Using fractional vortex beams to explore the mathematics of infinity

The mathematics of infinity can seem paradoxical and counterintuitive, but they have important real-world applications. In the early 20th century, the mathematician David Hilbert proposed a thought experiment to illustrate the bizarre nature of countable infinities. He imagined a hotel with infinite guests and infinite numbered rooms. While each room is initially occupied, this infinite hotel can always accommodate a new guest by shifting its existing occupants to the next highest room.

Kumar et al. achieved a real-world demonstration of Hilbert’s Hotel, recreating the outcome using a fractional order optical vortex beam in both the phase and polarization singularities.

In a vortex beam, the net topological charge’s value is determined by the number of positive and negative vortices. These vortices have integer charges and can only be created in pairs, leaving no easy means of shifting from one net charge value to another.

Here is where Hilbert’s Hotel appears. Fractional vortex beams change the fractional order between two integer values. As this occurs, vortex pairs are created until the half-integer point, where an infinite number of them exist. The positive and negative vortices then act like guests and rooms in Hilbert’s infinite hotel, as they shift places to create vacancies.

“When an infinite number of positive and negative vortices are present, the topological charge of the beam is no longer well-defined and can take on any value,” said author Anirban Ghosh. “This ambiguity apparently allows the beam to create a new topological charge out of nothing.”

In addition to providing a practical verification of a mathematical truth, the experiment also provides a way to further understand fractional vortex and vector beams and employ them in practical applications.

Source: “Simple experimental realization of optical Hilbert Hotel using scalar and vector fractional vortex beams,” by Subith Kumar, Anirban Ghosh, Chahat Kaushik, Arash Shiri, Greg Gbur, Sudhir Sharma, and G. K. Samanta, APL Photonics (2023). The article can be accessed at https://doi.org/10.1063/5.0150952 .