We consider the acceleration of charged particles near ultra-relativistic shocks , with Lorentz factor \Gamma _ { s } \gg 1 .
We present simulations of the acceleration process and compare these with results from semi-analytical calculations .
We show that the spectrum that results from acceleration near ultra-relativistic shocks is a power law , N ( E ) \propto E ^ { - s } , with a nearly universal value s \approx 2.2 - 2.3 for the slope of this power law .

We confirm that the ultra-relativistic equivalent of Fermi acceleration at a shock differs from its non-relativistic counterpart by the occurence of large anisotropies in the distribution of the accelerated particles near the shock .
In the rest frame of the upstream fluid , particles can only outrun the shock when their direction of motion lies within a small loss cone of opening angle \theta _ { c } \approx \Gamma _ { s } ^ { -1 } around the shock normal .

We also show that all physically plausible deflection or scattering mechanisms can change the upstream flight direction of relativistic particles originating from downstream by only a small amount : \Delta \theta \sim \Gamma _ { s } ^ { -1 } .
This limits the energy change per shock crossing cycle to \Delta E \sim E , except for the first cycle where particles originate upstream .
In that case the upstream energy is boosted by a factor \sim \Gamma _ { s } ^ { 2 } for those particles that are scattered back across the shock into the upstream region .