Asymmetry-plasma/main.tex

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  \includegraphics{head_foot/DOI} & \noindent\LARGE{\textbf{Self-Induced Plasma-Induced Symmetry Breaking in a Spherical Silicon Nanoparticle}} \\%Article title goes here instead of the text "This is the title"
  \vspace{0.3cm} & \vspace{0.3cm} \\
                                  & \noindent\large{Anton Rudenko,$^{\ast}$\textit{$^{a}$} Tatiana E. Itina,\textit{$^{a\ddag}$} Konstantin Ladutenko,\textit{$^{b}$} and Sergey Makarov\textit{$^{b}$}
 
                                    \textit{$^{a}$~Laboratoire Hubert Curien, UMR CNRS 5516, University of Lyon/UJM, 42000, Saint-Etienne, France }
                                    \textit{$^{b}$~ITMO University, Kronverksiy pr. 49, St. Petersburg, Russia}
 
                                    } \\%Author names go here instead of "Full name", etc.
 
 
  \includegraphics{head_foot/dates} & \noindent\normalsize {The concept
of nonlinear all-dielectric nanophotonics based on high refractive
index (e.g., silicon) nanoparticles supporting magnetic optical
response has recently emerged as a powerful tool for ultrafast
all-optical modulation at nanoscale. A strong modulation can be
achieved via photogeneration of dense electron-hole plasma in the
regime of simultaneous excitation of electric and magnetic optical
resonances, resulting in an effective transient reconfiguration of
nanoparticle scattering properties. However, previous works assumed
only homogenized plasma generation in the photoexcited nanoparticle,
neglecting all effects related to inhomogeneous plasma
distribution. Here numerical studying of the plasma photogeneration
allows us to propose a novel concept of deeply subwavelength
($\approx$$\lambda$$^3$/100) plasma-induced nanopatterning of
symmetrical silicon nanoparticle. More importantly, we reveal strong
symmetry breaking in the initially symmetrical nanoparticle during
ultrafast photoexcitation near the magnetic dipole resonance. The
ultrafast manipulation by nanoparticle inherent structure and symmetry 
paves the way to novel principles for nonlinear optical nanodevices.}

\end{tabular}

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\footnotetext{\textit{$^{a}$~Univ Lyon, UJM-St-Etienne, CNRS UMR 5516,
F-42000, Saint-Etienne, France}} \footnotetext{\textit{$^{b}$~ITMO
University, Kronverksiy pr. 49, St. Petersburg, Russia}}


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%%%MAIN TEXT%%%%
  
\section{Introduction}
 
All-dielectric nonlinear nanophotonics based on high refractive index
dielectric has become prospective paradigm in modern optics, owing to
recent advances in harmonics generation~\cite{shcherbakov2014enhanced,
yang2015nonlinear, makarov2016self, shorokhov2016multifold} and
ultrafast all-optical modulation~\cite{iyer2015reconfigurable,
makarov2015tuning, shcherbakov2015ultrafast, yang2015nonlinear,
baranov2016nonlinear, baranov2016tuning}. In fact, 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 magnetic optical
resonances in visible and near IR
ranges~\cite{kuznetsov2016optically}. For instance, even slight
variation of dielectric permittivity around magnetic dipole resonance
leads to significant changes of optical properties (transmittance or
reflectance) of all-dielectric nanoantennas~\cite{makarov2015tuning,
baranov2016nonlinear, baranov2016tuning} and
metasurfaces~\cite{iyer2015reconfigurable, shcherbakov2015ultrafast,
yang2015nonlinear}.

In all this works on all-dielectric nonlinear nanostructures, the
building blocks (nanoparticles) were considered as objects with
dielectric permittivity homogeneously distributed over
nanoparticle. Therefore, in order to manipulate by propagation angle
of transmitted light it is necessary to use complicated nanostructures
with reduced symmetry~\cite{albella2015switchable, baranov2016tuning,
shibanuma2016unidirectional}.

\begin{figure}[t] \centering
  \includegraphics[width=0.75\linewidth]{Concept}
  \caption{Schematic illustration of electron-hole plasma 2D and 1D
distributions in silicon nanoparticle around a magnetic resonance.}
  \label{fgr:concept}
\end{figure}

Recently, highly localized plasma inside the nanoparticles, irradiated by femtosecond laser, has been directly observed using plasma explosion imaging \cite{Hickstein2014}. Additionally, inhomogeneous resonant scattering patterns inside single silicon nanoparticles have been experimentally revealed \cite{Valuckas2017}.  

In this Letter, we show that electron-hole plasma (EHP) generation in
a spherical dielectric (e.g., silicon) nanoparticle leads to strongly
nonhomogeneous EHP distribution. To reveal and study this effect, we
for the first time provide full-wave numerical simulation of intensive
femtosecond (fs) laser pulse interaction with a dielectric
nanoparticle supporting Mie resonances and two-photon generation of
EHP. In particular, we couple finite-difference time-domain (FDTD)
method of the Maxwell's equations solving with equations describing
nonlinear EHP generation and its variation of material dielectric
permittivity. The obtained results propose a novel strategy to create
complicated nonsymmetrical nanostructures by using only a photoexcited
spherical silicon nanoparticle. Moreover, we show that dense EHP can
be generated at deeply subwavelength scale
($\approx$$\lambda$$^3$/100) supporting formation of small metallized
parts inside the nanoparticle which transforms all-dielectric
nanoparticle to a hybrid one that extends functionality of ultrafast
optical nanoantennas.


%Plan:
%\begin{itemize}
%\item Fig.1: Beautiful conceptual picture
%\item Fig.2: Temporal evolution of EHP in NP with different diameters
%at fixed intensity, in order to show that we have the highest
%asymmetry around magnetic dipole (MD) resonance. This would be really
%nice!
%\item Fig.3: Temporal evolution of EHP in NP with fixed diameter (at
%MD) at different intensities, in order to show possible regimes of
%plasma-patterning of NP volume. It would be nice, if we will show
%power patterns decencies on intensity for side probe pulse to show beam steering due to symmetry breaking.
%\item Fig.4: (a) Dependence on pulse duration is also interesting. We
%have to show at which duration the asymmetry factor is saturated. (b)
%2D map of asymmetry factor in false colors, where x-axis and y-axis correspond to intensity and NP diameter.
%\end{itemize} %Additionally, if you will manage to calculate
%evolution of scattering power pattern and show considerable effect of
% beam steering, we can try Nanoscale or LPR, because the novelty will
% be very high.


\section{Modeling details}
 
% TEI for Anton: please describe what you model first (what physical
% processes). Then, we speak about "pulse", but never "laser". Should
% we provide more details concerning irradiation source?

We focus out attention on silicon because this material is promising
for the implementation of nonlinear photonic devices thanks to a broad
range of optical nonlinearities, strong two-photon absorption and 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
optical properties have been extensively studied during last
decays~\cite{Van1987, Sokolowski2000, leuthold2010nonlinear}. High melting point for silicon preserves up to EHP densities of order $n_{cr} \approx 5\cdot{10}^{21}$ cm$^{-3}$ \cite{Korfiatis2007}, for which silicon acquires metallic properties ($Re(\epsilon) < 0$) and contributes to the EHP reconfiguration during ultrashort laser irradiation.

Since the 3D modeling of EHP photo-generation in a resonant silicon
nanoparticle has not been done before in time-domain, we develop a
model considering incidence of ultrashort light pulse on a silicon
sphere, where EHP is generated via one- and two-photon absorption
processes, and then decayed via Auger recombination. Importantly, we
also consider nonlinear feedback of the material by taking into
account intraband light absorption on the generated free carriers. In
order to simplify our model, we neglect diffusion of EHP, because the
aim of our work is to study EHP dynamics \textit{during} laser
interaction with the nanoparticle.

\subsection{Light propagation}

The propagation of light inside the silicon nanoparticle is modeled by solving the system of Maxwell's equations, written in the following way
\begin{align} \begin{cases} \label{Maxwell}$$
	\displaystyle{\frac{\partial{\vec{E}}}{\partial{t}}=\frac{\nabla\times\vec{H}}{\epsilon_0\epsilon}-\frac{1}{\epsilon_0\epsilon}(\vec{J}_p+\vec{J}_{Kerr})} \\ 		
	\displaystyle{\frac{\partial{\vec{H}}}{\partial{t}}=-\frac{\nabla\times\vec{E}}{\mu_0}},
	$$ \end{cases} \end{align} 
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 \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 band, described by the differential equation derived from the Drude model
\begin{equation} \label{Drude} \displaystyle{\frac{\partial{\vec{J_p}}}{\partial{t}} = - \nu_e\vec{J_p} + \frac{e^2n_e(t)}{m_e^*}\vec{E}}, \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 the electron collision frequency \cite{Sokolowski2000}. Silicon nanoparticle is surrounded by vacuum, where the light propagation is calculated by Maxwell's equations with $\vec{J} = 0$ and $\epsilon = 1$. The system of Maxwell's equations coupled with electron density equation is solved by the finite-difference numerical 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 absorbing boundary conditions related to convolutional perfectly matched layers (CPML) to avoid nonphysical reflections \cite{Roden2000}. Initial electric field is introduced as a Gaussian focused beam source as follows
\begin{align}
\begin{aligned}
\label{Gaussian}
{E_x}(t, r, z) = \frac{w_0}{w(z)}{exp}\left(i\omega{t} - \frac{r^2}{{w(z)}^2} - ikz - ik\frac{r^2}{2R(z)} + i\varsigma(z)\right)\\
\times\;{exp}\left(-\frac{(t-t_0)^2}{\theta^2}\right),
\end{aligned}
\end{align}
where $\theta$ is the pulse width at 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 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 curvature of the wavelength comprising the beam, and $\varsigma(z) = {\arctan}(\frac{z}{z_R}) $ is the Gouy phase shift. 

\subsection{Material ionization}

To account for material ionization, we couple Maxwell's equations with the kinetic equation for EHP generation and relaxation inside silicon nanoparticle.
\begin{figure*}[ht!]
\centering
\includegraphics[width=120mm]{fig2.png}
\caption{\label{fig2} Free carrier density snapshots of electron plasma evolution inside Si nanoparticle taken a) $30$ fs b) $10$ fs before the pulse peak, c) at the pulse peak, d) $10$ fs e) $30$ fs after the pulse peak. Pulse duration $50$ fs (FWHM). Wavelength $800$ nm in air. Radius of the nanoparticle $R \approx 105$ 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 nanoparticle. $n_{cr} = 5\cdot{10}^{21}$ cm$^{-3}$ is the critical plasma resonance electron density for silicon.}
\end{figure*}

The time-dependent conduction-band carrier density evolution is described with a rate equation, firstly proposed by van Driel \cite{Van1987}, taking into account photoionization, avalanche ionization and Auger recombination 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{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 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 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. Free carrier diffusion can be neglected during and shortly after the excitation \cite{Van1987, Sokolowski2000}.
\begin{figure*}[ht!]
\centering
\includegraphics[width=0.9\textwidth]{Ne_105nm_800}
\caption{\label{plasma-105nm} Split figure \ref{fig2} into two, this is first part.}
\end{figure*}

The changes of the real and imaginary parts of the permittivity associated with the time-dependent free carrier response \cite{Sokolowski2000} can be derived from equations (\ref{Maxwell}, \ref{Drude}) and are written as follows
\begin{align} \begin{cases}  \label{Index} $$
	\displaystyle{Re(\epsilon) = \epsilon -\frac{{e^2}n_e}{\epsilon_0m_e^*(\omega^2+{\nu_e}^2)}} \\
	\displaystyle{Im(\epsilon) = \frac{{e^2}n_e\nu_e}{\epsilon_0m_e^*\omega(\omega^2+{\nu_e}^2)}.}
 $$ \end{cases} \end{align}

\section{Results and discussion}

\subsection{Effect of the irradiation intensity on EHP generation}

Fig. \ref{fig2} demonstrates EHP temporal evolution inside silicon
nanoparticle of $R \approx 105$ nm during irradiation by
high-intensity ultrashort laser Gaussian pulse. Snapshots of electron
density taken at different times correspond to different total amount
of deposited energy and, therefore, different intensities of
irradiation. We introduce the parameter of EHP asymmetry $G$, defined
as a relation between the average electron density generated in the
front side of the nanoparticle and the average electron density in the
back side, as shown in Fig. \ref{fig2}. During the femtosecond pulse
interaction, this parameter significantly varies.

Far before the pulse peak in Fig. \ref{fig2}(a), the excitation
processes follow the intensity distribution, generating a low-density
electron plasma of a toroidal shape at magnetic dipole resonance
conditions. For higher intensities, the optical properties of silicon
change significantly according to equations (\ref{Index}), and
nonresonant electric dipole contributes to the forward shifting of EHP
density maximum. Therefore, EHP is localized in the front part of the
nanoparticle, increasing the asymmetry factor $G$ in
Fig. \ref{fig2}(b). Approximately at the pulse peak, the critical
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 electron plasma resonance \cite{Sokolowski2000}, is overcome. At the same time, $G$ factor
reaches the maximum value close to $2.5$ in
Fig. \ref{fig2}(c). Further irradiation leads to a decrease of the
asymmetry parameter down to $1$ for higher electron densities in
Fig. \ref{fig2}(d, e).

It is worth noting, that it is possible to achieve a formation of
deeply subwavelength EHP regions due to high field localization. In
particular, we observe very small EHP localization at magnetic dipole
resonant conditions for $R \approx 105$ nm. The EHP distribution in
Fig. \ref{fig2}(c) is the optimal for symmetry breaking in silicon
nanoparticle, as results in the bigger asymmetry factor $G$ and higher
electron densities $n_e$. We want to stress here that such regime
could be still safe for nanoparticle due to small volume of such high
EHP density, which should be diffused after some time.

\subsection{Effects of nanoparticle size/scattering efficiency factor
on scattering directions}

\begin{figure}[ht] \centering
\includegraphics[width=90mm]{fig3.png}
\caption{\label{fig3} a) Scattering efficiency factor $Q_{sca}$
dependence on the radius $R$ of non-excited silicon nanoparticle
calculated by Mie theory; b) Parameter of forward/backward scattering
dependence on the radius $R$ calculated by Mie theory for non-excited
silicon nanoparticle c) Optimization parameter $K$ dependence on the
average electron density $n_e^{front}$ in the front half of the
nanoparticle for indicated radii (1-7).}
\end{figure}

We have discussed the EHP kinetics for a silicon nanoparticle of a fixed radius $R \approx 105$ nm. In what follows, we investigate the influence of the nanoparticle size on
the EHP patterns and temporal evolution during ultrashort laser
irradiation. A brief analysis of the initial intensity distribution
inside the nanoparticle given by Mie theory for a spherical
homogenneous nanoparticle \cite{Mie1908} can be useful in
this case. Fig. \ref{fig3}(a, b) shows the scattering efficiency and
the asymmetry parameter for forward/backward scattering for
non-excited silicon nanoparticles of different radii calculated by Mie
theory \cite{Mie1908}. Scattering efficiency dependence gives
us the value of resonant sizes of nanoparticles, where the initial
electric fields are significantly enhanced and, therefore, we can
expect that the following conditions 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 fulfilled: the intensity is
enhanced $5-10$ times due to near-resonance conditions and its
distribution has a strong asymmetry.

In what follows, we present the calculation results obtained by using
Maxwell's equations coupled with electron kinetics for different
extremum radii for resonant and non-resonant conditions. One can note,
that the maximum asymmetry factor of EHP $G$ does not guarantee the
optimal asymmetry of intensity distribution, as the size of generated
plasma and the value of the electron density equally contribute to the
change of the modified nanoparticle optical response. For example, it
is easier to localize high electron densities inside smaller
nanoparticles, however, due to the negligible size of the generated
EHP with respect to laser wavelength in media, the intensity
distribution around the nanoparticle 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,
defined previously. The calculation results for different radii of
silicon nanoparticles and electron densities are presented in
Fig. \ref{fig3}(c). One can see, that the maximum value are achieved
for the nanoparticles, that satisfy both initial maximum forward
scattering and not far from the first resonant condition. For larger
nanoparticles, lower values of EHP asymmetry factor are obtained, as
the electron density evolves not only from the intensity patterns in
the front side of the nanoparticle but also in the back side.

To demonstrate the effect of symmetry breaking, we calculate the
intensity distribution around the nanoparticle for double-pulse
experiment. The first pulse of larger pulse energy and polarization
along $Ox$ generates asymmetric EHP inside silicon nanoparticle,
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
electron density will not have time to decrease significantly for
subpicosecond pulse separations. In our simulations, we use $\delta{t}
= 200$ fs pulse separation. The intensity distributions near the
silicon nanoparticle 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 presence. One can note, that the excited
nanoparticle is out of quasi-resonant condition and the intensity
enhancements in Fig. \ref{fig4}(c) are weaker than in
Fig. \ref{fig4}(b). Therefore, the generated nanoplasma acts like a quasi-metallic nonconcentric nanoshell inside the nanoparticle, providing a symmetry reduction \cite{Wang2006}.

\begin{figure}[ht] \centering
\includegraphics[width=90mm]{fig4.png}
\caption{\label{fig4} a) Electron plasma distribution inside Si
nanoparticle $R \approx 95$ nm $50$ fs after the pulse peak; (Kostya: Is it scattering field intensity snapshot  XXX fs after the second pulse maxima passed the particle?) Intensity
distributions around and inside the nanoparticle b) without plasma, c)
with electron plasma inside.}
\end{figure}

%\begin{figure} %\centering
% \includegraphics[height=0.7\linewidth]{Si-flow-R140-XYZ-Eabs}
% \caption{EHP distributions for nonres., MD, ED, and MQ nanoparticles
% at moderate photoexcitation. The aim is to show different possible
% EHP patterns and how strong could be symmetry breaking.
% \label{fgr:example}
%\end{figure}

\subsection{Asymmetry analysis: effects of pulse duration, intensity
and size} It is important to optimize asymmetry by varying pulse
duration, intensity and size.  TEI: for Konstantin L.: please change
Mie figure and add a discussion and how the results agree, what is the
difference TEI: for Anton R.: please add a discussion of the previous
experiments and how the results agree

TODO Kostya: add some discusion on rise-on time for optical switching
like this: Small size will give as a magnetic dipole b1 resonance with
Q-factor (ratio of wavelength to the resonance width at half-height)
of about 8, a1 Q approx 4, the larger particle will have b2 Q approx
40. For large particle we will have e.g. at R=238.4 second order b4
resonance with Q approx 800.  As soon as the period at WL=800nm is 2.6
fs, we need about 25 fs pulse to pump dipole response, about 150 fs
for quadrupole, and about 2000fs for b4. If we think of optical
switching applications this is a rise-on time.

TODO Kostya: Add discussion about mode selection due to the formation
of the plasma.

\section{Conclusions} We have considered light interaction with a
semiconductor nanoparticle under different irradiation conditions and
for various particle sizes. As a result of the presented
self-consistent calculations, we have obtained spatio-temporal EHP
evolution for different light intensities and pulse temporal
widths. It has been demonstrated that EHP generation strongly affects
nanoparticle scattering and, in particular, changes the preferable
scattering direction. In particular, the scattering efficiency factor
is used to define the optimum nanoparticle size for preferential
forward or backward scattering. Furthermore, a parameter has been
introduced to describe the scattering asymmetry as a ratio of the EHP
density in the front side to that in the back side of the
nanoparticle. This parameter can be then used for two-dimensional
scattering mapping, which is particularly important in numerous
photonics applications. The observed plasma-induced breaking symmetry
can be also useful for beam steering, or for the enhanced second
harmonics generation.

\section{Acknowledgments} TEI: need to add Acknowledgements The
authors gratefully acknowledge financial support from The French
Ministry of Science and Education and from France-Russia PHC
Kolmogorov project.  TEI: for Sergey M., please put Kivshar et al. and
other Refs TEI: for Anton R., please put much more references that you
have found describing experiments and previous calculations





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