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@@ -242,7 +242,7 @@ distributions in silicon nanoparticle around a magnetic resonance.}
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\label{fgr:concept}
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\end{figure}
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-On the other hand, plasma explosion imaging technique has been used to
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+Recently, plasma explosion imaging technique has been used to
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observe electron-hole plasmas (EHP), produced by femtosecond lasers,
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inside NPs~\cite{Hickstein2014}. Particularly, a strongly
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localized EHP in the front side\footnote{The incident wave propagates
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@@ -264,7 +264,7 @@ numerical simulation of the intense femtosecond (\textit{fs}) laser
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pulse interaction with a silicon NP supporting Mie
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resonances and two-photon free carrier generation. In particular, we
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couple finite-difference time-domain (FDTD) method used to solve
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-Maxwell equations with kinetic equations describing nonlinear EHP
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+three-dimensional Maxwell equations with kinetic equations describing nonlinear EHP
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generation. Three-dimensional transient variation of the material
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dielectric permittivity is calculated for NPs of several
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sizes. The obtained results propose a novel strategy to create
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@@ -333,7 +333,7 @@ evolution.
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\subsection{Light propagation}
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Ultra-short laser interaction and light propagation inside the silicon
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-NP are modeled by solving the system of Maxwell's equations
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+NP are modeled by solving the system of three-dimensional Maxwell's equations
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written in the following way
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\begin{align} \begin{cases} \label{Maxwell}$$
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\displaystyle{\frac{\partial{\vec{E}}}{\partial{t}}=\frac{\nabla\times\vec{H}}{\epsilon_0\epsilon}-\frac{1}{\epsilon_0\epsilon}(\vec{J}_p+\vec{J}_{Kerr})} \\
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@@ -375,10 +375,10 @@ Gaussian slightly focused beam as follows
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\begin{aligned}
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\label{Gaussian}
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{E_x}(t, r, z) = \frac{w_0}{w(z)}{exp}\left(i\omega{t} - \frac{r^2}{{w(z)}^2} - ikz - ik\frac{r^2}{2R(z)} + i\varsigma(z)\right)\\
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-\times\;{exp}\left(-\frac{(t-t_0)^2}{\theta^2}\right),
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+\times\;{exp}\left(-\frac{4\ln{2}(t-t_0)^2}{\theta^2}\right),
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\end{aligned}
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\end{align}
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-where $\theta$ is the temporal pulse width at the half maximum (FWHM),
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+where $\theta = 50$ fs is the temporal pulse width at the half maximum (FWHM),
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$t_0$ is a time delay, $w_0 = 3{\mu}m$ is the waist beam,
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$w(z) = {w_0}\sqrt{1+(\frac{z}{z_R})^2}$ is the Gaussian's beam spot
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size, $\omega = 2{\pi}c/{\lambda}$ is the angular frequency,
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@@ -424,19 +424,19 @@ $\alpha = 21.2$ cm$^2$/J is the avalanche ionization coefficient
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\cite{Pronko1998} at the wavelength $800$ nm in air. As we have noted,
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free carrier diffusion is neglected during and shortly after the laser
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excitation \cite{Van1987, Sokolowski2000}. In particular, from the
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-Einstein formula $D = k_B T_e \tau/m^* \approx (1-2){10}^5$ m/s
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+Einstein formula $D = k_B T_e \tau/m^* \approx (1--2){10}^{-3}$ m$^2$/s
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($k_B$ is the Boltzmann constant, $T_e$ is the electron temperature,
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$\tau=1$~\textit{fs} is the collision time, $m^* = 0.18 m_e$ is the effective
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-mass), where $T_e \approx 2*{10}^4$ K for $N_e$ close to $N_{cr}$. It
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+mass), where $T_e \approx 2*{10}^4$ K for $N_e$ close to $N_{cr}$ \cite{Ramer2014}. It
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means that during the pulse duration ($\approx$ 50~\textit{fs}) the diffusion
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-length will be around 5--10~nm for $N_e$ close to $N_{cr}$.
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+length will be around $5--10$~nm for $N_e$ close to $N_{cr}$.
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\begin{figure}[ht!]
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\centering
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\includegraphics[width=0.495\textwidth]{mie-fdtd-3}
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\caption{\label{mie-fdtd} (a, b) First four Lorentz-Mie coefficients
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($a_1$, $a_2$, $b_1$, $b_2$) and factors of asymmetry $G_I$, $G_{I^2}$
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- according to Mie theory at fixed wavelength 800~nm. (c, d) Intensity
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+ according to Mie theory at fixed wavelength $800$ nm. (c, d) Intensity
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distribution calculated by Mie theory and (e, f) EHP distribution
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for low free carrier densities $N_e \approx 10^{20}$ cm$^{-3}$ by
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Maxwell's equations (\ref{Maxwell}, \ref{Drude}) coupled with EHP
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@@ -501,35 +501,30 @@ license.
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\section{Results and discussion}
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-\begin{figure*}[ht!]
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+\begin{figure*}[p]
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\centering
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- \includegraphics[width=180mm]{plasma-grid.pdf}
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+ \includegraphics[width=145mm]{time-evolution-I-no-NP.pdf}
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+ \caption{\label{fig3} Temporal EHP (a, c, e) and asymmetry factor $G_{N_e}$ (b, d, f) evolution for different Si nanoparticle radii
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+ (a, b) $R = 75$ nm, (c, d) $R = 100$ nm, (e, f) $R = 115$ nm. Pulse
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+ duration $50$~\textit{fs} (FWHM). Wavelength $800$ nm in air. (b, d, f)
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+ Different stages of EHP evolution shown in Fig.~\ref{fig2} are
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+ indicated. The temporal evolution of Gaussian beam
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+ intensity is also shown. Peak laser fluence is fixed to be $0.125$J/cm$^2$.}
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+\vspace*{\floatsep}
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+ \centering
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+ \includegraphics[width=150mm]{plasma-grid.pdf}
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\caption{\label{fig2} EHP density snapshots inside Si nanoparticle of
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radii $R = 75$ nm (a-d), $R = 100$ nm (e-h) and $R = 115$ nm (i-l)
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taken at different times and conditions of excitation (stages
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$1-4$: (1) first optical cycle, (2) extremum at few optical cycles,
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- (3) Mie theory, (4) nonlinear effects). Pulse duration $50$~\textit{fs}
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+ (3) Mie theory, (4) nonlinear effects). $\Delta{Re(\epsilon)}$ indicates the real part change of the dielectric function defined by Equation (\ref{Index}). Pulse duration $50$~\textit{fs}
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(FWHM). Wavelength $800$ nm in air. Peak laser fluence is fixed to
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be $0.125$ J/cm$^2$.}
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\end{figure*}
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-
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- \begin{figure*}[ht!]
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- \centering
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- \includegraphics[width=160mm]{time-evolution-I-no-NP.pdf}
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- \caption{\label{fig3} Evolution of asymmetry factor $G$ as a function
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- of the average EHP density in the front part of the nanoparticle
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- (a, c, e) and time (b, d, f) for different Si nanoparticle radii
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- (a, b) $R = 75$ nm, (c, d) $R = 100$ nm, (e, f) $R = 115$ nm. Pulse
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- duration $50$~\textit{fs} (FWHM). Wavelength $800$ nm in air. (a, c, e)
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- Different stages of EHP evolution shown in Fig.~\ref{fig2} are
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- indicated. (b, d, f) The temporal evolution of Gaussian beam
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- intensity is also shown. Peak laser fluence is fixed to be $0.125$
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- J/cm$^2$.}
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- \end{figure*}
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%\subsection{Effect of the irradiation intensity on EHP generation}
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- Firstly, we analyze Mie coefficients (Fig.~\ref{mie-fdtd}(b) ) and
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+ Firstly, we analyze Mie coefficients (Fig.~\ref{mie-fdtd}) and
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the intensity distribution inside the non-excited Si NP as
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a function of its size for a fixed laser wavelength $\lambda = 800$
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nm. We introduce $G_I$ factor of asymmetry, corresponding to
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@@ -557,17 +552,17 @@ license.
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electric field made enough oscillations inside the Si NP. To achieve
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a qualitative description of the EHP distribution, we introduced
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another asymmetry factor
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- \red{$G_{N_e} = (N_e^{front}-N_e^{back})/(N_e^{front}+N_e^{back})$}
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+ $G_{N_e} = (N_e^{front}-N_e^{back})/(N_e^{front}+N_e^{back})$
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indicating the relationship between the average EHP densities in the
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- front and in the back parts of the NP. This way, $G = 0$ corresponds
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+ front and in the back parts of the NP. This way, $G_{N_e} = 0$ corresponds
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to the quasi-homogeneous case and the assumption of the NP
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homogeneous EHP distribution can be made to investigate the optical
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- response of the excited Si NP. However, in case $G$ significantly
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+ response of the excited Si NP. However, in case $G_{N_e}$ significantly
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differs from $0$, this assumption could not be proposed. In what
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follows, we discuss the results of the numerical modeling revealing
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the EHP evolution stages during pulse duration shown in
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Fig.~\ref{fig2} and the temporal/EHP dependent evolution of the
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- asymmetry factor $G$ in Fig.~\ref{fig3}.
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+ asymmetry factor $G_{N_e}$ in Fig.~\ref{fig3}.
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% Fig. \ref{Mie} demonstrates the temporal evolution of the EHP
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% generated inside the silicon NP of $R \approx 105$
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@@ -598,7 +593,7 @@ license.
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Si. The non-stationary intensity deposition results in different time
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delays for exciting electric and magnetic resonances inside Si NP
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because of different quality factors $Q$ of the resonances. In
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- particular, magnetic dipole resonance (\textit{b1}) has $Q \approx$8,
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+ particular, magnetic dipole resonance (\textit{b1}) has $Q \approx$ 8,
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whereas electric one (\textit{a1}) has $Q \approx$4. The larger
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particle supporting magnetic quadrupole resonance (\textit{b2})
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demonstrates \textit{Q} $\approx$ 40. As soon as the electromagnetic
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@@ -621,20 +616,19 @@ license.
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($t \approx 2--15$) leading to unstationery nature of the EHP evolution
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with a maximum of the EHP distribution on the front side of the Si NP
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owing to starting excitation of MD and MQ resonances, requiring more
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- time to be excited. At this stage, density of EHP ($< 10^{20}$cm$^2$)
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+ time to be excited. At this stage, density of EHP ($N_e < 10^{20}$cm$^2$)
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is still not high enough to affect significantly optical properties
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of the Si NP.
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A number of optical cycles ($>$10 or $t>$25~\textit{fs}) is necessary to
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achieve the stationary intensity pattern corresponding to the
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Mie-based intensity distribution at the \textit{'Stage $3$'} (see
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- Fig.~\ref{fig3}). The EHP density are still relatively not high to
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+ Fig.~\ref{fig3}). The EHP density is still relatively not high to
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influence the EHP evolution and strong diffusion rates but already
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enough to change the optical properties locally. Below the magnetic
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dipole resonance $R \approx 100$ nm, the EHP is mostly localized in
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the front side of the NP as shown in Fig.~\ref{fig2}(c). The highest
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- stationary asymmetry factor \red{$G_{N_e} \approx 3-4$ (should be
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- changed)} is achieved in this case. At the magnetic dipole
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+ stationary asymmetry factor $G_{N_e} \approx 0.5--0.6$ is achieved in this case. At the magnetic dipole
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resonance conditions, the EHP distribution has a toroidal shape and
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is much closer to homogeneous distribution. In contrast, above the
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magnetic dipole resonant size for $R = 115$ nm, the $G_{N_e} < 0$ due
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@@ -647,7 +641,7 @@ license.
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maximum. Therefore, EHP is localized in the front part of the NP,
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influencing the asymmetry factor $G_{N_e}$ in
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Fig.~\ref{fig3}. Approximately at the pulse peak, the critical
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- electron density $n_{cr} = 5\cdot{10}^{21}$ cm$^{-3}$ for silicon,
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+ electron density $N_{cr} = 5\cdot{10}^{21}$ cm$^{-3}$ for silicon,
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which corresponds to the transition to quasi-metallic state
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$Re(\epsilon) \approx 0$ and to the electron plasma resonance, is
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overcome. Further irradiation leads to a decrease in the asymmetry
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@@ -778,9 +772,6 @@ license.
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% and size} It is important to optimize asymmetry by varying pulse
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% duration, intensity and size.
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-TODO Kostya: Add discussion about mode selection due to the formation
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-of the plasma.
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-
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\section{Conclusions} We have considered ultra-short and sufficiently
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intense light interactions with a single semiconductor nanoparticle
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under different irradiation conditions and for various particle
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@@ -819,7 +810,7 @@ nano-bio-applications. The observed plasma-induced breaking symmetry
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can be also useful for beam steering, or for the enhanced second
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harmonics generation.
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-\section{Acknowledgments} A.R. gratefully acknowledges support from The French Ministry of Science and Education. S.V.M. is thankful to ITMO Fellowship Program and T.E.I to the ITMO Research Professorship Program and to the CINES of CNRS for computer support. The work was partially supported by Russian Foundation for Basic Researches (grants 17-03-00621, 17-02-00538, 16-29-05317).
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+\section{Acknowledgments} A. R. gratefully acknowledges support from The French Ministry of Science and Education. S. V. M. is thankful to ITMO Fellowship Program and T. E. I. to the ITMO Research Professorship Program and to the CINES of CNRS for computer support. The work was partially supported by Russian Foundation for Basic Researches (grants 17-03-00621, 17-02-00538, 16-29-05317).
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