Design considerations and experimental investigations on fiber ball lens systems for optical metrology

Optical fibers play an important role in general and, in particular, in the field of sensors. As part of a sensor system, quite often fibers are coupled by a ball lens. For efficient usage, the fiber ball lens systems (FBLSs) have to be optimized. The present work presents analytic expressions for the design parameters of such systems. FBLSs comprise sections of a single-mode optical fiber, a coreless fiber (CLF) and a ball lens. Their geometric dimensions have to be optimized for their use in different applications in optical metrology. The derived expressions facilitate the optimum parameter choice, which is usually done by expensive numerical simulations. For comparison and validation of the results by experiments, FBLSs with different ball lens radii and CLF sections have been prepared by a fusion splicer technique. Their characteristics were investigated in the optical spectral ranges at 630 nm and 1550 nm. Experimental methods comprising far-field, near-field and reflection/transmission measurements with optical fibers validate the theoretical considerations. To our knowledge, this is the first time that simple analytic approaches have been applied to the fabrication of FBLSs. This facilitates and quickens their general design for different applications in optical metrology significantly.


Introduction
While optical fibers were historically developed for high bit rate optical communications, their applications in optical metrology have dramatically increased in recent decades, especially with the perfection of microstructuring methods for silica fibers [1]. One of the big challenges for optical fibers, however, is still efficient light coupling from free space to the fiber. This is mainly due to the low numerical aperture of silica fibers. It becomes more critical with single-mode optical fibers, which are required for phase-sensitive measurements, e.g. in interferometers, and where a precise mode-field overlap must be achieved [2]. Standard methods to optimize the fiber coupling from free space comprise optical fiber benches in combination with lenses which, however, lead to bulky set-ups and are inconvenient for many applications.
Micro-scale refractive lens systems, where a lens is directly connected to the optical fiber, offer a great advantage with respect to compactness and efficiency and can be produced in different ways [3][4][5][6][7][8][9][10][11][12]. A very reliable and relatively simple fabrication of spherical end surfaces on silica fibers is achieved by using a commercial fiber splicer to control the shape of the fused fiber end, thus resulting in a ball lens section [6,7,9]. Here, different aspects of light coupling must be taken into consideration, as illustrated in figure 1 in a simplified way based on geometrical optics for a fiber ball lens system (FBLS) in air: if a sphere is formed at the end of a silica fiber, with a small core relative to the fiber diameter, then the light spot emerging from the core expands to the spherical fiber-air interface ( figure 1(a)). Due to the proximity of the spot to the refracting end surface, the emerging light beam after refraction is not converging but remains diverging. If such a system is to be used to collect light from free space to the fiber, its coupling efficiency is poor as it cannot change the curvature from an incoming light wave sufficiently to converge to the fiber core. Thus, for coupling purposes the divergence of the system must be reduced, which can be done by splicing a coreless fiber (CLF) section to the single-mode fiber (SMF). In this way, the characteristics of the emerging beam can be controlled by the length of the CLF section and are crucial for the imaging properties of the total system. Figure 1(b) illustrates the situation for a collimating FBLS, which is appropriate for coupling nearly parallel light from outside to the fiber, whereas figure 1(c) shows a converging FBLS. It should be noted that the use of a CLF section additionally ensures a well-defined location of the emerging light spot, unlike in figure 1(a) where, due to the fusion of the fiber end, glass material from the core and cladding are blended and the end of the guiding fiber section continuously fades away.
The optimum design for the FBLS of a given application requires the consideration of the optimum ball lens radius as well as the length of the CLF, which both determine the working distance of the FBLS. Due to the micro-scale dimensions of the fiber core, wave propagation and diffraction play an important role, which is not shown in the scheme of figure 1 based on simple geometrical optics. SMFs are optimized for light propagating in the fundamental transversal mode close to its cut-off frequency, where the transversal beam profile is nearly Gaussian-like. The optimum computation of the light path in the FBLS could be based on a beam propagation method, which not only takes into consideration diffraction effects when light exits from the fiber to the ball lens section [13]. It also accounts for the interference with higher order symmetric transversal modes, which are excited when the light enters the much larger ball section [14]. These modes are reflected at the ball's boundaries and may lead to complex intensity modulations due to interference. The computation, however, is expensive and requires very high precision for the high numbers of necessary iterations. Thus, for a quick design optimization of practical applications, it is less appropriate.
A much more appropriate computation is based on Gaussian beams (GBs) exiting the SMF, propagating through the coreless section and being refracted and reflected, respectively, at the spherical lens surface [7,15]. This method uses a complex matrix formalism. It gives very reliable results and has been verified with experimental results of high precision for optical interferometric sensors [7,15]. However, the method is predominantly based on numerical computations for the general design of FBLSs for various applications.
Our intention in this work has been to achieve analytic expressions for the FBLS design, like in a simplified approach when using geometrical optics. Geometrical optics can help to get an illustrative idea for the system design. This facilitates the calculation of some relationships between the dimensions of the ball lens and the required length of the CLF for collimating and converging the FBLS, respectively. However, the results may not deliver the required precision for optical systems with dimensions below 1 mm. Therefore, after starting with geometrical optics, we apply its matrix formalism to GBs to obtain complex analytic expressions, which yield a more realistic description of the experimental results, based on the design parameters, than geometrical optics. To the best of our knowledge, this has not been published for FBLSs so far.
In the experimental part of our work, different types of FBLSs have been prepared by a commercial fusion splicer and investigated for their near-field and far-field characteristics in the visible (630 nm) and infrared spectral ranges (1550 nm). The results are compared to the two different theoretical approaches and some key aspects of the system design are discussed. This is intended to validate our considerations for the simple choice of optimum design parameters for the FBLS.
In summary, in contrast to well-designed and expensive coupling optics, which are usually produced by external suppliers [12] and which commonly require well prepared equipments and some optics knowledge, the objective of our work has been to prepare ball lenses via a simple method, at low cost and by a simple fabrication procedure in a laboratory. Although their coupling efficiency to cleaved optical fibers is not perfect, due to a single spherical surface, it may help researchers with different scientific backgrounds when designing microoptical systems for various applications, such as, for instance, fiber optical interferometers and fiber sensors. Moreover, the derived formulas and design rules should facilitate their design without having to perform expensive mathematical calculations. Figure 2 illustrates some parameters which have to be taken into consideration for the fabrication of an FBLS. A CLF section of length l 1 is spliced to an SMF, which ensures a well-defined entry point of light from the SMF to the system. Both fibers have the same cladding radius r f . The ball lens with a radius R is then subsequently formed by fusion of a part of the CLF, leaving a non-fused CLF section of length l 2 between the ball lens and SMF. If we assume volume conservation during the ball formation process, the volume V 1 of the CLF before fusion must be equal to the sum of the volumes V 2 of the remaining straight CLF section and V bl of the ball lens after fusion, diminished by the overlapping segment V seg between the CLF and the ball lens. This can be expressed in the following way:

Geometrical optics consideration
The width b of the segment depends on the ball lens radius and the fiber radius: Equations (1)-(3) are rearranged to calculate the remaining length l 2 of the straight CLF section: If the ball lens radius is large compared to that of the fiber, the section V seg can be neglected with b approaching zero.
To determine if the FBLS is converging or diverging, its paraxial imaging characteristics can be calculated using the vertex equation for spherical surfaces from standard textbooks (see, e.g. [16,17]): Here, n f is the refractive index of the CLF in which the emerging light beam propagates along a f and is then refracted to the outside medium of the refractive index n i where the image is formed (figure 2). The back image distance a i measures the position of the real image of the SMF spot relative to the ball's vertex. For a converging system, a i is a positive quantity; for a diverging system, it is negative and represents a virtual image. For a collimating system, it approaches infinity. Rearranging equation (5) yields: There are different ways to prepare an FBLS. Usually, the CLF of length l 1 is first spliced to the SMF and in the next step cleaved by a high-precision cleaver set-up to the appropriate length. Next, the ball lens is formed by the fusion splicer [15]. For a system with defined imaging characteristics it is necessary to know the expected back image distance, which is directly related to the CLF length l 1 and the ball lens radius R after shaping by the fusion splicer. If we eliminate a f from equation (6) using we get: In our present work we use a different preparation, where the fusion splicer itself severs the CLF section before shaping the ball lens (see section 3). For this process, the main parameters for the splicer are the ball lens radius R and the splice-to-center distance s c , which is the distance from the SMF-CLF splice to the center of the ball lens. It can be seen from figure 2(c) that Substituting a f in equation (6) by the relation for s c according to equation (9), the relation for a i holds: Thus, for a converging FBLS with a real image, the denominator of equation (10) must be positive. If the denominator is close to zero, a i approaches infinity and we have the special case of a collimating system. Hence, with the denominator being zero we get: If we take into account that the typical refractive index of silica glass is slightly less than 1.5, a collimating lens system in air is achieved if the splice-to-center distance is nearly the same as the ball lens diameter, i.e. s c ∼ = 2 R and a f ∼ = 3 R. Larger s c and a f values, respectively, lead to a converging FBLS with the image distance becoming smaller. The shortest distance from the lens vertex is achieved with s c and a f , respectively, becoming very large. This is equivalent to parallel light in the fiber hitting the ball's surface, and the image point is at the distance of the back focus length a bf with The back focal length a bf is of the same quantity as s c for a collimating system. Thus, the shortest possible image distance with the smallest image size is about a i = a bf ∼ = 2 R. As a consequence, the working distance of an FBLS with a small image spot is directly governed by the size of the ball lens.

Matrix formalism and GB optics
So far we have simplified our considerations within the framework of geometrical optics that are valid for paraxial rays. However, some characteristics for the propagation of GBs have to be taken into account, even within the framework of geometrical optics. The nearly Gaussian spot at the end of the SMF can be described by its waist radius w 0 measured, e.g. at 1/e 2 of its peak intensity. The half-angle divergence θ 0 of the beam with a free-space wavelength λ, expanding in the far field of a medium of n f , is given by (see textbooks, e.g. [16]): To avoid beam obstruction in the expansion section of the CLF, θ 0 must be smaller than a limiting angle θ lim , which is fixed by the ratio of r f and l 2 , as shown in figure 2(c): This may become critical for large ball lens radii that require large l 2 values for converging and collimating an FBLS, respectively. To estimate a maximum allowable ball lens radius, we consider the case for commercial fibers we used in our investigations (see section 3). For the visible spectral range around 630 nm, the SMF SM600 by Fibercore is specified to have a 1/e 2 mode-field diameter of 4.3 µm, whereas the Corning SMF-28 has a mode-field diameter of 10.4 µm at 1550 nm in the infrared range. The refractive index of the CLF is 1.458 at 630 nm and 1.444 at 1550 nm [18]. Consequently, a GB of w 0 = 2.15 µm expands in the CLF with a far-field divergence of θ 0 = 0.064 rad or 3.7 • , respectively, according to (13), while the divergence at 1550 nm is θ 0 = 0.066 rad or 3.8 • , respectively. In both fibers the divergence is nearly identical; the ray can be considered nearly paraxial. The limiting length l 2 of a CLF, when obstruction occurs, is given if θ lim = θ 0 . Equating equation (13) with equation (14) yields the maximum value for l 2 : Assuming large l 2 of several hundreds of microns for a collimating system implies that R must be about one order of magnitude larger than r f . Hence, V seg and b in the above consideration can be neglected and s c ∼ = l 2 + R. Equating that with equation (11) gives us the estimate of the maximum acceptable ball radius R max for a collimating lens: Substituting l 2,max using equation (15) and rearranging the equation yields the maximum radius for the ball lens to avoid obstruction of the expanding GB: . (17) For the preparation of the collimating FBLS in the present work, we get R max ∼ = 786 µm in the visible range and R max ∼ = 758 µm in the infrared range based on the used fiber specifications. Thus, for collimating lenses, the ball lens diameter must not exceed 1.5 mm. For the converging FBLS, R should be somewhat smaller as the conditions for the maximum CLF section according to equation (15) must be maintained.
A more detailed analysis of the FBLS requires the application of the matrix formalism for the propagation of optical rays. This formalism, also termed the ABCD matrix method, has been developed for the paraxial regime propagation and is treated extensively in textbooks (see, e.g. [16,17]). The method is valid for simple geometrical optical rays (GORs) and has been expanded to the propagation of GBs [19]. As the fundamental mode in optical fibers can be approximated sufficiently well by a Gaussian transversal cross section, the propagation of rays with a Gaussian shape is more appropriate than using simple geometrical optics [6,7,15], yet is more expensive.
The path of a GOR through an optical system can be described by a parameter pair indicating its position h relative to the optical axis, as well as its slope angle θ with it. Figure 3 illustrates the situation for a GOR emerging from the center of the fiber core in a converging FBLS. It starts at the SMF-CLF fiber splice on the optical axis at z 0 with a ray slope of θ 0 . Then, it propagates through the fiber ball lens section with a refractive index n f along a distance a f , thus increasing its elevation from the optical axis, and intersects the ball surface at a height h. Note that due to the paraxial characteristics, this intersection takes place in close proximity to the ball vertex, with its position on the axis being approximately z V . This translation is described by the translation matrix T 1 : The refraction matrix R accounts for the change of the ray slope to θ i at the spherical ball surface of radius R, while the elevation remains constant: After refraction the ray propagates further along the back image distance a i from the vertex and intersects the optical axis in the image point at z i . This second translation is given by T 2 : The system matrix of the FBLS then results in M =T 2 ·R·T 1 after matrix multiplication: Hence, the ray parameters of the rays at the origin and in the image space, indicated by their subscripts 0 and i, respectively, are related to each other by the following equation: Now,we consider a light bundle of GORs, starting at the fiber splice on the optical axis with a half-angle aperture θ 0 and h 0 = 0. Then, in a converging FBLS, the outgoing light bundle has a half-angle aperture θ i and is intersecting the optical axis with h i = 0 in the image point at the back image distance a i from the vertex: Equation (23) shows that θ i is independent from a i and represents the outgoing aperture of the light bundle. Here, θ i = 0 is the limiting case for a collimating lens. Using this condition for equation (23), the calculation yields the same result for a f , as given by equation (11) for the design of a collimating lens system. From equation (24), the back image distance a i for a converging lens is obtained, which is identical to the results in equations (6) and (10).
The propagation of a GB can be described by the same matrices, but the beam parameters are defined in a different way. According to Kogelnik [20], a GB propagating in the z-direction is suitably parameterized by its beam radius w(z), measured at 1/e 2 of its intensity perpendicular to z, and its radius ρ(z) of the wavefront curvature. Both quantities are combined to yield the complex beam parameter q(z) [21], with i being the imaginary unit, λ is the free-space wavelength of light and n is the refractive index in the medium, respectively: The transformation of a GB originating at the splice position z 0 and with the complex beam parameter q 0 into the outgoing GB at position z i with the parameter q i is given by the following equation using the above-defined matrix elements: Unlike for GORs, the GB beam does not intersect the optical axis at the back image position z i but shows a minimum spot size, also termed the beam waist, and has a flat wave front at z i ( figure 3). This means that its curvature radius is infinite with 1/ρ(z i ) = 0. In front of z i and behind it, the spot widens up and the wave front is curved. The widening of the mode field w i (z − z i ) is given by: Here, a Ri is the Rayleigh length of the beam in the image space, which implies that the beam's waist radius widens up by a factor √ 2 if it propagates along a Ri from its image point at z i ( figure 3). The feature of the infinite curvature radius at z i is used to calculate the back image distance a i . Hence, we determine the spot radius w i in the beam waist as well as the half-angle aperture θ i of the outgoing beam in the far field. Using the parameters w 0 and 1/ρ(z 0 ) = 0 for the initial GB at z 0 , we have a purely imaginary starting value q 0 = i · n f π ·w 2 0 λ . After some lengthy calculations we obtain the following results for a converging FBLS (for more detailed calculations see the appendix): The back image distance a i as well as the image spot size w i depend on the Rayleigh length a Rf of the originating GB in the fiber. Both a i and w i , however, most strongly depend on the radius R of the ball lens as well as the CLF section l 2 in front of the ball. The influence of the CLF section on the parameters of the outgoing beam can be seen in figure 4, taking into account that s c = a f −R ∼ = l 2 + R. Here, the results as a function of s c are shown for the geometrical optics approach in comparison to the Gaussian consideration. In the computation for a ball lens radius of 150 µm in the visible range, a collimating FBLS in air is achieved with s c ∼ = 328 µm (w 0 = 2.15 µm, n f = 1.4577 at 630 nm). The image point of a GOR is at infinity, and its half-angle output divergence θ i is zero (dashed curves). With increasing s c , thus a longer CLF section, the image distance decreases steeply and shifts toward the lens vertex, while the beam divergence increases linearly with s c . With regard to the GB, a 'collimating' FBLS yields a back image distance a i of nearly zero; hence, its image point is very close to the vertex (solid curves). Its output divergence is minimum, but above zero, and the beam waist achieves its maximum value. With increasing s c , both the back image distance as well as the beam divergence of a GB asymptotically approach the curves of the GOR. The image waist radius w i decreases as well, which means an increasing resolution if, for instance, the FBLS is used as a sampling probe for optical coherence tomography [15].

Coupling between FBLSs and optical fibers
An intention of our investigations on FBLSs has also been the coupling with optical fibers, especially SMF. In our theoretical approach, we used the approximation of a Gaussian mode-field distribution for the fundamental mode of an optical fiber. The FBLS images this field again into a Gaussian distribution with a magnification factor, which is defined as the ratio w i /w 0 of output to input spot size. This magnification factor as a function of s c exhibits the same curve type as w i given by equation (30), and which is shown in figure 4. It has its largest value when s c is chosen to yield the 'collimating lens system' . This can be easily shown by equating the first derivative of w i with zero and rearranging it to yield a f , respectively s c . We then obtain the same result as given by equation (11) for geometrical optics. With increasing a f , respectively s c , the magnification factor decreases, which allows for better resolution of the FBLS, such as in optical coherence tomography (OCT) sensors [7,15], but the working distance also decreases. A free-space coupling from FBLS to FBLS or FBLS to SMF can be calculated by the overlap integral between different mode fields. If their alignment on the optical axis is done without tilt or lateral displacement, the longitudinal power-coupling efficiency T long between the mode-field radius w i (z) of the FBLS and w 2 of a second system, which could be an SMF or even an FBLS, is given by [22,23]: Here, L T is the coupling loss in dB. If the spot size of the second system is identical to that of the FBLS in their overlap position, i.e. w 2 = w i (z i ), we get a perfect coupling with zero loss. If the distance between the FBLS and system 2 is then increased or decreased away from the optimum position along the Rayleigh length on the image side a Ri , the spot size of the FBLS increases and we get w i (z i + a Ri ) = √ 2 · w i (z i ). Then, the coupling loss between both systems is only L T ∼ = 0.5 dB.
Therefore, we define a 0.5 dB loss range as the range of 2a Ri centered around the image position. A 3 dB coupling loss is achieved if the shift between the two systems in their optimum coupling distance is increased up to 2(1 + √ 2) · a Ri ∼ = 2.2 · a Ri . In an analogous manner, we define a 3 dB loss range with an extension of about 4.4·a Ri around the image position. As a consequence, if the Gaussian image spot size of the FBLS is relatively large, its Rayleigh length also becomes large. This allows for a larger longitudinal alignment tolerance, and larger working distances, usually of the order of some mm and, similarly to the case of collimated beams. This will be discussed below in our results.

FBLS preparation
All the FBLSs in our investigations were prepared using a Fujikura Arc Master FSM-100P+ fusion splicer with adaptable arc discharge parameters. This allowed us to splice disparate optical fibers with each other. Subsequently, after a cut of the spliced fiber by a light arc discharge to the appropriate length, a ball lens section at the end of the fiber tip was formed. Thus, the preparation of an FBLS, as shown in figures 1 and 2, could be achieved by a continuous splicing procedure without mechanically recleaving the spliced fibers, which usually requires a highly precise cleaving system [15]. Crucial parameters for the fusion splicer are the ball lens radius R as well as the distance s c from the fiber splice to the center of the ball lens. To ensure reproducible results, the fusion splicer was calibrated, and both parameters were checked by optical inspection using a LEICA DM4000 M microscope with digital image evaluation (figure 5). For the visible spectral range, we used a Thorlabs SM600 SMF with a mode-field diameter of 4.3 µm at 633 µm. For the infrared range, a Corning SMF-28 with 10.4 µm mode-field diameter at 1550 nm was chosen. In all cases, the CLF was a Thorlabs FG125LA, which was spliced to the SMF with the same cladding diameter of 125 µm, and which allowed for precise alignment of the optical axis. The refractive index of the CLF is specified by the supplier [18]. For 1550 nm, we have n f = 1.4444, and for 630 nm we assume n f = 1.4577. Ball lenses with diameters from 150 µm to 350 µm have been prepared and checked by optical means. For a detailed investigation of the influence of the CLF on the FBLS characteristics, as presented in this paper, ball lenses with a multitude of different CLF sections are required. For our purposes, a ball lens diameter of 270 µm seemed well qualified. Therefore, to ensure a reasonable amount of complexity, we prepared the FBLSs with a fixed ball lens diameter of 270 µm and varied the splice-to-center distance s c between about 200 µm and 800 µm. The most interesting part for that ball diameter is the range of s c from about 300 µm to about 500 µm, as here, the most significant distinctions between geometrical optics and GB approximation are expected. As the prepared FBLSs are quite fragile, not all of them could be used for the totality of the measurements presented in the following sections.

Far-field investigations in the visible spectral range
To check the beam quality as well as the divergence of light emission from the ball lens in the visible range, the profile of the outgoing beam in the far field was determined using a CINOGY Beam Profiler system in combination with a CCD camera. The advantage of this method is that the beam from the ball lens directly hits the camera without additional imaging optics required in between. The FBLS and the beam profiler were mounted on an optical bench, which allowed for a displacement of more than 100 mm between the FBLS and the beam profiler. As a light source we used a fiber-coupled laser source of 630 nm wavelength, launching 1 mW output power to the SMF of the FBLS. Figure 6(a) shows the beam profiles of a diverging FBLS, namely  with a very short CLF section, captured at different displacements. The beams of all the FBLSs exhibit typical Gaussian-like shapes and widen up with increasing displacement from the source. Gaussian-type functions could be well fitted to all the captured profiles of FBLSs with different splice-to-center distances s c and at all displacements. The fitted 1/e 2 value of the beam radius scales linearly with the displacement ( figure 6(b)). The slope of the straight lines is identical to the half-angle divergence θ i of the outgoing beam in the far field.
The evaluated values are shown in figure 7 (open circles) in comparison to the theoretical behavior expected for light bundles according to the geometrical optics approach (equation (23)), as well as for the Gaussian optics approach (equation (31)). It can be seen that a zero divergence was never achieved, as would be the case for an ideal collimating system according to geometrical optics. The minimum divergence values that we found are slightly below 10 mrad, which is in good accordance with the Gaussian optics approximation.

Backward reflections from mirror surfaces
A simple method to check the position of the imaged Gauss spot beyond the vertex of the ball lens is performed by measuring the backward reflected light power in the fiber. For this purpose, we used a set-up where the FBLS was mounted on an optical bench with a reflecting mirror perpendicular to the optical  axis ( figure 8). The position between the ball lens and the mirror could be shifted by a micrometer screw. The fiber-coupled laser source was connected to a single-mode 3 dB coupler, which allowed for a simultaneous measurement of the launched as well as the reflected power from the mirror. The measurements were performed in the visible range at the wavelength of 630 nm and in the infrared range at 1550 nm. In the case where the mirror surface is at the back image distance a i of the FBLS, the optical light path is reversed upon reflection, and the detected backward reflected power in the fiber is expected to be at maximum.
The reflected power as a function of the longitudinal shift is shown in figure 9 for FBLSs with various splice-to-center distances s c . The ball lens radius is R = 135 µm. The results in the visible range (left-hand side) are quite similar to those in the infrared range, which can be expected as the chromatic dispersion in the fiber glass of the lens has only a minor effect. The FBLS with s c = 210 µm and R = 135 µm is a diverging system; thus, no real image is formed, and the reflected power in the fiber decreases with increasing shift from the ball lens as the beam continuously widens up. It can be seen in the figure that for s c = 310 µm, a longitudinal shift over more than 3 mm in the visible range is possible, where the back reflected power shows a drop of less than 3 dB from its maximum position. Also, in the infrared range, a shift over more than 2 mm without less than 3 dB power drop is possible. For larger s c , where the FBLSs are all converging, the curves exhibit more pronounced maxima of the reflected power at a given distance from the lens. The maximum becomes narrower the larger the s c . This is due to the fact that with increasing s c , the image spot waist decreases and the output divergence increases as computed (see also figure 4). Thus, with decreasing waist, the Rayleigh length on the image side a Ri also decreases. The coupling efficiency quickly drops when shifting the mirror away from the position of the maximum reflected power. This requires more precise position control around this position.
The shift position at maximum backward reflected power yields the back image distance a i of the FBLS. After evaluating these data, the experimental a i are presented in figure 10 as a function of the s c and are marked by open dots. The diagram also shows the theoretical approximations by GBs (full lines) and geometrical optics (dashed lines). For large s c , the theoretical approximations become nearly identical. In this case, the measured image distances from the vertex are in good accordance with both theoretical computations.
The critical s c value, at which we get a 'collimating' FBLS after geometrical optics, can be calculated using equation (11) and yields s c = 295 µm for the visible range and s c = 304 µm for the infrared range. These values are given in figure 10 by the intersection of the geometrical optics approximation (dashed line) with the axis of abscissae. Here, the curve rises steeply and tends to infinity. Geometrical optics predicts a i at infinity for the critical s c , while our measurements are compatible with the Gaussian approximation, where the real image position remains finite and not too far away from the vertex. In addition to the approximation curves, the 0.5 dB loss-range limits are indicated by the two dot-and-dash-line curves. We may note that this range, for a given s c , is delimited in the vertical direction by an upper and, respectively, lower value, which are both at a distance of the Rayleigh length a Ri from a i . Within this range the beam radius widens up by not more than a factor of √ 2 compared to its minimum value at a i . As a Ri decreases with s c , the 0.5 dB loss range also narrows.

Optical fiber coupling
A further corroboration of the back image distance can be found from the direct light coupling from an FBLS to a cleaved optical fiber. We investigated the coupling characteristics using a similar set-up to that shown in figure 8, where the mirror has been replaced by a single-mode optical fiber connected to a power meter. By displacing the fiber away from the lens, the optimum coupling position could be found. Figure 11 illustrates the results for an FBLS with R = 135 and s c = 510 µm, taken in the infrared spectral range at 1550 nm. The expected back image distance is a i = 867 µm after equation (29), where we get the lowest spot waist radius of w i = 10.3 µm according to equation (30). The cleaved SMF-28 has a mode-field radius of 5.2 µm, which is nearly half the value of the image spot. Hence, the theoretical minimum coupling loss should be 1.9 dB after equation (32) plus 0.3 dB due to two Fresnel reflections at the ball-lens/air interface and the air/fiber interface, in total 2.2 dB. As can be seen from figure 11, the longitudinal shift position for maximum optical power in the fiber is virtually identical to the calculated a i , which is indicated by the vertical dotted line. The measured maximum power of 723 µW relative to 1830 µW reference power, however, means a loss of 4.0 dB and is higher than the theoretical value. The reason for the higher loss could be due to a larger spot size (see section 3.5). It should be noted that a relatively broad 3 dB loss range is found for the measured data. It is about 800 µm, while the theoretical value of 4.4·a Ri yields a somewhat higher value of 946 µm. As a consequence, a significant drop in the coupled fiber power in this case happens only along a few hundreds of µm. For comparison, the coupling between the two cleaved SMF-28 is significantly already reduced over a distance of a few µm [22], and the Rayleigh length is about only 55 µm. With regard to the 0.5 dB loss range, the discrepancy between the experimental value of 230 µm and the theoretical value, 2·a Ri = 430 µm, is more pronounced.

Near-field investigations
While the far-field investigation described in section 3.2 yields only information on the widened-up beam, direct information about the tiny beam close to its image position can only be achieved using magnifying optics. For our investigation we used a similar set-up to that illustrated in figure 8, however, with the mirror exchanged for a microscope objective lens (Melles Griot 60/0.65) in combination with CCD cameras  (Optronis VPC-175, Spiricon SP-1550 M). The objective/camera system was initially aligned to get a sharp image of the lens surface at the vertex. Subsequently, the objective/camera system was shifted away from the lens to longer distances while capturing images of the beam at positions along the optical axis. The images of the beam's cross section could be well described using a Gaussian intensity distribution. The 1/e 2 mode-field radius of an FBLS was evaluated as a function of the longitudinal position relative to the ball lens vertex. The position of the least beam radius yields the back image distance and the beam waist. Figure 12 shows the results for an FBLS with R = 135 and s c = 510 µm in the visible spectral range ( figure 12(a)) as well as in the infrared range ( figure 12(b)). The used FBLS is identical to the one of which the fiber coupling is illustrated in figure 11. In the visible range, its minimum mode-field radius is 9. The measured waist radii are significantly larger than the theoretical ones, namely by a factor of 2.2 for the visible range and a more moderate factor of 1.3 for the infrared range.
The larger experimental waist radius may be due to aspherical aberrations of the spherical lens surfaces. In a recent study on coupling efficiency improvement in single-mode optical fibers [12], the authors showed that imaging using a spherical lens surface on the optical fiber is strongly impaired by the aberration. As a consequence, the spot radius in the focal plane of a spherical lens may be up to three times larger than that of an aspheric lens surface with the same curvature radius at their apex as the spherical lens.
The larger waist in the infrared range at 1550 nm would yield a theoretical coupling loss of 3.6 dB with an SMF-28 plus 0.3 dB Fresnel reflection loss. This calculated overall loss of 3.9 dB is nearly identical to that measured for the lens (see above) and could explain the discrepancy described above. Further near-field measurements have been carried out in both spectral ranges. The evaluated back image positions are depicted in figure 10 by asterisks. They fit very well to the overall behavior, and especially to the curves of the GB approximations for the FBLS.

Discussion and applications
With regard to the converging FBLS, we should roughly differentiate between two different types with respect to their applications. The first type aims to get an enlarged beam spot at the output, which implies a relatively large Rayleigh length. This type is appropriate for free-space coupling of light between identical FBLSs. In geometrical optics, this is the classical case of a collimation lens with the ideal property of coupling over very long distances without widening up of the beam. In reality, this widening can never be avoided; thus, the efficient coupling range is limited and can only be calculated using GB approximations. Our experimental data show that a maximum working distance is achieved for an FBLS with the splice-to-center distance s c slightly longer than the critical s c for a collimating lens. The critical s c can be easily computed by the geometrical optics approximation in the present work. A collimating lens in the visible range is achieved for R = 135 µm with s c = 295 µm. All the FBLSs prepared with s c = 290 µm up to 360 µm exhibit their largest working distance in that range, while their coupling tolerance is wider the closer the s c is to that of a collimating lens. To compute the image position as well as the beam divergence, only the GB approximation yields exact results. The longitudinal shift tolerance of these FBLSs with regard to the measurement of reflections to the FBLS is several mm, both in the visible and in the infrared spectral range.
A second type of FBLS, namely one with a larger splice-to-center distance s c , is rather advantageous for applications which require narrower spot widths, as inthe case of direct coupling to SMFs or probes for OCT with good resolution. As a consequence, this leads to shorter working distances and narrower tolerance ranges for coupling. At relatively large s c in comparison to the Rayleigh length in the fiber, the latter being about 33 µm at 630 nm and about 80 µm at 1550 nm, the calculations of the back image distance as well as the beam divergence using geometrical optics approximation are nearly identical to those of the GB approximation. The difference becomes less important the larger the ball lens radius. For our FBLS with R = 135 µm, the difference can be nearly neglected when s c > 400 µm in the visible range and s c > 500 µm at 1550 nm. Hence, for a large s c , the computations of the back image distance as well as the output divergence can be simply performed using geometrical optics. For a not so large s c , a more precise consideration requires a GB approximation. This is also required for the calculation of light coupling from the FBLS to optical fibers. In all cases, for coupling over long distances of some mm, the use of FBLSs is definitely an advantage.
It should be noted, however, that the experimental mode-field radius may be larger than expected by theory due to spherical aberrations. This impairs the coupling efficiency caused by a mismatch in the mode-field overlap. Also, for a larger s c where, for instance, a small spot size is required in sensing applications, the resolution of a sensor probe may not be as high as expected.

Conclusion
In our present work, we have demonstrated a relatively simple method to prepare FBLSs for various applications via a fusion splicer technique. The crucial point is the length of the CLF section in front of the ball lens, which allows for sufficient beam expansion before refraction. To simplify the calculation of the appropriate fiber length for the FBLS design, we have derived analytic equations based on geometrical optics and GB approximation. They facilitate and quicken the choice of design parameters significantly without having to perform extensive numeric matrix simulations, which is the usual method.
Our investigations show that we can roughly differentiate between two types of approximations depending on the applications: if the free expanding section of the beam in the FBLS is relatively long, in our investigated cases about 4-5 times the ball radius, depending on the wavelength, the geometrical optics consideration is sufficient for the calculation of the back image distance and the output divergence. With regard to the input aperture angle of the system, it is assumed to be identical to the divergence of the GB as the light source launched by the SMF.
For shorter CLF sections, exact results can only be achieved on the basis of Gaussian optics. All investigations where the spot size of the beam is relevant require Gaussian optics. To validate our derived analytical descriptions, FBLSs with a ball lens diameter of 270 µm and varying lengths of CLF sections have been prepared. Image distances as well as output divergences have been evaluated based on far-field, near-field and reflection/transmission measurements with optical fibers at 630 nm and 1550 nm. The results are compatible with the described theoretical approximations. Although the FBLS cannot reach the high quality of optimized optical systems, which is due to their simplicity of having only one spherical refracting surface, the prepared FBLSs have the advantage of a less complex system and still exhibit sufficient imaging quality for various applications.

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.