Posts technical---or quite simplistic » research

Shake that grating

dnquark — Sun, 22 Jan 2012 06:00:00 +0000

Using phonon scattering and temporal degreees of freedom for superresolution at the nanoscale

(This means "we can see very small things if we shake them just right".)

Have you ever wondered why we can't see viruses under a microscope? The answer to this question is the diffraction limit. But what is the diffraction limit, exactly? In signals processing terms, it is the spatial frequency bandwidth for propagating optical waves. Using optical microscopes to look at a 130 nm flu virus is akin to trying to pick out the sounds of piccolos in a Strauss concerto, over a very bad cell phone connection. You can probably discern the piccolos if you are right in the concert hall, but the task is hopeless for someone on the other end of the line. Likewise, in optics, you can see arbitrarily small things – provided you are close enough to them so that you can pick up the signal from "the piccolos" (in physics, we call them evanescent waves). That is to say, your detector must be placed in the "near field", which is the area less than a micron from the object.

This, unfortunately, is not always practical or easy to do. But over the years researchers have come up with various tricks to get around the bandwidth problem. One conceptual approach is to take the high frequency information, encode it using low-frequency data, transmit it to the far-field detector, and decode it there. This idea has been around for several decades, but the encoding/decoding step is nontrivial to accomplish, and so this scheme hasn't seen adoption in applications.

Recently, we came up with an unusual new way to perform the high spatial frequency encoding, which may turn this around. In order to lower the spatial frequency of the light wave for transmission, it is necessary to scatter its evanescent components on some periodic subwavelength structure. We showed that this is possible to do with a dynamic phonon grating, validating Jacob Khurgin's original idea.

The phonon grating creates a periodic potential that converts evanescent waves to propagating optical signals. Because it's a dynamic structure, there is much flexibility in encoding and processing the signal. Most intriguingly, phonon scattering is associated with the temporal frequency (i.e. energy) shift of the incident photons. Because of this, it is possible to uniquely detect the scattered signal in the far field using coherent detection, greatly simplifying the decoding step.

This paper was a long time in the making, and it is finally getting some attention, perhaps due to the fact that nano-opto-mechanics with phonons is becoming a popular research area. It's about time the lattice vibration quasiparticles got some good publicity, they certainly deserve it!

A related project, in which we model a system where the scattering is performed by a chirped anisotropic nanostructure (but ultra-high frequency vibrations are still used to aid coherent detection) also got published very recently. In fact, it is the first article in the first 2012 issue of APL (mendeley link). I am currently taking the corresponding issue/article designation of {100,011101} as a divine sign that I should do more programming projects. (I believe that in the coming few months the Almighty will be pleased on that angle.)

Physics (v.4): a metamaterials manifesto

dnquark — Thu, 20 Jan 2011 06:00:00 +0000

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.

The sad state of scientific plotting software

dnquark — Wed, 24 Feb 2010 10:31:11 +0000

It's unfortunate that this blog is turning into rants about scientific computing. Perhaps I am simply ethnically predisposed to doing science and kvetching.

Today's rant is brought about by my collaborators' request to make my paper figures "presentable". The reason is simple -- current figure drafts are straight up Mathematica output, relying mostly on defaults, and they suck. Let's face it -- figures produced by Mathematica can be good, but rarely great, and trying to fine-tune the appearance of frames and axes can be a daunting task.

The problem isn't that Mathematica sucks at plots -- so does most other software. The problem is that to my knowledge there are no reasonable open-source alternatives to something like Origin. So right now, if you are unsatisfied with Mathematica's plotting capabilities, your options come down to this:

- get as far as you easily can with Mathematica, then switch to Illustrator and Photoshop. Pros: guaranteed to work. Cons: time-consuming, labor-intensive, and requires Illustrator, Photoshop, and Windows.

- import data into one of {R, Matlab, Python/Matplotlib} and hope it can do what you are after. Pros: you might get the output you want. Cons: you need to know R, Matlab, or Python.

What shocks me is that in this day and age, it seems that every few years, somebody sits down to write a plotting package, and ends up reinventing the wheel. So you have a bunch of wheels, none perfectly circular; there are all sorts of bumps and protrusions, all in different places. Within R, for instance, there are at least 3 different ways to create plots (base graphics, ggplot2, lattice), and more are being created (e.g. jjplot). However, despite all the man-hours spent, we still don't have such basic things as TeX integration. In a software package that claims to be the premier graphics solution for a statistician/applied mathematician! The syntax to add formulas to plots is revolting. Even Excel can do better.

All right, rant over. I need to go generate some figures.

More beam propagation eyecandy

dnquark — Wed, 08 Jul 2009 08:24:40 +0000

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.

While it is certainly lacking on the elegance part, it runs fast. If I had to do it over again, though, I would implement the numerics in Matlab, dump it all into a datafile, read into Mathematica and Manipulate[] away.

Beam propagation in 15 lines of Matlab

dnquark — Thu, 02 Jul 2009 10:33:55 +0000

I recently returned to examining transmission / reflection properties of anisotropic planar structures, and I figured that this time around I should do it the Right Way (TM), which boils down to:

1. Take legible notes and
2. Stay the hell away from Mathematica for numerics.

With both of these, I succeeded beautifully. Once again, I have some Matlab code which takes seconds to produce the same results my old Mathematica notebooks would take minutes or hours to compute -- and it's clean, legible, and ran on the first try!

function out = BeamProp1(x,z,kx,kxmask)
    kz0 = sqrt(1 - kx.^2);
    [zz,kk0] = meshgrid(z,kz0);
    fwdPropFieldK = diag(kxmask)*exp(1i.*kk0.*zz);
    clear('zz','kk0'); %free memory
    out = complex(zeros(length(x),length(z)));
    col = 1;
    [xx,kkx] = meshgrid(x,kx);
    for eKZslice=fwdPropFieldK
        [xx,fld] = meshgrid(x,eKZslice); % diff x's are in columns
        eKX = sum(fld.*exp(1i.*xx.*kkx),1).';
        out(:,col) = eKX;
        col=col+1;
    end
end

Couple of notes on the code: first, it does not handle evanescent fields gracefully: if you put evanescent fields (kx > 1) at the origin, they will exponentially grow for z < 0 and will clobber all other features of the plot. Second, the code is fully vectorized -- meaning that it sacrifices memory for speed. The good news is that even though there are 3 vectorized dimensions (x,z,kx), only two of them are put into meshgrid at the same time -- so no 3D matrices to store. Still, expect to have at least a few tens of megs available when dealing with large fields of view.