3-Dimensional heat analysis in short-length Er<sup>3+</sup>/Yb<sup>3+</sup> co-doped phosphate fiber laser with upconversion

T. Liu; Z. M. Yang; S. H. Xu

doi:10.1364/OE.17.000235

Optics Express
Vol. 17,
Issue 1,
pp. 235-247
(2009)
•https://doi.org/10.1364/OE.17.000235

3-Dimensional heat analysis in short-length Er³⁺/Yb³⁺ co-doped phosphate fiber laser with upconversion

T. Liu, Z. M. Yang, and S. H. Xu

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T. Liu, Z. M. Yang, and S. H. Xu, "3-Dimensional heat analysis in short-length Er³⁺/Yb³⁺ co-doped phosphate fiber laser with upconversion," Opt. Express 17, 235-247 (2009)

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About this Article
History
- Original Manuscript: November 7, 2008
- Revised Manuscript: December 21, 2008
- Manuscript Accepted: December 21, 2008
- Published: December 24, 2008

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.

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 ⁴I_11/2 level to the laser level ⁴I_13/2 [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:

{UC}_{1} : {Yb}^{3 +} (\overset{2}{F_{\frac{5}{2}}}) + {Er}^{3 +} (\overset{4}{I_{\frac{11}{2}}}) \to {Yb}^{3 +} (\overset{2}{F_{\frac{7}{2}}}) + {Er}^{3 +} (\overset{4}{F_{\frac{7}{2}}})

For all the upconversion processes above, a large part of populations on the upconversion level decays to the level ⁴I_13/2 by multiphonon relaxation step by step, generating heat in the active fiber. Because the fast nonradiative multiphonon relaxation from the ⁴I_11/2 to the ⁴I_13/2 level greatly decreases the back-energy transfer from Er³⁺: ⁴I_11/2 to Yb³⁺: ²F_7/2 and the upconversion losses occurring at the ⁴I_11/2 level of Er³⁺ ions [16], we neglect the back-energy transfer process and upconversion losses at this level in the calculation.

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 ⁴F_7/2, ²H_11/2, ⁴S_3/2 and ⁴F_9/2 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.

\frac{\partial N_{2 y}}{\partial t} = \frac{W_{12 y} N_{1 y} - W_{21 y} N_{2 y} - N_{2 y}}{τ_{y} - k_{1} N_{2 y} N_{1 e} - k_{2} N_{2 y} N_{2 e}} = 0

\frac{\partial N_{2 e}}{\partial t} = \frac{W_{12 e} N_{1 e} + A_{32 e} N_{3 e} - W_{21 e} N_{2 e} - k_{2} N_{2 e} N_{2 y} - N_{2 e}}{τ_{e} - 2 C_{up} N_{2 e}^{2}} = 0

\frac{\partial N_{3 e}}{\partial t} = k_{1} N_{1 e} N_{2 y} + A_{43 e} N_{4 e} - A_{32 e} N_{3 e} = 0

\frac{\partial N_{4 e}}{\partial t} = C_{up} N_{2 e}^{2} - A_{43 e} N_{4 e} = 0

N_{Er} = N_{1 e} + N_{2 e} + N_{3 e} + N_{4 e}

N_{Yb} = N_{1 y} + N_{2 y}

The signal absorption, signal emission, pump absorption and pump emission rates W _12e , W _21e , W _12y, and W _21y are given by:

W_{12 e} (z, t) = \frac{Γ_{s} σ_{as} {(υ_{s}) P}_{s} (z)}{h υ_{s} A_{core}} W_{21 e} (z, t) = \frac{Γ_{s} σ_{es} (υ_{s}) P_{s} (z)}{h υ_{s} A_{core}}

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

Table 1. The related parameters used in this paper

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3. Resolving of 3-dimensional temperature field

Figure 2 is the schematic illustration of Er³⁺/Yb³⁺ co-doped phosphate fiber laser. P_p ^± (z) and P_s ^± (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.

Fig. 2. Schematic illustration of phosphate fiber laser

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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]:

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial T (r, z)}{\partial r}] + \frac{\partial^{2} T (r, z)}{\partial z^{2}} = - \frac{Q (r, z)}{k}, 0 \leq r \leq r_{1}

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial T (r, z)}{\partial r}] + \frac{\partial^{2} T (r, z)}{\partial z^{2}} = 0, r_{1} \leq r \leq r_{2}

\frac{k \partial T (r, z)}{\partial r} = c [T_{c} - T (r, z)], r = r_{2}

T_{1} = T_{2}, \frac{\partial T_{1}}{\partial r} = \frac{\partial T_{2}}{\partial r}, r = r_{1}

\frac{\partial T (r, z)}{\partial z} = 0, z = 0, z = L

Where

Q (r, z) = \frac{2 (α_{a} η + α_{ps}) [P_{p}^{+} (z) + P_{p}^{-} (z)] + 2 α_{s} [P_{s}^{+} (z) + P_{s}^{-} (z)]}{π ω_{p}^{2}} \exp (\frac{- 2 r^{2}}{ω_{p}^{2}})

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. T_c is the heat sink temperature. T ₁ and T ₂ are the temperatures in fiber core and cladding regions, respectively. α_a is the absorption coefficient at the pump wavelength. α_ps and α_s are scattering loss coefficients at the pump wavelength and signal wavelength, respectively. η is the fractional thermal loading. ω_p is Gaussian radius of the pump light.

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

T_{1} (r, z) = T_{c} + C_{0} \ln r_{1} + D_{0} + \sum_{n = 1}^{\infty} \sum_{m = 1}^{\infty} A_{nm} \cos (\frac{mπ}{L} z) J_{0} (\frac{μ_{n}^{(0)}}{r_{1}} r) + \sum_{m = 1}^{\infty} [C_{m} I_{0} (\frac{mπ}{L} r_{1}) + D_{m} K_{0} (\frac{mπ}{L} r_{1})] \cos (\frac{mπ}{L} z)

T_{2} (r, z) = T_{c} + C_{0} \ln r + D_{0} + \sum_{m = 1}^{\infty} [C_{m} I_{0} (\frac{mπ}{L} r) + D_{m} K_{0} (\frac{mπ}{L} r)] \cos (\frac{mπ}{L} z)

Where

\frac{k C_{0}}{r_{2}} = - c (C_{0} \ln r_{2} + D_{0}), \frac{C_{0}}{r_{1}} = - \sum_{n = 1}^{\infty} A_{n 0} \frac{μ_{n}^{(0)}}{r_{1}} J_{1} (μ_{n}^{(0)})

A_{n 0} = \frac{4}{kL {(μ_{n}^{(0)})}^{2} J_{1}^{2} (μ_{n}^{(0)})} \int_{0}^{r_{1}} \exp (\frac{- 2 r^{2}}{ω_{p}^{2}}) J_{0} (\frac{μ_{n}^{(0)}}{r_{1}} r) r d r \times

A_{nm} = \frac{8 L}{k [m^{2} π^{2} r_{1}^{2} + L^{2} {(μ_{n}^{(0)})}^{2}] J_{1}^{2} (μ_{n}^{(0)})} \int_{0}^{L} \frac{(α_{a} η + α_{ps}) [P_{p}^{+} (z) + P_{p}^{-} (z)] + α_{s} [P_{s}^{+} (z) + P_{s}^{-} (z)]}{π ω_{p}^{2}} \cos (\frac{mπ}{L} z) dz \times

μ_n ⁽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 ⁴I_13/2 and also increase the fractional thermal load in the laser medium. The population-inversion density between the ⁴I_13/2 level and ⁴I_15/2 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]:

F_{ETU} = \frac{(n^{0} - n)}{n^{0}}

The multiphonon decay rate from the upconversion laser levels ²H_11/2, ⁴S_3/2, ⁴F_9/2, and ⁴I_9/2 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 = F_{ETU} + (1 - F_{ETU}) (\frac{1 - λ_{p}}{λ_{s}})

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

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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 Γ_p,s are given as Γ_p,s = 1-exp(-2r ₁ ²/ω_p,s ²), where ω_p,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 C_up. 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; C_up 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, C_up=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.

Fig. 4. The theoretical calculation and experimental results on signal laser output versus pump power. The best fit parameters k₁, k₂, and C_up 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, T_c = 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.

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.

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.

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

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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.

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

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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.

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

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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-T_c = θ, Eq. (7)-(11) are simplified as:

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial θ (r, z)}{\partial r}] + \frac{\partial^{2} θ (r, z)}{\partial z^{2}} = - \frac{Q (r, z)}{k}, 0 \leq r \leq r_{1}

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial θ (r, z)}{\partial r}] + \frac{\partial^{2} θ (r, z)}{\partial z^{2}} = 0, r_{1} \leq r \leq r_{2}

\frac{k \partial θ (r, z)}{\partial r} + c θ (r, z) = 0, r = r_{2}

θ_{1} = θ_{2}, \frac{\partial θ_{1}}{\partial r} = \frac{\partial θ_{2}}{\partial r}, r = r_{1}

\frac{\partial θ (r, z)}{\partial z} = 0, z = 0, z = L

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

θ_{2} (r, z) = C_{0} \ln r + D_{0} + \sum_{m = 1}^{\infty} [C_{m} I_{0} (\frac{mπ}{L} r) + D_{m} K_{0} (\frac{mπ}{L} r)] \cos (\frac{mπ}{L} z)

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

\frac{\partial^{2} t (r, z)}{\partial r^{2}} + \frac{1}{r} \frac{\partial t (r, z)}{\partial r} + \frac{\partial^{2} t (r, z)}{\partial z^{2}} = - \frac{Q (r, z)}{k} - \frac{\partial^{2} θ_{2} (r_{1}, z)}{\partial z^{2}}

t (r, z) = 0, \frac{\partial t}{\partial r} = \frac{\partial θ_{2}}{\partial r}, r = r_{1}

\frac{\partial t (z, r)}{\partial z} = 0, z = 0, z = L

θ_{2} (r_{1}, z) = C_{0} \ln r_{1} + D_{0} + \sum_{m = 1}^{\infty} [C_{m} I_{0} (\frac{mπ}{L} r_{1}) + D_{m} K_{0} (\frac{mπ}{L} r_{1})] \cos (\frac{mπ}{L} z)

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

t = \sum_{n = 1}^{\infty} \sum_{m = 0}^{\infty} A_{nm} \cos (\frac{mπ}{L} z) J_{0} (\frac{μ_{n}^{(0)}}{r_{1}} r)

Where A_nm 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:

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial T (r, z)}{\partial r}] = - \frac{Q (r, z)}{k}, (0 \leq r \leq r_{1})

Boundary conditions satisfy the following relations:

\frac{k \partial T (r, z)}{\partial r} = c [T_{c} - T (r, z)], r = r_{2}

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

T_{1} (r, z) = T_{c} + \frac{α_{a} η [P_{p}^{+} (z) + P_{p}^{-} (z)] + α_{s} [P_{s}^{+} (z) + P_{s}^{-} (z)]}{4 k π} \sum_{m = 1}^{\infty} \frac{{(- 2)}^{m}}{m!} [\frac{{(\frac{r}{r_{1}})}^{2 m} - 1}{m} + 2 \ln \frac{r_{1}}{r_{2}} - \frac{2 k}{h r_{2}}] (0 \leq r \leq r_{1})

T_{2} (r, z) = T_{c} + \frac{α_{a} η [P_{p}^{+} (z) + P_{p}^{-} (z)] + α_{s} [P_{s}^{+} (z) + P_{s}^{-} (z)]}{2 k π} \sum_{m = 1}^{\infty} \frac{{(- 2)}^{m}}{m!} (\ln \frac{r}{r_{2}} - \frac{k}{h r_{2}}) (r_{1} \leq r \leq r_{2})

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:

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial T (r, z)}{\partial r}] + \frac{\partial^{2} T (r, z)}{\partial z^{2}} = - \frac{Q (r, z)}{k}, (0 \leq r \leq r_{1})

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial T (r, z)}{\partial r}] + \frac{\partial^{2} T (r, z)}{\partial z^{2}} = 0, (r_{1} \leq r \leq r_{2})

\frac{\partial^{2} T (r, z)}{\partial r^{2}} + \frac{1}{r} \frac{\partial T (r, z)}{\partial r} + \frac{\partial^{2} T (r, z)}{\partial z^{2}} = 0, (r_{2} \leq r \leq r_{3})

Boundary conditions satisfy the following relations:

\frac{k_{cu} \partial T (r, z)}{\partial r} = c [T_{c} - T (r, z)], r = r_{3}

T_{2} = T_{3}, \frac{k \partial T_{2}}{\partial r} = \frac{k_{cu} \partial T_{3}}{\partial r}, r = r_{2}

T_{1} = T_{2}, \frac{\partial T_{1}}{\partial r} = \frac{\partial T_{2}}{\partial r}, r = r_{1}

\frac{\partial T (r, z)}{\partial z} = 0, z = 0, z = L

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

T_{1} (r, z) = T_{c} + C_{0} \ln r_{1} + D_{0} + \sum_{n = 1}^{\infty} \sum_{m = 0}^{\infty} A_{nm} \cos (\frac{mπ}{L} z) J_{0} (\frac{μ_{n}^{(0)}}{r_{1}} r) + \sum_{m = 1}^{\infty} [C_{m} I_{0} (\frac{mπ}{L} r_{1}) + D_{m} K_{0} (\frac{mπ}{L} r_{1})] \cos (\frac{mπ}{L} z)

T_{2} (r, z) = T_{c} + C_{0} \ln r + D_{0} + \sum_{m = 1}^{\infty} [C_{m} I_{0} (\frac{mπ}{L} r) + D_{m} K_{0} (\frac{mπ}{L} r)] \cos (\frac{mπ}{L} z)

T_{3} (r, z) = T_{c} + A_{0} \ln r + B_{0} + \sum_{m = 1}^{\infty} [A_{m} I_{0} (\frac{mπ}{L} r) + B_{m} K_{0} (\frac{mπ}{L} r)] \cos (\frac{mπ}{L} z)

Where

\frac{k_{cu} A_{0}}{r_{2}} = - c (A_{0} \ln r_{3} + B_{0}) A_{0} \ln r_{2} + B_{0} = C_{0} \ln r_{2} + D_{0}

k_{cu} A_{0} = k C_{0} \frac{C_{0}}{r_{1}} = - \sum_{n = 1}^{\infty} A_{n 0} \frac{μ_{n}^{(0)}}{r_{1}} J_{1} (μ_{n}^{(0)})

k_{cu} (\frac{mπ}{L}) [A_{m} I_{1} (\frac{mπ}{L} r_{3}) - B_{m} K_{1} (\frac{mπ}{L} r_{3})] = - c [A_{m} I_{0} (\frac{mπ}{L} r_{3}) + B_{m} K_{0} (\frac{mπ}{L} r_{3})]

A_{m} I_{0} (\frac{mπ}{L} r_{2}) + B_{m} K_{0} (\frac{mπ}{L} r_{2}) = C_{m} I_{0} (\frac{mπ}{L} r_{2}) + D_{m} K_{0} (\frac{mπ}{L} r_{2})

k_{cu} [A_{m} I_{1} (\frac{mπ}{L} r_{2}) - B_{m} K_{1} (\frac{mπ}{L} r_{2})] = k [C_{m} I_{1} (\frac{mπ}{L} r_{2}) - D_{m} K_{1} (\frac{mπ}{L} r_{2})]

(\frac{mπ}{L}) [C_{m} I_{1} (\frac{mπ}{L} r_{1}) - D_{m} K_{1} (\frac{mπ}{L} r_{1})] = - \sum_{n = 1}^{\infty} A_{nm} \frac{μ_{n}^{(0)}}{r_{1}} J_{1} (μ_{n}^{(0)})

r ₃ is the copper tube radius, k_cu 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).

References and links

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Fig. 2. Schematic illustration of phosphate fiber laser

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Fig. 5. Temperature field distribution in phosphate fiber

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

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

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

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

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

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

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Tables (2)

Table 1. The related parameters used in this paper

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

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Equations (61)

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{UC}_{1} : {Yb}^{3 +} (\overset{2}{F_{\frac{5}{2}}}) + {Er}^{3 +} (\overset{4}{I_{\frac{11}{2}}}) \to {Yb}^{3 +} (\overset{2}{F_{\frac{7}{2}}}) + {Er}^{3 +} (\overset{4}{F_{\frac{7}{2}}})

{UC}_{2} : {Yb}^{3 +} (\overset{2}{F_{\frac{5}{2}}}) + {Er}^{3 +} (\overset{4}{I_{\frac{13}{2}}}) \to {Yb}^{3 +} (\overset{2}{F_{\frac{7}{2}}}) + {Er}^{3 +} (\overset{4}{F_{\frac{9}{2}}})

{UC}_{3} : {Er}^{3 +} (\overset{4}{I_{\frac{13}{2}}}) + {Er}^{3 +} (\overset{4}{I_{\frac{13}{2}}}) \to {Er}^{3 +} (\overset{4}{I_{\frac{15}{2}}}) + {Er}^{3 +} (\overset{4}{I_{\frac{9}{2}}})

\frac{\partial N_{2 y}}{\partial t} = \frac{W_{12 y} N_{1 y} - W_{21 y} N_{2 y} - N_{2 y}}{τ_{y} - k_{1} N_{2 y} N_{1 e} - k_{2} N_{2 y} N_{2 e}} = 0

\frac{\partial N_{2 e}}{\partial t} = \frac{W_{12 e} N_{1 e} + A_{32 e} N_{3 e} - W_{21 e} N_{2 e} - k_{2} N_{2 e} N_{2 y} - N_{2 e}}{τ_{e} - 2 C_{up} N_{2 e}^{2}} = 0

\frac{\partial N_{3 e}}{\partial t} = k_{1} N_{1 e} N_{2 y} + A_{43 e} N_{4 e} - A_{32 e} N_{3 e} = 0

\frac{\partial N_{4 e}}{\partial t} = C_{up} N_{2 e}^{2} - A_{43 e} N_{4 e} = 0

N_{Er} = N_{1 e} + N_{2 e} + N_{3 e} + N_{4 e}

N_{Yb} = N_{1 y} + N_{2 y}

W_{12 e} (z, t) = \frac{Γ_{s} σ_{as} {(υ_{s}) P}_{s} (z)}{h υ_{s} A_{core}} W_{21 e} (z, t) = \frac{Γ_{s} σ_{es} (υ_{s}) P_{s} (z)}{h υ_{s} A_{core}}

W_{12 y} (z, t) = \frac{Γ_{p} σ_{ap} {(υ_{p}) P}_{p} (z)}{h υ_{p} A_{core}} W_{21 y} (z, t) = \frac{Γ_{p} σ_{ep} (υ_{p}) P_{p} (z)}{h υ_{p} A_{core}}

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial T (r, z)}{\partial r}] + \frac{\partial^{2} T (r, z)}{\partial z^{2}} = - \frac{Q (r, z)}{k}, 0 \leq r \leq r_{1}

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial T (r, z)}{\partial r}] + \frac{\partial^{2} T (r, z)}{\partial z^{2}} = 0, r_{1} \leq r \leq r_{2}

\frac{k \partial T (r, z)}{\partial r} = c [T_{c} - T (r, z)], r = r_{2}

T_{1} = T_{2}, \frac{\partial T_{1}}{\partial r} = \frac{\partial T_{2}}{\partial r}, r = r_{1}

\frac{\partial T (r, z)}{\partial z} = 0, z = 0, z = L

Q (r, z) = \frac{2 (α_{a} η + α_{ps}) [P_{p}^{+} (z) + P_{p}^{-} (z)] + 2 α_{s} [P_{s}^{+} (z) + P_{s}^{-} (z)]}{π ω_{p}^{2}} \exp (\frac{- 2 r^{2}}{ω_{p}^{2}})

T_{1} (r, z) = T_{c} + C_{0} \ln r_{1} + D_{0} + \sum_{n = 1}^{\infty} \sum_{m = 1}^{\infty} A_{nm} \cos (\frac{mπ}{L} z) J_{0} (\frac{μ_{n}^{(0)}}{r_{1}} r) + \sum_{m = 1}^{\infty} [C_{m} I_{0} (\frac{mπ}{L} r_{1}) + D_{m} K_{0} (\frac{mπ}{L} r_{1})] \cos (\frac{mπ}{L} z)

T_{2} (r, z) = T_{c} + C_{0} \ln r + D_{0} + \sum_{m = 1}^{\infty} [C_{m} I_{0} (\frac{mπ}{L} r) + D_{m} K_{0} (\frac{mπ}{L} r)] \cos (\frac{mπ}{L} z)

\frac{k C_{0}}{r_{2}} = - c (C_{0} \ln r_{2} + D_{0}), \frac{C_{0}}{r_{1}} = - \sum_{n = 1}^{\infty} A_{n 0} \frac{μ_{n}^{(0)}}{r_{1}} J_{1} (μ_{n}^{(0)})

k (\frac{mπ}{L}) [C_{m} I_{1} (\frac{mπ}{L} r_{2}) - D_{m} K_{1} (\frac{mπ}{L} r_{2})] = - c [C_{m} I_{0} (\frac{mπ}{L} r_{2}) + D_{m} K_{0} (\frac{mπ}{L} r_{2})]

(\frac{mπ}{L}) [C_{m} I_{1} (\frac{mπ}{L} r_{1}) - D_{m} K_{1} (\frac{mπ}{L} r_{1})] = - \sum_{n = 1}^{\infty} A_{nm} \frac{μ_{n}^{(0)}}{r_{1}} J_{1} (μ_{n}^{(0)})

A_{n 0} = \frac{4}{kL {(μ_{n}^{(0)})}^{2} J_{1}^{2} (μ_{n}^{(0)})} \int_{0}^{r_{1}} \exp (\frac{- 2 r^{2}}{ω_{p}^{2}}) J_{0} (\frac{μ_{n}^{(0)}}{r_{1}} r) r d r \times

\int_{0}^{L} \frac{(α_{a} η + α_{ps}) [P_{p}^{+} (z) + P_{p}^{-} (z)] + α_{s} [P_{s}^{+} (z) + P_{s}^{-} (z)]}{π ω_{p}^{2}} dz

A_{nm} = \frac{8 L}{k [m^{2} π^{2} r_{1}^{2} + L^{2} {(μ_{n}^{(0)})}^{2}] J_{1}^{2} (μ_{n}^{(0)})} \int_{0}^{L} \frac{(α_{a} η + α_{ps}) [P_{p}^{+} (z) + P_{p}^{-} (z)] + α_{s} [P_{s}^{+} (z) + P_{s}^{-} (z)]}{π ω_{p}^{2}} \cos (\frac{mπ}{L} z) dz \times

\int_{0}^{r_{1}} \exp (\frac{- 2 r^{2}}{ω_{p}^{2}}) J_{0} (\frac{μ_{n}^{(0)}}{r_{1}} r) rdr - \frac{2 r_{1}^{2} {(mπ)}^{2}}{[m^{2} π^{2} r_{1}^{2} + L^{2} {(μ_{n}^{(0)})}^{2}] J_{1} (μ_{n}^{(0)}) μ_{n}^{(0)}} [C_{m} I_{0} (\frac{mπ}{L} r_{1}) + D_{m} K_{0} (\frac{mπ}{L} r_{1})]

F_{ETU} = \frac{(n^{0} - n)}{n^{0}}

η = F_{ETU} + (1 - F_{ETU}) (\frac{1 - λ_{p}}{λ_{s}})

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial θ (r, z)}{\partial r}] + \frac{\partial^{2} θ (r, z)}{\partial z^{2}} = - \frac{Q (r, z)}{k}, 0 \leq r \leq r_{1}

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial θ (r, z)}{\partial r}] + \frac{\partial^{2} θ (r, z)}{\partial z^{2}} = 0, r_{1} \leq r \leq r_{2}

\frac{k \partial θ (r, z)}{\partial r} + c θ (r, z) = 0, r = r_{2}

θ_{1} = θ_{2}, \frac{\partial θ_{1}}{\partial r} = \frac{\partial θ_{2}}{\partial r}, r = r_{1}

\frac{\partial θ (r, z)}{\partial z} = 0, z = 0, z = L

θ_{2} (r, z) = C_{0} \ln r + D_{0} + \sum_{m = 1}^{\infty} [C_{m} I_{0} (\frac{mπ}{L} r) + D_{m} K_{0} (\frac{mπ}{L} r)] \cos (\frac{mπ}{L} z)

\frac{\partial^{2} t (r, z)}{\partial r^{2}} + \frac{1}{r} \frac{\partial t (r, z)}{\partial r} + \frac{\partial^{2} t (r, z)}{\partial z^{2}} = - \frac{Q (r, z)}{k} - \frac{\partial^{2} θ_{2} (r_{1}, z)}{\partial z^{2}}

t (r, z) = 0, \frac{\partial t}{\partial r} = \frac{\partial θ_{2}}{\partial r}, r = r_{1}

\frac{\partial t (z, r)}{\partial z} = 0, z = 0, z = L

θ_{2} (r_{1}, z) = C_{0} \ln r_{1} + D_{0} + \sum_{m = 1}^{\infty} [C_{m} I_{0} (\frac{mπ}{L} r_{1}) + D_{m} K_{0} (\frac{mπ}{L} r_{1})] \cos (\frac{mπ}{L} z)

t = \sum_{n = 1}^{\infty} \sum_{m = 0}^{\infty} A_{nm} \cos (\frac{mπ}{L} z) J_{0} (\frac{μ_{n}^{(0)}}{r_{1}} r)

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial T (r, z)}{\partial r}] = - \frac{Q (r, z)}{k}, (0 \leq r \leq r_{1})

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial T (r, z)}{\partial r}] = 0, (r_{1} \leq r \leq r_{2})

\frac{k \partial T (r, z)}{\partial r} = c [T_{c} - T (r, z)], r = r_{2}

T_{1} = T_{2}, \frac{\partial T_{1}}{\partial r} = \frac{\partial T_{2}}{\partial r}, r = r_{1}

T_{1} (r, z) = T_{c} + \frac{α_{a} η [P_{p}^{+} (z) + P_{p}^{-} (z)] + α_{s} [P_{s}^{+} (z) + P_{s}^{-} (z)]}{4 k π} \sum_{m = 1}^{\infty} \frac{{(- 2)}^{m}}{m!} [\frac{{(\frac{r}{r_{1}})}^{2 m} - 1}{m} + 2 \ln \frac{r_{1}}{r_{2}} - \frac{2 k}{h r_{2}}] (0 \leq r \leq r_{1})

T_{2} (r, z) = T_{c} + \frac{α_{a} η [P_{p}^{+} (z) + P_{p}^{-} (z)] + α_{s} [P_{s}^{+} (z) + P_{s}^{-} (z)]}{2 k π} \sum_{m = 1}^{\infty} \frac{{(- 2)}^{m}}{m!} (\ln \frac{r}{r_{2}} - \frac{k}{h r_{2}}) (r_{1} \leq r \leq r_{2})

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial T (r, z)}{\partial r}] + \frac{\partial^{2} T (r, z)}{\partial z^{2}} = - \frac{Q (r, z)}{k}, (0 \leq r \leq r_{1})

\frac{1}{r} \frac{\partial}{\partial r} [r \frac{\partial T (r, z)}{\partial r}] + \frac{\partial^{2} T (r, z)}{\partial z^{2}} = 0, (r_{1} \leq r \leq r_{2})

\frac{\partial^{2} T (r, z)}{\partial r^{2}} + \frac{1}{r} \frac{\partial T (r, z)}{\partial r} + \frac{\partial^{2} T (r, z)}{\partial z^{2}} = 0, (r_{2} \leq r \leq r_{3})

\frac{k_{cu} \partial T (r, z)}{\partial r} = c [T_{c} - T (r, z)], r = r_{3}

T_{2} = T_{3}, \frac{k \partial T_{2}}{\partial r} = \frac{k_{cu} \partial T_{3}}{\partial r}, r = r_{2}

T_{1} = T_{2}, \frac{\partial T_{1}}{\partial r} = \frac{\partial T_{2}}{\partial r}, r = r_{1}

\frac{\partial T (r, z)}{\partial z} = 0, z = 0, z = L

T_{1} (r, z) = T_{c} + C_{0} \ln r_{1} + D_{0} + \sum_{n = 1}^{\infty} \sum_{m = 0}^{\infty} A_{nm} \cos (\frac{mπ}{L} z) J_{0} (\frac{μ_{n}^{(0)}}{r_{1}} r) + \sum_{m = 1}^{\infty} [C_{m} I_{0} (\frac{mπ}{L} r_{1}) + D_{m} K_{0} (\frac{mπ}{L} r_{1})] \cos (\frac{mπ}{L} z)

T_{2} (r, z) = T_{c} + C_{0} \ln r + D_{0} + \sum_{m = 1}^{\infty} [C_{m} I_{0} (\frac{mπ}{L} r) + D_{m} K_{0} (\frac{mπ}{L} r)] \cos (\frac{mπ}{L} z)

T_{3} (r, z) = T_{c} + A_{0} \ln r + B_{0} + \sum_{m = 1}^{\infty} [A_{m} I_{0} (\frac{mπ}{L} r) + B_{m} K_{0} (\frac{mπ}{L} r)] \cos (\frac{mπ}{L} z)

\frac{k_{cu} A_{0}}{r_{2}} = - c (A_{0} \ln r_{3} + B_{0}) A_{0} \ln r_{2} + B_{0} = C_{0} \ln r_{2} + D_{0}

k_{cu} A_{0} = k C_{0} \frac{C_{0}}{r_{1}} = - \sum_{n = 1}^{\infty} A_{n 0} \frac{μ_{n}^{(0)}}{r_{1}} J_{1} (μ_{n}^{(0)})

k_{cu} (\frac{mπ}{L}) [A_{m} I_{1} (\frac{mπ}{L} r_{3}) - B_{m} K_{1} (\frac{mπ}{L} r_{3})] = - c [A_{m} I_{0} (\frac{mπ}{L} r_{3}) + B_{m} K_{0} (\frac{mπ}{L} r_{3})]

A_{m} I_{0} (\frac{mπ}{L} r_{2}) + B_{m} K_{0} (\frac{mπ}{L} r_{2}) = C_{m} I_{0} (\frac{mπ}{L} r_{2}) + D_{m} K_{0} (\frac{mπ}{L} r_{2})

k_{cu} [A_{m} I_{1} (\frac{mπ}{L} r_{2}) - B_{m} K_{1} (\frac{mπ}{L} r_{2})] = k [C_{m} I_{1} (\frac{mπ}{L} r_{2}) - D_{m} K_{1} (\frac{mπ}{L} r_{2})]

(\frac{mπ}{L}) [C_{m} I_{1} (\frac{mπ}{L} r_{1}) - D_{m} K_{1} (\frac{mπ}{L} r_{1})] = - \sum_{n = 1}^{\infty} A_{nm} \frac{μ_{n}^{(0)}}{r_{1}} J_{1} (μ_{n}^{(0)})

Symbol	Physical meaning	Value
N_Er/m^-3	the Er³⁺ concentration	2.57×10²⁶
N_yb/m^-3	the Yb³⁺ concentration	2.57×10²⁶
τ_y/S	the Yb³⁺ fluorescence lifetime	2×l0^-3
τ_e/S	the Er³⁺ fluorescence lifetime	8.4×l0^-3
A₄₃/S^-1	the Er³⁺ de-excitation rate from the ⁴I_9/2 level to ⁴I_11/2 level	7×10⁹
A₃₂/S^-1	the Er³⁺ de-excitation rate from the ⁴I_11/2 level to ⁴I_13/2 level	2.8×10⁶[19]
σ_ap/m^-2	the Yb³⁺ absorption cross-section at the 976nm pump wavelength	l.0l×l0^-24
σ_ep/m^-2	the Yb³⁺ emission cross-section at the 976nm pump wavelength	1.27×l0^-24
σ_as/ m	the Er³⁺ absorption cross-section at the 1535nm signal wavelength	4.99×l0^-25
σ_es/m^-2	the Er³⁺ emission cross-section at the 1535nm signal wavelength	6.02×l0^-25
Γ_P	the power filling factor of the pump power	0.9 4
Γ_s	the power filling factor of the signal power	0.82
C_up/m³·S^-1	the cooperative upconversion coefficient	1.67×l0^-24
k₂/m³·S^-1	the cumulative upconversion energy transfer coefficient	0.85×l0^-22
k₁/m³·S^-1	the energy transfer coefficient from the Yb³⁺ ²F_5/2 level to the Er³⁺ ⁴I_15/2 level	2.56×l0^-22
R₁	the reflectivity of the forward mirror at the lasing wavelength	1
R₂	the reflectivity of the backward mirror at the lasing wavelength	0.55
h/m²kg·S^-1	Planck’s constant	6.63×l0^-34
r₁/μm	the core radius	2.7
r₂/μm	the cladding radius	62.5

Upconversion energy level of Er³⁺ ions	Nonradiative transition rate [s^-1]	Total spontaneous radiative rate calculated from Judd-Ofelt date [s^-1]
²H_11/2	≈10⁹	6693.7
⁴S_3/2	≈10⁶	1298.2
⁴F_9/2	≈10⁷	1066.8
⁴I_9/2	≈10⁹	67.6

Symbol	Physical meaning	Value
N_Er/m^-3	the Er³⁺ concentration	2.57×10²⁶
N_yb/m^-3	the Yb³⁺ concentration	2.57×10²⁶
τ_y/S	the Yb³⁺ fluorescence lifetime	2×l0^-3
τ_e/S	the Er³⁺ fluorescence lifetime	8.4×l0^-3
A₄₃/S^-1	the Er³⁺ de-excitation rate from the ⁴I_9/2 level to ⁴I_11/2 level	7×10⁹
A₃₂/S^-1	the Er³⁺ de-excitation rate from the ⁴I_11/2 level to ⁴I_13/2 level	2.8×10⁶[19]
σ_ap/m^-2	the Yb³⁺ absorption cross-section at the 976nm pump wavelength	l.0l×l0^-24
σ_ep/m^-2	the Yb³⁺ emission cross-section at the 976nm pump wavelength	1.27×l0^-24
σ_as/ m	the Er³⁺ absorption cross-section at the 1535nm signal wavelength	4.99×l0^-25
σ_es/m^-2	the Er³⁺ emission cross-section at the 1535nm signal wavelength	6.02×l0^-25
Γ_P	the power filling factor of the pump power	0.9 4
Γ_s	the power filling factor of the signal power	0.82
C_up/m³·S^-1	the cooperative upconversion coefficient	1.67×l0^-24
k₂/m³·S^-1	the cumulative upconversion energy transfer coefficient	0.85×l0^-22
k₁/m³·S^-1	the energy transfer coefficient from the Yb³⁺ ²F_5/2 level to the Er³⁺ ⁴I_15/2 level	2.56×l0^-22
R₁	the reflectivity of the forward mirror at the lasing wavelength	1
R₂	the reflectivity of the backward mirror at the lasing wavelength	0.55
h/m²kg·S^-1	Planck’s constant	6.63×l0^-34
r₁/μm	the core radius	2.7
r₂/μm	the cladding radius	62.5

Upconversion energy level of Er³⁺ ions	Nonradiative transition rate [s^-1]	Total spontaneous radiative rate calculated from Judd-Ofelt date [s^-1]
²H_11/2	≈10⁹	6693.7
⁴S_3/2	≈10⁶	1298.2
⁴F_9/2	≈10⁷	1066.8
⁴I_9/2	≈10⁹	67.6

Abstract

1. Introduction

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

3. Resolving of 3-dimensional temperature field

4. Analytical results and discussions

5. Experimental validation on heat generation with upconversion

6. Conclusions

Appendix A

Appendix B

Appendix C

Acknowledgment

References and links

Cited By

Figures (13)

Tables (2)

Equations (61)

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