# The Low-Loss Spot Size Converter for Alignment with Cleaved Single Mode Fiber
## 2 Design of Materials and Methods
### GRIN Waveguide
\(P\) is represented by:  
\[P=\frac{2\pi}{\sqrt{A}} \tag{3}\]  
As shown in Figure 4a, when the light ray is incident vertically to the end face of the GRIN (with an incidence angle of 0), the periodic focus occurs at the focal position \(P_{l}\):  
\[P_{l}=\frac{(2l-1)}{4}P,l=1,2,3,\ldots \tag{4}\]  
Figure 3: The vacancies distribution and the refractive index profile in the vertical direction. (**a**) The vacancies distribution calculated by TRIM for multiple ion implantations; (**b**) comparison of the refractive index profile designed by us and the sum vacancies distribution calculated by TRIM. Obviously, the first focused position of light rays is at a distance of \(P/4\) according to Equation (4), and it is clear that the mode field size is the smallest at this position. Therefore, the shortest length of the GRIN waveguide should be set as \(P/4\). Combining Equations (2) and (3), the shortest length of the GRIN waveguide can be predicted by:  
\[l=\frac{P}{4}=\frac{\pi}{2\sqrt{A}}=h_{0}\frac{\pi}{4}\sqrt{\frac{2n_{max}}{n_{ max}-n_{0}}} \tag{5}\]  
The MFD of the light source was set to 8 \(\upmu\)m, and the simulation of the GRIN waveguide was performed via FDTD software. As shown in Figure 4b, the loss from the cleaved SMF to the beginning of the 2-D inversely-tapered waveguide can be as low as 0.24 dB when the GRIN waveguide length is 105 \(\upmu\)m. When \(h_{0}\) is 11.8 \(\upmu\)m, \(n_{max}\) is 1.46, and \(n_{0}\) is 1.458, the length calculated by Equation (5) is 99 \(\upmu\)m, which is close to the optimized length (105 \(\upmu\)m) by 3D-FDTD simulation. From the cleaved SMF to the taper tip position of the 2-D inversely-tapered waveguide, this part is the GRIN waveguide section, and the loss of the GRIN waveguide section consists of three parts. The first part of the loss is the reflection loss between the cleaved SMF and the GRIN waveguide, which could be minimized by the selection of a suitable refractive index matching liquids [8]. The second part of the loss is the transmission loss of the GRIN waveguide. The transmission loss is generally small and is neglected according to the 3D-FDTD simulations. The third part of the loss is the mode mismatch loss between the GRIN waveguide and the 2-D inversely-tapered waveguide, which is the main source of the loss. Therefore, reducing the mode mismatch loss is the focus of our discussion. Making the effective refractive indices of the two modes close to each other is an effective way to reduce the mode mismatch loss. The refractive indices and dimensions of Cladding 1 and Cladding 2 have been optimized in order to make the effective refractive index of the fundamental mode of the 2-D inversely-tapered waveguide close to that of the mode at the  
Figure 4: The 3D-FDTD simulation results of the GRIN waveguide. (**a**) The mode field distribution of the GRIN waveguide; (**b**) the loss of the GRIN waveguide section as a function of the length of the GRIN waveguide. end of the GRIN waveguide. In theory, the mode overlap integral can represent the mode mismatch loss to a certain extent. The modal overlap integral can be calculated by [5]:  
\[\eta=\frac{\left|\int E_{1}E_{2}dA\right|^{2}}{\int\left|E_{1}\right|^{2}dA\int \left|E_{2}\right|^{2}dA} \tag{6}\]  
where \(E_{1}\) and \(E_{2}\) are the complex electric fields for the mode at the output end of the GRIN waveguide (Figure 5a) and the TE fundamental mode at the starting tip of the 2-D inversely-tapered waveguide section (Figure 5b), respectively. Due to the fact that Equation (6) only considers the electric field and does not take into account other factors such as refractive indices or the contribution of higher-order modes, there is a certain error between the mode mismatch loss calculated by Equation (6) and the loss obtained by 3D-FDTD. The loss obtained by 3D-FDTD is 0.24 dB, while the mode mismatch loss calculated by Equation (6) is 0.41 dB.