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Control of the group-velocity differences between two distinct modes in a few-mode fiber can be used to define the spectral characteristics of long-period gratings written in them. Using this effect, we report the demonstration of strong mode conversion >99% with long-period fiber gratings over what is believed to be a record bandwidth of 63 nm. These novel spectra are obtained from gratings written in specially designed few-mode fibers in which the grating phase-matching condition is satisfied over a large spectral range. We show that the bandwidths of such mode converters can be tailored by suitably altering the design of the few-mode fibers. The polarization-dependent coupling for the mode converters varies by less than 0.004% over the entire spectrum.