# Tailorable and Broadband On-Chip Optical Power Splitter
## 3 Results
Based on the simulation results in Figure 3a, an upper gap size \(\mathrm{d}_{1}\) for a desired coupling ratio \(\mathrm{R}\) with a fixed lower gap size \(\mathrm{d}_{2}=150\) nm is given by  
\[d_{1}=\frac{1}{0.01}\times ln\Big{(}\frac{2.25}{R-0.004}\Big{)}, \tag{1}\]  
where the coupling ratio \(\mathrm{R}\) has a value between 0.004 and 0.5 and this equation is quoted for \(\mathrm{d}_{1}\) in nanometers. This equation can be used to determine an upper gap size for a desired coupling ratio. For example, to design a 90:10 beam splitter, the coupling ratio should be \(\mathrm{R}=0.1\), and the required upper gap size is \(\mathrm{d}_{1}=315\) nm with a fixed lower gap size of \(\mathrm{d}_{2}=150\) nm. Figure 3b shows the simulated insertion loss of the structure, \(\left[\left[\mathrm{P}_{\mathrm{in}}-\mathrm{P}_{\mathrm{up}}-\mathrm{P}_{ \mathrm{down}}\right]/\mathrm{P}_{\mathrm{in}}\right]\). By increasing the upper gap size from 150 nm to 600 nm, the insertion loss increases from 0.1% to 0.25% (\(\sim\)0.01 dB), which is negligibly low considering fabrication imperfections. In practice, light scattering from side-wall surfaces or fabrication error will be the dominant loss mechanisms. Figure 4 shows the transmittance through the upper and lower channels against input wavelength. The computational simulation is performed over a wavelength range of 1.5 \(\mathrm{\SIUnitSymbolMicro m}\) to 1.6 \(\mathrm{\SIUnitSymbolMicro m}\) for various upper gap sizes (\(\mathrm{d}_{1}=\) [150 nm, 200 nm, 315 nm, 590 nm]) corresponding to the power splitting ratios of 50:50, 70:30, 90:10, and 99:1, respectively. The simulated power split ratio has a deviation of less than 6.5% from its desired value within the overall simulation wavelength range. The profile of evanescent waves depends on the input wavelength, which may cause wavelength dependence. To reduce wavelength dependence, the lower gap size is optimized to 150 nm, which is larger than the characteristic decaying length of the input light but small enough to support slot waveguide modes  
Figure 3: Transmittance and insertion loss against the upper gap size, \(\mathrm{d}_{1}\). (**a**) The gray diamonds represent the total output power, and the red circles (blue squares) indicate the downward (upward) output, respectively. The theoretical fit obtained using an exponential function is shown as dotted lines atop the simulation data (red circles and blue squares). (**b**) Insertion loss versus upper gap size. While the upper gap size varies from 150 nm to 600 nm, the insertion loss of each gap is less than 0.3%. with weak wavelength dependence. The larger gap size reduces wavelength dependence, but leads to high optical power leakage. The coupling ratio deviation is defined by \(\left(\frac{\mathrm{abs}\big{(}\mathrm{P_{down,\ target}}\mathrm{-P_{down}}\big{)}}{ \mathrm{P_{down,\ target}}}\times 100\right)\%\), and the maximum deviations in wavelength range between 1.5 \(\upmu\)m and 1.6 \(\upmu\)m (1.2 \(\upmu\)m-1.8 \(\upmu\)m) are 0.49% (2.54%) for 50:50, 2.85% (12.30%) for 30:70, 0.27% (7.11%) for 10:90, and 0.86% (1.20%) for 1:99. Due to the slowly varying material refractive index in wavelength, the evanescent profile, as well as the splitting ratio, change slowly in wavelength. However, even in 1.2 \(\upmu\)m-1.8 \(\upmu\)m range, the maximum deviation is about 10%, showing ultra-broadband tailorability compared to recent methods of broadband power splitters [14; 18].