Maximum current density and beam brightness achievable by laser-driven electron sources

This paper discusses the extension to different electron beam aspect ratio of the Child-Langmuir law for the maximum achievable current density in electron guns. Using a simple model, we derive quantitative formulas in good agreement with simulation codes. The new scaling laws for the peak current density of temporally long and transversely narrow initial beam distributions can be used to estimate the maximum beam brightness and suggest new paths for injector optimization.

Figure 1

(a) Definition of 2D Child-Langmuir parameters. (b) On-axis electric field of a charged disk for infinite (blue dotted) and finite (red dashed) transverse dimensions. The surface charge density is the same in both cases. The on-axis coordinate $\zeta$ from the center of the disk is normalized by the disk radius $R$. The electric field is normalized by $\sigma /2{\epsilon }_0$.

Figure 2

Charge extracted as a function of radius. Disk model, three-dimensional simulations (Astra and GPT) and theory ($C_c=1$) for two different cases.

Figure 3

(a) Emitted charge vs electric field $E_0$ for a cigar aspect ratio beam, $C_c=1$. (b) Peak current vs cathode size in the short pancake-beam limit ($E_0=60\mathrm{MV}/\mathrm{m}$, $\mathrm{\Delta }t=500\mathrm{fs}$) and $C_s=1$ [see Eq. (6)].

Figure 4

Electron beam current extracted as function of charge injected (laser energy) into the diode. The example corresponds to the radius of $500\mu \mathrm{m}$ in Fig. 2. The inset shows the extracted charge vs the injected one. The laser pulse length was fixed to 60 ps. The ASTRA code was used in the simulations.

Figure 5

Electric field within the electron beam at the end of the laser pulse duration time $\mathrm{\Delta }t$ for a finite and infinite transverse dimension (left panel) and the relative electron beam charge density distribution (right panel) for the finite radius case simulated by GPT. The black line on the left panel shows the level of the external field.

Figure 6

Maximum brightness as a function of beam radius. We have assumed a typical value for the normalized rms transverse momentum spread $\frac{{\sigma }_{p_{\perp }}}{mc}=0.8\times {10}^{-3}$.