2.1. Accelerator stage

Most of the laser plasma acceleration experiments that have successfully demonstrated the production of quasi-monoenergetic electron beams with a narrow energy spread have been elucidated in terms of self-injection and acceleration mechanisms in the bubble regime^{[Reference Kostyukov, Pukhov and Kiselev26, Reference Lu, Tzoufras, Joshi, Tsung, Mori, Vieira, Fonseca and Silva27]}, where a drive laser pulse with wavelength $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\lambda _{L} $, peak power $P_{L} $, intensity $I_{L} $, and focused spot radius $r_{L} $ is characterized by a normalized vector potential $a_{0} \gg 1$ with respect to the electron rest energy $mc^{2}$, given for the linear polarization as

(1)$$\begin{eqnarray} a_{0} &=& \left (\frac {2e^{2}\lambda _{L}^{2} I_{L}}{\pi m^{2}c^{5}}\right )^{1/2} \nonumber \\ &\cong & 8.55\times 10^{-10}\sqrt {I_{L}({\rm W~cm}^{-2})} \lambda _{L}\;(\mu {\rm m}) \nonumber \\ &\approx & 6.82\sqrt {P_{L} ({\rm TW})} \frac {\lambda _{L}}{r_{L}}. \label {eq1} \end{eqnarray}$$

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 nonlinear wakefields in the bubble (blowout) regime^{[Reference Kostyukov, Pukhov and Kiselev26]} describes the accelerating wakefield $E_{z} (\xi )/E_{0} \approx (1/2)k_{p} \xi $ in the bubble frame moving in a plasma with velocity $v_{B} $, i.e., $\xi =z-v_{B} t$, where $k_{p} =\omega _{p} /c=(4\pi r_{{\rm e}} n_{e} )^{1/2}$ is the plasma wavenumber evaluated with a plasma frequency $\omega _{p} $, an unperturbed on-axis electron density $n_{e} $ and the classical electron radius $r_{{\rm e}} =e^{2}/mc^{2}$, and $E_{0} =mc\omega _{p} /e$ is the non-relativistic wave-breaking field, approximately given by $E_{0} \approx 96\ ({\rm GV}\ {\rm m}^{-1})(n_{e} /10^{18}\ ({\rm cm}^{-3}))^{1/ 2}$. In the bubble regime for $a_{0} \ge 2$, since an 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 $k_{p} R_{B} \approx 2\sqrt{a_{0}}$^{[Reference Lu, Tzoufras, Joshi, Tsung, Mori, Vieira, Fonseca and Silva27]}. Thus, the maximum accelerating field is given by $E_{z0} /E_{0} =(1/2)\alpha k_{p} R_{B} $, where $\alpha $ represents a factor taking into account the difference between the theoretical estimation and the accelerating field reduction due to the beam loading effects.

Here we consider the self-guided case, where a drive laser pulse propagates in a homogeneous density plasma. The equations of longitudinal motion of an electron with normalized energy $\gamma =E_{b} /mc^{2}$ and longitudinal velocity $\beta _{z} =v_{z} /c$ are written approximately as^{[Reference Nakajima, Lu, Zhao, Shen, Li and Xu28]}

(2)$$\begin{eqnarray} &&\frac {d\gamma }{dz}=\frac {1}{2}\alpha k_{p}^{2} R_{B} \left (1-\frac {\xi }{R_{B}}\right ),\nonumber \\ &&\frac {d\xi }{dz}=1-\frac {\beta _{B}}{\beta _{z}}\approx 1-\beta _{B} \approx \frac {3}{2\gamma _{g}^{2}}, \label {eqn2} \end{eqnarray}$$

where $\xi =z-v_{B} t$ ($0\le \xi \le R_{B}$) is the longitudinal coordinate of the bubble frame moving at a velocity of $v_{B} =c\beta _{B} \approx v_{g} -v_{{\rm etch}}$, taking into account diffraction at the laser pulse front that etches back at a velocity $v_{{\rm etch}} \sim ck_{p}^{2} /k^{2}$^{[Reference Lu, Tzoufras, Joshi, Tsung, Mori, Vieira, Fonseca and Silva27]} with laser wavenumber $k$, and $\gamma _{g} =(1-\beta _{g}^{2} )^{-1/2}\approx k/k_{p} \gg 1$ is assumed. Integrating Equations (2), the energy and phase of the electron can be calculated as^{[Reference Nakajima, Lu, Zhao, Shen, Li and Xu28]}

(3)$$\begin{eqnarray} &&\gamma (z)=\gamma _{0} +\frac {1}{3}\alpha \gamma _{g}^{2} k_{p}^{2} R_{B} \xi (z) \left (1-\frac {1}{2}\frac {\xi (z)}{R_{B} }\right ), \nonumber \\ &&\xi (z)=\frac {3}{2}\frac {z}{\gamma _{g}^{2}}, \end{eqnarray}$$

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

(4)$$\begin{eqnarray} \Delta \gamma _{\max } &=&\gamma _{\max } -\gamma _{0} \approx \frac {1}{6}\alpha \gamma _{g}^{2} k_{p}^{2} R_{B}^{2} \approx \frac {2}{3}\alpha a_{0} \gamma _{g}^{2} \nonumber \\ &=&\frac {2}{3}\alpha \kappa _{{\rm self}} a_{0} \frac {n_{c} }{n_{e}}, \label {eq2} \end{eqnarray}$$

where $\kappa _{{\rm self}}$ is the correction factor of the relativistic factor for the group velocity in a uniform plasma for a self-guided pulse, i.e., $\gamma _{g}^{2} =(1-\beta _{g}^{2} )^{-1}\approx \kappa _{{\rm self}} k^{2}/k_{p}^{2} =\kappa _{{\rm self}} n_{c} /n_{e}$, obtained from^{[Reference Nakajima, Lu, Zhao, Shen, Li and Xu28]}

(5)$$\begin{equation} \label {eq3} \kappa _{{\rm self}} =\frac {a_{0}^{2} }{8}\left (\sqrt {1+a_{0}^{2} /2} -1-\ln \frac {\sqrt {1+a_{0}^{2} /2} +1}{2}\right )^{-1}, \end{equation}$$

and $n_{c} =m\omega _{L}^{2} /4\pi e^{2}=\pi /(r_{{\rm e}} \lambda _{L}^{2}) \approx 1.115\times 10^{21}\ ({\rm cm}^{-3})$ $(\lambda _{L} /1\ \mu {\rm m})^{-2}$ is the critical plasma density. The dephasing length $L_{{\rm dp}} $ for the self-guided bubble regime is given by

(6)$$\begin{equation} \label {eq4} k_{p} L_{{\rm dp}} \approx \frac {2}{3}k_{p} R_{B} \gamma _{g}^{2} =\frac {4}{3}\sqrt {a_{0} } \kappa _{{\rm self}} \frac {n_{c} }{n_{e} }. \end{equation}$$

For a given energy gain $E_{b} $, the operating plasma density is determined from Equation (4) as

(7)$$\begin{eqnarray} n_{e} &=&\frac {2}{3}\alpha \kappa _{{\rm self}} a_{0} \frac {n_{c} } {\Delta \gamma _{\max }} \nonumber \\ &\approx & 1.9\times 10^{18}~({\rm cm}^{-3})\kappa _{{\rm self}} a_{0} \nonumber \\ &&\times \,\left (\frac {1~\mu {\rm m}}{\lambda _{L}}\right )^{2} \left (\frac {200~{\rm MeV}}{E_{b} /\alpha }\right ). \label {eq5} \end{eqnarray}$$

The accelerator length equal to the dephasing length becomes

(8)$$\begin{eqnarray} L_{{\rm acc}} &=& L_{{\rm dp}} \approx \sqrt {\frac {3}{2}} \frac {(\Delta \gamma _{\max } /\alpha )^{3/2}}{\pi \kappa _{{\rm self}}^{1/2} a_{0}} \lambda _{L} \nonumber \\ &\approx & \frac {3.1~({\rm mm})}{\kappa _{{\rm self}}^{1/2} a_{0}} \left (\frac {\lambda _{L} }{1~\mu {\rm m}}\right ) \left (\frac {E_{b} /\alpha }{200~{\rm MeV}}\right )^{3/2}, \label {eq6} \end{eqnarray}$$

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

(9)$$\begin{eqnarray} L_{{\rm pd}} &\approx & c\tau _{L} \frac {n_{c}}{n_{e}}=\frac {3}{2} \frac {c\tau _{L} \Delta \gamma _{\max } /\alpha }{\kappa _{{\rm self}} a_{0}} \nonumber \\ &\approx & \frac {5~({\rm mm})}{\kappa _{{\rm self}} a_{0}} \left (\frac {\tau _{L}}{30~{\rm fs}}\right ) \left (\frac {E_{b}/\alpha }{200~{\rm MeV}}\right ). \label {eq7} \end{eqnarray}$$

The dephasing length should be less than the pump depletion length, i.e., $L_{{\rm pd}} \ge L_{{\rm dp}} $. Thus, the required pulse duration for self-guiding of the drive laser pulse is given by

(10)$$\begin{equation} \label {eq8} \tau _{L} \ge 18~({\rm fs})\kappa _{{\rm self}}^{1/2} \left (\frac {\lambda _{L}}{1~\mu {\rm m}}\right ) \left (\frac {E_{b}/\alpha }{200~{\rm MeV}}\right )^{1/2}. \end{equation}$$

The matched spot radius becomes

(11)$$\begin{equation} \label {eq9} r_{m} \approx 3.9~(\mu {\rm m})\frac {R_{m}}{\sqrt {\kappa _{{\rm self}} a_{0} }} \left (\frac {\lambda _{L} }{1~\mu {\rm m}}\right )\left (\frac {E_{b} /\alpha } {200~{\rm MeV}}\right )^{1/2}, \end{equation}$$

where $R_{m} \equiv k_{p} r_{L} $ is the dimensionless matched spot radius given by^{[Reference Nakajima, Lu, Zhao, Shen, Li and Xu28]}

(12)$$\begin{equation} \label {eq10} R_{m} = \left \{\frac {\ln (1+a_{0}^{2} /2)}{\sqrt {1+a_{0}^{2} /2} -1-2\ln [(\sqrt {1+a_{0}^{2} /2} +1)/2]}\right \}^{1/2}. \end{equation}$$

The corresponding matched power is calculated as

(13)$$\begin{equation} \label {eq11} P_{L} =\frac {k_{p}^{2} r_{L}^{2} a_{0}^{2} }{32}P_{c} \approx 0.312~({\rm TW})\frac {a_{0} R_{m}^{2}}{\kappa _{{\rm self}}} \left (\frac {E_{b} /\alpha }{200~{\rm MeV}}\right ). \end{equation}$$

The required pulse energy becomes

(14)$$\begin{equation} \label {eq12} U_{L} =P_{L}\tau _{L}\ge 5.62~({\rm mJ}) \frac {a_{0} R_{m}^{2}}{\kappa _{{\rm self}}^{1/2}} \left (\frac {\lambda _{L}}{1~\mu {\rm m}}\right ) \left (\frac {E_{b}/\alpha }{200~{\rm MeV}}\right )^{3/2}. \end{equation}$$

2.2. 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 $\eta _{b} \equiv 1-E_{z}^{2} /E_{M}^{2} $ defined by the fraction of plasma wave energy absorbed by particles of the bunch with a root mean square (r.m.s.) radius $\sigma _{b} $, the beam-loaded field is given by $E_{z} =\sqrt{1-\eta _{b} } E_{M} =\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 $a_{0} \ge 2$. Thus, a loaded charge is calculated as^{[Reference Nakajima, Deng, Zhang, Shen, Liu, Li, Xu, Ostermayr, Petrovics, Klier, Iqbal, Ruhl and Tajima29]}

(15)$$\begin{eqnarray} Q_{b} &\simeq & \frac {e}{4k_{L} r_{{\rm e}} }\frac {\eta _{b} k_{p}^{2} \sigma _{b}^{2} }{1-\eta _{b}} \frac {E_{z} }{E_{0} }\left (\frac {n_{c} }{n_{e} }\right )^{1/2}\xmlpi {}\nonumber \\ &\approx & 76~({\rm pC})\frac {\eta _{b} a_{0}^{1/2} k_{p}^{2} \sigma _{b}^{2}}{\sqrt {1-\eta _{b}}} \left (\frac {n_{e} }{10^{18}~{\rm cm}^{-3}}\right )^{-1/2}. \label {eq13} \end{eqnarray}$$

Using the plasma density Equation (7), the loaded charge is given by

(16)$$\begin{equation} \label {eq14} Q_{b} \approx 55~({\rm pC})\frac {1-\alpha ^{2}}{\alpha ^{3/2}} \frac {k_{p}^{2} \sigma _{b}^{2} }{\kappa _{{\rm self}}^{1/2}} \left (\frac {\lambda _{L} }{1~\mu {\rm m}}\right ) \left (\frac {E_{b}}{200~{\rm MeV}}\right )^{1/2}. \end{equation}$$

Therefore, the field reduction factor $\alpha $ for accelerating charge $Q_{b} $ up to energy $E_{b} $ is obtained by solving the equation

(17)$$\begin{equation} \label {eq15} \alpha ^{2}+C\alpha ^{3/2}-1=0, \end{equation}$$

where the coefficient $C$ is defined as

(18)$$\begin{equation} \label {eq16} C\equiv \frac {Q_{b} }{55~({\rm pC})}\frac {\kappa _{{\rm self}}^{1/2}} {k_{p}^{2} \sigma _{b}^{2}}\left (\frac {1~\mu {\rm m}}{\lambda _{L}}\right ) \left (\frac {200~{\rm MeV}}{E_{b}}\right )^{1/2}. \end{equation}$$

2.3. Injector stage

Electron beams can be produced and accelerated in the injector stage driven by the same laser pulse as that in the accelerator stage, relying on a self-injection mechanism such as the expanding bubble self-injection mechanism^{[Reference Kalmykov, Yi, Khudik and Shvets30]} or an ionization-induced injection scheme with a short mixed gas cell^{[Reference Pak, Marsh, Martins, Lu, Mori and Joshi14–Reference Xia, Liu, Wang, Lu, Cheng, Deng, Li, Zhang, Liang, Leng, Lu, Wang, Wang, Nakajima, Li and Xu16, Reference Chen, Esarey, Schroeder, Geddes and Leemans31]}, where tunnel ionization leads to electron trapping near the centre of the laser wakefield. Here we consider the ionization-induced injection scheme. According to theoretical considerations in ionization-induced injection^{[Reference Chen, Esarey, Schroeder, Geddes and Leemans31]}, for trapping electrons ionized at the peak of the laser electric field, the minimum laser intensity is given by $1-\gamma _{g}^{-1} \le 0.64a_{\min }^{2} $. At a plasma density $n_{{\rm inj}} =10^{18}\ {\rm cm}^{-3}$ in the injector, the required minimum laser field is $a_{\min } \ge 1.23$. The maximum number of trapped electrons saturates at approximately $N_{e\max } \sim 5\times 10^{6}\ \mu {\rm m}^{-2}$ at a gas length $L_{{\rm inj}} \approx 1000\lambda _{L} $ for a plasma density $n_{{\rm inj}} =0.001n_{c}$ with a nitrogen concentration of $\alpha _{{\rm N}} =1\%$ and laser parameters of $a_{0} =2$ and $c\tau _{L} \approx 15\lambda _{L}$ due to the beam loading effects and initially trapped particle loss from the separatrix in the phase space. From the particle-in-cell (PIC)-simulation results^{[Reference Chen, Esarey, Schroeder, Geddes and Leemans31]}, the trapped electron density scales as

(19)$$\begin{eqnarray} N_{e}\;(\mu {\rm m}^{-2})&\sim & 8\times 10^{7}\alpha _{{\rm N}} k_{p} L_{{\rm inj}} \left (\frac {n_{{\rm inj}} }{n_{c}}\right )^{1/2} \nonumber \\ &\approx & 5\times 10^{8}\alpha _{{\rm N}} \left (\frac {L_{{\rm inj}}}{\lambda _{L}}\right ) \left (\frac {n_{{\rm inj}}}{n_{c}}\right ). \label {eq17} \end{eqnarray}$$

The energy spread is also proportional to both the mixed gas length and the nitrogen concentration. In a injector with gas length $L_{{\rm inj}} $, the electron charge $Q_{b} $ trapped inside a bunch with radius $r_{b} =1/k_{p} \approx 5.3\ (\mu {\rm m})$ at $n_{{\rm inj}} =10^{18}\ {\rm cm}^{-3}$ is estimated as

(20)$$\begin{equation} \label {eq18} Q_{b} \sim \frac {k_{p}^{2} r_{b}^{2} }{4r_{{\rm e}} n_{{\rm inj}}}eN_{e} \approx 6.4~({\rm pC})\alpha _{{\rm N}} k_{p}^{2} r_{b}^{2} \left (\frac {\lambda _{L}}{1~\mu {\rm m}}\right ) \left (\frac {L_{{\rm inj}}}{1~\mu {\rm m}}\right ). \end{equation}$$

An electron charge of 500 pC will be trapped via the ionization-induced injection mechanism in an injector gas cell with a 2 mm length and a nitrogen concentration of $\alpha _{{\rm N}} =4\%$.

2.4. Design of a SASE FEL

In the SASE FEL process, coupling the electron bunch with a co-propagating undulator radiation field induces an energy modulation of electrons that yields current modulation of the bunch due to the dispersion of the undulator dipole fields, known as microbunching. It means that the electrons are grouped into small bunches separated by a fixed distance that resonantly coincides with the wavelength of the radiation field. Consequently, the radiation field can be amplified coherently. In the absence of an initial resonant radiation field, a seed may build up from spontaneous incoherent emission in the SASE process.

The design of the FEL-based EUV light source is carried out using one-dimensional FEL theory as follows^{[Reference Elleaume, Chavanne and Faatz32]}. The FEL amplification takes place in an undulator with undulator period $\lambda _{u} $ and peak magnetic field $B_{u} $ at a resonant wavelength $\lambda _{X} $ given by

(21)$$\begin{equation} \label {eq19} \lambda _{X} =\frac {\lambda _{u}}{2\gamma ^{2}}\left (1+\frac {K^{2}}{2}\right ), \end{equation}$$

where $\gamma =E_{b} /m_{e} c^{2}$ is the relativistic factor of the electron beam energy $E_{b} $, and $K_{u} =0.934B_{u}\ ({\rm T})\lambda _{u}\ ({\rm cm})=\gamma \theta _{e} $ is the undulator parameter, which is related to the maximum electron deflection angle $\theta _{e} $. In the high-gain regime required for the operation of a SASE FEL, an important parameter is the Pierce parameter $\rho _{{\rm FEL}} $, given by

(22)$$\begin{equation} \label {eq20} \rho _{{\rm FEL}} =\frac {1}{2\gamma }\left [\frac {I_{b}}{I_{A}} \left (\frac {\lambda _{u} K_{u} A_{u}}{2\pi \sigma _{b}}\right )^{2}\right ]^{1/3}, \end{equation}$$

where $I_{b} $ is the beam current, $I_{A} =17\ {\rm kA}$ is the Alfven current, $\sigma _{b} $ is the r.m.s transverse size of the electron bunch, and the coupling factor is $A_{u} =1$ for a helical undulator and $A_{u} =J_{0} (\Xi )-J_{1} (\Xi )$ for a planar undulator, where $\Xi ={K_{u}^{2}}/[4(1+K_{u}^{2} /2)]$ and $J_{0} $ and $J_{1} $ are Bessel functions of the first kind. Another important dimensionless parameter is the longitudinal velocity spread $\Lambda $ of the beam normalized by the Pierce parameter:

(23)$$\begin{eqnarray} \Lambda ^{2}&=&\frac {1}{\rho _{{\rm FEL}}^{2}} \left [\left (\frac {\sigma _{\gamma }}{\gamma }\right )^{2}+ \left (\frac {\varepsilon \lambda _{u} }{4\lambda _{X} \beta }\right )^{2}\right ] \nonumber \\ &=& \frac {1}{\rho _{{\rm FEL}}^{2}}\left [\left (\frac {\sigma _{\gamma }}{\gamma }\right )^{2} +\left (\frac {\varepsilon _{n}^{2}}{2\sigma _{b}^{2} (1+K_{u}^{2}/2)}\right )^{2}\right ], \label {eq21} \end{eqnarray}$$

where $\sigma _{\gamma } /\gamma $ is the relativistic r.m.s. energy spread, $\varepsilon $ is the r.m.s. transverse emittance, $\beta =\sigma _{b}^{2} /\varepsilon $ is the beta function provided by the guiding field (undulator plus external focusing) and $\varepsilon _{n} $ is the normalized emittance, defined as $\varepsilon _{n} \equiv \gamma \varepsilon $, assuming that the beta function is constant along the length of the undulator. The $e$-folding gain length $L_{{\rm gain}} $ over which the power grows exponentially according to $\exp (2s/L_{{\rm gain}} )$ is given by

(24)$$\begin{equation} \label {eq22} L_{{\rm gain}} =\frac {\lambda _{u} }{4\pi \sqrt 3 \rho _{{\rm FEL}}}(1+\Lambda ^{2}). \end{equation}$$

In order to minimize the gain length, one needs a large Pierce parameter $\rho _{{\rm FEL}} $ and a normalized longitudinal velocity spread $\Lambda $ sufficiently low compared to unity, which means a sufficiently small energy spread $\sigma _{\gamma } /\gamma $ and $\varepsilon $. This expression applies to a moderately small beam size $\sigma _{b} $ such that the diffraction parameter $B\gg 1$, where $B$ is defined as

(25)$$\begin{equation} \label {eq23} B=\frac {16\pi ^{2}A_{u} \sigma _{b}^{2} }{\lambda _{X} \lambda _{u}} \left [\frac {K_{u}^{2} /2}{\gamma (1+K_{u}^{2} /2)}\frac {I_{b}}{I_{A}}\right ]^{1/2}. \end{equation}$$

The saturation length $L_{{\rm sat}} $ required to saturate the amplification can be expressed as

(26)$$\begin{equation} \label {eq24} L_{{\rm sat}} =L_{{\rm gain}} \ln \left [\left (\frac {\Lambda ^{2}+3/2}{\Lambda ^{2}+1/6}\right )\frac {P_{{\rm sat}}}{P_{{\rm in}}}\right ], \end{equation}$$

where $P_{{\rm in}} $ and $P_{{\rm sat}} $ are the input power and the saturated power, which are related to the electron beam power $P_{b} $ according to

(27)$$\begin{eqnarray} &&P_{b} =\gamma I_{b} m_{e} c^{2}=I_{b} E_{b} , \quad \nonumber \\ &&P_{{\rm sat}} \cong 1.37\rho _{{\rm FEL}} P_{{\rm b}} \exp (-0.82\Lambda ^{2}), \nonumber \\ &&P_{{\rm in}} \cong 3\sqrt {4\pi } \rho _{{\rm FEL}}^{2} P_{b} [N_{\lambda _{X}} \ln (N_{\lambda _{X}}/\rho _{{\rm FEL}})]^{-1/2}, \label {eq25} \end{eqnarray}$$

where $N_{\lambda _{X}}$ is the number of electrons per wavelength, given by $N_{\lambda _{X}} =I_{b}\lambda _{X}/(ec)$.

Table 1. Parameters for laser plasma accelerator-based EUV FEL light sources.

For an EUV light source based on a FEL, a planar undulator comprising alternating dipole magnets is used, e.g., a pure permanent magnet (PPM) undulator with ${\rm Nd}_{2}{\rm Fe}_{14}{\rm B}$ (Nd–Fe–B) blocks or a hybrid undulator comprising PPMs and ferromagnetic poles, e.g., a high saturation cobalt steel such as Vanadium Permendur or a simple iron. For a hybrid undulator, the thickness of the pole and magnet is optimized in order to maximize the peak field. The peak field $B_{u} $ of the gap is estimated in terms of the gap $g$ and period $\lambda _{u} $ according to $B_{u}=a\ ({\rm T})\exp [b(g/\lambda _{u})+c(g/\lambda _{u})^{2}]$ for a gap range $0.1<g/\lambda _{u} <1$, where $a=3.694\ {\rm T}$, $b=-5.068$ and $c=1.520$ for the hybrid undulator with Vanadium Permendur. Table 1 summarizes design examples for a fiber laser-driven laser plasma accelerator-based FEL-produced EUV radiation source at 13.5 nm wavelength using undulators with periods 5 mm (Case A), 10 mm (Case B), 15 mm (Case C), 20 mm (Case D), and 25 mm (Case E), all cases of which have the same gap:period ratio 0.2, e.g., $g=1$ mm (Case A), 2 mm (Case B), 3 mm (Case C), 4 mm (Case D), and 5 mm (Case E), respectively. The bunch duration of the electron beam in the injector stage at a plasma density of $n_{e} \approx 10^{18}\ {\rm cm}^{-3}$ is assumed to be $\sim $10 fs full-width at half-maximum (FWHM), based on a measurement of the electron bunch duration in a recent laser wakefield acceleration experiment^{[Reference Buck, Nicolai, Schmid, Sears, Sävert, Mikhailova, Krausz, Kaluza and Veisz33]}. The relative energy spread of the accelerated electron beam with an injection energy of $0.1E_{b}$, where $E_{b} $ is the final beam energy in the accelerator stage, is assumed to be of the order of 10% in the injector stage. After acceleration up to 10 times higher energy in the accelerator stage, the relative energy spread at the final beam energy is reduced to $\Delta E/E_{b} \sim 1\% $ due to adiabatic damping in the longitudinal beam dynamics. The transverse beam size is tuned by employing a beam focusing system. Figure 1 shows a schematic of the EUV light source based on a compact FEL driven by a fiber laser-based plasma accelerator.

Figure 1. Schematic of the EUV light source based on a compact FEL driven by a fiber laser-based plasma accelerator.

2.5. Design of all-optical Gamma-beam source

The design of a Gamma-beam source based on inverse Compton scattering is carried out by using a result of quantum electrodynamics on photon–electron interactions, namely, the Klein–Nishina formula, which gives the differential cross section of photons scattered from a single electron in the lowest order of quantum electrodynamics. In Compton scattering of a laser photon with energy $\hslash \omega _{L} $ ($\hslash \omega _{L}\ ({\rm eV})=1.240/\lambda _{L}\ (\mu {\rm m})$ for laser wavelength $\lambda _{L}\ (\mu {\rm m}))$ off a beam electron, the maximum energy of the scattered photon is given by $E_{\gamma \max } =4\gamma _{e}^{2} a\hslash \omega _{L} $, where $\gamma _{e} =E_{b} /m_{e} c^{2}$ is the relativistic factor for an electron beam energy $E_{b} $ with electron rest mass $m_{e} c^{2}\simeq 0.511\ {\rm MeV}$ and the factor $a=[1+4\gamma _{e} (\hslash \omega _{L} /m_{e} c^{2})]^{-1}$. In the laboratory frame, the differential cross section of Compton scattering^{[Reference Tolhokk34]} is given by

(28)$$\begin{equation} \label {eq26} \frac {d\sigma }{d\kappa }=2\pi ar_{{\rm e}}^{2} \left \{1+\frac {\kappa ^{2}(1-a)^{2}}{1-\kappa (1-a)}+\left [\frac {1-\kappa (1+a)}{1-\kappa (1-a)}\right ]^{2}\right \}, \end{equation}$$

where $\kappa =E_{\gamma } /E_{\gamma \max } $ is the energy of a scattered photon normalized by the maximum photon energy and $r_{{\rm e}}^{2} \simeq 79.4\ {\rm mb}$ ($1\ {\rm barn}=10^{-24}\ {\rm cm}^{2})$ with the classical electron radius $r_{{\rm e}} $. In the laboratory frame, the scattering angle $\theta $ of the photon is given by $\tan \theta =\gamma _{e}^{-1} \sqrt{(1-\kappa )/a\kappa } $. Integrating the differential cross section over $0\le \kappa \le 1$, the total cross section of Compton scattering becomes

(29)$$\begin{eqnarray} \sigma _{{\rm total}} &=&\pi r_{{\rm e}}^{2} a \left [\frac {2a^{2}+12a+2}{(1-a)^{2}}+a-1 \right .\nonumber \\ &&+\,\left .\frac {6a^{2}+12a-2}{(1-a)^{3}}\ln a\right ]. \label {eq27} \end{eqnarray}$$

This total cross section leads to a cross section of Thomson scattering $\sigma _{{\rm Thomson}} =8\pi r_{{\rm e}}^{2} /3=665\ {\rm mb}$ for an electron beam energy $E_{b} \to 0$. The fractional cross section for the photon energy range $E_{\gamma \max } -\Delta E_{\gamma } \le E_{\gamma } \le E_{\gamma \max } $ is given by

(30)$$\begin{eqnarray} \Delta \sigma &=&2\pi ar_{{\rm e}}^{2} \Delta \kappa \left [\left (\frac {1+a}{1-a}\right )^{2}+\frac {4}{(1-a)^{2}} \right .\nonumber \\ &&\times \,\left (1+\frac {1-a}{a}\Delta \kappa \right )^{-1} +(a-1)\left (1+\frac {\Delta \kappa }{2}\right ) \nonumber \\ &&+\left .\,\frac {1-6a-3a^{2}}{(1-a)^{3}\Delta \kappa } \ln \left (1+\frac {1-a}{a}\Delta \kappa \right )\right ], \label {eq28} \end{eqnarray}$$

with $\Delta \kappa =\Delta E_{\gamma } /E_{\gamma \max } \ll 1$. All photons in this energy range are scattered in the forward direction within a half-cone angle $\theta \sim \gamma _{e}^{-1} \sqrt{\Delta \kappa /a} $. For an electron beam interacting with a laser pulse at an angle of $\alpha _{{\rm int}}$ in the horizontal plane ($x$-plane), the luminosity representing the probability of collisions between electron and laser beams per unit cross section per unit time is obtained by $L\ ({\rm mb}^{-1}{\rm s}^{-1})=N_{e} N_{L} f_{L} /2\pi \Sigma $, where $N_{e} $ is the number of electrons contained in the electron bunch, $N_{L} $ is the number of photons per laser pulse, $f_{L} $ is the repetition rate of laser pulses, and $\Sigma $ is the area where the two beams overlap, given by

(31)$$\begin{eqnarray} \Sigma &=&(\sigma _{ey}^{2} +\sigma _{Ly}^{2} )^{1/2}[\cos ^{2}(\alpha _{{\rm int}} /2)(\sigma _{ex}^{2} +\sigma _{Lx}^{2} ) \nonumber \\ &&+\,\sin ^{2}(\alpha _{\mathrm {int}}/2)(\sigma _{ez}^{2} +\sigma _{Lz}^{2} )]^{1/2}, \label {eq29} \end{eqnarray}$$

where $\sigma _{ex} $ and $\sigma _{ey} $ are the r.m.s. horizontal and vertical sizes of the electron beam, $\sigma _{ez} $ is the r.m.s. bunch length of the electron beam, $\sigma _{Lx} $ and $\sigma _{Ly} $ are the r.m.s. horizontal and vertical spot sizes of the laser beam, and $\sigma _{Lz} $ is the r.m.s. pulse length of the laser beam. For a head-on collision providing efficient Gamma-beam production, the crossing angle between the electron and laser beams is chosen to be $\alpha _{{\rm int}} =0$. Tuning the beam focusing system and the interaction optics so as to give $\sigma _{ex} \approx \sigma _{ey} \approx \sigma _{Lx} \approx \sigma _{Ly} $, the luminosity turns out to be $L=N_{e} N_{L} f_{L} /(4\pi r_{{\rm int}}^{2})$, where $r_{{\rm int}}$ is the laser spot radius at the interaction point. Using $N_{e} =1.6022\times 10^{10}(Q_{e}/1\ {\rm nC})$ and $N_{L} =U_{LS} /\hslash \omega _{L} =5.0334\times 10^{18}U_{LS}\ ({\rm J})\lambda _{L} (\mu {\rm m})$, where $Q_{e} $ is the charge of the electron bunch and $U_{LS} =P_{LS} \tau _{LS} $ is the energy of a scatter pulse with peak power $P_{LS} $ and duration $\tau _{LS} $, the luminosity is calculated as

(32)$$\begin{eqnarray} L~({\rm mb}^{-1}{\rm s}^{-1})&=&Q_{e} I_{{\rm int}} f_{L} \tau _{LS}/(8e\hslash \omega _{L} ) \nonumber \\ &\approx & 1.0\times 10^{-14}f_{L}~({\rm s}^{-1}) Q_{e}~({\rm nC}) \nonumber \\ &&\times \, I_{{\rm int}} ({\rm W~cm}^{-2})\tau _{LS} ({\rm fs})\lambda _{L}\;(\mu {\rm m}), \label {eq30} \end{eqnarray}$$

where $I_{{\rm int}}$ is the focused intensity of the scatter pulse at the interaction point. Thus the Gamma-beam flux is given by

(33)$$\begin{eqnarray} N_{\gamma }~({\rm s}^{-1})&=&L\sigma _{{\rm tot}} \approx 1\times 10^{-14}\sigma _{{\rm tot}}~({\rm mb}) f_{L}~({\rm s}^{-1})Q_{e}~({\rm nC})\nonumber \\ &&\times \, I_{{\rm int}}\; ({\rm W~cm}^{-2}) \tau ~({\rm fs})\lambda _{L} (\mu {\rm m}). \label {eq31} \end{eqnarray}$$

The fractional Gamma-beam flux with photon energy spread $\Delta \kappa =\Delta E_{\gamma } /E_{\gamma \max } $ is estimated as

(34)$$\begin{eqnarray} \Delta N_{\gamma }~({\rm s}^{-1})&=&L\Delta \sigma \approx 1\times 10^{-14}\Delta \sigma ~({\rm mb})f_{L}~({\rm s}^{-1})Q_{e}~({\rm nC}) \nonumber \\ &&\times \, I_{L} ({\rm W~cm}^{-2})\tau ({\rm fs})\lambda _{L}\;(\mu {\rm m}). \label {eq32} \end{eqnarray}$$

Table 2 summarizes design examples for an all-optical laser plasma accelerator-based Gamma-beam source at photon energies 2.5 MeV (Case A), 5 MeV (Case B), 10 MeV (Case C), 15 MeV (Case D), and 20 MeV (Case E), respectively. Figure 2 is a schematic illustration of the Gamma-beam source based on inverse Compton scattering off relativistic electron beams driven by a laser plasma accelerator.

Figure 2. Schematic illustration of the Gamma-beam source based on inverse Compton scattering off relativistic electron beams driven by a laser plasma accelerator.

Table 2. Parameters for all-optical laser plasma accelerator-based Gamma-beam sources.