  With the advent of PW class lasers, the very large laser intensities
attainable on-target should enable the production of intense high order Doppler
harmonics from relativistic laser-plasma mirrors interactions. At present, the
modeling of these harmonics with Particle-In-Cell (PIC) codes is extremely
challenging as it implies an accurate description of tens of harmonic orders on
a a broad range of angles. In particular, we show here that standard Finite
Difference Time Domain (FDTD) Maxwell solvers used in most PIC codes partly
fail to model Doppler harmonic generation because they induce numerical
dispersion of electromagnetic waves in vacuum which is responsible for a
spurious angular deviation of harmonic beams. This effect was extensively
studied and a simple toy-model based on Snell-Descartes law was developed that
allows us to finely predict the angular deviation of harmonics depending on the
spatio-temporal resolution and the Maxwell solver used in the simulations. Our
model demonstrates that the mitigation of this numerical artifact with FDTD
solvers mandates very high spatio-temporal resolution preventing doing
realistic 3D simulations. We finally show that non dispersive pseudo-spectral
analytical time domain solvers can considerably reduce the spatio-temporal
resolution required to mitigate this spurious deviation and should enable in
the the near future 3D accurate modeling on supercomputers in a realistic
time-to-solution.
