Abstract

Temperature dependencies of stimulated emission cross section for Nd:YAG, Nd:YVO4, and Nd:GdVO4 was carefully evaluated. Our spectral evaluations with fine spectral resolution were carried out under the condition that the population inversion was induced into samples by a weak pumping field. Within the temperature range from 15°C to 65°C, the variation of emission cross section at 1.06 μm in Nd:YAG was −0.20%/°C, while those in Nd:YVO4 and Nd:GdVO4 for π-polarization were −0.50%/°C and −0.48%/°C, respectively. Consideration of measured temperature dependence gave the numerical model for temperature dependent emission cross sections of Nd-doped solid-state laser materials. We have also presented numerical approximations of this model for our samples by a simple polynomial, which can be applicable within the temperature range from 15°C to 350°C.

©2012 Optical Society of America

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References

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  1. T. Taira, “Domain-controlled laser ceramics toward Giant Micro-photonics,” Opt. Mater. Express 1(5), 1040–1050 (2011).
    [Crossref]
  2. H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008).
    [Crossref] [PubMed]
  3. R. Bhandari and T. Taira, “> 6 MW peak power at 532 nm from passively Q-switched Nd:YAG/ Cr4+:YAG microchip laser,” Opt. Express 19(20), 19135–19141 (2011).
    [Crossref] [PubMed]
  4. R. Bhandari and T. Taira, “Megawatt level UV output from [110] Cr4+:YAG passively Q-switched microchip laser,” Opt. Express 19(23), 22510–22514 (2011).
    [Crossref] [PubMed]
  5. S. Hayashi, K. Nawata, H. Sakai, T. Taira, H. Minamide, and K. Kawase, “High-power, single-longitudinal-mode terahertz-wave generation pumped by a microchip Nd:YAG laser [Invited],” Opt. Express 20(3), 2881–2886 (2012).
    [Crossref] [PubMed]
  6. M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
    [Crossref] [PubMed]
  7. M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
    [Crossref]
  8. T. Taira, A. Mukai, Y. Nozawa, and T. Kobayashi, “Single-mode oscillation of laser-diode-pumped Nd:YVO4 microchip lasers,” Opt. Lett. 16(24), 1955–1957 (1991).
    [Crossref] [PubMed]
  9. W. Koechner, Solid-State Laser Engineering, 6th revised and updated edition (Springer Science + Business Media, Inc., 2006), Chap. 7.
  10. S. Joly and T. Taira, “Novel method for pulse control in Nd:YVO4/Cr4+:YAG passively Q-switched microchip laser,” in Proceedings of the European Conference on Lasers and Electro-Optics, CA.8.1, Munich, Germany, May 22–26 (2011).
  11. M. Tsunekane and T. Taira, “High temperature operation of passively Q-switched, Cr:YAG/Nd:YAG micro-laser for ignition of engines,” in Proceedings of the European Conference on Lasers and Electro-Optics, CA.P.30, Munich, Germany, June 14–19 (2009).
  12. J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
    [Crossref]
  13. Y. Sato and T. Taira, “Variation of the stimulated emission cross section in Nd:YAG caused by the structural changes of Russell-Saunders manifolds,” Opt. Mater. Express 1(3), 514–522 (2011).
    [Crossref]
  14. G. Turri, H. P. Jenssen, F. Cornacchia, M. Tonelli, and M. Bass, “Temperature-dependent stimulated emission cross section in Nd3+:YVO4 crystals,” J. Opt. Soc. Am. B 26(11), 2084–2088 (2009).
    [Crossref]
  15. X. Délen, F. Balembois, and P. Georges, “Temperature dependence of the emission cross section of Nd:YVO4 around 1064 nm and consequences on laser operation,” J. Opt. Soc. Am. B 28(5), 972–976 (2011).
    [Crossref]
  16. Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
    [Crossref]
  17. P. Raybaut, F. Balembois, F. Druon, and P. Georges, “Numerical and experimental study of gain narrowing in ytterbium-based regenerative amplifiers,” IEEE J. Quantum Electron. 41(3), 415–425 (2005).
    [Crossref]
  18. Y. Sato and T. Taira, “Comparative study on the spectroscopic properties of Nd:GdVO4 and Nd:YVO4 with hybrid process,” IEEE J. Sel. Top. Quantum Electron. 11(3), 613–620 (2005).
    [Crossref]
  19. T. Kushida, “Linewidth and thermal shifts of spectral lines in neodymium-doped yttrium-aluminum-garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969).
    [Crossref]
  20. Y. Sato, H. Ishizuki, and T. Taira, “Novel model of thermal conductivity for optical materials,” Rev. Laser Eng. 36(APLS), 1081–1084 (2008).
    [Crossref]
  21. T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
    [Crossref]

2012 (1)

2011 (5)

2010 (1)

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

2009 (2)

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[Crossref] [PubMed]

G. Turri, H. P. Jenssen, F. Cornacchia, M. Tonelli, and M. Bass, “Temperature-dependent stimulated emission cross section in Nd3+:YVO4 crystals,” J. Opt. Soc. Am. B 26(11), 2084–2088 (2009).
[Crossref]

2008 (2)

H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008).
[Crossref] [PubMed]

Y. Sato, H. Ishizuki, and T. Taira, “Novel model of thermal conductivity for optical materials,” Rev. Laser Eng. 36(APLS), 1081–1084 (2008).
[Crossref]

2007 (1)

T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
[Crossref]

2005 (3)

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[Crossref]

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

Y. Sato and T. Taira, “Comparative study on the spectroscopic properties of Nd:GdVO4 and Nd:YVO4 with hybrid process,” IEEE J. Sel. Top. Quantum Electron. 11(3), 613–620 (2005).
[Crossref]

2004 (1)

Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
[Crossref]

1991 (1)

1969 (1)

T. Kushida, “Linewidth and thermal shifts of spectral lines in neodymium-doped yttrium-aluminum-garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969).
[Crossref]

Ando, A.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

Balembois, F.

X. Délen, F. Balembois, and P. Georges, “Temperature dependence of the emission cross section of Nd:YVO4 around 1064 nm and consequences on laser operation,” J. Opt. Soc. Am. B 28(5), 972–976 (2011).
[Crossref]

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

Bass, M.

G. Turri, H. P. Jenssen, F. Cornacchia, M. Tonelli, and M. Bass, “Temperature-dependent stimulated emission cross section in Nd3+:YVO4 crystals,” J. Opt. Soc. Am. B 26(11), 2084–2088 (2009).
[Crossref]

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[Crossref]

Bhandari, R.

Cornacchia, F.

Délen, X.

Dong, J.

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[Crossref]

Druon, F.

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

Fujii, M.

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[Crossref] [PubMed]

Georges, P.

X. Délen, F. Balembois, and P. Georges, “Temperature dependence of the emission cross section of Nd:YVO4 around 1064 nm and consequences on laser operation,” J. Opt. Soc. Am. B 28(5), 972–976 (2011).
[Crossref]

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

Hayashi, S.

Inohara, T.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

Ishizuki, H.

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[Crossref] [PubMed]

Y. Sato, H. Ishizuki, and T. Taira, “Novel model of thermal conductivity for optical materials,” Rev. Laser Eng. 36(APLS), 1081–1084 (2008).
[Crossref]

Jenssen, H. P.

Kan, H.

Kanehara, K.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

Kawase, K.

Kido, N.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

Kobayashi, T.

Kushida, T.

T. Kushida, “Linewidth and thermal shifts of spectral lines in neodymium-doped yttrium-aluminum-garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969).
[Crossref]

Minamide, H.

Miyazaki, M.

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[Crossref] [PubMed]

Mukai, A.

Nawata, K.

Nozawa, Y.

Rapaport, A.

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[Crossref]

Raybaut, P.

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

Saikawa, J.

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[Crossref] [PubMed]

Sakai, H.

Sato, Y.

Y. Sato and T. Taira, “Variation of the stimulated emission cross section in Nd:YAG caused by the structural changes of Russell-Saunders manifolds,” Opt. Mater. Express 1(3), 514–522 (2011).
[Crossref]

Y. Sato, H. Ishizuki, and T. Taira, “Novel model of thermal conductivity for optical materials,” Rev. Laser Eng. 36(APLS), 1081–1084 (2008).
[Crossref]

Y. Sato and T. Taira, “Comparative study on the spectroscopic properties of Nd:GdVO4 and Nd:YVO4 with hybrid process,” IEEE J. Sel. Top. Quantum Electron. 11(3), 613–620 (2005).
[Crossref]

Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
[Crossref]

Szipocs, F.

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[Crossref]

Taira, T.

S. Hayashi, K. Nawata, H. Sakai, T. Taira, H. Minamide, and K. Kawase, “High-power, single-longitudinal-mode terahertz-wave generation pumped by a microchip Nd:YAG laser [Invited],” Opt. Express 20(3), 2881–2886 (2012).
[Crossref] [PubMed]

R. Bhandari and T. Taira, “Megawatt level UV output from [110] Cr4+:YAG passively Q-switched microchip laser,” Opt. Express 19(23), 22510–22514 (2011).
[Crossref] [PubMed]

T. Taira, “Domain-controlled laser ceramics toward Giant Micro-photonics,” Opt. Mater. Express 1(5), 1040–1050 (2011).
[Crossref]

Y. Sato and T. Taira, “Variation of the stimulated emission cross section in Nd:YAG caused by the structural changes of Russell-Saunders manifolds,” Opt. Mater. Express 1(3), 514–522 (2011).
[Crossref]

R. Bhandari and T. Taira, “> 6 MW peak power at 532 nm from passively Q-switched Nd:YAG/ Cr4+:YAG microchip laser,” Opt. Express 19(20), 19135–19141 (2011).
[Crossref] [PubMed]

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[Crossref] [PubMed]

Y. Sato, H. Ishizuki, and T. Taira, “Novel model of thermal conductivity for optical materials,” Rev. Laser Eng. 36(APLS), 1081–1084 (2008).
[Crossref]

H. Sakai, H. Kan, and T. Taira, “>1 MW peak power single-mode high-brightness passively Q-switched Nd3+:YAG microchip laser,” Opt. Express 16(24), 19891–19899 (2008).
[Crossref] [PubMed]

T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
[Crossref]

Y. Sato and T. Taira, “Comparative study on the spectroscopic properties of Nd:GdVO4 and Nd:YVO4 with hybrid process,” IEEE J. Sel. Top. Quantum Electron. 11(3), 613–620 (2005).
[Crossref]

Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
[Crossref]

T. Taira, A. Mukai, Y. Nozawa, and T. Kobayashi, “Single-mode oscillation of laser-diode-pumped Nd:YVO4 microchip lasers,” Opt. Lett. 16(24), 1955–1957 (1991).
[Crossref] [PubMed]

Tonelli, M.

Tsunekane, M.

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

Turri, G.

Ueda, K.

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[Crossref]

IEEE J. Quantum Electron. (3)

M. Tsunekane, T. Inohara, A. Ando, N. Kido, K. Kanehara, and T. Taira, “High peak power, passively Q-switched microlaser for ignition of engines,” IEEE J. Quantum Electron. 46(2), 277–284 (2010).
[Crossref]

Y. Sato and T. Taira, “Saturation factors of pump absorption in solid-state lasers,” IEEE J. Quantum Electron. 40(3), 270–280 (2004).
[Crossref]

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

IEEE J. Sel. Top. Quantum Electron. (2)

Y. Sato and T. Taira, “Comparative study on the spectroscopic properties of Nd:GdVO4 and Nd:YVO4 with hybrid process,” IEEE J. Sel. Top. Quantum Electron. 11(3), 613–620 (2005).
[Crossref]

T. Taira, “RE3+-ion-doped YAG ceramic lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 798–809 (2007).
[Crossref]

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

Opt. Express (4)

Opt. Lett. (1)

Opt. Mater. Express (2)

Phys. Chem. Chem. Phys. (1)

M. Miyazaki, J. Saikawa, H. Ishizuki, T. Taira, and M. Fujii, “Isomer selective infrared spectroscopy of supersonically cooled cis- and trans-N-phenylamides in the region from the amide band to NH stretching vibration,” Phys. Chem. Chem. Phys. 11(29), 6098–6106 (2009).
[Crossref] [PubMed]

Phys. Rev. (1)

T. Kushida, “Linewidth and thermal shifts of spectral lines in neodymium-doped yttrium-aluminum-garnet and calcium fluorophosphate,” Phys. Rev. 185(2), 500–508 (1969).
[Crossref]

Phys. Status Solidi A (1)

J. Dong, A. Rapaport, M. Bass, F. Szipocs, and K. Ueda, “Temperature-dependent stimulated emission cross section andconcentration quenching in highly doped Nd3+:YAG crystals,” Phys. Status Solidi A 202(13), 2565–2573 (2005).
[Crossref]

Rev. Laser Eng. (1)

Y. Sato, H. Ishizuki, and T. Taira, “Novel model of thermal conductivity for optical materials,” Rev. Laser Eng. 36(APLS), 1081–1084 (2008).
[Crossref]

Other (3)

W. Koechner, Solid-State Laser Engineering, 6th revised and updated edition (Springer Science + Business Media, Inc., 2006), Chap. 7.

S. Joly and T. Taira, “Novel method for pulse control in Nd:YVO4/Cr4+:YAG passively Q-switched microchip laser,” in Proceedings of the European Conference on Lasers and Electro-Optics, CA.8.1, Munich, Germany, May 22–26 (2011).

M. Tsunekane and T. Taira, “High temperature operation of passively Q-switched, Cr:YAG/Nd:YAG micro-laser for ignition of engines,” in Proceedings of the European Conference on Lasers and Electro-Optics, CA.P.30, Munich, Germany, June 14–19 (2009).

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Figures (9)

Fig. 1
Fig. 1 Normalized emission intensities in fluorescence at 1.06 μm under 25°C: Nd:YAG (a), Nd:YVO4 π- and σ-polarization (b) and (c), and Nd:GdVO4 π- and σ-polarization (d) and (e).

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Fig. 2
Fig. 2 Temperature dependence of fluorescences from Nd-doped laser crystals: Nd:YAG (a), Nd:YVO4 π- and σ-polarization (b) and (c), and Nd:GdVO4 π- and σ-polarization (d) and (e).

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Fig. 3
Fig. 3 Temperature dependence of the normalized intensities at the emission peak.

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Fig. 4
Fig. 4 Temperature dependence of bif (a), νif (b), and Δνif (c) in Nd:YAG. In these figures markers show experimental results and lines show the fitting.

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Fig. 5
Fig. 5 I(ν) emitted from Nd:YAG at various temperatures. Solid lines are simulations calculated from Eq. (5) with parameters in Table 1, and dashed lines are experimentally measured under temperature tuning by heater.

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Fig. 6
Fig. 6 Predictions for the temperature-dependent I(ν) emitted from Nd:YAG (a), Nd:YVO4 and Nd:GdVO4 (b) by means of Eq. (5) and Tables 1-3.

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Fig. 7
Fig. 7 Temperature dependence of emission intensity of various Nd-doped laser media.

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Fig. 8
Fig. 8 Temperature dependence of the line-bandwidth of emission peaks in Nd-doped materials.

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Fig. 9
Fig. 9 Energy levels of the lowest terms of Nd trivalent. Only 4F3/2 become the emitting level under 808-nm pumping, and only 4I11/2 become the terminating level for 1-μm fluorescence.

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Tables (4)

Tables Icon

Table 1 Spectral parameters of representative transitions in Nd:YAG

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Table 2 Spectral parameters of representative transitions in Nd:YVO4 in π-polarization

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Tables Icon

Table 3 Spectral parameters of representative transitions in Nd:GdVO4 in π-polarization

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Table 4 ei for Nd:YAG, Nd:YVO4, and Nd:GdVO4 under T0 is 20°C

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Equations (19)

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

g= γ 1+γ N σ em ,
G tra = ∫ 0 l dkexp( −αk )exp[ ∫ k l γexp( −αz ) 1+γexp( −αz ) N σ em dz ] / ∫ 0 l dkexp( −αk ) ,
G ref = ∫ 0 l dkexp[ −α( l−k ) ]exp[ ∫ 0 k γexp( −αz ) 1+γexp( −αz ) N σ em dz ] / ∫ 0 l dkexp[ −α( l−k ) ] .
σ em ( ν,T )= λ 2 16 π 2 n 2 τ j ∑ i∈j f∈k a i b if f i Δ ν if ( Δ ν if /2 ) 2 + ( ν− ν if ) 2 ,
I( ν )∝ ν 3 ∑ i∈j f∈k a i b if f i Δ ν if ( Δ ν if /2 ) 2 + ( ν− ν if ) 2 ,
ν if ( T )= ν if ( 0 )− c if ( T Θ D ) 4 ∫ 0 Θ D T x 3 e x −1 dx ,
Δ ν if ( T )=Δ ν if ( 0 )+ d if ( T Θ D ) 7 ∫ 0 Θ D T x 6 e x ( e x −1 ) 2 dx ,
σ em ( T )= σ em ( T 0 )( e 0 − e 1 T+ e 2 T 2 − e 3 T 3 +⋯ ),
g if ( ν )= Δ ν if 2π 1 ( Δ ν if /2 ) 2 + ( ν− ν if ) 2 ,
g i k ( ν )= ∑ f∈k b if g if ( ν ) ,
∑ f∈k b if =1.
g jk ( ν )= ∑ i∈j a i f i g i k ( ν ) ,
a i = A i / ∑ i∈j A i f i .
σ jk ( ν )= λ 2 8π n 2 g jk ( ν ) ∑ i∈j A i ,
I jk ( ν )= η ν 3 c 2 g jk ( ν )= η ν 3 2π c 2 ∑ i∈j f∈k a i b if f i Δ ν if ( Δ ν if /2 ) 2 + ( ν− ν if ) 2 ,
η= ∫ λ I jk ( ν )dλ ,
σ jk ( ν )= λ 2 16 π 2 n 2 τ j ∑ i∈j f∈k a i b if f i Δ ν if ( Δ ν if /2 ) 2 + ( ν− ν if ) 2 .
f i ( T )= exp[ h ν if ( T ) kT ] / ∑ i∈j exp[ h ν if ( T ) kT ] ,
σ em ( T )= σ jk ( ν if ( T ) )= λ 2 4 π 2 n 2 a i b if f i Δ ν if ∑ i∈j A i ∝ a i b if f i Δ ν if .

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