# Cavity Design in Woodpile Based 3D Photonic Crystals
## 4 Numerical Results  
In this section, we use the finite-difference time-domain (FDTD) method to simulate the cavity mode estimating the resonant mode frequency and Q-factor in various defects when changing the size of the defect \(L\) from \(0.1c\)-\(2c\). A broadband dipole source whose bandwidth covers the complete photonic bandgap is positioned at the center of the cavity and along the \(z\)-axis. We demonstrate that effective mode volumes of the cavity mode could be suppressed by making the defect smaller, and Q-factors of cavities can become high when the resonant frequency is located around the center of the complete photonic bandgap. Figure 4 illustrates all the high Q-factor cavity modes we find near and inside the photonic bandgap as the defect size \(L\) is increased. The dashed lines indicate the boundary of the complete photonic bandgap and the position of the bandgap center related to the woodpile template without a defect. We find that high Q-factor cavity modes do not exist far outside the bandgap as expected although some of them appear at the band edge. Furthermore, the normalized resonant frequency of both fundamental modes (Mode 1) and high order modes (Mode 2-5) decreases when increasing the defect size \(L\). The calculated Q-factors for the fundamental modes (Mode 1, black square in Figure 4) are shown in Figure 5 as a function of normalized resonant frequency when defect size \(L\) ranges from 0.1\(c\)-0.8\(c\). The Q-factor reaches its maximum value near the middle of the photonic bandgap when defect size \(L\) = 0.25\(c\). As we use a finite structure in our simulation, the reflectivity drops when the frequency moves away from the bandgap center resulting in a drop of the Q-factor. For those cavity modes inside the complete photonic bandgap when the defect size \(0.15c\leq L\leq 0.6c\), the calculated normalized mode volumes are shown along with the Q-factors in Figure 6. It is shown that the calculated normalized mode volume reduces from 0.64 to 0.29 \((\lambda/n)^{3}\) when the defect size \(L\) decreases from 0.4\(c\) to 0.15\(c\), while Q-factor reaches the highest value of \(1.24\times 10^{5}\) when \(L\) = 0.25\(c\) and starts to drop quickly if the defect size is further reduced. As the cavity mode approaches the band edge the mode volume reaches a minimum because the field outside the bandgap is not confined. For \(L\) > 0.4\(c\) the mode volume appears to decrease slightly which may be a consequence of limited calculation space when the mode approaches the band edge and becomes leaky. Figure 7 illustrates the energy distribution of cavity modes when defect size \(L\) = 0.5\(c\), \(L\) = 0.25\(c\), \(L\) = 0.2\(c\) and \(L\) = 0.15\(c\), respectively. In Figure 7, we observe nearly perfect Gaussian confinement in such defects with \"wings\" (predominantly blue areas) floating inside the adjacent dielectric rods. Furthermore, we see  
Figure 3: The comparison between rod-connected diamond lattice (**a**) and woodpile lattice (**b**). Woodpile could be considered as a layered version of the rod-connected diamond structure [8]. that better confinement of the energy distribution of the cavity modes can be achieved by reducing the defect size \(L\) from \(0.5c\) to \(0.15c\) when the intensity of the \"wings\" becomes weaker and weaker. Figure 4: Normalized resonant frequency (\(c/\lambda\)) as a function of defect size \(L\). The dashed lines indicate the boundary of the complete photonic bandgap and the position of the bandgap center related to the woodpile template without a defect. Figure 5: Calculated Q factors for fundamental cavity modes (Mode 1) as a function of normalized resonant frequency when defect size \(L\) varies from \(0.1c\) to \(0.8c\). In Figure 4 we noticed that multiple modes can coexist in the cavity when defect size increase. To show the difference between these modes, we plot their energy distributions in Figure 8. The 2D snapshots are taken through the center of the cavity in _xy_-plane.