By Andrea Macchi

The continual development in the direction of better and better laser intensities has opened how you can new actual regimes and complex purposes of laser-plasma interactions, hence stimulating novel connections with ultrafast optics, astrophysics, particle physics, and biomedical purposes. This booklet is essentially orientated in the direction of scholars and younger researchers who have to gather quickly a simple wisdom of this lively and swiftly altering study box. To this goal, the presentation is targeted on a variety of simple types and encouraging examples, and contains themes which emerged lately comparable to ion acceleration, "relativistic engineering" and radiation friction. The contents are offered in a self-contained method assuming just a simple wisdom of classical electrodynamics, mechanics and relativistic dynamics on the undergraduate (Bachelor) point, with out requiring any earlier wisdom of plasma physics. accordingly, the e-book might serve in numerous methods: as a compact textbook for lecture classes, as a brief and obtainable creation for the newcomer, as a short reference for the skilled researcher, and in addition as an creation to a couple nonlinear mathematical equipment via examples in their program to laser-plasma modeling.

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E. e the laser wavelength. To put these assumptions in a rigorous frame, we introduce the method of multiple scales using an example from Bender and Orszag (1999), Chap. 11. e. 1. In the standard perturbative approach we write q(t) = q0 (t) + q1 (t) obtaining, to lowest order, q0 = cos t, while for the first order term dt2 q1 = −q1 − cos3 t = −q1 − ( /4)(cos 3t + 3 cos t). 23) The last term, albeit of small amplitude, is resonant with the natural frequency of the linear oscillator, so that we obtain a solution containing a secular term ∼t sin t growing with time.

2 Relativistic Electromagnetic Waves 41 Despite their limited validity Eqs. 14) give a sufficient basis to infer the occurrence of two main phenomena induced by relativistic effects, namely relativistic self-focusing (Sect. 3) and relativistic transparency (Sect. 4). Indeed, there are several other nonlinear phenomena where the relativistic effect cooperate with that of ponderomotive forces. g. Bulanov et al. 2001; Mourou et al. 2006). 14) lead to the prediction that an intense laser beam may undergo selffocusing (SF), overcoming the effect of diffraction.

32) Notice that from now on we may drop for simplicity the suffixes 1 and 2 for r1 and x2 , as other variables with different scale do not appear anymore. We must however remember that the original definitions r1 = αr and x2 = α 2 x with r and x in units of c/ω, now imply that in the following r is normalized to α −1 (c/ω) = c/ω p and x to α −2 (c/ω) = α −1 (c/ω p ) = cω/ω2p , respectively. 33) that is obtained by combining the second of Eq. 30) with Poisson’s equation 2 ϕ = n − 1. The Θ-function is inserted since n cannot become negative: whenever ∇⊥ 2 γ < 0, then n = 0 indicating that electron cavitation has occurred.