Anomalous collisional absorption of laser pulses in underdense plasma at low temperature

In a previous paper [M. Kundu, Phys. Plasmas 21, 013302 (2014)], fractional collisional absorption $\left(\alpha \right)$ of laser light in underdense plasma was studied by using a classical scattering model of electron-ion collision frequency ${\nu }_{\mathrm{ei}}$, where total velocity $v=\sqrt{v_{\mathrm{th}}^2+v_0^2}$ (with $v_{\mathrm{th}}$ and $v_0$ as the thermal and the ponderomotive velocity of an electron) dependent Coulomb logarithm $ln\Lambda \left(v\right)$ was shown to be responsible for the anomalous (unconventional) increase of ${\nu }_{\mathrm{ei}}$ and $\alpha \left(\propto {\nu }_{\mathrm{ei}}\right)$ with the laser intensity $I_0$ up to a maximum value about an intensity $I_{\mathrm{c}}$ in the low temperature $(T_{\mathrm{e}}<15\mathrm{eV})$ regime and a conventional $\approx I_0^{-3/2}$ decrease when $I_0\gg I_{\mathrm{c}}$. One may object that the anomalous increase in ${\nu }_{\mathrm{ei}}$ and $\alpha$ were partly due to the artifact introduced in $ln\Lambda$ through the maximum cutoff distance $b_{\max }\propto v$. In this work, we show similar anomalous increase in ${\nu }_{\mathrm{ei}}$ and $\alpha$ versus $I_0$ (in the low temperature and underdense density regime) with more accurate quantum and classical kinetic models of ${\nu }_{\mathrm{ei}}$ without using $ln\Lambda$, but with a proper choice of the total velocity dependent inverse cutoff length $k_{\max }\propto v^2$ (classical) or $k_{\max }\propto v$ (quantum). For a given $I_0<5\times {10}^{14}{\mathrm{W}\mathrm{cm}}^{-2}$, ${\nu }_{\mathrm{ei}}$ versus $T_{\mathrm{e}}$ also exhibits so far unnoticed identical anomalous increase as ${\nu }_{\mathrm{ei}}$ versus $I_0$, even if the conventional $k_{\max }\propto v_{\mathrm{th}}^2$ or $k_{\max }\propto v_{\mathrm{th}}$ (without $v_0$) is chosen. The total velocity dependent $k_{\max }$ in the kinetic models, as proposed here, is found to explain the anomalous increase of $\alpha$ with $I_0$ measured in some earlier laser-plasma experiments.

Figure 1

Variation of ${\nu }_{\mathrm{ei}}/{\omega }_{\mathrm{p}}$ with $v_0/v_{\mathrm{th}}$ in (a) log-linear and (b) log-log scale with $T_{\mathrm{e}}=25.86$ eV, $Z=1$, $n_{\mathrm{p}}={10}^{28}/m^3$, and $\omega /{\omega }_{\mathrm{p}}=5$, giving $\lambda \approx 66$ nm. Legends describe different expressions used for ${\nu }_{\mathrm{ei}}$, e.g., “Bornath-Q” [Eq. (2)], “Bornath-C” [Eq. (6)], “Dawson-C” [Eq. (7)], and the ballistic model [25] by “Mulser-C,” where we use $b_{\min }=1/k_{\max }=Z/v_{\mathrm{th}}^2$, $b_{\max }=v_{\mathrm{th}}/\omega$ for $ln\Lambda$.

Figure 2

Variation of ${\nu }_{\mathrm{ei}}/{\omega }_{\mathrm{p}}$ with $v_0/v_{\mathrm{th}}$ for (a) $T_{\mathrm{e}}=10$ eV and (b) $T_{\mathrm{e}}=15$ eV, using $k_{\max }=v_{\mathrm{th}}^2/Z$. Other parameters are the same as in Fig. 1.

Figure 3

Variation of ${\nu }_{\mathrm{ei}}/{\omega }_{\mathrm{p}}$ with $T_{\mathrm{e}}$ (in eV) using $k_{\max }=v_{\mathrm{th}}^2/Z$ and $I_0={10}^{14}{\mathrm{W}\mathrm{cm}}^{-2}$. Anomalous increase of ${\nu }_{\mathrm{ei}}/{\omega }_{\mathrm{p}}$ is indicated by a shaded region. The ${\nu }_{\mathrm{ei}}/{\omega }_{\mathrm{p}}\propto T_{\mathrm{e}}^{-3/2}$ line shows conventional scaling of ${\nu }_{\mathrm{ei}}/{\omega }_{\mathrm{p}}$ with $T_{\mathrm{e}}$, which does not obey kinetic results just after the frequency maximum with increasing $T_{\mathrm{e}}$, but it may be valid for higher $T_{\mathrm{e}}>80$ eV, where the $T_{\mathrm{e}}^{-3/2}$ line becomes parallel to the lines obtained by kinetic models. Other parameters are same as in Fig. 1.

Figure 4

Variation of ${\nu }_{\mathrm{ei}}/{\omega }_{\mathrm{p}}$ with the peak intensity for different kinetic models using total velocity dependent $k_{\max }=v^2/Z$. Other parameters are same as in Fig. 1. Frequency increases up to a maximum value; after that, it decreases as the intensity is increased.

Figure 5

Variation of ${\nu }_{\mathrm{ei}}/{\omega }_{\mathrm{p}}$ with intensity (using full quantum calculation, represented by “Bornath-Q” in Fig. 4) for successive $k$ values up to $k_{\text{max}}$ (represented by $m$ up to $m_{\text{max}}$) and $n_{\text{max}}=1$. Other parameters are the same as in Fig. 4.

Figure 6

$Q\left(k\right)$ versus $k$ (multiplied by $v_{\mathrm{e}}=v_{\mathrm{th}}=\sqrt{T_{\mathrm{e}}}$) for $T_{\mathrm{e}}=25,50,...,175$ eV with a fixed intensity ${10}^{14}{\mathrm{W}\mathrm{cm}}^{-2}$. We have chosen $k_{\max }=v_{\mathrm{th}}^2/Z,m_{\text{max}}=21$, and $n_{\text{max}}=1$. Other parameters are the same as in Fig. 4. With increasing $T_{\mathrm{e}}$ the peak value of $Q$, and the area under the corresponding $Q$ versus $k$, gradually increases to maximum value for $T_{\mathrm{e}}=75–100$ eV; then it drops for higher $T_{\mathrm{e}}>100$ eV.

Figure 7

Variation of ${\nu }_{\mathrm{ei}}/{\omega }_{\mathrm{p}}$ with $T_{\mathrm{e}}$ (in eV) using $k_{\max }=2v_{\mathrm{th}}$. Anomalous increase of ${\nu }_{\mathrm{ei}}/{\omega }_{\mathrm{p}}$ is indicated by a shaded region. The conventional scaling ${\nu }_{\mathrm{ei}}/{\omega }_{\mathrm{p}}\propto T_{\mathrm{e}}^{-3/2}$ line does not obey kinetic results just after the frequency maximum with increasing $T_{\mathrm{e}}$, but it seems to be valid for higher $T_{\mathrm{e}}>60$ eV, where the $T_{\mathrm{e}}^{-3/2}$ line becomes parallel to the lines obtained by kinetic models. Other parameters are the same as in Fig. 1.

Figure 8

Variation of ${\nu }_{\mathrm{ei}}/{\omega }_{\mathrm{p}}$ and $\alpha$ with laser intensity for a hydrogen like plasma of $Z=1$, $n_p={10}^{28}/m^3$, and $n_p=n_{\mathrm{c}}/25$. For (a), (b) $T_{\mathrm{e}}=15$ eV with classical cutoff $k_{\max }=v^2/Z$, and for (c), (d) $T_{\mathrm{e}}=5$ eV with quantum cutoff $k_{\max }=2v$ are used. Thickness of the plasma slab is $L=200\lambda =13.38\times {10}^{-6}$ m.

Figure 9

Comparison of experimental data (from Ref. [35]) of $\alpha$ versus $I_0$ with predictions from kinetic models for $\lambda =268$ nm, $n_p=0.3n_{\mathrm{c}}$, $T_{\mathrm{e}}=5$ eV, and $k_{\max }=v^2/Z$. Thickness of the plasma slab is chosen as $L=10\lambda =2.68\times {10}^{-6}$ m.

Figure 10

Comparison of experimental data (from Ref. [35]) of $\alpha$ versus $I_0$, with predictions from kinetic models, using a variable density profile $n_{\mathrm{p}}/n_{\mathrm{c}}=0.3\left[1+1.5tanh(I_0/I_{\mathrm{a}}.\mathrm{u}.)\right]$. Other parameters are as in Fig. 9.