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Special Considerations When Modeling Photonic Crystals with FDTD

Finite-difference time domain (FDTD) calculations require the computational domain to be terminated with boundaries that prevent non-physical 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 half-circles around the edge). Assuming it remains stable, one would get results like those presented below.

Here we see a large non-physical 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 non-physical 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  (UPML-PC) 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 UPML-PC 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 UPML-PC boundary must be used.