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Proposal of Laser-Driven Acceleration with Bessel Beam

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Proposal of Laser-Driven Acceleration with Bessel Beam 614 D. Li et al. / Proceedings of the 2004 FEL Conference, 614-617 PROPOSAL OF LASER-DRIVEN ACCELERATION WITH BESSEL BEAM D. Li*, K. Imasaki, Institute for Laser Technology, 2-6 yamada-oka, suita, Osaka 565-0871,Japan Abstract consequently, the accelerated particles, which slide A novel approach of laser-driven acceleration with behind the wave, are possible to receive slight Bessel beam is presented in this paper. Bessel beam, in deceleration as they undergo the decelerating phase of the contrast to the Gaussian beam, demonstrates “diffraction- wave field by just travelling in these regions. Because of free” characteristics in its propagation, which implies the “diffraction-free” characteristic of Bessel beam, potential in laser-driven acceleration. A configuration of multistage acceleration comes to be feasible. Bessel beam truncated by a set of annular slits makes several special regions in its travelling path, where the BESSEL BEAM laser field becomes very weak and the accelerated General Theory particles are possible to receive slight deceleration as they A solution of the Helmholtz wave equation for an undergo decelerating phase. Thus, multistage acceleration ω is realizable. With the help of numerical computation, we azimuthally symmetric wave of frequency that have shown the potential of multistage acceleration based propagates in the positive z direction gives the Bessel on a three-stage model. beam expression v i ( k cos( α ) z − ω t ) ψ ( r , t ) = J 0 ( k sin( α ) ρ ) e INTRODUCTION (1) The intense laser field provides an ultrastrong field An ideal Bessel beam maintains the same radial profile gradient implying an attractive potential in particles over arbitrary propagation distance and has maximum acceleration. Laser-driven accelerator, an active current amplitude on-axis and hence is called “diffraction-free”. research area, has received considerable attention in The following integral representation of Bessel function is recent years [1-7]. Many proposals, such as inverse free helpful to comprehend the angle α, i( k cos(α ) z− ω t ) electron laser [8], plasma-wave accelerator [9], J0 ( k sin(α ) ρ )e ponderomotive schemes [10], inverse Cherenkov 1 2π acceleration [11], and vacuum acceleration directly by the = dφ e i( k sinα x cos φ+ ksin α y sin φ+ kcos α z− ω t ) ∫0 longitudinal component of electric field [12,13] were 2π presented and extensively studied. (2) In general, laser beam from a laser cavity travels in the which exhibits that Bessel beam is superposition of an form of Gaussian mode, exhibiting manifest transverse infinite set of ordinary plane waves making angle α with spreading when it is focused down to small spot size. A respect to the z axis [18]. And this is also the principle to strongly focused Gaussian beam will show apparent generate Bessel beam, i.e., superpose plane waves with divergence after it passes a short certain distance, known equal amplitude and a common phase that make angle α as Rayleigh length. Thus, even there exist schemes to to the z axis. keep synchronization between accelerated particles and A simple geometrical relation tells that, if the Bessel the wave, the effective acceleration can only occur within beam is formed at z=0 plane and to propagate a distance z, a relative short range. In order to extend the interaction, extent of R=z tanα at z=0. That means an ideal Bessel “diffraction-free” Bessel beams are introduced to the laser beam needs arbitrarily large initial radial extent and driven accelerator [12,13]. arbitrarily large energy, to retain its “diffraction-free” Since Durnin experimentally demonstrated the character over an arbitrarily large distance, which is nondiffracting property of Bessel beam, it has been impossible to realize in practice. The physically meaning widely studied in various applications [14-16]. Hafizi et.al beam is an aperture-truncated one, which possesses nearly analysed the vacuum beat wave accelerator with using diffraction-free properties within some axial distance. The laser Bessel beam [17], and other Bessel beam driven generation of aperture-truncated Bessel beam was widely acceleration schemes were also discussed recently. studied elsewhere [14,15]. This paper is aimed at an innovative approach in The phase velocity of a Bessel beam relies on the angle vacuum laser-driven acceleration with using Bessel beam. α,i.e., vp= c/cos(α), which is larger than the light speed We truncate the Bessel beam by a set of annular slits c. If such a beam is used for direct laser-driven (hereafter, we call it slits-truncated Bessel beam), so that acceleration in vacuum, the particle will slide behind the several special regions are formed in its propagation, wave and cannot gain net energy during a wave period. where the laser field becomes rather weak, and Naturally, we conceive a scheme that after the particle is ___________________________________________ accelerated in the first half circle of the wave period, the laser field suddenly disappears to allow the particle avoid * [email protected] the following deceleration; and when the particle slides THPOS13 Available online at http://www.JACoW.org D. Li et al. / Proceedings of the 2004 FEL Conference, 614-617 615 into the next accelerating phase, the laser field takes on where R1=w1, R2=R1+d1, R3=R2+w2, R4=R3+d2, R5=R4+w3. again. In other words, we have to make “laser-on” and Together with the initial conditions, Eq. 3 can be “laser-off” regions alternatively appearing during laser numerically calculated to understand the propagation propagation and well arrange the intervals among them to properties. As an example, here we consider the on-axis realize required phase matching. case (r=0) and employ the following parameters: laser o wavelength λ=1 μm, α=1 , and w1 =4 mm, w2 =4 Annular Slits-truncated Bessel Beam A Bessel beam truncated by a set of annular slits can do the work, as schematically shown in Fig. 1, where laser Fig. 2 Propagation of aperture-truncated (blue curve) and slits-truncated (grey curve) Bessel beam Fig. 1 schematic configuration of annular slits- χ(z) truncated Bessel beam shown from z=16 cm. Synthesized function (red curve) gives out the approximation to the grey field disappears at those shadow regions. The lengths of curve “laser-on” and “laser-off” regions are related to the slits’ parameters, width wi (i=1,2,3) and the interval between mm, w =6 mm, d =2 mm, d =4 mm. The computation two slits di (i=1,2). Using the geometrical method we can 3 1 2 easily work out those lengths. Actually, the “laser-off” results for slits-truncated Bessel beam are illustrated in areas are not entirely free of laser field because of the Fig. 2, and for the sake of comparison, aperture-truncated diffraction effect induced by annular slits. To precisely Bessel beam (with circular aperture radius of a=20 mm, analysis Bessel beam diffraction, we employ scalar summation of wi and di ) is also presented. The blue and diffraction theory. grey curves represent aperture- and slits-truncated beams, In the Fresnel approximation the amplitude A(r, z) at a respectively. It is seen that, their amplitudes characterize distance z can be obtained from diffraction integral. For a oscillation and decay at a finite distance ~a/tan( α ). circularly symmetric incident amplitude A( ρ , 0), we Contrast to the aperture-truncated beam, the slits- have [19] truncated one exhibits two special zones where the ikr 2 k amplitude falls down by two orders in magnitude. With A ( r , z ) = exp( ikz + )( ) 2 z iz properly setting the parameters of annular slits, the 2 a k r ik ρ injected electrons are possible to spend the decelerating ρ × ρ A ( ρ ,0 ) J 0 ( ) exp( ) d ρ phase in those regions and receive slight deceleration. The ∫0 z 2 z (3) red curve is synthesized to approximate the grey curve where a is the radius of a circular aperture. For the case of with showing an average effect, in ignorance of the oscillation. It takes the below form, with the use of Bessel beam, we have A(ρ,0)=J (ksin(α)ρ) as the 0 exponential functions to fit the falling and rising edges, initial amplitude distribution. If the beam is truncated by a set of annular slits, e.g., for the case as shown in Fig. 1, χ (z) = [ H ( z− 0) − H ( z − 21.6)] the incident amplitude should be modified as + []H (z − 21.6) −H (z − 28.3) exp(− 0.056 × (z − 21.6)) A( ρ ,0) = ζ ( ρ ) J 0 (k sin αρ ) (4) + [H (z − 28.3) − H (z − 36.5)] exp( 0.056 × (z − 36.5)) and ζ (ρ) is given by + []H (z − 36.5) −H (z − 54.8) + [H ( z − 54 .8) − Η ( z − 69 .2)] exp( − 0.032 × (z − 54 .8)) 1 (0 ≤ ρ < R1) + [H (z − 69.2) − H (z − 83.4)] exp( 0.032 × (z − 83.4)) 0 (R1 ≤ ρ < R2) + []H (z − 83.4) −H (z − 111 .2) 1 (R ≤ ρ < R ) 2 3 ζ (ρ) = + [H ( z − 111 .2)]exp( − 0.02× (z − 111 .2)) , (5) 0 (R3 ≤ ρ < R4 ) 1 (R4 ≤ ρ < R5) where H is the Heaviside function. 0 (R5 ≤ ρ < ∞) New Concepts 616 D. Li et al. / Proceedings of the 2004 FEL Conference, 614-617 Laser Field acceleration scheme is shown in Fig. 3. From Fig. 3 (a) All field components satisfying Maxwell’s equation are we know that the electron is accelerated from 150 MeV to derived in reference [18] for ideal Bessel beam. For the more than 1 GeV just over ~90 cm distance.
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