# The Low-Loss Spot Size Converter for Alignment with Cleaved Single Mode Fiber
## 2 Design of Materials and Methods
### The 2-D Inversely-tapered Waveguide  
Inversely-tapered waveguides are usually used to couple light with mode fields of different spot sizes and shapes [2]. When the mode field from the cleaved SMF is incident on SSCs, the mode field will be smoothly guided into the Si waveguide as the Si waveguide cross-section increases. In order to reduce the loss, the design of the tapered Si waveguide needs to comply with the adiabatic transmission principle [18]. In addition to this, the roughness of the taper waveguide's surface could lead to scattering losses in fabrication. The scattering loss of the Si waveguide could be as low as 0.5 dB/cm [19]. The length of our designed device is relatively short, so the scattering loss is negligible in the simulation. The structure of the 2-D inversely-tapered waveguide is shown in Figure 6. The tapered waveguide tip width is 200 nm and the height is 60 nm in our design. Along the beam propagation direction, the width/height of the waveguide increase linearly to 500 nm/220 nm. The narrowest width of the tapered waveguide is set at 200 nm to be compatible with the level of lithography limitation at the current commercial silicon photonics fab [20]. The technology for fabricating linearly-tapered waveguides in the horizontal direction is mature. Silicon structures in the vertical direction can also be fabricated by gray-scale technology [21]. Grayscale technology refers to the introduction of gray-scale masks in reactive ion etching (RIE) to control the etching depth at different positions so as to realize the fabrication of Si waveguide structures in the vertical direction, which has been used in industrial manufacturing to fabricate micro-optical devices due to the low cost and the mature process [22]. Therefore, the linearly-tapered structure in the vertical direction in the 2-D inversely-tapered waveguide could be fabricated at a low cost by using gray-scale masks in RIE. Outside the 2-D inversely-tapered waveguide, two  
Figure 5: The field distributions of different modes. (**a**) The TE mode at the end face of the GRIN waveguide; (**b**) the fundamental TE mode at the beginning of the 2-D inversely-tapered waveguide. SiO\({}_{2}\) cladding layers with different refractive indices were designed, which can both be accurately defined by plasma-enhanced chemical vapor deposition (PECVD) [23; 24]. In our design, the length of the 2-D inversely-tapered waveguide has been optimized by eigenmode expansion (EME) simulation and was double-verified by 3D-FDTD simulation for comparison. The EME simulation could calculate the mode propagation coefficients for each part of the waveguide, which could optimize the length of the tapered waveguide to enable adiabatic transmission. As shown in Figure 7a, for the TM mode, the coupling loss is close to the minimum when the length of the tapered waveguide exceeds 200 \(\upmu\)m. In order to make the coupling loss also close to the minimum for the TE mode, the optimal length of the tapered waveguide can be set to 300 \(\upmu\)m. At this optimized length, it can be obtained that the loss of TE mode is 0.12 dB and the loss of TM mode is 0.05 dB for the full structure of the 2-D inversely-tapered waveguide. Figure 6: The scheme of the 2-D inversely-tapered waveguide. (**a**) With no cladding layer; (**b**) with Cladding 1 layer; (**c**) with Cladding 1 and Cladding 2 layers. Figure 7: The 3D-FDTD and EME simulation results of the 2-D inversely-tapered waveguide. (**a**) The coupling loss of the 2-D inversely-tapered waveguide as a function of the length of the 2-D inversely-tapered waveguide; (**b**) the coupling loss of the SSC as a function of \(\Delta n\). The refractive indices and dimensions of the Cladding 1 and Cladding 2 have been specially optimized in order to reduce the mode mismatch loss between the GRIN waveguide and the 2-D inversely-tapered waveguide. The optimized refractive index difference between Cladding 1 and Cladding 2 is as small as 0.002.