add unbreakable space before units

main.tex

@@ -249,7 +249,7 @@ 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
+$R = 100$~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
 NP. Much stronger enhancements can be achieved near electric
@@ -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.
@@ -343,13 +343,13 @@ written in the following way
 where $\vec{E}$ is the electric field, $\vec{H}$ is the magnetizing
 field, $\epsilon_0$ is the free space permittivity, $\mu_0$ is the
 permeability of free space, $\epsilon = n^2 = 3.681^2$ is the
-permittivity of non-excited silicon at $800$ nm wavelength
+permittivity of non-excited silicon at $800$~nm wavelength
 \cite{Green1995}, $\vec{J}_p$ and $\vec{J}_{Kerr}$ are the nonlinear
 currents, which include the contribution due to Kerr effect
 $\vec{J}_{Kerr} =
 \epsilon_0\epsilon_\infty\chi_3\frac{\partial\left(\left|\vec{E}\right|^2\vec{E}\right)}{\partial{t}}$,
 where $\chi_3 =4\cdot{10}^{-20}$ m$^2$/V$^2$ for laser wavelength
-$\lambda = 800$nm \cite{Bristow2007}, and heating of the conduction
+$\lambda = 800$~nm \cite{Bristow2007}, and heating of the conduction
 band, described by the differential equation derived from the Drude
 model
 \begin{equation} \label{Drude}
@@ -358,7 +358,7 @@ model
 \end{equation}
 where $e$ is the elementary charge, $m_e^* = 0.18m_e$ is the reduced
 electron-hole mass \cite{Sokolowski2000}, $N_e(t)$ is the
-time-dependent free carrier density and $\nu_e = 10^{15}$ s$^{-1}$ is
+time-dependent free carrier density and $\nu_e = 10^{15}$~s$^{-1}$ is
 the electron collision frequency \cite{Sokolowski2000}. Silicon
 NP is surrounded by vacuum, where the light propagation is
 calculated by Maxwell's equations with $\vec{J} = 0$ and
@@ -378,11 +378,11 @@ Gaussian slightly focused beam as follows
 \times\;{exp}\left(-\frac{4\ln{2}(t-t_0)^2}{\theta^2}\right),
 \end{aligned}
 \end{align}
-where $\theta = 50$ fs is the temporal pulse width at the half maximum (FWHM),
+where $\theta = 50$~\textit{fs} is the temporal pulse width at the half maximum (FWHM),
 $t_0$ is a time delay, $w_0 = 3{\mu}m$ is the waist beam,
 $w(z) = {w_0}\sqrt{1+(\frac{z}{z_R})^2}$ is the Gaussian's beam spot
 size, $\omega = 2{\pi}c/{\lambda}$ is the angular frequency,
-$\lambda = 800$ nm is the laser wavelength in air, $c$ is the speed of
+$\lambda = 800$~nm is the laser wavelength in air, $c$ is the speed of
 light, ${z_R} = \frac{\pi{{w_0}^2}n_0}{\lambda}$ is the Rayleigh
 length, $r = \sqrt{x^2 + y^2}$ is the radial distance from the beam's
 waist, $R_z = z\left(1+(\frac{z_R}{z})^2\right)$ is the radius of
@@ -398,7 +398,16 @@ 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.}
+% \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*}
 
 
@@ -413,22 +422,22 @@ written as
       \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
-$\sigma_2 = 0.1\cdot{10}^{-7}$ cm/W are the one-photon and two-photon
+is the intensity, $\sigma_1 = 1.021\cdot{10}^3$~cm$^{-1}$ and
+$\sigma_2 = 0.1\cdot{10}^{-7}$~cm/W are the one-photon and two-photon
 interband cross-sections \cite{Choi2002, Bristow2007, Derrien2013},
-$N_a = 5\cdot{10}^{22} cm^{-3}$ is the saturation particle density
-\cite{Derrien2013}, $C = 3.8\cdot{10}^{-31}$ cm$^6$/s is the Auger
-recombination rate \cite{Van1987}, $\tau_{rec} = 6\cdot{10}^{-12}$s is
+$N_a = 5\cdot{10}^{22}$~cm$^{-3}$ is the saturation particle density
+\cite{Derrien2013}, $C = 3.8\cdot{10}^{-31}$~cm$^6$/s is the Auger
+recombination rate \cite{Van1987}, $\tau_{rec} = 6\cdot{10}^{-12}$~s is
 the minimum Auger recombination time \cite{Yoffa1980}, and
-$\alpha = 21.2$ cm$^2$/J is the avalanche ionization coefficient
-\cite{Pronko1998} at the wavelength $800$ nm in air. As we have noted,
+$\alpha = 21.2$~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 $D = k_B T_e \tau/m^* \approx (1\div2)\cdot{10}^{-3}$ m$^2$/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 $N_{cr}$ \cite{Ramer2014}. It
-means that during the pulse duration ($\approx$ 50~\textit{fs}) the diffusion
+means that during the pulse duration ($\approx 50$~\textit{fs}) the diffusion
 length will be around $5\div10$~nm for $N_e$ close to $N_{cr}$.
 
 \begin{figure}[ht!] 
@@ -436,9 +445,9 @@ length will be around $5\div10$~nm for $N_e$ close to $N_{cr}$.
 \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
+  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
+  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
@@ -471,7 +480,7 @@ length will be around $5\div10$~nm for $N_e$ close to $N_{cr}$.
   %($\,$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$^2$.}
+  %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
@@ -506,23 +515,23 @@ license.
  \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
-   duration $50$~\textit{fs} (FWHM). Wavelength $800$ nm in air. (b,
+   (a, b) $R = 75$~nm, (c, d) $R = 100$~nm, (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}
    are indicated. The temporal evolution of Gaussian beam intensity is
-   also shown. Peak laser fluence is fixed to be $0.125$J/cm$^2$.}
+   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
-   radii $R = 75$ nm (a-d), $R = 100$ nm (e-h) and $R = 115$ nm (i-l)
+   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 $50$~\textit{fs}
-   (FWHM). Wavelength $800$ nm in air. Peak laser fluence is fixed to
-   be $0.125$ J/cm$^2$.}
+   (FWHM). Wavelength $800$~nm in air. Peak laser fluence is fixed to
+   be $0.125$~J/cm$^2$.}
  \end{figure*}
 
 %\subsection{Effect of the irradiation intensity on EHP generation}
@@ -547,7 +556,7 @@ 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
+ 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
@@ -568,13 +577,12 @@ license.
  shown in Fig.~\ref{fig2} and the temporal/EHP dependent evolution of
  the asymmetry factor $G_{N_e}$ in Fig.~\ref{fig3}.
 
-% Fig. \ref{Mie} demonstrates the temporal evolution of the EHP
-% generated inside the silicon NP of $R \approx 105$
-% nm. Here, irradiation by high-intensity, $I\approx $ from XXX to YYY
-% (???), ultrashort laser Gaussian pulse is considered. Snapshots of
-% free carrier density taken at different times correspond to
-% different total amount of the deposited energy (different laser
-% intensities).
+ % Fig. \ref{Mie} demonstrates the temporal evolution of the EHP
+ % generated inside the silicon NP of $R \approx 105$~nm. Here,
+ % irradiation by high-intensity, $I\approx $ from XXX to YYY (???),
+ % ultrashort laser Gaussian pulse is considered. Snapshots of free
+ % carrier density taken at different times correspond to different
+ % total amount of the deposited energy (different laser intensities).
 
 %To better analyze the degree of inhomogeneity, we introduce the EHP
 % asymmetry parameter, $G$, which is defined as a relation between the
@@ -597,10 +605,10 @@ license.
  Si. The non-stationary intensity deposition 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, magnetic dipole resonance (\textit{b1}) has $Q \approx$
- 8, whereas electric one (\textit{a1}) has $Q \approx$4. The larger
+ particular, magnetic dipole resonance (\textit{b1}) has $Q \approx
+ 8$, whereas electric one (\textit{a1}) has $Q \approx 4$. The larger
  particle supporting magnetic quadrupole resonance (\textit{b2})
- demonstrates \textit{Q} $\approx$ 40. As soon as the electromagnetic
+ demonstrates \textit{Q} $\approx 40$. As soon as the electromagnetic
  wave period at $\lambda$~=~800~nm is 2.6~\textit{fs}, one needs about
  10~\textit{fs} to pump the electric dipole, 20~\textit{fs} for the
  magnetic dipole, and about 100~\textit{fs} for the magnetic
@@ -619,7 +627,7 @@ license.
  ($t \approx 2\div15$) leading to the unstationery EHP evolution
  with a maximum of the EHP distribution in the front side of the Si NP
  owing to the starting excitation of MD and MQ resonances that require more
- time to be excited. At this stage, the density of EHP ($N_e < 10^{20}$cm$^2$)
+ time to be excited. 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.
 
@@ -645,7 +653,7 @@ license.
  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
- electron density $N_{cr} = 5\cdot{10}^{21}$ cm$^{-3}$ for silicon,
+ 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
@@ -653,7 +661,7 @@ license.
  observe in Fig.~\ref{fig2}(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
+ $N_e > 5\cdot{10}^{21}$~cm$^{-3}$, the EHP distribution evolves
  inside NPs because of the photoionization and avalanche ionization
  induced transient optical response and the effect of newly formed
  EHP. This way, the distribution becomes more homogeneous and the
@@ -685,7 +693,7 @@ license.
 % \end{figure}
 
 % We have discussed the EHP kinetics for a silicon NP of a
-% fixed radius $R \approx 105$ nm. In what follows, we investigate the
+% fixed radius $R \approx 105$~nm. In what follows, we investigate the
 % influence of the NP size on the EHP patterns and temporal
 % evolution during ultrashort laser irradiation. A brief analysis of
 % the initial intensity distribution inside the NP given by
@@ -700,7 +708,7 @@ license.
 % will result in a stronger electron density gradients. Additionally,
 % in the case of maximum forward or backward scattering, the initial
 % intensity distribution has the maximum of asymmetry. One can note,
-% that for $R \approx 100$ nm and $R \approx 150$ nm both criteria are
+% that for $R \approx 100$~nm and $R \approx 150$~nm both criteria are
 % fulfilled: the intensity is enhanced $5-10$ times due to
 % near-resonance conditions and its distribution has a strong
 % asymmetry.
@@ -718,8 +726,8 @@ license.
 % wavelength in media, the intensity distribution around the
 % NP will not change considerably. Therefore, we propose to
 % introduce the optimization factor
-% $K = \frac{n_e(G-1)R^2}{n_{cr}G{R_0^2}}$, where $R_0 = 100$ nm,
-% $n_{cr} = 5\cdot{10}^{21} cm^{-3}$, and $G$ is asymmetry of EHP,
+% $K = \frac{n_e(G-1)R^2}{n_{cr}G{R_0^2}}$, where $R_0 = 100$~nm,
+% $n_{cr} = 5\cdot{10}^{21}~cm^{-3}$, 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
@@ -739,11 +747,11 @@ license.
 % whereas the second pulse of lower pulse energy and polarization $Oz$
 % interacts with EHP after the first pulse is gone. The minimum
 % relaxation time of high electron density in silicon is
-% $\tau_{rec} = 6\cdot{10}^{-12}$ s \cite{Yoffa1980}, therefore, the
+% $\tau_{rec} = 6\cdot{10}^{-12}$~s \cite{Yoffa1980}, therefore, the
 % electron density will not have time to decrease significantly for
 % subpicosecond pulse separations. In our simulations, we use
 % $\delta{t} = 200\:f\!s$ pulse separation. The intensity
-% distributions near the silicon NP of $R = 95$ nm,
+% distributions near the silicon NP of $R = 95$~nm,
 % corresponding to maxima value of $K$ optimization factor, without
 % plasma and with generated plasma are shown in Fig. \ref{fig4}. The
 % intensity distribution is strongly asymmetric in the case of EHP
@@ -757,7 +765,7 @@ license.
 % \begin{figure}[ht] \centering
 % \includegraphics[width=90mm]{fig4.png}
 % \caption{\label{fig4} a) Electron plasma distribution inside Si
-% NP $R \approx 95$ nm $50\:f\!s$ after the pulse peak;
+% NP $R \approx 95$~nm 50~\textit{fs} after the pulse peak;
 % (Kostya: Is it scattering field intensity snapshot $XXX\:f\!s$ after
 % the second pulse maxima passed the particle?) Intensity
 % distributions around and inside the NP b) without plasma,