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Unlimited Energy Gain in the Laser-Driven Radiation Pressure Dominant Acceleration of Ions

The energy of the ions accelerated by an intense electromagnetic wave in the radiation pressure dominated regime can be greatly enhanced due to a transverse expansion of a thin target. The expansion decreases the number of accelerated ions in the irradiated region increasing the energy and the longitudinal velocity of remaining ions. In the relativistic limit, the ions become phase-locked with respect to the electromagnetic wave resulting in the unlimited ion energy gain. This effect and the use of optimal laser pulse shape provide a new approach for great enhancing the energy of laser accelerated ions.


Introduction
The radiation pressure of a super-intense electromagnetic pulse on a thin quasi-neutral plasma slab has been proposed in Ref. [1] as an acceleration mechanism able to provide ultrarelativistic ion beams. In this radiation pressure dominant acceleration (RPDA) regime (also called the "Laser Piston" or the "Light Sail" regime), the ions move forward with almost the same velocity as the electrons and thus have a kinetic energy well above that of the electrons.
This acceleration process is highly efficient, with the ion energy per nucleon being proportional in the ultrarelativistic limit to the electromagnetic pulse energy. The idea of transferring momentum from light to macroscopic objects goes back to [2]. In the mid '50s of the last century ion acceleration by a high intensity electromagnetic wave incident on an electron cloud carrying a small portion of ions was considered by V.I. Veksler [3] for conditions when the ion acceleration occurs in the collective electric field which is produced due to the radiation pressure acting on the electron component. An analytical description of a charged particle dynamics under the radiation pressure can be found in Ref. [4] (chapter 9, problem 6), where a solution is obtained for the motion of a charge under the action of the average force exerted upon it by the wave scattered by it. There is an analogy between the RPDA mechanism and the "Light Sail" scheme for spacecraft propulsion. This scheme, which uses the photon momentum transfer to the light-sail, has been proposed by F. A. Zander in 1924 [5]. The use of lasers for propelling the light-sail over interstellar distances has been proposed in Ref. [6]. (for details and further discussions see Ref. [7]).
Recently the RPDA regime of laser ion acceleration has attracted great attention (e.g. see review article [8]). In Refs. [9,10] the stability of the accelerated foil has been analyzed. Refs. [11,12] are devoted to extending its range of operation towards lower electromagnetic wave intensities. The interaction of a high intensity laser pulse with extended plasmas in the RPDA (or 'Laser Piston') regime has been simulated in [13]. In Refs. [1,14] effects of the foil transparency are considered. A foil accelerated to relativistic energies by a laser pulse can act as a relativistic flying mirror for frequency up-shift and intensification of a reflected counter-propagating light beam [15]. An indication of the effect of the radiation pressure on bulk target ions is obtained in experimental studies of plasma jets ejected from the rear side of thin solid targets irradiated by ultraintense laser pulses [16].
While publications develop regimes of energy enhancement of the accelerated ions by exploiting the dependence on the pulse polarization of the laser interaction with matter [11] and target structuring [17], in the present paper we propose to use targets expanding transversally in order to increase the energy of accelerated ions. The transverse expansion of the accelerated shell can be provided by the action of the ponderomotive force of a laser pulse with a finite waist. It can also occur as a result of the instability described in Ref. [9].

Mathematical model
The nonlinear dynamics of a laser accelerated foil is described within the framework of the thin shell approximation first formulated by E. Ott [18] and further generalized in Refs. [9,19].
In the electromagnetic wave interaction with a thin foil, the latter is modeled as an ideally reflecting mirror. The equations of motion of the surface element of an ideally reflecting mirror in the laboratory frame of reference can be written in the form where , , , ν p P and σ are the momentum, light pressure, unit vector normal to the shell surface element, and surface density, nl σ = , respectively. Here n and l are the plasma density and shell thickness. The geometry is illustrated in Fig. 1. We determine the surface element s Δ as being carrying N s σ Δ = Δ particles, which is constant in time. We take the shell initially to be at rest, at 0 t = , in the plane 0 x = . In order to describe how its shape and position change with time it is convenient to introduce the Lagrange coordinates η and ζ playing the role of the markers of the shell surface element. The shell shape and position are given by the equation is the shell normalized velocity.

Ion acceleration in the expanding foil
We consider the case when the accelerated shell moves in the longitudinal direction with an ultrarelativistic velocity, i.
which corresponds to local maxima or minima in the ( ) x t , where ( ) x t is locally independent of η and ζ . In this case the right hand side of Eq.
Inserting these expressions into the longitudinal component of Eq. (6), we obtain the equation where 2 2 In the case of no expansion of the shell we have 0 0 y π = , 0 0 z π = , and 1 y z Λ = Λ = , and Eq. (12) gives (for a constant amplitude pulse) the following asymptotic time dependence of the particle momentum [1] for t → ∞ i.e.

1/ 3 x p t
∝ , which is similar to the dependence on time of the momentum of a charged particle accelerated by the light radiation pressure [4]. Here In the case of transverse expansion along the y-axis only ( 0 0 y π > , 0 0 . According to (11) the shell areal density decreases as As found in Ref. [10], in the case of transverse expansion ( 0 0 , 0 y z π π > ), in the limit t → ∞ , Eq. (9) yields 3/ 5 where ( ) Introducing the wave phase as a new independent variable ( ) we find that in the limit x p → ∞ Eqs. (16,17) yield We can see that in the case of a non-expanding shell, when the momentum dependence on time is given by Eq. (14), the integral in the right hand side of expression (18) diverges, while for an expanding shell it follows from expressions (16) and (17) that the phase shift between the laser pulse and the accelerated ions remains finite. For a long enough laser pulse the ions at the pulse axis become trapped inside the pulse with their energy formally growing unlimitedly at the expense of the particle number decrease. We notice here that the unlimited electron acceleration regimes are well known for the electrons accelerated by the electromagnetic wave in the cyclotron autoresonance regime [21], when electrons are trapped by electrostatic wave propagating perpendicularly to the magnetic field [22] and in inhomogeneous plasmas with a downgrading density [23].
Substituting the dependence of the ion momentum on time in the form ( ) (17), we obtain 2 1 where 2 1 ( , , ; ) F z α β γ is the Gauss hypergeometric function [20]. Asymptotically expression (19) yields for t → ∞ where ( ) z Γ is the gamma function [20]. If the power index k is larger than 1/ 2 , the first term in the r.h.s. of Eq. (16) tends to zero for t → ∞ and the phase ψ remains finite being equal to the second term. When 1/ 2 k = , Eq. (17) yields i. e. the phase ψ diverges logarithmically with time.
For the ion momentum dependence on time given by Eq. (16) 3/ 5 k = . In this case we In order to fulfill the ion trapping conditions the laser pulse duration must be larger than The possibility of reaching an unlimited ion energy that is formally allowed by the mirror model adopted here is actually limited to a finite value when transparency effects are included (see Refs. [1,14]). Two effects compete in determining the transparency of the accelerated and expanding foil: as the foil momentum increases, in its proper frame of reference the frequency of the laser pulse decreases in the ultrarelativistic regime proportionally to 1/ 2γ (i. e., from Eqs. (11,16) as 3/ 5 t − ), while the foil surface density decreases (as 4 / 5 t − ). At fixed dimensionless pulse amplitude, the foil transparency depends on the ratio between the foil surface density and the pulse frequency (see Ref. [24]). For the expanding foil this ratio tends to zero as 1/ 5 t − , making the foil transparent to the pulse radiation. For the shell expanding in only one transverse dimension, the shell can be always opaque for the incident laser pulse.
If the condition (22) is not fulfilled, i.e.

( ) ( )
where E ⊥ is the laser field in the co-moving with the mirror frame of reference, the ion acceleration develops according to the first acceleration stage noticed in Ref. [1]. In this case all the electrons are displaced in the longitudinal direction, which results in the formation of the charge separation electric field || 2 E enl π > between the electron and ion layers. In this longitudinal electric field the ion momentum grows as 2 2 x p e nlt π = .
In a time of the order of ( )( the ion energy reaches the value when the condition (22) becomes valid. Then the ion acceleration will follow the above considered RPDA scenario.
In order to find the optimal conditions for unlimited acceleration we consider this regime in more detail. We represent the laser pulse shape by the function ( ) E ψ in terms of the phase ψ given by Eq. (17). Searching out this function is the problem of optimization. Changing the independent variable from t to ψ , we write Eq. (12) in the form where we introduced a normalized intensity of the laser pulse Here for simplicity we assume that the shell expansion is isotropic, i. e. 0 0 0 y z π π π ⊥ = = and y z Λ = Λ = Λ , so that from Eqs (10) and (17) we have System of Eqs. (23,25) can be cast in the form .
where 2 0 g = and 3 g are constant determined by initial conditions The Weierstrass elliptic function 2 3 ( ,{ , }) u g g z ℘ = gives the value of z for which [20] 3 2 3 According to Eq. (28) The solution to Eq. (23) with the initial condition ( 0) 0 x p ψ = = gives a dependence of the momentum on the phase: Here 2  The ion acceleration can be effectively optimized by tailoring the laser pulse shape.
Assuming the laser pulse shape to depend as we obtain the exact solution of Eq. (26) Using this dependence and integrating Eq. (23) Integration of Eq. (17) yields the time dependence of the phase ( ) t ψ . It reads or, asymptotically for This results in the momentum time dependence where the power ( The momentum per unit surface of the shell asymptotically depends on time as (1 2 ) If the power 5 4 1 2 m − < < − , the momentum per unit surface grows in time with the same asymptotic as the r.h.s. of the opaqueness condition Eq. (22). This means that the radiation pressure of the laser pulse with the shape being a power of the phase can accelerate an expanding shell in such a way that the shell remains opaque for the laser radiation.

Nonrelativistic limit
In This efficiency is by a factor ( ) 2 / ex τ las t higher than in the case of non-expanding foil [1], which points towards a way for enhancing the efficiency of the fast ion generation required within the framework of the concept of Fast Ignition with laser accelerated ions [25].
As we see the efficiency enhancement requires the laser pulse duration to be larger than the foil expansion time. Assuming the expansion time to be of the order of the inverse growth rate of the Raleigh-Taylor instability, ( ) [9], with the wavelength of transverse perturbations equal to the inverse laser pulse waist, 2 / q w π ⊥ = , we find

PIC simulations of the radiation pressure acceleration of a mass limited target
As an illustration of the realization of the RPDA regime in the interaction of a high intensity laser pulse with an expanding plasma shell we present the results of two-dimensional (2D) particle-in-cell (PIC) simulations of the dynamics of a mass limited target (MLT) in a strong laser field. We use the PIC code REMP [26]. As well known the MLT irradiated by strong laser light can behave similarly to the cluster. The Coulomb explosion of laser irradiated clusters is considered to be one of the basic mechanisms of ion acceleration [27]. When clusters of sufficiently small size are irradiated by a strong electromagnetic pulse, the ions can be accelerated together with the electrons by the RPDA mechanism up to an energy substantially higher than the energy achievable in the case of pure Coulomb explosion.
In Figs The value 0 125 a = of the dimensionless amplitude of the laser pulse approaches the threshold above which it is necessary to take into account the influence of radiation friction on the dynamics of the electron component [28,29]. It was noted in Refs. [1,30] that, as the target is accelerated, the influence of radiation friction decreases. Intensity of radiation emitted by electron is given by 2 2 3 2 3 e dp dp e I m c ds ds where p μ is the electron 4-vector, 0,1, 2,3 In the circularly polarized electromagnetic wave, whose amplitude is equal to 0 E the electron energy losses in the rest frame are given by The electromagnetic wave can provide energy gain rate not higher than in the case of the circularly polarized electromagnetic wave, and for linear polarization. For details see Refs. [29,31]. Here M λ is the radiation wavelength in the rest frame of reference of the accelerated mirror, i.e. it is equal to respectively. For the relativistic factor γ corresponding to an energy of 10 GeV, the threshold value of the dimensionless amplitude of the electromagnetic wave is above 1000, which is more than eight times larger than the amplitude used in the above numerical simulations.
As it is seen in the simulation, the laser pulse radiation pressure compresses the target in the longitudinal direction. The reflected light wavelength increases since the incident wave interacts with the receding relativistic mirror, as it is seen in Fig. 2, where a superposition of the ion density and the z-component of the electric field in the ( , ) x y plane is shown for 112.5 2 / t π ω = × .  [32,33]). The main part of the electrons and ions is compressed in the longitudinal direction. Behind the compressed shell we see a low-density plasma cloud.
The particles with the highest energy are localized in the high density shell as it is seen in   . This makes the laser pulse to interact with a thin dense shell expanding in the transverse direction in a regime close to one discussed above.
In Fig. 6 we present the electron and ion energy and the maximum ion density versus time.
At the initial stage the ion density increases and then tends to zero. The electron and ion energies grow being of the same order of magnitude. From Fig. 6 we see that at time 600 2 / t π ω = × , when the accelerated shell approaches the right hand side boundary of the computation box, the protons reach the energy of 14 GeV and the electron energy is equal to 27 GeV. In the inset we show the ion energy spectrum at 600 2 / t π ω = × , which demonstrates the mono-energetic peak with the width of the order of 5%.  If we estimate the energy of the accelerated ions according to expression (11) for the simulation parameters assuming that no deformation of the target occurs, we find it to be of the order of 3 GeV. The enhancement of the fast ion energy is due to the target expansion in the transverse direction as discussed above.

PIC simulations of radiation pressure acceleration of ions in the laser pulse interaction with a thin foil target
In order to compare the ion acceleration by the laser pulse interacting with the mass limited target with the case when the laser interacts with a thin foil target we performed particlein-cell simulations of the interaction of a circularly polarized electromagnetic pulse with a thin overdense plasma target consisting of electrons and protons (see also [29,30]). The target has the form of an ellipsoid in the ( , ) x y plane with horizontal and vertical semi-axes equal to 1λ and 50λ . It is initially localized at 50 , 0 x y λ = = . The dimensionless amplitude of the laser pulse is 125 a = , which corresponds to a laser intensity of The target is made of hydrogen plasma with the proton to electron mass ratio equal to 1836. The electron density corresponds to the ratio / 6 pe ω ω = .
Due to the interaction with the laser pulse, the target changes its shape and transforms into a cocoon, which traps the electromagnetic wave [Figs. 7 (a) -(d)], thereby providing ion acceleration over a distance longer than the Raleigh length. At the cocoon front a dense plasma clump is formed as was noticed in Ref. [9]. The clump propagates with a relativistic velocity. As a result, the coefficient of reflection of the electromagnetic wave from the supercritical plasma layer moving with a velocity approaching the speed of light progressively increases. It can be seen from Fig. 8 (b) that, at the instant of 112.5 2 / t π ω = × , the maximum proton energy is about 2.5 GeV, whereas the electron energy is about 2 GeV.   At later time the accelerating part of the foil target evolves to a transversely expanding shell, similarly to a mass-limited target (see Fig. 9, where the same quantities as in Fig. 7 are shown but for the case of the foil and for 250 2 / t π ω = × ). The laser pulse transforms the target into a new one thus changing the regime of its interaction, closer to the process described by our model and demonstrated above in the simulations with MLT. At the instant 250 2 / t π ω = × , the maximum proton energy is about 8.5 GeV.

Conclusion
In conclusion, the transverse expansion of a thin shell accelerated in the RPDA regime results in the increase of the ion energy and the acceleration efficiency at the expense of decreasing number of particles. In the relativistic limit, the ions become phase-locked with respect to the electromagnetic wave, which is the indication of an unlimited acceleration. This effect and the use of optimal laser pulse shape provide a new approach for great enhancing the energy of laser accelerated ions. Carried out [16] and forthcoming laboratory experiments on the laser plasma interaction in the RPDA regime will contribute to the development of the laboratory astrophysics discipline [34,35] and to studying of the "Light Sail" mechanism for spacecraft propulsion.