Modeling Photonic Crystals in the Frequency Domain
The first step in modeling photonic crystals is to
produce a dispersion diagram of the crystal without defects. This determines
if a band gap is present, and if so, its size and location in frequency.
While it is possible to produce dispersion diagrams by Fourier transforming
finite-difference time domain (FDTD) results, and then looking for the
frequencies where peaks in the frequency spectrum occur, I prefer to use the
plane wave expansion technique. Steven Johnson of MIT has written some
excellent software, Photonic Bands,
which is publicly available under the GNU license. I highly recommend this
software (it's only available for UNIX platforms).
As an example, we will look at some calculations I did
on a silicon slab (index n=3.478). All units are scaled in terms of the period, denoted here
by a. A triangular array of air holes (radius 0.3a) in the x-y plane
is etched into a silicon slab of thickness 0.55a. I start off modeling the
structure in 2D (z direction assumed uniform). Such a structure should
produce a gap for TE polarizations, and the dispersion diagram below does
show a band gap in the frequency range 0.2-0.28 c/a (where c is the speed of
light). The lower axis plots the propagation vector along the reduced
Brillouin zone for the triangular lattice.
The 2D calculations are trivial, and
take less than a minute to complete. In order to calculate the slab
structure, we will have to use the supercell approach since plane
wave methods assume infinite periodicity. We create an infinite stack of
slabs, each separated by a sufficient distance in the z-direction. The slab
modes confined by TIR will not interact with adjacent slabs if spacing
between the slabs is sufficient, resulting in the same modes as an
individual slab. The light not confined by TIR will interact
with adjacent slabs, so these radiation modes will not be the same as
for an individual slab. Fortunately, we usually care only about the slab
modes. If one needs to find the radiation modes, then the FDTD technique
will need to be employed.
From the dispersion diagram above, we
see a gap in the slab modes in the frequency range 0.26-0.34 c/a. This is
not a complete band gap as in the 2D case because of the radiation modes
(the yellow region). We can still create a linear defect in the
G-K direction so that a mode (or modes) exists in
the gap region, but if we make a sharp bend, there will be some coupling to
the radiation modes. So you can never get 100% transmission around a sharp
bend in a slab like you can in a 3D crystal with an omnidirectional band
We next model a line defect. One can
create either a reduced index defect (air holes larger than the
background holes, so the effective index of the defect is lower than the
background region), or an increased index defect (smaller holes).
Let's look at a reduced index defect. Again we start with 2D.
Since our modeling method requires
infinite periodicity, we will have to create a supercell in the x-y plane in
order to create the defect. While the supercell approach creates an infinite
array of line defects, they are separated by a sufficient amount of
background crystal region. Consequently, there is no interaction between
adjacent line defects within the gap frequencies (the region which we are
interested in). The above dispersion diagram shows the results of varying
the radius of the air holes in the defect in 2D.
We repeat the process for the slab.
Now we not only have a supercell in the x-y plane, but an infinite number of
slabs (separated by sufficient space) in the z direction. The yellow region
denotes the radiation modes where TIR confinement no longer occurs.
At this point we
have all of the information necessary to create our line defect in our
silicon slab. However, we will probably want to calculate some transmission
results for our photonic crystal wave guide. In order to do this, we must
employ time domain calculations.