Quasi-Bragg geometric-phase lens for high-numerical-aperture focusing

Geometric-phase lenses (GPLs) enable ultrathin wavefront shaping but suffer from a fundamental limitation in numerical aperture (NA) imposed by the Raman-Nath diffraction regime. In this work, we introduce a quasi-Bragg geometric-phase lens (QB-GPL) that overcomes this limitation by spatially engineering the diffraction regime within a single device. The proposed approach combines Raman-Nath-type diffraction near the lens center with Bragg-like diffraction in the outer aperture region, enabled by a three-dimensional liquid-crystal director distribution formed via a contact-transfer exposure method. By combining a GPL master with an isotropic plano-convex lens, a three-dimensional polarization field with a finite out-of-plane tilt is generated, allowing the formation of an inclined grating vector required for Bragg diffraction. Finite-difference time-domain simulations predict near-diffraction-limited focusing, and experimental results confirm high-NA operation and imaging functionality. Importantly, the high-NA performance demonstrated here originates not only from the numerical aperture value itself, but from the ability to maintain efficient diffraction under normal incidence even in the Bragg-dominant outer region. These results establish diffraction-regime engineering as a powerful design principle for high-performance flat optics.