In studying physics, asking very simple questions often puts one at the forefront of current research and, indeed, opens whole new research areas. The field of metamaterials was developed over the last decade by posing fundamentally simple queries: can light refract negatively? What are the basic limits on the wavelength of light inside materials? What would happen if different components of the electric field vector in a light beam experienced a radically different electromagnetic environment?
As one digs further, the questions become increasingly complex and specific; not all of them will have good answers, and not all of them will even be good questions. It is easy to start getting lost and discouraged. When this happens, you can run back up the rabbit hole to the gleaming edifice of one of the basic equations. For me, this refuge is the manifestly covariant form of Maxwell's electrodynamics.
I have worked with Maxwell's equations for many years, in many different forms. Down there, in the rabbit hole, they are in their work attire, sporting unsightly divergences and curls, with constituent parameters dangling awkwardly from the field components. But here, out in the open, they are barely recognizable, dressed up as tensors and 4-vectors, conversing in classical Greek. You'd be a fool to mistake the Levi-Civita symbol for the dielectric constant. In this form, they are the equations of Einstein and Feynman, they are shining peaks in the landscape of modern physics. And yet, they are also my equations. I'm adding my strokes, however tentative and insignificant, alongside those of the greats. This gives me the inspiration to carry on.
Back down the rabbit hole I go: I'm on a mission. Optics has always attracted me by the ease with which fundamental electromagnetic and quantum mechanical abstractions become reified – often, in pretty colors – both in a lab setting, as well as in many devices that have revolutionized our world. And yet, many beautiful designs and elegant ideas are doomed to failure, due to inherent limitations of optical materials. This revolts the avowed idealist in me.
The world of applied physics is rife with trade-offs. Indeed, many are codified in the fundamental laws, such as the Heisenberg uncertainty principle. Yet most limitations are mere caprices of Nature, which tailors material parameters to its fickle, oft-inscrutable specifications. Metamaterials offer a tantalizing escape from the status quo. By custom-designing material properties, we can strike down some of the vexing compromises that limit performance and capabilities of optical devices. My mission in the rabbit hole that became my Ph.D. thesis is to eliminate the compromise that is holding back all of nanophotonics.
Many of the trade-offs in nanophotonics involve the fundamental differences between metals and dielectrics. These differences make metals appealing as short-wavelength waveguides, or emission enhancers, yet woefully unsuitable for many other applications. Dielectrics suffer from a similar fate, in reverse. Can we create a material that would behave both like a metals and a dielectric? Yes. Such materials are called hyperbolic, and they can be fabricated using modern metamaterials techniques. Some rare examples of hyperbolic materials can even be found in nature, but this is of dubious benefit to the contemporary metamaterials ideologue.
Indeed, we are past the age when our building materials were logs and mud, and we are past the age when circuit switching elements operated by thermionic emission. In the 21st century, we should get past the age when we rely on a few serendipitously found crystals to determine what we can and cannot do with photons. To be sure, future metamaterials engineers might still need to emerge from the rabbit hole to comprehend the unblemished covariant beauty of Maxwell's equations. But now, their exact from inside materials will result from our manifest destiny rather than from an accident of fate.
I started playing around with running Mathematica inside Emacs. Once I develop the set of Mathematica <-> Matlab interoperability scripts, this functionality will be invaluable. An unexpected (pleasant) surprise was that Mathematica graphics is available at the console through Java -- you just have to load <<JavaGraphics`. The graphics performance is slow as molasses on my underpowered laptop, but it should be quite usable on a good workstation.
Mathematica's Manipulate[] functionality got me hooked on interactivity, so once I had the beam propagation code running smoothly, I decided to implement a primitive GUI for it -- how hard could it be?.. Wasn't hard, really, but somewhat time consuming and required lots of code. I mean, order of 100 lines of code to do something that's almost a one-liner in Mathematica.