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\includegraphics{head_foot/DOI} & \noindent\LARGE{\textbf{ Plasma-Induced Symmetry Breaking in a 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{$^{a}$} Tatiana E. Itina,\textit{$^{a\ddag}$} Konstantin Ladutenko,\textit{$^{b}$} and Sergey Makarov\textit{$^{b}$}
 
 \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 photogeneration 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. However, previous works assumed only homogenized plasma generation in the photoexcited nanoparticle, neglecting all effects related to inhomogeneous plasma distribution. Here numerical studying of the plasma photogeneration allows us to propose a novel concept of deeply subwavelength ($\approx$$\lambda$$^3$/100) plasma-induced nanopatterning of symmetrical silicon nanoparticle. More importantly, we reveal strong symmetry breaking in the initially symmetrical nanoparticle during ultrafast photoexcitation near the magnetic dipole resonance. The ultrafast manipulation by nanoparticle inherent structure and symmetry paves the way to novel principles 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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\footnotetext{\dag~Electronic Supplementary Information (ESI) available: [details of any supplementary information available should be included here]. See DOI: 10.1039/b000000x/}
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\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}

In this Letter, we show that electron-hole plasma (EHP) generation in a spherical dielectric (e.g., silicon) nanoparticle leads to strongly nonhomogeneous EHP distribution. To reveal and study this effect, we for the first time provide full-wave numerical simulation of intensive femtosecond (fs) laser pulse interaction with a dielectric nanoparticle supporting Mie resonances and two-photon generation of EHP. In particular, we couple finite-difference time-domain (FDTD) method of the Maxwell's equations solving with equations describing nonlinear EHP generation and its variation of material dielectric permittivity. The obtained results propose a novel strategy to create complicated nonsymmetrical nanostructures by using only a photoexcited spherical silicon nanoparticle. Moreover, we show that dense EHP can be generated at deeply subwavelength scale ($\approx$$\lambda$$^3$/100) supporting formation of small metallized parts inside the nanoparticle which transforms all-dielectric nanoparticle to a hybrid one that extends functionality of 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}
 
 %TEI for Anton: please describe what you model first (what physical processes). Then, we speak about "pulse", but never "laser". Should we provide more details concerning irradiation source?

We focus out attention on silicon because this material is promising for the implementation of nonlinear photonic devices thanks to a broad range of optical nonlinearities, strong two-photon absorption and 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}.

Since the 3D modeling of EHP photo-generation in a resonant silicon nanoparticle has not been done before in time-domain, we develop a model considering incidence of ultrashort light pulse on a silicon sphere, where EHP is generated via one- and two-photon absorption processes, and then decayed via Auger recombination. Importantly, we also consider nonlinear feedback of the material by taking into account intraband light absorption on the generated free carriers. In order to simplify our model, we neglect diffusion of EHP, because the aim of our work is to study EHP dynamics \textit{during} laser interaction with the nanoparticle.

\subsection{Light propagation}
The propagation of light inside the silicon nanoparticle is modeled by solving the system of Maxwell's equations, written in the following way
\begin{align} \begin{cases} \label{Maxwell}$$
	\displaystyle{\frac{\partial{\vec{E}}}{\partial{t}}=\frac{\nabla\times\vec{H}}{\epsilon_0\epsilon}-\frac{1}{\epsilon_0\epsilon}\vec{J}} \\ 		
	\displaystyle{\frac{\partial{\vec{H}}}{\partial{t}}=-\frac{\nabla\times\vec{E}}{\mu_0}},
	$$ \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 [green1995] \cite{Green1995}, and $\vec{J}$ is the nonlinear current, which includes the contribution due to heating of the conduction band, described by the differential equation derived from the Drude model
\begin{equation} \label{Drude} \displaystyle{\frac{\partial{\vec{J}}}{\partial{t}} = - \nu_e\vec{J} + \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 [sokolowski2000]\cite{Sokolowski2000}, $n_e(t)$ is the time-dependent free carrier density and $\nu_e = 10^{15} s^{-1}$ is the electron collision frequency [sokolowski2000]\cite{Sokolowski2000}. Silicon nanoparticle is surrounded by air, 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 equations is solved by the finite-difference numerical method [rudenko2016]%\cite{Rudenko2016}
, based on the finite-difference time-domain (FDTD) method [yee1966] \cite{Yee1966} and auxiliary-differential method for disperse media [taflove1995]\cite{Taflove1995}. At the edges of the grid, we apply absorbing boundary conditions related to convolutional perfect matched layers (CPML) to avoid nonphysical reflections [roden2000]% \cite{Roden2000}
. Initial electric field is introduced as a Gaussian focused beam source 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 pulse width at 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 material ionization, we couple Maxwell's equations with the kinetic equation for EHP generation and relaxation inside silicon nanoparticle.
\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) $30$ fs b) $10$ fs before the pulse peak, c) at the pulse peak, d) $10$ fs e) $30$ fs after the pulse peak. Pulse duration $50$ fs (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 with a rate equation, firstly proposed by van Driel [van1987] \cite{Van1987}, taking into account photoionization, avalanche ionization and Auger recombination 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 [choi2002, bristow2007, derrien2013] \cite{Choi2002, Bristow2007, Derrien2013}, $n_a = 5\cdot{10}^{22} cm^{-3}$ is the saturation particle density [derrien2013] \cite{Derrien2013}, $C = 3.8\cdot{10}^{-31} cm^6/s$ is the Auger recombination rate [van1987]\cite{Van1987}, $\tau_{rec} = 6\cdot{10}^{-12}$s is the minimum Auger recombination time [yoffa1980]\cite{Yoffa1980}, and $\alpha = 21.2 cm^2/J$ is the avalanche ionization coefficient [pronko1998] \cite{Pronko1998} at the wavelength $800$ nm in air. Free carrier diffusion can be neglected during and shortly after the excitation [van1987, sokolowski2000]\cite{Van1987, Sokolowski2000}.
\begin{figure*}[ht!]
\centering
\includegraphics[width=0.9\textwidth]{Ne_105nm_800}
\caption{\label{plasma-105nm} Split figure \ref{fig2} into two, this is first part.}
\end{figure*}

The changes of the real and imaginary parts of the permeability associated with the time-dependent free carrier response [sokolowski2000] \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 EHP temporal evolution inside silicon nanoparticle of $R \approx 105$ nm during irradiation by high-intensity ultrashort laser Gaussian pulse. Snapshots of electron density taken at different times correspond to different total amount of deposited energy and, therefore, different intensities of irradiation. We introduce the parameter of EHP asymmetry $G$, 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 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 equations (\ref{Index}), and nonresonant 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 electron plasma resonance [sokolowksi2000] \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 of the asymmetry parameter down to $1$ for higher electron densities 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 the optimal for symmetry breaking in silicon nanoparticle, as results in the bigger asymmetry factor $G$ and higher electron densities $n_e$. We want to stress here that such regime could be still safe for nanoparticle due to small volume of such high EHP density, which should be diffused after some time.  

\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}

Previously, the EHP kinetics has been demonstrated only for a silicon nanoparticle of a fixed radius $R \approx 105$ nm (TEI :REF ???). Here, 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 Mie theory for a spherical homogenneous nanoparticle [mie1908] \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 [mie1908]\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. 

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 [yoffa1980] \cite{Yoffa1980}, therefore, the electron density will not have time to decrease significantly for subpicosecond pulse separations. In our simulations, we use $\delta{t} = 200$ fs 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).

\begin{figure}[ht]
\centering
\includegraphics[width=90mm]{fig4.png}
\caption{\label{fig4} a) Electron plasma distribution inside Si nanoparticle $R \approx 95$ nm $50$ fs after the pulse peak; 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.
TEI: for Konstantin L.: please change Mie figure and add a discussion and how the results agree, what is the difference
TEI: for Anton R.: please add a discussion of the previous experiments and how the results agree

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 fs, we need about 25 fs pulse to pump dipole response, about 150 fs for quadrupole, and about 2000fs 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 light interaction with a 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 for different light intensities and pulse temporal widths. It has been demonstrated that 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}
TEI: need to add Acknowledgements
The authors gratefully acknowledge financial support from The French Ministry of Science and Education and from France-Russia PHC Kolmogorov project. 
TEI: for Sergey M., please put Kivshar et al. and other Refs 
TEI: for Anton R., please put much more references that you have found describing experiments and previous calculations





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