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1.5 μm波段激光对人眼损伤阈值较高，是人眼安全波段[1]的激光；位于大气窗口波段，穿透力较强，可用于激光雷达[2]及测距[3]；对水分子吸收性很强，广泛应用于医疗整容[4]以及大气通信[5]等领域。LD端面泵浦固体激光器以铒镱共掺晶体作为增益介质，钴尖晶石[6-7]作为调Q晶体产生1.5 μm波段的激光。常用的1.5 μm的介质有铒镱共掺硼酸盐[8-10]和磷酸盐玻璃[11-13]两类，其中Er:Yb:glass晶体具有短长度高增益、荧光寿命长、成本低等优点，其铒镱间能量转换效率高达95%。
与Nd3+和Yb3+等单掺材料相比，Er3+和Yb3+共掺杂材料在工作时产生的热量会更高，Er:Yb:glass不同于金刚石等[14–17]具有超高热导率，其导热系数很低，仅为0.85 W/(m·K)，导致激光系统的散热性很差，系统内的热负载加剧会产生严重的热效应[18-19]。激光增益介质产生热量的原因包括量子缺陷加热、激光淬火、上转换[20]和工作介质对泵浦光的吸收[21]，当激光与增益介质相互作用时，晶体吸收的一部分泵浦光会转换成热量储存在晶体内部[22]，导致其温度上升，这成为限制输出能量的因素之一。2015年，Belghachem等[23]用键合晶体实现了脉冲激光。与未键合晶体相比，峰值功率和效率提高。2016年，Mlyńczak等[24]对比940 nm和975 nm正反向泵浦对热效应的影响，发现940 nm反向泵浦可降低热效应影响。2019年，北京工业大学班晓娜等[25]提出双端键合结构，可以降低晶体内部温度梯度，单脉冲能量增加。2021年，河北工业大学齐月[26]建立铒玻璃模型，将温度场与应力场耦合，研究晶体温度和形变情况。热效应对激光的影响主要有两个方面：一方面，随着激光晶体温度的升高，荧光谱线会展宽且量子效率下降，导致阈值升高，转换效率下降；另一方面，温度梯度所产生的热应力和热透镜会严重影响激光器的输出稳定性和激光光束质量。因此，有必要研究Er:Yb:glass作为增益介质时的热处理能力，以便制定进一步优化输出性能的方案。
文中利用有限元分析方法详细计算了Er:Yb:glass作为增益介质内部的热积累过程，通过定量分析非键合和键合晶体及不同泵浦波长、功率和束腰半径对晶体热效应的影响，明确得出温度、应力和形变的具体数值和差值，可以根据具体情况选取相应参数。文中研究结果可为优化LD端面泵浦铒镱共掺磷酸盐玻璃1.5 μm被动调Q固体激光器[27]输出特性提供指导。
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图1为实验结构图，为分析其热效应，假设晶体表面温度与环境温度恒定，设定温度初始值为T0=293.15 K，介质内部产生的热量主要是通过热传导的方式来散热，外表面受到空气自然热对流的影响。当LD端面泵浦Er:Yb:glass时，泵浦源作为内部热源，工作时产生温度梯度，从而引发热传导过程。为了比较增益介质在不同情况下的散热性能，对其工作达到稳态后进行模拟仿真。
图 1 实验结构图
Figure 1. Diagram of experimental structure
为提高泵浦效率，LD通过耦合透镜组准直聚焦进入晶体端面的中心(见图2)，并以中心位置为原点建立坐标系，晶体横截面为3 mm×3 mm，厚度分别为2.5 mm和1.5 mm，LD入射端面作为前表面，远离泵浦侧为后表面，按照超细化网格对晶体模型进行划分。
图 2 晶体建模图
Figure 2. Diagram of crystal modeling
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当介质达到输出功率稳定、温度分布趋于稳态时，各部分温度分布均匀且不再随时间发生变化，此时增益介质的热传导方程为：
$$ \frac{{\partial }^{2}{T}\left({x},{y},{z}\right)}{\partial {{x}}^{2}}+\frac{{\partial }^{2}{T}\left({x},{y},{z}\right)}{\partial {{y}}^{2}}+\frac{{\partial }^{2}{T}\left({x},{y},{z}\right)}{\partial {{z}}^{2}}+\frac{{q}\left({x},{y},{z}\right)}{{K}}=0 $$ (1) 式中：$ K $为晶体的热导系数；$ q\left(x,y,z\right) $为热源函数；$ T\left(x,y,z\right) $为晶体的温度分布。
Er:Yb:glass与空气之间进行热对流交换，即：
$$ {{q}}_{0}={h}\left({{T}}_{0}-{T}\right) $$ (2) 式中：$ h $为传热系数(W/(K·m2))；$ {T}_{0} $为外部环境温度；$ T $为介质温度。
泵浦光入射到晶体端面上，在晶体内部产生了热传导性质的内热源，内热源产生热从而引起内部的温度分布变化。假设泵浦光在端面方向上近似为高斯分布，部分光在轴向上被增益介质吸收，呈指数衰减趋势，泵浦光在激光晶体中传播，光强分布可以近似高斯分布，光强分布可表示为：
$$ {I}_{p}\left(x,y,z,t\right)=\frac{2{P}_{in}}{\pi {{\omega }_{p}}^{2}}{{\rm{e}}}^{-\frac{2{x}^{2}+{y}^{2}+{z}^{2}}{{{\omega }_{p}}^{2}}}{{\rm{e}}}^{-\alpha z} $$ (3) 进而产生的热量$ Q\left(x,y,z,t\right) $可以表示为：
$$ Q\left(x,y,z,t\right)=\xi \alpha I\left(x,y,z,t\right) $$ (4) 得到：
$$ Q\left(x,y,z,t\right)=\frac{2\alpha {P}_{in}\xi }{\pi {{\omega }_{p}}^{2}}{{\rm{e}}}^{-\tfrac{2{x}^{2}+{y}^{2}+{z}^{2}}{{{\omega }_{p}}^{2}}}{{\rm{e}}}^{-\alpha z} $$ (5) 式中：$ {P}_{in} $为入射到增益介质端面的泵浦功率；$ {\omega }_{p} $为泵浦束腰半径；$ \alpha $为增益介质的吸收系数；$ \xi $为热转换系数，$ \xi =1-{\lambda }_{p}/{\lambda }_{L} $，$ {\lambda }_{p} $为泵浦光波长，$ {\lambda }_{L} $为振荡光波长。具体参数如表1所示。
表 1 材料参数
Table 1. Material parameters
Parameters Symbol Er:Yb:glass Co:MALO Size a×b/mm2 3×3 3×3 Length L/mm 2.5 1.5 Thermal conductivity K/W·m−1·K−1 0.745 15 Thermal expansion αT/K−1 7.2×10−6 8.0×10−6 Temperature coefficient
of the refractive indexdn/dT/K−1 −2.7×10−6 3.0×10−6 Density ρ/kg·m−3 2633 3500 Refractive index n0 1.53 1.67 Poisson's ratio ν 0.24 0.24 Young's modulus E/Pa 70×109 2.3×105 Quantum defects ξ 0.3648 - Absorption coefficient
at 940 nmα/cm−1 4.28 0.0763 Absorption coefficient
at 975 nmα/cm−1 19.4 0.0783 -
利用有限元分析法对Er:Yb:glass晶体的温度场分布进行分析，当泵浦功率为600 mW，泵浦束腰半径为125 μm，图3和图4对比了在非键合和键合情况下的温度场分布。
图 3 非键合晶体温度分布。(a)前表面；(b)切面
Figure 3. Temperature distribution of unbonded crystal. (a) Front surface; (b) Section
如图3(a)和4(a)所示，前表面的最高温度都在中心处，由切面温度分布得知晶体最高温度在前表面附近，在键合情况下，由于Co:MALO远离泵浦侧，对磷酸盐玻璃的温度分布影响很小，Co:MALO不吸收泵浦光辐射，因此不会被显著加热。
由图5可知，两模型前表面温度分布基本相同，最高温度分别为349 K和348 K，但在Er:Yb:glass后表面差异较明显，非键合方式后表面温度为318.62 K，而键合晶体在键合后表面处温度已经下降至301.86 K，采用键合晶体显著降低了热效应的影响，故引入键合晶体的模型来降低晶体内部的温度分布。
图 4 键合晶体温度分布。(a)前表面；(b)切面
Figure 4. Bonded crystal temperature distribution. (a) Front surface; (b) Section
图 5 温度分布。(a)沿x轴；(b)沿z轴
Figure 5. Temperature distribution. (a) Along the x-axis; (b) Along the z-axis
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对于LD端面泵浦Er:Yb:glass/Co:MALO键合晶体，增益介质对泵浦波长940 nm和976 nm表现出不同的吸收系数，故改变泵浦波长也会导致晶体的温度场发生变化，在泵浦功率为600 mW，束腰半径为125 μm时，采用976 nm中心波长的泵浦情况下温度分布如图6所示。
图 6 976 nm泵浦波长下温度分布。(a) 前表面；(b)切面
Figure 6. Temperature distribution at 976 nm pump wavelength. (a) Front surface; (b) Section
采用中心波长为976 nm泵浦的情况下晶体的最高温度为475 K，比940 nm情况下的温度高出127 K。由图6(b)可知940 nm波长对增益介质的穿透性要远高于 976 nm，但在976 nm情况下温度扩散范围较小。
由图7(a)可知，当采用976 nm泵浦时，晶体在x轴方向上的温度远高于采用940 nm泵浦的情况。由图7(b)可知，晶体在中心波长976 nm下的温度下降的速度更快，在晶体长度1.2 mm附近，两种情况温度达到相同，在此波长下，温度上升造成的最危险的影响在前表面。
图 7 不同泵浦波长下的温度分布。(a)沿x轴；(b)沿z轴
Figure 7. Temperature distribution at different pump wavelengths. (a) Along x-axis; (b) Along z-axis
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泵浦功率变化也会影响晶体的温度分布，当泵浦波长为940 nm，束腰半径为125 μm时，泵浦功率对温度分布的影响如图8所示。当泵浦功率为500 mW、600 mW、700 mW、800 mW和900 mW时，晶体的最高温度分别为339 K、348 K、358 K、367 K和376 K。泵浦功率的增加会导致晶体吸收更多的能量，从而产生更多热量，随着热量的积累，晶体的温度逐渐上升。泵浦功率的变化会引起晶体内温度梯度场分布的变化，从而产生热透镜效应，影响激光的输出质量。
图 8 不同泵浦功率下的温度分布。(a)沿x轴；(b)沿z轴
Figure 8. Temperature distribution under different pump power. (a) Along x-axis; (b) Along z-axis
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当泵浦波长为940 nm、泵浦功率为600 mW时，在不同泵浦束腰半径下，键合晶体的温度分布曲线如图9所示。
当束腰半径分别为85 μm、105 μm、125 μm和150 μm时，晶体的最高温度为357 K、352 K、348 K和345 K，平均增加4 K。随着泵浦束腰半径的增大，光功率密度减小，键合晶体的温度减小。在晶体前表面，只有中心处的温度差异较大，以中心处为基准，沿x轴方向的温度基本相同；而沿晶体的z轴方向，温度变化较明显，因此需要考虑泵浦束腰半径对温度分布的影响。
图 9 不同束腰半径下的温度分布。 (a)沿x轴；(b)沿z轴
Figure 9. Temperature distribution under different waist size. (a) Along x-axis; (b) Along z-axis
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在键合晶体的基础上，当Co:MALO键合在Er:Yb:glass晶体前，靠近泵浦侧，当泵浦波长为940 nm，功率为600 mW，束腰半径为125 μm时，键合位置对晶体的温度分布影响如图10所示。
由图10可知，前表面最高温度在中心处为310 K，由切面温度分布图知晶体最高温度在增益介质靠近键合位置附近，由于Co:MALO靠近泵浦侧，明显减弱了热效应对增益介质的影响，降低了晶体温度。
图 10 温度分布。(a)前表面；(b)切面
Figure 10. Temperature distribution. (a) Front surface; (b) Section
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利用有限元分析法对晶体应力场进行分析，当泵浦功率为600 mW，束腰半径为125 μm时，对比了两种情况下的热应力分布。由图11可知，非键合和键合晶体应力最大值分别为8.94 MPa和8.98 MPa，位置均在中心处。键合晶体的应力略大于非键合晶体，这是由于当介质中产生热量时，Er:Yb:glass与Co:MALO相接触，二者的热导系数相差较大，键合面会由于挤压而形成压力，这种压力会延伸至靠近泵浦的一侧，导致键合晶体的应力增加，这个应力值应小于断裂应力阈值。
图 11 热应力分布。(a)非键合晶体；(b)键合晶体
Figure 11. Thermal stress distribution.(a) Non bonded crystal; (b) Bonded crystal
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增益介质对不同泵浦波长的吸收系数不同，导致在976 nm和940 nm处的热应力大小不同。在键合晶体的基础上，当泵浦功率为600 mW，束腰半径为125 μm时，由图12得知，在泵浦波长为976 nm的情况下，晶体的最大应力值为25.7 MPa，约为在940 nm波长下最大应力值的三倍。增益介质对976 nm泵浦的吸收系数高，导致应力的延展性很差，晶体前表面应力值较大，而边缘和侧面的应力值很小，此时是扩散键合较为安全的情况。
图 12 976 nm泵浦波长下热应力分布
Figure 12. Thermal stress distribution at 976 nm pump wavelength
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图13为键合晶体在不同泵浦功率和束腰半径下的热应力大小的曲线，随着泵浦功率的增加，晶体的温度梯度增大，热应力逐渐增大。当泵浦束腰半径为125 μm时，随着泵浦功率从500 mW增加至900 mW，热应力分别为7.54 MPa、8.94 MPa、11.5 MPa、13.2 MPa和15.9 MPa。由图13(b)可知，在相同泵浦功率下，随着泵浦束腰半径增加，热应力呈现减小趋势。减小泵浦功率且增大束腰半径情况下，晶体内部的热应力最小。
图 13 热应力。(a)不同泵浦功率；(b)不同束腰半径
Figure 13. Thermal stress. (a) Different pump powers; (b) Different waist size
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在键合晶体的基础上，当Co:MALO键合在Er:Yb:glass晶体前，靠近泵浦侧，泵浦波长为940 nm，功率为600 mW，束腰半径为125 μm时，键合位置对晶体的热应力影响如图14所示。此时最大应力为7.02 MPa，小于Co:MALO键合在远离泵浦侧键合晶体的最大应力8.98 MPa。
图 14 热应力分布
Figure 14. Thermal stress distribution
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当泵浦功率为600 mW，束腰半径为125 μm时，非键合和键合晶体的热形变分布如图15所示。最大热形变出现在晶体端面中心处，形状是以泵浦光为中心的圆形凸起。两种情况下晶体的最大热形变量分别为0.29 μm和0.32 μm，键合晶体显著减小了增益介质远离泵浦侧的热形变量。
图 15 热形变分布。(a)非键合晶体；(b)键合晶体
Figure 15. Thermal deformation distribution. (a) Non-bonded crystal; (b) Bonded crystal
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在键合晶体中，采用976 nm的泵浦，泵浦功率为600 mW，束腰半径为125 μm时，形变量分布如图16所示。最大形变量为0.73 μm，远大于在940 nm泵浦下的0.32 μm，圆形凸起更加明显。沿晶体z轴方向，热形变迅速减小，铒玻璃后表面没有热形变产生。
图 16 976 nm泵浦波长下的热形变分布
Figure 16. Thermal deformation distribution at 976 nm pump wavelength
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键合晶体在不同泵浦功率和束腰半径下的形变量如图17所示。在940 nm泵浦下，泵浦束腰半径相同时，增大泵浦功率，热形变会明显增大，泵浦功率和热形变量呈正比，泵浦功率每增加100 mW，晶体的热形变量增加0.5 μm。在泵浦功率相同时，增大泵浦束腰半径，热形变量会减小，但产生的变化相较于泵浦功率并不明显。当泵浦功率为600 mW，泵浦束腰半径分别为85 μm、105 μm、125 μm和150 μm时，晶体的热形变量分别为0.34 μm、0.33 μm、0.32 μm和0.32 μm，形变量变化并不明显。
图 17 热形变。 (a)不同泵浦功率；(b)不同束腰半径
Figure 17. Thermal deformation. (a) Different pump powers; (b) Different waist sizes
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在键合晶体的基础上，当Co:MALO键合在Er: Yb:glass晶体前，靠近泵浦侧，当泵浦波长为940 nm，功率为600 mW，束腰半径为125 μm时，键合位置对晶体的热应力影响如图18所示。此时形变量为0.23 μm，可以有效减少晶体的热形变量，改善激光器的输出特性。
图 18 热形变分布
Figure 18. Thermal deformation distribution
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文中采用有限元分析方法，根据热传导理论研究LD泵浦Er: Yb: glass/Co: MALO热效应，由于Co: MALO的热沉效应，键合晶体可以降低激光晶体的最高温度、热应力和热形变量。在此基础上，采用中心波长为940 nm的泵浦会降低激光器的最高温度，而976 nm泵浦结构是扩散键合较安全的结构，增加泵浦功率和减小泵浦束腰半径同样会导致晶体的热效应加剧。在设计激光系统时，应防止晶体的温度过高和形变量过大，减轻热效应对输出特性的影响。该研究为进一步设计热性能更好的Er:Yb:glass激光器提供了优化条件，也为输出高功率、高光束质量的1.5 μm激光提供了理论依据。
Thermal effect analysis of LD end-pumped Er:Yb:glass/Co:MALO crystal
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摘要: 为了减弱激光二极管端面泵浦固体激光器热效应的影响，提高激光谐振腔稳定性及改善激光器的输出特性，以激光二极管端面泵浦Er:Yb:glass/Co:MALO晶体为研究对象，采用有限元分析方法，并根据热传导理论对其进行多物理场耦合热效应分析，系统分析了非键合和键合晶体、泵浦中心波长、功率以及束腰半径对激光晶体的温度场、热应力场以及形变量的影响。结果表明：键合晶体中Co:MALO晶体不仅起到了被动调Q的作用，还起到了热沉的效果，可以有效改善晶体内部的温度分布、热应力和形变量。中心波长为940 nm的泵浦对晶体的穿透性远高于976 nm，采用940 nm的泵浦可以改善晶体的最高温度，但976 nm泵浦结构是扩散键合较安全的结构。由于增大泵浦功率会导致晶体单位面积功率密度分布的增加，热效应也会加剧，泵浦功率增大100 mW对应温度增加9 K，热应力增加1.5 MPa，热形变增加0.5 μm。减小泵浦光束半径也会导致热效应的增加，但影响相较于功率不明显。理论分析结果可为激光二极管端面泵浦铒镱共掺磷酸盐玻璃1.5 μm固体激光器减小热效应的合理优化设计提供数据理论支持。
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关键词:
- LD端面泵浦 /
- Er:Yb:glass/Co:MALO /
- 调Q激光器 /
- 热效应 /
- 热应力
Abstract:Objective The poor thermal conductivity of Er:Yb:glass leads to poor heat dissipation in the laser system and the increased thermal load leads to severe thermal effect, which limits the laser output of high power, but there are few studies on the thermal effect of Er:Yb:glass. When the laser interacts with the gain medium, a portion of the pump light absorbed by the crystal will be converted to heat and stored in the crystal, resulting in a rise in its temperature, which becomes one of the factors limiting the output energy. The influence of thermal effect on the laser has two main aspects. On the one hand, as the laser crystal temperature increases, the fluorescence spectral line will broaden and the quantum efficiency will decrease, which will eventually lead to the decrease of conversion efficiency. On the other hand, the thermal stress and thermal lens effect generated by the temperature gradient will seriously affect the output stability and beam quality of the laser. Therefore, it is necessary to investigate the heat treatment capability of Er:Yb:glass as gain medium in order to develop a scheme for further optimization of the output performance. Methods The heat accumulation process inside Er:Yb:glass as the gain medium is calculated in detail by finite element analysis method. Assuming that the surface temperature of the crystal is constant with the ambient temperature, the heat generated inside the medium is mainly dissipated by heat conduction, and the outer surface is affected by natural heat convection of air. When the LD is end-pumped with Er:Yb:glass, the pump source acts as an internal heat source and operates with a temperature gradient, which triggers the heat conduction process (Fig.2). In order to compare the heat dissipation performance of the gain medium under different conditions, the simulation is carried out after reaching steady state. The crystal model was divided according to the ultra-fine grid, and the effects of non-bonded and bonded crystals as well as different pump wavelengths, powers and beam waist radius on crystal temperature distribution, thermal stress and deformation were quantitatively analyzed. Results and Discussions Co:MALO in the bonded crystal not only acts as a Q-switched crystal, but also as a heat sink. The bonded crystal significantly reduces the influence of thermal effects, especially the temperature of the surface after the gain medium (Fig.4). The pump wavelength of 940 nm is more penetrating to the gain medium than 976 nm, but the temperature diffusion range at 976 nm is less than 940 nm (Fig.6). The temperature of the crystal pumped at 976 nm is higher than that of 940 nm (Fig.7), but the temperature drops faster at the central wavelength of 976 nm. At this wavelength, the most dangerous effect of the temperature rise is on the surface. The increase of pump power of 100 mW corresponds to the increase of temperature of 9 K (Fig.8).The increase of the beam waist radius leads to the decrease of the optical power density and the decrease of the temperature of the bonded crystal, with only a large difference in temperature at the center of the crystal surface. The pump power is proportional to the thermal deformation. The increase of pump power of 100 mW corresponds to the increases of the thermal deformation of the crystal of 0.5 μm (Fig.17). At the same pump power, the thermal deformation will decrease with the increase of the beam waist radius, but the resulting change is not significant compared to the pump power. Co:MALO crystal bonding on the pump side can effectively reduce temperature (Fig.10), thermal stress (Fig.14) and deformation variables (Fig.18). Conclusions The finite element method was used to study the LD pumped Er:Yb:glass/Co:MALO based on the theory of heat conduction. Due to the heat sink effect of CO:MALO, the bonded crystal can reduce the maximum temperature, thermal stress and thermal deformation of the laser crystal. On this basis, the pump with a central wavelength of 940 nm will reduce the maximum laser temperature, while the 976 nm pump structure is a safer structure for diffusion bonding. Increasing the pump power and decreasing the beam waist radius will also lead to the thermal effect of the crystal. In the design of the laser system, the thermal effects on the output characteristics should be mitigated by preventing excessive crystal temperatures and deformations. This study provides an optimized condition for the further design of Er:Yb:glass laser with better thermal performance, and also provides a theoretical basis for the output of 1.5 μm laser with high power and beam quality. -
Key words:
- LD end-pumped /
- Er:Yb:glass/Co:MALO /
- Q-switched laser /
- thermal effect /
- thermal stress
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图 3 非键合晶体温度分布。(a)前表面；(b)切面
Figure 3. Temperature distribution of unbonded crystal. (a) Front surface; (b) Section
图 4 键合晶体温度分布。(a)前表面；(b)切面
Figure 4. Bonded crystal temperature distribution. (a) Front surface; (b) Section
图 5 温度分布。(a)沿x轴；(b)沿z轴
Figure 5. Temperature distribution. (a) Along the x-axis; (b) Along the z-axis
图 6 976 nm泵浦波长下温度分布。(a) 前表面；(b)切面
Figure 6. Temperature distribution at 976 nm pump wavelength. (a) Front surface; (b) Section
图 7 不同泵浦波长下的温度分布。(a)沿x轴；(b)沿z轴
Figure 7. Temperature distribution at different pump wavelengths. (a) Along x-axis; (b) Along z-axis
8 不同泵浦功率下的温度分布。(a)沿x轴；(b)沿z轴
8. Temperature distribution under different pump power. (a) Along x-axis; (b) Along z-axis
9 不同束腰半径下的温度分布。 (a)沿x轴；(b)沿z轴
9. Temperature distribution under different waist size. (a) Along x-axis; (b) Along z-axis
图 10 温度分布。(a)前表面；(b)切面
Figure 10. Temperature distribution. (a) Front surface; (b) Section
图 11 热应力分布。(a)非键合晶体；(b)键合晶体
Figure 11. Thermal stress distribution.(a) Non bonded crystal; (b) Bonded crystal
图 12 976 nm泵浦波长下热应力分布
Figure 12. Thermal stress distribution at 976 nm pump wavelength
图 13 热应力。(a)不同泵浦功率；(b)不同束腰半径
Figure 13. Thermal stress. (a) Different pump powers; (b) Different waist size
图 15 热形变分布。(a)非键合晶体；(b)键合晶体
Figure 15. Thermal deformation distribution. (a) Non-bonded crystal; (b) Bonded crystal
图 16 976 nm泵浦波长下的热形变分布
Figure 16. Thermal deformation distribution at 976 nm pump wavelength
图 17 热形变。 (a)不同泵浦功率；(b)不同束腰半径
Figure 17. Thermal deformation. (a) Different pump powers; (b) Different waist sizes
表 1 材料参数
Table 1. Material parameters
Parameters Symbol Er:Yb:glass Co:MALO Size a×b/mm2 3×3 3×3 Length L/mm 2.5 1.5 Thermal conductivity K/W·m−1·K−1 0.745 15 Thermal expansion αT/K−1 7.2×10−6 8.0×10−6 Temperature coefficient
of the refractive indexdn/dT/K−1 −2.7×10−6 3.0×10−6 Density ρ/kg·m−3 2633 3500 Refractive index n0 1.53 1.67 Poisson's ratio ν 0.24 0.24 Young's modulus E/Pa 70×109 2.3×105 Quantum defects ξ 0.3648 - Absorption coefficient
at 940 nmα/cm−1 4.28 0.0763 Absorption coefficient
at 975 nmα/cm−1 19.4 0.0783 
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