Abstract

The absorption and the kinetics of the emission in the mid-infrared (mid-IR) were investigated in AgClxBr1x crystals doped with Dy3+ ions. Strong emission bands were detected at 3, 4.4, and 5.5μm and attributed to the H13/26H15/26, H11/26H13/26, and F11/26+H9/26H11/26 transitions. Various optical parameters were calculated for the Dy3+ doped crystals, using the Judd–Ofelt approximation and the rate equations. The measured results and the calculated parameters indicate that these doped crystals could be used for the development of mid-IR solid-state lasers or mid-IR fiber lasers.

© 2011 Optical Society of America

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References

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  1. J. Mulrooney, J. Clifford, C. Fitzpatrick, P. Chambers, and E. Lewis, “Monitoring of carbon dioxide exhaust emissions using mid-infrared spectroscopy,” J. Opt. A Pure Appl. Opt. 9, S87–S91 (2007).
    [CrossRef]
  2. I. Shafir, L. Nagli, and A. Katzir, “Raman spectroscopy of rare earth doped silver halide crystals,” Appl. Phys. Lett. 94, 231907 (2009).
    [CrossRef]
  3. L. Nagli, O. Gayer, and A. Katzir, “Middle-infrared luminescence of praseodymium ions in silver halide crystals and fibers,” Opt. Lett. 30, 1831–1833 (2005).
    [CrossRef] [PubMed]
  4. G. Brodetzki, O. Gayer, I. Shafir, L. Nagli, and A. Katzir, “Middle infrared luminescence of Tb3+ in silver halide crystals and fibers,” J. Lumin. 128, 1323–1330 (2008).
    [CrossRef]
  5. I. Shafir, O. Gayer, L. Nagli, S. Shalem, and A. Katzir, “Middle infrared luminescence of Nd-ions in silver halide crystals,” J. Lumin. 126, 541–546 (2007).
    [CrossRef]
  6. S. Browman, J. Ganem, B. Feldman, and A. Kueny, “Infrared laser characteristics of praseodymium-doped lanthanum trichloride,” IEEE J. Quantum Electron. 30, 2925–2928 (1994).
    [CrossRef]
  7. S. Browman, L. Shaw, B. Feldman, and J. Ganem, “A 7 μmpraseodymium-based solid-state laser,” IEEE J. Quantum Electron. 32, 646–649 (1996).
    [CrossRef]
  8. N. Djeu, V. E. Hartwell, A. A. Kaminskii, and A. V. Butashin, “Room-temperature 3.4 μmDy:BaYb2F8 laser,” Opt. Lett. 22, 997–999 (1997).
    [CrossRef] [PubMed]
  9. N. P. Barnes and R. E. Allen, “Room temperature Dy:YLF laser operation at 4.34 μm,” IEEE J. Quantum Electron. 27, 277–282 (1991).
    [CrossRef]
  10. M. C. Nostrand, R. H. Page, S. A. Payne, W. F. Krupke, and P. G. Schunemann, “Room-temperature laser action at 4.3–4.4 μm in CaGa2S4:Dy3+,” Opt. Lett. 24, 1215–1217(1999).
    [CrossRef]
  11. R. S. Quimby, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Modeling of cascade lasing in Dy: chalcogenide glass fiber laser with efficient output at 4.5 μm,” IEEE Photon. Technol. Lett. 20, 123–125 (2008).
    [CrossRef]
  12. B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127, 750–761 (1962).
    [CrossRef]
  13. G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37, 511–520 (1962).
    [CrossRef]
  14. M. J. Weber, “Probabilities for radiative and nonradiative decay of Er3+ in LaF3,” Phys. Rev. 157, 262–272 (1967).
    [CrossRef]
  15. L. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn, dn and fn Configuration (Massachusetts Institute of Technology, 1963).
    [PubMed]
  16. M. J. Weber, “Selective excitation and decay of Er3+ fluorescence in LaF3,” Phys. Rev. 156, 231–241 (1967)
    [CrossRef]
  17. The Art and Science of Growing Crystals, J.J.Gilman, ed. (Wiley, 1963).
  18. I. Shafir, L. Nagli, and A. Katzir, “Temperature dependence of middle infrared absorption lines in silver halide crystals doped with Pr3+, Dy3+, and Nd3+ ions,” J. Appl. Phys. 108, 083106 (2010)
    [CrossRef]

2010 (1)

I. Shafir, L. Nagli, and A. Katzir, “Temperature dependence of middle infrared absorption lines in silver halide crystals doped with Pr3+, Dy3+, and Nd3+ ions,” J. Appl. Phys. 108, 083106 (2010)
[CrossRef]

2009 (1)

I. Shafir, L. Nagli, and A. Katzir, “Raman spectroscopy of rare earth doped silver halide crystals,” Appl. Phys. Lett. 94, 231907 (2009).
[CrossRef]

2008 (2)

G. Brodetzki, O. Gayer, I. Shafir, L. Nagli, and A. Katzir, “Middle infrared luminescence of Tb3+ in silver halide crystals and fibers,” J. Lumin. 128, 1323–1330 (2008).
[CrossRef]

R. S. Quimby, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Modeling of cascade lasing in Dy: chalcogenide glass fiber laser with efficient output at 4.5 μm,” IEEE Photon. Technol. Lett. 20, 123–125 (2008).
[CrossRef]

2007 (2)

I. Shafir, O. Gayer, L. Nagli, S. Shalem, and A. Katzir, “Middle infrared luminescence of Nd-ions in silver halide crystals,” J. Lumin. 126, 541–546 (2007).
[CrossRef]

J. Mulrooney, J. Clifford, C. Fitzpatrick, P. Chambers, and E. Lewis, “Monitoring of carbon dioxide exhaust emissions using mid-infrared spectroscopy,” J. Opt. A Pure Appl. Opt. 9, S87–S91 (2007).
[CrossRef]

2005 (1)

1999 (1)

1997 (1)

1996 (1)

S. Browman, L. Shaw, B. Feldman, and J. Ganem, “A 7 μmpraseodymium-based solid-state laser,” IEEE J. Quantum Electron. 32, 646–649 (1996).
[CrossRef]

1994 (1)

S. Browman, J. Ganem, B. Feldman, and A. Kueny, “Infrared laser characteristics of praseodymium-doped lanthanum trichloride,” IEEE J. Quantum Electron. 30, 2925–2928 (1994).
[CrossRef]

1991 (1)

N. P. Barnes and R. E. Allen, “Room temperature Dy:YLF laser operation at 4.34 μm,” IEEE J. Quantum Electron. 27, 277–282 (1991).
[CrossRef]

1967 (2)

M. J. Weber, “Probabilities for radiative and nonradiative decay of Er3+ in LaF3,” Phys. Rev. 157, 262–272 (1967).
[CrossRef]

M. J. Weber, “Selective excitation and decay of Er3+ fluorescence in LaF3,” Phys. Rev. 156, 231–241 (1967)
[CrossRef]

1962 (2)

B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127, 750–761 (1962).
[CrossRef]

G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37, 511–520 (1962).
[CrossRef]

Aggarwal, I. D.

R. S. Quimby, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Modeling of cascade lasing in Dy: chalcogenide glass fiber laser with efficient output at 4.5 μm,” IEEE Photon. Technol. Lett. 20, 123–125 (2008).
[CrossRef]

Allen, R. E.

N. P. Barnes and R. E. Allen, “Room temperature Dy:YLF laser operation at 4.34 μm,” IEEE J. Quantum Electron. 27, 277–282 (1991).
[CrossRef]

Barnes, N. P.

N. P. Barnes and R. E. Allen, “Room temperature Dy:YLF laser operation at 4.34 μm,” IEEE J. Quantum Electron. 27, 277–282 (1991).
[CrossRef]

Brodetzki, G.

G. Brodetzki, O. Gayer, I. Shafir, L. Nagli, and A. Katzir, “Middle infrared luminescence of Tb3+ in silver halide crystals and fibers,” J. Lumin. 128, 1323–1330 (2008).
[CrossRef]

Browman, S.

S. Browman, L. Shaw, B. Feldman, and J. Ganem, “A 7 μmpraseodymium-based solid-state laser,” IEEE J. Quantum Electron. 32, 646–649 (1996).
[CrossRef]

S. Browman, J. Ganem, B. Feldman, and A. Kueny, “Infrared laser characteristics of praseodymium-doped lanthanum trichloride,” IEEE J. Quantum Electron. 30, 2925–2928 (1994).
[CrossRef]

Butashin, A. V.

Chambers, P.

J. Mulrooney, J. Clifford, C. Fitzpatrick, P. Chambers, and E. Lewis, “Monitoring of carbon dioxide exhaust emissions using mid-infrared spectroscopy,” J. Opt. A Pure Appl. Opt. 9, S87–S91 (2007).
[CrossRef]

Clifford, J.

J. Mulrooney, J. Clifford, C. Fitzpatrick, P. Chambers, and E. Lewis, “Monitoring of carbon dioxide exhaust emissions using mid-infrared spectroscopy,” J. Opt. A Pure Appl. Opt. 9, S87–S91 (2007).
[CrossRef]

Djeu, N.

Feldman, B.

S. Browman, L. Shaw, B. Feldman, and J. Ganem, “A 7 μmpraseodymium-based solid-state laser,” IEEE J. Quantum Electron. 32, 646–649 (1996).
[CrossRef]

S. Browman, J. Ganem, B. Feldman, and A. Kueny, “Infrared laser characteristics of praseodymium-doped lanthanum trichloride,” IEEE J. Quantum Electron. 30, 2925–2928 (1994).
[CrossRef]

Fitzpatrick, C.

J. Mulrooney, J. Clifford, C. Fitzpatrick, P. Chambers, and E. Lewis, “Monitoring of carbon dioxide exhaust emissions using mid-infrared spectroscopy,” J. Opt. A Pure Appl. Opt. 9, S87–S91 (2007).
[CrossRef]

Ganem, J.

S. Browman, L. Shaw, B. Feldman, and J. Ganem, “A 7 μmpraseodymium-based solid-state laser,” IEEE J. Quantum Electron. 32, 646–649 (1996).
[CrossRef]

S. Browman, J. Ganem, B. Feldman, and A. Kueny, “Infrared laser characteristics of praseodymium-doped lanthanum trichloride,” IEEE J. Quantum Electron. 30, 2925–2928 (1994).
[CrossRef]

Gayer, O.

G. Brodetzki, O. Gayer, I. Shafir, L. Nagli, and A. Katzir, “Middle infrared luminescence of Tb3+ in silver halide crystals and fibers,” J. Lumin. 128, 1323–1330 (2008).
[CrossRef]

I. Shafir, O. Gayer, L. Nagli, S. Shalem, and A. Katzir, “Middle infrared luminescence of Nd-ions in silver halide crystals,” J. Lumin. 126, 541–546 (2007).
[CrossRef]

L. Nagli, O. Gayer, and A. Katzir, “Middle-infrared luminescence of praseodymium ions in silver halide crystals and fibers,” Opt. Lett. 30, 1831–1833 (2005).
[CrossRef] [PubMed]

Hartwell, V. E.

Judd, B. R.

B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127, 750–761 (1962).
[CrossRef]

Kaminskii, A. A.

Katzir, A.

I. Shafir, L. Nagli, and A. Katzir, “Temperature dependence of middle infrared absorption lines in silver halide crystals doped with Pr3+, Dy3+, and Nd3+ ions,” J. Appl. Phys. 108, 083106 (2010)
[CrossRef]

I. Shafir, L. Nagli, and A. Katzir, “Raman spectroscopy of rare earth doped silver halide crystals,” Appl. Phys. Lett. 94, 231907 (2009).
[CrossRef]

G. Brodetzki, O. Gayer, I. Shafir, L. Nagli, and A. Katzir, “Middle infrared luminescence of Tb3+ in silver halide crystals and fibers,” J. Lumin. 128, 1323–1330 (2008).
[CrossRef]

I. Shafir, O. Gayer, L. Nagli, S. Shalem, and A. Katzir, “Middle infrared luminescence of Nd-ions in silver halide crystals,” J. Lumin. 126, 541–546 (2007).
[CrossRef]

L. Nagli, O. Gayer, and A. Katzir, “Middle-infrared luminescence of praseodymium ions in silver halide crystals and fibers,” Opt. Lett. 30, 1831–1833 (2005).
[CrossRef] [PubMed]

Koster, G. F.

L. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn, dn and fn Configuration (Massachusetts Institute of Technology, 1963).
[PubMed]

Krupke, W. F.

Kueny, A.

S. Browman, J. Ganem, B. Feldman, and A. Kueny, “Infrared laser characteristics of praseodymium-doped lanthanum trichloride,” IEEE J. Quantum Electron. 30, 2925–2928 (1994).
[CrossRef]

Lewis, E.

J. Mulrooney, J. Clifford, C. Fitzpatrick, P. Chambers, and E. Lewis, “Monitoring of carbon dioxide exhaust emissions using mid-infrared spectroscopy,” J. Opt. A Pure Appl. Opt. 9, S87–S91 (2007).
[CrossRef]

Mulrooney, J.

J. Mulrooney, J. Clifford, C. Fitzpatrick, P. Chambers, and E. Lewis, “Monitoring of carbon dioxide exhaust emissions using mid-infrared spectroscopy,” J. Opt. A Pure Appl. Opt. 9, S87–S91 (2007).
[CrossRef]

Nagli, L.

I. Shafir, L. Nagli, and A. Katzir, “Temperature dependence of middle infrared absorption lines in silver halide crystals doped with Pr3+, Dy3+, and Nd3+ ions,” J. Appl. Phys. 108, 083106 (2010)
[CrossRef]

I. Shafir, L. Nagli, and A. Katzir, “Raman spectroscopy of rare earth doped silver halide crystals,” Appl. Phys. Lett. 94, 231907 (2009).
[CrossRef]

G. Brodetzki, O. Gayer, I. Shafir, L. Nagli, and A. Katzir, “Middle infrared luminescence of Tb3+ in silver halide crystals and fibers,” J. Lumin. 128, 1323–1330 (2008).
[CrossRef]

I. Shafir, O. Gayer, L. Nagli, S. Shalem, and A. Katzir, “Middle infrared luminescence of Nd-ions in silver halide crystals,” J. Lumin. 126, 541–546 (2007).
[CrossRef]

L. Nagli, O. Gayer, and A. Katzir, “Middle-infrared luminescence of praseodymium ions in silver halide crystals and fibers,” Opt. Lett. 30, 1831–1833 (2005).
[CrossRef] [PubMed]

Nielson, L. W.

L. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn, dn and fn Configuration (Massachusetts Institute of Technology, 1963).
[PubMed]

Nostrand, M. C.

Ofelt, G. S.

G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37, 511–520 (1962).
[CrossRef]

Page, R. H.

Payne, S. A.

Quimby, R. S.

R. S. Quimby, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Modeling of cascade lasing in Dy: chalcogenide glass fiber laser with efficient output at 4.5 μm,” IEEE Photon. Technol. Lett. 20, 123–125 (2008).
[CrossRef]

Sanghera, J. S.

R. S. Quimby, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Modeling of cascade lasing in Dy: chalcogenide glass fiber laser with efficient output at 4.5 μm,” IEEE Photon. Technol. Lett. 20, 123–125 (2008).
[CrossRef]

Schunemann, P. G.

Shafir, I.

I. Shafir, L. Nagli, and A. Katzir, “Temperature dependence of middle infrared absorption lines in silver halide crystals doped with Pr3+, Dy3+, and Nd3+ ions,” J. Appl. Phys. 108, 083106 (2010)
[CrossRef]

I. Shafir, L. Nagli, and A. Katzir, “Raman spectroscopy of rare earth doped silver halide crystals,” Appl. Phys. Lett. 94, 231907 (2009).
[CrossRef]

G. Brodetzki, O. Gayer, I. Shafir, L. Nagli, and A. Katzir, “Middle infrared luminescence of Tb3+ in silver halide crystals and fibers,” J. Lumin. 128, 1323–1330 (2008).
[CrossRef]

I. Shafir, O. Gayer, L. Nagli, S. Shalem, and A. Katzir, “Middle infrared luminescence of Nd-ions in silver halide crystals,” J. Lumin. 126, 541–546 (2007).
[CrossRef]

Shalem, S.

I. Shafir, O. Gayer, L. Nagli, S. Shalem, and A. Katzir, “Middle infrared luminescence of Nd-ions in silver halide crystals,” J. Lumin. 126, 541–546 (2007).
[CrossRef]

Shaw, L.

S. Browman, L. Shaw, B. Feldman, and J. Ganem, “A 7 μmpraseodymium-based solid-state laser,” IEEE J. Quantum Electron. 32, 646–649 (1996).
[CrossRef]

Shaw, L. B.

R. S. Quimby, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Modeling of cascade lasing in Dy: chalcogenide glass fiber laser with efficient output at 4.5 μm,” IEEE Photon. Technol. Lett. 20, 123–125 (2008).
[CrossRef]

Weber, M. J.

M. J. Weber, “Selective excitation and decay of Er3+ fluorescence in LaF3,” Phys. Rev. 156, 231–241 (1967)
[CrossRef]

M. J. Weber, “Probabilities for radiative and nonradiative decay of Er3+ in LaF3,” Phys. Rev. 157, 262–272 (1967).
[CrossRef]

Appl. Phys. Lett. (1)

I. Shafir, L. Nagli, and A. Katzir, “Raman spectroscopy of rare earth doped silver halide crystals,” Appl. Phys. Lett. 94, 231907 (2009).
[CrossRef]

IEEE J. Quantum Electron. (3)

S. Browman, J. Ganem, B. Feldman, and A. Kueny, “Infrared laser characteristics of praseodymium-doped lanthanum trichloride,” IEEE J. Quantum Electron. 30, 2925–2928 (1994).
[CrossRef]

S. Browman, L. Shaw, B. Feldman, and J. Ganem, “A 7 μmpraseodymium-based solid-state laser,” IEEE J. Quantum Electron. 32, 646–649 (1996).
[CrossRef]

N. P. Barnes and R. E. Allen, “Room temperature Dy:YLF laser operation at 4.34 μm,” IEEE J. Quantum Electron. 27, 277–282 (1991).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

R. S. Quimby, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Modeling of cascade lasing in Dy: chalcogenide glass fiber laser with efficient output at 4.5 μm,” IEEE Photon. Technol. Lett. 20, 123–125 (2008).
[CrossRef]

J. Appl. Phys. (1)

I. Shafir, L. Nagli, and A. Katzir, “Temperature dependence of middle infrared absorption lines in silver halide crystals doped with Pr3+, Dy3+, and Nd3+ ions,” J. Appl. Phys. 108, 083106 (2010)
[CrossRef]

J. Chem. Phys. (1)

G. S. Ofelt, “Intensities of crystal spectra of rare-earth ions,” J. Chem. Phys. 37, 511–520 (1962).
[CrossRef]

J. Lumin. (2)

G. Brodetzki, O. Gayer, I. Shafir, L. Nagli, and A. Katzir, “Middle infrared luminescence of Tb3+ in silver halide crystals and fibers,” J. Lumin. 128, 1323–1330 (2008).
[CrossRef]

I. Shafir, O. Gayer, L. Nagli, S. Shalem, and A. Katzir, “Middle infrared luminescence of Nd-ions in silver halide crystals,” J. Lumin. 126, 541–546 (2007).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

J. Mulrooney, J. Clifford, C. Fitzpatrick, P. Chambers, and E. Lewis, “Monitoring of carbon dioxide exhaust emissions using mid-infrared spectroscopy,” J. Opt. A Pure Appl. Opt. 9, S87–S91 (2007).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. (3)

M. J. Weber, “Selective excitation and decay of Er3+ fluorescence in LaF3,” Phys. Rev. 156, 231–241 (1967)
[CrossRef]

M. J. Weber, “Probabilities for radiative and nonradiative decay of Er3+ in LaF3,” Phys. Rev. 157, 262–272 (1967).
[CrossRef]

B. R. Judd, “Optical absorption intensities of rare-earth ions,” Phys. Rev. 127, 750–761 (1962).
[CrossRef]

Other (2)

L. W. Nielson and G. F. Koster, Spectroscopic Coefficients for the pn, dn and fn Configuration (Massachusetts Institute of Technology, 1963).
[PubMed]

The Art and Science of Growing Crystals, J.J.Gilman, ed. (Wiley, 1963).

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Figures (9)

Fig. 1
Fig. 1

Room temperature absorption spectrum of Dy-doped AgBr crystal.

Fig. 2
Fig. 2

Room temperature and low-temperature absorption cross sections of a Dy-doped AgBr 0.5 Cl 0.5 crystal.

Fig. 3
Fig. 3

Room temperature emission cross section of a Dy:AgBr crystal excited by an Nd:YAG laser at λ = 1.29 μm . (a)  H 11 / 2 6 H 15 / 2 6 ; (b)  F 6 11 / 2 + H 6 9 / 2 H 6 13 / 2 ; (c)  H 6 13 / 2 H 6 15 / 2 ; (d)  H 6 11 / 2 H 6 13 / 2 ; (e)  F 6 11 / 2 + H 6 9 / 2 H 6 11 / 2 .

Fig. 4
Fig. 4

Room temperature and low-temperature emission cross section of Dy : AgBr 0.5 Cl 0.5 crystal, excited by an Nd:YAG laser at λ = 1.29 μm .

Fig. 5
Fig. 5

Time dependence of the luminescence emission at 4 5 μm for Dy-doped AgBr 0.5 Cl 0.5 , AgBr, and AgCl crystals that were excited at room temperature by an OPO at λ = 1.3 μm .

Fig. 6
Fig. 6

Time dependence of the luminescence emission at 4 5 μm for Dy : AgBr 0.5 Cl 0.5 crystals that were excited at room temperature and at a low temperature by an OPO emitting at 1.3 μm .

Fig. 7
Fig. 7

Time dependence of the luminescence emission at 3 μm for a Dy-doped AgBr 0.5 Cl 0.5 crystal, which was excited at 1.3 and at 1.6 μm by an OPO at room temperature, and the fit results.

Fig. 8
Fig. 8

Time dependence of the luminescence emission at 4 5 μm for Dy-doped AgBr 0.5 Cl 0.5 crystals that were excited at 1.3 μm by an OPO at room temperature and the fit results.

Fig. 9
Fig. 9

Energy scheme of Dy.

Tables (2)

Tables Icon

Table 1 Luminescence Parameters for the Dy : AgBr 0.5 Cl 0.5 Crystal

Tables Icon

Table 2 Kinetics Parameters for Dy-Doped Silver Halide Crystals

Equations (8)

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

S meas ( J , J ) = 3 c h ( 2 J + 1 ) 8 π 3 N λ ¯ e 2 9 n ( n 2 + 2 ) 2 α ( λ ) d λ .
S ed = i = 2 , 4 , 6 Ω i | S , L , J | | U ( i ) | | S , L , J | 2 ,
S md = ( e h 4 m π c ) 2 | S , L , J | | L + 2 S | | S , L , J | 2 .
A J J = 64 π 2 e 2 3 h ( 2 J + 1 ) λ ¯ 3 n ( n 2 + 2 ) 2 9 S calc ( J , J ) .
τ rad = 1 / Σ J A J J , β = τ rad A J J .
σ a , b ( λ ) = A a , b 8 π n 2 c λ 5 I ( λ ) λ I ( λ ) d λ .
d N 4 d t = ( W 43 + W 42 + W 41 ) N 4 , d N 3 d t = ( W 32 + W 31 ) N 3 + W 43 N 4 , d N 2 d t = W 21 N 2 + W 32 N 3 + W 42 N 4 , d N 1 d t = W 21 N 2 + W 31 N 3 + W 41 N 4 .
W i j [ 1 / sec ] = [ 0 0 0 0 49 0 0 0 70.2 14.8 0 0 351 53 50.5 0 ] .

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