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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 11 — Jun. 2, 2014
  • pp: 13988–14003
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Algorithm for evaluation of temperature distribution of a vapor cell in a diode-pumped alkali laser system: part I

Juhong Han, You Wang, He Cai, Wei Zhang, Liangping Xue, and Hongyuan Wang  »View Author Affiliations


Optics Express, Vol. 22, Issue 11, pp. 13988-14003 (2014)
http://dx.doi.org/10.1364/OE.22.013988


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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

1. Introduction

Since W. F. Krupke in Lawrence Livermore National Laboratory (LLNL) invented the diode-pumped alkali laser (DPAL) at the beginning of the 21th century, such a new type of laser has been rapidly developed in the last decade [1

1. W. F. Krupke, “Diode Pumped Alkali Laser,” US Patent Application US 2003/0099272 Al, (2003).

3

3. W. F. Krupke, R. J. Beach, S. A. Payne, V. K. Kanz, and 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).

].

Unlike other types of lasers, the density of a gain medium inside a vapor cell is extremely sensitive to the ambient temperature [17

17. D. A. Steck, Rubidium 85 D Line Data. Available: http://steck.us/alkalidata.

19

19. Z. N. Yang, H. Y. Wang, Q. S. Lu, W. H. Hua, and X. J. Xu, “Modeling of an optically side-pumped alkali vapor amplifier with consideration of amplified spontaneous emission,” Opt. Express 19(23), 23118–23131 (2011). [CrossRef] [PubMed]

]. When the pump power for a DPAL is small enough, heat generated in the vapor cell provides almost no effects on laser properties and is usually neglected. However, when a high-powered diode is used to pump the vapor cell, the laser features might become somewhat strange, and some physical characteristics, e.g. the transverse gain distribution, might be different from those for an ordinary solid-state laser.

Therefore, there is a necessity for DPAL technicians to investigate generated heat as well as the temperature distribution in the vapor cell for construction of a DPAL system with high beam quality. One of the greatest difficulties for examination of thermal characteristics of a DAPL cell is that the population status exhibits an inhomogeneous distribution due to the temperature gradient inside an alkali cell. Simultaneous evaluations of both the population and the temperature distributions cannot be simply completed only with the analyses of thermal conductivity or kinetic calculations. Until now, two teams have carried out studies on the temperature distribution inside an alkali vapor cell. One is the team headed by B. L. Pan, whose interests concentrate on the heat transfer and the optical path difference (OPD) under different pump powers [20

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

, 21

21. Y. F. Liu, B. L. Pan, J. Yang, Y. J. Wang, and 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]

]. They calculated the temperature distribution by using an assumed absorption coefficient, and the lasing process was not taken into account in their scheme. The other is B. D. Barmashenko's team and their calculation model was only based on the kinetic evaluation while the temperature distribution was not discussed [22

22. B. D. Barmashenko and 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]

25

25. S. Rosenwaks, B. D. Barmashenko and Waichman, “Semi-analytical and 3D CFD DPAL modeling: feasibility of supersonic operation”, Proc. SPIE 8962, High Energy/Average Power Lasers and Intense Beam Applications VII, 896209, 1–9 (2014).

]. In their model, the temperature is assumed as a constant inside the lasing region.

To deduce an accurate temperature distribution inside a vapor cell, it is essential to create an analytic system by considering both the laser kinetics and the heat transfer. In our report, the heat generating dynamics and heat transfer are investigated by two interrelated theoretical procedures. The results reveal that the radial temperature gradient inside the vapor cell cannot be simply ignored for a real DPAL. To the best of our knowledge, there have not been any similar reports on this topic so far.

2. Theory and method

For an end-pumped configuration, the optical axis of a pump laser diode coincides with that of a DPAL. As shown in Fig. 1
Fig. 1 Schematic illustration of a segmented configuration of a vapor cell.
, we divide a cylindrical vapor cell into many cylindrical annuli whose axes are same. Every cylindrical annulus is thought as a heat source. These coaxial cylindrical annuli will be used in the segmental accumulation procedure as introduced next.

During the evaluation, we made the following assumptions:

  • (1) The diameter of the DPAL beam is approximately treated to be unchanged along the optical axis;
  • (2) The transverse pump distribution holds out a Gaussian intensity profile and keeps unchanged along the optical axis;
  • (3) The temperature of every cylindrical annulus is a constant along the optical axis;
  • (4) The effects of both end-windows of the enclosed vapor cell are ignored.

Actually, the variations in diameter of both the DPAL and the pump beams are very tiny inside a 2.5cm-long cell. For example, the variation of the beam size is only about 2.2% for a pump beam with the waist size of 150 μm. In this research, the temperature distribution along the axial direction is ignored with purpose of the algorithm simplification. We will undertake the evaluation of an enclosed cell to deduce the realistic 3-dimensional temperature distribution, in which the effects of both end-windows are taken into account in the evaluation regime.

Heat transfer in a gas-state medium exhibits three types: heat radiation, convection, and conduction. Since the thickness of every cylindrical annulus is very small, heat transfer between the annulus gap is mainly dominated by heat conduction and the effects of natural convection is neglected in such a narrow gap [26

26. S. Kiwan and 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]

, 27

27. P. Teerstra and M. M. Yovanovich, “ Comprehensive review of natural convection in horizontal circular annuli,” in 7th AIAA/ ASME Joint Thermophysics and Heat Transfer Conference, Albuquerque, New Mexico, 15 –18 June 1998 (AIAA, 1998), pp. 141–152.

]. Additionally, heat radiation is so small that it can also be ignored here.

2.1. Analyses of laser kinetics

Generally, a DPAL is thought to be a typical three-level laser source with a tiny quantum defect. As shown in Fig. 2
Fig. 2 Diagram of energy levels of an alkali atom.
, the stimulated radiation transition n2P1/2n2S1/2 is called D1 line and the stimulated absorption transition (pump transition) n 2S1/2n 2P3/2 is called D2 line, where n = 4, 5 and 6 for K, Rb and Cs, respectively [12

12. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004). [CrossRef]

]. The D2 line can be collisionally broadened to achieve spectrally homogeneous transition by using a buffer gas such as helium. In the presence of helium buffer gas, the D2 pump transition line-shape changes from Gaussian to Lorentzian. Thus, pump energy absorbed in the spectral wings of the pump transition can dramatically enhance the laser gain by comparing the case of no helium buffer gas. The effectively collision-broadened linewidth is generally at least ten times the Doppler linewidth. The relaxation rate of the fine-structure can be enhanced by adding some alkanes with small hydrocarbon molecules [18

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

, 19

19. Z. N. Yang, H. Y. Wang, Q. S. Lu, W. H. Hua, and X. J. Xu, “Modeling of an optically side-pumped alkali vapor amplifier with consideration of amplified spontaneous emission,” Opt. Express 19(23), 23118–23131 (2011). [CrossRef] [PubMed]

, 28

28. Z. N. Yang, H. Y. Wang, Q. S. Lu, Y. D. Li, W. H. Hua, X. J. Xu, and J. B. Chen, “Modeling, numerical approach, and power scaling of alkali vapor lasers in side-pumped configuration with flowing medium,” J. Opt. Soc. Am. B 28(6), 1353–1364 (2011). [CrossRef]

]. In the theoretical simulation of this report, we choose cesium as the laser gain medium, and helium and ethane as buffer gases.

As shown in Fig. 3
Fig. 3 Transverse view of a vapor cell.
, we select an arbitrarily cylindrical annulus (jth) among the segments. The outside radius rj and the inner radius rj + 1 of this cylindrical annulus can be simply expressed by
rj=R(j1)·R/N,rj+1=Rj·R/N.
(1)
where R is the radius of a vapor cell, N is the total number of segmented cylindrical annuli, respectively.

Next, we calculate the population distribution in the three energy-level system of the jth cylindrical annulus (see Fig. 3) by using the well-known rate equations as follows [12

12. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004). [CrossRef]

]:
dn1jdt=Γpj+ΓLj+n2jτD1+n3jτD2,dn2jdt=ΓLj+γ32(Tj)[n3j2n2jexp(ΔEkBTj)]n2jτD1,dn3jdt=Γpjγ32(Tj)[n3j2n2jexp(ΔEkBTj)]n3jτD2,
(2)
where kB is the Boltzmann constant, τD1 is the D1 radiative lifetime, τD2 is the D2 radiative lifetime, ΔE is the energy gap between the 2P3/2 and the 2P1/2 states, Tj is the temperature of the jth cylindrical annulus, γ32(Tj) is the fine-structure relaxation rate as given by
γ32(Tj)=nethaneσ32ethanevrCsethane(Tj)+nHeσ32HevrCsHe(Tj),
(3)
where nethane is the number density of ethane in the cell, σ32ethaneis the ethane cross-section with the value of 5.2 × 10−15 cm2, σ32Heis the cross-section with the value for He with the value of 2.25 × 10−19 cm2, νγCs-He(Tj) and νγCs-ethane(Tj) are respectively the root mean square thermally averaged relative velocities between cesium atoms and He atoms as well as ethane molecules as given by [12

12. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004). [CrossRef]

]
νγethane(Tj)=3kBTj(1mCs+1methane),νγHe(Tj)=3kBTj(1mCs+1mHe),
(4)
where mCs, mHe and methane are qualities of a cesium atom, a helium atom and an ethane molecule, respectively. ΓPj is the stimulated absorption transition rate caused by pump photons as given by [12

12. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004). [CrossRef]

]
ΓPj=ηdelVLjλhcPj(λ)×{1exp[(n1j12n3j)σD2(Tj,λ)l]}×{1+RPexp[(n1j12n3j)σD2(Tj,λ)l]},
(5)
where ηdel is the fraction of the pump power delivered from the pump excitation source to the input end of the laser gain medium, RP is the reflectance of the pump light at the output coupler of a laser cavity resonator, VLj is the volume of the jth cylindrical annulus as given by
VLj=π(rj2rj+12)l,
(6)
Pj(λ) is the spectrally resolved partial pump power of the jth cylindrical annulus as expressed by [29

29. S. W. Smith, The Scientist and Engineer's Guide to Digital Signal Processing[J], II ed, (California Technical Publishing, 1999), Chapter 25.

]
Pj(λ)=Pjpeak2π(ΔλD2FWHM/22ln2)2exp((λλ0)22(ΔλD2FWHM/22ln2)2),
(7)
where is the spectrally resolved FWHM linewidth, is the peak pump power of the jth cylindrical annulus as given by
Pjpeak=Ij(r)dSSjImaxexp(2rj2ωP2),
(8)
where Ij(r) is the intensity distribution of the pump power of the jth cylindrical annulus, r is the radius at the cross-section, ωP is the pump waist radius, Sj is the cross-section area of the jth cylindrical annulus:
Sj=π(rj2rj+12).
(9)
The maximal intensity of the pump power of the jth cylindrical annulus Imaxcan be calculated by
Imax=PPexp(2r2ωP2)dS=PP0Rexp(2r2ωP2)2πrdrPPj=1Nexp(2rj2ωP2)Sj.
(10)
σD2(Tj,λ) is the pump absorption cross-section as expressed by [12

12. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004). [CrossRef]

]
σD2(Tj,λ)=σD2Hebroadened(Tj)1+(λλD2ΔλD2FWHM/2)2,
(11)
where
σD2Hebroadened(Tj)=σD2radiative2πτD2nHeamagat(19.3GHzamagat)Tj294K,
(12)
where nHe-amagat is the number density of helium in the vapor cell, σD2radiative is the atomic cross section of the D2 line with the value of 2.31 × 10 −9 cm2, respectively. ΓLj is the transition rate of laser emission as expressed by [12

12. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004). [CrossRef]

]
ΓLj={1VLjPLjhυLRoc1Roc×{exp[(n2jn1j)σD1Hebroadened(Tj)l]1}×{1+TT2exp[(n2jn1j)σD1Hebroadened(Tj)l]},withlaseroutput0,withoutlaseroutput,
(13)
where PLj is the output alkali laser power, Roc is the reflectance of the output coupler, and TT is the one-way cavity transmittance by neglecting the ground-state absorption as well as the output coupler loss. σD1Hebroadened(Tj) is collisionally-broadened cross section as given by [12

12. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004). [CrossRef]

]
σD1Hebroadened(Tj)=σD1radiative2πτD1nHeamagat(21.5GHzamagat)Tj294K,
(14)
where σD1radiativeis the atomic cross section of the D1 line.

In addition, the number density of every energy-level must satisfy the following two equations for a steady-state laser emission [12

12. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004). [CrossRef]

]:
exp[2(n2j(Tj)-n1j(Tj))σD1He-broadened(Tj)l]×TT2Roc=1,
(15)
n0j(Tj)=n1j(Tj)+n2j(Tj)+n3j(Tj),
(16)
where is the total alkali number density of the jth cylindrical annulus as expressed by [22

22. B. D. Barmashenko and 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]

]
n0j(Tj)={n01(Tw),j=1n01(Tw)(TwTj),j>1,
(17)
where Tw is the temperature of the cell wall, n01(Tw) is the saturated alkali number density inside the first cylindrical annulus which is adjacent to the inner surface cell wall as given by [30

30. D. A. Steck, “Cesium D line data,” Available: http://steck.us/alkalidata

]
n01(Tw)=133.322NARTw(108.221274006.048Tw0.00060194Tw0.19623log10Tw),
(18)
where R is a constant of proportionality with the value of 8.3143 J/(mol·K), PV is the saturation pressure of the cesium vapor in Torr and NA is Avogadro number, respectively.

Next, we calculate the volume density of generated heat of the jth cylindrical annulus by using the following formula [12

12. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004). [CrossRef]

]:
Ωj=γ32(Tj)[n3j2n2jexp(ΔEkBTj)]ΔE,
(19)
where ΔE is the energy gap between 6 2S1/2 and 6 2P3/2 levels with the value of 554 cm−1.

Thus, the generated heat of the jth cylindrical annulus can be obtained by the following calculation:

Qj=VLiΩj.
(20)

2.2. Theoretical analyses of heat transfer

2.2.1. Calculation of a Transverse Section Except the Central Core

Generally, the differential equation of thermal conductivity in the cylindrical coordinate system is given by [31

31. M. J. Latif, Heat Conduction, III ed., (Verlag Berlin and Heidelberg GmbH & Co. K, 2009), Chapter 1,.

]
ddr(rdTdr)+ΩrK(T)=0,
(21)
where Ω stands for the volume density of generated heat and K(T) denotes the coefficient of thermal conductivity, respectively. When the thickness of the segmented cylindrical annulus is small enough, the volume density of the jth cylindrical annulus can be approximately expressed as of the volume density Ωj at the exterior side of the cylindrical annulus (j = 1, 2, … (N-1)). Similarly, K(T) can also be approximately expressed as K(Tj), which is the thermal conductivity at the exterior side of the jth cylindrical annulus as expressed by [24

24. B. D. Barmashenko and 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]

]
K(Tj)=PHePHe+PC2H6KHe(Tj)+PC2H6PHe+PC2H6KC2H6(Tj),
(22)
wherePHeand PC2H6are respectively the partial pressures of helium and ethane. KHe(Tj)and KHe(Tj)are respectively the thermal conductivities of helium and ethane as given by [32

32. C. L. Yaws, Matheson Gas Data Book, VII ed., (McGraw-Hill & Matheson Tri-Gas, 2001), Appendix 23.

]

KHe(Tj)=0.05516+3.2540×104Tj2.2723×108Tj2,KC2H6(Tj)=0.01936+1.2547×104Tj+3.8298×108Tj2.
(23)

Next, we get the following formula by undertaking the integral calculation on both sides of Eq. (21):
dTdr+Ωjr2K(Tj)=Cj1r,
(24)
where Cj1 is a constant for the jth cylindrical annulus. According to Fourier’s Law, the quantity of the transferred heat Фj from the jth cylindrical annulus to the (j-1)th one can be expressed by [31

31. M. J. Latif, Heat Conduction, III ed., (Verlag Berlin and Heidelberg GmbH & Co. K, 2009), Chapter 1,.

]
Φj=[K(T)AdTdr]|T=Tj,
(25)
where A stands for the lateral area of the jth cylindrical annulus as given by
A=2πrjl,
(26)
where l is the cell length. After substituting Eqs. (24) and (26) into Eq. (25), can be calculated by the following formula:
Cj1=Ωjrj22K(Tj)ΦjrjK(Tj)Aj.
(27)
Note Ωj is calculated by use of Eq. (18) of Subsection 2.1. Then, we make a further integral calculation on both sides of Eq. (24) and the temperature distribution inside the jth cylindrical annulus can be given by
T(r)=Cj1lnrΩjr24K(Tj)+Cj0,
(28)
where Cj0 is another constant for the jth cylindrical annulus which can be solved by substituting r = rj and T = Tj into Eq. (28):
Cj0=TjCj1lnrj+Ωjrj24K(Tj).
(29)
By substituting Cj1 and Cj0into Eq. (28), one can obtain the temperature distribution in the transverse section of the jth cylindrical annulus. The temperature of the inner side of the jth cylindrical annulus, Tj+1, can also be deduced and is then used as the board condition in the calculation of the (j + 1)th cylindrical annulus. By employing a circulatory calculation, we can therefore obtain T2, T3, …,TN.

2.2.2. Calculation of Temperature of the Central Core

The exterior radius rN of the central cylinder (core) can be simply expressed by

rN=R/N,
(30)

We can calculate out CN1=0 by substituting the boundary condition dTdr|r=0=0 into Eq. (24) .Then we simplify Eq. (24) as the following style:

dTdr+ΩNr2K(TN)=0.
(31)

The temperature distribution inside the central core can be given by

T(r)=ΩNr24K(TN)+CN0,
(32)

can be solved by substituting r = rN, T = TN into Eq. (32),

CN0=TN+ΩNrN24K(TN).
(33)

The central temperature of the vapor cell, TN + 1, can be obtained by substituting into Eq. (32). By combing the results of Eqs. (28) and (32) together, it is possible to acquire the whole temperature distribution at the cross-section of the vapor cell.

2.3. Calculation of radial temperature distribution

The flowchart for evaluation of the temperature distribution is diagramed in Fig. 4
Fig. 4 Flowchart of evaluating the distributions of temperature and population of a vapor cell.
. First, we assume that the total heat transferred out from a vapor cell is PThermal . By considering a fact that the heat delivered from the first cylindrical annulus to the cell wall, Φ1, is equal to the total heat release, the following relationship is tenable:
Φ1=PThermal.
(34)
The temperature at the outside of the first cylindrical annulus, which is equal to Tw in this case, is approximately assigned as the temperature of this cylindrical annulus during the kinetic calculation if the thickness of the segmented thickness is small enough. Therefore, the volume heat density Ω1 as well as the generated heat Q1 can be deduced by employing the approach introduced in Subsection 2.1. By using Φ1 and Ω1, we can evaluate the temperature distribution inside the first cylindrical annulus with Eq. (28).

The temperature of the inner side of the first cylindrical annulus, T2, can be then evaluated and is utilized as the initial conditions in calculating the temperature distribution of the second cylindrical annulus. As depictured in 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.
, the heat transferred from the second cylindrical annulus to the first one, Φ2, is thus calculated by

Φ2=Φ1Q1=PThermalQ1.
(35)

Therefore, through a circulatory calculation of Q1, Q2, …, QN-1, we can obtain heat transferred from the jth cylindrical annulus to the (j-1)th one as expressed by
Φj=PThermali=1j1Qi,
(36)
where j = 1, 2, …, N.

3. Results and discussions

3.1. Population distributions

3.1.1. Different waists of pump beams

We first analyze the population distributions inside a cesium vapor cell for different waists of pump beams. By using the approach introduced in Section 2, we calculate the population density distributions inside the cell as illustrated in 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.
. During the evaluation, the pump power is fixed to 10 W and the other parameters are listed in Table 1

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

table-icon
View This Table
. The waist radii are assumed as 150 μm, 300 μm, 500 μm, and 700 μm corresponding to (a), (b), (c), and (d) in Fig. 7, respectively. It is observed that the total density of the cesium vapor n0 increases with the radial position r. Such a phenomenon is due to the fact that the temperature at the central area is higher than that near the wall of the vapor cell. Some significant variation of the population densities n1 and n2 can be seen in the figure. The inflection points in the legend give rise to discontinuity of the first derivative or “angles” on the curves for n1 and n2. Such inflection points located in the lasing boundary line. In the lasing region, n2 is always larger than n1 because of population inversion. We also find that, the bigger the spot size of a pump beam is, the lower n3 becomes. It means that more electrons will be stimulated into the 6 2P3/2 level under a higher pump density. However, it also leads to a population accumulation at the top energy-level. One can realize that the higher pump density brings about a relative weak relaxing capability by comparing n3 and n2 in (a), (b), (c), and (d) of Fig. 7.

3.1.2. Different pump power

Next, we discuss the population distributions inside a cesium vapor cell for different pump power when the beam waist radius is fixed to 500 μm. By using the method mentioned above, we get the results when the pump power is 1 W, 20 W and 50 W, respectively. As diagramed in Fig. 8
Fig. 8 Population distributions with the pump power of 1 W (a), 20 W (b) and 50 W (c), respectively.
, the total number population n0 decreases with the pump power. The reason is that the central temperature increases with the pump power and the total population density generally exhibits a degressive tendency with the temperature rising by referring Eq. (17). In the calculation, lasing output corresponding to the population distribution in Fig. 8(a) cannot be achieved because the threshold condition is unsatisfied.

3.2. Temperature distributions

By use of the approach introduced in Section 2, the radial temperature distributions are obtained. As shown in 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.
, the temperature at the cross section exhibits a distinct gradient and achieves the maximum values at the central axis for every case. Note that the waist radius of the pump beam in Fig. 9(a) is 150, 300, 500 and 700 μm, respectively, when the pump power is set to 10 W. We can observe that the curve for ωp = 500 μm is almost located at the lowest position at the diagram. It means that a big pump density will not always cause a high heat generation. To make the expression clear, we also produced a 3-Dmensional drawing for ωp = 500 μm as depictured in Fig. 9(b). In Fig. 10
Fig. 10 Temperature distributions with the pump power of 1, 10, 100 and 500 W, respectively.
, the pump powers is set to 1, 10, 100, and 500 W, respectively, when the waist radius of the pump beam is 500 μm. It is obvious that the temperature gradient increases rapidly with the pump power. Such tendencies can be explained by a fact that the thermal conductivity of a gas-state medium is so small that the generated heat cannot be transferred outside efficiently.

It is interesting to find that the peak position of the optical-optical conversion efficiency corresponds to a relatively low output. The optimum pump power is around 30 W for the optical-optical conversion efficiency but the laser output is only 13.4 W. The results are different from some traditional solid-state lasers. Thus, the output characteristics of a DPAL are not only dependent on the energy levels, but also determined by the thermal features.

4. Conclusion

Until now, the temperature inside the vapor cell has been considered as a constant in most of the literatures of DPALs. Actually, it is impossible completely eliminate the temperature gradient of a heated gas-state medium because of its small thermal conductivity. Since the saturated density of the alkali vapor is strongly dependent on the temperature, the kinetics process should be taken into account during evaluating the thermal characteristics. In this study, we developed a scheme by using both the heat transfer and the laser kinetics to analyze the temperature distribution inside an enclosed vapor cell. In our theoretical model, a cell is segregated to many coaxial cylindrical annuli. The population density and temperature in every cylindrical annulus have been finally calculated. It is seen that the temperature in the center of a vapor cell is higher than the other places at the cross section. The temperature gradient becomes serious with increase of the pump intensity. However, unlike some of the solid-state lasers, a stronger pump power does not always result in a higher optical-optical conversion efficiency for a DPAL. To solve such a problem, one of the effective methods to achieve a high-powered DPAL might be adopting a flowing-gas system. The thermal effects should be dramatically decreased by using such a dynamic procedure. The mathematical model will be much more complicated than that used in this report. In fact, the theoretical model in this paper can also be applied to the other two types of DPALs if the D1 and D2 radiative lifetimes, the pump absorption cross section, the collisionally-broadened cross section, and the thermally averaged relative velocities are changed to the new values corresponding to rubidium and potassium.

Additionally, the temperature distribution will inevitably bring about some changes in refractive index with temperature dn/dT. Decreasing the thermally-induced wave front distortion and thermal lensing effects should be important in realization of a DPAL with high beam quality. We will also introduce our results on this issue afterwards.

Acknowledgments

We are very grateful to Prof. Salman Rosenwaks and Dr. Boris D. Barmashenko at Ben-Gurion University of the Negev of Israel for their valuable helps in calculation of saturated alkali number densities inside a static alkali vapor cell.

References and links

1.

W. F. Krupke, “Diode Pumped Alkali Laser,” US Patent Application US 2003/0099272 Al, (2003).

2.

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

3.

W. F. Krupke, R. J. Beach, S. A. Payne, V. K. Kanz, and 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).

4.

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

5.

Y. Wang, T. Kasamatsu, Y. Zheng, H. Miyajima, H. Fukuoka, S. Matsuoka, M. Niigaki, H. Kubomura, T. Hiruma, and 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]

6.

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

7.

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

8.

Y. Wang and 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]

9.

Y. Wang, M. Niigaki, H. Fukuoka, Y. Zheng, H. Miyajima, S. Matsuoka, H. Kubomura, T. Hiruma, and 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]

10.

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

11.

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

12.

R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, and L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004). [CrossRef]

13.

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

14.

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

15.

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

16.

C. C. Lai, K. Y. Huang, H. J. Tsai, K. Y. Hsu, S. K. Liu, C. T. Cheng, K. D. Ji, C. P. Ke, S. R. Lin, and S. L. Huang, “Yb3+:YAG silica fiber laser,” Opt. Lett. 34(15), 2357–2359 (2009). [CrossRef] [PubMed]

17.

D. A. Steck, Rubidium 85 D Line Data. Available: http://steck.us/alkalidata.

18.

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

19.

Z. N. Yang, H. Y. Wang, Q. S. Lu, W. H. Hua, and X. J. Xu, “Modeling of an optically side-pumped alkali vapor amplifier with consideration of amplified spontaneous emission,” Opt. Express 19(23), 23118–23131 (2011). [CrossRef] [PubMed]

20.

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

21.

Y. F. Liu, B. L. Pan, J. Yang, Y. J. Wang, and 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]

22.

B. D. Barmashenko and 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]

23.

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

24.

B. D. Barmashenko and 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]

25.

S. Rosenwaks, B. D. Barmashenko and Waichman, “Semi-analytical and 3D CFD DPAL modeling: feasibility of supersonic operation”, Proc. SPIE 8962, High Energy/Average Power Lasers and Intense Beam Applications VII, 896209, 1–9 (2014).

26.

S. Kiwan and 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]

27.

P. Teerstra and M. M. Yovanovich, “ Comprehensive review of natural convection in horizontal circular annuli,” in 7th AIAA/ ASME Joint Thermophysics and Heat Transfer Conference, Albuquerque, New Mexico, 15 –18 June 1998 (AIAA, 1998), pp. 141–152.

28.

Z. N. Yang, H. Y. Wang, Q. S. Lu, Y. D. Li, W. H. Hua, X. J. Xu, and J. B. Chen, “Modeling, numerical approach, and power scaling of alkali vapor lasers in side-pumped configuration with flowing medium,” J. Opt. Soc. Am. B 28(6), 1353–1364 (2011). [CrossRef]

29.

S. W. Smith, The Scientist and Engineer's Guide to Digital Signal Processing[J], II ed, (California Technical Publishing, 1999), Chapter 25.

30.

D. A. Steck, “Cesium D line data,” Available: http://steck.us/alkalidata

31.

M. J. Latif, Heat Conduction, III ed., (Verlag Berlin and Heidelberg GmbH & Co. K, 2009), Chapter 1,.

32.

C. L. Yaws, Matheson Gas Data Book, VII ed., (McGraw-Hill & Matheson Tri-Gas, 2001), Appendix 23.

33.

H. Cai, Y. Wang, W. Zhang, L. P. Xue, H. Y. Wang, J. H. Han, and Z. Y. Liao, “Characteristic analyses of a diode-pumped rubidium vapor laser using a kinetic algorithm,” Opt. & Laser Technol., to be submitted.

34.

N. D. Zameroski, G. D. Hager, W. Rudolph, and 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]

OCIS Codes
(140.1340) Lasers and laser optics : Atomic gas lasers
(140.3430) Lasers and laser optics : Laser theory
(140.3460) Lasers and laser optics : Lasers
(140.3480) Lasers and laser optics : Lasers, diode-pumped

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: April 18, 2014
Revised Manuscript: May 18, 2014
Manuscript Accepted: May 18, 2014
Published: May 30, 2014

Citation
Juhong Han, You Wang, He Cai, Wei Zhang, Liangping Xue, and Hongyuan Wang, "Algorithm for evaluation of temperature distribution of a vapor cell in a diode-pumped alkali laser system: part I," Opt. Express 22, 13988-14003 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13988


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References

  1. W. F. Krupke, “Diode Pumped Alkali Laser,” US Patent Application US 2003/0099272 Al, (2003).
  2. 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]
  3. 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).
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. 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]
  10. B. V. Zhdanov, R. J. Knize, “Diode-pumped 10 W continuous wave cesium laser,” Opt. Lett. 32(15), 2167–2169 (2007). [CrossRef] [PubMed]
  11. W. F. Krupke, “Diode pumped alkali lasers (DPALs)—A review (rev1),” Prog. Quantum Electron. 36(1), 4–28 (2012). [CrossRef]
  12. R. J. Beach, W. F. Krupke, V. K. Kanz, S. A. Payne, M. A. Dubinskii, L. D. Merkle, “End-pumped continuous-wave alkali vapor lasers: experiment, model, and power scaling,” J. Opt. Soc. Am. B 21(12), 2151–2163 (2004). [CrossRef]
  13. 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]
  14. 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]
  15. R. J. Garman, “Modelling of the intracavity optical fields in a copper vapour laser,” Opt. Commun. 119(3-4), 415–423 (1995). [CrossRef]
  16. C. C. Lai, K. Y. Huang, H. J. Tsai, K. Y. Hsu, S. K. Liu, C. T. Cheng, K. D. Ji, C. P. Ke, S. R. Lin, S. L. Huang, “Yb3+:YAG silica fiber laser,” Opt. Lett. 34(15), 2357–2359 (2009). [CrossRef] [PubMed]
  17. D. A. Steck, Rubidium 85 D Line Data. Available: http://steck.us/alkalidata .
  18. 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]
  19. Z. N. Yang, H. Y. Wang, Q. S. Lu, W. H. Hua, X. J. Xu, “Modeling of an optically side-pumped alkali vapor amplifier with consideration of amplified spontaneous emission,” Opt. Express 19(23), 23118–23131 (2011). [CrossRef] [PubMed]
  20. 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]
  21. 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]
  22. 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]
  23. B. D. Barmashenko, S. Rosenwaks, “Feasibility of supersonic diode pumped alkali lasers: Model calculations,” Appl. Phys. Lett. 102(14), 141108 (2013). [CrossRef]
  24. 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]
  25. S. Rosenwaks, B. D. Barmashenko and Waichman, “Semi-analytical and 3D CFD DPAL modeling: feasibility of supersonic operation”, Proc. SPIE 8962, High Energy/Average Power Lasers and Intense Beam Applications VII, 896209, 1–9 (2014).
  26. 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]
  27. P. Teerstra and M. M. Yovanovich, “ Comprehensive review of natural convection in horizontal circular annuli,” in 7th AIAA/ ASME Joint Thermophysics and Heat Transfer Conference, Albuquerque, New Mexico, 15 –18 June 1998 (AIAA, 1998), pp. 141–152.
  28. Z. N. Yang, H. Y. Wang, Q. S. Lu, Y. D. Li, W. H. Hua, X. J. Xu, J. B. Chen, “Modeling, numerical approach, and power scaling of alkali vapor lasers in side-pumped configuration with flowing medium,” J. Opt. Soc. Am. B 28(6), 1353–1364 (2011). [CrossRef]
  29. S. W. Smith, The Scientist and Engineer's Guide to Digital Signal Processing[J], II ed, (California Technical Publishing, 1999), Chapter 25.
  30. D. A. Steck, “Cesium D line data,” Available: http://steck.us/alkalidata
  31. M. J. Latif, Heat Conduction, III ed., (Verlag Berlin and Heidelberg GmbH & Co. K, 2009), Chapter 1,.
  32. C. L. Yaws, Matheson Gas Data Book, VII ed., (McGraw-Hill & Matheson Tri-Gas, 2001), Appendix 23.
  33. H. Cai, Y. Wang, W. Zhang, L. P. Xue, H. Y. Wang, J. H. Han, and Z. Y. Liao, “Characteristic analyses of a diode-pumped rubidium vapor laser using a kinetic algorithm,” Opt. & Laser Technol., to be submitted.
  34. 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]

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