 
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 finitedifference 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 xdirection) 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. 
