changes

main.tex

@@ -423,14 +423,15 @@ length will be around 5--10~nm for $N_e$ close to $N_{cr}$.
 \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) Intensity
-  distribution calculated by Mie theory and (e, f) EHP distribution
-  for low free carrier 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 right along $Z$ axis, electric field polarization
-  $\vec{E}$ is along $X$ axis.}
+  ($a_1$, $a_2$, $b_1$, $b_2$) 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
+  $N_e \approx 10^{20}$~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
+  right along $Z$ axis, electric field polarization $\vec{E}$ is along
+  $X$ axis.}
 \end{figure}
 
 The changes of the real and imaginary parts of the permittivity
@@ -486,15 +487,22 @@ license.
 
 %\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:
+ We start with a pure electromagnetic problem without EHP
+ generation. We plot in Fig.~\ref{mie-fdtd}(a) Mie coefficients of a
+ Si NP as a function of its size for a fixed laser wavelength
+ $\lambda = 800$~nm.  For the NP sizes under consideration most of
+ contribution to the electromagnetic response originates from electric
+ 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 $G_I$
+ 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
+ $I^{back}=\int_{(z<0)} |E|^2d{\mathrm{v}}$ are expressed via
+ amplitude of the electric field $|E|$. 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
@@ -506,26 +514,28 @@ license.
  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
+ excitation with EHP concentration of $N_e \approx
+ 10^{20}$~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, transient
+ analysis reveal much more details.
+
+ To achieve a qualitative description for evolution of the EHP
+ distribution during the \textit{fs} pulse, 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, $G_{N_e} = 0$
+ front and in the back halfs 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{plasma-grid} and the temporal/EHP dependent evolution of
- the asymmetry factor $G_{N_e}$ in Fig.~\ref{time-evolution}.
+ modeling of the temporal evolution of the asymmetry factor $G_{N_e}$
+ in Fig.~\ref{time-evolution} revealing the EHP evolution stages during pulse
+ duration shown in Fig.~\ref{plasma-grid}.
 
  % Fig. \ref{Mie} demonstrates the temporal evolution of the EHP
  % generated inside the silicon NP of $R \approx 105$~nm. Here,