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在光纤放大器中,由稳态情况下的速率方程可得铒、镱共掺光纤(EYDF)内激光功率的分布,其中粒子数对应的能级图如图1所示。
基于图1粒子能级图,结合对应的粒子数方程,可得稳态下的功率传输方程为:
$$ \begin{split} \frac{\partial {P}_{p}\left(z,{\lambda }_{p}\right)}{\partial {z}}=&{\varGamma }_{p}[{\sigma }_{65}\left({\lambda }_{p}\right){N}_{6}\left(z\right)-{\sigma }_{56}{\left({\lambda }_{p}\right)N}_{5}\left(z\right)-\\ &{\sigma }_{13}\left({\lambda }_{p}\right){N}_{1}\left(z\right)]{P}_{p}(z,{\lambda }_{p})-{\alpha }_{p}{P}_{p}(z,{\lambda }_{p}) \end{split} $$ (1) $$ \begin{split} \frac{\partial {{P}}_{{s}}\left({z},{{\lambda }}_{{s}}\right)}{\partial {z}}=&{{\varGamma }}_{{s}}\left[{{{\sigma }}_{21}{\left({{\lambda }}_{{s}}\right){N}}_{2}\left({z}\right)-{\sigma }}_{12}\left({{\lambda }}_{{s}}\right){{N}}_{1}\left({z}\right)\right]{{P}}_{{s}}({z},{{\lambda }}_{{s}})-\\ &{{\alpha }}_{{s}}{{P}}_{{s}}({z},{{\lambda }}_{{s}})\\[-10pt] \end{split} $$ (2) $$ \begin{split} \frac{\partial {P}_{ASE,Er}\left(z,{\lambda }_{k1}\right)}{\partial z}=&{\varGamma }_{s}\left[{{\sigma }_{21}\left({\lambda }_{k1}\right){N}_{2}\left(z\right)-\sigma }_{12}\left({\lambda }_{k1}\right){N}_{1}\left(z\right)\right]\cdot\\ &{P}_{ASE,Er}(z,{\lambda }_{k1})-{\alpha }_{ASE,Er}{P}_{ASE,Er}(z,{\lambda }_{k1}) \end{split} $$ (3) $$ \begin{split} \frac{\partial {P}_{ASE,Yb}\left(z,{\lambda }_{k2}\right)}{\partial z}=&{\varGamma }_{p}[{\sigma }_{65}\left({\lambda }_{k2}\right){N}_{6}\left(z\right)-{\sigma }_{56}\left({\lambda }_{k2}\right){N}_{5}\left(z\right)-\\ &{\sigma }_{13}\left({\lambda }_{k2}\right){N}_{1}\left(z\right)]{P}_{ASE,Yb}\left(z,{\lambda }_{k2}\right)-\\ &{\alpha }_{ASE,Yb}{P}_{ASE,Yb}(z,{\lambda }_{k2})\\[-10pt] \end{split} $$ (4) 式中:
$ {N}_{1} $ 、$ {N}_{2} $ 、$ {N}_{3} $ 分别为铒离子不同能级上的粒子数密度;$ {N}_{5} $ 、$ {N}_{6} $ 分别为镱离子对应能级上的粒子数密度,其与功率对应的关系可由图1对应的粒子数方程求得,$ {C}_{cr} $ 为镱离子向铒离子能量转换的交叉弛豫系数,$ {A}_{21}=1/{\tau }_{21} $ 、$ {A}_{65}=1/{\tau }_{65} $ 、$ {S}_{32}=1/{\tau }_{32} $ 、其中$ {\tau }_{21} $ 、$ {\tau }_{65} $ 分别为对应能级粒子的自发辐射寿命,$ {\tau }_{32} $ 为无辐射跃迁寿命,$ {W}_{12} $ 、$ {W}_{13} $ 、$ {W}_{56} $ 、$ {W}_{21} $ 和$ {W}_{65} $ 分别为对应能级的受激吸收和受激发射概率。${{\varGamma }}_{p}$ 和${{\varGamma }}_{s}$ 分别为泵浦光和信号光的功率填充因子;$ {P}_{s} $ 、$ {P}_{p} $ 、$ {P}_{ASE,Er} $ 和$ {P}_{ASE,Yb} $ 分别为信号光、泵浦光、铒和镱离子放大自发辐射功率;$ {\sigma }_{12} $ 、$ {\sigma }_{21} $ 、$ {\sigma }_{13} $ 、$ {\sigma }_{56} $ 和$ {\sigma }_{65} $ 分别为对应的吸收、发射截面;$ h $ 为普朗克常量;$ {A}_{eff} $ 为有效模场面积;$ {\lambda }_{s} $ 、$ {\lambda }_{p} $ 、$ {\lambda }_{ASE,Er} $ 和$ {\lambda }_{ASE,Yb} $ 分别为信号光、泵浦光、铒和镱离子放大自发辐射波长;$ v=c/\lambda $ ,其中$ v $ 为输出光的波长;$ c $ 为光速。联立公式(1)~(4),采用数值求解的方式便可达到$ {P}_{s} $ 泵浦光、$ {P}_{p} $ 信号光、$ {P}_{ASE,Er} $ 铒和$ {P}_{ASE,Yb} $ 镱放大自发辐射光功率在光纤中的具体分布。在MOPA结构的放大器中,耦合到EYDF的一部分泵浦光将转化为热量,进而影响增益光纤的折射率。设局部光纤中转化的热量与入射泵浦功率成正比,则双包层光纤单位时间内吸收的热量为:
$$ {Q}\left({z},{r},{t}\right)={{\alpha }}_{{{{a}}}{{{p}}}}{\eta }{P}_{p}\left(t\right)h\left(z\right){\left|e\left(r\right)\right|}^{2} $$ (5) 式中:
${{\alpha }}_{{a}{p}}$ 为泵浦光的吸收系数;$ \mathrm{\eta } $ 为吸收泵浦能量的热转换系数;$ {P}_{p}\left(t\right) $ 为泵浦功率,泵浦电源的波动可通过影响泵浦效率进而影响泵浦光的时间稳定性[12],即泵浦光随时间$ t $ 波动;$ {\left|e\left(r\right)\right|}^{2} $ 为泵浦横向分布函数;$ h\left(z\right) $ 为泵浦纵向分布函数,其可由公式(1)~(4)得出。泵浦光的热转换系数$ \eta $ 由泵浦光与信号光之间的量子亏损以及上能级粒子快速非辐射衰减决定[13],即:$$ \eta ={F}_{ETU}+\left[1-{F}_{ETU}\right](1-{\lambda }_{p}/{\lambda }_{s}) $$ (6) 式中:
$ {\lambda }_{p} $ 和$ {\lambda }_{s} $ 分别为泵浦光和信号光的波长;$ {F}_{ETU} $ 由受激吸收的粒子由能量传递上转换效应造成的反转粒子数衰减决定,即:$$ {F}_{ETU}=\left|\frac{\mathrm{u}\mathrm{p}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}{\mathrm{a}\mathrm{b}\mathrm{s}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}\right|\approx 1-\frac{{P}_{s}\left(l\right)}{{P}_{p}\left(0\right)-{P}_{p}\left(l\right)} ( {\lambda }_{s}/{\lambda }_{p} ) $$ (7) 为得到中频段输出光的噪声分布情况,假设该归一化横向模场为高斯分布[10],则泵浦横向分布函数为:
$$ {\left|e\left(r\right)\right|}^{2}=\frac{2}{\pi {{\omega }_{p}}^{2}}{\rm{exp}}\left(-\frac{2{r}^{2}}{{{\omega }_{p}}^{2}}\right) $$ (8) 式中:
$ {\omega }_{p} $ 为高斯分布的半径。将公式(5)代入热传导方程,对其进行傅里叶变化,并采用圆柱坐标系可得:$$ {k}^{2}{T}_{k}(z,f)+\frac{i{C}_{v}f}{{k}_{t}}{T}_{k}(z,f)=\frac{N\left(z\right)}{{k}_{t}}F\left(k\right){P}_{p}\left(f\right) $$ (9) 式中:
$ {C}_{v} $ 为单位体积的比热容;$ {k}_{t} $ 为介质内的热导率;$ T $ 为对应的温度;$ f $ 为时间$ t $ 的傅里叶频率。其中:$$ N\left(z\right) = {\alpha }_{ap}\eta h\left(z\right) $$ (10) $$ {T}_{k}(z,f)={\int }_{0}^{\infty }T(z,r,f){J}_{1}\left(kr\right)r{\rm{d}}r $$ (11) $$ F\left(k\right)={\int }_{0}^{\infty }{\left|e\left(r\right)\right|}^{2}{J}_{1}\left(kr\right)r{\rm{d}}r=\frac{1}{2\pi }{\rm{exp}}\left(-\frac{{{\omega }_{p}}^{2}{k}^{2}}{8}\right) $$ (12) 公式(11)~(12)中的
$ {J}_{1} $ 为一阶贝塞尔函数。对公式(9)进行汉克尔逆变换来求解$ {T}_{k} $ ,并在横向上对温度波动进行平均,得到泵浦功率与温度场之间的表达式为:$$ T(z,f)=\frac{N\left(z\right){P}_{p}\left(f\right)}{4{\pi }^{2}{k}_{t}}{\int }_{0}^{\infty }\frac{k{\rm{exp}}\left(-\dfrac{{{\omega }_{p}}^{2}{k}^{2}}{4}\right)}{{k}^{2}+2i{{k}_{1}}^{2}}{\rm{d}}k $$ (13) 泵浦扰动引起温度变化的传递函数为:
$$ \varTheta \left(f\right)=\frac{{\rm{exp}}(i{{\omega }_{p}}^{2}{C}_{v}f/4{k}_{t})}{8\pi {k}_{t}}{E}_{1}\left[i{{\omega }_{p}}^{2}{C}_{v}f/4{k}_{t}\right] $$ (14) 式中:
$ {k}_{1} $ =$\sqrt{\dfrac{{C}_{v}f}{2{k}_{t}}}$ ;$ {E}_{1} $ 为指数积分函数。泵浦扰动在增益光纤内引起的温度变化将影响增益光纤的光程,进而作用于输出激光的相位[14]。在光纤长度为
$ l $ 的光纤内,因自热效应引起的激光频率变化为:$$ \Delta \nu \left(f\right)=-\nu \mathcal{E}\varTheta \left(f\right)\Delta {P}_{p}\left(f\right){\int }_{0}^{l}N\left(z\right){\rm{d}}z $$ (15) 故泵浦激光引起的相位噪声为:
$$ S\left(f\right)={\nu }^{2}{\mathcal{E}}^{2}{\left[\varTheta \left(f\right)\right]}^{2}{{P}_{p}}^{2}{\left[{\alpha }_{ap}\eta h\left(z\right)\right]}^{2}{RIN}_{P}\left(f\right) $$ (16) 式中:
$ \nu $ 为输出激光的中心频率;$ \mathcal{E} $ 为材料的热光系数;$ \varTheta \left(f\right) $ 泵浦扰动引起温度变化的传递函数;$ {P}_{p} $ 为泵浦的功率;${\mathrm{\alpha }}_{{a}{p}}$ 为泵浦光的吸收系数;$ \mathrm{\eta } $ 为吸收泵浦能量的热转换系数;$ {RIN}_{P}\left(f\right) $ 为泵浦光的相对强度噪声;$ h\left(z\right) $ 为泵浦纵向分布函数。 -
由实验可得,泵浦激光的强度噪声为
$ 1/f $ 噪声和脉冲尖峰噪声的混合,设泵浦激光的$ {RIN}_{P}\left(f\right) $ [15]为:$$ {RIN}_{P}\left(f\right)=K\left[\left(1+a\sigma \left({f}_{k1}\right)+b\sigma \left({f}_{k2}\right)\right)/{f}^{3/2}\right] $$ (17) 式中:
$ {f}_{k1} $ 分别为150、250、350、450、550、650、750、850、950 Hz;$ {f}_{k2} $ 为200、300、400、500、600、700、800、900 Hz;$ a=1\times {10}^{3} $ ;$b =1\times {10}^{1} $ ;$ K=1\times {10}^{-11} $ 。若考虑种子光本底
$ 1/f $ 噪声的影响[16],则输出端的相位噪声$ S\left(f\right) $ 为:$$ \begin{split} {S}_{\varphi }\left(f\right)=&{\nu }^{2}{\mathcal{E}}^{2}{\left[\varTheta \left(f\right)\right]}^{2}{\left[{\alpha }_{ap}\eta {P}_{p}h\left(z\right)\right]}^{2}{RIN}_{P}\left(f\right)+\\ &{K}_{1}(1/{f}^{3})+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m}\left({f}\right) \end{split} $$ (18) 式中:
$ {K}_{1}=1\times {10}^{10} $ ;$ \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m}\left({f}\right) $ 为随机白噪声。基于表1的实验参数,由公式(16)、(18)可分别对EYDF放大器中泵浦光产生的相位噪声和增加本底的相位噪声进行仿真计算。表 1 激光相位噪声仿真的实验参数
Table 1. Experimental parameters for laser phase noise simulation
Parameter Value Parameter Value $ {\lambda }_{s} $ 1550 nm $ {\sigma }_{12}\left({\lambda }_{s}\right) $ $ 1.75\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ {\lambda }_{p} $ 915 nm $ {\sigma }_{21}\left({\lambda }_{s}\right) $ $ 2.45\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ {\lambda }_{k1} $ 1575 nm $ {\sigma }_{13}\left({\lambda }_{p}\right) $ $ 4.13\times {10}^{-28} $ $ {\rm{m}}^{2} $ $ {\lambda }_{k2} $ 975 nm $ {\sigma }_{56}\left({\lambda }_{p}\right) $ $ 2.2\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ h $ $6.626\times {10}^{-34}\;{\rm{Js }}$ $ {\sigma }_{65}\left({\lambda }_{p}\right) $ $ 4.38\times {10}^{-27} $ $ {\rm{m}}^{2} $ $ c $ $3\times {10}^{8}\;{\rm{m} }/{\rm{s}}$ $ {\sigma }_{12}\left({\lambda }_{k1}\right) $ $ 1.75\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ {\mathrm{\Gamma }}_{p} $ $ 0.007\;885 $ $ {\sigma }_{21}\left({\lambda }_{k1}\right) $ $ 2.35\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ {\mathrm{\Gamma }}_{s} $ $ 0.925\;4 $ $ {\sigma }_{13}\left({\lambda }_{k2}\right) $ $ 1.35\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ {A}_{eff} $ $418.64\;{\text{μ}{\rm{m} } }^{2}$ $ {\sigma }_{56}\left({\lambda }_{k2}\right) $ $ 1.1\times {10}^{-24} $ $ {\rm{m}}^{2} $ $ {\alpha }_{s} $ $ 0.07\;\mathrm{d}\mathrm{B}/\mathrm{\rm{m}} $ $ {\sigma }_{65}\left({\lambda }_{k2}\right) $ $ 5.84\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ {\alpha }_{p} $ $ 0.22\;\mathrm{d}\mathrm{B}/\mathrm{\rm{m}} $ $ {C}_{cr} $ $ 5.54\times {10}^{-22}\;{\mathrm{\rm{m}}}^{3}/\mathrm{s} $ $ {N}_{Er} $ $ 1.2\times {10}^{25}/{\mathrm{\rm{m}}}^{3} $ $ {\tau }_{21} $ $ 10\;\mathrm{\rm{m}}\mathrm{s} $ $ {N}_{Yb} $ $ 2.16\times {10}^{26}/{\mathrm{\rm{m}}}^{3} $ $ {\tau }_{32} $ $ 1\;\mathrm{n}\mathrm{s} $ $ {\alpha }_{ASE,Yb} $ $ 0.25\;\mathrm{d}\mathrm{B}/\mathrm{\rm{m}} $ $ {\tau }_{65} $ $ 1.5\;\mathrm{\rm{m}}\mathrm{s} $ $ {\alpha }_{ASE,Er} $ $ 0.15\;\mathrm{d}\mathrm{B}/\mathrm{\rm{m}} $ $ {\omega }_{p} $ $6.5\times {10}^{-6}\;{\rm{m}}$ $ \mathcal{E} $ $1.008\times {10}^{-5}/{\rm{K } }$ $ {C}_{v} $ $2.2\times {10}^{6}\;{\rm{J}}\cdot {({\rm{m} }^{3}\cdot {\rm{K} })}^{-1}$ $ {\alpha }_{ap} $ $ 3\;{\rm{m}}^{-1} $ $ {k}_{t} $ $0.84 \;{\rm{W}}\cdot {({\rm{m} }\cdot {\rm{K} })}^{-1}$ 当增益光纤的长度为5 m、泵浦波长为915 nm时,不同泵浦功率通过热波动引起的相位噪声如图2所示。由图2(a)可知,随着泵浦功率的增大,泵浦功率波动引起激光的相位噪声逐渐增加,但其增长速率逐渐降低。由于整个光热转换过程发生在放大器中,其输出功率不能实现锁定,使光纤末端泄露的光功率与泵浦功率成正相关,故输出光的相位噪声不随泵浦功率线性增加。在考虑种子光的本底噪声时,整个相位噪声如图2(b)所示。此时100 Hz~1 kHz范围内的电子尖峰噪声随泵浦功率的增大而增大,但对应的基底噪声却基本不随泵浦功率而改变。这是由于热噪声的基底太小,被种子本底的
$ 1/f $ 噪声所淹没,对应的基底噪声变化不明显,而在100 Hz~1 kHz范围内的尖峰噪声较强,故能在本底上显露出来。由公式(1)~(4)和公式(6)可得,泵浦波长
$ {\lambda }_{p} $ 会影响热转换效率$ \eta $ 和泵浦功率的分布$ {P}_{p}h\left(z\right) $ 。其通过增益光纤中泵浦光与信号光的光-光转换效率影响输出光的相位噪声,如图3所示。由图3(a)可知,当泵浦波长$ {\lambda }_{p} $ 增加时,由热波动引起的相位噪声逐渐减小。由于泵浦波长$ {\lambda }_{p} $ 的不同,激光对应的吸收、发射截面$ \sigma $ 也不同。在相同泵浦功率和增益光纤长度的条件下,975 nm的泵浦光产生的热噪声小于915 nm的泵浦光。在考虑种子本底的$ 1/f $ 噪声时,975 nm泵浦的相位噪声基本被淹没,图3(b)无法观测到100 Hz~1 kHz频段内的电子尖峰噪声。故根据对应的吸收发射截面$ \sigma $ 优化泵浦光的波长,便可降低输出光的相位噪声。图 2 5 m光纤、915 nm泵浦下不同泵浦功率的相位噪声。(a)泵浦噪声影响的相位噪声;(b)泵浦噪声结合本底的相位噪声
Figure 2. Phase noise of different pump powers under 5 m fiber and 915 nm pump. (a) Phase noise influenced by pump noise; (b) Phase noise combined with pump noise and background
在放大器中,增益光纤的长度是影响热噪声的重要因素。文中仿真计算了泵浦功率为482 mW、泵浦波长为915 nm时,不同增益光纤长度下通过热波动引起的相位噪声,如图4所示。由图4(a)可知,泵浦产生的相位噪声与增益光纤长度成正相关。当光纤长度越长时,增益光纤对泵浦光的吸收便越强,故泵浦转化的热噪声也将越强。然而泵浦相位噪声的增长速率却与增益光纤长度成反比。在考虑本底的
$ 1/f $ 噪声时,随着增益光纤长度的缩减,对应的100 Hz~1 kHz频段内的电子尖峰噪声逐渐被淹没,如图4(b)所示。倘若在保证输出功率的条件下,根据速率方程减短增益光纤的长度,便可降低输出光的相位噪声特性,从而提高激光放大器的输出性能。
Phase noise of pumping in single-frequency fiber amplifier
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摘要: 研究了铒镱共掺单频光纤放大器中100 Hz~1 kHz频段内的相位噪声,并通过实验证明该尖峰噪声为泵浦电源产生的相位噪声。从铒镱共掺光纤放大器的功率传输方程出发、结合泵浦激光的热传递函数,数值分析了泵浦功率、泵浦波长、增益光纤长度对100 Hz~1 kHz频段内相位噪声的影响。通过二级光放大结构对输出激光的相位噪声进行测量,并将实验结果与数值仿真结果进行对比,证明了理论模型的可靠性。该研究优化了主谐振功率放大结构铒镱共掺单频光纤放大器的相位噪声特性、并为提高相干合成时的合束效率提供指导。以上所得结果普遍适用于主谐振功率放大结构的光纤放大器。Abstract: The phase noise in the 100 Hz-1 kHz band of erbium-ytterbium co-doped single-frequency fiber amplifier was studied, and it was proved by experiments that the peak noise was the phase noise produced by the pump power supply. Based on the power transfer equations of erbium-ytterbium co-doped fiber amplifier and the heat transfer function of pump laser, the effects of pump power, pump wavelength and gain fiber length on the phase noise in 100 Hz-1 kHz band were analyzed numerically. The phase noise of the output laser was measured by using two-stage optical amplifier structure, and the experimental results were compared with the numerical simulation results, which proved the reliability of the theoretical model. This study optimized the phase noise characteristics of erbium-ytterbium co-doped single-frequency fiber amplifier with main resonance power amplifier structure and provided guidance for improving beam combining efficiency in coherent combination. In general, the aforementioned findings hold true for fiber amplifiers with main oscillation power amplification structure.
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图 6 输出电流1.1 A,不同电源的相位噪声。(a) IT6332 A电流源的相位噪声;(b) LPS-305电流源的相位噪声;(c)不同泵浦电源尖峰噪声分析图
Figure 6. Output current is 1.1 A, and the phase noise of different power supplies. (a) Phase noise of the IT6332 A current source; (b) Phase noise of the LPS-305 current source; (c) Analysis graph of the spike noise of different pump power supplies
图 11 不同增益光纤长度下的相位噪声。(a)增益光纤长度为5 m;(b)增益光纤长度为2.5 m;(c)增益光纤长度为1 m;(d)不同光纤长度尖峰噪声分析图(插图:强度噪声)
Figure 11. Phase noise at different gain fiber lengths. (a) Gain fiber length is 5 m; (b) Gain fiber length is 2.5 m; (c) Gain fiber length is 1 m; (d) Spikes with different fiber lengths noise analysis chart (Inset: Intensity noise)
表 1 激光相位噪声仿真的实验参数
Table 1. Experimental parameters for laser phase noise simulation
Parameter Value Parameter Value $ {\lambda }_{s} $ 1550 nm $ {\sigma }_{12}\left({\lambda }_{s}\right) $ $ 1.75\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ {\lambda }_{p} $ 915 nm $ {\sigma }_{21}\left({\lambda }_{s}\right) $ $ 2.45\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ {\lambda }_{k1} $ 1575 nm $ {\sigma }_{13}\left({\lambda }_{p}\right) $ $ 4.13\times {10}^{-28} $ $ {\rm{m}}^{2} $ $ {\lambda }_{k2} $ 975 nm $ {\sigma }_{56}\left({\lambda }_{p}\right) $ $ 2.2\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ h $ $6.626\times {10}^{-34}\;{\rm{Js }}$ $ {\sigma }_{65}\left({\lambda }_{p}\right) $ $ 4.38\times {10}^{-27} $ $ {\rm{m}}^{2} $ $ c $ $3\times {10}^{8}\;{\rm{m} }/{\rm{s}}$ $ {\sigma }_{12}\left({\lambda }_{k1}\right) $ $ 1.75\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ {\mathrm{\Gamma }}_{p} $ $ 0.007\;885 $ $ {\sigma }_{21}\left({\lambda }_{k1}\right) $ $ 2.35\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ {\mathrm{\Gamma }}_{s} $ $ 0.925\;4 $ $ {\sigma }_{13}\left({\lambda }_{k2}\right) $ $ 1.35\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ {A}_{eff} $ $418.64\;{\text{μ}{\rm{m} } }^{2}$ $ {\sigma }_{56}\left({\lambda }_{k2}\right) $ $ 1.1\times {10}^{-24} $ $ {\rm{m}}^{2} $ $ {\alpha }_{s} $ $ 0.07\;\mathrm{d}\mathrm{B}/\mathrm{\rm{m}} $ $ {\sigma }_{65}\left({\lambda }_{k2}\right) $ $ 5.84\times {10}^{-25} $ $ {\rm{m}}^{2} $ $ {\alpha }_{p} $ $ 0.22\;\mathrm{d}\mathrm{B}/\mathrm{\rm{m}} $ $ {C}_{cr} $ $ 5.54\times {10}^{-22}\;{\mathrm{\rm{m}}}^{3}/\mathrm{s} $ $ {N}_{Er} $ $ 1.2\times {10}^{25}/{\mathrm{\rm{m}}}^{3} $ $ {\tau }_{21} $ $ 10\;\mathrm{\rm{m}}\mathrm{s} $ $ {N}_{Yb} $ $ 2.16\times {10}^{26}/{\mathrm{\rm{m}}}^{3} $ $ {\tau }_{32} $ $ 1\;\mathrm{n}\mathrm{s} $ $ {\alpha }_{ASE,Yb} $ $ 0.25\;\mathrm{d}\mathrm{B}/\mathrm{\rm{m}} $ $ {\tau }_{65} $ $ 1.5\;\mathrm{\rm{m}}\mathrm{s} $ $ {\alpha }_{ASE,Er} $ $ 0.15\;\mathrm{d}\mathrm{B}/\mathrm{\rm{m}} $ $ {\omega }_{p} $ $6.5\times {10}^{-6}\;{\rm{m}}$ $ \mathcal{E} $ $1.008\times {10}^{-5}/{\rm{K } }$ $ {C}_{v} $ $2.2\times {10}^{6}\;{\rm{J}}\cdot {({\rm{m} }^{3}\cdot {\rm{K} })}^{-1}$ $ {\alpha }_{ap} $ $ 3\;{\rm{m}}^{-1} $ $ {k}_{t} $ $0.84 \;{\rm{W}}\cdot {({\rm{m} }\cdot {\rm{K} })}^{-1}$ -
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