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    Journals >High Power Laser Science and Engineering >Volume 9 >Issue 4 >Page 04000e58 > Article
    • High Power Laser Science and Engineering
    • Vol. 9, Issue 4, 04000e58 (2021)
    Static and dynamic mode evolution in high-power distributed side-coupled cladding-pumped fiber amplifiers
    Rumao Tao*, Yu Liu, Lianghua Xie, Cong Gao, Min Li, Benjian Shen, Shan Huang, Honghuan Lin, Jianjun Wang, and Feng Jing
    Author Affiliations
  • Laser Fusion Research Center, China Academy of Engineering Physics, Mianyang 621900, China
    • show less
    DOI: 10.1017/hpl.2021.44 Cite this Article Set citation alerts
    Rumao Tao, Yu Liu, Lianghua Xie, Cong Gao, Min Li, Benjian Shen, Shan Huang, Honghuan Lin, Jianjun Wang, Feng Jing. Static and dynamic mode evolution in high-power distributed side-coupled cladding-pumped fiber amplifiers[J]. High Power Laser Science and Engineering, 2021, 9(4): 04000e58 Copy Citation Text show less

    Abstract

    We present a theoretical study of mode evolution in high-power distributed side-coupled cladding-pumped (DSCCP) fiber amplifiers. A semi-analytical model taking the side-pumping schemes, transverse mode competition, and stimulated thermal Rayleigh scattering into consideration has been built, which can model the static and dynamic mode evolution in high-power DSCCP fiber amplifiers. The mode evolution behavior has been investigated with variation of the fiber amplifier parameters, such as the pump power distribution, the length of the DSCCP fiber, the averaged coupling coefficient, the number of the pump cores and the arrangement of the pump cores. Interestingly, it revealed that static mode evolution induced by transverse mode competition is different from the dynamic evolution induced by stimulated thermal Rayleigh scattering. This shows that the high-order mode experiences a slightly higher gain in DSCCP fiber amplifiers, but the mode instability thresholds for DSCCP fiber amplifiers are higher than those for their end-coupled counterparts. By increasing the pump core number and reducing the averaged coupling coefficient, the mode instability threshold can be increased, which indicates that DSCCP fibers can provide additional mitigation strategies of dynamic mode instability.

    Keywords

    \begin{align}Q\left(r,\phi, z,t\right)&\cong g\left(r,\phi, z,t\right)\left(\frac{v_{\mathrm{p}}-{v}_{\mathrm{s}}}{v_{\mathrm{s}}}\right){I}_{\mathrm{s}}\left(r,\phi, z,t\right)\notag\\&\quad+\gamma \left(r,\phi, z\right){I}_{\mathrm{s}}\left(r,\phi, z,t\right),\end{align} ((1))

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    \begin{align}g\left(r,\phi, z,t\right)=\left[\left({\sigma}_{\mathrm{s}}^{\mathrm{a}}+{\sigma}_{\mathrm{s}}^{\mathrm{e}}\right){n}_{\mathrm{u}}\left(r,\phi, z,t\right)-{\sigma}_{\mathrm{s}}^{\mathrm{a}}\right]{N}_{\mathrm{Yb}}\left(r,\phi \right),\end{align} ((2))

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    \begin{align}{n}_{\mathrm{u}}\left(r,\phi, z,t\right)&=\left\{\left[{P}_{\mathrm{p}}^{+}\left(z,t\right)+{P}_{\mathrm{p}}^{-}\left(z,t\right)\right]{\sigma}_{\mathrm{p}}^{\mathrm{a}}/{hv}_{\mathrm{p}}{A}_{\mathrm{p}}\right.\notag\\&\quad\left.+\sum {P}_{\mathrm{s}i}\left(z,t\right){\Phi}_i{\sigma}_{\mathrm{s}}^{\mathrm{a}}/{hv}_{\mathrm{s}}\right\}\notag\\&\quad\times\left\{\left[{P}_{\mathrm{p}}^{+}\left(z,t\right)+{P}_{\mathrm{p}}^{-}\left(z,t\right)\right]\left({\sigma}_{\mathrm{p}}^{\mathrm{a}}+{\sigma}_{\mathrm{p}}^{\mathrm{e}}\right)/ {hv}_{\mathrm{p}}{A}_{\mathrm{p}}\right.\notag\\&\quad\left.+\sum {P}_{\mathrm{s}i}\left(z,t\right){\Phi}_1\left({\sigma}_{\mathrm{s}}^{\mathrm{a}}+{\sigma}_{\mathrm{s}}^{\mathrm{e}}\right)/{hv}_{\mathrm{s}}+1/\tau\right\}^{-1},\end{align} ((3a))

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    \begin{align}\frac{\mathrm{d}{P}_{\mathrm{p}\mathrm{p}}^{\pm}\left(z,t\right)}{\mathrm{d}z}=-{k}_1{P}_{\mathrm{p}\mathrm{p}}^{\pm}\left(z,t\right)+{k}_2{P}_{\mathrm{p}}^{\pm}\left(z,t\right),\end{align} ((3b))

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    \begin{align}\frac{\mathrm{d}{P}_{\mathrm{p}}^{\pm}\left(z,t\right)}{\mathrm{d}z}&=\frac{P_{\mathrm{p}}^{\pm}\left(z,t\right)}{A_{\mathrm{p}}}\int \int \left[\left({\sigma}_{\mathrm{p}}^{\mathrm{a}}+{\sigma}_{\mathrm{p}}^{\mathrm{e}}\right){n}_{\mathrm{u}}\left(r,\phi, z,t\right)-{\sigma}_{\mathrm{p}}^{\mathrm{a}}\right]\notag\\&\quad\times{N}_{\mathrm{Yb}}\left(r,\phi \right)r\;\mathrm{d}\phi\;\mathrm{d}r+{k}_1{P}_{\mathrm{p}\mathrm{p}}^{\pm}\left(z,t\right)-{k}_2{P}_{\mathrm{p}}^{\pm}\left(z,t\right),\end{align} ((3c))

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    \begin{align}\frac{\mathrm{d}{P}_{\mathrm{s}i}\left(z,t\right)}{\mathrm{d}z}&={P}_{\mathrm{s}i}\left(z,t\right)\int \int \left[\left({\sigma}_{\mathrm{s}}^{\mathrm{a}}+{\sigma}_{\mathrm{s}}^{\mathrm{e}}\right){n}_{\mathrm{u}}\left(r,\phi, z,t\right)-{\sigma}_{\mathrm{s}}^{\mathrm{a}}\right]\notag\\&\quad\times{N}_{\mathrm{Yb}}\left(r,\phi \right){\Phi}_i\;\mathrm{d}\phi\;\mathrm{d}r,\end{align} ((3d))

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    \begin{align}\frac{\mathrm{d}{P}_{\mathrm{p}\mathrm{p}}^{\pm}\left(z,t\right)}{\mathrm{d}z}=-{k}_1{P}_{\mathrm{p}\mathrm{p}}^{\pm}\left(z,t\right)+{k}_2{P}_{\mathrm{p}}^{\pm}\left(z,t\right),\end{align} ((3e))

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    \begin{align}&\xi (L)\approx {\xi}_0\exp \left[{\int}_0^{{L}}\mathrm{d}z\int \int g\left(r,\phi, z\right)\left({\psi}_2{\psi}_2-{\psi}_1{\psi}_1\right)r\;\mathrm{d}r\;\mathrm{d}\phi \right]\notag\\&\quad+\frac{\xi_0}{4}\sqrt{\frac{2\pi }{\int_0^{{L}}{P}_1(z)\left|\chi {''} \left({\Omega}_0,z\right)\right|\;\mathrm{d}z}}\notag\\&\quad\times\exp \left[{\int}_0^{{L}}\mathrm{d}z\int \int g\left(r,\phi, z\right)\left({\psi}_2{\psi}_2-{\psi}_1{\psi}_1\right)r\;\mathrm{d}r\;\mathrm{d}\phi \right]\notag\\&\quad{}\times {R}_{\mathrm{N}}\left({\Omega}_0\right)\exp \left[{\int}_0^{{L}}{P}_1(z)\chi \left({\Omega}_0,z\right)\;\mathrm{d}z\right],\end{align} ((4))

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    \begin{align}\chi \left(\Omega \right)=2\frac{n_0{\omega}_2^2}{c^2{\beta}_2}\operatorname{Im}\left[\int \int \left({\overline{h}}_{12}+{\overline{h}}_{12} {'}\right){\psi}_1{\psi}_2r\;\mathrm{d}r\;\mathrm{d}\phi \right],\end{align} ((5a))

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    \begin{align}{\overline{h}}_{kl}\left(r,\phi, z\right)=\frac{\alpha {n}_2}{\pi}\sum \limits_v\sum \limits_{m=1}^{\infty}\frac{J_v\left({\delta}_m,r\right)}{N\left({\delta}_m\right)}\frac{B_{kl}\left(\phi, z\right)}{{\alpha \delta}_m^2-j\Omega},\end{align} ((5b))

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    \begin{align}{\overline{h}}_{kl} {'}\left(r,\phi, z\right)=\frac{\eta \alpha}{\pi \kappa}\sum \limits_v\sum \limits_{m=1}^{\infty}\frac{J_v\left({\delta}_m,r\right)}{N\left({\delta}_m\right)}\frac{B_{kl} {'}\left(\phi, z\right)}{{\alpha \delta}_m^2-j\Omega},\end{align} ((5c))

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    \begin{align}{B}_{kl}\left(\phi, z\right)&={\int}_0^{2\pi} \hbox{d}\phi {'}{\int}_0^R{g}_0{J}_v\left({\delta}_m,r {'}\right)\cos v\left(\phi -\phi {'}\right)\notag\\&\quad\times\frac{\psi_k\left(r {'},\phi {'}\right){\psi}_{\mathrm{l}}\left(r {'},\phi {'}\right)}{{\left(1+{I}_0/{I}_{\mathrm{saturation}}\right)}^2}\;\mathrm{d}r {'},\end{align} ((5d))

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    \begin{align}{B}_{kl} {'}\left(\phi, z\right)&={\int}_0^{2\pi} \hbox{d}\phi {'}{\int}_0^R\gamma \left(r {'},\phi {'}\right){J}_v\left({\delta}_m,r {'}\right)\cos v\notag\\&\quad\times\left(\phi -\phi {'}\right){\psi}_k\left(r {'},\phi {'}\right){\psi}_{\mathrm{l}}\left(r {'},\phi {'}\right)\;\mathrm{d}r {'},\end{align} ((5e))

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    \begin{align}\frac{1}{N\left({\delta}_m\right)}=\frac{1}{\int_0^R{rJ}_v^2\left({\delta}_m,r\right)\;\mathrm{d}r},\end{align} ((5f))

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    \begin{align}{n}_2=\frac{\eta }{\kappa_{\mathrm{T}}}\left(\frac{v_{\mathrm{p}}-{v}_{\mathrm{s}}}{v_{\mathrm{s}}}\right),\end{align} ((5g))

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    \begin{align}\Omega ={\omega}_1\hbox{--} {\omega}_2,\ \alpha =\kappa /\rho C,\end{align} ((5h))

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    \begin{align}\frac{\mathrm{d}{P}_{\mathrm{p}\mathrm{p}i}^{\pm}\left(z,t\right)}{\mathrm{d}z}=- {k}_1{P}_{\mathrm{p}\mathrm{p}i}^{\pm}\left(z,t\right)+{k}_2{P}_{\mathrm{p}}^{\pm}\left(z,t\right),\end{align} ((6a))

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    \begin{align}&\frac{\mathrm{d}{P}_{\mathrm{p}}^{\pm}\left(z,t\right)}{\mathrm{d}z}=\frac{P_{\mathrm{p}}^{\pm}\left(z,t\right)}{A_{\mathrm{p}}}\int \int \left[\left({\sigma}_{\mathrm{p}}^{\mathrm{a}}+{\sigma}_{\mathrm{p}}^{\mathrm{e}}\right){n}_{\mathrm{u}}\left(r,\phi, z,t\right)-{\sigma}_{\mathrm{p}}^{\mathrm{a}}\right]\notag\\&\enspace\times{N}_{\mathrm{Yb}}\left(r,\phi \right)r\;\mathrm{d}\phi\;\mathrm{d}r+{k}_1\sum \limits_i{P}_{\mathrm{p}\mathrm{p}i}^{\pm}\left(z,t\right)-2{k}_2{P}_{\mathrm{p}}^{\pm}\left(z,t\right),\end{align} ((6b))

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    \begin{align}\frac{\mathrm{d}{P}_{\mathrm{p}\mathrm{p}1}^{\pm}\left(z,t\right)}{\mathrm{d}z}&=- {k}_1{P}_{\mathrm{p}\mathrm{p}1}^{\pm}\left(z,t\right)+{k}_2{P}_{\mathrm{p}}^{\pm}\left(z,t\right)\notag\\&\quad+{k}_3\left[- {P}_{\mathrm{p}\mathrm{p}1}^{\pm}\left(z,t\right)+{P}_{\mathrm{p}\mathrm{p}2}^{\pm}\left(z,t\right)\right],\end{align} ((7a))

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    \begin{align}\frac{\mathrm{d}{P}_{\mathrm{p}\mathrm{p}2}^{\pm}\left(z,t\right)}{\mathrm{d}z}&=-{k}_1{P}_{\mathrm{p}\mathrm{p}2}^{\pm}\left(z,t\right)+{k}_2{P}_{\mathrm{p}}^{\pm}\left(z,t\right)\nonumber\\&\quad{}+{k}_3\left[- {P}_{\mathrm{p}\mathrm{p}2}^{\pm}\left(z,t\right)+{P}_{\mathrm{p}\mathrm{p}1}^{\pm}\left(z,t\right)\right],\end{align} ((7b))

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    \begin{align}\frac{\mathrm{d}{P}_{\mathrm{p}}^{\pm}\left(z,t\right)}{\mathrm{d}z}&=\frac{P_{\mathrm{p}}^{\pm}\left(z,t\right)}{A_{\mathrm{p}}}\int \int \left[\left({\sigma}_{\mathrm{p}}^{\mathrm{a}}+{\sigma}_{\mathrm{p}}^{\mathrm{e}}\right){n}_{\mathrm{u}}\left(r,\phi, z,t\right)-{\sigma}_{\mathrm{p}}^{\mathrm{a}}\right]\notag\\&\quad\times {N}_{\mathrm{Yb}}\left(r,\phi \right)r\;\mathrm{d}\phi\;\mathrm{d}r\notag\\&\quad{}+{k}_1{P}_{\mathrm{p}\mathrm{p}1}^{\pm}\left(z,t\right)+{k}_1{P}_{\mathrm{p}\mathrm{p}2}^{\pm}\left(z,t\right)-2{k}_2{P}_{\mathrm{p}}^{\pm} \left(z,t\right),\end{align} ((7c))

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    Rumao Tao, Yu Liu, Lianghua Xie, Cong Gao, Min Li, Benjian Shen, Shan Huang, Honghuan Lin, Jianjun Wang, Feng Jing. Static and dynamic mode evolution in high-power distributed side-coupled cladding-pumped fiber amplifiers[J]. High Power Laser Science and Engineering, 2021, 9(4): 04000e58
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