  We explore the regime of operation of the modulator stage of a recently
proposed laser-plasma accelerator scheme [Phys. Rev. Lett. 127, 184801 (2021)],
dubbed the Plasma-Modulated Plasma Accelerator (P-MoPA). The P-MoPA scheme
offers a potential route to high-repetition-rate, GeV-scale plasma accelerators
driven by picosecond-duration laser pulses from, for example, kilohertz
thin-disk lasers. The first stage of the P-MoPA scheme is a plasma modulator in
which a long, high-energy 'drive' pulse is spectrally modulated by
co-propagating in a plasma channel with the low-amplitude plasma wave driven by
a short, low-energy 'seed' pulse. The spectrally modulated drive pulse is
converted to a train of short pulses, by introducing dispersion, which can
resonantly drive a large wakefield in a subsequent accelerator stage with the
same on-axis plasma density as the modulator. In this paper we derive the 3D
analytic theory for the evolution of the drive pulse in the plasma modulator
and show that the spectral modulation is independent of transverse coordinate,
which is ideal for compression into a pulse train. We then identify a
transverse mode instability (TMI), similar to the TMI observed in optical fiber
lasers, which sets limits on the energy of the drive pulse for a given set of
laser-plasma parameters. We compare this analytic theory with particle-in-cell
(PIC) simulations and find that even higher energy drive pulses can be
modulated than those demonstrated in the original proposal.
