# Photonic Crystal Cavity with a Thin Low-Index Layer for Silicon-Compatible Nanolight Source
## 3 Dispersion Curves of Slot Slab Waveguide  
First, to determine the total height (\(H\)) of the cylinder for the strong vertical confinement of lights, we investigated the effect of Si slab height on the effective index. We calculated the dispersion  
Figure 1: **(a)** Schematic diagram of the proposed photonic crystal (PhC) cavity consisting of 19 \(\times\) 19 square lattice of cylinders; **(b)** In each cylinder with a radius of r and total height of \(H\), a thin SiO\({}_{2}\) layer, indicated in cyan, with a thickness of \(t\) is sandwiched between thick Si layers. The central hole has a smaller radius of \(r_{d}\), and the four nearest neighbor rods are shifted into the center over a distance of \(d\) from their original positions, as denoted by red dotted circles. curves of a transverse magnetic (TM) Si slab waveguide mode by increasing the height (\(H\)) of the slab, as shown in Figure 2a. Here, the dominant electric field is aligned in the height direction. When the height is 200 nm (red), the dispersion curve follows the light line for small wavevectors and long wavelengths because of weak vertical confinement in the slab. As the height increases, the dispersion curve approaches the dispersion line of an infinitely thick Si slab waveguide (dotted blue line). By using the dispersion curves of Figure 2a, we obtain the effective index of the Si slab waveguide mode as a function of the height (\(H\)) for a wavelength of 1550 nm, as shown in Figure 2b. As height increases, the effective index increases, and when height is larger than 800 nm, the index approaches the bulk index of Si, 3.48. Since the effective index of the slab with a height of 1000 nm is 3.4, which is almost the same value of bulk Si, we chose a total rod height of 1000 nm. In addition, the fabrication of Si rods with a height of 1000 nm can be accomplished conventionally using standard Si CMOS fabrication technology. Next, we calculated the effective index of the SiO\({}_{2}\) slot waveguide as a function of slot thickness (t), as shown in Figure 3a, where the low-index SiO\({}_{2}\) slot is sandwiched between high-index Si materials at the center of the Si slab with a height (\(H\)) of 1000 nm. Here, a TM-like waveguide mode with the dominant electric field in the vertical direction to the slab is considered. As the slot thickness increases, the effective index decreases monotonically, because the waveguide mode experiences more low-index material (SiO\({}_{2}\)) than high-index material (Si). For example, the effective indexes in the slots with thicknesses of 10 nm and 50 nm are 3.29 and the 3.16, respectively. Figure 3b shows the strong confinement of the z-directional electric field intensity profiles for the waveguides of the 10-nm and 50-nm thick slots, where the total height is fixed at 1000 nm. The electric field is more strongly confined within the 10-nm thick slot than in the slot thickness of the 50-nm thick slot. In the electric field intensity line graphs of Figure 3c for different slot thickness, the high electric field intensity is strongly confined in the slot region, as shown in the mode profiles of Figure 3b. In addition, the 10-nm thick slot shows the highest electric field intensity in the slot, and the peak intensity decreases with increasing slot thickness. Here, the electric field intensity is normalized by the total electric field energy of the waveguide mode for each slot structure. The electric field intensity is related to enhance the emission of the rare earth emitter embedded in the low-index material by the Purcell effect. In addition to the electric field intensity, the confinement factor denoted by Equation (1) is also considered for the proposed structure as a candidate nanolaser, because the confinement factor can yield an effective modal gain by Equation (2). Therefore, we calculated the confinement factor by  
Figure 2: (**a**) Dispersion curves of a transverse magnetic (TM) silicon (Si) slab waveguide mode for different heights (\(H\)) of the slab, ranging from 200 nm to 1000 nm.