Enhancement of Electron Bunch Acceleration by the Wakefield Generated with Periodical Chirped Laser Pulse in under Dense Plasma

This paper presents a two dimensional analytical model for a linear laser wakefield generation by the chirped laser pulse and electron bunch acceleration has been studied numerically. It is found that the negative periodical chirp leads to increase the longitudinal and transverse components of the wakefield amplitude by one order of magnitude. In this status, we inject an electron bunch with the initial energy of about 2MeV in front of the periodical chirped laser pulse and we saw the final energy of the trapped electrons arrive to about 0.9Gev in 17mm of plasma length, with an energy spread about 1% and a the emmitance of the order of 0.12 mm mrad.

In this work the effect of the linear and periodic profiles for frequency variation of the laser pulse are investigated for the 2D linear laser wakefield generation.We show that appropriate chirped laser pulse improves the wakefield amplitude in 2D case and so, increases accelerating and focusing forces.Moreover, we examined the external electron bunch injection in front of the laser pulse by a numerical simulation.This paper is organized as follows.In sec 2, analytical calculations of the 2D linear wakefield are provided.In sec 3, the numerical results and discussions of the wakefield generation and electron bunch acceleration are considered, and in sec 4 some concluding remarks are noted.

2D Wakefield in the Linear Regime
To describe the excitation of plasma waves by a laser pulse, we use the Maxwell's and hydrodynamic equations for cold plasma electrons: (Andreev, Gorbunov, Kirsanov, Nakajima, & Ogata, 1997 Where , , is the normalized amplitude of the vector of potential of the laser field. For plasma with uniform density, combining these equations with assumption cylindrical symmetric, we can obtain following equations for the evolution of the 2D linear wakefield in cylindrical coordinate: In the quasi static approximation (Sprangle, Esarey, & Ting, 1990) by using coordinates moving with the laser pulse variables , and for simplicity the spatial component are normalized to being the plasma wave number and g v is the group velocity of the laser pulse in plasma.
The Equations ( 6) and ( 7) can be written in the form:

~
are the normalized axial and radial components of the wakefield is the normalized plasma density.
If charged particles are trapped in the appropriate position in the wakefield the axial and radial forces accelerate and focus charge particles in the laser wakefield, respectively.

Periodical Chirp
The transverse electric field component of the chirped laser pulse can be described by have been used in previous works, where b is the linear chirp parameter (Schroeder et al., 2003;Tóth et al., 2003;Khachtryan et al., 2005;Mirzanejhad et al., 2010 ).We use periodical function for the chirp relationship, In next section wakefield amplitude is compared for linear and periodical chirp, numerically.

Wakefield Generation
In this section generation of the 2D linear wakefield is investigated, for the linear and periodical chirped laser pulse.In all figures, we assume a laser pulse with an intensity of ) and unchirped laser pulses.It is seen that the periodically chirp can increase wakefield amplitude considerably and growth the wakefield amplitude for the linear chirp is very small, also shifted slightly compare with unchirped case.(gray solid line), and the constant frequency laser pulse (dash line) Due to increasing amplitude of the longitudinal and transverse electric fields via periodic chirp rather than linear chirp, it seems the periodic chirp suitable candidate for high energy electron bunch generation.More, we will investigate efficient parameters of the periodic chirp pulse to generate the wakefield.
An effective parameter governing on the wakefield amplitude is the chirp parameter.The normalized longitudinal electric field of the wakefield in term of the different chirp parameters for periodic chirp pulse is shown in Figure 3 ) (Esarey et al., 2009).Also the optimum value of  and 0  correspond to 1 and 0, respectively.In Figure 5, the normalized longitudinal and transverse fields distribution z E and r E are plotted.We know that there are regions in the wakefield that overlap occurs between accelerating region and focusing region.Thus, a trapped electron in these regions can be accelerated to the ultra relativistic energies while the focusing force keeps the electron near the wakefield axis.Clearly, the spread of overlap region is about 4 /  In the next section we will inject an electron bunch in front of the laser pulse.In this scheme electrons accelerate to the ultra-relativistic velocities in the first cycle of the wakefield behind the laser pulse (Khachtryan, 2001).

Electron Bunch Acceleration
The normalized relativistic equation of motion for an electron is  is relativistic factor of the electron.The last term of right-hand-side represents the ponderomotive force (Quesnel & Mora, 1998).For convenience, we have introduced the dimensionless time , and we study the dynamic of electrons in laser wakefield in Cartesian coordinate.
We decompose Equation (12) into its three components: . 4 Equation ( 13) are coupled ordinary differential equations.We solve them numerically by the forth order Range-Kutta method and the results are discussed as follows.
In this simulation we choose a bunch with Gaussian distribution in both coordinates and velocities which consist of 5 10 electrons. .Average energy of electrons for the periodical chirp (soiled line) and unchirp laser pulse (dashed line) are shown in Fig. ( 6) in terms of z k p .Clearly, periodical chirp increases mean energy of the electrons to 0.9GeV after 17 mm, i.e., ~45 GeV/m acceleration gradient, whereas the maximum energy of electron for unchirped case arrived about 100MeV.Initial and final structures of electron bunch before and after acceleration are plotted in Figures 7a and 7b, respectively.In this case 46% of the electrons are trapped and accelerated.The emittance is important quantity of an electron bunch.This is a measure of the divergence of the bunch.The final emmitance of electron bunch is 0.12 mm mrad and its final energy spread is 1%.

Conclusions
We have studied the effect of the frequency variation of the laser pulse on the 2D linear laser wakefield acceleration.It was indicated that the effect of the periodically frequency variation profile is better than conventional linear chirp and unchirped case.We trust that employing the appropriate negative periodical chirp can excite a strong wakefield (about one order of magnitude) toward linear chirp and unchirped case.
Finally we tested electron bunch acceleration by the periodical chirp laser wake field with numerical simulation.Results of this simulation indicate accelerating gradient of the electron beam is about 45GeV/m with an energy spread about 1% and the emmitance 0.12mm mrad, with the help of periodical frequency variation laser pulse.

Figure 1 .
Figure 1.The normalized electric field amplitude and the normalized local frequency variation of the periodical chirp pulse is shown as a function of  p k , for 20 /   p d   , 5 .0 / 0 0   c m eE e x 9 are solved numerically for linear and periodically chirp pulse.In Figure2aand 2b the longitudinal wakefield amplitude on the axis Figure 2. The normalized (a) longitudinal ) / ) 0 , ( ( p g e z v m r z eE 

Figure 4 .
Figure 4. Normalized longitudinal wakefield amplitude vs the normalized pulse length

Figure 5 .
Figure 5.The normalized longitudinal and transverse fields distribution z E and r E versus the  p k

Figure 6 .
Figure 6.Average energy of electrons for injection in front of the periodical chirp (soiled line) and unchirp laser pulse (dashed line) as a function of z k p

Figure 7 .
Figure 7. Electron bunch spatial distribution (a) before acceleration in 3   z k p , (b) after acceleration in 3469  z k p . It is clear from this figure that the suitable value is Figure.3. Normalized longitudinal wakefield amplitude vs the normalized periodically chirp parameter Another important parameter for increment the wakefield amplitude is the pulse length.Effect of this parameter on the acceleration field amplitude is shown in Figure (4).Clearly, the optimum value is 3  with unchirp laser pulse, because the final results of the linear chirp are not suitable.