Jitender Singh¹, Jyoti Rajput¹, Harjit Singh Ghotra^1,2, Niti Kant^,1,∗1Department of Physics, Lovely Professional University, G.T. Road, Phagwara (144411), Punjab, India
²Extreme Light Infrastructure-Nuclear Physics(ELI-NP), Horia Hulubei NIPNE, 30 Reactorului Street, 077125 Magurele, Jud.Iifov, Romania

First author contact: ^∗Author to whom any correspondence should be addressed
Received:2021-03-25Revised:2021-05-18Accepted:2021-05-19Online:2021-07-16


Abstract
In this paper, a radially polarised cosh-Gaussian laser beam (CGLB) is used to study the electron acceleration produced in vacuum. A highly energetic electron beam can be achieved by a CGLB, even with comparatively low-powered lasers. The properties of a CGLB cause it to focus earlier, over a shorter duration than a Gaussian laser beam, which makes it suitable for obtaining high energies over small durations. It is found that the energy gained by the electrons strongly depends upon the decentering parameter of the laser profile. It is also observed that for a fixed value of energy gain, if the decentering parameter is increased, then the intensity of the laser field decreases. The dependence of the energy gained by electrons on the laser intensity and the laser-spot size is also studied.
Keywords： cosh-Gaussian laser beam;vacuum;electron acceleration;decentered parameter


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Jitender Singh, Jyoti Rajput, Harjit Singh Ghotra, Niti Kant. Electron Acceleration by a radially polarised cosh-Gaussian laser beam in vacuum. Communications in Theoretical Physics, 2021, 73(9): 095502- doi:10.1088/1572-9494/ac02b6

1. Introduction

The development of mechanisms that can yield highly energetic particles in very small spaces and times is the most fascinating challenge the scientific community is addressing today. High-energy particles are considered to be a potential way to decode the mysteries of how basic material particles and our whole universe originated. This would not only improve our tools for understanding various chemical, physical, and biological process, but also has the potential to modify the all the processes involved in the medical sciences. The acceleration of charged particles involves the simple mechanism of subjecting the charged particles to electric fields. The acceleration obtained depends upon the magnitude of the electric field applied. The various facilities involved in high-energy particle physics use very large structures to obtain continuously increasing electric fields to accelerate the charged particles to very high energies. A huge amount of money and large amounts of space have been devoted to this, and there is a constant need to improve the quality of the energetic beams obtained.

In 1979, the whole concept changed, when Tajima and Dawson proposed that by aiming an intense electromagnetic pulse into a plasma, a wake field could be generated and that the electrons trapped in the wake could be made to accelerate to very high energies [1]. Sugihara et al [2] showed that when an electromagnetic wave was suddenly applied to a plasma, it then heated the plasma particles, which showed a large acceleration. Strickland et al [3] demonstrated a mechanism in which a very high-powered pulse laser was obtained by the amplification of any short pulse. Using this technique, it is possible to compress the pulse to a few attoseconds, or even to zepto-seconds, to obtain laser intensities of the order of (10²²–10²⁴) W cm⁻² [4]. Malka et al [5] reported the experimental results of MeV electron generation by an ultra-intense linearly polarised laser pulse in vacuum. Their initial laser plasma experiments used just one laser, which was solely responsible for creating the plasma and generating the wake field. Improvements have now been made to the design; additional high-power laser systems are used, which increase the transverse plasma density gradient by a large amount, yielding higher energies of up to 8 GeV [6].

The main aim of all these mechanisms is to obtain electron beams which are monoenergetic or have a very small energy spread. Important results for high-quality electron beams were reported in 2004 [7–9].

Using different laser-beam profiles and polarizations, the effects of an externally applied magnetic field, laser-spot size, laser-pulse duration, laser intensity, and the initial phase in improving the quality and energies of the accelerated beams have been reported [10–14].

Many complex laser-beam profiles, such as complex Hermite–Gaussian beams [15], Laguerre–Gaussian beams [16, 17] and Bessel–Gaussian beams [18] have been used for electron acceleration.

In this work, the CGLB profile is used to study electron acceleration in vacuum. It has been shown that CGLB possesses more power and has ability to focus earlier than the Gaussian beam [19–22]. Also, these beams are found to be potentially much more efficient in extracting the energy from conventional laser amplifiers [23]. These properties of CGLB make it suitable for electron acceleration. The effects of strong self-focusing on the acceleration of electrons in plasma were reported in [24]. The effects of laser intensity, laser-spot size, and the laser beam’s width parameter on the energy gained by electrons in vacuum are presented in this paper.

2. Electron dynamics

Consider a cosh-Gaussian laser beam propagating along the z-axis in vacuum. The electric-field vector is given by(1)$\begin{eqnarray}\overrightarrow{E}=\hat{r}{E}_{r}+\hat{z}{E}_{z}.\end{eqnarray}$The transverse component of a cosh-Gaussian radially polarized laser beam is expressed as(2)$\begin{eqnarray}{E}_{r}={E}_{0}\,\cosh \left(br/{r}_{0}\right)\,\exp \left(-{r}^{2}/{r}_{0}^{2}\right)\exp \left\{{\rm{i}}\left(kz-\omega t\right)\right\}.\end{eqnarray}$The longitudinal component can be obtained by using Maxwell’s equations in vacuum, $\overrightarrow{{\rm{\nabla }}}.\overrightarrow{E}=0$ and is represented as(3)$\begin{eqnarray}{E}_{z}=-({\rm{i}}/kr){E}_{r}\left[1+\left(br/{r}_{0}\right)\tanh \left(br/{r}_{0}\right)-2{r}^{2}/{r}_{0}^{2}\right],\end{eqnarray}$where $\omega $ is the laser frequency, $k=\omega /c$ is the propagation constant, $c$ is the velocity of light in vacuum, ${E}_{0}$ is the amplitude of the electric field at a central position of z, i.e., at z=0, ${r}_{0}$ is the waist width of the Gaussian amplitude distribution, and b is the normalized modal parameter or the decentered parameter. The components of the magnetic field of the laser pulse are obtained using Maxwell’s equations and can be expressed as(4)$\begin{eqnarray}{B}_{L}=\left({E}_{r}/c\right)\hat{\theta }.\end{eqnarray}$The Lorentz force equation is given by(5)$\begin{eqnarray}\overrightarrow{F}={\rm{d}}\overrightarrow{p}/{\rm{d}}t={\rm{d}}\left(\gamma {m}_{0}v\right)/{\rm{d}}t=-e\left[\overrightarrow{E}+\left(\overrightarrow{v}\times \overrightarrow{B}\right)\right],\end{eqnarray}$where e and ${m}_{0}$ are the electron’s charge and mass, respectively. The equations governing electron momentum and energy are(6)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{p}_{r}}{{\rm{d}}t}=-e{E}_{r}\left(1-\displaystyle \frac{{v}_{z}}{c}\right),\end{eqnarray}$(7)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{p}_{z}}{{\rm{d}}t}=-e\left({E}_{z}+\displaystyle \frac{{v}_{r}{E}_{r}}{c}\right),\end{eqnarray}$(8)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{p}_{\theta }}{{\rm{d}}t}=0,\end{eqnarray}$(9)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}\gamma }{{\rm{d}}t}=-\displaystyle \frac{e}{{m}_{0}{c}^{2}}\left[{E}_{r}{v}_{r}+{E}_{z}{v}_{z}\right],\end{eqnarray}$where $\gamma =\sqrt{1+\left({p}_{r}^{2}+{p}_{\theta }^{2}+{p}_{z}^{2}\right)/{m}_{0}^{2}}{c}^{2}$ is the Lorentz factor.

The normalized equations are given by(10)$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}\gamma }{{\rm{d}}t^{\prime} }=-{a}_{0}\,\cosh \left(\displaystyle \frac{br^{\prime} }{{r^{\prime} }_{0}}\right)\exp \left(\displaystyle \frac{-{r^{\prime} }^{2}}{{r^{\prime} }_{0}^{2}}\right)\left[\cos \left(\phi \right){\beta }_{r}\right.\\ \,+\left.\sin \left(\phi \right)\left(\displaystyle \frac{{\beta }_{z}}{k^{\prime} r^{\prime} }\right)\left\{1+\left(\displaystyle \frac{br^{\prime} }{{r^{\prime} }_{0}}\right)\tanh \left(\displaystyle \frac{br^{\prime} }{{r^{\prime} }_{0}}\right)-\displaystyle \frac{2{r^{\prime} }^{2}}{{r^{\prime} }_{0}^{2}}\right\}\right],\end{array}\end{eqnarray}$(11)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{\beta }_{r}}{{\rm{d}}t^{\prime} }=-\displaystyle \frac{{a}_{0}}{{c}_{1}}\,\cosh \left(\displaystyle \frac{br^{\prime} }{{r^{\prime} }_{0}}\right)\exp \left(-\displaystyle \frac{{r^{\prime} }^{2}}{{r^{\prime} }_{0}^{2}}\right)\cos \left(\phi \right)\left(1-{\beta }_{z}\right),\end{eqnarray}$(12)$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}{\beta }_{z}}{{\rm{d}}t^{\prime} }=-\displaystyle \frac{{a}_{0}}{{c}_{2}}\,\cosh \left(\displaystyle \frac{br^{\prime} }{{r^{\prime} }_{0}}\right)\exp \left(-\displaystyle \frac{{r^{\prime} }^{2}}{{r^{\prime} }_{0}^{2}}\right)\,\left[\cos \left(\phi \right){\beta }_{r}\right.\\ \,+\left.\sin \left(\phi \right)\left(\displaystyle \frac{1}{k^{\prime} r^{\prime} }\right)\left\{1+\left(\displaystyle \frac{br^{\prime} }{{r^{\prime} }_{0}}\right)\tanh \left(\displaystyle \frac{br^{\prime} }{{r^{\prime} }_{0}}\right)-\displaystyle \frac{2{r^{\prime} }^{2}}{{r^{\prime} }_{0}^{2}}\right\}\right],\end{array}\end{eqnarray}$where $\begin{array}{l}{c}_{1}=\left[1/\sqrt{1-{\beta }_{r}^{2}}+{\beta }_{r}^{2}/{\left(1-{\beta }_{r}^{2}\right)}^{3/2}\right],\\ {c}_{2}=\left[1/\sqrt{1-{\beta }_{z}^{2}}+{\beta }_{z}^{2}/{\left(1-{\beta }_{z}^{2}\right)}^{3/2}\right],\end{array}$ and $\phi =\left(t^{\prime} -k^{\prime} z^{\prime} \right).$

The various normalized parameters used throughout this paper are as follows:$\begin{eqnarray*}\begin{array}{l}{a}_{0}=e{E}_{0}/{m}_{0}\omega \,c,t^{\prime} =\omega \,t,{\beta }_{r}={v}_{r}/c,\\ {\beta }_{z}={v}_{z}/c,r^{\prime} =r\omega /c,z^{\prime} =z\omega /c,\\ k^{\prime} =k\,c/\omega ,{r^{\prime} }_{0}={r}_{0}\,\omega /c.\end{array}\end{eqnarray*}$Equations (10)–(12) are ordinary coupled differential equations. These equations are solved using different numerical values for the variables and constants and the results are plotted using computer simulations for the gain in energy with time.

3. Results and discussions

Equations (10)–(12) are solved numerically for electron-energy gain. ${p}_{r}$ ${p}_{\varsigma }$ and ${p}_{z}$ are the r, θ, and z components of momentum. The momentum is given by $p=\gamma {m}_{0}v.$ β_r and β_z are the radial and longitudinal components of normalized velocity, which is given by β=v/c. The following laser parameter values are used for the numerical simulations. The angular frequency of the laser beam is $\omega =1.88\times {10}^{15}$ rad/s, corresponding to a laser wavelength of $\lambda =1$ μm. The value of the normalized propagation constant $k^{\prime} =1$ (corresponding to $k=6.28\times {10}^{6}$ m⁻¹). The initial electron energy is taken to be 1 MeV for $z=0.$ If the electrons are pre-accelerated or given some initial energy, then there is an effective exchange of energy between the laser and the electrons, and the beam-collimation effect accelerates electrons to relativistic energies. Also, the normalized beam-waist width is ${r^{\prime} }_{0}=70$ (corresponding to a laser-spot size ${r}_{0}=11.13$ μm).

The values of the normalized initial laser-intensity parameters used are ${a}_{0}=1$ (with a corresponding intensity $I=1.37\times {10}^{18}$ W cm⁻²), ${a}_{0}=2.5$ ($I=8.56\times {10}^{18}$ W cm⁻²), ${a}_{0}=5$ ($I=3.43\times {10}^{19}$ W cm⁻²), and ${a}_{0}=7.5$ ($I=7.70\times {10}^{20}$ W cm⁻²).

Figure 1 shows the variations of electron-energy gain ($\gamma {m}_{0}{c}^{2}$) with normalized time obtained by using different values of the normalized laser intensity ${a}_{0}.$ The beam-waist width is ${r^{\prime} }_{0}=70$ and the decentering parameter $b=0.5.$ It is observed that with an increase in the intensity of laser beam, the energy gained by the electrons increases along the direction of propagation of the beam. For ${a}_{0}=1,$ an energy gain of about 3.37 GeV is observed, and for ${a}_{0}=7.5,$ the energy gain is increased to 4.23 GeV. With an increase in the intensity of the laser beam, the traversal of longitudinal distance by the electrons and their interaction with the laser pulse take place over a comparatively larger duration. The ponderomotive force on the electrons increases with laser intensity due to an increase in the longitudinal component of the laser field, so the electrons’ energy increases. The ponderomotive force is a nonlinear force that depends upon the intensity distribution of the electric field, and it causes the particles to move towards areas of weaker field strength. The ponderomotive force generally includes the Lorentz force experienced by free charges and the forces exerted by an electromagnetic field on a medium.

Figure 1.

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Figure 1.Electron-energy gain with normalized time for specific values of the laser-intensity parameter ${a}_{0}$= 1, 2.5, 5, and 7.5 for ${r}_{0}^{{\prime} }=70$ and $b=2.5.$


Electrons are trapped and accelerated to high energies while interacting with the leading part of the laser pulse. When they reach the trailing part of the pulse, their energy gain almost saturates [15].

Figure 2 shows the variation of electron-energy gain with different values of the decentering parameter b for fixed values of laser intensity ${a}_{0}=2.5$ and laser-spot size $\left({r^{\prime} }_{0}=70\right).$ Plots are obtained for b=0.2, 0.5, 0.8, and 1.1. It can be observed that the energy gain strongly depends upon the decentering parameter of the laser beam. An energy gain of 3.26 GeV is observed for b=0.2, which increases to 6.06 GeV for b=1.1. With an increase in the decentering parameter, the distribution of intensity in the axial direction increases. This indicates an increase in the focal depth and the shrinkage of the focal spot [25], which strongly increases the energy gained by the electrons.

Figure 2.

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Figure 2.Electron-energy gain with normalized time for specific values of the decentering parameter b=0.2, 0.5, 0.8, and 1.1 at ${a}_{0}=2.5,$ and ${r}_{0}^{{\prime} }=70.$


Figure 3 shows the dependence of the energy gained by the electrons on the laser-spot size and the beam waist (${r}_{0}$). Graphs are plotted for different values of the normalized beam-waist width ${r^{\prime} }_{0}$= 50 (${r}_{0}$= 8.4 μm), ${r^{\prime} }_{0}$ =70 (${r}_{0}$= 11.8 μm), ${r^{\prime} }_{0}$ =90 (${r}_{0}$ =15.2 μm), and ${r^{\prime} }_{0}$ =110 (${r}_{0}$= 18.6 μm), by keeping fixed values for ${a}_{0}=2.5,$ and $b=0.5.$ The beam waist is an important parameter which determines the confocal parameter. This means that the smaller the value of the beam waist, the shorter the distance over which the laser beam is relatively collimated.

Figure 3.

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Figure 3.Electron-energy gain with the normalized time for different values of the laser-beam width ${r^{\prime} }_{0}$= 50, 70, 90, and 110 at ${a}_{0}=2.5$ and $b=0.5.$


It can be observed that the gain in energy increases with an increase in beam waist for a given fixed intensity of the laser beam. An energy gain of 1.81 GeV is observed for ${r^{\prime} }_{0}$ =50, which increases to 9.19 GeV for ${r^{\prime} }_{0}$ =110. With an increase in the beam waist, the divergence of the laser beam decreases. Certain valleys are observed with increasing spot size. There is a decrease in energy for electrons that originate away from the axis. The valley in the curve is due to the complex trajectory of electrons under the influence of the laser field and due to the dependence of energy on phase [26]. The intensity of the laser beam is maintained up to several Rayleigh lengths. This results in higher energy gain with an increase in the laser-spot size.

Figure 4 shows the variation of the decentering parameter with the intensity of the laser beam for a fixed value of energy gain. For ${a}_{0}$=1.0, 2.5, 5.0, and 7.5, values of b=0.83, 0.73, 0.57, and 0.50 are observed for the same value of energy gain of 4.237 GeV. It can be seen that with an increase in the intensity of the CGLB, the value of the decentering parameter decreases for the same value of the energy gained. The self-focusing of the CGLB is highly sensitive to the value of its decentering parameter. With an increase in intensity, the self-focusing is reported to become stronger with decreasing values of b [27]. This helps electrons to gain large energies for comparatively lower values of the incident laser-beam intensity.

Figure 4.

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Figure 4.Variation of the decentering parameter b with the normalized incident laser-beam intensity a₀ for a fixed value of the energy gained by electrons (4.237 GeV) and ${r^{\prime} }_{0}=70.$

4. Conclusions

This work demonstrates that the cosh-Gaussian laser-beam profile possess higher power than the previously used Gaussian beams. The tendency of the cosh-Gaussian beams to focus earlier enables them to accelerate the charged particles to very high energies with relatively lower values of laser intensity. The decentering parameter of the CGLB strongly influences the maximum energy gain, and by keeping this parameter at high values, a larger energy gain can be obtained, even by a low-power laser. Laser-spot size also plays a significant role in accelerating the electron beams. Small spot sizes result in large scattering of the beam so that it diverges quickly and electrons cannot gain sufficient energy, but when the spot size increases, the beam remains collimated up to a greater length, electrons spend longer in the accelerating fields, and attain high energies. Since low-power laser sources can easily be realised, by using suitable values of the other parameters, we can effectively use cosh-Gaussian laser beams for efficient electron acceleration.

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