Abstract

A 3-dimenstional (3D) heat flow model of laser diode (LD) pumped short-length Er3+/Yb3+ heavily co-doped phosphate fiber laser that includes the energy-transfer upconversion (ETU) effects has been developed. The fully 3D analytical solution with the consideration of longitudinal heat flow has been given to describe the temperature distribution in phosphate fiber. The calculated results show that both ETU processes and longitudinal heat conduction have a great influence on the fiber laser performance. Finally, we have validated the analytical expression by measuring the temperature distribution of an end-pumped short-length Er3+/Yb3+ co-doped fiber laser, which was placed into the copper tube.

©2009 Optical Society of America

1. Introduction

Phosphate glasses exhibit excellent solubility for rare-earth ions, and are usually pulled into fibers with high gain coefficient per unit length. Among phosphate glass fibers Er3+/Yb3+ co-doped fiber has broad absorption band and more than two orders of magnitude of pump absorption than single Er3+-doped fibers at 976nm, which make it very attractive for constructing short fiber resonant cavity lasers operating at 1.5μm [1–3]. A diode-pumped Er3+/Yb3+ co-doped glass fiber laser generating more than 1 W/mm power has been reported [3]. However, in such fiber lasers heat accumulation from the quantum defect between pump and laser photons and the fast nonradiative decay from the 4I11/2 level to the laser level 4I13/2 [4] becomes a serious problem. Meanwhile, the energy-transfer upconversion (ETU) between Er3+ and Yb3+ ions due to the high doping concentration is not neglectable and thus gives rise to an extra heat load into the active fiber. Serious heat accumulation in phosphate glass fiber often leads to thermal stress fracture, thermal birefringence, thermal lensing [5] detrimental to laser performance.

Many theoretical researches have concentrated on the thermal effects caused by ETU processes in Nd-doped and Er-doped solid-state lasers [6, 7]. However, the upconversion-induced heat load in Er3+/Yb3+ co-doped fiber lasers has received little attention due to the low rare-earth doping concentration in the traditional silica fiber. For the heavily doped phosphate glass fiber, upconversion-induced heat accumulation has become an extremely important issue. A key job in analyzing the upconversion-induced thermal load is to accurately obtain the temperature field distribution within the fiber. In previous numerical simulation works, the longitudinal heat flow along fiber length was usually neglected due to the low pump absorption coefficient, low signal gain coefficient and long fiber length [8, 9]. Such assumptions may introduce a big error in estimating the heat load of a high gain coefficient, short resonant cavity phosphate fiber laser. Recently, a numerical method has been used to seek 3D thermal distribution in phosphate fiber [10]. However, the upconversion-induced heat load has not been considered yet. In addition, numerical methods are difficult to be used for analyzing the thermal stress, thermal lensing, and thermal birefringence of active materials, while only analytical solutions can do. Up to now, 3D analytical calculations have all been performed for solid-state lasers [11, 12]. In view that the analytical methods can play a significant role on accurately evaluating the laser performance. Thus a 3D analytical investigation on thermal effect is necessary for a short-length high-power heavily doped fiber laser.

In this paper, we establish a 3D heat analytical model and analytical calculations on the temperature field in fibers are performed based on the space-dependent rate equations including the energy transfer upconversion as one of main heat sources. To our best knowledge, this is the first 3D thermal analysis for fiber lasers with analytical solutions including ETU processes. This thermal flow model is confirmed simply by measuring the temperature distribution of an end-pumped short Er3+/Yb3+ co-doped fiber laser, which was placed into the copper tube.

2. Rate equation analysis on populations including energy-transfer upconversion

An energy-level scheme of the Er3+/Yb3+ co-doped system including ETU processes is illustrated in Fig. 1 under the excitation of a 976nm LD [13–15]. There are three upconversion processes that mainly contribute to the heat accumulation, as depicted in Fig. 1:

UC1:Yb3+(F522)+Er3+(I1124)Yb3+(F722)+Er3+(F724)
UC2:Yb3+(F522)+Er3+(I1324)Yb3+(F722)+Er3+(F924)
UC3:Er3+(I1324)+Er3+(I1324)Er3+(I1524)+Er3+(I924)

For all the upconversion processes above, a large part of populations on the upconversion level decays to the level 4I13/2 by multiphonon relaxation step by step, generating heat in the active fiber. Because the fast nonradiative multiphonon relaxation from the 4I11/2 to the 4I13/2 level greatly decreases the back-energy transfer from Er3+: 4I11/2 to Yb3+: 2F7/2 and the upconversion losses occurring at the 4I11/2 level of Er3+ ions [16], we neglect the back-energy transfer process and upconversion losses at this level in the calculation.

 figure: Fig. 1.

Fig. 1. Energy-level scheme of the Er3+/Yb3+ co-doped system including ETU processes: UC1, UC2, cumulative upconversion; UC3, cooperative upconversion; M, multiphonon relaxation; F, fluorescence process; L, 1535 nm laser emission.

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The space-dependent rate equations for the Er3+ and Yb3+ population densities are written as follows [17, 18]. The symbols denoting population density of each level are labeled in Fig. 1. As the 4F7/2, 2H11/2, 4S3/2 and 4F9/2 levels in Er3+ ions relax rapidly to the next lower-lying level by multiphonon decay [13, 14], these states have been excluded in rate equations.

N2yt=W12yN1yW21yN2yN2yτyk1N2yN1ek2N2yN2e=0
N2et=W12eN1e+A32eN3eW21eN2ek2N2eN2yN2eτe2CupN2e2=0
N3et=k1N1eN2y+A43eN4eA32eN3e=0
N4et=CupN2e2A43eN4e=0
NEr=N1e+N2e+N3e+N4e
NYb=N1y+N2y

The signal absorption, signal emission, pump absorption and pump emission rates W 12e , W 21e , W 12y, and W 21y are given by:

W12ezt=Γsσas(υs)Ps(z)hυsAcoreW21ezt=Γsσes(υs)Ps(z)hυsAcore
W12yzt=Γpσap(υp)Pp(z)hυpAcoreW21yzt=Γpσep(υp)Pp(z)hυpAcore

Where the physical meanings of parameters are presented in Table 1.

Tables Icon

Table 1. The related parameters used in this paper

3. Resolving of 3-dimensional temperature field

Figure 2 is the schematic illustration of Er3+/Yb3+ co-doped phosphate fiber laser. Pp ± (z) and Ps ± (z) are the pump power and the signal power in the positive and negative z direction, respectively. They can be described by the steady-state power transmission equations [18, 20, 21] and can be solved numerically. Figure 3 shows the pump and signal powers as a function of the position along the fiber length as the input pump power is 100 mW.

 figure: Fig. 2.

Fig. 2. Schematic illustration of phosphate fiber laser

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 figure: Fig. 3.

Fig. 3. Pump and signal powers as a function of the position along the fiber length: (a) pump power along the fiber length; (b) signal power in positive and negative directions along the fiber length.

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The general steady-state heat equations for fiber lasers under air cooling are given by [22]:

1rr[rTrzr]+2Trzz2=Qrzk,0rr1
1rr[rTrzr]+2Trzz2=0,r1rr2
kTrzr=c[TcTrz],r=r2
T1=T2,T1r=T2r,r=r1
Trzz=0,z=0,z=L

Where

Qrz=2(αaη+αps)[Pp+(z)+Pp(z)]+2αs[Ps+(z)+Ps(z)]πωp2exp(2r2ωp2)

is thermal power density within fiber. r 1 and r 2 are the core and cladding radius, respectively. k denotes the fiber thermal conductivity. c is the convective coefficient. Tc is the heat sink temperature. T 1 and T 2 are the temperatures in fiber core and cladding regions, respectively. αa is the absorption coefficient at the pump wavelength. αps and αs are scattering loss coefficients at the pump wavelength and signal wavelength, respectively. η is the fractional thermal loading. ωp is Gaussian radius of the pump light.

The equation system (7)–(11) can be solved as follows:

T1rz=Tc+C0lnr1+D0+n=1m=1Anmcos(Lz)J0(μn(0)r1r)+m=1[CmI0(Lr1)+DmK0(Lr1)]cos(Lz)
T2rz=Tc+C0lnr+D0+m=1[CmI0(Lr)+DmK0(Lr)]cos(Lz)

Where

kC0r2=c(C0lnr2+D0),C0r1=n=1An0μn(0)r1J1(μn(0))
k(L)[CmI1(Lr2)DmK1(Lr2)]=c[CmI0(Lr2)+DmK0(Lr2)]
(L)[CmI1(Lr1)DmK1(Lr1)]=n=1Anmμn(0)r1J1(μn(0))
An0=4kL(μn(0))2J12(μn(0))0r1exp(2r2ωp2)J0(μn(0)r1r)rdr×
0L(αaη+αps)[Pp+(z)+Pp(z)]+αs[Ps+(z)+Ps(z)]πωp2dz
Anm=8Lk[m2π2r12+L2(μn(0))2]J12(μn(0))0L(αaη+αps)[Pp+(z)+Pp(z)]+αs[Ps+(z)+Ps(z)]πωp2cos(Lz)dz×
0r1exp(2r2ωp2)J0(μn(0)r1r)rdr2r12()2[m2π2r12+L2(μn(0))2]J1(μn(0))μn(0)[CmI0(Lr1)+DmK0(Lr1)]

μn (0) is the nth zero point of the zero order Bessel function of the first kind. J 0 and J 1 are the zero and first order Bessel function of the first kind, I 0, I 1 and K 0, K 1 are the zero and first order modified Bessel function of the first kind and second kind, respectively. More detailed deductions are shown in Appendix A.

4. Analytical results and discussions

Upconversion processes consume the absorbed pump photons, decrease the population inversion at the laser level 4I13/2 and also increase the fractional thermal load in the laser medium. The population-inversion density between the 4I13/2 level and 4I15/2 level is denoted as n 0 and n without and with ETU effects, respectively. So the fractional reduction of the population inversion due to ETU can be expressed as [23, 24]:

FETU=(n0n)n0

The multiphonon decay rate from the upconversion laser levels 2H11/2, 4S3/2, 4F9/2, and 4I9/2 is much larger than that of the spontaneous radiative transition rate, as shown in Table 2. Thus it is reasonable to neglect the radiative rate from the upconversion laser levels. Hence the fractional thermal loading can be expressed as:

η=FETU+(1FETU)(1λpλs)
Tables Icon

Table 2. Nonradiative transition rate and total spontaneous radiative transitions rate of Er3+ upconversion levels

The value of related parameters of Er3+/Yb3+ co-doped phosphate fiber are listed in Table 1. The fluorescence lifetimes, absorption and emission cross sections used as input parameters in the calculations are from the measurements on our home-made phosphate glass. The glass samples were cut and polished to less than 1mm thick in order to minimize the reabsorption effect. The power filling factors Γp,s are given as Γp,s = 1-exp(-2r 1 2/ωp,s 2), where ωp,s = r 1(0.616 + 1.660/V 1.5 + 0.987/V 6) is the Gaussian beam waist radius at the given wavelength λ; V = 2πr 1 NA/λ is the normalized frequency for core radius r 1, numerical aperture NA and wavelength λ [25]. The upconversion coefficient and energy transfer coefficients are the fitting parameters. Different approaches [19, 26] have been used to determine the energy transfer coefficient k1 and the cooperative upconversion coefficient Cup. k1 is often assumed to be independent of Er3+ concentration and ranges between 1×10-22 and 5×10-22 m3 s-1 depending on Yb 3+ concentration; Cup slightly depends on Er3+ concentration and is of the order of ~10-24 m3 s-1. We used k1=2.56×10-22 m3 s-1, Cup=1.67×10-24 m3 s-1, k2=0.85×10-22 m3 s-1 to fit the curve of signal laser output versus pump power as shown in Fig. 4. To eliminate the thermal effects on signal power, the laser was actively refrigerated.

 figure: Fig. 4.

Fig. 4. The theoretical calculation and experimental results on signal laser output versus pump power. The best fit parameters k1, k2, and Cup are 2.56×10-22 m3 s-1 , 0.85×10-22 m3 s-1, 1.67×10-24 m3 s-1, respectively.

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The other parameters used in the analytical calculations are: k = 0.55 W·m-1·K-1, c = 10W·m-2K-1, Tc = 300K.

When the fiber length is 1cm, and the input pump power is 100 mW, the distribution of calculated temperature field considering ETU effects in fiber is shown in Fig. 5.

 figure: Fig. 5.

Fig. 5. Temperature field distribution in phosphate fiber

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Figure 6 shows the temperature distribution of fiber end surface at the pump side along the active fiber radial coordinate with and without considering longitudinal heat conduction, respectively. The maximum temperatures in active fiber are 479.85K and 571.38K, respectively. The maximum temperature can be reduced by 19.07% when the longitudinal heat conduction is considered. So the 2-dimensional approximate calculations bring a rather big error in estimating the internal temperature field of short-length fiber lasers. The analytical solution of 2-dimensional approximate with only radial heat flow is shown in Appendix B.

 figure: Fig. 6.

Fig. 6. The end-surface temperature distribution along active fiber radial coordinate: (a) without considering longitudinal heat conduction; (b) with longitudinal heat conduction.

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Figure 7 shows the temperature distribution of fiber end surface at the pump side along the active fiber radial coordinate with and without the consideration of upconversion when the pump power is 100 mW. Compared with the maximum temperatures of the two cases, 15.91% extra heat is generated in the active fiber by ETU processes. With increasing the pump power, the temperature differences increase as shown in Fig. 8, indicating that the influence of ETU processes increases, and more and more extra heat induced by ETU accumulates in the active fiber. However, the increasing rate of maximum temperature (namely the slope of curve with consideration of ETU processes) in active fiber reduces with the increasing pump power and it tend to be a constant as the pump power is sufficiently high, e.g. more than 200mW. The reason is that the fractional thermal loading η decreases with increasing pump power, as illustrated by Fig. 9.

 figure: Fig. 7.

Fig. 7. The end-surface temperature distribution along radial coordinate when the pump power is 100 mW: (a) with upconversion; (b) without upconversion.

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 figure: Fig. 8.

Fig. 8. The maximum temperatures in active fiber core vs. pump power with and without considering ETU.

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 figure: Fig. 9.

Fig. 9. The fractional thermal loading at different pump powers.

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5. Experimental validation on heat generation with upconversion

In order to verify our 3D heat model, an improved passive cooling construction is used, which can remove heat load rapidly. As is shown in Fig. 10, the active fiber was inserted into a ceramic ferrule, and then placed into the copper tube whose inner diameter is slightly larger than the ceramic ferrule. The ceramic ferrule is just for protecting the fiber and its thermal conductivity is a bit higher than phosphate fiber, which can be regarded as a whole with active fiber. A thermocouple was inserted into the copper tube along a fine groove in the inner surface of the copper and closely contacted the ceramic ferrule to test its temperature during laser operation. The whole setup is under air cooling, and the surrounding temperature is kept at 295.95K.

 figure: Fig. 10.

Fig. 10. The fiber cross-section inside the tube

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 figure: Fig. 11.

Fig. 11. The signal vs. absorbed pump power plot

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In the experiment, a 1-cm-long Er3+/Yb3+ co-doped phosphate fiber was used to construct a resonant cavity and end-pumped configuration was adopted. The parameters of the fiber are the same as described in Table 1. The signal laser vs. pump power plot is shown in Fig. 11. The laser slope efficiency was 20.4%. We compared the tested temperature with the analytical solution based on the present experimental configuration. The detailed deduction process is shown in Appendix C.

 figure: Fig. 12.

Fig. 12. The inner surface temperature of the copper tube at different pump power.

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 figure: Fig. 13.

Fig. 13. Calculated values vs. experimental values at different pump power.

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Figure 12 shows the inner surface temperature of the copper tube along fiber axial direction, the pump power is 172.5mW, 160.4mW, 138.2mW, 128.5mW, 100.8mW, 71.7mW, 35.1mW from up to down, respectively. With increasing the pump power, the heat generated increases and along the fiber length the temperature slightly decreases from the pump end to the other. Figure 13 shows the calculated values and experimental values at different pump powers. The values calculated by analytical expression have good agreements with experimental values. The standard error between them is 0.56%, indicating that our analytical expressions can be used to optimize rare-earth ions doping concentration and resonator cavity parameters of short-length fiber lasers.

6. Conclusions

A theoretical heat flow model of LD pumped Er3+/Yb3+ co-doped phosphate fiber laser that includes ETU effects has been developed. The 3D analytical solution of the temperature field in short fiber laser is given. The calculated results demonstrate that the ETU processes and the longitudinal heat conduction have a great influence on heat accumulation in short-length heavily doped fiber laser. The theoretical model is verified experimentally, indicating that this model can be used to optimize rare-earth ions doping concentration and resonator cavity parameters of short-length fiber lasers.

Appendix A

Let T-Tc = θ, Eq. (7)-(11) are simplified as:

1rr[rθrzr]+2θrzz2=Qrzk,0rr1
1rr[rθrzr]+2θrzz2=0,r1rr2
kθrzr+cθrz=0,r=r2
θ1=θ2,θ1r=θ2r,r=r1
θrzz=0,z=0,z=L

Equation (18) can be solved as follows by separating the variables:

θ2rz=C0lnr+D0+m=1[CmI0(Lr)+DmK0(Lr)]cos(Lz)

Let θ 1 = t + θ 2(r 1,z), Eq. (17), Eq. (20) and Eq. (21) are written by:

2trzr2+1rtrzr+2trzz2=Qrzk2θ2r1zz2
trz=0,tr=θ2r,r=r1
tzrz=0,z=0,z=L
θ2r1z=C0lnr1+D0+m=1[CmI0(Lr1)+DmK0(Lr1)]cos(Lz)

According to the homogeneous boundary conditions, the solution of Eq. (23) can be obtained:

t=n=1m=0Anmcos(Lz)J0(μn(0)r1r)

Where Anm is a constant to be determined, which can be gotten by substituting Eq. (27) into Eq. (23) according to the orthogonality of cosine function and Bessel function of the first kind.

Appendix B

The steady-state heat equations of radial heat flow without longitudinal heat flow along fiber length under air cooling are given by:

1rr[rTrzr]=Qrzk,(0rr1)
1rr[rTrzr]=0,(r1rr2)

Boundary conditions satisfy the following relations:

kTrzr=c[TcTrz],r=r2
T1=T2,T1r=T2r,r=r1

According to the Taylor Series, the analytical solution is written as follows:

T1rz=Tc+αaη[Pp+(z)+Pp(z)]+αs[Ps+(z)+Ps(z)]4kπm=1(2)mm![(rr1)2m1m+2lnr1r22khr2](0rr1)
T2rz=Tc+αaη[Pp+(z)+Pp(z)]+αs[Ps+(z)+Ps(z)]2kπm=1(2)mm!(lnrr2khr2)(r1rr2)

Appendix C

As the active fiber was inserted into the copper tube, the steady-state heat equations of this model under air cooling are given by:

1rr[rTrzr]+2Trzz2=Qrzk,(0rr1)
1rr[rTrzr]+2Trzz2=0,(r1rr2)
2Trzr2+1rTrzr+2Trzz2=0,(r2rr3)

Boundary conditions satisfy the following relations:

kcuTrzr=c[TcTrz],r=r3
T2=T3,kT2r=kcuT3r,r=r2
T1=T2,T1r=T2r,r=r1
Trzz=0,z=0,z=L

With the same method, the analytical solution is written as follows:

T1rz=Tc+C0lnr1+D0+n=1m=0Anmcos(Lz)J0(μn(0)r1r)+m=1[CmI0(Lr1)+DmK0(Lr1)]cos(Lz)
T2rz=Tc+C0lnr+D0+m=1[CmI0(Lr)+DmK0(Lr)]cos(Lz)
T3rz=Tc+A0lnr+B0+m=1[AmI0(Lr)+BmK0(Lr)]cos(Lz)

Where

kcuA0r2=c(A0lnr3+B0)A0lnr2+B0=C0lnr2+D0
kcuA0=kC0C0r1=n=1An0μn(0)r1J1(μn(0))
kcu(L)[AmI1(Lr3)BmK1(Lr3)]=c[AmI0(Lr3)+BmK0(Lr3)]
AmI0(Lr2)+BmK0(Lr2)=CmI0(Lr2)+DmK0(Lr2)
kcu[AmI1(Lr2)BmK1(Lr2)]=k[CmI1(Lr2)DmK1(Lr2)]
(L)[CmI1(Lr1)DmK1(Lr1)]=n=1Anmμn(0)r1J1(μn(0))

r 3 is the copper tube radius, kcu denotes the copper tube thermal conductivity, T 3 is the temperatures in copper tube region.

Acknowledgment

This research was supported by the GuangDong Science and Technology Program (2005A10602001), the GuangZhou Science and Technology Program (2006Z2-D0161).

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References

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  1. J. T. Kringlebotn, P. R. Morkel, L. Reekie, J. L. Archambault, and D. N. Payne, “Efficient diode-pumped single frequency erbium: ytterbium fiber laser,” IEEE Photon. Technol. Lett. 5, 1162–1164 (1993).
    [Crossref]
  2. P. Polynkin, V. Temyanko, J. Moloney, and N. Peyghambarian, “Dramatic change of guiding properties in heavily Yb-doped, soft-glass active fibers caused by optical pumping,” Appl. Phys. Lett. 90, 2411061–2411063 (2007).
    [Crossref]
  3. J. Mayers, R. Wu, T. Chen, M. Myers, C. Hardy, J. Driver, and R. Tate, “New high-power rare-earth-doped fiber laser materials and architectures,” Proc. SPIE 4974, 177–183 (2003)
    [Crossref]
  4. W. L. Barnes, S. B. Poole, J. E. Townsend, L. Reekie, D. J. Taylor, and D. N. Payne, “Er3+-Yb3+ and Er3+ doped fiber lasers,” IEEE J. Lightwave Technol. 7, 1461–1465 (1989).
    [Crossref]
  5. V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90, 369–372 (2008)
    [Crossref]
  6. 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]
  7. M. Pollnau, “Analysis of Heat Generation and Thermal Lensing in Erbium 3-μm Lasers,” IEEE J. Quantum. Electron. 39, 350–357 (2003)
    [Crossref]
  8. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum. Electron. 37, 207–217 (2001).
    [Crossref]
  9. P. X. Li, C. Zhu, S. Z. Zou, H. Zhao, D. S. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40, 360–364 (2008).
    [Crossref]
  10. L. Li, H. Li, T. Qiu, V. L. Temyanko, M. M. Morrell, and A. Schulzgen, “3-Dimensional thermal analysis and active cooling of short-length high-power fiber lasers,” Opt. Express 13, 3420–3428 (2005).
    [Crossref] [PubMed]
  11. X. Huai and Z. G. Li, “Thermal stress analysis of Nd:YVO4 laser medium end pumped by a Gaussian beam,” Appl. Phys. Lett. 92, 1121–1122 (2008).
    [Crossref]
  12. M. Sabaeian, H. Nadgaran, and L. Mousave, “Analytical solution of the heat equation in a longitudinally pumped cubic solid-state laser,” Appl. Opt. 47, 2317–2325 (2008).
    [Crossref] [PubMed]
  13. F. Song, S. Liu, Z. H. Wu, H. Cai, X. Zhang, L. Teng, and J. G. Tian, “Determination of the thermal loading in laser-diode-pumped erbium-ytterbium-codoped phosphate glass microchip laser, ” J. Opt. Soc. Am. B 24, 2327–2332 (2007)
    [Crossref]
  14. S. Bjurshagen, J. E. Hellstrom, V. Pasiskevicius, M. C. Pujol, M. Aguilo, and F. D½az, “Fluorescence dynamics and rate equation analysis in Er3+ and Yb3+ doped double tungstates, ” Appl. Opt. 45, 4715–4725 (2006)
    [Crossref] [PubMed]
  15. F. Song, G. Y. Zhang, M. R. Shang, H. Tan, J. Yang, and F. Meng, “Three-photon phenomena in the upconversion luminescence of erbirum-ytterbium-codoped phosphate glass,” Appl. Phys. Lett. 79, 1748–1750 (2001)
    [Crossref]
  16. G. Karlsson, F. Laurell, J. Tellefsen, B. Denker, B. Galagan, V. Osiko, and S. Sverchkov, “Development and characterization of Yb-Er laser glass for high average power laser diode pumping,” Appl. Phys. B 75, 41–46 (2002)
    [Crossref]
  17. Z. H. Wu, F. Song, S. J. Liu, B. Qin, J. Su, J. G. Tian, and D. Y. Zhang, “Er3+, Yb3+ co-doped phosphate glass lasers, ” Acta Phys. Sin. 54, 5637–5641 (2005)
  18. M. Karasek, “Optimum design of Er3+-Yb3+ co-doped fibers for large-signal high-pump-power applications,” IEEE J. Quantum. Electron. 33, 1699–1705 (1997).
    [Crossref]
  19. D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
    [Crossref]
  20. C. E. Chryssou, F. D. Pasquale, and C. W. Pitt, “Improved gain performance in Yb3+-sensitized Er3+-doped alumina (Al2O3) channel optical waveguide amplifiers,” J. Lightwave Technol. 19, 345–349 (2001).
    [Crossref]
  21. I. Kelson and A. A. Hardy, “Strongly Pumped Fiber Lasers,” IEEE J. Quantum. Electron. 34, 1570–1577 (1998).
    [Crossref]
  22. Y. Wang, C. Q. Xu, and H. Po, “Analysis of Raman and thermal effects in kilowatt fiber lasers,” Opt. Commun. 242, 487–502 (2004).
    [Crossref]
  23. S. Bjurshagen and R. Koch, “Modeling of energy-transfer upconversion and thermal effects in end-pumped quasi-three-level lasers,” Appl. Opt. 43, 4753–4767 (2004).
    [Crossref] [PubMed]
  24. Y. P. Lan, Y. F. Chen, and S. C. Wang, “Repetition-rate dependence of thermal loading in diode-end-pumped Q-switched lasers: influence of energy-transfer upconversion,” Appl. Phys. B 71, 27–31 (2000)
  25. P. Myslinski, D. Nguyen, and J. Chrostowski, “Effects of concentration on the performance of erbium-doped fiber amplifiers,” IEEE J. Lightwave Technol. 15, 112–120 (1997).
    [Crossref]
  26. S. Taccheo, G. Sorbello, S. Longhi, and P. Laporta, “Measurement of the energy transfer and upconversion constants in Er-Yb-doped phosphate glass,” Opt. Quantum Electron. 31, 249–262 (1999).
    [Crossref]

2008 (4)

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90, 369–372 (2008)
[Crossref]

P. X. Li, C. Zhu, S. Z. Zou, H. Zhao, D. S. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40, 360–364 (2008).
[Crossref]

X. Huai and Z. G. Li, “Thermal stress analysis of Nd:YVO4 laser medium end pumped by a Gaussian beam,” Appl. Phys. Lett. 92, 1121–1122 (2008).
[Crossref]

M. Sabaeian, H. Nadgaran, and L. Mousave, “Analytical solution of the heat equation in a longitudinally pumped cubic solid-state laser,” Appl. Opt. 47, 2317–2325 (2008).
[Crossref] [PubMed]

2007 (2)

F. Song, S. Liu, Z. H. Wu, H. Cai, X. Zhang, L. Teng, and J. G. Tian, “Determination of the thermal loading in laser-diode-pumped erbium-ytterbium-codoped phosphate glass microchip laser, ” J. Opt. Soc. Am. B 24, 2327–2332 (2007)
[Crossref]

P. Polynkin, V. Temyanko, J. Moloney, and N. Peyghambarian, “Dramatic change of guiding properties in heavily Yb-doped, soft-glass active fibers caused by optical pumping,” Appl. Phys. Lett. 90, 2411061–2411063 (2007).
[Crossref]

2006 (1)

2005 (2)

Z. H. Wu, F. Song, S. J. Liu, B. Qin, J. Su, J. G. Tian, and D. Y. Zhang, “Er3+, Yb3+ co-doped phosphate glass lasers, ” Acta Phys. Sin. 54, 5637–5641 (2005)

L. Li, H. Li, T. Qiu, V. L. Temyanko, M. M. Morrell, and A. Schulzgen, “3-Dimensional thermal analysis and active cooling of short-length high-power fiber lasers,” Opt. Express 13, 3420–3428 (2005).
[Crossref] [PubMed]

2004 (2)

Y. Wang, C. Q. Xu, and H. Po, “Analysis of Raman and thermal effects in kilowatt fiber lasers,” Opt. Commun. 242, 487–502 (2004).
[Crossref]

S. Bjurshagen and R. Koch, “Modeling of energy-transfer upconversion and thermal effects in end-pumped quasi-three-level lasers,” Appl. Opt. 43, 4753–4767 (2004).
[Crossref] [PubMed]

2003 (2)

M. Pollnau, “Analysis of Heat Generation and Thermal Lensing in Erbium 3-μm Lasers,” IEEE J. Quantum. Electron. 39, 350–357 (2003)
[Crossref]

J. Mayers, R. Wu, T. Chen, M. Myers, C. Hardy, J. Driver, and R. Tate, “New high-power rare-earth-doped fiber laser materials and architectures,” Proc. SPIE 4974, 177–183 (2003)
[Crossref]

2002 (1)

G. Karlsson, F. Laurell, J. Tellefsen, B. Denker, B. Galagan, V. Osiko, and S. Sverchkov, “Development and characterization of Yb-Er laser glass for high average power laser diode pumping,” Appl. Phys. B 75, 41–46 (2002)
[Crossref]

2001 (3)

F. Song, G. Y. Zhang, M. R. Shang, H. Tan, J. Yang, and F. Meng, “Three-photon phenomena in the upconversion luminescence of erbirum-ytterbium-codoped phosphate glass,” Appl. Phys. Lett. 79, 1748–1750 (2001)
[Crossref]

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum. Electron. 37, 207–217 (2001).
[Crossref]

C. E. Chryssou, F. D. Pasquale, and C. W. Pitt, “Improved gain performance in Yb3+-sensitized Er3+-doped alumina (Al2O3) channel optical waveguide amplifiers,” J. Lightwave Technol. 19, 345–349 (2001).
[Crossref]

2000 (2)

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

Y. P. Lan, Y. F. Chen, and S. C. Wang, “Repetition-rate dependence of thermal loading in diode-end-pumped Q-switched lasers: influence of energy-transfer upconversion,” Appl. Phys. B 71, 27–31 (2000)

1999 (1)

S. Taccheo, G. Sorbello, S. Longhi, and P. Laporta, “Measurement of the energy transfer and upconversion constants in Er-Yb-doped phosphate glass,” Opt. Quantum Electron. 31, 249–262 (1999).
[Crossref]

1998 (2)

I. Kelson and A. A. Hardy, “Strongly Pumped Fiber Lasers,” IEEE J. Quantum. Electron. 34, 1570–1577 (1998).
[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]

1997 (2)

M. Karasek, “Optimum design of Er3+-Yb3+ co-doped fibers for large-signal high-pump-power applications,” IEEE J. Quantum. Electron. 33, 1699–1705 (1997).
[Crossref]

P. Myslinski, D. Nguyen, and J. Chrostowski, “Effects of concentration on the performance of erbium-doped fiber amplifiers,” IEEE J. Lightwave Technol. 15, 112–120 (1997).
[Crossref]

1993 (1)

J. T. Kringlebotn, P. R. Morkel, L. Reekie, J. L. Archambault, and D. N. Payne, “Efficient diode-pumped single frequency erbium: ytterbium fiber laser,” IEEE Photon. Technol. Lett. 5, 1162–1164 (1993).
[Crossref]

1989 (1)

W. L. Barnes, S. B. Poole, J. E. Townsend, L. Reekie, D. J. Taylor, and D. N. Payne, “Er3+-Yb3+ and Er3+ doped fiber lasers,” IEEE J. Lightwave Technol. 7, 1461–1465 (1989).
[Crossref]

Aguilo, M.

Archambault, J. L.

J. T. Kringlebotn, P. R. Morkel, L. Reekie, J. L. Archambault, and D. N. Payne, “Efficient diode-pumped single frequency erbium: ytterbium fiber laser,” IEEE Photon. Technol. Lett. 5, 1162–1164 (1993).
[Crossref]

Ballato, J.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90, 369–372 (2008)
[Crossref]

Barnes, W. L.

W. L. Barnes, S. B. Poole, J. E. Townsend, L. Reekie, D. J. Taylor, and D. N. Payne, “Er3+-Yb3+ and Er3+ doped fiber lasers,” IEEE J. Lightwave Technol. 7, 1461–1465 (1989).
[Crossref]

Bass, M.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90, 369–372 (2008)
[Crossref]

Bjurshagen, S.

Brown, D. C.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum. Electron. 37, 207–217 (2001).
[Crossref]

Cai, H.

Chen, M.

P. X. Li, C. Zhu, S. Z. Zou, H. Zhao, D. S. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40, 360–364 (2008).
[Crossref]

Chen, T.

J. Mayers, R. Wu, T. Chen, M. Myers, C. Hardy, J. Driver, and R. Tate, “New high-power rare-earth-doped fiber laser materials and architectures,” Proc. SPIE 4974, 177–183 (2003)
[Crossref]

Chen, Y.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90, 369–372 (2008)
[Crossref]

Chen, Y. F.

Y. P. Lan, Y. F. Chen, and S. C. Wang, “Repetition-rate dependence of thermal loading in diode-end-pumped Q-switched lasers: influence of energy-transfer upconversion,” Appl. Phys. B 71, 27–31 (2000)

Chrostowski, J.

P. Myslinski, D. Nguyen, and J. Chrostowski, “Effects of concentration on the performance of erbium-doped fiber amplifiers,” IEEE J. Lightwave Technol. 15, 112–120 (1997).
[Crossref]

Chryssou, C. E.

Clarkson, W. 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]

D½az, F.

Denker, B.

G. Karlsson, F. Laurell, J. Tellefsen, B. Denker, B. Galagan, V. Osiko, and S. Sverchkov, “Development and characterization of Yb-Er laser glass for high average power laser diode pumping,” Appl. Phys. B 75, 41–46 (2002)
[Crossref]

Driver, J.

J. Mayers, R. Wu, T. Chen, M. Myers, C. Hardy, J. Driver, and R. Tate, “New high-power rare-earth-doped fiber laser materials and architectures,” Proc. SPIE 4974, 177–183 (2003)
[Crossref]

Fontaine, N.

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

Funk, D. S.

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

Galagan, B.

G. Karlsson, F. Laurell, J. Tellefsen, B. Denker, B. Galagan, V. Osiko, and S. Sverchkov, “Development and characterization of Yb-Er laser glass for high average power laser diode pumping,” Appl. Phys. B 75, 41–46 (2002)
[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]

Hardy, A. A.

I. Kelson and A. A. Hardy, “Strongly Pumped Fiber Lasers,” IEEE J. Quantum. Electron. 34, 1570–1577 (1998).
[Crossref]

Hardy, C.

J. Mayers, R. Wu, T. Chen, M. Myers, C. Hardy, J. Driver, and R. Tate, “New high-power rare-earth-doped fiber laser materials and architectures,” Proc. SPIE 4974, 177–183 (2003)
[Crossref]

Hayden, J. S.

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

Hellstrom, J. E.

Hoffman, H. J.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum. Electron. 37, 207–217 (2001).
[Crossref]

Huai, X.

X. Huai and Z. G. Li, “Thermal stress analysis of Nd:YVO4 laser medium end pumped by a Gaussian beam,” Appl. Phys. Lett. 92, 1121–1122 (2008).
[Crossref]

Jiang, D. S.

P. X. Li, C. Zhu, S. Z. Zou, H. Zhao, D. S. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40, 360–364 (2008).
[Crossref]

Karasek, M.

M. Karasek, “Optimum design of Er3+-Yb3+ co-doped fibers for large-signal high-pump-power applications,” IEEE J. Quantum. Electron. 33, 1699–1705 (1997).
[Crossref]

Karlsson, G.

G. Karlsson, F. Laurell, J. Tellefsen, B. Denker, B. Galagan, V. Osiko, and S. Sverchkov, “Development and characterization of Yb-Er laser glass for high average power laser diode pumping,” Appl. Phys. B 75, 41–46 (2002)
[Crossref]

Kelson, I.

I. Kelson and A. A. Hardy, “Strongly Pumped Fiber Lasers,” IEEE J. Quantum. Electron. 34, 1570–1577 (1998).
[Crossref]

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]

Koch, R.

Kringlebotn, J. T.

J. T. Kringlebotn, P. R. Morkel, L. Reekie, J. L. Archambault, and D. N. Payne, “Efficient diode-pumped single frequency erbium: ytterbium fiber laser,” IEEE Photon. Technol. Lett. 5, 1162–1164 (1993).
[Crossref]

Lan, Y. P.

Y. P. Lan, Y. F. Chen, and S. C. Wang, “Repetition-rate dependence of thermal loading in diode-end-pumped Q-switched lasers: influence of energy-transfer upconversion,” Appl. Phys. B 71, 27–31 (2000)

Laporta, P.

S. Taccheo, G. Sorbello, S. Longhi, and P. Laporta, “Measurement of the energy transfer and upconversion constants in Er-Yb-doped phosphate glass,” Opt. Quantum Electron. 31, 249–262 (1999).
[Crossref]

Laurell, F.

G. Karlsson, F. Laurell, J. Tellefsen, B. Denker, B. Galagan, V. Osiko, and S. Sverchkov, “Development and characterization of Yb-Er laser glass for high average power laser diode pumping,” Appl. Phys. B 75, 41–46 (2002)
[Crossref]

Li, G.

P. X. Li, C. Zhu, S. Z. Zou, H. Zhao, D. S. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40, 360–364 (2008).
[Crossref]

Li, H.

Li, L.

Li, P. X.

P. X. Li, C. Zhu, S. Z. Zou, H. Zhao, D. S. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40, 360–364 (2008).
[Crossref]

Li, Z. G.

X. Huai and Z. G. Li, “Thermal stress analysis of Nd:YVO4 laser medium end pumped by a Gaussian beam,” Appl. Phys. Lett. 92, 1121–1122 (2008).
[Crossref]

Liu, S.

Liu, S. J.

Z. H. Wu, F. Song, S. J. Liu, B. Qin, J. Su, J. G. Tian, and D. Y. Zhang, “Er3+, Yb3+ co-doped phosphate glass lasers, ” Acta Phys. Sin. 54, 5637–5641 (2005)

Liu, W. C.

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

Longhi, S.

S. Taccheo, G. Sorbello, S. Longhi, and P. Laporta, “Measurement of the energy transfer and upconversion constants in Er-Yb-doped phosphate glass,” Opt. Quantum Electron. 31, 249–262 (1999).
[Crossref]

Mayers, J.

J. Mayers, R. Wu, T. Chen, M. Myers, C. Hardy, J. Driver, and R. Tate, “New high-power rare-earth-doped fiber laser materials and architectures,” Proc. SPIE 4974, 177–183 (2003)
[Crossref]

Mccomb, T.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90, 369–372 (2008)
[Crossref]

Meng, F.

F. Song, G. Y. Zhang, M. R. Shang, H. Tan, J. Yang, and F. Meng, “Three-photon phenomena in the upconversion luminescence of erbirum-ytterbium-codoped phosphate glass,” Appl. Phys. Lett. 79, 1748–1750 (2001)
[Crossref]

Moloney, J.

P. Polynkin, V. Temyanko, J. Moloney, and N. Peyghambarian, “Dramatic change of guiding properties in heavily Yb-doped, soft-glass active fibers caused by optical pumping,” Appl. Phys. Lett. 90, 2411061–2411063 (2007).
[Crossref]

Morkel, P. R.

J. T. Kringlebotn, P. R. Morkel, L. Reekie, J. L. Archambault, and D. N. Payne, “Efficient diode-pumped single frequency erbium: ytterbium fiber laser,” IEEE Photon. Technol. Lett. 5, 1162–1164 (1993).
[Crossref]

Morrell, M. M.

Mousave, L.

Myers, M.

J. Mayers, R. Wu, T. Chen, M. Myers, C. Hardy, J. Driver, and R. Tate, “New high-power rare-earth-doped fiber laser materials and architectures,” Proc. SPIE 4974, 177–183 (2003)
[Crossref]

Myslinski, P.

P. Myslinski, D. Nguyen, and J. Chrostowski, “Effects of concentration on the performance of erbium-doped fiber amplifiers,” IEEE J. Lightwave Technol. 15, 112–120 (1997).
[Crossref]

Nadgaran, H.

Nguyen, D.

P. Myslinski, D. Nguyen, and J. Chrostowski, “Effects of concentration on the performance of erbium-doped fiber amplifiers,” IEEE J. Lightwave Technol. 15, 112–120 (1997).
[Crossref]

Obarski, G. E.

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

Osiko, V.

G. Karlsson, F. Laurell, J. Tellefsen, B. Denker, B. Galagan, V. Osiko, and S. Sverchkov, “Development and characterization of Yb-Er laser glass for high average power laser diode pumping,” Appl. Phys. B 75, 41–46 (2002)
[Crossref]

Pasiskevicius, V.

Pasquale, F. D.

Payne, D. N.

J. T. Kringlebotn, P. R. Morkel, L. Reekie, J. L. Archambault, and D. N. Payne, “Efficient diode-pumped single frequency erbium: ytterbium fiber laser,” IEEE Photon. Technol. Lett. 5, 1162–1164 (1993).
[Crossref]

W. L. Barnes, S. B. Poole, J. E. Townsend, L. Reekie, D. J. Taylor, and D. N. Payne, “Er3+-Yb3+ and Er3+ doped fiber lasers,” IEEE J. Lightwave Technol. 7, 1461–1465 (1989).
[Crossref]

Peskin, A. P.

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

Peters, P. M.

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

Peyghambarian, N.

P. Polynkin, V. Temyanko, J. Moloney, and N. Peyghambarian, “Dramatic change of guiding properties in heavily Yb-doped, soft-glass active fibers caused by optical pumping,” Appl. Phys. Lett. 90, 2411061–2411063 (2007).
[Crossref]

Pitt, C. W.

Po, H.

Y. Wang, C. Q. Xu, and H. Po, “Analysis of Raman and thermal effects in kilowatt fiber lasers,” Opt. Commun. 242, 487–502 (2004).
[Crossref]

Pollnau, M.

M. Pollnau, “Analysis of Heat Generation and Thermal Lensing in Erbium 3-μm Lasers,” IEEE J. Quantum. Electron. 39, 350–357 (2003)
[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]

Polynkin, P.

P. Polynkin, V. Temyanko, J. Moloney, and N. Peyghambarian, “Dramatic change of guiding properties in heavily Yb-doped, soft-glass active fibers caused by optical pumping,” Appl. Phys. Lett. 90, 2411061–2411063 (2007).
[Crossref]

Poole, S. B.

W. L. Barnes, S. B. Poole, J. E. Townsend, L. Reekie, D. J. Taylor, and D. N. Payne, “Er3+-Yb3+ and Er3+ doped fiber lasers,” IEEE J. Lightwave Technol. 7, 1461–1465 (1989).
[Crossref]

Pujol, M. C.

Qin, B.

Z. H. Wu, F. Song, S. J. Liu, B. Qin, J. Su, J. G. Tian, and D. Y. Zhang, “Er3+, Yb3+ co-doped phosphate glass lasers, ” Acta Phys. Sin. 54, 5637–5641 (2005)

Qiu, T.

Reekie, L.

J. T. Kringlebotn, P. R. Morkel, L. Reekie, J. L. Archambault, and D. N. Payne, “Efficient diode-pumped single frequency erbium: ytterbium fiber laser,” IEEE Photon. Technol. Lett. 5, 1162–1164 (1993).
[Crossref]

W. L. Barnes, S. B. Poole, J. E. Townsend, L. Reekie, D. J. Taylor, and D. N. Payne, “Er3+-Yb3+ and Er3+ doped fiber lasers,” IEEE J. Lightwave Technol. 7, 1461–1465 (1989).
[Crossref]

Richardson, M.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90, 369–372 (2008)
[Crossref]

Sabaeian, M.

Sanford, N. A.

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

Schulzgen, A.

Shang, M. R.

F. Song, G. Y. Zhang, M. R. Shang, H. Tan, J. Yang, and F. Meng, “Three-photon phenomena in the upconversion luminescence of erbirum-ytterbium-codoped phosphate glass,” Appl. Phys. Lett. 79, 1748–1750 (2001)
[Crossref]

Siegman, A. E.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90, 369–372 (2008)
[Crossref]

Song, F.

F. Song, S. Liu, Z. H. Wu, H. Cai, X. Zhang, L. Teng, and J. G. Tian, “Determination of the thermal loading in laser-diode-pumped erbium-ytterbium-codoped phosphate glass microchip laser, ” J. Opt. Soc. Am. B 24, 2327–2332 (2007)
[Crossref]

Z. H. Wu, F. Song, S. J. Liu, B. Qin, J. Su, J. G. Tian, and D. Y. Zhang, “Er3+, Yb3+ co-doped phosphate glass lasers, ” Acta Phys. Sin. 54, 5637–5641 (2005)

F. Song, G. Y. Zhang, M. R. Shang, H. Tan, J. Yang, and F. Meng, “Three-photon phenomena in the upconversion luminescence of erbirum-ytterbium-codoped phosphate glass,” Appl. Phys. Lett. 79, 1748–1750 (2001)
[Crossref]

Sorbello, G.

S. Taccheo, G. Sorbello, S. Longhi, and P. Laporta, “Measurement of the energy transfer and upconversion constants in Er-Yb-doped phosphate glass,” Opt. Quantum Electron. 31, 249–262 (1999).
[Crossref]

Su, J.

Z. H. Wu, F. Song, S. J. Liu, B. Qin, J. Su, J. G. Tian, and D. Y. Zhang, “Er3+, Yb3+ co-doped phosphate glass lasers, ” Acta Phys. Sin. 54, 5637–5641 (2005)

Sudesh, V.

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90, 369–372 (2008)
[Crossref]

Sverchkov, S.

G. Karlsson, F. Laurell, J. Tellefsen, B. Denker, B. Galagan, V. Osiko, and S. Sverchkov, “Development and characterization of Yb-Er laser glass for high average power laser diode pumping,” Appl. Phys. B 75, 41–46 (2002)
[Crossref]

Taccheo, S.

S. Taccheo, G. Sorbello, S. Longhi, and P. Laporta, “Measurement of the energy transfer and upconversion constants in Er-Yb-doped phosphate glass,” Opt. Quantum Electron. 31, 249–262 (1999).
[Crossref]

Tan, H.

F. Song, G. Y. Zhang, M. R. Shang, H. Tan, J. Yang, and F. Meng, “Three-photon phenomena in the upconversion luminescence of erbirum-ytterbium-codoped phosphate glass,” Appl. Phys. Lett. 79, 1748–1750 (2001)
[Crossref]

Tate, R.

J. Mayers, R. Wu, T. Chen, M. Myers, C. Hardy, J. Driver, and R. Tate, “New high-power rare-earth-doped fiber laser materials and architectures,” Proc. SPIE 4974, 177–183 (2003)
[Crossref]

Taylor, D. J.

W. L. Barnes, S. B. Poole, J. E. Townsend, L. Reekie, D. J. Taylor, and D. N. Payne, “Er3+-Yb3+ and Er3+ doped fiber lasers,” IEEE J. Lightwave Technol. 7, 1461–1465 (1989).
[Crossref]

Tellefsen, J.

G. Karlsson, F. Laurell, J. Tellefsen, B. Denker, B. Galagan, V. Osiko, and S. Sverchkov, “Development and characterization of Yb-Er laser glass for high average power laser diode pumping,” Appl. Phys. B 75, 41–46 (2002)
[Crossref]

Temyanko, V.

P. Polynkin, V. Temyanko, J. Moloney, and N. Peyghambarian, “Dramatic change of guiding properties in heavily Yb-doped, soft-glass active fibers caused by optical pumping,” Appl. Phys. Lett. 90, 2411061–2411063 (2007).
[Crossref]

Temyanko, V. L.

Teng, L.

Tian, J. G.

F. Song, S. Liu, Z. H. Wu, H. Cai, X. Zhang, L. Teng, and J. G. Tian, “Determination of the thermal loading in laser-diode-pumped erbium-ytterbium-codoped phosphate glass microchip laser, ” J. Opt. Soc. Am. B 24, 2327–2332 (2007)
[Crossref]

Z. H. Wu, F. Song, S. J. Liu, B. Qin, J. Su, J. G. Tian, and D. Y. Zhang, “Er3+, Yb3+ co-doped phosphate glass lasers, ” Acta Phys. Sin. 54, 5637–5641 (2005)

Townsend, J. E.

W. L. Barnes, S. B. Poole, J. E. Townsend, L. Reekie, D. J. Taylor, and D. N. Payne, “Er3+-Yb3+ and Er3+ doped fiber lasers,” IEEE J. Lightwave Technol. 7, 1461–1465 (1989).
[Crossref]

Veasey, D. L.

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

Walter, S.N. H.

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

Wang, S. C.

Y. P. Lan, Y. F. Chen, and S. C. Wang, “Repetition-rate dependence of thermal loading in diode-end-pumped Q-switched lasers: influence of energy-transfer upconversion,” Appl. Phys. B 71, 27–31 (2000)

Wang, Y.

Y. Wang, C. Q. Xu, and H. Po, “Analysis of Raman and thermal effects in kilowatt fiber lasers,” Opt. Commun. 242, 487–502 (2004).
[Crossref]

Wu, R.

J. Mayers, R. Wu, T. Chen, M. Myers, C. Hardy, J. Driver, and R. Tate, “New high-power rare-earth-doped fiber laser materials and architectures,” Proc. SPIE 4974, 177–183 (2003)
[Crossref]

Wu, Z. H.

F. Song, S. Liu, Z. H. Wu, H. Cai, X. Zhang, L. Teng, and J. G. Tian, “Determination of the thermal loading in laser-diode-pumped erbium-ytterbium-codoped phosphate glass microchip laser, ” J. Opt. Soc. Am. B 24, 2327–2332 (2007)
[Crossref]

Z. H. Wu, F. Song, S. J. Liu, B. Qin, J. Su, J. G. Tian, and D. Y. Zhang, “Er3+, Yb3+ co-doped phosphate glass lasers, ” Acta Phys. Sin. 54, 5637–5641 (2005)

Xu, C. Q.

Y. Wang, C. Q. Xu, and H. Po, “Analysis of Raman and thermal effects in kilowatt fiber lasers,” Opt. Commun. 242, 487–502 (2004).
[Crossref]

Yang, J.

F. Song, G. Y. Zhang, M. R. Shang, H. Tan, J. Yang, and F. Meng, “Three-photon phenomena in the upconversion luminescence of erbirum-ytterbium-codoped phosphate glass,” Appl. Phys. Lett. 79, 1748–1750 (2001)
[Crossref]

Young, M.

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

Zhang, D. Y.

Z. H. Wu, F. Song, S. J. Liu, B. Qin, J. Su, J. G. Tian, and D. Y. Zhang, “Er3+, Yb3+ co-doped phosphate glass lasers, ” Acta Phys. Sin. 54, 5637–5641 (2005)

Zhang, G. Y.

F. Song, G. Y. Zhang, M. R. Shang, H. Tan, J. Yang, and F. Meng, “Three-photon phenomena in the upconversion luminescence of erbirum-ytterbium-codoped phosphate glass,” Appl. Phys. Lett. 79, 1748–1750 (2001)
[Crossref]

Zhang, X.

Zhao, H.

P. X. Li, C. Zhu, S. Z. Zou, H. Zhao, D. S. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40, 360–364 (2008).
[Crossref]

Zhu, C.

P. X. Li, C. Zhu, S. Z. Zou, H. Zhao, D. S. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40, 360–364 (2008).
[Crossref]

Zou, S. Z.

P. X. Li, C. Zhu, S. Z. Zou, H. Zhao, D. S. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40, 360–364 (2008).
[Crossref]

Acta Phys. Sin. (1)

Z. H. Wu, F. Song, S. J. Liu, B. Qin, J. Su, J. G. Tian, and D. Y. Zhang, “Er3+, Yb3+ co-doped phosphate glass lasers, ” Acta Phys. Sin. 54, 5637–5641 (2005)

Appl. Opt. (3)

Appl. Phys. B (3)

Y. P. Lan, Y. F. Chen, and S. C. Wang, “Repetition-rate dependence of thermal loading in diode-end-pumped Q-switched lasers: influence of energy-transfer upconversion,” Appl. Phys. B 71, 27–31 (2000)

G. Karlsson, F. Laurell, J. Tellefsen, B. Denker, B. Galagan, V. Osiko, and S. Sverchkov, “Development and characterization of Yb-Er laser glass for high average power laser diode pumping,” Appl. Phys. B 75, 41–46 (2002)
[Crossref]

V. Sudesh, T. Mccomb, Y. Chen, M. Bass, M. Richardson, J. Ballato, and A. E. Siegman, “Diode-pumped 200μm diameter core, gain-guided, index-antiguided single mode fiber laser,” Appl. Phys. B 90, 369–372 (2008)
[Crossref]

Appl. Phys. Lett. (3)

P. Polynkin, V. Temyanko, J. Moloney, and N. Peyghambarian, “Dramatic change of guiding properties in heavily Yb-doped, soft-glass active fibers caused by optical pumping,” Appl. Phys. Lett. 90, 2411061–2411063 (2007).
[Crossref]

X. Huai and Z. G. Li, “Thermal stress analysis of Nd:YVO4 laser medium end pumped by a Gaussian beam,” Appl. Phys. Lett. 92, 1121–1122 (2008).
[Crossref]

F. Song, G. Y. Zhang, M. R. Shang, H. Tan, J. Yang, and F. Meng, “Three-photon phenomena in the upconversion luminescence of erbirum-ytterbium-codoped phosphate glass,” Appl. Phys. Lett. 79, 1748–1750 (2001)
[Crossref]

IEEE J. Lightwave Technol. (2)

W. L. Barnes, S. B. Poole, J. E. Townsend, L. Reekie, D. J. Taylor, and D. N. Payne, “Er3+-Yb3+ and Er3+ doped fiber lasers,” IEEE J. Lightwave Technol. 7, 1461–1465 (1989).
[Crossref]

P. Myslinski, D. Nguyen, and J. Chrostowski, “Effects of concentration on the performance of erbium-doped fiber amplifiers,” IEEE J. Lightwave Technol. 15, 112–120 (1997).
[Crossref]

IEEE J. Quantum. Electron. (4)

I. Kelson and A. A. Hardy, “Strongly Pumped Fiber Lasers,” IEEE J. Quantum. Electron. 34, 1570–1577 (1998).
[Crossref]

M. Pollnau, “Analysis of Heat Generation and Thermal Lensing in Erbium 3-μm Lasers,” IEEE J. Quantum. Electron. 39, 350–357 (2003)
[Crossref]

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum. Electron. 37, 207–217 (2001).
[Crossref]

M. Karasek, “Optimum design of Er3+-Yb3+ co-doped fibers for large-signal high-pump-power applications,” IEEE J. Quantum. Electron. 33, 1699–1705 (1997).
[Crossref]

IEEE Photon. Technol. Lett. (1)

J. T. Kringlebotn, P. R. Morkel, L. Reekie, J. L. Archambault, and D. N. Payne, “Efficient diode-pumped single frequency erbium: ytterbium fiber laser,” IEEE Photon. Technol. Lett. 5, 1162–1164 (1993).
[Crossref]

J. Lightwave Technol. (1)

J. Non-Cryst. Silids (1)

D. L. Veasey, D. S. Funk, P. M. Peters, N. A. Sanford, G. E. Obarski, N. Fontaine, M. Young, A. P. Peskin, W. C. Liu, S.N. H. Walter, and J. S. Hayden, “ Yb/Er-codoped and Yb-doped waveguide lasers in phosphate glass,” J. Non-Cryst. Silids 263, 369–381 (2000)
[Crossref]

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

Opt. Commun. (1)

Y. Wang, C. Q. Xu, and H. Po, “Analysis of Raman and thermal effects in kilowatt fiber lasers,” Opt. Commun. 242, 487–502 (2004).
[Crossref]

Opt. Express (1)

Opt. Laser Technol. (1)

P. X. Li, C. Zhu, S. Z. Zou, H. Zhao, D. S. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40, 360–364 (2008).
[Crossref]

Opt. Quantum Electron. (1)

S. Taccheo, G. Sorbello, S. Longhi, and P. Laporta, “Measurement of the energy transfer and upconversion constants in Er-Yb-doped phosphate glass,” Opt. Quantum Electron. 31, 249–262 (1999).
[Crossref]

Phys. Rev. B (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]

Proc. SPIE (1)

J. Mayers, R. Wu, T. Chen, M. Myers, C. Hardy, J. Driver, and R. Tate, “New high-power rare-earth-doped fiber laser materials and architectures,” Proc. SPIE 4974, 177–183 (2003)
[Crossref]

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Figures (13)

Fig. 1.
Fig. 1. Energy-level scheme of the Er3+/Yb3+ co-doped system including ETU processes: UC1, UC2, cumulative upconversion; UC3, cooperative upconversion; M, multiphonon relaxation; F, fluorescence process; L, 1535 nm laser emission.
Fig. 2.
Fig. 2. Schematic illustration of phosphate fiber laser
Fig. 3.
Fig. 3. Pump and signal powers as a function of the position along the fiber length: (a) pump power along the fiber length; (b) signal power in positive and negative directions along the fiber length.
Fig. 4.
Fig. 4. The theoretical calculation and experimental results on signal laser output versus pump power. The best fit parameters k1, k2, and Cup are 2.56×10-22 m3 s-1 , 0.85×10-22 m3 s-1, 1.67×10-24 m3 s-1, respectively.
Fig. 5.
Fig. 5. Temperature field distribution in phosphate fiber
Fig. 6.
Fig. 6. The end-surface temperature distribution along active fiber radial coordinate: (a) without considering longitudinal heat conduction; (b) with longitudinal heat conduction.
Fig. 7.
Fig. 7. The end-surface temperature distribution along radial coordinate when the pump power is 100 mW: (a) with upconversion; (b) without upconversion.
Fig. 8.
Fig. 8. The maximum temperatures in active fiber core vs. pump power with and without considering ETU.
Fig. 9.
Fig. 9. The fractional thermal loading at different pump powers.
Fig. 10.
Fig. 10. The fiber cross-section inside the tube
Fig. 11.
Fig. 11. The signal vs. absorbed pump power plot
Fig. 12.
Fig. 12. The inner surface temperature of the copper tube at different pump power.
Fig. 13.
Fig. 13. Calculated values vs. experimental values at different pump power.

Tables (2)

Tables Icon

Table 1. The related parameters used in this paper

Tables Icon

Table 2. Nonradiative transition rate and total spontaneous radiative transitions rate of Er3+ upconversion levels

Equations (61)

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

UC1:Yb3+(F522)+Er3+(I1124)Yb3+(F722)+Er3+(F724)
UC2:Yb3+(F522)+Er3+(I1324)Yb3+(F722)+Er3+(F924)
UC3:Er3+(I1324)+Er3+(I1324)Er3+(I1524)+Er3+(I924)
N2yt=W12yN1yW21yN2yN2yτyk1N2yN1ek2N2yN2e=0
N2et=W12eN1e+A32eN3eW21eN2ek2N2eN2yN2eτe2CupN2e2=0
N3et=k1N1eN2y+A43eN4eA32eN3e=0
N4et=CupN2e2A43eN4e=0
NEr=N1e+N2e+N3e+N4e
NYb=N1y+N2y
W12ezt=Γsσas(υs)Ps(z)hυsAcoreW21ezt=Γsσes(υs)Ps(z)hυsAcore
W12yzt=Γpσap(υp)Pp(z)hυpAcoreW21yzt=Γpσep(υp)Pp(z)hυpAcore
1rr[rTrzr]+2Trzz2=Qrzk,0rr1
1rr[rTrzr]+2Trzz2=0,r1rr2
kTrzr=c[TcTrz],r=r2
T1=T2,T1r=T2r,r=r1
Trzz=0,z=0,z=L
Qrz=2(αaη+αps)[Pp+(z)+Pp(z)]+2αs[Ps+(z)+Ps(z)]πωp2exp(2r2ωp2)
T1rz=Tc+C0lnr1+D0+n=1m=1Anmcos(Lz)J0(μn(0)r1r)+m=1[CmI0(Lr1)+DmK0(Lr1)]cos(Lz)
T2rz=Tc+C0lnr+D0+m=1[CmI0(Lr)+DmK0(Lr)]cos(Lz)
kC0r2=c(C0lnr2+D0),C0r1=n=1An0μn(0)r1 J1 (μn(0))
k(L)[CmI1(Lr2)DmK1(Lr2)]=c[CmI0(Lr2)+DmK0(Lr2)]
(L)[CmI1(Lr1)DmK1(Lr1)]=n=1Anmμn(0)r1J1(μn(0))
An0=4kL(μn(0))2J12(μn(0))0r1exp(2r2ωp2)J0(μn(0)r1r)rdr×
0L (αaη+αps)[Pp+(z)+Pp(z)]+αs[Ps+(z)+Ps(z)]πωp2 dz
Anm=8Lk[m2π2r12+L2(μn(0))2]J12(μn(0))0L(αaη+αps)[Pp+(z)+Pp(z)]+αs[Ps+(z)+Ps(z)]πωp2cos(Lz)dz×
0r1exp(2r2ωp2)J0(μn(0)r1r)rdr2r12()2[m2π2r12+L2(μn(0))2]J1(μn(0))μn(0)[CmI0(Lr1)+DmK0(Lr1)]
FETU=(n0n)n0
η=FETU+(1FETU)(1λpλs)
1rr[rθrzr]+2θrzz2=Qrzk,0rr1
1rr[rθrzr]+2θrzz2=0,r1rr2
kθrzr+cθrz=0,r=r2
θ1=θ2,θ1r=θ2r,r=r1
θrzz=0,z=0,z=L
θ2rz=C0lnr+D0+m=1[CmI0(Lr)+DmK0(Lr)]cos(Lz)
2trzr2+1rtrzr+2trzz2=Qrzk2θ2r1zz2
trz=0,tr=θ2r,r=r1
tzrz=0,z=0,z=L
θ2r1z=C0lnr1+D0+m=1[CmI0(Lr1)+DmK0(Lr1)]cos(Lz)
t=n=1m=0Anmcos(Lz)J0(μn(0)r1r)
1rr[rTrzr]=Qrzk , (0rr1)
1rr[rTrzr]=0,(r1rr2)
kTrzr=c[TcTrz],r=r2
T1=T2,T1r=T2r,r=r1
T1rz=Tc+αaη[Pp+(z)+Pp(z)]+αs[Ps+(z)+Ps(z)]4kπm=1(2)mm![(rr1)2m1m+2lnr1r22khr2](0rr1)
T2rz=Tc+αaη[Pp+(z)+Pp(z)]+αs[Ps+(z)+Ps(z)]2kπm=1(2)mm!(lnrr2khr2)(r1rr2)
1rr[rTrzr]+2Trzz2=Qrzk,(0rr1 )
1rr[rTrzr]+2Trzz2=0,(r1rr2 )
2Trzr2+1rTrzr+2Trzz2=0,(r2rr3)
kcuTrzr=c[TcTrz],r=r3
T2=T3,kT2r=kcuT3r,r=r2
T1=T2,T1r=T2r,r=r1
Trzz=0,z=0,z=L
T1rz=Tc+C0lnr1+D0+n=1m=0Anmcos(Lz)J0(μn(0)r1r)+m=1[CmI0(Lr1)+DmK0(Lr1)]cos(Lz)
T2rz=Tc+C0lnr+D0+m=1[CmI0(Lr)+DmK0(Lr)]cos(Lz)
T3rz=Tc+A0lnr+B0+m=1[AmI0(Lr)+BmK0(Lr)]cos(Lz)
kcuA0r2=c(A0lnr3+B0)A0lnr2+B0=C0lnr2+D0
kcuA0=kC0C0r1=n=1An0μn(0)r1J1(μn(0))
kcu(L)[AmI1(Lr3)BmK1(Lr3)]=c[AmI0(Lr3)+BmK0(Lr3)]
AmI0(Lr2)+BmK0(Lr2)=CmI0(Lr2)+DmK0(Lr2)
kcu[AmI1(Lr2)BmK1(Lr2)]=k[CmI1(Lr2)DmK1(Lr2)]
(L)[CmI1(Lr1)DmK1(Lr1)]=n=1Anmμn(0)r1J1(μn(0))

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