• High Power Laser Science and Engineering
  • Vol. 9, Issue 3, 03000e40 (2021)
M. Cipriani1、*, S. Yu. Gus’kov2, F. Consoli1, R. De Angelis1, A. A. Rupasov2, P. Andreoli1, G. Cristofari1, G. Di Giorgio1, and M. Salvadori1、3、4
Author Affiliations
  • 1ENEA, Fusion and Technologies for Nuclear Safety Department, C.R. Frascati, Frascati (Rome), Italy
  • 2Lebedev Physical Institute, Moscow, Russia
  • 3Università di Roma La Sapienza, Roma, Italy
  • 4INRS-EMT, Varennes, Québec, Canada
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    DOI: 10.1017/hpl.2021.27 Cite this Article Set citation alerts
    M. Cipriani, S. Yu. Gus’kov, F. Consoli, R. De Angelis, A. A. Rupasov, P. Andreoli, G. Cristofari, G. Di Giorgio, M. Salvadori. Time-dependent measurement of high-power laser light reflection by low-Z foam plasma[J]. High Power Laser Science and Engineering, 2021, 9(3): 03000e40 Copy Citation Text show less

    Abstract

    Porous materials have many applications for laser–matter interaction experiments related to inertial confinement fusion. Obtaining new knowledge about the properties of the laser-produced plasma of porous media is a challenging task. In this work, we report, for the first time to the best of our knowledge, the time-dependent measurement of the reflected light of a terawatt laser pulse from the laser-produced plasma of low-Z foam material of overcritical density. The experiments have been performed with the ABC laser, with targets constituted by foam of overcritical density and by solid media of the same chemical composition. We implemented in the MULTI-FM code a model for the light reflection to reproduce and interpret the experimental results. Using the simulations together with the experimental results, we indicate a criterion for estimating the homogenization time of the laser-produced plasma, whose measurement is challenging with direct diagnostic techniques and still not achieved.
    $$\begin{align}\mathrm{IsFoam}\left(x,t\right)=1-\frac{H\left(x,t\right)}{H_c},\end{align}$$ ((1))

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    $$\begin{align}H\left(x,t\right)=2{\int}_{0}^{t}\frac{\hbox{d}t^{\prime }}{\tau_0\left(x,t^{\prime}\right)},\kern1.2em 0\le H\left(x,t\right)\le H\left(x,{t}_{{h}}\right),\end{align}$$ ((2))

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    $$\begin{align}{\tau}_0\left(x,t^{\prime}\right)&\approx 2.4\cdot {10}^{-3}\frac{Z^4{\left({\delta}_0-{b}_0\right)}^2{\rho}_p}{A^{1/2}\;T{\left(x,t^\prime \right)}^{5/2}}\notag\\[4pt]&\approx 2.4\cdot {10}^{-3}\frac{Z^4{\delta}_0^2{\rho}_p}{A^{1/2}\;T{\left(x,t^\prime \right)}^{5/2}}\end{align}$$ ((3))

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    $$\begin{align}H\left(x,{t}_{{h}}\right)={H}_c=\left\{\displaystyle\begin{array}{c}1,\kern5.039997em {\rho}_p\ge {\rho}_{\mathrm{cr},}\\[7pt] {}1-{\left[\frac{{\left(1-{\rho}_p/{\rho}_{\mathrm{cr}}\right)}^{\alpha }}{{\left(1-{\rho}_p/{\rho}_{{s}}\right)}^{\alpha }}\right]}^2,\kern0.24em {\rho}_p\le {\rho}_{\mathrm{cr}},\end{array}\right.\end{align}$$ ((4))

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    $$\begin{align}\frac{\delta_0}{b_0}\approx {\left(\frac{\rho_{{s}}}{\rho_p}\right)}^{\alpha }.\end{align}$$ ((5))

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    $$\begin{align}K\left(x,t\right)=\left\{\begin{array}{ll}\begin{array}{l}\left\{\vphantom{\left[{K}_{{f}}\left(x,t\right)\cdot\mathrm{IsFoam}\Big(x,t\Big)\right]^2}\left[{K}_{{f}}\left(x,t\right)\cdot\mathrm{IsFoam}\left(x,t\right)\right]^2\right.\\+\left.\left\{{K}_{{b}}\left(x,t\right)\cdot\left[1-\mathrm{IsFoam}\left(x,t\right)\right]\right\}\vphantom{\left[{K}_{{f}}\left(x,t\right)\cdot\mathrm{IsFoam}\Big(x,t\Big)\right]^2}\right\}^{1/2},\end{array}&\\& \kern-6pc\rho \left(x,t\right)<{\rho}_{\mathrm{cr}},\\[6pt]{}{K}_{{f}}\left(x,t\right)\cdot \mathrm{IsFoam}\left(x,t\right),&\\& \kern-6pc\rho \left(x,t\right)\ge {\rho}_{\mathrm{cr}},\end{array} \right.\end{align}$$ ((6))

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    $$\begin{align}{K}_{{f}}\left(x,t\right)&=\frac{\delta_0}{b_0{L}_{p0}}\left\{\frac{1}{\left(1-\frac{b_0}{\delta_0}\right){\left[1-H\left(x,t\right)\right]}^{1/2}}-1\right\},\notag\\&\kern10pc0\le t\le {t}_{{h}},\end{align}$$ ((7))

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    $$\begin{align}{L}_{p0}\approx \frac{\pi^2}{2}\frac{\rho_{{s}}}{\rho_p}{b}_0,\end{align}$$ ((8))

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    $$\begin{align*}q\left(x,t\right)=-\left[1-\mathrm{IsFoam}\left(x,t\right)\right]\left[\kappa \left(x,t\right)\frac{\partial {T}_{{e}}\left(x,t\right)}{\partial x}\right],\end{align*}$$ ()

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    $$\begin{align}\frac{\partial }{\partial t}v\left(x,t\right)=-\left[1-\mathrm{IsFoam}\left(x,t\right)\right]\frac{\partial }{\partial m}P\left(x,t\right),\end{align}$$ ((9))

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    $$\begin{align*}S={\partial}_x{I}_{+}-{\partial}_x{I}_{-}.\end{align*}$$ ()

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    $$\begin{align*}{\left(\frac{S}{\rho}\right)}_i=\frac{I_{+,i+1}-{I}_{+,i}-{I}_{-,i+1}+{I}_{-,i}}{{\Delta m}_i},\end{align*}$$ ()

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    $$\begin{align}{I}_{+,i}={I}_{+,i+1}\;\exp \left(-{K}_i\cdot {\Delta x}_i\right),\end{align}$$ ((10))

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    $$\begin{align}{I_{{w}}}^{\mathrm{R}}&=\left(\chi \cdot {\mathrm{IsFoam}}_{{w}+1/2}\right)\;{I}_{{w}},\notag\\[-2pt]{I_{{w}}}^{\mathrm{T}}&=\left(1-\chi \cdot {\mathrm{IsFoam}}_{{w}+1/2}\right)\;{I}_{{w}},\end{align}$$ ((11))

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    $$\begin{align}\chi \approx \frac{2}{\pi^2}{\left(\frac{\rho_p}{\rho_{{s}}}\right)}^{\left(1-\alpha \right)}=\frac{2}{\pi^2}{\left(\frac{b_0}{\delta_0}\right)}^{\left(1-\alpha \right)/\alpha }.\end{align}$$ ((12))

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    $$\begin{align*}{I_{{w}}}^{\mathrm{R}}+{I_{{w}}}^{\mathrm{T}}={I}_{{w}},\end{align*}$$ ()

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    M. Cipriani, S. Yu. Gus’kov, F. Consoli, R. De Angelis, A. A. Rupasov, P. Andreoli, G. Cristofari, G. Di Giorgio, M. Salvadori. Time-dependent measurement of high-power laser light reflection by low-Z foam plasma[J]. High Power Laser Science and Engineering, 2021, 9(3): 03000e40
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