  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
(sub-relativistic) 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.
