Plasma Physics II

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Y[Y ]' = ω β = -ω βp f p f2 dβ -u τfin which we found that the right-hand side is indeed a first τ derivative. Integrating we obtain21 d 2 2 2 2u))][ (γ (1-β = -ω β γ+C=ω β (γ -γ)f p f p f m2 d τin which we introduce the constant γ; it has a physical meaning, since the right-hand side must be greater ormequal to zero, and we can identify a maximum limit for the fluid velocity writing2 2 1/ 2– uγ =(1 /c )m mso that -u ≤u≤um mThis means that the motion of the electron fluid is bounded, oscillating between -u and u , eventuallym mreaching an inversion point in which u=0.Now, taking the square root and integrating the expression above, we getd u)][ γ(1-β;∫ f √2=± ω β τp f1 /2(γ -γ)mwhich allows us to determine u(τ), as well as the number density, n, and the electric field,

E. This integral is, in general, hard to be solved, while it can be studied in limiting cases, such as the non-relativistic one. In this approximation, we expect to find the linear solution we already developed; in particular, we obtain that $3\; \backslash omega^2\; =\; \backslash omega\_p^2\; (1\; -\; \backslash frac\{u^2\}\{16\})$ which reduces to $\backslash omega\; =\; \backslash omega\_p$ if we consider . Another interesting condition is the one in which $u\; \backslash rightarrow\; 1$, from which we find $\backslash omega\; =\; \backslash frac\{\backslash omega\_p\}\{\backslash sqrt\{2\}\}$ as $u$ approaches unity. This can be interpreted saying that if the electrons are relativistic, their inertia grows and the frequency of oscillation goes to zero; in this particular condition, the number density of the electrons shows some distortions (peaks) with respect to the linear sinusoidal trend, as we can see below. The longitudinal component of the electric field can reach much larger amplitude than the electric field in conventional radiofrequency cavities; this can be exploited to accelerate charged particles.

particles..Purely transverse modes

The equation of motion becomes

2² d² p/ dω βx , y p f u+ =0x , y

2² d² τ β −1f

which can be rewritten as

2² ω βd p f p[ + ] =0x , y

2² d² τ (β −1) γf

if we divide and multiply the second term by the γ factor. Furthermore, since the longitudinal component of the electric field is, by assumption, equal to zero, we obtain that

d γ =0/d τ

thus that the γ factor is a constant of the motion

γ(τ )=γ(0)≡γ 0

For this reason, we might as well write

2² ω βd p f p[ + ] =0x , y

2² d² τ (β −1) γf 0

which resembles a lot the equation for the linear transverse case. Thus, looking for solution of the form

p , p≈cos ω τ ≈sin ω τx y

and get, by substitution

2β² ω =ω p 2(β −1) γf 0

We can define an effective plasma frequency which, when used, makes the description identical to the

linearcase: 2ω²p'ω = γp₀ as well as the related dielectric tensor and refraction index. Notice that 2²N > Nnon lin non lin, non rel

A correspondence can be found also in the concept of cut-off frequency; in particular, if we look for the critical density we would find that n = n_γcr, non lin cr , lin 0

This opens up a range of densities to the propagation of the wave, forbidden to the linear non-relativistic one, since we can manipulate the velocity of the wave to change the γ factor and, thus, the critical density. This phenomenon is called relativistic self-induced transparency (RIT).

Another interesting phenomenon is the so-called relativistic self-focusing; this effect can happen if, inside the plasma, two different conditions are induced: one in which the mode can propagate as purely transverse mode, and one in which it cannot. Such effect can be exploited in a columnar plasma to focus light, exploiting Snell's law for total reflection.

CHAPTER

Emission of Radiation”) is a device that emits a coherent and monochromatic beam of light through the process of stimulated emission. It consists of three main components: an active medium, an energy source, and an optical resonator. The active medium is a material that can amplify light through stimulated emission. It can be a gas, a liquid, a solid, or a semiconductor. When the active medium is excited by an external energy source, such as an electrical discharge or another laser, it emits photons of light. The energy source provides the necessary energy to excite the active medium. It can be a flash lamp, an electrical discharge, or another laser. The energy is absorbed by the active medium, causing the atoms or molecules to transition to a higher energy state. The optical resonator consists of two mirrors placed at opposite ends of the active medium. One mirror is partially reflective, allowing some of the light to escape, while the other mirror is fully reflective, reflecting all the light back into the active medium. This creates a feedback loop, where the light is amplified as it bounces back and forth between the mirrors. The laser beam produced by the device is characterized by its coherence and monochromaticity. Coherence means that the light waves are in phase with each other, resulting in a narrow beam with minimal divergence. Monochromaticity means that the light has a single wavelength, resulting in a pure color. 2.2 LASER-PLASMA INTERACTION When an intense laser pulse interacts with a surface, it can generate a plasma under certain conditions. A plasma is a state of matter in which atoms or molecules are ionized, creating a collection of positively charged ions and negatively charged electrons. The interaction between the laser pulse and the surface can lead to various phenomena, depending on the plasma density. These include the excitation of collective modes, the emission of electromagnetic radiation, and the confinement of the plasma for inertial fusion. Studying the interaction between lasers and plasmas is important for a wide range of applications. It can be used to understand the behavior of plasmas under extreme conditions, such as those found in fusion reactors or astrophysical phenomena. It can also be used to develop new technologies, such as laser-driven particle accelerators or high-energy-density physics experiments. In the next sections, we will explore the basic concepts of lasers and plasmas in more detail, with the aim of using our knowledge to study their interaction.

"Emission of Radiation" is an electromagnetic wave featured by an high space-time coherence, that is a wave for which we can clearly identify a unique wavevector, k, and a frequency, ω - that is the closer we can get to a real monochromatic wave. A laser, to be more precise, is a wave which maintains these characteristics over a large distance and for long period of time, without losing coherence - while a monochromatic wave is forever coherent. Said that, a laser can be, on practice, modeled as a monochromatic wave with space varying amplitude: E(x, t) = E(x)cos(ωt - k·x) where E is called envelop function.

Such wave have to be maintained by a material able to sustain a sufficiently large range of frequencies, in particular such that the following condition is met: ωτ ≥ 2π.

Such quantity can vary greatly, from the nanoseconds range to pico- or even femtoseconds one, depending on the pulse.

Application and on the material employed. The typical ones are: titanium-sapphire (Ti:Al₂O₃), able to sustain lasers of 0.8μm wavelength; neodymium-doped yttrium aluminum garnet (Nd:YAG), used for the production of lasers of 1μm in wavelength; carbon dioxide (CO₂), for lasers 10μm in wavelength. Another quantity related to the wavelength of the laser pulse is the so-called focal spot dimensions, that is the smallest spot size that we can obtain for a laser; this dimensions are of the order of the wavelength squared.

Lasers can also span greatly in terms of energy per pulse, that is in terms of power, from the terawatts (1J/1ps) level to the petawatts (1J/1fs). The energy of the pulse can be derived by means of the electromagnetic theory, integrating the energy density over the volume of the pulse, that is

E = ∫ E² dV ≈ cλτ pulse² Vπ pulse

assuming equal contribution from the electric and the magnetic parts of the wave. Analogously,

The intensity of the laser pulse can be found: c²I|E|² ≈ 4π, which is the Poynting vector module when |E|=|B|; the intensity can reach very high values too, up to 22,210 W/cm². This quantity is of interest when we deal with the interaction between lasers and matter, since, given the intensity of the laser, we can derive the electric field amplitude and compare it with the typical values of the atomic electric field.

In particular, if the laser intensity is higher than the atomic one, we expect ionizations to be produced; actually, using a laser we can ionize matter even if the pulsation is not high enough, since we must consider the absorption of multiple photons by the atomic electrons. Furthermore, if the laser electric field is high enough, the atoms can be ionized by means of tunneling effect; then the electron is accelerated by the laser, gaining energy and returning to the atom, releasing the excess energy in the form of a photon, hence producing higher harmonics.

2.2 THE

PONDEROMOTIVE FORCE

Let us consider an electric field propagating in a plasma as E(t) = Ex(t)(x)cos(k·x-ω0) and consider the action of such field on a single charged particle; as we know, Lorentz's force will be applied on the particle: 2d x q vE = (×B)2mc dt

Nonlinearities can arise from this equation if the cross product is considered - that is if the amplitudes of the electric and the magnetic fields are not small enough - or if the amplitude of the electric field depends on space.

In particular, as we saw in the first part of the course, the resulting oscillatory motion can be decomposed in the motion of the particle around the guiding center and the motion of the guiding center itself. Herein, we will assume to know the solution to the zeroth order equation of motion: 2(0)d x q E with t = (x)cos ϕ, ϕ = k·x-ω0/cc2mdt and to be able to write a first order expansion of the electric field: [ ]Ex-x E(x) ≈ E(x) + [( )·Δ]cc

x = cexpanding them around the guiding center position, x . The equation above is simply integrated once, giving E₀(0)v sin sin=− ϕ=−v ϕosmω in which we introduce the oscillation velocity, v , that depends on the electromagnetic wave (i.e. the laser)osfrequency and wavelength. Integrating once more, we obtain E₀(0)x x cos= − ϕc 2m ω that represent an oscillatory motion at frequency ω. This derivation can also be done using a corresponding vector potential given by 1 A_E=− , with A= A sin ϕ0c t making a correspondence between E and A : 0 0 ωE A=0 0c By doing this, we can write the oscillation velocity as q A₀v =os m or, analogously, v q A_os0= =a 0²c m c called normalized vector potential -notice the dependency on the rest mass of the charged particle. This quantity can be closely related to the intensity of the laser pulse; in fact: 2E² 0 2 2 2a E I∝ ∝ λ ∝ λ0 0²ω Notice that if we want to reach arelativistic regime in the plasma, i.e. if we want v /c=a ~1, we need thisos 018 -2 2product to be of the order of 10 Wcm μm, which correspond to