## 2 Ray Tracing Using Gaussian ABCD Matrix  
Figure 1 shows the propagation of a beam through a lensed optical fiber composed of an SMF and a CSF section. The beam propagating through the SMF with a core diameter of 9 \(\upmu\)m can be approximated as a Gaussian paraxial beam, and therefore, we can apply the ABCD matrix law with complex parameter q(z) to define the beam waist (\(\omega\)) and the radius of wavefront curvature (R) [18] of the beam as a function of the axial distance z. Consequently, we can determine a focal length L\({}_{\ell}\), the distance from the end of the lens to the plane where the beam has a minimum beam waist (\(\omega_{2}\)), as shown in Figure 1.  
Figure 1: Schematic of Gaussian-beam propagation through a lensed optical fiber.  
The evolution of the q-parameter can be calculated with the ABCD matrix, whose elements are given in matrix M. Parameter q\({}_{1}\) of the beam at the end of the SMF is related to the parameter q\({}_{2}\) at the focal plane as:  
\[\text{q}_{2}=\frac{\text{Aq}_{1}+\text{B}}{\text{Cq}_{1}+\text{D}}\text{ and }\text{M}=\text{M}_{34}\text{M}_{23}\text{M}_{12}\text{M}_{01}=\left[\begin{array}[] {cc}\text{A}&\text{B}\\ \text{C}&\text{D}\end{array}\right], \tag{1}\]  
where matrices M\({}_{01}\), M\({}_{12}\), M\({}_{23}\), and M\({}_{34}\) track the ray paths as per the paraxial optical approximation. Here, each of the matrices was calculated as [19; 20]  
\[\text{M}_{01}=\left[\begin{array}{cc}1&0\\ 0&\frac{\text{n}_{0}}{\text{n}_{1}}\end{array}\right],\text{M}_{12}=\left[ \begin{array}{cc}1&\text{L}_{\text{C}}\\ 0&1\end{array}\right],\text{M}_{23}=\left[\begin{array}{cc}1&0\\ \frac{\text{n}_{2}-\text{n}_{1}}{\text{n}_{2}\text{R}}&1\end{array}\right], \text{M}_{34}=\left[\begin{array}{cc}1&\text{L}_{\text{f}}\\ 0&1\end{array}\right] \tag{2}\]  
Here, M\({}_{01}\) denotes the ray matrix of the beam diverging from the boundary between the SMF and CSF, with n\({}_{0}\) = n\({}_{1}\). Moreover, M\({}_{12}\) describes the propagation of the beam through the CSF along geometrical path L\({}_{\text{C}}\) instead of optical path nL\({}_{\text{C}}\), which is the length L\({}_{\text{C}}\) between the exits of the SMF and the CSF in the case of the paraxial approximation. In addition, M\({}_{23}\) describes the path of the ray refracted by the lens and M\({}_{34}\) the ray path from the curvature of the lensed optical fiber with focal length L\({}_{\text{f}}\), with n\({}_{1}\) being the refractive index of the optical fiber of the CSF and n\({}_{2}\) the refractive index of air.  
At the focal plane, because the Gaussian beam has an infinite radius of curvature there, we can calculate L\({}_{\text{f}}\) and \(\omega_{2}\) as a function of R and L\({}_{\text{C}}\):  
\[\text{L}_{\text{f}}=\frac{-\alpha-\text{a}^{2}\Big{(}\alpha\text{(L}_{\text{ C}})^{2}+\beta\text{L}_{\text{C}}\Big{)}}{\alpha^{2}+\text{a}^{2}\Big{(} \alpha\text{L}_{\text{C}})^{2}+2\alpha\beta\text{L}_{\text{C}}+\beta^{2} \Big{)}}\text{ and }\omega_{2}=\omega_{1}\sqrt{\left(\frac{\text{n}_{1}}{\text{n}_{2}} \right)\frac{\text{A}^{2}+\text{a}^{2}\text{B}^{2}}{\text{AD}-\text{BC}}}, \tag{3}\]  
where \(a=-i\frac{\lambda}{n_{1}\tau_{\text{A}0}\omega_{2}}\), \(\alpha=\frac{\text{n}_{2}-\text{n}_{1}}{\text{n}_{2}\text{R}}\) and \(\beta=\frac{\text{n}_{1}}{\text{n}_{2}}\).  
Figure 2 shows the behavior of L\({}_{\text{f}}\) calculated as functions of R and L  
From Figure 2, we can estimate that the R value must lie between \(\sim\)70 \(\upmu\)m and 120 \(\upmu\)m and the LC value must lie between \(\sim\)275 \(\upmu\)m and 400 \(\upmu\)m to obtain the desired focal length of 375 \(\upmu\)m (as indicated by the thick red contour line). The yellow region inside the red contour line in Figure 2 shows the LC and R regime required for converging focal lengths instead of diverging negative focal lengths.