## 1 Introduction  
Optical probes [1, 2] are widely used in optical coherence tomography (OCT) and other sensing and imaging systems for investigating the internal structures of biological tissues and the intraluminal imaging of various organs. However, conventional probes are bulky and unwieldy, and thus, researchers have considered replacing them with lensed optical fibers [3, 4, 5]. In particular, a set of two lensed optical fibers can replace existing bulky optical lens systems for constructing compact microelectromechanical systems (MEMS) that can afford significantly improved imaging performances [6].  
There are several conventional methods for fabricating the lensed optical fibers used in such compact MEMS setups. Current methods for introducing surface curvature at the optical-fiber tip include curing a photoreactive polymer with ultraviolet light [7], polishing the end tip of the fiber [8], melting the optical-fiber tail via arc discharge [6, 9], and fusing a GRIN lens with the optical fiber [10].  
Two of the abovementioned methods have found wide usage in the fabrication of lensed optical fibers. The first method involves fusing a GRIN lens with a gradually varying refractive index into an optical fiber, which requires very precise cleaving. In this method, a section of the GRIN lens is precisely cut off to focus a beam at the right position; however, the method requires precise calculations of the optical beam path within the GRIN lens. The second method involves attaching a coreless silica fiber (CSF) to an optical fiber and using an arc discharge to shape the radius of curvature at the end of the CSF to form a lens [9]. With this approach, a beam from a lensed optical fiber can be made to converge or diverge depending on the CSF length and the radius of curvature of the CSF tip. This method also requires the precise cleaving of optical fibers to precisely adjust the CSF length. In addition to precise cleaving, it is also necessary to make the CSF end suitably curved. Thus, it is difficult to fabricate a lensed fiber using a CSF when compared with GRIN-lens-based fabrication. However, a specific focal length of the GRIN lens for a lensed optical fiber cannot be easily obtained for customized needs. Furthermore, precise cleaving is also necessary for using a GRIN lens when compared with the CSF-based method.  
Against this backdrop, in this work, we apply a new method to fabricate lensed optical fibers with desired focal lengths by using femtosecond and CO\({}_{2}\) lasers. Here, the femtosecond laser is used to precisely cleave a CSF [11; 12; 13] directly without being heavily affected by thermal effects. We note that our precise CSF cleaving technique using a femtosecond laser has been implemented previously, demonstrating its superiority over conventional micromachining, which cannot be used to precisely fabricate an optical fiber [12]. Furthermore, the proposing method is simpler than the conventional method involving polishing processes together with thermal processing [14; 15].  
The CO\({}_{2}\) laser with a wavelength of 10.6 \(\upmu\)m and having a suitable absorption rate for silicon material is used to melt the CSF tip to form the lens curvature. The method also yields a more uniform and consistent radius of curvature with a precision of \(\leq\)1 \(\upmu\)m at the end of the optical fiber along with the desired focal length when compared with the widely used conventional arc discharge method [6; 9]. The combination of cleaving a fiber and forming a lens on the optical-fiber tip described above provides a versatile approach to fabricate lensed optical fibers with precise focal lengths.  
In our study, the focal length and size of the beam emerging from the manufactured lensed optical fibers are measured using a knife-edge method [16] and compared with the counterpart theoretical values. For calculation, we use the ABCD method for a Gaussian beam [17] to analyze beam propagation at the CSF section after beam divergence from the lead single-mode fiber (SMF) and focusing by the lensed fiber.