  Achieving large bandwidth at strong mode conversion strengths is a central
problem in few-mode optical waveguides. The most uncomplicated modification of
a standard mode-converting grating is to incorporate a single extra gap. Such
simple design has not been applied yet for broadening conversion spectrum
between core modes of any few-mode waveguide e.g. few-mode fiber, possibly due
to lacking knowledge of critical parameter control: (a) phase shift value at
resonance wavelength, and (b) deterministic method for placement of the phase
shift(s) accommodating the change in waveguide properties during grating
writing. From semi-analytical considerations, we show in full generality that
our design of a single-phase-shifted grating increases the bandwidth at 99 %
conversion strength by more than 2.9 times, together with elucidating crucial
parameter tolerances. The phase-shift placement is determinable on-the-go from
the conversion spectra, thereby obviating any estimation of modified waveguide
properties during grating-writing. This offers the longed on-demand in-fiber
solution for dispersion control in fiber laser cavities and communication
systems.
