The slingshot effect: A possible new laser-driven high energy acceleration mechanism for electrons

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Tajima

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Dawson

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Gorbunov

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Kirsanov

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M. J. H.

 

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A. G.

 

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Faure

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Glinec

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Pukhov

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Kiselev

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Kalmykov

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Shvets

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 et al.,  , ().

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 et al.,  , – ().

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Korn

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Eremin

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Kim

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Korzhimanov

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A. V.

 

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A. M.

 

Sergeev

,  , ().

$τ=τ1+τ2$⁠, with $τ1,τ2$ the time lapses spent during the oscillation respectively in the domain of the elastic and of the constant force; due to the size of the domains it is $τ1≥T Hnr/2$⁠. Hence, τ depends on Z and on the oscillation amplitude, but is at the least of the order of $T Hnr$⁠.

The longitudinal equation of motion of the $Z=0$ electrons is $dpz̃dt=m[Fpz (ct−ze)−ωp2ze], Fpz (ξ) ≡ (eλ)2η∂ξϵs2(ξ)8(mπ)2c3γ(ξ),$ where $η=1,1/2$⁠, respectively, for circular, linear polarization, and $ωp2=4πe2n0/m$ is the square of the plasma frequency. In the nonrelativistic regime we find $γ̃≃1, ze(t) ≪ ct$⁠, $p̃z≃mże$⁠, hence the maximum of $ϵs2(ξ)$ and the end of the pulse reach these electrons respectively at $t≃t¯=ξ0/c$⁠, $t≃l/c$⁠, and the equation of motion in the bulk reduces to that of a forced harmonic oscillator: $z̈e=Fpz(ct)−ωp2ze$⁠; the solution with initial conditions $ze(0)=że(0)=0$ is $ze(t)=∫0t dt′Fpz(ct′) sin[ωp(t−t′)]/ωp$⁠. Electrons initially located at small $Z>0$ will experience approximately the same longitudinal displacement with a time delay Z/c. $F̃pzṽz=Fpzże$ keeps nonnegative if during the pulse $że(t)=∫0t dt′Fpz(ct′) cos[ωp(t−t′)]$ switches from positive to negative at $t=t¯=ξ0/c$ (only). In particular, it must be $ze(t)≥0$ for all $t≤l/c$⁠, and $ξ0,n0,l$ must fulfill $π/4<ξ0ωp/c<3π/4$⁠. If the pulse is so intense to make electrons relativistic the conditions for maximal slingshot loading are more complicated; for an estimate in the proposed experimental conditions see the comments after (A7).

In (6), we have chosen the additive constant so that the minima of U w.r.t. z_e are equal to zero for all Z.

To perform the work integral, we have used the relation $z2+R2=12∂z [ zz2+R2+R2sinh−1 zR ]$⁠. We have chosen the additive constant equal to the last three terms, so that $UR(0−, Z)=U(0+, Z)$⁠.

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R. A.

 

Fonseca

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As $∂T H/∂ζ|ζ=0=0$⁠, only in the nonrelativistic limit $TH≃ζ→0 T H nr=π/cK=πm/e2n0$ is independent of ζ (nonrelativistic harmonic oscillator).

In fact, (B3) implies $t<(z′+|x−x′|)/c$ for all $x′ ∉ CR$⁠; $z′/c$ is the time when the $R=∞$ pulse reaches $x′, |x−x′|/c$ is the time lapse necessary for a EM signal to travel from $x′$ to $x$⁠, hence their sum is the time when the information that the charge distribution does not extend indefinitely in the xy plane, but is confined in $CR$⁠, arrives at $x$⁠.

In fact, by the assumptions we find $−2ctz′≤−2zz′$⁠, $(ct−z′)2=(ct)2−2ctz′+z′2≤R2+z2−2zz′+z′2=R2+(z−z′)2$ and, taking the square root, $ctr≤z′$⁠.

In fact, $eẼρ$ decreases as Z grows and when the electrons penetrate inside $CR$⁠, as correspondingly the negative contribution due to the electrons displaced beyond $S0(t)$ grows, while the positive one by the ions decreases.