Update on Overleaf.

main.tex

@@ -205,7 +205,7 @@ University, Kronverksiy pr. 49, St. Petersburg, Russia}}
 \section{Introduction}
  
 All-dielectric nonlinear nanophotonics based on high refractive index
-dielectric has become prospective paradigm in modern optics, owing to
+dielectric materials has become prospective paradigm in modern optics, owing to
 recent advances in harmonics generation~\cite{shcherbakov2014enhanced,
   yang2015nonlinear, makarov2016self, shorokhov2016multifold,
   makarov2017efficient} and ultrafast all-optical
@@ -216,17 +216,17 @@ 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
+all-dielectric nanodevices are due to the existance 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 of optical properties (transmittance or
+to significant changes in optical properties (transmittance or
 reflectance) of all-dielectric nanoantennas~\cite{makarov2015tuning,
   baranov2016nonlinear, baranov2016tuning} and
 metasurfaces~\cite{iyer2015reconfigurable, shcherbakov2015ultrafast,
   yang2015nonlinear, shcherbakov2017ultrafast, makarov2017light}.
 
-In these works on all-dielectric nonlinear nanostructures, the
+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
@@ -248,9 +248,7 @@ 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 in this case
-was attributed to a nanolensing effect inside the NP and the
-intensity enhancement as low as $10\%$ on the far side of the
+$R = 100$~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.
@@ -385,13 +383,13 @@ $\varsigma(z) = {\arctan}(\frac{z}{z_R}) $ is the Gouy phase shift.
 
 \subsection{Material ionization}
 
-To account for the material ionization that is induced by a
+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 a rate equation that was proposed by van Driel
+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
@@ -529,7 +527,7 @@ 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 \red{TODO: insert inline equation for $N_e^{front}$ and $N_e^{back}$}. This way, $G_{N_e} = 0$
+ front and in the back halves of the NP \red{TODO: insert inline equation for $N_e^{front}$ and $N_e^{back}$}. 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
  optical response of the excited Si NP. When $G_{N_e}$ significantly
@@ -540,8 +538,8 @@ license.
  evolution stages during pulse duration. Typical change of the
  permittivity corresponding to each stage is shown in
  Fig.~\ref{plasma-grid}.  For better visual representation of time
- scale of a single optical cycle we put a
- squared electric field profile on all plots in
+ 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).
@@ -579,7 +577,7 @@ license.
  \textit{'Stage~2'} corresponds to further electric field oscillations
  ($t \approx 5$--$15$) leading to the formation of ED field pattern in
  the center of the NP as it can be seen in
- Fig.~\ref{plasma-grid}(f,j). We would like to stress the
+ Fig.~\ref{plasma-grid}(f,j). We stress the
  unstationery nature of field pattern at this stage. The energy
  balance between extinction and pumping is not set, moreover, there is
  a simultaneous growth of the incident pulse amplitude. This leads to
@@ -592,7 +590,7 @@ license.
  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.
 
- When the number of optical cycles is big enough ($t>20$~\textit{fs})
+ 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 the \textit{'Stage~3'} (see
@@ -608,8 +606,8 @@ 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 presence of continuous pumping the Stage~3 is
- superposed with Stage~1 field pattern, resulting in the additional
+ Once again, due to the presence of a continuous 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
  full 3D simulation in Fig.~\ref{mie-fdtd}(f). Note that pumping of NP
@@ -619,33 +617,31 @@ license.
 
  The explain this we need to consider time evolution of mean EHP
  densities $N_e$ in the front and back halves of NP presented in
- Fig.~\ref{time-evolution}(a,c,e). As soon as recombination and
+ 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. Front and back
- halves are separated in space, which obviously leads to the presence of
+ 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 with the same
- optical cycle of the incident wave. This delay causes a large value
- of the asymmetry factor during first stage. However, as soon as mean
+ optical cycle of the incident wave. This delay causes a large asymmetry factor during first stage. However, as soon as mean
  EHP density increases the relative contribution of this pumping steps to
  the resulting asymmetry becomes smaller. This way variations of asymmetry
  $G_{N_e}$ synced with the period of incident light decreases.
 
  Higher excitation conditions are followed with large values of
- electric field amplitude, which lead to appearance of high EHP
+ electric field amplitude, which lead to the appearance of high EHP
  densities causing a significant change of optical properties of
- silicon according to the equations (\ref{Index}). As it follows from
- Mie theory the initial (at the end of Stage~3) space pattern of
+ silicon according to the equations (\ref{Index}). From 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
  refer to these strong nonlinear phenomena as \textit{'Stage~4'}. In
- general the reconfiguration of the electric field is obligatory as
+ 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.
 
  This way evolution of EHP density during Stage~4 depends on the
- result of multipole modes superposition in the end of Stage~3 and is
+ 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,
  the origin of it is quite different.  The $R=75$~nm NP is out of
@@ -653,8 +649,8 @@ license.
  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, presence of EHP increases absorption in accordance with
- (\ref{Index}). This effectively leads to a kind of screening, it
- becomes harder for incident wave to penetrate deep into EHP. Finally,
+ (\ref{Index}). This 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).
 
@@ -662,7 +658,7 @@ license.
  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 was actually observed in
+ can be increased. This result was observed in
  Fig.~\ref{time-evolution}(d) with a local maximum near 100~\textit{fs}
  mark.
  
@@ -671,11 +667,11 @@ license.
  Stage~1 results into the presence of two EHP spatial maxima, back and
  front shifted. They serve to be a starting seed for EHP formation,
  the interplay between them forms a complex behavior of the asymmetry
- factor curve. Namely, it changes the sign from negative to positive
+ 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. However, it is interesting to note, that in
+ 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 ruin ED and MD resonances, responsible for the back-shifted
  EHP. As soon as this EHP region is quite visible on the last snapshot
@@ -692,7 +688,7 @@ license.
  effect is likely to be enhanced by electron diffusion inside Si
  NPs. 
      
- It is worth noting that it is possible to achieve a formation of
+ 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.
@@ -818,8 +814,7 @@ NPs and investigated the asymmetry of EHP distributions. % for different laser i
 %NP scattering and, in particular, changes the preferable
 %scattering direction. 
 Different pathways of EHP evolution from the front side to the back
-side have been revealed, depending on the NP sizes, and the
-origins of different behavior have been explained by the
+side have been revealed, depending on the NP sizes, and different behaviors have been explained by the
 non-stationarity of the energy deposition and different quality
 resonant factors for exciting the electric and magnetic dipole
 resonances, intensity distribution by Mie theory and newly
@@ -828,10 +823,9 @@ electric dipole resonance on the EHP asymmetric distribution during
 first optical cycles has been revealed for different size
 parameters. The higher EHP asymmetry is established for NPs
 of smaller sizes below the first magnetic dipole
-resonance. Essentially different EHP evolution and lower asymmetry is
-achieved for larger NPs due to the stationary intensity
+resonance. Essentially different EHP evolution and lower asymmetry has been achieved for larger NPs due to the stationary intensity
 enhancement in the back side of the NP. The EHP densities
-above the critical value were shown to lead to the EHP distribution
+above the critical value have been shown to lead to the EHP distribution
 homogenization.
 % In particular, the scattering efficiency factor is used to define
 % the optimum NP size for preferential forward or backward