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\title{Plasmas in  extreme electromagnetic fields%\thanks{Grants or other notes
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\author{Francesco Pegoraro         %\and  Second Author %etc.
}

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\institute{F. Pegoraro \at
     Physics Department,  Pisa University, largo Pontecorvo 3, 56127 Pisa, Italy\\National Research Council, National Institute of Optics,  via G. Moruzzi 1,  Pisa, Italy\\
              Tel.: +39  050  2214844\\
              Fax:  +39  050  2214333\\
              \email{pegoraro@df.unipi.it}           %  \\
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\date{Received: date / Accepted: date}
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\maketitle

\begin{abstract}
The recent developments in the generation of laser pulses with
ultra-high power (presently petawatt and progressing) have opened up  a new frontier in plasma research  by making 
 it possible to obtain and to study ``mesoscopic''  amounts of
relativistic (ionized) matter  in compact-size experiments in the laboratory. 
This  will make it possible to investigate
in a  controlled environment    the nonlinear dynamics 
of collective relativistic systems, to enter the Quantum Electrodynamics  plasma regime,  and  to explore conditions that are of
interest for high energy astrophysics and beyond.

\keywords{Relativistic plasmas \and Ultraintense electromagnetic fields \and High energy laboratory astrophysics}
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\end{abstract}

\section{Introduction}

The Nobel Prize in Physics 2018 was awarded \cite{nobel} ``for groundbreaking inventions in the field of laser physics'' with one half to Arthur Ashkin``for the optical tweezers and their application to biological systems", the other half jointly to G\'erard Mourou and Donna Strickland ``for their method of generating high-intensity, ultra-short optical pulses.''\\ Strickland   and Mourou's   invention, the so-called ``chirped pulse amplification'' in the optical range \cite{StMou}, has driven an unprecedented development in the study of the interaction of ultra-intense electromagnetic radiation with matter \cite{MTB}. In the short time  period of two decades  this  has raced through the nonlinear optics of bound electrons in strong fields, the  relativistic optics where the quiver velocity of the free electrons  nears  the speed of light,  reaching  the ultrarelativistic regime where the  kinetic energy  of the electrons  is  very much larger  than their  rest mass energy and massive ions enter  the relativistic domain, and  finally now aiming at  the yet unexplored plasma regime where Quantum Electrodynamics effects and particle pair  creation play  a major role \cite{square}.
\, This race  has opened up a new frontier in plasma research   (see  the ``Extreme Light Infrastructure'' project  \cite{ELI})  directing   it towards the study of novel  fundamental physics phenomena, such as the nonlinear dynamics  of collective relativistic systems  made up of high energy particles and photons that interact  according to the rules of multi-particle Quantum Electrodynamics \cite{MSh},  and making it possible to investigate  matter  conditions  \cite{astr}  that are expected to occur in extremely energetic astrophysical objects  such as pulsar atmospheres or black hole neighbourhoods. In addition it  has  made possible   the realization of  novel compact  ultra-intense sources of high frequency electromagnetic radiation  up to the gamma range  (see e.g. \cite{MPeg})  and  the development of  high field gradient  particle accelerators \cite {Esa,Macc} that bring promise both for high energy particle physics \cite{Sif} and for medical applications  such as widely  distributed  centres for hadron therapy \cite{BulPeg}.


\section{High energy plasmas}



 Plasmas take a variety of forms,  from magnetic  loops  \cite{nasas} in the solar corona to giant relativistic jets \cite{nasaj} in galaxies, forms  that are shaped by the collective dynamics of fields and particles \cite{DOE}. In high temperature diluted (fully ionized) plasmas
  this dynamics occurs on timescales that can be very much  shorter than those of the binary processes among individual particles. This dynamics is  dominated by nonlinearities that under proper conditions can produce
extremely large energy densities (see e.g., \cite{PSc,weber}), on mesoscopic scales   in the laboratory  (spatial size of the order of  tens of microns,  particle density $10^{21} - 10^{22} \, \, cm^{-3}$ 
close to solid density, electron energy in the $GeV$ range, ion
energy presently in the tens of $MeV$  range) and on huge spatial scales in the Universe.
 
  
  \subsection{High energy density} 
  
A plasma is a medium that is very effective both for  producing and for  manipulating \cite{TT} high energy density  electromagnetic fields  and electromagnetic  radiation.   Relativistic electromagnetic   solitons  \cite{sol1,sol2} generated in laboratory plasmas  
 by ultraintense laser pulses provide a good example. In these plasma structures  the radiation pressure  of the oscillatory   electromagnetic  fields trapped inside the soliton acts on the  plasma electrons piled up in the soliton  walls. Here   the radiation pressure  is balanced by the Maxwell stress tension produced by  the stationary  electric field
 due to  the charge separation that  is created inside the soliton by the displacement of the electrons with respect to the much heavier ions. \, 
A similar  phenomenon is the so-called relativistic self-focusing \cite{lit}  now routinely observed  \cite{sfoc} in  ultra-intense laser plasma interaction experiments. Relativistic self-focusing allows a laser pulse to  propagate without spreading over distances  significantly longer than the ``Rayleigh length'' because  light diffraction is  counteracted by  nonlinear effects  (see \cite{Ask})  and in particular by the nonlinearity  introduced  by the 
 relativistic kinematics  into the plasma dynamics through the relationship between  particle velocities  and particle momenta. This nonlinearity makes the electrons in the plasma  effectively ``heavier'' where the laser pulse is stronger, thus causing a reduction of the local value of the effective Langmuir frequency.
  
      \subsection{Transport} 
      
      
  The fact that plasmas can store large energy densities lies at the  heart of magnetic fusion research \cite{Nfus}  that aims to produce net thermonuclear energy in a controlled way in the
laboratory at relatively low particle densities. However, the effectiveness of the collective interactions between
the particles and the electromagnetic fields in plasmas can cause a fast energy transfer between particles and fields: in almost all
conditions the transport properties of a fusion plasma are determined by collective plasma excitations \cite{coppi}, not by binary processes.
Being far  from thermodynamic equilibrium  conditions, the amplitude of these particle and field fluctuations is orders of magnitude larger than that
of the thermal fluctuations one encounters in regimes of local equilibrium, giving rise to anomalous, i.e. enhanced, transport \cite{Bal}.
 We may just briefly recall here  that 
  the physics of thermonuclear fusion in the laboratory is very
different from that inside a star because of its miniaturization \cite{LC}.
A star is dense and thus opaque to the electromagnetic radiation: it is at
local thermodynamic equilibrium and loses energy mostly
through Black Body radiation. These 
 losses are proportional to the star
surface while gains (thermonuclear fusion power) are
proportional to the star volume.
To be hot and confined, a plasma in a laboratory magnetic fusion
experiment must be dilute: it is transparent to its own radiation
(mainly Bremsstrahlung) and suffers intrinsic losses that are
proportional to the plasma volume as  are its fusion gains.
  

    
  The transport properties of astrophysical plasmas too can be
controlled by collective plasma excitations, not by binary
processes. Accretion disks \cite{Pr} are an example.
Conservation of angular momentum would prevent accretion of
matter rotating in a disk around, e.g., a compact stellar object.
Collisional (so-called molecular) viscosity is  insufficient to explain the rate of
accretion required in order to account for the emitted electromagnetic 
radiation: angular momentum must be transported outwards
much more efficiently \cite{SS,Lyn}.
Collective plasma excitations, such as the so-called Magneto-Rotational (Velikhov) instability \cite{MR,vel}, provide the mechanism that
may  account for the anomalous  transport of angular momentum.

Collective instabilities  can  also provide  an effective mechanism of   particle acceleration under both laboratory and astrophysical conditions.  
For example, a  relativistic reformulation of  the reconnection instability of magnetic  field lines, which  in nonrelativistic regimes is a major player in determining the magnetic structure of  laboratory \cite{Yamada} and solar plasmas \cite{Priest},  
has been invoked  \cite{Uzd}  as  the   acceleration mechanism of electrons up to $PeV$energies  in  the gamma-ray flares \cite{Tav}  observed in the Crab Nebula. 

   \subsection{The Weibel instability} 

  However, in determining transport  properties collective excitations work differently from collisions.
The Weibel instability  \cite{Weib} and its related quasistatic  magnetic field generation provide  a good example.
This well known instability  feeds on the
energy difference between the different degrees of freedom in
momentum space, just as a thermal machine would do by
extracting heat from bodies at different temperatures.
The major difference is in the ``final'' state.
Collisions would equalize the temperature of the two bodies.
In contrast, the Weibel instability  transforms part (corresponding
approximately to energy equipartition) of the energy difference
between the different degrees of freedom into (steady) magnetic
field energy (in a sense, this  corresponds to the work that is produced  by the thermal machine).
Moreover a real steady state is not reached after the instability amplitude  has  ceased growing  and the plasma
continues to evolve transferring e.g., energy to smaller scale
excitations as  it does in a turbulent cascade process \cite{pal}.\\

The Weibel instability  is an important mechanism of magnetic field generation in space in conditions where  collisional effects such as the so-called Biermann battery \cite{greg}  are ineffective. It may provide the seed field that feeds the dynamo process that can produce
magnetic fields on large spatial scales \cite{seed}.  \,
 A variant of the Weibel instability  is of major importance for laser-produced relativistic plasmas  and it occurs in plasmas 
 where  an effective  anisotropy in momentum space can be associated to the presence of two counterstreaming 
 particle  populations. In this  case it goes under the more appropriate name of ``current filamentation'' instability \cite{fil,filbis}.
Counterstreaming electron populations occur naturally in the
interaction of ultraintense laser pulses with the plasma that is formed
almost instantaneously when such pulses interact with a gas  or
with solid material: electrons are accelerated to relativistic energies
by the pulse  and penetrate   the material and the resulting charge
imbalance induces a neutralizing return current. In particular, this filamentation is of primary importance  for electron transport in inertial fusion experiments 
and  for  the so-called fast ignition \cite{fast} where the generated magnetic field can channel the electrons. In very different conditions,  having been reformulated  within a Quantum-ChromoDynamic framework \cite{chromo},
 it has been reconsidered under the name of ``Chromo-Weibel instability''
 in relativistic heavy  ion collision experiments.

\subsection{Relativistic mirrors} 

Possibly the most impressive  example of how relativistic plasmas  can be used to concentrate electromagnetic energy is provided 
by relativistic plasma mirrors \cite{bulm} moving 
at a speed close to that of light,  finally performing in the laboratory  the ``gedanken'' experiment described by 
A. Einstein more than a century ago \cite{ein}.
The coherent interaction between a high intensity electromagnetic  pulse and a
foil of relativistic electrons counterpropagating at a relativistic velocity  \cite{bulf} provides a  method for reaching
unprecedented electromagnetic  energy densities.
The foil of  counterpropagating relativistic electrons consists of  the electrons accelerated at
wave breaking by the electric field of  a nonlinear Langmuir wake wave
generated in the plasma by a second laser pulse propagating in the opposite direction with respect to the first  \cite{bulm}.
 In a longitudinal nonlinear wave (such as the  large amplitude Langmuir wake wave
with phase velocity close to the speed of light $c$ induced in the  plasma by the second laser  pulse),
if electrons  are accelerated along the direction of the wave propagation up to velocities larger
than the wave phase velocity, the wave breaks \cite{break}.
At the wave break position a cusp develops
in the electron density and the electron oscillatory
motion in the wake wave is transformed into a net relativistic
motion in the direction of   breaking.
There are important issues  that need to be addressed in the use of such relativistic mirrors, such as the reflectivity of the mirror  formed by the electron  foil \cite{sail}, its parabolic shape that can focus the reflected pulse, etc.. However 
 the main physics point to be noted is that the reflected pulse is not the result of the incoherent superposition of (inverse) Thomson
(Compton) scattered waves but of the coherent response of the
plasma electrons that, as in a Free Electron Laser, satisfy the density-wavelength condition 
$n\lambda^3 >> 1$. In these conditions electromagnetic  processes are described by the coherent plasma response and not by single particle
scattering cross sections.

Plasma relativistic mirrors 
provide an innovative    tool  for  reaching   higher and higher electromagnetic field intensities: this tool  is based on 
the  plasma's ability   to concentrate the electromagnetic
 energy of pulses produced by available laser systems  \cite{bulf}
without the need  to resort to  larger and larger laser  systems.
An example of this capability  is offered by 
the interaction of regular nonlinear structures in the plasma, such as  solitons or magnetic vortices,  with a strong wake wave \cite{tennis}.
 The electromagnetic fields of the nonlinear structures  are partially reflected by
the electron density modulations of the wake wave incident with a  phase velocity close to the speed of light and a single-cycle, high-intensity electromagnetic
pulse is formed in the reflection process. Due to the Doppler effect,  the size of the reflected pulse is much shorter (it can reach the attosecond range) than that of the initial  nonlinear
structure \cite{tennis}.   As noted in \cite{atto}, ``the measurement and control of subcycle field evolution of few-cycle
light have opened the door to a radically new approach to exploring and controlling processes of the microcosm.''



 \section{A new plasma physics regime}
 

These theoretical and experimental developments will make it  possible to investigate a new regime of plasma dynamics where photons   form an additional  particle population and particle pairs can be created
together with incoherent,  high frequency radiation \cite{tamb}.
Using, as mentioned above,  plasma mirrors to intensify light it will be possible  to  achieve
extremely high electromagnetic fields  \cite{bulf} with amplitudes that 
approach the critical field of Quantum Electrodynamics, the so-called  Schwinger field \cite{Sauter}, and to create  globally neutral electron-positron-photon  plasmas \cite{Bell},
 turning the vacuum into a medium  that  can affect the propagation of the laser pulses  themselves  \cite{Koga}.

The nonlinear dynamics of the electromagnetic field in vacuum is described in the long wavelength limit by the Euler-Heisenberg 
Lagrangian \cite{EH} that  depends on the electromagnetic Lorentz invariants $|{\bf  E} |^2 - |{\bf B}|^2$  and ${\bf  E} \cdot {\bf  B}$,  which  vanish for a plane wave
in vacuum. Strongly focused  and multiple  intersecting pulse configurations  are thus of special interest   (see e.g. \cite{Nar,SSmb}) and an extensive  research effort  
has  flourished  in recent years.   
 Vacuum  birefringence  \cite{Roz}, high harmonic generation    \cite{harmsq,harmfed},
   pair production avalanches \cite{paircasc}  and  photon splitting \cite{photsplit} are among the phenomena that have been intensively investigated.
 \,\,  In a plasma  it will be possible  to study  under controlled conditions  the collective 
dynamics  of   matter made  of  relativistic   electrons and  positrons \cite{Sarri}, of  interest for high energy astrophysics,  in the presence of extremely large electromagnetic fields   and  more generally the collective
dynamics of matter   in a Quantum Electrodynamics regime.



  
  
  \section{Conclusions} 
  The availability in the laboratory of relativistic plasmas that can be
studied in  compact experiments  has opened up  the way to
 rapid new developments in  a wide range of diverse fields of science.
To name a few, these  range from nonlinear system dynamics and nonlinear relativistic optics  to novel powerful  sources of  high energy  radiation,
X-ray lasers, attosecond pulses,  high field gradient  techniques for  particle acceleration  and 
plasmas dominated by high frequency incoherent radiation.
These developments  will establish a strong connection with  high energy astrophysics in  fields  such as  radiation pressure  acceleration,  relativistic  shocks and relativistic magnetic reconnection
in the search for mechanisms of extreme particle acceleration (e.g., in the context of  cosmic ray physics). 
\, Using plasmas as tools,  it will be possible to  generate electromagnetic  fields 
 approaching the Schwinger field to
investigate    nonlinear optics in vacuum  in the laboratory.

The miniaturization  of high energy physics  experiments down to ``table top'' size 
 in  high energy density    plasmas  produced by ultra-intense laser pulses 
has  represented   a major  breakthrough   in plasma physics research,
making    fast   progress possible 
  in yet unexplored regimes   with  a vast  participation  of scientists working  in  research institutions in different countries.



In this paper an effort has been made to provide an overall view of a vast set of topics and their intertwining relationships, together with an adequate list of references 
to guide the interested reader. Of course, the selection of the papers quoted reflects the  personal view of the author and is not meant to be exhaustive.



\section*{Acknowledgement}

I would  like to thank the {\it Accademia Nazionale dei Lincei}  and the {\it Fondazione Guido Donegani}  for hosting  the International conference ``Classical and quantum plasmas: matter under extreme conditions" (Rome, Palazzo Corsini,
April  5 - 6,  2018) where these physics  issues were discussed.

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