Abstract

A time-dependent analytical thermal model of the temperature and the corresponding induced thermal stresses on the pump face of quasi-continuous wave (qcw) end-pumped laser rods is derived. We apply the model to qcw diode-end-pumped rods and show the maximum peak pump power that can be utilized without fracturing the rod. To illustrate an application of the model, it is applied to a qcw pumped Tm:YLF rod and found to be in very good agreement with published experimental results.The results indicate new criteria to avoid fracture when operating Tm:YLF rods at low qcw pump duty cycles.

© 2008 Optical Society of America

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References

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  1. W. KoechnerSolid-State Laser Engineering 4th ed., (Springer-Verlag Berlin, Heidelberg, Germany,1996).
  2. C. Pfistner, R. Weber, H. P. Weber, S Merazzi, and R. Gruber, “Thermal Beam Distortions in End-Pumped Nd:YAG, Nd:GSGG and Nd:YLF,” IEEE J. Quantum Electron. 30, 1605–1615 (1994).
    [CrossRef]
  3. A. K. Cousins, “Temperature and Thermal Stress Scaling in Finite-Length End-Pumped Lasers Rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992).
    [CrossRef]
  4. L. Yan and C. H. Lee, “Thermal Effects in End-Pumped Nd:phosphate Glasses,” J. Appl. Phys. 75, 1286–1292 (1994).
    [CrossRef]
  5. S. Chénais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and Absolute Temperature Mapping and Heat Transfer Measurements in Diode-End-Pumped Yb:YAG,” Appl. Phys. B  79, 221–224 (2004).
    [CrossRef]
  6. X. Peng, L. Xu, and A. Asundi, “High-power efficient continous-wave TEM00 intracavity frequencydoubled diode-pumped Nd:YLF laser,” Appl. Opt. 44, 800–807 (2005).
    [CrossRef] [PubMed]
  7. G. Barton, Elements of Green’s Functions and Propagation, (Oxford University Press, Oxford,1995).
  8. H. S. Carslaw and J. C. Jaeger, Conduction of Heats in Solids, (Oxford University Press, Oxford, 1959).
  9. T. Y. Fan, “Heat Generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
    [CrossRef]
  10. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity 3rd ed., (McGraw-Hill, New York, 1970).
  11. B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, (Courier Dover Publications, New York, 1997).
  12. LASCAD Manual, (LASCAD, 2008), https://www.las-cad.com/files/PP_FEA.pdf.
  13. ABAQUS Finite Element Software Package (ABAQUS, 2008) http://www.simulia.com/products/abaqus_fea.html.
  14. E. H. Bernhardi, C. Bollig, L. Harris, M. J. D. Esser, and A. Forbes, “Investigating thermal stresses in quasicw pumped Tm:YLF laser crystals,” in Proceedings of Advanced Solid-State Photonics, (Nara, Japan,2008), Poster WB11.
  15. S. So, J. I. Mackenzie, D. P. Shepherd, W. A. Clarkson, J. G. Betterson, and E. K. Gorton, “A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers,” Appl. Phys. B  84, 389–393 (2006).
    [CrossRef]
  16. M. Pollnau, P. J. Hardman, M. A. Kern, W. A. Clarkson, and D. C. Hanna, “Upconversion-induced heat generation and thermal lensing in Nd:YLF and Nd:YAG,” Phys. Rev. B  58, 16076–16092 (1998).
    [CrossRef]
  17. B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95, 3255–3271 (2004).
    [CrossRef]

2006 (1)

S. So, J. I. Mackenzie, D. P. Shepherd, W. A. Clarkson, J. G. Betterson, and E. K. Gorton, “A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers,” Appl. Phys. B  84, 389–393 (2006).
[CrossRef]

2005 (1)

2004 (2)

S. Chénais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and Absolute Temperature Mapping and Heat Transfer Measurements in Diode-End-Pumped Yb:YAG,” Appl. Phys. B  79, 221–224 (2004).
[CrossRef]

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95, 3255–3271 (2004).
[CrossRef]

1998 (1)

M. Pollnau, P. J. Hardman, M. A. Kern, W. A. Clarkson, and D. C. Hanna, “Upconversion-induced heat generation and thermal lensing in Nd:YLF and Nd:YAG,” Phys. Rev. B  58, 16076–16092 (1998).
[CrossRef]

1994 (2)

L. Yan and C. H. Lee, “Thermal Effects in End-Pumped Nd:phosphate Glasses,” J. Appl. Phys. 75, 1286–1292 (1994).
[CrossRef]

C. Pfistner, R. Weber, H. P. Weber, S Merazzi, and R. Gruber, “Thermal Beam Distortions in End-Pumped Nd:YAG, Nd:GSGG and Nd:YLF,” IEEE J. Quantum Electron. 30, 1605–1615 (1994).
[CrossRef]

1993 (1)

T. Y. Fan, “Heat Generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
[CrossRef]

1992 (1)

A. K. Cousins, “Temperature and Thermal Stress Scaling in Finite-Length End-Pumped Lasers Rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992).
[CrossRef]

Asundi, A.

Balembois, F.

S. Chénais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and Absolute Temperature Mapping and Heat Transfer Measurements in Diode-End-Pumped Yb:YAG,” Appl. Phys. B  79, 221–224 (2004).
[CrossRef]

Barnes, N. P.

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95, 3255–3271 (2004).
[CrossRef]

Barton, G.

G. Barton, Elements of Green’s Functions and Propagation, (Oxford University Press, Oxford,1995).

Bernhardi, E. H.

E. H. Bernhardi, C. Bollig, L. Harris, M. J. D. Esser, and A. Forbes, “Investigating thermal stresses in quasicw pumped Tm:YLF laser crystals,” in Proceedings of Advanced Solid-State Photonics, (Nara, Japan,2008), Poster WB11.

Betterson, J. G.

S. So, J. I. Mackenzie, D. P. Shepherd, W. A. Clarkson, J. G. Betterson, and E. K. Gorton, “A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers,” Appl. Phys. B  84, 389–393 (2006).
[CrossRef]

Boley, B. A.

B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, (Courier Dover Publications, New York, 1997).

Bollig, C.

E. H. Bernhardi, C. Bollig, L. Harris, M. J. D. Esser, and A. Forbes, “Investigating thermal stresses in quasicw pumped Tm:YLF laser crystals,” in Proceedings of Advanced Solid-State Photonics, (Nara, Japan,2008), Poster WB11.

Carslaw, H. S.

H. S. Carslaw and J. C. Jaeger, Conduction of Heats in Solids, (Oxford University Press, Oxford, 1959).

Chénais, S.

S. Chénais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and Absolute Temperature Mapping and Heat Transfer Measurements in Diode-End-Pumped Yb:YAG,” Appl. Phys. B  79, 221–224 (2004).
[CrossRef]

Clarkson, W. A.

S. So, J. I. Mackenzie, D. P. Shepherd, W. A. Clarkson, J. G. Betterson, and E. K. Gorton, “A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers,” Appl. Phys. B  84, 389–393 (2006).
[CrossRef]

M. Pollnau, P. J. Hardman, M. A. Kern, W. A. Clarkson, and D. C. Hanna, “Upconversion-induced heat generation and thermal lensing in Nd:YLF and Nd:YAG,” Phys. Rev. B  58, 16076–16092 (1998).
[CrossRef]

Cousins, A. K.

A. K. Cousins, “Temperature and Thermal Stress Scaling in Finite-Length End-Pumped Lasers Rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992).
[CrossRef]

Druon, F.

S. Chénais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and Absolute Temperature Mapping and Heat Transfer Measurements in Diode-End-Pumped Yb:YAG,” Appl. Phys. B  79, 221–224 (2004).
[CrossRef]

Esser, M. J. D.

E. H. Bernhardi, C. Bollig, L. Harris, M. J. D. Esser, and A. Forbes, “Investigating thermal stresses in quasicw pumped Tm:YLF laser crystals,” in Proceedings of Advanced Solid-State Photonics, (Nara, Japan,2008), Poster WB11.

Fan, T. Y.

T. Y. Fan, “Heat Generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
[CrossRef]

Forbes, A.

E. H. Bernhardi, C. Bollig, L. Harris, M. J. D. Esser, and A. Forbes, “Investigating thermal stresses in quasicw pumped Tm:YLF laser crystals,” in Proceedings of Advanced Solid-State Photonics, (Nara, Japan,2008), Poster WB11.

Forget, S.

S. Chénais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and Absolute Temperature Mapping and Heat Transfer Measurements in Diode-End-Pumped Yb:YAG,” Appl. Phys. B  79, 221–224 (2004).
[CrossRef]

Georges, P.

S. Chénais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and Absolute Temperature Mapping and Heat Transfer Measurements in Diode-End-Pumped Yb:YAG,” Appl. Phys. B  79, 221–224 (2004).
[CrossRef]

Goodier, J. N.

S. P. Timoshenko and J. N. Goodier, Theory of Elasticity 3rd ed., (McGraw-Hill, New York, 1970).

Gorton, E. K.

S. So, J. I. Mackenzie, D. P. Shepherd, W. A. Clarkson, J. G. Betterson, and E. K. Gorton, “A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers,” Appl. Phys. B  84, 389–393 (2006).
[CrossRef]

Gruber, R.

C. Pfistner, R. Weber, H. P. Weber, S Merazzi, and R. Gruber, “Thermal Beam Distortions in End-Pumped Nd:YAG, Nd:GSGG and Nd:YLF,” IEEE J. Quantum Electron. 30, 1605–1615 (1994).
[CrossRef]

Hanna, D. C.

M. Pollnau, P. J. Hardman, M. A. Kern, W. A. Clarkson, and D. C. Hanna, “Upconversion-induced heat generation and thermal lensing in Nd:YLF and Nd:YAG,” Phys. Rev. B  58, 16076–16092 (1998).
[CrossRef]

Hardman, P. J.

M. Pollnau, P. J. Hardman, M. A. Kern, W. A. Clarkson, and D. C. Hanna, “Upconversion-induced heat generation and thermal lensing in Nd:YLF and Nd:YAG,” Phys. Rev. B  58, 16076–16092 (1998).
[CrossRef]

Harris, L.

E. H. Bernhardi, C. Bollig, L. Harris, M. J. D. Esser, and A. Forbes, “Investigating thermal stresses in quasicw pumped Tm:YLF laser crystals,” in Proceedings of Advanced Solid-State Photonics, (Nara, Japan,2008), Poster WB11.

Jaeger, J. C.

H. S. Carslaw and J. C. Jaeger, Conduction of Heats in Solids, (Oxford University Press, Oxford, 1959).

Kern, M. A.

M. Pollnau, P. J. Hardman, M. A. Kern, W. A. Clarkson, and D. C. Hanna, “Upconversion-induced heat generation and thermal lensing in Nd:YLF and Nd:YAG,” Phys. Rev. B  58, 16076–16092 (1998).
[CrossRef]

Koechner, W.

W. KoechnerSolid-State Laser Engineering 4th ed., (Springer-Verlag Berlin, Heidelberg, Germany,1996).

Lee, C. H.

L. Yan and C. H. Lee, “Thermal Effects in End-Pumped Nd:phosphate Glasses,” J. Appl. Phys. 75, 1286–1292 (1994).
[CrossRef]

Mackenzie, J. I.

S. So, J. I. Mackenzie, D. P. Shepherd, W. A. Clarkson, J. G. Betterson, and E. K. Gorton, “A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers,” Appl. Phys. B  84, 389–393 (2006).
[CrossRef]

Merazzi, S

C. Pfistner, R. Weber, H. P. Weber, S Merazzi, and R. Gruber, “Thermal Beam Distortions in End-Pumped Nd:YAG, Nd:GSGG and Nd:YLF,” IEEE J. Quantum Electron. 30, 1605–1615 (1994).
[CrossRef]

Peng, X.

Petros, M.

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95, 3255–3271 (2004).
[CrossRef]

Pfistner, C.

C. Pfistner, R. Weber, H. P. Weber, S Merazzi, and R. Gruber, “Thermal Beam Distortions in End-Pumped Nd:YAG, Nd:GSGG and Nd:YLF,” IEEE J. Quantum Electron. 30, 1605–1615 (1994).
[CrossRef]

Pollnau, M.

M. Pollnau, P. J. Hardman, M. A. Kern, W. A. Clarkson, and D. C. Hanna, “Upconversion-induced heat generation and thermal lensing in Nd:YLF and Nd:YAG,” Phys. Rev. B  58, 16076–16092 (1998).
[CrossRef]

Shepherd, D. P.

S. So, J. I. Mackenzie, D. P. Shepherd, W. A. Clarkson, J. G. Betterson, and E. K. Gorton, “A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers,” Appl. Phys. B  84, 389–393 (2006).
[CrossRef]

Singh, U. N.

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95, 3255–3271 (2004).
[CrossRef]

So, S.

S. So, J. I. Mackenzie, D. P. Shepherd, W. A. Clarkson, J. G. Betterson, and E. K. Gorton, “A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers,” Appl. Phys. B  84, 389–393 (2006).
[CrossRef]

Timoshenko, S. P.

S. P. Timoshenko and J. N. Goodier, Theory of Elasticity 3rd ed., (McGraw-Hill, New York, 1970).

Walsh, B. M.

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95, 3255–3271 (2004).
[CrossRef]

Weber, H. P.

C. Pfistner, R. Weber, H. P. Weber, S Merazzi, and R. Gruber, “Thermal Beam Distortions in End-Pumped Nd:YAG, Nd:GSGG and Nd:YLF,” IEEE J. Quantum Electron. 30, 1605–1615 (1994).
[CrossRef]

Weber, R.

C. Pfistner, R. Weber, H. P. Weber, S Merazzi, and R. Gruber, “Thermal Beam Distortions in End-Pumped Nd:YAG, Nd:GSGG and Nd:YLF,” IEEE J. Quantum Electron. 30, 1605–1615 (1994).
[CrossRef]

Weiner, J. H.

B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, (Courier Dover Publications, New York, 1997).

Xu, L.

Yan, L.

L. Yan and C. H. Lee, “Thermal Effects in End-Pumped Nd:phosphate Glasses,” J. Appl. Phys. 75, 1286–1292 (1994).
[CrossRef]

Yu, J.

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95, 3255–3271 (2004).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (2)

S. So, J. I. Mackenzie, D. P. Shepherd, W. A. Clarkson, J. G. Betterson, and E. K. Gorton, “A power-scaling strategy for longitudinally diode-pumped Tm:YLF lasers,” Appl. Phys. B  84, 389–393 (2006).
[CrossRef]

S. Chénais, S. Forget, F. Druon, F. Balembois, and P. Georges, “Direct and Absolute Temperature Mapping and Heat Transfer Measurements in Diode-End-Pumped Yb:YAG,” Appl. Phys. B  79, 221–224 (2004).
[CrossRef]

IEEE J. Quantum Electron. (3)

C. Pfistner, R. Weber, H. P. Weber, S Merazzi, and R. Gruber, “Thermal Beam Distortions in End-Pumped Nd:YAG, Nd:GSGG and Nd:YLF,” IEEE J. Quantum Electron. 30, 1605–1615 (1994).
[CrossRef]

A. K. Cousins, “Temperature and Thermal Stress Scaling in Finite-Length End-Pumped Lasers Rods,” IEEE J. Quantum Electron. 28, 1057–1069 (1992).
[CrossRef]

T. Y. Fan, “Heat Generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993).
[CrossRef]

J. Appl. Phys. (2)

L. Yan and C. H. Lee, “Thermal Effects in End-Pumped Nd:phosphate Glasses,” J. Appl. Phys. 75, 1286–1292 (1994).
[CrossRef]

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95, 3255–3271 (2004).
[CrossRef]

Phys. Rev. (1)

M. Pollnau, P. J. Hardman, M. A. Kern, W. A. Clarkson, and D. C. Hanna, “Upconversion-induced heat generation and thermal lensing in Nd:YLF and Nd:YAG,” Phys. Rev. B  58, 16076–16092 (1998).
[CrossRef]

Other (8)

G. Barton, Elements of Green’s Functions and Propagation, (Oxford University Press, Oxford,1995).

H. S. Carslaw and J. C. Jaeger, Conduction of Heats in Solids, (Oxford University Press, Oxford, 1959).

W. KoechnerSolid-State Laser Engineering 4th ed., (Springer-Verlag Berlin, Heidelberg, Germany,1996).

S. P. Timoshenko and J. N. Goodier, Theory of Elasticity 3rd ed., (McGraw-Hill, New York, 1970).

B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, (Courier Dover Publications, New York, 1997).

LASCAD Manual, (LASCAD, 2008), https://www.las-cad.com/files/PP_FEA.pdf.

ABAQUS Finite Element Software Package (ABAQUS, 2008) http://www.simulia.com/products/abaqus_fea.html.

E. H. Bernhardi, C. Bollig, L. Harris, M. J. D. Esser, and A. Forbes, “Investigating thermal stresses in quasicw pumped Tm:YLF laser crystals,” in Proceedings of Advanced Solid-State Photonics, (Nara, Japan,2008), Poster WB11.

Supplementary Material (2)

» Media 1: MOV (754 KB)     
» Media 2: MOV (329 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

An example of a measured top-hat transverse intensity profile produced by a fibre coupled diode laser pump (own experimental results).

Fig. 2.
Fig. 2.

The analytically (red) and numerically (black) predicted temperature in the centre of the Tm:YLF rod as a function of time while the rod is subjected to a qcw pump with a peak power of (a) 200 W at 10 Hz (τon =10 ms) and, (b) 90 W at 50 Hz (τon =10 ms).

Fig. 3.
Fig. 3.

The maximum stress on the pump face of the Tm:YLF rod as a function of time while the rod is subjected to a qcw pump with a peak power of (a) 200 W at 10 Hz (τon = 10 ms), and (b) 90 W at 50 Hz (τon = 10 ms). The analytical (red) and numerical (black) solutions are shown.

Fig. 4.
Fig. 4.

(0.75 MB and 0.33 MB respectively) Animations of (a) the analytical stress distribution on the pump face [Media 1] and (b) the numerical stress distribution in volume of the Tm:YLF rod while it is subjected to a 90 W peak power qcw pump beam at 50 Hz (τon = 10 ms) [Media 2].

Fig. 5.
Fig. 5.

(a). The average pump power (as a fraction of the cw fracture power Pcw ) at which fracture of the Tm:YLF rod occurs as a function of qcw pump duty cycle (τon =10 ms). The green shaded region indicates the average pump power at which the Tm:YLF rod can be pumped without fracturing according to the analytical model. The yellow shaded region indicates the difference between the analytical model and Pcw . (b) The same notation as in (a) but for the peak pump power (in units of Pcw ) at which fracture of the Tm:YLF rod occurs as a function of qcw pump duty cycle.

Tables (1)

Tables Icon

Table 1. Parameter values of the pumped Tm:YLF rod that were implemented in the simulations.

Equations (13)

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

u ( r , t ) t D 2 u ( r , t ) = Q ( r , t ) ,
u ( r , t ) = 0 t 0 R Q ( ξ , τ ) G ( r , ξ , t τ ) dξdτ ,
G ( r , ξ , t ) = m = 1 2 ξ R 2 J 1 2 ( μ m ) J 0 ( μ m r R ) J 0 ( μ m r R ) exp ( D μ m 2 t R 2 ) .
Q r t = { αηE π w 2 ρ C p τ on 0 ; nT t nT + τ on ; nT + τ on t ( n + 1 ) T ,
u ( r , pT + t ) = 2 αηER kπw τ on m = 1 J 0 ( μ m r R ) J 1 ( μ m w R ) f ( p , t , μ m ) μ m 3 J 1 2 ( μ m ) ,
f ( p , t , μ m ) = exp ( μ m 2 t τ D ) { [ exp ( μ m 2 τ on τ D ) 1 ] [ exp ( μ m 2 pT τ D ) 1 ] 1 exp ( μ m 2 T τ D ) - [ 1 exp ( μ m 2 τ τ D ) ] } ,
σ r ( r , t ) = c [ 1 R 2 0 R u ( r , t ) rdr 1 r 2 0 r u ( r , t ) rdr ] ;
σ θ ( r , t ) = C [ 1 R 2 0 R u ( r , t ) rdr + 1 r 2 0 r u ( r , t ) rdr u ( r , t ) ] ,
σ r ( r , pT + t ) = 2 CαηER kπw τ on m = 1 J 1 ( μ m w R ) μ m 3 J 1 2 ( μ m ) [ J 1 ( μ m ) μ m - R J 1 ( μ m r R ) m ] f ( p , t , μ m ) ;
σ θ ( r , pT + t ) = 2 CαηER kπw τ on m = 1 J 1 ( μ m w R ) μ m 3 J 1 2 ( μ m ) [ J 1 ( μ m ) μ m + R J 1 ( μ m r R ) m J 0 ( μ m r R ) ] f ( p , t , μ m ) .
σ T ( r , pT + t ) = σ θ σ r
= 2 CαηER kπw τ on m = 1 J 1 ( μ m w R ) μ m 3 J 1 2 ( μ m ) J 2 ( μ m r R ) f ( p , t , μ m ) ,
P av P cw .

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