Hydrodynamic regime and cold plasmas hit by short laser pulses

We briefly report and elaborate on some conditions allowing a hydrodynamic description of the impact of a very short and arbitrarily intense laser pulse onto a cold plasma, as well as the localization of the first wave-breaking due to the plasma inhomogeneity. We use a recently developed fully relativistic plane model whereby we reduce the system of the Lorentz-Maxwell and continuity PDEs into a 1-parameter family of decoupled systems of non-autonomous Hamilton equations in dimension 1, with the light-like coordinate $\xi=ct\!-\!z$ replacing time $t$ as an independent variable. Apriori estimates on the Jacobian $\hat J$ of the change from Lagrangian to Eulerian coordinates in terms of the input data (initial density and pulse profile) are obtained applying Liapunov direct method to an associated family of pairs of ODEs; wave-breaking is pinpointed by the inequality $\hat J\le 0$. These results may help in drastically simplifying the study of extreme acceleration mechanisms of electrons, which have very important applications.


Introduction and plane model
Ultraintense laser-plasma interactions lead to exciting phenomena [22,25,27,6,24], notably plasma compression for inertial fusion [23], laser wakefield acceleration (LWFA) [28,26,29] and other extremely compact acceleration mechanisms of charged particles, which hopefully will allow the production of new, table-top accelerators.Huge investments are presently devoted to the development of the latter 1 , because their small size would drastically facilitate  (2).In a) we also illustrate the meaning of the functions n u (z), n d (z) defined in (19).c): Projections onto the z, ct plane of sample particle worldlines (WLs) λ 1 , λ 2 in Minkowski space [14]; they intersect the support (pink) of a plane EM wave of total length l moving in the positive z direction.Since each WL intersects once every hyperplane ξ = const (beside every hyperplane t = const), we can use ξ rather than t as a parameter along it.While the t-instants of intersection with the front and the end of the EM wave (e.g.t 1i , t 1f for λ 1 ) depend on the particular WL, the corresponding ξ-intstants are the same for all WLs: ξ i = 0, ξ f = l.
the extremely important applications of accelerators in particle physics, medicine, material science, industry, inertial fusion, environmental remediation, etc.In general, these phenomena are ruled by the equations of a relativistic kinetic theory coupled to Maxwell equations, which today can be solved numerically via increasingly powerful particle-in-cell (PIC) codes.However, since the simulations involve huge costs for each choice of the input data, exploring the data space blindly to single out interesting regions remains prohibitive.All analytical insights that can simplify the work, at least in special cases or in a limited space-time region, are welcome.Sometimes, good predictions can be obtained also by a hydrodynamic description (HD) of the plasma, i.e. treating it as a multicomponent (electron and ions) fluid, and by numerically solving the (simpler) associated hydrodynamic equations via multifluid (such as QFluid [30]) or hybrid kinetic/fluid codes; but in general it is not known a priori in which conditions, or spacetime regions, this is possible.
Here we summarize and slightly elaborate on a set of conditions [16] enabling a rather simple HD of the impact of a very short (and possibly very intense) laser pulse onto a cold diluted plasma at rest and the localization after the impact of the first wave-breakings (WBs) of the plasma wave (PW) [1,20] due to inhomogeneities of the initial density [5].Our analysis is based on a fully relativistic plane Lagrangian model [7,10,14] and very little computational power.We recall that small WBs are not necessarily undesired where the initial density decreases: they may be used [4] to inject and trap a small bunch of plasma electrons as test electrons in the PW trailing the pulse (self-injection), so that these undergo LWFA in the forward direction.The impact of very short laser pulses on suitable initial plasma profiles may allow also the slingshot effect [19,17,18], i.e. the backward acceleration and expulsion of (less) energetic electrons from the vacuum-plasma interface, during or just after the impact.
The plane model is as follows.One assumes that the plasma is initially neutral, unmagnetized and at rest with zero densities in the region z < 0.More precisely, the t = 0 initial conditions for the electron fluid Eulerian density n e and velocity v e are v e (0,x) = 0, n e (0,x) = n 0 (z), where the initial electron (as well as proton) density n 0 (z) fulfills for some n b > 0 (two examples of such densities are reported in fig.1).One assumes that before the impact the laser pulse is a free plane transverse wave travelling in the z-direction, i.e. the electric and magnetic fields E, B are of the form (given a vector w, we denote by w ⊥ its component ⊥ k ≡ ∇z), where the support of ϵ ⊥ (ξ) is a suitable interval [0, l] (ξ = 0 as the left extreme means that the pulse reaches the plasma at t = 0; l is constrained below).The input data of a specific problem are the functions n 0 (z), ϵ ⊥ (ξ); it is useful to introduce also the related functions −e, m are the electron charge and mass, c is the speed of light, K ≡ 4πe 2 mc 2 .By definition, v is dimensionless and nonnegative, N (z) strictly grows with z.When reached by the pulse, electrons start oscillating transversely (i.e. in the x, y directions) and drifting in the positive z-direction, respectively pushed by the electric and magnetic parts of the Lorentz force due to the pulse; thereafter, electrons start oscillating also longitudinally (i.e. in z-direction), pushed by the restoring electric force due to charge separation.We shall assume that the length l of the pulse makes the latter essentially short (ES) w.r.t. the density n 0 , in the sense of definition (13), implying that the pulse overcomes each electron before the z-displacement ∆ of the latter reaches a negative minimum for the first time.Most applications use slowly modulated monochromatic (SMM) waves where i = ∇x, j = ∇y, and the length λ = 2π/k of the carrier wave is much smaller than the length l of the support [0, l] of ϵ(ξ).Then α ⊥ (ξ) = −ϵ ⊥ o (ξ+λ/4)ϵ(ξ)/k up to terms O (λ/l) 2 (see appendix 5.4 in [10] for details), whence α ⊥ (ξ), v(ξ) ≃ 0 for ξ ≥ l.As we recall below, if v(ξ) ≪ 1 for all ξ then electrons keep nonrelativistic (NR); by Proposition 1 of [16], the pulse is ES if the modulation is symmetric about its center ξ = l/2 (i.e.ϵ(ξ) = ϵ(l−ξ)) and its duration l/c does not exceed the NR plasma oscillation period t nr H ≡ πm/n b e 2 associated to the maximum n b of n 0 (z), i.e. if (whence 4πe 2 mc 2 n b λ 2 ≪ 1, and the plasma is underdense).A general sufficient condition [16] for a pulse to be ES will be recalled in formula (24) below; it may be G b > 1.
One describes the plasma as a fully relativistic collisionless fluid of electrons and a static fluid of ions (as usual, in the short time lapse of interest here the motion of the much heavier ions is negligible), with E, B and the plasma dynamic variables fulfilling the Lorentz-Maxwell and continuity equations.Since at the impact time t = 0 the plasma is made of two static fluids, by continuity such a hydrodynamic description (HD) is justified and one can neglect the depletion of the pulse at least for small t > 0; the specific time lapse is determined a posteriori, by self-consistency (see e.g.[13]).This allows us to reduce (see [7,10], or [8,9,15,11] for shorter presentations) the system of Lorentz-Maxwell and continuity partial differential equations (PDEs) into ordinary ones, more precisely into a continuous family of decoupled Hamilton equations for systems with one degree of freedom.Each system rules the Lagrangian (in the sense of non-Eulerian) description of the motion of the electrons having a same initial longitudinal coordinate Z > 0 (the Z electrons, for brevity), and reads it is equipped with the initial conditions Here the unknowns ∆(ξ, Z), ŝ(ξ, Z) are respectivey the present longitudinal displacement and s-factor2 of the Z electrons espressed as functions of ξ, Z, while ẑe (ξ, Z) is the present longitudinal coordinate of the Z electrons; we express all dynamic variables f (t, Z) (in the Lagrangian description) as functions f of ξ, Z; f ′ stands for the total derivative Z plays the role of the family parameter.The lightlike coordinate ξ = ct−z in Minkowski spacetime can be adopted instead of time t as an independent variable because all particles must travel at a speed lower than c, see fig.1.c; at the end, to express the solution as a function of t, Z one just needs to replace everywhere ξ by the inverses ξ(t, Z) of the strictly increasing (in ξ) functions t(ξ, Z) ≡ (ξ + ẑe (ξ, Z))/c, with Z ≥ 0. All the electron dynamic variables can be expressed in terms of the basic ones ∆, ŝ and the initial coordinates X ≡ (X, Y, Z) of the generic electron fluid element.In particular, the electrons' transverse momentum in mc units is given by û⊥ = p⊥ /mc = eα ⊥ mc 2 , and v = û⊥2 .Ultra-intense pulses are characterized by max ξ∈[0,l] {v(ξ)} ≫ 1 and induce ultra-relativistic electron motions.Eq. ( 8) are Hamilton equations with ξ, ∆, −ŝ playing the role of the usual t, q, p and (dimensionless) Hamiltonian the first term gives the electron relativistic factor γ, while U plays the role of a potential energy due to the electric charges' mutual interaction.For ξ ≥ l eqs (8) are autonomous and can be solved also by quadrature, since the Hamiltonian Ĥ(ξ, Z) ≡ Ȟ ∆(ξ, Z), ŝ(ξ, Z), ξ; Z becomes h(Z) ≡ Ĥ(l, Z) = const.The solutions of (8-9) yield the motions of the Z electrons' fluid elements, which are fully represented by their worldlines (WLs) in Minkowski space.
In fig. 3 we display the projections onto the z, ct plane of a set of WLs for two specific sets of input data; as evident, the PW emerges from them as a collective effect.Mathematically, the PW features are derived passing to the Eulerian description of the electron fluid; the resulting flow is laminar with xy plane symmetry.The Jacobian of the transformation X → xe ≡ (x e , ŷe , ẑe ) from the Lagrangian to the Eulerian coordinates reduces to Ĵ(ξ, Z) = ∂ ẑe (ξ, Z)/∂Z, because x⊥ e −X ⊥ does not depend on X ⊥ .The HD breaks where WLs intersect, leading to WB of the PW.No WB occurs as long as Ĵ > 0 for all Z ≥ 0. If the initial density is uniform, then (8)(9), and hence also their solutions, are Z-independent, and Ĵ ≡ 1 for all ξ, Z. Otherwise, WB occurs after a sufficiently long time [5].
In section 2 we present upper and lower bounds on ŝ, ∆ [16] that provide useful approximations of these dynamic variables in the interval 0 ≤ ξ ≤ l.In section 3 we use these bounds to formulate sufficient conditions on the input data n 0 (z), ϵ ⊥ (ξ) guaranteeing that Ĵ(ξ, Z) > 0 for all Z > 0 and ξ ∈ [0, l], so that there is no wave-breaking during the laserplasma interaction (WBDLPI).These conditions are derived [16] with the help of a suitable Liapunov function and now can be more easily checked where n 0 is concave, thanks to the new results of Proposition 1 and Corollary 2. Qualitatively, n 0 (z) and/or its local relative variations must be sufficiently small.For ξ ≥ l, while ∆ and ŝ are periodic with a suitable period ξ H , Ĵ satisfies [13] (section 3) where a, b are periodic in ξ with period ξ H (Z), and b has zero average over a period.As b oscillates between positive and negative values, so does the second term, which dominates as ξ → ∞, with ξ acting as a modulating amplitude.Localizing WBs after the laserplasma interaction is best investigated via (11) [13].In section 4 we briefly compare the dynamics of ŝ, ∆, Ĵ induced by the same pulse on two different n 0 s having the same upper bound n b .Their behaviour for z ≃ 0 is crucial; WBDLPI can be excluded under rather broad conditions for typical LWFA experiments.We also comment on the spacetime region R where the model's predictions are reliable.Other typical phenomena of plasma physics (turbulent flows, diffusion, heating, moving ions,...) can be excluded inside R, but can and will occur outside.
for all Z ≥ 0. Clearly, a SS pulse is also ES.As we now see, ES pulses are recommendable because they allow useful apriori bounds on ŝ, ∆, Ĥ, Ĵ and thus simplify the control of the PW and its WB; moreover, a suitable ES pulse with l ∼ ξ2 (Z) maximizes the energy transfer from the pulse to the Z electrons [27,19].
, where M ≡ Kn 0 .In fig. 2 we plot a monochromatic laser pulse slowly modulated by a Gaussian and the corresponding solution (s, ∆).The qualitative behaviour of the solution remains the same also if n 0 (z) ̸ = const.The above functions simplify: ∆ (1) Hence, a lower bound ξ(1) 3 for ξ3 is the smallest ξ > 0 such that f (ξ) = 0, and the sufficient condition (24) ensuring that the pulse is ES boils down to f (l) ≥ 0. b) Corresponding solution of (8-9).As expected: ŝ is insensitive to the rapid oscillations of ϵ ⊥ ; for ξ ≥ l the energy Ĥ is conserved, and the solution is periodic.The pulse length l is determined on physical grounds; if e.g. the plasma is created locally by the impact of the front of the pulse on a gas (e.g.hydrogen or helium), then [0, l] consists of all points ξ where the pulse intensity is sufficient to transform the gas into a plasma by ionization.3 Hydrodynamic regime up to wave-breaking As said, the map xe (ξ, •) : X → x is invertible, and the HR is justified, as long as If Ĵ(ξ,Z) ≤ 0 then ẑe (ξ,Z ′ ) = ẑe (ξ,Z) for some Z ′ ̸ = Z, i.e. the layer of Z ′ electrons crosses the layer of Z electrons, and WB takes place.Let κ ≡ (1+v)/ŝ 3 .Differentiating (8) with respect to (w.r.t.) Z we find that ε, σ ≡ ∂ŝ/∂Z fulfill the Cauchy problem Differentiating the periodicity identity ẑe [ξ +nξ H (Z),Z] = ẑe (ξ,Z) w.r.t.Z, ξ yields [13] Ĵ so that (11) holds with b ≡ − ∆′ ∂ log ξ H ∂Z , a ≡ Ĵ − ξb.This is consistent [13] with Floquet theorem applied to (29).Known Ĵ, σ in [l, l+ξ H [ we extend them to all ξ ≥ l via (30).
If more realistically the pulse is not a plane wave, but cylindrically symmetric around ⃗ z with a finite spot radius R (which we assume to stay constant in the plasma, by self-focusing), then -by causality -our results hold strictly inside the causal cone (of axis ⃗ z, radius R) trailing the pulse, and approximately in a neighbourhood thereof, as far as the pulse is not significantly affected by its interaction with the plasma; for typical LWFA experiments this means travelling many l in the z-direction [13,12].

Figure 1 :
Figure 1: a), b): Examples of initial plasma densities of the type(2).In a) we also illustrate the meaning of the functions n u (z), n d (z) defined in(19).c): Projections onto the z, ct plane of sample particle worldlines (WLs) λ 1 , λ 2 in Minkowski space[14]; they intersect the support (pink) of a plane EM wave of total length l moving in the positive z direction.Since each WL intersects once every hyperplane ξ = const (beside every hyperplane t = const), we can use ξ rather than t as a parameter along it.While the t-instants of intersection with the front and the end of the EM wave (e.g.t 1i , t 1f for λ 1 ) depend on the particular WL, the corresponding ξ-intstants are the same for all WLs: ξ i = 0, ξ f = l.

Figure 2 :
Figure 2: a) Normalized amplitude of a linearly polarized [ψ = 0 in (6)] SMM laser pulse, modulated by a Gaussian with full width at half maximum l ′ and peak amplitude a 0 ≡ λeE ⊥ M /mc 2 = 1.3; this makes electrons moderately relativistic, and ∆ u ≡ ∆ (0) (l) ≃ 0.45l ′ .The pulse is ES w.r.t.n 0 (Z) ≡ n 0 = 4/Kl ′2 .b)Corresponding solution of (8-9).As expected: ŝ is insensitive to the rapid oscillations of ϵ ⊥ ; for ξ ≥ l the energy Ĥ is conserved, and the solution is periodic.The pulse length l is determined on physical grounds; if e.g. the plasma is created locally by the impact of the front of the pulse on a gas (e.g.hydrogen or helium), then [0, l] consists of all points ξ where the pulse intensity is sufficient to transform the gas into a plasma by ionization.Here instead we conventionally fix l = 4l ′ , what makes G b = √ Kn 0 l/2π ≃ 1.27.If l ′ = 7.5µm, λ = 0.8µm, then n 0 = 2 × 10 18 cm −3 and the peak intensity is I = 7.25×10 18 W/cm 2 ; these are typical values in LWFA experiments with Ti:Sapphire lasers.
Figure 2: a) Normalized amplitude of a linearly polarized [ψ = 0 in (6)] SMM laser pulse, modulated by a Gaussian with full width at half maximum l ′ and peak amplitude a 0 ≡ λeE ⊥ M /mc 2 = 1.3; this makes electrons moderately relativistic, and ∆ u ≡ ∆ (0) (l) ≃ 0.45l ′ .The pulse is ES w.r.t.n 0 (Z) ≡ n 0 = 4/Kl ′2 .b)Corresponding solution of (8-9).As expected: ŝ is insensitive to the rapid oscillations of ϵ ⊥ ; for ξ ≥ l the energy Ĥ is conserved, and the solution is periodic.The pulse length l is determined on physical grounds; if e.g. the plasma is created locally by the impact of the front of the pulse on a gas (e.g.hydrogen or helium), then [0, l] consists of all points ξ where the pulse intensity is sufficient to transform the gas into a plasma by ionization.Here instead we conventionally fix l = 4l ′ , what makes G b = √ Kn 0 l/2π ≃ 1.27.If l ′ = 7.5µm, λ = 0.8µm, then n 0 = 2 × 10 18 cm −3 and the peak intensity is I = 7.25×10 18 W/cm 2 ; these are typical values in LWFA experiments with Ti:Sapphire lasers.