Abstract

The dynamical behavior of a green-emitting upconversion-pumped Er(0.5 at.%):YAG laser with reabsorption losses by an excited-state absorption process is studied. Stability analysis is used to identify the various regimes of laser emission and the bifurcations leading to them. The results are illustrated by Runge–Kutta integration of the rate equations. An analytical expression of the laser threshold is also presented. Calculated values of the laser threshold and experimental results available in the literature are discussed. The model can be applied to the prediction and enhancement of the performances of the green Er3+ upconversion lasers.

© 2004 Optical Society of America

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References

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  1. F. Sanchez, P. Le Boudec, P.-L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
    [CrossRef] [PubMed]
  2. E. Lacot, F. Stoeckel, and M. Chenevier, “Dynamics of an erbium-doped fiber laser,” Phys. Rev. A 49, 3997–4007 (1994).
    [CrossRef] [PubMed]
  3. S. Colin, E. Contesse, P. Le Boudec, G. Stephan, and F. Sanchez, “Evidence of a saturable-absorption effect in heavily erbium-doped fibers,” Opt. Lett. 21, 1987–1989 (1996).
    [CrossRef] [PubMed]
  4. F. Sanchez and A. Kellou, “Laser dynamics with excited-state absorption,” J. Opt. Soc. Am. B 14, 209–213 (1997).
    [CrossRef]
  5. R. A. McFarlane, “Dual wavelength visible upconversion laser,” Appl. Phys. Lett. 54, 2301–2302 (1989).
    [CrossRef]
  6. L. F. Johnson and H. J. Guggenheim, “New laser lines in the visible from Er3+ ions in BaY2F8,” Appl. Phys. Lett. 20, 474–477 (1972).
    [CrossRef]
  7. R. Brede, E. Heumann, J. Koetke, T. Danger, G. Huber, and B. Chai, “Green up-conversion laser emission in Er-doped crystals at room temperature,” Appl. Phys. Lett. 63, 2030–2031 (1993).
    [CrossRef]
  8. M. Pollnau, W. Lüthy, and H. P. Weber, “Population mechanisms of the green Er3+:LiYF4 laser,” J. Appl. Phys. 77, 6128–6134 (1995).
    [CrossRef]
  9. S. Georgescu, O. Toma, C. Florea, and C. Naud, “ESA processes responsible for infrared-pumped, green and violet luminescence in low-concentrated Er: YAG,” J. Lumin. 101, 87–99 (2003).
    [CrossRef]
  10. J. D. Crawford, “Introduction to bifurcation theory,” Rev. Mod. Phys. 63, 991–1037 (1991).
    [CrossRef]
  11. B. N. Parlett, “The QR algorithm,” Comput. Sci. Eng. 2, 38–42 (2000).
    [CrossRef]
  12. P. LeBoulanger, J.-L. Doualan, S. Girard, J. Margerie, and R. Moncorgé, “Excited-state absorption spectroscopy of Er3+-doped Y3Al5O12, YVO4, and phosphate glass,” Phys. Rev. B 60, 11380–11390 (1999).
    [CrossRef]
  13. T. Danger, J. Koetke, R. Brede, E. Heumann, G. Huber, and B. H. T. Chai, “Spectroscopy and green upconversion laser emission of Er3+-doped crystals at room temperature,” J. Appl. Phys. 76, 1413–1422 (1994).
    [CrossRef]
  14. R. Brede, T. Danger, E. Heumann, G. Huber, and B. Chai, “Room-temperature green laser emission of Er:LiYF4,” Appl. Phys. Lett. 63, 729–730 (1993).
    [CrossRef]

2003 (1)

S. Georgescu, O. Toma, C. Florea, and C. Naud, “ESA processes responsible for infrared-pumped, green and violet luminescence in low-concentrated Er: YAG,” J. Lumin. 101, 87–99 (2003).
[CrossRef]

2000 (1)

B. N. Parlett, “The QR algorithm,” Comput. Sci. Eng. 2, 38–42 (2000).
[CrossRef]

1999 (1)

P. LeBoulanger, J.-L. Doualan, S. Girard, J. Margerie, and R. Moncorgé, “Excited-state absorption spectroscopy of Er3+-doped Y3Al5O12, YVO4, and phosphate glass,” Phys. Rev. B 60, 11380–11390 (1999).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

M. Pollnau, W. Lüthy, and H. P. Weber, “Population mechanisms of the green Er3+:LiYF4 laser,” J. Appl. Phys. 77, 6128–6134 (1995).
[CrossRef]

1994 (2)

T. Danger, J. Koetke, R. Brede, E. Heumann, G. Huber, and B. H. T. Chai, “Spectroscopy and green upconversion laser emission of Er3+-doped crystals at room temperature,” J. Appl. Phys. 76, 1413–1422 (1994).
[CrossRef]

E. Lacot, F. Stoeckel, and M. Chenevier, “Dynamics of an erbium-doped fiber laser,” Phys. Rev. A 49, 3997–4007 (1994).
[CrossRef] [PubMed]

1993 (3)

R. Brede, T. Danger, E. Heumann, G. Huber, and B. Chai, “Room-temperature green laser emission of Er:LiYF4,” Appl. Phys. Lett. 63, 729–730 (1993).
[CrossRef]

F. Sanchez, P. Le Boudec, P.-L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

R. Brede, E. Heumann, J. Koetke, T. Danger, G. Huber, and B. Chai, “Green up-conversion laser emission in Er-doped crystals at room temperature,” Appl. Phys. Lett. 63, 2030–2031 (1993).
[CrossRef]

1991 (1)

J. D. Crawford, “Introduction to bifurcation theory,” Rev. Mod. Phys. 63, 991–1037 (1991).
[CrossRef]

1989 (1)

R. A. McFarlane, “Dual wavelength visible upconversion laser,” Appl. Phys. Lett. 54, 2301–2302 (1989).
[CrossRef]

1972 (1)

L. F. Johnson and H. J. Guggenheim, “New laser lines in the visible from Er3+ ions in BaY2F8,” Appl. Phys. Lett. 20, 474–477 (1972).
[CrossRef]

Brede, R.

T. Danger, J. Koetke, R. Brede, E. Heumann, G. Huber, and B. H. T. Chai, “Spectroscopy and green upconversion laser emission of Er3+-doped crystals at room temperature,” J. Appl. Phys. 76, 1413–1422 (1994).
[CrossRef]

R. Brede, T. Danger, E. Heumann, G. Huber, and B. Chai, “Room-temperature green laser emission of Er:LiYF4,” Appl. Phys. Lett. 63, 729–730 (1993).
[CrossRef]

R. Brede, E. Heumann, J. Koetke, T. Danger, G. Huber, and B. Chai, “Green up-conversion laser emission in Er-doped crystals at room temperature,” Appl. Phys. Lett. 63, 2030–2031 (1993).
[CrossRef]

Chai, B.

R. Brede, E. Heumann, J. Koetke, T. Danger, G. Huber, and B. Chai, “Green up-conversion laser emission in Er-doped crystals at room temperature,” Appl. Phys. Lett. 63, 2030–2031 (1993).
[CrossRef]

R. Brede, T. Danger, E. Heumann, G. Huber, and B. Chai, “Room-temperature green laser emission of Er:LiYF4,” Appl. Phys. Lett. 63, 729–730 (1993).
[CrossRef]

Chai, B. H. T.

T. Danger, J. Koetke, R. Brede, E. Heumann, G. Huber, and B. H. T. Chai, “Spectroscopy and green upconversion laser emission of Er3+-doped crystals at room temperature,” J. Appl. Phys. 76, 1413–1422 (1994).
[CrossRef]

Chenevier, M.

E. Lacot, F. Stoeckel, and M. Chenevier, “Dynamics of an erbium-doped fiber laser,” Phys. Rev. A 49, 3997–4007 (1994).
[CrossRef] [PubMed]

Colin, S.

Contesse, E.

Crawford, J. D.

J. D. Crawford, “Introduction to bifurcation theory,” Rev. Mod. Phys. 63, 991–1037 (1991).
[CrossRef]

Danger, T.

T. Danger, J. Koetke, R. Brede, E. Heumann, G. Huber, and B. H. T. Chai, “Spectroscopy and green upconversion laser emission of Er3+-doped crystals at room temperature,” J. Appl. Phys. 76, 1413–1422 (1994).
[CrossRef]

R. Brede, T. Danger, E. Heumann, G. Huber, and B. Chai, “Room-temperature green laser emission of Er:LiYF4,” Appl. Phys. Lett. 63, 729–730 (1993).
[CrossRef]

R. Brede, E. Heumann, J. Koetke, T. Danger, G. Huber, and B. Chai, “Green up-conversion laser emission in Er-doped crystals at room temperature,” Appl. Phys. Lett. 63, 2030–2031 (1993).
[CrossRef]

Doualan, J.-L.

P. LeBoulanger, J.-L. Doualan, S. Girard, J. Margerie, and R. Moncorgé, “Excited-state absorption spectroscopy of Er3+-doped Y3Al5O12, YVO4, and phosphate glass,” Phys. Rev. B 60, 11380–11390 (1999).
[CrossRef]

Florea, C.

S. Georgescu, O. Toma, C. Florea, and C. Naud, “ESA processes responsible for infrared-pumped, green and violet luminescence in low-concentrated Er: YAG,” J. Lumin. 101, 87–99 (2003).
[CrossRef]

François, P.-L.

F. Sanchez, P. Le Boudec, P.-L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

Georgescu, S.

S. Georgescu, O. Toma, C. Florea, and C. Naud, “ESA processes responsible for infrared-pumped, green and violet luminescence in low-concentrated Er: YAG,” J. Lumin. 101, 87–99 (2003).
[CrossRef]

Girard, S.

P. LeBoulanger, J.-L. Doualan, S. Girard, J. Margerie, and R. Moncorgé, “Excited-state absorption spectroscopy of Er3+-doped Y3Al5O12, YVO4, and phosphate glass,” Phys. Rev. B 60, 11380–11390 (1999).
[CrossRef]

Guggenheim, H. J.

L. F. Johnson and H. J. Guggenheim, “New laser lines in the visible from Er3+ ions in BaY2F8,” Appl. Phys. Lett. 20, 474–477 (1972).
[CrossRef]

Heumann, E.

T. Danger, J. Koetke, R. Brede, E. Heumann, G. Huber, and B. H. T. Chai, “Spectroscopy and green upconversion laser emission of Er3+-doped crystals at room temperature,” J. Appl. Phys. 76, 1413–1422 (1994).
[CrossRef]

R. Brede, T. Danger, E. Heumann, G. Huber, and B. Chai, “Room-temperature green laser emission of Er:LiYF4,” Appl. Phys. Lett. 63, 729–730 (1993).
[CrossRef]

R. Brede, E. Heumann, J. Koetke, T. Danger, G. Huber, and B. Chai, “Green up-conversion laser emission in Er-doped crystals at room temperature,” Appl. Phys. Lett. 63, 2030–2031 (1993).
[CrossRef]

Huber, G.

T. Danger, J. Koetke, R. Brede, E. Heumann, G. Huber, and B. H. T. Chai, “Spectroscopy and green upconversion laser emission of Er3+-doped crystals at room temperature,” J. Appl. Phys. 76, 1413–1422 (1994).
[CrossRef]

R. Brede, T. Danger, E. Heumann, G. Huber, and B. Chai, “Room-temperature green laser emission of Er:LiYF4,” Appl. Phys. Lett. 63, 729–730 (1993).
[CrossRef]

R. Brede, E. Heumann, J. Koetke, T. Danger, G. Huber, and B. Chai, “Green up-conversion laser emission in Er-doped crystals at room temperature,” Appl. Phys. Lett. 63, 2030–2031 (1993).
[CrossRef]

Johnson, L. F.

L. F. Johnson and H. J. Guggenheim, “New laser lines in the visible from Er3+ ions in BaY2F8,” Appl. Phys. Lett. 20, 474–477 (1972).
[CrossRef]

Kellou, A.

Koetke, J.

T. Danger, J. Koetke, R. Brede, E. Heumann, G. Huber, and B. H. T. Chai, “Spectroscopy and green upconversion laser emission of Er3+-doped crystals at room temperature,” J. Appl. Phys. 76, 1413–1422 (1994).
[CrossRef]

R. Brede, E. Heumann, J. Koetke, T. Danger, G. Huber, and B. Chai, “Green up-conversion laser emission in Er-doped crystals at room temperature,” Appl. Phys. Lett. 63, 2030–2031 (1993).
[CrossRef]

Lacot, E.

E. Lacot, F. Stoeckel, and M. Chenevier, “Dynamics of an erbium-doped fiber laser,” Phys. Rev. A 49, 3997–4007 (1994).
[CrossRef] [PubMed]

Le Boudec, P.

S. Colin, E. Contesse, P. Le Boudec, G. Stephan, and F. Sanchez, “Evidence of a saturable-absorption effect in heavily erbium-doped fibers,” Opt. Lett. 21, 1987–1989 (1996).
[CrossRef] [PubMed]

F. Sanchez, P. Le Boudec, P.-L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

LeBoulanger, P.

P. LeBoulanger, J.-L. Doualan, S. Girard, J. Margerie, and R. Moncorgé, “Excited-state absorption spectroscopy of Er3+-doped Y3Al5O12, YVO4, and phosphate glass,” Phys. Rev. B 60, 11380–11390 (1999).
[CrossRef]

Lüthy, W.

M. Pollnau, W. Lüthy, and H. P. Weber, “Population mechanisms of the green Er3+:LiYF4 laser,” J. Appl. Phys. 77, 6128–6134 (1995).
[CrossRef]

Margerie, J.

P. LeBoulanger, J.-L. Doualan, S. Girard, J. Margerie, and R. Moncorgé, “Excited-state absorption spectroscopy of Er3+-doped Y3Al5O12, YVO4, and phosphate glass,” Phys. Rev. B 60, 11380–11390 (1999).
[CrossRef]

McFarlane, R. A.

R. A. McFarlane, “Dual wavelength visible upconversion laser,” Appl. Phys. Lett. 54, 2301–2302 (1989).
[CrossRef]

Moncorgé, R.

P. LeBoulanger, J.-L. Doualan, S. Girard, J. Margerie, and R. Moncorgé, “Excited-state absorption spectroscopy of Er3+-doped Y3Al5O12, YVO4, and phosphate glass,” Phys. Rev. B 60, 11380–11390 (1999).
[CrossRef]

Naud, C.

S. Georgescu, O. Toma, C. Florea, and C. Naud, “ESA processes responsible for infrared-pumped, green and violet luminescence in low-concentrated Er: YAG,” J. Lumin. 101, 87–99 (2003).
[CrossRef]

Parlett, B. N.

B. N. Parlett, “The QR algorithm,” Comput. Sci. Eng. 2, 38–42 (2000).
[CrossRef]

Pollnau, M.

M. Pollnau, W. Lüthy, and H. P. Weber, “Population mechanisms of the green Er3+:LiYF4 laser,” J. Appl. Phys. 77, 6128–6134 (1995).
[CrossRef]

Sanchez, F.

Stephan, G.

S. Colin, E. Contesse, P. Le Boudec, G. Stephan, and F. Sanchez, “Evidence of a saturable-absorption effect in heavily erbium-doped fibers,” Opt. Lett. 21, 1987–1989 (1996).
[CrossRef] [PubMed]

F. Sanchez, P. Le Boudec, P.-L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

Stoeckel, F.

E. Lacot, F. Stoeckel, and M. Chenevier, “Dynamics of an erbium-doped fiber laser,” Phys. Rev. A 49, 3997–4007 (1994).
[CrossRef] [PubMed]

Toma, O.

S. Georgescu, O. Toma, C. Florea, and C. Naud, “ESA processes responsible for infrared-pumped, green and violet luminescence in low-concentrated Er: YAG,” J. Lumin. 101, 87–99 (2003).
[CrossRef]

Weber, H. P.

M. Pollnau, W. Lüthy, and H. P. Weber, “Population mechanisms of the green Er3+:LiYF4 laser,” J. Appl. Phys. 77, 6128–6134 (1995).
[CrossRef]

Appl. Phys. Lett. (4)

R. A. McFarlane, “Dual wavelength visible upconversion laser,” Appl. Phys. Lett. 54, 2301–2302 (1989).
[CrossRef]

L. F. Johnson and H. J. Guggenheim, “New laser lines in the visible from Er3+ ions in BaY2F8,” Appl. Phys. Lett. 20, 474–477 (1972).
[CrossRef]

R. Brede, E. Heumann, J. Koetke, T. Danger, G. Huber, and B. Chai, “Green up-conversion laser emission in Er-doped crystals at room temperature,” Appl. Phys. Lett. 63, 2030–2031 (1993).
[CrossRef]

R. Brede, T. Danger, E. Heumann, G. Huber, and B. Chai, “Room-temperature green laser emission of Er:LiYF4,” Appl. Phys. Lett. 63, 729–730 (1993).
[CrossRef]

Comput. Sci. Eng. (1)

B. N. Parlett, “The QR algorithm,” Comput. Sci. Eng. 2, 38–42 (2000).
[CrossRef]

J. Appl. Phys. (2)

M. Pollnau, W. Lüthy, and H. P. Weber, “Population mechanisms of the green Er3+:LiYF4 laser,” J. Appl. Phys. 77, 6128–6134 (1995).
[CrossRef]

T. Danger, J. Koetke, R. Brede, E. Heumann, G. Huber, and B. H. T. Chai, “Spectroscopy and green upconversion laser emission of Er3+-doped crystals at room temperature,” J. Appl. Phys. 76, 1413–1422 (1994).
[CrossRef]

J. Lumin. (1)

S. Georgescu, O. Toma, C. Florea, and C. Naud, “ESA processes responsible for infrared-pumped, green and violet luminescence in low-concentrated Er: YAG,” J. Lumin. 101, 87–99 (2003).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Phys. Rev. A (2)

F. Sanchez, P. Le Boudec, P.-L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A 48, 2220–2229 (1993).
[CrossRef] [PubMed]

E. Lacot, F. Stoeckel, and M. Chenevier, “Dynamics of an erbium-doped fiber laser,” Phys. Rev. A 49, 3997–4007 (1994).
[CrossRef] [PubMed]

Phys. Rev. B (1)

P. LeBoulanger, J.-L. Doualan, S. Girard, J. Margerie, and R. Moncorgé, “Excited-state absorption spectroscopy of Er3+-doped Y3Al5O12, YVO4, and phosphate glass,” Phys. Rev. B 60, 11380–11390 (1999).
[CrossRef]

Rev. Mod. Phys. (1)

J. D. Crawford, “Introduction to bifurcation theory,” Rev. Mod. Phys. 63, 991–1037 (1991).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Energy-level scheme of Er:YAG showing the various processes considered in our model. The ESA processes [a], [b], and [c] are various candidates for the second step of the pumping process. The ESA process of cross section σ1 is the parasitic ESA process. The laser transition  4S3/24I15/2 is also shown.

Fig. 2
Fig. 2

(a) Stability diagram showing the domains in parameter space that correspond to various laser regimes. For our configuration, the critical value of the resonator losses is ρc=6.2×10-3 cm-1. (b) Detail of (a).

Fig. 3
Fig. 3

Real part of the eigenvalues of matrix J for ρ=7.1×10-3 cm-1>ρc. The constant eigenvalues λ5=0, λ6=-0.667 µs-1, and λ7=-20 µs-1 have been omitted. The steady-state bifurcation point at ϕth=4.359×1018 cm-2 µs-1.

Fig. 4
Fig. 4

Relaxation oscillations obtained by numerical integration of rate-equation-system Eqs. (1), for ϕ=5.5×1018 cm-2 µs-1 and ρ=7.1×10-3 cm-1.

Fig. 5
Fig. 5

(a) Real part of the eigenvalues of matrix J for ρ=4.5×10-3 cm-1<ρc. There is visible a Hopf bifurcation point at ϕ3=4.843×1018 cm-2 µs-1. We omit λ5=0, λ6=-0.667 µs-1, and λ7=-20 µs-1 for the same reason as above. (b) Detail of (a), showing another two bifurcations: a steady-state bifurcation point at ϕ1=4.116×1018 cm-2 µs-1 and a Hopf bifurcation point at ϕ2=4.117×1018 cm-2 µs-1.

Fig. 6
Fig. 6

Laser pulses obtained by numerical integration of rate-equation-system Eq. (1), for ϕ=4.4×1018 cm-2 µs-1 and ρ=4.5×10-3 cm-1.

Tables (1)

Tables Icon

Table 1 Spectroscopic Parameters of Er:YAG

Equations (43)

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

dN0dt=σem(f51N5-f08N0)φ+N1T1-σ0ϕN0,
dN1dt=-N1T1+N2T2-σbϕN1-σ1N1φ,
dN2dt=-N2T2+N3T3-σcϕN2,
dN3dt=-N3T3+N4T4+σ0ϕN0-σaϕN3,
dN4dt=-N4T4+N5T5,
dN5dt=-N5T5-σem(f51N5-f08N0)φ+σaϕN3+σbϕN1+σcϕN2+σ1N1φ,
dφdt=v[σem(f51N5-f08N0)-ρ-σ1N1]×φ+k N5T5,
ρ=ρ0-log(R1R2)2lp.
N0+N1+N2+N3+N4+N5=Nt.
N¯0=Nt+ρσ1-1+σemσ1f51+T2+T3+T4T5N¯51-σemσ1f08+σ0ϕ(T2+T3),
N¯1=-σemσ1f08N¯0+σemσ1f51N¯5-ρσ1,
N¯2=σ0ϕN¯0+N¯5T5,
N¯3=T3T2N¯2,
N¯4=T4T5N¯5,
φ¯=1ρ σemσ1σbϕA-1T5N¯5+σbϕσ1 σemρB-1,
A=(f51+f08)+σ0ϕ(T2+T3)f51+T2+T3+T4T5f081-σemσ1f08+σ0ϕ(T2+T3),
B=-f08Nt+ρσ11-σemσ1f08+σ0ϕ(T2+T3).
σemρA1T5-σemσ1σbϕAN52+σemσ1 σbϕ-1T1-σ0ϕf08A-2 σem2σ1ρσbϕAB+σemρT5B+σ0ϕ f51f08N5+ρσ1T1+σemσ1 σbϕ-1T1-σ0ϕf08B-σem2σ1ρσbϕB2=0.
J=-n30-n50n01000n05n060-n01-n51n12000n1600-n12n23000n3000-n23n34000000-n34n450n50n51000-n05-n45-n06-n16-vn50n61000vn05v(n06+n16-ρ),
n01=1T1,
n05=σemf51φ¯,
n06=σem(f51N¯5-f08N¯0),
n12=1T2,
n16=-σ1N¯1,
n23=1T3,
n30=σ0ϕ,
n34=1T4,
n45=1T5,
n50=σemf08φ¯,
n51=σbϕ+σ1φ¯,
n61=-vσ1φ¯.
f51-ρσemNt T2+T3+T4+T5T5ϕ2-σ1σemσbT5
+1σbT5ρσemNt T1+T2+T3T1ϕ
-1σ0σbT1T5 f08+ρσemNt=0.
ξ=ρσemNt,
ζ=σ1σemσbT5.
Δ=1σbT1T52 (T1+T2+T3)2σbT1-4(T2+T3+T4+T5)σ0ξ2+1σbT1T5 2(T1+T2+T3)ζ-4f08σ0T5(T2+T3+T4+T5)+4f51σ0ξ+ζ2+4f08f51σ0σbT1T5.
ξf51 T5T2+T3+T4+T5.
ϕth=ζ+1σbT5 T1+T2+T3T1ξ+Δ2f51-T2+T3+T4+T5T5ξ.
ϕth(ρ=0)=12 f51 ζ+ζ2+4f08f51σ0σbT1T51/2.
ϕth(ρ=0;σ1=0)=f08f51σ0σbT1T51/2.
ϕth(ρ=0)=12 f51 ζ+ζ2+4f08f51σ0σcT2T51/2.
ϕth(ρ=0;σ1=0)=f08f51σ0σcT2T51/2.

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