Electron dynamic in externally magnetized ion channel by considering self-magnetic field effects

In the scenario of the high-power laser-plasma interaction, a large-amplitude plasma wave is driven while the duration of the laser pulse is comparable to the plasma wave period. On the other hand, an ion channel can be created when the electrons are expelled from the high-intensity region making a density depression on the laser axis. Furthermore, during the laser pulse propagation through the plasma, a large self-magnetic field can be generated by the produced energetic electrons which carry an electric current (Tatarakis et al., Reference Tatarakis, Gopal, Watts, Beg, Dangor, Krushelnick, Wagner, Norreys, Clark, Zepf and Evans2002a, Reference Tatarakis, Watts, Beg, Clark, Dangor, Gopal, Haines, Norreys, Wagner, Wei, Zepf and Krushelnick2002b; Gopal et al., Reference Gopal, Gupta, Kim, Hur and Suk2016). This self-magnetic field could have an important role on the electron acceleration mechanism in these systems. In this study, the electron acceleration in the laser-plasma interaction system is investigated in the presence of an external axial guiding magnetic field by taking into account the self-magnetic field effects in the calculations. We consider the self-magnetic field generated in the wake of the laser as follows (Singh et al., Reference Singh, Gupta, Bhasin and Tripathi2003; Gupta et al., Reference Gupta, Kaur, Gopal and Suk2016):

(1)$$B_{\rm s} = \lpar B_{\rm s}/r_0\rpar e^{{-}x^2/2r_0^2 }\lpar {-}x\overset{\frown}{y}\rpar \comma $$

where B _s and r ₀ represent to the self-magnetic field amplitude and the spot size of laser pulse, respectively. Also, we assume an external axial magnetic field with the amplitude B ₀ as follows:

(2)$$B_z = B_0\overset{{\frown}}z.$$

In this system, it is considered a nonlinear profile for the generated plasma wave as follows (Yadav et al., Reference Yadav, Sharma and Gupta2018):

(3)$$\eqalign{E = & \hat{x}A_{\rm p}{\rm exp}\left({\displaystyle{{-x^2} \over {r_{\rm p}^2 }}} \right)\displaystyle{{2x} \over {kr_{\rm p}^2 }}{\rm sin}\lpar {\rm \omega }t-kz + {\rm \theta} _0\rpar \cr + & \overset{{\frown}}zA_{\rm p}{\rm exp}\left({\displaystyle{{-x^2} \over {r_{\rm p}^2 }}} \right){\rm cos}\lpar {\rm \omega }t-kz + {\rm \theta} _0\rpar \comma } $$

where A _p is the electron wave amplitude, r _p is the wakefield radius, and θ₀ is the initial wave phase. An electron density depleted ion channel with a radial electric field affecting the dynamics of the injected electrons is also considered. The electrostatic force generated by the performed ion channel can be written as follows:

(4)$$E_r = \nabla {\rm \varphi } = \nabla \left[{{\rm \varphi }_{\bf 0}\displaystyle{{\lpar 1-r^2\rpar } \over {r_0^2 }}} \right] = {-}2{\rm \varphi }_{\bf 0}\displaystyle{r \over {r_0^2 }}\overset{{\frown}}r\comma$$

where φ₀ is the ion channel potential amplitude.

The energy and momentum equations for the relativistic electron are given by

(5)$$\displaystyle{{dP} \over {dt}} = {-}e\lpar E + E_r\rpar + \displaystyle{{V \times B} \over c}\comma \;$$

(6)$$\displaystyle{{d\gamma } \over {dt}} = \displaystyle{{-e} \over {m_0c^2}}\lpar E + E_r\rpar \cdot V\comma \;$$

where B = B _s + B _z and P = γmv with ${\rm \gamma} = 1 + \lpar \lpar p_x^2 + p_y^2 + p_z^2 \rpar /m^2c^2\rpar$ is the momentum of the electron. Substituting the plasma wave field, ion channel potential, the external- and the self-magnetic fields from Eqs. (1)–(4) in momentum and energy equations and then by rewriting them into components, we have

(7)$$\eqalign{\displaystyle{{dp_x} \over {dt}} = & -eA_{\rm p}{\rm exp}\left({\displaystyle{{-x^2} \over {r_{\rm p}^2 }}} \right)\displaystyle{{2x} \over {kr_{\rm p}^2 }}{\rm sin}\lpar {\rm \omega }t-kz + {\rm \theta} _0\rpar \cr -& e{\rm \varphi }_0\displaystyle{{2x} \over {r_0^2 }}-\displaystyle{{eV_yB_0} \over c}-x\displaystyle{{eV_zB_s{\rm exp}\lpar {-}x^2/2r_0^2 \rpar } \over {r_0c}}\comma } $$

(8)$$\displaystyle{{dp_y} \over {dt}} = {-}e{\rm \varphi }_0\displaystyle{{2y} \over {r_0^2 }} + \displaystyle{{eV_xB_0} \over c}\comma $$

(9)$$\eqalign{\displaystyle{{dp_z} \over {dt}} = & -eA_{\rm p}{\rm exp}\left({\displaystyle{{-x^2} \over {r_{\rm p}^2 }}} \right){\rm cos}\lpar {\rm \omega }t-kz + {\rm \theta} _0\rpar \cr + & x\displaystyle{{eV_XB_s{\rm exp}\lpar {-}x^2/2r_0^2 \rpar } \over {r_0c}}\comma } $$

(10)$$\eqalign{\displaystyle{{d\gamma } \over {dt}} = & \displaystyle{{-eA_{\rm p}v_z} \over {m_0c^2}}{\rm exp}\left({\displaystyle{{-x^2} \over {r_{\rm p}^2 }}} \right){\rm cos}\lpar {\rm \omega }t-kz + {\rm \theta} _0\rpar \cr -& \displaystyle{{eA_{\rm p}v_x} \over {m_0c^2}}{\rm exp}\left({\displaystyle{{-x^2} \over {r_{\rm p}^2 }}} \right)\displaystyle{{2x} \over {kr_{\rm p}^2 }}{\rm sin}\lpar {\rm \omega }t-kz + {\rm \theta} _0\rpar \cr -& e{\rm \varphi }_0\displaystyle{{2xv_x} \over {m_0c^2r_0^2 }}-e{\rm \varphi }_0\displaystyle{{2yv_y} \over {m_0c^2r_0^2 }}.} $$

We normalize Eqs. (7)–(10) using the dimensionless physical quantities as: x′ → kx, t′ → ωt, P′ → P/mc, k′ → kc/ω, a _p → eA _p/m ₀ωc, z′ → kz, $r_1^2 \to k^2r_0^2$, $r_2^2 \to k^2r_{\rm p}^2$, φ₀ → eφ₀/m ₀c ², ${\rm \Omega }_{\rm \iota} \to eB_0/m_0{\rm \omega }$, Ω_s → eB _s/m ₀ω.

Thus, Eqs. (7)–(10) can be written as follows:

(11)$$\eqalign{\displaystyle{{dp_x^{\prime} } \over {dt^{\prime}}} = & -a_{\rm p}{\rm exp}\left({\displaystyle{{-x^{{\prime}2}} \over {r_2^2 }}} \right)\displaystyle{{2x^{\prime}} \over {kr_2^2 }}{\rm sin}\lpar t^{\prime}-z^{\prime} + {\rm \theta} _0\rpar \cr -& k^{\prime}{\rm \varphi} _0\displaystyle{{2x^{\prime}} \over {r_1^2 }}-\displaystyle{{ep^{\prime}_y\Omega _{\rm \iota} } \over {\rm \gamma} } + \displaystyle{{ep^{\prime}_z\Omega _{\rm s}x^{\prime}{\rm exp}\lpar {-}x^{{\prime}2}/2r_1^2 \rpar } \over {r_1{\rm \gamma} }}\comma }$$

(12)$$\displaystyle{{dp_y^{\prime} } \over {dt^{\prime}}} = {-}k^{\prime}{\rm \varphi} _0\displaystyle{{2y^{\prime}} \over {r_1^2 }} + \displaystyle{{ep^{\prime}_x\Omega _{\rm \iota} } \over {\rm \gamma }}\comma $$

(13)$$\eqalign{\displaystyle{{dp_z^{\prime} } \over {dt^{\prime}}} = & -a_{\rm p}\exp \left({\displaystyle{{-x^{{\prime}2}} \over {r_2^2 }}} \right)\cos \lpar t^{\prime}-z^{\prime} + {\rm \theta} _0\rpar \cr -& \displaystyle{{ep_x^{\prime} \Omega _{\rm s}x^{\prime}\exp \lpar {-}x^{{\prime}2}/2r_1^2 \rpar } \over {r_1{\rm \gamma} }} \comma} $$

(14)$$\eqalign{\displaystyle{{d{\rm \gamma }} \over {dt^{\prime}}} = & \displaystyle{{-a_{\rm p}p^{\prime}_z} \over {\rm \gamma }}{\rm exp}\left({\displaystyle{{-x^{{\prime}2}} \over {r_2^2 }}} \right)\displaystyle{{2x^{\prime}} \over {kr_2^2 }}{\rm cos}\lpar t^{\prime}-z^{\prime} + {\rm \theta} _0\rpar \cr -& \displaystyle{{a_{\rm p}} \over {\rm \gamma} }p^{\prime}_x{\rm exp}\left({\displaystyle{{-x^{{\prime}2}} \over {r_2^2 }}} \right){\rm sin}\lpar t^{\prime}-z^{\prime} + {\rm \theta} _0\rpar -\displaystyle{{a_{\rm p}} \over \gamma }p^{\prime}_y \cr & {\rm exp}\lpar \displaystyle{{-x^{{\prime}2}} \over {r_2^2 }}\rpar {\rm sin}\lpar t^{\prime}-z^{\prime} + {\rm \theta} _0\rpar -k^{\prime}{\rm \varphi} _0\displaystyle{{2p^{\prime}_xx^{\prime}} \over {{\rm \gamma} r_1^2 }}-k^{\prime}{\rm \varphi} _0\displaystyle{{2p^{\prime}_yy^{\prime}} \over {{\rm \gamma} r_1^2 }}.} $$

Equations (11)–(14) are the coupled differential equations that describe the dynamics of the relativistic electron in an axial magnetized ion channel by considering the self-magnetic field effects. These equations are solved numerically using a relativistic 3D single-particle code with a fourth-order Runge–Kutta algorithm.

Numerical results and discussion

A numerical model is proposed to describe the relativistic electron dynamics in an ion channel. In this system, the electron is affected by the radial field of the channel in the presence of the plasma wave and an external axial guiding magnetic field by taking into account the influence of the self-magnetic field. The self-magnetic field is produced by energetic electrons carrying an electric current. We have numerically solved the relativistic equations of motion and the energy equation using a 3D single-particle simulation code containing the fourth-order Runge–Kutta algorithm. The electron energy gain has been estimated for different parameters of the plasma wave, ion channel, external magnetic field, as well as the self-magnetic field. These parameters are also optimized to obtain the most appropriate system for the best electron energy gain. For calculations, we use the initial dimensionless parameters as k = 1.01, θ₀ = π/2, r ₁ = 4, r ₂ = 2, X ₀ = 0.5, Y ₀ = 0.0, Z ₀ = 0.0, V _x0 = 0.3, V _y0 = 0.1, V _z0 = 0.8. The effects of the axial guiding magnetic field on the electron acceleration and its trajectory in the ion channel have been indicated in Figures 1a and 1b. An additional axial guiding magnetic field can modify the acceleration gradient by confining the electrons motion. In an electron acceleration process, a number of electrons escape from the interaction zone quickly and do not receive enough energy from the plasma wave. In order to confine the energetic electrons in the accelerating zone, we applied an external axial field along with the wave propagation direction. Figure 1a shows the variation of the electron energy γ with the normalized distance z for different axial guiding magnetic field amplitudes Ω_s = 0.0, 0.3, 0.5 for a _p = 0.5, φ₀ = 0.6, and Ω_ι = 0.0. As shown in the figure, γ increases with increasing the amplitude of the magnetic field. The guiding magnetic field causes the electrons to rotate with the cyclotron frequency about the force lines. In other words, this applied guiding magnetic field helps electron to be maintained in the propagation direction of the plasma wave which can resonantly accelerate the electrons to high-level energies. To show this convergence role of externally applied magnetic field, the electron trajectory in the x–z plane in the both absence (${\rm \Omega _\iota} = 0.0$) and presence of the external magnetic field (${\rm \Omega _\iota} = 0.5$) has been illustrated in Figure 1b. As clearly seen in the figure, the axial guiding magnetic field converges the electron trajectory in the perpendicular direction (x-direction). The same scenario will be taken for the electron trajectory in another perpendicular direction (y-direction).

Fig. 1. (a) Electron energy gain γ versus the normalized distance (z) for different axial guiding magnetic field amplitudes Ω_ι = 0.0, 0.1, 0.3 and (b) the trajectory of electron in the absence (${\rm \Omega _\iota} = 0.0$) and presence of the external magnetic field (${\rm \Omega _\iota} = 0.5$). Other parameters are a _p = 0.5, k = 1.01, θ₀ = π/2, r ₁ = 2, r ₂ = 4, X ₀ = 0.5, Y ₀ = 0.0, Z ₀ = 0.0, V _x0 = 0.3, V _y0 = 0.1, V _z0 = 0.8, and φ₀ = 0.6.

One may note that there is a limitation for the electron energy gain in the high amplitudes of magnetic field because of resonance mismatch. Therefore, it can be found that there is an optimum value of the external axial magnetic field strength for maximizing the electron energy gain. We have illustrated the maximum electron energy γ_max as a function of the axial guiding magnetic field amplitude ${\rm \Omega }_{\rm \iota}$ in Figure 2. As seen in the figure, the maximum of the electron energy gain increases with magnetic field amplitude until an optimum strength of the magnetic field and beyond it, the energy gain starts to be reduced.

Fig. 2. The maximum of the electron energy gain γ_max versus the axial guiding magnetic field amplitude (Ω_ι). Other parameters are the same as in Fig. 1.

In the mechanism of the electron acceleration by the plasma wave, a major limitation of this scheme arises from the wave-particle dephasing. It is shown that this limitation can be overcome by considering the influence of the self-magnetic field which is perpendicular to the wave propagation direction. This field can considerably affect the electron acceleration during the interaction with the plasma wave. Thus, here, we investigate the role of the self-magnetic field [as shown in Eq. (1)] on the electron energy gain. Figure 3a illustrates the variation of the energy gain of electron with the normalized distance z for different self-magnetic field amplitudes Ω_s = 0.0, 0.3, 0.5. The figure has been plotted for the optimum amplitude of the external axial guiding magnetic field ${\rm \Omega _\iota} = 0.25$ and other parameters are as kept in Figure 1. It is observed that the electron energy gain increases from γ = 95 to γ = 131 for the self-magnetic field with amplitude Ω_s = 0.3. In this case, energy gain has increased more than 36% than one obtained in the absence of self-magnetic field amplitude. It is turns out that γ increases significantly in systems containing stronger self-magnetic field. This can be attributed to the enhancement of the electron momentum due to the v × B force appeared in the propagation direction of the plasma wave which can resonantly accelerate the electrons. On the other hand, as the self-magnetic field increases, the electron trajectory is stabilized as it moves further along the propagation direction and so the electron energy gain is improved with stronger magnetic field. One can clearly see that considering the self-magnetic field has a more significant role on the enhancement of the electron energy gain in an axially magnetized plasma. The presence of the external axial magnetic field causes the saturation of the energy gain to occur at the longer distances [compare the curve ${\rm \Omega _\iota} = 0.0$ with other curves in Fig. 1(a)]. In other words, in such systems the gradient of electron energy gain increases as compared to the systems that considered the effect of the self-magnetic field in the absence of axial magnetic field (Singh et al., Reference Singh, Gupta, Bhasin and Tripathi2003; Gupta et al., Reference Gupta, Kaur, Gopal and Suk2016). Figure 3b indicates the electron trajectory in the x–z plane for the same parameters in Figure 3a. As illustrated in the figure, the self-magnetic field not only can affect the electron trajectory in the perpendicular direction (a weak convergence in the x-direction) but also it significantly changes the electron trajectory in the longitudinal direction (a strong convergence in the z-direction). As clearly seen, by increasing the strength of the self-magnetic field the electrons return to the z-axis more frequently and so can gain more energy from the plasma wave.

Fig. 3. (a) Electron energy gain γ and (b) the trajectory of electron versus the normalized distance (z) for different self-magnetic field amplitude Ω_s = 0.0, 0.3, 0.5 for Ω_ι = 0.25.

In the following, we investigate the effect of the initial kinetic energy γ₀ on the electron energy gain during the acceleration by the plasma wave. For this purpose, the electron energy gain γ with the distance z for different initial kinetic energy (${\rm \gamma }_0{\rm} = 1{\rm.71\comma \;\ 1}{\rm.96\comma \;\ 2}.6$) has indicated in Figure 4. As seen in the figure, the preaccelerated electron gains more energy as compared to the electron with low initial energy. It is observed that for the preaccelerated electrons with initial energy ${\rm \gamma }_0{\rm} = 2.6$, the electron energy gain has increased to ${\rm \gamma }{\rm} = 180$, while for electron initial energy ${\rm \gamma }_0{\rm} = 1.71$, it has increased to ${\rm \gamma }{\rm} = 84$. This is because the electron with higher initial energy can interact with the plasma wave for longer time because of the less scattering. In other words, by reduction of the ponderomotive scattering the electron remains confined in the ion channel for longer distances and consequently gains more energy from the plasma wave.

Fig. 4. Electron energy gain γ versus the normalized distance (z) for different initial kinetic energy of electron γ₀ = 1.71, 1.96, 2.6 for Ω_s = 0.3, ${\rm \Omega _\iota} = 0.25$, and θ₀ = π/2.

As in the plasma wakefield electron acceleration mechanism, the plasma wave is one of the major sources, the plasma wave phase θ₀ and amplitude a _p, are the key parameters in this scenario. We have investigated the effects of these factors on the electron energy gain. Figure 5 indicates the variation of the electron energy γ with the normalized distance z for different plasma wave amplitudes. As clearly seen in the figure, by increasing the plasma wave amplitude the electron energy gain increases significantly. In this case, the electrons trapped in high-intensity plasma waves can experience a stronger longitudinal force and consequently be accelerated to high-energy levels due to the dominant ponderomotive effects. The figure shows that in this configuration, the electron energy gain increases more than three times (from ${\rm \gamma }{\rm} = 80$ to ${\rm \gamma }{\rm} = 285$) by increasing the plasma wave amplitude from a _p = 0.4 to a _p = 0.6. However, for very high plasma wave amplitudes the electron acceleration is limited due to the wave-breaking phenomenon related to the nonlinearity.

Fig. 5. Electron energy gain γ versus the normalized distance (z) for different plasma wave amplitudes a _p = 0.4, 0.5, 0.6 for γ₀ = 2.6, ${\rm \Omega _\iota} = 0.25$, θ₀ = π/2, and Ω_s = 0.3.

The electron can gain energy when it is trapped in the accelerating phase of the plasma wave. By deviation of the plasma wave phase, the electron goes into the defocusing and decelerating region and thereby it loses its energy. The variation of the electron energy γ with the normalized distance z for different plasma wave phases θ₀ has shown in Figure 6. The figure illustrates that the electron can gain higher energy from the plasma wave for smaller phases. This is because the electron is in the plasma wave accelerating phase in smaller initial wave phases and, so, it can gain more energy from the plasma wave. This results show that there is an optimum value for the initial phase of plasma wave for maximizing the electron energy gain.

Fig. 6. Electron energy gain γ versus the normalized distance (z) for different the plasma wave phases θ₀ = 0, π/4, π/2 for γ₀ = 2.6, $a_{\rm p}{\rm} = 5$, ${\rm \Omega _\iota} = 0.25$, and Ω_s = 0.3.

It is worth mentioning the role of the ion channel field as a substantial factor on the electron dynamics in this configuration. The radial electric field of the electron density depleted ion channel can significantly affect the acceleration of the injected electrons. It can confine the electron to propagate for longer distances along the longitudinal direction. To study the effect of the ion channel field, the electron energy γ with distance z has illustrated in Figure 7 for different normalized potential amplitudes of the ion channel. As illustrated in the figure, the electron energy gain mainly increases in the presence of the ion channel. In this state, the electron energy gain is improved during the acceleration by the plasma wave because the space-charge field stabilizes and focuses the electron motion in the acceleration region. A combined effect of the longitudinal electric field of the plasma wave and the transverse electric field of the ion channel can produce the energetic electrons. It may note that in the systems containing an ion channel with much stronger field the energy gain may be reduced due to the strong restoring force corresponding to the large space-charge fields.

Fig. 7. Electron energy gain γ versus the normalized distance (z) for different ion channel potential amplitudes φ₀ = 0.3, 0.5, 0.6 for γ₀ = 2.6, $a_{\rm p}{\rm} = 5$, $\Omega _{\rm \iota} = 0.25$, and Ω_s = 0.3.

At last but not least, it should be noted that although our results show that in most cases, the electron energy gain linearly increases by channel length, but in the long distances, where the nonlinear phenomena such as electron dephasing and resonance mismatch are important, the gradient of the electron energy gain decreases and finally becomes saturated.