### Modeling Photonic Crystals in the Time Domain

Using frequency domain calculations, we constructed the dispersion diagram for the photonic crystal, and then the dispersion diagram for the line defect. From these calculations, we have all of the parameters needed to construct our photonic crystal wave guide. However, it would be beneficial to model the transmission through the line defect. Does it propagate light at all, and if so, at 100% transmission as we predicted?

In order to find the transmission, one needs to use a finite-difference time domain (FDTD) calculation. We will begin as usual in 2D. Using our results from the frequency domain calculations, we assume a triangular array (with lattice constant a) of air holes (r=0.3a) in silicon. A reduced index line defect is created by using larger air holes (r=0.45a).

The computational domain is shown below, with 16 Yee cells per period. A dipole source (in the x-direction) is placed near one end of the line defect, and turned on for 5000 time steps, with its intensity following a Gaussian profile in time. The  frequency flux through the line defect region is monitored at several points along the line defect. The ratio of these flux values will determine if there is any transmission loss along the line defect (we also monitor frequency intensity at points along the defect as a check). The simulation is repeated several times with pulses of different frequency, as to cover the entire frequency range that was predicted from our dispersion diagram for the line defect.

When we take the ratio of the flux at the first and second monitoring points, we get the results plotted below. Here we have scaled the period value a so that the center frequency of the line defect lies at the corresponding wavelength of 1550 nm. As we can see from the plot, there is no transmission loss for our 2D example.

Here is a picture in the time domain of the pulse propagating down the defect.

We next run FDTD on the full 3D slab. Our computational domain is now 170 times larger than the 2D example. With just under 2.5 million Yee cells, it will require around 2.5 GB of memory. While it would be possible to run such a calculation of a very high end workstation, I chose to run it on a parallel cluster of computers.

We notice that loss begins to occur at wavelengths greater than 1580 nm. This corresponds exactly to where the line defect mode intersects the region of the radiation modes in our dispersion diagram. At wavelengths greater than 1580 nm, losses are occurring because of a lack of TIR confinement above and below the slab.

These results also show a slight loss at wavelengths below 1510 nm. This is not a true loss, but rather a breakdown in our modeling method caused by dispersion. This wavelength range corresponds to where the defect mode is near the Brillouin zone edge, and as a result, the group velocity declines rapidly with increase in frequency (propagation constant near 0.5 in our dispersion diagram). If the pulse is too wide in frequency (i.e. too short in time), our time domain pulse spreads out, and we get inaccurate results. To correct it, we need to rerun the simulation with a very narrow pulse in frequency (or very long in time duration).

The corresponding picture of the time domain pulse propagating down the line defect in the slab is shown below (the picture is from the plane at the center of slab).

The next logical step would be an investigation of transmission around sharp bends, and how to optimize transmission around those bends. For these calculations, we will need the full power of our parallel cluster, as they will require on the order of 100 million Yee cells.

We know that we can get 100% transmission around a sharp bend in a 2D structure because it possesses a complete band gap (for one polarization state). However, for a slab, there will be some coupling to radiation modes, and only a transmission calculation will tell us how much. Unfortunately, I can't post these results now as they are part of a paper I'm currently working on. Once the paper is completed, I will post some of the results here.