Quadrifocal diffractive lenses based on aperiodically structured Lucas sequences

In this work, we present a new family of aperiodic diffractive lenses based on the Lucas sequence, implemented in both binary-phase and kinoform configurations. These lenses generate four focal planes along the optical axis with a partially balanced energy distribution, offering enhanced control over axial light distribution. A comparative analysis reveals that the kinoform design provides higher irradiance at the principal focal planes, while the axial positions between focal planes are directly related to the golden ratio, reflecting the intrinsic properties of the Lucas sequence and its close relationship with the Fibonacci sequence. The proposed lenses show strong potential for applications in optical trapping, where multiple particles can be confined simultaneously, and in multifocal imaging systems, where parallel acquisition can reduce scanning time and maintain uniform resolution across planes, as well as in quadrifocal intraocular lenses. This study demonstrates the versatility of Lucas-sequence-based diffractive lenses for advanced optical and laser-based technologies.