Asymmetry-plasma/main.tex

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  \includegraphics{head_foot/DOI} & \noindent\LARGE{\textbf{Photogenerated Electron-Hole Plasma Induced Symmetry Breaking in Spherical Silicon Nanoparticle}} \\%Article title goes here instead of the text "This is the title"
  \vspace{0.3cm} & \vspace{0.3cm} \\
                                  & \noindent\large{Anton Rudenko,$^{\ast}$\textit{$^{a}$}  Konstantin Ladutenko,\textit{$^{b}$}  Sergey Makarov\textit{$^{b}$} and Tatiana E. Itina\textit{$^{a}$$^{b}$}
 
                                    \textit{$^{a}$~Laboratoire Hubert Curien, UMR CNRS 5516, University of Lyon/UJM, 42000, Saint-Etienne, France }
                                    \textit{$^{b}$~ITMO University, Kronverksiy pr. 49, St. Petersburg, Russia}
 
                                    } \\%Author names go here instead of "Full name", etc.
 
 
  \includegraphics{head_foot/dates} & \noindent\normalsize
{The concept of nonlinear all-dielectric nanophotonics based on high
refractive index (e.g., silicon) nanoparticles supporting magnetic
optical response has recently emerged as a powerful tool for ultrafast
all-optical modulation at nanoscale. A strong modulation can be
achieved via photo-generation of dense electron-hole plasma in the
regime of simultaneous excitation of electric and magnetic optical
resonances, resulting in an effective transient reconfiguration of
nanoparticle scattering properties. Because only homogeneous plasma
generation was previously considered in the photo-excited
nanoparticle, a possibility of symmetry breaking, however, remains
unexplored. To examine these effects, numerical modeling is
performed. Based on the simulation results, we propose an original
concept of a deeply subwavelength
($\approx$$\lambda$$^3$/100) plasma-induced nanopatterning of
spherical silicon nanoparticles. In particular, the revealed strong
symmetry breaking in the initially symmetrical nanoparticle, which is
observed during ultrafast photoexcitation near the magnetic dipole
resonance, enables a considerable increase in the precision of
laser-induced nanotreatment. Importantly, the proposed ultrafast
manipulation of the nanoparticle inherent structure and symmetry paves
a way to the novel principles that are also promising for nonlinear
optical nanodevices.}

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\footnotetext{\textit{$^{a}$~Univ Lyon, UJM-St-Etienne, CNRS UMR 5516,
F-42000, Saint-Etienne, France}} \footnotetext{\textit{$^{b}$~ITMO
University, Kronverksiy pr. 49, St. Petersburg, Russia}}


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%%%MAIN TEXT%%%%
  
\section{Introduction}
 
All-dielectric nonlinear nanophotonics based on high refractive index
dielectric has become prospective paradigm in modern optics, owing to
recent advances in harmonics generation~\cite{shcherbakov2014enhanced,
  yang2015nonlinear, makarov2016self, shorokhov2016multifold,
  makarov2017efficient} and ultrafast all-optical
modulation~\cite{iyer2015reconfigurable, makarov2015tuning,
  shcherbakov2015ultrafast, yang2015nonlinear, baranov2016nonlinear,
  baranov2016tuning, shcherbakov2017ultrafast}. In fact,
all-dielectric nanoantennas and metasurfaces possess much smaller
parasitic Joule losses at high intensities as compared with their
plasmonic counterparts, whereas their nonlinear properties are
comparable. More importantly, the unique properties of the nonlinear
all-dielectric nanodevices are due to existing of both electric and
magnetic optical resonances in visible and near IR
ranges~\cite{kuznetsov2016optically}. For instance, even slight
variation of dielectric permittivity around optical resonances leads
to significant changes of optical properties (transmittance or
reflectance) of all-dielectric nanoantennas~\cite{makarov2015tuning,
  baranov2016nonlinear, baranov2016tuning} and
metasurfaces~\cite{iyer2015reconfigurable, shcherbakov2015ultrafast,
  yang2015nonlinear, shcherbakov2017ultrafast}.

In these works on all-dielectric nonlinear nanostructures, the
building blocks (nanoparticles) were considered as objects with
dielectric permittivity \textit{homogeneously} distributed over
nanoparticle. Therefore, in order to manipulate the propagation angle
of the transmitted light it was proposed to use complicated
nanostructures with reduced symmetry~\cite{albella2015switchable,
  baranov2016tuning, shibanuma2016unidirectional}.

\begin{figure}[t] \centering
  \includegraphics[width=0.75\linewidth]{Concept.pdf}
  \caption{Schematic illustration of electron-hole plasma 2D and 1D
distributions in silicon nanoparticle around a magnetic resonance.}
  \label{fgr:concept}
\end{figure}

On the other hand, plasma explosion imaging technique has been used to
observe electron-hole plasmas (EHP), produced by femtosecond lasers,
inside nanoparticles~\cite{Hickstein2014}. Particularly, a strongly
localized EHP in the front side~\footnote{The incident wave propagate
  in positive direction of $z$ axis. Geometric center of the nanoparticle is
  located at $z=0$, front side corresponds to nanoparticle volume with
$z>0$ and back side for $z<0$} of NaCl nanocrystals of $R = 100$ nm
was revealed. The forward ejection of ions in this case was attributed
to a nanolensing effect inside the nanoparticle and the intensity
enhancement as low as $10\%$ on the far side of the nanoparticle. Much
stronger enhancements can be achieved near electric and magnetic
dipole resonances excited in single semiconductor nanoparticles, such
as silicon (Si), germanium (Ge) etc.

In this Letter, we show that ultra-short laser-based EHP
photo-excitation in a spherical semiconductor (e.g., silicon)
nanoparticle leads to a strongly inhomogeneous carrier
distribution. To reveal and study this effect, we perform a full-wave
numerical simulation of the intense femtosecond (\textit{fs}) laser
pulse interaction with a silicon nanoparticle supporting Mie
resonances and two-photon free carrier generation. In particular, we
couple finite-difference time-domain (FDTD) method used to solve
Maxwell equations with kinetic equations describing nonlinear EHP
generation.  Three-dimensional transient variation of the material
dielectric permittivity is calculated for nanoparticles of several
sizes. The obtained results propose a novel strategy to create
complicated non-symmetrical nanostructures by using single photo-excited
spherical silicon nanoparticles. Moreover, we show that a dense
EHP can be generated at deeply subwavelength scale
($\approx$$\lambda$$^3$/100) supporting the formation of small
metalized parts inside the nanoparticle. In fact, such effects
transform an all-dielectric nanoparticle to a hybrid one strongly
extending functionality of the ultrafast optical nanoantennas.


%Plan:
%\begin{itemize}
%\item Fig.1: Beautiful conceptual picture
%\item Fig.2: Temporal evolution of EHP in NP with different diameters
%at fixed intensity, in order to show that we have the highest
%asymmetry around magnetic dipole (MD) resonance. This would be really
%nice!
%\item Fig.3: Temporal evolution of EHP in NP with fixed diameter (at
%MD) at different intensities, in order to show possible regimes of
%plasma-patterning of NP volume. It would be nice, if we will show
%power patterns decencies on intensity for side probe pulse to show beam steering due to symmetry breaking.
%\item Fig.4: (a) Dependence on pulse duration is also interesting. We
%have to show at which duration the asymmetry factor is saturated. (b)
%2D map of asymmetry factor in false colors, where x-axis and y-axis correspond to intensity and NP diameter.
%\end{itemize} %Additionally, if you will manage to calculate
%evolution of scattering power pattern and show considerable effect of
% beam steering, we can try Nanoscale or LPR, because the novelty will
% be very high.


\section{Modeling details}

We focus attention on silicon because this material is promising for
the implementation of numerous nonlinear photonic devices. This
advantage is based on a broad range of optical nonlinearities, strong
two-photon absorption, as well as a possibility of the photo-induced
EHP excitation~\cite{leuthold2010nonlinear}. Furthermore, silicon
nanoantennas demonstrate a sufficiently high damage threshold due to
the large melting temperature ($\approx$1690~K), whereas its nonlinear
optical properties have been extensively studied during last
decays~\cite{Van1987, Sokolowski2000, leuthold2010nonlinear}. High
silicon melting point typically preserves structures formed from this
material up to the EHP densities on the order of the critical value
$n_{cr} \approx 5\cdot{10}^{21}$ cm$^{-3}$ \cite{Korfiatis2007}. At
the critical density and above, silicon acquires metallic properties
($Re(\epsilon) < 0$) and contributes to the EHP reconfiguration during
ultrashort laser irradiation.

The process of three-dimensional photo-generation of the EHP in
silicon nanoparticles has not been modeled before in
time-domain. Therefore, herein we propose a model considering
ultrashort laser interactions with a resonant silicon sphere, where
the EHP is generated via one- and two-photon absorption processes.
Importantly, we also consider nonlinear feedback of the material by
taking into account the intraband light absorption on the generated
free carriers. To simplify our model, we neglect free carrier
diffusion at the considered short time scales. In fact, the aim of the
present work is to study the EHP dynamics \textit{during} ultra-short
laser interaction with the nanoparticle. The created electron-hole
plasma then will recombine, however, as its existence modifies both
laser-particle interaction and, hence, the following particle
evolution.

\subsection{Light propagation}

Ultra-short laser interaction and light propagation inside the silicon
nanoparticle are modeled by solving the system of Maxwell's equations
written in the following way
\begin{align} \begin{cases} \label{Maxwell}$$
    \displaystyle{\frac{\partial{\vec{E}}}{\partial{t}}=\frac{\nabla\times\vec{H}}{\epsilon_0\epsilon}-\frac{1}{\epsilon_0\epsilon}(\vec{J}_p+\vec{J}_{Kerr})} \\
    \displaystyle{\frac{\partial{\vec{H}}}{\partial{t}}=-\frac{\nabla\times\vec{E}}{\mu_0}},
    $$ \end{cases}
\end{align} 
where $\vec{E}$ is the electric field, $\vec{H}$ is the magnetizing
field, $\epsilon_0$ is the free space permittivity, $\mu_0$ is the
permeability of free space, $\epsilon = n^2 = 3.681^2$ is the
permittivity of non-excited silicon at $800$ nm wavelength
\cite{Green1995}, $\vec{J}_p$ and $\vec{J}_{Kerr}$ are the nonlinear
currents, which include the contribution due to Kerr effect
$\vec{J}_{Kerr} =
\epsilon_0\epsilon_\infty\chi_3\frac{\partial\left(\left|\vec{E}\right|^2\vec{E}\right)}{\partial{t}}$,
where $\chi_3 =4\cdot{10}^{-20}$ m$^2$/V$^2$ for laser wavelength
$\lambda = 800$nm \cite{Bristow2007}, and heating of the conduction
band, described by the differential equation derived from the Drude
model
\begin{equation} \label{Drude}
  \displaystyle{\frac{\partial{\vec{J_p}}}{\partial{t}} = -
    \nu_e\vec{J_p} + \frac{e^2N_e(t)}{m_e^*}\vec{E}},
\end{equation}
where $e$ is the elementary charge, $m_e^* = 0.18m_e$ is the reduced
electron-hole mass \cite{Sokolowski2000}, $N_e(t)$ is the
time-dependent free carrier density and $\nu_e = 10^{15}$ s$^{-1}$ is
the electron collision frequency \cite{Sokolowski2000}. Silicon
nanoparticle is surrounded by vacuum, where the light propagation is
calculated by Maxwell's equations with $\vec{J} = 0$ and
$\epsilon = 1$. The system of Maxwell's equations coupled with
electron density equation is solved by the finite-difference numerical
method \cite{Rudenko2016}, based on the finite-difference time-domain
(FDTD) \cite{Yee1966} and auxiliary-differential methods for
dispersive media \cite{Taflove1995}. At the edges of the grid, we
apply the absorbing boundary conditions related to convolutional
perfectly matched layers (CPML) to avoid nonphysical reflections
\cite{Roden2000}. The initial electric field is introduced as a
Gaussian slightly focused beam as follows
\begin{align}
\begin{aligned}
\label{Gaussian}
{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)\\
\times\;{exp}\left(-\frac{(t-t_0)^2}{\theta^2}\right),
\end{aligned}
\end{align}
where $\theta$ is the temporal pulse width at the half maximum (FWHM),
$t_0$ is a time delay, $w_0 = 3{\mu}m$ is the waist beam,
$w(z) = {w_0}\sqrt{1+(\frac{z}{z_R})^2}$ is the Gaussian's beam spot
size, $\omega = 2{\pi}c/{\lambda}$ is the angular frequency,
$\lambda = 800$ nm is the laser wavelength in air, $c$ is the speed of
light, ${z_R} = \frac{\pi{{w_0}^2}n_0}{\lambda}$ is the Rayleigh
length, $r = \sqrt{x^2 + y^2}$ is the radial distance from the beam's
waist, $R_z = z\left(1+(\frac{z_R}{z})^2\right)$ is the radius of
curvature of the wavelength comprising the beam, and
$\varsigma(z) = {\arctan}(\frac{z}{z_R}) $ is the Gouy phase shift.

\subsection{Material ionization}

To account for the material ionization that is induced by a
sufficiently intense laser field inside the particle, we couple
Maxwell's equations with the kinetic equation for the electron-hole
plasma as described below.
% \begin{figure*}[ht!]
% \centering
% \includegraphics[width=120mm]{fig2.png}
% \caption{\label{fig2} Free carrier density snapshots of electron plasma evolution inside Si nanoparticle taken a) $30\:f\!s$  b) $10\:f\!s$ before the pulse peak, c) at the pulse peak, d) $10\:f\!s$ e) $30\:f\!s$  after the pulse peak. Pulse duration $50\:f\!s$ (FWHM). Wavelength $800$ nm in air. Radius of the nanoparticle $R \approx 105$ nm, corresponding to the resonance condition. Graph shows the dependence of the asymmetric parameter of electron plasma density on the average electron density in the front half of the nanoparticle. $n_{cr} = 5\cdot{10}^{21}$ cm$^{-3}$ is the critical plasma resonance electron density for silicon.}
% \end{figure*}


The time-dependent conduction-band carrier density evolution is
described by a rate equation that was proposed by van Driel
\cite{Van1987}. This equation takes into account such processes as
photoionization, avalanche ionization and Auger recombination, and is
written as
\begin{equation} \label{Dens}
  \displaystyle{\frac{\partial{N_e}}{\partial t} =
    \frac{N_a-N_e}{N_a}\left(\frac{\sigma_1I}{\hbar\omega} +
      \frac{\sigma_2I^2}{2\hbar\omega}\right) + \alpha{I}n_e -
    \frac{C\cdot{N_e}^3}{C\tau_{rec}N_e^2+1},} \end{equation} where
$I=\frac{n}{2}\sqrt{\frac{\epsilon_0}{\mu_0}}\left|\vec{E}\right|^2$
is the intensity, $\sigma_1 = 1.021\cdot{10}^3$ cm$^{-1}$ and
$\sigma_2 = 0.1\cdot{10}^{-7}$ cm/W are the one-photon and two-photon
interband cross-sections \cite{Choi2002, Bristow2007, Derrien2013},
$N_a = 5\cdot{10}^{22} cm^{-3}$ is the saturation particle density
\cite{Derrien2013}, $C = 3.8\cdot{10}^{-31}$ cm$^6$/s is the Auger
recombination rate \cite{Van1987}, $\tau_{rec} = 6\cdot{10}^{-12}$s is
the minimum Auger recombination time \cite{Yoffa1980}, and
$\alpha = 21.2$ cm$^2$/J is the avalanche ionization coefficient
\cite{Pronko1998} at the wavelength $800$ nm in air. As we have noted,
free carrier diffusion is neglected during and shortly after the laser
excitation \cite{Van1987, Sokolowski2000}. In particular, from the
Einstein formula $D = k_B T_e \tau/m^*$ $\approx$ (1-2)10$^5$ m/s
(k$_B$ is the Boltzmann constant, T$_e$ is the electron temperature,
$\tau$=1~\textit{fs} is the collision time, $m^*$ = 0.18$m_e$ is the effective
mass), where T$_e$ $\approx$ 2*10$^4$~K for N$_e$ close to N$_cr$. It
means that during the pulse duration ($\approx$ 50~\textit{fs}) the diffusion
length will be around 5--10~nm for N$_e$ close to N$_cr$.

\begin{figure}[ht!] 
\centering
\includegraphics[width=0.495\textwidth]{mie-fdtd-3}
\caption{\label{mie-fdtd} (a, b) First four Lorentz-Mie coefficients
  ($a_1$, $a_2$, $b_1$, $b_2$) and factors of asymmetry $G_I$, $G_{I^2}$ 
  according to Mie theory at fixed wavelength 800~nm. (c, d) Intensity
  distribution calculated by Mie theory and (e, f) EHP distribution
  for low free carrier densities $N_e \approx 10^{20}$ cm$^{-3}$ by
  Maxwell's equations (\ref{Maxwell}, \ref{Drude}) coupled with EHP
  density equation (\ref{Dens}). (c-f) Incident light propagates from
  the left to the right along $Z$ axis, electric field polarization
  $\vec{E}$ is along $X$ axis.}
\end{figure}

%\begin{figure*}[ht!] \label{EHP}
%\centering
%\begin{tabular*}{0.76\textwidth}{ c@{\extracolsep{\fill}} c@{\extracolsep{\fill}} c@{\extracolsep{\fill}} c@{\extracolsep{\fill}} c}
%$-80\:f\!s$&$-60\:f\!s$&$-30\:f\!s$&$ -20\:f\!s$&$ -10\:f\!s$\\
%\end{tabular*}
%{\setlength\topsep{-1pt}
%\begin{flushleft}
%$R=75$~nm
%\end{flushleft}}
%\includegraphics[width=0.9\textwidth]{2nm_75}
%{\setlength\topsep{-1pt}
%\begin{flushleft}
%$R=100$~nm
%\end{flushleft}}
%\includegraphics[width=0.9\textwidth]{2nm_100}
%{\setlength\topsep{-1pt}
%\begin{flushleft}
%$R=115$~nm
%\end{flushleft}}
%\includegraphics[width=0.9\textwidth]{2nm_115}
%\caption{\label{plasma-105nm} Evolution of electron density $n_e$
 % (using $10^{\,20} \ {\rm cm}^{-3}$ units) for
  %(a$\,$--$\,$e)~$R=75$~nm, (f$\,$--$\,$j$\,$)~$R=100$~nm, and
  %($\,$k$\,$--$\,$o)~$R=115$~nm. Gaussian pulse duration $80\:f\!s$,
  %snapshots are taken before the pulse maxima, the corresponding
  %time-shifts are shown in the top of each column. Laser irradiation
  %fluences are (a-e) $0.12$ J/cm$^2$, (f-o) $0.16$ J/cm$^2$.}
%\end{figure*}

The changes of the real and imaginary parts of the permittivity
associated with the time-dependent free carrier response
\cite{Sokolowski2000} can be derived from equations (\ref{Maxwell},
\ref{Drude}) and are written as follows
\begin{align} \begin{cases}  \label{Index} $$
	\displaystyle{Re(\epsilon^\ast) = \epsilon -\frac{{e^2}N_e}{\epsilon_0m_e^*(\omega^2+{\nu_e}^2)}} \\
	\displaystyle{Im(\epsilon^\ast) = \frac{{e^2}N_e\nu_e}{\epsilon_0m_e^*\omega(\omega^2+{\nu_e}^2)}.}
 $$ \end{cases} \end{align}

\subsection{Mie calculations}
A steady-state interaction of a plain electromagnetic wave with a
spherical particle has a well-known analytical solution described by a
Mie theory~\cite{Bohren1983}. It is only valid in the absence of
nonlinear optical response, thus we can compare it against
above-mentioned FDTD-EHP model only for small plasma densities, where
we can neglect EHP impact to the refractive index. Non-stationary
nature of a femtosecond pulse increase the complexity of the
analysis. A detailed discussion on the relation between Mie theory and
FDTD-EHP model will be provided in the next section.

We used Scattnlay program to evaluate calculations of Mie coefficients
and near-field distribution~\cite{Ladutenko2017}.  This program is
available online at GitHub~\cite{Scattnlay-web} under open source
license.

\section{Results and discussion}

\begin{figure*}[ht!]
 \centering
 \includegraphics[width=180mm]{Figure_2.pdf}
 \caption{\label{fig2} EHP density snapshots inside Si nanoparticle of
   radii $R = 75$ nm (a-d), $R = 100$ nm (e-h) and $R = 115$ nm (i-l)
   taken at different times and conditions of excitation (stages
   $1-4$: (1) first optical cycle, (2) extremum at few optical cycles,
   (3) Mie theory, (4) nonlinear effects). Pulse duration $50\:f\!s$
   (FWHM). Wavelength $800$ nm in air. Peak laser fluence is fixed to
   be $0.125$ J/cm$^2$.}
 \end{figure*}
 
 \begin{figure*}[ht!]
 \centering
 \includegraphics[width=120mm]{Figure_3.pdf}
 \caption{\label{fig3} Evolution of asymmetry factor $G$ as a function
   of the average EHP density in the front part of the nanoparticle
   (a, c, e) and time (b, d, f) for different Si nanoparticle radii
   (a, b) $R = 75$ nm, (c, d) $R = 100$ nm, (e, f) $R = 115$ nm. Pulse
   duration $50\:f\!s$ (FWHM). Wavelength $800$ nm in air. (a, c, e)
   Different stages of EHP evolution shown in \ref{fig2} are
   indicated. (b, d, f) The temporal evolution of Gaussian beam
   intensity is also shown. Peak laser fluence is fixed to be $0.125$
   J/cm$^2$.}
 \end{figure*}

%\subsection{Effect of the irradiation intensity on EHP generation}

 Firstly, we analyze Mie coefficients (Fig.~\ref{mie-fdtd}(b) ) and
 the intensity distribution inside the non-excited Si nanoparticle as
 a function of its size for a fixed laser wavelength $\lambda = 800$
 nm.  We introduce $G_I$ factor of asymmetry, corresponding to
 difference between the volume integral of intensity in the front side
 of the nanoparticle to that in the back side normalized to their sum:
 $G_I = (I^{front}-I^{back})/(I^{front}+I^{back})$, where
 $I^{front}=\int_{(z>0)}|E|^2d{\mathrm{v}}$ and
 $I^{back}=\int_{(z<0)} |E|^2d{\mathrm{v}}$. The factor $G_{I^2}$ was
 introduced in a similar way using volume integrals of squared
 intensity as a better option to predict EHP asymmetry due to
 two-photon absorption.  Fig.~\ref{mie-fdtd}(b) shows $G$ factors
 as a function of the nanoparticle size. For the nanoparticles of
 sizes below the first magnetic dipole resonance, the intensity is
 enhanced in the front side as in Fig. \ref{mie-fdtd}(c) and
 $G_I > 0$. The behavior changes near the size resonance value,
 corresponding to $R \approx 105$ nm. In contrast, for larger sizes,
 the intensity is enhanced in the back side of the nanoparticle as
 demonstrated in Fig. \ref{mie-fdtd}(d). In fact, the similar EHP
 distributions can be obtained by applying Maxwell's equations coupled
 with the rate equation for relatively weak excitation
 $N_e \approx 10^{20}$ cm$^{-3}$. The optical properties do not change
 considerably due to excitation according to (\ref{Index}). Therefore,
 the excitation processes follow the intensity distribution. However,
 such coincidence was achieved in quasi-stationary conditions, when
 electric field made enough oscillations inside the Si NP. To achieve
 a qualitative description of the EHP distribution, we introduced
 another asymmetry factor
 \red{$G_{N_e} = (N_e^{front}-N_e^{back})/(N_e^{front}+N_e^{back})$}
 indicating the relationship between the average EHP densities in the
 front and in the back parts of the NP. This way, $G = 0$ corresponds
 to the quasi-homogeneous case and the assumption of the NP
 homogeneous EHP distribution can be made to investigate the optical
 response of the excited Si NP. However, in case $G$ significantly
 differs from $0$, this assumption could not be proposed. In what
 follows, we discuss the results of the numerical modeling revealing
 the EHP evolution stages during pulse duration shown in
 Fig. \ref{fig2} and the temporal/EHP dependent evolution of the
 asymmetry factor $G$ in Fig. \ref{fig3}.

% Fig. \ref{Mie} demonstrates the temporal evolution of the EHP
% generated inside the silicon nanoparticle of $R \approx 105$
% nm. Here, irradiation by high-intensity, $I\approx $ from XXX to YYY
% (???), ultrashort laser Gaussian pulse is considered. Snapshots of
% free carrier density taken at different times correspond to
% different total amount of the deposited energy (different laser
% intensities).

%To better analyze the degree of inhomogeneity, we introduce the EHP
% asymmetry parameter, $G$, which is defined as a relation between the
% average electron density generated in the front side of the
% nanoparticle and the average electron density in the back side, as
% shown in Fig. \ref{fig2}. During the femtosecond pulse interaction,
% this parameter significantly varies.




 In order to describe all stages of strong interaction of light with
 Si NP, we present the calculation results obtained by using Maxwell's
 equations coupled with electron kinetics equations for different
 radii for resonant and non-resonant conditions. In this case, the
 geometry of the EHP distribution can strongly deviate from the
 intensity distribution given by Mie theory. Two main reasons cause
 the deviation: (i) non-stationarity of the energy deposition and (ii)
 nonlinear effects, taking place due to transient optical changes in
 Si. The non-stationary intensity deposition results in different time
 delays for exciting electric and magnetic resonances inside Si NP
 because of different quality factors $Q$ of the resonances. In
 particular, magnetic dipole resonance (\textit{b1}) has $Q \approx$8,
 whereas electric one (\textit{a1}) has $Q \approx$4. The larger
 particle supporting magnetic quadrupole resonance (\textit{b2})
 demonstrates \textit{Q} $\approx$ 40. As soon as the electromagnetic
 wave period at $\lambda$~=~800~nm is 2.6~\textit{fs}, we need about 10~\textit{fs}
 pulse to pump the electric dipole, 20~\textit{fs} for the magnetic dipole, and
 about 100~\textit{fs} for the magnetic quadrupole.

 According to these estimations, the first optical cycles taking place
 on few-femtosecond scale result in the excitation of the
 low-\textit{Q} electric dipole resonance independently on the exact
 size of NPs and with the EHP concentration mostly on the front side
 of the NPs. We address to this phenomena as \textit{'Stage 1'}, as
 shown in Figs.~\ref{fig2} and~\ref{fig2}. The first stage at the
 first optical cycle demonstrates the dominant electric dipole
 resonance effect on the intensity/EHP density distribution inside the
 NPs in Fig.~\ref{fig2}(a,e,j) and \ref{fig3}. The larger the NPs size
 is, the higher the NP asymmetry $G_{N_e}$ is achieved.

 \textit{'Stage 2'} corresponds to further electric field oscillations
 (t$\approx$2--15) leading to unstationery nature of the EHP evolution
 with a maximum of the EHP distribution on the front side of the Si NP
 owing to starting excitation of MD and MQ resonances, requiring more
 time to be excited. At this stage, density of EHP ($< 10^{20}$cm$^2$)
 is still not high enough to affect significantly optical properties
 of the Si NP.

 A number of optical cycles ($>$10 or $t>$25~\textit{fs}) is necessary to
 achieve the stationary intensity pattern corresponding to the
 Mie-based intensity distribution at the \textit{'Stage $3$'} (see
 Fig.~\ref{fig3}). The EHP density are still relatively not high to
 influence the EHP evolution and strong diffusion rates but already
 enough to change the optical properties locally. Below the magnetic
 dipole resonance $R \approx 100$ nm, the EHP is mostly localized in
 the front side of the NP as shown in Fig. \ref{fig2}(c). The highest
 stationary asymmetry factor \red{$G_{N_e} \approx 3-4$ (should be
   changed)} is achieved in this case. At the magnetic dipole
 resonance conditions, the EHP distribution has a toroidal shape and
 is much closer to homogeneous distribution. In contrast, above the
 magnetic dipole resonant size for $R = 115$ nm, the $G_{N_e} < 0$ due
 to dominantly EHP localized in the back side of the NP.

 For the higher excitation conditions, the optical properties of
 silicon change significantly according to the equations
 (\ref{Index}). As a result, the non-resonant electric dipole
 contributes to the forward shifting of EHP density
 maximum. Therefore, EHP is localized in the front part of the NP,
 influencing the asymmetry factor $G_{N_e}$ in
 Fig.~\ref{fig3}. Approximately at the pulse peak, the critical
 electron density $n_{cr} = 5\cdot{10}^{21}$ cm$^{-3}$ for silicon,
 which corresponds to the transition to quasi-metallic state
 $Re(\epsilon) \approx 0$ and to the electron plasma resonance, is
 overcome. Further irradiation leads to a decrease in the asymmetry
 parameter down to $G_{N_e} = 0$ for higher EHP densities, as one can
 observe in Fig. \ref{fig2}(d, h, l).

 As the EHP acquires quasi-metallic properties at stronger excitation
 $N_e > 5\cdot{10}^{21}$ cm$^{-3}$, the EHP distribution evolves
 inside NPs because of the photoionization and avalanche ionization
 induced transient optical response and the effect of newly formed
 EHP. This way, the distribution becomes more homogeneous and the
 effect is likely to be enhanced by electron diffusion inside Si
 NPs. We refer to these nonlinear phenomena as \textit{'Stage $4$'}.
     
 It is worth noting that it is possible to achieve a formation of
 deeply subwavelength EHP regions due to high field localization. The
 smallest EHP localization and the larger asymmetry factor are
 achieved below the magnetic dipole resonant conditions for $R < 100$
 nm. Thus, the EHP distribution in Fig. \ref{fig2}(c) is optimal for
 symmetry breaking in Si NP, as it results in the larger asymmetry
 factor $G_{N_e}$ and higher electron densities $n_e$. We stress here
 that such regime could be still safe for NP due to the very small
 volume where such high EHP density is formed.

% \subsection{Effects of nanoparticle size and scattering efficiency
% factor on scattering directions}

% \begin{figure}[ht] \centering
% \includegraphics[width=90mm]{fig3.png}
% \caption{\label{fig3} a) Scattering efficiency factor $Q_{sca}$
% dependence on the radius $R$ of non-excited silicon nanoparticle
% calculated by Mie theory; b) Parameter of forward/backward scattering
% dependence on the radius $R$ calculated by Mie theory for non-excited
% silicon nanoparticle c) Optimization parameter $K$ dependence on the
% average electron density $n_e^{front}$ in the front half of the
% nanoparticle for indicated radii (1-7).}
% \end{figure}

% We have discussed the EHP kinetics for a silicon nanoparticle of a
% fixed radius $R \approx 105$ nm. In what follows, we investigate the
% influence of the nanoparticle size on the EHP patterns and temporal
% evolution during ultrashort laser irradiation. A brief analysis of
% the initial intensity distribution inside the nanoparticle given by
% the classical Mie theory for homogeneous spherical particles
% \cite{Mie1908} can be useful in this case. Fig. \ref{fig3}(a, b)
% shows the scattering efficiency and the asymmetry parameter for
% forward/backward scattering for non-excited silicon nanoparticles of
% different radii calculated by Mie theory \cite{Mie1908}. Scattering
% efficiency dependence gives us the value of resonant sizes of
% nanoparticles, where the initial electric fields are significantly
% enhanced and, therefore, we can expect that the following conditions
% will result in a stronger electron density gradients. Additionally,
% in the case of maximum forward or backward scattering, the initial
% intensity distribution has the maximum of asymmetry. One can note,
% that for $R \approx 100$ nm and $R \approx 150$ nm both criteria are
% fulfilled: the intensity is enhanced $5-10$ times due to
% near-resonance conditions and its distribution has a strong
% asymmetry.

% In what follows, we present the calculation results obtained by
% using Maxwell's equations coupled with electron kinetics for
% different extremum radii for resonant and non-resonant
% conditions. One can note, that the maximum asymmetry factor of EHP
% $G$ does not guarantee the optimal asymmetry of intensity
% distribution, as the size of generated plasma and the value of the
% electron density equally contribute to the change of the modified
% nanoparticle optical response. For example, it is easier to localize
% high electron densities inside smaller nanoparticles, however, due
% to the negligible size of the generated EHP with respect to laser
% wavelength in media, the intensity distribution around the
% nanoparticle will not change considerably. Therefore, we propose to
% introduce the optimization factor
% $K = \frac{n_e(G-1)R^2}{n_{cr}G{R_0^2}}$, where $R_0 = 100$ nm,
% $n_{cr} = 5\cdot{10}^{21} cm^{-3}$, and $G$ is asymmetry of EHP,
% defined previously. The calculation results for different radii of
% silicon nanoparticles and electron densities are presented in
% Fig. \ref{fig3}(c). One can see, that the maximum value are achieved
% for the nanoparticles, that satisfy both initial maximum forward
% scattering and not far from the first resonant condition. For larger
% nanoparticles, lower values of EHP asymmetry factor are obtained, as
% the electron density evolves not only from the intensity patterns in
% the front side of the nanoparticle but also in the back side.

%TODO: 
%Need to discuss agreement/differences between Mie and FDTD+rate. Anton ?!

% To demonstrate the effect of symmetry breaking, we calculate the
% intensity distribution around the nanoparticle for double-pulse
% experiment. The first pulse of larger pulse energy and polarization
% along $Ox$ generates asymmetric EHP inside silicon nanoparticle,
% whereas the second pulse of lower pulse energy and polarization $Oz$
% interacts with EHP after the first pulse is gone. The minimum
% relaxation time of high electron density in silicon is
% $\tau_{rec} = 6\cdot{10}^{-12}$ s \cite{Yoffa1980}, therefore, the
% electron density will not have time to decrease significantly for
% subpicosecond pulse separations. In our simulations, we use
% $\delta{t} = 200\:f\!s$ pulse separation. The intensity
% distributions near the silicon nanoparticle of $R = 95$ nm,
% corresponding to maxima value of $K$ optimization factor, without
% plasma and with generated plasma are shown in Fig. \ref{fig4}. The
% intensity distribution is strongly asymmetric in the case of EHP
% presence. One can note, that the excited nanoparticle is out of
% quasi-resonant condition and the intensity enhancements in
% Fig. \ref{fig4}(c) are weaker than in Fig. \ref{fig4}(b). Therefore,
% the generated nanoplasma acts like a quasi-metallic nonconcentric
% nanoshell inside the nanoparticle, providing a symmetry reduction
% \cite{Wang2006}.

% \begin{figure}[ht] \centering
% \includegraphics[width=90mm]{fig4.png}
% \caption{\label{fig4} a) Electron plasma distribution inside Si
% nanoparticle $R \approx 95$ nm $50\:f\!s$ after the pulse peak;
% (Kostya: Is it scattering field intensity snapshot $XXX\:f\!s$ after
% the second pulse maxima passed the particle?) Intensity
% distributions around and inside the nanoparticle b) without plasma,
% c) with electron plasma inside.}
% \end{figure}

%\begin{figure} %\centering
% \includegraphics[height=0.7\linewidth]{Si-flow-R140-XYZ-Eabs}
% \caption{EHP distributions for nonres., MD, ED, and MQ nanoparticles
% at moderate photoexcitation. The aim is to show different possible
% EHP patterns and how strong could be symmetry breaking.
% \label{fgr:example}
%\end{figure}

%\subsection{Asymmetry analysis: effects of pulse duration, intensity
% and size} It is important to optimize asymmetry by varying pulse
% duration, intensity and size.

TODO Kostya: Add discussion about mode selection due to the formation
of the plasma.

\section{Conclusions} We have considered ultra-short and sufficiently
intense light interactions with a single semiconductor nanoparticle
under different irradiation conditions and for various particle
sizes. As a result of the presented self-consistent calculations, we
have obtained spatio-temporal EHP evolution inside the
nanoparticles and investigated the asymmetry of EHP distributions. % for different laser intensities. % and temporal pulse widths.
%It has been demonstrated that the EHP generation strongly affects
%nanoparticle scattering and, in particular, changes the preferable
%scattering direction. 
Different pathways of EHP evolution from the front side to the back
side have been revealed, depending on the nanoparticle sizes, and the
origins of different behavior have been explained by the
non-stationarity of the energy deposition and different quality
resonant factors for exciting the electric and magnetic dipole
resonances, intensity distribution by Mie theory and newly
plasma-induced nonlinear effects. The effect of the strong broadband
electric dipole resonance on the EHP asymmetric distribution during
first optical cycles has been revealed for different size
parameters. The higher EHP asymmetry is established for nanoparticles
of smaller sizes below the first magnetic dipole
resonance. Essentially different EHP evolution and lower asymmetry is
achieved for larger nanoparticles due to the stationary intensity
enhancement in the back side of the nanoparticle. The EHP densities
above the critical value were shown to lead to the EHP distribution
homogenization.
% In particular, the scattering efficiency factor is used to define
% the optimum nanoparticle size for preferential forward or backward
% scattering. Furthermore, a parameter has been introduced to describe
% the scattering asymmetry as a ratio of the EHP density in the front
% side to that in the back side of the nanoparticle. This parameter
% can be then used for two-dimensional scattering mapping, which is
% particularly important in numerous photonics applications.
The EHP asymmetry opens a wide range of applications in nanoparticle
nanomashining/manipulation at nanoscale, catalysis as well as
nano-bio-applications.  The observed plasma-induced breaking symmetry
can be also useful for beam steering, or for the enhanced second
harmonics generation.

\section{Acknowledgments} We gratefully acknowledge support from The
French Ministry of Science and Education, from the French Center of
Scientific Research (CNRS) and from the PHC Kolmogorov project
"FORMALAS". S.V.M. is thankful to ITMO Fellowship Program. 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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