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The cavity is constructed by perforating a slab of Ta2O5 (which has an index of 2.0995) with air holes forming a hexagonal lattice. The lattice constant is 575 nm, and the air hole radius within the photonic crystal lattice is 194 nm. The cavity itself is formed by removing a central hole, reducing the radius of the inner six holes to 100 nm, and removing holes along the outer edge of the cavity to form the structure shown in the figures below.

 

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When we construct this structure in FDTD, we create the slab out of a rectangle. Then we add cylinders to the simulation for the holes. In the locations where the holes and the slab overlap, we use the mesh order property to make sure that FDTD uses the refractive index data from the holes (cylinders) instead of from the slab (rectangle). For more details about mesh order, refer to the mesh order page in the Reference Guide.

 

Two dipole sources (depicted with green arrows to show the direction of the H field) are used to excite the modes of the cavity. These dipoles are not located in the center of the PC structure in order to reduce the chance that they are located at a zero of the cavity mode. The dipole sources are used to inject energy into the simulation volume. Some of the dipole radiation will be coupled into the cavity modes and decay slowly. The radiation which does not get coupled into the cavity modes will be scattered and quickly exit the simulation volume.

 

The frequency domain monitors in FDTD simulations calculate the mode profile by taking a discrete Fourier Transform of the time domain data. Obviously, we do not want to include the portion of the time signal at the beginning of the simulation, since it contains radiation which does not excite the modes; we are only interested in the later portion of the time signal when all the energy left in the cavity is in the resonant modes. As you can see in the modeling instructions on the next page, we can use monitor apodization to select only the portion of the time data at which all the energy left in the simulation is in the resonant modes. A more in depth discussion of monitor apodization (also based on a PC cavity) can be found in the Apodization.

 

The rest of this setup section discusses some important simulation settings, namely the boundary conditions, the mesh and the simulation time.

 

Boundary Conditions

The orange boundaries which can be seen in the screenshot above are Perfectly Matched Layer (PML) boundary conditions. PML boundaries absorb incident radiation, and are intended to absorb all radiation propagating away from the cavity.  It is important to leave some distance between the cavity and the PML boundaries. If the boundaries are too close to the cavity, they will start to absorb the non-propagating local evanescent fields that exist within the cavity.  A simple rule is to leave at least half a wavelength of distance above and below the structure.

 

Next, notice that the lower half of the simulation (z<0) is shaded blue. This is because we used a symmetric boundary condition on the z min boundary in order to reduce the computation time by a factor of two. The drawback of using the symmetric boundary condition is that it will forbid certain modes from appearing in the results (modes that do not exhibit the same symmetry relation as the boundary condition). For this PC cavity, there is a plane of symmetry through the center of the slab (z=0 plane). Using a symmetric boundary condition on this plane will only allow TE-like modes and eliminate TM-like modes from the results.

 

Note that the reason the dipoles are located z=0 is because this is the ideal location for the sources. Magnetic dipoles will have an E field pointing along the z=0 plane. The blue color of the symmetric boundary condition is intentional; it indicates that the E field should lie along (be parallel to) this boundary. Most sources in FDTD contain blue arrows which show the E field. These should always lie on blue boundaries.

 

Mesh

To get good results for cavity simulation, it is important to include an integer number of mesh cells per lattice constant in the two directions. The wavelength in the material is on the order of λ = c / f / n = 3e8 / 160e12 / 2.0995 = 890 nm. We can expect reasonable accuracy with a λ/10 mesh. 8 points per period in the x direction gives a mesh size of 575nm/ 8 = 71.875nm. The mesh size will be smaller in the y direction since the lattice constant is 575*sin(60) nm. This should give reasonable accuracy while keeping the simulation time to a minimum.

 

To make sure that the mesh can actually be an integer number of mesh cells, the simulation (FDTD) span has to fit exactly an integer number of mesh cells inside the boundaries. For this reason we set the x span of the FDTD region to 575*12 nm and the y span of the FDTD region to 575*sin(60)*12 nm.

 

By clicking the 'View mesh' button, it is possible to view the mesh around the structure. It is important that each hole is meshed in the same way. If the mesh lines fall at different locations with respect to the holes, each hole will have a slightly different size and shape in the simulation.

 

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Before running simulations with periodic structures, it is good practice to use an index monitor to check that the structure actually looks periodic when it is meshed. The index monitor results below show that each of the holes looks like a cross. This is because the mesh is a bit coarse. However, each cross (except for the 6 inner holes) looks identical. It is important to make sure that all the holes are meshed the same way, if we want to obtain good results. It is possible to view the meshed structure with the index monitor in layout mode (i.e. before the simulation has been run).

The modeling instructions section on the next page contains step by step instructions of how to create index monitor plots.  

 

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Simulation time

To obtain accurate frequency domain data, it is generally required to run the simulation until the time domain fields have decayed to zero. High quality factor cavity simulations are one exception to this rule. This is fortunate, since high Q modes decay very slowly.  Running high Q cavity simulations long enough for the fields to fully decay would be very slow.  

 

A combination of time domain analysis and frequency domain apodization allow us to accurately calculate the Q-factor and profile of cavity modes without running the simulation until the fields decay. However, some care should still be taken when using this technique.  Other measurements like power transmission and field amplitudes will not necessarily be correct when the simulation is stopped early.

 

Simulation Results

The simulation contains a Q analysis group, which contains a script that finds the resonance peaks and Q factors of the cavity modes. We have placed the Q factor group, which contains a time monitor, away from the origin of the simulation for exactly the same reason the dipoles are not placed at the origin.

 

We can use the analysis script to obtain the largest two resonance peaks in the source bandwidth and their Q factors. It is easy to obtain more resonance peaks: simply change the "number_resonances" parameter in the Analysis -> Variables tab of the Q analysis object.

 

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We can also obtain a plot of the E fields as a function of time (below right) and a plot of the resonance peaks. The Q factor calculations are discussed in detail in the Cavities and Resonators section of the Online Help.

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Once we know the resonance frequencies, the corresponding E field profiles can be plotted.

 

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The image below shows real(Ey) for the mode at 201 THz.

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Further analysis: Symmetric boundary conditions

We will need to run quite a few simulations in order to find the inner hole radius which maximizes the Q factor of the mode at 201 THz. Since the mode profile possesses symmetry about the x and y axes, we can use anti-symmetric/symmetric boundary conditions to reduce the simulation time by an additional factor of 4.

 

From the above image and the discussion on the Choosing between symmetric and anti-symmetric BCs page in the User Guide - Simulation section, we can see that the E fields for the mode  of interest have a plane of anti-symmetry at X=0, and a plane of symmetry at Y=0.

 

Whenever the EM fields have a plane of symmetry through the middle of the simulation region, using symmetrical boundary conditions will give the same results as running the full simulation. The plot below shows real(Ey) after we set the x min and y min boundary conditions to anti-symmetric/symmetric. This plot looks identical to the one above except for the magnitude. The change in magnitude arises because the sources have been mirrored, i,e in the full simulation there are only two sources, but by using symmetry we have mirrored these sources so that there are 8 now.

 

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Further analysis: Optimization of inner hole radius

FDTD contains a built in optimizer. We choose to use the particle swarm optimization algorithm which is included with the optimizer, but it is also possible to define a different optimization algorithm. For more details about the optimizer, see the Optimization section of the online User Guide.

 

In this case, we choose to try to find the radius of the 6 inner holes of the PC Cavity which optimizes the Q factor of the first resonance. The sweep below shows an optimal radius corresponding to 0.167*0.575 um = 96 nm. Before running the optimization it is important to set the number of resonances in the Q analysis group to be 1; for more details, see the Modeling instructions page.

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