Many proposals
for using inverse Compton scattering to produce high brightness, short
wavelength radiation require that one simultaneously obtain both large
numbers of scattered photons, and that these photons have a nearly monochromatic
energy spectrum. Utilizing a straightforward physical model, we derive
the constraints on scattered photon number due bandwidth limits arising
from: angular spread in the laser beam, Fourier transform limits of
the laser pulse, lowest order laser intensity effects, and angular acceptance.
It is shown that, for bandwidths at the percent level, it is difficult
to obtain more than one scattered photon per incident electron. Implications
of this result on ultra-fast sources are discussed, as are ways of mitigating
the needed laser energy and further raising the photon by use of laser-plasma
guiding.