Abstract

Intrinsic linewidth formula modified by taking account of fluctuation-dissipation balance for thermal photons in a THz quantum-cascade laser (QCL) is exhibited. The linewidth formula based on the model that counts explicitly the influence of noisy stimulated emissions due to thermal photons existing inside the laser cavity interprets experimental results on intrinsic linewidth, ~91.1 Hz reported recently with a 2.5 THz bound-to-continuum QCL. The line-broadening induced by thermal photons is estimated to be ~22.4 Hz, i.e., 34% broadening. The modified linewidth formula is utilized as a bench mark in engineering of THz thermal photons inside laser cavities.

© 2012 OSA

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  1. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics1(2), 97–105 (2007).
    [CrossRef]
  2. S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express19(19), 17996–18003 (2011).
    [CrossRef] [PubMed]
  3. M. S. Vitiello, L. Consolino, S. Bartalini, A. Taschin, A. Tredicucci, M. Inguscio, and P. De Natale, “Quantum-limited frequency fluctuations in a terahertz laser,” Nat. Photonics6(8), 525–528 (2012) (and “Supplementary information” for the reference).
    [CrossRef]
  4. M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron.44(1), 12–29 (2008).
    [CrossRef]
  5. H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys.204(3), 262–275 (1967).
    [CrossRef]
  6. T. Liu, K. E. Lee, and Q. J. Wang, “Effects of resonant tunneling and dynamics of coherent interaction on intrinsic linewidth of quantum cascade lasers,” Opt. Express20(15), 17145–17159 (2012).
    [CrossRef]
  7. U. Herzog and A. Bergou, “Quantum-limited linewidth of a good-cavity laser: an analytical theory from near to above threshold,” Phys. Rev. A62(6), 063814 (2000).
    [CrossRef]
  8. K. Shimoda, H. Takahashi, and C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn.12(6), 686–700 (1957).
    [CrossRef]
  9. For an explicit form of LW of a maser oscillator, see Eq. (10).62) in Chap. 10, “Fundamentals of noise processes” by Yoshihisa Yamamoto in quantum information lecture series (2010), http://first-quantum.net/e/forStudents/lecture/index.html . It is noted that the correction factor 1/2 due to stabilization of amplitude fluctuations above threshold is not included in the equation and nsp = 1 is also assumed.
  10. C. Jirauschek, “Monte Carlo study of intrinsic linewidths in terahertz quantum cascade lasers,” Opt. Express18(25), 25922–25927 (2010).
    [CrossRef] [PubMed]
  11. Equation (3).14) in reference [7] is applicable to THz-QCLs even near above thresholds since amplitude fluctuations are stabilized by very fast nonradiative relaxation of upper laser-state electrons as pointed out in Appendix B in reference [4] and also in Appendix A of this article.
  12. M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett.88(24), 241109 (2006).
    [CrossRef]
  13. C. H. Henry, “Line broadening of semiconductor lasers,” in Coherence, amplification, and quantum effects in semiconductor lasers, Y. Yamamoto, Ed. (Wiley, 1991) pp. 5–76.
  14. M. S. Vitiello, private communication on optical and electronic parameters.
  15. R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, 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]
  16. M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett.89(13), 131114 (2006).
    [CrossRef]
  17. M. Ravaro, S. Barbieri, G. Santarelli, V. Jagtap, C. Manquest, C. Sirtori, S. P. Khanna, and E. H. Linfield, “Measurement of the intrinsic linewidth of terahertz quantum cascade lasers using a near-infrared frequency comb,” Opt. Express20(23), 25654–25661 (2012).
    [CrossRef] [PubMed]
  18. S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron.47(7), 984–988 (2011).
    [CrossRef]
  19. M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett.86(11), 111115 (2005).
    [CrossRef]
  20. M. S. Vitiello, J.-H. Xu, M. Kumar, F. Beltram, A. Tredicucci, O. Mitrofanov, H. E. Beere, and D. A. Ritchie, “High efficiency coupling of Terahertz micro-ring quantum cascade lasers to the low-loss optical modes of hollow metallic waveguides,” Opt. Express19(2), 1122–1130 (2011).
    [CrossRef] [PubMed]
  21. The substantial reduction of dark currents by suppression of incoming thermal photons in a cooled low loss fiber coupled to a near infrared photomultiplier tube was in fact confirmed experimentally; T. Hirohata, Y. Negi, and M. Niigaki, Japan patent application No. 2009–047909 (2009) (will be published elsewhere).
  22. The Si doping level in injectors of the THz BTC-QCL is designed to be Ninj/S~1010 1/cm2 [3]. However, in the analysis, the lower doping density 5 × 109 1/cm2 is used by taking account of compensation by residual defect states. In fact, the assumed (effective) doping density 5 × 109 1/cm2 leads to good agreements in threshold and maximum currents between theory and experiments.
  23. M. S. Vitiello, G. Scamarcio, J. Faist, G. Scalari, C. Walther, H. E. Beere, and D. A. Ritchie, “Probing quantum efficiency by laser-induced hot-electron cooling,” Appl. Phys. Lett.94(2), 021115 (2009).
    [CrossRef]
  24. G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, and G. Davies, “Far-infrared (λ~87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett.82(19), 3165–3167 (2003).
    [CrossRef]
  25. The band-structure computations for the 2.5 THz and 3.5 THz BTC-QCLs were performed by solving Schrödinger/Poisson equations; K. Fujita, private communication.

2012 (3)

2011 (3)

2010 (1)

2009 (1)

M. S. Vitiello, G. Scamarcio, J. Faist, G. Scalari, C. Walther, H. E. Beere, and D. A. Ritchie, “Probing quantum efficiency by laser-induced hot-electron cooling,” Appl. Phys. Lett.94(2), 021115 (2009).
[CrossRef]

2008 (2)

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, 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]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron.44(1), 12–29 (2008).
[CrossRef]

2007 (1)

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics1(2), 97–105 (2007).
[CrossRef]

2006 (2)

M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett.89(13), 131114 (2006).
[CrossRef]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett.88(24), 241109 (2006).
[CrossRef]

2005 (1)

M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett.86(11), 111115 (2005).
[CrossRef]

2003 (1)

G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, and G. Davies, “Far-infrared (λ~87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett.82(19), 3165–3167 (2003).
[CrossRef]

2000 (1)

U. Herzog and A. Bergou, “Quantum-limited linewidth of a good-cavity laser: an analytical theory from near to above threshold,” Phys. Rev. A62(6), 063814 (2000).
[CrossRef]

1967 (1)

H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys.204(3), 262–275 (1967).
[CrossRef]

1957 (1)

K. Shimoda, H. Takahashi, and C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn.12(6), 686–700 (1957).
[CrossRef]

Ajili, L.

G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, and G. Davies, “Far-infrared (λ~87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett.82(19), 3165–3167 (2003).
[CrossRef]

Akikusa, N.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron.44(1), 12–29 (2008).
[CrossRef]

Alto, J.

M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett.89(13), 131114 (2006).
[CrossRef]

Barbieri, S.

M. Ravaro, S. Barbieri, G. Santarelli, V. Jagtap, C. Manquest, C. Sirtori, S. P. Khanna, and E. H. Linfield, “Measurement of the intrinsic linewidth of terahertz quantum cascade lasers using a near-infrared frequency comb,” Opt. Express20(23), 25654–25661 (2012).
[CrossRef] [PubMed]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett.89(13), 131114 (2006).
[CrossRef]

Bartalini, S.

M. S. Vitiello, L. Consolino, S. Bartalini, A. Taschin, A. Tredicucci, M. Inguscio, and P. De Natale, “Quantum-limited frequency fluctuations in a terahertz laser,” Nat. Photonics6(8), 525–528 (2012) (and “Supplementary information” for the reference).
[CrossRef]

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron.47(7), 984–988 (2011).
[CrossRef]

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

Beere, H.

G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, and G. Davies, “Far-infrared (λ~87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett.82(19), 3165–3167 (2003).
[CrossRef]

Beere, H. E.

M. S. Vitiello, J.-H. Xu, M. Kumar, F. Beltram, A. Tredicucci, O. Mitrofanov, H. E. Beere, and D. A. Ritchie, “High efficiency coupling of Terahertz micro-ring quantum cascade lasers to the low-loss optical modes of hollow metallic waveguides,” Opt. Express19(2), 1122–1130 (2011).
[CrossRef] [PubMed]

M. S. Vitiello, G. Scamarcio, J. Faist, G. Scalari, C. Walther, H. E. Beere, and D. A. Ritchie, “Probing quantum efficiency by laser-induced hot-electron cooling,” Appl. Phys. Lett.94(2), 021115 (2009).
[CrossRef]

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, 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]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett.88(24), 241109 (2006).
[CrossRef]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett.89(13), 131114 (2006).
[CrossRef]

Beltram, F.

Beltran, F.

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, 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]

Bergou, A.

U. Herzog and A. Bergou, “Quantum-limited linewidth of a good-cavity laser: an analytical theory from near to above threshold,” Phys. Rev. A62(6), 063814 (2000).
[CrossRef]

Borri, S.

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron.47(7), 984–988 (2011).
[CrossRef]

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

Consolino, L.

M. S. Vitiello, L. Consolino, S. Bartalini, A. Taschin, A. Tredicucci, M. Inguscio, and P. De Natale, “Quantum-limited frequency fluctuations in a terahertz laser,” Nat. Photonics6(8), 525–528 (2012) (and “Supplementary information” for the reference).
[CrossRef]

Davies, G.

G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, and G. Davies, “Far-infrared (λ~87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett.82(19), 3165–3167 (2003).
[CrossRef]

De Natale, P.

M. S. Vitiello, L. Consolino, S. Bartalini, A. Taschin, A. Tredicucci, M. Inguscio, and P. De Natale, “Quantum-limited frequency fluctuations in a terahertz laser,” Nat. Photonics6(8), 525–528 (2012) (and “Supplementary information” for the reference).
[CrossRef]

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron.47(7), 984–988 (2011).
[CrossRef]

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

Edamura, T.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron.44(1), 12–29 (2008).
[CrossRef]

Faist, J.

M. S. Vitiello, G. Scamarcio, J. Faist, G. Scalari, C. Walther, H. E. Beere, and D. A. Ritchie, “Probing quantum efficiency by laser-induced hot-electron cooling,” Appl. Phys. Lett.94(2), 021115 (2009).
[CrossRef]

G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, and G. Davies, “Far-infrared (λ~87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett.82(19), 3165–3167 (2003).
[CrossRef]

Fujita, K.

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron.44(1), 12–29 (2008).
[CrossRef]

Galli, I.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron.47(7), 984–988 (2011).
[CrossRef]

Giuliani, G.

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, 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]

Giusfredi, G.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron.47(7), 984–988 (2011).
[CrossRef]

Green, R. P.

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, 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]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett.88(24), 241109 (2006).
[CrossRef]

Haken, H.

H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys.204(3), 262–275 (1967).
[CrossRef]

Haug, H.

H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys.204(3), 262–275 (1967).
[CrossRef]

Herzog, U.

U. Herzog and A. Bergou, “Quantum-limited linewidth of a good-cavity laser: an analytical theory from near to above threshold,” Phys. Rev. A62(6), 063814 (2000).
[CrossRef]

Hu, Q.

M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett.86(11), 111115 (2005).
[CrossRef]

Inguscio, M.

M. S. Vitiello, L. Consolino, S. Bartalini, A. Taschin, A. Tredicucci, M. Inguscio, and P. De Natale, “Quantum-limited frequency fluctuations in a terahertz laser,” Nat. Photonics6(8), 525–528 (2012) (and “Supplementary information” for the reference).
[CrossRef]

Jagtap, V.

Jirauschek, C.

Kan, H.

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron.44(1), 12–29 (2008).
[CrossRef]

Khanna, S. P.

Kumar, M.

Kumar, S.

M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett.86(11), 111115 (2005).
[CrossRef]

Lee, K. E.

Linfield, E.

G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, and G. Davies, “Far-infrared (λ~87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett.82(19), 3165–3167 (2003).
[CrossRef]

Linfield, E. H.

Liu, T.

Losco, T.

M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett.88(24), 241109 (2006).
[CrossRef]

Mahler, L.

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, 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]

Manquest, C.

Mazzotti, D.

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron.47(7), 984–988 (2011).
[CrossRef]

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

Mitrofanov, O.

Pastor, P. C.

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron.47(7), 984–988 (2011).
[CrossRef]

Ravaro, M.

Reno, J. L.

M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett.86(11), 111115 (2005).
[CrossRef]

Ritchie, D.

G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, and G. Davies, “Far-infrared (λ~87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett.82(19), 3165–3167 (2003).
[CrossRef]

Ritchie, D. A.

M. S. Vitiello, J.-H. Xu, M. Kumar, F. Beltram, A. Tredicucci, O. Mitrofanov, H. E. Beere, and D. A. Ritchie, “High efficiency coupling of Terahertz micro-ring quantum cascade lasers to the low-loss optical modes of hollow metallic waveguides,” Opt. Express19(2), 1122–1130 (2011).
[CrossRef] [PubMed]

M. S. Vitiello, G. Scamarcio, J. Faist, G. Scalari, C. Walther, H. E. Beere, and D. A. Ritchie, “Probing quantum efficiency by laser-induced hot-electron cooling,” Appl. Phys. Lett.94(2), 021115 (2009).
[CrossRef]

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, 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]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett.88(24), 241109 (2006).
[CrossRef]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett.89(13), 131114 (2006).
[CrossRef]

Santarelli, G.

Scalari, G.

M. S. Vitiello, G. Scamarcio, J. Faist, G. Scalari, C. Walther, H. E. Beere, and D. A. Ritchie, “Probing quantum efficiency by laser-induced hot-electron cooling,” Appl. Phys. Lett.94(2), 021115 (2009).
[CrossRef]

G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, and G. Davies, “Far-infrared (λ~87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett.82(19), 3165–3167 (2003).
[CrossRef]

Scamarcio, G.

M. S. Vitiello, G. Scamarcio, J. Faist, G. Scalari, C. Walther, H. E. Beere, and D. A. Ritchie, “Probing quantum efficiency by laser-induced hot-electron cooling,” Appl. Phys. Lett.94(2), 021115 (2009).
[CrossRef]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett.88(24), 241109 (2006).
[CrossRef]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett.89(13), 131114 (2006).
[CrossRef]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett.86(11), 111115 (2005).
[CrossRef]

Shimoda, K.

K. Shimoda, H. Takahashi, and C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn.12(6), 686–700 (1957).
[CrossRef]

Sirtori, C.

M. Ravaro, S. Barbieri, G. Santarelli, V. Jagtap, C. Manquest, C. Sirtori, S. P. Khanna, and E. H. Linfield, “Measurement of the intrinsic linewidth of terahertz quantum cascade lasers using a near-infrared frequency comb,” Opt. Express20(23), 25654–25661 (2012).
[CrossRef] [PubMed]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett.89(13), 131114 (2006).
[CrossRef]

Spagnolo, V.

M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett.89(13), 131114 (2006).
[CrossRef]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett.88(24), 241109 (2006).
[CrossRef]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett.86(11), 111115 (2005).
[CrossRef]

Takahashi, H.

K. Shimoda, H. Takahashi, and C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn.12(6), 686–700 (1957).
[CrossRef]

Taschin, A.

M. S. Vitiello, L. Consolino, S. Bartalini, A. Taschin, A. Tredicucci, M. Inguscio, and P. De Natale, “Quantum-limited frequency fluctuations in a terahertz laser,” Nat. Photonics6(8), 525–528 (2012) (and “Supplementary information” for the reference).
[CrossRef]

Tonouchi, M.

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics1(2), 97–105 (2007).
[CrossRef]

Townes, C. H.

K. Shimoda, H. Takahashi, and C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn.12(6), 686–700 (1957).
[CrossRef]

Tredicucci, A.

M. S. Vitiello, L. Consolino, S. Bartalini, A. Taschin, A. Tredicucci, M. Inguscio, and P. De Natale, “Quantum-limited frequency fluctuations in a terahertz laser,” Nat. Photonics6(8), 525–528 (2012) (and “Supplementary information” for the reference).
[CrossRef]

M. S. Vitiello, J.-H. Xu, M. Kumar, F. Beltram, A. Tredicucci, O. Mitrofanov, H. E. Beere, and D. A. Ritchie, “High efficiency coupling of Terahertz micro-ring quantum cascade lasers to the low-loss optical modes of hollow metallic waveguides,” Opt. Express19(2), 1122–1130 (2011).
[CrossRef] [PubMed]

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, 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]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett.88(24), 241109 (2006).
[CrossRef]

Vitiello, M. S.

M. S. Vitiello, L. Consolino, S. Bartalini, A. Taschin, A. Tredicucci, M. Inguscio, and P. De Natale, “Quantum-limited frequency fluctuations in a terahertz laser,” Nat. Photonics6(8), 525–528 (2012) (and “Supplementary information” for the reference).
[CrossRef]

M. S. Vitiello, J.-H. Xu, M. Kumar, F. Beltram, A. Tredicucci, O. Mitrofanov, H. E. Beere, and D. A. Ritchie, “High efficiency coupling of Terahertz micro-ring quantum cascade lasers to the low-loss optical modes of hollow metallic waveguides,” Opt. Express19(2), 1122–1130 (2011).
[CrossRef] [PubMed]

M. S. Vitiello, G. Scamarcio, J. Faist, G. Scalari, C. Walther, H. E. Beere, and D. A. Ritchie, “Probing quantum efficiency by laser-induced hot-electron cooling,” Appl. Phys. Lett.94(2), 021115 (2009).
[CrossRef]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett.88(24), 241109 (2006).
[CrossRef]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett.89(13), 131114 (2006).
[CrossRef]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett.86(11), 111115 (2005).
[CrossRef]

Walther, C.

M. S. Vitiello, G. Scamarcio, J. Faist, G. Scalari, C. Walther, H. E. Beere, and D. A. Ritchie, “Probing quantum efficiency by laser-induced hot-electron cooling,” Appl. Phys. Lett.94(2), 021115 (2009).
[CrossRef]

Wang, Q. J.

Williams, B. S.

M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett.86(11), 111115 (2005).
[CrossRef]

Worrall, C.

M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett.89(13), 131114 (2006).
[CrossRef]

Xu, J.

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, 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]

Xu, J.-H.

Yamanishi, M.

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron.47(7), 984–988 (2011).
[CrossRef]

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express19(19), 17996–18003 (2011).
[CrossRef] [PubMed]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron.44(1), 12–29 (2008).
[CrossRef]

Appl. Phys. Lett. (6)

M. S. Vitiello, G. Scamarcio, V. Spagnolo, T. Losco, R. P. Green, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Electron-lattice coupling in bound-to-continuum THz quantum-cascade lasers,” Appl. Phys. Lett.88(24), 241109 (2006).
[CrossRef]

R. P. Green, J. Xu, L. Mahler, A. Tredicucci, F. Beltran, 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]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, C. Worrall, H. E. Beere, D. A. Ritchie, C. Sirtori, J. Alto, and S. Barbieri, “Subband electronic temperatures and electron-lattice energy relaxation in terahertz quantum cascade lasers with different conduction band offsets,” Appl. Phys. Lett.89(13), 131114 (2006).
[CrossRef]

M. S. Vitiello, G. Scamarcio, V. Spagnolo, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, “Measurement of subband electronic temperatures and population inversion in THz quantum-cascade lasers,” Appl. Phys. Lett.86(11), 111115 (2005).
[CrossRef]

M. S. Vitiello, G. Scamarcio, J. Faist, G. Scalari, C. Walther, H. E. Beere, and D. A. Ritchie, “Probing quantum efficiency by laser-induced hot-electron cooling,” Appl. Phys. Lett.94(2), 021115 (2009).
[CrossRef]

G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie, and G. Davies, “Far-infrared (λ~87 μm) bound-to-continuum quantum-cascade lasers operating up to 90 K,” Appl. Phys. Lett.82(19), 3165–3167 (2003).
[CrossRef]

IEEE J. Quantum Electron. (2)

S. Borri, S. Bartalini, P. C. Pastor, I. Galli, G. Giusfredi, D. Mazzotti, M. Yamanishi, and P. De Natale, “Frequency-noise dynamics of mid-infrared quantum cascade lasers,” IEEE J. Quantum Electron.47(7), 984–988 (2011).
[CrossRef]

M. Yamanishi, T. Edamura, K. Fujita, N. Akikusa, and H. Kan, “Theory of the intrinsic linewidth of quantum-cascade lasers: hidden reason for the narrow linewidth and line-broadening by thermal photons,” IEEE J. Quantum Electron.44(1), 12–29 (2008).
[CrossRef]

J. Phys. Soc. Jpn. (1)

K. Shimoda, H. Takahashi, and C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn.12(6), 686–700 (1957).
[CrossRef]

Nat. Photonics (2)

M. S. Vitiello, L. Consolino, S. Bartalini, A. Taschin, A. Tredicucci, M. Inguscio, and P. De Natale, “Quantum-limited frequency fluctuations in a terahertz laser,” Nat. Photonics6(8), 525–528 (2012) (and “Supplementary information” for the reference).
[CrossRef]

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics1(2), 97–105 (2007).
[CrossRef]

Opt. Express (5)

Phys. Rev. A (1)

U. Herzog and A. Bergou, “Quantum-limited linewidth of a good-cavity laser: an analytical theory from near to above threshold,” Phys. Rev. A62(6), 063814 (2000).
[CrossRef]

Z. Phys. (1)

H. Haug and H. Haken, “Theory of noise in semiconductor laser emission,” Z. Phys.204(3), 262–275 (1967).
[CrossRef]

Other (7)

For an explicit form of LW of a maser oscillator, see Eq. (10).62) in Chap. 10, “Fundamentals of noise processes” by Yoshihisa Yamamoto in quantum information lecture series (2010), http://first-quantum.net/e/forStudents/lecture/index.html . It is noted that the correction factor 1/2 due to stabilization of amplitude fluctuations above threshold is not included in the equation and nsp = 1 is also assumed.

Equation (3).14) in reference [7] is applicable to THz-QCLs even near above thresholds since amplitude fluctuations are stabilized by very fast nonradiative relaxation of upper laser-state electrons as pointed out in Appendix B in reference [4] and also in Appendix A of this article.

C. H. Henry, “Line broadening of semiconductor lasers,” in Coherence, amplification, and quantum effects in semiconductor lasers, Y. Yamamoto, Ed. (Wiley, 1991) pp. 5–76.

M. S. Vitiello, private communication on optical and electronic parameters.

The substantial reduction of dark currents by suppression of incoming thermal photons in a cooled low loss fiber coupled to a near infrared photomultiplier tube was in fact confirmed experimentally; T. Hirohata, Y. Negi, and M. Niigaki, Japan patent application No. 2009–047909 (2009) (will be published elsewhere).

The Si doping level in injectors of the THz BTC-QCL is designed to be Ninj/S~1010 1/cm2 [3]. However, in the analysis, the lower doping density 5 × 109 1/cm2 is used by taking account of compensation by residual defect states. In fact, the assumed (effective) doping density 5 × 109 1/cm2 leads to good agreements in threshold and maximum currents between theory and experiments.

The band-structure computations for the 2.5 THz and 3.5 THz BTC-QCLs were performed by solving Schrödinger/Poisson equations; K. Fujita, private communication.

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Figures (4)

Fig. 1
Fig. 1

Band diagram of the active region in a THz BTC-QCL [3].

Fig. 2
Fig. 2

Illustration for fluctuation and dissipation flows of thermal photons in the THz QCL.

Fig. 3
Fig. 3

Population-inverted two-level system consisting of upper and lower laser-states subjected to quantum fluctuations.

Fig. 4
Fig. 4

Three-level model in a module of the THz BTC-QCL used in the rate equation analysis.

Equations (25)

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δf= (1+ α c 2 )γ[ n sp +( n sp +1) N bb ] 4π N photon = (1+ α c 2 )( ω photon ) γ 2 [ n sp +( n sp +1) N bb ] 4π P int .
δf= (1+ α c 2 )( ω photon ) γ 2 [1+2 N bb ] 4π P int ,
γ active N bb,0 ( T e )+ γ clad N bb,0 ( T L )+ γ mf N bb,a-f ( T a-f )+ γ mb N bb,a-b ( T a-b ) =( γ active + γ clad + γ mf + γ mb ) N bb,cavity ,
N bb,cavity =[ γ active N bb,0 ( T e )+ γ clad N bb,0 ( T L )+ γ mf N bb,a-f ( T a-f )+ γ mb N bb,a-b ( T a-b )]/γ,  
δf={ (1+ α c 2 )( ω photon ) γ 2 4π P int }{ n sp + n sp N bb,cavity + [ ( γ active /γ) N bb,,0 ( T e ) +( γ clad /γ) N bb,,0 ( T L )+( γ mf /γ) N bb,a-f ( T a -f )+( γ mb /γ) N bb,a-b ( T a-b ) ] } ={ (1+ α c 2 )( ω photon ) γ 2 4π P int }{ n sp +( n sp +1) [ ( γ active /γ) N bb,0 ( T e ) +( γ clad /γ) N bb,0 ( T L )+( γ mf /γ) N bb,a-f ( T a -f )+( γ mb /γ) N bb,a-b ( T a-b ) ] }.
δf={ (1+ α c 2 )( ω photon ) γ 2 4π P int }{ n sp + [ ( γ active /γ) N bb,,0 ( T e ) +( γ clad /γ) N bb,,0 ( T L )+( γ mf /γ) N bb,a-f ( T a -f )+( γ mb /γ) N bb,a-b ( T a-b ) ] }.
d( N photon + N bb,cavity )/dt =[γM(β/ τ r )( N 3 N 2 )] N photon +Mβ N 3 / τ r +M(β/ τ r )( N 3 N 2 ) N bb,cavity +γ N bb,eff γ N bb,cavity =[γM(β/ τ r )( N 3 N 2 )]( N photon + N bb,cavity )+Mβ N 3 / τ r +γ N bb,eff .
β N 30 / τ r +β N 30 N bb,cavity / τ r =β N 2 N bb, cavity / τ r .
d( N photon + N bb,cavity )/dt=[γM(β/ τ r )( N 3 N 2 )] N photon +Mβ( N 3 N 30 )(1+ N bb,cavity )/ τ r +γ N bb, eff γ N bb,cavity .
η int ( I 0 /e )= N 3 / τ 31 N 1 / τ 13 + N 3 / τ 32 N 2 / τ 23 +(β/ τ r )( N 3 N 2 ) N photon , 
(1 η int )( I 0 /e )+ N 3 / τ 31 + N 2 / τ 21 =( I 0 /e )+ N 1 / τ 13 + N 1 / τ 12 . 
N inj = N 1 + N 2 + N 3 .
N 3 = η int ( I 0 /e ) τ 31 [1+( τ 21 / τ 12 )]+( τ 31 / τ 13 ) N inj 1+( τ 21 / τ 12 )+( τ 31 / τ 13 ) ,
N 2 =( τ 21 / τ 12 ) N 1 = N 2therm .
N 1 =( k B T e )( ρ 2D S)exp[ E F / k B T e ] i=1 i=6 exp[(i1)ΔE/ k B T e ] = ( k B T e )( ρ 2D S)exp[ E F / k B T e ]{1exp[6ΔE/ k B T e ]} 1exp[ΔE/ k B T e ] ,
N 2 = N 2therm =( k B T e )( ρ 2D S)exp[ E F / k B T e ]exp[ E M / k B T e ].
f bf2 = τ 21 / τ 12 = N 2therm / N 1 = exp[ E M / k B T e ]{1exp[ΔE/ k B T e ]} 1exp[ E M / k B T e ] ,  
f bf3 = τ 31 / τ 13 = N 3therm / N 1 = exp[( E M + E 32 )/ k B T e ]{1exp[ΔE/ k B T e ]} 1exp[ E M / k B T e ] .
I th =( e η int ){ ( γ τ r Mβ τ 31 )( 1+ f bf2 + f bf3 1+2 f bf2 )+( N inj ( f bf2 f bf3 ) τ 31 (1+2 f bf2 ) ) },
N 3th =( γ τ r Mβ )( 1+ f bf2 1+2 f bf2 )+( N inj f bf2 1+2 f bf2 ), 
n spth = N 3th ( N 3 N 2 ) th =( 1+ f bf2 1+2 f bf2 )+( Mβ γ τ r )( N inj f bf2 1+2 f bf2 ).
γ N photon = M[ η int ( I 0 I th )/e][1+2 f bf2 ] (1+2 f bf2 )+( τ 21 / τ 31 )(1+2 f bf3 ) , 
N 3 N 3th = [ η int ( I 0 I th )/e] τ 21 ( 1+2 f bf2 )+( τ 21 / τ 31 )( 1+2 f bf3 )
Δ n sp = n sp n spth = (Mβ/ τ r γ)[ η int ( I 0 I th )/e] τ 21 ( 1+2 f bf2 )+( τ 21 / τ 31 )( 1+2 f bf3 ) ,
β/ τ r = 4π e 2 f 0 Z 32 2 Γ conf ε 0 n eff 2 V c ( Γ FWHM ) ~2.12× 10 4 μs 1 , 

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