In a LMA optical fiber the index of the core region is graded (i.e., as viewed in a radial cross-section) and has a grading depth of Δng, as measured from a central maximum at or near the axis to a lower level that is not greater than the central maximum and not less than the index of the cladding region. When the fiber is to be bent at a bend radius, the grading depth, the radius of the core region, and the difference between the central maximum index and the cladding region index are configured to reduce bend distortion. They may also advantageously be configured to maximize the effective mode-field area of the fundamental mode, suppress higher order modes, and reduce bend loss. In a preferred embodiment, the core region includes a centralized gain region, which in turn includes a dark region that is no more than 30% of the area of the gain region. Also described is a method of making such LMA fibers.