Plasma waves in a different frame

It is worth noticing that the frequency of a longitudinal wave is a zero of $ε(ω)$ also for more general expressions than Eq. (20). For example, longitudinal waves also exist in a dielectric medium that can be described (at least in some frequency range) by $ε(ω)=1−ωp2/(ω2−ω02)$ (see, e.g., Ref. 1, Sec. 32-8; Ref. 2, Sec. ST9) where ω₀ is the frequency of bound electrons and friction has been neglected for simplicity; posing $ε(ω)=0$ yields $ω=(ω02+ωp2)1/2$ for the frequency of these waves, which provide a first simple classical representation of bulk polaritons in solid state physics.

Although not strictly relevant in the present context, it is curious to notice that ultracold plasmas with temperatures as low as $10−2$ K, or $10−7$ eV, can be created via field ionization (Ref. 40).

It may be interesting to notice that for waves with super-luminal phase speed $vp>c$⁠, in a moving frame with $β=c/vp$ the fields are space-independent. The Lorentz transformation to such frame can also be helpful to simplify calculations (Refs. 26 and 41).

In Chian's paper, an immobile ion background is assumed and this is also the only case here considered. A generalization of Chian's equation to mobile ions is given by Decoster (Ref. 26).

For the reader's curiosity, more recently the co-moving frame technique has been also used for the nonlinear analysis of a plasma wave in the context of Born-Infeld electrodynamics (Ref. 42).

We use the term “test electron” with the usual meaning in electrodynamics, i.e., the electron moves in the wave field without affecting it.

The issue of electrons “injection,” i.e., controlling the initial conditions in order to achieve the maximum energy gain, is crucial to laser-plasma accelerators but it will be not discussed here. Electron “extraction” when the maximum energy has been reached occurs naturally if the electron has got up to the end of the plasma wave in the laboratory; in fact, it must be kept in mind that traveling for half a wavelength in $S′$ corresponds to a much longer distance traveled in S (see Sec. V).

The number of ions along a segment of length L oriented along the boost direction x must be the same in both frames, hence $n0L=n′0L′$⁠. Since the length $L′$ is contracted to $L/γp$⁠, it follows that $n′0=γpn0$⁠.