Update on Overleaf.

main.tex

@@ -305,12 +305,12 @@ advantage is based on a broad range of optical nonlinearities, strong
 two-photon absorption, as well as a possibility of the photo-induced
 EHP excitation~\cite{leuthold2010nonlinear}. Furthermore, silicon
 nanoantennas demonstrate a sufficiently high damage threshold due to
-the large melting temperature ($\approx$1690~K), whereas its nonlinear
+the large melting temperature ($\approx 1690$K), whereas its nonlinear
 optical properties have been extensively studied during last
 decays~\cite{Van1987, Sokolowski2000, leuthold2010nonlinear}. High
 silicon melting point typically preserves structures formed from this
 material up to the EHP densities on the order of the critical value
-$n_{cr} \approx 5\cdot{10}^{21}$ cm$^{-3}$ \cite{Korfiatis2007}. At
+$N_{cr} \approx 5\cdot{10}^{21}$ cm$^{-3}$ \cite{Korfiatis2007}. At
 the critical density and above, silicon acquires metallic properties
 ($Re(\epsilon) < 0$) and contributes to the EHP reconfiguration during
 ultrashort laser irradiation.
@@ -410,7 +410,7 @@ written as
 \begin{equation} \label{Dens}
   \displaystyle{\frac{\partial{N_e}}{\partial t} =
     \frac{N_a-N_e}{N_a}\left(\frac{\sigma_1I}{\hbar\omega} +
-      \frac{\sigma_2I^2}{2\hbar\omega}\right) + \alpha{I}n_e -
+      \frac{\sigma_2I^2}{2\hbar\omega}\right) + \alpha{I}N	_e -
     \frac{C\cdot{N_e}^3}{C\tau_{rec}N_e^2+1},} \end{equation} where
 $I=\frac{n}{2}\sqrt{\frac{\epsilon_0}{\mu_0}}\left|\vec{E}\right|^2$
 is the intensity, $\sigma_1 = 1.021\cdot{10}^3$ cm$^{-1}$ and
@@ -424,12 +424,12 @@ $\alpha = 21.2$ cm$^2$/J is the avalanche ionization coefficient
 \cite{Pronko1998} at the wavelength $800$ nm in air. As we have noted,
 free carrier diffusion is neglected during and shortly after the laser
 excitation \cite{Van1987, Sokolowski2000}. In particular, from the
-Einstein formula $D = k_B T_e \tau/m^*$ $\approx$ (1-2)10$^5$ m/s
-(k$_B$ is the Boltzmann constant, T$_e$ is the electron temperature,
-$\tau$=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$_cr$. It
+Einstein formula $D = k_B T_e \tau/m^* \approx (1-2){10}^5$ m/s
+($k_B$ is the Boltzmann constant, $T_e$ is the electron temperature,
+$\tau=1$~\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 $N_{cr}$. It
 means that during the pulse duration ($\approx$ 50~\textit{fs}) the diffusion
-length will be around 5--10~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
@@ -618,7 +618,7 @@ license.
  is, the higher the NP asymmetry $G_{N_e}$ is achieved.
 
  \textit{'Stage 2'} corresponds to further electric field oscillations
- (t$\approx$2--15) leading to unstationery nature of the EHP evolution
+ ($t \approx 2--15$) leading to unstationery nature of the EHP evolution
  with a maximum of the EHP distribution on the front side of the Si NP
  owing to starting excitation of MD and MQ resonances, requiring more
  time to be excited. At this stage, density of EHP ($< 10^{20}$cm$^2$)
@@ -668,7 +668,7 @@ license.
  achieved below the magnetic dipole resonant conditions for $R < 100$
  nm. Thus, the EHP distribution in Fig.~\ref{fig2}(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
+ 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
  volume where such high EHP density is formed.