Abstract

We present a generally applicable theoretical model describing excited-state decay lifetime analysis of metal ions in a host crystal matrix. In contrast to common practice, we include multi-phonon non-radiative transitions competitively to the radiative one. We have applied our theory to Co2+ ions in a mixed AgCl0.5Br0.5 crystal, and as opposed to a previous analysis, find excellent agreement between theory and experiment over the entire measured temperature range. The fit predicts a zero absolute temperature radiative lifetime τrad(0) = 5.5 ms, more than three times longer than the measured effective low-temperature one τeff(0) = 1.48 ms. Furthermore, the fit configuration potential dissociation energy has been estimated as D = 2500 cm−1 and the lattice vibrational cutoff frequency as ħωco = 180 cm−1. We have experimentally verified the latter by optical reflection measurement in the far-IR.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. S. B. Mirov, V. V. Fedorov, D. Martyshkin, I. S. Moskalev, M. Mirov, and S. Vasilyev, “Progress in mid-IR lasers based on Cr and Fe-doped II-VI chalcogenides,” IEEE J. Sel. Top. Quantum Electron,  21, 1601719 (2015).
    [Crossref]
  2. I. T. Sorokina and E. Sorokin, “Femtosecond Cr2+- based lasers”, IEEE J. Sel. Top. Quantum Electron. 21, 1601519 (2015).
    [Crossref]
  3. Y. Kalisky, The Physics and Engineering of Solid State Lasers(SPIE, Bellingham, WA, 2006).
    [Crossref]
  4. Y. V. Orlovskii, K. K. Pukhov, T. T. Basiev, and T. Tsuboi, “Nonlinear mechanism of multiphonon relaxation of the energy of electronic excitation in optical crystals doped with rare-earth ions,” Opt. Mater. 4, 583–595 (1995).
    [Crossref]
  5. Y. V. Orlovskii, R. J. Reeves, R. C. Powell, T. T. Basiev, and K. K. Pukhov, “Multiple-phonon nonradiative relaxation: Experimental rates in fluoride crystals doped with Er3+ and Nd3+ ions and a theoretical model,” Phys. Rev. B 49, 3821–3830 (1994).
    [Crossref]
  6. Y. V. Orlovskii, T. T. Basiev, I. N. Vorobiev, E. O. Orlovskaya, N. P. Barnes, and S. B. Mirov, “Temperature dependencies of excited states lifetimes and relaxation rates of 3–5 phonon (4–6 µm) transitions in the YAG, LuAG and YLF crystals doped with trivalent holmium, thulium, and erbium,” Opt. Mater. 18, 355–365 (2002).
    [Crossref]
  7. L. A. Riseberg and H. W. Moos, “Multiphonon orbit-lattice relaxation of excited states of rare-earth ions in crystals,” Phys. Rev. 174, 429–438 (1968).
    [Crossref]
  8. M. J. Weber, “Multiphonon relaxation of rare-earth ions in yttrium orthoaluminate,” Phys. Rev. B 8, 54–64 (1973).
    [Crossref]
  9. C. B. Layne, W. H. Lowdermilk, and M. J. Weber, “Multiphonon relaxation of rare-earth ions in oxide glasses,” Phys. Rev. B 16, 10–20 (1977).
    [Crossref]
  10. Z. Burshtein, “Radiative, nonradiative, and mixed-decay transitions of rare-earth ions in dielectric media,” Opt. Eng. 49, 091005 (2010).
    [Crossref]
  11. Y. Tsur, S. Goldring, E. Galun, and A. Katzir, “Ground state depletion - a step towards mid-IR lasing of doped silver halides,” J. Luminescence 175, 113–116 (2016).
    [Crossref]
  12. V. B. Fowler and D. L. Dexter, “Relation between absorption and emission probabilities in luminescent centers in ionic solids,” Phys. Rev. 128, 2154–2165 (1962).
    [Crossref]
  13. P. M. Morse, “Diatomic molecules according to the wave mechanics. II. Vibrational levels,” Phys. Rev. 34, 57–64 (1929).
    [Crossref]
  14. C. Layne, W. Lowdermilk, and M. Weber, “Multiphonon relaxation of Rare-Earth ions in oxide glasses,” Phys. Rev. B 16, 10–20 (1977).
    [Crossref]
  15. I. Zakosky-Neuberger, I. Shafir, L. Nagli, and A. Katzir, “Optical and luminescence properties of Co2+:AgCl0.2Br0.8 crystals and their potential applications as gain media for middle-infrared lasers,” Appl. Phys. Lett. 99, 201111 (2011).
    [Crossref]
  16. H. Hecht, Z. Burshtein, A. Katzir, S. Noach, M. Sokol, E. Frumker, E. Galun, and A. A. Ishaaya, “Passive Q-switching of a Tm:YLF laser with a Co2+-doped silver halide saturable absorber,” Opt. Mater. 64, 64–69 (2017).
    [Crossref]
  17. A. Hadni, Essentials of Modern Physics Applied to the Study of the Infrared(Pergamon, 1967).
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    [Crossref] [PubMed]
  19. A. S. Korsakov, D. S. Vrublevsky, A. E. Lvov, and L. V. Zhukova, “Refractive index dispersion of AgCl1−xBrx (0<x<1) and Ag1−xTlxBr1−xlx(0<x<0.05),” Opt. Mater. 64, 40–46 (2017).
    [Crossref]
  20. A. Fujii, H. Stolz, and W. von der Osten, “Excitons and phonons in mixed silver halides studied by resonant Raman scattering,” J. Phys. C 16, 1713–1728 (1983).
    [Crossref]
  21. C. R. Berry, “Physical defects in silver halides,” Phys. Rev. 97, 677–679 (1955).
    [Crossref]
  22. A. Yariv, Introduction to Optical Electronics, (Holt-Reinhart-Winston, 1971).
  23. L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron,  32, 885–895 (1996).
    [Crossref]
  24. B. Henderson and R. H. Bartran, Crystal-Field Engineering of Solid-State Laser Materials(Cambridge University, 2000).
    [Crossref]
  25. A. Goldstein, P. Loiko, Z. Burshtein, N. Skoptsov, I. Glazunov, E. Galun, N. Kuleshov, and K. Yumashev, “Development of saturable absorbers for laser passive Q-switching near 1.5 µm based on transparent ceramic Co2+:MgAl2O4,” J. Am. Ceram. Soc. 99, 1324–1330 (2016).
    [Crossref]
  26. F. Seitz, The Modern Theory of Solids (McGraw-Hill, New York, 1940).
  27. W. Koechner, Solid-State Laser Engineering(Springer, 2006).
  28. S. Perets, R. Z. Shneck, R. Gajic, A. Golubovic, and Z. Burshtein, “Vibrational spectra of sodium gadolinium tungstate NaGd(WO4)2 single crystals: Observation od spatial dispersion,” Vib. Spectrosc. 49, 110–117 (2009).
    [Crossref]

2017 (2)

H. Hecht, Z. Burshtein, A. Katzir, S. Noach, M. Sokol, E. Frumker, E. Galun, and A. A. Ishaaya, “Passive Q-switching of a Tm:YLF laser with a Co2+-doped silver halide saturable absorber,” Opt. Mater. 64, 64–69 (2017).
[Crossref]

A. S. Korsakov, D. S. Vrublevsky, A. E. Lvov, and L. V. Zhukova, “Refractive index dispersion of AgCl1−xBrx (0<x<1) and Ag1−xTlxBr1−xlx(0<x<0.05),” Opt. Mater. 64, 40–46 (2017).
[Crossref]

2016 (2)

Y. Tsur, S. Goldring, E. Galun, and A. Katzir, “Ground state depletion - a step towards mid-IR lasing of doped silver halides,” J. Luminescence 175, 113–116 (2016).
[Crossref]

A. Goldstein, P. Loiko, Z. Burshtein, N. Skoptsov, I. Glazunov, E. Galun, N. Kuleshov, and K. Yumashev, “Development of saturable absorbers for laser passive Q-switching near 1.5 µm based on transparent ceramic Co2+:MgAl2O4,” J. Am. Ceram. Soc. 99, 1324–1330 (2016).
[Crossref]

2015 (2)

S. B. Mirov, V. V. Fedorov, D. Martyshkin, I. S. Moskalev, M. Mirov, and S. Vasilyev, “Progress in mid-IR lasers based on Cr and Fe-doped II-VI chalcogenides,” IEEE J. Sel. Top. Quantum Electron,  21, 1601719 (2015).
[Crossref]

I. T. Sorokina and E. Sorokin, “Femtosecond Cr2+- based lasers”, IEEE J. Sel. Top. Quantum Electron. 21, 1601519 (2015).
[Crossref]

2011 (1)

I. Zakosky-Neuberger, I. Shafir, L. Nagli, and A. Katzir, “Optical and luminescence properties of Co2+:AgCl0.2Br0.8 crystals and their potential applications as gain media for middle-infrared lasers,” Appl. Phys. Lett. 99, 201111 (2011).
[Crossref]

2010 (1)

Z. Burshtein, “Radiative, nonradiative, and mixed-decay transitions of rare-earth ions in dielectric media,” Opt. Eng. 49, 091005 (2010).
[Crossref]

2009 (1)

S. Perets, R. Z. Shneck, R. Gajic, A. Golubovic, and Z. Burshtein, “Vibrational spectra of sodium gadolinium tungstate NaGd(WO4)2 single crystals: Observation od spatial dispersion,” Vib. Spectrosc. 49, 110–117 (2009).
[Crossref]

2002 (1)

Y. V. Orlovskii, T. T. Basiev, I. N. Vorobiev, E. O. Orlovskaya, N. P. Barnes, and S. B. Mirov, “Temperature dependencies of excited states lifetimes and relaxation rates of 3–5 phonon (4–6 µm) transitions in the YAG, LuAG and YLF crystals doped with trivalent holmium, thulium, and erbium,” Opt. Mater. 18, 355–365 (2002).
[Crossref]

1996 (1)

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron,  32, 885–895 (1996).
[Crossref]

1995 (1)

Y. V. Orlovskii, K. K. Pukhov, T. T. Basiev, and T. Tsuboi, “Nonlinear mechanism of multiphonon relaxation of the energy of electronic excitation in optical crystals doped with rare-earth ions,” Opt. Mater. 4, 583–595 (1995).
[Crossref]

1994 (1)

Y. V. Orlovskii, R. J. Reeves, R. C. Powell, T. T. Basiev, and K. K. Pukhov, “Multiple-phonon nonradiative relaxation: Experimental rates in fluoride crystals doped with Er3+ and Nd3+ ions and a theoretical model,” Phys. Rev. B 49, 3821–3830 (1994).
[Crossref]

1993 (1)

1983 (1)

A. Fujii, H. Stolz, and W. von der Osten, “Excitons and phonons in mixed silver halides studied by resonant Raman scattering,” J. Phys. C 16, 1713–1728 (1983).
[Crossref]

1977 (2)

C. Layne, W. Lowdermilk, and M. Weber, “Multiphonon relaxation of Rare-Earth ions in oxide glasses,” Phys. Rev. B 16, 10–20 (1977).
[Crossref]

C. B. Layne, W. H. Lowdermilk, and M. J. Weber, “Multiphonon relaxation of rare-earth ions in oxide glasses,” Phys. Rev. B 16, 10–20 (1977).
[Crossref]

1973 (1)

M. J. Weber, “Multiphonon relaxation of rare-earth ions in yttrium orthoaluminate,” Phys. Rev. B 8, 54–64 (1973).
[Crossref]

1968 (1)

L. A. Riseberg and H. W. Moos, “Multiphonon orbit-lattice relaxation of excited states of rare-earth ions in crystals,” Phys. Rev. 174, 429–438 (1968).
[Crossref]

1962 (1)

V. B. Fowler and D. L. Dexter, “Relation between absorption and emission probabilities in luminescent centers in ionic solids,” Phys. Rev. 128, 2154–2165 (1962).
[Crossref]

1955 (1)

C. R. Berry, “Physical defects in silver halides,” Phys. Rev. 97, 677–679 (1955).
[Crossref]

1929 (1)

P. M. Morse, “Diatomic molecules according to the wave mechanics. II. Vibrational levels,” Phys. Rev. 34, 57–64 (1929).
[Crossref]

Barnes, N. P.

Y. V. Orlovskii, T. T. Basiev, I. N. Vorobiev, E. O. Orlovskaya, N. P. Barnes, and S. B. Mirov, “Temperature dependencies of excited states lifetimes and relaxation rates of 3–5 phonon (4–6 µm) transitions in the YAG, LuAG and YLF crystals doped with trivalent holmium, thulium, and erbium,” Opt. Mater. 18, 355–365 (2002).
[Crossref]

Bartran, R. H.

B. Henderson and R. H. Bartran, Crystal-Field Engineering of Solid-State Laser Materials(Cambridge University, 2000).
[Crossref]

Basiev, T. T.

Y. V. Orlovskii, T. T. Basiev, I. N. Vorobiev, E. O. Orlovskaya, N. P. Barnes, and S. B. Mirov, “Temperature dependencies of excited states lifetimes and relaxation rates of 3–5 phonon (4–6 µm) transitions in the YAG, LuAG and YLF crystals doped with trivalent holmium, thulium, and erbium,” Opt. Mater. 18, 355–365 (2002).
[Crossref]

Y. V. Orlovskii, K. K. Pukhov, T. T. Basiev, and T. Tsuboi, “Nonlinear mechanism of multiphonon relaxation of the energy of electronic excitation in optical crystals doped with rare-earth ions,” Opt. Mater. 4, 583–595 (1995).
[Crossref]

Y. V. Orlovskii, R. J. Reeves, R. C. Powell, T. T. Basiev, and K. K. Pukhov, “Multiple-phonon nonradiative relaxation: Experimental rates in fluoride crystals doped with Er3+ and Nd3+ ions and a theoretical model,” Phys. Rev. B 49, 3821–3830 (1994).
[Crossref]

Berry, C. R.

C. R. Berry, “Physical defects in silver halides,” Phys. Rev. 97, 677–679 (1955).
[Crossref]

Bunimovich, D.

Burshtein, Z.

H. Hecht, Z. Burshtein, A. Katzir, S. Noach, M. Sokol, E. Frumker, E. Galun, and A. A. Ishaaya, “Passive Q-switching of a Tm:YLF laser with a Co2+-doped silver halide saturable absorber,” Opt. Mater. 64, 64–69 (2017).
[Crossref]

A. Goldstein, P. Loiko, Z. Burshtein, N. Skoptsov, I. Glazunov, E. Galun, N. Kuleshov, and K. Yumashev, “Development of saturable absorbers for laser passive Q-switching near 1.5 µm based on transparent ceramic Co2+:MgAl2O4,” J. Am. Ceram. Soc. 99, 1324–1330 (2016).
[Crossref]

Z. Burshtein, “Radiative, nonradiative, and mixed-decay transitions of rare-earth ions in dielectric media,” Opt. Eng. 49, 091005 (2010).
[Crossref]

S. Perets, R. Z. Shneck, R. Gajic, A. Golubovic, and Z. Burshtein, “Vibrational spectra of sodium gadolinium tungstate NaGd(WO4)2 single crystals: Observation od spatial dispersion,” Vib. Spectrosc. 49, 110–117 (2009).
[Crossref]

DeLoach, L. D.

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron,  32, 885–895 (1996).
[Crossref]

Dexter, D. L.

V. B. Fowler and D. L. Dexter, “Relation between absorption and emission probabilities in luminescent centers in ionic solids,” Phys. Rev. 128, 2154–2165 (1962).
[Crossref]

Fedorov, V. V.

S. B. Mirov, V. V. Fedorov, D. Martyshkin, I. S. Moskalev, M. Mirov, and S. Vasilyev, “Progress in mid-IR lasers based on Cr and Fe-doped II-VI chalcogenides,” IEEE J. Sel. Top. Quantum Electron,  21, 1601719 (2015).
[Crossref]

Fowler, V. B.

V. B. Fowler and D. L. Dexter, “Relation between absorption and emission probabilities in luminescent centers in ionic solids,” Phys. Rev. 128, 2154–2165 (1962).
[Crossref]

Frumker, E.

H. Hecht, Z. Burshtein, A. Katzir, S. Noach, M. Sokol, E. Frumker, E. Galun, and A. A. Ishaaya, “Passive Q-switching of a Tm:YLF laser with a Co2+-doped silver halide saturable absorber,” Opt. Mater. 64, 64–69 (2017).
[Crossref]

Fujii, A.

A. Fujii, H. Stolz, and W. von der Osten, “Excitons and phonons in mixed silver halides studied by resonant Raman scattering,” J. Phys. C 16, 1713–1728 (1983).
[Crossref]

Gajic, R.

S. Perets, R. Z. Shneck, R. Gajic, A. Golubovic, and Z. Burshtein, “Vibrational spectra of sodium gadolinium tungstate NaGd(WO4)2 single crystals: Observation od spatial dispersion,” Vib. Spectrosc. 49, 110–117 (2009).
[Crossref]

Galun, E.

H. Hecht, Z. Burshtein, A. Katzir, S. Noach, M. Sokol, E. Frumker, E. Galun, and A. A. Ishaaya, “Passive Q-switching of a Tm:YLF laser with a Co2+-doped silver halide saturable absorber,” Opt. Mater. 64, 64–69 (2017).
[Crossref]

Y. Tsur, S. Goldring, E. Galun, and A. Katzir, “Ground state depletion - a step towards mid-IR lasing of doped silver halides,” J. Luminescence 175, 113–116 (2016).
[Crossref]

A. Goldstein, P. Loiko, Z. Burshtein, N. Skoptsov, I. Glazunov, E. Galun, N. Kuleshov, and K. Yumashev, “Development of saturable absorbers for laser passive Q-switching near 1.5 µm based on transparent ceramic Co2+:MgAl2O4,” J. Am. Ceram. Soc. 99, 1324–1330 (2016).
[Crossref]

Glazunov, I.

A. Goldstein, P. Loiko, Z. Burshtein, N. Skoptsov, I. Glazunov, E. Galun, N. Kuleshov, and K. Yumashev, “Development of saturable absorbers for laser passive Q-switching near 1.5 µm based on transparent ceramic Co2+:MgAl2O4,” J. Am. Ceram. Soc. 99, 1324–1330 (2016).
[Crossref]

Goldring, S.

Y. Tsur, S. Goldring, E. Galun, and A. Katzir, “Ground state depletion - a step towards mid-IR lasing of doped silver halides,” J. Luminescence 175, 113–116 (2016).
[Crossref]

Goldstein, A.

A. Goldstein, P. Loiko, Z. Burshtein, N. Skoptsov, I. Glazunov, E. Galun, N. Kuleshov, and K. Yumashev, “Development of saturable absorbers for laser passive Q-switching near 1.5 µm based on transparent ceramic Co2+:MgAl2O4,” J. Am. Ceram. Soc. 99, 1324–1330 (2016).
[Crossref]

Golubovic, A.

S. Perets, R. Z. Shneck, R. Gajic, A. Golubovic, and Z. Burshtein, “Vibrational spectra of sodium gadolinium tungstate NaGd(WO4)2 single crystals: Observation od spatial dispersion,” Vib. Spectrosc. 49, 110–117 (2009).
[Crossref]

Hadni, A.

A. Hadni, Essentials of Modern Physics Applied to the Study of the Infrared(Pergamon, 1967).

Hecht, H.

H. Hecht, Z. Burshtein, A. Katzir, S. Noach, M. Sokol, E. Frumker, E. Galun, and A. A. Ishaaya, “Passive Q-switching of a Tm:YLF laser with a Co2+-doped silver halide saturable absorber,” Opt. Mater. 64, 64–69 (2017).
[Crossref]

Henderson, B.

B. Henderson and R. H. Bartran, Crystal-Field Engineering of Solid-State Laser Materials(Cambridge University, 2000).
[Crossref]

Ishaaya, A. A.

H. Hecht, Z. Burshtein, A. Katzir, S. Noach, M. Sokol, E. Frumker, E. Galun, and A. A. Ishaaya, “Passive Q-switching of a Tm:YLF laser with a Co2+-doped silver halide saturable absorber,” Opt. Mater. 64, 64–69 (2017).
[Crossref]

Kalisky, Y.

Y. Kalisky, The Physics and Engineering of Solid State Lasers(SPIE, Bellingham, WA, 2006).
[Crossref]

Katzir, A.

H. Hecht, Z. Burshtein, A. Katzir, S. Noach, M. Sokol, E. Frumker, E. Galun, and A. A. Ishaaya, “Passive Q-switching of a Tm:YLF laser with a Co2+-doped silver halide saturable absorber,” Opt. Mater. 64, 64–69 (2017).
[Crossref]

Y. Tsur, S. Goldring, E. Galun, and A. Katzir, “Ground state depletion - a step towards mid-IR lasing of doped silver halides,” J. Luminescence 175, 113–116 (2016).
[Crossref]

I. Zakosky-Neuberger, I. Shafir, L. Nagli, and A. Katzir, “Optical and luminescence properties of Co2+:AgCl0.2Br0.8 crystals and their potential applications as gain media for middle-infrared lasers,” Appl. Phys. Lett. 99, 201111 (2011).
[Crossref]

D. Bunimovich and A. Katzir, “Dielectric properties of silver halide and potassium halide crystals,” Appl. Opt. 32, 2045–2048 (1993).
[Crossref] [PubMed]

Koechner, W.

W. Koechner, Solid-State Laser Engineering(Springer, 2006).

Korsakov, A. S.

A. S. Korsakov, D. S. Vrublevsky, A. E. Lvov, and L. V. Zhukova, “Refractive index dispersion of AgCl1−xBrx (0<x<1) and Ag1−xTlxBr1−xlx(0<x<0.05),” Opt. Mater. 64, 40–46 (2017).
[Crossref]

Krupke, W. F.

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron,  32, 885–895 (1996).
[Crossref]

Kuleshov, N.

A. Goldstein, P. Loiko, Z. Burshtein, N. Skoptsov, I. Glazunov, E. Galun, N. Kuleshov, and K. Yumashev, “Development of saturable absorbers for laser passive Q-switching near 1.5 µm based on transparent ceramic Co2+:MgAl2O4,” J. Am. Ceram. Soc. 99, 1324–1330 (2016).
[Crossref]

Layne, C.

C. Layne, W. Lowdermilk, and M. Weber, “Multiphonon relaxation of Rare-Earth ions in oxide glasses,” Phys. Rev. B 16, 10–20 (1977).
[Crossref]

Layne, C. B.

C. B. Layne, W. H. Lowdermilk, and M. J. Weber, “Multiphonon relaxation of rare-earth ions in oxide glasses,” Phys. Rev. B 16, 10–20 (1977).
[Crossref]

Loiko, P.

A. Goldstein, P. Loiko, Z. Burshtein, N. Skoptsov, I. Glazunov, E. Galun, N. Kuleshov, and K. Yumashev, “Development of saturable absorbers for laser passive Q-switching near 1.5 µm based on transparent ceramic Co2+:MgAl2O4,” J. Am. Ceram. Soc. 99, 1324–1330 (2016).
[Crossref]

Lowdermilk, W.

C. Layne, W. Lowdermilk, and M. Weber, “Multiphonon relaxation of Rare-Earth ions in oxide glasses,” Phys. Rev. B 16, 10–20 (1977).
[Crossref]

Lowdermilk, W. H.

C. B. Layne, W. H. Lowdermilk, and M. J. Weber, “Multiphonon relaxation of rare-earth ions in oxide glasses,” Phys. Rev. B 16, 10–20 (1977).
[Crossref]

Lvov, A. E.

A. S. Korsakov, D. S. Vrublevsky, A. E. Lvov, and L. V. Zhukova, “Refractive index dispersion of AgCl1−xBrx (0<x<1) and Ag1−xTlxBr1−xlx(0<x<0.05),” Opt. Mater. 64, 40–46 (2017).
[Crossref]

Martyshkin, D.

S. B. Mirov, V. V. Fedorov, D. Martyshkin, I. S. Moskalev, M. Mirov, and S. Vasilyev, “Progress in mid-IR lasers based on Cr and Fe-doped II-VI chalcogenides,” IEEE J. Sel. Top. Quantum Electron,  21, 1601719 (2015).
[Crossref]

Mirov, M.

S. B. Mirov, V. V. Fedorov, D. Martyshkin, I. S. Moskalev, M. Mirov, and S. Vasilyev, “Progress in mid-IR lasers based on Cr and Fe-doped II-VI chalcogenides,” IEEE J. Sel. Top. Quantum Electron,  21, 1601719 (2015).
[Crossref]

Mirov, S. B.

S. B. Mirov, V. V. Fedorov, D. Martyshkin, I. S. Moskalev, M. Mirov, and S. Vasilyev, “Progress in mid-IR lasers based on Cr and Fe-doped II-VI chalcogenides,” IEEE J. Sel. Top. Quantum Electron,  21, 1601719 (2015).
[Crossref]

Y. V. Orlovskii, T. T. Basiev, I. N. Vorobiev, E. O. Orlovskaya, N. P. Barnes, and S. B. Mirov, “Temperature dependencies of excited states lifetimes and relaxation rates of 3–5 phonon (4–6 µm) transitions in the YAG, LuAG and YLF crystals doped with trivalent holmium, thulium, and erbium,” Opt. Mater. 18, 355–365 (2002).
[Crossref]

Moos, H. W.

L. A. Riseberg and H. W. Moos, “Multiphonon orbit-lattice relaxation of excited states of rare-earth ions in crystals,” Phys. Rev. 174, 429–438 (1968).
[Crossref]

Morse, P. M.

P. M. Morse, “Diatomic molecules according to the wave mechanics. II. Vibrational levels,” Phys. Rev. 34, 57–64 (1929).
[Crossref]

Moskalev, I. S.

S. B. Mirov, V. V. Fedorov, D. Martyshkin, I. S. Moskalev, M. Mirov, and S. Vasilyev, “Progress in mid-IR lasers based on Cr and Fe-doped II-VI chalcogenides,” IEEE J. Sel. Top. Quantum Electron,  21, 1601719 (2015).
[Crossref]

Nagli, L.

I. Zakosky-Neuberger, I. Shafir, L. Nagli, and A. Katzir, “Optical and luminescence properties of Co2+:AgCl0.2Br0.8 crystals and their potential applications as gain media for middle-infrared lasers,” Appl. Phys. Lett. 99, 201111 (2011).
[Crossref]

Noach, S.

H. Hecht, Z. Burshtein, A. Katzir, S. Noach, M. Sokol, E. Frumker, E. Galun, and A. A. Ishaaya, “Passive Q-switching of a Tm:YLF laser with a Co2+-doped silver halide saturable absorber,” Opt. Mater. 64, 64–69 (2017).
[Crossref]

Orlovskaya, E. O.

Y. V. Orlovskii, T. T. Basiev, I. N. Vorobiev, E. O. Orlovskaya, N. P. Barnes, and S. B. Mirov, “Temperature dependencies of excited states lifetimes and relaxation rates of 3–5 phonon (4–6 µm) transitions in the YAG, LuAG and YLF crystals doped with trivalent holmium, thulium, and erbium,” Opt. Mater. 18, 355–365 (2002).
[Crossref]

Orlovskii, Y. V.

Y. V. Orlovskii, T. T. Basiev, I. N. Vorobiev, E. O. Orlovskaya, N. P. Barnes, and S. B. Mirov, “Temperature dependencies of excited states lifetimes and relaxation rates of 3–5 phonon (4–6 µm) transitions in the YAG, LuAG and YLF crystals doped with trivalent holmium, thulium, and erbium,” Opt. Mater. 18, 355–365 (2002).
[Crossref]

Y. V. Orlovskii, K. K. Pukhov, T. T. Basiev, and T. Tsuboi, “Nonlinear mechanism of multiphonon relaxation of the energy of electronic excitation in optical crystals doped with rare-earth ions,” Opt. Mater. 4, 583–595 (1995).
[Crossref]

Y. V. Orlovskii, R. J. Reeves, R. C. Powell, T. T. Basiev, and K. K. Pukhov, “Multiple-phonon nonradiative relaxation: Experimental rates in fluoride crystals doped with Er3+ and Nd3+ ions and a theoretical model,” Phys. Rev. B 49, 3821–3830 (1994).
[Crossref]

Page, R. H.

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron,  32, 885–895 (1996).
[Crossref]

Payne, S. A.

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron,  32, 885–895 (1996).
[Crossref]

Perets, S.

S. Perets, R. Z. Shneck, R. Gajic, A. Golubovic, and Z. Burshtein, “Vibrational spectra of sodium gadolinium tungstate NaGd(WO4)2 single crystals: Observation od spatial dispersion,” Vib. Spectrosc. 49, 110–117 (2009).
[Crossref]

Powell, R. C.

Y. V. Orlovskii, R. J. Reeves, R. C. Powell, T. T. Basiev, and K. K. Pukhov, “Multiple-phonon nonradiative relaxation: Experimental rates in fluoride crystals doped with Er3+ and Nd3+ ions and a theoretical model,” Phys. Rev. B 49, 3821–3830 (1994).
[Crossref]

Pukhov, K. K.

Y. V. Orlovskii, K. K. Pukhov, T. T. Basiev, and T. Tsuboi, “Nonlinear mechanism of multiphonon relaxation of the energy of electronic excitation in optical crystals doped with rare-earth ions,” Opt. Mater. 4, 583–595 (1995).
[Crossref]

Y. V. Orlovskii, R. J. Reeves, R. C. Powell, T. T. Basiev, and K. K. Pukhov, “Multiple-phonon nonradiative relaxation: Experimental rates in fluoride crystals doped with Er3+ and Nd3+ ions and a theoretical model,” Phys. Rev. B 49, 3821–3830 (1994).
[Crossref]

Reeves, R. J.

Y. V. Orlovskii, R. J. Reeves, R. C. Powell, T. T. Basiev, and K. K. Pukhov, “Multiple-phonon nonradiative relaxation: Experimental rates in fluoride crystals doped with Er3+ and Nd3+ ions and a theoretical model,” Phys. Rev. B 49, 3821–3830 (1994).
[Crossref]

Riseberg, L. A.

L. A. Riseberg and H. W. Moos, “Multiphonon orbit-lattice relaxation of excited states of rare-earth ions in crystals,” Phys. Rev. 174, 429–438 (1968).
[Crossref]

Seitz, F.

F. Seitz, The Modern Theory of Solids (McGraw-Hill, New York, 1940).

Shafir, I.

I. Zakosky-Neuberger, I. Shafir, L. Nagli, and A. Katzir, “Optical and luminescence properties of Co2+:AgCl0.2Br0.8 crystals and their potential applications as gain media for middle-infrared lasers,” Appl. Phys. Lett. 99, 201111 (2011).
[Crossref]

Shneck, R. Z.

S. Perets, R. Z. Shneck, R. Gajic, A. Golubovic, and Z. Burshtein, “Vibrational spectra of sodium gadolinium tungstate NaGd(WO4)2 single crystals: Observation od spatial dispersion,” Vib. Spectrosc. 49, 110–117 (2009).
[Crossref]

Skoptsov, N.

A. Goldstein, P. Loiko, Z. Burshtein, N. Skoptsov, I. Glazunov, E. Galun, N. Kuleshov, and K. Yumashev, “Development of saturable absorbers for laser passive Q-switching near 1.5 µm based on transparent ceramic Co2+:MgAl2O4,” J. Am. Ceram. Soc. 99, 1324–1330 (2016).
[Crossref]

Sokol, M.

H. Hecht, Z. Burshtein, A. Katzir, S. Noach, M. Sokol, E. Frumker, E. Galun, and A. A. Ishaaya, “Passive Q-switching of a Tm:YLF laser with a Co2+-doped silver halide saturable absorber,” Opt. Mater. 64, 64–69 (2017).
[Crossref]

Sorokin, E.

I. T. Sorokina and E. Sorokin, “Femtosecond Cr2+- based lasers”, IEEE J. Sel. Top. Quantum Electron. 21, 1601519 (2015).
[Crossref]

Sorokina, I. T.

I. T. Sorokina and E. Sorokin, “Femtosecond Cr2+- based lasers”, IEEE J. Sel. Top. Quantum Electron. 21, 1601519 (2015).
[Crossref]

Stolz, H.

A. Fujii, H. Stolz, and W. von der Osten, “Excitons and phonons in mixed silver halides studied by resonant Raman scattering,” J. Phys. C 16, 1713–1728 (1983).
[Crossref]

Tsuboi, T.

Y. V. Orlovskii, K. K. Pukhov, T. T. Basiev, and T. Tsuboi, “Nonlinear mechanism of multiphonon relaxation of the energy of electronic excitation in optical crystals doped with rare-earth ions,” Opt. Mater. 4, 583–595 (1995).
[Crossref]

Tsur, Y.

Y. Tsur, S. Goldring, E. Galun, and A. Katzir, “Ground state depletion - a step towards mid-IR lasing of doped silver halides,” J. Luminescence 175, 113–116 (2016).
[Crossref]

Vasilyev, S.

S. B. Mirov, V. V. Fedorov, D. Martyshkin, I. S. Moskalev, M. Mirov, and S. Vasilyev, “Progress in mid-IR lasers based on Cr and Fe-doped II-VI chalcogenides,” IEEE J. Sel. Top. Quantum Electron,  21, 1601719 (2015).
[Crossref]

von der Osten, W.

A. Fujii, H. Stolz, and W. von der Osten, “Excitons and phonons in mixed silver halides studied by resonant Raman scattering,” J. Phys. C 16, 1713–1728 (1983).
[Crossref]

Vorobiev, I. N.

Y. V. Orlovskii, T. T. Basiev, I. N. Vorobiev, E. O. Orlovskaya, N. P. Barnes, and S. B. Mirov, “Temperature dependencies of excited states lifetimes and relaxation rates of 3–5 phonon (4–6 µm) transitions in the YAG, LuAG and YLF crystals doped with trivalent holmium, thulium, and erbium,” Opt. Mater. 18, 355–365 (2002).
[Crossref]

Vrublevsky, D. S.

A. S. Korsakov, D. S. Vrublevsky, A. E. Lvov, and L. V. Zhukova, “Refractive index dispersion of AgCl1−xBrx (0<x<1) and Ag1−xTlxBr1−xlx(0<x<0.05),” Opt. Mater. 64, 40–46 (2017).
[Crossref]

Weber, M.

C. Layne, W. Lowdermilk, and M. Weber, “Multiphonon relaxation of Rare-Earth ions in oxide glasses,” Phys. Rev. B 16, 10–20 (1977).
[Crossref]

Weber, M. J.

C. B. Layne, W. H. Lowdermilk, and M. J. Weber, “Multiphonon relaxation of rare-earth ions in oxide glasses,” Phys. Rev. B 16, 10–20 (1977).
[Crossref]

M. J. Weber, “Multiphonon relaxation of rare-earth ions in yttrium orthoaluminate,” Phys. Rev. B 8, 54–64 (1973).
[Crossref]

Wilke, G. D.

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron,  32, 885–895 (1996).
[Crossref]

Yariv, A.

A. Yariv, Introduction to Optical Electronics, (Holt-Reinhart-Winston, 1971).

Yumashev, K.

A. Goldstein, P. Loiko, Z. Burshtein, N. Skoptsov, I. Glazunov, E. Galun, N. Kuleshov, and K. Yumashev, “Development of saturable absorbers for laser passive Q-switching near 1.5 µm based on transparent ceramic Co2+:MgAl2O4,” J. Am. Ceram. Soc. 99, 1324–1330 (2016).
[Crossref]

Zakosky-Neuberger, I.

I. Zakosky-Neuberger, I. Shafir, L. Nagli, and A. Katzir, “Optical and luminescence properties of Co2+:AgCl0.2Br0.8 crystals and their potential applications as gain media for middle-infrared lasers,” Appl. Phys. Lett. 99, 201111 (2011).
[Crossref]

Zhukova, L. V.

A. S. Korsakov, D. S. Vrublevsky, A. E. Lvov, and L. V. Zhukova, “Refractive index dispersion of AgCl1−xBrx (0<x<1) and Ag1−xTlxBr1−xlx(0<x<0.05),” Opt. Mater. 64, 40–46 (2017).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

I. Zakosky-Neuberger, I. Shafir, L. Nagli, and A. Katzir, “Optical and luminescence properties of Co2+:AgCl0.2Br0.8 crystals and their potential applications as gain media for middle-infrared lasers,” Appl. Phys. Lett. 99, 201111 (2011).
[Crossref]

IEEE J. Quantum Electron (1)

L. D. DeLoach, R. H. Page, G. D. Wilke, S. A. Payne, and W. F. Krupke, “Transition metal-doped zinc chalcogenides: spectroscopy and laser demonstration of a new class of gain media,” IEEE J. Quantum Electron,  32, 885–895 (1996).
[Crossref]

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

S. B. Mirov, V. V. Fedorov, D. Martyshkin, I. S. Moskalev, M. Mirov, and S. Vasilyev, “Progress in mid-IR lasers based on Cr and Fe-doped II-VI chalcogenides,” IEEE J. Sel. Top. Quantum Electron,  21, 1601719 (2015).
[Crossref]

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

I. T. Sorokina and E. Sorokin, “Femtosecond Cr2+- based lasers”, IEEE J. Sel. Top. Quantum Electron. 21, 1601519 (2015).
[Crossref]

J. Am. Ceram. Soc. (1)

A. Goldstein, P. Loiko, Z. Burshtein, N. Skoptsov, I. Glazunov, E. Galun, N. Kuleshov, and K. Yumashev, “Development of saturable absorbers for laser passive Q-switching near 1.5 µm based on transparent ceramic Co2+:MgAl2O4,” J. Am. Ceram. Soc. 99, 1324–1330 (2016).
[Crossref]

J. Luminescence (1)

Y. Tsur, S. Goldring, E. Galun, and A. Katzir, “Ground state depletion - a step towards mid-IR lasing of doped silver halides,” J. Luminescence 175, 113–116 (2016).
[Crossref]

J. Phys. C (1)

A. Fujii, H. Stolz, and W. von der Osten, “Excitons and phonons in mixed silver halides studied by resonant Raman scattering,” J. Phys. C 16, 1713–1728 (1983).
[Crossref]

Opt. Eng. (1)

Z. Burshtein, “Radiative, nonradiative, and mixed-decay transitions of rare-earth ions in dielectric media,” Opt. Eng. 49, 091005 (2010).
[Crossref]

Opt. Mater. (4)

H. Hecht, Z. Burshtein, A. Katzir, S. Noach, M. Sokol, E. Frumker, E. Galun, and A. A. Ishaaya, “Passive Q-switching of a Tm:YLF laser with a Co2+-doped silver halide saturable absorber,” Opt. Mater. 64, 64–69 (2017).
[Crossref]

A. S. Korsakov, D. S. Vrublevsky, A. E. Lvov, and L. V. Zhukova, “Refractive index dispersion of AgCl1−xBrx (0<x<1) and Ag1−xTlxBr1−xlx(0<x<0.05),” Opt. Mater. 64, 40–46 (2017).
[Crossref]

Y. V. Orlovskii, K. K. Pukhov, T. T. Basiev, and T. Tsuboi, “Nonlinear mechanism of multiphonon relaxation of the energy of electronic excitation in optical crystals doped with rare-earth ions,” Opt. Mater. 4, 583–595 (1995).
[Crossref]

Y. V. Orlovskii, T. T. Basiev, I. N. Vorobiev, E. O. Orlovskaya, N. P. Barnes, and S. B. Mirov, “Temperature dependencies of excited states lifetimes and relaxation rates of 3–5 phonon (4–6 µm) transitions in the YAG, LuAG and YLF crystals doped with trivalent holmium, thulium, and erbium,” Opt. Mater. 18, 355–365 (2002).
[Crossref]

Phys. Rev. (4)

L. A. Riseberg and H. W. Moos, “Multiphonon orbit-lattice relaxation of excited states of rare-earth ions in crystals,” Phys. Rev. 174, 429–438 (1968).
[Crossref]

V. B. Fowler and D. L. Dexter, “Relation between absorption and emission probabilities in luminescent centers in ionic solids,” Phys. Rev. 128, 2154–2165 (1962).
[Crossref]

P. M. Morse, “Diatomic molecules according to the wave mechanics. II. Vibrational levels,” Phys. Rev. 34, 57–64 (1929).
[Crossref]

C. R. Berry, “Physical defects in silver halides,” Phys. Rev. 97, 677–679 (1955).
[Crossref]

Phys. Rev. B (4)

C. Layne, W. Lowdermilk, and M. Weber, “Multiphonon relaxation of Rare-Earth ions in oxide glasses,” Phys. Rev. B 16, 10–20 (1977).
[Crossref]

M. J. Weber, “Multiphonon relaxation of rare-earth ions in yttrium orthoaluminate,” Phys. Rev. B 8, 54–64 (1973).
[Crossref]

C. B. Layne, W. H. Lowdermilk, and M. J. Weber, “Multiphonon relaxation of rare-earth ions in oxide glasses,” Phys. Rev. B 16, 10–20 (1977).
[Crossref]

Y. V. Orlovskii, R. J. Reeves, R. C. Powell, T. T. Basiev, and K. K. Pukhov, “Multiple-phonon nonradiative relaxation: Experimental rates in fluoride crystals doped with Er3+ and Nd3+ ions and a theoretical model,” Phys. Rev. B 49, 3821–3830 (1994).
[Crossref]

Vib. Spectrosc. (1)

S. Perets, R. Z. Shneck, R. Gajic, A. Golubovic, and Z. Burshtein, “Vibrational spectra of sodium gadolinium tungstate NaGd(WO4)2 single crystals: Observation od spatial dispersion,” Vib. Spectrosc. 49, 110–117 (2009).
[Crossref]

Other (6)

F. Seitz, The Modern Theory of Solids (McGraw-Hill, New York, 1940).

W. Koechner, Solid-State Laser Engineering(Springer, 2006).

A. Yariv, Introduction to Optical Electronics, (Holt-Reinhart-Winston, 1971).

B. Henderson and R. H. Bartran, Crystal-Field Engineering of Solid-State Laser Materials(Cambridge University, 2000).
[Crossref]

Y. Kalisky, The Physics and Engineering of Solid State Lasers(SPIE, Bellingham, WA, 2006).
[Crossref]

A. Hadni, Essentials of Modern Physics Applied to the Study of the Infrared(Pergamon, 1967).

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Figures (9)

Fig. 1
Fig. 1 Energy scheme describing simultanzand spontaneous emission of several phonons (nonradiative multi-phonon), indicated by wiggled arrows.
Fig. 2
Fig. 2 Variation of the B ( υ ) = [ 2 υ 1 ] 2 υ υ 2 [ Γ ( υ + 1 ) ] 2 function of Eq. (5) between υ = 1 and υ = 16.
Fig. 3
Fig. 3 Energy scheme describing competing radiative and multi-phonon non-radiative decay probabilities of an occupied higher electronic state to a multiplet of lower states. The partially wiggled arrows symbolize the competing nature within each inter-state decay transition. Enhanced brightness of the wiggled part symbolizes the reduced non-radiative probability for larger energy gaps. For more details, see text.
Fig. 4
Fig. 4 Energy scheme describing competing radiative and multi-phonon non-radiative decay probabilities of a higher occupied electronic multiplet to a multiplet of lower states at a finite temperature. The partially wiggled arrows symbolize the competing nature within each inter-state decay transition. Enhanced grayness of a wiggled part symbolizes the reduced non-radiative probability for larger energy gaps. Enhanced brightness of a higher multiplet state symbolizes its actual lower population compared to the lower states at the quasi-thermal equilibrium. For more details, see text.
Fig. 5
Fig. 5 Experimental absorption cross section spectrum of cobalt Co2+ in a silver chloride-bromide AgCl0.5Br0.5 crystal at near absolute zero temperature per Fig. 1 in [11], resolved into Gaussian single-peak components. Parameters relevant for further analysis are inset: Eu in cm−1 units; standard deviation width w0,u in cm−1 units; and S0,u in cm2cm−1 units.
Fig. 6
Fig. 6 Experimental fluorescence emission-shape spectrum of cobalt Co2+ in a silver chloride-bromide AgCl0.5Br0.5 crystal at near absolute zero temperature per Fig. 1 in [11], resolved into Gaussian single-peak components. Parameters relevant for further analysis are inset: g(); E in cm−1 units; and standard deviation width w0∗, in cm−1 units. The emission cross-section scale was added to comply with our revised estimate of τrad(0), per Fig. 8.
Fig. 7
Fig. 7 Energy-level scheme of absorption and fluorescence emission transitions in doubly ionized cobalt Co2+ in a silver chloride-bromide AgCl0.5Br0.5 crystal, based on experimental results provided in [11] and [16] (Figs. 5 and 6 above).
Fig. 8
Fig. 8 Effective fluorescence lifetime of an excited cobalt Co2+ in a silver chloride-bromide AgCl0.5Br0.5 crystal as function of absolute temperature. Full circles are experimental results from [11], Fig. 2. Dashed line is a fit to our Eq. (11). For detailed fit parameters see text and Table 1. Dotted curve replicates the original fitting attempt by Tsur et al using Eq. (1) in [11], presented here as Eq. (17).
Fig. 9
Fig. 9 Front surface reflectance of an AgCl0.5Br0.5 single crystal silver chloride-bromide, corrected for second surface reflection, as function of frequency. Solid line - experimental result; dashed line - fit to Eq. (28). Fit parameters are inset in the figure frame.

Tables (2)

Tables Icon

Table 1 Fluorescence parameters and adjustable material and process parameters obtained by the fit to Eq. (11). The adjustable parameters are emphasized by a bold-faced lettering text.

Tables Icon

Table 2 Fit parameters for reflection measurement of Fig. 9. The numerical j value represents the subscript index of ħω0 frequencies.

Equations (28)

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

τ r a d 1 ( 0 ) = 8 π n 2 ν 0 2 c 2 0 σ e m ( ν ) d ν ,
f = 3 m c π n e 2 0 σ e m ( ν ) d ν ,
f = 3 m c 8 π 2 e 2 n 3 ν 0 2 τ r e d 1 ( 0 ) .
τ n r 1 ( 0 ) = fB ( υ ) N c 1 N c 2 4 π 2 ρ M a 3 D 3 m ( ω c o D ) E g / ω c o ,
B ( υ ) = [ 2 υ 1 ] 2 υ υ 2 [ Γ ( υ + 1 ) ] 2 .
τ n r 1 ( 0 ) = τ r a d 1 ( 0 ) c e 2 N c 1 N c 2 π 2 n 3 [ M ¯ a t c 2 ] D E g 2 B ( υ ) ( ω c o D ) υ .
τ e f f 1 ( 0 ) = τ r a d 1 ( 0 ) + τ n r 1 ( 0 ) = τ r a d 1 ( 0 ) [ 1 + c e 2 N c 1 N c 2 π 2 n 3 [ M ¯ a t c 2 ] D E g 2 B ( υ ) ( ω c o D ) υ ] τ r a d 1 ( 0 ) [ 1 + C n r ( E g , ω c o , D ) ] ,
C n r ( E g , ω c o , D ) = τ n r 1 ( 0 ) = c e 2 N c 1 N c 2 π 2 n 3 [ M ¯ a t c 2 ] D E g 2 B ( υ ) ( ω c o D ) υ ,
τ e f f 1 ( T ) = τ r a d 1 ( 0 ) [ 1 + 1 exp ( E g / k B T ) 1 + C n r ( 1 + 1 exp ( ω c o / k B T ) 1 ) υ ] .
τ e f f 1 ( T ) = τ r a d 1 ( 0 ) [ 1 + C n r ( 1 + 1 exp ( ω c o / k B T ) 1 ) υ ] .
τ e f f 1 ( T ) = τ r a d 1 ( 0 ) [ 1 + g ( ) C n r ( 1 + 1 exp ( ω c o / k B T ) 1 ) υ ( ) ] ,
C n r ( ) c e 2 N c 1 N c 2 π 2 n 3 [ M ¯ a t c 2 ] D E g 2 B ( υ ( ) ) ( ω c o D ) υ ( ) ,
η = τ e f f ( 0 ) / τ r a d ( 0 ) = [ 1 + g ( ) C n r ( ) ( E g ω c o , D ) ] 1 .
τ r a d , u 1 ( T ) = τ r a d 1 ( 0 ) S 0 , u S 0 , 0 , * d u Z ( T ) e E u / k B T ,
τ e f f 1 ( T ) = u τ r a d , u 1 ( T ) [ 1 + g ( ) C n r , u ( 1 + 1 exp ( ω c o / k B T ) 1 ) υ u ( ) ] ,
C n r , u ( ) c e 2 N c 1 N c 2 π 2 n 3 [ M ¯ a t c 2 ] D ( E g + E u ) 2 B ( υ u ( ) ) ( ω c o D ) υ u ( ) ,
τ e f f 1 ( 0 ) = τ r a d 1 + W a 1 exp ( E a c t / k B T ) ,
ω L O ( j ) = 2 ω 0 j 2 + Q 0 j / ε ( ) ,
f e f f ( j ) = 3 m 2 π 2 e 2 V c 2 Q 0 j ε ( ) ,
γ ( λ ) = Δ N λ 4 8 π n 2 c τ r a d g ( λ ) ,
Γ S L J = 1 2 ξ L S [ J ( J + 1 ) L ( L + 1 ) S ( S 1 ) ] ,
R e x = 1 ( 1 R 1 ) ( 1 R 2 ) ( 1 R 1 R 2 ) ,
R 1 = ( 1 + β ) ( 1 + β ) 2 4 β R e x ( 2 R e x ) 2 β ( 2 R e x ) .
ε ( ω ) = ε ( ) + j Q 0 j ( ω 0 j 2 ω 2 ) ( ω 0 j 2 ω 2 ) + γ j 2 ω 2 ,
ε ( ω ) = j Q 0 j γ j ω ( ω 0 j 2 ω 2 ) + γ j 2 ω 2 ,
n 2 = 1 2 ε 2 + ε 2 + 1 2 ε ,
κ 2 = 1 2 ε 2 + ε 2 1 2 ε ,
R ( ω ) = | n 1 + i κ | 2 | n + 1 + i κ | 2 .

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