Stintz, H. Li, K. J. Malloy, L. F. Lester, “Extremely low room-temperature threshold current density diode lasers using lnAs dots in In0.15Ga0.85As quantum well,” Electron. Lett. 35(14), 1163–1165 (1999).
[CrossRef]

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photonics Technol. Lett. 11(12), 1527–1529 (1999).
[CrossRef]

Markus, A.

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J. Provost, C. Paranthoen, A. Fiore, “Impact of intraband relaxation on the performance of a quantum-dot laser,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1308–1314 (2003).
[CrossRef]

Martinez, A.

A. Martinez, Y. Li, L. F. Lester, A. L. Gray, “Microwave frequency characterization of undoped and p-doped quantum dot lasers,” Appl. Phys. Lett. 90(25), 251101 (2007).
[CrossRef]

Matthews, D. R.

D. R. Matthews, H. D. Summers, P. M. Smowton, M. Hopkinson, “Experimental investigation of the effect of wetting-layer states on the gain-current characteristic of quantum-dot lasers,” Appl. Phys. Lett. 81(26), 4904–4906 (2002).
[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(2), 272–274 (2004).
[CrossRef]

Mi, Z.

Z. Mi, P. Bhattacharya, “DC and dynamic characteristics of P-doped and tunnel injection 1.65-μm InAs quantum-dash lasers grown on InP (001),” IEEE J. Quantum Electron. 42, 1224–1232 (2006).
[CrossRef]

Mittelstein, M.

K. Y. Lau, S. Xin, W. I. Wang, N. Bar-Chaim, M. Mittelstein, “Enhancement of modulation bandwidth in InGaAs strained-layer single quantum well lasers,” Appl. Phys. Lett. 55(12), 1173–1175 (1989).
[CrossRef]

Miyamoto, Y.

M. Asada, Y. Miyamoto, Y. Suematsu, “Gain and the threshold of three-dimensional quantum-box lasers,” IEEE J. Quantum Electron. 22(9), 1915–1921 (1986).
[CrossRef]

Muszalski, J.

J. Muszalski, J. Houlihan, G. Huyet, B. Corbett, “Measurement of linewidth enhancement factor in self-assembled quantum dot semiconductor lasers emitting at 1310nm,” Electron. Lett. 40(7), 428–430 (2004).
[CrossRef]

Naderi, N. A.

L. F. Lester, F. Grillot, N. A. Naderi, V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE 8619, 861907 (2013).
[CrossRef]

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22(11), 763–765 (2010).
[CrossRef]

M. Pochet, N. A. Naderi, N. Terry, V. Kovanis, L. F. Lester, “Dynamic behavior of an injection-locked quantum-dash Fabry-Perot laser at zero-detuning,” Opt. Express 17(23), 20623–20630 (2009).
[CrossRef] [PubMed]

N. A. Naderi, M. Pochet, F. Grillot, N. B. Terry, V. Kovanis, L. F. Lester, “Modeling the injection-locked behavior of a quantum dash semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 15(3), 563–571 (2009).
[CrossRef]

Newell, T. C.

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photonics Technol. Lett. 11(12), 1527–1529 (1999).
[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(2), 272–274 (2004).
[CrossRef]

Paranthoen, C.

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J. Provost, C. Paranthoen, A. Fiore, “Impact of intraband relaxation on the performance of a quantum-dot laser,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1308–1314 (2003).
[CrossRef]

Pearce, E. J.

P. M. Smowton, E. J. Pearce, H. C. Schneider, W. W. Chow, M. Hopkinson, “Filamentation and linewidth enhancement factor in InGaAs quantum dot lasers,” Appl. Phys. Lett. 81(17), 3251–3253 (2002).
[CrossRef]

Pochet, M.

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22(11), 763–765 (2010).
[CrossRef]

N. A. Naderi, M. Pochet, F. Grillot, N. B. Terry, V. Kovanis, L. F. Lester, “Modeling the injection-locked behavior of a quantum dash semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 15(3), 563–571 (2009).
[CrossRef]

M. Pochet, N. A. Naderi, N. Terry, V. Kovanis, L. F. Lester, “Dynamic behavior of an injection-locked quantum-dash Fabry-Perot laser at zero-detuning,” Opt. Express 17(23), 20623–20630 (2009).
[CrossRef] [PubMed]

Provost, J.

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J. Provost, C. Paranthoen, A. Fiore, “Impact of intraband relaxation on the performance of a quantum-dot laser,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1308–1314 (2003).
[CrossRef]

Schneider, H. C.

P. M. Smowton, E. J. Pearce, H. C. Schneider, W. W. Chow, M. Hopkinson, “Filamentation and linewidth enhancement factor in InGaAs quantum dot lasers,” Appl. Phys. Lett. 81(17), 3251–3253 (2002).
[CrossRef]

Shchekin, O. B.

O. B. Shchekin, J. Ahn, D. G. Deppe, “High temperature performance of self-organized quantum dot laser with stacked p-doped active region,” Electron. Lett. 38(14), 712–713 (2002).
[CrossRef]

Simpson, T. B.

T. B. Simpson, J. M. Liu, A. Gavrielides, “Small-signal analysis of modulation characteristics in a semiconductor laser subject to strong optical injection,” IEEE J. Quantum Electron. 32(8), 1456–1468 (1996).
[CrossRef]

T. B. Simpson, J.-M. Liu, M. AlMulla, N. G. Usechak, V. Kovanis, “Tunable photonic microwave oscillator self-locked by polarization- rotated optical feedback,” in Proc. IEEE Int. Freq. Control Symp., May 2012, pp. 1–5.
[CrossRef]

Smowton, P. M.

P. M. Smowton, E. J. Pearce, H. C. Schneider, W. W. Chow, M. Hopkinson, “Filamentation and linewidth enhancement factor in InGaAs quantum dot lasers,” Appl. Phys. Lett. 81(17), 3251–3253 (2002).
[CrossRef]

D. R. Matthews, H. D. Summers, P. M. Smowton, M. Hopkinson, “Experimental investigation of the effect of wetting-layer states on the gain-current characteristic of quantum-dot lasers,” Appl. Phys. Lett. 81(26), 4904–4906 (2002).
[CrossRef]

Stintz, A.

G. T. Liu, A. Stintz, H. Li, K. J. Malloy, L. F. Lester, “Extremely low room-temperature threshold current density diode lasers using lnAs dots in In0.15Ga0.85As quantum well,” Electron. Lett. 35(14), 1163–1165 (1999).
[CrossRef]

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photonics Technol. Lett. 11(12), 1527–1529 (1999).
[CrossRef]

Suematsu, Y.

M. Asada, Y. Miyamoto, Y. Suematsu, “Gain and the threshold of three-dimensional quantum-box lasers,” IEEE J. Quantum Electron. 22(9), 1915–1921 (1986).
[CrossRef]

Summers, H. D.

D. R. Matthews, H. D. Summers, P. M. Smowton, M. Hopkinson, “Experimental investigation of the effect of wetting-layer states on the gain-current characteristic of quantum-dot lasers,” Appl. Phys. Lett. 81(26), 4904–4906 (2002).
[CrossRef]

Sung, H. K.

E. K. Lau, H. K. Sung, M. C. Wu, “Frequency response enhancement of optical injection locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008).
[CrossRef]

Terry, N.

Terry, N. B.

N. A. Naderi, M. Pochet, F. Grillot, N. B. Terry, V. Kovanis, L. F. Lester, “Modeling the injection-locked behavior of a quantum dash semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 15(3), 563–571 (2009).
[CrossRef]

Usechak, N. G.

T. B. Simpson, J.-M. Liu, M. AlMulla, N. G. Usechak, V. Kovanis, “Tunable photonic microwave oscillator self-locked by polarization- rotated optical feedback,” in Proc. IEEE Int. Freq. Control Symp., May 2012, pp. 1–5.
[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(2), 272–274 (2004).
[CrossRef]

Vahala, K.

C. Harder, K. Vahala, A. Yariv, “Measurement of the linewidth enhancement factor α of semiconductor lasers,” Appl. Phys. Lett. 42(4), 328–330 (1983).
[CrossRef]

Vahala, K. J.

K. J. Vahala, C. E. Zah, “Effect of doping on the optical gain and the spontaneous noise enhancement factor in quantum well amplifiers and lasers studied by simple analytical expressions,” Appl. Phys. Lett. 52(23), 1945–1947 (1988).
[CrossRef]

Wang, W. I.

K. Y. Lau, S. Xin, W. I. Wang, N. Bar-Chaim, M. Mittelstein, “Enhancement of modulation bandwidth in InGaAs strained-layer single quantum well lasers,” Appl. Phys. Lett. 55(12), 1173–1175 (1989).
[CrossRef]

Wu, M. C.

E. K. Lau, H. K. Sung, M. C. Wu, “Frequency response enhancement of optical injection locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008).
[CrossRef]

Xin, S.

K. Y. Lau, S. Xin, W. I. Wang, N. Bar-Chaim, M. Mittelstein, “Enhancement of modulation bandwidth in InGaAs strained-layer single quantum well lasers,” Appl. Phys. Lett. 55(12), 1173–1175 (1989).
[CrossRef]

Yariv, A.

Y. Arakawa, A. Yariv, “Theory of gain, modulation response, and spectral linewidth in AlGaAs quantum well lasers,” IEEE J. Quantum Electron. 21(10), 1666–1674 (1985).
[CrossRef]

C. Harder, K. Vahala, A. Yariv, “Measurement of the linewidth enhancement factor α of semiconductor lasers,” Appl. Phys. Lett. 42(4), 328–330 (1983).
[CrossRef]

Zah, C. E.

K. J. Vahala, C. E. Zah, “Effect of doping on the optical gain and the spontaneous noise enhancement factor in quantum well amplifiers and lasers studied by simple analytical expressions,” Appl. Phys. Lett. 52(23), 1945–1947 (1988).
[CrossRef]

Appl. Phys. Lett. (7)

K. Y. Lau, S. Xin, W. I. Wang, N. Bar-Chaim, M. Mittelstein, “Enhancement of modulation bandwidth in InGaAs strained-layer single quantum well lasers,” Appl. Phys. Lett. 55(12), 1173–1175 (1989).
[CrossRef]

K. J. Vahala, C. E. Zah, “Effect of doping on the optical gain and the spontaneous noise enhancement factor in quantum well amplifiers and lasers studied by simple analytical expressions,” Appl. Phys. Lett. 52(23), 1945–1947 (1988).
[CrossRef]

D. R. Matthews, H. D. Summers, P. M. Smowton, M. Hopkinson, “Experimental investigation of the effect of wetting-layer states on the gain-current characteristic of quantum-dot lasers,” Appl. Phys. Lett. 81(26), 4904–4906 (2002).
[CrossRef]

P. M. Smowton, E. J. Pearce, H. C. Schneider, W. W. Chow, M. Hopkinson, “Filamentation and linewidth enhancement factor in InGaAs quantum dot lasers,” Appl. Phys. Lett. 81(17), 3251–3253 (2002).
[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(2), 272–274 (2004).
[CrossRef]

A. Martinez, Y. Li, L. F. Lester, A. L. Gray, “Microwave frequency characterization of undoped and p-doped quantum dot lasers,” Appl. Phys. Lett. 90(25), 251101 (2007).
[CrossRef]

C. Harder, K. Vahala, A. Yariv, “Measurement of the linewidth enhancement factor α of semiconductor lasers,” Appl. Phys. Lett. 42(4), 328–330 (1983).
[CrossRef]

Electron. Lett. (4)

D. J. Bossert, D. Gallant, “Improved method for gain/index measurements of semiconductor lasers,” Electron. Lett. 32(4), 338–339 (1996).
[CrossRef]

J. Muszalski, J. Houlihan, G. Huyet, B. Corbett, “Measurement of linewidth enhancement factor in self-assembled quantum dot semiconductor lasers emitting at 1310nm,” Electron. Lett. 40(7), 428–430 (2004).
[CrossRef]

G. T. Liu, A. Stintz, H. Li, K. J. Malloy, L. F. Lester, “Extremely low room-temperature threshold current density diode lasers using lnAs dots in In0.15Ga0.85As quantum well,” Electron. Lett. 35(14), 1163–1165 (1999).
[CrossRef]

O. B. Shchekin, J. Ahn, D. G. Deppe, “High temperature performance of self-organized quantum dot laser with stacked p-doped active region,” Electron. Lett. 38(14), 712–713 (2002).
[CrossRef]

IEEE J. Quantum Electron. (6)

M. Asada, Y. Miyamoto, Y. Suematsu, “Gain and the threshold of three-dimensional quantum-box lasers,” IEEE J. Quantum Electron. 22(9), 1915–1921 (1986).
[CrossRef]

Z. Mi, P. Bhattacharya, “DC and dynamic characteristics of P-doped and tunnel injection 1.65-μm InAs quantum-dash lasers grown on InP (001),” IEEE J. Quantum Electron. 42, 1224–1232 (2006).
[CrossRef]

Y. Arakawa, A. Yariv, “Theory of gain, modulation response, and spectral linewidth in AlGaAs quantum well lasers,” IEEE J. Quantum Electron. 21(10), 1666–1674 (1985).
[CrossRef]

T. B. Simpson, J. M. Liu, A. Gavrielides, “Small-signal analysis of modulation characteristics in a semiconductor laser subject to strong optical injection,” IEEE J. Quantum Electron. 32(8), 1456–1468 (1996).
[CrossRef]

E. K. Lau, H. K. Sung, M. C. Wu, “Frequency response enhancement of optical injection locked lasers,” IEEE J. Quantum Electron. 44(1), 90–99 (2008).
[CrossRef]

S.-Z. Chan, “Analysis of an optically injected semiconductor laser for microwave generation,” IEEE J. Quantum Electron. 46(3), 421–428 (2010).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (3)

S. Chan, J.-M. Liu, “Tunable narrow-linewidth photonic microwave generation using semiconductor laser dynamics,” IEEE J. Sel. Top. Quantum Electron. 10(5), 1025–1032 (2004).
[CrossRef]

N. A. Naderi, M. Pochet, F. Grillot, N. B. Terry, V. Kovanis, L. F. Lester, “Modeling the injection-locked behavior of a quantum dash semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 15(3), 563–571 (2009).
[CrossRef]

A. Markus, J. X. Chen, O. Gauthier-Lafaye, J. Provost, C. Paranthoen, A. Fiore, “Impact of intraband relaxation on the performance of a quantum-dot laser,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1308–1314 (2003).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

T. C. Newell, D. J. Bossert, A. Stintz, B. Fuchs, K. J. Malloy, L. F. Lester, “Gain and linewidth enhancement factor in InAs quantum-dot laser diodes,” IEEE Photonics Technol. Lett. 11(12), 1527–1529 (1999).
[CrossRef]

M. Pochet, N. A. Naderi, Y. Li, V. Kovanis, L. F. Lester, “Tunable photonic oscillators using optically injected quantum-dash diode lasers,” IEEE Photonics Technol. Lett. 22(11), 763–765 (2010).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (1)

M. Bayer, A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65(4), 041308 (2002).
[CrossRef]

Proc. SPIE (1)

L. F. Lester, F. Grillot, N. A. Naderi, V. Kovanis, “Differential gain enhancement in a quantum dash laser using strong optical injection,” Proc. SPIE 8619, 861907 (2013).
[CrossRef]

Other (3)

M. T. Crowley, N. A. Naderi, H. Su, F. Grillot, and L. F. Lester, Semiconductors and Semimetals: Advances in Semiconductor Lasers (Academic, 2012), vol. 86, Chap. 10.

B. Riou, N. Trenado, F. Grillot, F. Mallecot, V. Colson, M. F. Martineau, B. Thédrez, L. Silvestre, D. Meichenin, K. Merghem, and A. Ramdane, “High performance strained-layer InGaAsP/InP laser with low linewidth enhancement factor over 30 nm,” in Proceedings of IEEE European Conference on Optical Communication (ECOC) (2003), paper We4.P.85, Rimini, Italy.

T. B. Simpson, J.-M. Liu, M. AlMulla, N. G. Usechak, V. Kovanis, “Tunable photonic microwave oscillator self-locked by polarization- rotated optical feedback,” in Proc. IEEE Int. Freq. Control Symp., May 2012, pp. 1–5.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

The measured net modal gain curves versus wavelength for a set of seven current densities at or below the threshold value. Qualitatively, one can see that the differential gain increases for shorter wavelengths. Injection locking (IL) on the nearest Fabry-Perot mode was undertaken at 1535, 1550 and 1580 nm.

Fig. 2
Fig. 2

Small-signal modulation response of the slave diode laser at zero optical detuning for three injection wavelengths. The data for the free-running laser is also included. Notice that at 1535 nm the modulation response is flat, spanning a 16.5 GHz bandwidth.

Fig. 3
Fig. 3

a) linewidth enhancement factor and b) threshold gain shift as a function of external power injection ratio, Rext for the QDash slave laser at 1535, 1550 and 1580 nm. The solid lines in (a) correspond to the FM/AM data, and the dotted lines correspond to values extracted from the modulation responses. The error analyses for the extracted values of the LEF and threshold gain shift shown in (b) are calculated based on a one standard deviation confidence interval.

Tables (1)

Tables Icon

Table 1 Differential Gains of Quantum Dash Laser Comparing the Free-running Case to Strong Injection Cases at 1550 and 1535 nm for which the LEF is Essentially Zero

Equations (6)

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| H R | 2 =10log[ ( C Z ) 2 ( ω 2 + Z 2 ) ( CA ω 2 ) 2 + ( Bω ω 3 ) 2 1 ( 1+ ( ω/ γ c ) 2 ) ]
A=( γ fr γ N ) R FE 2 + γ N + γ th
B=( ω r 2 + γ N γ th )+ ( η 0 / R FE ) 2 + γ th [ ( γ fr γ N ) R FE 2 + γ N ]
C= ( η 0 / R FE ) 2 [ ( γ fr γ N ) R FE 2 + γ N ]( ω r 2 + γ N γ th )Z
Z= η 0 R FE 1+ α 2 γ th = 2 η 0 R FE 1+ α 2
R FE 4 [ ( 1/ γ p )( 2 η 0 / ( 1+ α 2 ) 1 2 )+1 ] R FE 2 ( γ N / ω r 2 )( 2 η 0 / ( 1+ α 2 ) 1 2 )=0

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