Relativistic Nonlinear Waves in Plasmas
In this chapter we focus on waves in a relativistic plasma. For electromagnetic waves, we introduce the nonlinear refractive index and the two most prominent phenomena of “relativistic optics”, i.e. self-focusing and transparency. For both phenomena, an a
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Relativistic Nonlinear Waves in Plasmas
Abstract In this chapter we focus on waves in a relativistic plasma. For electromagnetic waves, we introduce the nonlinear refractive index and the two most prominent phenomena of “relativistic optics”, i.e. self-focusing and transparency. For both phenomena, an account of a more complete theoretical description is presented along with an introduction to some methods of nonlinear physics, such as the multiple scale expansion, the nonlinear Schrödinger equation, and the Lagrangian approach. A brief description of standing nonlinear solutions, i.e. cavitons or (post-)solitons, is also given. For electrostatic waves we discuss the wave-breaking limit and focus on properties relevant to electron accelerators and field amplification schemes that will be described in the following chapters.
3.1 Linear Waves The starting point of our analysis is the wave equation for the electric field 4π 1 ∇ 2 − 2 ∂t2 E − ∇(∇ · E) = 2 ∂t J, c c
(3.1)
which is obtained from Maxwell’s equation by eliminating B. Here we consider an unmagnetized, “cold” plasma whose response to highfrequency fields is due to electrons only. The current density J = −en e ue may thus be obtained by the cold fluid equations of Sect. 2.2.2. First we review linear waves in a homogeneous plasma with uniform and constant electron density n e , neglecting all nonlinear and “relativistic” terms. Thus we linearize Eq. (2.71) obtaining ∂t J −(e/m e )n e E. For monochromatic fields, using the same notation as in Eq. (2.38) we obtain i ω2p ˜ n 0 e2 ˜ E=− E, J˜ = −i meω 4π ω A. Macchi, A Superintense Laser-Plasma Interaction Theory Primer, SpringerBriefs in Physics, DOI: 10.1007/978-94-007-6125-4_3, © The Author(s) 2013
(3.2) 37
38
3 Relativistic Nonlinear Waves in Plasmas
with the plasma frequency
ωp ≡
4π e2 n e me
1/2 .
(3.3)
By substituting into (3.1) we obtain the inhomogeneous Helmholtz equation 2 ω2 ˜ = ∇ 2 + n2 (ω) ω E˜ − ∇(∇ · E) ˜ = 0, (3.4) ∇ 2 + ε(ω) 2 E˜ − ∇(∇ · E) c c2 where we introduced the well-known expressions for the dielectric function ε(ω) and the refraction index n(ω) of a cold plasma ε(ω) = n2 (ω) = 1 −
ω2p ω2
.
(3.5)
˜ Now consider transverse EM waves with ∇ ·E = 0. For a plane wave E(r) = E0 eik·r with k · E0 = 0, we obtain by direct substitution the dispersion relation − k 2 c2 + ε(ω)ω2 = −k 2 c2 + ω2 − ω2p = 0.
(3.6)
The propagation of the wave requires k = |k| to be a real number, which occurs when ω > ω p . Thus, the plasma frequency is a cut-off frequency for EM transverse waves. For a given frequency ω this condition can be also written as a condition on the plasma density ne < nc ≡
m e ω2 = 1.1 × 1021 cm−3 (λ/1 µm)−2 , 4π e2
(3.7)
where n c is called the cut-off or, more often, the “critical” density. A plasma having electron density n e < n c for a particular wavelength λ, allowing the propagation of transverse EM waves of such wavelength, is said to be underdense. In the opposite condition of an overdense plasma having n e > n c , ε = 1 − n e /n c < 0, thus n a
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