%% График от E^2 и E^4 в зависимости от концентрации (меняем eplion)
%%  FDTD Lumerical ?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%This is the LaTeX ARTICLE template for RSC journals
%Copyright The Royal Society of Chemistry 2014
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\documentclass[twoside,twocolumn,9pt]{article}
\usepackage{extsizes}
\usepackage[super,sort&compress,comma]{natbib} 
\usepackage[version=3]{mhchem}
\usepackage[left=1.5cm, right=1.5cm, top=1.785cm, bottom=2.0cm]{geometry}
\usepackage{balance}
\usepackage{widetext}
\usepackage{times,mathptmx}
\usepackage{sectsty}
\usepackage{graphicx} 
\usepackage{lastpage}
\usepackage[format=plain,justification=raggedright,singlelinecheck=false,font={stretch=1.125,small,sf},labelfont=bf,labelsep=space]{caption}
\usepackage{float}
\usepackage{fancyhdr}
\usepackage{fnpos}
\usepackage{tabularx}
\usepackage[english]{babel}
\usepackage{array}
\usepackage{droidsans}
\usepackage{charter}
\usepackage[T1]{fontenc}
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{setspace}
\usepackage[compact]{titlesec}
%%% Please don't disable any packages in the preamble, as this may
%%% cause the template to display incorrectly.%%%
\usepackage{amsmath}

\usepackage{epstopdf}%This line makes .eps figures into .pdf - please
                     %comment out if not required.

\definecolor{cream}{RGB}{222,217,201}

\begin{document}

\pagestyle{fancy}
\thispagestyle{plain}
\fancypagestyle{plain}{

%%%HEADER%%%
\fancyhead[C]{\includegraphics[width=18.5cm]{head_foot/header_bar}}
\fancyhead[L]{\hspace{0cm}\vspace{1.5cm}\includegraphics[height=30pt]{head_foot/journal_name}}
\fancyhead[R]{\hspace{0cm}\vspace{1.7cm}\includegraphics[height=55pt]{head_foot/RSC_LOGO_CMYK}}
\renewcommand{\headrulewidth}{0pt}
}
%%%END OF HEADER%%%

%%%PAGE SETUP - Please do not change any commands within this
%%% section%%%
\makeFNbottom
\makeatletter
\renewcommand\LARGE{\@setfontsize\LARGE{15pt}{17}}
\renewcommand\Large{\@setfontsize\Large{12pt}{14}}
\renewcommand\large{\@setfontsize\large{10pt}{12}}
\renewcommand\footnotesize{\@setfontsize\footnotesize{7pt}{10}}
\makeatother

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\renewcommand\footnoterule{\vspace*{1pt}% 
\color{cream}\hrule width 3.5in height 0.4pt \color{black}\vspace*{5pt}} 
\setcounter{secnumdepth}{5}

\makeatletter 


\renewcommand\@biblabel[1]{#1}            
\renewcommand\@makefntext[1]% 
{\noindent\makebox[0pt][r]{\@thefnmark\,}#1}
\makeatother 
\renewcommand{\figurename}{\small{Fig.}~}
\sectionfont{\sffamily\Large}
\subsectionfont{\normalsize}
\subsubsectionfont{\bf}
\setstretch{1.125} %In particular, please do not alter this line.
\setlength{\skip\footins}{0.8cm}
\setlength{\footnotesep}{0.25cm}
\setlength{\jot}{10pt}
\titlespacing*{\section}{0pt}{4pt}{4pt}
\titlespacing*{\subsection}{0pt}{15pt}{1pt}
%%%END OF PAGE SETUP%%%

%%%FOOTER%%%
\fancyfoot{}
\fancyfoot[LO,RE]{\vspace{-7.1pt}\includegraphics[height=9pt]{head_foot/LF}}
\fancyfoot[CO]{\vspace{-7.1pt}\hspace{13.2cm}\includegraphics{head_foot/RF}}
\fancyfoot[CE]{\vspace{-7.2pt}\hspace{-14.2cm}\includegraphics{head_foot/RF}}
\fancyfoot[RO]{\footnotesize{\sffamily{1--\pageref{LastPage} ~\textbar  \hspace{2pt}\thepage}}}
\fancyfoot[LE]{\footnotesize{\sffamily{\thepage~\textbar\hspace{3.45cm} 1--\pageref{LastPage}}}}
\fancyhead{}
\renewcommand{\headrulewidth}{0pt} 
\renewcommand{\footrulewidth}{0pt}
\setlength{\arrayrulewidth}{1pt}
\setlength{\columnsep}{6.5mm}
\setlength\bibsep{1pt}
%%%END OF FOOTER%%%

%%%FIGURE SETUP - please do not change any commands within this
%%% section%%%
\makeatletter 
\newlength{\figrulesep} 
\setlength{\figrulesep}{0.5\textfloatsep} 

\newcommand{\topfigrule}{\vspace*{-1pt}% 
\noindent{\color{cream}\rule[-\figrulesep]{\columnwidth}{1.5pt}} }

\newcommand{\botfigrule}{\vspace*{-2pt}% 
\noindent{\color{cream}\rule[\figrulesep]{\columnwidth}{1.5pt}} }

\newcommand{\dblfigrule}{\vspace*{-1pt}% 
\noindent{\color{cream}\rule[-\figrulesep]{\textwidth}{1.5pt}} }

\makeatother
%%%END OF FIGURE SETUP%%%

%%%TITLE, AUTHORS AND ABSTRACT%%%
\twocolumn[
  \begin{@twocolumnfalse}
\vspace{3cm}
\sffamily
\begin{tabular}{m{4.5cm} p{13.5cm} }
  \includegraphics{head_foot/DOI} & \noindent\LARGE{\textbf{Photo-generated Electron-Hole Plasma-Induced Symmetry Breaking in Spherical Silicon Nanoparticles}} \\%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, remain unexplored. To examine these effects, numerical modeling is performed. Based on the simulation results, we propose an original concept of a well-controlled 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.}

\end{tabular}

 \end{@twocolumnfalse} \vspace{0.6cm}

]%%%END OF TITLE, AUTHORS AND ABSTRACT%%%

%%%FONT SETUP - please do not change any commands within this section
\renewcommand*\rmdefault{bch}\normalfont\upshape \rmfamily
\section*{} \vspace{-1cm}


%%%FOOTNOTES%%%

\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}}


% Please use \dag to cite the ESI in the main text of the article.

% If you article does not have ESI please remove the the \dag symbol
% from the title and the footnotetext below.
% \footnotetext{\dag~Electronic Supplementary Information (ESI)
% available: [details of any supplementary information available
% should be included here]. See DOI:10.1039/b000000x/} %additional
% addresses can be cited as above using the lower-case letters, c, d,
% e... If all authors are from the same address, no letter is required

% \footnotetext{\ddag~Additional footnotes to the title and authors can
% be included \emph{e.g.}\ `Present address:' or `These authors
% contributed equally to this work' as above using the symbols: \ddag,
% \textsection, and \P. Please place the appropriate symbol next to the
% author's name and include a \texttt{\textbackslash footnotetext} entry
% in the the correct place in the list.}


%%%END OF FOOTNOTES%%%

%%%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} and
ultrafast all-optical modulation~\cite{iyer2015reconfigurable,
makarov2015tuning, shcherbakov2015ultrafast, yang2015nonlinear,
baranov2016nonlinear, baranov2016tuning}. 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 magnetic dipole resonance
leads to significant changes of optical properties (transmittance or
reflectance) of all-dielectric nanoantennas~\cite{makarov2015tuning,
baranov2016nonlinear, baranov2016tuning} and
metasurfaces~\cite{iyer2015reconfigurable, shcherbakov2015ultrafast,
yang2015nonlinear}.

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

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

Recently, femtosecond lasers have been used to ionize nanoparticles locally and to produce electron-hole plasmas (EHP) inside them, which have been directly observed by using plasma explosion imaging\cite{Hickstein2014}. Interestingly, inhomogeneous resonant scattering patterns have been experimentally revealed inside a single silicon nanoparticle\cite{Valuckas2017}.  

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 ($f\!s$) 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  photo-excited
single spherical silicon nanoparticles. Moreover, we show that dense EHP can
be generated at deeply subwavelength scale
($\approx$$\lambda$$^3$/100) supporting formation of small metalized
parts inside the nanoparticle, which transforms all-dielectric
nanoparticle to a hybrid one that extends 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 out 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 follows
% \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}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.495\textwidth]{mie-fdtd-3}
\caption{\label{mie-fdtd} Mie and FDTD comparison. (c-f) Incident light goes from the left.}
\end{figure}

\begin{figure*}[ht!]
\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*}
\includegraphics[width=0.9\textwidth]{2nm_75}
\includegraphics[width=0.9\textwidth]{2nm_100}
\includegraphics[width=0.9\textwidth]{2nm_115}
\caption{\label{plasma-105nm} Evolution of lectron density $n_e$ (using $10^{\,20} \ {\rm cm}^{-3}$ units) for  (a--e)~$R=75$, (f--j)~$R=100$~nm, and (k--o)~$R=115$~nm. Gaussian pulse duration $80\:f\!s$, snapshot are taken before the pulse maxima, the corresponding time-shifts are shown in the top of each column.}
\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} $$
	\displaystyle{Re(\epsilon) = \epsilon -\frac{{e^2}n_e}{\epsilon_0m_e^*(\omega^2+{\nu_e}^2)}} \\
	\displaystyle{Im(\epsilon) = \frac{{e^2}n_e\nu_e}{\epsilon_0m_e^*\omega(\omega^2+{\nu_e}^2)}.}
 $$ \end{cases} \end{align}

\section{Results and discussion}

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

Fig. \ref{fig2} 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.

Far before the pulse peak shown in Fig. \ref{fig2}(a), the excitation
processes follow the intensity distribution, generating a low-density
electron plasma of a toroidal shape at magnetic dipole resonance
conditions. For higher intensities, 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
nanoparticle, increasing the asymmetry factor $G$ in
Fig. \ref{fig2}(b). 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 (so-called "volume plasmons", and/or "localized plasmons" typical for metallic nanoparticles !??) \cite{Sokolowski2000}, is overcome. 
At the same time, $G$ factor
reaches the maximum value close to $2.5$ in
Fig. \ref{fig2}(c). Further irradiation leads to a decrease in the
asymmetry parameter down to $1$ for higher electron densities, as one may observe in
Fig. \ref{fig2}(d, e).

It is worth noting that it is possible to achieve a formation of
deeply subwavelength EHP regions due to high field localization. In
particular, we observe very small EHP localization at magnetic dipole
resonant conditions for $R \approx 105$ nm. The EHP distribution in
Fig. \ref{fig2}(c) is optimal for symmetry breaking in silicon
nanoparticle, as it results in the larger asymmetry factor $G$ and higher
electron densities $n_e$. We stress here that such regime
could be still safe for nanoparticle due to the very small volume where such high
EHP density is formed. 

TODO: need to discuss this -
Of cause, this plasma is expected to diffuse after the considered time and turn to the homogeneously distributed one over the nanoparticle volume with a smaller density. Part of the electrons can also be ejected/injected into the surrounding medium, the process known to depend on the Shottky barrier at the particle border.

\subsection{Effects of nanoparticle size/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 some discusion on rise-on time for optical switching
like this: Small size will give as a magnetic dipole b1 resonance with
Q-factor (ratio of wavelength to the resonance width at half-height)
of about 8, a1 Q approx 4, the larger particle will have b2 Q approx
40. For large particle we will have e.g. at R=238.4 second order b4
resonance with Q approx 800.  As soon as the period at WL=800nm is $2.6\:f\!s$, we need about $25\:f\!s$ pulse to pump dipole response, about $150\:f\!s$
for quadrupole, and about $2000\:f\!s$ for b4. If we think of optical
switching applications this is a rise-on time.

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 particle 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. 
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 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".  




%%%END OF MAIN TEXT%%%

%The \balance command can be used to balance the columns on the final
%page if desired. It should be placed anywhere within the first column
%of the last page.

%\balance

%If notes are included in your references you can change the title
% from 'References' to 'Notes and references' using the following
% command:

% \renewcommand\refname{Notes and references}

%%%REFERENCES%%%

\bibliography{References.bib} %You need to replace "rsc" on this line
                              %with the name of your .bib file
\bibliographystyle{rsc} %the RSC's .bst file

\end{document}