Effect of Electromagnetic Pulse Transverse Inhomogeneity on the Ion Acceleration by Radiation Pressure
K. V. Lezhnin, F. F. Kamenets, V. S. Beskin, M. Kando, T. Zh. Esirkepov, S. V. Bulanov
EEffect of Electromagnetic Pulse Transverse Inhomogeneityon the Ion Acceleration by Radiation Pressure
K. V. Lezhnin, F. F. Kamenets, V. S. Beskin,
1, 2
M. Kando, T. Zh. Esirkepov, and S. V. Bulanov
1, 3, ∗ Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Moscow Region, 141700, Russia Russian Acad. Sci., P. N. Lebedev Phys. Inst., Leninskii Prosp 53, Moscow 119991, Russia Japan Atomic Energy Agency, Kansai Photon Science Institute,8-1-7 Umemidai, Kizugawa-shi, Kyoto, 619-0215 Japan (Dated: August 10, 2018)In the ion acceleration by radiation pressure a transverse inhomogeneity of the electromagneticpulse results in the displacement of the irradiated target in the off-axis direction limiting achievableion energy. This effect is described analytically within the framework of the thin foil target modeland with the particle-in-cell simulations showing that the maximum energy of accelerated ionsdecreases while the displacement from the axis of the target initial position increases. The resultsobtained can be applied for optimization of the ion acceleration by the laser radiation pressure withthe mass limited targets.Keywords: Relativistic laser plasmas, Ion acceleration, Radiation pressure
PACS numbers: 52.38.Kd, 52.65.Rr
INTRODUCTION
Studies of the high energy ion generation in the in-teraction between an ultraintense laser pulse and a smalloverdense targets, are of fundamental importance for var-ious research fields ranging from the developing the ionsources for thermonuclear fusion and medical applica-tions to the investigation of high energy density phenom-ena in relativistic astrophysics (see review articles [1–7]and the literature cited therein).Theory and experiments on laser acceleration can clar-ify the basic features of particle acceleration in astro-physical objects. Indeed, according to common point ofview, activity of radio pulsars, active galactic nuclei, andeven gamma-bursters connects with the highly magne-tized wind in which electric field is approximately equalto magnetic one [8]. Charged particles produced in sucha wind can get the energy of the order of m α c γ w thatis much higher than the energy of the wind ( ≈ mc γ w )[9, 10]. Here γ w is the Lorentz-factor associated withthe wind velocity. Moreover, interacting with the exter-nal environment (the companion star in a close binarysystem, the current sheets in the pulsar wind), a regionwhere the electric field is greater than the magnetic canform where therefore the acceleration of particles can beeven more effective [12].Depending on the laser and target parameters differ-ent regimes of acceleration appear – from accelerationat the target surface called the Target Normal SheathAcceleration (TNSA) [13, 15, 16] through the Coulombexplosion [17–20] to radiation pressure dominance accel-eration (RPDA) regime [21–24]. The ion accelerationregimes are shown in the plane of the laser intensity –the surface density n e l of the target in Fig. 1 (see also[6]). Here n e is the electron density in the target and l is its thickness. At the intensity above 10 W / cm theplasma electron energy becomes relativistic. The dashedline, is given by the formula a = n e r e λl , (1)where a = eE /m e ωc is the normalized laser pulse am-plitude, ω and λ = 2 πc/ω are the laser frequency andwavelength, respectively, and r e = e /m e c = 2 . × − cm is the classical electron radius; m e and e are theelectron mass and charge, and c is the speed of light invacuum. This line separates the intensity – surface den-sity plane into two domains. In the domain below the linethe plasma is opaque and above it is transparent for thelaser radiation [25]. When the laser radiation interactswith the opaque target a relatively small portion of hotelectrons can escape forming a sheath with strong elec-tric charge separation electric field where the accelerationoccurs in the TNSA regime. Above the dashed line thelaser radiation is so intense that it blows out almost allelectrons from the target irradiated region. The remain-ing ions undergo fast expansion, the Coulomb explosion,due repelling of noncompensated electric charges. At theopaqueness-transparency threshold, in the vicinity of thedashed line in Fig. 1 the optimal conditions for the ionacceleration in the RPDA regime are realized [22, 27].A fundamental feature of the RPDA acceleration pro-cess, proposed by Veksler [21], is its high efficiency, asthe ion energy per nucleon turns out to be proportionalin the ultrarelativistic limit to the electromagnetic pulseenergy. As far it concerns the experimental evidence ofthe RPDA mechanism there are indications on its real-ization in the laser thin foil interaction reported in Refs.[29–31].The usage of a finite transverse size target, it is calledthe Mass Limited Target (MLT) or the Reduced Mass a r X i v : . [ phy s i c s . p l a s m - ph ] D ec FIG. 1. The ion acceleration regimes in the plane of the laserintensity [W/cm ]–the surface density n e l of the target. Target [32–38], including the cluster targets [18, 39], pro-vide a way for enhancement of the ion energy and accel-eration efficiency and a way for high brightness X-raygeneration [40]. The irradiation of MLT by enough highintensity lasers is one of the most perspective approachesto develop compact ion accelerators [28, 41].In the present paper, we discuss the RPDA regime un-der the conditions when a transverse inhomogeneous laserpulse irradiates a MLT positioned not precisely at thelaser pulse axis. This situation natually occurs due to afinite pointing stability of the laser systems. As a resultthe transverse component of the radiation pressure leadsto the displacement of the irradiated target in the off-axisdirection. Apparently, after a finite interval of time thetarget leaves the laser pulse preventing from the furtherion acceleration. Below on the ground of a theoreticalmodel of relativistic mirror [22, 28, 41] we calculate theacceleration time and hence the achieved ion energy de-pendence on the laser pulse amplitude and transversesize and on the initial displacement of the target fromthe laser axis. According to recently published papers,various instabilities of the target plasma appear in theRPDA regime, for instance, the Rayleigh-Taylor-like in-stability [43] leads to the target modulation forming thelow density bubbles and high density clumps resulting inthe broadening of the accelerated ion energy spectrum.In order to elucidate the kinetic, nonlinear and instabil-ity effects we carry out the PIC simulations of the finitewaist laser pulse interaction with the MLT by using theREMP code [44].
DYNAMICS OF THE MASS LIMITED TARGETPOSITIONED SLIGHTLY OFF-AXISThe Equations of Motion
We describe the nonlinear dynamics of a laser acceler-ated target within the framework of the thin shell approx-imation formulated by Ott [45] and further generalizedon the 3D geometry in Refs. [46, 47] and extended to therelativistic case in Refs. [41, 43]. In a way of Refs. [41–43] here we derive of the mo-tion equations required for further consideration of theMLT dynamics. The equations of motion of the surfaceelement of a thin foil target in the laboratory frame ofreference can be written in the form d p dt = P νσ , (2)where p , P , ν , and σ are the momentum, light pressure,unit vector normal to the shell surface element, and sur-face density, σ = nl , respectively. Here n and l are theplasma ion density and shell thickness. We determinethe surface element ∆ s as carrying ∆ N = σ ∆ s particles,with ∆ N constant in time. We take the shell initiallyto be at rest, at t = 0, in the plane x = 0. In orderto describe how its shape and position change with timeit is convenient to introduce the Lagrange coordinates η and ζ playing the role of the markers of the shell surfaceelement. The shell shape and position are given by theequation M = M ( η, ζ, t ) ≡ { x ( η, ζ, t ) , y ( η, ζ, t ) , z ( η, ζ, t ) } . (3)At a regular point, the surface area of a shell element andthe unit vector normal to the shell are equal to ν ∆ s = ∂ η M × ∂ ζ M d η d ζ (4)and ν = ∂ η M × ∂ ζ M | ∂ η M × ∂ ζ M | , (5)respectively (see e.g., [48]). The particle number con-servation implies σ ∆ s = σ ∆ s , where σ = n l . Thisyields σ = σ | ∂ η M × ∂ ζ M | . (6)Using these relationships and representing the coordi-nates x i as x = ξ x ( η, ζ, t ) , (7) y = η + ξ y ( η, ζ, t ) , (8) z = ζ + ξ z ( η, ζ, t ) (9)with initial conditions: ξ i ( η, ζ,
0) = 0 and ˙ ξ i ( η, ζ,
0) = v i ( η, ζ, σ ∂ t p x = P (1 + ∂ η ξ y + ∂ ζ ξ z + { ξ y , ξ z } ) , (10) σ ∂ t p y = P ( − ∂ η ξ x + { ξ z , ξ x } ) , (11) σ ∂ t p z = P ( − ∂ ζ ξ x + { ξ x , ξ y } ) , (12) ∂ t ξ i = c p i ( m α c + p k p k ) / , (13)Here m α is the ion mass, i = 1 , ,
3, and summation overrepeated indices is assumed, { ξ j , ξ k } = ∂ η ξ j ∂ ζ ξ k − ∂ ζ ξ j ∂ η ξ k (14)are Poisson’s brackets. This form of the equations is par-ticularly convenient for analysing small but finite dis-placement of the target elements from the axis.The radiation pressure on the shell exerted by a circu-larly polarized electromagnetic wave propagating alongthe x -axis with amplitude E = E ( t − x/c ) is P = K E π (cid:18) − β x β x (cid:19) , (15)where β x = p x ( m α c + p x ) − / is the shell normalizedvelocity in the x -direction. The coefficient K equal to K = 2 | ρ | + | α | (16)depends on | ρ | and | α | which are the light reflection andabsorption coefficients, respectively (see also Ref. [50]).Effects of the reflection coefficient dependence on the ionenergy due to the relativistic transparency has been dis-cussed in Refs. [27, 51]. Below we shall not consider therelativistic transparency effects assuming ideally reflect-ing light target with K = 2.We note here that in Eqs. (10–12) there is no a forceacting between the target surface elements, i. e. wecan consider a finite transverse size MLT for which theLagrange coordinates η and ζ belong to a finite domain: η ∈ [ η , η ] and ζ ∈ [ ζ , ζ ].For homogeneous laser pulse, E =constant, the flatMLT is accelerated along the x -axis with p y = 0, p z = 0, ξ y = 0, and ξ z = 0. The ion momentum and displace-ment in the x -direction are given by dependences on time[22]: p (0) x ( t ) = m α c (cid:18) tt / (cid:19) / , (17) ξ (0) x ( t ) = ct − ct / / t / , (18)where the characteristic time is t / = 8 πσ m α c E . (19)Here we have assumed that the target energy is ultrarel-ativistic, p (0) x /m α c (cid:29) t las , laser pulse accelerates theions up to the energy E = m α c γ max with the gamma-factor given by γ max = E t las πσ m α c . (20) FIG. 2. Laser pulse LP and the target T at t = 0 and t = t acc According to Eq. (17) the acceleration time, t acc , canbe defined via γ max = ( t acc /t / ) / . We find it takinginto account that the laser pulse rear reaches the targetat t = t acc , as it is illustrated in Fig. 2. The accelerationtime is determined by equation t las = (cid:90) t acc (cid:18) − v ( t ) c (cid:19) dt ≈ (cid:90) t acc dtγ ( t ) dt (21)This and Eq. (20) yield t acc = 23 γ max t las . (22) The Mass Limited Target Irradiated by GaussianLaser Pulse
In order to analyse the transverse motion of the MLTirradiated by the laser pulse we consider the pulse whoseenvelope has a Gaussian form, E ( y, z ) = E exp (cid:18) − y l y − z l z (cid:19) (23)with the laser pulse width equal to l y and l z in the y -and z -direction, respectively.Assuming a smallness of the transverse displacement, ξ y (cid:28) η , ξ z (cid:28) ζ , and considering the near-axis region, η (cid:28) l y , ζ (cid:28) l z , we obtain from Eqs. (10–12) the lin-earized system of equations, ∂ t (cid:16) ( γ (0) ( t )) ∂ t ξ (1) x (cid:17) = c ( γ (0) ( t )) t (0)1 / (cid:18) ∂ η ξ (1) y + ∂ ζ ξ (1) z − η l y − ζ l z (cid:19) , (24) ∂ t (cid:16) γ (0) ( t ) ∂ t ξ (1) y (cid:17) = − c ( γ (0) ( t )) t (0)1 / ∂ η ξ (1) x , (25) ∂ t (cid:16) γ (0) ( t ) ∂ t ξ (1) z (cid:17) = − c ( γ (0) ( t )) t (0)1 / ∂ ζ ξ (1) x (26)with given dependence on time of the ion gamma-factor γ (0) ( t ) = tt (0)1 / / , (27)which is found within the framework of the 1D modelof the RPDA thin foil acceleration [22]. The approachused corresponds to so-called betatron approximationwell known in the theory of standard accelerators ofcharged particles [52]. In these expressions the charac-teristic time is t (0)1 / = 8 πσ m α c/ E .In order to find the solution to the system of equationsin partial derivatives (24–26) we use the anzatz ξ (1) x ( η, ζ, t ) = Ξ x ( t ) + Ξ xηη ( t ) η + Ξ xζζ ( t ) ζ , (28) ξ (1) y ( η, ζ, t ) = Ξ yη ( t ) η, (29) ξ (1) z ( η, ζ, t ) = Ξ zζ ( t ) ζ, (30)which is a self-similar solution reducing Eqs. (24–26) toordinary differential equations for the functions Ξ x ( t ),Ξ xηη ( t ), Ξ xζζ ( t ), Ξ yη ( t ), and Ξ zζ ( t ): ddτ (cid:18) γ (0) d Ξ xηη dτ (cid:19) = − l y , (31) ddτ (cid:18) γ (0) d Ξ xζζ dτ (cid:19) = − l z , (32) ddτ (cid:18) γ (0) d Ξ yη dτ (cid:19) = − xηη , (33) ddτ (cid:18) γ (0) d Ξ zζ dτ (cid:19) = − xζζ , (34) ddτ (cid:18) γ (0) d Ξ x dτ (cid:19) = Ξ yζ + Ξ zζ . (35)We introduced a new independent variable equal to τ = ct (0)1 / / (cid:90) t dt ( γ (0) ( t )) ≈ c / ( t (0)1 / ) / t / . (36)For initial conditions ξ (1) x ( η, ζ,
0) = 0 and ˙ ξ (1) x ( η, ζ,
0) =0, solution to Eqs. (31–35) readsΞ xηη = − ctl y (cid:18) t / t (cid:19) / , Ξ xζζ = − ctl z (cid:18) t / t (cid:19) / , (37)Ξ yη = 81( ct ) l y (cid:18) t / t (cid:19) / , Ξ zζ = 81( ct ) l z (cid:18) t / t (cid:19) / , (38)Ξ x = − ct ) (cid:18) t / t (cid:19) / (cid:18) l y + 1 l z (cid:19) . (39)As it is seen from Eqs. (29) and (38) the target elementwith initial coordinates η and ζ moves in the transversedirection with the displacement proportional to t / . Wecan estimate the time required to leave the region withstrong laser field as δt ⊥ = (cid:18) l ⊥ δr (cid:19) / c / t / / (40) with l ⊥ = min { l y , l z } and δr = max { η, ζ } . Accordingto Eqs. (20) and (40) the achieved ion energy is of theorder of E α = m α c (cid:18) δt ⊥ t / (cid:19) / , (41)which implies δt ⊥ < t acc . The opposite case realized forsmall enough initial position of the MLT centroid, δr ,and/or wide enough laser pulse corresponds to the perfectlaser-target alignment.Using obtained above relationships we can write thecharacteristic time t / as t / = 2 ω pe ω m α m e l c a , (42)which for the solid density target, ω pe /ω ≈ , of thethickness l = 0 . µ m for the laser intensity of the orderof 10 W/cm corresponding to a = 300, m α = m p ,yields t / ≈ . m α c ( t las / t / ) for 100 fs laser pulse duration is about20 GeV with the acceleration time given by Eq. (22)equal to 10 ps. The perfect alignment condition implies t ⊥ > t acc . Super Gaussian Laser Pulse Interaction with MassLimited Target
Here we analyse the case when the laser pulse when itsenvelope has Super-Gaussian form, E ( y, z ) = E exp (cid:18) − y l y (cid:19) , (43)with the index equal to 4. For the sake of brevity we con-sider two-dimensional geometry. Generalization to the3D case is straightforward.For small transverse displacement, ξ y (cid:28) η , in the near-axis region, η (cid:28) l y , within the framework of the betatronapproximation the target dynamics is described by thelinearized system of equations, ∂ τ (cid:16) γ (0) ∂ τ ξ (1) x (cid:17) = ∂ η ξ (1) y − η l y , (44) ∂ τ (cid:18) γ (0) ∂ τ ξ (1) y (cid:19) = ∂ η ξ (1) x (45)with the independent variable τ defined by Eq. (36) andthe ion gamma-factor γ (0) given by Eq. (27).The self-similar solution to Eqs. (44–45) has a form ξ (1) x ( η, τ ) = Ξ x ( τ ) + Ξ xηη ( τ ) η + Ξ xηηηη ( τ ) η , (46) ξ (1) y ( η, τ ) = Ξ yη ( τ ) η + Ξ yηηη ( τ ) η . (47)Substituting these functions to Eqs. (44–45) we obtainordinary differential equations: ddτ (cid:18) γ (0) d Ξ xηηηη dτ (cid:19) = − l y , (48) ddτ (cid:18) γ (0) d Ξ yηηη dτ (cid:19) = − xηηηη , (49) ddτ (cid:18) γ (0) d Ξ xηη dτ (cid:19) = 3 Ξ yηηη , (50) ddτ (cid:18) γ (0) d Ξ yη dτ (cid:19) = − xηη , (51) ddτ (cid:18) γ (0) d Ξ x dτ (cid:19) = Ξ yη . (52)For zero initial conditions for the displacement ξ (1) i ( η,
0) = 0 and its time derivative ˙ ξ (1) i ( η,
0) = 0, solu-tion to Eqs. (48–52) readsΞ xηηηη = − ctl y (cid:18) t / t (cid:19) / , Ξ yηηη = 81( ct ) l y (cid:18) t / t (cid:19) / , (53)Ξ xηη = 4374( ct ) l y (cid:18) t / t (cid:19) / , Ξ yη = − ct ) l y (cid:18) t / t (cid:19) / , (54)Ξ x = − ct ) l y (cid:18) t / t (cid:19) . (55)As it follows from expressions (44–45) and (53–55) thetarget is deformed in such the way that the peripheryexpands and the near-axis region contracts. This para-doxical behaviour can be explained by the fact that dueto the density decreasing in the peripheral regions thetarget elements there move forward faster modulating thefoil curvature, which results in contraction of the near-axis elements, which is distinctly seen in Fig. 3. Thelongitudinal, along the x -axis velocity has two maxima,the transverse, y -component velocity gradient is positiveat large y , which corresponds to the foil expansion, andit is negative near the axis corresponding to the foil com-pression. RESULTS OF PARTICLE-IN-SELLSIMULATIONS
Theoretical analysis of the target off-axis displacementeffects has been carried within the framework of the lin-earized model equations (24–26). In order to take intoaccount the nonlinear and kinetic effects, the target de-formation and instability we have conducted a series of2D-PIC simulations using the two-dimensional version ofrelativistic electromagnetic code REMP [44].The simulation box is 300 λ × λ with mesh resolutionof 20 cells per wavelength. The total number of particles FIG. 3. Thin target deformation by the super-Gaussian laserpulse. The curves x , v x and v y show the target position and x - and y -components of the target element velocity in the x, y -plane for l y /ct / = 2 at t/t / = 0 .
5. Inset: close-up ofthe near-axis region.FIG. 4. a) and b) Distribution of the x - and z - components ofthe electric field; c) and d) of the electron and ion density inthe ( x, y ) plane; e) and f) phase planes ( x, p x ) of the electronsand ion ions, respectively, at t = 100. The initial off-axisdisplacement equals δy = 0 . is equal to 7 × . The target has the form of an ellipsoidin the ( x, y ) plane with horizontal and vertical semiaxesequal to 1 λ and 3 . λ . It is initially located at x = 50 λ inthe near axis region with its y -coordinate varying from 0 λ to 1 . λ . The target comprises of hydrogen plasma withproton-to-electron mass ratio equal to 1836. The electrondensity corresponds to the ratio ω pe /ω = 10. A circularlypolarized laser pulse is excited in the vacuum region atthe left-hand side of the computation domain. The laserpulse has a Gaussian shape with a length of l x = 20 λ and l y = 25 λ , and with dimensionless amplitude a = eE/m e ωc varies from to 250 to 325. Under the simulationconditions, the accelerated ion energy according to Eq.(20) is equal to 4.5 GeV. The acceleration length l acc = ct acc is equal to 135 λ .The aim of the PIC simulations is to investigate thedependence of the energy of accelerated ions on the initialdisplacement of the target along the y -axis.In Figs. 4 a) - d) we present electromagnetic field andelectron and ion density distribution in the ( x, y ) plane at t = 100. Here and below the laser period 2 π/ω and wave-length λ are time and space units. Fig. 4 b), with thedistribution of the z -component of the electromagneticfield in the ( x, y ) plane, shows the laser pulse reflectionfrom the receding with relativistic velocity target. Dueto the double Doppler effect the wavelength of the re-flected light is substantially longer than the wavelengthof the incident radiation. The laser field interaction withthe plasma target is accompanied by the high order har-monics radiation distinctly seen in the short-wavelengthscattered radiation. The up-down asymmetry of E x ( x, y )appears due asymmetry of the initial position of the tar-get with respect to the laser pulse axis. There also itcan be seen a strong longitudinal quasistatic (long wave-length) electric field formed at the rear side of the tar-get. In this field the positively charged ion accelerationoccurs. As it follows from Figs. 4 c) and d), where theelectron and ion density distribution in the ( x, y ) planeis shown, the electrons pushed forward by the laser radi-ation move almost together with the ions pulled by theelectric charge separation electric field. In Figs. e) and f)we present the phase planes ( x, p x ) of the electrons andion ions, respectively, which demonstrate that the high-est energy electrons and ions are localised in the sameregion.In the process of nonlinear interaction with the MLTthe laser pulse becomes modulated in the transverse di-rection as we can see in Fig. 4 b). This makes the in-teraction with the target of initially Gaussian pulse tobe similar to that of the super-Gaussian pulse. As a re-sult, the dependences of the x - and y -components of theion and electron momentum on the y -coordinate shownin Fig. 5 are in qualitative agreement with theoreticalcurves in Fig.3. Here it is possible to see a character-istic double maximum profile in the ion distribution inthe ( y, p x ) plane. The ( y, p y ) distribution clearly shows FIG. 5. a) and b) Electron phase planes ( x, p x ) and ( y, p y )and c) and d) of the ions, at t = 100 for initial y coordinateequal to 0 . the target expansion at its perifery and the contractionin the near-axis region.In Figs. 6 a), b), where we plot distribution of theelectron and ion density in the ( x, y ) plane at t = 250,we see, although the target is strongly deformed and dis-placed in the vertical direction, the ions and electrons aremostly localized in the same region. The c) and d) framespresent the phase planes ( x, p x ) of the electrons and ionions with the insets showing the electron and ion energyspectra, respectively. The electron component has a flatenergy distribution with the maximum energy of the or-der of 8 GeV. The accelerated ion energy distributionshows a relatively narrow, approximately of 20%, peakat the energy of the order of 4 GeV. The ion phase plane( y, p x ) and ion phase plane ( y, p y ) in Figs. 6 e) and f)demonstrate that the high energy ions remain localizedin the near-axis region.Dependence of the accelerated ion energy, E α , on thetarget initial position, δr , is presented in Fig. 7 fordifferent laser pulse amplitude. At the simulation con-ditions the accelerated ions reach their maximum energyat the time approximately equal to 250 fs, so all of thegraphs are presented at that moment of time. Here weplot the theoretical curves (dashed lines) calculated byusing Eqs. (40–42) and the energy value obtained insimulations (dots in color). The theoretical dependenceof the ion energy on the inititial target position followsfrom Eqs. (40) and (41). It reads E α,δr = m α c / l / ⊥ c / t / / δr / . (56)This expression is valid in the limit of substantially large δr . When δr → FIG. 6. a) and b) Distribution of the electron and ion den-sity in the ( x, y ) plane; c) and d) phase planes ( x, p x ) of theelectrons and ion ions (the insets show the electron and ionenergy spectra), respectively; e) ion phase plane ( y, p x ); f) ionphase plane ( y, p y ), at t = 250 for initial y coordinate equalto 0 . parently, the ion energy from the off-axis localized targetcannot be larger that the ion energy in the case of thetarget positioned exactly on the axis, E α,max . In orderto take this into account we shall use the interpolationformula 1 E sα = 1 E sα,max + 1 E sα,δr (57)with the fitting parameter s >>
1. In the limit of small δr the ion energy is equal to E α,max . For large initialvertical coordinate it is proportional to δr − / accortdingto Eq. (56). In Fig. 7 we plot the normalized ion energy γ p achieved with the MLT initially shifted in the verti-cal direction versus the initial target coordinate δy fordifferent amplitudes of the Gaussian laser pulse The plotmarkers are the 2D PIC simulation results and the curvescorrespond to theoretical dependences given by Eq. (57)for 1. a = 325; 2. a = 300; 3. a = 275; 4. a = 250).For small the initial target coordinate the ion energydecreases with δy more slowly than it is predicted bythe theory due to self-modulation of the laser pulse intransverse direction, which is distinctly seen in the elec- FIG. 7. Normalized proton energy gained with the MLT ini-tially shifted in the vertical direction versus the initial targetcoordinate for different amplitudes of the Gaussian laser pulse(1. a = 325; 2. a = 300; 3. a = 275; 4. a = 250). The plotmarkers are the simulation results and the curves correspondto theoretical dependences. tromagnetic field didstribution in Fig. 4 b). The laserpulse self-modulation prevents the target from sleapageout off the acceleration phase providing the fast ion colli-mation seen in Fig. 6 e). As it follows from dependencespresented in Fig. 7, the laser pulse modulation effectsare significant for δy < . µ m. CONCLUSIONS AND DISCUSSIONS
We have studied the effects of the laser pulse trans-verse inhomogeneity on the ion acceleration in the RPDAregime. Within the framework of a thin foil approxima-tion we found the dependence of the accelerated ion max-imum energy on the off-axis displacement of the mass lim-ited target for Gaussian and super-Gaussian laser pulseprofiles. When the target is irradiated by the Gaussianlaser pulse it is pushed away from the pulse by the pon-deromotive pressure of electromagnetic radiation, whilein the case super-Gaussian the central part of the targetmay undergo self-contraction provided its initial of-axisdisplacement is small enough. The 2D particle in cell sim-ulations affirm the theoretical calculations at large initialcoordinate of the target in the vertical direction, δy . Ifthe target is positioned in the vicinity of the axis, theself-modulation of the laser pulse in transverse directionprevents the target from sleapage out off the accelerationphase providing the fast ion collimation.The results obtained can be used for determining therequired laser-target alignment parameters and/or diag-nostics of the ion acceleration by the laser radiation pres-sure with mass limited targets, widely used in the exper-iments. ∗ Also at the ITMO University, Saint-Petersburg 197101,Russia; Russian Acad. Sci., A. M. Prokhorov GeneralPhys. Inst., Vavilov Str. 38, Moscow, 119991, Russia[1] M. Borghesi, J. Fuchs, S. V. Bulanov, A. J. Mackinnon,P. Patel, and M. Roth,
Fus. Sci. and Technology , 412(2006)[2] A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov,and A. M. Sergeev, Phys. Usp. , 9 (2011)[3] H. Daido, M. Nishiuchi, and A. S. Pirozhkov, Rep. Prog.Phys. , 056401 (2012)[4] A. Macchi, M. Passoni, and M. Borghesi, Rev. Mod.Phys. , 751 (2013)[5] S. Yu. Gus’kov, Plasma Phys. Rep. , 1 (2013)[6] S. V. Bulanov, J. J. Wilkens, M. Molls, T. Zh. Esirkepov,G. Korn, G. Kraft, S. D. Kraft, and V. S. Khoroshkov, Phys. Usp. , 1265 (2014)[7] S. V. Bulanov, T. Zh. Esirkepov, M. Kando, J. Koga, K.Kondo, and G. Korn, Plasma Phys. Rep. , 1 (2015)[8] V. S. Beskin, MHD Flows in Compact Astrophysical Ob-jects . (Springer, Berlin, 2010)[9] E. V. Derishev, V. V. Kocharovsky, and Vl. V.Kocharovsky
ApJ , , 640 (1999)[10] B. E. Stern and J. Poutanen, MNRAS , 1695 (2008)[11] D. Khangulyan, F. Aharonian, and V. Bosch-Ramon
MNRAS , , 467 (2008)[12] B. Cerutti, A. Philippov, K. Parfrey, and A. Spitkovsky MNRAS (in press) http://arxiv.org/abs/1410.3757[13] A. V. Gurevich, L. V. Pariskaya, and L. P. Pitaevskii,
Sov. Phys. JETP , 449 (1966)[14] P. Mora, Phys. Rev. Lett. , 185002 (2003)[15] S. Wilks, W. L. Kruer, M. Tabak, and A. B. Langdon, Phys. Rev. Lett. , 1383 (1992)[16] S. P. Hatchett, C. G. Brown, T. E. Cowan, E. A. Henry, J.S. Johnson, M. H. Key, J. A. Koch, A. B. Langdon, B. F.Lasinski, R. W. Lee, A. J. Mackinnon, D. M. Pennington,M. D. Perry, T. W. Phillips, M. Roth, T. C. Sangster,M. S. Singh, R. A. Snavely, M. A. Stoyer, S. C. Wilks,and K. Yasuike, Phys. Plasmas , 2076 (2000)[17] I. Last, I. Schek, and J. Jortner, J. Chem. Phys. ,6685 (1997)[18] K. Nishihara, H. Amitani, M. Murakami, S. V. Bulanov,and T. Zh. Esirkepov,
Nucl. Instrum. Meth. Phys. Res.A , 98 (2001)[19] V. F. Kovalev and V. Yu. Bychenkov,
Phys. Rev. Lett. , 185004 (2003)[20] M. Murakami and M. M. Basko, Phys. Plasmas ,012105 (2006)[21] V. I. Veksler, At. Energ. , 427 (1957)[22] T. Zh. Esirkepov, M. Borghesi, S. V. Bulanov, G.Mourou, and T. Tajima, Phys. Rev. Lett. , 175003(2004)[23] O. Klimo, J. Psikal, J. Limpouch, and V. T. Tikhonchuk, Phys. Rev. ST Accel. Beams , 031301 (2008)[24] A. P. L. Robinson, M. Zepf, S. Kar, R. G. Evans, and C.Bellei, New J. Phys. , 013021 (2008)[25] A. V. Vshivkov, N. M. Naumova, F. Pegoraro, and S. V.Bulanov, Phys. Plasmas , 2752 (1998)[26] S. V. Bulanov, T. Zh. Esirkepov, M. Kando, S. S. Bu-lanov, S. G. Rykovanov, and F. Pegoraro, Phys. Plasmas , 123114 (2013)[27] S. S. Bulanov, C. B. Schroeder, E. Esarey, and W. P. Leemans, Phys. Plasmas , 093112 (2012)[28] S. V. Bulanov, T. Zh. Esirkepov, M. Kando, A. S.Pirozhkov, N. N. Rosanov, Phys. Usp. , 429 (2013)[29] S. Kar, M. Borghesi, S. V. Bulanov, A. Macchi, M. H.Key, T. V. Liseykina, A. J. Mackinnon, P. K. Patel, L.Romagnani, A. Schiavi, and O. Willi, Phys. Rev. Lett. , 225004 (2008)[30] A. Henig, S. Steinke, M. Schnuerer, T. Sokollik, R. Ho-erlein, D. Kiefer, D. Jung, J. Schreiber, B. M. Hegelich,X. Q. Yan, J. Meyer-ter-Vehn, T. Tajima, P. V. Nickles,W. Sandner, and D. Habs,
Phys. Rev. Lett. , 245003(2009)[31] S. Kar, K. F. Kakolee, B. Qiao, A. Macchi, M. Cerchez,D. Doria, M. Geissler, P. McKenna, D. Neely, J. Oster-holz, R. Prasad, K. Quinn, B. Ramakrishna, G. Sarri, O.Willi, X. Y. Yuan, M. Zepf, and M. Borghesi,
Phys. Rev.Lett. , 185006 (2012)[32] J. Limpouch, J. Psikal, A.A. Andreev, K. Yu. Platonov,and S. Kawata,
Laser and Particle Beams , 225 (2008)[33] A. A. Andreev, J. Limpouch, J. Psikal, K. Yu. Platonov,and V. T. Tikhonchuk, Eur. Phys. J. ST , 123 (2009)[34] T. Kluge, W. Enghardt, S. D. Kraft, U. Schramm, K.Zeil, T. E. Cowan and M. Bussmann,
Phys. Plasmas ,123103 (2010)[35] K. Zeil, J. Metzkes, T. Kluge, M. Bussmann, T. E Cowan,S. D. Kraft, R. Sauerbrey, B. Schmidt, M. Zier, andU. Schramm, Plasma Phys. Control. Fusion , 084004(2014)[36] A. Zigler, S. Eisenman, M. Botton, E. Nahum, E.Schleifer, A. Baspaly, I. Pomerantz, F. Abicht, J.Branzel, G. Priebe, S. Steinke, A. Andreev, M.Schnuerer, W. Sandner, D. Gordon, P. Sprangle, and K.W. D. Ledingham, Phys. Rev. Lett. , 215004 (2013)[37] J. W. Wang, M. Murakami, S. M. Weng, H. Xu, J. J.Ju, S. X. Luan, and W. Yu,
Phys. Plasmas , 123103(2014).[38] T. P. Yu, Z. M. Sheng, Y. Yin, H. B. Zhuo, Y. Y. Ma, F.Q. Shao and A. Pukhov Phys. Plasmas , 053105 (2014)[39] Y. Fukuda, A. Ya. Faenov, M. Tampo, T. A. Pikuz, T.Nakamura, M. Kando, Y. Hayashi, A. Yogo, H. Sakaki,T. Kameshima, A. S. Pirozhkov, K. Ogura, M. Mori, T.Zh. Esirkepov, J. Koga, A. S. Boldarev, V. A. Gasilov,A. I. Magunov, T. Yamauchi, R. Kodama, P. R. Bolton,Y. Kato, T. Tajima, H. Daido, and S. V. Bulanov, Phys.Rev. Lett. , 165002 (2009)[40] T.-P. Yu, A. M. Pukhov, Z.-M. Sheng, F. Liu, and G.Shvets,
Phys. Rev. Lett. , 045001 (2013)[41] S. V. Bulanov, E. Yu. Echkina, T. Zh. Esirkepov, I. N.Inovenkov, M. Kando, F. Pegoraro, and G. Korn,
Phys.Rev. Lett. , 135003 (2010)[42] S. V. Bulanov, E. Yu. Echkina, T. Zh. Esirkepov, I. N.Inovenkov, M. Kando, F. Pegoraro, and G. Korn,
Phys.Plasmas , 063102 (2010)[43] F. Pegoraro and S. V. Bulanov, Phys. Rev. Lett. ,065002 (2007)[44] T. Zh. Esirkepov, Comput. Phys. Commun. , 144(2001)[45] E. Ott,
Phys. Rev. Lett. , 142 (1972)[46] W. Manheimer, D. Colombait, and E. Ott, Phys. Fluids , 2164 (1984)[47] T. Taguchi and K. Mima, Phys. Plasmas , 2790 (1995)[48] G. A. Korn and T. M. Korn, Mathematical Handbook forScientists and Engineers. (Dover Publ., New York, 2000)[49] E. Yu. Echkina, I. N. Inovenkov, T. Zh. Esirkepov, F.
Pegoraro, M. Borghesi, and S. V. Bulanov,
Plasma Phys.Rep. , 15 (2010)[50] S. V. Bulanov, T. Zh. Esirkepov, M. Kando, F. Pegoraro,S. S. Bulanov, C. G. R. Geddes, C. B. Schroeder, E.Esarey, and W. P. Leemans, Phys. Plasmas , 103105(2012) [51] A. Macchi, S. Veghini, and F. Pegoraro, Phys. Rev. Lett. , 085003 (2009)[52] S. Humphries, Jr.,