In Hilbert’s hotel paradox, even though each of an infinite number of rooms may be occupied (top), a new guest can always be accommodated by moving all existing guests up by one room (bottom). [Image: Greg Gbur] [Enlarge image]
The classical optics of continuous, finite beams is not a place one would commonly go looking to find the mathematics of infinite, countable sets. But Greg Gbur, a specialist in theoretical optics at the University of North Carolina, Charlotte (USA), has shown a surprising realization of one celebrated bit of transfinite math—David Hilbert’s “hotel paradox”—in a completely classical light field (Optica, doi: 10.1364/OPTICA.3.000222).
The strange world of Hilbert’s Hotel
Hilbert first framed the hotel paradox in a 1924 lecture, in which he asked his audience to envision a hotel with a “countably infinite” number of rooms—essentially, discrete, natural-number-like elements that can be counted one by one, even though the counting itself might take infinite time. Each room is occupied, making the entire hotel filled. Yet a new guest can always be accommodated by moving all existing guests to the next-highest-numbered room, and putting the new guest in the first room. In Hilbert’s Hotel, it’s always “No Vacancy”—yet management can always find you a room.
Hilbert offered the paradox as an entry into the strange world of transfinite numbers and infinite set theory, in which statements such as ℵ0 + 1 = ℵ0 and even ℵ0 + N = ℵ0 become provably true. Since then, researchers in quantum electrodynamics and quantum optics have demonstrated systems that realize Hilbert’s Hotel in countable parameters such as quantum number.
The case of optical vortices
In an optical vortex beam, the wavefronts form a helical structure, with the relative phase varying continuously around the central propagation axis. The axis itself is a zone of singularity, where the intensity is zero and the phase is undefined. [Image: Greg Gbur]
Could a classical light field also manifest Hilbert’s Hotel behavior? Gbur found that the answer is yes—and, he says, he “sort of accidentally stumbled on the connection” while doing some unrelated work on an unusual phenomenon called optical vortices. These are singularities in a light field that consist of areas of zero intensity in which phase is undefined, and around which the light propagates in a helical or corkscrew beam, with the helicity commonly, though not always, associated with orbital angular momentum (OAM).
Vortices are characterized in particular by a quantity called “topological charge”—the integer multiple of 2π that gives the phase increase or decrease of the light as it twists in a closed circuit around the vortex singularity. Like OAM, topological charge in a vortex beam is a conserved quantity. What this means is that new vortices in an existing beam must be created or destroyed in pairs, with one positively charged vortex offset by an equally negatively charged vortex. Gbur likens the situation to the well-known case of pair production, in which a particle is created simultaneously with an offsetting antiparticle.
In his research on vortices, Gbur came across some theoretical work from 2004 by M.V. Barry, suggesting that beams could jump from one topological charge to another in certain circumstances—but that the jump would be highly discontinuous, and would occur at the precise midpoint between two integer phase offsets (i.e., two topological-charge values). The jump could be made, Barry suggested, through the creation of “an infinite chain of alternating-strength vortices” close to the vortex axis, which would annihilate each other—leaving a single new vortex behind, and raising the system’s topological charge by one.
Vortex beams can be created by passing an optical beam through a spiral phase plate (top). But the changes are not continuous; instead, topological charge jumps in step-function fashion (bottom). What happens at the singularities? [Image: Greg Gbur]
Subsequent experimental work later that same year by Jonathan Leach, Eric Yao, and Miles Padgett confirmed that passing a beam through a half-integer spiral phase plate did indeed produce such a chain of optical vortex pairs.
Hilbert’s Hotel springs out of nowhere
The vortex system thus provided two key parameters—a countable value, and an infinite set—that, to Gbur, made the system ripe for analysis using the transfinite math of Hilbert’s Hotel.
To make the jump between two consecutive topological-charge values—between, say, 4 and 5—somehow a single new vortex must be created, Gbur explains. Yet ordinarily, vortices are created only in pairs to avoid violating conservation of topological charge. The system gets around this at the midpoint between the topological charge states, 4.5, by essentially creating Hilbert’s Hotel, in the form of an infinite chain of paired vortices. At that point, the beam’s topological charge becomes undefined, opening up a degree of freedom and allowing the otherwise forbidden creation of a single, unpaired vortex.
“Just like in Hilbert’s Hotel,” says Gbur, “you can free up as many hotel rooms—as many unpaired vortices—as you want.” In a sense, he says, “the system creates an infinite number of vortex pairs so it can cheat. It has this rule that it has to satisfy of conserving topological charge, and it gets around it by creating infinite pairs of plus and minus vortices, so nobody can actually say what the charge is.”
Surprising optical relevance
One of the key outcomes of this work, suggests Gbur in the paper, is simply that transfinite math can crop up in surprising and unexpected ways in optics—and that such math “may be hidden in even more optical systems,” especially those involving vortices. He also noted in an interview that the work could provide insight on some practical problems. For example, topological charge and OAM have been suggested as possible information carriers in optical communications. The insights gained from looking at these phenomena from the point of view of transfinite math, Gbur suggests, could in turn provide insight into more efficient or faster ways of changing the OAM state of a beam, for just such applications.