# Electro-magnetic (EM) waves inside a plasma

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## Plasma with a constant density gradient

We will now consider a plane electromagnetic wave normally incident onto a plasma slab whose density is not homogeneous. The expressions of the fields inside a density gradient will be derived, which will be useful in section 1.4.1, where we describe the resonance absorption of EM radiation. In fact resonant absorption occurs when a density gradient is present which is parallel to the incoming wave electric field, so that a resonant field is driven.
In the following we will use a Cartesian coordinate system where the plasma fills the x > 0 semi-plane and we will assume variations only in the x direction. The wave equation for the electric field E = E(x)^y is d2E dx2 + !2 c2 (!; x)E = 0 (1.13) Assuming that the plasma density is a linear function of the position of the form ne = ncx=L, we find d2E dx2 + !2 c2 (1 􀀀 x=L)E = 0 (1.14) which, after a change of variable x ! = (!2=c2L)1=3(x 􀀀 L), becomes d2E d2 􀀀 E = 0.

### Laser absorption in over-dense plasmas: non collisional mechanisms

As the preservation of the surface profile is a critical issue for the surface wave excitation, the laser pulse has to be ultra-short (< 100fs) in order to limit plasma expansion. In such a plasma the thermal velocity of electrons is of the order of some hundreds of eV and collisional processes are not assumed to play an important role on the short time-scales (< 1ps) which are typical of the interaction. For these reasons we will present only the non-collisional absorption mechanisms which are present in over-dense plasmas.

#### Resonant absorption

Resonant absorption is a non-collisional absorption mechanism which takes place at the critical surface (defined as the surface where ne = nc = !2me=4e2, where ! is the frequency of the incident radiation) of a plasma having a density gradient. In order to have resonant absorption it is necessary to have a laser pulse with an electric field component parallel to the plasma gradient, that is Erne 6= 0. In fact this component induces electron oscillations along the direction of the density gradient, generating fluctuations in the plasma density. The oscillations can be amplified if a resonance is present in such a way to transfer an important part of the laser electro-magnetic energy into an electro-static oscillation, that is an electron plasma wave. We can describe this process with a simplified 2D model, following [8]: a plane plasma target, having a density gradient described by the function ne = ncx=L fills the x > 0 half-plane. The wave vector of the laser field has to be in the plane of incidence in order to have a component parallel to the gradient, thus E = Ex^x+Ey ^y. Electrons oscillating between regions of different density create a density fluctuation n such that: n = ne(r + rosc) 􀀀 ne(r) ‘ rosc rne.

Experimental results on laser absorption

Several experiments with multi-terawatt lasers have been performed in which the dominant mechanism is believed to be a mixture of vacuum heating, resonance absorption and J B absorption. We will briefly mention here only some experimental results to give an idea of the typical absorption percentage in the regime of high intensity, short laser pulses which is the regime that we will investigate in the present work. For example in the measurements carried out at Rutherford Labs [31] in the UK, it was found that the fraction of energy absorbed was about 40% at I20 = 1019W cm􀀀2m2 for a 45 angle of incidenceand a pulse duration of L = 400fs. In a similar experiment performed at LLNL in the USA [33], where the angle of incidence was 22, it was found that the absorbed energy was lower, i.e. ‘ 30%. However for ultra-short pulses (< 100fs) the absorption mechanisms cited in this section are less efficient and laser reflection is higher, reaching 80% [22].

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Plasma expansion and ion acceleration

In this section we will present the mechanisms of ion acceleration which are relevant for the case of a super-intense laser pulse interacting with a dense (solid) target. Initially the plasma expansion will be treated analytically using a self-similar model, in order to introduce the first acceleration mechanism, target normal sheath acceleration (TNSA). This mechanism, which is efficient both at the irradiated and rear side of the target, is particularly important for ion acceleration occurring at the rear, i.e. the side that is not irradiated, where it is driven by the electrons generated at the front (irradiated) side which travels through the target. The second mechanism is shock acceleration, which is ponderomotively driven by the intense pulse at the irradiated side. While TNSA accelerates ions outside the target (both at the front and the rear), shock acceleration accelerates ions inside the target.

Target Normal Sheath Acceleration

According to this model ions are accelerated normally to the target surface by the electric field associated with an electron plasma sheath. The electron sheath is set by the hot electrons accelerated by the laser pulse at the front surface and which have propagated through the target. After this initial phase the acceleration proceeds as a plasma expansion into a vacuum. The expansion is driven by the electric field due to space-charge separation at the plasma front: electrons transfer their energy to the ions via the electric field, which progressively decreases until the acceleration ceases. We will first calculate the electric field of the sheath assuming ions are not moving during the sheath formation and fill the half space x < 0. Electrons are assumed to be at equilibrium with a temperature Te, such that the electron density is given by ne = n0 exp(e=kBTe).

Introduction
1 Relevant physics issues
1.1 Electro-magnetic (EM) waves inside a plasma
1.1.1 Plasma with a constant density gradient
1.2 Relativistic effects
1.3 Ponderomotive force
1.4 Laser absorption in over-dense plasmas
1.4.1 Resonant absorption
1.4.2 Vacuum heating
1.4.3 JB heating
1.4.4 Experimental results on laser absorption
1.5 Plasma expansion and ion acceleration
1.5.1 Target Normal Sheath Acceleration
1.5.2 Shocks
1.6 Surface waves: analytical model
1.6.1 First order fields
1.6.2 Surface waves at the plasma-vacuum interface
1.6.3 Electrostatic and Electromagnetic limits
1.6.4 Effects of electron temperature
2 Resonant excitation of a SPW
2.1 Laser-plasma coupling via SPW
2.1.1 Local field amplification
2.1.2 Enhanced absorption via SPW
2.2 Effect of density variation
2.3 Incidence angles out of resonance
2.4 Effects of the modulation depth
2.5 Effects of the target thickness
2.6 Conclusion
3 Magnetic field generation
3.1 Analytical model for magnetic field generation
3.2 Quasi-static magnetic field in PIC simulations
3.2.1 Magnetic field when the SPW is excited
3.2.2 Magnetic field for the flat target
3.3 Conclusion
4 Electron heating
4.1 Electron energy
4.1.1 Non-resonant angles
4.1.2 Modulation depth
4.2 Electron emission angles
4.2.1 Non-resonant angle of incidences
4.2.2 Effects of the modulation depth
4.3 Laminar targets
4.3.1 Electron heating
4.3.2 Non-resonant incidence and reduced modulation depth
4.4 Conclusion
5 Ion acceleration
5.1 Characteristics of ion emission
5.2 The shock front
5.3 Non-resonant angles
5.4 Reduced modulation depth
5.5 The laminar target
5.5.1 Non-resonant angles and reduced modulation depth
5.6 Ion energy at the end of the interaction
5.7 Conclusion
6 The case of a Gaussian laser pulse
6.1 SPW excitation
6.2 Electron heating
6.3 Ion acceleration
6.4 The magnetic field
6.5 Conclusion
Conclusions
Appendices
.1 Particle In Cell codes
.1.1 The principle of PIC technique
.1.2 Technical aspects
.2 Modifications, diagnostics and tests
.3 Angle-energy relation
.4 Laserlab project (June 2012)

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