# The Low-Loss Spot Size Converter for Alignment with Cleaved Single Mode Fiber
## 2 Design of Materials and Methods
### GRIN Waveguide  
GRIN waveguides have a gradual refractive index distribution, and light beams transmitted within GRIN waveguides could be compressed or enlarged [12]. In this paper, the refractive index profile of the ion-implanted GRIN waveguide is a symmetric Gaussian profile in the horizontal direction, as shown in Figure 2b. The refractive index profile  
Figure 1: The scheme of the proposed SSC. is a half-parabolic profile in the range of 0-9 \(\upmu\)m and a Gaussian profile in the range of 9-11 \(\upmu\)m in the vertical direction as shown in Figure 2c. Here, we give the theory of using ion implantation technology to fabricate the GRIN waveguide of our design. Ion implantation is a mature technique for waveguide fabrication in the field of integrated optics, that is compatible with current commercial silicon photonics technology [13]. Ion implantation can precisely change the refractive index of the material by controlling the type of ions, ions energy, and ions dose. Ion energy can range from several thousand electron volts to over ten million electron volts [14]. In the vertical direction, a single ion implantation can form a Gaussian profile of the refractive index distribution, and with multiple ion implantations, it is possible to achieve a customized refractive index profile [15]. The lateral Gaussian distribution can be naturally formed by using a lithograph-defined window within the ion implantation [16]. Therefore, the control of the refractive index in the vertical direction is more important for the GRIN waveguide. As shown in Figure 3, the distribution of vacancies formed after multiple implantations of N\({}^{+}\) into SiO\({}_{2}\) simulated by the transport of ions in matter (TRIM) software could fit the refractive index profile of our designed GRIN waveguide, which demonstrates that our designed GRIN waveguide can be fabricated by multiple implantations of N\({}^{+}\) into SiO\({}_{2}\). The vacancy distribution could linearly fit the refractive index distribution within the error tolerance after annealing [17]. Therefore, the refractive index distribution formed by the N\({}^{+}\) implanted into the SiO\({}_{2}\) can be predicted by the vacancy distribution obtained from the TRIM simulation. For the type of ions, N\({}^{+}\) implanted into SiO\({}_{2}\) can increase the refractive index of SiO\({}_{2}\) by up to 6%. Therefore, the refractive index range is from 1.458 to 1.482 in our design. In addition to this, ion implantation introduces a high-temperature annealing process [17]. Therefore, the subsequent high-temperature environment will not affect the  
Figure 2: Schematic and the refractive index transverse distribution of the GRIN waveguide. (**a**) The 3D view of the GRIN waveguide; (**b**) the refractive index profile in the horizontal direction (\(\gamma\)-axis); (**c**) the refractive index profile in the vertical direction (\(Z\)-axis). refractive index profile of the GRIN waveguide fabricated by ion implantation, which is beneficial for achieving optical integration. Here, the route of the light rays in the GRIN waveguide with a parabolic refractive index profile could be analyzed according to the geometric optics propagation theory [12]:  
\[y=y_{0}cos\left(\sqrt{A}z\right)+\frac{tan\theta}{\sqrt{A}}sin\left(\sqrt{A}z\right) \tag{1}\]  
where \(y_{0}\) and \(\theta\) are the incident position and the angle of incidence of the light rays, respectively. \(A\) is a constant related to \(n_{max}\), \(n_{0}\), and \(h_{0}\). Their relationship is described by:  
\[A=\frac{2}{h_{0}2}\left(1-\frac{n_{0}}{n_{max}}\right) \tag{2}\]  
where \(n_{max}\) is the maximum refractive index, and \(n_{0}\) is the refractive index of SiO\({}_{2}\), \(h_{0}\) is the thickness of the waveguide. According to Equation (1), when the angle of incidence of light rays is zero radians, regardless of whether the refractive index profile of the GRIN medium is parabolic or half-parabolic, the route of light rays within the GRIN medium is a sinusoidal function with a period \(P\)[12].