On a `time' reparametrization in relativistic electrodynamics with travelling waves

We briefly report on our method [Fiore JPA 2017] of simplifying the equations of motion of charged particles in an electromagnetic field that is the sum of a plane travelling wave and a static part; it is based on changes of the dependent variables and the independent one (light-like coordinate instead of time t). We sketch its application to a few cases of extreme laser-induced accelerations, both in vacuum and in plane problems at the vacuum-plasma interface, where we are able to reduce the system of the (Lorentz-Maxwell and continuity) partial differential equations into a family of decoupled systems of Hamilton equations in 1 dimension. Since Fourier analysis plays no role, the method can be applied to all kind of travelling waves, ranging from almost monochromatic to socalled"impulse".


I. INTRODUCTION AND SET-UP
The equation of motion of a particle with charge q in external electric and magnetic fields E(x), B(x) [x ≡ (ct, x)] in its general form is non-autonomous and highly nonlinear: here β ≡ v/c, p ≡ mv/ 1−β 2 is its relativistic momentum.
Usually, (1) is simplified assuming: 1. E, B are constant or vary "slowly" in space/time; or 2. E, B are "small" (so that nonlinear effects in E, B are negligible); or 3. E, B are monochromatic waves, or slow modulations of; or 4. the motion of the particle keeps non-relativistic.
The astonishing developments of Laser technologies (especially Chirped Pulse Amplification [2,3]) today allow the construction of compact sources of extremely intense (up to 10 23 W/cm 2 ) coherent EM waves, possibly concentrated in very short laser pulses ( fs). Even more intense/short (or cheaper) laser pulses by new technologies (thin film compression [4], etc.) will be soon available. In particular, these lasers can be used for making small particle-accelerators based on Laser Wake Field Acceleration (LWFA) [5] in plasmas. Extreme conditions are present also in several violent astrophysical processes (see e.g. [6] and references therein). In either case the effects are so fast, huge, highly nonlinear, ultra-relativistic that conditions 1-4 are not fulfilled. Alternative simplifying approaches are therefore desirable.
Here we summarize a new approach [1] that is especially fruitful if in the spacetime region Ω of interest (i.e., where we wish to follow the charged particles' worldlines) E, B can be decomposed into a static part and a plane transverse travelling wave propagating in the z direction: x = xi+yj+zk, ⊥ ⊥ k. We decompose vectors as u = u ⊥ + u z k. We assume only that ⊥ (ξ) is piecewise continuous and a) ⊥ has a compact support [0, l], α ⊥ is the travelling-wave part of the transverse EM po- We can treat on the same footing all such ⊥ , in particular: 1. A modulated monochromatic wave fulfilling (3): modul.

2.
A superposition of waves of type 1.
3. An 'impulse' (few cycles, or even a fraction of). The idea is: as no particle can reach the speed of light c,ξ(t) = ct−z(t) is strictly growing, and we can adopt ξ = ct−z as a parameter on the worldline λ (see fig. 1) and in the action functional of the particle:

II. GENERAL RESULTS FOR ONE PARTICLE
To parametrize λ by ξ we have to replace dτ /dt = 1/γ = 1−ẋ 2 /c 2 (τ is the particle proper time) by From p = mdx/dτ , γ = dt/dτ we find that the sfactorŝ is the light-like componentû − =γ −û z of the 4-velocity u = (u 0 , u) ≡ (γ,γβ) = p 0 mc 2 , p mc (all these are dimensionless), andû =ŝx .γ,û z ,β,x can be expressed as rational functions ofû ⊥ ,ŝ: By Hamilton's principle, any extremum λ of S is the worldline of a possible motion of the particle with initial position x 0 at time t 0 and final position x 1 at time t 1 .
If in addition B s ≡ 0, then A s ≡ 0 (in the Coulomb gauge), ] andv =û ⊥2 are already known. The system (15) to be solved simplifies tô Some remarkables properties of the solutions are [1]: (17) is solved by quadrature.
• Since u z ≥ 0, the z-drift is positive-definite. Rescaling ⊥ → a ⊥ ,x ⊥ ,û ⊥ scale like a, whereasẑ,û z scale like a 2 (hence the trajectory goes to a straight line in the limit a → ∞). This is due to magnetic force qβ ∧ B.
• Corollary The final u and energy gain read Both are very small if the pulse modulation is slow [extremely small if ∈ S(R) or ∈ C ∞ c (R)]. Recall the Lawson-Woodward Theorem [10][11][12][13] (an outgrowth of the original Woodward-Lawson Theorem [14,15]): in spite of large energy variations during the interaction, the final energy gain E f of a charged particle P interacting with an EM field is zero if: i) the interaction occurs in R 3 vacuum (no boundaries); ii) E s = B s = 0 and ⊥ is slowly modulated; iii) v z c along the whole acceleration path; iv) nonlinear (in ⊥ ) effects qβ∧B are negligible; v) the power radiated by P is negligible. Our Corollary, as Ref. [9], states the same result if we relax iii), iv), but the EM field is a plane travelling wave.
To obtain a non-zero E f one has to violate some other conditions of the theorem, as e.g. we see in next cases. where w(ξ) ≡ q (ξ)/kmc 2 ; clearly W (ξ) grows with ξ. In particular if ⊥ (ξ) = 0 for ξ ≥ l ≡, then for such ξ Then the solution (19) reduces toŝ(ξ) = 1−κ ξ, (23) If ⊥ is slowly modulated the energy gain (13) E f is negative if κ > 0, positive if κ ≤ 0 and has a unique maximum point κ M < 0 if (ξ) has a finite support with a unique maximum. Here is an acceleration device based on this solution: at t = 0 the particle initially lies at rest with z 0 0, just at the left of a metallic grating G contained in the z = 0 plane and set at zero electric potential; another metallic plate P contained in a plane z = z p > 0 is set at electric potential V = V p . A short laser pulse ⊥ hitting the particle boosts it into the latter region through the ponderomotive force; choosing qV p > 0 implies κ = −qV p /z p mc 2 < 0, and a backward longitudinal electric force qE z s . If qV p is large enough, then z(t) will reach a maximum smaller than z p , then is accelerated backwards and exits the grating with energy E f and negligible transverse momentum. A large E f requires extremely large |V p |, far beyond the material breakdown threshold, what prevents its realization as a static field (namely, sparks between G, P would arise and rapidly reduce |V p |). A way out is to make the pulse itself generate such large |E z s | within a plasma at the right time so as to induce the it slingshot effect, as sketchily explained at the end of next section.

IV. PLANE PLASMA PROBLEMS
Assume that the plasma is initially in hydrodynamic conditions with all initial data [velocities, densities n h , EM fields of the form (2)] not depending on x ⊥ . Then also the solutions for B, E, u h , n h , ∆x h ≡ x h (t, X) − X (displacements) do not depend on x ⊥ . Here x h (t,X) is the position at t of the h-th fluid material element with initial position X ≡ (X,Y,Z); X h (t, x) is the inverse (at fixed t). More specifically, we consider the impact of an EM plane wave with a pump of the type (3.a) on a cold plasma at equilibrium (figure below); the initial conditions are: Then Maxwell eq.s where N h (Z) ≡ Z 0 dζ n h (0,ζ): we thus reduce by one the number of unknowns, expressing E z in terms of the (still unknown) longitudinal motion. A ⊥ is coupled to the currents through A ⊥ = 4πj ⊥ (in the Landau gauges). Including (24) this amounts to the integral equation The right-hand side (rhs) is zero for t ≤ 0, because t = 0 is the beginning of the laser-plasma interaction. Within short time intervals [0, t ] (to be determined a posteriori) we can approximate A ⊥ (t, z) α ⊥ (ct−z)+ Bs 2 ∧x; we also neglect the motion of ions with respect to that of electrons. Then the Hamilton equations for the electron fluid with 'time' ξ and the initial conditions amount to (9) and x e (0,X) = X,û e (0,X) = 0 ⇒ŝ e (0,X) = 1. (28) this is a family parametrized by Z of decoupled ODEs which can be solved numerically. The approximation on A ⊥ (t,z) is acceptable as long as the so determined motion makes |rhs(26)| |α ⊥ + Bs 2 ∧x|; otherwise rhs(26) determines the first correction to A ⊥ ; and so on.
The above results are based on a laser spot size R = ∞ (plane wave). When including corrections due to the finite R (based on causality and heuristic estimates), they imply: the impact of a very short and intense laser pulse on the surface of a cold low-density