Abstract

A model of transient modal instability in fiber amplifiers is presented. This model combines an optical beam propagation method that incorporates laser gain through local solution of the rate equations and refractive index perturbations caused by the thermo-optic effect with a time-dependent thermal solver with a quantum defect heating source term. This model predicts modal instability a fiber amplifier operating at 241, 270, and 287 Watts of output power characterized by power coupling to un-seeded modes, the presence of stable and unstable regions within the fiber, and rapid intensity variations along the fiber. The instability becomes more severe as the power is increased.

© 2013 OSA

Full Article  |  PDF Article
OSA Recommended Articles
Origin of thermal modal instabilities in large mode area fiber amplifiers

B. Ward, C. Robin, and I. Dajani
Opt. Express 20(10) 11407-11422 (2012)

Optimizing high-power Yb-doped fiber amplifier systems in the presence of transverse mode instabilities

Cesar Jauregui, Hans-Jürgen Otto, Sven Breitkopf, J. Limpert, and A. Tünnermann
Opt. Express 24(8) 7879-7892 (2016)

Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers

Kristian Rymann Hansen and Jesper Lægsgaard
Opt. Express 22(9) 11267-11278 (2014)

References

  • View by:
  • |
  • |
  • |

  1. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13218 .
    [Crossref] [PubMed]
  2. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3258 .
    [Crossref] [PubMed]
  3. H. J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20(14), 15710–15722 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15710 .
    [Crossref] [PubMed]
  4. M. Karow, H. Tünnermann, J. Neumann, D. Kracht, and P. Weßels, “Beam quality degradation of a single-frequency Yb-doped photonic crystal fiber amplifier with low mode instability threshold power,” Opt. Lett. 37(20), 4242–4244 (2012), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-37-20-4242 .
    [Crossref] [PubMed]
  5. N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20(12), 13274–13283 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13274 .
    [Crossref] [PubMed]
  6. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10180 .
    [Crossref] [PubMed]
  7. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-11407 .
    [Crossref] [PubMed]
  8. C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20(12), 12912–12925 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-12912 .
    [Crossref] [PubMed]
  9. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express 21(2), 1944–1971 (2013), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-1944 .
    [Crossref] [PubMed]
  10. L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21(3), 2642–2656 (2013), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2642 .
    [Crossref] [PubMed]
  11. A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20(22), 24545–24558 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24545 .
    [Crossref] [PubMed]
  12. A. V. Smith and J. J. Smith, “Steady-periodic method for modeling mode instability in fiber amplifiers,” Opt. Express 21(3), 2606–2623 (2013), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2606 .
    [Crossref] [PubMed]
  13. S. A. Shakir, R. A. Motes, and R. W. Berdine, “Efficient scalar beam propagation method,” IEEE J. Quantum Electron. 47(4), 486–491 (2011), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5730173 .
    [Crossref]
  14. P. D. Dragic and B. G. Ward, “Accurate modeling of the intrinsic Brillouin linewidth via finite-element analysis,” IEEE Photon. Technol. Lett. 22(22), 1698–1700 (2010), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5590281 .
    [Crossref]
  15. S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, PETSc Users Manual, (ANL-95/11 - Revision 3.3, Argonne National Laboratory, 2012), http://www.mcs.anl.gov/petsc/ .
  16. R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4300920&isnumber=4294077 .
    [Crossref]
  17. G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 0.363–370 (1992), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=119536&isnumber=3419 .
    [Crossref]

2013 (3)

2012 (6)

A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20(22), 24545–24558 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24545 .
[Crossref] [PubMed]

H. J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20(14), 15710–15722 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15710 .
[Crossref] [PubMed]

M. Karow, H. Tünnermann, J. Neumann, D. Kracht, and P. Weßels, “Beam quality degradation of a single-frequency Yb-doped photonic crystal fiber amplifier with low mode instability threshold power,” Opt. Lett. 37(20), 4242–4244 (2012), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-37-20-4242 .
[Crossref] [PubMed]

N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20(12), 13274–13283 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13274 .
[Crossref] [PubMed]

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-11407 .
[Crossref] [PubMed]

C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20(12), 12912–12925 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-12912 .
[Crossref] [PubMed]

2011 (4)

2010 (1)

P. D. Dragic and B. G. Ward, “Accurate modeling of the intrinsic Brillouin linewidth via finite-element analysis,” IEEE Photon. Technol. Lett. 22(22), 1698–1700 (2010), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5590281 .
[Crossref]

2007 (1)

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4300920&isnumber=4294077 .
[Crossref]

Alkeskjold, T. T.

Berdine, R. W.

S. A. Shakir, R. A. Motes, and R. W. Berdine, “Efficient scalar beam propagation method,” IEEE J. Quantum Electron. 47(4), 486–491 (2011), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5730173 .
[Crossref]

Broeng, J.

Cole, J. H.

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4300920&isnumber=4294077 .
[Crossref]

Dajani, I.

de Vries, O.

Dong, L.

Dragic, P. D.

P. D. Dragic and B. G. Ward, “Accurate modeling of the intrinsic Brillouin linewidth via finite-element analysis,” IEEE Photon. Technol. Lett. 22(22), 1698–1700 (2010), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5590281 .
[Crossref]

Eberhardt, R.

Eidam, T.

Haarlammert, N.

Hansen, K. R.

Jansen, F.

Jauregui, C.

Karow, M.

Kliner, A.

Kracht, D.

Lægsgaard, J.

Liem, A.

Limpert, J.

Motes, R. A.

S. A. Shakir, R. A. Motes, and R. W. Berdine, “Efficient scalar beam propagation method,” IEEE J. Quantum Electron. 47(4), 486–491 (2011), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5730173 .
[Crossref]

Neumann, J.

Otto, H. J.

Peschel, T.

Robin, C.

Schermer, R. T.

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4300920&isnumber=4294077 .
[Crossref]

Schmidt, O.

Schreiber, T.

Shakir, S. A.

S. A. Shakir, R. A. Motes, and R. W. Berdine, “Efficient scalar beam propagation method,” IEEE J. Quantum Electron. 47(4), 486–491 (2011), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5730173 .
[Crossref]

Smith, A. V.

Smith, J. J.

Stutzki, F.

Tünnermann, A.

C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20(12), 12912–12925 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-12912 .
[Crossref] [PubMed]

N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20(12), 13274–13283 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13274 .
[Crossref] [PubMed]

H. J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20(14), 15710–15722 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15710 .
[Crossref] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13218 .
[Crossref] [PubMed]

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3258 .
[Crossref] [PubMed]

Tünnermann, H.

Ward, B.

Ward, B. G.

P. D. Dragic and B. G. Ward, “Accurate modeling of the intrinsic Brillouin linewidth via finite-element analysis,” IEEE Photon. Technol. Lett. 22(22), 1698–1700 (2010), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5590281 .
[Crossref]

Weßels, P.

Wirth, C.

IEEE J. Quantum Electron. (2)

S. A. Shakir, R. A. Motes, and R. W. Berdine, “Efficient scalar beam propagation method,” IEEE J. Quantum Electron. 47(4), 486–491 (2011), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5730173 .
[Crossref]

R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4300920&isnumber=4294077 .
[Crossref]

IEEE Photon. Technol. Lett. (1)

P. D. Dragic and B. G. Ward, “Accurate modeling of the intrinsic Brillouin linewidth via finite-element analysis,” IEEE Photon. Technol. Lett. 22(22), 1698–1700 (2010), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5590281 .
[Crossref]

Opt. Express (11)

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13218 .
[Crossref] [PubMed]

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3258 .
[Crossref] [PubMed]

H. J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20(14), 15710–15722 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15710 .
[Crossref] [PubMed]

N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20(12), 13274–13283 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13274 .
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10180 .
[Crossref] [PubMed]

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-11407 .
[Crossref] [PubMed]

C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20(12), 12912–12925 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-12912 .
[Crossref] [PubMed]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express 21(2), 1944–1971 (2013), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-1944 .
[Crossref] [PubMed]

L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21(3), 2642–2656 (2013), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2642 .
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20(22), 24545–24558 (2012), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24545 .
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Steady-periodic method for modeling mode instability in fiber amplifiers,” Opt. Express 21(3), 2606–2623 (2013), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2606 .
[Crossref] [PubMed]

Opt. Lett. (1)

Other (2)

S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, PETSc Users Manual, (ANL-95/11 - Revision 3.3, Argonne National Laboratory, 2012), http://www.mcs.anl.gov/petsc/ .

G. R. Hadley, “Transparent boundary condition for the beam propagation method,” IEEE J. Quantum Electron. 28, 0.363–370 (1992), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=119536&isnumber=3419 .
[Crossref]

Supplementary Material (2)

» Media 1: MOV (2438 KB)     
» Media 2: MOV (2533 KB)     

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Sequence of steps required to carry out the time dependent amplifier simulation.

Fig. 2
Fig. 2

Plot of the optical intensity as a function of length at the center of the fiber core and at points offset from the center of the fiber core by half the core radius after a simulation time of 14 ms (a) and 20 ms (b) for an output power of 287 Watts. The accompanying movie (Media 1) shows the evolution of these intensity probes over a time period of 40 ms. The insets shows the optical field at the output. The markers in the insets show the locations of the probes.

Fig. 3
Fig. 3

Modal content of the amplifier output over time as well as total power from the integrated optical output intensity and the sum of the first 8 modal powers for amplifier output powers of 241 (a), 270 (b), and 287 (c) Watts. Plot (d) is a zoomed in view of (c) to show the details of the rapid fluctuations between 10 and 22 ms. The modal content was calculated with respect to the cold fiber modes. The pump was ramped on linearly in time over the first 10 ms in each case.

Fig. 4
Fig. 4

Plots of the intensity profiles of the first 8 guided modes of the cold fiber incorporated in the simulated amplifier: LP01(a), LP11o(b), LP11e(c), LP21o(d), LP21e(e), LP02(f), LP31o(g), LP31e(h).

Fig. 5
Fig. 5

Frequency spectra of the first 8 modes of the simulated fiber calculated over the simulation period of 100 ms for output powers of 241 (a), 270 (b), and 287 (c) Watts.

Fig. 6
Fig. 6

Temperature distribution as a function of length at the fiber core and at points offset from the center of the fiber core by half the core radius at a time of 14 ms or the amplifier operating at 287 Watts of output power. The accompanying movie (Media 2) shows the evolution of these temperature probes over a time period of 40 ms.

Tables (1)

Tables Icon

Table 1 Simulated Fiber Parameters

Equations (28)

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

2 E( r )+ n 2 ( r ) k 0 2 E( r )=0,
[ 2iβ z + t 2 + n 2 ( r,φ,z ) k 0 2 β 2 ]E( r,φ,z )=0
[ t 2 + n 2 ( r,φ ) k 0 2 β 2 ]E( r,φ )=0.
S= 1 2 Ω [ | t E(r,φ) | 2 + n 2 ( r,φ ) k 0 2 | E(r,φ) | 2 β 2 | E(r,φ) | 2 ] dA
E(r,φ)= q=Q Q E q (r)exp[ iqφ ]
t E(r,φ)= E r r ^ + 1 r E φ φ ^ = q=Q Q [ E q r r ^ + iq r E q φ ^ ] exp[ iqφ ]
n 2 (r,φ)= n 0 2 + q=Q Q V q (r)exp[ iqφ ]
V q (r) 1 2π 0 2π δ n 2 (r,φ)exp[ iqφ ] dφ
S= 1 2 q=Q Q Ω [ | E q r | 2 +( q 2 r 2 k 0 2 n 0 2 + β 2 ) | E q | 2 k 0 2 q =Q Q V q q E q * E q ] dA.
Ω f( r ) exp[ iqφ ]dAπδ( q ) k=1 N1 f( r k + r k+1 2 ) ( r k+1 2 r k 2 )
Ω [ f ( r ) ] 2 exp[ iqφ ]dAπδ( q ) k=1 N1 ( f( r k+1 )f( r k ) r k+1 r k ) 2 ( r k+1 2 r k 2 ).
S= 1 2 E * [ K β 2 M ]E
[ K β 2 M ]E=0
[ 2ikM d dz +K k 2 M ]E=0.
E m+1 = ( M+ iΔz 4k ( K m+1 k 2 M ) ) 1 ( M iΔz 4k ( K m k 2 M ) ) E m
δ n 2 ( r,φ,z )=δ n wg 2 ( r,φ )+i n 0 g( r,φ,z )+2 n 0 dn dT ΔT(r,φ,z)
f( I s , I p )= ( I s σ as τ h ν s + I p σ ap τ h ν p ) / ( 1+ I s ( σ as + σ es )τ h ν s + I p ( σ ap + σ ep )τ h ν p )
g( r,φ,z )= N ion [ f( r,φ,z ) σ es ( 1f( r,φ,z ) σ as ) ]
g p ( r,φ,z )= N ion [ f( r,φ,z ) σ ep ( 1f( r,φ,z ) σ ap ) ].
d dz I p ( z )=± 1 A clad Ω g p ( r,φ,z ) I p ( r,φ,z )dA
Q(r,φ,z)=( ν p ν s 1 )g(r,φ,z) I s (r,φ,z).
t T( r,φ,z,t )= K ρC 2 T( r,φ,z,t )+ 1 ρC Q( r,φ,z,t )
2 z 2 T( r,φ,z ) | m 1 Δ z 2 [ T m+1 ( r,φ )2 T m ( r,φ )+ T m1 ( r,φ ) ].
T q,l+1 = [ M TH + Δt 2 K ρC K q,TH ] 1 [ ( M TH Δt 2 K ρC K q,TH ) T q,l + Δt 2ρC M TH ( Q q,l+1 + Q q,l ) ]
M TH = k=1 N1 m=1 M Δ A k [ v k,m v k,m T ]
K q,TH = k=1 N1 m=1 M Δ A k [ 1 Δ r k 2 u k,m u k,m T + 4 q 2 ( r k + r k+1 ) 2 v k,m v k,m T + 1 Δ z 2 ( v k,m+1 v k,m+1 T 2 v k,m v k,m T + v k,m1 v k,m1 T ) ]
Q q,lzm = 1 2π 0 2π Q( r k ,φ, z m , t l )exp[ iqφ ] dφ
E( r,φ,0,t )= n=1 N P n E n ( r,φ )exp[ i f n ( t ) ]

Metrics