# Cavity Design in Woodpile Based 3D Photonic Crystals
## 3 Woodpile Parameters and Cavity Design  
In Figure 1a we show the woodpile structure we are modelling. The woodpile forms an FCC lattice when the height of the rods \(h\) is equal to \(c/4\)[1], and the distance between two adjacent rods \(d=c/\sqrt{2}\) where \(c\) indicates the vertical period which is also the height of four stacking layers. Before we show the numerical results and the defect design, we choose the optimized woodpile with widest complete photonic bandgap simply by adjusting the width \(w\) of the rods [20]. Throughout this work the refractive index of logs is \(n_{rad}=3.3\), corresponding to common semiconductor materials such as Gallium Phosphide (GaP), while the background material is air \(n_{air}=1\). Figure 1b shows the band structure calculated using plane-wave expansion (PWE) [52] method. The green region indicates the location of the optimized complete photonic bandgap which ranges from normalized frequency \(c/\lambda=0.485\) to \(0.569\) (where \(\lambda\) is the wavelength) and the gap/midgap ratio reaches a maximum value of \(15.88\%\) when the width of rods \(w=0.21c\)[20]. All the simulations shown in this paper are based on this woodpile template with optimized parameter specified in Figure 1.  
Since we have the optimized woodpile, we start to introduce a rectangular parallelepiped defect to create a cavity. The whole simulated woodpile template has 37 stacking layers in the z direction and 13 rods in each layer. The location of the defect is placed between the 19 th layer and the 20 th layer counting from the bottom. The refractive index of the defect is the same as that of the rods \(n_{defect}=3.3\) in this paper. Figure 2 shows the location of the defect in one vertical period (four stacking layers in \(z\) axis). The defect is placed between the middle layers and the height of the defect is \(2h\) which makes it connect with both rods in the upper and lower layer. In both middle layers, the center of defect is also between two adjacent rods.  
Figure 1: (**a**) Parameters of the woodpile structure defined in the simulation: vertical period \(c\), height of rods \(h=c/4\), distance between adjacent logs \(d=c/\sqrt{2}\) and width of rods \(w\). The refractive index of rods is \(n_{rad}=3.3\), while the background material is air \(n_{air}=1\); (**b**) The calculated band structure by using plane-wave expansion (PWE) method [52] with optimized parameters. The complete photonic bandgap reaches a maximum value of 15.88% when \(w=0.21c\)[20]. The insets illustrate the symmetric points in Brillouin zone of the face-centered-cubic (FCC) lattice relative to the woodpile and the calculation routes to obtain the complete photonic bandgap.  
Figure 2: The location of the defect in one vertical period (four stacking layers in the \(z\) direction). (**a**) view of the \(xz\)-plane (**b**) view of the \(yz\)-plane (**c**) view of the \(xy\)-plane (from the top) and (**d**) the free 3D view. \(L\) is the size of the defect in \(x\) and \(y\) direction while the height is always \(2h\) to make it connect with both rods at the bottom and top.  
Woodpile is face-center-cubic (FCC) structure, which could be considered as a rod-connected diamond lattice. As shown in Figure 3, rods in different colors in rod-connected diamond lattice (a) stand for different layers in woodpile lattice (b) [8]. The woodpile structure is obtained by substituting short rods in rod-connected diamond lattice with a single long rod. The position of defects shown in this paper is thus chosen from the body center of the lattice to get the most symmetrical position.