Angularly homogenizing gradient index optical fiber having a refractive index profile that is non-quadratic to a degree sufficient to enhance precession of light as it is propagated through the fiber. Deviation from the quadratic may be limited to avoid profoundly changing the radial boundary within the fiber. Beam asymmetry, for example, associated with small aperture sources launched into a fiber off axis, may be made more symmetric as the beam is propagated through the homogenizing gradient index optical fiber. A refractive index profile may be manufactured to avoid a pure quadratic profile, or a fiber having a refractive index profile that is quadratic in only some orientations about the fiber axis may be twisted during draw to induce a refractive index profile path that enhances propagation precession.