Photonic Honeycomb Lattice

Conical Diffraction and Gap Solitons in Honeycomb Photonic Lattices

Or Peleg, Guy Bartal, Barak Freedman, Ofer manela, Mordechai Segev, Demetrios N. Christodoulides

PRL 98, 103901 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e103901


(For anyone who's not me reading this, let me add at this point that I am a condensed matter theorists studying carbon nanotubes and graphene. These low-dimensional carbon systems have a lot of interesting properties due to the interesting band structure of the honeycomb lattice.)

Interesting article. The authors have created a photonic honeycomb lattice. I don't understand how it works. In the authors' words, "We use the optical induction technique to induce a honeycomb lattice on a photorefractive SBN:75 crystal." Is the crystal similar to a diffraction grating? Perhaps interference of reflections from different parts of the crystal generate a standing wave pattern whose maxima are the vertices of a honeycomb lattice.

The authors report both theoretical and experimental results. Their theoretical work suggests that the linear dispersion near the Dirac points (diabolical points) of a honeycomb lattice would lead to a phenomena called conical diffraction. Apparently, Hamilton discovered conical points back in 1837! They are a hot topic in modern research now; it's hard to believe they've been know for so long.

Apparently, the diabolical points discovered by Hamilton exist in the space of possible polarizations in a biaxial crystal. If a randomly polarized beam strikes a biaxial crystal at the proper angle, it will be refracted into a cone. The diabolic points of the honeycomb are quite different, as the authors point out. They arise from the symmetry of the lattice. Moreover, the conical points in the honeycomb lattice occur in reciprocal space, not the space of possible polarizations.

I'd have to review the paraxial approximation before I made any deeper investigations.

It's interesting to me how big the photonic lattice is compared to the carbon lattices I study. The lattice constant of graphene is about 2.5 angstroms, or a quarter of a nanometer. The lattice spacing in the authors' experiment is 8 microns --- it's larger by a factor of 32 000! The crystal they use to generate the photonic lattice is measured in millimeters: you could see it sitting on a table, unlike a graphene sample. A nanotube can be centimeters long, but it's only a nanometer wide --- smaller than the wavelength of visible light. You'll never be able to see a nanotube on the table.

There are apparently some unique features in the soliton spectrum of the honeycomb lattice, but I know very little about solitons. Looking at the graphs, it is clear to me that the solitons inherit the threefold rotational symmetry of the underlying lattice. Solitons sound similar to excitons. They only exist in bandgaps, so you only see them for higher bands in graphene and metallic nanotubes.

The key feature of the honeycomb lattice, at least with respect to the optics experiments here, is conical diffraction. You send in a Gaussian, bell-shaped beam, and what comes out is a ring of constant thickness whose radius grows linearly as is propagates through the lattice.

That's a pretty amazing result. Could something similar happen in graphene? If you sent in an electron wave packet with a gaussian profile, would you see some interesting conical diffraction? One can also consider the converse: do any of the interesting properties of graphene have an analog in the photonic lattice? There are many possibilities: quantized conductance, relativistic quantum hall spectrum, antilocalization, absence of backscattering. Obviously light couples to fields very differently than electrons, but it seems that some of the scattering and transport properties might be common to both systems.