Electron Acceleration by Beating of Two Intense Cross-Focused Hollow Gaussian Laser Beams in Plasma

Commun. Theor. Phys. 69 018 86 94 Vol. 69, No. 1, January 1, 018 Electron Acceleration by Beating of Two Intense Cross-Focused Hollow Gaussian Laser Beams in Plasma Saleh T. Mahmoud, 1 Rakhi Gauniyal, Nafis Ahmad, 1 Priyanka Rawat, and Gunjan Purohit, 1 Department of Physics, College of Science, UAE University, P.O. Box 15551 Al-Ain, United Arab Emirates Laser Plasma Computational Laboratory, Department of Physics, DAV PG College, Dehradun, Uttarakhand-48001, India Received August 8, 017; revised manuscript received November 6, 017 Abstract This paper presents propagation of two cross-focused intense hollow Gaussian laser beams HGBs in collisionless plasma and its effect on the generation of electron plasma wave EPW and electron acceleration process, when relativistic and ponderomotive nonlinearities are simultaneously operative. Nonlinear differential equations have been set up for beamwidth of laser beams, power of generated EPW, and energy gain by electrons using WKB and paraxial approximations. Numerical simulations have been carried out to investigate the effect of typical laser-plasma parameters on the focusing of laser beams in plasmas and further its effect on power of excited EPW and acceleration of electrons. It is observed that focusing of two laser beams in plasma increases for higher order of hollow Gaussian beams, which significantly enhanced the power of generated EPW and energy gain. The amplitude of EPW and energy gain by electrons is found to enhance with an increase in the intensity of laser beams and plasma density. This study will be useful to plasma beat wave accelerator and in other applications requiring multiple laser beams. PACS numbers: 5.38.-r, 5.38.Hb, 5.35.Mw DOI: 10.1088/053-610/69/1/86 Key words: hollow Gaussian beams HGBs, collisionless plasma, cross-focusing, electron plasma wave, particle acceleration 1 Introduction Laser driven plasma-based accelerators are of great interest because of their ability to sustain extremely large acceleration gradients. [1 4] They have widespread applications in various fields such as particle physics, materials science, medicine ranging from x-ray diagnostics to particle beam therapies, manufacturing industry and to produce extremely short electron bunches. In laser plasma interaction, various schemes are operative for particle acceleration such as laser wakefield accelerator LWFA, [5 7] plasma beat wave accelerator PBWA, [5,8] multiple laser pulses, [9 10] self-modulated LWFA [11 1] etc. in which very high intense laser beams are used. The key requirement for achieving higher acceleration is the large amplitude of the electron plasma wave. A large amplitude plasma wave with phase velocity near the speed of light is driven by intense laser pulse in which plasma electrons can be trapped and accelerated to relativistic energies. Therefore, the excitation of electron plasma wave by intense laser beams in plasmas has been an active field of research for charged particle acceleration. There has been significant interest in the beat wave excitation of electron plasma wave and its application in plasma based particle accelerators. The plasma beat-wave accelerator scheme is one of the most mature methods for plasma acceleration in laser plasma interaction. [5] In this scheme, two intense laser beams of slightly different frequencies ω 1 & ω and corresponding wave numbers k 1 & k are used to resonantly excite an electron plasma wave. The resonance conditions for beat wave excitation of electron plasma wave are: ω = ω 1 ω = ω p0 and k = k 1 k = k p0. If the beat frequency ω is close to the plasma frequency ω p0, a very large amplitude relativistic electron plasma wave can be generated, which can be used to accelerate electrons efficiently to high energies in short distances. In this process, self-focusing of an intense laser beam in plasmas plays an important role and arises due to increase of the on-axis index of refraction relative to the edge of the laser beam. When two intense laser beams with slightly different frequencies simultaneously propagates in plasma, self-focusing of one laser beam is affected by the self-focusing of another laser beam. Due to mutual interaction of two laser beams, cross-focusing takes place in plasma. This method is used to generate large amplitude electron plasma wave for ultrahigh gradient electron acceleration in PBWA scheme. A lot of theoretical and experimental work has been reported on the excitation of electron plasma wave and electron acceleration by beating of two laser beams in plasma. [5,13 37] It has been observed that the focusing of laser beams in plasma and the yield of electron acceleration depends on the spatial profile of laser beams and the nonlinearities associated with plasma. Most of the earlier investigations on excitation of electron plasma wave and electron acceleration have been carried out by taking Gaussian intensity distribution of laser beams with ponderomo- Supported by United Arab Emirates University for Financial under Grant No. UPAR 014-31S164 Corresponding author, E-mail: gunjan75@gmail.com c 018 Chinese Physical Society and IOP Publishing Ltd http://www.iopscience.iop.org/ctp http://ctp.itp.ac.cn

No. 1 Communications in Theoretical Physics 87 tive/relativistic nonlinearity. In contrast with Gaussian profile of laser beams, currently, the hollow Gaussian intensity profile of laser beams with minimum field intensity at the center has attracted much more attentions because of their unique physical properties and applications. [38 40] The main feature of considered hollow Gaussian laser beams is having the same power at different beam orders with null intensity at the centre. The propagation dynamics of hollow Gaussian beams HGBs have been widely studied both experimentally and theoretically. [41 46] Various techniques [47 50] have been used to generate HGBs. These beams can be expressed as a superposition of a series of Lagurerre-Gaussian modes. [41] Moreover, when an ultra-intense laser beams propagates in plasma, both relativistic and ponderomotive nonlinearities arise simultaneously due to electron mass variation and electron density perturbations respectively, which depend on the time scale of the laser pulse. [51 5] Therefore, in comparison with only relativistic/ponderomotive nonlinearity, the dynamics of the propagation of laser beams in plasma is expected to be drastically affected by cumulative effects of relativistic-ponderomotive nonlinearity. It is also important to mention that the relativistic effect and ponderomotive nonlinearity contribute to focusing on a femtosecond time scale at very high intensity. Cross-focusing of HGBs with relativistic/ponderomotive or relativisticponderomotive nonlinearities in plasma have been investigated in detail [53 55] but no one has studied the excitation of EPW and electron acceleration. Moreover, donut wakefields generation for particle acceleration by Laguerre-Gaussian laser pulses carrying a finite amount of orbital angular momentum using particle-incell simulations have also been reported in under-dense plasma. [56 57] In the present study, we have considered the propagation of two intense hollow Gaussian beams in collisionless plasma. The intensity distribution of the beams along the axis is zero and the maximum is away from the axis. We have studied the cross-focusing of two intense hollow Gaussian laser beams in collisionless plasma with relativistic and ponderomotive nonlinearities and further its effect on the excitation of EPW and electron acceleration. The paraxial-ray approximation [58 59] is used to describe the focal region of the laser beam where all the relevant parameters correspond to a narrow range around the maximum irradiance of the HGBs. Section presents the relativistic-ponderomotive focusing of two HGBs in plasma under paraxial-ray approximation. The effect of the cross-focusing of the HGBs on the excitation of the electron plasma wave and electron acceleration is studied n d n 0 = 1 + c ω p0 a 1 r10 4 f 1 8 { a1 r 10 f 4 1 + a r 4 0 f 8 + a r0 f 4 in Sec. 3. Section 4 deals with the results and discussion. The conclusions drawn from the results of present investigation have been summarized in Sec. 5. Propagation of Hollow Gaussian Laser Beams in Collisional Plasma We consider two intense linearly polarized co-axial hollow Gaussian laser beams of frequencies ω 1 and ω propagating along the z-direction in the collisional plasma. The initial electric field distribution of the beams are given by r E 1 E1 n = E 10 exp r, 1 r 10 E E = E 0 r r 0 r 10 n exp r r 0, where r is the radial coordinate of the cylindrical coordinate system, r 10 and r 0 are the initial spot size of the laser beams, E 10 and E 0 are the amplitude of HGBs maximum r 10 = r 0 = r max = r 1, n, n is the order of the HGBs n = 0 corresponds to the fundamental Gaussian laser beam and a positive integer, characterizing the shape of the HGBs an position of its irradiance maximum. For hollow Gaussian laser beams, transforming the r, z coordinate in to η, z coordinate by the relation [44] η = [r/r 10, 0 f 1, n], 3 where η is a reduced radial coordinate, f 1, is the beam width of HGBs, r = r 10,0 f 1, n 1/ is the position of the maximum irradiance for the propagating beams. Most of the power of HGBs are concentrated in the region around η = 0. The ponderomotive force acting on the electrons in the presence of relativistic nonlinearity is given by [5 53] F P = m 0 c γ 1, 4 where γ is the relativistic Lorentz factor and is given by here 1 γ n η + n 4n exp[ η + n ] 1 γ 3 4n η + n 8n exp[ η + n ] γ = [1 + α 1 E 1 E 1 + α E E ] 1/, 5 e α 1, = m 0 c ω1,. This force modifies the background electron density because electrons are expelled away from the region of higher electric field. The modified electron density n d due to relativistic-ponderomotive force is given as [5] n d = n 0 [ 1 + c ω p γ γ γ ]. 6 Using Eq. 3 into Eq. 7, the total electron density can be written as 8n η + n + η + n 8n 4n η + n + η + } n 4n.

88 Communications in Theoretical Physics Vol. 69 Here a 1 = α 1 E 10 and a = α E 0 are the intensities of first and second laser beams. The dielectric constant of the plasma is given by ε 10,0 = 1 ω p0 ω1,, 7 where ω p0 [= 4πn 0 e /m 0 γ 1/ ] is the plasma frequency with e is the charge of an electron, m 0 is its rest mass, and n 0 is the density of plasma electrons in the absence of laser beam. The effective dielectric constant ϵ 1, of the plasma in the presence of relativistic-ponderomotive nonlinearity at frequencies ω 1 and ω respectively is given by ε 1, r, z = ε 10,0 + ω p0 ω1, 1 n d, 8 n 0 γ where ϵ 10,0 is the linear part of dielectric constant of the plasma. The dielectric constant may be expended around the maximum of HGBs i.e. at η = 0. For HGBs, one can express the dielectric constant ϵ 1, η, z, in terms of ϵ 1, r, z. Expanding dielectric constant around η = 0 by Taylor expansion, one can write where ε 1, η, z = ε 10.0 z η ε 1, z, 9 ε 10,0 z = ε 10,0 η, z η=0 = 1 ω p0 1 ω1, 1 + g c a1 1/ ω1, 1 + g r10 f 1 4 + a r0 f 4 n n e n, ε10,0 η, z ε 1, z = η = ω p0 g 4c g a1 η=0 ω1, 1 + g 3/ ω1, 1 + g r10 f 1 4 + a r0 f 4 n n e n, with g = a 1 /f 1 + a /f n n e n, ϵ 10,0, and ϵ 1, are the coefficients associated with η 0 and η in the expansion of ϵ 1, η, z around η = 0. The wave equation governing the electric vectors of the two laser beams in plasma can be written as E 1, + 1 E 1, + E 1, r r r + ω 1, c ε 1,r, ze 1, = 0. 10 In Eq. 10 the second term E can be neglected, which is justified as long as ωp0/ω 1, 1/ε 1, in ε 1, 1. One can thus write the wave equation for the two laser beams in plasma is as, The solution of Eq. 11 can be written as [58] E 1, + ω 1, c ε 1,r, ze 1, = 0. 11 E 1, r, z = A 1, r, z exp i k 1, zdz, 1 where A 1, r, z is the complex amplitude of the electric field E 1,, k 1, z = ω 1, /c ε 10,0 z and ε 10,0 z is the dielectric function, equivalent to the highest electric field on the wave front of the HGBs. Substituting E 1, r, z from Eq. 1 into Eq. 11, the wave equation becomes ik 1, A 1, + ia 1, k 1, The complex amplitude A 1, r, z may be expressed as = A 1, r + 1 r A 1, + ω 1, r k 1, c ε ε 0. 13 A 1, r, z = A 01, r, z exp ik 1, zs 1, r, z, 14 where A 01, is a complex function of space and S 1, r, z is the eikonal associated with the HGBs. Substituting Eq. 14 into Eq. 13 and separating the real and imaginary parts of the resulting equation, the following set of equations is obtained. S 1, k 1, + S 1, + k 1, A 01, + A 01, S 1, r S1, + 1 r = 1 A 01, r A 01, + ω 1, r c k1, ε ε 0, 15 k1, A + 1 01, r S 1, + A 01, S 1, + A 01, k 1, = 0. 16 r r r k 1, In the paraxial approximation i.e. when η n, Eqs. 15 and 16 lead to the maintenance of the HGBs character during propagation. Using Eq. 3, Eqs. 15 and 16 in terms of η can be expressed as S 1, + S 1, k 1, 1 S1, 1 A 01, 1 + = k 1, r10,0 f 1, η k1, r 10,0 f 1, A 01, η + η + A 01, η ω 1,ε n η c k1,, 17 A 01, + A 01, S 1, 1 r10,0 f 1, η + S 1, 1 A 01, S 1, + n + η η r10,0 f 1, η η + A 01, k 1, = 0. 18 k 1,

No. 1 Communications in Theoretical Physics 89 The solution of Eq. 17 can be written as [58 59] A 01, = E 1, n f1, n + η 4n exp[ n + η ], 19 S 1, η, z = n + η βz + φz, 0 where βz = r1,f 1, df 1, /dz Here φz is an arbitrary function of z and f 1, z is the beam width parameter for first and second HGBs respectively. Substituting from Eqs. 19 and 0 into Eq. 17 and equating the coefficients of η 0 and η on both sides of the resulting equation, one obtains d f 1, 1 { 4 [ ω dξ = ε 10,0 f 1, f1, ρ p0 g 4c g a1 10,0 ω1, 1 + g 3/ ω1, 1 + g r10 f 1 4 + a r0 f 4 n n e n]}, 1 where ξ = cz/r 10,0ω 1, is the dimensionless distance of propagation ρ 10,0 = r 10,0 ω 1, /c is the dimensionless initial beam width. 3 Beat Wave Excitation of Plasma Wave and Electron Acceleration Due to beating of two hollow Gaussian laser beams in plasma and modification in plasma density via relativisticponderomotive force, electron plasma wave EPW is generated. In this process, the contribution of ions is negligible because they only provide a positive background, i.e., only plasma electrons are responsible for excitation of EPW. The amplitude of electron plasma wave which depends on the background electron density gets strongly coupled to the laser beams. To study the effect of relativistic and ponderomotive nonlinearities on the generation of the plasma wave by the beat wave process in paraxial region, we start with the following fluid equations: i The continuity equation: N e + N e V = 0. t ii The momentum equation: [ V ] m t + V V = ee e c V B Γ em 0 V 3k 0 T e N e. 3 N e iii Poisson s equation: E = 4πeN e, 4 where N e is the total electron density, V is the sum of the drift velocities of the electron in the EM field and selfconsistent field, Γ e is the Landau damping factor, [60] E and B are the self-consistent electric and magnetic fields of laser beams corresponding to the electron plasma wave; and other symbol have their usual meaning. Using Eq. to Eq. 4, we obtain the following equation governing the electron plasma wave N t + Γ N e t υ th N e [ N m NE = N ] V V V, 5 t where υ th is the electron thermal speed. At the difference frequency ω = ω 1 ω, Eq. 5 reduces to ω ω 1 ω N 1 + iγ e ω 1 ω N 1 υth p0 n d N 1 + N 1 1 = γ n 0 4 n T V 1 V, 6 where N 1 is the component of electron density oscillating at frequency ω. The drift velocities of electron in the pump field at the frequency ω 1, are V 1, = ee 1, /imω 1,. Therefore, one can write V 1 V e E 10 E 0 = m 0 ω 1ω f 1 f n η + n 4n exp{ η + n exp i[k 1 k z + k 1 S 1 k S ]}. 7 Equation 6 contains two plasma waves both have different frequencies. The first one is supported by hot plasma and the second by the source term at the different frequency. If ω p0 / ω 1, the phase velocity is almost equal to the thermal velocity of the electron and Landau damping occur. But as ω p0 / ω 1, the phase velocity is very large as compared to the electron thermal velocity and Landau damping is negligible. The solution of Eq. 9 with in the WKB approximation can be expressed as N 1 = N 10 r, z exp[ ikz + S] + N 0 exp[ ik 1 k z + k 1 S 1 k S ], 8 where k = [ω 1 ω ωp0]/v th, N 10 and N 0 are the slowly varying real function of the space coordinates. Substituting for N 1 from Eq. 8 into Eq. 6 the equation for N 10 and N 0 can be obtained by equating the coefficient of exp[ ikz + S] and exp[ ik 1 k z + k 1 S 1 k S ] in the resulting equation. ω N 10 + iγ e ωn 10 υ th [ N 10 k + S i k + S N10 S in 10 + N 10

90 Communications in Theoretical Physics Vol. 69 S S N 10 S N 10 i in 10 r r r r + N 10 r + 1 S N 10 r r + N ] 10 r ω N 0 + iω Γ e N 0 + k υthn 0 + ωp0 n d n 0 γ N 0 = n de E 01 E 0 4m ω 1 ω f 1 f n r1, f 1, η + n 4n n d + ωp0 n 0 γ = 0, 9 exp[ η + n { k r 1,f 1, + 16η + n + 16n η + n 3n 8}]. 30 After separating the real and imaginary parts of Eq. 9, we obtain 1 S S + k r1, f 1, η = 1 [ N 10 N 10 r1, f 1, r + 1 N 10 r r k N 10 S η + 1 η + S 1 + n η + N 10 r1, f 1, The equation for N 0 is given by N 0 = r 1, f 1, ] + 1 [ vth S N10 η η n d e E 01 E 0 4m 0 ω 1ω f 1 f n r1, f 1, η + n 4n exp The solution of Eqs. 31 and 3 can be written as [58 59] N 10 = n d ω k υth ωp0 n 0 γ ], 31 Γ e ω υth N10 = 0. 3 η + n [ k r 1,f 1, 16η + n 16n η + n + 3n + 8] [ ω + iγ e ω + k v th + ω p0 n d/n 0 γ] B f p n η + n 4n r 1, f 1, a 0 f p 4n exp { Sη, z = η + n βz + ϕz, r 1,f1, a 0 f p η + } n ki z,. 33 βz = 1 f p df p dz, 34 where k i = Γ e ω/kυth, a 0 and B are the constants to be determined by the boundary condition that the magnitude of the generated plasma wave at z = 0 is zero. This gives B = e E 01 E 0 n d 1 a0 n { η + n n 4m 0 ω 1ω n r1, exp η + n r } 1, r 1, a 0 [ k r 1, 16n η + n 16η + n + 3n + 8] [ ω iγ e ω k v th ω p0 n d/n 0 γ z=0 ], 35 and a 0 = r10r 0/r 10 + r0. In Eq. 34, f p is the dimensionless beam width parameter of the plasma wave given by d f p ρ dξ = 0 f { p 1 r1, f 4n 1, ω [ p0 g c a1 f1, k r1, f 1, a 0 f p k vth 1 + g 3/ ωp0 r1 f 1 4 + a r f 4 n n e n 4g ]} 1 + g. 36 Using Poisson s equation one can obtain the electric vector E ω of the plasma wave excited at the difference frequency: 4πei { f1, n [ r 1, E ω = G 1 exp k f p a η + n 1 f ] 1, 0 fp exp[ ikz + S] } G exp[ ik 1 k z + k 1 S 1 k S ] η + n 4n exp[ η + n ], 37 where e E 01 E 0 n d [ k r10,0 16n η + n 16η + n + 3n + 8] exp k i z G 1 = 4m ω 1 ω n f p r10,0 [ ω iγ e ω k vth ω p0 n, d/n 0 γ z=0 ] e E 01 E 0 n d [ k r10,0f 1, 16n η + n 16η + n + 3n + 8] G = 4m ω 1 ω n f 1 f r10,0 f 1, [ ω iγ e ω k vth ω p0 n. d/n 0 γ] The power associated with electron wave is given by [9] Vg k m c 4 P EPW = 16πe E ωe ωπrdr, 38 0 where V g = υ th 1 ω p0/ω 1, 1/ is the group velocity. Equation 38 has been solved numerically with the help of Eqs. 1, 36, and 37. The excited electron plasma wave transfers its energy to accelerate the electrons at the difference frequency of laser beams. The equation governing electron momentum

No. 1 Communications in Theoretical Physics 91 and energy are dp dt = ee ω, dγ dt = ee ω V m 0 c, where γ = 1 + P /m 0c is the energy gain by the electron or electron energy. This equation can be written as dγ dz = ee ω m 0 c, 39 where E ω is given by Eqs. 37 and 39 gives the electron energy. Equation 39 has been solved numerically, where we have used f 1 and f by Eq. 1. 4 Numerical Results and Discussion In order to study the cross-focusing of two intense hollow Gaussian laser beams in collisonal plasma with dominant relativistic-ponderomotive nonlinearity and its effect on the generation of electron plasma wave and electron acceleration in paraxial-ray approximation, numerical computation of Eqs. 19, 1, 38, and 39 has been performed for the typical laser plasma parameters. The following set of the parameters has been used in the numerical calculation: ω 1 = 1.776 10 15 rad/s, ω = 1.716 10 15 rad/s, r 1 = 15 µm, r = 0 µm, ω p0 = 0.3ω, a 1 = 3, a =,.4,.8, and n = 1,, 3. For initial plane wave fronts of the beams, the initial conditions for f 1, are f 1, = 1 and df 1, /dz = 0 at z = 0. When two intense hollow Gaussian laser beams of slightly different frequencies simultaneously propagate through the plasma, the background electron density of plasma is modified due to ponderomotive force. Equation 19 describes the intensity profile of HGBs in plasma along the radial direction in presence of relativistic and ponderomotive nonlinearities, while Eq. 1 determines the focusing/defocusing of the hollow Gaussian beams along the distance of propagation in the plasma, where the first term on the right hand side shows the diffraction behaviour of laser beams and remaining terms on the right hand side show the converging behaviour that arises due to the relativistic and ponderomotive nonlinearity. The focusing/defocusing behaviour of laser beams depends on the magnitudes of the nonlinear coupling terms. It is clear from Eq. 19, the intensity profile of both laser beams depends on the focusing/defocusing of the laser beams i.e. beam width parameters f 1,. Fig. 1 Color online Variation of the beam width parameters f 1 and f with the normalized distance of propagation ξ of first and second HGBs: a For different orders of n red, black and blue colors are for n = 1, and 3 with a 1 = 3, a =.4 and ω p0 = 0.3ω. b For different values of a red, black, and blue colors are for a =,.4,.8 with a 1 = 3, n = and ω p0 = 0.3ω. c For different values of ω p0 red, black, and blue colors are for ω p0 = 0.ω, 0.3ω, and 0.4ω with a 1 = 3, a =.4. and n =, respectively, when relativistic and ponderomotive nonlinearities are operative. The solid line represents f 1 and the dotted line represents f.

9 Communications in Theoretical Physics Vol. 69 due to the strong focusing of both HGBs at higher values of n and plasma density respectively. Such kind of highly self-focused intense laser beams interact with each other and generate large amplitude electron plasma wave. Fig. Color online Variation of the normalized intensity of HGBs with the normalized distance of propagation ξ for different orders of n red, black and blue colors are for n = 1, and 3 keeping a 1 = 3, a =.4, ω p0 = 0.3ω, when relativistic and ponderomotive nonlinearities are operative. a and b are for first and second laser beam. The results of Eqs. 19 and 1 are presented in Figs. 1 and respectively. Figures 1a 1c illustrate the focusing behavior of two hollow Gaussian laser beams in plasma with normalized distance of propagation. It is clear that both of the beams show oscillatory self focusing. One can see from Fig. 1a as the order of the hollow Gaussian beam n increases focusing of both beams also increases, which implies that with the increase in n the hollow space across the beam also increases. It is observed from Figs. 1b and 1c that with an increase in the intensity of incident laser beam and plasma density there is an increase in the extent of self-focusing of both the laser beams. This is due to the fact that with an increase in plasma density the number of electrons contributing to the pondermotive-relativistic nonlinearity also increases. These results indicate that focusing of one beam is significantly affected by the presence of another beam. Figures a and b represent the normalized intensity of first and second laser beam in plasma respectively along with the normalized distance of propagation for different n. It is observed from the figures that with an increase in n, normalized intensity of both laser beams also increases. It has also been observed that the intensity of laser beams is higher at higher plasma density and incident laser beam intensity results not shown here. This is Fig. 3 Color online Variation of the normalized power P of electron plasma wave with normalized distance ξ: a For different orders of n red, black and blue colors are for n = 1, and 3 with a 1 = 3, a =.4, and ω p0 = 0.3ω. b For different values of a red, black, and blue colors are for a =,.4,.8 with a 1 = 3, n =, and ω p0 = 0.3ω, respectively, when relativistic and ponderomotive nonlinearities are operative. To see the effect of cross-focusing of two intense hollow Gaussian laser beams on the generation of electron plasma wave, we have studied the excitation of the electron plasma wave by the beat wave process in the presence of relativistic and ponderomotive nonlinearities. It is important to mention here that beating between two copropagating intense laser beams in plasma can generate a longitudinal electron plasma wave with a high electric field and a relativistic phase velocity. The beat plasma wave is very efficient for electron acceleration up to ultrarelativistic energies. Equation 38 describes the normalized power of electron plasma wave generated as a result of beating of the two laser beams. It is evident from Eq. 38, the power of electron plasma wave depends upon the focusing of laser beams in plasma f 1,, focusing of electron plasma wave f p and the electric field associated with electron plasma wave E ω. The same set of parameters has been used for numerical calculations as in Figs. 1. Figures 3a and 3b show the effect of changing the order of HGBs and the initial intensity of second laser beam a on the power of generated electron plasma wave, at the position

No. 1 Communications in Theoretical Physics 93 of optimum irradiance of the beams η = 0. By increasing laser beam orders and the intensity of second laser beam, the power of generated electron plasma wave increases. This is because the power of generated electron plasma wave depends on the focusing behaviour of laser beams f 1,, f p and intensity of laser beams. Fig. 4 Color online Variation of the γ of the electron with normalized distance ξ: a For different orders of n red, black and blue colors are for n = 1, and 3 with a 1 = 3, a =.4, and ω p0 = 0.3ω. b For different values of a red, black, and blue colors are for a =,.4,.8 with a 1 = 3, n =, and ω p0 = 0.3ω, respectively, when relativistic and ponderomotive nonlinearities are operative. The large amplitude electron plasma wave can be used to accelerate the electrons in plasma beat wave accelerator scheme. Equation 39 gives the expression for energy gain by the electrons. This equation has been solved numerically with the help of Eq. 37 i.e. energy gain depends on the electric field associated with excited electron plasma wave. Figures 4a and 4b respectively illustrate the effect of changing the value of nand the intensity of second laser beam a on energy gain by the electrons. These figures clearly indicate that the maximum energy gain by electrons is significantly increased by increasing the value of b and a. Although, the nature of these results are similar to Figs. 3a and 3b due to the same reasons as discussed above. Thus, we see that cross-focusing of laser beams plays a crucial role for efficient generation of electron plasma wave and electron acceleration. 5 Conclusions In conclusion, plasma wave generation and electron acceleration by beating of two intense HGBs in collisionless plasma with dominant relativistic-ponderomotive nonlinearity has been discussed. Paraxial-ray approximation has been used to establish the given formulation. Effects of laser and plasma parameters on the focusing of two HGBs in plasma at difference frequency, generation of electron plasma wave and particle acceleration process is examined. Following conclusions are made from the results: i The order of the hollow Gaussian laser beams plays an important role in plasma beat wave accelerator scheme. ii Focusing of both hollow Gaussian laser beams in plasma is enhanced by increasing the laser beam orders, incident laser intensity, and plasma density. iii The intensity of both laser beams in plasma is higher for higher orders of the HGBs. iv Maximum power of generated electron plasma wave depends on the extent of focusing of laser beams in plasma and the electric field associated with electron plasma wave respectively. v The power of electron plasma wave increases by increasing the laser beam orders and the initial intensity of second/first laser beams. vi The maximum energy gain also depends on the electric field associated with electron plasma wave and increases by increasing the laser beam orders and the initial intensity of second/first laser beams. The results of the present investigation are relevant to laser beat wave based particle accelerators, terahertz generation and in other applications requiring multiple laser beams. References [1] S. M. Hooker, Nature Photonics 7 013 775. [] E. Esarey, C. B. Schroeder, and W. P. Leemans, Rev. Mod. Phys. 81 009 19. [3] V. Malka, J. Faure, Y. A. Gauduel, et al., Nature Physics 4 008 447. [4] R. Bingham, Phil. Trans. R. Soc. A. 364 006 559. [5] T. Tajima and J. M. Dawson, Phys. Rev. Lett. 43 1979 67. [6] A. Pukhov and J. Meyer-ter-vehn, Appl. Phys. B 74 00 355. [7] F. Albert, A. G. R. Thomas, S. P. D. Mangles, et al., Plasma Phys. Control. Fusion 56 014 084015. [8] C. V. Filip, R. Narang, S. Ya. Tochitsky, et al., Phys. Rev. E 69 004 06404. [9] K. Nakajima, Phys. Rev. A 45 199 1149. [10] D. Umstadter, E. Esarey, and J. Kim, Phys. Rev. Lett. 7 1994 14.

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