3.1. Propagation of relativistic laser pulses in plasma

The wave equation for the normalized vector potential describing the evolution of a laser pulse with laser wavelength ${\it\lambda}_{L}$ and duration ${\it\tau}_{L}$ (full width at half maximum, FWHM) in a plasma channel can be written as^{[Reference Hafizi, Ting, Sprangle and Hubbard48]}

(1)$$\begin{eqnarray}\left({\rm\nabla}^{2}-\frac{\partial ^{2}}{c^{2}\partial t^{2}}\right)\mathbf{a}=k^{2}\left(1-{\it\eta}^{2}\right)\mathbf{a},\end{eqnarray}$$

where $\mathbf{a}\equiv e\mathbf{A}/m_{e}c^{2}$ is the vector potential $\mathbf{A}$ of the laser pulse normalized with respect to the electron rest energy $m_{e}c^{2}$, satisfying the Coulomb gauge ${\rm\nabla}\boldsymbol{\cdot }\mathbf{a}=0$, and $k={\it\omega}/c=2{\it\pi}/{\it\lambda}_{L}$ is the free-space wavenumber along the propagation direction. The (squared) refractive index for linearly polarized electromagnetic waves in the long-pulse limit ($ck_{p}{\it\tau}_{L}\gg 1$) is given by^{[Reference Hafizi, Ting, Hubbard, Sprangle and Penano49]}

(2)$$\begin{eqnarray}{\it\eta}^{2}(r,z)=1-\frac{k_{p}^{2}}{k^{2}{\it\gamma}_{L}}\left[1+\frac{1}{k_{p}^{2}}{\rm\nabla}_{\bot }^{2}{\it\gamma}_{L}+\frac{{\rm\Delta}n}{n_{0}}\frac{r^{2}}{r_{0}^{2}}\right],\end{eqnarray}$$

where $k_{p}={\it\omega}_{p}/c=(4{\it\pi}r_{e}n_{0})^{1/2}$ is the plasma wavenumber evaluated with the unperturbed on-axis density $n_{0}$ and the classical electron radius $r_{e}=e^{2}/mc^{2}$, and ${\it\gamma}_{L}=(1+a^{2}/2)^{1/2}$ is the relativistic factor of the laser intensity for linear polarization. In Equation (2), the first term represents free-space propagation, and the three terms in the square brackets correspond to relativistic self-focusing, ponderomotive channeling and a preformed plasma channel with a parabolic density distribution of the form $n(r)=n_{0}[1+({\rm\Delta}n/n_{0})(r^{2}/r_{L}^{2})]$, where $r_{L}$ is the laser spot radius and ${\rm\Delta}n$ is the channel depth. Analysis of the wave equation with the standard paraxial form provides the matched spot radius $r_{m}$ under the condition of a beam propagating with a constant spot size $r_{L}$, i.e., $k_{p}r_{m}=k_{p}r_{L}=R_{m}$, given by^{[Reference Nakajima, Lu, Zhao, Shen, Li and Xu21]}

(3)$$\begin{eqnarray}\displaystyle k_{p}^{2}r_{m}^{2} & \equiv & \displaystyle R_{m}^{2}(a_{0}^{2},{\rm\Delta}n/n_{0})\nonumber\\ \displaystyle & = & \displaystyle 2\ln ({\it\gamma}_{L0})\{\!{\it\gamma}_{L0}-1{-}2\ln ((1+{\it\gamma}_{L0})/2)\nonumber\\ \displaystyle & & \displaystyle +\,({\rm\Delta}n/n_{0})[\!1-{\it\gamma}_{L0}+\ln ((1+{\it\gamma}_{L0})/2)\nonumber\\ \displaystyle & & \displaystyle +\,(a_{0}^{2}/4)\,_{4}F_{3}(1/2,1,1,1;2,2,2;-a_{0}^{2}/2)\!]\!\}\!^{-1},\quad\end{eqnarray}$$

where ${\it\gamma}_{L0}=(1+a_{0}^{2}/2)^{1/2}$ is the relativistic factor for a Gaussian laser beam with peak amplitude $a_{0}$ and $_{p}F_{q}$ denotes the generalized hypergeometric series of order $q$ and class $q-p+1$.

Under the matched condition that no phase shift of the laser pulse occurs, the group velocity is written as ${\it\beta}_{g}^{2}\approx 1-k_{p}^{2}/({\it\kappa}_{\text{ch}}k^{2})$, where a correction factor for the group velocity is defined as^{[Reference Nakajima, Lu, Zhao, Shen, Li and Xu21]}

(4)$$\begin{eqnarray}\displaystyle & & \displaystyle {\it\kappa}_{\text{ch}}(a_{0}^{2},{\rm\Delta}n/n_{0})=\frac{a_{0}^{2}}{8} [\!{\it\gamma}_{L0}-1-\ln ((1+{\it\gamma}_{L0})/2)\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{a_{0}^{2}}{8}\frac{{\rm\Delta}n}{n_{0}}_{4}F_{3}(1/2,1,1,1;2,2,2;-a_{0}^{2}/2)\!]\,\!^{-1}.\end{eqnarray}$$

Under the matched condition for self-guiding, the spot radius and the group velocity correction factor are given by Equations (3) and (4), respectively, where the channel depth is set to be ${\rm\Delta}n=0$.

3.2. Laser plasma acceleration in the quasi-linear regime

In the linear laser wakefield with the accelerating field $E_{z}=E_{z0}\cos {\rm\Psi}$, the equations of the longitudinal motion of an electron with the normalized velocity ${\it\beta}_{z}=v_{z}/c\approx 1$ and electron energy ${\it\gamma}=E_{e}/m_{e}c^{2}$ are given by^{[Reference Chiu, Cheshkov and Tajima50]}

(5)$$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\displaystyle \frac{d{\it\gamma}}{dz}=k_{p}\frac{E_{z0}}{E_{0}}\cos {\rm\Psi}\quad \text{and}\\ \displaystyle \frac{d{\rm\Psi}}{dz}=k_{p}\left(1-\frac{{\it\beta}_{p}}{{\it\beta}_{z}}\right)\approx \frac{k_{p}}{2{\it\gamma}_{g}^{2}},\end{array} & & \displaystyle\end{eqnarray}$$

where ${\rm\Psi}=k_{p}(z-v_{p}t)+{\rm\Psi}_{0}$ is the phase of the plasma wave, $E_{0}=mc{\it\omega}_{p}/e$ is the non-relativistic wave-breaking field approximately given by $E_{0}\approx 96~[\text{GV m}^{-1}]$ $(n_{e}/10^{18}~[\text{cm}^{-3}])^{1/2}$, ${\it\beta}_{p}=v_{p}/c\approx v_{g}/c={\it\beta}_{g}$ is the phase velocity $v_{p}$ of the plasma wave normalized to $c$, and ${\it\gamma}_{g}=(1-{\it\beta}_{g}^{2})^{-1/2}\gg 1$ is assumed. By integrating Equations (5), the energy and phase of the electron can be calculated as

(6)$$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\displaystyle {\it\gamma}(z)={\it\gamma}_{0}+2{\it\gamma}_{g}^{2}\frac{E_{z0}}{E_{0}}[\sin {\rm\Psi}(z)-\sin {\rm\Psi}_{0}]\quad \text{and}\quad \\ \displaystyle {\rm\Psi}(z)\approx \frac{k_{p}z}{2{\it\gamma}_{g}^{2}}+{\rm\Psi}_{0}.\end{array} & & \displaystyle\end{eqnarray}$$

Setting the initial electron phase ${\rm\Psi}_{0}=0$ at $z=0$, the maximum energy gain is given by

(7)$$\begin{eqnarray}{\rm\Delta}{\it\gamma}_{\max }={\it\gamma}_{\max }-{\it\gamma}_{0}=2{\it\gamma}_{g}^{2}\frac{E_{z0}}{E_{0}},\end{eqnarray}$$

at $k_{p}z={\it\pi}{\it\gamma}_{g}^{2}$ or $z={\it\lambda}_{p}{\it\gamma}_{g}^{2}/2$. As shown from Equations (6), setting ${\rm\Psi}_{0}=-{\it\pi}/2$, the maximum energy gain reaches ${\rm\Delta}{\it\gamma}_{\max }=4{\it\gamma}_{g}^{2}E_{z0}/E_{0}$ at $k_{p}z=2{\it\pi}{\it\gamma}_{g}^{2}$ or $z={\it\lambda}_{p}{\it\gamma}_{g}^{2}$. However, electrons undergo both acceleration and focusing only for the phase $0\leqslant {\rm\Psi}\leqslant {\it\pi}/2$. Hence, we define the dephasing length as $L_{\text{dp}}={\it\lambda}_{p}{\it\gamma}_{g}^{2}/2$. Considering a driving laser pulse of normalized intensity $a_{0}^{2}$ moving in a plasma channel with channel depth ${\rm\Delta}n$ at the group velocity ${\it\beta}_{g}=v_{g}/c$ with the corresponding relativistic factor ${\it\gamma}_{g}^{2}=(1-{\it\beta}_{g}^{2})^{-1}\approx {\it\kappa}_{\text{ch}}({\it\omega}^{2}/{\it\omega}_{p}^{2})={\it\kappa}_{\text{ch}}(n_{c}/n_{e})={\it\kappa}_{\text{ch}}{\it\gamma}_{g0}^{2}$, where ${\it\gamma}_{g0}={\it\omega}/{\it\omega}_{p}$ is the relativistic factor for the group velocity in a uniform plasma and its correction factor ${\it\kappa}_{\text{ch}}$ is given by Equation (4), the maximum energy gain and the dephasing length are written as ${\rm\Delta}{\it\gamma}_{\max }=2{\it\kappa}_{\text{ch}}{\it\gamma}_{g0}^{2}(E_{z0}/E_{0})$ and $L_{\text{dp}}=({\it\lambda}_{p}/2){\it\kappa}_{\text{ch}}{\it\gamma}_{go}^{2}$, respectively. In the limit of $a_{0}^{2}\ll 1$, ${\it\kappa}_{\text{ch}}\approx (1+{\rm\Delta}n/n_{0})^{-1}$. In the quasi-linear regime, taking the beam loading effect into account, the maximum accelerating field driven by a Gaussian laser pulse is given by^{[Reference Nakajima, Deng, Zhang, Shen, Liu, Li, Xu, Ostermayr, Petrovics, Klier, Iqbal, Ruhl and Tajima20]}

(8)$$\begin{eqnarray}\frac{E_{z0}}{E_{0}}\simeq \sqrt{{\it\pi}}{\it\alpha}a_{0}^{2}\left(\frac{k_{p}{\it\sigma}_{L}}{4}\right)\exp \left(-\frac{k_{p}^{2}{\it\sigma}_{L}^{2}}{4}\right)\approx 0.38{\it\alpha}a_{0}^{2},\end{eqnarray}$$

where ${\it\alpha}$ denotes a factor of accelerating field reduction due to the beam loading effect and ${\it\sigma}_{L}$ is the rms pulse length of the Gaussian temporal profile with the FWHM length $c{\it\tau}_{L}\sim 0.375{\it\lambda}_{p}$ for $k_{p}{\it\sigma}_{L}=\sqrt{2}$.

3.3. Laser plasma acceleration in the bubble regime

Previous laser plasma acceleration experiments that successfully demonstrated the production of quasi-monoenergetic electron beams with narrow energy spread have been elucidated in terms of self-injection and an acceleration mechanism in the bubble regime^{[Reference Kostyukov, Pukhov and Kiselev37, Reference Lu, Tzoufras, Joshi, Tsung, Mori, Vieira, Fonseca and Silva38]}. In these experiments, electrons are self-injected into a nonlinear wake, often referred to as a bubble, i.e., a cavity void of plasma electrons consisting of a spherical ion column surrounded by a narrow electron sheath, formed behind the laser pulse instead of a periodic plasma wave in the linear regime. The phenomenological theory of a nonlinear wakefield in the bubble regime describes the accelerating wakefield $E_{z}({\it\xi})/E_{0}\approx (1/2)k_{p}{\it\xi}$ in the bubble frame moving in the plasma with velocity $v_{B}$, i.e., ${\it\xi}=z-v_{B}t$. In the bubble (blowout) regime for $a_{0}\geqslant 2$, since the electron-evacuated cavity shape is determined by balancing the Lorentz force of the ion sphere exerted on the electron sheath with the ponderomotive force of the laser pulse, the bubble radius $R_{B}$ is approximately given as $k_{p}R_{B}\approx 2\sqrt{a_{0}}$ (Ref. [Reference Lu, Tzoufras, Joshi, Tsung, Mori, Vieira, Fonseca and Silva38]). Thus, the maximum accelerating field is obtained as $E_{z0}/E_{0}=(1/2){\it\alpha}k_{p}R_{B}$, where ${\it\alpha}$ represents a factor that takes into account the difference between the simulation and theoretical estimation, and more significantly the accelerating field reduction due to the beam loading effects.

In self-guided laser wakefield acceleration, where a driving laser pulse propagates by means of self-channeling, the equations of longitudinal motion of an electron are approximately written as^{[Reference Nakajima, Lu, Zhao, Shen, Li and Xu21]}

(9)$$\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\displaystyle \frac{d{\it\gamma}}{dz}=k_{p}\frac{E_{z0}}{E_{0}}\left(1-\frac{{\it\xi}}{R_{B}}\right)=\frac{1}{2}{\it\alpha}k_{p}^{2}R_{B}\left(1-\frac{{\it\xi}}{R_{B}}\right)\\ \text{and}\quad \displaystyle \frac{d{\it\xi}}{dz}=1-\frac{{\it\beta}_{B}}{{\it\beta}_{z}}\approx 1-{\it\beta}_{B}\approx \frac{3}{2{\it\gamma}_{g}^{2}},\end{array} & & \displaystyle\end{eqnarray}$$

where ${\it\xi}=z-v_{B}t\,(0\leqslant {\it\xi}\leqslant R_{B})$ is the longitudinal coordinate of the bubble frame moving at the velocity $v_{B}=c{\it\beta}_{B}\approx v_{g}-v_{\text{etch}}$ and taking into account the diffraction at the laser front that etches back at the velocity $v_{\text{etch}}\simeq c({\it\omega}_{p}/{\it\omega})^{2}$ (Ref. [Reference Lu, Tzoufras, Joshi, Tsung, Mori, Vieira, Fonseca and Silva38]). Integrating Equations (5), the energy and phase of the electron can be calculated as

(10)$$\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\displaystyle {\it\gamma}(z)={\it\gamma}_{0}+\frac{1}{3}{\it\alpha}{\it\gamma}_{g}^{2}k_{p}^{2}R_{B}{\it\xi}(z)\left(1-\frac{1}{2}\frac{{\it\xi}(z)}{R_{B}}\right)\\ \text{and}\quad \displaystyle {\it\xi}(z)=\frac{3}{2}\frac{z}{{\it\gamma}_{g}^{2}},\end{array} & & \displaystyle\end{eqnarray}$$

where ${\it\gamma}_{0}={\it\gamma}(0)$ is the injection energy. Hence, the maximum energy gain is obtained at ${\it\xi}=R_{B}$ as

(11)$$\begin{eqnarray}\displaystyle {\rm\Delta}{\it\gamma}_{\max } & = & \displaystyle {\it\gamma}_{\max }-{\it\gamma}_{0}\approx \frac{1}{6}{\it\alpha}{\it\gamma}_{g}^{2}k_{p}^{2}R_{B}^{2}\approx \frac{2}{3}{\it\alpha}a_{0}{\it\gamma}_{g}^{2}\nonumber\\ \displaystyle & = & \displaystyle \frac{2}{3}{\it\alpha}{\it\kappa}_{\text{self}}a_{0}\frac{n_{c}}{n_{e}}.\end{eqnarray}$$

The dephasing length $L_{\text{dp}}$ for the self-guided bubble regime is given by

(12)$$\begin{eqnarray}k_{p}L_{\text{dp}}\approx \frac{2}{3}k_{p}R_{B}{\it\gamma}_{g}^{2}=\frac{4}{3}\sqrt{a_{0}}{\it\kappa}_{\text{self}}\frac{n_{c}}{n_{e}},\end{eqnarray}$$

i.e.,

(13)$$\begin{eqnarray}\displaystyle L_{\text{dp}} & \simeq & \displaystyle \frac{4}{3{\it\pi}}{\it\lambda}_{L}\sqrt{a_{0}}{\it\kappa}_{\text{self}}\left(\frac{n_{c}}{n_{e}}\right)^{3/2}\nonumber\\ \displaystyle & {\approx} & \displaystyle 24.7\,[\text{mm}]~\sqrt{a_{0}}{\it\kappa}_{\text{self}}\left(\frac{0.8\,{\rm\mu}\text{m}}{{\it\lambda}_{L}}\right)^{2}\left(\frac{10^{18}\,\text{cm}^{-3}}{n_{e}}\right)^{3/2},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

while the pump depletion length due to pulse-front erosion is given by

(14)$$\begin{eqnarray}L_{\text{pd}}\approx c{\it\tau}_{L}\frac{n_{c}}{n_{e}},\end{eqnarray}$$

where ${\it\kappa}_{\text{self}}\equiv {\it\kappa}_{\text{ch}}(a_{0}^{2},0)$ is the correction factor of the group velocity for the self-guided pulse in Equation (4).

Table 1. Parameters of experiments on GeV-class laser wakefield acceleration.

For a driving laser pulse propagating in a plasma channel, the equations of electron motion are given by setting $v_{B}=c{\it\beta}_{B}\approx v_{g}$ in Equations (5), i.e., $d{\it\xi}/dz\approx 1-{\it\beta}_{B}\approx 1/2{\it\gamma}_{g}^{2}$. Hence, the maximum energy gain is

(15)$$\begin{eqnarray}\displaystyle {\rm\Delta}{\it\gamma}_{\max } & = & \displaystyle {\it\gamma}_{\max }-{\it\gamma}_{0}\approx \frac{1}{2}{\it\alpha}{\it\gamma}_{g}^{2}k_{p}^{2}R_{B}^{2}\approx 2{\it\alpha}a_{0}{\it\gamma}_{g}^{2}\nonumber\\ \displaystyle & = & \displaystyle 2{\it\alpha}{\it\kappa}_{\text{ch}}a_{0}\frac{n_{c}}{n_{e}},\end{eqnarray}$$

and the dephasing length $L_{\text{dp}}$ is

(16)$$\begin{eqnarray}k_{p}L_{\text{dp}}\approx 2k_{p}R_{B}{\it\gamma}_{g}^{2}=4\sqrt{a_{0}}{\it\kappa}_{\text{ch}}\frac{n_{c}}{n_{e}},\end{eqnarray}$$

where ${\it\kappa}_{\text{ch}}$ is given by Equation (4). For channel-guided laser plasma acceleration, the pump depletion length, at which the total field energy becomes half of the initial laser energy, is given by

(17)$$\begin{eqnarray}L_{\text{pd}}\approx \frac{\sqrt{{\it\pi}}}{2{\it\alpha}^{2}}a_{0}{\it\sigma}_{L}\frac{n_{c}}{n_{e}}=\frac{1}{4}\sqrt{\frac{{\it\pi}}{\ln 2}}\frac{a_{0}c{\it\tau}_{L}}{{\it\alpha}^{2}}\frac{n_{c}}{n_{e}},\end{eqnarray}$$

where ${\it\sigma}_{L}=c{\it\tau}_{L}/(2\sqrt{\ln 2})\approx 0.6c{\it\tau}_{L}$ is the rms pulse length.

The matched power $P_{m}$ corresponding to the matched spot size $r_{L}$ is calculated as

(18)$$\begin{eqnarray}P_{L}=\frac{k_{p}^{2}r_{L}^{2}a_{0}^{2}}{32}P_{c},\end{eqnarray}$$

where $P_{c}=17n_{c}/n_{0}\;[\text{GW}]$ is the critical power for the relativistic self-focusing at the plasma density $n_{0}$ and the required pulse energy is $U_{L}=P_{L}{\it\tau}_{L}$.

3.4. Beam loading effects

In laser wakefield acceleration, an accelerated electron beam induces its own wakefield and cancels the laser-driven wakefield. Assuming the beam loading efficiency ${\it\eta}_{b}\equiv 1-E_{z}^{2}/E_{M}^{2}$ defined by the fraction of the plasma wave energy absorbed by particles of the bunch with the rms radius ${\it\sigma}_{b}$, the beam-loaded field is given by $E_{z}=\sqrt{1-{\it\eta}_{b}}E_{M}={\it\alpha}E_{M}$, where $E_{M}$ is the accelerating field without beam loading, given by $E_{M}\approx a_{0}^{1/2}E_{0}$ for the bubble regime $a_{0}\geqslant 2$. Thus, the loaded charge is calculated as^{[Reference Nakajima, Deng, Zhang, Shen, Liu, Li, Xu, Ostermayr, Petrovics, Klier, Iqbal, Ruhl and Tajima20]}

(19)$$\begin{eqnarray}\displaystyle Q_{b} & \simeq & \displaystyle \frac{e}{4k_{L}r_{e}}\frac{{\it\eta}_{b}k_{p}^{2}{\it\sigma}_{b}^{2}}{1-{\it\eta}_{b}}\frac{E_{z}}{E_{0}}\left(\frac{n_{c}}{n_{e}}\right)^{1/2}\nonumber\\ \displaystyle & {\approx} & \displaystyle 76\left[\text{pC}\right]\!\frac{(1-{\it\alpha}^{2})a_{0}^{1/2}k_{p}^{2}{\it\sigma}_{b}^{2}}{{\it\alpha}}\!\left(\frac{n_{e}}{10^{18}\,\text{cm}^{-3}}\!\right)\!^{-1/2}.\quad\end{eqnarray}$$

Here, the field reduction factor ${\it\alpha}$ for accelerating charge $Q_{b}$ in the operating plasma density $n_{e}$ is obtained by solving the equation

(20)$$\begin{eqnarray}{\it\alpha}^{2}+B{\it\alpha}-1=0,\end{eqnarray}$$

where the coefficient $B$ is defined as

(21)$$\begin{eqnarray}B\equiv \frac{1}{a_{0}^{1/2}k_{p}^{2}{\it\sigma}_{b}^{2}}\left(\frac{Q_{b}}{\text{76 pC}}\right)\left(\frac{n_{e}}{10^{18}\,\text{cm}^{-3}}\right)^{1/2}.\end{eqnarray}$$

Thus, the field reduction factor is obtained by

(22)$$\begin{eqnarray}{\it\alpha}=\frac{B}{2}\left[\left(1+\frac{4}{B^{2}}\right)^{1/2}-1\right].\end{eqnarray}$$

For $B\ll 1$, the field reduction factor becomes ${\it\alpha}\approx 1$ and contrarily, for $B\gg 1$, ${\it\alpha}\approx 0$.

3.5. Comparison with experimental results on GeV-class electron beams

Table 1 summarizes the parameters for experiments on laser wakefield acceleration driven by a self-guided laser pulse with channel depth ${\rm\Delta}n/n_{e}=0$ (Refs. [Reference Clayton, Ralph, Albert, Fonseca, Glenzer, Joshi, Lu, Marsh, Martins, Mori, Pak, Tsung, Pollock, Ross, Silva and Froula3, Reference Wang, Zgadzaj, Fazel, Li, Yi, Zhang, Henderson, Chang, Korzekwa, Tsai, Pai, Quevedo, Dyer, Gaul, Martinez, Bernstein, Borger, Spinks, Donovan, Khudik, Shvets, Ditmire and Downer5, Reference Kim, Pae, Cha, Kim, Yu, Sung, Lee, Jeong and Lee6, Reference Liu, Xia, Wang, Lu, Wang, Deng, Li, Zhang, Liang, Leng, Lu, Wang, Wang, Nakajima, Li and Xu11, Reference Pollock, Clayton, Ralph, Albert, Davidson, Divol, Filip, Glenzer, Herpoldt, Lu, Marsh, Meinecke, Mori, Pak, Rensink, Ross, Shaw, Tynan, Joshi and Froula12, Reference Kneip, Nagel, Martins, Mangles, Bellei, Chekhlov, Clarke, Delerue, Divall, Doucas, Ertel, Fiuza, Fonseca, Foster, Hawkes, Hooker, Krushelnick, Mori, Palmer, Ta Phuoc, Rajeev, Schreiber, Streeter, Urner, Vieira, Silva and Najmudin51, Reference Froula, Clayton, Döppner, Marsh, Barty, Divol, Fonseca, Glenzer, Joshi, Lu, Martins, Michel, Mori, Palastro, Pollock, Pak, Ralph, Ross, Siders, Silva and Wang52]) and a channel-guided laser pulse with ${\rm\Delta}n/n_{e}\neq 0$ (Refs. [Reference Leemans, Nagler, Gonsalves, Toth, Nakamura, Geddes, Esarey, Schroeder and Hooker2, Reference Lu, Liu, Wang, Wang, Liu, Deng, Xu, Xia, Li, Zhang, Lu, Wang, Wang, Liang, Leng, Shen, Nakajima, Li and Xu4, Reference Kameshima, Hong, Sugiyama, Wen, Wu, Tang, Zhu, Gu, Zhang, Peng, Kurokawa, Chen, Tajima, Kumita and Nakajima7, Reference Karsch, Osterhoff, Popp, Rowlands-Rees, Major, Fuchs, Marx, Hörlein, Schmid, Veisz, Becker, Schramm, Hidding, Pretzler, Habs, Grüner, Krausz and Hooker8, Reference Ibbotson, Bourgeois, Rowlands-Rees, Caballero, Bajlekov, Walker, Kneip, Mangles, Nagel, Palmer, Delerue, Doucas, Urner, Chekhlov, Clarke, Divall, Ertel, Foster, Hawkes, Hooker, Parry, Rajeev, Streeter and Hooker53]). Since the maximum energy gain scales as ${\rm\Delta}{\it\gamma}_{\max }\propto n_{c}\propto {\it\lambda}_{L}^{-2}$ for a given $a_{0}$, most previous experiments have employed chirped pulse amplification lasers with wavelength ${\it\lambda}_{L}=800\,\text{nm}$ and pulse duration ${\it\tau}_{L}\leqslant 80\,\text{fs}$, except for the case of Ref. [Reference Wang, Zgadzaj, Fazel, Li, Yi, Zhang, Henderson, Chang, Korzekwa, Tsai, Pai, Quevedo, Dyer, Gaul, Martinez, Bernstein, Borger, Spinks, Donovan, Khudik, Shvets, Ditmire and Downer5], where a PW-class laser with wavelength ${\it\lambda}_{L}=1057\,\text{nm}$ and ${\it\tau}_{L}\sim 150\,\text{fs}$ was employed. The validity of the energy scaling formulas (Equations (7), (11) and (15)) based on the present analytical methods may be verified by comparison with these experimental results. Figure 1 shows the comparison of measured electron beam energies in laser wakefield acceleration with the energy scaling on the operating plasma density for the self-guided case (Figure 1(a)) in the bubble regime and the channel-guided case (Figure 1(b)) in both the quasi-linear regime and the bubble regime. In the energy scaling formulas, we assume that the field reduction due to beam loading and the group velocity correction due to relativistic effects cancel each other out, i.e., ${\it\alpha}{\it\kappa}_{\text{self}}\sim 1$ or ${\it\alpha}{\it\kappa}_{\text{ch}}\sim 1$.

For self-guided laser wakefield acceleration, the multi-GeV acceleration results reported in Refs. [Reference Wang, Zgadzaj, Fazel, Li, Yi, Zhang, Henderson, Chang, Korzekwa, Tsai, Pai, Quevedo, Dyer, Gaul, Martinez, Bernstein, Borger, Spinks, Donovan, Khudik, Shvets, Ditmire and Downer5, Reference Kim, Pae, Cha, Kim, Yu, Sung, Lee, Jeong and Lee6] provide us with informative examples for testing the energy scaling formula (Equation (11)). In Ref. [Reference Wang, Zgadzaj, Fazel, Li, Yi, Zhang, Henderson, Chang, Korzekwa, Tsai, Pai, Quevedo, Dyer, Gaul, Martinez, Bernstein, Borger, Spinks, Donovan, Khudik, Shvets, Ditmire and Downer5], a 625-TW, 160-fs laser pulse with wavelength ${\it\lambda}_{L}=1057\,\text{nm}$ is focused onto a $1/e^{2}$ spot radius of $r_{L}=50\,{\rm\mu}\text{m}$, producing $a_{0}=3.6$ at the entrance of a 7-cm long gas cell with $n_{e}=4.8\times 10^{17}\,\text{cm}^{-3}$. The accelerated electron beam has a quasi-monoenergetic peak at 2.0 GeV with a relative energy spread of 10% (FWHM), containing a total charge of 540 pC in a bunch. In this case, from Equations (3) and (4), the dimensionless matched spot radius $R_{m}=k_{p}r_{m}$ and the correction factor of the group velocity ${\it\kappa}_{\text{self}}={\it\kappa}_{\text{ch}}(a_{0}^{2},0)$ are $R_{m}=2.0$ and ${\it\kappa}_{\text{self}}=1.46$, respectively, at $a_{0}=3.6$. Thus, since the matched spot radius is $r_{m}=15\,{\rm\mu}\text{m}$, the laser pulse undergoes self-focusing after propagating through the gas cell. Using Equations (20) and (21) and assuming the electron beam size $k_{p}{\it\sigma}_{b}=1$, the field reduction factor ${\it\alpha}$ is calculated as ${\it\alpha}\approx 0.32$ ($B\approx 2.84$). From Equation (13), the dephasing length is $L_{\text{dp}}\approx 118\,\text{mm}$, while from Equation (14), the pump depletion length due to pulse-front erosion is $L_{\text{pd}}\approx 100\,\text{mm}$. From Equation (10), the electron beam energy at the accelerator length $z=L_{\text{acc}}$ is estimated as

(23)$$\begin{eqnarray}\displaystyle E_{b}(L_{\text{acc}}) & = & \displaystyle E_{b0}+m_{e}c^{2}{\it\alpha}\sqrt{a_{0}}k_{p}L_{\text{acc}}\left(1-\frac{L_{\text{acc}}}{2L_{\text{dp}}}\right)\nonumber\\ \displaystyle & {\approx} & \displaystyle E_{b0}+96\,[\text{MeV}]\,{\it\alpha}\sqrt{a_{0}}\left(1-\frac{L_{\text{acc}}}{2L_{\text{dp}}}\right)\nonumber\\ \displaystyle & & \displaystyle \times \,\left(\frac{L_{\text{acc}}}{1\,\text{mm}}\right)\left(\frac{n_{e}}{10^{18}\,\text{cm}^{-3}}\right)^{1/2},\end{eqnarray}$$

where $E_{b0}$ is the injection beam energy. For $L_{\text{acc}}=70\,\text{mm}$, the beam energy is evaluated to be $E_{b}=2.0\;\text{GeV}$. This estimate is in good agreement with the measured beam energy of $2.0\pm 0.1~\text{GeV}$^{[Reference Wang, Zgadzaj, Fazel, Li, Yi, Zhang, Henderson, Chang, Korzekwa, Tsai, Pai, Quevedo, Dyer, Gaul, Martinez, Bernstein, Borger, Spinks, Donovan, Khudik, Shvets, Ditmire and Downer5]}, taking into account the field reduction factor ${\it\alpha}$ due to beam loading.

Figure 1. A comparison of measured electron beam energies in laser wakefield acceleration with the energy scaling as a function of the operating plasma density for (a) the self-guided case in the bubble regime at laser wavelengths of 800 nm (solid line) and 1057 nm (dashed line) and (b) the channel-guided case in both the quasi-linear regime (dashed line) and the bubble regime (solid line). The experimental data are plotted with filled squares for ${\it\lambda}_{L}=800\,\text{nm}$ and the open square for ${\it\lambda}_{L}=1057\,\text{nm}$ in (a), and with filled circles for ${\it\lambda}_{L}=800\,\text{nm}$ in (b).

Figure 2. Electron beam energy spectra obtained from the experiment^{[Reference Kim, Pae, Cha, Kim, Yu, Sung, Lee, Jeong and Lee6]}, where a 212-TW, 60-fs laser pulse is focused on a $1/e^{2}$ spot radius of $r_{L}=21\,{\rm\mu}\text{m}$ producing $a_{0}=3.7$ at the entrance of a gas jet for three cases consisting of (a) a 4-mm long single stage with $n_{e}=2.1\times 10^{18}\,\text{cm}^{-3}$, (b) a 10-mm long single stage with $n_{e}=1.3\times 10^{18}\,\text{cm}^{-3}$ and (c) two stages comprising a 4-mm long injector with $n_{e}=2\times 10^{18}\,\text{cm}^{-3}$ and a 10-mm long accelerator with $n_{e}=0.8\times 10^{18}\,\text{cm}^{-3}$.

In Ref. [Reference Kim, Pae, Cha, Kim, Yu, Sung, Lee, Jeong and Lee6], a 212-TW, 60-fs laser pulse is focused on a $1/e^{2}$ spot radius of $r_{L}=21\,{\rm\mu}\text{m}$, producing $a_{0}=3.7$ at the entrance of a gas jet for three cases consisting of a 4-mm long single stage with $n_{e}=2.1\times 10^{18}\,\text{cm}^{-3}$, a 10-mm long single stage with $n_{e}=1.3\times 10^{18}\,\text{cm}^{-3}$, and two stages comprising a 4-mm long injector with $n_{e}=2\times 10^{18}\,\text{cm}^{-3}$ and a 10-mm long accelerator with $n_{e}=0.8\times 10^{18}\,\text{cm}^{-3}$. As shown in Figure 2, the accelerated electron beams for the three cases have peak energies of 0.35, 0.87 and 3 GeV, respectively, and total loaded charges of 88, 110 and 80 pC, respectively. From Equations (3) and (4), the dimensionless matched spot radius $R_{m}=k_{p}r_{m}$ and the correction factor of the group velocity ${\it\kappa}_{\text{self}}={\it\kappa}_{\text{ch}}(a_{0}^{2},0)$ are $R_{m}=2.0$ and ${\it\kappa}_{\text{self}}=1.48$, respectively, at $a_{0}=3.7$. Since the matched spot radius is $r_{m}=7.3\,{\rm\mu}\text{m}$ for the 4-mm single-stage case, $r_{m}=9.3\,{\rm\mu}\text{m}$ for the 10-mm single-stage case and $r_{m}=7.5\,{\rm\mu}\text{m}$ in the injector jet and $r_{m}=12\,{\rm\mu}\text{m}$ in the accelerator jet for the two-stage case, respectively, it is inferred that a laser pulse with a focused spot radius of $r_{L}=21\,{\rm\mu}\text{m}$ is initially self-focused down to the matched spot radius. Using Equations (20) and (21) and assuming the electron beam size $k_{p}{\it\sigma}_{b}=1$, the field reduction factor ${\it\alpha}$ is calculated as ${\it\alpha}\approx 0.66\,(B=0.86)$ for the 4-mm single-stage case, ${\it\alpha}=0.66\,(B=0.86)$ for the 10-mm single-stage case and ${\it\alpha}=0.66\,(B=0.86)$ in the injector/${\it\alpha}\approx 0.78\,(B=0.49)$ in the accelerator for the two-stage case. From Equation (13), the dephasing length is $L_{\text{dp}}\approx 23.1\,\text{mm}$ for the 4-mm single-stage case, $L_{\text{dp}}\approx 47.4\,\text{mm}$ for the 10-mm single-stage case and $L_{\text{dp}}\approx 24.9\,\text{mm}$ in the injector/$L_{\text{dp}}\approx 98.3\,\text{mm}$ in the accelerator for the two-stage case. The pump depletion length due to pulse-front erosion is $L_{\text{pd}}\approx 14.9\,\text{mm}$ for the 4-mm single-stage case, $L_{\text{pd}}\approx 24.1\,\text{mm}$ for the 10-mm single-stage case and $L_{\text{pd}}\approx 15.7\,\text{mm}$ in the injector/$L_{\text{pd}}\approx 39.2\,\text{mm}$ in the accelerator for the two-stage case. From Equation (23), the beam energy is estimated to be $E_{b}\approx 353~\text{MeV}$ with an effective acceleration length of $L_{\text{acc}}=2\,\text{mm}$ for the 4-mm single-stage case and $E_{b}\approx 1.02\;\text{GeV}$ with $L_{\text{acc}}=8\,\text{mm}$ for the 10-mm single-stage case. For the two-stage case, the output energy of the injector is $E_{b}\approx 409\;\text{MeV}$ with $L_{\text{acc}}\approx 2.5\,\text{mm}$ and the output energy of the accelerator stage reaches $E_{b}\approx 2.46~\text{GeV}$ with $L_{\text{acc}}\approx 10\,\text{mm}$, assuming that the injection energy is $E_{b0}\approx 409\;\text{MeV}$ and the focused spot size at the entrance of the accelerator stage is decreased to $r_{L}=7.5\,{\rm\mu}\text{m}$ due to self-focusing in the injector stage, increasing the normalized vector potential up to $a_{0}\approx 10.4$. In this experiment, most of the charge produced in the injector is injected into the accelerator stage, while the large energy spread is attributed to the fact that the accelerator length is shorter than the dephasing length at which the energy compression takes place in the phase space as well as the maximum energy.

4.1. Self-guided laser plasma accelerator in the bubble regime

For a given energy gain $W$, the operating plasma density is determined from Equation (11) as

(24)$$\begin{eqnarray}\displaystyle n_{e} & = & \displaystyle \frac{2}{3}{\it\alpha}{\it\kappa}_{\text{self}}a_{0}\frac{n_{c}}{{\rm\Delta}{\it\gamma}_{\max }}\nonumber\\ \displaystyle & {\approx} & \displaystyle 5.94\times 10^{16}\,[\text{cm}^{-3}]\,{\it\kappa}_{\text{self}}a_{0}\left(\frac{0.8\,{\rm\mu}\text{m}}{{\it\lambda}_{L}}\right)^{2}\left(\frac{10~\text{GeV}}{W/{\it\alpha}}\right).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The accelerator length, equal to the dephasing length, becomes

(25)$$\begin{eqnarray}\displaystyle L_{\text{acc}} & = & \displaystyle L_{\text{dp}}\approx \sqrt{\frac{3}{2}}\frac{({\rm\Delta}{\it\gamma}_{\max }/{\it\alpha})^{3/2}}{{\it\pi}a_{0}{\it\kappa}_{\text{self}}^{1/2}}{\it\lambda}_{L}\nonumber\\ \displaystyle & {\approx} & \displaystyle \frac{0.9\,[\text{m}]}{a_{0}{\it\kappa}_{\text{self}}^{1/2}}\left(\frac{{\it\lambda}_{L}}{0.8\,{\rm\mu}\text{m}}\right)\left(\frac{W/{\it\alpha}}{10~\text{GeV}}\right)^{3/2},\end{eqnarray}$$

while the pump depletion length due to pulse-front erosion is given by $L_{\text{pd}}\approx c{\it\tau}_{L}n_{c}/n_{e}$. The dephasing length should be less than the pump depletion length, i.e., $L_{\text{pd}}\geqslant L_{\text{dp}}$. Therefore, the pulse length is set to be $c{\it\tau}_{L}\geqslant (2/3{\it\pi})\sqrt{a_{0}}{\it\kappa}_{\text{self}}{\it\lambda}_{p}$.

(26)$$\begin{eqnarray}\displaystyle {\it\tau}_{L} & {\geqslant} & \displaystyle \sqrt{\frac{2}{3}}\frac{{\it\lambda}_{L}}{{\it\pi}c}{\it\kappa}_{\text{self}}^{1/2}\left(\frac{{\rm\Delta}{\it\gamma}_{\max }}{{\it\alpha}}\right)^{1/2}\nonumber\\ \displaystyle & {\geqslant} & \displaystyle 97\,[\text{fs}]\,{\it\kappa}_{\text{self}}^{1/2}\left(\frac{{\it\lambda}_{L}}{0.8\,{\rm\mu}\text{m}}\right)\left(\frac{W/{\it\alpha}}{10~\text{GeV}}\right)^{1/2}.\end{eqnarray}$$

The matched spot radius becomes

(27)$$\begin{eqnarray}\displaystyle r_{L} & = & \displaystyle \frac{1}{2{\it\pi}}\sqrt{\frac{3}{2}}{\it\lambda}_{L}\frac{R_{m}}{\sqrt{a_{0}{\it\kappa}_{\text{self}}}}\left(\frac{{\rm\Delta}{\it\gamma}_{\max }}{{\it\alpha}}\right)^{1/2}\nonumber\\ \displaystyle & {\approx} & \displaystyle 22\;[{\rm\mu}\text{m}]\frac{R_{m}}{\sqrt{a_{0}{\it\kappa}_{\text{self}}}}\left(\frac{{\it\lambda}_{L}}{0.8\,{\rm\mu}\text{m}}\right)\left(\frac{W/{\it\alpha}}{10~\text{GeV}}\right)^{1/2},\quad\end{eqnarray}$$

where $R_{m}(a_{0}^{2},0)=k_{p}r_{L}$ is the dimensionless matched spot radius given by Equation (3) for self-guiding, ${\rm\Delta}n/n_{0}=0$. The matched power is calculated as

(28)$$\begin{eqnarray}\displaystyle P_{L} & = & \displaystyle \frac{k_{p}^{2}r_{L}^{2}a_{0}^{2}}{32}P_{c}=\frac{51}{64}\,[\text{GW}]\,\frac{a_{0}R_{m}^{2}}{{\it\kappa}_{\text{self}}}\frac{{\rm\Delta}{\it\gamma}_{\max }}{{\it\alpha}}\nonumber\\ \displaystyle & {\approx} & \displaystyle 15.6\,[\text{TW}]\,\frac{a_{0}R_{m}^{2}}{{\it\kappa}_{\text{self}}}\left(\frac{W/{\it\alpha}}{10~\text{GeV}}\right).\end{eqnarray}$$

The required pulse energy is

(29)$$\begin{eqnarray}U_{L}=P_{L}{\it\tau}_{L}\geqslant 1.51\,[\text{J}]\,\frac{a_{0}R_{m}^{2}}{{\it\kappa}_{\text{self}}^{1/2}}\left(\frac{{\it\lambda}_{L}}{0.8\,{\rm\mu}\text{m}}\right)\left(\frac{W/{\it\alpha}}{10~\text{GeV}}\right)^{3/2}.\end{eqnarray}$$

The field reduction factor ${\it\alpha}$ for loading charge $Q_{b}$ up to a given energy $W$ is obtained by solving the equation

(30)$$\begin{eqnarray}{\it\alpha}^{2}+C{\it\alpha}^{3/2}-1=0,\end{eqnarray}$$

where the coefficient $C$ is given as

(31)$$\begin{eqnarray}C=\frac{Q_{b}}{312\,[\text{pC}]}\frac{{\it\kappa}_{\text{self}}^{1/2}}{k_{p}^{2}{\it\sigma}_{b}^{2}}\left(\frac{0.8\,{\rm\mu}\text{m}}{{\it\lambda}_{L}}\right)\left(\frac{10~\text{GeV}}{W}\right)^{1/2}.\end{eqnarray}$$

4.2. Channel-guided laser plasma accelerator in the bubble regime

The operating plasma density is determined by

(32)$$\begin{eqnarray}\displaystyle n_{e} & = & \displaystyle 2{\it\alpha}{\it\kappa}_{\text{ch}}a_{0}\frac{n_{c}}{{\rm\Delta}{\it\gamma}_{\max }}\nonumber\\ \displaystyle & {\approx} & \displaystyle 1.78\times 10^{17}\,[\text{cm}^{-3}]\,{\it\kappa}_{\text{ch}}a_{0}\left(\frac{0.8\,{\rm\mu}\text{m}}{{\it\lambda}_{L}}\right)^{2}\left(\frac{10~\text{GeV}}{W/{\it\alpha}}\right),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and the accelerator length becomes

(33)$$\begin{eqnarray}\displaystyle L_{\text{stage}} & = & \displaystyle L_{\text{dp}}\approx \frac{{\it\lambda}_{L}}{\sqrt{2}{\it\pi}{\it\kappa}_{\text{ch}}^{1/2}a_{0}}\left(\frac{{\rm\Delta}{\it\gamma}_{\max }}{{\it\alpha}}\right)^{3/2}\nonumber\\ \displaystyle & {\approx} & \displaystyle \frac{0.5\,[\text{m}]}{{\it\kappa}_{\text{ch}}^{1/2}a_{0}}\left(\frac{{\it\lambda}_{L}}{0.8\,{\rm\mu}\text{m}}\right)\left(\frac{W/{\it\alpha}}{10~\text{GeV}}\right)^{3/2}.\end{eqnarray}$$

The pump depletion length, at which the total field energy becomes half of the initial laser energy, is given by

(34)$$\begin{eqnarray}\displaystyle L_{\text{pd}} & {\approx} & \displaystyle \frac{\sqrt{{\it\pi}}}{2{\it\alpha}^{2}}a_{0}{\it\sigma}_{L}\frac{n_{c}}{n_{e}}=\frac{1}{4}\sqrt{\frac{{\it\pi}}{\ln 2}}\frac{a_{0}c{\it\tau}_{L}}{{\it\alpha}^{2}}\frac{n_{c}}{n_{e}}\nonumber\\ \displaystyle & {\approx} & \displaystyle \frac{0.156\,[\text{m}]}{{\it\alpha}^{2}{\it\kappa}_{\text{ch}}}\left(\frac{W/{\it\alpha}}{10~\text{GeV}}\right)\left(\frac{{\it\tau}_{L}}{100\,\text{fs}}\right),\end{eqnarray}$$

where ${\it\sigma}_{L}=c{\it\tau}_{L}/(2\sqrt{\ln 2})\approx 0.6c{\it\tau}_{L}$ is the rms pulse length. The requirement for the accelerator length $L_{\text{acc}}=L_{\text{dp}}\leqslant L_{\text{pd}}$ bounds the minimum pulse duration as

(35)$$\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\displaystyle c{\it\tau}_{L}\geqslant \frac{8}{{\it\pi}}\left(\frac{\ln 2}{{\it\pi}}\right)^{1/2}\frac{{\it\alpha}^{2}{\it\kappa}_{\text{ch}}}{\sqrt{a_{0}}}{\it\lambda}_{p}\quad \text{or}\quad \\ \displaystyle {\it\tau}_{L}\geqslant 320\,[\text{fs}]\frac{{\it\alpha}^{2}{\it\kappa}_{\text{ch}}^{1/2}}{a_{0}}\left(\frac{{\it\lambda}_{L}}{0.8\,{\rm\mu}\text{m}}\right)\left(\frac{W/{\it\alpha}}{10~\text{GeV}}\right)^{1/2}.\end{array} & & \displaystyle\end{eqnarray}$$

The matched spot radius becomes

(36)$$\begin{eqnarray}\displaystyle r_{L} & = & \displaystyle \frac{{\it\lambda}_{L}}{2{\it\pi}}\left(\frac{n_{c}}{n_{e}}\right)^{1/2}R_{m}\nonumber\\ \displaystyle & {\approx} & \displaystyle 12.6\,[{\rm\mu}\text{m}]\frac{R_{m}}{\sqrt{{\it\kappa}_{\text{ch}}a_{0}}}\left(\frac{{\it\lambda}_{L}}{0.8\,{\rm\mu}\text{m}}\right)\!\left(\frac{W/{\it\alpha}}{10~\text{GeV}}\right)^{1/2},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $R_{m}(a_{0}^{2},{\rm\Delta}n/n_{0})=k_{p}r_{L}$ is the dimensionless matched spot radius given by Equation (3). The matched power is calculated as

(37)$$\begin{eqnarray}P_{L}=\frac{k_{p}^{2}r_{L}^{2}a_{0}^{2}}{32}P_{c}\approx 5.2\,[\text{TW}]\frac{a_{0}R_{m}^{2}}{{\it\kappa}_{\text{ch}}}\left(\frac{W/{\it\alpha}}{10~\text{GeV}}\right),\end{eqnarray}$$

and the required pulse energy is

(38)$$\begin{eqnarray}U_{L}=P_{L}{\it\tau}_{L}\geqslant 1.66\,[\text{J}]\frac{{\it\alpha}^{2}R_{m}^{2}}{{\it\kappa}_{\text{ch}}^{1/2}}\left(\frac{{\it\lambda}_{L}}{0.8\,{\rm\mu}\text{m}}\right)\left(\frac{W/{\it\alpha}}{10~\text{GeV}}\right)^{3/2}.\end{eqnarray}$$

The field reduction factor ${\it\alpha}$ is determined from the coefficient $C$, given by

(39)$$\begin{eqnarray}C\equiv \frac{Q_{b}}{180\,[\text{pC}]}\frac{{\it\kappa}_{\text{self}}^{1/2}}{k_{p}^{2}{\it\sigma}_{b}^{2}}\left(\frac{0.8\,{\rm\mu}\text{m}}{{\it\lambda}_{L}}\right)\left(\frac{10\,\text{GeV}}{W}\right)^{1/2}.\end{eqnarray}$$

4.3. Channel-guided laser plasma accelerator in the quasi-linear regime

For a given $a_{0}\leqslant 1.7$, the pulse duration is given by

(40)$$\begin{eqnarray}{\it\tau}_{L}\approx \frac{160\,[\text{fs}]}{a_{0}\sqrt{{\it\kappa}_{\text{ch}}}}\left(\frac{{\it\lambda}_{L}}{0.8\,{\rm\mu}\text{m}}\right)\left(\frac{W/{\it\alpha}}{10\,\text{GeV}}\right)^{1/2}.\end{eqnarray}$$

For this pulse duration, the operating plasma density becomes

(41)$$\begin{eqnarray}n_{e}\approx 6.8\times 10^{16}\,[\text{cm}^{-3}]\,{\it\kappa}_{\text{ch}}a_{0}^{2}\left(\frac{0.8\,{\rm\mu}\text{m}}{{\it\lambda}_{L}}\right)^{2}\left(\frac{10\,\text{GeV}}{W/{\it\alpha}}\right).\end{eqnarray}$$

Thus, the required accelerator length $L_{\text{acc}}$ can be set to be

(42)$$\begin{eqnarray}L_{\text{acc}}=L_{\text{dp}}\approx \frac{1.65\,[\text{m}]}{a_{0}^{3}{\it\kappa}_{\text{ch}}^{1/2}}\left(\frac{{\it\lambda}_{L}}{0.8\,{\rm\mu}\text{m}}\right)\left(\frac{W/{\it\alpha}}{10\,\text{GeV}}\right)^{3/2},\end{eqnarray}$$

where $n_{c}/n_{e}$ is given by Equation (9). The matched spot radius is calculated from

(43)$$\begin{eqnarray}r_{L}\approx 20\,[{\rm\mu}\text{m}]\frac{R_{m}}{{\it\kappa}_{\text{ch}}^{1/2}a_{0}}\left(\frac{{\it\lambda}_{L}}{0.8\,{\rm\mu}\text{m}}\right)\left(\frac{W/{\it\alpha}}{10~\text{GeV}}\right)^{1/2},\end{eqnarray}$$

where $R_{m}=k_{p}r_{L}$ is the dimensionless matched spot radius for a given $a_{0}$ and ${\rm\Delta}n/n_{0}$, given by Equation (3). The required peak power of the laser pulse is given by

(44)$$\begin{eqnarray}P_{L}=\frac{k_{p}^{2}r_{L}^{2}a_{0}^{2}}{32}P_{c}\approx 13.7\,[\text{TW}]\frac{R_{m}^{2}}{{\it\kappa}_{\text{ch}}}\left(\frac{W/{\it\alpha}}{10\,\text{GeV}}\right),\end{eqnarray}$$

and the required pulse energy becomes

(45)$$\begin{eqnarray}U_{L}=P_{L}{\it\tau}_{L}\approx 2.2\,[\text{J}]\frac{R_{m}^{2}}{a_{0}{\it\kappa}_{\text{ch}}^{3/2}}\left(\frac{{\it\lambda}_{L}}{0.8\,{\rm\mu}\text{m}}\right)\left(\frac{W/{\it\alpha}}{10\,\text{GeV}}\right)^{3/2}.\end{eqnarray}$$

The field reduction factor ${\it\alpha}$ is determined from the coefficient $C$, given by

(46)$$\begin{eqnarray}C\equiv \frac{Q_{b}}{111\,[\text{pC}]}\frac{{\it\kappa}_{\text{self}}^{1/2}}{a_{0}k_{p}^{2}{\it\sigma}_{b}^{2}}\left(\frac{0.8\,{\rm\mu}\text{m}}{{\it\lambda}_{L}}\right)\left(\frac{10\,\text{GeV}}{W}\right)^{1/2}.\end{eqnarray}$$