small changes and cleanup

main.tex

@@ -139,7 +139,6 @@
  
   \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
@@ -243,7 +242,7 @@ distributions in silicon nanoparticle around a magnetic resonance.}
 \end{figure}
 
 Recently, plasma explosion imaging technique has been used to
-observe electron-hole plasmas (EHP), produced by femtosecond lasers,
+observe electron-hole plasma (EHP), produced by femtosecond lasers,
 inside NPs~\cite{Hickstein2014}. Particularly, a strongly
 localized EHP in the front side\footnote{The incident wave propagates
   in positive direction of $z$ axis. For the NP with
@@ -259,8 +258,8 @@ 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
+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 free carrier
 generation. In particular, we couple finite-difference time-domain
 (FDTD) method used to solve three-dimensional Maxwell equations with
@@ -270,20 +269,15 @@ 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$/100) supporting the formation of small
-metalized parts inside the NP. In fact, such effects transform an
-all-dielectric NP to a hybrid one strongly extending functionality of
-the ultrafast optical nanoantennas.
+subwavelength scale ($< \lambda / 10$) supporting the formation of
+small metalized parts inside the NP. In fact, such effects transform
+an all-dielectric NP to a hybrid metall-dielectric one strongly
+extending functionality of the ultrafast optical nanoantennas.
 
 
 %Plan:
 %\begin{itemize}
-%\item Fig.1: Beautiful conceptual picture
-%\item Fig.2: Temporal evolution of EHP in NP with different diameters
-%at fixed intensity, in order to show that we have the highest
-%asymmetry around magnetic dipole (MD) resonance. This would be really
-%nice!
+
 %\item Fig.3: Temporal evolution of EHP in NP with fixed diameter (at
 %MD) at different intensities, in order to show possible regimes of
 %plasma-patterning of NP volume. It would be nice, if we will show
@@ -316,19 +310,19 @@ the critical density and above, silicon acquires metallic properties
 ultrashort laser irradiation.
 
 The process of three-dimensional photo-generation of the EHP in
-silicon NPs 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 NP. The created electron-hole
-plasma then will recombine, however, as its existence modifies both
+silicon NPs 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 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.
+evolution. However, the plasma then will recombine at picosecond time
+scale.
 
 \subsection{Light propagation}
 
@@ -395,21 +389,6 @@ To account for the material ionization that is induced by a
 sufficiently intense laser field inside the particle, we couple
 Maxwell's equations with the kinetic equation for the electron-hole
 plasma as described below.
-% \begin{figure*}[ht!]
-% \centering
-% \includegraphics[width=120mm]{fig2.png}
-% \caption{\label{fig2} Free carrier density snapshots of electron
-% plasma evolution inside Si NP 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 NP
-% $R \approx 105$~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 NP.
-% $n_{cr} = 5\cdot{10}^{21}$~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
@@ -438,7 +417,7 @@ Einstein formula $D = k_B T_e \tau/m^* \approx (1\div2)\cdot{10}^{-3}$ m$^2$/s
 $\tau=1$~\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 ($\approx 50$~\textit{fs}) the diffusion
-length will be around $5\div10$~nm for $N_e$ close to $N_{cr}$.
+length will be around $5--10$~nm for $N_e$ close to $N_{cr}$.
 
 \begin{figure}[ht!] 
 \centering
@@ -454,35 +433,6 @@ length will be around $5\div10$~nm for $N_e$ close to $N_{cr}$.
   $\vec{E}$ is along $X$ axis.}
 \end{figure}
 
-%\begin{figure*}[ht!] \label{EHP}
-%\centering
-%\begin{tabular*}{0.76\textwidth}{ c@{\extracolsep{\fill}} c@{\extracolsep{\fill}} c@{\extracolsep{\fill}} c@{\extracolsep{\fill}} c}
-%$-80\:f\!s$&$-60\:f\!s$&$-30\:f\!s$&$ -20\:f\!s$&$ -10\:f\!s$\\
-%\end{tabular*}
-%{\setlength\topsep{-1pt}
-%\begin{flushleft}
-%$R=75$~nm
-%\end{flushleft}}
-%\includegraphics[width=0.9\textwidth]{2nm_75}
-%{\setlength\topsep{-1pt}
-%\begin{flushleft}
-%$R=100$~nm
-%\end{flushleft}}
-%\includegraphics[width=0.9\textwidth]{2nm_100}
-%{\setlength\topsep{-1pt}
-%\begin{flushleft}
-%$R=115$~nm
-%\end{flushleft}}
-%\includegraphics[width=0.9\textwidth]{2nm_115}
-%\caption{\label{plasma-105nm} Evolution of electron density $n_e$
- % (using $10^{\,20} \ {\rm cm}^{-3}$ units) for
-  %(a$\,$--$\,$e)~$R=75$~nm, (f$\,$--$\,$j$\,$)~$R=100$~nm, and
-  %($\,$k$\,$--$\,$o)~$R=115$~nm. Gaussian pulse duration $80\:f\!s$,
-  %snapshots are taken before the pulse maxima, the corresponding
-  %time-shifts are shown in the top of each column. Laser irradiation
-  %fluences are (a-e) $0.12$~J/cm$^2$, (f-o) $0.16$~J/cm$^2$.}
-%\end{figure*}
-
 The changes of the real and imaginary parts of the permittivity
 associated with the time-dependent free carrier response
 \cite{Sokolowski2000} can be derived from equations (\ref{Maxwell},
@@ -513,17 +463,17 @@ 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
-   (a, b) $R = 75$~nm, (c, d) $R = 100$~nm, (e, f) $R = 115$~nm. Pulse
+ \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 $50$~\textit{fs} (FWHM). Wavelength $800$~nm in air. (b,
-   d, f) Different stages of EHP evolution shown in Fig.~\ref{fig2}
+   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 fluence is fixed to be $0.125$~J/cm$^2$.}
 \vspace*{\floatsep}
  \centering
  \includegraphics[width=150mm]{plasma-grid.pdf}
- \caption{\label{fig2} EHP density snapshots inside Si nanoparticle of
+ \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) extremum at few optical cycles,
@@ -574,8 +524,8 @@ license.
  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}.
+ shown in Fig.~\ref{plasma-grid} and the temporal/EHP dependent evolution of
+ the asymmetry factor $G_{N_e}$ in Fig.~\ref{time-evolution}.
 
  % Fig. \ref{Mie} demonstrates the temporal evolution of the EHP
  % generated inside the silicon NP of $R \approx 105$~nm. Here,
@@ -588,7 +538,7 @@ license.
 % asymmetry parameter, $G$, which is defined as a relation between the
 % average electron density generated in the front side of the
 % NP and the average electron density in the back side, as
-% shown in Fig. \ref{fig2}. During the femtosecond pulse interaction,
+% shown in Fig. \ref{plasma-grid}. During the femtosecond pulse interaction,
 % this parameter significantly varies.
 
 
@@ -617,10 +567,10 @@ license.
  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
+ as shown in Figs.~\ref{plasma-grid} and~\ref{plasma-grid}. 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
+ NPs in Fig.~\ref{plasma-grid}(a,e,j) and~\ref{time-evolution}. The larger the NPs size
  is, the higher the NP asymmetry $G_{N_e}$ is achieved.
 
  \textit{'Stage 2'} corresponds to further electric field oscillations
@@ -634,11 +584,11 @@ license.
  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
+ Fig.~\ref{time-evolution}). 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
+ front side of the NP as shown in Fig.~\ref{plasma-grid}(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
@@ -652,13 +602,13 @@ license.
  contributes to the forward shifting of EHP density
  maximum. Therefore, EHP is localized in the front part of the NP,
  influencing the asymmetry factor $G_{N_e}$ in
- Fig.~\ref{fig3}. Approximately at the pulse peak, the critical
+ Fig.~\ref{time-evolution}. Approximately at the pulse peak, the critical
  electron density $N_{cr} = 5\cdot{10}^{21}$~cm$^{-3}$ for silicon,
  which corresponds to the transition to quasi-metallic state
  $Re(\epsilon) \approx 0$ and to the electron plasma resonance, is
  overcome. Further irradiation leads to a decrease in the asymmetry
  parameter down to $G_{N_e} = 0$ for higher EHP densities, as one can
- observe in Fig.~\ref{fig2}(d, h, l).
+ observe in Fig.~\ref{plasma-grid}(d, h, l).
 
  As the EHP acquires quasi-metallic properties at stronger excitation
  $N_e > 5\cdot{10}^{21}$~cm$^{-3}$, the EHP distribution evolves
@@ -672,7 +622,7 @@ license.
  deeply subwavelength EHP regions due to high field localization. The
  smallest EHP localization and the larger asymmetry factor are
  achieved below the magnetic dipole resonant conditions for $R < 100$~nm.
- Thus, the EHP distribution in Fig.~\ref{fig2}(c) is optimal for
+ 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
@@ -682,8 +632,8 @@ license.
 % factor on scattering directions}
 
 % \begin{figure}[ht] \centering
-% \includegraphics[width=90mm]{fig3.png}
-% \caption{\label{fig3} a) Scattering efficiency factor $Q_{sca}$
+% \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
@@ -698,7 +648,7 @@ license.
 % 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{fig3}(a, b)
+% \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
@@ -730,7 +680,7 @@ license.
 % $n_{cr} = 5\cdot{10}^{21}~cm^{-3}$, 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{fig3}(c). One can see, that the maximum value are achieved
+% 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