Design method for <i>C</i> ² -continuous tiled freeform lens arrays

This work presents a design method for creating second-order smooth (<i>C</i>²-continuous) freeform lens arrays. In particular, it introduces a design method of a single lens tile that can be seamlessly tiled to form a globally smooth array. Conventional freeform lens arrays rely on sharp transitions or discontinuities between tiles, which correspond to infinite curvatures that cannot be perfectly manufactured and are therefore a source of stray light. The proposed algorithm demonstrates that by introducing saddle-shaped lenses between convex and concave elements, a <i>C</i>²-continuous surface can be achieved while maintaining control over the target irradiance pattern. Because the surfaces contain no kinks or discontinuities, they minimize glare and stray light in illumination applications and are considerably easier to manufacture.