Figure 2 illustrates some parameters which have to be taken into consideration for the fabrication of an FBLS. A CLF section of length l1 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 rf. 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 l2 between the ball lens and SMF. If we assume volume conservation during the ball formation process, the volume V1 of the CLF before fusion must be equal to the sum of the volumes V2 of the remaining straight CLF section and Vbl of the ball lens after fusion, diminished by the overlapping segment Vseg between the CLF and the ball lens. This can be expressed in the following way:

Figure 2. Sections of the FBLS; (a) the single-mode and CLF before ball lens shaping by fusion; (b) the dimensions after ball lens shaping by fusion; (c) image formation in the FBLS (geometrical optics approximation); the maximum aperture given by θlim is indicated by a dotted ray.

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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 l2 of the straight CLF section:

If the ball lens radius is large compared to that of the fiber, the section Vseg 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, nf is the refractive index of the CLF in which the emerging light beam propagates along af and is then refracted to the outside medium of the refractive index ni where the image is formed (figure 2). The back image distance ai measures the position of the real image of the SMF spot relative to the ball's vertex. For a converging system, ai 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 l1 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 l1 and the ball lens radius R after shaping by the fusion splicer. If we eliminate af 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 sc, 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 af in equation (6) by the relation for sc according to equation (9), the relation for ai 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, ai 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. sc≅ 2 R and af≅ 3 R. Larger sc and af values, respectively, lead to a converging FBLS with the image distance becoming smaller. The shortest distance from the lens vertex is achieved with sc and af, 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 abf with

The back focal length abf is of the same quantity as sc for a collimating system. Thus, the shortest possible image distance with the smallest image size is about ai= abf ≅ 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.

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 w0 measured, e.g. at 1/e2 of its peak intensity. The half-angle divergence θ0 of the beam with a free-space wavelength λ, expanding in the far field of a medium of nf, is given by (see textbooks, e.g. [16]):

To avoid beam obstruction in the expansion section of the CLF, θ0 must be smaller than a limiting angle θlim, which is fixed by the ratio of rf and l2, as shown in figure 2(c):

This may become critical for large ball lens radii that require large l2 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/e2 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 w0 = 2.15 µm expands in the CLF with a far-field divergence of θ0 = 0.064 rad or 3.7°, respectively, according to (13), while the divergence at 1550 nm is θ0 = 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 l2 of a CLF, when obstruction occurs, is given if θlim = θ0. Equating equation (13) with equation (14) yields the maximum value for l2:

Assuming large l2 of several hundreds of microns for a collimating system implies that R must be about one order of magnitude larger than rf. Hence, Vseg and b in the above consideration can be neglected and sc ≅ l2 + R. Equating that with equation (11) gives us the estimate of the maximum acceptable ball radius Rmax for a collimating lens:

Substituting l2,max using equation (15) and rearranging the equation yields the maximum radius for the ball lens to avoid obstruction of the expanding GB:

For the preparation of the collimating FBLS in the present work, we get Rmax ≅ 786 µm in the visible range and Rmax ≅ 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 z0 with a ray slope of θ0. Then, it propagates through the fiber ball lens section with a refractive index nf along a distance af, 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 zV. This translation is described by the translation matrix T1:

Figure 3. Image formation in the FBLS (Gaussian optics approximation).

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The refraction matrix R accounts for the change of the ray slope to θi at the spherical ball surface of radius R, while the elevation remains constant:

After refraction the ray propagates further along the back image distance ai from the vertex and intersects the optical axis in the image point at zi. This second translation is given by T2:

The system matrix of the FBLS then results in M =T2⋅R⋅T1 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 θ0 and h0 = 0. Then, in a converging FBLS, the outgoing light bundle has a half-angle aperture θi and is intersecting the optical axis with hi= 0 in the image point at the back image distance ai from the vertex:

Equation (23) shows that θi is independent from ai and represents the outgoing aperture of the light bundle. Here, θi= 0 is the limiting case for a collimating lens. Using this condition for equation (23), the calculation yields the same result for af, as given by equation (11) for the design of a collimating lens system. From equation (24), the back image distance ai 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/e2 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 z0 and with the complex beam parameter q0 into the outgoing GB at position zi with the parameter qi 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 zi but shows a minimum spot size, also termed the beam waist, and has a flat wave front at zi (figure 3). This means that its curvature radius is infinite with 1/ρ(zi) = 0. In front of zi and behind it, the spot widens up and the wave front is curved. The widening of the mode field wi(z − zi) is given by:

where

Here, aRi is the Rayleigh length of the beam in the image space, which implies that the beam's waist radius widens up by a factor if it propagates along aRi from its image point at zi (figure 3).

The feature of the infinite curvature radius at zi is used to calculate the back image distance ai. Hence, we determine the spot radius wi in the beam waist as well as the half-angle aperture θi of the outgoing beam in the far field. Using the parameters w0 and 1/ρ(z0) = 0 for the initial GB at z0, we have a purely imaginary starting value . After some lengthy calculations we obtain the following results for a converging FBLS (for more detailed calculations see the appendix):

where

The back image distance ai as well as the image spot size wi depend on the Rayleigh length aRf of the originating GB in the fiber. Both ai and wi, however, most strongly depend on the radius R of the ball lens as well as the CLF section l2 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 sc = af−R≅ l2+ R. Here, the results as a function of sc 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 sc ≅ 328 µm (w0= 2.15 µm, nf= 1.4577 at 630 nm). The image point of a GOR is at infinity, and its half-angle output divergence θi is zero (dashed curves). With increasing sc, thus a longer CLF section, the image distance decreases steeply and shifts toward the lens vertex, while the beam divergence increases linearly with sc. With regard to the GB, a 'collimating' FBLS yields a back image distance ai 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 sc, 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 wi decreases as well, which means an increasing resolution if, for instance, the FBLS is used as a sampling probe for optical coherence tomography [15].

Figure 4. The back image distance ai, image waist wi and output divergence θi of an FBLS (R= 150 µm, w0= 2.15 µm, nf = 1.4577 at wavelength 630 nm) as a function of the splice-to-center distance sc.

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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 wi/w0 of output to input spot size. This magnification factor as a function of sc exhibits the same curve type as wi given by equation (30), and which is shown in figure 4. It has its largest value when sc is chosen to yield the 'collimating lens system'. This can be easily shown by equating the first derivative of wi with zero and rearranging it to yield af, respectively sc. We then obtain the same result as given by equation (11) for geometrical optics. With increasing af, respectively sc, 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 Tlong between the mode-field radius wi(z) of the FBLS and w2 of a second system, which could be an SMF or even an FBLS, is given by [22, 23]:

Here, LT 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. w2 = wi(zi), 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 aRi, the spot size of the FBLS increases and we get . Then, the coupling loss between both systems is only LT≅ 0.5 dB.

Therefore, we define a 0.5 dB loss range as the range of 2aRi 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 . In an analogous manner, we define a 3 dB loss range with an extension of about 4.4·aRi 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.