  In nonlinear Thomson scattering, a relativistic electron reflects and
re-radiates the photons of a laser pulse, converting optical light to x rays or
beyond. While this extreme frequency conversion offers a promising source for
probing high-energy-density materials and driving uncharted regimes of
nonlinear quantum electrodynamics, conventional nonlinear Thomson scattering
has inherent tradeoffs in its scaling with laser intensity. Here we discover
that the ponderomotive control afforded by spatiotemporal pulse shaping enables
novel regimes of nonlinear Thomson scattering that substantially enhance the
scaling of the radiated power, emission angle, and frequency with laser
intensity. By appropriately setting the velocity of the intensity peak, a
spatiotemporally shaped pulse can increase the power radiated by orders of
magnitude. The enhanced scaling with laser intensity allows for operation at
significantly lower electron energies and can eliminate the need for a
high-energy electron accelerator.
