Abstract

Optical properties including radiative quantum efficiencies, cross-relaxation coefficients, refractive index, energy-gap law parameters, and maximum phonon energy are presented for a new low-phonon-frequency, nonhygroscopic host crystal potassium lead chloride (KPb2Cl5) doped with Dy3+ and Nd3+. Assuming that the total decay rate (W) from each level is composed of radiative (Arad), multiphonon (WMP), and concentration-dependent cross-relaxation (Wc) rates (W=Arad+WMP+Wc), we determined radiative quantum efficiencies (ηrad=Arad/W) from emission data for five samples of various Dy3+ concentrations (N0). These results were compared with values calculated from a Judd–Ofelt analysis of the absorption spectrum. This technique required identification of cross-relaxation pathways. A cross-relaxation coefficient k=1.83×10-37 cm6 s-1 (where Wc=kN02) was measured for the Dy3+  6H9/2+6F11/2 level. The measured multiphonon decay rates yielded energy-gap law (WMP[ΔE]B exp[-βΔE]) parameters B=3.72×109 s-1 and β=1.16×10-2 cm, indicating that laser action should be possible to near 9 µm (ΔE=1100 cm-1) in this laser host at room temperature.

© 2001 Optical Society of America

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    [CrossRef]

1999 (6)

Y. Guimond, J. L. Adam, A. M. Jurdyc, J. Mugnier, B. Jacquier, and X. H. Zhang, “Dy3+-doped stabilized GeGaS glasses for 1.3 μm optical fiber amplfiers,” Opt. Mater. 12, 467–471 (1999).
[CrossRef]

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]

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission and ion–ion interactions in thulium- and terbium-doped gallium lanthanum sulfide glass,” J. Opt. Soc. Am. B 16, 308–316 (1999).
[CrossRef]

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission from holmium doped gallium lanthanum sulphide glass,” Infrared Phys. Technol. 40, 329–335 (1999).
[CrossRef]

A. M. Tkachuk, S. E. Ivanova, L. I. Isaenko, A. P. Eliseev, W. F. Krupke, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Dy3+-doped crystals of double chlorides and double fluorides as the active media of IR solid-state lasers and telecommunication amplifiers,” J. Opt. Technol. 66, 460–462 (1999).
[CrossRef]

A. Tkachuk, S. Ivanova, L. Isaenko, A. Yelisseyev, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Comparative spectroscopic study of the Dy3+ doped double chloride and double fluoride crystals for telecommunication amplifiers and IR lasers,” Acta Phys. Pol. A 95, 381–394 (1999).

1998 (3)

L. I. Isaenko, A. Yelisseyev, V. A. Nadolinny, V. I. Pashkov, M. C. Nostrand, S. A. Payne, R. H. Page, and R. W. Solarz, “Spectroscopic investigation of rare-earth-doped chloride single crystals for telecommunications amplifiers,” in Solid State Lasers VII, R. Scheps, ed., Proc. SPIE 3265, 242–249 (1998).
[CrossRef]

T. Schaafsma, L. B. Shaw, B. Cole, J. S. Sanghera, and I. D. Aggarwal, “Modeling of Dy3+-doped GeAsSe glass 1.3-μm optical fiber amplifiers,” IEEE Photon. Technol. Lett. 10, 1548–1550 (1998).
[CrossRef]

A. Braud, S. Girard, J. L. Doualan, and R. Moncorge, “Spectroscopy and fluorescence dynamics of (Tm3+, Tb3+) and (Tm3+, Eu3+) doped LiYF4 single crystals for 1.5-μm laser operation,” IEEE J. Quantum Electron. 34, 2246–2255 (1998).
[CrossRef]

1997 (1)

R. H. Page, K. I. Schaffers, S. A. Payne, and W. F. Krupke, “Dy-doped chlorides as gain media for 1.3 μm telecommunications amplifiers,” J. Lightwave Technol. 15, 786–793 (1997).
[CrossRef]

1996 (1)

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

1995 (3)

S. Tanabe, T. Hanada, M. Watanabe, T. Hayashi, and N. Soga, “Optical properties of dysprosium-doped low-phonon-energy glasses for a potential 1.3-μm optical amplifier,” J. Am. Ceram. Soc. 78, 2917–2922 (1995).
[CrossRef]

K. Nitsch, M. Dusek, M. Nikl, K. Polak, and M. Rodova, “Ternary alkali lead chlorides–crystal-growth, crystal-structure, absorption and emission properties,” Prog. Cryst. Growth Charact. 30, 1–22 (1995).
[CrossRef]

R. S. Quimby, K. T. Gahagan, B. G. Aitken, and M. A. Newhouse, “Self-calibrating quantum efficiency measurement technique and application to Pr3+-doped sulfide glass,” Opt. Lett. 20, 2021–2023 (1995).
[CrossRef] [PubMed]

1994 (2)

K. Wei, D. P. Machewirth, J. Wenzel, E. Snitzer, and G. H. Sigel, Jr., “Spectroscopy Of Dy3+ in Ge–Ga–S glass and its suitability for 1.3-μm fiber-optic amplifier applications,” Opt. Lett. 19, 904–906 (1994).
[CrossRef] [PubMed]

D. W. Hewak, B. N. Samson, J. A. M. Neto, R. I. Laming, and D. N. Payne, “Emission at 1.3-μm from dysprosium-doped GaLaS glass,” Electron. Lett. 30, 968–970 (1994).
[CrossRef]

1991 (2)

M. Nikl, K. Nitsch, I. Velicka, J. Hybler, K. Polak, and T. Fabian, “Photoluminescence of KPb2Cl5,” Phys. Status Solidi B 168, K37–K42 (1991).
[CrossRef]

J. A. Caird, A. J. Ramponi, and P. R. Staver, “Quantum efficiency and excited-state relaxation dynamics in neodymium-doped phosphate laser glasses,” J. Opt. Soc. Am. B 8, 1391–1403 (1991).
[CrossRef]

1976 (1)

F. Auzel, “Multiphonon-assisted anti-Stokes and Stokes fluorescence of triply ionized rare-earth ions,” Phys. Rev. B 13, 2809–2817 (1976).
[CrossRef]

1972 (1)

A. I. Burshtein, “Hopping mechanism of energy transfer,” Sov. Phys. JETP 35, 882–885 (1972).

1971 (1)

W. F. Krupke, “Radiative transition probabilities within the 4f3 ground configuration of Nd: YAG,” IEEE J. Quantum Electron. QE-7, 153–159 (1971).
[CrossRef]

1964 (1)

C. K. Jorgensen and B. R. Judd, “Hypersensitive pseudoquadrupole transitions in lanthanides,” Mol. Phys. 8, 281–290 (1964).
[CrossRef]

1962 (3)

B. G. Wybourne, “Structure of fn-configurations. II. f5 and f9 configurations,” J. Chem. Phys. 36, 2301–2310 (1962).
[CrossRef]

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]

Adam, J. L.

Y. Guimond, J. L. Adam, A. M. Jurdyc, J. Mugnier, B. Jacquier, and X. H. Zhang, “Dy3+-doped stabilized GeGaS glasses for 1.3 μm optical fiber amplfiers,” Opt. Mater. 12, 467–471 (1999).
[CrossRef]

Aggarwal, I. D.

T. Schaafsma, L. B. Shaw, B. Cole, J. S. Sanghera, and I. D. Aggarwal, “Modeling of Dy3+-doped GeAsSe glass 1.3-μm optical fiber amplifiers,” IEEE Photon. Technol. Lett. 10, 1548–1550 (1998).
[CrossRef]

Aitken, B. G.

Auzel, F.

F. Auzel, “Multiphonon-assisted anti-Stokes and Stokes fluorescence of triply ionized rare-earth ions,” Phys. Rev. B 13, 2809–2817 (1976).
[CrossRef]

Bowman, S. R.

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

Braud, A.

A. Braud, S. Girard, J. L. Doualan, and R. Moncorge, “Spectroscopy and fluorescence dynamics of (Tm3+, Tb3+) and (Tm3+, Eu3+) doped LiYF4 single crystals for 1.5-μm laser operation,” IEEE J. Quantum Electron. 34, 2246–2255 (1998).
[CrossRef]

Brocklesby, W. S.

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission from holmium doped gallium lanthanum sulphide glass,” Infrared Phys. Technol. 40, 329–335 (1999).
[CrossRef]

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission and ion–ion interactions in thulium- and terbium-doped gallium lanthanum sulfide glass,” J. Opt. Soc. Am. B 16, 308–316 (1999).
[CrossRef]

Burshtein, A. I.

A. I. Burshtein, “Hopping mechanism of energy transfer,” Sov. Phys. JETP 35, 882–885 (1972).

Caird, J. A.

Cole, B.

T. Schaafsma, L. B. Shaw, B. Cole, J. S. Sanghera, and I. D. Aggarwal, “Modeling of Dy3+-doped GeAsSe glass 1.3-μm optical fiber amplifiers,” IEEE Photon. Technol. Lett. 10, 1548–1550 (1998).
[CrossRef]

Doualan, J. L.

A. Braud, S. Girard, J. L. Doualan, and R. Moncorge, “Spectroscopy and fluorescence dynamics of (Tm3+, Tb3+) and (Tm3+, Eu3+) doped LiYF4 single crystals for 1.5-μm laser operation,” IEEE J. Quantum Electron. 34, 2246–2255 (1998).
[CrossRef]

Dusek, M.

K. Nitsch, M. Dusek, M. Nikl, K. Polak, and M. Rodova, “Ternary alkali lead chlorides–crystal-growth, crystal-structure, absorption and emission properties,” Prog. Cryst. Growth Charact. 30, 1–22 (1995).
[CrossRef]

Eliseev, A. P.

Fabian, T.

M. Nikl, K. Nitsch, I. Velicka, J. Hybler, K. Polak, and T. Fabian, “Photoluminescence of KPb2Cl5,” Phys. Status Solidi B 168, K37–K42 (1991).
[CrossRef]

Feldman, B. J.

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

Gahagan, K. T.

Ganem, J.

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

Girard, S.

A. Braud, S. Girard, J. L. Doualan, and R. Moncorge, “Spectroscopy and fluorescence dynamics of (Tm3+, Tb3+) and (Tm3+, Eu3+) doped LiYF4 single crystals for 1.5-μm laser operation,” IEEE J. Quantum Electron. 34, 2246–2255 (1998).
[CrossRef]

Guimond, Y.

Y. Guimond, J. L. Adam, A. M. Jurdyc, J. Mugnier, B. Jacquier, and X. H. Zhang, “Dy3+-doped stabilized GeGaS glasses for 1.3 μm optical fiber amplfiers,” Opt. Mater. 12, 467–471 (1999).
[CrossRef]

Hanada, T.

S. Tanabe, T. Hanada, M. Watanabe, T. Hayashi, and N. Soga, “Optical properties of dysprosium-doped low-phonon-energy glasses for a potential 1.3-μm optical amplifier,” J. Am. Ceram. Soc. 78, 2917–2922 (1995).
[CrossRef]

Hayashi, T.

S. Tanabe, T. Hanada, M. Watanabe, T. Hayashi, and N. Soga, “Optical properties of dysprosium-doped low-phonon-energy glasses for a potential 1.3-μm optical amplifier,” J. Am. Ceram. Soc. 78, 2917–2922 (1995).
[CrossRef]

Hector, J. R.

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission and ion–ion interactions in thulium- and terbium-doped gallium lanthanum sulfide glass,” J. Opt. Soc. Am. B 16, 308–316 (1999).
[CrossRef]

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission from holmium doped gallium lanthanum sulphide glass,” Infrared Phys. Technol. 40, 329–335 (1999).
[CrossRef]

Hewak, D. W.

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission from holmium doped gallium lanthanum sulphide glass,” Infrared Phys. Technol. 40, 329–335 (1999).
[CrossRef]

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission and ion–ion interactions in thulium- and terbium-doped gallium lanthanum sulfide glass,” J. Opt. Soc. Am. B 16, 308–316 (1999).
[CrossRef]

D. W. Hewak, B. N. Samson, J. A. M. Neto, R. I. Laming, and D. N. Payne, “Emission at 1.3-μm from dysprosium-doped GaLaS glass,” Electron. Lett. 30, 968–970 (1994).
[CrossRef]

Hybler, J.

M. Nikl, K. Nitsch, I. Velicka, J. Hybler, K. Polak, and T. Fabian, “Photoluminescence of KPb2Cl5,” Phys. Status Solidi B 168, K37–K42 (1991).
[CrossRef]

Isaenko, L.

A. Tkachuk, S. Ivanova, L. Isaenko, A. Yelisseyev, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Comparative spectroscopic study of the Dy3+ doped double chloride and double fluoride crystals for telecommunication amplifiers and IR lasers,” Acta Phys. Pol. A 95, 381–394 (1999).

Isaenko, L. I.

A. M. Tkachuk, S. E. Ivanova, L. I. Isaenko, A. P. Eliseev, W. F. Krupke, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Dy3+-doped crystals of double chlorides and double fluorides as the active media of IR solid-state lasers and telecommunication amplifiers,” J. Opt. Technol. 66, 460–462 (1999).
[CrossRef]

L. I. Isaenko, A. Yelisseyev, V. A. Nadolinny, V. I. Pashkov, M. C. Nostrand, S. A. Payne, R. H. Page, and R. W. Solarz, “Spectroscopic investigation of rare-earth-doped chloride single crystals for telecommunications amplifiers,” in Solid State Lasers VII, R. Scheps, ed., Proc. SPIE 3265, 242–249 (1998).
[CrossRef]

Ivanova, S.

A. Tkachuk, S. Ivanova, L. Isaenko, A. Yelisseyev, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Comparative spectroscopic study of the Dy3+ doped double chloride and double fluoride crystals for telecommunication amplifiers and IR lasers,” Acta Phys. Pol. A 95, 381–394 (1999).

Ivanova, S. E.

Jacquier, B.

Y. Guimond, J. L. Adam, A. M. Jurdyc, J. Mugnier, B. Jacquier, and X. H. Zhang, “Dy3+-doped stabilized GeGaS glasses for 1.3 μm optical fiber amplfiers,” Opt. Mater. 12, 467–471 (1999).
[CrossRef]

Jorgensen, C. K.

C. K. Jorgensen and B. R. Judd, “Hypersensitive pseudoquadrupole transitions in lanthanides,” Mol. Phys. 8, 281–290 (1964).
[CrossRef]

Judd, B. R.

C. K. Jorgensen and B. R. Judd, “Hypersensitive pseudoquadrupole transitions in lanthanides,” Mol. Phys. 8, 281–290 (1964).
[CrossRef]

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

Jurdyc, A. M.

Y. Guimond, J. L. Adam, A. M. Jurdyc, J. Mugnier, B. Jacquier, and X. H. Zhang, “Dy3+-doped stabilized GeGaS glasses for 1.3 μm optical fiber amplfiers,” Opt. Mater. 12, 467–471 (1999).
[CrossRef]

Krupke, W. F.

Laming, R. I.

D. W. Hewak, B. N. Samson, J. A. M. Neto, R. I. Laming, and D. N. Payne, “Emission at 1.3-μm from dysprosium-doped GaLaS glass,” Electron. Lett. 30, 968–970 (1994).
[CrossRef]

Machewirth, D. P.

Moncorge, R.

A. Braud, S. Girard, J. L. Doualan, and R. Moncorge, “Spectroscopy and fluorescence dynamics of (Tm3+, Tb3+) and (Tm3+, Eu3+) doped LiYF4 single crystals for 1.5-μm laser operation,” IEEE J. Quantum Electron. 34, 2246–2255 (1998).
[CrossRef]

Mugnier, J.

Y. Guimond, J. L. Adam, A. M. Jurdyc, J. Mugnier, B. Jacquier, and X. H. Zhang, “Dy3+-doped stabilized GeGaS glasses for 1.3 μm optical fiber amplfiers,” Opt. Mater. 12, 467–471 (1999).
[CrossRef]

Nadolinny, V. A.

L. I. Isaenko, A. Yelisseyev, V. A. Nadolinny, V. I. Pashkov, M. C. Nostrand, S. A. Payne, R. H. Page, and R. W. Solarz, “Spectroscopic investigation of rare-earth-doped chloride single crystals for telecommunications amplifiers,” in Solid State Lasers VII, R. Scheps, ed., Proc. SPIE 3265, 242–249 (1998).
[CrossRef]

Neto, J. A. M.

D. W. Hewak, B. N. Samson, J. A. M. Neto, R. I. Laming, and D. N. Payne, “Emission at 1.3-μm from dysprosium-doped GaLaS glass,” Electron. Lett. 30, 968–970 (1994).
[CrossRef]

Newhouse, M. A.

Nikl, M.

K. Nitsch, M. Dusek, M. Nikl, K. Polak, and M. Rodova, “Ternary alkali lead chlorides–crystal-growth, crystal-structure, absorption and emission properties,” Prog. Cryst. Growth Charact. 30, 1–22 (1995).
[CrossRef]

M. Nikl, K. Nitsch, I. Velicka, J. Hybler, K. Polak, and T. Fabian, “Photoluminescence of KPb2Cl5,” Phys. Status Solidi B 168, K37–K42 (1991).
[CrossRef]

Nitsch, K.

K. Nitsch, M. Dusek, M. Nikl, K. Polak, and M. Rodova, “Ternary alkali lead chlorides–crystal-growth, crystal-structure, absorption and emission properties,” Prog. Cryst. Growth Charact. 30, 1–22 (1995).
[CrossRef]

M. Nikl, K. Nitsch, I. Velicka, J. Hybler, K. Polak, and T. Fabian, “Photoluminescence of KPb2Cl5,” Phys. Status Solidi B 168, K37–K42 (1991).
[CrossRef]

Nostrand, M. C.

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]

A. M. Tkachuk, S. E. Ivanova, L. I. Isaenko, A. P. Eliseev, W. F. Krupke, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Dy3+-doped crystals of double chlorides and double fluorides as the active media of IR solid-state lasers and telecommunication amplifiers,” J. Opt. Technol. 66, 460–462 (1999).
[CrossRef]

A. Tkachuk, S. Ivanova, L. Isaenko, A. Yelisseyev, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Comparative spectroscopic study of the Dy3+ doped double chloride and double fluoride crystals for telecommunication amplifiers and IR lasers,” Acta Phys. Pol. A 95, 381–394 (1999).

L. I. Isaenko, A. Yelisseyev, V. A. Nadolinny, V. I. Pashkov, M. C. Nostrand, S. A. Payne, R. H. Page, and R. W. Solarz, “Spectroscopic investigation of rare-earth-doped chloride single crystals for telecommunications amplifiers,” in Solid State Lasers VII, R. Scheps, ed., Proc. SPIE 3265, 242–249 (1998).
[CrossRef]

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.

A. Tkachuk, S. Ivanova, L. Isaenko, A. Yelisseyev, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Comparative spectroscopic study of the Dy3+ doped double chloride and double fluoride crystals for telecommunication amplifiers and IR lasers,” Acta Phys. Pol. A 95, 381–394 (1999).

A. M. Tkachuk, S. E. Ivanova, L. I. Isaenko, A. P. Eliseev, W. F. Krupke, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Dy3+-doped crystals of double chlorides and double fluorides as the active media of IR solid-state lasers and telecommunication amplifiers,” J. Opt. Technol. 66, 460–462 (1999).
[CrossRef]

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]

L. I. Isaenko, A. Yelisseyev, V. A. Nadolinny, V. I. Pashkov, M. C. Nostrand, S. A. Payne, R. H. Page, and R. W. Solarz, “Spectroscopic investigation of rare-earth-doped chloride single crystals for telecommunications amplifiers,” in Solid State Lasers VII, R. Scheps, ed., Proc. SPIE 3265, 242–249 (1998).
[CrossRef]

R. H. Page, K. I. Schaffers, S. A. Payne, and W. F. Krupke, “Dy-doped chlorides as gain media for 1.3 μm telecommunications amplifiers,” J. Lightwave Technol. 15, 786–793 (1997).
[CrossRef]

Pashkov, V. I.

L. I. Isaenko, A. Yelisseyev, V. A. Nadolinny, V. I. Pashkov, M. C. Nostrand, S. A. Payne, R. H. Page, and R. W. Solarz, “Spectroscopic investigation of rare-earth-doped chloride single crystals for telecommunications amplifiers,” in Solid State Lasers VII, R. Scheps, ed., Proc. SPIE 3265, 242–249 (1998).
[CrossRef]

Payne, D. N.

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission from holmium doped gallium lanthanum sulphide glass,” Infrared Phys. Technol. 40, 329–335 (1999).
[CrossRef]

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission and ion–ion interactions in thulium- and terbium-doped gallium lanthanum sulfide glass,” J. Opt. Soc. Am. B 16, 308–316 (1999).
[CrossRef]

D. W. Hewak, B. N. Samson, J. A. M. Neto, R. I. Laming, and D. N. Payne, “Emission at 1.3-μm from dysprosium-doped GaLaS glass,” Electron. Lett. 30, 968–970 (1994).
[CrossRef]

Payne, S. A.

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]

A. M. Tkachuk, S. E. Ivanova, L. I. Isaenko, A. P. Eliseev, W. F. Krupke, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Dy3+-doped crystals of double chlorides and double fluorides as the active media of IR solid-state lasers and telecommunication amplifiers,” J. Opt. Technol. 66, 460–462 (1999).
[CrossRef]

A. Tkachuk, S. Ivanova, L. Isaenko, A. Yelisseyev, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Comparative spectroscopic study of the Dy3+ doped double chloride and double fluoride crystals for telecommunication amplifiers and IR lasers,” Acta Phys. Pol. A 95, 381–394 (1999).

L. I. Isaenko, A. Yelisseyev, V. A. Nadolinny, V. I. Pashkov, M. C. Nostrand, S. A. Payne, R. H. Page, and R. W. Solarz, “Spectroscopic investigation of rare-earth-doped chloride single crystals for telecommunications amplifiers,” in Solid State Lasers VII, R. Scheps, ed., Proc. SPIE 3265, 242–249 (1998).
[CrossRef]

R. H. Page, K. I. Schaffers, S. A. Payne, and W. F. Krupke, “Dy-doped chlorides as gain media for 1.3 μm telecommunications amplifiers,” J. Lightwave Technol. 15, 786–793 (1997).
[CrossRef]

Polak, K.

K. Nitsch, M. Dusek, M. Nikl, K. Polak, and M. Rodova, “Ternary alkali lead chlorides–crystal-growth, crystal-structure, absorption and emission properties,” Prog. Cryst. Growth Charact. 30, 1–22 (1995).
[CrossRef]

M. Nikl, K. Nitsch, I. Velicka, J. Hybler, K. Polak, and T. Fabian, “Photoluminescence of KPb2Cl5,” Phys. Status Solidi B 168, K37–K42 (1991).
[CrossRef]

Quimby, R. S.

Ramponi, A. J.

Rodova, M.

K. Nitsch, M. Dusek, M. Nikl, K. Polak, and M. Rodova, “Ternary alkali lead chlorides–crystal-growth, crystal-structure, absorption and emission properties,” Prog. Cryst. Growth Charact. 30, 1–22 (1995).
[CrossRef]

Samson, B. N.

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission from holmium doped gallium lanthanum sulphide glass,” Infrared Phys. Technol. 40, 329–335 (1999).
[CrossRef]

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission and ion–ion interactions in thulium- and terbium-doped gallium lanthanum sulfide glass,” J. Opt. Soc. Am. B 16, 308–316 (1999).
[CrossRef]

D. W. Hewak, B. N. Samson, J. A. M. Neto, R. I. Laming, and D. N. Payne, “Emission at 1.3-μm from dysprosium-doped GaLaS glass,” Electron. Lett. 30, 968–970 (1994).
[CrossRef]

Sanghera, J. S.

T. Schaafsma, L. B. Shaw, B. Cole, J. S. Sanghera, and I. D. Aggarwal, “Modeling of Dy3+-doped GeAsSe glass 1.3-μm optical fiber amplifiers,” IEEE Photon. Technol. Lett. 10, 1548–1550 (1998).
[CrossRef]

Schaafsma, T.

T. Schaafsma, L. B. Shaw, B. Cole, J. S. Sanghera, and I. D. Aggarwal, “Modeling of Dy3+-doped GeAsSe glass 1.3-μm optical fiber amplifiers,” IEEE Photon. Technol. Lett. 10, 1548–1550 (1998).
[CrossRef]

Schaffers, K. I.

R. H. Page, K. I. Schaffers, S. A. Payne, and W. F. Krupke, “Dy-doped chlorides as gain media for 1.3 μm telecommunications amplifiers,” J. Lightwave Technol. 15, 786–793 (1997).
[CrossRef]

Schunemann, P. G.

Schweizer, T.

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission and ion–ion interactions in thulium- and terbium-doped gallium lanthanum sulfide glass,” J. Opt. Soc. Am. B 16, 308–316 (1999).
[CrossRef]

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission from holmium doped gallium lanthanum sulphide glass,” Infrared Phys. Technol. 40, 329–335 (1999).
[CrossRef]

Shaw, L. B.

T. Schaafsma, L. B. Shaw, B. Cole, J. S. Sanghera, and I. D. Aggarwal, “Modeling of Dy3+-doped GeAsSe glass 1.3-μm optical fiber amplifiers,” IEEE Photon. Technol. Lett. 10, 1548–1550 (1998).
[CrossRef]

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

Sigel Jr., G. H.

Snitzer, E.

Soga, N.

S. Tanabe, T. Hanada, M. Watanabe, T. Hayashi, and N. Soga, “Optical properties of dysprosium-doped low-phonon-energy glasses for a potential 1.3-μm optical amplifier,” J. Am. Ceram. Soc. 78, 2917–2922 (1995).
[CrossRef]

Solarz, R. W.

A. Tkachuk, S. Ivanova, L. Isaenko, A. Yelisseyev, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Comparative spectroscopic study of the Dy3+ doped double chloride and double fluoride crystals for telecommunication amplifiers and IR lasers,” Acta Phys. Pol. A 95, 381–394 (1999).

A. M. Tkachuk, S. E. Ivanova, L. I. Isaenko, A. P. Eliseev, W. F. Krupke, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Dy3+-doped crystals of double chlorides and double fluorides as the active media of IR solid-state lasers and telecommunication amplifiers,” J. Opt. Technol. 66, 460–462 (1999).
[CrossRef]

L. I. Isaenko, A. Yelisseyev, V. A. Nadolinny, V. I. Pashkov, M. C. Nostrand, S. A. Payne, R. H. Page, and R. W. Solarz, “Spectroscopic investigation of rare-earth-doped chloride single crystals for telecommunications amplifiers,” in Solid State Lasers VII, R. Scheps, ed., Proc. SPIE 3265, 242–249 (1998).
[CrossRef]

Staver, P. R.

Tanabe, S.

S. Tanabe, T. Hanada, M. Watanabe, T. Hayashi, and N. Soga, “Optical properties of dysprosium-doped low-phonon-energy glasses for a potential 1.3-μm optical amplifier,” J. Am. Ceram. Soc. 78, 2917–2922 (1995).
[CrossRef]

Tkachuk, A.

A. Tkachuk, S. Ivanova, L. Isaenko, A. Yelisseyev, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Comparative spectroscopic study of the Dy3+ doped double chloride and double fluoride crystals for telecommunication amplifiers and IR lasers,” Acta Phys. Pol. A 95, 381–394 (1999).

Tkachuk, A. M.

Velicka, I.

M. Nikl, K. Nitsch, I. Velicka, J. Hybler, K. Polak, and T. Fabian, “Photoluminescence of KPb2Cl5,” Phys. Status Solidi B 168, K37–K42 (1991).
[CrossRef]

Watanabe, M.

S. Tanabe, T. Hanada, M. Watanabe, T. Hayashi, and N. Soga, “Optical properties of dysprosium-doped low-phonon-energy glasses for a potential 1.3-μm optical amplifier,” J. Am. Ceram. Soc. 78, 2917–2922 (1995).
[CrossRef]

Wei, K.

Wenzel, J.

Wybourne, B. G.

B. G. Wybourne, “Structure of fn-configurations. II. f5 and f9 configurations,” J. Chem. Phys. 36, 2301–2310 (1962).
[CrossRef]

Yelisseyev, A.

A. Tkachuk, S. Ivanova, L. Isaenko, A. Yelisseyev, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Comparative spectroscopic study of the Dy3+ doped double chloride and double fluoride crystals for telecommunication amplifiers and IR lasers,” Acta Phys. Pol. A 95, 381–394 (1999).

L. I. Isaenko, A. Yelisseyev, V. A. Nadolinny, V. I. Pashkov, M. C. Nostrand, S. A. Payne, R. H. Page, and R. W. Solarz, “Spectroscopic investigation of rare-earth-doped chloride single crystals for telecommunications amplifiers,” in Solid State Lasers VII, R. Scheps, ed., Proc. SPIE 3265, 242–249 (1998).
[CrossRef]

Zhang, X. H.

Y. Guimond, J. L. Adam, A. M. Jurdyc, J. Mugnier, B. Jacquier, and X. H. Zhang, “Dy3+-doped stabilized GeGaS glasses for 1.3 μm optical fiber amplfiers,” Opt. Mater. 12, 467–471 (1999).
[CrossRef]

Acta Phys. Pol. A (1)

A. Tkachuk, S. Ivanova, L. Isaenko, A. Yelisseyev, S. A. Payne, R. W. Solarz, M. C. Nostrand, and R. H. Page, “Comparative spectroscopic study of the Dy3+ doped double chloride and double fluoride crystals for telecommunication amplifiers and IR lasers,” Acta Phys. Pol. A 95, 381–394 (1999).

Electron. Lett. (1)

D. W. Hewak, B. N. Samson, J. A. M. Neto, R. I. Laming, and D. N. Payne, “Emission at 1.3-μm from dysprosium-doped GaLaS glass,” Electron. Lett. 30, 968–970 (1994).
[CrossRef]

IEEE J. Quantum Electron. (3)

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

W. F. Krupke, “Radiative transition probabilities within the 4f3 ground configuration of Nd: YAG,” IEEE J. Quantum Electron. QE-7, 153–159 (1971).
[CrossRef]

A. Braud, S. Girard, J. L. Doualan, and R. Moncorge, “Spectroscopy and fluorescence dynamics of (Tm3+, Tb3+) and (Tm3+, Eu3+) doped LiYF4 single crystals for 1.5-μm laser operation,” IEEE J. Quantum Electron. 34, 2246–2255 (1998).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

T. Schaafsma, L. B. Shaw, B. Cole, J. S. Sanghera, and I. D. Aggarwal, “Modeling of Dy3+-doped GeAsSe glass 1.3-μm optical fiber amplifiers,” IEEE Photon. Technol. Lett. 10, 1548–1550 (1998).
[CrossRef]

Infrared Phys. Technol. (1)

T. Schweizer, B. N. Samson, J. R. Hector, W. S. Brocklesby, D. W. Hewak, and D. N. Payne, “Infrared emission from holmium doped gallium lanthanum sulphide glass,” Infrared Phys. Technol. 40, 329–335 (1999).
[CrossRef]

J. Am. Ceram. Soc. (1)

S. Tanabe, T. Hanada, M. Watanabe, T. Hayashi, and N. Soga, “Optical properties of dysprosium-doped low-phonon-energy glasses for a potential 1.3-μm optical amplifier,” J. Am. Ceram. Soc. 78, 2917–2922 (1995).
[CrossRef]

J. Chem. Phys. (2)

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

B. G. Wybourne, “Structure of fn-configurations. II. f5 and f9 configurations,” J. Chem. Phys. 36, 2301–2310 (1962).
[CrossRef]

J. Lightwave Technol. (1)

R. H. Page, K. I. Schaffers, S. A. Payne, and W. F. Krupke, “Dy-doped chlorides as gain media for 1.3 μm telecommunications amplifiers,” J. Lightwave Technol. 15, 786–793 (1997).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Opt. Technol. (1)

Mol. Phys. (1)

C. K. Jorgensen and B. R. Judd, “Hypersensitive pseudoquadrupole transitions in lanthanides,” Mol. Phys. 8, 281–290 (1964).
[CrossRef]

Opt. Lett. (3)

Opt. Mater. (1)

Y. Guimond, J. L. Adam, A. M. Jurdyc, J. Mugnier, B. Jacquier, and X. H. Zhang, “Dy3+-doped stabilized GeGaS glasses for 1.3 μm optical fiber amplfiers,” Opt. Mater. 12, 467–471 (1999).
[CrossRef]

Phys. Rev. (1)

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

Phys. Rev. B (1)

F. Auzel, “Multiphonon-assisted anti-Stokes and Stokes fluorescence of triply ionized rare-earth ions,” Phys. Rev. B 13, 2809–2817 (1976).
[CrossRef]

Phys. Status Solidi B (1)

M. Nikl, K. Nitsch, I. Velicka, J. Hybler, K. Polak, and T. Fabian, “Photoluminescence of KPb2Cl5,” Phys. Status Solidi B 168, K37–K42 (1991).
[CrossRef]

Proc. SPIE (1)

L. I. Isaenko, A. Yelisseyev, V. A. Nadolinny, V. I. Pashkov, M. C. Nostrand, S. A. Payne, R. H. Page, and R. W. Solarz, “Spectroscopic investigation of rare-earth-doped chloride single crystals for telecommunications amplifiers,” in Solid State Lasers VII, R. Scheps, ed., Proc. SPIE 3265, 242–249 (1998).
[CrossRef]

Prog. Cryst. Growth Charact. (1)

K. Nitsch, M. Dusek, M. Nikl, K. Polak, and M. Rodova, “Ternary alkali lead chlorides–crystal-growth, crystal-structure, absorption and emission properties,” Prog. Cryst. Growth Charact. 30, 1–22 (1995).
[CrossRef]

Sov. Phys. JETP (1)

A. I. Burshtein, “Hopping mechanism of energy transfer,” Sov. Phys. JETP 35, 882–885 (1972).

Other (9)

J. A. Skidmore, B. L. Freitas, C. E. Reinhardt, E. J. Utterback, R. H. Page, and M. A. Emanuel, “High-power operation of InGaAsP-InP laser diode array at 1.73 μm,” IEEE Photon. Technol. Lett. 9, 1334–1336 (1997); D. T. Schaafsma, L. B. Shaw, B. Cole, J. S. Sanghera, and I. D. Aggarwal, “Modeling of Dy3+-doped GeAsSe glass 1.3-μm optical fiber amplifiers,” IEEE Photon. Technol. Lett. 10, 1548–1550 (1998).
[CrossRef]

A. A. Kaminskii, Crystalline Lasers: Physical Processes and Operating Schemes (CRC Press, New York, 1996).

R. S. Quimby, “Active phenomena in doped halide glasses,” in Fluoride Glass Fiber Optics, I. D. Aggarwal and G. Lu, eds. (Academic, San Diego, Calif., 1991), p. 356.

B. G. Wybourne, Spectroscopic Properties of Rare Earths (Interscience, New York, 1965).

S. R. Bowman, S. K. Searles, J. Ganem, and P. Schmidt, “Further investigations of potential 4 μm laser materials,” in Advanced Solid-State Lasers, M. M. Fejer, H. Injeyan, and U. Keller, eds., Vol. 26 of OSA Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 487–490.

M. C. Nostrand, R. H. Page, S. A. Payne, W. F. Krupke, P. G. Schunemann, and L. I. Isaenko, “Room temperature CaGa2S4:Dy3+ laser action at 2.43 and 4.31 μm and KPb2Cl5:Dy3+ laser action at 2.43 μm,” in Advanced Solid-State Lasers, M. M. Fejer, H. Injeyan, and U. Keller, eds., Vol. 26 of OSA Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 441–449.

M. C. Nostrand, R. H. Page, S. A. Payne, W. F. Krupke, P. G. Schunemann, and L. I. Isaenko, “Laser demonstrations of rare-earth ions in low-phonon chloride and sulfide crystals,” in Advanced Solid-State Lasers, H. Injeyan, U. Keller, and C. Marshall, eds., Vol. 34 of OSA Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 459–463.

M. C. Nostrand, R. H. Page, S. A. Payne, W. F. Krupke, P. G. Schunemann, and L. I. Isaenko, “Spectroscopic data for infrared transitions in CaGa2S4:Dy3+ and KPb2Cl5:Dy3+,” in Advanced Solid-State Lasers, W. R. Bosenberg and M. M. Fejer, eds., Vol. 19 of OSA Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 524–528.

G. Dieke, Spectra and Energy Levels of Rare-Earth Ions in Crystals (Interscience, New York, 1968).

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Figures (13)

Fig. 1
Fig. 1

Mid-IR transmission spectrum through 5 mm of KPb2Cl5. Long-wave cutoff owing to multiphonon absorption. A transmission of ≈80% indicates an average index n2.0 in the 4–10-µm region.

Fig. 2
Fig. 2

Absorption cross section of KPb2Cl5:Dy3+ at 15 K and at room temperature (300 K), showing line broadening with temperature. Energy-level values were determined based on peak-to-peak values of the 15-K spectrum and appear in Fig. 3.

Fig. 3
Fig. 3

Energy-level diagram of trivalent Dy. The letter designations of the levels are taken after Dieke.20 The proposed phonon-assisted energy-transfer pathway is indicated. Note that this pathway actually involves two neighboring ions: a donor ion in the W level and an acceptor ion in the Z level interacting to promote both ions into the Y level.

Fig. 4
Fig. 4

Raman phonon spectrum of KPb2Cl5. The highest feature occurs near 203 cm-1, which we assign as νmax.

Fig. 5
Fig. 5

Temporal decay of the W level on 1.3-µm excitation. The decay rate increases as the sample concentration increases, which is indicative of concentration-dependent cross relaxation. Notice that the decay is largely singly exponential for each sample.

Fig. 6
Fig. 6

Spectral overlap of (a) two donor ions and (b) a donor and an acceptor ion. The relatively large overlap of the donor ion absorption and emission suggests that hopping is likely to occur. Bottom (note the log scale), way in which multiphonon Stokes sideband contributions to the cross sections can increase the spectral overlap.

Fig. 7
Fig. 7

Total decay rate for the W level (WW) plotted against the square of the Dy3+ concentration (N02). Also shown is the fit WW=W0+kWcN02, where kWc is the cross-relaxation coefficient. The slope implies that kWc=1.83×10-37 cm6 s-1. WW=2W0 when N0=0.8×1020 cm-3.

Fig. 8
Fig. 8

Pedagogical illustration of the concept of the radiative quantum efficiency model. (a) Level 3 is pumped directly, and we assume that η2rad=1 and β32rad=0. The wavy arrow indicates nonradiative decay (NR). Hypothetical emission rates ϕ (in arbitrary photons per second) are shown as a function of wavelength λ for (b) η3rad=1, (c) η3rad=0.5, and (d) η3rad=0.

Fig. 9
Fig. 9

Emission data used to determine branching ratios βJrad and fluorescence ratios  PϕJ/K for samples 1 and 4. The vertical scales have been adjusted to conserve the total emission rate for each sample. (a) Emission spectrum  WI(λ) obtained by direct excitation of the W level. Note the scale change near 1.3 µm and beyond 4 µm. The transfer of population from the W level to the Y level in the higher doped sample (4) is shown by the relative peak heights of the WZ and YZ transitions. (b) Emission spectrum obtained by direct excitation of the X level. The 1.7-µm (XZ) feature is not shown. Note that the 2.9-µm (YZ) feature is dramatically reduced compared with that of the W-level-pumped spectrum in (a). (c) Emission spectrum obtained by direct excitation of the A level. Only the AZ and WZ transition are shown.

Fig. 10
Fig. 10

W, X, and Y level radiative quantum efficiencies plotted as a function of the W-level cross-relaxation fraction for sample 1 according to Eqs. (30), (31), and (32), respectively. The dashed vertical line indicates the value of ηWc determined from temporal measurements according to Eq. (21). The η>1 region is inaccessible.

Fig. 11
Fig. 11

Nd3+ absorption spectrum and energy levels. Note the letter designations of the  4F5/2 and  4F3/2 levels. The  4F3/2 R level is assumed to have unit quantum efficiency. The  4F5/2 S level was pumped for quantum efficiency experiments.

Fig. 12
Fig. 12

Emission spectrum obtained by direct excitation of the S level of KPb2Cl5:Nd3+. Emission from both the  4F3/2 R and the  4F5/2 S (energy gap, ≈1000 cm-1) levels is observed. The data imply a fluorescence ratio  SϕR/S of 75.56.

Fig. 13
Fig. 13

Multiphonon relaxation rate versus energy gap in KPb2Cl5 (filled diamonds). The source of each data point is indicated. The curves for YAG, YLF, LaCl3, and LaBr3 are shown for comparison. These data are taken from Ref. 23.

Tables (7)

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Table 1 Dy3+ Concentrations of the Five Samples Used in Our Studya

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Table 2 Results of the Judd–Ofelt Analysis for Sample 1a

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Table 3 Measured Radiative Branching Ratios According To Eq. (26)a

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Table 4 Experimental Values for the Fluorescence Ratios As Defined in Eqs. (27)–(29)a

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Table 5 Radiative, Multiphonon, and Cross-Relaxation Quantum Efficiencies as Determined by the Analysis of Section 4a

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Table 6 Total Branching Ratios As Defined in Eq. (20)

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Table 7 Measured Data from Tables 3 and 5 Compared with Judd–Ofelt Data from Table 2a

Equations (36)

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SED=t=2,4,6Ωt|fn[SL]JU(t)fn[SL]J|2,
σ(λ) dλλ=4π23 e2c 12J+1 (n2+2)29nSED+nSMD.
SMD=(/2mc)2|fn[SL]JL+2Sfn[SL]J|2,
|4fn[SL]J=S,L C(S, L)|fnSLJ.
AJJ=32π33λ3 e2c c2J+1 n(n2+2)29SED+n3SMD,
AJ=J AJJ=1/τJrad,
βJJrad=AJJ/AJ,
ηJrad=τJmeas/τJrad,
WJ=AJrad+WJMP+WJcWJ0+WJc,
WJc=kJcNDNA=kJcN02,
RDX6=3c8π4n2AJrad  σDem(λ)σXabs(λ)dλ,
kJc=π(2π/3)5/2RDA3RDD3AJrad.
σJJem(λ)=λ5βJJradAJrad8πcn2 I(λ)JJI(λ)λdλ,
σStokes=σelec exp(-αsΔE),
αs=(hνmax)-1(ln{(N¯/S0)[1-exp(-hνmax/kT)]}-1),
WW=WW0+kWcN02.
d(WNW)dt=0=WR-WNW/τW,
d(WNX)dt=0=bWXWNW/τW-WNX/τX,
d(WNY)dt=0=(bWY+ηWcβWYc)WNW/τW+bXYWNX/τX-WNY/τY,
bJJ=ηJradβJJ+ηJMPβJJMP+ηJcβJJc
ηWcWWc/WW=1-WW0/WW.
[WNW/τW]ηWrad=WRηWrad,
[WNX/τX]ηXrad=WR bWXηXrad,
[WNY/τY]ηYrad=WR(bWY+ηWcβWYc+bWXbXY)ηYrad.
[PNJ/τJ]ηJradJ JJ  PI(λ)hc/λ dλPϕJ,
βJJrad=JJPI(λ)λdλJ JJ PI(λ)λdλ.
WϕX/W WϕX WϕW=bWXηXradηXrad,
 WϕY/W WϕY WϕW=(bWY+ηWcβWYc+bWXbXY)ηYradηWrad.
 XϕY/X XϕY XϕX=bXYηYradηXrad.
ηYrad=(1-ηWc)-WϕX/YXϕY/X(1+ηWc)2ηWc WϕW/Y(1-βWXrad)+2ηWc WϕX/Y(1-βXYrad)+(1-ηWc)WϕW/YβWYrad,
ηXrad=ηYrad XϕY/X+ηYrad(1-βXYrad),
ηWrad=ηXrad(1-ηWc) WϕX/W+ηXrad(1-βWXrad).
ηArad=ηWrad AϕW/A+ηWrad(1-βAWrad).
ηYrad(ηWc0)=1-WϕX/YXϕY/X WϕW/YβWYrad.
ηSrad=1 SϕR/S+1.
WJMP=B exp(-βΔE)[1-exp(-hνmax/kT)]-p,

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