Numerical investigations of electron-self-injection in different shaped bubbles in wakefield acceleration

الموضوعات : Journal of Theoretical and Applied Physics

Sonu Kumar ¹ (Department of Physics, Indian Institute of Technology Delhi, New Delhi, India.)
Dhananjay Singh ² (Department of Physics, PKRM College, Dhanbad, Jharkhand, India.)
Hitendra K. Malik ³ (Department of Physics, Indian Institute of Technology Delhi, New Delhi, India.)

تاريخ الإرسال : 09 الخميس , ربيع الثاني, 1444 تاريخ التأكيد : 21 الثلاثاء , ربيع الثاني, 1444 تاريخ الإصدار : 11 الإثنين , شوال, 1444

الکلمات المفتاحية: energy gain, bubble shape, self-injection, Bubble wakefield acceleration,

ملخص المقالة :

Electron-self-injection in bubble wakefield acceleration is the new concept for acceleration of electrons inside bubble. In this technique, self-injected plasma electrons have been used for acceleration whose advantage is that there is no need of external source of electrons. In our case, we have carried out numerical investigations of self-injected plasma electrons in different shaped bubbles such as spherical, longitudinal ellipsoid and transverse ellipsoid bubble. For these numerical investigations, by carrying out relativistic Hamiltonian analysis of plasma electrons, we have used 4th order Runge-Kutta (RK) method by employing MATLAB ode45, a nonstiff differential equations solver. We have discussed different parameters such as impact parameter, radius of bubble, bubble velocity for their effect on the formation of bubbles with different shapes and self-injection of the electrons.

نص كامل:



Numerical investigations of electron-self-injection in different shaped bubbles in wakefield acceleration

 

 Sonu Kumar1, Dhanajay. K. Singh2 and Hitendra K. Malik1*

1Plasma Waves and Particle Acceleration Laboratory, Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi - 110016, India

2University Department of Physics, PKRM College, Dhanbad, Jharkhand -826004, India

*Email: [e-mail]

 

Abstract

Electron-self-injection in bubble wakefield acceleration is the new concept for acceleration of electrons inside bubble. In this technique, self-injected plasma electrons have been used for acceleration whose advantage is that there is no need of external source of electrons. In our case, we have carried out numerical investigations of self-injected plasma electrons in different shaped bubbles such as spherical, longitudinal ellipsoid and transverse ellipsoid bubble. For these numerical investigations, by carrying out relativistic Hamiltonian analysis of plasma electrons, we have used 4th order Runge-Kutta (RK) method by employing MATLAB ode45, a nonstiff differential equations solver. We have discussed different parameters such as impact parameter, radius of bubble, bubble velocity for their effect on the formation of bubbles with different shapes and self-injection of the electrons.

Keywords: Bubble wakefield acceleration, plasma, impact parameter, bubble velocity, bubble radius.

1. Introduction

Accelerated charged particles with very high velocity comparable to the light speed, when collide with other moving particles or stationary target, create a huge amount of energy. Today, Large Hadron Collider called as LHC accelerate charged particles in circular track and allow them collide with other charged particles. Sometimes ago, Big-bang concept was tried to be achieved in LHC laboratory for creating the condition of birth of our universe with the help of accelerator concept. Hence, new dimensions of research are possible based on the concept of particle acceleration that has other applications in fusion energy, synchrotron radiation production, nuclear energy, medical treatment and many more. However, such accelerators are huge in size and have very high cost. For example, LHC occupies 27 km of circumference, and its maintenance cost is very high involving a lot of manpower too. The conventional accelerators employ a radiofrequency cavity and have an electrical breakdown problem that limits their efficiency. For removal of such disadvantage, researchers have used the concept of wakefield acceleration, thanks to our pioneer researchers Tajima and Dawson who conceived this acceleration technique. Tajima and Dawson [1] in 1979 created an oscillatory wake in the plasma by introducing electromagnetic field driver called intense short laser pulse; specifically they generated an electron plasma wave corresponding field to which was called wakefield. This wakefield had an advantage of producing three times more accelerator gradient than the one achieved in conventional particle accelerators. Here plasma is used as a medium because it is ionized and is already in breakdown stage. In comparison to size and cost of conventional accelerators, it takes only a few meters in size and has a low cost. For the understanding point of view of the wakefield, the example of a surfer is the best example that uses the wake created by a moving boat in the lake for moving forward. One can visualize the plasma as lake, laser pulse as a driver, and the surfers as charged particles.

In the concept of laser wakefield acceleration (LWFA) [2–4], plasma electrons are expelled by energetic laser photons, creating an electron free region (a region of almost stationary ions). The expelled electrons move back to their positions because of the restoring force due to background ions and overshoot there initial positions because of their inertia and create an electron plasma wave [5]. The frequency of this wave is in the range of electron plasma frequency and accelerator gradient is in GeV/m. There are other methods also for the particle acceleration; for example, the plasma wakefield acceleration (PWFA) where a beam of charged particles drives the wakefield, self-modulated laser wakefield acceleration (SMLWFA) where the laser pulse gets self-modulated and the plasma beat wave wakefield acceleration (PBWA) where two laser beams drive the wakefield  [6–9]. Some other wakefield acceleration techniques are their such as proton driven wakefield [10]. In LWFA, a nonlinear plasma wave is generated when the laser’s intensity is quite high, and here a multidimensional wake is created, expelled plasma electrons are trapped around the boundary of the electron free region, i.e. the ion-cavity or the bubble [11–15]. Advantage of this type of wakefield acceleration is that there is no need of externally injected electrons for witness bunch. These electrons are trapped in the first wake at the tail of the bubble and self-injected into the bubble [16–18]. Self-injection of the plasma electrons in bubble wakefield is recent development for the generation of high quality electrons beam [19–28]. These high quality electron beam has been used for many applications such as material characterization [29], generation of synchrotron radiations such as betatron radiations which have been used in medical treatments [30–35] and many more [36–46].

Kostyukov et al. [47] and Lu et al. [14] have developed the model of bubble wakefield and used the spherical shape of the bubble with bubble velocity equal to one. They used a scalar gauge condition and wakefield potential with relativistic Hamiltonian equations analysis, but they did not give the concept of different shapes of bubble during self-injection of the electrons. Trajectory of the plasma electrons has been developed by Lu et al. [48] and bubble structure has been evaluated by Toosi et al. [49]. Li et al. [50] worked for the shape of the bubble by using the electromagnetic field produced inside the bubble and introduced a geometrical parameter  such that  defined the spherical bubble and  defined the longitudinal ellipsoid bubble and  defined the transverse ellipsoid bubble. Further work was done by Kumar et al. [17] on different gauges in bubble wakefield acceleration. But no investigation has been done numerically revealing the self-injection process of the plasma electrons and longitudinal and transverse phase-spaces in different shaped bubble regime along with a role of impact parameter, bubble radius and bubble velocity. In the present work, hence, we consider different shapes of the bubble and discuss the phenomenon concerning the electron-self-injection inside the bubbles. We deal with slower as well as faster bubble velocities on the trajectory of the trapped electrons. Our analysis with consideration of the impact parameter defines the radial distance between the trapped electrons and the laser driver.

 

2. Numerical investigation

We have considered Maxwell’s equations and solved the d’Alembert differential equations by using Gauge condition  and wakefield potential . Finally, we obtain

 

where  is the geometrical coefficient, a parameter, defining the shape of the bubble. We consider space-time distribution [17] of the electromagnetic field inside the different bubbles (2D) with propagation direction , where  is the bubble velocity; the transverse direction is along -axis. We carry out Hamiltonian analysis [51] of the trapped electrons as , where  and  are longitudinal and transverse canonical momenta of the electrons and  be the longitudinal and transverse kinetic momenta of the electrons.  is the relativistic Lorentz factor of the electrons. and . and  represents the functions of  and .  is normalized with , is normalized with , where  is the mass of electron. and  are scalar and vector potentials of the electromagnetic field. Time is normalized with  and the length by .

Equation of motion of the trapped plasma electrons inside the bubble (cavity) is determined by Hamiltonian equations of motion.

 

 

 

 

 

Now putting the expression (1) in (3) and (4), we find

 

 

Now final four coupled differential equations are

 

 

 

 

 

3. Results and Discussion

Above four coupled differential equations are solved using fourth order Runge-Kutta method by using ode45 MATLAB numerical method. Here we use initial conditions as , where is an impact parameter called the radial distance between the electrons and laser pulse. For under-dense plasma, initial value of Lorentz factor depends on the bubble velocity that is  [15] which is equal to the laser pulse group velocity. Bubble radius and impact parameter have been defined in Figure 1.

 

                                                                        

 

Figure 1: This figure shows the parameters related with bubble and  is bubble radius, where  and  are longitudinal and transverse coordinates.

 

Shape of the bubble 

We have used wake potential of the generated bubble via equation (12). Hence,

                                                            (13)

The values of geometrical coefficient C vary from . When , we get spherical bubble, at , we get longitudinal ellipsoid bubble and at , we get transverse ellipsoid bubble and these investigations on bubble shape has been already carried out by Kumar et al. [17]. Now Hamiltonian analysis of the trapped electrons is carried out with bubble velocity V = 0.969, i.e. slower than the speed of light, impact parameter  and radius of the bubble as . Here we consider different values of  for the trajectory of the trapped electrons, and longitudinal and transverse phase-space portrait.

 

Figure 2: Trajectory of electrons with variation of shapes of the bubble with C = 0 (spherical bubble), C= - 0.2 (transverse ellipsoid bubble) and C= 0.2 (longitudinal ellipsoid bubble) when  and bubble velocity  for Lorentz factor Here radius of the bubble .

 

Trajectory of self-injected plasma electrons in different shaped bubble

Self-injection scheme of the plasma electrons has been understood by finding trajectory of these electrons in  direction following the process developed by Lu et al. [48] but they did the work only for spherical bubble regime. So, we have extended their work and found the trajectory in different shaped bubble as shown in Figure 2. We have found the trajectory of the self-injected plasma electrons with different shaped bubble such as spherical bubble regime at , longitudinal ellipsoid bubble regime at  and transverse ellipsoid bubble at . The results show that the electrons in transverse ellipsoid retrace their path greater than with the cases of longitudinal ellipsoid and spherical bubble.

 

 

 

Figure 3: Longitudinal phase-space diagram of trapped electrons in different shaped bubble with C = 0 for spherical bubble (a), with C = 0.2 for longitudinal ellipsoid bubble (b) and with C = -- 0.2 for transverse ellipsoid bubble (c), when V = 0.969 for  with impact parameter of the trapped electron as  and radius of the bubble as R = 10.

 

Longitudinal phase-space of self-injected plasma electrons in different shaped bubble

In Figure 3, we have plotted longitudinal phase-space diagram of the trapped plasma electrons. Here we observe that with transverse bubble regime at  as shown in Figure 3(c), the electrons move more times than with spherical bubble at  as shown in Figure 3(a) and with longitudinal ellipsoid bubble at  as shown in Figure 3(b).

 

 

 

Figure 4: Transverse phase-space diagram of the trapped electrons in different shaped bubble when C = 0 (spherical bubble) in (a), C = 0.2 (longitudinal ellipsoid bubble) in (b) and C = - 0.2 (transverse ellipsoid bubble) in (c) V = 0.969 for  with impact parameter of the trapped electron as  and radius of the bubble as R = 10.

 

Transverse phase-space of self-injected plasma electrons in different shaped bubble

In Figure 4, after plotting the transverse phase-space of the trapped or self-injected plasma electrons, we have understood that with the transverse ellipsoid bubble at C = - 0.2 as shown in Figure 4(c), the electrons retrace their path better than with the spherical bubble case at C = 0 as shown in Figure 4(a) and with the longitudinal ellipsoid bubble at C = 0.2 as shown in Figure 4(b).

 

Effect of impact parameter on electron trajectory

Impact parameter is defined as the radial distance between the trapped electrons and the laser pulse as the driver. If the value of the impact parameter is too small, then the trapped electrons get deflected by the generated wake. Here we consider different values of the impact parameter in different shapes of the bubble, such as longitudinal and transverse ellipsoid shaped bubble. The impact parameter is defined as the transverse component . We have investigated the effect of  on spherical bubble at , longitudinal ellipsoid bubble at  and transverse ellipsoid bubble at . For these investigations, we have used three different values of the impact parameter as  as shown in Figures 5, 6 and 7, respectively. We have found that with transverse ellipsoid bubble case, self-injected plasma electrons travel large distance in longitudinal as well as in transverse directions (Figure 7) as compared to the case of spherical bubble (Figure 5) and longitudinal ellipsoid bubble (Figure 6).

 

Figure 5: Trajectory of the electrons in spherical bubble at C = 0 and with initial electrons momentum value  and  for Lorentz factor Here radius of the bubble is  with different values of impact parameter .

 

Figure 6: Trajectory of the electrons in longitudinal ellipsoid bubble at C = 0.2 and with initial electrons momentum value  and  for Lorentz factor Here radius of the bubble is  with different values of impact parameter .

 

Figure 7: Trajectory of the electrons in transverse ellipsoid bubble at C = 0.2 and with initial electrons momentum value  and  for Lorentz factor Here radius of the bubble is  with different values of impact parameter .

 

Effect of bubble radius on electron trajectory

Bubble radius is an important parameter for defining the size of the bubble. Here we have used different size of the bubble by defining different values of radius of the different shape of bubble and for this, we have first considered the spherical bubble at  with different values of bubble radius . Figure 8 shows the trajectory of the electrons in spherical bubble when  with fixed value of the bubble velocity as  and fixed impact parameter . We have found that with bubble radius , the trapped electrons after the self-injection process move again for another self-injection process as compared to the cases of  and . Now for the second case, at  for longitudinal ellipsoid bubble, no second self-injection of the electrons has been seen for different size of the bubble (Figure 9). But, for the transverse ellipsoid bubble at  as shown in Figure 10, the second self-injection has been possible with different size of the bubble at  and  and this behaviour has not been possible with spherical bubble and longitudinal ellipsoid bubble.

 

 

 

Figure 8: Trajectory of the electrons in spherical bubble at C = 0.0 and with bubble velocity  for Lorentz factor Here the impact parameter is   with different values of bubble radius .

 

 

Figure 9: Trajectory of the electrons in longitudinal ellipsoid bubble at C = 0.2 and with bubble velocity  for Lorentz factor Here the impact parameter is   with .

 

 

 

Figure 10: Trajectory of the electrons in transverse ellipsoid bubble at C = - 0.2 and with bubble velocity  for Lorentz factor Here the impact parameter is  with .

 

Effect of bubble Velocity

Bubble velocity is a very important parameter for controlling the bubble shape and trapping of the electrons. We consider here different values of bubble velocity for determining the trajectory of the self-injected plasma electrons. We discuss the effect of slower bubble velocity in three different bubble shapes. For the first case, we use three different values of the bubble velocity as  for ,  for  and  for  in spherical bubble at  as shown in Figure 11. We observe that with slower bubble velocity , the self-injected plasma electrons travel in larger spherical region as compared to the case with  and  (Figure 11) with fixed value of bubble radius  and impact parameter . Now for longitudinal ellipsoid bubble at , large distance is only covered in longitudinal direction at  but not in transverse direction and no further self-injection has been possible for  and also we observe that with  and , longitudinal as well as transverse distances covered by the electrons are lower (Figure 12). When the bubble shape is changed from the spherical and longitudinal ellipsoid to transverse ellipsoid at  (Figure 13), we see that the self-injected plasma electrons cover a large distance in transverse ellipsoid with slower bubble velocity with  in comparison with the spherical bubble and longitudinal ellipsoid bubble. With  and , we find that the electrons make larger excursion in transverse ellipsoid bubble than in the spherical bubble and longitudinal ellipsoid bubble.

 

Figure 11: Trajectory of the electrons in spherical bubble at  with bubble radius  and impact parameter . Here, different bubble velocities have been considered as .

 

 

Figure 12: Trajectory of the electrons in longitudinal ellipsoid bubble at  with bubble radius  and with impact parameter . Here, different bubble velocities have been considered as .

 

 

Figure 13: Trajectory of the electrons in transverse ellipsoid bubble at  with bubble radius  and impact parameter . Here, different bubble velocities have been considered as .

 

 

4. Conclusions

We investigated the self- injection process of the trapped plasma electrons inside different shaped bubbles named as spherical bubble, longitudinal ellipsoid bubble, and transverse ellipsoid bubble by using 4th order Runge – Kutta method with ode45 MATLAB code. We found that the trapping is more efficient in the transverse ellipsoid bubble than the spherical bubble and the longitudinal ellipsoid bubble. We showed the plots for visualising the longitudinal and transverse phase-spaces of the trapped electrons inside the bubbles of each shape. More phase-space is found to be created in the transverse ellipsoid bubble than in the spherical and longitudinal ellipsoid bubbles, which is better for the trapped electrons for acquiring larger energy and the efficient acceleration.

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