Abstract

A detailed analysis is presented of the possibility of an Er3+ and Pr3+ codoped fiber as an amplifier for 1.3 and 1.5μm transmission windows. The numerical models with 10101480nm and 9861480nm pump lasers are proposed, the rate and power propagation equations are solved numerically, and the dependence of the gains at 1310, 1530, and 1600nm windows on pump wavelength and power, active ion concentrations, and signal wavelength are calculated. The results show that with 10101480nm pumps of 200100mW and with fixed Pr3+ and Er3+ concentrations at 8.0×1024 and 2.0×1024  ionsm3, the signals at 1310 and 1530nm windows may be equally amplified with a gain of 26.0dB in the active fiber with a length of 7.0m, and with a 1480986nm pump power of 100300mW and Pr3+ and Er3+ ion concentrations of 2.0×1024 and 2.0×1024  ionsm3, the signals at 1310 and 1530nm may be nearly equally amplified with a gain of 8.0dB in a 10.0m long active fiber.

© 2009 Optical Society of America

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References

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  1. Y. Choi and K. Kim, “1.6 μm emission from Pr3+: (F33,F43)-->H43 transition in Pr3+- and Pr3+/Er3+-doped selenide glasses,” Appl. Phys. Lett. 78, 1249-1251 (2001).
    [CrossRef]
  2. P. Golding, S. Jackson, T. King, and M. Pollnau, “Energy transfer process in Er3+-doped and Er3+, Pr3+-codoped ZBLAN glasses,” Phys. Rev. B 62, 856-864 (2000).
    [CrossRef]
  3. S. Park, D. Lee, and J. Heo, “Energy transfer between Er3+ and Pr3+ in chalcogenide glasses for dual wavelength fiber optical amplifiers,” J. Appl. Phys. 91, 9072-9077 (2002).
    [CrossRef]
  4. Y. Choi, B. Park, K. Kim, and J. Heo, “Pr3+- and Pr3+/Er3+-doped selenide glasses for potential 1.6 μm optical amplifier materials,” ETRI J. 23, 97-105 (2001).
    [CrossRef]
  5. S. Park, D. Lee, J. Heo, and H. Kim, “Pr3+/Er3+ co-doped Ge-As-Ga-S glasses as dual-wavelength fiber-optic amplifiers at 1.31 and 1.55 μm window,” J. Am. Ceram. Soc. 83, 1284-1286 (2000).
    [CrossRef]
  6. C. Yeh, C. Lee, and S. Chi, “120-nm bandwidth erbium-doped fiber amplifier in parallel configuration,” IEEE Photonics Technol. Lett. 16, 1637-1639 (2004).
    [CrossRef]
  7. T. Naito, T. Tanaka, and K. Torii, “A broadband distributed Raman amplifier for bandwidth beyond 100 nm,” in Optical Fiber Communication Conference (OFC), A.Sawchuk, ed., Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), paper TuR1.
  8. C. Jiang and W. Hu, “Multiband-fiber Raman amplifier,” in Proceedings of the Ninth Optoelectronics and Communications Conference (OECC) and the Third International Conference on Optical Internet (COIN) (SPIE2004).
    [PubMed]
  9. T. Whitley and R. Wyatt, “Alternative Gaussian spot size polynomial for use with doped fiber amplifier,” IEEE Photonics Technol. Lett. 11, 1325-1327 (1993).
    [CrossRef]
  10. E. Delevaque, T. Georges, and M. Monerie, “Modeling of pair-induced quenching in erbium-doped silicate fibers,” IEEE Photonics Technol. Lett. 5, 73-75 (1993).
    [CrossRef]
  11. F. Pasquale and M. Federighi, “Improved gain characteristics in high concentration Er3+/Yb3+ codoped glass waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2127-2131 (1994).
    [CrossRef]
  12. M. Karasek, “Optimum design of Er3+-Yb3+ codoped fibers for large-signal high-pump-power applications,” IEEE J. Quantum Electron. 33, 1699-1705 (1997).
    [CrossRef]
  13. E. Yahel and A. Hendy, “Modeling and optimization of short Er3+-Yb3+ co-doped fiber lasers,” IEEE J. Quantum Electron. 39, 1444-1451 (2003).
    [CrossRef]
  14. S. Shen, A. Jha, X. Liu, and M. Nataly, “Telluride glasses for broadband amplifiers and integrated optics,” J. Am. Ceram. Soc. 85, 1391-1395 (2002).
    [CrossRef]
  15. F. Gan, Optical and Spectroscopic Properties of Glasses (Shanghai Science and Technology Press, 1992).

2004 (1)

C. Yeh, C. Lee, and S. Chi, “120-nm bandwidth erbium-doped fiber amplifier in parallel configuration,” IEEE Photonics Technol. Lett. 16, 1637-1639 (2004).
[CrossRef]

2003 (1)

E. Yahel and A. Hendy, “Modeling and optimization of short Er3+-Yb3+ co-doped fiber lasers,” IEEE J. Quantum Electron. 39, 1444-1451 (2003).
[CrossRef]

2002 (2)

S. Shen, A. Jha, X. Liu, and M. Nataly, “Telluride glasses for broadband amplifiers and integrated optics,” J. Am. Ceram. Soc. 85, 1391-1395 (2002).
[CrossRef]

S. Park, D. Lee, and J. Heo, “Energy transfer between Er3+ and Pr3+ in chalcogenide glasses for dual wavelength fiber optical amplifiers,” J. Appl. Phys. 91, 9072-9077 (2002).
[CrossRef]

2001 (2)

Y. Choi, B. Park, K. Kim, and J. Heo, “Pr3+- and Pr3+/Er3+-doped selenide glasses for potential 1.6 μm optical amplifier materials,” ETRI J. 23, 97-105 (2001).
[CrossRef]

Y. Choi and K. Kim, “1.6 μm emission from Pr3+: (F33,F43)-->H43 transition in Pr3+- and Pr3+/Er3+-doped selenide glasses,” Appl. Phys. Lett. 78, 1249-1251 (2001).
[CrossRef]

2000 (2)

P. Golding, S. Jackson, T. King, and M. Pollnau, “Energy transfer process in Er3+-doped and Er3+, Pr3+-codoped ZBLAN glasses,” Phys. Rev. B 62, 856-864 (2000).
[CrossRef]

S. Park, D. Lee, J. Heo, and H. Kim, “Pr3+/Er3+ co-doped Ge-As-Ga-S glasses as dual-wavelength fiber-optic amplifiers at 1.31 and 1.55 μm window,” J. Am. Ceram. Soc. 83, 1284-1286 (2000).
[CrossRef]

1997 (1)

M. Karasek, “Optimum design of Er3+-Yb3+ codoped fibers for large-signal high-pump-power applications,” IEEE J. Quantum Electron. 33, 1699-1705 (1997).
[CrossRef]

1994 (1)

F. Pasquale and M. Federighi, “Improved gain characteristics in high concentration Er3+/Yb3+ codoped glass waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2127-2131 (1994).
[CrossRef]

1993 (2)

T. Whitley and R. Wyatt, “Alternative Gaussian spot size polynomial for use with doped fiber amplifier,” IEEE Photonics Technol. Lett. 11, 1325-1327 (1993).
[CrossRef]

E. Delevaque, T. Georges, and M. Monerie, “Modeling of pair-induced quenching in erbium-doped silicate fibers,” IEEE Photonics Technol. Lett. 5, 73-75 (1993).
[CrossRef]

Chi, S.

C. Yeh, C. Lee, and S. Chi, “120-nm bandwidth erbium-doped fiber amplifier in parallel configuration,” IEEE Photonics Technol. Lett. 16, 1637-1639 (2004).
[CrossRef]

Choi, Y.

Y. Choi and K. Kim, “1.6 μm emission from Pr3+: (F33,F43)-->H43 transition in Pr3+- and Pr3+/Er3+-doped selenide glasses,” Appl. Phys. Lett. 78, 1249-1251 (2001).
[CrossRef]

Y. Choi, B. Park, K. Kim, and J. Heo, “Pr3+- and Pr3+/Er3+-doped selenide glasses for potential 1.6 μm optical amplifier materials,” ETRI J. 23, 97-105 (2001).
[CrossRef]

Delevaque, E.

E. Delevaque, T. Georges, and M. Monerie, “Modeling of pair-induced quenching in erbium-doped silicate fibers,” IEEE Photonics Technol. Lett. 5, 73-75 (1993).
[CrossRef]

Federighi, M.

F. Pasquale and M. Federighi, “Improved gain characteristics in high concentration Er3+/Yb3+ codoped glass waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2127-2131 (1994).
[CrossRef]

Gan, F.

F. Gan, Optical and Spectroscopic Properties of Glasses (Shanghai Science and Technology Press, 1992).

Georges, T.

E. Delevaque, T. Georges, and M. Monerie, “Modeling of pair-induced quenching in erbium-doped silicate fibers,” IEEE Photonics Technol. Lett. 5, 73-75 (1993).
[CrossRef]

Golding, P.

P. Golding, S. Jackson, T. King, and M. Pollnau, “Energy transfer process in Er3+-doped and Er3+, Pr3+-codoped ZBLAN glasses,” Phys. Rev. B 62, 856-864 (2000).
[CrossRef]

Hendy, A.

E. Yahel and A. Hendy, “Modeling and optimization of short Er3+-Yb3+ co-doped fiber lasers,” IEEE J. Quantum Electron. 39, 1444-1451 (2003).
[CrossRef]

Heo, J.

S. Park, D. Lee, and J. Heo, “Energy transfer between Er3+ and Pr3+ in chalcogenide glasses for dual wavelength fiber optical amplifiers,” J. Appl. Phys. 91, 9072-9077 (2002).
[CrossRef]

Y. Choi, B. Park, K. Kim, and J. Heo, “Pr3+- and Pr3+/Er3+-doped selenide glasses for potential 1.6 μm optical amplifier materials,” ETRI J. 23, 97-105 (2001).
[CrossRef]

S. Park, D. Lee, J. Heo, and H. Kim, “Pr3+/Er3+ co-doped Ge-As-Ga-S glasses as dual-wavelength fiber-optic amplifiers at 1.31 and 1.55 μm window,” J. Am. Ceram. Soc. 83, 1284-1286 (2000).
[CrossRef]

Hu, W.

C. Jiang and W. Hu, “Multiband-fiber Raman amplifier,” in Proceedings of the Ninth Optoelectronics and Communications Conference (OECC) and the Third International Conference on Optical Internet (COIN) (SPIE2004).
[PubMed]

Jackson, S.

P. Golding, S. Jackson, T. King, and M. Pollnau, “Energy transfer process in Er3+-doped and Er3+, Pr3+-codoped ZBLAN glasses,” Phys. Rev. B 62, 856-864 (2000).
[CrossRef]

Jha, A.

S. Shen, A. Jha, X. Liu, and M. Nataly, “Telluride glasses for broadband amplifiers and integrated optics,” J. Am. Ceram. Soc. 85, 1391-1395 (2002).
[CrossRef]

Jiang, C.

C. Jiang and W. Hu, “Multiband-fiber Raman amplifier,” in Proceedings of the Ninth Optoelectronics and Communications Conference (OECC) and the Third International Conference on Optical Internet (COIN) (SPIE2004).
[PubMed]

Karasek, M.

M. Karasek, “Optimum design of Er3+-Yb3+ codoped fibers for large-signal high-pump-power applications,” IEEE J. Quantum Electron. 33, 1699-1705 (1997).
[CrossRef]

Kim, H.

S. Park, D. Lee, J. Heo, and H. Kim, “Pr3+/Er3+ co-doped Ge-As-Ga-S glasses as dual-wavelength fiber-optic amplifiers at 1.31 and 1.55 μm window,” J. Am. Ceram. Soc. 83, 1284-1286 (2000).
[CrossRef]

Kim, K.

Y. Choi, B. Park, K. Kim, and J. Heo, “Pr3+- and Pr3+/Er3+-doped selenide glasses for potential 1.6 μm optical amplifier materials,” ETRI J. 23, 97-105 (2001).
[CrossRef]

Y. Choi and K. Kim, “1.6 μm emission from Pr3+: (F33,F43)-->H43 transition in Pr3+- and Pr3+/Er3+-doped selenide glasses,” Appl. Phys. Lett. 78, 1249-1251 (2001).
[CrossRef]

King, T.

P. Golding, S. Jackson, T. King, and M. Pollnau, “Energy transfer process in Er3+-doped and Er3+, Pr3+-codoped ZBLAN glasses,” Phys. Rev. B 62, 856-864 (2000).
[CrossRef]

Lee, C.

C. Yeh, C. Lee, and S. Chi, “120-nm bandwidth erbium-doped fiber amplifier in parallel configuration,” IEEE Photonics Technol. Lett. 16, 1637-1639 (2004).
[CrossRef]

Lee, D.

S. Park, D. Lee, and J. Heo, “Energy transfer between Er3+ and Pr3+ in chalcogenide glasses for dual wavelength fiber optical amplifiers,” J. Appl. Phys. 91, 9072-9077 (2002).
[CrossRef]

S. Park, D. Lee, J. Heo, and H. Kim, “Pr3+/Er3+ co-doped Ge-As-Ga-S glasses as dual-wavelength fiber-optic amplifiers at 1.31 and 1.55 μm window,” J. Am. Ceram. Soc. 83, 1284-1286 (2000).
[CrossRef]

Liu, X.

S. Shen, A. Jha, X. Liu, and M. Nataly, “Telluride glasses for broadband amplifiers and integrated optics,” J. Am. Ceram. Soc. 85, 1391-1395 (2002).
[CrossRef]

Monerie, M.

E. Delevaque, T. Georges, and M. Monerie, “Modeling of pair-induced quenching in erbium-doped silicate fibers,” IEEE Photonics Technol. Lett. 5, 73-75 (1993).
[CrossRef]

Naito, T.

T. Naito, T. Tanaka, and K. Torii, “A broadband distributed Raman amplifier for bandwidth beyond 100 nm,” in Optical Fiber Communication Conference (OFC), A.Sawchuk, ed., Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), paper TuR1.

Nataly, M.

S. Shen, A. Jha, X. Liu, and M. Nataly, “Telluride glasses for broadband amplifiers and integrated optics,” J. Am. Ceram. Soc. 85, 1391-1395 (2002).
[CrossRef]

Park, B.

Y. Choi, B. Park, K. Kim, and J. Heo, “Pr3+- and Pr3+/Er3+-doped selenide glasses for potential 1.6 μm optical amplifier materials,” ETRI J. 23, 97-105 (2001).
[CrossRef]

Park, S.

S. Park, D. Lee, and J. Heo, “Energy transfer between Er3+ and Pr3+ in chalcogenide glasses for dual wavelength fiber optical amplifiers,” J. Appl. Phys. 91, 9072-9077 (2002).
[CrossRef]

S. Park, D. Lee, J. Heo, and H. Kim, “Pr3+/Er3+ co-doped Ge-As-Ga-S glasses as dual-wavelength fiber-optic amplifiers at 1.31 and 1.55 μm window,” J. Am. Ceram. Soc. 83, 1284-1286 (2000).
[CrossRef]

Pasquale, F.

F. Pasquale and M. Federighi, “Improved gain characteristics in high concentration Er3+/Yb3+ codoped glass waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2127-2131 (1994).
[CrossRef]

Pollnau, M.

P. Golding, S. Jackson, T. King, and M. Pollnau, “Energy transfer process in Er3+-doped and Er3+, Pr3+-codoped ZBLAN glasses,” Phys. Rev. B 62, 856-864 (2000).
[CrossRef]

Shen, S.

S. Shen, A. Jha, X. Liu, and M. Nataly, “Telluride glasses for broadband amplifiers and integrated optics,” J. Am. Ceram. Soc. 85, 1391-1395 (2002).
[CrossRef]

Tanaka, T.

T. Naito, T. Tanaka, and K. Torii, “A broadband distributed Raman amplifier for bandwidth beyond 100 nm,” in Optical Fiber Communication Conference (OFC), A.Sawchuk, ed., Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), paper TuR1.

Torii, K.

T. Naito, T. Tanaka, and K. Torii, “A broadband distributed Raman amplifier for bandwidth beyond 100 nm,” in Optical Fiber Communication Conference (OFC), A.Sawchuk, ed., Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), paper TuR1.

Whitley, T.

T. Whitley and R. Wyatt, “Alternative Gaussian spot size polynomial for use with doped fiber amplifier,” IEEE Photonics Technol. Lett. 11, 1325-1327 (1993).
[CrossRef]

Wyatt, R.

T. Whitley and R. Wyatt, “Alternative Gaussian spot size polynomial for use with doped fiber amplifier,” IEEE Photonics Technol. Lett. 11, 1325-1327 (1993).
[CrossRef]

Yahel, E.

E. Yahel and A. Hendy, “Modeling and optimization of short Er3+-Yb3+ co-doped fiber lasers,” IEEE J. Quantum Electron. 39, 1444-1451 (2003).
[CrossRef]

Yeh, C.

C. Yeh, C. Lee, and S. Chi, “120-nm bandwidth erbium-doped fiber amplifier in parallel configuration,” IEEE Photonics Technol. Lett. 16, 1637-1639 (2004).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Choi and K. Kim, “1.6 μm emission from Pr3+: (F33,F43)-->H43 transition in Pr3+- and Pr3+/Er3+-doped selenide glasses,” Appl. Phys. Lett. 78, 1249-1251 (2001).
[CrossRef]

ETRI J. (1)

Y. Choi, B. Park, K. Kim, and J. Heo, “Pr3+- and Pr3+/Er3+-doped selenide glasses for potential 1.6 μm optical amplifier materials,” ETRI J. 23, 97-105 (2001).
[CrossRef]

IEEE J. Quantum Electron. (3)

F. Pasquale and M. Federighi, “Improved gain characteristics in high concentration Er3+/Yb3+ codoped glass waveguide amplifiers,” IEEE J. Quantum Electron. 30, 2127-2131 (1994).
[CrossRef]

M. Karasek, “Optimum design of Er3+-Yb3+ codoped fibers for large-signal high-pump-power applications,” IEEE J. Quantum Electron. 33, 1699-1705 (1997).
[CrossRef]

E. Yahel and A. Hendy, “Modeling and optimization of short Er3+-Yb3+ co-doped fiber lasers,” IEEE J. Quantum Electron. 39, 1444-1451 (2003).
[CrossRef]

IEEE Photonics Technol. Lett. (3)

C. Yeh, C. Lee, and S. Chi, “120-nm bandwidth erbium-doped fiber amplifier in parallel configuration,” IEEE Photonics Technol. Lett. 16, 1637-1639 (2004).
[CrossRef]

T. Whitley and R. Wyatt, “Alternative Gaussian spot size polynomial for use with doped fiber amplifier,” IEEE Photonics Technol. Lett. 11, 1325-1327 (1993).
[CrossRef]

E. Delevaque, T. Georges, and M. Monerie, “Modeling of pair-induced quenching in erbium-doped silicate fibers,” IEEE Photonics Technol. Lett. 5, 73-75 (1993).
[CrossRef]

J. Am. Ceram. Soc. (2)

S. Park, D. Lee, J. Heo, and H. Kim, “Pr3+/Er3+ co-doped Ge-As-Ga-S glasses as dual-wavelength fiber-optic amplifiers at 1.31 and 1.55 μm window,” J. Am. Ceram. Soc. 83, 1284-1286 (2000).
[CrossRef]

S. Shen, A. Jha, X. Liu, and M. Nataly, “Telluride glasses for broadband amplifiers and integrated optics,” J. Am. Ceram. Soc. 85, 1391-1395 (2002).
[CrossRef]

J. Appl. Phys. (1)

S. Park, D. Lee, and J. Heo, “Energy transfer between Er3+ and Pr3+ in chalcogenide glasses for dual wavelength fiber optical amplifiers,” J. Appl. Phys. 91, 9072-9077 (2002).
[CrossRef]

Phys. Rev. B (1)

P. Golding, S. Jackson, T. King, and M. Pollnau, “Energy transfer process in Er3+-doped and Er3+, Pr3+-codoped ZBLAN glasses,” Phys. Rev. B 62, 856-864 (2000).
[CrossRef]

Other (3)

T. Naito, T. Tanaka, and K. Torii, “A broadband distributed Raman amplifier for bandwidth beyond 100 nm,” in Optical Fiber Communication Conference (OFC), A.Sawchuk, ed., Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), paper TuR1.

C. Jiang and W. Hu, “Multiband-fiber Raman amplifier,” in Proceedings of the Ninth Optoelectronics and Communications Conference (OECC) and the Third International Conference on Optical Internet (COIN) (SPIE2004).
[PubMed]

F. Gan, Optical and Spectroscopic Properties of Glasses (Shanghai Science and Technology Press, 1992).

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Figures (7)

Fig. 1
Fig. 1

Schematic of energy levels and transition configuration of Er 3 + Pr 3 + codoped fiber amplifier pumped with 1010 and 1480 nm LDs.

Fig. 2
Fig. 2

Schematic of energy level and transition configuration of Er 3 + Pr 3 + codoped fiber amplifier pumped with 986 and 1480 nm LDs.

Fig. 3
Fig. 3

Variation of the gain at 1310, 1530, and 1600 nm with active ion concentration and fiber length. Pump wavelength, pump power, and input signal power are 1010 1480 nm , 100 100 mW , and 30 dBm , respectively. (a) Pr 3 + = 2.0 × 10 24 and Er 3 + = 2.0 × 10 24   ions m 3 ; (b) Pr 3 + = 4.0 × 10 24 and Er 3 + = 2.0 × 10 24   ions m 3 ; (c) Pr 3 + = 6.0 × 10 24 and Er 3 + = 2.0 × 10 24   ions m 3 .

Fig. 4
Fig. 4

Variation of the gain at 1310, 1530, and 1600 nm with pump power and fiber length. Pump wavelength and input signal power are 1010 1480 nm and 30 dBm , respectively. Pump power is (a) 120 100 mW , (b) 160 100 mW , and (c) 200 100 mW .

Fig. 5
Fig. 5

Variation of the gain at 1310, 1530, and 1600 nm with active ion concentration and fiber length. Pump wavelength, pump power, and input signal power are 986 1480 nm , 100 100 mW , and 30 dBm , respectively. (a) Pr 3 + = 2.0 × 10 24 and Er 3 + = 2.0 × 10 24   ions m 3 ; (b) Pr 3 + = 3.0 × 10 24 and Er 3 + = 2.0 × 10 24   ions m 3 ; (c) Pr 3 + = 4.0 × 10 24 and Er 3 + = 2.0 × 10 24   ions m 3 .

Fig. 6
Fig. 6

Variation of the gain at 1310, 1530, and 1600 nm with pump power and fiber length. Pump wavelength and input signal power are 1010 1480 nm and 30 dBm , respectively. Pump power is (a) 120 100 mW , (b) 200 100 mW , and (c) 300 100 mW .

Fig. 7
Fig. 7

Variation of the gain spectra with fiber length. Pump wavelength and input signal power are 986 1480 nm , 200 200 mW , and 30 dBm , respectively.

Tables (1)

Tables Icon

Table 1 Spectroscopic Parameters of Er 3 + Doped and Pr 3 + Doped Telluride Fiber for Numerical Calculation

Equations (28)

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

N p r 1 t = ( W p r 13 + W p r 14 + C 23 N e r 2 + C 34 N e r 3 ) N p r 1 + A p r 21 N p r 2 + ( W p r 31 + A p r 31 ) N p r 3 + ( A p r 41 + W p r 41 ) N p r 4 ,
N p r 2 t = ( W p r 24 + A p r 21 ) N p r 2 + ( W p r 42 + A p r 42 ) N p r 4 ,
N p r 3 t = W p r 13 N p r 1 ( W p r 31 + A p r 31 ) N p r 3 + C 23 N p r 1 N e r 2 ,
N p r 4 t = W p r 14 N p r 1 + W p r 24 N p r 2 ( W p r 41 + W p r 42 + A p r 41 + A p r 42 ) N p r 4 + C 34 N p r 1 N e r 3 ,
N e r 1 t = ( W e r p 12 + W e r s 12 ) N e r 1 + ( W e r p 21 + W e r s 21 ) N e r 2 + A e r 21 N e r 2 + A e r 31 N e r 3 + A e r 41 N e r 4 + ( C 23 N e r 2 + C 34 N e r 3 ) N p r 1 + C e r 24 N e r 2 2 ,
N e r 2 t = ( W e r p 12 + W e r s 12 ) N e r 1 ( W e r p 21 + W e r s 21 + A p r 21 ) N e r 2 + A e r 32 N e r 3 2 C e r 24 N e r 2 2 C 23 N p r 1 N e r 2 ,
N e r 3 t = A e r 43 N e r 4 ( A e r 32 + A e r 31 ) N e r 3 C 34 N p r 1 N e r 3 ,
N e r 4 t = ( A e r 43 + A e r 41 ) N e r 4 + C e r 24 N e r 2 2 ,
N Er t = N e r 1 + N e r 2 + N e r 3 + N e r 4 ,
N Pr t = N p r 1 + N p r 2 + N p r 3 + N p r 4 .
P 1480 z = Γ 1480 [ ( σ e r 21 N e r 2 σ e r 12 N e r 1 ) ] P 1480 + α p P 1480 ,
P 1010 z = Γ 1010 [ ( σ p r 41 N p r 4 σ p r 14 N p r 1 ) ] P 1010 + α p P 1010 ,
P 1310 z = Γ 1310 [ σ p r 42 N p r 4 σ p r 24 N p r 2 ] P 1310 α s P 1310 ,
P 1530 z = Γ 1530 [ σ e r 21 N e r 2 σ e r 12 N e r 1 ] P 1530 α s P 1530 ,
P 1600 z = Γ 1600 [ σ p r 31 N p r 3 σ p r 13 N p r 1 ] P 1600 α s P 1600 ,
P a s e 1 ± z = Γ a s e ( σ p r 42 N p r 4 σ p r 24 N p r 2 ) P a s e 1 ± ± 2 h ν Δ ν σ p r 42 + α a s e 1 P a s e 1 ± ,
P a s e 2 ± z = Γ a s e ( σ e r 21 N e r 2 σ e r 12 N e r 1 ) P a s e 2 ± ± 2 h ν Δ ν σ e r 21 + α a s e 2 P a s e 2 ± ,
P a s e 3 ± z = Γ a s e ( σ p r 31 N p r 3 σ p r 13 N p r 1 ) P a s e 1 ± ± 2 h ν Δ ν σ p r 31 + α a s e 3 P a s e 3 ± ,
C up = 3.5 × 10 24 + 2.41 × 10 49 ( N Er t 4.4 × 10 25 )
C cr = 1.0 × 10 22 + 4.0 × 10 49 [ ( N Pr N Er ) 1 2 1.0 × 10 25 ] .
N p r 1 t = ( W p r 13 + W p r 14 + C 23 N e r 2 + C 34 N e r 3 ) N p r 1 + A p r 21 N p r 2 + ( W p r 31 + A p r 31 ) N p r 3 + ( A p r 41 + W p r 41 ) N p r 4 ,
N p r 2 t = ( W p r 24 + A p r 21 ) N p r 2 + ( W p r 42 + A p r 42 ) N p r 4 ,
N p r 3 t = W p r 13 N p r 1 ( W p r 31 + A p r 31 ) N p r 3 + C 23 N p r 1 N e r 2 ,
N p r 4 t = W p r 14 N p r 1 + W p r 24 N p r 2 ( W p r 41 + W p r 42 + A p r 41 + A p r 42 ) N p r 4 + C 34 N p r 1 N e r 3 ,
N e r 1 t = ( W e r p 12 + W e r p 13 + W e r s 12 ) N e r 1 + ( W e r s 21 + W e r p 21 + A e r 21 ) N e r 2 + ( W e r p 31 + A e r 31 ) N e r 3 + A e r 41 N e r 4 + ( C 23 N e r 2 + C 34 N e r 3 ) N p r 1 + C e r 24 N e r 2 2 ,
N e r 2 t = ( W e r p 12 + W e r s 12 ) N e r 1 ( W e r p 21 + W e r s 21 + A p r 21 ) N e r 2 + A e r 32 N e r 3 2 C e r 24 N e r 2 2 C 23 N p r 1 N e r 2 ,
N e r 3 t = A e r 43 N e r 4 + ( W e r p 13 A e r 32 A e r 31 ) N e r 3 C 34 N p r 1 N e r 3 ,
N e r 4 t = ( A e r 43 + A e r 43 ) N e r 4 + C e r 24 N e r 2 2 ,

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