Frequency Domain

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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 gap.

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.