Abstract

Microscopic density matrix analysis on the linewidth enhancement factor (LEF) of both mid-infrared (mid-IR) and Terahertz (THz) quantum cascade lasers (QCLs) is reported, taking into account of the many body Coulomb interactions, coherence of resonant-tunneling transport and non-parabolicity. A non-zero LEF at the gain peak is obtained due to these combined microscopic effects. The results show that, for mid-IR QCLs, the many body Coulomb interaction and non-parabolicity contribute greatly to the non-zero LEF. In contrast, for THz QCLs, the many body Coulomb interactions and the resonant-tunneling effects greatly influence the LEF resulting in a non-zero value at the gain peak. This microscopic model not only partially explains the non-zero LEF of QCLs at the gain peak, which observed in the experiments for a while but cannot be explicitly explained, but also can be employed to improve the active region designs so as to reduce the LEF by optimizing the corresponding parameters.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
    [CrossRef] [PubMed]
  2. J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264(5158), 553–556 (1994).
    [CrossRef] [PubMed]
  3. C. H. Henry, “Theory of the linewidth of semiconductor-lasers,” IEEE J. Quantum Electron.18(2), 259–264 (1982).
    [CrossRef]
  4. T. Chattopadhyay and P. Bhattacharyya, “Role of linewidth enhancement factor on the frequency response of the synchronized quantum cascade laser,” Opt. Commun.309, 349–354 (2013).
    [CrossRef]
  5. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor-lasers - an overview,” IEEE J. Quantum Electron.23(1), 9–29 (1987).
    [CrossRef]
  6. M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys.94(8), 5426–5428 (2003).
    [CrossRef]
  7. R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett.92(7), 071106 (2008).
    [CrossRef]
  8. T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006).
    [CrossRef]
  9. J. von Staden, T. Gensty, W. Elsässer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett.31(17), 2574–2576 (2006).
    [CrossRef] [PubMed]
  10. N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett.92(12), 121104 (2008).
    [CrossRef]
  11. J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron.40(12), 1663–1674 (2004).
    [CrossRef]
  12. T. Liu, K. E. Lee, and Q. J. Wang, “Microscopic density matrix model for optical gain of terahertz quantum cascade lasers: Many-body, nonparabolicity, and resonant tunneling effects,” Phys. Rev. B86(23), 235306 (2012).
    [CrossRef]
  13. C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998).
    [CrossRef]
  14. W. W. Chow, S. W. Koch, and M. S. I. I. I. Semiconductor-Laser Physics, (Springer-Verlag, Berlin, 1994).
  15. S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B80(24), 245316 (2009).
    [CrossRef]
  16. C. Gmachl, F. Capasso, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho, “High-power, continuous-wave, current-tunable, single-mode quantum-cascade distributed-feedback lasers at lambda ~5.2 and lambda ~7.95 μm,” Opt. Lett.25(4), 230–232 (2000).
    [CrossRef] [PubMed]
  17. S. Fathololoumi, E. Dupont, C. W. I. Chan, Z. R. Wasilewski, S. R. Laframboise, D. Ban, A. Mátyás, C. Jirauschek, Q. Hu, and H. C. Liu, “Terahertz quantum cascade lasers operating up to ~200 K with optimized oscillator strength and improved injection tunneling,” Opt. Express20(4), 3866–3876 (2012).
    [CrossRef] [PubMed]
  18. U. Ekenberg, “Nonparabolicity effects in a quantum well: sublevel shift, parallel mass, and Landau levels,” Phys. Rev. B Condens. Matter40(11), 7714–7726 (1989).
    [CrossRef] [PubMed]
  19. S. Panda, B. K. Panda, and S. Fung, “Effect of conduction band nonparabolicity on the dark current in a quantum well infrared detector,” J. Appl. Phys.101(4), 043705 (2007).
    [CrossRef]
  20. E. Dupont, S. Fathololoumi, and H. C. Liu, “Simplified density-matrix model applied to three-well terahertz quantum cascade lasers,” Phys. Rev. B81(20), 205311 (2010).
    [CrossRef]

2013

T. Chattopadhyay and P. Bhattacharyya, “Role of linewidth enhancement factor on the frequency response of the synchronized quantum cascade laser,” Opt. Commun.309, 349–354 (2013).
[CrossRef]

2012

2010

E. Dupont, S. Fathololoumi, and H. C. Liu, “Simplified density-matrix model applied to three-well terahertz quantum cascade lasers,” Phys. Rev. B81(20), 205311 (2010).
[CrossRef]

2009

S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B80(24), 245316 (2009).
[CrossRef]

2008

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett.92(7), 071106 (2008).
[CrossRef]

N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett.92(12), 121104 (2008).
[CrossRef]

2007

S. Panda, B. K. Panda, and S. Fung, “Effect of conduction band nonparabolicity on the dark current in a quantum well infrared detector,” J. Appl. Phys.101(4), 043705 (2007).
[CrossRef]

2006

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006).
[CrossRef]

J. von Staden, T. Gensty, W. Elsässer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett.31(17), 2574–2576 (2006).
[CrossRef] [PubMed]

2004

J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron.40(12), 1663–1674 (2004).
[CrossRef]

2003

M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys.94(8), 5426–5428 (2003).
[CrossRef]

2002

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

2000

1998

C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998).
[CrossRef]

1994

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264(5158), 553–556 (1994).
[CrossRef] [PubMed]

1989

U. Ekenberg, “Nonparabolicity effects in a quantum well: sublevel shift, parallel mass, and Landau levels,” Phys. Rev. B Condens. Matter40(11), 7714–7726 (1989).
[CrossRef] [PubMed]

1987

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor-lasers - an overview,” IEEE J. Quantum Electron.23(1), 9–29 (1987).
[CrossRef]

1982

C. H. Henry, “Theory of the linewidth of semiconductor-lasers,” IEEE J. Quantum Electron.18(2), 259–264 (1982).
[CrossRef]

Aellen, T.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006).
[CrossRef]

Akikusa, N.

N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett.92(12), 121104 (2008).
[CrossRef]

Baillargeon, J. N.

C. Gmachl, F. Capasso, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho, “High-power, continuous-wave, current-tunable, single-mode quantum-cascade distributed-feedback lasers at lambda ~5.2 and lambda ~7.95 μm,” Opt. Lett.25(4), 230–232 (2000).
[CrossRef] [PubMed]

C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998).
[CrossRef]

Ban, D.

Beere, H. E.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett.92(7), 071106 (2008).
[CrossRef]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Beltram, F.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett.92(7), 071106 (2008).
[CrossRef]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Bhattacharyya, P.

T. Chattopadhyay and P. Bhattacharyya, “Role of linewidth enhancement factor on the frequency response of the synchronized quantum cascade laser,” Opt. Commun.309, 349–354 (2013).
[CrossRef]

Blaser, S.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006).
[CrossRef]

Buus, J.

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor-lasers - an overview,” IEEE J. Quantum Electron.23(1), 9–29 (1987).
[CrossRef]

Capasso, F.

J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron.40(12), 1663–1674 (2004).
[CrossRef]

M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys.94(8), 5426–5428 (2003).
[CrossRef]

C. Gmachl, F. Capasso, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho, “High-power, continuous-wave, current-tunable, single-mode quantum-cascade distributed-feedback lasers at lambda ~5.2 and lambda ~7.95 μm,” Opt. Lett.25(4), 230–232 (2000).
[CrossRef] [PubMed]

C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Chan, C. W. I.

Chattopadhyay, T.

T. Chattopadhyay and P. Bhattacharyya, “Role of linewidth enhancement factor on the frequency response of the synchronized quantum cascade laser,” Opt. Commun.309, 349–354 (2013).
[CrossRef]

Cho, A. Y.

J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron.40(12), 1663–1674 (2004).
[CrossRef]

M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys.94(8), 5426–5428 (2003).
[CrossRef]

C. Gmachl, F. Capasso, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho, “High-power, continuous-wave, current-tunable, single-mode quantum-cascade distributed-feedback lasers at lambda ~5.2 and lambda ~7.95 μm,” Opt. Lett.25(4), 230–232 (2000).
[CrossRef] [PubMed]

C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Chu, S. N. G.

C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998).
[CrossRef]

Chuang, S. L.

J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron.40(12), 1663–1674 (2004).
[CrossRef]

M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys.94(8), 5426–5428 (2003).
[CrossRef]

Davies, A. G.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Dupont, E.

Edamura, T.

N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett.92(12), 121104 (2008).
[CrossRef]

Ekenberg, U.

U. Ekenberg, “Nonparabolicity effects in a quantum well: sublevel shift, parallel mass, and Landau levels,” Phys. Rev. B Condens. Matter40(11), 7714–7726 (1989).
[CrossRef] [PubMed]

Elsässer, W.

Faist, J.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006).
[CrossRef]

C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Fathololoumi, S.

Fung, S.

S. Panda, B. K. Panda, and S. Fung, “Effect of conduction band nonparabolicity on the dark current in a quantum well infrared detector,” J. Appl. Phys.101(4), 043705 (2007).
[CrossRef]

Gensty, T.

Giovannini, M.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006).
[CrossRef]

Giuliani, G.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett.92(7), 071106 (2008).
[CrossRef]

J. von Staden, T. Gensty, W. Elsässer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett.31(17), 2574–2576 (2006).
[CrossRef] [PubMed]

Gmachl, C.

J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron.40(12), 1663–1674 (2004).
[CrossRef]

M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys.94(8), 5426–5428 (2003).
[CrossRef]

C. Gmachl, F. Capasso, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho, “High-power, continuous-wave, current-tunable, single-mode quantum-cascade distributed-feedback lasers at lambda ~5.2 and lambda ~7.95 μm,” Opt. Lett.25(4), 230–232 (2000).
[CrossRef] [PubMed]

C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998).
[CrossRef]

Green, R. P.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett.92(7), 071106 (2008).
[CrossRef]

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor-lasers,” IEEE J. Quantum Electron.18(2), 259–264 (1982).
[CrossRef]

Hoyler, N.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006).
[CrossRef]

Hu, Q.

Hutchinson, A. L.

C. Gmachl, F. Capasso, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho, “High-power, continuous-wave, current-tunable, single-mode quantum-cascade distributed-feedback lasers at lambda ~5.2 and lambda ~7.95 μm,” Opt. Lett.25(4), 230–232 (2000).
[CrossRef] [PubMed]

C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Hvozdara, L.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006).
[CrossRef]

Iotti, R. C.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Ishihara, M.

N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett.92(12), 121104 (2008).
[CrossRef]

Jirauschek, C.

Kasahara, K.

N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett.92(12), 121104 (2008).
[CrossRef]

Kim, J.

J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron.40(12), 1663–1674 (2004).
[CrossRef]

Köhler, R.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Kumar, S.

S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B80(24), 245316 (2009).
[CrossRef]

Kumazaki, N.

N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett.92(12), 121104 (2008).
[CrossRef]

Laframboise, S. R.

Lee, K. E.

T. Liu, K. E. Lee, and Q. J. Wang, “Microscopic density matrix model for optical gain of terahertz quantum cascade lasers: Many-body, nonparabolicity, and resonant tunneling effects,” Phys. Rev. B86(23), 235306 (2012).
[CrossRef]

Lerttamrab, M.

J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron.40(12), 1663–1674 (2004).
[CrossRef]

M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys.94(8), 5426–5428 (2003).
[CrossRef]

Linfield, E. H.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Liu, H. C.

Liu, T.

T. Liu, K. E. Lee, and Q. J. Wang, “Microscopic density matrix model for optical gain of terahertz quantum cascade lasers: Many-body, nonparabolicity, and resonant tunneling effects,” Phys. Rev. B86(23), 235306 (2012).
[CrossRef]

Mahler, L.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett.92(7), 071106 (2008).
[CrossRef]

Mann, C.

Mátyás, A.

Maulini, R.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006).
[CrossRef]

Osinski, M.

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor-lasers - an overview,” IEEE J. Quantum Electron.23(1), 9–29 (1987).
[CrossRef]

Panda, B. K.

S. Panda, B. K. Panda, and S. Fung, “Effect of conduction band nonparabolicity on the dark current in a quantum well infrared detector,” J. Appl. Phys.101(4), 043705 (2007).
[CrossRef]

Panda, S.

S. Panda, B. K. Panda, and S. Fung, “Effect of conduction band nonparabolicity on the dark current in a quantum well infrared detector,” J. Appl. Phys.101(4), 043705 (2007).
[CrossRef]

Ritchie, D. A.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett.92(7), 071106 (2008).
[CrossRef]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Rossi, F.

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Sirtori, C.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Sivco, D. L.

J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron.40(12), 1663–1674 (2004).
[CrossRef]

M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys.94(8), 5426–5428 (2003).
[CrossRef]

C. Gmachl, F. Capasso, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho, “High-power, continuous-wave, current-tunable, single-mode quantum-cascade distributed-feedback lasers at lambda ~5.2 and lambda ~7.95 μm,” Opt. Lett.25(4), 230–232 (2000).
[CrossRef] [PubMed]

C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998).
[CrossRef]

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Sugiyama, A.

N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett.92(12), 121104 (2008).
[CrossRef]

Takagi, Y.

N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett.92(12), 121104 (2008).
[CrossRef]

Terazzi, R.

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006).
[CrossRef]

Tredicucci, A.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett.92(7), 071106 (2008).
[CrossRef]

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

C. Gmachl, F. Capasso, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho, “High-power, continuous-wave, current-tunable, single-mode quantum-cascade distributed-feedback lasers at lambda ~5.2 and lambda ~7.95 μm,” Opt. Lett.25(4), 230–232 (2000).
[CrossRef] [PubMed]

C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998).
[CrossRef]

von Staden, J.

Wang, Q. J.

T. Liu, K. E. Lee, and Q. J. Wang, “Microscopic density matrix model for optical gain of terahertz quantum cascade lasers: Many-body, nonparabolicity, and resonant tunneling effects,” Phys. Rev. B86(23), 235306 (2012).
[CrossRef]

Wasilewski, Z. R.

Xu, J. H.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett.92(7), 071106 (2008).
[CrossRef]

Appl. Phys. Lett.

R. P. Green, J. H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett.92(7), 071106 (2008).
[CrossRef]

T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, “Direct measurement of the linewidth enhancement factor by optical heterodyning of an amplitude-modulated quantum cascade laser,” Appl. Phys. Lett.89(9), 091121 (2006).
[CrossRef]

N. Kumazaki, Y. Takagi, M. Ishihara, K. Kasahara, A. Sugiyama, N. Akikusa, and T. Edamura, “Detuning characteristics of the linewidth enhancement factor of a midinfrared quantum cascade laser,” Appl. Phys. Lett.92(12), 121104 (2008).
[CrossRef]

C. Gmachl, F. Capasso, J. Faist, A. L. Hutchinson, A. Tredicucci, D. L. Sivco, J. N. Baillargeon, S. N. G. Chu, and A. Y. Cho, “Continuous-wave and high-power pulsed operation of index-coupled distributed feedback quantum cascade laser at λ≈8.5 μm,” Appl. Phys. Lett.72(12), 1430–1432 (1998).
[CrossRef]

IEEE J. Quantum Electron.

J. Kim, M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Theoretical and experimental study of optical gain and linewidth enhancement factor of type-I quantum-cascade lasers,” IEEE J. Quantum Electron.40(12), 1663–1674 (2004).
[CrossRef]

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor-lasers - an overview,” IEEE J. Quantum Electron.23(1), 9–29 (1987).
[CrossRef]

C. H. Henry, “Theory of the linewidth of semiconductor-lasers,” IEEE J. Quantum Electron.18(2), 259–264 (1982).
[CrossRef]

J. Appl. Phys.

M. Lerttamrab, S. L. Chuang, C. Gmachl, D. L. Sivco, F. Capasso, and A. Y. Cho, “Linewidth enhancement factor of a type-I quantum-cascade laser,” J. Appl. Phys.94(8), 5426–5428 (2003).
[CrossRef]

S. Panda, B. K. Panda, and S. Fung, “Effect of conduction band nonparabolicity on the dark current in a quantum well infrared detector,” J. Appl. Phys.101(4), 043705 (2007).
[CrossRef]

Nature

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature417(6885), 156–159 (2002).
[CrossRef] [PubMed]

Opt. Commun.

T. Chattopadhyay and P. Bhattacharyya, “Role of linewidth enhancement factor on the frequency response of the synchronized quantum cascade laser,” Opt. Commun.309, 349–354 (2013).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B80(24), 245316 (2009).
[CrossRef]

T. Liu, K. E. Lee, and Q. J. Wang, “Microscopic density matrix model for optical gain of terahertz quantum cascade lasers: Many-body, nonparabolicity, and resonant tunneling effects,” Phys. Rev. B86(23), 235306 (2012).
[CrossRef]

E. Dupont, S. Fathololoumi, and H. C. Liu, “Simplified density-matrix model applied to three-well terahertz quantum cascade lasers,” Phys. Rev. B81(20), 205311 (2010).
[CrossRef]

Phys. Rev. B Condens. Matter

U. Ekenberg, “Nonparabolicity effects in a quantum well: sublevel shift, parallel mass, and Landau levels,” Phys. Rev. B Condens. Matter40(11), 7714–7726 (1989).
[CrossRef] [PubMed]

Science

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science264(5158), 553–556 (1994).
[CrossRef] [PubMed]

Other

W. W. Chow, S. W. Koch, and M. S. I. I. I. Semiconductor-Laser Physics, (Springer-Verlag, Berlin, 1994).

Cited By

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

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Schematic conduction band diagram of a two-phonon resonance gain region designed at 60 kV/cm in the “tight-binding” scheme [13]. The coupling between the periods, achieved by resonant tunneling (Ω51 is the coupling strength), is shown through the injector barrier. The layer sequence of the structure, in Angstrom, and starting from the injection barrier, is as follows:40/25/15/74/11/60/34/39/11/34/11/34/12/37/17/41. In0.52Al0.48As barrier layers are in bold, In0.53Ga0.47As well layers are in roman, and n-doped layers (2.5 × 1017 cm−3) are underlined.

Fig. 2
Fig. 2

Linewidth enhancement factor (LEF) including many body Coulomb interactions, coherence of resonant tunneling transport and non-parabolicity for different biases at 100 K. The points indicate the values of LEF at the gain peak.

Fig. 3
Fig. 3

(a) LEF as a function of the applied bias at the gain peak, 0.18 μm redshift and blueshift points, respectively, at 100 K. From the left to the right, the curves correspond to microscopic model “many body + non-parabolicity”, microscopic free-carrier model, macroscopic model with and without resonant tunneling, respectively. The lines are meant to guide the eye. (b) The gain spectra calculated from microscopic model “many-body + non-parabolicity” (solid line), microscopic model “many-body + parabolicity” (dotted lines), microscopic model “free carriers” (dot-dashed lines) and macroscopic matrix density model (dashed lines).

Fig. 4
Fig. 4

(a) LEF at the two effective mass ratios m5/m4 of electrons of 1.5 and ~1.3 using the microscopic model “many body + non-parabolicity”. (b) Gain spectra at the two effective mass ratios m5/m4 of electrons of 1.5 and ~1.3 using the microscopic model “many body + non-parabolicity”.

Fig. 5
Fig. 5

Conduction band diagram of a four level resonant-phonon THz QCL with a diagonal design at 12.3 kV/cm in the “tight-binding” scheme. Ω41, Ω23 and Ω31 are the injection, extraction and parasitic coupling strength, respectively. The thickness in angstrom of each layer is given as 49/88/27/82/42/160 starting from the injector barrier. The barriers Al0.15Ga 0.85As are indicated in bold fonts. The widest well is doped at 3 × 1010 cm−2.

Fig. 6
Fig. 6

LEF including many body Coulomb interactions, coherence of resonant-tunneling transport and non-parabolicity for different biases at 100 K. The points indicate the values of LEF at the gain peak. The simulation parameters can be found in [12].

Fig. 7
Fig. 7

LEF as a function of the applied bias at the gain peak, 0.2 THz redshift and 0.2 THz blueshift points, respectively, at 100 K. From the left to the right, the curves correspond to microscopic model “many body + non-parabolicity”, microscopic free-carrier model, macroscopic model with and without resonant tunneling, respectively.

Fig. 8
Fig. 8

The gain spectra at resonance and 100 K calculated from microscopic model “many-body + non-parabolicity” (solid line), microscopic model “many-body + parabolicity” (dotted lines), microscopic model “free carriers” (dot-dashed lines) and macroscopic matrix density model (dashed lines).

Equations (37)

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

H = j = 3 , 4 k ( μ j 5 E b 5 , k b j , k + c . c ) k [ ( Δ 5 1 / 2 ) b 1 , k b 5 , k + c . c ] + j = 1 , 3 , 4 , 5 k ε j , k b j , k b j , k + 1 2 u v v u 1 , 3 , 4 , 5 k k q V q u v v u b u , k + q b v , k q b v , k b u , k ,
G= 2 ω λ ε 0 nc V m ξ Im[ k ( μ 45 p 45,k + μ 35 p 35,k ) ],
δn= μ ε 0 n V m ξ Re( k ( μ 45 p 45,k + μ 35 p 35,k ) ),
α=2 ω λ c d( δn ) d N 0 / dG d N 0 ,
H = k ( μ E b 4 , k b 3 , k + c . c ) k [ ( Δ 4 1 / 2 ) b 1 , k b 4 , k + c . c ] k [ ( Δ 23 / 2 ) b 3 , k b 2 , k + c . c ] k [ ( Δ 3 1 / 2 ) b 1 , k b 3 , k + c . c ] + j = 1 , 2 , 3 , 4 k ε j , k b j , k b j , k + 1 2 u v v u 1 , 2 , 3 , 4 k k q V q u v v u b u , k + q b v , k q b v , k b u , k ,
G = 2 ω λ ε 0 n c V m ξ Im ( k μ 34 p ˜ 34 , k ) ,
δ n = μ ε 0 n V m ξ Re ( k μ 34 p ˜ 34 , k ) ,
dO dt = i [H,O],
[ b i,k , b j, k ] + = δ ij,k k , [ b i,k , b j, k ] + = [ b i,k , b j, k ] + =0,
d p 5 1 ,k dt = γ 5 1 p p 5 1 ,k i ε ˜ 1 5,k p 5 1 ,k +i Ω ˜ 5 1 ( n 5,k n 1 ,k )i Ω ˜ 45 p 4 1 ,k i Ω ˜ 35 p 3 1 ,k ,
d p 45,k dt = γ 45p p 45,k i( ε ˜ 54,k ω λ ) p 45,k i Ω ˜ 45 ( n 5,k n 4,k )+i Ω ˜ 35 p 34,k +i Ω ˜ 5 1 p 4 1 ,k ,
d p 35,k dt = γ 35p p 35,k i( ε ˜ 53,k ω λ ) p 35,k i Ω ˜ 35 ( n 5,k n 3,k )+i Ω ˜ 45 p 34,k +i Ω ˜ 5 1 p 3 1 ,k ,
d p 4 1 ,k dt = γ 4 1 p p 4 1 ,k i( ε ˜ 1 4,k ω λ ) p 4 1 ,k i Ω ˜ 45 p 5 1 ,k +i Ω ˜ 5 1 p 45,k ,
d p 34,k dt = γ 34p p 34,k i ε ˜ 43,k p 34,k +i Ω ˜ 45 p 35,k i Ω ˜ 35 p 45,k ,
d p 3 1 ,k dt = γ 3 1 p p 3 1 ,k i( ε ˜ 1 3,k ω λ ) p 3 1 ,k i Ω ˜ 35 p 5 1 ,k +i Ω ˜ 5 1 p 35,k ,
d n 1 ,k dt =i( Ω ˜ 5 1 p 5 1 ,k Ω ˜ 5 1 p 5 1 ,k ) γ 1 [ n 1 ,k f 1 ,k ( μ 1 ,e , T 1,e ) ] γ 3 1 [ n 1 ,k f 1 ,k ( μ 3 1 , T l ) ],
d n 5,k dt =i( Ω ˜ 45 p 45,k Ω ˜ 45 p 45,k )+i( Ω ˜ 35 p 35,k Ω ˜ 35 p 35,k ) +i( Ω ˜ 5 1 p 5 1 ,k Ω ˜ 5 1 p 5 1 ,k ) γ 5 [ n 5,k f 5,k ( μ 4,e , T 4,e ) ] j=4,3 γ 5j [ n 5,k f 5,k ( μ 5j , T l ) ] ,
d n 4,k dt =i( Ω ˜ 45 p 45,k Ω ˜ 45 p 45,k ) γ 4 [ n 4,k f 4,k ( μ 4,e , T 4,e ) ] j=5,3 γ 4j [ n 4,k f 4,k ( μ 4j , T l ) ] ,
d n 3,k dt =i( Ω ˜ 35 p 35,k Ω ˜ 35 p 35,k ) γ 3 [ n 3,k f 3,k ( μ 3,e , T 3,e ) ] j=5,4 γ 3j [ n 3,k f 3,k ( μ 3j , T l ) ] γ 3 1 [ n 3,k f 3,k ( μ 3 1 , T l ) ],
ε ˜ uv,k = ε u,k ε v,k k k ( V k k uuuu n u, k V k k vvvv n v, k ) + k k ( n u, k n v, k ) V k k uvuv ,
Ω ˜ uv = μ uv ξ 2 + 1 k k V k k uvvu p uv, k 2 V 0 uvuv k p uv, k ,
Ω ˜ 5 1 = Δ 5 1 2 + 1 k k V k k 5 1 1 5 p 5 1 , k 2 V 0 5 1 5 1 k p 5 1 , k ,
d p 34,k dt = γ 34p p 34,k i ε ˜ 43,k p 34,k i Ω ˜ 0 ( n 4,k n 3,k )+i Ω ˜ 4 1 p 3 1 ,k i Ω ˜ 23 p 24,k i Ω ˜ 3 1 p 4 1 ,k ,
d p 4 1 ,k dt = γ 4 1 p p 4 1 ,k i ε ˜ 1 4,k p 4 1 ,k i Ω ˜ 4 1 ( n 1,k n 4,k )i Ω ˜ 0 p 3 1 ,k +i Ω ˜ 3 1 p 34,k ,
d p 23,k dt = γ 23p p 23,k i ε ˜ 32,k p 23,k i Ω ˜ 23 ( n 3,k n 2,k )+i Ω ˜ 0 p 24,k +i Ω ˜ 3 1 p 2 1 ,k ,
d p 3 1 ,k dt = γ 3 1 p p 3 1 ,k i ε ˜ 1 3,k p 3 1 ,k i Ω ˜ 0 p 4 1 ,k +i Ω ˜ 4 1 p 34,k i Ω ˜ 23 p 2 1 ,k i Ω ˜ 3 1 ( n 1 ,k n 3,k ),
d p 24,k dt = γ 24p p 24,k i ε ˜ 42,k p 24,k +i Ω ˜ 0 p 23,k +i Ω ˜ 4 1 p 2 1 ,k i Ω ˜ 23 p 34,k ,
d p 2 1 ,k dt = γ 2 1 p p 2 1 ,k i ε ˜ 1 2,k p 2 1 ,k +i Ω ˜ 4 1 p 24,k i Ω ˜ 23 p 3 1 ,k +i Ω ˜ 3 1 p 23,k ,
d n 4,k dt =i( Ω ˜ 0 p 34,k Ω ˜ 0 p 34,k )i( Ω ˜ 4 1 p 4 1 ,k Ω ˜ 4 1 p 4 1 ,k ) γ 4 [ n 4,k f 4,k ( μ 4,e , T 4,e ) ] γ 43 [ n 4,k f 4,k ( μ 43 , T l ) ] γ sp n 4,k ,
d n 3,k dt =i( Ω ˜ 0 p 34,k Ω ˜ 0 p 34,k )i( Ω ˜ 23 p 23,k Ω ˜ 23 p 23,k ) γ 3 [ n 3,k f 3,k ( μ 3,e , T 3,e ) ] γ 43 [ n 3,k f 3,k ( μ 43 , T l ) ]+ γ sp n 4,k i( Ω ˜ 3 1 p 3 1 ,k Ω ˜ 3 1 p 3 1 ,k ),
d n 2,k dt =i( Ω ˜ 23 p 23,k Ω ˜ 23 p 23,k ) γ 2 [ n 2,k f 2,k ( μ 2,e , T 2,e ) ] γ 2 1 [ n 2,k f 2,k ( μ 2 1 , T l ) ],
d n 1 ,k dt =i( Ω ˜ 4 1 p 4 1 ,k Ω ˜ 4 1 p 4 1 ,k ) γ 1 [ n 1 ,k f 1 ,k ( μ 1 ,e , T 1,e ) ] γ 2 1 [ n 1 ,k f 1 ,k ( μ 2 1 , T l ) ] i( Ω ˜ 3 1 p 3 1 ,k Ω ˜ 3 1 p 3 1 ,k ),
ε ˜ uv,k = ε u,k ε v,k k k ( V k k uuuu n u, k V k k vvvv n v, k ) + k k ( n u, k n v, k ) V k k uvuv ,
Ω ˜ 0 = μξ 2 + 1 k k V k k 4334 p 43, k 2 V 0 4343 k p 43, k ,
Ω ˜ uv = Δ uv 2 + 1 k k V k k uvvu p uv, k 2 V 0 uvuv k p uv, k ,
p 4 1 ,k = p 4 1 ,k (0) , p 34,k = p 34,k (0) + p ˜ 34,k e i ω λ t , p 3 1 ,k = p 3 1 ,k (0) + p ˜ 3 1 ,k e i ω λ t ,
p 23,k = p 23,k (0) , p 24,k = p 24,k (0) + p ˜ 24,k e i ω λ t , p 2 1 ,k = p 2 1 ,k (0) + p ˜ 2 1 ,k e i ω λ t .

Metrics