Beam dynamics analysis of dielectric laser acceleration using a fast 6D tracking scheme

A six-dimensional symplectic tracking approach exploiting the periodicity properties of dielectric laser acceleration (DLA) gratings is presented. The longitudinal kick is obtained from the spatial Fourier harmonics of the laser field within the structure, and the transverse kicks are obtained using the Panofsky-Wenzel theorem. Additionally to the usual, strictly longitudinally periodic gratings, our approach is also applicable to periodicity chirped (subrelativistic) and tilted (deflection) gratings. In the limit of small kicks and short periods we obtain the 6D Hamiltonian, which allows, for example, to obtain matched beam distributions in DLAs. The scheme is applied to beam and grating parameters similar to recently performed experiments. The paper concludes with an outlook to laser based focusing schemes, which are promising to overcome fundamental interaction length limitations, in order to build an entire microchip-sized laser driven accelerator.

Figure 1

Example Bragg cavity grating with a Bragg mirror on one side.

Figure 2

Tilted Bragg cavity grating, where the dual drive laser comes from top and bottom and is polarized in the electron beam direction (left to right).

Figure 3

One period of a symmetric Bragg mirror cavity structure.

Figure 4

Longitudinal electric field on the beam axis for the Bragg mirror structure in Fig. 3 and spatial Fourier harmonics for incident laser field normalized to $1\mathrm{V}/\mathrm{m}$.

Figure 5

Tilted grating with periodic boundary conditions in $x$ and $z$ direction and $\tan \alpha ={\lambda }_{\mathrm{gz}}/{\lambda }_{\mathrm{gx}}=v/u$.

Figure 6

Single cell of the tilted grating deflector structure and enlargement of kick integration curves array. The integrated kicks are displayed in Fig. 7.

Figure 7

Contour lines of ${\underset{\_}{e}}_1(x,y)$ and kick field ${\underset{\_}{\overrightarrow{f}}}_1(x,y)$ for the tilted grating with $\alpha =30\mathrm{deg}$. The fields have been obtained both analytically and numerically, the bottom plots show the relative difference.

Figure 8

The potential (Eq. (36) as function of $\phi$ and $y$ at the synchronous phase indicated by the solid vertical line. The longitudinally unstable fixed point (dashed line) flips the sign of the transverse potential. The color scale indicates the adiabatic change of the potential with $\beta$.

Figure 9

Overview of electron acceleration and focusing properties for an x-invariant grating.

Figure 10

Longitudinal emittance evolution for a linearly matched Gaussian beam in linearized fields, with the full fields, and with 10% excess energy spread.

Figure 11

Acceleration ramp according to Eqs. (56) and (58).

Figure 12

Longitudinal (bottom) and transverse (top) phase space for 16 cells of the chirped grating with an initially unbunched beam, focused into the structure. The axes are $-100\mathrm{nm}<y<100\mathrm{nm}$ and $0<\phi <2\pi$. The color represents the phase space density, where the initial plots are normalized to their maximum and all other plots are normalized to the respective maxima of the second column.

Figure 13

Energy spectra and particle survival rate for the unbunched beam.

Figure 14

Longitudinal phase space evolution for zero transverse emittance. The vertical axis is $0<\phi <2\pi$.

Figure 15

Longitudinal (bottom) and transverse (top) phase space for 16 cells of the chirped grating with the beam initially focused into the structure. The axes are $-100\mathrm{nm}<y<100\mathrm{nm}$ and $1.5<\phi <2.6$. Again, the color represents the phase space density, normalized to the maximum of the second column.

Figure 16

Energy spectra and particle survival rate for a short low energy bunch.

Figure 17

Evolution of the y and z phase spaces. The axes are $-400\mathrm{nm}<y<400\mathrm{nm}$ and $0<\phi <2\pi$. Again, the color represents the phase space density, normalized to the second column.

Figure 18

Energy spectra with 1 nm (solid lines) and zero (dashed lines) transverse emittance normalized to the initial number of particles $10^6$ (left) and particle survival rate (right).

Figure 19

Horizontal, vertical, and longitudinal phase space projection for every tenth cell. The axes are $-1.5\mu \mathrm{m}<x<1.5\mu \mathrm{m}$, $-0.4\mu \mathrm{m}<y<0.4\mu \mathrm{m}$ and $1.3<\phi <1.8$. Again, the color represents the phase space density, normalized to the second column.

Figure 20

Evolution of the energy spectrum and particle loss.

Figure 21

Horizontal and longitudinal phase space for every fortieth cell for ${ϵ}_y=0$. The axes are $-1.5\mu \mathrm{m}<x<1.5\mu \mathrm{m}$ and $1.3<\phi <1.8$. Two periods of transverse oscillations are visible [cf. Eq. (54)].