2.1 Decay of magnetic field reversal configuration in collisionless plasma

First analyze the case of the magnetic field annihilation in a plasma. We assume that the initial (at $t=0$) magnetic field in homogeneous density ${n}_0$ plasma has a form $\mathbf{B}={B}_0\kern0.1em \operatorname{sgn}(x){\mathbf{e}}_y$ and there is no electric current in the plane $x=0$.

In the limit of small amplitude field the solution of Equation (4) is for the electric field

(6)$$\begin{align}{E}_z={B}_0{J}_0\left[{\omega}_{pe}\sqrt{t^2-{\left(x/c\right)}^2}\right]\theta \left( ct-|x|\right),\end{align}$$

and for the electron momentum in the region $\mid x\mid \ll ct$

(7)$$\begin{align}{p}_z={\frac{eB_0t}{\omega_{pe}}}\ {}_1F_2\left(\left\{\frac{1}{2}\right\},\left\{1,\frac{3}{2}\right\},-\frac{1}{4}{\omega}_{pe}^2{t}^2\right).\end{align}$$

Here $\theta (x)$ is the Heaviside step function, ${J}_0(x)$ is the Bessel function of zeroth order and ${}_p{F}_q\big(\left\{{a}_1,\dots, {a}_p\right\},$ $\left\{{b}_1,\dots, {b}_p\right\},x\big)$ is the generalized hypergeometric function^[Reference Olver, Lozier, Boisvert and Clark^90] (in Equation (7) $p=1$ and $q=2$). On both sides of the initial discontinuity the electric field pulse propagates outwards at the velocity equal to $c$. The pulse width is inversely proportional to the time. Over a time of $1/{\omega}_{pe}$ the current sheet of the width $c/{\omega}_{pe}$ is formed. Asymptotically, at $t\to \infty$ the electric field vanishes and the electron momentum tends to ${eB}_0/{\omega}_{pe}$.

In the limit of strong magnetic field, when Equation (3) is satisfied, we may use the ultra-relativistic approximation assuming ${p}_z/{m}_ec>>1$ and neglecting the electric current effects in Equation (4) for sufficiently short time interval. From Equation (4) follows the wave equation for the electric field. D’Alembert solution of this equation shows that in the region $\mid x\mid < ct$, instead of initial magnetic field, an electric field $E={B}_0$ directed along the $z$ axis appears. The electron momentum grows here as

(8)$$\begin{align}{p}_z=-{eB}_0\left(t-|x|/c\right).\end{align}$$

The solution is valid at time less than ${\omega}_{Be}/{\omega}_{pe}^2$. From Equation (8) follows that the electrons become ultra-relativistic provided ${\omega}_{Be}/{\omega}_{pe}>>1$, which corresponds to the condition given by Equation (3).

Such an annihilation process is demonstrated via the 1D kinetic simulations. A pair of opposite magnetic fields are initially applied in a homogenous plasma as shown in Figure 1(a). A sharp magnetic boundary is formed around $x=0$. According to Faraday’s law,

(9)$$\begin{align}\nabla \times \mathbf{E}=-\frac{1}{c}{\partial}_t\mathbf{B},\end{align}$$

an electric field is induced with the evolution of the local magnetic field. In the 1D case, only ${B}_y$ component varies along $x$-direction, and therefore the variation of electric field is obtained as

(10)$$\begin{align}{E}_z=-\frac{1}{c}{\partial}_t{A}_z.\end{align}$$

Figure 1 (a) The initial condition of 1D model. (b) Magnetic field annihilation and electric field growing.

The expected electric field growing is shown in Figure 1(b). The strength of magnetic field significantly decreases near the critical point of $x=0$ and ${E}_z$ grows in the corresponding region with a transverse expansion along $x$-direction. Notice that such magnetic annihilation and electric field generation are valid in both highly-conducting collisional and collisionless plasmas and even in the vacuum. In this review, the magnetic annihilation and reconnection in the collisionless plasmas are focused.

2.2 Finite amplitude opposite polarity electromagnetic wave interaction with thin current layer

Now we consider the finite amplitude electromagnetic wave collision with the electric current carrying thin plasma layer. This current sheet in the $yz$ plane has the electric current density described by the expression

(11)$$\begin{align}\mathbf{j}={I}_0\delta (x){\mathbf{e}}_z={n}_0 lc\delta (x){\mathbf{e}}_z\sum \limits_{j=e,i}{e}_j\frac{p_{j,0}}{\sqrt{m_j^2{c}^2+{p}_{0,j}^2}}.\end{align}$$

An electromagnetic wave of amplitude ${E}_w$, given at $t<0$ by the electromagnetic potential $\mathbf{A}=-{E}_w\left( ct+|x|\right)\cdot \theta ( ct+|x|){\mathbf{e}}_z$, is incident on the current sheet. The equation for the field (4) takes the form

(12)$$\begin{align}&{c}^2{\partial}_{tt}a-{\partial}_{xx}a=-2{\omega}_{0e}\delta (x)\notag\\&\quad\times{}\left[\frac{a+{\pi}_e}{\sqrt{1+{\left(a+{\pi}_e\right)}^2}}+\mu \frac{a-{\pi}_i}{\sqrt{1+{\mu}^2{\left(a-{\pi}_i\right)}^2}}\right]\notag\\&\quad{}+\frac{2e}{m_e{c}^2}\left({E}_w+{B}_0\right){\delta}^{\prime }(t)+\frac{2e}{m_e}{E}_w\delta (t),\end{align}$$

where

(13)$$\begin{align}{\omega}_{0e}=\frac{2\pi {n}_0{e}^2l}{m_ec},\end{align}$$

the normalized field equals $a={eA}_z/{m}_e{c}^2$, the normalized initial electron and ion momenta are ${\pi}_e={p}_{e,0}/{m}_ec$ and ${\pi}_i={p}_{i,0}/{m}_ec$, respectively, and $\mu ={m}_e/{m}_p$ is the electron to ion mass ratio. We assume that ${m}_i={m}_p$.

Equation (12) can be written by using the d’Alembert relationship, giving the solution of the Cauchy problem for the wave equation, as

(14)$$\begin{align}&a\left(x,t\right)=\notag\\&\quad{}-{\omega}_{0e}{\int}_0^{t-\mid x\mid /c}\left\{\frac{a\left(0,{t}^{\prime}\right)+{\pi}_e}{\sqrt{1+{\left[a\left(0,{t}^{\prime}\right)+{\pi}_e\right]}^2}}\right.\notag\\&\quad{}\!\left.+\mu \frac{a\left(0,{t}^{\prime}\right)-{\pi}_i}{\sqrt{1+{\mu}^2{\left[a\left(0,{t}^{\prime}\right)-{\pi}_i\right]}^2}}\right\}\hbox{d}{t}^{\prime}\notag\\&\quad{}+\frac{2\pi {eI}_0}{m_e{c}^2}[\mid\! x\!\mid +\theta ( ct-|x|)( ct-|x|)]\notag\\&\quad{}-\frac{eE_w}{m_e{c}^3}\left[\left( ct+x\right)\theta\! \left( ct+x\right)+\left( ct-x\right)\theta\! \left( ct-x\right)\right].\end{align}$$

To find the function $a\left(0,t\right)$ we set $x=0$ in Equation (14) and differentiate it with respect to time $t$. The result is an ordinary differential equation of the first order whose solution is

(15)$$\begin{align}&{\int}_0^a\left(0,t\right)\left[\frac{a^{\prime }+{\pi}_e}{\sqrt{1+{\left({a}^{\prime }+{\pi}_e\right)}^2}}+\mu \frac{a^{\prime }-{\pi}_i}{\sqrt{1+{\mu}^2{\left({a}^{\prime }-{\pi}_i\right)}^2}}\right.\notag\\&\quad{}{\left.-\frac{2\pi {eI}_0}{m_e{c}^2{\omega}_{0e}}-\frac{2{eE}_w}{m_e{c}^2{\omega}_{0e}}\right]}^{-1}\hbox{d}{a}^{\prime }=-{\omega}_{0e}t.\end{align}$$

One may see from Equation (15) that if the amplitude of the electromagnetic wave is substantially large, ${E}_w>4\pi {en}_0l-\pi {I}_0/c$, there can be no steady state solution. In the limit $t\to \infty$ the electromagnetic potential at $x=0$ grows proportionally to time as

(16)$$\begin{align}a\left(0,t\right)\approx -\frac{et}{m_ec}\left[{E}_w-\frac{\pi }{c}\left(4{en}_0 lc-{I}_0\right)\right].\end{align}$$

Consequently, the unlimited acceleration of charged particles occurs with the particle momentum growing as

(17)$$\begin{align}{p}_z\left(0,t\right)\approx - et\left[{E}_w-\frac{\pi }{c}\left(4{en}_0 lc-{I}_0\right)\right].\end{align}$$

Substituting the function $a\left(0,t\right)$ found from Equation (15) to (14) we obtain the expression describing the electromagnetic field outside the current sheet. In the case of super-strong electromagnetic field, when ${E}_w>4\pi {en}_0l-\pi {I}_0/c$, the electric field has a form of two almost constant waves with the amplitude ${E}_w-4\pi {en}_0l+\pi {I}_0/c$ propagating outwards in two directions. For the electromagnetic wave incident on the current sheet with moderate amplitude, ${E}_w<4\pi {en}_0l-\pi {I}_0/c$, the wave is reflected from the current sheet. In the region $\mid x\mid < ct$ the electric field vanishes and the magnetic field is equal to ${B}_y=\left({B}_0+{E}_w\right)\operatorname{sgn}(x)$. The momentum of charged particles increases a finite value.

We note that the solution found here corresponds to the paradigm of the thin foil nonlinear electrodynamics developed in Refs. [Reference Bulanov and Syrovatskii89,Reference Vshivkov, Naumova, Pegoraro and Bulanov91–Reference Bratman and Samsonov94].

2.3 Tearing mode instability of thin current layer

The 2D dynamics of thin current sheet is considered within the framework of the tearing mode instability concept. The electromagnetic tearing mode instability of thin current sheet has been studied in Refs. [Reference Bulanov37,Reference Bulanov and Syrovatskii89,Reference Neil95–Reference Bulanov and Sasorov97]. The electromagnetic field given by the vector $\mathbf{A}=\left({m}_e{c}^2a/e\right){\mathbf{e}}_z$ and electrostatic potential $\varphi ={m}_e{c}^2\phi /e$ is governed by the equations

(18)$$\begin{align}{W} a=\delta (x)J,\end{align}$$

(19)$$\begin{align}{W} \varphi =\delta (x)R,\end{align}$$

where ${W} ={\partial}_{xx}+{\partial}_{yy}-{c}^{-2}{\partial}_{tt}$ is the d’Alambertian,

(20)$$\begin{align}J=4\pi \sum \limits_{j=e,i}{e}_j{n}_j{lv}_{z,j}/{m}_j{c}^2\end{align}$$

and

(21)$$\begin{align}R=4\pi \sum \limits_{j=e,i}{e}_j{n}_jl/{m}_j{c}^2\end{align}$$

are the surface electric current and electric charge density, respectively. Here it is assumed that the variables $a$ and $\varphi$ depend on the coordinates $x$, $y$ and time $t$. Linearizing Equations (18) and (19) and presenting dependence of the potentials on time and coordinates in the form ${a}^{(1)}={a}^{(1)}(x)\exp \left(-\omega t+ ikx\right)$ and ${\varphi}^{(1)}={\varphi}^{(1)}\exp \left(-\omega t+ ikx\right)$, we can write the Equations (18) and (19) as

(22)$$\begin{align}\frac{\hbox{d}^2{a}^{(1)}}{\hbox{d}x^2}-{Q}^2{a}^{(1)}=\delta (x)\left({J}_a{a}^{(1)}+{J}_{\varphi }{\varphi}^{(1)}\right),\end{align}$$

(23)$$\begin{align}\frac{\hbox{d}^2{\varphi}^{(1)}}{\hbox{d}x^2}-{Q}^2{\varphi}^{(1)}=\delta (x)\left({R}_a{a}^{(1)}+{R}_{\varphi }{\varphi}^{(1)}\right),\end{align}$$

with ${J}_a={\partial}_aJ$, ${J}_{\varphi }={\partial}_{\varphi }J$, ${R}_a={\partial}_aR$, and ${R}_{\varphi }={\partial}_{\varphi }R$, where

(24)$$\begin{align}Q=\sqrt{k^2-{\omega}^2/{c}^2}.\end{align}$$

The solution of Equations (22) and (23) yields

(25)$$\begin{align}{a}^{(1)}(x)=-\frac{\exp \left(-Q|x|\right)}{2Q}\left[{J}_a{a}^{(1)}(0)+{J}_{\varphi }{\varphi}^{(1)}(0)\right],\end{align}$$

(26)$$\begin{align}{\varphi}^{(1)}(x)=-\frac{\exp \left(-Q|x|\right)}{2Q}\left[{R}_a{a}^{(1)}(0)+{R}_{\varphi }{\varphi}^{(1)}(0)\right].\end{align}$$

Setting $x=0$ in both sides of Equations (25) and (26) we obtain the system of algebraic equations. The condition of solvability of the system obtained results in the equation

(27)$$\begin{align}1+\frac{J_a+{R}_{\varphi }}{2Q}+\frac{J_a{R}_{\varphi }-{R}_a{J}_{\varphi }}{4{Q}^2}=0.\end{align}$$

We assume that the unperturbed electric current density and electric charge density are ${J}^{(0)}=-4\pi {en}_e^{(0)}{lv}_{z,e}^{(0)}/c$ and ${R}^{(0)}=0$, respectively. Here the electron velocity along the $z$ axis is ${v}_{z,e}^{(0)}={\beta}_0c$.

Obtained within the framework of cold electron and ion hydrodynamics approximation ${J}_a$, ${J}_{\varphi }$, ${R}_a$, and ${R}_{\varphi }$ are given by

(28)$$\begin{align}{J}_a=\frac{2{\omega}_{0e}}{c}\left(\frac{1}{\gamma^{(0)3}}+\mu +\frac{k^2{c}^2}{\omega^2{\gamma}^{(0)}}{\beta}^{(0)}\right),\end{align}$$

(29)$$\begin{align}{J}_{\varphi }=-\frac{2{\omega}_{0e}}{c}\frac{k^2{c}^2}{\omega^2{\gamma}^{(0)}}{\beta}^{(0)},\end{align}$$

(30)$$\begin{align}{R}_a=\frac{2{\omega}_{0e}}{c}\frac{k^2{c}^2}{\omega^2{\gamma}^{(0)}}{\beta}^{(0)},\end{align}$$

(31)$$\begin{align}{R}_{\varphi }=-\frac{2{\omega}_{0e}}{c}\frac{k^2{c}^2}{\omega^2}\left(\frac{1}{\gamma^{(0)}}+\mu \right),\end{align}$$

with ${\gamma}^{(0)}=1/\sqrt{1-{\beta}^{(0)2}}$. Substituting these expressions for ${J}_a$, ${J}_{\varphi }$, ${R}_a$, and ${R}_{\varphi }$ into Equation (27) we obtain the dispersion equation for the frequency $\omega$ and wave number $k$. It is

(32)$$\begin{align}&{W}^2\left[1+\frac{\omega_{0e}}{Qc}\left(\frac{1}{\gamma^{(0)3}}+\mu \right)\right]+W\frac{\omega_{0e}}{Qc}\left[\frac{\beta^{(0)2}}{\gamma^{(0)2}}\right.\notag\\&\quad{}\left.-\frac{1}{\gamma^{(0)}}-\mu -\frac{\omega_{0e}}{Qc}\left(\frac{1}{\gamma^{(0)3}}+\mu \right)\left(\frac{1}{\gamma^{(0)}}+\mu \right)\right]\notag\\&\quad{}-{\left(\frac{\omega_{0e}}{Qc}\right)}^2\frac{{\mu \beta}^{(0)2}}{\gamma^{(0)2}}=0,\end{align}$$

where $W={\left(\omega / kc\right)}^2$.

In the short wavelength limit, when $kc/{\omega}_{0e} \gg 1$ the growth rate of the tearing instability is approximately equal to

(33)

In the long wavelength limit, when $kc/{\omega}_{0e}\ll 1$ we have the growth rate of the tearing instability

(34)

For high enough electron energy, in the range ${\mu}^{-1}\gg {\gamma}^{(0)}\gg {\mu}^{-1/3}$, does not depend on the electron to ion mass ratio, $\mu$, being approximately equal to $kc{\beta}^{(0)}{\gamma}^{(0)}$.

Following from Equation (33) in the non-relativistic limit, when ${\beta}^{(0)}\ll 1$ and ${\gamma}^{(0)}\approx 1$, the growth rate of the tearing mode instability of thin current sheet is $kc{\beta}^{(0)}\sqrt{\mu }$ (see also Ref. [Reference Bulanov and Syrovatskii89]).

2.4 The electric and magnetic field configuration arising during decay of thin current sheet

In the case of a thin current sheet localized in the $y=0$ plane carrying the surface electric current density $2{en}_0 lc{\beta}^{(0)}\delta (y)$, the magnetic field produced by this current is $\mathbf{B}=\left({B}_0\kern0.1em \operatorname{sgn}(x),\mathrm{0,0}\right)$ with ${B}_0=4\pi {en}_0 lc{\beta}^{(0)}$. At the nonlinear stage of the tearing mode thin current sheet can break forming a finite length hole in the density distribution^[Reference Bulanov and Sasorov^97]. Accordingly, it is worthwhile to analyze in detail the electromagnetic field configuration which arise during the current sheet decay. Here we examine a model for the current sheet decay assuming the time evolution of the electric current to be given: at $t>0$ the electric current density is described by the dependence

(35)$$\begin{align}\mathbf{j}&={I}_0f\left(x,t\right)\delta (y){\mathbf{e}}_z,\notag\\f\left(x,t\right)&=1-\theta \left[x(t)-|x|\right].\end{align}$$

In other words the current sheet breaks into halves, which move apart along the $x$ axis with their ends coordinates given by $x=\pm x(t)$, i.e., the width of the gap formed in the current sheet increases as $2x(t)$.

The electromagnetic field (it is the normalized electromagnetic potential $a={eA}_z/{m}_e{c}^2$) is described by the wave equation

(36)$$\begin{align}{\partial}_{xx}a+{\partial}_{yy}a-{c}^{-2}{\partial}_{tt}a=-\frac{\omega_{0e}}{c}{I}_0f\left(x,t\right)\delta (y)\end{align}$$

with the initial conditions

(37)$$\begin{align}&a\left(t=0\right)=-\frac{\omega_{0e}}{c}\mid y\mid,&{\partial}_ta\left(t=0\right)=0.\end{align}$$

Solving the Cauchy problem for Equations (36) and (37) with the surface electric current density given by Equation (35), we find the electric field ${E}_z=-{\partial}_t{A}_z/c$ and magnetic field ${B}_y=-{\partial}_x{A}_z/c$ (the normalized fields equal $d={eE}_z/{m}_ec=-\dot{a}$ and $b={eB}_y/{m}_ec=-{a}^{\prime }$). Here and below dot and prime stand for derivatives with respect to time and $x$ coordinate.

If the coordinate $x$ of the boundary of a gap in the current sheet is a linear function of time,

(38)$$\begin{align}x(t)=\pm \beta ct,\end{align}$$

the electric field dependence on time and coordinates is described by the expression

(39)$$\begin{align}&{E}_z\left(x,y,t\right)=\frac{I_0\beta }{2c\sqrt{1-{\beta}^2}}\notag\\&\quad{}\times \ln \left[\frac{\left({t}_{+}+\sqrt{t^2-{r}^2/{c}^2}\right)\left({t}_{-}+\sqrt{t^2-{r}^2/{c}^2}\right)}{\left({t}_{+}-\sqrt{t^2-{r}^2/{c}^2}\right)\left({t}_{-}-\sqrt{t^2-{r}^2/{c}^2}\right)}\right],\end{align}$$

where

(40)$$\begin{align}{t}_{\pm }=\frac{t\pm x\beta /c}{\sqrt{1-{\beta}^2}}\end{align}$$

and ${r}^2={x}^2+{y}^2$. In Figure 2 we show contours of equal value of the electric field in the $xy$ plane at $t=7.5$. As can be seen, the cylindrical electromagnetic wave propagates outwards leaving almost constant electric field in the vicinity of the center.

Figure 2 Contours of equal value of the electric field in the xy plane at $t=7.5$.

The $y$ component of the magnetic field is given by

(41)$$\begin{align}&{B}_y\left(x,y,t\right)=\frac{I_0}{2c\sqrt{1-{\beta}^2}}\notag\\&\quad{}\times \ln \left[\frac{\left({t}_{+}+\sqrt{t^2-{r}^2/{c}^2}\right)\left({t}_{-}-\sqrt{t^2-{r}^2/{c}^2}\right)}{\left({t}_{+}-\sqrt{t^2-{r}^2/{c}^2}\right)\left({t}_{-}+\sqrt{t^2-{r}^2/{c}^2}\right)}\right].\end{align}$$

The $x$ component of the magnetic field vanishes in the plane $y=0$ due to symmetry of the problem under consideration.

At the point $r=0$ the magnetic field is equal to zero, ${B}=0$, for all $t$. The electric field here is constant, given by

(42)$$\begin{align}{E}_z\left(r=0,t\right)=\frac{I_0\beta }{c\sqrt{1-{\beta}^2}}\ln \left(\frac{1+\sqrt{1-{\beta}^2}}{1-\sqrt{1-{\beta}^2}}\right).\end{align}$$

In the non-relativistic limit of the current sheet evolution for $\beta = 1$ the electric field is

(43)$$\begin{align}{E}_z\left(r=0,t\right)\approx \frac{2{I}_0\beta }{c}\ln \left(\frac{2}{\beta}\right).\end{align}$$

In the ultra-relativistic case, with $\beta \to 1$, we have the electric field ${E}_z\left(r=0,t\right)\approx 4{I}_0$.

In the vicinity of the point $r=0$ the magnetic field is a linear function of the coordinates. According to Equation (41) here the magnetic field is

(44)$$\begin{align}{B}_y\left(r=0,t\right)\approx \frac{4{I}_0}{c\beta t}x.\end{align}$$

We see that the electric field $E\approx \beta {B}_0$ is generated as the current sheet decays. Here ${B}_0$ is the magnetic field far outside of the current sheet and $\beta =v/c$ with $v$ equal to the characteristic plasma velocity.

Using the relationships obtained above for the electric and magnetic fields near the point $r=0$ (i.e., in the region $r<\beta ct$) the electromagnetic potential can be approximated as

(45)$$\begin{align}{A}_z\left(x,y,t\right)\approx -\frac{1}{\pi }{B}_0\beta ct+\frac{B_0}{\pi \beta ct}\left({x}^2-{y}^2\right).\end{align}$$

This expression describes nonstationary zero line of the magnetic field of hyperbolic type. The electric field directed along the zero line is constant, while the gradient of the magnetic field decreases inversely proportional to the time.

3.1 Thermoelectric field and ponderomotive driven field

Considering unmagnetized plasmas with ideal MHD condition, the electrons move quickly along the temperature and density gradients due to their small inertia. The non-neutral charge distribution generates an electric field. Combine the ideal Ohm’s law (Equation (46)) and Cauchy momentum Equation (47),

(46)$$\begin{align}\mathbf{E}+\frac{\mathbf{u}\times \mathbf{B}}{c}=0,\end{align}$$

(47)$$\begin{align}{n}_e{m}_e{\partial}_t\mathbf{u}=\frac{\mathbf{j}\times \mathbf{B}}{c}-\nabla P,\end{align}$$

where $P$ is the electron pressure and ${n}_e$ is the number density of the electron component. Taking the quasi-stationary condition (${\partial}_t\mathbf{u}=0$) into account, the self-generated electric field is obtained,

(48)$$\begin{align}\mathbf{E}=-\frac{\nabla P}{en_e}.\end{align}$$

Such a field is the response of the plasma to prevent the electrons motion according to the gradients in electron pressure. Applying curl on Equation (48) and plugging in Faraday’s law (Equation (9)), it yields

(49)$$\begin{align}{\partial}_t\mathbf{B}=-c\frac{\nabla {n}_e\times \nabla P}{en_e^2}.\end{align}$$

Using the equation of state,

(50)$$\begin{align}P={n}_e{k}_B{T}_e,\end{align}$$

Equation (49) can be further extended to

(51)$$\begin{align}{\partial}_t\mathbf{B}=-\frac{ck_B}{en_e}\left(\nabla {n}_e\times \nabla {T}_e\right),\end{align}$$

where ${k}_B$ is the Boltzmann constant and ${T}_e$ is the electron temperature. When the laser irradiates on the solid surface, the density varies mainly along the longitudinal direction and the temperature gradient is significant in the transverse direction. Therefore, a toroidal magnetic field is generated around the laser axis due to the misalignment between the gradients of the density and temperature. Such a spontaneous magnetic field generation process is often called as the Biermann battery effect^[Reference Biermann^51]. It is proposed to be the origin source of magnetic field in astrophysics and amplified as the seed fields^[Reference Naoz and Narayan^98]. Biermann battery effect has also been applied in the early laser driven MR experiments^[Reference Nilson, Willingale, Kaluza, Kamperidis, Minardi, Wei, Fernandes, Notley, Bandyopadhyay, Sherlock, Kingham, Tatarakis, Najmudin, Rozmus, Evans, Haines, Dangor and Krushelnick^50, Reference Li, Seguin, Frenje, Rygg, Petrasso, Town, Landen, Knauer and Smalyuk^52] with moderate intensities (${\sim}{10}^{15}\ \mathrm{W}/{\mathrm{cm}}^2$) and long pulse length (${\sim} \hbox{ns}$), mega-gauss magnetic fields in plasmas at the focal spot edges were obtained.

When the laser intensity becomes higher, the ponderomotive force of the laser field expels the electrons away from the focal spot and significantly changes the local density distributions forming sharp density gradients. By the continuous radiation pushing, the background electrons are prevented from returning and shielding against the charge separation field. A ponderomotive electric current is formed $\mathbf{J}={n}_ee\mathbf{v}\sim {n}_e\nabla I$ and it induces the magnetic field as $\textbf{B}\sim \nabla {n}_e\times \nabla I$. Theoretical and numerical studies have proved that magnetic field with 100 MG can be generated via the interaction of overdense plasma and intense ($\mathit{I}\sim {10}^{18}\ \mathrm{W}/{\mathrm{cm}}^2$) laser pulse^[Reference Wilks, Kruer, Tabak and Langdon^99–Reference Das, Kumar, Shukla, Bera, Verma, Mandal, Vashishta, Patel, Hayashi, Tanaka, Chatterjee, Lad, Kumar and Kaw^103]. Comparing with Biermann battery effect, the ponderomotive driven magnetic field is much stronger. However, it decays quickly after the laser-matter interaction, which indicates the field has a life time with the scale of the pulse duration ($t\sim {\tau}_L$). In contrast, the magnetic field in Bierman battery effect decays in a long time scale and can be regarded as a quasi-steady field.

3.2 Static magnetic field driven by energetic electron beam

The generation of magnetic field based on ultra intense laser-plasma interactions has been investigated intensively in recent years^[Reference Nakamura and Mima^104–Reference Murakami, Honrubia, Weichman, Arefiev and Bulanov^106]. Furthermore, the interactions of beam-plasma are also discussed for strong magnetic dipoles generation^[Reference Jia, Mima, Cai, Taguchi, Nagatomo and He^107, Reference Naseri, Bochkarev, Ruan, Bychenkov, Khudik and Shvets^108]. The magnetic field generation under the condition of ultra-intense laser interacting with underdense or near critical density plasma is discussed in this section. The ultra-intense laser considered here means the corresponding intensity exceeds the relativistic threshold ($I\gg {10}^{18}\ \mathrm{W}/{\mathrm{cm}}^2$) and the laser power is beyond the characteristic power of self-focusing (${\mathcal{P}}_c=2{m}_e^2{c}^5/{e}^2=17\ \mathrm{GW}$)^[Reference Sun, Ott, Lee and Guzdar^109]. The laser is assumed to have a short pulse duration ($c{\tau}_L\sim {\lambda}_p$), in which ${\lambda}_p$ is the plasma wavelength.

The strong ponderomotive force quickly expels electrons from the laser path and leaves an electron plasma channel which has a positive charge background since ions with heavier inertia response slowly. Balancing the electron energy gain and loss from the laser and the electrostatic potential,

(52)$$\begin{align}e{\phi}_{static}={\varepsilon}_{pond},\end{align}$$

(53)$$\begin{align}\pi {n}_e{e}^2{R}_{ch}^2={a}_{ch}{m}_e{c}^2,\end{align}$$

the radius of the plasma channel can be estimated,

(54)$$\begin{align}{R}_{ch}=\sqrt{\frac{a_{ch}{n}_c}{n_e}}\frac{\lambda }{\pi },\end{align}$$

where ${n}_c=m{\omega}_L^2/4\pi {e}^2$ is the critical density of plasmas and $\lambda$ is the laser wavelength. ${a}_{ch}$ is the laser pulse amplitude inside the plasma channel^[Reference Bulanov, Esarey, Schroeder, Leemans, Bulanov, Margarone, Korn and Haberer^110], which is proportional to the laser power and plasma density,

(55)$$\begin{align}{a}_{ch}={\left(\frac{2}{K}\frac{\mathcal{P}}{{\mathcal{P}}_c}\frac{n_e}{n_c}\right)}^{1/3},\end{align}$$

where $K$ is a constant factor coming from the integration over the duration of the laser pulse.

The charge separation field in the plasma channel appears at the wake of the laser pulse and forms the so-called wakefield^[Reference Tajima and Dawson^111]. In a relatively low density collisionless plasma, the induced wakefield, following the laser pulse, propagates with a phase velocity close to the speed of light in vacuum. The electrons injected into the acceleration phase of the wakefield with the initial velocity close to the wakefield phase velocity can be trapped and accelerated. The strength of the wakefield behind the laser pulse has been analytically studied^[Reference Sprangle, Esarey, Ting and Joyce^112],

(56)$$\begin{align}{E}_x\left(\zeta, r\right)=-\frac{2{\pi}^2{k}_p{\phi}_L(r)}{4{\pi}^2-{k}_p^2{\left(c{\tau}_L\right)}^2}\times \left[\sin {k}_p\left(c{\tau}_L-\zeta \right)+\sin {k}_p\zeta \right],\end{align}$$

where the laser is supposed to propagate in $x$-direction, $\zeta =x- ct$ is the transformation to the light frame, ${\phi}_L$ is the ponderomotive potential of the laser pulse, and ${k}_p=2\pi /{\lambda}_p$ is the wave number of the Langmuir waves.

The trapped and accelerated electron beam generates a current along the plasma channel. In the case of near critical density plasma, the accelerated beam contains a large electric charge resulting in a strong direct current. Consequently, the azimuthal magnetic field confined in the plasma channel is produced. From Ampere-Maxwell law,

(57)$$\begin{align}\nabla \times \mathbf{B}=\frac{4\pi }{c}\mathbf{j}+\frac{1}{c}{\partial}_t\mathbf{E},\end{align}$$

the strength of the magnetic field can be estimated provided the quasi-static condition of ${\partial}_t\mathbf{E}=0$. Since the magnetic field has the same size as the radius of the plasma channel, one obtains the maximum amplitude,

(58)$$\begin{align}\mid \mathbf{B}\mid =4\pi {n}_e{eR}_{ch}.\end{align}$$

An example of kinetic simulation is presented here to demonstrate the details of the process. Linear polarized laser with the intensity ${I}_0={10}^{21}\ \mathrm{W}/{\mathrm{cm}}^2$ is incident into the plasma target with the peak density of ${n}_e=0.25{n}_c$ and propagates along $x$-direction. The laser pulse has Gaussian profiles in both transverse and longitudinal directions with the full width at half maximum (FWHM) spot size of and pulse duration ${\tau}_L=30\ \mathrm{fs}$.

The electron density distribution with a plasma channel feature is shown in Figure 3(a). The radius of the channel is about , which is well consistent with the theoretical prediction according to Equation (54). Due to the self-focusing effect, the laser amplitude inside the channel is much larger than that of the initial field in vacuum. The maximum amplitude of the wakefield excited by the enhanced laser pulse therefore can be estimated to^[Reference Bulanov, Esirkepov, Hayashi, Kiriyama, Koga, Kotaki, Mori and Kando^113]

(59)$$\begin{align}{E}_x=\frac{m_e{c}^2}{2\pi \lambda e}{\left(\frac{\omega_{pe}}{\omega_0}\right)}^{2/3}{\left(\frac{\mathcal{P}}{{\mathcal{P}}_c}\right)}^{1/6}.\end{align}$$

Figure 3 The numerical demonstration of static magnetic field driven by energetic electron beam. (a) The electron density distribution with the plasma channel formation. (b) The blue and red curves represent the longitudinal electric field and the electron density profile on the laser axis ($y=0$). (c) The average energy distribution of the electrons. (d) The z component of the azimuthal magnetic field induced by the energetic electron beam.

The profile of the longitudinal electric field along the laser axis ($y=0$) is presented in Figure 3(b) and the maximum of the acceleration phase reaches about ${E}_{x,\mathit{\max}}\approx 65\ \mathrm{GV}/\mathrm{cm}$. The corresponding electron density profile (blue in Figure 3(b)) indicates a bunch of the electrons are trapped and accelerated in the wakefield. With such a high acceleration gradient, the electrons in the channel are expected to gain energy quickly. The average energy distribution of the electrons,

(60)$$\begin{align}<{\varepsilon}_e>=\frac{\int {\varepsilon}_e\left(x,y\right)f\left(x,y\right) \hbox{d}x\hbox{d}y}{\int f\left(x,y\right) \hbox{d}x\hbox{d}y},\end{align}$$

is shown in Figure 3(c). The energy of the electron beam reaches hundreds MeV and it corresponds to an electric current density of $\approx 0.2{n}_c ec$. According to Ampere-Maxwell law, an azimuthal magnetic field is generated with the same radius of the plasma channel as presented in Figure 3(d). The amplitude reaches about 0.6 GG which agrees with the estimate obtained from Equation (58). The magnetic dipole propagates together with the laser pulse in the plasma channel, which is quasi-steady till the laser depletion and the electron beam dissipation. The depletion length of the pumping laser can be estimated when the laser energy is completely dumped to the excited wakefield (${\varepsilon}_L\sim {\varepsilon}_w$),

(61)$$\begin{align}\frac{E_L^2}{4\pi }c{\tau}_L={n}_e{m}_e{c}^2\frac{a_{ch}^2}{2}{l}_{dp},\end{align}$$

which can be rewritten in the form of

(62)$$\begin{align}{l}_{dp}=\frac{2c{\tau}_L}{a_0^2}{\left(\frac{2}{K}\frac{\mathcal{P}}{{\mathcal{P}}_c}\frac{\omega_0}{\omega_{pe}}\right)}^{2/3}.\end{align}$$

Equation (62) indicates that the magnetic field has a relatively long time scale in the ultra-intense laser case, which is important for designing the corresponding experiment.

4.1 Magnetic dipole expansion

According to Equation (54), the radius of the magnetic dipole ${R}_{ch}$ is the function of local plasma density ${n}_e$. Combining Equations (54) and (55), ${R}_{ch}$ is then expressed as

(63)$$\begin{align}{R}_{ch}={\left(\frac{2}{K}\frac{\mathcal{P}}{{\mathcal{P}}_c}\right)}^{1/6}{\left(\frac{n_c}{n_e}\right)}^{1/3}\frac{\lambda }{\pi }.\end{align}$$

Suppose the density distribution is inhomogeneous in the longitudinal direction; then the size of the magnetic dipole varies with the propagation as

(64)$$\begin{align}\frac{\hbox{d}R_{ch}}{R_{ch}}+\frac{\lambda }{3\pi}\frac{v_g}{n_e}\frac{\partial {n}_e}{\partial x} \hbox{d}t=0,\end{align}$$

where ${v}_g$ is the group velocity of the pumping laser in the plasma which is close to the speed of light since it is ultra-intense and the plasma target is underdense. Equation (64) implies that a longitudinal density down gradient will force the magnetic dipole to expand in the transverse direction. In Figure 4(a), an example of such B-field expansion is presented. Here the plasma has a density down ramp in $x$-direction from $0.25{n}_c$ at $x=55 \lambda$ to $0.17{n}_c$ at $x=90 \lambda$. In the region of $x<55\lambda$, the density is homogeneous. The magnetic dipole structures at different snapshots are combined together to present the expansion effect clearly, in which the dashed curves indicate the radius of the magnetic dipole. From Region I to II, the dipole radius grows in the density plateau region until it saturates at the maximum radius according to Equation (54). In the density downramp from Region II to IV, it expands significantly and doubles the size from ${R}_{ch}\approx 5\lambda$ to ${R}_{ch}^{\prime}\approx 12\lambda$. The transverse profiles of ${B}_z$ along different $x$-coordinates are shown in Figure 4(b). Based on the Ertel’s theorem^[Reference Zakharov and Kuznetsov^114], the circulation of magnetic field,

(65)$$\begin{align}\oint \mathbf{B}\cdot \hbox{d}\mathbf{r}=\frac{4\pi }{c}\int \mathbf{j}\cdot \hbox{d}\mathbf{S},\end{align}$$

is a Poincare invariant. Therefore, accompanying the size growing, the amplitude of the magnetic field decreases, which is also consistent with the numerical results shown here.

Figure 4 (a) The transverse expansion of the magnetic dipole along a density downramp region. The distributions of ${B}_z$ at different snapshots are combined here. (b) The profiles of ${B}_z$ along different $x$-coordinates.

The expansion of the magnetic dipole in the density downramp provides an optimal condition for the study of MR. By sending two parallel identical laser pulses into the plasma (laser axes distributed at ${y}_1$ and ${y}_2$ along $y$-direction), a pair of azimuthal magnetic dipoles are generated by the parallel accelerated electron beams. In the ideal case, the two magnetic dipoles are translational symmetry along the transverse direction,

(66)$$\begin{align}\mathbf{B}\left(x,y,z\right)=\mathbf{B}\left(x,y+d,z\right),\end{align}$$

where $d$ is the separation distance of the two laser axes. It should be noted that the separation distance between the two pulses should be large enough to avoid the two plasma channels interference with each other in the initial stage, i.e., $d>2{R}_{ch}$. Due to the feature of magnetic dipole in each channel, the total magnetic field has an antisymmetry distribution on the two sides of the system center ${y}_c=\left({y}_1+{y}_2\right)/2$, i.e.,

(67)$$\begin{align}\mathbf{B}\left(x,{y}_c+y,z\right)=-\mathbf{B}\left(x,{y}_c-y,z\right).\end{align}$$

Therefore, with the expansion of the two magnetic dipoles, the opposite magnetic polarities approach at the central region in transverse, where the magnetic field annihilation and the field line reconnection are expected to occur.

The double magnetic dipole structure and the expansion are shown in the following dimensionless simulation. The surf in Figure 5(a) represents the energy density distribution ($<{n}_e{\varepsilon}_e>$) of the electrons. High density energetic electrons driven by the double parallel laser pulses correspond to the bright regions. The azimuthal magnetic fields around the accelerated electron beams are depicted by the circles. The projections of ${B}_z$ components in the $xy$ plane shown in Figure 5(b) refer to the early stage in which the two magnetic dipoles are still propagating in the uniform density region without expansion. In the density downramp region as shown in Figure 5(c), significant expansion of the magnetic dipoles forces the opposite polarities to touch each other along the center as indicated by the dot and cross signs. It also shows the decrease of the magnetic field amplitude with the transverse expansion. Both the size increase and the amplitude decay are consistent with the theoretical predictions. The dashed line here indicates the region of magnetic field annihilation and a current sheet is developed locally during MR process.

Figure 5 (a) The energy density distribution ($<{n}_e{\varepsilon}_e>$) of electrons. The round circles represent the azimuthal magnetic fields. The projections of ${B}_z$ components in (b) the uniform density region and (c) the density downramp region.

4.2 Magnetic field annihilation and field line reconnection

The 2D magnetic configuration in the transverse plane ($yz$ plane) can be described by a complex function^[Reference Syrovatskii^115],

(68)$$\begin{align}B\left(y,z\right)={B}_y-{iB}_z,\end{align}$$

with a complex variable $\zeta =y+ iz$. In the vicinity of the reconnection region (the dashed line in Figure 5(c)), the magnetic field lines form a hyperbolic structure in the transverse plane and the constant potential surface is

(69)$$\begin{align}\mid \mathbf{A}\left(y,z\right)\mid =\Re \left\{{B}_0{\zeta}^2/2\right\}.\end{align}$$

Here $\mathbf{A}$ is the local vector potential. The magnetic fields with opposite directions but equal amplitude meet at the so-called zeroth point (X-point) where the sum of magnetic field is zero. Figure 6(a) presents the hyperbolic magnetic field lines distribution close to the zeroth point in the transverse plane. With the approaching of the field lines to the $X$-point, the lines reconnection occurs and the magnetic fields vanish associated with the generation of electric field according to

(70)$$\begin{align}\mathbf{E}=-\frac{1}{c}{\partial}_t\mathbf{A}.\end{align}$$

Figure 6 (a) and (b) are contours of the constant vector potentials around the $X$-point based on theoretical model. (a) refers to the initial stage when the opposite magnetic fields just begin to vanish. (b) refers to the moment when the current sheet in MR has formed and bifurcated. (c) and (d) are the corresponding distributions demonstrated by numerical simulations.

Therefore the charged particles can be accelerated during MR via such an electric field produced by the rapid change of magnetic field, which will be discussed in detail in the next section. The current driven by the accelerated particles generates an extra magnetic field represented by the blue curve in Figure 6(b). The new magnetic field changes the field topology and the $X$-point breaks into two new zeroth points as depicted by ${X}^{\prime }$. The $X$-point bifurcation continues and the $X$-point grows to be a current layer shown by the red region in Figure 6(b), which is known as the current sheet. It can also be understood as the electric current filamentation and breaks up to separated pieces due to the tearing mode instability. The development of such a process takes about 15 laser periods. According to the theoretical estimate in Equation (33) under the short wavelength limit, the growth rate of the tearing instability is about ${\tau}^{-1}$, the reciprocal of the pulse duration, which is about 11 laser periods. The broken configuration of magnetic field is given by

(71)$$\begin{align}B\left(y,z\right)=\frac{B_0\zeta }{\sqrt{s{(t)}^2-{\zeta}^2}}.\end{align}$$

Here $s$ is the spatial scale of current sheet modulation^[Reference Syrovatskii^115, Reference Bulanov, Sasorov and Syrovatskii^116]. The corresponding vector potential is

(72)$$\begin{align}\mid \mathbf{A}\left(y,z\right)\mid =\Re \left\{{B}_0\sqrt{s{(t)}^2-{\zeta}^2}\right\}.\end{align}$$

Recalling Equation (70), the electric field is given as

(73)$$\begin{align}E(t)=-\frac{1}{c}\frac{B_0s(t)\dot{s}(t)}{\sqrt{s{(t)}^2-{\zeta}^2}}.\end{align}$$

The 2D configuration of magnetic field lines and the constant vector potential is demonstrated by a 3D numerical simulation. The curves and the vectors in Figures 6(c) and 6(d) present the contour magnetic field strength and the magnetic field lines with directions. The formation of $X$-point corresponding to the situation of Figure 6(a) is shown in Figure 6(c) in which the magnetic field loops produced by the parallel currents form an eight-like curve with a single zeroth-point localized in the center. The opposite magnetic field lines in the vicinity of the zeroth-point are not vanishing each other at this moment.

With the process of MR and the development of tearing instability, the $X$-point breaks and a thin current sheet grows as seen in Figure 6(d). The new closed loops in the middle of the current sheet can be regarded as the magnetic islands which are the results of tearing driven current filamentation. The $X$-point also separates along $z$-direction and the magnetic field lines around the bifurcated ${X}^{\prime}$-points change their directions to connect the lines generated from the other current (indicated by the vectors). The evolution of the magnetic field structures shown in Figure 6(d) is well consistent with the theoretical model displayed in Figure 6(b).

4.3 Relativistic magnetic field dissipation and inductive electric field generation

Recalling Ampere-Maxwell law in Equation (57), the term of displacement current (${\mathbf{j}}_D={\partial}_t\mathbf{E}$) is normally neglected in the context of MHD approximation. In the vicinity of the current sheet, the curl of magnetic field can be estimated as $B/{r}_{cs}$, where ${r}_{cs}$ is the characteristic size of the current sheet. In this case, the Ampere-Maxwell equation excluding displacement current can be rewritten as

(74)$$\begin{align}\frac{B}{r_{cs}}=\frac{4\pi }{c}{n}_{cs} eu,\end{align}$$

where ${n}_{cs}$ and $u$ are the density and velocity of the charged particles inside current sheet. Due to the relativistic constraint that the particle velocity cannot exceed the speed of light, the following relation is obtained to satisfy the MHD approximation,

(75)$$\begin{align}\frac{B}{4\pi {en}_{cs}{r}_{cs}}=\frac{u}{c}\le 1.\end{align}$$

However, in the relativistic regime of MR, such a relation is not always satisfied. Considering the situation of ultra-intense laser interacting with underdense or near critical density plasma, substituting the magnetic field with Equation (58), one obtains

(76)$$\begin{align}\frac{n_e{R}_{ch}}{n_{cs}{r}_{cs}}=\frac{u}{c}\approx 1.\end{align}$$

Here the velocities of the relativistic charged particles are assumed to reach their upper limits of the light speed. As mentioned above, the current sheet is formed in the density downramp region where the magnetic dipoles are expanding, which implies the local density is much lower than the density plateau where the energetic electron beams and static magnetic fields are formed, i.e., ${n}_{cs}\ll {n}_e$. On the other hand, the size of the current sheet is obviously smaller than the radius of magnetic dipoles, i.e., ${r}_{cs}\ll {R}_{ch}$. Therefore, under the relativistic MR regime, the above relation changes to

(77)$$\begin{align}\frac{n_e{R}_{ch}}{n_{cs}{r}_{cs}}\gg 1,\end{align}$$

and the restrained condition for MHD approximation in Equation (75) becomes invalid. This means the limited conduction current (${n}_e ec$) is not able to sustain the variation of magnetic field ($\nabla \times B$) and the effect of the displacement current (${j}_D={\partial}_tE$) must be taken into account. The displacement current refers to the growth rate of an inductive electric field. To compensate the difference of $c\nabla \times B$ and the conduction current, substituting magnetic field with Equation (58) and the local current in the current sheet with ${n}_{cs} ec$, the inductive electric field is obtained,

(78)$$\begin{align}{\partial}_tE=4\pi {n}_e ec\left(\frac{R_{ch}}{r_{cs}}-\frac{n_{cs}}{n_e}\right).\end{align}$$

The displacement current plays an important role here to balance the fast magnetic field variation and converts the energy from magnetic field to electric field. The growing inductive electric field indicates the relativistic MR is a non-stationary process violating the freezing-in condition and the corresponding Ohm law should be modified^[Reference Pegoraro^73, Reference Pegoraro^74]. Due to the non-stationary feature, the relativistic MR is also proposed as dynamic dissipation of magnetic field^[Reference Syrovatskii^75].

The growth of displacement current and the generation of the inductive electric field via magnetic field annihilation are presented by kinetic simulations. Figure 7(a) shows the distribution of the magnetic dipoles during the opposite polarities vanishing. The relativistic MR is expected to occur in the current sheet region of $80<x\left(\lambda \right)<110$ and $y\sim 0$. The components of Ampere-Maxwell law in Equation (57) along $y=0$ are plotted in Figure 7(b) in which the surface represents the distributions of the longitudinal electric field. Two clear peaks of displacement current (red curve) can be found in the region of $100<x\left(\lambda \right)<110$ and $80<x\left(\lambda \right)<90$. Both of the regions have relatively low conduction currents (blue), which is consistent with the theory model mentioned above that the appearance of displacement current is to balance the gap between the curl of magnetic field (black curve) and the conduction current. The longitudinal electric field (green curve) always grows behind the peaks of the displacement currents and the interval between the peaks of ${E}_x$ and ${j}_D$ indicates the growth rate. On the surface of ${E}_x$ distribution, the two bright peaks around $100<x\left(\lambda \right)<110$ are the laser driven wakefields. In the region close to $y=0$, the fields are induced by the displacement current in MR, which is lower than the laser driven wakefield but with the same order of magnitude. The heavier mass of the ions limits the acceleration effect. Under the interactions with state-of-the-art high power laser and plasmas, the out-jets here are mainly produced by the accelerated electrons. It is noted that the inductive electric fields accelerate electrons in the counter-propagation direction of the laser pulse. Therefore, the electrons are accelerated in the backward direction and only experience the inductive electric field in a short range since they are not co-moving with the magnetic dipoles.

Figure 7 (a) The magnetic field ${B}_z$ distributions in the simulation when MR is occurring. (b) The surface represents the distribution of longitudinal electric field (${E}_x$). The curves are the profiles of all the components in Ampere-Maxwell law (Equation (57)).

4.4 Charged particle acceleration in relativistic MR

Charged particles in the current sheet experiencing strong inductive electric field will be accelerated to leave the reconnection region with relativistic energy forming high energetic out-jets. Such violent acceleration is independent of the Joule dissipation since the energy is much higher than the thermal energy. The out-jets in MR are commonly observed in astrophysics. It also relates to the explanation of the narrow radiation spectrum obtained from the Crab Nebula^[Reference Abdo, Ackermann and Ajello^40–Reference Cerutti, Werner, Uzdensky and Begelman^43]. A typical supra thermal electron spectrum is presented in Figure 8 based on the setup discussed in Section 4.3. The incident parallel laser pulses have the intensity of ${10}^{21}\ \mathrm{W}/{\mathrm{cm}}^2$ and the plasma plateau density is $0.2{{n}}_{{c}}$. The cutoff energy increases from several MeV to ${\sim} 40\ \mathrm{MeV}$ during the electrons propagating through the reconnection region. The energy dumping from magnetic field to the bunch of supra thermal electrons is about 1%. The energy of the backward jets from MR is not comparable to the wakefield acceleration. The propagation direction of the inductive electric field is opposite to the direction of the out-jets. Therefore the electrons experience only short acceleration stage in the electric field and the electrons are not trapped.

Figure 8 The energy distributions of the electrons inside current sheet before and after magnetic field reconnection.

Here we discuss the details of the particle acceleration dynamics in the regime of relativistic MR. Suppose there is a hyperbolic distribution of magnetic field in the vicinity of the $X$-point as shown in Figure 9(a). The inductive electric field is generated according to the variation of magnetic field in the transverse plane which results in the growth of the electric field along the longitudinal direction. Therefore the fields in the vicinity of the $X$-point are described as

(79)$$\begin{align}\mathbf{E}={E}_0\hat{\mathbf{x}},\end{align}$$

Figure 9 (a) Schematic of the theoretical model in the vicinity of $X$-point. (b) The analytical solutions of particles motion with the expressions in Equations (96) and (97). (c) and (d) are the trajectories of charged particles given by the solutions of Equations (98) and (99) for the initial conditions of ${B}_0/{E}_0\beta =3$, $y(1)\in \left[-2,2\right],\dot{y}(1)=0$ and $z(1)\in \left[-2,2\right],\dot{z}(1)=0$. (e) and (f) show the typical accelerated particle trajectories obtained in the kinetic simulations.

(80)$$\begin{align}\mathbf{B}={b}_0\left(-y\hat{\mathbf{z}}-z\hat{\mathbf{y}}\right).\end{align}$$

The charged particle motion is dominated by Lorentz force $\hbox{d}\mathbf{p}/ \hbox{d}t=q\left(\mathbf{E}+\mathbf{v}\times \mathbf{B}/c\right)$. The dynamic equations for different components can be expressed as

(81)$$\begin{align}\frac{\hbox{d}p_x}{\hbox{d}t}={qE}_0+\frac{q}{c}{\left(\mathbf{v}\times \mathbf{B}\right)}_x=q\left[{E}_0+\frac{b_0}{2c}\frac{\hbox{d}\left({z}^2-{y}^2\right)}{\hbox{d}t}\right],\end{align}$$

(82)$$\begin{align}\frac{\hbox{d}p_y}{\hbox{d}t}=\frac{q}{c}{\left(\mathbf{v}\times \mathbf{B}\right)}_y=q\frac{b_0}{c}y\frac{\hbox{d}x}{\hbox{d}t},\end{align}$$

(83)$$\begin{align}\frac{\hbox{d}p_z}{\hbox{d}t}=\frac{q}{c}{\left(\mathbf{v}\times \mathbf{B}\right)}_z=-q\frac{b_0}{c}z\frac{\hbox{d}x}{\hbox{d}t}.\end{align}$$

According to Equation (81), the particle motion in the longitudinal direction is homogeneous and can be obtained with time integral,

(84)$$\begin{align}{p}_x=q\left\{{E}_0t+\frac{b_0}{2c}\left[\left({z}^2-{z}_0^2\right)-\left({y}^2-{y}_0^2\right)\right]\right\}.\end{align}$$

The acceleration process in MR can be derived with respect to non-relativistic and relativistic conditions^[Reference Bulanov and Sasorov^117]. Here we shall focus on the relativistic case of charged particle acceleration driven by inductive electric field. The characteristic scale length of electron motion in MR region can be estimated as the Larmor radius

(85)$$\begin{align}R=\frac{cp}{e\mid B\mid }=\frac{cp}{qb_0{r}_{cs}}.\end{align}$$

Considering the particles are only accelerated by the longitudinal electric field,

(86)$$\begin{align}cp={qE}_0{r}_{cs},\end{align}$$

the characteristic scale length can be written as

(87)$$\begin{align}R={E}_0/{b}_0,\end{align}$$

and the corresponding characteristic time when the electrons are accelerated before leaving MR region is

(88)$$\begin{align}T=R/c={E}_0/{cb}_0.\end{align}$$

The longitudinal motion of the accelerated charged particle in relativistic case is simply as

(89)$$\begin{align}x(t)= ct,\end{align}$$

and the corresponding momentum is

(90)$$\begin{align}{p}_x=\gamma {mv}_x=\frac{\varepsilon \dot{\varepsilon}}{qE_0{c}^2}.\end{align}$$

Here $\varepsilon$ is the kinetic energy of the particles. Comparing with the general solution of longitudinal momentum in Equation (84), one obtains

(91)$$\begin{align}\frac{\varepsilon \dot{\varepsilon}}{qE_0{c}^2}=q\left\{{E}_0t+\frac{b_0}{2c}\left[\left({z}^2-{z}_0^2\right)-\left({y}^2-{y}_0^2\right)\right]\right\}.\end{align}$$

In the region of current sheet, where $\left({y}^2+{z}^2\right)<{R}^2$, the second term on RHS of Equation (91) can be neglected. Therefore the kinetic energy of the accelerated particles can be simplified as

(92)$$\begin{align}\varepsilon (t)={\left({\varepsilon}_{t=0}^2+{q}^2{E}_0^2{c}^2{t}^2\right)}^{1/2}.\end{align}$$

From the equation of motion for $\hbox{d}\mathbf{p}/ \hbox{d}t=\hbox{d}\left(\gamma m\mathbf{v}/ \hbox{d}t\right)$, the following relation can be derived,

(93)$$\begin{align}{\ddot{x}}_{\alpha }+{\dot{x}}_{\alpha}\frac{\dot{\varepsilon}}{\varepsilon }={\dot{p}}_{\alpha}\frac{c^2}{\varepsilon },\end{align}$$

here ${x}_{\alpha }$ refers to the $x$, $y$ and $z$ directions in Cartesian coordinates. Recalling Equations (82) and (83), the particle motions in the transverse plane are obtained,

(94)$$\begin{align}\ddot{y}+\dot{y}\frac{\dot{\varepsilon}}{\varepsilon }=-\frac{cb_0}{E_0}\frac{\dot{\varepsilon}}{\varepsilon }y,\end{align}$$

(95)$$\begin{align}\ddot{z}+\dot{z}\frac{\dot{\varepsilon}}{\varepsilon }=\frac{cb_0}{E_0}\frac{\dot{\varepsilon}}{\varepsilon }z.\end{align}$$

Since the particles are accelerated to relativistic energy which is much higher than its initial energy ${\varepsilon}_{t=0}$, it is reasonable to estimate the particle kinetic energy in Equation (92) as $\varepsilon (t)={qE}_0 ct$. Therefore, the solutions of Equations (94) and (95), i.e., the particle motions in the transverse plane can be written as

(96)$$\begin{align}y(t)={y}_0{J}_0\left(\sqrt{\frac{4{b}_0 ct}{E_0}}\right),\end{align}$$

(97)$$\begin{align}z(t)={z}_0{I}_0\left(\sqrt{\frac{4{b}_0 ct}{E_0}}\right),\end{align}$$

in which ${J}_0(x)$ and ${I}_0(x)$ are the ordinary and the modified Bessel function of zeroth order. The analytical solutions of particles motion in $y$ and $z$ directions are plotted in Figure 9(b). It presents an oscillation along $y$-direction and an exponential expansion in $z$-direction.

In the above presented analysis of the charged particle acceleration in the vicinity of hyperbolic zero line of the magnetic field, Equations (79) and (80) describe the configuration with constant gradient of the magnetic field, i.e., ${b}_0$ is a constant. This corresponds to the case when the charged particle moves faster than the magnetic configuration changes. In ultra-relativistic limit the characteristic velocity of both the charged particles and of the magnetic field modulations is close to the speed of light in vacuum. Using Equations (44) and (71) showing that the magnetic field gradient is inversely proportional to time, ${b}_0={B}_0/\beta ct$ and $\varepsilon ={qE}_0 ct$ (see Equation (91) in the limit $t\to \infty $), we can modify the equations of the electron motion to the form

(98)$$\begin{align}\ddot{y}+\dot{y}\frac{1}{t}=-\frac{B_0}{E_0\beta}\frac{1}{t^2}y,\end{align}$$

(99)$$\begin{align}\ddot{z}+\dot{z}\frac{1}{t}=-\frac{B_0}{E_0\beta}\frac{1}{t^2}z,\end{align}$$

i.e., they are the Cauchy-Euler equations. These equations have the solutions

(100)$$\begin{align}y(t)={y}_1\cos \left(\sqrt{\frac{B_0}{E_0\beta }}\ln t\right)+{y}_2\sin \left(\sqrt{\frac{B_0}{E_0\beta }}\ln t\right),\end{align}$$

(101)$$\begin{align}z(t)={z}_1{t}^{\sqrt{\frac{B_0}{E_0\beta }}}+{z}_2{t}^{-\sqrt{\frac{B_0}{E_0\beta }}},\end{align}$$

where constants ${y}_1$, ${y}_2$, ${z}_1$, and ${z}_2$ are determined by initial conditions.

The solutions of the analytical model for the accelerated particle trajectories as functions of time along $y$ and $z$ directions are presented in Figures 9(c) and 9(d), respectively. The corresponding results obtained from 3D kinetic simulations are shown in Figures 9(e) and 9(f). The electrons from the current sheet experiencing the inductive electric field acceleration are randomly sampled here. As mentioned above, the accelerated electrons move opposite to the laser propagation direction from the current sheet. Here the starting point of the backward electrons motion is set to be $x=0$. Figure 9(e) reflects the fact that the in-flow electrons are squeezed into the current sheet along the $y$ direction pushed by the expanding magnetic dipoles. The candidate electrons can actually come from the region far from the current sheet. Once applied by the growing inductive electric field, the electrons move backwards with a clear oscillation. However, in the $z$-direction, all the candidate electrons must locate in the region close to $z=0$ and there are no in-flows along this direction. These electrons expand in the $z$-direction afterwards while being accelerated. The simulation results show well consistency with the analytical model shown in Figures 9(c) and 9(d).

5.1 Relativistic MR with higher order mode laser

A Hermite-Gaussian mode beam in the transverse plane propagating in $x$-direction can be described as

(102)$$\begin{align}{E}_{l,m}&={E}_0\frac{w_0}{w(x)}{H}_l\left[\frac{\sqrt{2}y}{w(x)}\right]{H}_m\left[\frac{\sqrt{2}z}{w(x)}\right]\notag\\&\quad \times \exp \left[-\frac{r^2}{w^2(x)}\right]\exp \left[-i\frac{kr^2}{2R(x)}\right]\notag\\&\quad \times \exp \left\{-i\left[ kx-\left(l+m+1\right)\varphi (x)\right]\right\}.\end{align}$$

Here ${H}_{l,m}$ are the Hermite polynomials, $w(x)$ is the Gaussian beam radius, $\varphi (x)$ is the Gouy phase at $x$, and $R(x)$ is the radius of curvature of the beam’s wavefronts. In the case of $l=m=0$, the fundamental Gaussian beam is obtained. Now considering the ${\mathrm{TEM}}_{10}$ mode with $l=1$ and $m=0$, Equation (102) becomes

(103)$$\begin{align}{E}_{1,0}&={E}_0\frac{w_0}{w(x)}\frac{2\sqrt{2}y}{w(x)}\notag\\&\quad \times \exp \left[-\frac{r^2}{w^2(x)}\right]\exp \left[-i\frac{kr^2}{2R(x)}\right]\notag\\&\quad \times \exp \left\{-i\left[ kx-2\varphi (x)\right]\right\}.\end{align}$$

On the focused plane with $x=0$, the intensity distribution is

(104)$$\begin{align}{\left|{E}_{1,0}\right|}^2=\mid {E}_0^2\mid \frac{8{y}^2}{w_0^2}\exp \left(-\frac{2{r}^2}{w_0^2}\right),\end{align}$$

which is presented in Figure 10(a) in logarithmic scale. The corresponding profile in the polarization direction $(y)$ is shown in Figure 10(b). It has double peaks in the intensity distribution which becomes comparable to the case with two laser pulses. However, in the higher order mode case, the two intensity peaks are naturally synchronized. The electron density distribution in the interaction of ${\mathrm{TEM}}_{10}$ and plasma is presented in Figure 10(c). The kinetic simulation presented here has the following parameters, ${I}=5\times {10}^{21}\ \mathrm{W}/{\mathrm{cm}}^2$, transverse size $16\lambda$ and the pulse duration of 20fs. The plasma target is near critical hydrogen with the density of $0.1{n}_c$. Two identical electron bubbles are formed and propagating in parallel. Wakefield acceleration occurs in each bubble and generates two energetic electron beams inside the bubbles. The corresponding magnetic dipoles produced by the accelerated electron beams are shown in Figure 10(d), which is similar to the situation presented in Figure 5(c). In the case with ${\mathrm{TEM}}_{10}$ mode laser, the magnetic dipoles are tightly attached after generation, which forces the magnetic field to start annihilation in the very beginning stage. The following MR process has been discussed thoroughly above. The advantage by employing ${\mathrm{TEM}}_{10}$ mode laser is the simultaneous production of the parallel magnetic dipoles, which overcomes the difficulties of laser synchronization.

Figure 10 (a) The intensity distribution of ${\mathrm{TEM}}_{10}$ mode laser on the focused plane and (b) the corresponding profile. (c) The electron density distribution and (d) the ${B}_z$ distribution obtained from numerical simulations in the interaction of ${\mathrm{TEM}}_{10}$ mode laser with plasma.

5.2 Laser split by solid cone target

When the high power laser has a relative broad spot size, it is possible to use a solid cone target to split the laser forming a donut shape pulse. By placing such a target in front of the near critical density plasma, the donut shape pulse after interaction generates a forward accelerated electron ring. The corresponding magnetic field has opposite polarities in the center which are expected to annihilate and induce the charged particle acceleration via inductive electric field. The reconnection rate for the laser split case is affected by the geometry of the cone target. The opening angle of the target determines the separation distance between the pulses after split. However, if the angle is too large, it reflects a significant portion energy of the incident laser. The detailed process is presented by the following numerical simulation results with the parameters of ${I}={10}^{21}\ \mathrm{W}/{\mathrm{cm}}^2$, transverse size $20\,\lambda,$ and the pulse duration of $15\,\mathrm{fs}$. The cone target has the solid density of $50\,{{n}}_{{c}}$ and the following near critical density plasma is $0.3\,{{n}}_{{c}}$. Figure 11(a) presents the laser intensity before and after passing through the solid cone target. An initial Gaussian speckle is shown at $t=6{T}_0$ and it is partially reflected in the center by the splitter as shown in the plane of $t=13{T}_0$. At $t=20{T}_0$, the multi-rings correspond to the diffracted fields by the cone target. Due to the self-focusing effect, the field amplitude is enhanced again and forms the donut shape propagating forward into the near critical density target as presented in the $t=27{T}_0$ plane.

Figure 11 (a) The evolution of incident laser intensity before and after interacting with the solid cone target. A loop structure (donut shape) is formed. (b) The electron density distribution driven by the donut shape field. The rear plane corresponds to the density distribution slice of $y=0$. The left plane is the projection of the slice of $x=85\lambda$. The bottom plane is the projection of magnetic field ${B}_z$ distribution, which shows the magnetic dipoles are formed.

The electron density distribution driven by the fields after splitting also presents a ring structure in the transverse plane as shown in Figure 11(b) the left plane. In the longitudinal plane ($xy$ plane), the parallel electron channels are formed in which the trapped electron beams are accelerated as discussed above. Magnetic dipoles (shown in the bottom plane) are formed with a similar structure as mentioned above. Annihilation and reconnection of the magnetic fields are expected in the center plane. By using the solid cone target, if the laser pulse is well collimated on the cone, the difficulty of multi laser synchronization is also overcome.