# Photonic Crystal Cavity with a Thin Low-Index Layer for Silicon-Compatible Nanolight Source
## 4 Design of Photonic Crystal Cavity Mode
In the XZ mode profile, the inset shows the electric field confinement in the low-index SiO\({}_{2}\) layer of the defect cylinder; (**c**) Resonant wavelengths (black) and \(Q\) factors (blue) as functions of the shifted distance of four nearest cylinders while the center defect radius is fixed at 90 nm. The highest \(Q\) factor of 15,000 is obtained when the nearest cylinders shifted to the inner side over a distance of 60 nm. Based on the optimized cavity structure, we investigated the emission enhancement of rare-earth dopants that can be embedded in the thin SiO\({}_{2}\) layer by calculating the Purcell factor. The magnitude of the enhancement is given by the following Purcell factor \(F_{P}\) when the emitter is placed at the intensity maximum of the mode profile [21], which corresponds to the proposed cavity. The dopant emitter is placed at the 10-nm thick SiO\({}_{2}\) layer of the center cylinder, in which the spatial overlap between the cavity mode and the emitter can be achieved:  
\[F_{P}=\frac{3}{4\pi^{2}}\bigg{(}\frac{\lambda_{c}}{n_{\text{SiO}_{2}}}\bigg{)} ^{3}\bigg{(}\frac{Q}{V}\bigg{)} \tag{3}\]  
where \(\lambda_{c}\) is the resonant wavelength of the cavity, \(n_{\text{SiO}_{2}}\) is the refractive index of silica, \(V\) is the mode volume, and \(Q\) is the \(Q\) factor. To calculate the Purcell factor, we need to know the mode volume of the cavity mode. The mode volume can be calculated by under the following formula [13]:  
\[V_{\text{mod}}^{\text{eff}}=\frac{\int_{V}e\bigg{(}\overset{\rightarrow}{r} \bigg{)}\bigg{|}\overset{\rightarrow}{E}\bigg{(}\overset{\rightarrow}{r} \bigg{)}\bigg{|}^{2}d^{3}r}{\max\bigg{[}e\bigg{(}\overset{\rightarrow}{r} \bigg{)}\bigg{|}\overset{\rightarrow}{E}\bigg{(}\overset{\rightarrow}{r} \bigg{)}\bigg{|}^{2}\bigg{]}} \tag{4}\]  
where \(e\bigg{(}\overset{\rightarrow}{r}\bigg{)}\) is the permittivity as a function of the radial direction, and \(\overset{\rightarrow}{E}\bigg{(}\overset{\rightarrow}{r}\bigg{)}\) is the electric field of the cavity mode as a function of the radial direction at the cavity center. On the other hand, in case the maximum electric field is found in a low-index region, the mode volume can be represented as a normalized unitless effective mode volume [22]. The normalized mode volume is defined as follows:  
\[\widetilde{V}_{\text{mod}}^{\text{eff}}=\frac{\int_{V}e\bigg{(}\overset{ \rightarrow}{r}\bigg{)}\bigg{|}\overset{\rightarrow}{E}\bigg{(}\overset{ \rightarrow}{r}\bigg{)}\bigg{|}^{2}d^{3}r}{e\bigg{(}\overset{ \rightarrow}{r}\max\bigg{)}\max\bigg{[}\bigg{|}\overset{\rightarrow}{E} \bigg{(}\overset{\rightarrow}{r}\bigg{)}\bigg{|}^{2}\bigg{]}}\Bigg{(}\frac{2n \bigg{(}\overset{\rightarrow}{r}\max\bigg{)}}{\lambda}\Bigg{)}^{3} \tag{5}\]  
\[\widetilde{V}_{\text{mod}}^{\text{eff}}=V_{\text{eff}}\Bigg{(}\frac{2n\bigg{(} \overset{\rightarrow}{r}\max\bigg{)}}{\lambda}\Bigg{)}^{3} \tag{6}\]  
where \(\overset{\rightarrow}{r}\max\) is the position of the maximum electric field. Therefore, \(n(\overset{\rightarrow}{r}\max)\) is \(n_{\text{SiO}_{2}}=1.48\). \(e\bigg{(}\overset{\rightarrow}{r}\max\bigg{)}\) is the maximum permittivity of the radial direction. The normalized unitless effective mode volume is 0.64. For the optimized cavity structure with \(r_{d}\) = 90 nm and \(d\) = 60 nm, the mode volume is obtained as \(V_{\text{mod}}^{\text{eff}}\) = 0.01 \(\mu\)m\({}^{3}\) (= 0.0089\(\frac{\lambda_{c}}{n_{\text{SiO}_{2}}}\))\({}^{3}\)\(\mu\)m\({}^{3}\)), where, \(\lambda_{c}\) is 1537 nm and \(n_{\text{SiO}_{2}}\) is 1.48. By using a wavelength of 1537 nm, \(Q\) factor of 15,000, and refractive index of SiO\({}_{2}\) of 1.48, the Purcell factor \(F_{P}\) is 127,000. The large Purcell factor means that the proposed cavity structure can enhance the emission of the rare-earth dopants in the low-index layer and will be useful as a silicon-compatible nanolight source.