## 4 Fabrication and Measurement of GRIN Fiber Probe Samples  
To study the coupling efficiency of a GRIN fiber probe coupled with HC-PCF further, and to demonstrate the relationship between the waist radius and working distance of a GRIN fiber probe, GRIN fiber probe samples that meet requirements should be fabricated first. Following the fabricated steps of an ultra-small GRIN fiber probe in the author's research group, samples can be fabricated and measured using the fabricated system and measuring system [21]. The measuring system is the Beam Analyzer produced by Duma Optronics, which has a beam width/position resolution of about 1 \(\upmu\)m with a position accuracy of \(\pm\)15 \(\upmu\)m and power resolution of about 0.1 \(\upmu\)W with a power measurement accuracy of \(\pm\)10%.  
The parameters of materials are as follows. The center wavelength \(\lambda\) of the light source beam is 1.55 \(\upmu\)m, and the SMF core radius \(\omega_{0}\) is 4.5 \(\upmu\)m. The no-core fiber has a refractive index \(n_{0}\) of 1.486. The 50/125 GRIN fiber core has a core diameter of 50 \(\upmu\)m, outer diameter of 125 \(\upmu\)m, refractive index \(n_{1}\) at the axis of 1.497, and gradient constant \(g\) of 5.587 mm\({}^{-1}\). The selected no-core fiber and GRIN fiber are both produced by Prime Optical Fiber Corporation (POFC). The refractive index \(n_{2}\) of air is set as 1. The max length of \(L_{\text{max}}\) and \(L_{0\text{max}}\) can be calculated as 0.57 mm and 0.29 mm, respectively, according to Section 3 under these parameters. NKT Photonics' HC19-1550-01 hollow core photonic crystal fiber with a mode field diameter of 13 \(\upmu\)m is used. Five probes with different waist radii \(\omega_{f}\) were designed and fabricated.  
Theoretical and experimental parameters of the fabricated probes are shown in Table 1. The data of No. 1-No. 5 in Table 1 is the theoretical value. The corresponding length of the no-core fiber \(L_{\text{0 opt}}\) and length of the GRIN fiber \(L_{\text{opt}}\) can be calculated for each \(\gamma\) according to Equation (9). The specified lengths of no-core fibers and GRIN fibers were cut as calculated above and the actual length of fibers was recorded in No. 6-No. 10. After that, GRIN fiber probe samples were fabricated with these cut fibers added to SMF using the fabricated system and the beam of these samples was analyzed by the measuring system. Then, the measured working distance \(z_{\omega}\) and the probe waist radius \(\omega_{f}\) were recorded in Table 1 from No. 6-No. 10.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Types** & **No.** & \(\gamma\) & **(L\({}_{0}\)) opt (mm)** & **(L)\({}_{\text{opt}}\) (mm)** & \(\omega_{f}\) (\(\upmu\)m)** & **Max(\(z_{\omega}\)) (mm)** \\ \hline \multirow{6}{*}{Theoretical results} & 1 & 0.62 & 0.290 & 0.213 & 4.00 & 0.165 \\  & 2 & 0.92 & 0.290 & 0.171 & 6.00 & 0.278 \\ \cline{1-1}  & 3 & 1.23 & 0.290 & 0.151 & 8.00 & 0.376 \\ \cline{1-1}  & 4 & 1.54 & 0.290 & 0.139 & 10.00 & 0.463 \\ \cline{1-1}  & 5 & 1.85 & 0.290 & 0.131 & 12.00 & 0.538 \\ \hline \multirow{6}{*}{Experimental results} & 6 & 0.95 & 0.290 & 0.213 & 6.15 & 0.188 \\  & 7 & 1.35 & 0.290 & 0.171 & 8.80 & 0.270 \\ \cline{1-1}  & 8 & 1.56 & 0.290 & 0.151 & 10.14 & 0.320 \\ \cline{1-1}  & 9 & 1.82 & 0.290 & 0.139 & 11.80 & 0.400 \\ \cline{1-1}  & 10 & 2.33 & 0.290 & 0.131 & 15.16 & 0.530 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the GRIN fiber probe with a specified waist radius.  
The data in Table 1 shows that the actual waist radius and working distance of the probe are slightly different from the theoretical calculation. These errors may come from the error of the measuring device, the limitation of the fabrication device, and human operation error.