Stable Charged-Particle Acceleration and Focusing in a Laser Accelerator Using Spatial Harmonics

Regarding the laser-driven acceleration of charged particles in photonic systems, a central unmet challenge is the achievement of simultaneous transverse and longitudinal stability at nonultrarelativistic energies. At such energies, Earnshaw’s theorem [S. Earnshaw, Trans. Cambridge Philos. Soc. 7, 97 (1842)] indicates that a synchronous accelerating wave gives a defocusing effect. We present a scheme in which particles are accelerated by interaction with a resonant spatial harmonic and are focused by strong ponderomotive interaction with nonresonant spatial harmonics. We show that this scheme exhibits net transverse focusing and longitudinal stability, and we discuss its use in a compact laser accelerator.

Figure 1

Stability diagram for a single nonresonant spatial harmonic $a_{n_0+1}\ne 0$. Increasing the slow modulation length $N$ and increasing the particle energy ${\gamma }_{n_0}$ improves transverse stability and gives higher values of the accelerating field $E_0$ for a fixed net field $E_0(1+|a_{n_0+1}|)=6\mathrm{GV}/\mathrm{m}$. Locations of the two example cases discussed in the text are shown with dot markers. The region below the dotted line ($\eta =0.03$) is longitudinally unstable. ($a=1.15\mu \mathrm{m}$ and $\lambda =5\mu \mathrm{m}$).

Figure 2

Longitudinal phase-space trajectories for a bucket having acceleration gradient $1\mathrm{GeV}/\mathrm{m}$ being perturbed by a strong nonresonant wave. The blue orbits are obtained by averaging the phase space motion over a single period $\Delta \tilde{t}=1$. The dotted line is the raw phase-space motion for the outermost blue orbit, showing the rather violent nonresonant longitudinal motion that provides approximately 40% of the transverse stability in this case. The black line is the separatrix obtained from the unperturbed Hamiltonian. [$N=250$, ${\gamma }_{n_0}=6$, $E_{n_0}=\sqrt{2}\mathrm{GV}/\mathrm{m}$, $E_0(1+a_{n_0+1})=6\mathrm{GV}/\mathrm{m}$, ${\varphi }_0=-\pi /4$].

Figure 3

Accelerator eigenmode simultaneously providing both transverse focusing and longitudinal acceleration. Arrows indicate electric field, and color shading indicates magnetic field in silicon. Unshaded channel in center and holes represent vacuum. Slow modulation of the waveguide fingers ($N=8$) produces two sidebands about the unmodulated waveguide mode. Beam is synchronized to the fast sideband. Here, we take the design particles to be positrons, so that the synchronous phase ${\varphi }_0$ is $3\pi /4$. See movies in Supplemental Material [16]. ($a=1.15\mu \mathrm{m}$ and $\lambda =5\mu \mathrm{m}$)

Figure 4

Spectral properties of the accelerator eigenmode shown in Fig. 3. (a) Band diagram. Three spatial harmonics are shown with thick lines. The fast sideband, $q=0$, is the resonant spatial harmonic and is approximately speed-of-light. The strongest nonresonant harmonic, $q=1$, is slightly slower and provides the majority of the transverse focusing. (b) Spatial harmonic spectrum. The strength of the two sidebands, $q=0$ and $q=2$, relative to the unmodulated spatial harmonic, $q=1$, can be increased by increasing the depth of the slow modulation.