Researching | Magnetic field annihilation and charged particle acceleration in ultra-relativistic laser plasmas

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High Power Laser Science and Engineering
Vol. 9, Issue 1, 010000e2 (2021)

Magnetic field annihilation and charged particle acceleration in ultra-relativistic laser plasmas | Editors' Pick

Yan-Jun Gu^1、2、* and Sergei V. Bulanov^1、3

Author Affiliations

¹Institute of Physics of the ASCR, ELI-Beamlines, Na Slovance 2, 18221 Prague, Czech Republic

²Institute of Laser Engineering, Osaka University, Osaka565-0871, Japan

³Kansai Photon Research Institute, National Institutes for Quantum and Radiological Science and Technology, 8-1-7 Kizugawa-shi, Kyoto 619-0215, Japan

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DOI: 10.1017/hpl.2020.45 Cite this Article Set citation alerts

Yan-Jun Gu, Sergei V. Bulanov. Magnetic field annihilation and charged particle acceleration in ultra-relativistic laser plasmas[J]. High Power Laser Science and Engineering, 2021, 9(1): 010000e2 Copy Citation Text

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Abstract

Magnetic reconnection driven by laser plasma interactions attracts great interests in the recent decades. Motivated by the rapid development of the laser technology, the ultra strong magnetic field generated by the laser-plasma accelerated electrons provides unique environment to investigate the relativistic magnetic field annihilation and reconnection. It opens a new way for understanding relativistic regimes of fast magnetic field dissipation particularly in space plasmas, where the large scale magnetic field energy is converted to the energy of the nonthermal charged particles. Here we review the recent results in relativistic magnetic reconnection based on the laser and collisionless plasma interactions. The basic mechanism and the theoretical model are discussed. Several proposed experimental setups for relativistic reconnection research are presented.

Keywords

high energy density physics laboratory astrophysics laser plasmas interactions particle acceleration

$$\begin{align}{j}_{rel}= enc\end{align}$$

((1))

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$$\begin{align}B=4\pi enl.\end{align}$$

((2))

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$$\begin{align}\frac{B^2}{4\pi {nm}_e{c}^2}>1.\end{align}$$

((3))

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$$\begin{align}\frac{1}{c^2}{\partial}_{tt}{A}_z-{\partial}_{xx}{A}_z&=\frac{4\pi }{c}{j}_z+\frac{1}{c^2}{A}_z\left(x,t=0\right){\delta}^{\prime}(t)\nonumber\\&\quad+\frac{1}{c^2}{\partial}_t{A}_z\left(x,t=0\right)\delta (t).\end{align}$$

((4))

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$$\begin{align}{j}_z=\sum \limits_{j=e,p}{e}_j^2{n}_j\left(x,t\right)\frac{A_z}{{m}_j^2{c}^4+{e}_j^2{A}_z^2{c}^2},\end{align}$$

((5))

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

((6))

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

((7))

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

((8))

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

((9))

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

((10))

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

((11))

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

((12))

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

((13))

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

((14))

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

((15))

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

((16))

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

((17))

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

((18))

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

((19))

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

((20))

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

((21))

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

((22))

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

((23))

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

((24))

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

((25))

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

((26))

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

((27))

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

((28))

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

((29))

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

((30))

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

((31))

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

((32))

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((33))

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((34))

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

((35))

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

((36))

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

((37))

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

((38))

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

((39))

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

((40))

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

((41))

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

((42))

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

((43))

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

((44))

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

((45))

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

((46))

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

((47))

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

((48))

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

((49))

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

((50))

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

((51))

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

((52))

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

((53))

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

((54))

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

((55))

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

((56))

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

((57))

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

((58))

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

((59))

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

((60))

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

((61))

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

((62))

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

((63))

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

((64))

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

((65))

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

((66))

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

((67))

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

((68))

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

((69))

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

((70))

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

((71))

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

((72))

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

((73))

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

((74))

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

((75))

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

((76))

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

((77))

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

((78))

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

((79))

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

((80))

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

((81))

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

((82))

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

((83))

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

((84))

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

((85))

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

((86))

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

((87))

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

((88))

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

((89))

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

((90))

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

((91))

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

((92))

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

((93))

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

((94))

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

((95))

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

((96))

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

((97))

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

((98))

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

((99))

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

((100))

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

((101))

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

((102))

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

((103))

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

((104))

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Yan-Jun Gu, Sergei V. Bulanov. Magnetic field annihilation and charged particle acceleration in ultra-relativistic laser plasmas[J]. High Power Laser Science and Engineering, 2021, 9(1): 010000e2

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