Update on Overleaf.

main.tex

@@ -226,13 +226,7 @@ reflectance) of all-dielectric nanoantennas~\cite{makarov2015tuning,
 metasurfaces~\cite{iyer2015reconfigurable, shcherbakov2015ultrafast,
   yang2015nonlinear, shcherbakov2017ultrafast}.
 
-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}.
+
 
 \begin{figure}[t] \centering
   \includegraphics[width=0.75\linewidth]{Concept.pdf}
@@ -241,24 +235,20 @@ distributions in silicon nanoparticle around a magnetic resonance.}
   \label{fgr:concept}
 \end{figure}
 
-Recently, plasma explosion imaging technique has been used to
-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
-  geometric center located at $z=0$ front side corresponds to the
-  volume $z>0$ and back side for $z<0$} of NaCl nanocrystals of
-$R = 100$~nm was revealed. 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 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. We consider
+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 schematically shown 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 free carrier
+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.
@@ -269,7 +259,7 @@ 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
-an all-dielectric NP to a hybrid metall-dielectric one strongly
+a dielectric NP to a hybrid metall-dielectric one strongly
 extending functionality of the ultrafast optical nanoantennas.
 
 
@@ -307,8 +297,8 @@ the critical density and above, silicon acquires metallic properties
 ($Re(\epsilon) < 0$) and contributes to the EHP reconfiguration during
 ultrashort laser irradiation.
 
-The process of three-dimensional photo-generation of the EHP in
-silicon NPs has not been modeled before in time-domain. Therefore,
+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
@@ -324,9 +314,7 @@ scale.
 
 \subsection{Light propagation}
 
-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
+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}},
@@ -360,7 +348,7 @@ method \cite{Rudenko2016}, based on the finite-difference time-domain
 (FDTD) \cite{Yee1966} and auxiliary-differential methods for
 dispersive media \cite{Taflove1995}. At the edges of the grid, we
 apply the absorbing boundary conditions related to convolutional
-perfectly matched layers (CPML) to avoid nonphysical reflections
+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}
@@ -420,14 +408,14 @@ 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, b) First four Lorentz-Mie coefficients
-  ($a_1$, $a_2$, $b_1$, $b_2$) and factors of asymmetry $G_I$,
-  $G_{I^2}$ according to Mie theory at fixed wavelength $800$~nm. (c,
-  d) Squared intensity distribution calculated by Mie theory and (e,
-  f) EHP distribution for low free carrier densities
+\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
+  (\ref{Dens}). \red{Please, write $G_I$ and $G_{I^2}$ instead of \textit{G} as a title to axis-Y in (b)} (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}
@@ -448,17 +436,19 @@ 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 increase the complexity of the
-analysis. A detailed discussion on the relation between Mie theory and
+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 Scattnlay program to evaluate calculations of Mie coefficients
-and near-field distribution~\cite{Ladutenko2017}.  This program is
+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=145mm]{time-evolution-I-no-NP.pdf}
@@ -468,20 +458,20 @@ license.
   $R = 115$~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 fluence is fixed
-  to be $0.125$~J/cm$^2$.}
+  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). \red{Please, remove italic style of font for 't, fs' on titles of X-axises. Also, all fonts should be the same: please don't mix Times and Arial.}}
 \vspace*{\floatsep}
  \centering
  \includegraphics[width=150mm]{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) 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 $130$~\textit{fs}
-   (FWHM). Wavelength $800$~nm in air. Peak laser fluence is fixed to
-   be $0.125$~J/cm$^2$.}
+   $1-4$: (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~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}
@@ -494,10 +484,10 @@ license.
  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 of the NP.  We introduce $G_I$
+ distribution of electric field inside the NP. 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:
+ 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
@@ -527,37 +517,30 @@ license.
  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 quasi-homogeneous case and the assumption of the
- NP homogeneous EHP distribution can be made to investigate the
+ $N_e^{back}=\frac{2}{V}\int_{(z<0)} {N_e}d{\mathrm{v}}$, 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 (see Fig.~\ref{time-evolution}) of the temporal evolution of
- EHP densities and the asymmetry factor $G_{N_e}$. It reveals the EHP
- evolution stages during pulse duration. Typical change of the
+ 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 of the
  permittivity corresponding to each stage is shown in
- Fig.~\ref{plasma-grid}.  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).
-
- To describe all the stages of non-linear light interaction with Si
+ 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. In this case, the
+ 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 Mie theory. Two main reasons cause
+ 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
+ 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,
@@ -566,7 +549,7 @@ license.
  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 of NP, which results
+ there is no multipole modes structure inside NP, which results
  into 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
@@ -585,7 +568,7 @@ license.
  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 of NP predicted by Mie theory). At this
+ 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.
 
@@ -605,7 +588,7 @@ license.
  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.
 
- Once again, due to the presence of a continuous pumping the Stage~3 is
+ In other words, due to the presence of a quasi-stationary pumping the Stage~3 is
  superposed with the 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) and result of
@@ -630,10 +613,10 @@ license.
  Higher excitation conditions are followed by larger values of
  electric field amplitude, which lead to the appearance of high EHP
  densities causing a significant change in the optical properties of
- silicon according to the equations (\ref{Index}). From Mie theory, the initial (at the end of Stage~3) space pattern of
+ silicon according to the equations (\ref{Index}). From the Mie theory, the initial (at the end of Stage~3) space pattern of
  optical properties is non-homogeneous. When non-homogeneity of
  optical properties becomes strong enough it leads to the
- reconfiguration of the electric field inside of NP and vice versa. We
+ reconfiguration of the electric field inside NP and vice versa. 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