Abstract

We present a new chromatic numerical approach to simulate the amplification of laser pulses in multipass laser amplifiers. This enables studies on spectral effects such as gain narrowing and spectral shaping with optical elements expressed by a transmission transfer function. We observe good agreement between our simulations and measurements with a Ho:YLF regenerative amplifier (RA). To demonstrate the capabilities of our simulation model, we numerically integrate an intra-cavity etalon in this laser and find optimum etalon parameters that enhance the peak power of the output pulses up to 65%.

© 2016 Optical Society of America

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References

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  1. A. D. DiChiara, S. Ghmire, D. A. Reis, L. F. DiMauro, and P. Agostini, Strong-field and Attosecond Physics with Mid-infrared Lasers in Attosecond Physics (Springer-Verlag, 2013), pp. 81–98.
  2. K.-H. Hong, C.-J. Lai, J. P. Siqueira, P. Krogen, J. Moses, C.-L. Chang, G. J. Stein, L. E. Zapata, and F. X. Kärtner, “Multi-mJ, kHz, 2.1 μm optical parametric chirped-pulse amplifier and high-flux soft x-ray high-harmonic generation,” Opt. Lett. 39(11), 3145–3148 (2014).
    [Crossref] [PubMed]
  3. M. Hemmer, D. Sánchez, M. Jelínek, V. Smirnov, H. Jelinkova, V. Kubeček, and J. Biegert, “2-μm wavelength, high-energy Ho:YLF chirped-pulse amplifier for mid-infrared OPCPA,” Opt. Lett. 40(4), 451–454 (2015).
    [Crossref] [PubMed]
  4. L. von Grafenstein, M. Bock, D. Ueberschaer, U. Griebner, and T. Elsaesser, “Picosecond 34 mJ pulses at kHz repetition rates from a Ho:YLF amplifier at 2 µm wavelength,” Opt. Express 23(26), 33142–33149 (2015).
    [Crossref] [PubMed]
  5. P. Kroetz, A. Ruehl, G. Chatterjee, A.-L. Calendron, K. Murari, H. Cankaya, P. Li, F. X. Kärtner, I. Hartl, and R. J. D. Miller, “Overcoming bifurcation instability in high-repetition-rate Ho:YLF regenerative amplifiers,” Opt. Lett. 40(23), 5427–5430 (2015).
    [Crossref] [PubMed]
  6. S. Klingebiel, C. Wandt, C. Skrobol, I. Ahmad, S. A. Trushin, Z. Major, F. Krausz, and S. Karsch, “High energy picosecond Yb:YAG CPA system at 10 Hz repetition rate for pumping optical parametric amplifiers,” Opt. Express 19(6), 5357–5363 (2011).
    [Crossref] [PubMed]
  7. C. P. J. Barty, G. Korn, F. Raksi, C. Rose-Petruck, J. Squier, A.-C. Tien, K. R. Wilson, V. V. Yakovlev, and K. Yamakawa, “Regenerative pulse shaping and amplification of ultrabroadband optical pulses,” Opt. Lett. 21(3), 219–221 (1996).
    [Crossref] [PubMed]
  8. P. Malevich, G. Andriukaitis, T. Flöry, A. J. Verhoef, A. Fernández, S. Ališauskas, A. Pugžlys, A. Baltuška, L. H. Tan, C. F. Chua, and P. B. Phua, “High energy and average power femtosecond laser for driving mid-infrared optical parametric amplifiers,” Opt. Lett. 38(15), 2746–2749 (2013).
    [Crossref] [PubMed]
  9. K. Murari, H. Cankaya, P. Kroetz, G. Cirmi, P. Li, A. Ruehl, I. Hartl, and F. X. Kärtner, “Intracavity gain shaping in millijoule-level, high gain Ho:YLF regenerative amplifiers,” Opt. Lett. 41(6), 1114–1117 (2016).
    [Crossref] [PubMed]
  10. K. Yamakawa, M. Aoyama, S. Matsuoka, H. Takuma, D. N. Fittinghoff, and C. P. J. Barty, “Ultrahigh-peak and high-average power chirped-pulse amplification of sub-20-fs pulses with Ti:Sapphire amplifiers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 385–394 (1998).
    [Crossref]
  11. W. F. Krupke and L. L. Chase, “Ground-state depleted solid-state lasers: principles, characteristics and scaling,” Opt. Quantum Electron. 22(S1), S1–S22 (1990).
    [Crossref]
  12. A. Dergachev, “High-energy, kHz-rate, picosecond, 2-µm laser pump source for mid-IR nonlinear optical devices,” Proc. SPIE 8599, 85990B (2013).
    [Crossref]
  13. L. M. Frantz and J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34(8), 2346–2349 (1963).
    [Crossref]
  14. W. Koechner, Solid-State Laser Engineering, 4th ed. (Springer-Verlag, 1996), pp. 158–183.
  15. P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in Ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
    [Crossref]
  16. A. E. Siegman, Lasers (University Science Books, 1986), p. 14.
  17. R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, “Optical properties of Nd3+ pairs in LiYF4-existence of a short range interaction,” J. Lumin. 34(6), 295–305 (1986).
    [Crossref]
  18. K. Mecseki, D. Bigourd, S. Patankar, N. H. Stuart, and R. A. Smith, “Flat-top picosecond pulses generated by chirped spectral modulation from a Nd:YLF regenerative amplifier for pumping few-cycle optical parametric amplifiers,” Appl. Opt. 53(10), 2229–2235 (2014).
    [Crossref] [PubMed]
  19. E. Sorokin, G. Tempea, and T. Brabec, “Measurement of the root-mean-square width and the root-mean-square chirp in ultrafast optics,” J. Opt. Soc. Am. B 17(1), 146–150 (2000).
    [Crossref]
  20. O. J. P. Collet, “Modeling of end-pumped Ho:YLF amplifiers,” M.Sc. thesis (University Stellenbosch, 2013).

2016 (1)

2015 (3)

2014 (2)

2013 (2)

2011 (1)

2005 (1)

P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in Ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
[Crossref]

2000 (1)

1998 (1)

K. Yamakawa, M. Aoyama, S. Matsuoka, H. Takuma, D. N. Fittinghoff, and C. P. J. Barty, “Ultrahigh-peak and high-average power chirped-pulse amplification of sub-20-fs pulses with Ti:Sapphire amplifiers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 385–394 (1998).
[Crossref]

1996 (1)

1990 (1)

W. F. Krupke and L. L. Chase, “Ground-state depleted solid-state lasers: principles, characteristics and scaling,” Opt. Quantum Electron. 22(S1), S1–S22 (1990).
[Crossref]

1986 (1)

R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, “Optical properties of Nd3+ pairs in LiYF4-existence of a short range interaction,” J. Lumin. 34(6), 295–305 (1986).
[Crossref]

1963 (1)

L. M. Frantz and J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34(8), 2346–2349 (1963).
[Crossref]

Ahmad, I.

Ališauskas, S.

Andriukaitis, G.

Aoyama, M.

K. Yamakawa, M. Aoyama, S. Matsuoka, H. Takuma, D. N. Fittinghoff, and C. P. J. Barty, “Ultrahigh-peak and high-average power chirped-pulse amplification of sub-20-fs pulses with Ti:Sapphire amplifiers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 385–394 (1998).
[Crossref]

Balembois, F.

P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in Ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
[Crossref]

Baltuška, A.

Barthem, R. B.

R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, “Optical properties of Nd3+ pairs in LiYF4-existence of a short range interaction,” J. Lumin. 34(6), 295–305 (1986).
[Crossref]

Barty, C. P. J.

K. Yamakawa, M. Aoyama, S. Matsuoka, H. Takuma, D. N. Fittinghoff, and C. P. J. Barty, “Ultrahigh-peak and high-average power chirped-pulse amplification of sub-20-fs pulses with Ti:Sapphire amplifiers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 385–394 (1998).
[Crossref]

C. P. J. Barty, G. Korn, F. Raksi, C. Rose-Petruck, J. Squier, A.-C. Tien, K. R. Wilson, V. V. Yakovlev, and K. Yamakawa, “Regenerative pulse shaping and amplification of ultrabroadband optical pulses,” Opt. Lett. 21(3), 219–221 (1996).
[Crossref] [PubMed]

Biegert, J.

Bigourd, D.

Bock, M.

Brabec, T.

Buisson, R.

R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, “Optical properties of Nd3+ pairs in LiYF4-existence of a short range interaction,” J. Lumin. 34(6), 295–305 (1986).
[Crossref]

Calendron, A.-L.

Cankaya, H.

Chang, C.-L.

Chase, L. L.

W. F. Krupke and L. L. Chase, “Ground-state depleted solid-state lasers: principles, characteristics and scaling,” Opt. Quantum Electron. 22(S1), S1–S22 (1990).
[Crossref]

Chatterjee, G.

Chua, C. F.

Cirmi, G.

Dergachev, A.

A. Dergachev, “High-energy, kHz-rate, picosecond, 2-µm laser pump source for mid-IR nonlinear optical devices,” Proc. SPIE 8599, 85990B (2013).
[Crossref]

Druon, F.

P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in Ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
[Crossref]

Elsaesser, T.

Fernández, A.

Fittinghoff, D. N.

K. Yamakawa, M. Aoyama, S. Matsuoka, H. Takuma, D. N. Fittinghoff, and C. P. J. Barty, “Ultrahigh-peak and high-average power chirped-pulse amplification of sub-20-fs pulses with Ti:Sapphire amplifiers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 385–394 (1998).
[Crossref]

Flöry, T.

Frantz, L. M.

L. M. Frantz and J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34(8), 2346–2349 (1963).
[Crossref]

Georges, P.

P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in Ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
[Crossref]

Griebner, U.

Harmand, H.

R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, “Optical properties of Nd3+ pairs in LiYF4-existence of a short range interaction,” J. Lumin. 34(6), 295–305 (1986).
[Crossref]

Hartl, I.

Hemmer, M.

Hong, K.-H.

Jelínek, M.

Jelinkova, H.

Karsch, S.

Kärtner, F. X.

Klingebiel, S.

Korn, G.

Krausz, F.

Kroetz, P.

Krogen, P.

Krupke, W. F.

W. F. Krupke and L. L. Chase, “Ground-state depleted solid-state lasers: principles, characteristics and scaling,” Opt. Quantum Electron. 22(S1), S1–S22 (1990).
[Crossref]

Kubecek, V.

Lai, C.-J.

Li, P.

Major, Z.

Malevich, P.

Matsuoka, S.

K. Yamakawa, M. Aoyama, S. Matsuoka, H. Takuma, D. N. Fittinghoff, and C. P. J. Barty, “Ultrahigh-peak and high-average power chirped-pulse amplification of sub-20-fs pulses with Ti:Sapphire amplifiers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 385–394 (1998).
[Crossref]

Mecseki, K.

Miller, R. J. D.

Moses, J.

Murari, K.

Nodvik, J. S.

L. M. Frantz and J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34(8), 2346–2349 (1963).
[Crossref]

Patankar, S.

Phua, P. B.

Pugžlys, A.

Raksi, F.

Raybaut, P.

P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in Ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
[Crossref]

Rose-Petruck, C.

Ruehl, A.

Sánchez, D.

Siqueira, J. P.

Skrobol, C.

Smirnov, V.

Smith, R. A.

Sorokin, E.

Squier, J.

Stein, G. J.

Stuart, N. H.

Takuma, H.

K. Yamakawa, M. Aoyama, S. Matsuoka, H. Takuma, D. N. Fittinghoff, and C. P. J. Barty, “Ultrahigh-peak and high-average power chirped-pulse amplification of sub-20-fs pulses with Ti:Sapphire amplifiers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 385–394 (1998).
[Crossref]

Tan, L. H.

Tempea, G.

Tien, A.-C.

Trushin, S. A.

Ueberschaer, D.

Verhoef, A. J.

Vial, J. C.

R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, “Optical properties of Nd3+ pairs in LiYF4-existence of a short range interaction,” J. Lumin. 34(6), 295–305 (1986).
[Crossref]

von Grafenstein, L.

Wandt, C.

Wilson, K. R.

Yakovlev, V. V.

Yamakawa, K.

K. Yamakawa, M. Aoyama, S. Matsuoka, H. Takuma, D. N. Fittinghoff, and C. P. J. Barty, “Ultrahigh-peak and high-average power chirped-pulse amplification of sub-20-fs pulses with Ti:Sapphire amplifiers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 385–394 (1998).
[Crossref]

C. P. J. Barty, G. Korn, F. Raksi, C. Rose-Petruck, J. Squier, A.-C. Tien, K. R. Wilson, V. V. Yakovlev, and K. Yamakawa, “Regenerative pulse shaping and amplification of ultrabroadband optical pulses,” Opt. Lett. 21(3), 219–221 (1996).
[Crossref] [PubMed]

Zapata, L. E.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in Ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
[Crossref]

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

K. Yamakawa, M. Aoyama, S. Matsuoka, H. Takuma, D. N. Fittinghoff, and C. P. J. Barty, “Ultrahigh-peak and high-average power chirped-pulse amplification of sub-20-fs pulses with Ti:Sapphire amplifiers,” IEEE J. Sel. Top. Quantum Electron. 4(2), 385–394 (1998).
[Crossref]

J. Appl. Phys. (1)

L. M. Frantz and J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34(8), 2346–2349 (1963).
[Crossref]

J. Lumin. (1)

R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, “Optical properties of Nd3+ pairs in LiYF4-existence of a short range interaction,” J. Lumin. 34(6), 295–305 (1986).
[Crossref]

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

Opt. Express (2)

Opt. Lett. (6)

C. P. J. Barty, G. Korn, F. Raksi, C. Rose-Petruck, J. Squier, A.-C. Tien, K. R. Wilson, V. V. Yakovlev, and K. Yamakawa, “Regenerative pulse shaping and amplification of ultrabroadband optical pulses,” Opt. Lett. 21(3), 219–221 (1996).
[Crossref] [PubMed]

P. Malevich, G. Andriukaitis, T. Flöry, A. J. Verhoef, A. Fernández, S. Ališauskas, A. Pugžlys, A. Baltuška, L. H. Tan, C. F. Chua, and P. B. Phua, “High energy and average power femtosecond laser for driving mid-infrared optical parametric amplifiers,” Opt. Lett. 38(15), 2746–2749 (2013).
[Crossref] [PubMed]

K. Murari, H. Cankaya, P. Kroetz, G. Cirmi, P. Li, A. Ruehl, I. Hartl, and F. X. Kärtner, “Intracavity gain shaping in millijoule-level, high gain Ho:YLF regenerative amplifiers,” Opt. Lett. 41(6), 1114–1117 (2016).
[Crossref] [PubMed]

P. Kroetz, A. Ruehl, G. Chatterjee, A.-L. Calendron, K. Murari, H. Cankaya, P. Li, F. X. Kärtner, I. Hartl, and R. J. D. Miller, “Overcoming bifurcation instability in high-repetition-rate Ho:YLF regenerative amplifiers,” Opt. Lett. 40(23), 5427–5430 (2015).
[Crossref] [PubMed]

K.-H. Hong, C.-J. Lai, J. P. Siqueira, P. Krogen, J. Moses, C.-L. Chang, G. J. Stein, L. E. Zapata, and F. X. Kärtner, “Multi-mJ, kHz, 2.1 μm optical parametric chirped-pulse amplifier and high-flux soft x-ray high-harmonic generation,” Opt. Lett. 39(11), 3145–3148 (2014).
[Crossref] [PubMed]

M. Hemmer, D. Sánchez, M. Jelínek, V. Smirnov, H. Jelinkova, V. Kubeček, and J. Biegert, “2-μm wavelength, high-energy Ho:YLF chirped-pulse amplifier for mid-infrared OPCPA,” Opt. Lett. 40(4), 451–454 (2015).
[Crossref] [PubMed]

Opt. Quantum Electron. (1)

W. F. Krupke and L. L. Chase, “Ground-state depleted solid-state lasers: principles, characteristics and scaling,” Opt. Quantum Electron. 22(S1), S1–S22 (1990).
[Crossref]

Proc. SPIE (1)

A. Dergachev, “High-energy, kHz-rate, picosecond, 2-µm laser pump source for mid-IR nonlinear optical devices,” Proc. SPIE 8599, 85990B (2013).
[Crossref]

Other (4)

W. Koechner, Solid-State Laser Engineering, 4th ed. (Springer-Verlag, 1996), pp. 158–183.

A. D. DiChiara, S. Ghmire, D. A. Reis, L. F. DiMauro, and P. Agostini, Strong-field and Attosecond Physics with Mid-infrared Lasers in Attosecond Physics (Springer-Verlag, 2013), pp. 81–98.

A. E. Siegman, Lasers (University Science Books, 1986), p. 14.

O. J. P. Collet, “Modeling of end-pumped Ho:YLF amplifiers,” M.Sc. thesis (University Stellenbosch, 2013).

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Figures (11)

Fig. 1
Fig. 1 (a) Absorption and emission cross sections σabs and σem of Ho:YLF [12] at room temperature. (b) Gain cross sections σg for different inverted fractions. Values below and above zero cause absorption and amplification, respectively. Calculated with Eq. (3) and the cross sections from (a).
Fig. 2
Fig. 2 Comparison of (a) the classical and (b) the modified FN equations. While the classical equations define and utilize a stored fluence Jstor the modified equations circumvent this expression by using the inverted fraction β.
Fig. 3
Fig. 3 Simulation of a single pass of the fluence Ji-1 through a gain medium. (1): The incoming fluence is sliced into n equal fluence slices. (2): Each fluence slice consecutively passes the gain medium, in which the inverted fraction β is updated between the passes. (3): The sum of all fluence slices results in the total output fluence.
Fig. 4
Fig. 4 (a) Comparison between the build-up of the inverted fraction β simulated with rate equations and FN equations with and without correction of the inversion decay losses. To evaluate the effect of the life time, (b) and (c) presents simulation results with varied life times.
Fig. 5
Fig. 5 (a) Flow-chart for consecutive pump and amplification cycles. For single pass pumping, the pump fluence only passes once the gain medium, whereas the seed fluence is amplified for 2N amplification passes (N round trips). (b) The incoming seed or pump fluence is sliced into n slices and each slice individually passes the gain medium. After each slice, the inverted fraction is consecutively updated.
Fig. 6
Fig. 6 (a) Setup of our Ho:YLF RA including intra-cavity etalon. (b) Consecutive order of passes through the gain shaping elements during amplification in the RA. TFP, thin film polarizer; PC, Pockels cell.
Fig. 7
Fig. 7 (a) Measured and simulated output spectra of the Ho:YLF RA without intra-cavity etalon and (b) and (c) with different intra-cavity etalons. The black dash-dotted curve shows the transmission function of the etalon that was used for the corresponding simulation. Experimental data taken from [9].
Fig. 8
Fig. 8 Change in the spectral FWHM of the output pulses in dependence of the pump power for three different system configurations. Pulses are coupled out with a constant pulse energy of 0.5 mJ, which was adjusted by the the pump power and round trip number RT. An increased pump power follows an increased inverted fraction β in the gain medium. The numbers in the figure represent the number of round trip at which the pulses are coupled out.
Fig. 9
Fig. 9 (a) Parameter scan over the etalon parameters etalon thickness L and etalon angle Θ (for an etalon surface reflection of 3.3%) and (d) over surface reflection R and etalon angle (for an etalon thickness of 239.46 µm). Color represents the spectral FWHM λFWHM of the output pulses. Cuts through (a) along the etalon angle and along the etalon thickness are presented in (b) and (c), respectively. Two selected output spectra (marked with the two crosses in (d)) are presented in (e) and the calculated Fourier limited pulse shapes of these two spectra in (f).
Fig. 10
Fig. 10 (a) Calculated Fourier limited FWHM pulse duration as a function of the etalon thickness for 4 different surface reflections. (b) Normalized pulse contrast, as a measure for the pulse quality, for the same surface reflections.
Fig. 11
Fig. 11 Comparison of three etalon thicknesses (a) 240 µm, (b) 200 µm and (c) 150 µm as a function of the surface reflection. The analysis has been conducted in terms of the rms and FWHM pulse duration (upper row), and in terms of normalized pulse contrast and peak power (lower row).

Tables (1)

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Table 1 Variables and Constants Used in This Paper

Equations (23)

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β= n e N total ,
G i1 (λ)=exp( σ g,i1 (λ)Nl),
σ g,i1 (λ)= β i1 ( σ em (λ)+ σ abs (λ)) σ abs (λ),
J i (λ)= J sat (λ)T(λ)ln(1+ G i1 (λ)exp( J i1 (λ) J sat (λ) )1)),
J sat (λ)= hc λ( σ abs (λ)+ σ em (λ)) ,
β i = β i1 [ λ( J i (λ) T(λ) J i1 (λ) ) ]dλ hclN .
β i * = β i exp( Δt τ gain ),
T etalon (λ)= 1 1+ 4R (1R) 2 sin 2 ( π λ nLcos(Θ) ) ,
T * (λ)=T(λ) T etalon (λ).
τ rms = t 2 t 2 ,
with t n = 1 N t n I(t)dt and N= I(t)dt ,
NPC= τ FWHM τ rms 2 2ln2 τ FWHM τ rms 2.355 .
g i = g i1 +Δ g i = g i1 + Δ J stor,i J sat l ,
β i = β i1 Δ β i = β i1 λΔ J i hclN .
Δ J stor = J stor,i J stor,i1 =( J i J i1 )=Δ J i ,
g i = g i1 Δ J i J sat l ,
J sat = hc λ( σ em + σ abs ) ,
g x =N( β x ( σ em + σ abs ) σ abs ),
β i = β i1 λΔ J i hclN = β i1 Δ β i .
Δ β i λΔ J i (λ)dλ .
β i = β i1 [ λ( J i (λ) T(λ) J i1 (λ) ) ]dλ hclN .
β i = β i1 Δ E i / E Ph NlA .
β i = β i1 Δ N Ph,i N total = β i1 + Δ n e,i N total .

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