Design study of a dielectric laser undulator

Dielectric laser acceleration (DLA) achieves remarkable gradients from the optical near fields of a grating structure. Tilting the dielectric grating with respect to the electron beam leads to deflection forces, and the DLA structure can be utilized as a microchip undulator. We investigate the beam dynamics in such structures analytically and by numerical simulations. A crucial challenge is to keep the beam focused, especially in the direction of the narrow channel. An alternating phase focusing scheme is optimized for this purpose and matched lattice functions are obtained. We distinguish synchronous operation with phase jumps in the grating and asynchronous operation with a strictly periodic grating and well-designed synchronicity mismatch. Especially, the asynchronous DLA undulator is a promising approach, since a simple, commercially available grating suffices for the focusing lattice design. We pave the way toward experiments of radiation generation in these structures and provide estimates of the emitted radiation wavelength and power. The analytical models are validated by numerical simulations in the dedicated DLA simulation tool dlatrack6d and astra, where the underlying laser fields are computed by cst studio.

Figure 1

cst studio simulation [27] of the field between two opposing dielectric gratings for a $\widehat{\mathit{z}}$ polarized drive-laser with amplitude $E_0$ in (a). In (b), the field strength is compared to a cosine wave with $e_1(0,0)$ complex amplitude, showing that the fundamental Fourier coefficient is dominant.

Figure 2

cst studio simulation [27] for the structure coefficient $|e_1(\alpha )|$ of a silica grating at different tilt angles $\alpha$ for a $\widehat{\mathit{z}}$ polarized drive-laser with amplitude $E_0$ in (a). The grating design in (b) uses a tooth width of $w_t=0.6{\lambda }_g$, a tooth height of $h_t=0.75{\lambda }_0$, and a gap width of $\mathrm{\Delta }y=0.6{\lambda }_0$.

Figure 3

Lattice design and beam trajectory (amplitude not to scale) of a phase-synchronous DLA undulator for a $\widehat{\mathit{z}}$ polarized drive-laser with amplitude $E_0$. Each segment consisting of $n=2$ tilted DLA cells imposes a constant deflection force on the beam. A subsequent drift space ${\lambda }_z/2$ shifts the phase by $\mathrm{\Delta }\phi =\pi$ which flips the direction of the force.

Figure 4

Transverse momentum modulation of the reference particle in a tilted grating undulator. In a synchronous DLA lattice, the trajectory is a sequence of constant acceleration with alternating sign. The asynchronous DLA lattice imposes a sinusoidal varying deflection.

Figure 5

Numerical and $\mathcal{O}({a_0}^{-2})$ analytically approximated solutions for the reference trajectories $x(z)$ in an asynchronous DLA undulator with $\mathrm{\Delta }{k}_0/k_0=0.36\%$ detuning from synchronous operation, $\alpha =27°$ tilt angle, and $E_0=10\mathrm{GV}/\mathrm{m}$ laser field strength. Both the laser detuning $\mathrm{\Delta }k$ and the initial phase $\mathrm{\Delta }\phi$ influence the dynamics.

Figure 6

Tilt angle–dependent quadrupole focusing of a synchronous DLA undulator at maximum $K(\alpha )$ for ${\phi }_s=\pi /2$ and $E_0=10\mathrm{GV}/\mathrm{m}$ field amplitude with an optimum at $\alpha \approx 27°$.

Figure 7

Maximum of the betafunction ${\widehat{\beta }}_{\max }$ for a matched beam in a synchronous DLA undulator with $E_0=10\mathrm{GV}/\mathrm{m}$ laser field strength at optimum FODO cell length. The absolute minimum of ${\widehat{\beta }}_{\max }$ at ${\phi }_s=\pi /2$ implies $K_{\mathrm{und}}^{\mathrm{sync}}=0$.

Figure 8

Maximum ${\widehat{\beta }}_{\max }$ for a matched beam in an asynchronous DLA undulator with $E_0=10\mathrm{GV}/\mathrm{m}$ laser field strength at optimum FODO cell length. In comparison to the synchronous design the minimum of ${\widehat{\beta }}_{\max }$ at ${\lambda }_{\mathrm{und}}\approx 0.55\mathrm{mm}$ is larger, but the undulator parameter $K_{\mathrm{und}}^{\text{async}}$ does not vanish.

Figure 9

Comparison of the optimum ${\widehat{\beta }}_{\max }$ for $\alpha =27°$, the radiation wavelength, and the radiation peak power for both the synchronous and the asynchronous DLA undulator design at $E_0=10\mathrm{GV}/\mathrm{m}$ drive-laser field amplitude.

Figure 10

dlatrack6d simulations of the reference trajectory in (a) and transverse rms size in (b) of an electron beam with ${ϵ}_y=25\mathrm{pm}$ and ${ϵ}_x={ϵ}_z=0$ emittance for three different setups of the synchronous DLA undulator lattice. The setup for ${\phi }_s=90°$ synchronous phase with ${\lambda }_{\mathrm{und}}=491\mu \mathrm{m}$ provides maximum focusing but no transverse deflection. The other setups compare the beam dynamics for ${\phi }_s=38.2°$ in two lattice designs with different undulator wavelengths ${\lambda }_{\mathrm{und}}$.

Figure 11

Fourier analysis of the reference trajectory $x^{\prime }(z)$ in order to determine the contributing oscillation wavelengths ${\lambda }_{\mathrm{und}}$ and the effective undulator parameter $K_{\mathrm{und}}$ for the synchronous DLA undulator lattices.

Figure 12

Poincaré plots for the three synchronous DLA undulator lattices based on the phase space coordinates $(y,y^{\prime })$ of particles with ${ϵ}_y=25\mathrm{pm}$ single-particle emittance evaluated after each FODO period ${\lambda }_{\mathrm{und}}$.

Figure 13

dlatrack6d simulations of the reference trajectory in (a) and transverse rms. size in (b) of an electron beam with ${ϵ}_y=25\mathrm{pm}$ and ${ϵ}_x={ϵ}_z=0$ emittance for two operation modes of the asynchronous DLA undulator with linear laser field ramp-up shown in (c). The astra [43] simulations show the results of a fully numeric, time-dependent beam dynamics model calculated with particle tracking in a 3D field map of the DLA cell from cst studio [27].

Figure 14

dlatrack6d simulation of the ${\lambda }_{\mathrm{und}}=546\mu \mathrm{m}$ asynchronous DLA undulator with realistic ARES injection parameters. The plots show the transverse momentum vs the relative phase [cf. (14)] at two different $z$ positions.

Figure 15

Comparison of horizontal phase space from dlatrack6d and astra at $z\approx 9.3\mathrm{mm}$ in the asynchronous DLA undulator lattice for ${\lambda }_{\mathrm{und}}=546\mu \mathrm{m}$.

Figure 16

Comparison of vertical phase space from dlatrack6d and astra at $z\approx 9.3\mathrm{mm}$ in the asynchronous DLA undulator lattice for ${\lambda }_{\mathrm{und}}=546\mu \mathrm{m}$.

Figure 17

3D field map of the laser distribution in the beam channel of a tilted silica DLA grating unit cell. The simulations conducted with the particle tracking code astra [43] use a periodic extension in order to model the field distribution of the full DLA lattice.