Pell–Lucas diffractive lenses for trifocal multi-trap optical tweezers

In this work, we introduce a new family of trifocal diffractive lenses based on the Pell–Lucas number sequence. Two phase configurations are considered: binary-phase and kinoform-type. Numerical simulations of the irradiance distributions show that both designs generate three well-defined focal points along the optical axis with identical spatial positions. The kinoform-type lens, however, demonstrates significantly higher energy efficiency by concentrating nearly all the diffracted energy into the first diffraction order of the Pell–Lucas lens. This represents the first reports of trifocal diffractive lenses based on an aperiodic mathematical sequence. The intensity profile is characterized by a dominant central focus accompanied by two additional lower-intensity focal positions, whose axial arrangement follows the characteristic behavior imposed by the Pell–Lucas sequence and exhibits a direct relationship with the silver ratio. Experimental validation of the kinoform Pell–Lucas zone plate shows good agreement with numerical predictions. Furthermore, the proposed lenses are implemented in a multi-trap optical tweezers configuration, enabling simultaneous particles manipulation at three distinct axial planes.