The ideas of topology, a branch of mathematics studying the preserved properties of objects under continuous deformations, have found tremendous success in closed physical systems. From a topological point of view, a coffee mug and a doughnut are the same. Since each object has one hole, it is possible for the doughnut to morph into a mug through continuous deformations: One would simply make a depression into one side of the doughnut to make the mug and shrink the doughnut's hole to make the mug's handle. This sort of abstraction gives way to new properties in physics and interesting applications in practice.
A new study led by University of Pennsylvania physicist Bo Zhen investigated topological phenomena in open, or non-Hermitian, physical systems which could potentially lead to novel methods in optical communications and better optical sensors. Optical communications use light to carry signals instead of electrical current and are a staple of many modern data centers. This research, published in Science, connects the fields of topological photonics, non-Hermitian physics, and singular optics, providing a framework to explore more complex non-Hermitian topological systems.
In classical mechanics, Zhen explained, there is a relationship between the energy and momentum in a system, called dispersion. One example of this is someone riding a bicycle. The momentum and kinetic energy, both associated with this person, follow the dispersion relationship. But if one were to ride a bike through many bumps or other obstacles, the dispersion relationship between momentum and energy would start to change. With this in mind, one might be able to engineer the arrangement of the bumps to tune this dispersion relationship.
The same is true for photons, particles of light. When transmitting in air, the energy of photons is linearly proportional to momentum. But in this research, Zhen and his collaborators showed that when light is sent through purely transparent objects, but with nanoscale periodic patterns in them, the dispersion becomes rather exotic, instead of being linear.
Another way to look at this dispersion relationship is that rather figuring out the energy from the momentum, one can fix the energy and find out all the corresponding momentum points. If you trace out all these momentum points and connect all these dots, the result is a contour, which is usually called an iso-frequency contour. Iso-frequency contours can be simple or complicated, but they are often closed objects, like circles.
“You can go back to the example of the bike rider,” Zhen says. “When you give the energy, all you need to do is basically draw a circle and all the points on the circle have exactly the same energy. When you go to life, you have these periodic patterns, such as an array of bumps and obstacles, and the dispersion becomes complicated. Rather than one circle you can have multiple circles or even more weird structures. But you can actually take these contours and turn it into a coloring problem, turning the inside of the circle red and the outside as blue.”
In this study, the researchers found that in the momentum space, their iso-frequency contour was an open-ended arc. This created a problem if someone wants to color the graph; the left and right side are always connected to each other.
Previously, researchers had found this sort of phenomenon in electronic systems, usually on the two-dimensional surfaces of three-dimensional objects called Weyl systems. In this work, the researchers found the phenomenon in the bulk material itself, a 2D bulk material exhibiting this open-ended arc.
“Usually people studied closed, or Hermitian, electronic systems and found these open arcs on the surface of three dimensional objects,” Zhen said. “Here we study photons in a two-dimensional system, but with radiation losses into open space, and find this phenomenon is also allowed to occur.”
To investigate this, Zhen and his collaborators needed to create periodic patterns to modulate the dispersion relationship of light. They started with a structure called a bulk slab, that they would shine light through, but every few hundred nanometers they “punched” a hole in it using a technique called interference lithography. Just as placing obstacles, such as bumps, in front of the bike rider would change their dispersion relationships, as the light traveled through this environment, its dispersion relationship changed.
“Usually if you want to map out these iso-frequency contours,” Zhen said, “it’s quite tedious to do if possible at all. But experimentally, based on a previous paper, we actually developed a new technology that can immediately show us the iso-frequency contour of the sample at any given frequency. It’s a very nice and convenient way of mapping out these contours.”
Although Zhen is most excited about the fundamental aspects of this research, he says it could potentially provide a new way of doing optical communications: rather than having integer topological charges, these channels could be given fractional topological charges. Other potential applications of this systems are to build better optical sensors for chemical and biological sensing, which are being pursued in Zhen’s group.
In addition to trying to expand the fractional numbers possible in these systems, Zhen is also collaborating with theorists at Penn to try to find more interesting topological physics they can infer on the fundamental natures of non-Hermitian topological physics.