## 5 Coupling Efficiency Experiment and Discussion
It can be seen from Figure 6 that the trend of experimental results is consistent with the theoretical results, but there is little discrepancy between the experimental and theoretical results due to some sources of error. The biggest reason for the discrepancy is that the simplified model only takes distributions of the input beam into consideration, without the distributions of HC-PCF. The second  
Figure 5: Initial distance measurement between the GRIN fiber probe and HC-PCF: (**a**) original image, (**b**) binarization processing, and (**c**) image in pixels. Figure 6: Theoretical and experimental values of coupling efficiency. reason is that the MFD of HC-PCF is used as the integral range and the contribution of other areas within the core diameter is neglected. Fresnel reflection, operating error, and straightness error of micro-displacement can also cause deviation. Figure 6 shows that the coupling efficiency of the probe is lower than that of SMF when the coupling distance is less than 0.1 mm. However, with the increase of coupling distance, the coupling efficiency of SMF decreases gradually, while the coupling efficiency of the probe shows a trend of increasing first and then decreasing. The coupling efficiency of the GRIN fiber probe reaches a maximum value of 80.22% at 180 \(\upmu\)m, where the coupling distance is equal to the working distance of the probe, which corresponds to the theoretical calculation in Equation (5). As a comparison, the coupling efficiency of SMF at the same position has decreased to 33.45%. With the increase of coupling distance, the probe can obtain a higher coupling efficiency than SMF and gain a similar value when the distance is longer than 0.8 mm. Considering this, the incident end of HC-PCF can be placed at the range of 0.1 mm to 0.8 mm for this GRIN fiber probe, where the GRIN fiber probe can acquire a higher coupling efficiency than SMF when it is coupled with HC-PCF for gas sensing. The experimental results show that the coupling model of the GRIN fiber probe can have a much longer working distance than SMF, so a larger gap between two coupled fibers can be achieved. The larger gap may make it easier for gas to enter the HC-PCF, so it takes less time to let the gas fill the HC-PCF before measurement, thereby reducing the system response time. In addition, the model of the GRIN fiber probe can obtain a higher coupling efficiency at the same distance, which can improve the sensitivity of the system. The higher coupling efficiency of the model means the light entering HC-PCF has a higher power, and the power of the output light from HC-PCF passing through the gas is proportional to the power of input light if the gas concentration is constant according to Lambert-Beer law. Therefore, the model of the GRIN fiber probe may display a bigger change in light power than the model of SMF when the concentration changes the same value; that is, the model of the GRIN fiber probe has a higher sensitivity. The optimization problem used to obtain the maximum working distance for each specified waist radius is proposed in Equation (9). To verify the relationship between the working distance and maximum coupling efficiency, the coupling efficiency experiments for five fabricated fiber probes in Table 1 have been conducted. Theoretical and experimental results are plotted in Figure 7 and measurement errors are shown as error bars. The maximum coupling efficiency in Figure 7 is slightly different from theoretical value due to similar errors in Figure 6, and the working distance of the probe is slightly less than the theoretical value because of some errors expounded in Section 4. The results show that the maximum coupling efficiency of the probe decreases and the working distance of the GRIN fiber probe increases with the  
Figure 7: Theoretical and experimental working distance and maximum coupling efficiency versus \(\gamma\). increase of the coupling distance; that is, the maximum coupling efficiency and working distance are contradictory.