Web edition: September 12, 2008
Sounds preposterous, but a pair of physicists has shown that light can do just this — at least in theory. Visible light, along with all other forms of electromagnetic radiation, is governed by Maxwell’s equations, and the researchers have found a new solution to these equations in which light forms linked knots. The team is now working to create light in this form experimentally.
“This is very exciting,” says Antti Niemi of
The story begins with a mathematical discovery in 1931. Heinz Hopf found a way of filling up all of space with circles. (More precisely, he made a map from the analogue of a sphere in four dimensions to the circle.) He started with a donut shape, which mathematicians call a torus. He imagined taking a piece of string and wrapping it smoothly around the torus so that the string passes through the “donut hole” once and around the outside once as well. Enough pieces of string placed alongside this first one could cover the entire surface of the torus.
Now he just had to fill all of space with tori. He packed them like Russian dolls, extending forever both inward and outward from the starting torus. The smallest torus would be so skinny that it would simply be a circle. The biggest torus would be so fat that the “donut hole” on the torus wouldn’t be a hole at all — it would form a line extending up so far that its two ends would meet only “at infinity.” By filling space with tori and covering tori with circles, Hopf put every point in space on some circle.
Mathematicians were excited about Hopf’s discovery (called
the “Hopf fibration”) because it showed that high-dimensional spheres were more
complex than imagined. But it wasn’t until 20 years ago that physicists
realized the Hopf fibration had implications for electromagnetism: Antonio
William Irvine of New York University and Dirk Bouwmeester of the University of
California, Santa Barbara stumbled across Rañada’s work around 10 years ago and realized that it might describe a form
light could actually take. “The main thing we did is that we took this solution
In special situations, however, the loops might be stable, such
as if light travels through plasma instead of through free space. One of the
problems that has plagued experimental nuclear fusion reactors is that the
plasma at the heart of them moves faster and faster and tends to escape. That
motion can be controlled with magnetic fields, but current methods to generate
those fields still don’t do the job. If
Rañada, whose work Bouwmeester and
Irvine, W.T.M. and Bouwmeester, D. 2008. Linked and knotted beams of light. Nature Physics 4 (Sept.): 716 – 720.
Dimensions Math Videos: www.dimensions-math.org.