Skulls in the Stars

Some months ago, I wrote a post introducing the subfield of optics known as singular optics.  Singular optics is concerned with the behavior of wavefields in the neighborhood of regions where the intensity of the wave is zero, and the “phase” of the wave is therefore singular.  The zeros typically take the form of lines in three-dimensional space, and surfaces of constant phase often form a spiral around this line, and circulate around it as time evolves:

This circulation of the phase has led to such structures being known as “optical vortices”.  It can be shown that such vortices are stable features of a wavefield; that is, they are resistant to distortions of the wave induced by focusing, propagation through atmospheric turbulence, etc.

One of the fascinating aspects of the development of singular optics is that it provides a different perspective on optical waves.  Instead of considering light as an extended field “flowing” through space, singular optics allows us to view it as a topological structure, and to characterize any field by its structure.  What that means, roughly, is that we can in a sense talk about the “shape” of a wavefield, and look at the possibility of creating wavefields of unusual shape.

This was done for optical vortices in 2001, when Dennis and Berry demonstrated¹ theoretically that the zero lines of optical vortices can be produced in the form of knots or links in a wavefield. Not long after, Dennis showed that zero lines could also be braided in the form of a pigtail braid².

More recently, other authors have started considering what other sorts of topological features might be achievable in wavefields.  Topology is a branch of mathematics that is concerned with what properties of an object are preserved under distortions that don’t include tearing or gluing of the object.  A typical way to highlight this is to note that, in topology, a sphere and a cube are in a sense equivalent shapes:

If we imagine the cube to be fashioned of clay, we can squish and shape the clay to make a cube without tearing or gluing the clay at any point.  Similarly, a coffee cup and a torus (donut) are equivalent:

Both the cup and the donut have a single hole in them, and one can be deformed into the other preserving the hole.  However, a sphere and a donut are not topologically equivalent objects, because one must tear a hole in a sphere to make the donut hole or one must glue shut a hole in a donut to make a solid sphere.  Some shapes are fundamentally different from one another, in that they must be ripped in order to be made to match.

The archetypical example of this fundamental difference is a one-sided surface, known as a Möbius strip (picture from Wikipedia):

Whereas “ordinary” surfaces have two sides, like a sheet of paper, a Möbius strip has only one side: if you start on one side of the strip and follow a path along it, you will eventually find yourself on the other side of the strip!  Recently, Möbius strips were in the news, as a nanoscale strip was constructed by researchers out of DNA.

With the advent of singular optics and its emphasis on the structural properties of wavefields, it was perhaps inevitable for someone to investigate whether it is possible to make Möbius strips in optics.  It turns that that it is possible, but one must take advantage of a different sort of optical singularity of a wavefield, known as a polarization singularity.

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