If you had asked me yesterday morning if there was any room left to play with the theory that governs the spontaneous emission properties of simple atoms, I would have said no. But then I encountered a colleague of mine, Willem Vos. He showed me a picture of an atom, surrounded by 6 blobs of material, and asked me to predict which direction the atom was most likely to spontaneously emit. Needless to say, I got the answer wrong, and Willem took great pleasure in explaining to me exactly why I was wrong.

Now, I hear the practical among you ask: why should I care which direction atoms will emit? The answer lies in photonic crystals—a very popular buzzword in the optics community at the moment. Photonic crystals are our attempt to overcome the shortcomings of nature. You see, nature only provides us with materials that have a limited range of optical properties, and those limits restrict the performance of optics and limit the types of optical components that are practical.

Photonic crystals overcome this limitation by structuring materials on the same scale as the wavelength of light. So, since green light has a wavelength of around 500nm, photonic crystal structures for manipulating green light require physical features around 250nm or less in size. For instance, if a red dye is placed in a photonic crystal that simply doesn't allow red light to propagate through it—the crystal acts as a mirror for red light—the dye is forced to emit green or blue light instead.

This control—either the inhibition or increase in emission at a particular wavelength—is reasonably well understood. But the question of whether these structures can encourage spontaneous emission in specific directions remained open. Well, in truth, not totally open. For instance, if you place an excited atom in the right orientation between two very high reflectivity mirrors, then it is much more likely to emit light that is trapped between the mirrors, rather than at some angle that allows the photon to escape.

This is because the excited atom will always emit into some vacuum state, chosen randomly from among all the locally available states, called the local density of states. Between two mirrors, there are many more states available; once the mirrors are reflective enough, there are almost no other states available to emit into.

So, clearly, the structure around the atom changes the emission pattern—although I haven't discussed it above, the orientation of the atom also changes the emission pattern. But it's not clear just how much freedom to we have to play with this emission pattern.

To investigate this problem, Vos, Koenderink, and Nikolaev examined how the local density of states was influenced by the local environment and the orientation of the emitter. Looking at a very general expression for the local density of states—any emitter surrounded by a structure made from materials that meet certain requirements—they realized that there is a certain symmetry in the mathematics. Consequently, the emission rate in any direction can be described as a weighted sum of just three rates.

Combined with the fact that the emission has to occur in some direction and that the three rates must sum to the total emission rate, and we end up with a very constrained system. Basically, the average radiation pattern of an emitter has to be described by spheres, ellipsoids, donuts, or peanuts. Everything else violates the symmetry of the mathematics and physical constraints.

To calculate the emission pattern we need to know three rates (as well as their direction) and their relative weights, something that is pretty much impossible to know about any real structure. However, Vos rescues us from that particular bit of pain by showing a relatively simple way to calculate these weightings for structures.

This procedure makes use of the fact that the functions that create the emission patterns all lie on the surface of a sphere—this is a consequence of requiring that the weighting factors make the probability of emission in some direction exactly one. Any function that is constrained like this can be described as a small set of simple functions; in this case, the simple functions are directly related to the three emission rates.

To cap the paper off, there are calculations showing the emission patterns of emitters at different locations within a photonic crystal. Depending on the location, the emission can be quite highly directional, nearly spherical, or anywhere in between.

One of the things that I often miss when writing for Ars is that I rarely have personal contact with the scientists directly involved in the research. There are two consequences to this: I have to interpret the results all by myself—with sometimes hilariously bad results. More importantly, the papers don't give you a sense of the researcher's feeling of accomplishment, or even the degree of involvement of the authors. I can assure you that Vos was very excited about having solved a difficult problem in a very elegant way.

Physical Review A, 2009, DOI:10.1103/PhysRevA.80.053802