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main.tex

@@ -137,22 +137,29 @@
                                     } \\%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. However, only homogeneous plasma
-generation was previously considered in the 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.}
+  \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. However, only homogeneous plasma
+  generation was previously considered in the 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}
 
@@ -223,8 +230,8 @@ metasurfaces~\cite{iyer2015reconfigurable, shcherbakov2015ultrafast,
 In these 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
+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}.
 
@@ -250,24 +257,24 @@ and magnetic dipole resonances excited in single semiconductor
 NPs, such as silicon (Si), germanium (Ge) etc.
 
 In this Letter, we show that ultra-short laser-based EHP
-photo-excitation in a spherical semiconductor (e.g., silicon)
-NP leads to a strongly inhomogeneous carrier
-distribution. To reveal and study this effect, we perform a full-wave
-numerical simulation of the intense femtosecond (\textit{fs}) laser
-pulse interaction with a silicon NP supporting Mie
-resonances and two-photon free carrier 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
+photo-excitation in a spherical semiconductor (e.g., silicon) NP leads
+to a strongly inhomogeneous carrier distribution. To reveal and study
+this effect, we perform a full-wave numerical simulation of the
+intense femtosecond (\textit{fs}) laser pulse interaction with a
+silicon NP supporting Mie resonances and two-photon free carrier
+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
 ($\approx$$\lambda$$^3$/100) supporting the formation of small
-metalized parts inside the NP. In fact, such effects
-transform an all-dielectric NP to a hybrid one strongly
-extending functionality of the ultrafast optical nanoantennas.
+metalized parts inside the NP. In fact, such effects transform an
+all-dielectric NP to a hybrid one strongly extending functionality of
+the ultrafast optical nanoantennas.
 
 
 %Plan:
@@ -497,12 +504,13 @@ license.
 \begin{figure*}[p]
  \centering
  \includegraphics[width=145mm]{time-evolution-I-no-NP.pdf}
- \caption{\label{fig3} Temporal EHP (a, c, e) and asymmetry factor $G_{N_e}$ (b, d, f) evolution for different Si nanoparticle radii
+ \caption{\label{fig3} Temporal EHP (a, c, e) and asymmetry factor
+   $G_{N_e}$ (b, d, f) evolution for different Si nanoparticle radii
    (a, b) $R = 75$ nm, (c, d) $R = 100$ nm, (e, f) $R = 115$ nm. Pulse
-   duration $50$~\textit{fs} (FWHM). Wavelength $800$ nm in air. (b, d, f)
-   Different stages of EHP evolution shown in Fig.~\ref{fig2} are
-   indicated. The temporal evolution of Gaussian beam
-   intensity is also shown. Peak laser fluence is fixed to be $0.125$J/cm$^2$.}
+   duration $50$~\textit{fs} (FWHM). Wavelength $800$ nm in air. (b,
+   d, f) Different stages of EHP evolution shown in Fig.~\ref{fig2}
+   are indicated. The temporal evolution of Gaussian beam intensity is
+   also shown. Peak laser fluence is fixed to be $0.125$J/cm$^2$.}
 \vspace*{\floatsep}
  \centering
  \includegraphics[width=150mm]{plasma-grid.pdf}
@@ -510,53 +518,55 @@ license.
    radii $R = 75$ nm (a-d), $R = 100$ nm (e-h) and $R = 115$ nm (i-l)
    taken at different times and conditions of excitation (stages
    $1-4$: (1) first optical cycle, (2) extremum at few optical cycles,
-   (3) Mie theory, (4) nonlinear effects). $\Delta{Re(\epsilon)}$ indicates the real part change of the dielectric function defined by Equation (\ref{Index}). Pulse duration $50$~\textit{fs}
+   (3) Mie theory, (4) nonlinear effects). $\Delta{Re(\epsilon)}$
+   indicates the real part change of the dielectric function defined
+   by Equation (\ref{Index}). Pulse duration $50$~\textit{fs}
    (FWHM). Wavelength $800$ nm in air. Peak laser fluence is fixed to
    be $0.125$ J/cm$^2$.}
  \end{figure*}
 
 %\subsection{Effect of the irradiation intensity on EHP generation}
 
- Firstly, we analyze Mie coefficients (Fig.~\ref{mie-fdtd}) and
- the intensity distribution inside the non-excited Si NP as
- a function of its size for a fixed laser wavelength $\lambda = 800$~nm.  We introduce $G_I$ factor of asymmetry, corresponding to
- difference between the volume integral of intensity in the front side
- of the NP to that in the back side normalized to their sum:
+ Firstly, we analyze Mie coefficients (Fig.~\ref{mie-fdtd}) and the
+ intensity distribution inside the non-excited Si NP as a function of
+ its size for a fixed laser wavelength $\lambda = 800$~nm.  We
+ introduce $G_I$ factor of asymmetry, corresponding to difference
+ between the volume integral of intensity in the front side of the NP
+ to that in the back side normalized to their sum:
  $G_I = (I^{front}-I^{back})/(I^{front}+I^{back})$, where
  $I^{front}=\int_{(z>0)}|E|^2d{\mathrm{v}}$ and
  $I^{back}=\int_{(z<0)} |E|^2d{\mathrm{v}}$. The factor $G_{I^2}$ was
  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 magnetic dipole resonance, the intensity is
- enhanced in the front side as in Fig.~\ref{mie-fdtd}(c) and
- $G_I > 0$. The behavior changes near the size resonance value,
- corresponding to $R \approx 105$~nm. In contrast, for larger sizes,
- the intensity is enhanced in the back side of the NP as
- demonstrated in Fig.~\ref{mie-fdtd}(d). In fact, very similar EHP
- distributions can be obtained by applying Maxwell's equations coupled
- with the rate equation for relatively weak excitation
- $N_e \approx 10^{20}$ cm$^{-3}$. The optical properties do not change
- considerably due to 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. 
+ 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 magnetic dipole resonance,
+ the intensity is enhanced in the front side as in
+ Fig.~\ref{mie-fdtd}(c) and $G_I > 0$. The behavior changes near the
+ size resonance value, corresponding to $R \approx 105$~nm. In
+ contrast, for larger sizes, the intensity is enhanced in the back
+ side of the NP as demonstrated in Fig.~\ref{mie-fdtd}(d). In fact,
+ very similar EHP distributions can be obtained by applying Maxwell's
+ equations coupled with the rate equation for relatively weak
+ excitation $N_e \approx 10^{20}$ cm$^{-3}$. The optical properties do
+ not change considerably due to 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.
  
- To achieve
- a qualitative description of the EHP distribution, we introduced
- another asymmetry factor
+ To achieve a qualitative description of the EHP distribution, 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 parts of the NP. This way, $G_{N_e} = 0$ corresponds
- to the quasi-homogeneous case and the assumption of the NP
- homogeneous EHP distribution can be made to investigate the optical
- response of the excited Si NP. 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 revealing
- the EHP evolution stages during pulse duration shown in
- Fig.~\ref{fig2} and the temporal/EHP dependent evolution of the
- asymmetry factor $G_{N_e}$ in Fig.~\ref{fig3}.
+ front and in the back parts of the NP. This way, $G_{N_e} = 0$
+ corresponds to the quasi-homogeneous case and the assumption of the
+ NP homogeneous EHP distribution can be made to investigate the
+ optical response of the excited Si NP. 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 revealing the EHP evolution stages during pulse duration
+ shown in Fig.~\ref{fig2} and the temporal/EHP dependent evolution of
+ the asymmetry factor $G_{N_e}$ in Fig.~\ref{fig3}.
 
 % Fig. \ref{Mie} demonstrates the temporal evolution of the EHP
 % generated inside the silicon NP of $R \approx 105$
@@ -587,17 +597,19 @@ license.
  Si. The non-stationary intensity deposition results in different time
  delays for exciting electric and magnetic resonances inside Si NP
  because of different quality factors $Q$ of the resonances. In
- particular, magnetic dipole resonance (\textit{b1}) has $Q \approx$ 8,
- whereas electric one (\textit{a1}) has $Q \approx$4. The larger
+ particular, magnetic dipole resonance (\textit{b1}) has $Q \approx$
+ 8, whereas electric one (\textit{a1}) has $Q \approx$4. The larger
  particle supporting magnetic quadrupole resonance (\textit{b2})
  demonstrates \textit{Q} $\approx$ 40. As soon as the electromagnetic
- wave period at $\lambda$~=~800~nm is 2.6~\textit{fs}, one needs about 10~\textit{fs}
- to pump the electric dipole, 20~\textit{fs} for the magnetic dipole, and
- about 100~\textit{fs} for the magnetic quadrupole. According to these considerations, the first optical cycles taking place on few-femtosecond scale result in the excitation of the
- low-\textit{Q} electric dipole resonance independently on the exact
- size of NPs and with the EHP concentration mostly on the front side
- of the NPs. We address to this phenomena as \textit{'Stage 1'}, as
- shown in Figs.~\ref{fig2} and~\ref{fig2}. The first stage at the
+ wave period at $\lambda$~=~800~nm is 2.6~\textit{fs}, one needs about
+ 10~\textit{fs} to pump the electric dipole, 20~\textit{fs} for the
+ magnetic dipole, and about 100~\textit{fs} for the magnetic
+ quadrupole. According to these considerations, the first optical
+ cycles taking place on few-femtosecond scale result in the excitation
+ of the low-\textit{Q} electric dipole resonance independently on the
+ exact size of NPs and with the EHP concentration mostly on the front
+ side of the NPs. We address to this phenomena as \textit{'Stage 1'},
+ as shown in Figs.~\ref{fig2} and~\ref{fig2}. The first stage at the
  first optical cycle demonstrates the dominant electric dipole
  resonance effect on the intensity/EHP density distribution inside the
  NPs in Fig.~\ref{fig2}(a,e,j) and~\ref{fig3}. The larger the NPs size
@@ -611,19 +623,20 @@ license.
  is still not high enough to significantly affect the optical properties
  of the NP.
 
- A number of optical cycles ($>$10 or $t>$25~\textit{fs}) is necessary to
- achieve the stationary intensity pattern corresponding to the
+ A number of optical cycles ($>$10 or $t>$25~\textit{fs}) is necessary
+ to achieve the stationary intensity pattern corresponding to the
  Mie-based intensity distribution at the \textit{'Stage $3$'} (see
- Fig.~\ref{fig3}). The EHP density is still relatively small to
- affect the EHP evolution or for diffusion, but is already high
- enough to change the local optical properties. Below the magnetic
- dipole resonance $R \approx 100$~nm, the EHP is mostly localized in
- the front side of the NP as shown in Fig.~\ref{fig2}(c). The highest
- stationary asymmetry factor $G_{N_e} \approx 0.5\div0.6$ is achieved in this case. At the magnetic dipole
- resonance conditions, the EHP distribution has a toroidal shape and
- is much closer to the homogeneous distribution. In contrast, above the
- magnetic dipole resonant size for $R = 115$~nm, and the $G_{N_e} < 0$ due
- to the fact that EHP is dominantly localized in the back side of the NP.
+ Fig.~\ref{fig3}). The EHP density is still relatively small to affect
+ the EHP evolution or for diffusion, but is already high enough to
+ change the local optical properties. Below the magnetic dipole
+ resonance $R \approx 100$~nm, the EHP is mostly localized in the
+ front side of the NP as shown in Fig.~\ref{fig2}(c). The highest
+ stationary asymmetry factor $G_{N_e} \approx 0.5\div0.6$ is achieved
+ in this case. At the magnetic dipole resonance conditions, the EHP
+ distribution has a toroidal shape and is much closer to the
+ homogeneous distribution. In contrast, above the magnetic dipole
+ resonant size for $R = 115$~nm, and the $G_{N_e} < 0$ due to the fact
+ that EHP is dominantly localized in the back side of the NP.
 
  For the higher excitation conditions, the optical properties of
  silicon change significantly according to the equations
@@ -801,7 +814,11 @@ nano-bio-applications.  The observed plasma-induced breaking symmetry
 can be also useful for beam steering, or for the enhanced second
 harmonics generation.
 
-\section{Acknowledgments} 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).
+\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).