Abstract

A diode-pumped alkali laser (DPAL) is one of the most hopeful candidates to achieve high power performances. As the laser medium is in a gas-state, populations of energy-levels of a DPAL are strongly dependent on the vapor temperature. Thus, the temperature distribution directly determines the output characteristics of a DPAL. In this report, we developed a systematic model by combining the procedures of heat transfer and laser kinetics together to explore the radial temperature distribution in the transverse section of a cesium vapor cell. A cyclic iterative approach is adopted to calculate the population densities. The corresponding temperature distributions have been obtained for different beam waists and pump powers. The conclusion is thought to be useful for realizing a DPAL with high output power.

© 2014 Optical Society of America

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    [CrossRef]

2013

B. D. Barmashenko, S. Rosenwaks, “Feasibility of supersonic diode pumped alkali lasers: Model calculations,” Appl. Phys. Lett. 102(14), 141108 (2013).
[CrossRef]

B. D. Barmashenko, S. Rosenwaks, “Detailed analysis of kinetic and fluid dynamic processes in diode-pumped alkali lasers,” J. Opt. Soc. Am. B 30(5), 1118–1126 (2013).
[CrossRef]

2012

B. D. Barmashenko, S. Rosenwaks, “Modeling of flowing gas diode pumped alkali lasers: dependence of the operation on the gas velocity and on the nature of the buffer gas,” Opt. Lett. 37(17), 3615–3617 (2012).
[CrossRef] [PubMed]

Y. F. Liu, B. L. Pan, J. Yang, Y. J. Wang, M. H. Li, “Thermal Effects in High-Power Double Diode-End-Pumped Cs Vapor Lasers,” IEEE J. Quantum Electron. 48(4), 485–489 (2012).
[CrossRef]

W. F. Krupke, “Diode pumped alkali lasers (DPALs)—A review (rev1),” Prog. Quantum Electron. 36(1), 4–28 (2012).
[CrossRef]

2011

2010

Y. Wang, K. Inoue, H. Kan, T. Ogawa, S. Wada, “A MOPA with double-end pumped configuration using total internal reflection,” Laser Phys. 20(2), 447–453 (2010).
[CrossRef]

C. V. Sulham, G. P. Perram, M. P. Wilkinson, D. A. Hostutler, “A pulsed, optically pumped rubidium laser at high pump intensity,” Opt. Commun. 283(21), 4328–4332 (2010).
[CrossRef]

Q. Zhu, B. L. Pan, L. Chen, Y. J. Wang, X. Y. Zhang, “Analysis of temperature distributions in diode-pumped alkali vapor lasers,” Opt. Commun. 283(11), 2406–2410 (2010).
[CrossRef]

2009

2008

B. V. Zhdanov, A. Stooke, G. Boyadjian, A. Voci, R. J. Knize, “Rubidium vapor laser pumped by two laser diode arrays,” Opt. Lett. 33(5), 414–415 (2008).
[CrossRef] [PubMed]

S. Kiwan, O. Zeitoun, “Natural convection in a horizontal cylindrical annulus using porous fins,” Int. J. Numer. Methods Heat Fluid Flow 18(5), 618–634 (2008).
[CrossRef]

2007

Y. Wang, M. Niigaki, H. Fukuoka, Y. Zheng, H. Miyajima, S. Matsuoka, H. Kubomura, T. Hiruma, H. Kan, “Approaches of output improvement for cesium vapor laser pumped by a volume-Bragg-grating coupled laser-diode-array,” Phys. Lett. A 360(4-5), 659–663 (2007).
[CrossRef]

B. V. Zhdanov, R. J. Knize, “Diode-pumped 10 W continuous wave cesium laser,” Opt. Lett. 32(15), 2167–2169 (2007).
[CrossRef] [PubMed]

2006

R. H. Page, R. J. Beach, V. K. Kanz, W. F. Krupke, “Multimode-diode-pumped gas (alkali-vapor) laser,” Opt. Lett. 31(3), 353–355 (2006).
[CrossRef] [PubMed]

Y. Wang, T. Kasamatsu, Y. Zheng, H. Miyajima, H. Fukuoka, S. Matsuoka, M. Niigaki, H. Kubomura, T. Hiruma, H. Kan, “Cesium vapor laser pumped by a volume-Bragg-grating coupled quasi-continuous-wave laser-diode array,” Appl. Phys. Lett. 88(14), 141112 (2006).
[CrossRef]

2004

2003

W. F. Krupke, R. J. Beach, V. K. Kanz, S. A. Payne, “Resonance transition 795-nm rubidium laser,” Opt. Lett. 28(23), 2336–2338 (2003).
[CrossRef] [PubMed]

Y. Wang, H. Kan, “Improvement on evaluating absorption efficiency of a medium rod for LD side-pumped solid-state lasers,” Opt. Commun. 226(1-6), 303–316 (2003).
[CrossRef]

2001

1996

M. Stanghini, M. Basso, R. Genesio, A. Tesi, R. Meucci, M. Ciofini, “A new three-equation model for the CO2 laser,” IEEE J. Quantum Electron. 32(7), 1126–1131 (1996).
[CrossRef]

1995

R. J. Garman, “Modelling of the intracavity optical fields in a copper vapour laser,” Opt. Commun. 119(3-4), 415–423 (1995).
[CrossRef]

Barmashenko, B. D.

Basso, M.

M. Stanghini, M. Basso, R. Genesio, A. Tesi, R. Meucci, M. Ciofini, “A new three-equation model for the CO2 laser,” IEEE J. Quantum Electron. 32(7), 1126–1131 (1996).
[CrossRef]

Beach, R. J.

Boyadjian, G.

Chen, J. B.

Chen, L.

Q. Zhu, B. L. Pan, L. Chen, Y. J. Wang, X. Y. Zhang, “Analysis of temperature distributions in diode-pumped alkali vapor lasers,” Opt. Commun. 283(11), 2406–2410 (2010).
[CrossRef]

Cheng, C. T.

Ciofini, M.

M. Stanghini, M. Basso, R. Genesio, A. Tesi, R. Meucci, M. Ciofini, “A new three-equation model for the CO2 laser,” IEEE J. Quantum Electron. 32(7), 1126–1131 (1996).
[CrossRef]

Dubinskii, M. A.

Early, J. T.

W. F. Krupke, R. J. Beach, S. A. Payne, V. K. Kanz, J. T. Early, “DPAL: A new class of lasers for CW power beaming at ideal photovoltaic cell wavelengths,” 2nd International Symposium on Beamed Energy Propulsion (Japan), (2003).

Fukuoka, H.

Y. Wang, M. Niigaki, H. Fukuoka, Y. Zheng, H. Miyajima, S. Matsuoka, H. Kubomura, T. Hiruma, H. Kan, “Approaches of output improvement for cesium vapor laser pumped by a volume-Bragg-grating coupled laser-diode-array,” Phys. Lett. A 360(4-5), 659–663 (2007).
[CrossRef]

Y. Wang, T. Kasamatsu, Y. Zheng, H. Miyajima, H. Fukuoka, S. Matsuoka, M. Niigaki, H. Kubomura, T. Hiruma, H. Kan, “Cesium vapor laser pumped by a volume-Bragg-grating coupled quasi-continuous-wave laser-diode array,” Appl. Phys. Lett. 88(14), 141112 (2006).
[CrossRef]

Garman, R. J.

R. J. Garman, “Modelling of the intracavity optical fields in a copper vapour laser,” Opt. Commun. 119(3-4), 415–423 (1995).
[CrossRef]

Genesio, R.

M. Stanghini, M. Basso, R. Genesio, A. Tesi, R. Meucci, M. Ciofini, “A new three-equation model for the CO2 laser,” IEEE J. Quantum Electron. 32(7), 1126–1131 (1996).
[CrossRef]

Hager, G. D.

Hiruma, T.

Y. Wang, M. Niigaki, H. Fukuoka, Y. Zheng, H. Miyajima, S. Matsuoka, H. Kubomura, T. Hiruma, H. Kan, “Approaches of output improvement for cesium vapor laser pumped by a volume-Bragg-grating coupled laser-diode-array,” Phys. Lett. A 360(4-5), 659–663 (2007).
[CrossRef]

Y. Wang, T. Kasamatsu, Y. Zheng, H. Miyajima, H. Fukuoka, S. Matsuoka, M. Niigaki, H. Kubomura, T. Hiruma, H. Kan, “Cesium vapor laser pumped by a volume-Bragg-grating coupled quasi-continuous-wave laser-diode array,” Appl. Phys. Lett. 88(14), 141112 (2006).
[CrossRef]

Hostutler, D. A.

N. D. Zameroski, G. D. Hager, W. Rudolph, D. A. Hostutler, “Experimental and numerical modeling studies of a pulsed rubidium optically pumped alkali metal vapor laser,” J. Opt. Soc. Am. B 28(5), 1088–1099 (2011).
[CrossRef]

C. V. Sulham, G. P. Perram, M. P. Wilkinson, D. A. Hostutler, “A pulsed, optically pumped rubidium laser at high pump intensity,” Opt. Commun. 283(21), 4328–4332 (2010).
[CrossRef]

Hsu, K. Y.

Hua, R. Z.

Hua, W. H.

Huang, K. Y.

Huang, S. L.

Inoue, K.

Y. Wang, K. Inoue, H. Kan, T. Ogawa, S. Wada, “A MOPA with double-end pumped configuration using total internal reflection,” Laser Phys. 20(2), 447–453 (2010).
[CrossRef]

Ji, K. D.

Kan, H.

Y. Wang, K. Inoue, H. Kan, T. Ogawa, S. Wada, “A MOPA with double-end pumped configuration using total internal reflection,” Laser Phys. 20(2), 447–453 (2010).
[CrossRef]

Y. Wang, M. Niigaki, H. Fukuoka, Y. Zheng, H. Miyajima, S. Matsuoka, H. Kubomura, T. Hiruma, H. Kan, “Approaches of output improvement for cesium vapor laser pumped by a volume-Bragg-grating coupled laser-diode-array,” Phys. Lett. A 360(4-5), 659–663 (2007).
[CrossRef]

Y. Wang, T. Kasamatsu, Y. Zheng, H. Miyajima, H. Fukuoka, S. Matsuoka, M. Niigaki, H. Kubomura, T. Hiruma, H. Kan, “Cesium vapor laser pumped by a volume-Bragg-grating coupled quasi-continuous-wave laser-diode array,” Appl. Phys. Lett. 88(14), 141112 (2006).
[CrossRef]

Y. Wang, H. Kan, “Improvement on evaluating absorption efficiency of a medium rod for LD side-pumped solid-state lasers,” Opt. Commun. 226(1-6), 303–316 (2003).
[CrossRef]

Kanz, V. K.

Kasamatsu, T.

Y. Wang, T. Kasamatsu, Y. Zheng, H. Miyajima, H. Fukuoka, S. Matsuoka, M. Niigaki, H. Kubomura, T. Hiruma, H. Kan, “Cesium vapor laser pumped by a volume-Bragg-grating coupled quasi-continuous-wave laser-diode array,” Appl. Phys. Lett. 88(14), 141112 (2006).
[CrossRef]

Ke, C. P.

Kiwan, S.

S. Kiwan, O. Zeitoun, “Natural convection in a horizontal cylindrical annulus using porous fins,” Int. J. Numer. Methods Heat Fluid Flow 18(5), 618–634 (2008).
[CrossRef]

Knize, R. J.

Krupke, W. F.

Kubomura, H.

Y. Wang, M. Niigaki, H. Fukuoka, Y. Zheng, H. Miyajima, S. Matsuoka, H. Kubomura, T. Hiruma, H. Kan, “Approaches of output improvement for cesium vapor laser pumped by a volume-Bragg-grating coupled laser-diode-array,” Phys. Lett. A 360(4-5), 659–663 (2007).
[CrossRef]

Y. Wang, T. Kasamatsu, Y. Zheng, H. Miyajima, H. Fukuoka, S. Matsuoka, M. Niigaki, H. Kubomura, T. Hiruma, H. Kan, “Cesium vapor laser pumped by a volume-Bragg-grating coupled quasi-continuous-wave laser-diode array,” Appl. Phys. Lett. 88(14), 141112 (2006).
[CrossRef]

Lai, C. C.

Li, M. H.

Y. F. Liu, B. L. Pan, J. Yang, Y. J. Wang, M. H. Li, “Thermal Effects in High-Power Double Diode-End-Pumped Cs Vapor Lasers,” IEEE J. Quantum Electron. 48(4), 485–489 (2012).
[CrossRef]

Li, Y. D.

Lin, S. R.

Liu, S. K.

Liu, Y. F.

Y. F. Liu, B. L. Pan, J. Yang, Y. J. Wang, M. H. Li, “Thermal Effects in High-Power Double Diode-End-Pumped Cs Vapor Lasers,” IEEE J. Quantum Electron. 48(4), 485–489 (2012).
[CrossRef]

Lu, Q. S.

Matsuoka, S.

Y. Wang, M. Niigaki, H. Fukuoka, Y. Zheng, H. Miyajima, S. Matsuoka, H. Kubomura, T. Hiruma, H. Kan, “Approaches of output improvement for cesium vapor laser pumped by a volume-Bragg-grating coupled laser-diode-array,” Phys. Lett. A 360(4-5), 659–663 (2007).
[CrossRef]

Y. Wang, T. Kasamatsu, Y. Zheng, H. Miyajima, H. Fukuoka, S. Matsuoka, M. Niigaki, H. Kubomura, T. Hiruma, H. Kan, “Cesium vapor laser pumped by a volume-Bragg-grating coupled quasi-continuous-wave laser-diode array,” Appl. Phys. Lett. 88(14), 141112 (2006).
[CrossRef]

Merkle, L. D.

Meucci, R.

M. Stanghini, M. Basso, R. Genesio, A. Tesi, R. Meucci, M. Ciofini, “A new three-equation model for the CO2 laser,” IEEE J. Quantum Electron. 32(7), 1126–1131 (1996).
[CrossRef]

Miyajima, H.

Y. Wang, M. Niigaki, H. Fukuoka, Y. Zheng, H. Miyajima, S. Matsuoka, H. Kubomura, T. Hiruma, H. Kan, “Approaches of output improvement for cesium vapor laser pumped by a volume-Bragg-grating coupled laser-diode-array,” Phys. Lett. A 360(4-5), 659–663 (2007).
[CrossRef]

Y. Wang, T. Kasamatsu, Y. Zheng, H. Miyajima, H. Fukuoka, S. Matsuoka, M. Niigaki, H. Kubomura, T. Hiruma, H. Kan, “Cesium vapor laser pumped by a volume-Bragg-grating coupled quasi-continuous-wave laser-diode array,” Appl. Phys. Lett. 88(14), 141112 (2006).
[CrossRef]

Niigaki, M.

Y. Wang, M. Niigaki, H. Fukuoka, Y. Zheng, H. Miyajima, S. Matsuoka, H. Kubomura, T. Hiruma, H. Kan, “Approaches of output improvement for cesium vapor laser pumped by a volume-Bragg-grating coupled laser-diode-array,” Phys. Lett. A 360(4-5), 659–663 (2007).
[CrossRef]

Y. Wang, T. Kasamatsu, Y. Zheng, H. Miyajima, H. Fukuoka, S. Matsuoka, M. Niigaki, H. Kubomura, T. Hiruma, H. Kan, “Cesium vapor laser pumped by a volume-Bragg-grating coupled quasi-continuous-wave laser-diode array,” Appl. Phys. Lett. 88(14), 141112 (2006).
[CrossRef]

Ogawa, T.

Y. Wang, K. Inoue, H. Kan, T. Ogawa, S. Wada, “A MOPA with double-end pumped configuration using total internal reflection,” Laser Phys. 20(2), 447–453 (2010).
[CrossRef]

Page, R. H.

Pan, B. L.

Y. F. Liu, B. L. Pan, J. Yang, Y. J. Wang, M. H. Li, “Thermal Effects in High-Power Double Diode-End-Pumped Cs Vapor Lasers,” IEEE J. Quantum Electron. 48(4), 485–489 (2012).
[CrossRef]

Q. Zhu, B. L. Pan, L. Chen, Y. J. Wang, X. Y. Zhang, “Analysis of temperature distributions in diode-pumped alkali vapor lasers,” Opt. Commun. 283(11), 2406–2410 (2010).
[CrossRef]

Payne, S. A.

Perram, G. P.

C. V. Sulham, G. P. Perram, M. P. Wilkinson, D. A. Hostutler, “A pulsed, optically pumped rubidium laser at high pump intensity,” Opt. Commun. 283(21), 4328–4332 (2010).
[CrossRef]

Rosenwaks, S.

Rudolph, W.

Stanghini, M.

M. Stanghini, M. Basso, R. Genesio, A. Tesi, R. Meucci, M. Ciofini, “A new three-equation model for the CO2 laser,” IEEE J. Quantum Electron. 32(7), 1126–1131 (1996).
[CrossRef]

Stooke, A.

Sulham, C. V.

C. V. Sulham, G. P. Perram, M. P. Wilkinson, D. A. Hostutler, “A pulsed, optically pumped rubidium laser at high pump intensity,” Opt. Commun. 283(21), 4328–4332 (2010).
[CrossRef]

Tashiro, H.

Tesi, A.

M. Stanghini, M. Basso, R. Genesio, A. Tesi, R. Meucci, M. Ciofini, “A new three-equation model for the CO2 laser,” IEEE J. Quantum Electron. 32(7), 1126–1131 (1996).
[CrossRef]

Tsai, H. J.

Voci, A.

Wada, S.

Y. Wang, K. Inoue, H. Kan, T. Ogawa, S. Wada, “A MOPA with double-end pumped configuration using total internal reflection,” Laser Phys. 20(2), 447–453 (2010).
[CrossRef]

R. Z. Hua, S. Wada, H. Tashiro, “Versatile, compact, TEM00-mode resonator for side-pumped single-rod solid-state lasers,” Appl. Opt. 40(15), 2468–2474 (2001).
[CrossRef] [PubMed]

Wang, H. Y.

Wang, Y.

Y. Wang, K. Inoue, H. Kan, T. Ogawa, S. Wada, “A MOPA with double-end pumped configuration using total internal reflection,” Laser Phys. 20(2), 447–453 (2010).
[CrossRef]

Y. Wang, M. Niigaki, H. Fukuoka, Y. Zheng, H. Miyajima, S. Matsuoka, H. Kubomura, T. Hiruma, H. Kan, “Approaches of output improvement for cesium vapor laser pumped by a volume-Bragg-grating coupled laser-diode-array,” Phys. Lett. A 360(4-5), 659–663 (2007).
[CrossRef]

Y. Wang, T. Kasamatsu, Y. Zheng, H. Miyajima, H. Fukuoka, S. Matsuoka, M. Niigaki, H. Kubomura, T. Hiruma, H. Kan, “Cesium vapor laser pumped by a volume-Bragg-grating coupled quasi-continuous-wave laser-diode array,” Appl. Phys. Lett. 88(14), 141112 (2006).
[CrossRef]

Y. Wang, H. Kan, “Improvement on evaluating absorption efficiency of a medium rod for LD side-pumped solid-state lasers,” Opt. Commun. 226(1-6), 303–316 (2003).
[CrossRef]

Wang, Y. J.

Y. F. Liu, B. L. Pan, J. Yang, Y. J. Wang, M. H. Li, “Thermal Effects in High-Power Double Diode-End-Pumped Cs Vapor Lasers,” IEEE J. Quantum Electron. 48(4), 485–489 (2012).
[CrossRef]

Q. Zhu, B. L. Pan, L. Chen, Y. J. Wang, X. Y. Zhang, “Analysis of temperature distributions in diode-pumped alkali vapor lasers,” Opt. Commun. 283(11), 2406–2410 (2010).
[CrossRef]

Wilkinson, M. P.

C. V. Sulham, G. P. Perram, M. P. Wilkinson, D. A. Hostutler, “A pulsed, optically pumped rubidium laser at high pump intensity,” Opt. Commun. 283(21), 4328–4332 (2010).
[CrossRef]

Xu, X. J.

Yang, J.

Y. F. Liu, B. L. Pan, J. Yang, Y. J. Wang, M. H. Li, “Thermal Effects in High-Power Double Diode-End-Pumped Cs Vapor Lasers,” IEEE J. Quantum Electron. 48(4), 485–489 (2012).
[CrossRef]

Yang, Z. N.

Zameroski, N. D.

Zeitoun, O.

S. Kiwan, O. Zeitoun, “Natural convection in a horizontal cylindrical annulus using porous fins,” Int. J. Numer. Methods Heat Fluid Flow 18(5), 618–634 (2008).
[CrossRef]

Zhang, X. Y.

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Figures (11)

Fig. 1
Fig. 1

Schematic illustration of a segmented configuration of a vapor cell.

Fig. 2
Fig. 2

Diagram of energy levels of an alkali atom.

Fig. 3
Fig. 3

Transverse view of a vapor cell.

Fig. 4
Fig. 4

Flowchart of evaluating the distributions of temperature and population of a vapor cell.

Fig. 5
Fig. 5

Drawing of illustrating heat generated and transferred for the first cylindrical annulus and the jth cylindrical annulus of a vapor cell.

Fig. 6
Fig. 6

Sketch for determining the value of PThermal.

Fig. 7
Fig. 7

Population distributions with different waists of a pump beam. The waist radii are 150 μm (a), 300 μm (b), 500 μm (c) and 700 μm (d), respectively.

Fig. 8
Fig. 8

Population distributions with the pump power of 1 W (a), 20 W (b) and 50 W (c), respectively.

Fig. 9
Fig. 9

(a) Temperature distributions with the waist of a pump beam of 150, 300, 500 and 700 μm, respectively. (b) 3-dmensional diagram for ωp = 500 μm.

Fig. 10
Fig. 10

Temperature distributions with the pump power of 1, 10, 100 and 500 W, respectively.

Fig. 11
Fig. 11

Absorbed power and generated heat versus the pump power (a). Optical-optical conversion efficiency and laser output power versus the pump power (b).

Tables (1)

Tables Icon

Table 1 Parameters for evaluating temperature distribution of a Cs vapor cell

Equations (37)

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r j =R( j1 )·R/N, r j+1 =Rj·R/N.
d n 1 j dt = Γ p j + Γ L j + n 2 j τ D 1 + n 3 j τ D 2 , d n 2 j dt = Γ L j + γ 32 ( T j )[ n 3 j 2 n 2 j exp( ΔE k B T j ) ] n 2 j τ D 1 , d n 3 j dt = Γ p j γ 32 ( T j )[ n 3 j 2 n 2 j exp( ΔE k B T j ) ] n 3 j τ D 2 ,
γ 32 ( T j )= n ethane σ 32 ethane v r Csethane ( T j )+ n He σ 32 He v r CsHe ( T j ) ,
ν γ ethane ( T j )= 3 k B T j ( 1 m Cs + 1 m ethane ) , ν γ He ( T j )= 3 k B T j ( 1 m Cs + 1 m He ) ,
Γ P j = η del V L j λ hc P j (λ)×{ 1exp[ ( n 1 j 1 2 n 3 j ) σ D 2 ( T j ,λ )l ] } ×{ 1+ R P exp[ ( n 1 j 1 2 n 3 j ) σ D 2 ( T j ,λ )l ] },
V L j =π( r j 2 r j+1 2 )l ,
P j (λ)= P j peak 2π (Δ λ D 2 FWHM /2 2ln2 ) 2 exp( (λ λ 0 ) 2 2 (Δ λ D 2 FWHM /2 2ln2 ) 2 ) ,
P j peak = I j ( r )d S S j I max exp( 2 r j 2 ω P 2 ),
S j =π( r j 2 r j+1 2 ) .
I max = P P exp( 2 r 2 ω P 2 ) dS = P P 0 R exp( 2 r 2 ω P 2 ) 2πrdr P P j=1 N exp( 2 r j 2 ω P 2 ) S j .
σ D 2 ( T j ,λ)= σ D 2 Hebroadened ( T j ) 1+ ( λ λ D 2 Δ λ D 2 FWHM /2 ) 2 ,
σ D 2 Hebroadened ( T j )= σ D 2 radiative 2π τ D 2 n Heamagat ( 19.3 GHz amagat ) T j 294K ,
Γ L j ={ 1 V L j P L j h υ L R oc 1 R oc ×{ exp[ ( n 2 j n 1 j ) σ D 1 Hebroadened ( T j )l ]1 } ×{ 1+T T 2 exp[ ( n 2 j n 1 j ) σ D 1 Hebroadened ( T j )l ] },withlaseroutput 0,withoutlaseroutput ,
σ D 1 Hebroadened ( T j )= σ D 1 radiative 2π τ D 1 n Heamagat ( 21.5 GHz amagat ) T j 294K ,
exp[ 2( n 2 j ( T j )- n 1 j ( T j ) ) σ D 1 He-broadened ( T j )l ]×T T 2 R oc =1 ,
n 0 j ( T j )= n 1 j ( T j )+ n 2 j ( T j )+ n 3 j ( T j ) ,
n 0 j ( T j )={ n 0 1 ( T w ),j=1 n 0 1 ( T w )( T w T j ),j>1 ,
n 0 1 ( T w )= 133.322 N A R T w ( 10 8.22127 4006.048 T w 0.00060194 T w 0.19623 log 10 T w ) ,
Ω j = γ 32 ( T j )[ n 3 j 2 n 2 j exp( ΔE k B T j )]ΔE ,
Q j = V L i Ω j .
d dr ( r dT dr )+ Ωr K(T) =0 ,
K( T j )= P He P He + P C 2 H 6 K He ( T j )+ P C 2 H 6 P He + P C 2 H 6 K C 2 H 6 ( T j ) ,
K He ( T j )=0.05516+3.2540× 10 4 T j 2.2723× 10 8 T j 2 , K C 2 H 6 ( T j )=0.01936+1.2547× 10 4 T j +3.8298× 10 8 T j 2 .
dT dr + Ω j r 2K( T j ) = C j 1 r ,
Φ j =[ K( T )A dT dr ] | T= T j ,
A=2π r j l ,
C j 1 = Ω j r j 2 2K( T j ) Φ j r j K( T j ) A j .
T(r)= C j 1 lnr Ω j r 2 4K( T j ) + C j 0 ,
C j 0 = T j C j 1 ln r j + Ω j r j 2 4K( T j ) .
r N =R/N ,
dT dr + Ω N r 2K( T N ) =0 .
T(r)= Ω N r 2 4K( T N ) + C N 0 ,
C N 0 = T N + Ω N r N 2 4K( T N ) .
Φ 1 = P Thermal .
Φ 2 = Φ 1 Q 1 = P Thermal Q 1 .
Φ j = P Thermal i=1 j1 Q i ,
P Thermal = j=1 N Q j .

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