Title: Dimensional Crossover of Microscopic Magnetic Metasurfaces for Magnetic Field Amplification

Abstract: Transformation optics applied to low frequency magnetic systems has been recently implemented to design magnetic field concentrators and cloaks with superior performance. Although this achievement has been amply demonstrated theoretically and experimentally in bulk 3D macrostructures, the performance of these devices at low dimensions remains an open question. In this work, we numerically investigate the non-monotonic evolution of the gain of a magnetic metamaterial field concentrator as the axial dimension is progressively shrunk. In particular, we show that in planar structures the role played by the diamagnetic components becomes negligible, whereas the paramagnetic elements increase their magnetic field channeling efficiency. This is further demonstrated experimentally by tracking the gain of superconductor-ferromagnet concentrators through the superconducting transition. Interestingly, for thicknesses where the diamagnetic petals play an important role for the concentration gain, they also help to reduce the stray field of the concentrator, thus limiting the perturbation of the external field (invisibility). Our findings establish a roadmap and set clear geometrical limits for designing low dimensional magnetic field concentrators.