Abstract
This chapter provides an introduction to the optics and plasma physics concepts that will be used throughout the rest of the book. The optics section focuses on basic concepts of light wave propagation and Fourier optics, which will be useful when discussing optical smoothing techniques in Chap. 9. After a general introduction of plasma concepts, the plasma physics section emphasizes the description of plasma waves: it presents their fluid and kinetic descriptions, the wave energy, and action concepts and discusses the properties of acoustic waves in multi-species plasmas which can play a major role in laser–plasma instabilities and their mitigation, as discussed in Chap. 7. Electron-ion collisions, which will later play a central role in laser absorption (Chap. 4) and in the saturation of nonlinear kinetic effects (Chap. 10), are introduced next. Finally, the last section introduces the isothermal expansion of plasma in vacuum, which represents a type of plasma profiles often encountered in laser–plasma experiments.
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Notes
- 1.
The factor 2π comes from our choice of a unitary Fourier transform definition for angular frequencies, verifying \(\mathcal {F}^{-1}\{\mathcal {F}[f]\}=f\).
- 2.
To deal with temperature units in formulas, it will often be easy to introduce the ratio T∕mc2 with m the electron mass: since mc2 ≈ 511 keV, the ratio is simply expressed in practical units as T [keV]/511.
- 3.
This is only true in the absence of collisions, since collisions are random and will apply differently for each particle, leading to diffusion in phase-space.
- 4.
Expressing the dyadic in matrix form, we have
$$\displaystyle \begin{aligned} \begin{array}{rcl} (\mathbf{v} \mathbf{v})\cdot \nabla f_s = \begin{pmatrix} v_x v_x & v_x v_y &\displaystyle v_x v_z \\ v_y v_x & v_y v_y &\displaystyle v_y v_z \\ v_z v_x & v_z v_y &\displaystyle v_z v_z \end{pmatrix} \cdot \nabla f_s = \begin{pmatrix} v_x v_x \partial_x f_s + v_x v_y \partial_y f_s + v_x v_z \partial_z f_s \\ v_y v_x \partial_x f_s + v_y v_y \partial_y f_s + v_y v_z \partial_z f_s \\ v_z v_x \partial_x f_s + v_z v_y \partial_y f_s + v_z v_z \partial_z f_s\end{pmatrix} \,; \end{array} \end{aligned} $$the j-th vector element can be expressed as follows, with an implicit summation over i ∈{x, y, z}: [(vv) ⋅∇fs]j = vjvi∂ifs = ∂ivjvifs = [∇⋅ (vvfs)]j, hence the result (keep in mind that v is a variable independent of r, i.e., ∇ does not act on v).
- 5.
The j-th vector component [∇⋅ (nsvsvs)]j = ∂i(nsvsjvsi) = (nsvsj)∂ivsi + vsi∂i(nsvsj), from which we easily get by identification ∇⋅ (nsvsvs) = nsvs(∇⋅vs) + (vs ⋅∇)(nsvs).
- 6.
The permittivity is usually defined as ε = ε0εr, with εr the relative permittivity. However, for consistency with the rest of the plasma physics literature, we will refer to the relative permittivity as ε through this book, and call it the dielectric constant to avoid confusion.
- 7.
There is, however, a finite residual electric field over a “skin depth” region, as will be discussed in Sect. 3.2.
- 8.
Separating a variable X into a longitudinal and transverse part,
, we can write 
, where the first equality used the fact that k ⋅X⊥ = 0, the second used the vector identity a(b ⋅c) = c(a ⋅b) + b × (a ×c) and the third used 
. The equivalent relation in real space is simply 
. - 9.
- 10.
While the Z function is rarely included in scientific programing languages, the error function erf(z) usually is, and the two are connected by the relation
$$\displaystyle \begin{aligned} \begin{array}{rcl} Z(\zeta)=i\sqrt{\pi}e^{-\zeta^2}[1+\text{erf}(i\zeta)] \,. \end{array} \end{aligned} $$(1.163) - 11.
Recall that ε = 1 + iσ∕(ε0ω), so the real part of the conductivity corresponds to the wave dam**, i.e., the imaginary part of the dielectric constant.
- 12.
We merely used the conductivity for convenience, in order to avoid expanding a term ∝ ωε(ω) after Eq. (1.193).
- 13.
Here kee** the full expression for the susceptibility of the heavy ion does not complicate the algebra, unlike for the fast mode.
, we can write 
, where the first equality used the fact that k ⋅X⊥ = 0, the second used the vector identity a(b ⋅c) = c(a ⋅b) + b × (a ×c) and the third used 
. The equivalent relation in real space is simply 
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Michel, P. (2023). Fundamentals of Optics and Plasma Physics. In: Introduction to Laser-Plasma Interactions. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-031-23424-8_1
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