I’ve noticed there seems to be a general unspoken rule about the relationship between mathematics and science: any mathematics, no matter how abstract or seemingly disconnected from reality, eventually finds use or representation in the natural world. For example, most people are probably familiar with the idea of an “imaginary number,” namely the square root of -1. The term “imaginary” was coined by famed mathematician René Descartes, who really meant it as a derogatory term: imaginary numbers were thought to be useless artifacts of the imagination. Today, such numbers are a fundamental part of physics, used in every branch from quantum mechanics to optics to mechanics to describe the properties of physical systems. They are almost the opposite of useless.
Other examples abound. Number theory, which is the branch of mathematics devoted to the study of the integers, i.e. 1,2,3,4,…, and their relationships, would seem to be completely devoid of practical interest. However, it plays an important role in modern cryptography, helping to keep our data secure in the information age. Another example is the study of quaternions, objects which may be considered three-dimensional generalizations of imaginary numbers. These quantities were almost forgotten by the early 20th century but have become extremely useful in computer graphics and robotics, among other technologies.
Even with a knowledge that even the most abstract math can be of real-world relevance, I still can find myself caught of guard, even stunned, when such math peeks out from within a very physical problem. Today, I was working on an optics problem when I realized that the problem in question was a direct demonstration of the strangeness of infinite sets! The optics problem in question involves light beams with so-called optical vortices in them, something I’ve talked about on the blog before. And the property of infinite sets in question is the very strange booking practices of a hotel with infinite rooms!
I’ve talked a lot about the strangeness of infinity on this blog; see, for instance, my four-part series on the different “sizes” of infinity: part 1, part 2, part 3, part 4. Though we tend to think of “infinity” as a single concept, there are in fact an infinite number of infinities, each bigger than the one before. That is quite weird, but we can find plenty of weirdness even if we stick with the smallest of infinities, the aforementioned set of integers, or “countable numbers”: 1,2,3,4,5,6,…
A classic illustration of the weirdness of the countable numbers is given by Hilbert’s Grand Hotel. It was supposedly introduced by the famed mathematician David Hilbert in a lecture in 1924, but was popularized in George Gamow’s 1947 book One Two Three… Infinity.
We may describe it as follows. Let us suppose that there is a hotel which has a countably infinite number of rooms. In other words, there is one room for each of the countable numbers: Room 1, Room 2, Room 3, Room 4, and so on. It might look as pictured below.
Now let us suppose that there’s a convention on the mathematics of infinity in town and Hilbert’s Hotel has been completely booked — there is no vacancy in the hotel. However, another mathematician arrives late to town and needs a room: what is one to do?
Not a problem! Everyone simply moves to the next room up. The person in Room 1 moves to Room 2, the person in Room 2 moves to Room 3, and so forth. Because there are a countably infinite number of rooms, everyone gets a room, yet Room 1 is left vacant. The new arrival can move in. In fact, we can repeat this process indefinitely — there is always a vacancy at the Hilbert Hotel, no matter how many new guests arrive, even a countably infinite number of new guests.
This works due to the nature of countable infinities, as discussed by Cantor (and blogged about by me). We can’t count an infinite set of objects, so we measure the size of two infinite sets by establishing a one-to-one correspondence between the sets. We can even do this with finite sets, for instance in comparing the size of two piles of candy.
But if we add an element a to the set of countable numbers, we can still match every element of the new set to the original set of countable numbers: everything shifts over, but since the set is infinite, we can still make a one-to-one correspondence.
This means that, in essence, “infinity plus one” is equal to “infinity,” and the strangeness of this is illustrated by Hilbert’s Hotel. It is so strange, in fact, that one would not expect to find any phenomenon in physics that is analogous — would we?
With this not-so-subtle segue, let us turn to the phenomenon of vortices in optics. The study of such vortices and related phenomena is now its own subfield of optics known as singular optics. What, exactly, do I mean by a vortex of light? This requires a little explanation.
You are probably familiar with the idea that light has wave properties. Similar to water waves, a light wave consists of little up-and-down “wiggles” that propagate at the speed of light. The simplest sort of wave one can imagine is called a plane wave, as pictured below.
In this case, the wave is traveling to the left, and when the wave is “up,” it is up everywhere in the plane perpendicular to the direction of travel. To visualize the wave, we explicitly draw these surfaces and refer to them as the wavefronts.
A light wave with an optical vortex in it, however, has its wavefronts joined together to form a helical structure.
As the light wave propagates, the wavefronts effectively look like they are twisting around the central axis like a drill bit turns; this swirling motion is what causes us to dub them “optical vortices.”
We usually only look at vortices in a cross-section of a light wave; we then color this cross-section according to the relative phase, or position in the up-and-down of the wave. In a cross-section of the above “screw,” the phase would look as follows.
As we travel counterclockwise around the central axis, the phase increases continuously from 0° to 360°: at the central point, all the different phase values converge and we have what we call a phase singularity.
We can also, however, have higher-order vortices, in which the phase increases or decreases by some multiple of 360° as one travels around the central axis; that multiple, which is always an integer, is known as the topological charge. Some examples are shown below.
The choice of the word “charge” to describe the behavior of a vortex is not accidental. These vortices act very much like discrete particles, making them interesting objects of study and potentially useful for applications. For instance: the topological charge is a stable quantity, and is resistant to perturbations of the light wave containing it, such as propagation of light through atmospheric turbulence. In fact, since topological charge is conserved, vortices can in general only be created or destroyed in pairs of opposite sign, oddly analogous to the particle/anti-particle creation/annihilation in high-energy particle physics.
So how does one create beams of light with vortices in them? One of the easiest ways is to shine a non-vortex beam through a so-called spiral phase plate, a transparent piece of glass fashioned into a ramp, as illustrated below.
When light passes into the glass, it slows down. The part of the wavefronts passing through a thick portion of glass get slowed down more than those passing through a thinner portion. If the refractive index of the glass and the height of the ramp is chosen just right, one can force the transmitted light beam to have an integer topological charge: the output ends up being a pure vortex beam.
However, it actually takes careful planning to design a spiral phase plate to create a perfect vortex beam. What happens if it is designed to effectively make a fractional topological charge, say 3.5 or 4.1235? The topological charge is physically required to be an integer value, so we will not get a fractional vortex out of a fractional spiral phase plate. What happens?
This question caught the attention of a number of researchers in 2004. Among them was Professor Michael Berry, who co-founded the field of singular optics with John Nye in a 1974 paper. Berry constructed* a mathematical model to describe the behavior of a beam of light after being transmitted through a spiral phase plate, whether fractional or integer valued. What he found is that the topological charge of the transmitted beam “jumps” whenever the charge of the phase plate is of half-integer value, as depicted below.
This is somewhat peculiar, as we have already noted that vortices are created in pairs of opposite topological charge — this plot indicates that a single positive vortex appears, seemingly out of nowhere, as soon as the phase plate takes on a half-integer value.
So what is happening? Let’s actually look at the phase of the field when the phase plate transitions continuously from being charge 3 to being charge 4. The first two plots below show the field for plate charge 3 and charge 3.1.
I’ve left the plot labels in this time to show that we’re looking at a square region just to the right of the central axis; the reason for this will be clear in a moment. For plate charge 3.0, we have an ordinary vortex of topological charge 3; as the plate charge is increased, it breaks into three vortices of topological charge 1, but nothing else happens.
Now let’s increase the plate charge and get really close to 3.5. At 3.44, we find that a pair of vortices have been created close to the origin — they are labeled in purple — and what appears to be another pair have been created next to them, labeled with a red arrow. When the plate charge is increased to 3.46, we find that these vortices separate in space, and another pair are created next to them.
Here’s where it gets crazy. As we get arbitrarily close to the value 3.5, the number of vortex pairs increases rapidly; Berry has shown that, in fact, an infinite number of vortex pairs are created along that horizontal line. We can’t, of course, show you the entire infinite line of vortices, but here’s what the phase looks like at exactly 3.5.
The fact that we’re looking at an infinite, countable set of vortices, all lined up in a neat row, should already start making you think of Hilbert’s Hotel. But let’s go on and see what happens as we increase the plate charge further…
Now we find that pairs of vortices are re-annihilating, from the outside inward. In each picture, the red arrow indicates a pair of vortices that are about to annihilate.
Even at this point, though, we can see that the vortices are now unbalanced: there’s going to be one extra left over when all the annihilations complete. We see this below.
Finally, as the plate charge approaches 4, we will find that the four separate charge 1 vortices coalesce into a single charge 4 vortex.
But where did the extra charge come from? We note the following: the pairs of opposite vortices that were created when the plate charge < 3.5 are joined along a white-black interface. However, the pairs of opposite vortices that annihilate are ones on adjacent white-black interfaces. Without all the colors confusing things, the process looks as below.
By changing the order in which we annihilate the infinite set of charges that exist when the plate charge = 3.5, we are able to squeeze out a single extra positive charge.
Hopefully it is becoming clear how this connects to Hilbert’s Hotel, but to see it explicitly, let’s introduce a variant we can call “Hilbert’s roommate problem.” Let us assume that every room in the hotel is booked and has two people sharing. However, a single person in one of the rooms decides to leave. Can we still have all the rooms filled, and allow that person to leave? Sure! We just ask one of the roommates in each room to move one room to their left.
Though it may be awkward for all of the guests involved, we end up with a situation where every room still has two people, but one person has left the hotel.
This seems to be precisely what is happening at the critical juncture when the plate charge = 3.5: an infinite “hotel” of vortex pairs is created. Then, when the plate charge increases beyond 3.5, those charges are paired differently; each vortex gets a different “roommate” to annihilate with. To put it in simpler terms, this is equivalent to saying that “infinity minus one equals infinity.”
This trick, the creation of a “new” vortex from seemingly nothing, can only happen because of the existence of an infinite set of pairs. In essence, it appears that the transition of the light wave from topological charge 3 to topological charge 4 requires the peculiar properties of a countable infinity in order to work. This, quite frankly, leaves me somewhat flabbergasted: I never expected to see the mathematics of transfinite numbers play a role in any physical problem. As far as I know, it is the first physical system ever seen to exhibit such properties.
Hilbert’s Hotel gives us another insight into the peculiarity of the system when the plate charge = 3.5. Just as we can add or remove as many guests from the hotel as we like without making or breaking any vacancies, we can “pull out” as many positive or negative charges from the infinite set of pairs and still pair up every charge that remains. When the plate charge is 3.5, then, the topological charge of the field can be considered to be 3, 4, 5, 50, -1000, 1 billion, or whatever we want. The topological charge of the field is truly undefined in this state, and may be considered whatever we like. We have what I would call a singularity of topological charge, even though topological charge itself is a measure of the singularity of the light wave!
All of these observations are fine details, however; what I really hope has been made clear at this point is that we have now seen an optical realization of Hilbert’s Hotel!
I don’t know if Michael Berry made this connection when writing his paper; he doesn’t mention it, but he is undoubtedly clever enough to have made the connection himself. It is not necessary to understand the physics of his paper, and so he may just not have bothered to mention it; I’ll have to ask him if I get the chance.
There are, of course, physical caveats to be made about the system in question. It is not possible to see the infinite set of vortex pairs in any realistic experimental apparatus. For one thing, it appears that the infinity of vortices only appears when the plate charge is exactly 3.5, which could never be matched in practice. Furthermore, any physical light beam will be of finite transverse cross-section, which means that the infinite set will be lost in the darkness of the edge of the beam in a real experiment. This is not surprising; nature tends to hide the weird infinities that could potentially crop up in physical systems. I’ve talked before about how strange infinity mirrors can be, and how we can only approximately see the infinities hidden within them.
Berry’s overall model has, however, been confirmed in experiments**, which produced images of the phase similar to those theoretical plots shown above.
I should add a final caveat that there may be other ways to think about this optical system without relying on infinities and Hilbert’s Hotel; in my experience, whenever infinity appears in a physical problem, it plays an ambiguous role that can lead to multiple interpretations. But, as far as I can tell, the explanation given above is an acceptable one, and it shows again how seemingly “imaginary” mathematics can sneak its way into very realistic physical problems.
* M.V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6 (2004), 259-268.
** J. Leach and E. Yao and M.J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6 (2004), 71.