Con respecto a la Teoría Musical

with γ(r) =p1 +a2_{(r)/2 for a laser with beam profilea(r) peaking on the laser}

axis for r= 0 (which is typically a Gaussian or super-Gaussian beam profile). In both cases a local decrease of the plasma frequency (see Eq.2.21) atr= 0 results in an increased refractive index; while the~v_×B~ term causes the electron density to locally decrease, the transverse beam profile causes γ to locally increase. The phase velocityvp(r) = c/N(r) is hence larger off-axis than on-axis, resulting in the

self-focusing (see Fig. 2.3). For the relativistic self-focusing, the threshold value is a2

0(ωpw0)/c ≥ 8 [54], where the self-focusing overcomes the laser divergence

(with w0 the beam waist, see Chap.3.1.4).

Self-focusing can have a significant and also unpredictable effect on the overall laser-matter interaction due to the fact that it results in higher intensities in the axis of the laser-propagation direction. In addition, relativistic channels might form in the (relativistic) transparent plasma [64] that can guide the laser pulse tightly focused over many times the Rayleigh length zr (see Chap.3.1.4) along

the plasma.

2.2.5

Laser absorption and energy transfer in plasmas

The laser absorption is probably the most important process during the laser- plasma interaction, as it transfers energy from the laser to the plasma. With the laser intensities available today, the laser electric fields are still to low to directly act on the plasma ions; hence, laser energy transfer is mediated by the plasma electrons, i.e, electrons gain energy (Th) in the laser field, which is “distributed”

a(r)

Phase fronts in plasma

Figure 2.3: Self-focusing in a relativistic plasma due to a transverse gradient in the refractive index causing slower group velocity vp on-axis than off-axis (Adaption

of Ref. [65])

to the plasma ions and atoms through subsequent thermalization (collisions) of the electrons (Te) or other processes like instabilities. The laser absorption by

electrons in a plasma can be separated into collisional and collisionless processes. In order to identify the dominant process it is useful to introduce the ion-electron collision frequency νei, which is given by [59, 66]

νei∝

neZ

Te3/2

. (2.29)

For high density plasmas (and high Z) at low electron temperatures collisional absorption is the dominant mechanism. The collisional absorption, which is also called “inverse Bremsstrahlung”, is the collision of an electron with an ion under the presence of an electric field, i.e., the electron absorbs a photon during the collision process. Moreover, as shown in the last section, the highest densities accessible by the laser are found at the critical surface of the plasma, so that the collisional absorption is most effective here.

With increasing laser intensities the electron temperature rises and collisional absorption becomes more and more ineffective. Typically, in interactions where the laser intensity exceeds _≫ 1015_W/cm2_{, collisionless absorption is the domi-}

nant process once the electron temperature is _≈ 103_Z2_{eV[61]. The collisionless}

absorption cannot be treated analytically in general, but has been studied with computer simulations for many years now. As a result, a large number of mech- anisms has been identified in this regime, where the most prominent are the

resonance absorption [67], the~j_×B~ heating [68] and the vacuum/Brunel

anism are stochastic heating [70], Landau damping [71] and the anomalous skin layer absorption [72]. These mechanisms typically result in the generation of very “hot” electrons in contrast to the collisional absorption.

Resonance absorption

The resonance absorption is a process that in general requires oblique laser inci- dence of a p-polarized light wave and a long density scale length. The laser, that is incident on the plasma at an angle θ is reflected near the critical surface of the plasma (where ne =nccos2(θ)). Due to the p-polarization of the light wave, the

(tangential) electric field of the laser can reach/tunnel into the plasma and excite electron oscillations at the critical surface. These oscillations can grow resonantly over several laser oscillations and drive an electron plasma wave into the plasma. Energy can be transfered to the plasma through dampening of this wave by wave breaking or collisions [59].

Vacuum/Brunel heating

For steep or step-like plasmas, where the amplitude of the electron oscillations exceed the plasma scale length [54], resonantly driven plasma waves are no longer supported. Here, the vacuum heating plays an important role in the laser absorp- tion, which has been introduced by Brunel in 1987 [69] as the “not-so-resonant, resonant absorption”. In particular, an electron at this sharp plasma-vacuum interface is first accelerated by the laser electric field into the vacuum. When the field of the laser reverses, the electron is pushed back into the plasma. However, the laser electric field can only penetrate the steep and highly overdense plasma to its skin depth ls (see Sect. 2.2.2 and Eq. 2.24). Now that the electric field is

screened by the plasma, the electron can penetrate deeper into the plasma with- out being dragged out by the laser again and eventually thermalize by subsequent collisions.

Relativistic~j_×B~ heating

In the~j_×B~ heating the electron is directly accelerated in the laser field, where for relativistic intensities, i.e., fora0 >1, the electron motion is dominated by the

along x, the (ponderomotive) force is given as [54] fx=− me 4 ∂v2_(x) ∂x (1−cos(2ωt)). (2.30)

Here, the first term describes the motion of the electron in the laser propagation direction and the second one the motion under the fast oscillating vector potential of the LP wave. For an electron in a plasma, the latter term is responsible for the heating of the plasma similar to the vacuum heating. However, the j _×B

heating works best with normal incidence and is mostly independent of the laser polarization; only for CP, where the vector potential has no oscillating component,

~j _×B~ _{heating ceases to work. This is an important effect, that is sometimes}

used to reduce plasma heating due the laser light and alter the overall laser- plasma interaction; an example is given in Sect. 2.3.2 and in Chap. 4.4, where the reduced plasma heating results in a strongly modified energy spectrum of the laser-accelerated ions.