Progress during the last decade in wall-plug to light efficiency of lasers makes them serious competitors to current microwave driven accelerators for high energy physics as well as for medical applications. One of the aspects which makes the optical system significantly different comparing to microwave machines is the trapping condition. 

For realistic gradients (E0=1-10GV/m), the typical value of the normalized longitudinal field $a=eE0 \lambda /mc^2$ is much smaller than unity, as compared with the case of a conventional RF photo-injector ($\lambda$=10 cm), where typically a=~1. Consider as an example the case of  ($\lambda$=1 um) and zero initial energy, the required gradient to capture sub-relativistic particles in a uniform structure is of the order of 3 TV/m. For comparison, in the microwave regime the trapping gradient is ~20MV/m, for which, evidently, the electrons become relativistic within a few wavelengths. Consequently, in the optical regime, there is a need for a tapered structure to maintain local phase synchronicity with the particles.

In this work, we present a longitudinal tapering of the structure to maximize the trapping efficiency. We further solve the dynamics of the trapping process with\without longitudinal space-charge effects for both the resonant particle and a distribution of particles.  Throughout our analysis we assume that the vacuum tunnel's radius (internal radius) is constant and only single mode operation is considered. Also, since we are interested not only on the effect of the field on the electrons but also vice-versa, it is assumed that the relation between interaction impedance and the phase-velocity is known and is determined locally (adiabatic tapering).

We investigate the resonant particle dynamics, and analyze the effect of the initial phase and energy distribution on the trapped electrons for a distribution of particles. We suggest a few different trapping criteria, together with several types of phase and energy distributions. The figure below describes two potential types of tapered structures, each keeping either the external or the internal radius uniform.