  Interest in the nonlinear properties of multi-mode optical waveguides has
seen a recent resurgence on account of the large dimensionality afforded by the
platform. However, a perceived fundamental limitation of intermodal parametric
interactions - that they are impractically narrowband - has yet to be solved.
Here we show that by engineering the relative group velocity within the
discrete spatial degree of freedom, we can tailor the phase matching bandwidth
of intermodal parametric nonlinearities. We demonstrate group-velocity-tailored
four-wave mixing between the $LP_{0,4}$ and $LP_{0,5}$ modes of a multi-mode
fiber with unprecedented gain bandwidths (>60 nm at ~1550 nm). As evidence of
the technological utility of this methodology, we seed this process to generate
a high-peak-power wavelength-tunable fiber laser in the Ti:Sapphire wavelength
regime. More generally, with the combination of intermodal interactions, which
dramatically expand the phase matching degrees of freedom for nonlinear optics,
and intermodal group velocity engineering, which enables tailoring the
bandwidth of such interactions, we showcase a platform for nonlinear optics
that can be broadband while being wavelength agnostic.
