Introduction
A broad range of applications, such as GeV electrons sources (Wang et al., Reference Wang, Zgadzaj, Fazel, Li, Yi, Zhang, Henderson, Chang, Korzekwa, Tsai, Pai, Quevedo, Dyer, Gaul, Martinez, Bernstein, Borger, Spinks, Donovan, Khudik, Shvets, Ditmire and Downer2013), multi-MeV protons, ion, and gamma ray sources (Fuchs et al., Reference Fuchs, Antici, d'Humieres, Lefebvre, Borghesi, Brambrink, Cecchetti, Kaluza, Malka, Manclossi, Meyroneinc, Mora, Schreiber, Toncian, Pepin and Audebert2006; Kneip et al., Reference Kneip, Nagel, Bellei, Bourgeois, Dangor, Gopal, Heathcote, Mangles, Marquès, Maksimchuk, Nilson, Ta Phuoc, Reed, Tzoufras, Tsung, Willingale, Mori, Rousse, Krushelnick and Najmudin2008; Flippo et al., Reference Flippo, Bartal, Beg, Chawla, Cobble, Gaillard, Hey, MacKinnon, Macphee, Nilson, Offermann, Le Pape and Schmitt2010; Cipiccia et al., Reference Cipiccia, Islam, Ersfeld, Shanks, Brunetti, Vieux, Yang, Issac, Wiggins, Welsh, Anania, Maneuski, Montgomery, Smith, Hoek, Hamilton, Lemos, Symes, Rajeev, Shea, Dias and Jaroszynski2011), have been employed in relativistic laser target interaction in a properly chosen experimental setup. These applications reveal the importance of investigation of the laser interaction with plasmas and matters (Wilks et al., Reference Wilks, Kruer, Tabak and Langdon1992; Brandl et al., Reference Brandl, Hidding, Osterholz, Hemmers, Karmakar, Pukhov and Pretzler2009; Arefiev et al., Reference Arefiev, Breizman, Schollmeier and Khudik2012; Kemp and Divol, Reference Kemp and Divol2012; Breuer and Hommelhoff, Reference Breuer and Hommelhoff2013; Chen et al., Reference Chen, Nakai, Sentoku, Arikawa, Azechi, Fujioka, Keane, Kojima, Goldstein, Maddox, Miyanaga, Morita, Nagai, Nishimura, Ozaki, Park, Sakawa, Takabe, Williams and Zhang2013; Liu et al., Reference Liu, Wang, Liu, Fu, Xu, Yan and He2013; Braenzel et al., Reference Braenzel, Andreev, Abicht, Ehrentraut, Platonov and Schnürer2017; Cheng et al., Reference Cheng, Yao, Zhang and Xue2017). The most important question is how we can transfer the enormous fraction of laser energy into the generation of relativistic electrons. In the laser interaction with underdense plasma, the electron acceleration mechanism strongly depends on the laser pulse duration. In the wakefield acceleration, the laser pulse duration should be shorter or comparable to the plasma wave period (Esarey et al., Reference Esarey, Schroeder and Leemans2009), and this mechanism becomes less efficient by long laser pulses (Malka et al., Reference Malka, Faure, Marques, Amiranoff, Rousseau, Ranc, Chambaret, Najmudin, Walton, Mora and Solodov2001; Mangles et al., Reference Mangles, Walton, Tzoufras, Najmudin, Clarke, Dangor, Evans, Fritzler, Gopal, Hernandez-Gomez, Mori, Rozmus, Tatarakis, Thomas, Tsung, Wei and Krushelnick2005).
For generation more energetic electrons by making use of a laser beam, different additional concept have been used, such as fast electron beams generation from plasma mirrors (Bocoum et al., Reference Bocoum, Thévenet, Böhle, Beaurepaire, Vernier, Jullien, Faure and Lopez-Martens2016), vacuum laser acceleration of relativistic electron (MeLikian, Reference Melikian2014; Thevenet et al., Reference Thévenet, Leblanc, Kahaly, Vincenti, Vernier, Quéré and Faure2016; Zhang et al., Reference Zhang, Jiao, Zhang, Zhang and Gu2017; Pandari et al., Reference Pandari, Jahangiri and Niknam2019), relativistic electron mirrors from nanoscale foils (Kiefer et al., Reference Kiefer, Yeung, Dzelzainis, Foster, Rykovanov, Lewis, Marjoribanks, Ruhl, Habs, Schreiber, Zepf and Dromey2013), boosted high harmonic pulse from a double-sided relativistic mirror (Esirkepov et al., Reference Esirkepov, Bulanov, Kando, Pirozhkov and Zhidkov2009), and electron acceleration by relativistic surface plasmons (Fedeli et al., Reference Fedeli, Sgattoni, Cantono, Garzella, Réau, Prencipe, Passoni, Raynaud, Květoň, Proska, Macchi and Ceccotti2016).
We can avoid deceleration when the light field is reflected out of the electron trajectory (Singh, Reference Singh2004; Mulser et al., Reference Mulser, Bauer and Ruhl2008; Wu et al., Reference Wu, Meyer-ter-Vehn, Fernández and Hegelich2010) and or by electron injection into the light field with a specific initial momentum (Dodin and Fisch, Reference Dodin and Fisch2003; Andreev and Platonov, Reference Andreev and Platonov2013; Bocoum et al., Reference Bocoum, Thévenet, Böhle, Beaurepaire, Vernier, Jullien, Faure and Lopez-Martens2016; Thevenet et al., Reference Thévenet, Leblanc, Kahaly, Vincenti, Vernier, Quéré and Faure2016). Moreover, the electron acceleration has already been enhanced by making use of an additional external electric field (Plettner et al., Reference Plettner, Byer, Colby, Cowan, Sears, Spencer and Siemann2005; Dimant and Oppenheim, Reference Dimant and Oppenheim2010; He et al., Reference He, Hou, Nees, Easter, Faure, Krushelnick and Thomas2013; Cheng et al., Reference Cheng, Xue and Liu2016).
It is well known that the non-linear plasma properties are intensified by magnetic field due to the Lorentz force, so it is reasonable the plasma dynamics affected by magnetic field. Therefore, generation of magnetic fields is one of the more interesting debatable topics among the scientists in laser–plasma interaction. This issue has a wide range of applications in fast ignition scheme for inertial confinement fusion (Tabak et al., Reference Tabak, Hammer, Glinsky, Kruer, Wilks, Woodworth, Campbell, Perry and Mason1994), laboratory astrophysics (Ripin et al., Reference Ripin, Manka, Peyser, McLean, Stamper, Mostovych, Grun, Kearney, Crawford and Huba1990), and particle acceleration (Schmitz and Kull, Reference Schmitz and Kull2002; Yu et al., Reference Yu, Yu, Chen, Zhang, Yin, Cao, Lu and Xu2003; Liu et al., Reference Liu, He and Chen2004; Qiao et al., Reference Qiao, He, Zhu and Zheng2005). The experiments have been reported the magnetic field strength in the order of about tens of mega-gauss (MG) (Fuchs et al., Reference Fuchs, Malka, Adam, Amiranoff, Baton, Blanchot, Héron, Laval, Miquel, Mora, Pépin and Rousseaux1998; Najmudin et al., Reference Najmudin, Tatarakis, Pukhov, Clark, Clarke, Dangor, Faure, Malka, Neely, Santala and Krushelnick2001); however, the recent 3D particle-in-cell (PIC) simulation and analytical investigations predicted generation of the super strong magnetic field up to 100 MG in laser interaction with an overdense plasma channel (Pukhov and Meyer-ter-Vehn, Reference Pukhov and Meyer-ter-Vehn1996; Qiao et al., Reference Qiao, He and Zhu2006; Wang et al., Reference Wang, Lin, Sheng, Liu, Zhao, Guo, Lu, He, Chen and Yan2011; Stark et al., Reference Stark, Toncian and Arefiev2016).
The ponderomotive force plays an important role in a long laser pulse interaction with plasma. In this case, the ponderomotive pressure repels the electrons away from the axis of the beam in the transverse direction, while the heavy ions remain at rest. As a result, an ion channel is created with a quasi-static transverse electric field due to the charge separation. The electron acceleration in an ion channel has been experimentally demonstrated in the past decades (Gahn et al., Reference Gahn, Tsakiris, Pukhov, Meyer-ter-Vehn, Pretzler, Thirolf, Habs and Witte1999, Reference Gahn, Tsakiris, Pretzler, Witte, Thirolf, Habs, Delfin and Wahlström2002; Kitagawa et al., Reference Kitagawa, Sentoku, Akamatsu, Sakamoto, Kodama, Tanaka, Azumi, Norimatsu, Matsuoka, Fujita and Yoshida2004; Walton et al., Reference Walton, Mangles, Najmudin, Tatarakis, Wei, Gopal, Marle, Dangor, Krushelnick, Fritzler and Malka2006; Kneip et al., Reference Kneip, Nagel, Bellei, Bourgeois, Dangor, Gopal, Heathcote, Mangles, Marquès, Maksimchuk, Nilson, Ta Phuoc, Reed, Tzoufras, Tsung, Willingale, Mori, Rousse, Krushelnick and Najmudin2008, Reference Kneip, McGuffey, Nagel, Palmer, Bellei, Schreiber, Huntington, Dollar, Matsuoka, Chvykov, Kalintchenko, Yanovsky, Maksimchuk, Ta Phuoc, Mangles, Krushelnick and Najmudin2009); however, the test electron acceleration by strong laser field in an ion channel has already been studied theoretically by Reference Arefiev, Khudik and SchollmeierArefiev et al. (Reference Arefiev, Khudik and Schollmeier2014).
In this paper, we will follow the method is used by Arefiev et al., by taking into account the fact that the plasma channel should be magnetized (Pukhov and Meyer-ter-Vehn, Reference Pukhov and Meyer-ter-Vehn1996; Stark et al., Reference Stark, Toncian and Arefiev2016). Therefore, we would like to examine the electron direct laser acceleration (DLA) in a two-dimensional magnetized plasma channel. We will focus our attention on the role played by the self-generated quasi-static magnetic field on the electron acceleration process.
The paper is organized in the following fashion. In the "Model description" section, a single electron model is formulated for an electron irradiated by an incoming laser pulse in a steady-state two-dimensional transversely magnetized ion channel. The purpose of this section is to justify the necessity of including the self-generated magnetic field during the plasma channel creation, to explain how such a plasma channel can be modeled. We try to explain the effect of the self-generated magnetic field on the dephasing rate in this section. In the “Numerical results and discussion” section, the closed set of electron motion of equations in the presence of the laser pulse and self-generated magnetic field is presented and solved with proper initial condition. Finally, the summary and conclusion are given in the last section.
Model description
In this section, we describe a single electron dynamics irradiated by a laser field in a homogenous transversely magnetized plasma channel. The origin of the self-generated magnetic field is the current induced in the channel. The PIC simulations show that the quasi-static magnetic field increases proportional at distance from axis and modulates in the direction of laser field propagation with spatial variation in the order of the laser wavelength λ (Stark et al., Reference Stark, Toncian and Arefiev2016; Pukhov and Meyer-ter-Vehn, Reference Pukhov and Meyer-ter-Vehn1996).
It is well known in the interaction of high intensity laser field with a plasma, since the longitudinal momentum dominates the kinetic energy of electrons, we can assume that all of the electrons moving forward with relativistic velocity. However, a few number of electrons move under the angle with respect to the channel axis; therefore, if we assume the radius of channel is R, it is easy to show that the magnetic field for y < R is given by B x = 2πJ zy/c, where J z = en ec. On the other hand, if we include the magnetic field modulation in z direction, we can present the quasi-static magnetic field in the form as follows:
$$B_x/B_0=(n_e/2n_c)(2{\rm \pi} \mid y\mid/{\rm \lambda}) \sin 2{\rm \pi} z/{\rm \lambda},$$where B 0 = mcω/e and n c = πmc 2/(eλ)2 are the magnetic field of the laser field and critical density, respectively. Figure 1 depicts a two-dimensional spatial setup, with a Cartesian system of coordinates (x, y, z). The figure shows the channel is a slab of heavy immobile ions with density n 0. The static electric field 
$E_c = {\bf e}_y{\rm \omega} _{pe}^2m_ey/|e|$ induced due to the charge separation, where ey is a unit vector in y direction and ωpe = (4πn 0e 2/m e)1/2 refers to the plasma frequency (m e and e are electron mass and charge). The laser field is linearly polarized by polarization angle θ and propagates in the z direction. In addition, the self-generated magnetic field is perpendicular to y–z plane.
Fig. 1. Schematic setup of a single electron model in an ion channel.
It is worthwhile to note that the plasma channel generally is cylindrical; however, because of azimuthal symmetry and with no loss of generality, we can use a two-dimensional (2D) plasma channel. On the other hand, the 2D plasma channel has some advantages: first for simplicity and the second is that for a given laser amplitude, the amplitude of laser field that responsible to direct acceleration of betatron oscillation is changed by changing the polarization angle. Therefore, we can examine the interplay between the oscillating electric field and the electrostatic field of the channel.
At this stage, we consider a single electron dynamics placed in an ion channel (Fig. 1) irradiated by a laser field. We assume that the laser field has the same parameters with the laser which produced the channel. We introduced the laser field by normalized vector potential as follows:
$${\bf a}(z,t)=a(\xi)[{\bf e}_x\cos{\rm \theta} + {\bf e}_y\sin {\rm \theta}],$$where ξ = ω(t − z/c) is the wave phase in which ω is wave frequency and c is the speed of light. Also, t is the time in the channel frame of reference, and ex is a unit vector in x direction. We consider pulse amplitude in the form of 
$a(\xi )=a_*(\xi )\sin \xi$, where 
$0\leq a_*(\xi )\leq a_0$ is a given slowly varying envelope. The total electric and magnetic fields inside the plasma channel are as follows:
$${\bf E} = {\bf E}_c + {\bf E}_L,\quad {\bf B} = {\bf B}_L + {\bf B}_x,$$where Bx is the quasi-static self-generated magnetic field directed along the x-axis, and the electric and magnetic fields corresponding to the laser field can be expressed as follows:
$${\bf E}_L=-{m_ec \over |e|}{{\rm \partial} {\bf a} \over {\rm \partial} t},\quad {\bf B}_L={m_ec^2 \over |e|}\nabla\times {\bf a}.$$The equation of motion for an electron inside the plasma channel is given by the following equation:
$${d{\bf p} \over d t}= {-|e|{\bf E} \over m_ec}- {|e| \over {\rm \gamma} m_ec}{\bf p}\times {\bf B},$$where p is the dimensionless electron momentum normalized to m ec, and 
${\rm \gamma} =\sqrt {1+{\bf p}^2}$ is the relativistic factor. By making use of Eqs (1)–(5), we obtain the closed set of equations to describe the electron dynamics in the plasma channel as follows:
$$\eqalign{& {d \over d{\rm \tau}}[\,p_y-a\sin {\rm \theta}]= -{\rm \gamma} {{\rm \omega}_{\,pe}^2 \over {\rm \omega}^2}Y_B-{{\rm \omega}_{ce} \over {\rm \omega}}p_z,\cr & {d p_z \over d{\rm \tau}} = p_x\left(\cos{\rm \theta} {{\rm \partial} a \over {\rm \partial} \xi}\right) +p_y\left(\sin {\rm \theta} {{\rm \partial} a \over {\rm \partial}\xi}+{{\rm \omega}_{ce} \over {\rm \omega}}\right),\cr &{d\xi \over d{\rm \tau}}={\rm \gamma}-p_z,\cr &{d Y_B \over d{\rm \tau}}=p_y,\cr &p_x=a\cos{\rm \theta},}$$where τ is a dimensionless proper time defined by the relation dτ/dt = ω/γ, and Y B = ωy/c is dimensionless displacement across the channel. Moreover, ωce = eB x/m ec and 
${\rm \omega} _{pe}=(4{\rm \pi} n_e^2/m_e)^{1/2}$ are electron cyclotron and plasma frequencies, where n e shows the electron density. The third relation in Eq. (6) represents a dephasing rate and usually defines as R B = γ − p z. Using Eq. (6), we can show that
$${d \over d{\rm \tau}}\left({\rm \gamma}-p_z+{1 \over 2}{{\rm \omega}_{\,pe}^2 \over {\rm \omega}^2}Y_B^2 +Y_B{{\rm \omega}_{ce} \over {\rm \omega}}\right)=0.$$Integrating Eq. (7), we get
$$R_B={\rm \gamma}-p_z=I_B-{1 \over 2}\Omega_B^2Y_B^2-\Omega_{ce}Y_B,$$where I B is a constant and determined by initial conditions, and ΩB = ωpe/ω and Ωce = ωce/ω are normalized plasma and cyclotron frequencies, respectively. For example, if we assume the electron is initially at rest (i.e., R → 1) and is placed on Y 0B, we can find I B as follows:
$$I_B=1+{1 \over 2}\Omega_B^2Y_{0B}^2+\Omega^\prime_{ce}Y_{0B},$$where 
$\Omega ^\prime _{ce}$ refers to normalized cyclotron frequency at Y = Y 0B. It should be noted that the last term on the right-hand side of Eq. (8) shows the effect of the self-generated magnetic field on the dephasing rate. It is well known that the axial part of the electron momentum larger than the transverse part at ultra-relativistic wave amplitude (a ≫ 1); therefore, we expect that R B → 0 when the electron efficiently accelerated by the laser field.
In order to determine the electron maximum displacement across to the channel, the dephasing rate should becomes small vanishingly in Eq. (8). Therefore, substituting R B = 0 in Eq. (8), we get
$$Y_{B*}={-\Omega_{ce}\pm\sqrt{\Omega_{ce}^2+2I_B\Omega_B^2} \over \Omega_B^2}.$$Equation (10) is in complete agreement with the previous knowledge (Arefiev et al., Reference Arefiev, Khudik and Schollmeier2014) in the limit of Ωce = 0, for a non-magnetized plasma channel.
Numerical results and discussion
In this section, we investigate the electron dynamics in the plasma channel for a laser irradiated with polarization angle θ in the presence of the self-generated static magnetic field in the form of Eq. (1).
The electron dynamics is studied by the numerical solution of the coupled relations in Eq. (6) for a Gaussian profile pulse laser 
$a=a_*(\xi )\sin (\xi )$ is defined as follows:
$$a_*(\xi)=a_0\exp\left[-{(\xi-\xi_0)^2 \over 2{\rm \sigma}^2}\right],$$where a 0 = 10, ξ0 = 400, and σ = 100. Here, a 0 is the normalized laser field amplitude, and parameters ξ0 and σ characterize the laser beam initial position with respect to the electron and the beam duration. In the course of this paper, we consider the initial condition for electron at τ = 0, with p y(0) = 0, p z(0) = 0, Y 0 = 0.05, and ξ = 0.
Figure 2 shows the variation of γmax/γvac (where γmax is the maximum γ-factor and 
${\rm \gamma} _{{\rm vac}}=1+a_0^2/2$) as a function of normalized plasma frequency in the magnetized and non-magnetized plasma channels. Figure 2a and 2b are plotted for polarization angles θ = π/2 and θ = 0, respectively. In addition, Figure 2c and 2d indicate the maximum strength of magnetic field would feel by the test electron for polarization angles θ = π/2 and θ = 0. The figure shows the strength of self-generated magnetic field rises as the plasma density increases. We find that the role of magnetic field is more effective in the enhancement of electron acceleration for polarization angle θ = π/2.
Fig. 2. Maximum normalized γ-factor as a function of ωpe/ω for a magnetized (red line) and non-magnetized plasma channel. (a) Polarization angle θ = π/2 and (b) polarization angle θ = 0. (c,d) Indicate the normalized maximum self-generated magnetic field variation as a function of ωpe/ω for polarization angles θ = π/2 and θ = 0, respectively.
Actually, the electron acceleration is affected by plasma density, laser field amplitude and polarization angle, and self-generated magnetic field strength. The previous investigation have been revealed the efficient electron acceleration by low density plasma channels and also the maximum enhancement when laser electric field and the field of the channel are collinear (Arefiev et al., Reference Arefiev, Khudik and Schollmeier2014). Figure 2 shows when the effect of self-generated magnetic field is included not only the maximum γ-factor increases but also the electron acceleration enhanced for a relatively high-density plasma channel.
The electron has two oscillating motions: (a) oscillation with laser field frequency in polarization plane and (b) betatron oscillating motion around magnetic field with frequency ωc. Two oscillatory motions are in y–z plane for polarization angle θ = π/2. Therefore, we expect the resonance takes place when the electron betatron oscillation closes to the laser field frequency. For example, the resonant absorption for polarization angle θ = π/2 is clearly shown in Figure 2a and 2c for ωpe/ω = 0.356 when the (B x)max ≈ 0.9B 0. This process is governed mechanism for the enhancement of electron acceleration for polarization angle in the vicinity of θ ≈ π/2.
Figure 3a and 3b demonstrate the normalized γ-factor variation as a function of ξ/2π for polarization angle θ = π/2 in the magnetized and non-magnetized plasma channels, respectively. The figure compares the electron acceleration in the presence of the plasma channel and vacuum. Thus, even for γ < γvac, the electron is accelerated (γvac = 51). It is understood from Figure 3a that γ > γvac for wide range of ξ/2π, means that electron acceleration is enhanced in the presence of the magnetized plasma channel. This is not true for a non-magnetized plasma channel in Figure 3b. The maximum γ-factor for electron driven by laser beam is around γmax/γvac ≈ 5.7 which is in good agreement with Figure 2a.
Fig. 3. Relative γ-factor at θ = π/2 and ωp/ω = 0.353 (a) for a magnetized and (b) for a non-magnetized plasma channel as a function of ξ/2π.
In order to determine simultaneous effects of the polarization angle and channel density, we plotted the variation of maximum γ-factor with respect to the θ and ωpe/ω in Figures 4 and 5. In Fig. 4, the influence of self-generated magnetic field is included, while Fig. 5 are plotted for a non-magnetized plasma channel. It seems for a magnetized plasma channel and for polarization angles 0.39π ≤ θ ≤ 0.5π electron acceleration enhanced up to γmax ≈ 12γvac and the required threshold density for electron acceleration decreases to n e = 0.0036n c. While for non-magnetized plasma, the maximum acceleration for polarization angles 0.32π ≤ θ ≤ 0.5π is around γmax ≈ 8γvac for threshold density n e = 0.0064n c. Therefore, for polarization angles near to θ ≈ π/2, resonant absorption plays an important role in acceleration enhancement. As we mentioned, the enhancement occurs when the frequency of betatron oscillation closes to the laser field frequency. For a magnetized plasma channel, the resonance condition is satisfied for different plasma densities at different polarization angles in the range 0.39π ≤ θ ≤ 0.5π. Figure 4 indicates that the electron acceleration enhanced in magnetized plasma specially for the relatively dense plasma channel.
Fig. 4. Variation of maximum γ-factor as a function of ωpe/ω and polarization angle θ for a magnetized plasma channel.
Fig. 5. Variation of maximum γ-factor as a function of ωpe/ω and polarization angle θ for a non-magnetized plasma channel.
Summary and conclusion
In summary, we have analyzed the dynamics of an electron irradiated by a linearly polarized laser pulse in a two-dimensional transversely magnetized ion channel. Taking into account that a real plasma channel is magnetized, we expect magnetic field makes affect the test electron acceleration through plasma channel. Therefore, the outcomes of the present paper help to the reader to study a real physical system. We supposed that the magnetic field is generated due the current induced inside the channel and presented a simple model for a magnetized plasma channel. The results revealed that the self-generated magnetic field sustains the electron acceleration for polarization angle near to θ = π/2. For small polarization angle, the effect of magnetic field was no perceptible. It is shown that for polarization angle near to θ ≈ π/2, when the electron betatron oscillation frequency closes to the laser field frequency, the resonance condition is achieved and electron acceleration enhanced.
