An optical element is disclosed, for receiving relatively narrow light from a planar light-emitting diode (LED) source, and for redistributing the light into a relatively wide range of output angles that span a full 360 degrees. The element may be used to retrofit existing fixtures that were originally designed for incandescent bulbs with LED-based light sources that have similar emission profiles. The element is small enough so that it may be packaged along with an LED module and its control electronics in the volume envelope of an incandescent light bulb. An exemplary element is a single, transparent, rotationally-symmetric lens that has a batwing shape in cross-section, extending angularly away from a longitudinal axis. The lens also includes a variety of curved, straight, specular and optionally diffuse portions on its longitudinal and transverse faces, all of which cause a variety of internal and external reflections, refractions, and optionally scattering. As such, many of the specific lens features cannot be directly linked to specific optical effects at a particular angle; rather, the features all interact with each other to produce the wide-angle light output.