Micro-lens fibers function as both light steering and transmission media without the need for spatial light optics. Fiber lenses with high numerical aperture (NA) are particularly in demand in imaging, integrated photonic devices, and medical optics. For example, a higher NA fiber lens can resolve finer details during imaging and achieve higher quality imaging [1–3]. High-resolution in situ imaging of atoms in optical lattices can provide the ability to study quantum effects at the single particle level in modern quantum optics experiments [4,5]. In addition, high NA optical tweezers enable strong optical trapping of microparticles [6–8], which is essential for manipulating microlevel elements in medical optics.

Conventional fiber lenses are realized by melting the fiber facet or using UV adhesives to integrate spherical lenses on the fiber facet [9–12]. However, the NA of the conventional fiber lens can only go up to 0.6 due to significant fabrication difficulties [10–12]. Recently, various flat lenses, such as meta-lenses [13–15] and flat diffractive lenses [16,17], have seen a rapid development. They are considered to be good substitutes for spherical fiber lenses [18–22]. A variety of meta-lenses have been integrated on the fiber facet [23–27]. However, the typical value of their full width at half maximum (FWHM) of the focal spot size is about λ to 5λ with a comparatively low NA (up to 0.6) [27–30]. In addition, there exist significant challenges in the practical development of fiber meta-lenses due to low integrability and high fabrication complexity and cost [26,31]. Polymer fiber diffractive lenses have also been proposed [6,17,32,33]. Although they can achieve comparatively higher NA, such lenses are as thick as 5 μm with a large aperture of about 100 μm [6,17,33] due to their limited refractive index difference [34].

In recent years, graphene oxide (GO) and reduced graphene oxide (rGO) films with novel optical and physical properties have been proposed as promising materials for the fabrication of flat lenses. The wide bandwidth, ranging from the ultraviolet to terahertz regime, makes graphene family materials an ideal choice for broadband device design [35]. Furthermore, they have excellent mechanical robustness and outstanding temperature, chemical, and biological stability enabling applications in harsh environments such as strong acid, alkaline, high temperatures, high moisture, and lower earth orbital conditions. [36–39]. In addition, unlike the multi-step nanofabrication, etching, and transfer processes required for other meta-lenses, the fiber facet GO lens can potentially be flexibly and effectively fabricated without a mask using femtosecond laser direct writing (FLDW) technology. During the laser fabrication, the GO film can be photoreduced to rGO, and a large refractive index contrast of 0.8, which is two orders of magnitude larger than that of normal refractive materials in the visible band, large extinction coefficient, and thickness difference can be achieved [2,40–42]. Recent works have demonstrated exciting progress in the design method and fabrication of GO lenses on the glass or flexible substrates with various properties such as being coma aberration free and having a low fabrication cost [36,42–46].

In this work, we demonstrate the design and FLDW fabrication of a GO lens on the facet of a standard single-mode fiber (SMF) [Fig. 1(a)]. Ultra-high NA (0.89) is achieved with an ultra-thin GO lens (400 nm) with a diameter of only 12 μm, and the insertion loss is 2.1 dB, perfect for integration. Meanwhile, the near diffraction-limited focal spot (FWHM=0.68λ) has been verified theoretically and experimentally. Our proposed ultra-thin fiber GO lens has potential applications in high-resolution imaging, medical optics, and on-chip integration.

The fabrication scheme of the proposed fiber GO lens is shown in Fig. 1. Thin GO films can be reduced to rGO by a femtosecond laser. The amplitude and phase of the incident optical field are modulated by the rGO zones with a much higher refractive index and optical absorption. Designing the number and radii of the concentric rGO zones can realize constructive interference of the incident single-mode optical field in the focal region, generating a subwavelength focal spot. Rayleigh–Sommerfeld (RS) diffraction theory was applied to design and simulate the fiber GO lens. The study shows that it can accurately and quickly determine the radius and number of rings of the GO lens with desired NA and focal lengths without optimization [47]. As shown in Fig. 1(b), the incident fiber mode field U0 propagates in the fiber along the z direction. The flat GO lens attached to the fiber facet works as a diffractive lens. The incident mode field will be partly absorbed and diffracted by the GO and rGO zones as it passes through the GO lens. The modulated electrical field U1 can be expressed as follows: U1(x1,y1)=U0t(x1,y1)exp(−ikΦ(x1,y1)),(1)where k=2π/λ is the wave vector, λ is the wavelength of the incident beam, and t(x1,y1) and Φ(x1,y1) are the transmission coefficient and phase modulation factor of the GO lens. As shown in Fig. 1(b), the phase and intensity modulation factors will be different in the rGO and GO zones. According to the RS diffraction theory, when the incident mode field passes through a diffraction plane, the output far field in the observation plane (x2,y2) at a distance z can be written as U2(x2,y2,z)=12π∬x1O1y1U1(x1,y1)(−ik−1δ)exp(−ikδ)δ2zds.(2)

In this diffraction integral, δ stands for distance: δ=(x1−x2)2+(y1−y2)2+z2. To design the GO lens on the fiber facet, the incident electrical field U0 is set as the mode field of an SMF, and parameters such as the diameter D and focal length f need to be considered simultaneously. Considering the intensity distribution at the zero point at the observation plane at distance f, substituting x2=0,y2=0,z=f into Eq. (2), we get If=|U2(0,0,f)|2=|12π∬x1O1y1U1(x1,y1)(−ik−1δf)exp(−ikδf)δf2fds|2,(3)where δf stands for distance: δf=x12+y12+f2. To achieve the best modulation of the fiber mode field, the design of the key parameters, i.e., the ring number and the ring radius of the GO lens, should be limited by the diameter of the incident optical field. Meanwhile, to achieve the maximum constructive interference at the focal point, each ring position on the diffraction plane is given by two adjacent points satisfying dI/dr1=0. Finally, the number of rings and each ring radius of the GO lens with the diameter D and focal length f can be designed for a given mode field.

Taking advantage of the large refractive index difference between rGO and GO, the GO lens shows an extraordinary ability to modulate the incident light. Numerical simulation shows that the GO lens can generate subwavelength focal spot, i.e., a high NA (>0.7), which is close to the diffraction limit. To realize the fiber GO lens with the largest NA, the parameters, such as the focal length, the ring number, and the ring width, must be seriously considered during the lens design.

The micro scale of the fiber mode field severely limits the design of GO lenses. For a long focal length GO lens (>10μm), the radius of the rGO rings must be much larger than the effective mode field radius in SMF. In this case, almost no rGO rings can effectively modulate the incident mode field, resulting in a larger light spot and low NA. To design the fiber GO lens with the maximum NA working below 850 nm, the relationship between the designed focal lengths and the FWHM of the focal spot has been simulated (shown in Fig. 2). It is easy to see that a shorter focal length results in a smaller focal spot. If the designed focal length is set below 6 μm, the corresponding FWHM will be in the subwavelength scale. The shorter the focal length, the tighter the focusing is required and the smaller the minimum gap between the adjacent rGO rings will be (shown in Fig. 2). Considering the minimum rGO linewidth we can achieve, the focal length is set to be 3 μm to generate a smaller focal spot, which is 720 nm (0.85λ). This design allows the fiber GO lens to achieve the NA as high as 0.72.

The FLDW system (Innofocus Nanoprint3D) was used to fabricate the fiber GO lens. Studies have shown that femtosecond laser pulses can remove the oxygen functional groups of GO and reduce GO to rGO [2]. During the GO to rGO transition, the thickness of the rGO film decreases to about half of the initial GO film thickness, and dramatic refractive index change (Δn∼0.8) between rGO and GO over a wide wavelength range is more than one or two orders of magnitude larger than that of the conventional materials such as glass or polymers, indicating a large phase modulation range of the incident field [2,40,41,43]. Meanwhile, the significant change of the extinction coefficient provides high contrast in light absorption, which achieves efficient amplitude modulation. These properties make GO film an excellent candidate for flat diffractive devices.

In our fabrication, we use the vacuum filtration method to prepare the GO film. Filter membranes with 30 nm microspores ensure that the GO films will stay on the membranes and water can pass through the filter. By controlling the concentration and the amount of GO suspension, GO films with different thicknesses can be prepared. After filtration, the membrane with the GO film was dried at 80°C for 10 min. By using ethanol, the GO film can be peeled off and transferred to glass substrate or facet of SMF with physical contact ceramic ferrule (PCCF).

The schematic of the FLDW system is shown in Fig. 3. Polarizers 1 and 2 and a half-wave plate were used to control the laser power. A charge-coupled device (CCD) was used to monitor the fabrication. A high NA objective lens (100×, 0.8 NA) focused the femtosecond laser (1030 nm, 290 fs, 25 kHz) into a narrow spot for high resolution fabrication. Samples were placed on a nanometric scanning stage. The entire system was controlled by a personal computer (PC).

GO films are sensitive to laser parameters, such as laser repetition rate, laser power, and fabrication speed. Higher repetition rate can generate more heat, and the rGO zones will be wider as thermal diffusion occurs [41–43,46]. Studies of the relationship between the fabricated rGO lines and the laser power as well as the fabrication speed are shown in Fig. 4 (insets: micrographs of photoreduced rGO lines under different fabrication parameters). Based on the relationship between laser parameters and rGO linewidth, the 900 nm linewidth of the rGO zone was chosen, and the corresponding laser parameters were 150 μW laser power, 5 μm/s fabrication speed, and 25 kHz laser repetition rate. Figure 5 shows the microscopic image of the fabricated SMF GO lens working at 850 nm with 3 μm focal length and five rings. The radii are 2.07, 3.33, 4.39, 5.40, and 6.34 μm, respectively.

The fabrication time of a fiber GO lens is a few minutes, so the laser parameters can easily remain stable in such a short time. The alignment between the fiber core and the GO lens is the key to successful fabrication, and we have developed an in situ fabrication method using an 850 nm laser to couple to the fiber. This illumination can facilitate the precise positioning of the fiber during GO lens fabrication. The in situ fabrication is shown in Fig. 5. The atomic force microscope (AFM) image [Figs. 6(a) and 6(b)] shows that the thickness difference between rGO and GO is 200 nm and the narrowest linewidth of the rGO zone is 900 nm. The film thickness was measured by a step profiler and is about 407 nm [Fig. 6(c)].

A setup was used to characterize the focusing property of the fiber GO lens (shown in Fig. 7). A distributed Bragg reflector (DBR) laser (850 nm) was coupled into the SMF. The fiber facet with the GO lens was fixed by a home-made fiber holder. A 100× objective lens with an NA of 0.8 was used to collect the far-field intensity distribution of the fiber GO lens. High precision stage scanned along the optical axis was used to collect the intensity distribution at different cross sections along the z direction. Complete far-field patterns can be reconstructed by scanning the focal field along the optical axis. The whole system was controlled by a PC.

The experimental and theoretical results of the fiber GO lens are shown in Fig. 8. Figure 8(a) is the calculated far-field intensity distribution in the y−z plane. The focal length is calculated to be 3.05 μm. The transverse optical field at the focal region is shown in Fig. 8. The measured focal length is 2.91 μm, which is in excellent agreement with the theoretical results. The transverse optical field of the focal plane is shown in Fig. 8(e). The FWHM of the focal spot is measured to be 582 nm (0.68λ), and the corresponding NA is 0.89 (NA=0.61λ/FWHM). Thus, the near diffraction-limited focal spot has been experimentally confirmed, and the measured optical field agrees with the results predicted by RS diffractive theory. The differences between theoretical and measured FWHMs may be due to underestimation of the refractive index in the rGO zone of the theoretical model. We used a femtosecond laser pulse with a higher energy than the reference value [43], which may have produced a reduced GO zone with higher refractive index.

Figures 8(c) and 8(f) compare the normalized intensity distributions along the optical axis and at the y=0 axis in the focal plane, respectively, where the red line is the theoretical result and the blue line is the experimental result. The normalized intensity distributions in Figs. 8(c) and 8(f) show great consistency with the simulated results only showing slight deviation in the focal plane. Such a slight deviation may be due to the surface roughness of the GO film and the inhomogeneous areas created by the femtosecond laser processing. The slight variation of the refractive index n and extinction coefficient k of the rGO zone may also contribute to such deviation. In addition, the influence of the higher asymmetric mode in the SMF occurring in the off-axis regions cannot be ignored.

We measured the insertion loss of fiber GO lens at 850 nm. The output powers of SMF and SMF with fiber GO lens were measured, respectively, and the insertion loss is 2.1 dB. The fiber GO lens also exhibits high stability. After six months of storage at room temperature, its focal length changed by only 0.3%, while its NA remained the same at 0.89.

A fiber GO lens with a thickness of 400 nm has been successfully designed and fabricated. The RS diffraction theory was applied to design and simulate the fiber GO lens, which accurately determines the radius and number of the rGO zones without the need for optimization. The FLDW method was applied to convert GO to rGO by photoreduction process. The fiber GO lens working at 850 nm demonstrates the capability of generating near diffraction-limited focal spot (582 nm, 0.68λ) and achieves a high NA of 0.89 with the 12 μm diameter GO lens. The insertion loss was measured to be 2.1 dB. The theoretical design and experimental results support the effectiveness of the fiber GO lens in modulating the fiber mode field. Our fiber GO lens can be integrated into many devices. With the advantages of high NA, super-miniaturization, and subwavelength focusing capability, it enables broad potential applications in super-resolution imaging, medical optical tweezers, endoscopes, and integrated photonics chips.