This is the first in a series of posts about the upcoming OSA Frontiers in Optics meeting in Orlando. This post covers research related to the presentation FM4C.5: Mohammad-Ali Miri; Matthias Heinrich; Demetrios N. Christodoulides, SUSY-generated complex optical potentials with real-valued spectra. To be (hopefully) cross-posted at the Frontiers in Optics blog.
(Edited to make a few additional observations.)
One of the most fruitful strategies in optics research is to investigate the implications of concepts and mathematics used in seemingly very different fields of physics. The most dramatic example of this today is the foundation of the field of transformation optics, which uses the mathematical tools of general relativity to create novel optical devices. As I’ve discussed in previous posts, treating matter as an effective “warping” of space has led to the theoretical development of exotic objects such as invisibility cloaks, “perfect” optical illusions, and even optical wormholes.
With this in mind, it was probably inevitable that scientists would tap even more unlikely fields for inspiration. In a recent paper*, researchers at CREOL and the Max Planck Institute for the Physics of Complex Systems apply the mathematics of supersymmetry in the design of optical structures.
If you’re not familiar with supersymmetry**, it is best known as a hypothesis in particle physics that literally doubles the number of elementary particles that exist in nature, and serves as one possible extension of the standard model of physics that attempts to provide a unified “theory of almost everything.” It turns out, however, that supersymmetric math can be applied to more mundane problems, including quantum mechanics and optics, the latter of which we consider in this post.
Let’s start with a brief discussion of supersymmetry in particle physics, highlighting those pieces that are relevant for our optics discussion. In the standard model of physics, all of nature can be characterized by the collection of particles illustrated in the figure below, as well as their antiparticles.
The quarks and leptons make up most of ordinary matter — in fact, ordinary matter is made up of the “up” (u) and “down” (d) quarks, which atomic nuclei are made of, and the electron “e”. The gauge bosons are particles associated with the fundamental forces of nature: the gluons (g) are responsible for the strong nuclear force that holds nuclei together, the photon (γ) is responsible for the electromagnetic force and, consequently, light, and the W and Z bosons are responsible for the weak nuclear force, connected with a number of forms of radioactive decay. Topping off the list is the extremely heavy Higgs boson, long predicted but only confirmed experimentally this year. The Higgs is required in the standard model to provide mass for many of the other particles, which otherwise should be massless.
I mention this list to emphasize that the standard model consists of a bewildering number of “elementary” particles — and we haven’t even included the antimatter twins that nearly all of them possess! Supersymmetry hypothesizes that each of these particles — matter and antimatter alike — has a supersymmetric twin which is similar to but not exactly the same as its ordinary sibling (like fraternal twins, really).
Among the differences: these supersymmetric (SUSY) particles are all expected to be significantly heavier than their ordinary counterparts.
Why would one need to double the number of so-called “fundamental” particles? There are a number of reasons, but perhaps the easiest to understand is analogous to the role of antimatter in physical theory. Before antimatter was discovered, physics had a problem related to the “self-energy” of the electron***. Because electric charges of the same sign repel one another, it takes energy to assemble a collection of charges together, including the electron. However, the electron is a point particle, and according to classical theory it therefore takes an infinite amount of energy to create one. This is obviously an unsatisfying result.
The existence of anti-electrons (positrons) solves the problem. It turns out that antimatter causes the self-energy of an electron to be finite — the simple existence of an “evil twin” to ordinary matter causes the infinite self-energy to mostly cancel out.
Returning to the standard model, naive calculations suggest that the Higgs boson and all of the ordinary particles should be much, much heavier than they actually are — close to a billion-billion times heavier! In analogy with the antimatter case, however, one can show that the interactions of the ordinary particles with their “fraternal SUSY twins” result in most of this mass cancelling out.
There are many more reasons why SUSY has been extensively studied as an extended theory of nature: see Starts With a Bang for an excellent summary of the hypothesis and its current state. Unfortunate for SUSY, it does not seem to be in agreement with the most recent experimental data, and may end up eventually being discarded as untrue.
However, the idea of SUSY — that we can construct mathematical “twins” of particles (or more general objects) that can interact with one another — can be adopted into other fields, such as optics. The next natural question to ask: “twins” of what? We don’t work with quarks, leptons or Higgs bosons in optics! The answer is that SUSY can be used to construct pairs of very different optical structures whose interactions with light are nevertheless very much the same. In the work considered here, the researchers considered the effect of using SUSY to make pairs of closely related optical waveguides.
A few words about waveguides are probably necessary here for those who are not optical scientists! A waveguide is, in essence, a “tunnel for light,” which allows light to enter at one end and keeps it confined until it emerges at the other end. The most familiar example of a waveguide to most people is a fiber optic cable, used to transmit internet and phone communications from one location to another using light.
How does an optical fiber, which is made of transparent glass, manage to trap light within it? The answer is the phenomenon of total internal reflection, which I have discussed in a previous “basics” post. In short: when light travels from an optically dense medium (such as glass) into an optically rare medium (such as air), the direction of the light gets bent towards the surface. If light is incident upon the interface at an angle nearly parallel to the surface, it will completely reflect and no light will be transmitted.
When light is passed into an optical fiber, then, it tends to “bounce” in the fiber, completely trapped, until it reaches the other end.
This picture is a little too simple, however, for a variety of reasons. The most important of these is that light has wavelike properties, and this means that light cannot follow just any bouncy path through the fiber, even if it satisfies total internal reflection. In fact, there are only a finite number of discrete directions of propagation that light can take through a fiber, and these different ways that light can propagate in the fiber are known as the modes of the fiber.
The number of modes a fiber can hold is dependent upon the material of the fiber, the structure, and its size. The total number of modes is limited by total internal reflection (TIR): the higher-order modes bounce around at steeper angles, and rays outside the TIR limit aren’t confined at all. The picture below is a rough sketch of what a set of modes in a multimode fiber might look like.
It should be noted that the higher-order modes take a longer path through the fiber; they effectively travel slower through the fiber for a given wavelength (color) of light. This may be considered their fundamental property: how fast they propagate along the length of the fiber.
We finally, finally have enough information to return to discuss SUSY optics! The researchers from CREOL and the Max Planck Institute essentially applied supersymmetry to a different type of waveguide known as a slab waveguide, illustrated below.
Light is confined to a core region sandwiched between two thicker layers and propagates along the z direction. The optical properties of the core itself vary along the x direction and, similar to the optical fiber, light can only propagate in certain discrete modes along the z direction.
Here is where the supersymmetry comes in! Using SUSY mathematics, the researchers were able to design a second waveguide, with a different core structure, that has almost exactly the same set of mode speeds.
We can say that the modes in optics play the role analogous to particles in particle physics. But what good does this do for us? In the picture above, I numbered the modes that exist in each waveguide. You will see that the SUSY waveguide is missing the 0th order, or fundamental, mode. This was not a mistake! Working through the mathematics, the SUSY waveguide cannot allow a fundamental mode to propagate. This can actually be used to make novel optical devices, as the researchers demonstrated.
Suppose we put the original waveguide and the SUSY waveguide side by side and couple light only into the original waveguide. Thanks to the proximity of the waveguides and the fact that the higher-order mode speeds are identical, some light will inevitably “jump” from the original guide to the SUSY one.
If nothing else is done, the light will eventually “jump” back, and bounce back and forth. Suppose, however, we also design the SUSY waveguide to absorb light, then anything that jumps into it will be quickly quenched. However, the fundamental mode (the 0th mode) has no twin in the SUSY waveguide, and no light will jump over and be absorbed. We may therefore use the SUSY waveguide to filter out all light except the fundamental mode. In a sense, the SUSY waveguide becomes an automatic “railroad side track” that diverts all traffic except the fundamental one off the “main track”. this curious effect may be useful in the design of novel laser sources.
It is in this application that we get something that is close to the spirit of SUSY in particle physics: by the use of additional modes (particles), we get new optical effects that do not exist with the original modes (particles) alone!
Other unusual possibilities exist for SUSY in optics. In the same paper, the researchers demonstrate that it is possible to use SUSY to make pairs of objects with very different structures but which scatter light in exactly the same way; this is reminiscent of the illusion optics concept mentioned earlier. Furthermore, they demonstrate that it is possible to make pairs of SUSY fiber structures that couple light from one fiber to another and change the angular momentum of the light in the process. In essence, the transition to the SUSY fiber will impart a “twist” onto the light.
It is interesting to note that this is not the first work to consider the implications of SUSY in optics. In fact, a concise paper nearly two decades earlier by researchers in Mexico**** lays out the basic formalism of SUSY in wave optics and suggests other curious possibilities, such as having light waves of two different frequencies propagating in the same fiber without producing an interference pattern!
This idea of creating “twins” of optical structures is only now being really investigated, however. Like transformation optics before it, I suspect that supersymmetric optics will lead to a variety of novel and quite unusual optical designs and optical phenomena.
Curiously, supersymmetric optics illustrates another point: good mathematical theories should never be discarded, even if they fail in their original purpose. Though supersymmetry may not be the solution to mysteries in the standard model, it is already paying dividends, so to speak, in optics.
* M-A. Miri, M. Heinrich, R. El-Ganainy and D.N. Christodoulides, “Supersymmetric optical structures,” Phys. Rev. Lett. 110 (2013), 233902.
** To be fair, there are relatively few people on the planet who can say they are “familiar” with supersymmetry!
*** Summarizing an argument on Hitoshi Murayama’s webpage.
**** S.M. Chumakov and K.B. Wolf, “Supersymmetry in Helmholtz optics,” Phys. Lett. A 193 (1994), 51-53.