  Dispersion-free ultra-high order FFT-based Maxwell solvers have recently
proven to be paramount to a large range of applications, including the
high-fidelity modeling of high-intensity laser-matter interactions with
Particle-In-Cell (PIC) codes. To enable a massively parallel scaling of these
solvers, a novel parallelization technique was recently proposed, which
consists in splitting the simulation domain into several processor sub-domains,
with guard regions appended at each sub-domain boundaries. Maxwell's equations
are advanced independently on each sub-domain using local shared-memory FFTs
(instead of a single distributed global FFT). This implies small truncation
errors at sub-domain boundaries, the amplitude of which depends on guard
regions sizes and order of the Maxwell solver. For a moderate number of guard
cells (<10) , this 'local' technique proved to be highly scalable on up to a
million cores. Yet, depending on the targeted applications, the number of guard
cells required to mitigate truncations errors might be very large, which would
severely limit the parallel efficiency of this technique due to the large
volume of guard cells to be exchanged between sub-domains. In this context, we
propose a novel 'hybrid' parallelization technique that ensures very good
scaling of FFT-based solvers with an arbitrarily high number of guard cells. It
consists in performing distributed FFTs on local groups of processors with
guard regions now appended to boundaries of each group of processors. A dual
domain decomposition method is used for the Maxwell solver and other parts of
the PIC cycle to keep the simulation load-balanced. This 'hybrid' technique was
implemented in the open source exascale library PICSAR. Benchmarks show that
for a large number of guard cells ($>16$), the 'hybrid' technique offers a
$\times 3$ speed-up and $\times 8$ memory savings compared to the 'local' one.
