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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, Saint-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 Mie
  resonances, resulting in an effective transient reconfiguration of
  nanoparticle scattering properties. However, only homogeneous plasma
  generation was previously considered in a photo-excited
  nanoparticle, remaining unexplored any effects related to the
  plasma-induced optical inhomogeneities. Here we examine these
  effects by using 3D numerical modeling of coupled electrodynamic and
  material ionization equations. Based on the simulation results, we
  observed a deeply subwavelength plasma-induced nanopatterning of
  spherical silicon nanoparticles. In particular, we revealed strong
  symmetry breaking in the initially symmetrical nanoparticle, which
  arises during ultrafast photoexcitation near the magnetic dipole 
  resonance. The proposed ultrafast breaking of the nanoparticle
  symmetry paves the way to the novel opportunities for nonlinear
  optical nanodevices.} 

\end{tabular}



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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 materials has become prospective paradigm in modern optics, owing to
recent advances in harmonics generation~\cite{shcherbakov2014enhanced,
  yang2015nonlinear, makarov2016self, shorokhov2016multifold,
  makarov2017efficient, makarov2017light} 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 nonlinear
all-dielectric nanodevices are due to the existence 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 in optical properties (transmittance or
reflectance) of all-dielectric nanoantennas~\cite{makarov2015tuning,
  baranov2016nonlinear, baranov2016tuning} and
metasurfaces~\cite{iyer2015reconfigurable, shcherbakov2015ultrafast,
  yang2015nonlinear, shcherbakov2017ultrafast}.



\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}

In previous works on all-dielectric nonlinear nanostructures, the
building blocks (nanoparticles) were considered as objects with
dielectric permittivity \textit{homogeneously} distributed over
nanoparticle (NP). 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}. On the other hand,
plasma explosion imaging technique~\cite{Hickstein2014} revealed
\textit{in situ} strongly asymmetrical electron-hole plasma (EHP)
distribution in various dielectric NPs during their pumping by
femtosecond laser pulses. Therefore, local permittivity in the
strongly photoexcited NPs can be significantly inhomogeneous, and
symmetry of nanoparticles can be reduced.

%The forward ejection of ions was attributed in this case to a nanolensing effect inside the NP and to intensity enhancement as low as $10\%$ on the far side of the NP. Much stronger enhancements can be achieved near electric and magnetic dipole resonances excited in single semiconductor NPs, such as silicon (Si), germanium (Ge) etc.

In this Letter, we show theoretically that ultra-fast photo-excitation
in a spherical silicon NP leads to a strongly inhomogeneous EHP
distribution, as is shown schematically in Fig.~\ref{fgr:concept}. To
reveal and analyze this effect, we perform a full-wave numerical
simulation. We consider an intense femtosecond (\textit{fs}) laser
pulse to interact with a silicon NP supporting Mie resonances and
two-photon EHP generation. In particular, we couple finite-difference
time-domain (FDTD) method used to solve three-dimensional Maxwell
equations with kinetic equations describing nonlinear EHP generation.
Three-dimensional transient variation of the material dielectric
permittivity is calculated for NPs of several sizes. The obtained
results propose a novel strategy to create complicated non-symmetrical
nanostructures by using single photo-excited spherical silicon
NPs. Moreover, we show that a dense EHP can be generated at deeply
subwavelength scale ($< \lambda / 10$) supporting the formation of
small metalized parts inside the NP. In fact, such effects transform a
dielectric NP to a hybrid metall-dielectric one strongly extending
functionality of the ultrafast optical nanoantennas.


%Plan:
%\begin{itemize}

%\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 high melting temperature ($\approx 1690$~K), whereas its nonlinear
optical properties were extensively studied during the last
decades~\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~\cite{Korfiatis2007}
$N_{cr} \approx 5\cdot{10}^{21}$~cm$^{-3}$. 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 and temporal
evolution of EHP in silicon NPs has not been modeled
before. 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 due
to the considered short time scales. In fact, the aim of the present
work is to study the EHP dynamics \textit{during} ultra-short
(\textit{fs}) laser interaction with the NP. The created electron-hole
modifies both laser-particle interaction and, hence, the following
particle evolution. However, the plasma then will recombine at
picosecond time scale.

\subsection{Light propagation}

The incident wave propagates in positive direction of $z$ axis, the NP
geometric center located at $z=0$ front side corresponds to the volume
$z>0$ and back side for $z<0$, as shown in
Fig.~\ref{fgr:concept}. Ultra-short laser interaction and light
propagation inside the silicon NP are modeled by solving the system of
three-dimensional 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
NP 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 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{4\ln{2}(t-t_0)^2}{\theta^2}\right),
\end{aligned}
\end{align}
where $\theta \approx 130$~\textit{fs} is the temporal pulse width at
the half maximum (FWHM), $t_0$ is the time delay, $w_0 = 3{\mu}m$ is the
beam waist, $w(z) = {w_0}\sqrt{1+(\frac{z}{z_R})^2}$ 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 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.

The time-dependent conduction-band carrier density evolution is
described by the rate equation 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 particle saturation 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
$D = k_B T_e \tau/m^* \approx (1$--$\,2)\cdot{10}^{-3}$ m$^2$/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
$N_{cr}$ \cite{Ramer2014}. It means that during the pulse duration
($\approx 130$~\textit{fs}) the diffusion length will be around
10$\,$--15~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) First four Lorentz-Mie coefficients
  ($a_1$, $a_2$, $b_1$, $b_2$) and (b) factors of asymmetry $G_I$,
  $G_{I^2}$ according to the Mie theory at fixed wavelength
  $800$~nm. (c, d) Squared intensity distributions at different radii
  \textit{R} calculated by the Mie theory and (e, f) EHP distribution
  for low 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}

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~\textit{fs} pulse increases the complexity of the
analysis. A detailed discussion on the relation between the Mie theory
and FDTD-EHP model will be provided in the next section.

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

\section{Results and discussion}

\subsection{Asymmetry factors}

\begin{figure*}[p]
 \centering
 \includegraphics[width=140mm]{time-evolution-I-no-NP.pdf}
 \caption{\label{time-evolution} Temporal EHP (a, c, e) and asymmetry
   factor $G_{N_e}$ (b, d, f) evolution for different Si nanoparticle
   radii of (a, b) $R = 75$~nm, (c, d) $R = 100$~nm, and (e, f)
   $R = 115$~nm. Pulse duration $130$~\textit{fs} (FWHM).  Wavelength
   $800$~nm in air. (b, d, f) Different stages of EHP evolution shown
   in Fig.~\ref{plasma-grid} are indicated. The temporal evolution of
   Gaussian beam intensity is also shown. Peak laser intensity is
   fixed to be 10$^{12}$~W/cm$^2$. For better visual representation of
   time scale at a single optical cycle we put a squared electric
   field profile in all plots in Fig.~\ref{time-evolution} in gray
   color as a background image (note linear time scale on the left
   column and logarithmic scale on the right one).}
    \end{figure*}
    
\begin{figure*}   
 \centering
 \includegraphics[width=145mm]{plasma-grid.pdf}
 \caption{\label{plasma-grid} 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) maximum
   during few optical cycles, (3) quasi-stationary regime, (4)
   strongly nonlinear regime). $\Delta{Re(\epsilon)}$ indicates the
   real part change of the local permittivity defined by Equation
   (\ref{Index}). Pulse duration 130~\textit{fs} (FWHM). Wavelength 800~nm in
   air. Peak laser intensity is fixed to be 10$^{12}$~W/cm$^2$. }
 \end{figure*}

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

 We start with a pure electromagnetic problem without EHP
 generation. We plot in Fig.~\ref{mie-fdtd}(a) Mie coefficients of a
 Si NP as a function of its size for a fixed laser wavelength
 $\lambda = 800$~nm.  For the NP sizes under consideration most of
 contribution to the electromagnetic response originates from electric
 and magnetic dipole (ED and MD), while for sizes near
 $R \approx 150$~nm a magnetic quadrupole (MQ) response turns to be
 the main one. The superposition of multipoles defines the
 distribution of electric field inside the NP. We introduce $G_I$
 factor of asymmetry, corresponding to difference between the volume
 integral of squared electric field in the front side ($I^{front}$) of
 the NP to that in the back side ($I^{back}$) 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
 $I^{back}=\int_{(z<0)} |E|^2d{\mathrm{v}}$ are expressed via
 amplitude of the electric field $|E|$. The factor $G_{I^2}$ was
 determined in a similar way by using volume integrals of squared
 intensity to predict EHP asymmetry due to two-photon absorption.
 Fig.~\ref{mie-fdtd}(b) shows $G$ factors as a function of the NP
 size. For the NPs of sizes below the first MD 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 NP as demonstrated
 in Fig.~\ref{mie-fdtd}(d). In fact, rather similar EHP distributions
 can be obtained by applying Maxwell's equations coupled with the rate
 equation for a relatively weak excitation with EHP concentration of
 $N_e \approx 10^{20}$~cm$^{-3}$, see Fig.~\ref{mie-fdtd}(e,f). The
 optical properties do not change considerably due to the excitation
 according to (\ref{Index}). Therefore, the excitation processes
 follow the intensity distribution. However, such coincidence was
 achieved under quasi-stationary conditions, after the electric field
 made enough oscillations inside the Si NP. Further on, we present
 transient analysis, which reveals much more details.

 To achieve a quantitative description for evolution of the EHP
 distribution during the \textit{fs} pulse, we introduced another
 asymmetry factor
 $G_{N_e} = (N_e^{front}-N_e^{back})/(N_e^{front}+N_e^{back})$
 indicating the relationship between the average EHP densities in the
 front and in the back halves of the NP, defined as
 $N_e^{front}=\frac{2}{V}\int_{(z>0)} {N_e}d{\mathrm{v}}$ and
 $N_e^{back}=\frac{2}{V}\int_{(z<0)} {N_e}d{\mathrm{v}}$, where
 $V = \frac{4}{3}\pi{R}^3$ is the volume of the nanosphere. In this
 way, $G_{N_e} = 0$ corresponds to the symmetrical case and the
 assumption of the NP homogeneous EHP distribution can be made to
 investigate the optical response of the excited Si NP. When $G_{N_e}$
 significantly differs from $0$, this assumption, however, could not
 be justified. In what follows, we discuss the results of the
 numerical modeling of the temporal evolution of EHP densities and the
 asymmetry factors $G_{N_e}$ for different sizes of Si NP, as shown in
 Fig.~\ref{time-evolution}). Typical change in permittivity
 corresponding to each stage is shown in Fig.~\ref{plasma-grid}. It
 reveals the EHP evolution stages during interaction of femtosecond
 laser pulse with the Si NPs.
 
 \subsection{Stages of transient Si nanoparticle photoexcitation}

 To describe all the stages of non-linear light interaction 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 at peak intensity
 10$^{12}$~W/cm$^2$ and $\lambda = 800~nm$. In this case, the geometry
 of the EHP distribution can strongly deviate from the intensity
 distribution given by the Mie theory. Two main reasons cause the
 deviation: (i) non-stationarity of interaction between
 electromagnetic pulse and NP (ii) nonlinear effects, taking place due
 to transient optical changes in Si. The non-stationary intensity
 deposition during \textit{fs} pulse 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, MD
 resonance (\textit{b1}) has $Q \approx 8$, whereas electric one
 (\textit{a1}) has $Q \approx 4$. The larger particle supporting MQ
 resonance (\textit{b2}) demonstrates $ Q \approx 40$. As soon as the
 electromagnetic wave period at $\lambda = 800$~nm is
 $\approx 2.7$~\textit{fs}, one needs about 10~\textit{fs} to pump the
 ED, 20~\textit{fs} for the MD, and about 100~\textit{fs} for the MQ.
 According to these considerations, after few optical cycles taking
 place on a 10~\textit{fs} scale it results in the excitation of the
 low-\textit{Q} ED resonance, which dominates MD and MQ independently
 on the exact size of NPs. Moreover, during the first optical cycle
 there is no multipole modes structure inside NP, which results in a
 very similar field distribution for all sizes of NP under
 consideration as shown in Fig.~\ref{plasma-grid}(a,e,i). We address
 to this phenomena as \textit{'Stage~1'}. This stage demonstrates the
 initial penetration of electromagnetic field into the NP during the
 first optical cycle. Resulting factors $G_{N_e}$ exhibit sharp
 increase at this stage (Fig.~\ref{time-evolution}(b,d,f), yielding
 strong and ultrafast symmetry breaking.
 
 \textit{'Stage~2'} corresponds to further electric field oscillations
 ($t \approx 5$--$15$~\textit{fs}) leading to the formation of ED E-field
 pattern in the center of the Si NP as shown in
 Fig.~\ref{plasma-grid}(f,j). We stress the nonstationary nature of
 E-field pattern at this stage, whereas there is a simultaneous growth
 of the incident pulse amplitude. This leads to a superposition of ED
 near-field pattern with that from Stage 1, resulting in EHP
 concentration in the front side of the Si NP. This effect dominates
 for the smallest NP with $R=75$~nm in Fig.~\ref{plasma-grid}(b),
 where ED mode is tuned far away from the resonance (see
 Fig.~\ref{mie-fdtd}(c) for field suppression inside NP predicted by
 the Mie theory). At this stage, the density of EHP
 ($N_e < 10^{20}$~cm$^2$) is still not high enough to significantly
 affect the optical properties of the NP
 (Figs.~\ref{time-evolution}(a,c,e)).

 When the number of optical cycles is large enough
 ($t>20$~\textit{fs}) both ED and MD modes can be exited to the level
 necessary to achieve the stationary intensity pattern corresponding
 to the Mie-based intensity distribution at \textit{'Stage~3'}
 (see Fig.~\ref{plasma-grid}(c,g,k)). The EHP density for the most
 volume of NP is still relatively small to affect the EHP evolution,
 but is already high enough to change the local optical properties,
 i.e. real part of permittivity. Below the MD resonance ($R = 75$~nm),
 the EHP is mostly localized in the front side of the NP as shown in
 Fig.~\ref{plasma-grid}(c). The highest quasi-stationary asymmetry
 factor $G_{N_e} \approx 0.5$--$0.6$ is achieved in this case
 (Fig~\ref{time-evolution}(b)). At the MD resonance conditions
 ($R = 100$~nm), the EHP distribution has a toroidal shape and is much
 closer to the homogeneous distribution. In contrast, above the MD
 resonant size for $R = 115$~nm the $G_{N_e} < 0$ due to the fact that
 EHP is dominantly localized in the back side of the NP.

 Diue to a quasi-stationary pumping during Stage~3,
 it is superposed with Stage~1 field pattern, resulting in an
 additional EHP localized in the front side. This can be seen when
 comparing result from the Mie theory in Fig.~\ref{mie-fdtd}(d) with
 that of full 3D simulation in Fig.~\ref{mie-fdtd}(f). Note that
 pumping of NP significantly changes during a single optical cycle,
 this leads to a large variation of asymmetry factor $G_{N_e}$ at the
 first stage. This variation steadily decreases as it goes to Stage~3,
 as shown in Fig.~\ref{time-evolution}(b,d,f).

 To explain this effect, we consider the time evolution of average EHP
 densities $N_e$ in the front and back halves of the NP presented in
 Fig.~\ref{time-evolution}(a,c,e). As soon as the recombination and
 diffusion processes are negligible at \textit{fs} time scale, both
 $N_e^{front}$ and $N_e^{back}$ curves experience monotonous behavior
 with small pumping steps synced to the incident pulse. The front and
 the back halves of NP are separated in space, which obviously leads
 to the presence of time delay between pumping steps in each curve
 caused by the same optical cycle of the incident wave. This delay
 causes a large asymmetry factor during the first stage. However, as soon
 as average EHP density increases, the relative contribution of these
 pumping steps to the resulting asymmetry becomes smaller. This way
 variations of asymmetry $G_{N_e}$ synced with the period of incident
 light decrease.

 Higher excitation conditions are followed by larger values of
 electric field amplitude, which leads to the appearance of high EHP
 densities causing a significant change in the optical properties of
 silicon according to the equations (\ref{Index}). From the Mie
 theory, the initial (at the end of Stage~3) spatial pattern of the
 optical properties is non-homogeneous. When non-homogeneity of the
 optical properties becomes strong enough it leads to the
 reconfiguration of the E-field inside NP, which in turn strongly
 affects further reconfiguration of the optical properties. We refer
 to these strong nonlinear phenomena as \textit{'Stage~4'}. In
 general, the reconfiguration of the electric field is unavoidable as
 far as the result from the Mie theory comes with the assumption of
 homogeneous optical properties in a spherical NP.

 Thus, the evolution of EHP density during Stage~4 depends on the
 result of multipole modes superposition at the end of Stage~3 and is
 quite different as we change the size of NP. For $R=75$~nm and
 $R=100$~nm, we observe a front side asymmetry before Stage~4,
 however, its origin is quite different. The $R=75$~nm NP is out
 of resonance, moreover, Mie field pattern and the one, which comes
 from Stage~1 are quite similar. As soon as EHP density becomes
 high enough to change optical properties, the NP is still out of
 resonance, however, the presence of EHP increases absorption in
 agreement with (\ref{Index}), decreases Q-factor, and destroys
 optical modes.
 % This effect effectively leads to a partial screening, and it
 % becomes harder for the incident wave to penetrate deeper into
 % EHP. Finally, this finishes spilling the NP`s volume with plasma
 % reducing the asymmetry, see Fig.~\ref{plasma-grid}(d).
 
 For $R=100$~nm, the evolution during the final stage goes in a
 similar way, with a notable exception regarding MD resonance.  As
 soon as presence of EHP increases the absorption, it suppresses the
 MD resonance with symmetric filed pattern, thus, the asymmetry factor
 can be increased. This result was observed in
 Fig.~\ref{time-evolution}(d) with a local maximum near 100~\textit{fs}
 mark.
 
 The last NP with $R=115$~nm shows the most complex behavior during
 Stage~4. The superposition of Mie-like E-field pattern with that
 from Stage~1 results in the presence of two EHP spatial maxima, back
 and front shifted. They serve as starting seeds for the EHP
 formation, and an interplay between them forms a complex behavior of
 the asymmetry factor curve. Namely, the sign is changed from negative
 to positive and back during the last stage. This numerical result can
 hardly be explained in a simple qualitative manner, it is too complex
 to account all near-field interaction of incident light with two EHP
 regions inside a single NP. It is interesting to note, however, that
 in a similar way as it was for $R=100$~nm the increased absorption
 should destroy ED and MD resonances, which are responsible for the
 back-shifted EHP. As soon as this EHP region is quite visible on the
 last snapshot in Fig.~\ref{plasma-grid}(l), this means that EHP seeds
 are self-supporting.
 
 %A bookmark by Kostya

 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. 
     
 It is worth noting that it is possible to achieve the formation of
 deeply subwavelength EHP regions due to high field localization. The
 smallest EHP localization and the larger asymmetry factor are
 achieved below the MD resonant conditions for $R < 100$~nm.
 Thus, the EHP distribution in Fig.~\ref{plasma-grid}(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 NP size and scattering efficiency
% factor on scattering directions}

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

% We have discussed the EHP kinetics for a silicon NP of a
% fixed radius $R \approx 105$~nm. In what follows, we investigate the
% influence of the NP size on the EHP patterns and temporal
% evolution during ultrashort laser irradiation. A brief analysis of
% the initial intensity distribution inside the NP given by
% the classical Mie theory for homogeneous spherical particles
% \cite{Mie1908} can be useful in this case. Fig. \ref{time-evolution}(a, b)
% shows the scattering efficiency and the asymmetry parameter for
% forward/backward scattering for non-excited silicon NPs of
% different radii calculated by Mie theory \cite{Mie1908}. Scattering
% efficiency dependence gives us the value of resonant sizes of
% NPs, 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
% NP optical response. For example, it is easier to localize
% high electron densities inside smaller NPs, however, due
% to the negligible size of the generated EHP with respect to laser
% wavelength in media, the intensity distribution around the
% NP 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 NPs and electron densities are presented in
% Fig. \ref{time-evolution}(c). One can see, that the maximum value are achieved
% for the NPs, that satisfy both initial maximum forward
% scattering and not far from the first resonant condition. For larger
% NPs, 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 NP 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 NP for double-pulse
% experiment. The first pulse of larger pulse energy and polarization
% along $Ox$ generates asymmetric EHP inside silicon NP,
% 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 NP 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 NP 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 NP, providing a symmetry reduction
% \cite{Wang2006}.

% \begin{figure}[ht] \centering
% \includegraphics[width=90mm]{fig4.png}
% \caption{\label{fig4} a) Electron plasma distribution inside Si
% NP $R \approx 95$~nm 50~\textit{fs} 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 NP 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 NPs
% 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.

 \section{Conclusions} We have rigorously modeled and studied
 ultra-fast and intense light interaction with a single silicon
 nanoparticle of various sizes for the first time to our best
 knowledge. As a result of the presented self-consistent nonlinear
 calculations, we have obtained spatio-temporal EHP evolution inside
 the NPs and investigated the asymmetry of the EHP distributions.  We
 have revealed EHP strong asymmetric distribution during the first optical
 cycle for different sizes. The highest average EHP asymmetry has
 been observed for NPs of smaller sizes below the first magnetic
 dipole resonance, when EHP is concentrated in the front side mostly
 during the laser pulse absorption. Essentially different EHP
 evolution and lower asymmetry has been achieved for larger NPs due to
 the intensity enhancement in the back side of the NP. The EHP
 densities above the critical value have been shown to lead to
 homogenization of the EHP distribution.  The observed plasma-induced
 breaking symmetry can be useful for creation of nonsymmetrical
 nanophotonic designes, e.g. for beam steering or enhanced second
 harmonics generation. Also, the asymmetric EHP opens a wide range of
 applications in NP nanomashining at deeply subwavelength scale.

\section{Acknowledgments}
A. R. and T. E. I. gratefully acknowledge the CINES of CNRS for
computer support. S. V. M. is thankful to ITMO Fellowship
Program. This work was partially supported by Russian Foundation for
Basic Researches (grants 17-03-00621, 17-02-00538, 16-29-05317).




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