Photonic Jet-Shaped Optical Fiber Tips versus Lensed Fibers

: Shaped optical ﬁber tips have recently attracted a lot of interest for photonic jet light focusing due to their easy manipulation to scan a sample. However, lensed optical ﬁbers are not new. This study analyzes how ﬁber tip parameters can be used to control focusing properties. Our study shows that the conﬁgurations to generate a photonic jet (PJ) can clearly be distinguished from more classical-lensed ﬁbers focusing. PJ is a highly concentrated, propagative light beam, with a full width at half maximum (FWHM) that can be lower than the diffraction limit. According to the simulations, the PJs are obtained when light is coupled in the guide fundamental mode and when the base diameter of the microlens is close to the core diameter. For single mode ﬁbers or ﬁbers with a low number of modes, long tips with a relatively sharp shape achieve PJ with smaller widths. On the contrary, when the base diameter of the microlens is larger than the ﬁber core, the focus point tends to move away from the external surface of the ﬁber and has a larger width. In other words, the optical system (ﬁber/microlens) behaves in this case like a classical-lensed ﬁber with a larger focus spot size. The results of this study can be used as guidelines for the tailored fabrication of shaped optical ﬁber tips according to the targeted application.


Introduction
The term photonic jet (PJ) was first coined by Z. Chen et al. in 2004 when Modelling cylindrical dielectric structures under plane wave illumination; simulations showed that light can be concentrated at the mesoscale with a full width at half maximum (FWHM) lower than the diffraction limit (<λ/2, where λ is the free space wavelength), and with a power density significantly higher than the incident wave [1][2][3]. The key properties of the PJ include: its position (working distance W D : distance from the tip end to the maximum intensity of the photonic jet), FWHM, light intensity peak value and intensity decay length. The determination of these properties has been the aim of numerous theoretical and experimental studies [4][5][6][7][8][9]. The properties of the PJ depend on the material refractive indexes, the dielectric object geometric shape and size, and on the incidence wavelength [2,10,11]. PJ can be obtained using cyilnders, spheres or non-spherical dielectric objects [11]. The interest of PJ has been demonstrated for applications such as enhanced Raman scattering [12][13][14], fluorescence enhancement [15,16], improvement in optical-disk data storage technology [17,18], in sub-micrometer laser etching process on semiconductors and metals [19][20][21][22], to achieve optical tweezers [23] and many other applications [24].
Unfortunately, the technique is not easy to implement in some industrial processes such as lithography or spectral analysis, where the PJ must scan a sample with a complex pattern. For such applications, the sphere-like object cannot be directly placed on the sample. A mechanical micro-holder or optical tweezers are required to manipulate it accurately [19]. For these reasons, more recently, attention has been paid to PJ generation with waveguides [25] and optical fibers; solutions have been proposed to put the sphere at the end of a hollow-core optical fiber [26,27] or to use an optical fiber with a shaped tip [21,[28][29][30][31]. Shaped fiber tips have several advantages: easy to move, no necessary contact with the sample and possibility to collect the backscattered light. Owing to the aforementioned arguments, this technique can become a major solution in industrial processes and characterization such as sub-micron laser processing or high-resolution spectral analysis. Nevertheless, the concept of shaped fiber tips has already been known as microlensed fibers for the past 40 years. Initially, microlensed fibers were extensively exploited to improve the coupling between light sources and optical fibers [32][33][34][35][36]. Microlensed fibers are also used in Optical Coherence Tomography OCT [37], scanning microscopy and spectroscopy [38]. Lensed fibers have been fabricated using a large range of methods including: chemical etching [39], laser heating [40] and electric arc discharge melting [41].
The aim of the present work is to investigate shaped optical fiber tips to identify what distinguishes photonic jet generation from classical-lensed fibers focusing. This study also makes it possible to control the properties of the PJ by optimizing the different parameters of a shaped optical fiber tip.

Numerical Approach
First a multimode silica optical fiber with a low number of modes is considered for 2D simulation: core diameter 2a = 20 µm and cladding diameter of 125 µm (20/125 µm fiber), cladding and core refractive index, 1.44 and 1.457 respectively, corresponding to a numerical aperture (NA) around 0.22. The fiber is exited at a wavelength of 1064 nm by its fundamental mode, approximated by a Gaussian function. The propagation direction is set along the x-axis. The source wave is linearly polarized along the z-axis (see axes in Figure 1). To describe the tip shape, the standard form of the rational quadratic Bézier curve is used, set by the tip height (h), the base diameter (D) and the Bézier weight (W 0 ) [31]. The tip is considered to have the same refractive index as the fiber core. This minimizes the reflection. Light propagate along the fiber, then through the shaped fiber tip and then in the air. The electric field distribution is obtained numerically with a 2D finite element method (FEM), COMSOL Multiphysics©, using a free triangular mesh having a maximum mesh size of λ/10. Scattering boundary conditions are used along the fiber cladding and perfectly matched layers (PMLs) around the free space surrounding the tip.
As illustrated in Figure 1-top, several microlens base diameters (30 µm, 50 µm, 70 µm and 80 µm) were considered in order to study their influence on the PJ using the same shape Bézier weight W 0 = 1.1 and height h = 26 µm.The influence of the tip height (h) and sharpness (described by the Bézier weight W 0 , see Figure 1-down) have also been studied.  More generally, Figure 2 shows the evolution of the 2D simulated intensity distribution when the tip base diameters increases from 30 µm up to 80 µm. For each configuration the maximum electrical field value outside the tip (for an unitary incident fundamental mode), the corresponding FWHM (evaluate on the intensity) and working distance are given in Table 1.

Results
When the tip diameter increases, the PJ position (W D ) is further, its electrical field maximum decreases and FWHM increases. Figure 2d corresponds to the longer and the largest PJ, with a working distance 200% longer than in Figure 2a. The FWHM is 260% larger and the electric field norm is 46% lower than the PJ obtained with the lowest base diameter. The corresponding electric field norm |E| profile on the photonic jet planes has been plotted for the 20 µm core fiber but also for optical fibers with core diameter 50 µm (NA = 0.22) and 10 µm (NA = 0.12). Compared with the 20/125 fiber, a longer and not so sharp tip (h = 36 µm, W 0 = 0.7) is required for the 50 µm core fiber and inversely, a smaller and sharper tip (h = 14 µm, W 0 = 1.1) is required for the 10 µm core fiber, thus justifying our choices. For the three fibers, the simulated base diameters have been chosen to be first close to the core diameter and then larger. Figure 3 illustrates the corresponding electric field norm |E| profile on the photonic jet planes. For the three fibers, a PJ is obtained when the microlens base diameter is close to the core diameter. The trend of the simulations are in good agreement: when the base diameter of the tip increases, the maximum electric field decreases, the PJ FWHM and working distance increase. In other words, the optical system (fiber/microlens) acts more and more as a classical-lensed fiber.
We now consider the influence of microlens Bézier weight (sharpness) and height on the focusing properties. As illustrated in Figure 4a with the 20/125 silica fiber, an increase of the Bézier weight (blunter tip) pushes the PJ (W D ) away from the fiber tip end and increases the PJ FWHM (as a lens having a larger focal distance would). The narrower PJ FWHM, around 0.55 µm, is obtained for the the lower weight W 0 = 0.3 (sharper tip) at a working distance W D of 3 µm. Compared with W 0 = 1.1, the PJ width is divided by 2. Thus, the present investigation confirms that the Bézier curvature has a primordial role in controlling the focusing parameters.
In Figure 4c,d, we can also observe that the FWHM and W D decrease with the increase of the tip height.

Conclusions
The present study was designed to control photonic jet generation by assessing the effect of the optical fiber tip parameters and to clarify the differences between classical fiber lens and the shaped fiber tips making possible to obtain PJ. A first already known difference is that PJ is only due to the fundamental mode, namely, the other modes being off-axis focused [31]. We show in this paper that a second important difference is that PJs occur when the microlens base diameter is close to the fiber core diameter, whereas classical fiber lenses stand generally on the whole optical fiber cladding. When the tip is shaped by thermoforming, this can explain why a PJ is easier to obtain with multimode fibers (having a core diameter close to the optical cladding one). PJs achieved using a shaped fiber tip may be used to scan a sample with a complex pattern and also collect the backscattered light with the same fiber. They are, therefore, a promising solution for many applications from laser nano-abalation to local spectrocopy.