Relativistic  Laser  Plasma  Interaction

J. Meyer-ter-Vehn, A. Pukhov, Rel. Las. Plas. Interaction, part I: Analytical Tools
A. Pukhov and J. Meyer-ter-Vehn,  Rel. Las. Plas. Interaction, part II: Particle-in-Cell Simulation
in Relativistic Optics
, eds. G. A. Mourou, C. P. J. Barty, M. D. Perry (Springer Verlag, under preparation)
Recent work in the  Laser Plasma Theory Group at MPQ was devoted to the interpretation of  laser plasma experiments at relativistic intensities, using 3D-PIC simulation. A most prominent feature is the generation of highly collimated electron beams with energies in the 10 - 100 MeV regime. These beams may produce secondary beams of   γ-rays and other nuclear radiation (positrons, neutron, etc.) with table-top lasers. They open a wide field of applications e.g. in medicine and material research. In recent publications, we have contributed to the basic understanding of the underlying relativistic laser plasma interaction.

Relativistic Non-Linear Optics

The MPQ  ATLAS  laser  produces focussed laser intensities up to few times 1019 W/cm2 on target material, which ionizes and turns into plasma.  At intensities above 1018 W/cm2, the laser light accelerates target electrons almost to the velocity of light  such that their masses increase by the relativistic factor γ = (1- v2/c 2 )-1/2 . In ATLAS experiments we encounter electrons which are 10 - 100 times heavier than electrons at rest. This strongly changes  laser plasma interaction.

Induced Transparency

rel. dispersion  Light of frequency ω propagates in plasma according to the dispersion relation ω2p2/<γ>+k 2c2 , which is plotted here. It depends on the plasma frequency ωp2=4πe2ne /m   and the average <γ>-factor. In dense plasma  with  ω <  ω p , light cannot propagate and is reflected from the surface. However, for relativistic intensities generating large <γ> -factors, the plasma becomes transparent. We call this induced transparency.

Relativistic Self-Focussing

self-focus Due to the transverse intensity profile of the light beam, the relativistic effects are strongest on the axis and modulate the index of refraction  n=(1-(ωp2/<γ>)/ω 2)1/2 accordingly.  An initially planar wavefront is deformed in a plasma as shown in the figure. Since the phase velocity  vph=c/n  is smaller on the axis, the plasma acts like a positive lens and leads to self-foccusing for laser powers beyond a critical level.

Profile Steepening

prof-steep Another important effect is the steepening of pulse envelopes propagating with group velocity  vgr=cn. The peak region with high intensity runs faster than those with low intensity at the pulse head, and this leads to optical shock formation. Pulse shapes with steeply rising fronts are interesting for studying high intensity effects in matter.

Three-dimensional Particle-In-Cell Simulation

The Virtual Laser Plasma Laboratory

pukhov A. Pukhov, J. Plas. Phys. 61, 425 (1999)
Particle-in-Cell (PIC) simulations solve the laser plasma interaction at the fundamental level of Maxwell´s equations and the equation of motion for relativistic particles moving in the electromagnetic fields which are averaged over cells. The three-dimensional PIC code VLPL (Virtual Laser Plasma Laboratory) has been developed by A. Pukhov at MPQ. It is well adjusted to parallel computors with some 100 processors and typically handles 109 particles in 108 cells distributed over a three-dimensional volume. Examples of  VLPL simulations are given below and on the page Laser Wake Field Acceleration. PIC

Relativistic channeling and electron beam generation

A. Pukhov, J. Meyer-ter-Vehn, PRL 76, 3975 (1996)

Beyond a critical power Pcrit = 17.4 nc/n e GW, a laser pulse propagating  in plasma undergoes self-focussing as it is seen in the figure below. Here ne /nc is the electron  density normalized to the critical density nc. In three-dimensional space the laser beam self-focusses to a super-channel just 1-2 wavelengths in diameter. An outstanding feature is the relativistic electron beam accelerated in the channel in the direction of laser propagation. With a density of order n c ~ 1021 cm-3 , it produces a current density of order 1012 A/cm2 and total currents of some 10 kA, which generate a quasi-stationary magnetic field in the order of 100 MegaGauss.  The pinching effect of the magnetic field adds to the self-focussing.


Electron and ion spectra

A. Pukhov, Zh.M.  Sheng, J. Meyer-ter-Vehn, Phys. Plasm. 6, 2847 (1999) .
The energy spectra of the electrons show a characteristic exponential decay and the corresponding effective temperatures scale  according to Teff ~ 1.5  I1/2MeV  with intensity I in units of 1018 W/cm2. This is in agreement with measured spectra. Since electrons are expelled from the channel, a radial electric field is created which accelerates ions in radial direction. Depending on laser intensity, multi-MeV ion energies are found in  simulation as well as experiment. In deuterium plasma, these energetic ions cause fusion reactions, and the corresponding 2.45 MeV neutrons have been detected experimentally.  
electron spectrum            ion spectrum

3D-PIC simulation compared to experiment

C. Gahn et al. Phys. Rev. Lett. 83, 4772 (1999)
Self-focussing and electron beam generation have been observed in MPQ experiments, using gas jet targets and a 150 fs laser pulse with focussed intensity 6 x 10 19 W/cm2 . The measured electron  spectra were found to be in excellent agreement with the corresponding 3D-PIC simulation. This opened the possibility to investigate the electron acceleration mechanism in more detail. The electron phase space is shown below on the right-hand side as a snapshot  after 300 fs propagation time. The pulse propagates from left to right. The longitudinal E z field reveals some self-modulated laser wakefield excitation near the laser head and wakefield acceleration in the γ-plot, but apparently the plasma wave breaks after a few oscillations. Nevertheless, strong electron acceleration with γ ~  40 - 50 is visible in the broken-wave region, and the question arises what is the acceleration mechanism here. Zooming the phase space in the region z/λ ~ 270 - 280, one finds that it is modulated with the laser period and shows large transverse momenta px, indicating that direct laser acceleration takes place.
Gahn phasespace

How do the electrons gain energy?

Electrons can gain energy only from the electric field, either the transverse component mainly originating from the laser pulse or from  the longitudinal component mainly originating from plasma waves. To find out which mechanism dominates, we have determined the total longitudinal and transverse gain for each electron and show the result in the figure. Surprisingly, most electrons in this simulation gained their final energy from the transverse laser field, and the longitudinal field had rather a decelerating than an accelerating effect. The deceleration can be attributed to the negative longitudinal component of the laser field occuring in narrow channels.

Relativistic channels as Inverse Free Electron Lasers

A. Pukhov, Zh.M.  Sheng, J. Meyer-ter-Vehn, Phys. Plasm. 6, 2847 (1999)
IFEL The result obtained above can be understood in terms of an Inverse Free Electron Laser model. The azimuthal magnetic and the radial electric field of the self-focussed channel acts like the wiggler of  a free electron laser (FEL), causing transverse oscillations of relativistic electrons with betatron frequency ωβ2= ωp 2/(2γ) when moving along the channel axis. This is exactly the configuration of an FEL. At resonance when the Doppler-shifted laser frequency coincides with ωβ = ωL ( 1 - v|| /  v ph ) , the electron can experience acceleration from the laser field over many laser periods, and this explains the large transverse momenta. It is then the magnetic laser field which turns the transverse motion into longitudinal motion without adding further energy.

Laser hole boring into overdense plasma

A. Pukhov, J. Meyer-ter-Vehn, PRL 79, 2686 (1997)holebore
In the case of overdense plasma (here ne/nc = 10), the laser light cannot penetrate into the plasma initially, but the light pressure starts to bore a hole into the overdense region. This is observed in the ion density plot at 330 fs and 660 fs. Matter is pushed to the side and forms a conical shock. Electrons are accelerated in the hole region and corresponding strong currents are seen in the magnetic field pattern. At the surface of the hole the current is directed outwards, while in the inner regions of the hole it is directed into the plasma. A particular interesting feature is seen in the overdense part of the plasma which has not yet been reached by the hole boring  and into which only the electron current can penetrate. Here the electron current is seen to disintegrate into current filaments at 330 fs, but these filaments have apparently reunited  in a single thick current filament at 660 fs. Filamentation is due to Weibel instability.

Current filamentation and filament coalescence

M. Honda, J. Meyer-ter-Vehn, A. Pukhov, PRL  85, 2128 (2000)
filaments  We have also studied current filamentation by 2D PIC simulation in the plane transverse to the current. At time ωpt=0, a uniform relativistic electron current is assumed having 10% of the plasma density. Initially it is completely compensated by a uniform return current. This two stream configuration quickly decays into many filaments, which, in a later phase, coalesce and form a few thick filaments. The process of coalescence is found to be highly dissipative leading to strong anomalous stopping of the initial beam. These features may be relevant to the concept of fast ignition of fusion targets.

Confined electron-positron plasma

B. Shen and J. MtV, Phys. Rev. E65, 016405 (2002).
B. Shen and J. MtV, Phys. Plasmas 8, 1003 (2001)
e+e- Electron-positron and γ-photon production by high-intensity laser pulses has been investigated for a special target geometry, in which two pulses irradiate a very thin foil (10-100 nm < skin depth) with same intensity from opposite sides. A stationary solution is derived describing foil compression between the two pulses. Circular polarization is chosen such that all electrons and positrons rotate in same direction in the plane of the foil. We discuss the laser and target parameters required in order to optimize the γ-photon and pair production rate. We find a   γ -photon intensity of  7 x 10 27/ (sr sec) and a positron density of  5 x 1022 cm -3 when using two 330 fs , 7 x 1021 W/cm2 laser pulses.