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Historically, infinity was long considered a vague concept - boundless,endless, larger than the largest - without any quantifiable mathematicalfoundation. This view changed in the 1800s through the pioneering work of GeorgCantor showing that infinite sets follow their own seemingly paradoxicalmathematical rules. In 1924, David Hilbert highlighted the strangeness ofinfinity through a thought experiment now referred to as the Hilbert Hotelparadox, or simply Hilbert's Hotel. The paradox describes an "fully" occupiedimaginary hotel having infinite number of single-occupancy rooms, the managercan always find a room for new guest by simply shifting current guests to thenext highest room, leaving first room vacant. The investigation of wavefieldsingularities has uncovered the existence of a direct optical analogy toHilbert's thought experiment. Since then, efforts have been made to investigatethe properties of Hilbert's Hotel by controlling the dynamics of phasesingularities in``fractional'' order optical vortex beams. Here, we have takensuch proposals to the next level and experimentally demonstrated Hilbert'sHotel using both phase and polarization singularities of optical fields. Usinga multi-ramped spiral-phase-plate and a supercontinuum source, we generated andcontrolled fractional order vortex beams for the practical implementation ofHilbert's Hotel in scalar and vector vortex beams. Using a multi-rampedspiral-phase-plate, we show the possibility for complicated transitions of thegeneralized Hilbert's Hotel. The generic experimental scheme illustrates theusefulness of structured beams in visualizing unusual mathematical concepts andalso for fractional vector beams driven fundamental and applied research.

Simple experimental realization of optical Hilbert Hotel using scalar and vector fractional vortex beams