Abstract

The modulation bandwidth of quantum well nanoLED and nanolaser devices is calculated from the laser rate equations using a detailed model for the Purcell enhanced spontaneous emission. It is found that the Purcell enhancement saturates when the cavity quality-factor is increased, which limits the maximum achievable spontaneous recombination rate. The modulation bandwidth is thereby limited to a few tens of GHz for realistic devices.

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References

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  1. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Letters to Nature 425(6961), 944–947 (2003).
    [CrossRef]
  2. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
  3. Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44(1), 657–668 (1991).
    [CrossRef] [PubMed]
  4. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
    [CrossRef] [PubMed]
  5. H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. 2(7), 484–488 (2006).
    [CrossRef]
  6. E. K. Lau, A. Lakhani, R. S. Tucker, and M. C. Wu, “Enhanced modulation bandwidth of nanocavity light emitting devices,” Opt. Express 17(10), 7790–7799 (2009).
    [CrossRef] [PubMed]
  7. L. A. Coldren, and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (John Wiley & Sons, inc., New York, 1995).
  8. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, inc., New York, 1998)
  9. A. Mecozzi and J. Mørk, “Saturation induced by picosecond pulses in semiconductor optical amplifiers,” J. Opt. Soc. Am. B 14(4), 761–770 (1997).
    [CrossRef]
  10. J. D. Joannopoulos, S. G. Johnson, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, New Jersey, 2008).
  11. J.-M. Gerard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots”, in Single Quantum Dots, Fundamentals, Applications and New Concepts, P. Michler (Springer, Berlin, 2003), pp. 269–314.
  12. H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66(10), 4801 (1989).
    [CrossRef]
  13. T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Top. Quantum Electron. 3(3), 808 (1997).
    [CrossRef]
  14. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005).
    [CrossRef] [PubMed]
  15. S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. 77(12), 2444–2446 (1996).
    [CrossRef] [PubMed]
  16. T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. 85(18), 3989–3991 (2004).
    [CrossRef]
  17. M. P. Marder, Condensed Matter Physics (John Wiley & Sons, inc., New York, 2000).
    [PubMed]
  18. R. S. Tucker, “High-speed modulation of semiconductor lasers,” J. Lightwave Technol. 3(6), 1180–1192 (1985).
    [CrossRef]

2009 (1)

2006 (1)

H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. 2(7), 484–488 (2006).
[CrossRef]

2005 (1)

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005).
[CrossRef] [PubMed]

2004 (1)

T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. 85(18), 3989–3991 (2004).
[CrossRef]

2003 (1)

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Letters to Nature 425(6961), 944–947 (2003).
[CrossRef]

1999 (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

1997 (2)

T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Top. Quantum Electron. 3(3), 808 (1997).
[CrossRef]

A. Mecozzi and J. Mørk, “Saturation induced by picosecond pulses in semiconductor optical amplifiers,” J. Opt. Soc. Am. B 14(4), 761–770 (1997).
[CrossRef]

1996 (1)

S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. 77(12), 2444–2446 (1996).
[CrossRef] [PubMed]

1991 (1)

Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44(1), 657–668 (1991).
[CrossRef] [PubMed]

1989 (1)

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66(10), 4801 (1989).
[CrossRef]

1985 (1)

R. S. Tucker, “High-speed modulation of semiconductor lasers,” J. Lightwave Technol. 3(6), 1180–1192 (1985).
[CrossRef]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Akahane, Y.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Letters to Nature 425(6961), 944–947 (2003).
[CrossRef]

Altug, H.

H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. 2(7), 484–488 (2006).
[CrossRef]

Arakawa, Y.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005).
[CrossRef] [PubMed]

Asano, T.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Letters to Nature 425(6961), 944–947 (2003).
[CrossRef]

Baba, T.

T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. 85(18), 3989–3991 (2004).
[CrossRef]

T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Top. Quantum Electron. 3(3), 808 (1997).
[CrossRef]

Barnett, S. M.

S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. 77(12), 2444–2446 (1996).
[CrossRef] [PubMed]

Björk, G.

Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44(1), 657–668 (1991).
[CrossRef] [PubMed]

Brorson, S. D.

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66(10), 4801 (1989).
[CrossRef]

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Englund, D.

H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. 2(7), 484–488 (2006).
[CrossRef]

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005).
[CrossRef] [PubMed]

Fattal, D.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005).
[CrossRef] [PubMed]

Inoshita, K.

T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. 85(18), 3989–3991 (2004).
[CrossRef]

Kim, I.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Koyama, F.

T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. 85(18), 3989–3991 (2004).
[CrossRef]

Kuroki, Y.

T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. 85(18), 3989–3991 (2004).
[CrossRef]

Lakhani, A.

Lau, E. K.

Lee, R. K.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Loudon, R.

S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. 77(12), 2444–2446 (1996).
[CrossRef] [PubMed]

Machida, S.

Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44(1), 657–668 (1991).
[CrossRef] [PubMed]

Mecozzi, A.

Mørk, J.

Nakaoka, T.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005).
[CrossRef] [PubMed]

Noda, S.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Letters to Nature 425(6961), 944–947 (2003).
[CrossRef]

Nozaki, K.

T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. 85(18), 3989–3991 (2004).
[CrossRef]

O’Brien, J. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Painter, O.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Sano, D.

T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. 85(18), 3989–3991 (2004).
[CrossRef]

Scherer, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Solomon, G.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005).
[CrossRef] [PubMed]

Song, B.-S.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Letters to Nature 425(6961), 944–947 (2003).
[CrossRef]

Tucker, R. S.

Vuckovic, J.

H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. 2(7), 484–488 (2006).
[CrossRef]

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005).
[CrossRef] [PubMed]

Waks, E.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005).
[CrossRef] [PubMed]

Wu, M. C.

Yamamoto, Y.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44(1), 657–668 (1991).
[CrossRef] [PubMed]

Yariv, A.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Yokoyama, H.

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66(10), 4801 (1989).
[CrossRef]

Zhang, B.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, and F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. 85(18), 3989–3991 (2004).
[CrossRef]

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

T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Top. Quantum Electron. 3(3), 808 (1997).
[CrossRef]

J. Appl. Phys. (1)

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66(10), 4801 (1989).
[CrossRef]

J. Lightwave Technol. (1)

R. S. Tucker, “High-speed modulation of semiconductor lasers,” J. Lightwave Technol. 3(6), 1180–1192 (1985).
[CrossRef]

J. Opt. Soc. Am. B (1)

Letters to Nature (1)

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Letters to Nature 425(6961), 944–947 (2003).
[CrossRef]

Nat. Phys. (1)

H. Altug, D. Englund, and J. Vučković, “Ultrafast photonic crystal nanocavity laser,” Nat. Phys. 2(7), 484–488 (2006).
[CrossRef]

Opt. Express (1)

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Phys. Rev. A (1)

Y. Yamamoto, S. Machida, and G. Björk, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44(1), 657–668 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95(1), 013904 (2005).
[CrossRef] [PubMed]

S. M. Barnett and R. Loudon, “Sum rule for modified spontaneous emission rates,” Phys. Rev. Lett. 77(12), 2444–2446 (1996).
[CrossRef] [PubMed]

Science (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[CrossRef] [PubMed]

Other (5)

L. A. Coldren, and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (John Wiley & Sons, inc., New York, 1995).

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, inc., New York, 1998)

J. D. Joannopoulos, S. G. Johnson, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, New Jersey, 2008).

J.-M. Gerard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots”, in Single Quantum Dots, Fundamentals, Applications and New Concepts, P. Michler (Springer, Berlin, 2003), pp. 269–314.

M. P. Marder, Condensed Matter Physics (John Wiley & Sons, inc., New York, 2000).
[PubMed]

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Figures (4)

Fig. 1
Fig. 1

The relative positions of the background optical DOS (black), the cavity optical DOS (red) and the electronic well DOS (green).

Fig. 2
Fig. 2

a) The calculated spontaneous recombination rates for Q equal to 100 (red), 1000 (blue) and 10000 (green). In the full model the emission into the cavity (solid lines) is several orders of magnitude lower than in the linear model (dashed lines). Also shown is the emission into the background in the full model (dotted line) and Rel for Q = 100 (dot-dashed line). β is 0.94 and Vn is 0.1. b) The effective Purcell factor in the full model (solid lines) and in the linear model (dashed lines) for Q equal to 100 (red), 1000 (blue) and 10000 (green). The effective Purcell factor for Q = 10 is also shown.

Fig. 3
Fig. 3

Results from device A with Q = 104 and Vn = 10 and device B with Q = 102 and Vn = 0.1 plotted against pump (in units of J0 = Ntr /τsp ). a) The carrier (solid) and photon (dashed) densities for device A, b) the spontaneous (solid) and stimulated (dashed) emission for device A and c) the 3dB-bandwidth for device A. d-f) are the same as a-c) but for device B. Both the full model (red) and the linear model (blue) are shown.

Fig. 4
Fig. 4

The 3dB-bandwidth for a) J = J0 and b) J = 100 J0 . The red lines are contours of equal 3dB-bandwidth in GHz. The black line separates the potential laser devices from the LED devices and the white line indicates which of the laser devices that have larger stimulated emission than spontaneous emission at the given pump.

Tables (1)

Tables Icon

Table 1 Definition and standard values of the parameters used in this paper. If nothing else is specified in the text, the value stated here has been used in the calculations.

Equations (23)

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

N ˙ = J R c R b R s t R n r
S ˙ = Γ ( R c + R s t ) N τ p
R s t = G S = v g G 0 1 + ε S ln ( N + N s N t r + N s ) S
H ( ω ) = ω R 2 ω R 2 ω 2 + i ω γ R
f 3 d B = 1 2 π ω R 2 γ R 2 2 + ( ω R 2 γ R 2 2 ) 2 + ω R 4
ω R 2 = 1 τ p ( a S 0 + R c , N ) + Γ ( 1 τ n r + R b , N ) ( a p S 0 + R c S 0 )
γ R = 1 τ n r + R c , N + R b , N + S 0 ( a + Γ a p ) + Γ R c S 0
F = 6 π 2 Q V n
R c = β F N τ s p
R b = ( 1 β ) N τ s p
R s p = ρ e l ( E ) f 2 ( E ) ( 1 f 1 ( E ) ) A 21 ( h ν ) L ( E h ν ) dh ν dE
ρ o p = 8 π ν 2 v g 3 ( H ( h ν L h ν ) + H ( h ν h ν U ) ) + F B 21 h ν τ 21 Γ p 2 ( h ν h ν 0 ) 2 + Γ p 2
A 21 ( h ν ) = 1 τ 21 [ ( h ν h ν 0 ) 3 ( H ( h ν L h ν ) + H ( h ν h ν U ) ) + F Γ p 2 ( h ν h ν 0 ) 2 + Γ p 2 ]
ρ e l ( E ) = m r π 2 W n = 1 H ( E ( E g + n 2 E 1 ) )
R e l = ρ e l ( E ) f 2 ( E ) ( 1 f 1 ( E ) ) F τ 21 L ( E h ν ) dh ν dE F τ 21 ρ e l ( E ) f 2 ( E ) ( 1 f 1 ( E ) ) dE
F τ 21 Γ p 2 ( h ν h ν 0 ) 2 + Γ p 2 δ ( E h ν ) d h ν = F τ 21 Γ p 2 ( E E 0 ) 2 + Γ p 2
R c = F τ 21 Γ p 2 ( E E 0 ) 2 + Γ p 2 ρ e l ( E ) f 2 ( E ) ( 1 f 1 ( E ) ) A ( E ) dE
R c = F τ 21 Γ p π ρ e l ( E 0 ) f 2 ( E 0 ) ( 1 f 1 ( E 0 ) ) = 12 ω 0 π τ 21 V n ρ e l ( E 0 ) f 2 ( E 0 ) ( 1 f 1 ( E 0 ) )
F e f f R c R b u l k
( linear model ) R b u l k = N τ s p (full model)     R b u l k = 1 τ s p ρ e l ( E ) f 2 ( E ) ( 1 f 1 ( E ) ) ( h ν h ν 0 ) 3 L ( E h ν ) dh ν dE 
f 3 d B 1 2 π 1 τ p 2 + τ e f f 2
1 τ e f f = F e f f τ s p + 1 β τ s p + 1 τ n r F e f f τ s p
1 τ e f f = F e f f d R b u l k d N + d F e f f d N R b u l k + d R b d N + 1 τ n r

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