## 5 Conclusions  
We propose and demonstrate a new method for fabricating precisely a lensed fiber with a desired focal length by cleaving a CSF using an ultrafast femtosecond laser without appreciable thermal effects and forming the radius of curvature at the end of this optical fiber using a CO\({}_{2}\) laser. In our study, a precisely cleaved segment of a CSF was obtained using a femtosecond laser and the segment was attached to a long SMF. The CO\({}_{2}\) laser was used to melt the end of the coreless fiber to precisely shape the radius of curvature with a precision of \(\leq\)1 \(\upmu\)m as a function of the exposure time and the power of the CO\({}_{2}\) laser beam, thereby resulting in a focal length as close as possible to the design value using simulated parameters in Figure 2. This technique forms an alternative to other conventional methods for forming fiber lenses for biosensing applications. The method can be used to realize a broad range of focal lengths with variable CSF lengths and curvature radii of the lenses for application to various imaging requirements.  
In the study, we used a CO\({}_{2}\) laser to fabricate a lensed optical fiber with L\({}_{\text{C}}\) = 308 \(\upmu\)m and R = 72.8 \(\upmu\)m at the end of the CSF. The simulated and calculated focal lengths were compared with the actually measured focal length. Furthermore, the MFD of the SMF-28 communication optical fiber was varied from 9.9 \(\upmu\)m to 10.9 \(\upmu\)m, and the obtained focal length of 406 \(\upmu\)m with the mean MFD differed from the designed and theoretical focal distance of 375 \(\upmu\)m. This difference between the measured and the calculated focal lengths can be attributed to the uncertainty associated with the MFD.  
Figure 8: Variation in the focal length as a function of the MFD size.  
The beam size at the focal length of 406 \(\upmu\)m was also measured using the knife-edge method, and it was estimated as 27 \(\upmu\)m at 1/e\({}^{2}\) of the peak intensity. The theoretical beam size was 23 \(\upmu\)m at the 2\(\upomega_{\text{z}}\) point, corresponding to an error of 4 \(\upmu\)m with respect to the experimental value. These findings indicate that it is possible to fabricate a lensed optical fiber with a desired focal length by using femtosecond and CO\({}_{2}\) lasers. Furthermore, lenses with optimum converging and diverging focal lengths can be obtained by varying L\({}_{\text{C}}\) and R for diverging or converging the light. Therefore, these lenses can be used as fine optical elements in OCT or MEMS.  
In the study, we applied the paraxial approximation based on the ABCD matrix to trace the focal point; however, in future, it would be worthwhile to confirm the difference between the measured and the calculated focal lengths with general approaches using ray-tracing methods. We also plan to examine the systematic statistical repeatability and precision of fabricated micro-lenses using a large number of fabricated samples to determine the shortest range of focal lengths possible with the method.  
Conceptualization, [PERSON]-D.L., [PERSON]-[PERSON] and [PERSON]-[PERSON]; methodology, [PERSON]-D.L., [PERSON]-[PERSON], [PERSON]-[PERSON] and [PERSON]-[PERSON]; software, [PERSON]-D.L. and [PERSON]-T.K.; validation, [PERSON]-B.S., [PERSON]-H.L. and [PERSON]-T.K.; formal analysis, [PERSON]-D.L. and [PERSON]-T.K.; investigation, [PERSON]-D.L. and [PERSON]-T.K.; writing--original draft preparation, J.-T.K.; writing--review and editing, B.-H.L. and [PERSON]-T.K.; visualization, J.-T.K.; supervision, [PERSON]-[PERSON] and [PERSON]-[PERSON]; project administration, [PERSON]-B.S. and [PERSON]-T.K. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.