 
Special Considerations When Modeling
Photonic Crystals with FDTD
Finitedifference time domain (FDTD)
calculations require the computational domain to be terminated with
boundaries that prevent nonphysical reflections from bouncing back into the
computational domain. Two of the most popular types are Mur absorbing
boundaries and PML boundaries. Unfortunately, the periodic nature
of a photonic crystal presents a problem for both types of boundaries.
First we will look at an example where
a line defect in a photonic crystal is terminated with Mur absorbing
boundaries. Mur boundaries work best for a normally incident plane wave at
an uniform boundary. Here we have neither a plane wave, nor an uniform
boundary. In fact, for many geometries, the structure will not be stable for Mur boundaries (we will need to get rid of the halfcircles around the
edge). Assuming it remains stable, one would get results like those
presented below. 


Here we see a large nonphysical pulse
reflected from the boundary at the end of the line defect, and it propagates
back up the defect to the monitoring point. Unfortunately, it overlaps the
incident pulse, so when we Fourier transform the pulse, we will get
incorrect results.
Using standard PML boundaries reduces
the spurious reflection, but it does not eliminate it (up to 20% of the
pulse can be reflected).
One solution is to extend the computational domain (and terminate it with either Mur or standard PML boundaries). 


Since the computational domain is extended, the
incident and nonphysical reflection no longer overlap. When the incident pulse
is Fourier transformed, we now get very accurate results. Additionally, we can use this
method with Mur absorbing boundaries (which have very low computational
overhead compared to PML boundaries). The downside is that we must use a
larger computational domain (which requires more computer resources).
If we wish to absorb the pulse
propagating down the line defect, we must employ a special type of PML
boundary. We extend the photonic crystal region several periods (usually
around 10 periods) into the boundary region. By adjusting the
absorption parameter so that it remains constant in planes perpendicular to
the line defect, we can create special uniaxial perfectly matched layer
photonic crystal (UPMLPC) regions, which will absorb the pulse
traveling down the line defect. 


As the result above shows, the pulse
was nearly totally absorbed. Unfortunately, this method requires a large UPMLPC region (10 or more periods), which happens to be
very computationally intensive compared to the normal computational
domain. Also, PML boundaries are harder to implement in a
parallel computing cluster than Mur boundaries.
However, sometimes the extended computational domain method is not feasible,
so the UPMLPC boundary must be used. 
