1. Introduction

There is increasing demand for low bend loss optical fibers, particularly for fiber-to-the-home (FTTH) optical networks, and for sensing applications [1,2]. Indeed, theoretical and experimental investigations of bend loss have been a feature of optical fiber research for many years [3–8]. Generally speaking, the bend loss arises from two distinct physical contributions: transition loss and pure bend loss [6].

When an uncoated fiber is bent, the apparent refractive index profile is changed. Light propagating on the outside of the bend has further to travel, and this effect can be modeled as a tilt applied to the refractive index profile of the fiber [8], as shown in Fig. 1. In addition, there is a bend-dependent elasto-optic contribution to the refractive index change. Referring to Fig. 1(a), the fiber’s core and the outer rim of the cladding can play the roles of arms in a directional coupler structure. Based on coupled mode theory, the fundamental core-guided mode can couple with a cladding mode peaked at the outer rim of the fiber, which propagates along the fiber like a “whispering-gallery” mode. With increasing bend curvature, the refractive index profile becomes more tilted and the effective index of the cladding mode increases. At a critical bend radius, it equals the effective index of the core-guided mode. If there is sufficient overlap between the core mode and the cladding mode at this point, the two modes will be coupled together. For yet tighter fiber bends, the cladding mode becomes the fundamental mode of the fiber. Light coupled into the cladding mode will be absorbed by the fiber’s coating, if it has one. However, in the absence of a coating, this mode can propagate with reasonably low attenuation over macroscopic distances.

In [7], Love and Durniak pointed that the total bend loss could be minimized by ensuring that the fundamental mode propagates through the bend approximately adiabatically with minimal coupling to any other mode. They provided a criterion giving an approximate upper bound on the rate of change of curvature for a given curvature. In this report, we used knowledge of a specially-fabricated fiber with a small outer diameter and a large core to design a grooved mandrel, in which the groove follows an adiabatic transition path for the fundamental mode through the critical bend radius. When the fiber is wound around the mandrel, transition loss induced by the modes’ coupling is decreased sharply and light is efficiently transferred to the fundamental cladding mode of the bent fiber. The fiber can then be bent more tightly than the critical radius without further losses, provided that the exit path from the bent region is also adiabatic.

 

Fig. 1. (a) Refractive index profile across the fabricated fiber (blue solid) and the index profile when the fiber is bent at the critical bend point (red dashed). (b) Geometry of a bent fiber, x origin in the fiber axis.

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2 Numerical analyses

2.1 Mode coupled theory

When an optical fiber is bent, the refractive index profile of the fiber can be presented by [8]:

n₀ is the refractive index profile when the fiber is straight. R is the bend radius. x is a transverse co-ordinate along a line joining the center of curvature and the centre of the fiber, origin in the centre of the fiber and positive outwards as in Fig. 1(b). χ = -0.22 (for silica) [8] accommodates the elasto-optic effect.

Therefore for a bent optical fiber the refractive index profile is tilted as shown in Fig. 1(a). Figure 2 shows the calculated dispersion of modes in our fiber as function of the bend radius, including the core mode (at low bend) and the anti-crossings with the first few cladding modes. This was calculated using a full-vector finite element method with transparent boundary conditions. Computed mode field patterns at selected values of bend radius are also presented. The figure shows that the fundamental mode, which is the core mode at low bends, becomes a cladding mode at tighter bends after the first anti-crossing (see red line in Fig. 2). Directional coupling between the two modes is expected to be strongest around the first anti-crossing point. As described in [7–9], the maximum rate of change of bend curvature at every position along the transition path should be smaller than a critical value to reduce the coupling-induced bend loss. This critical value is related to the difference in mode indices between adjacent modes. The difference is smallest at the anti-crossing point, so this is the critical bend radius. Before the critical point, the power is concentrated in the core and is approximately a Gaussian shape. After the critical bend radius point, the power is concentrated in the rim of the fiber, and is confined by reflection from the glass-air interface. The fundamental mode thus propagates in the cladding when the fiber is more tightly bent.

Exactly the same occurs for higher order supermodes at smaller values of bend radius. At these tighter bends, the index profile becomes even more tilted, so that higher-order cladding modes also anticross with the core mode. However, in this work we aim to make the transition through the first critical point adiabatic, and so the optical power should follow the solid red curve in Fig. 2. Hence, the anti-crossings with higher cladding modes are not relevant.

 

Fig. 2. Effective mode indices of the 4 lowest-order modes as a function of bend curvature 1/R. The dispersive lines correspond to the first few cladding modes, confined to the outer edge of the fiber by the tilted index profile. The anti-crossings correspond to coupling points at which light can be transferred from one mode to another. The mode patterns pointing to certain positions are plots of the z-component of the power flow.

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2.2 Adiabatic bend profile

Coupled local mode theory [4,9] can be applied to the two lowest order modes (corresponding to the solid red and green dashed curves in Fig. 2) found above, to calculate the critical rate of change of curvature as a function of local curvature 1/R for low loss [7]. However, we found it more convenient to model the fiber as a directional coupler of varying asymmetry first. Then we applied coupled local mode theory to derive an analytic expression for the critical rate of change of curvature in terms of three parameters δn₀ (vertical width of the anti-crossing), 1/R ₀ (critical curvature) and m (relative slope of the two curves far from the anti-crossing) that could simply be fitted to the effective index data of Fig. 2:

where k is the free-space wave number and P is the fraction of power loss we are prepared to tolerate. The fit was excellent where it matters, i.e. around the anti-crossing, and thus avoided the need to evaluate numerical integrals of mode fields.

The optimum, or critical, profile for the transition – i.e. the path with the shortest possible length over which one can maintain adiabaticity – can be found by solving the differential equation formed by taking the equality in Eq. (2) at each point:

where the length scale z ₀ = 3/kδn ₀ if we take P = 1/10. This profile is plotted as the solid red curve in Fig. 3 for our fiber. However, in practice, realization of this precise optimum profile is very difficult since it is sensitive to details of the index distribution and indeed the wavelength. Instead we have used an Archimedean spiral profile, which determines the grooved curve on the taper cone part of the actual mandrel in Fig. 4(a). This design is convenient for fabrication, because the Archimedean spirals have a constant pitch, although the required fiber length is much longer.

 

Fig. 3. (a) Bend curvature in the transition path for the Archimedean spiral profile (blue solid) and the optimum profile (red dashed), with the z origin at the critical bend radius point. (b) Rate of change of curvature versus curvature for the Archimedean spiral profile (blue solid) and the optimum profile (red dashed).

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Figure 3(a) shows that we are able to tolerate a huge change of bend curvature far away from the critical bend radius point. From Fig. 3(b), it can be seen that the whole Archimedean spiral profile is an adiabatic path for the fundamental mode, because the rate of change of bend curvature for the Archimedean spiral profile is always below that for the optimum profile, approaching closest to the optimum profile at the critical bend radius point. However, this optimum profile is for the fiber we used, which has an unusually small outer diameter. For commercial 125-μm-diameter standard single mode fiber, it is much more difficult to get such an adiabatic path, because the critical rate of change of curvature turns out to be over 5 orders of magnitude smaller! An adiabatic transition would be commensurately longer.

3. Experiments

 

Fig. 4. (a) A schematic of the mandrel (b) A photograph of bent coated fiber wound around the mandrel. Light is incident through the fiber from the right.

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A graded index optical fiber preform was drawn down for this experiment. The preform was intended for production of multimode fiber, but was drawn to a smaller size than originally intended (outer diameter of 35 μm). Based on the measured index profile of the preform, we anticipate that the second-mode cut-off wavelength should be around 1840 nm, and so our measurements around 1550 nm are performed in a regime where the fiber is slightly overmoded. The refractive index profile is shown as the blue solid curve in Fig. 1(a). The red dashed curve represents the index profile when the fiber is bent at the critical bend radius point.

We also designed a grooved mandrel, which consists of three parts. Taper cones on either side are joined to an untapered grooved rod in the middle, as in Fig. 4(a). The geometry of the grooved curves on the taper cones follow the Archimedean spiral profile mentioned above, with dimensions as shown in Fig. 4(a).

A photograph of the bent, coated fiber wound around the mandrel is shown in Fig. 4(b), to give a visual impression of the experiment. The fiber looks bright because it was left coated for the photograph. For the measurements, the coating was locally removed before it was wound around the mandrel. The input source is a PCF-based supercontinuum light source, which is pumped by a 1064 nm Neodymium microchip laser. A long length of fiber (roughly 150 m) was used between the input and the bending experiment. With the excitation conditions used, the light in the fiber was predominantly in the fundamental mode, which we verified using near-field imaging. The output transmission spectra were recorded using an optical spectrum analyzer (OSA).

 

Fig. 5. Bend loss spectra for fiber bent around the mandrel without (black solid) and with (green dashed) index matching gel, and straight fiber with index matching gel (red dotted)

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Fig. 6. Bend loss spectra for fiber bent around the whole mandrel (black solid) and fiber bent just around the middle part of the mandrel (red dashed)

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Transmission spectra recorded through the fiber are shown in Fig. 5. First, the fiber was locally stripped of its coating and carefully cleaned, and a transmission spectrum was recorded. The uncoated fiber was then wound around the mandrel, and another transmission spectrum was recorded. This was normalized with respect to the original spectrum, and the result is plotted as the black solid line in Fig. 5. In the next step, the uncoated fiber on the mandrel was covered in index-matching gel, to remove the effects of waveguiding at the external glass-air interface. The normalized transmission spectrum is shown as the green dashed curve. Finally, the fiber was removed from the mandrel, still covered in index-matching gel, and straightened out on the optical table, and the normalized transmission spectrum is shown as the red dotted curve. When the uncoated fiber is wound around the mandrel, bend loss is massively increased by the application of index-matching gel (at the long wavelengths being studied here). However, when the fiber is laid straight again, the transmission loss is nearly zero. This means the huge loss of the green curve is not because the cladding is too thin to isolate the field from the environment, but because the fundamental mode field propagates into the cladding as the bend curvature increases, as explained in Fig. 2. The index matching gel strongly attenuates the cladding mode due to its being guided at the external fiber surface. We can conclude that, in the bent fiber, virtually all of the light is guided by the external surface as a cladding mode. However, in the absence of index-matching gel, the total transmission through the bent fiber is only slightly less than through the unbent fiber, and so the light which propagates in the cladding in the bent region is almost all returned to the core when the fiber is goes through the second adiabatic transition back to an unbent state. This can only happen if the light is predominantly transferred to a single cladding mode, by following the red line in Fig. 2.

For further study, the uncoated fiber was wound without the adiabatic transition. For the results presented in Fig. 5, it was wound around the tapered and untapered parts of the mandrel. The black solid curve of Fig. 5 is reproduced in Fig. 6, along with the result of a second measurement in which the fiber was just wound around the middle, untapered part of the mandrel. To keep the bend transition as abrupt as possible the fiber was kept taut as it came off the mandrel using weights (11.5 g on each side). Figure 6 shows that there is almost 10 dB more loss when the transition was non-adiabatic. We conclude that the adiabatic transition from unbent to bent fiber enables efficient transfer of power from the core mode to a single cladding mode. Transition losses in the case of winding without the tapered section would be far higher were it not for the fact that our fiber was expressly designed to make it easy to achieve an adiabatic transition. Control of the rate of change of curvature is difficult in an uncoated 35μm fiber.

4. Conclusions

We have demonstrated low-loss transmission through a tightly-bent optical fiber by adiabatic transfer of power to a cladding mode in the bent region. The fiber was intentionally designed to have a large overlap between the guided mode and the fundamental cladding mode of the bent fiber, by decreasing the outer diameter. The rate of change of bend radius required for adiabaticity was established using numerical calculations and was controlled experimentally through the use of a purpose-built mandrel based on the known fiber design. We verified the performance of the transition by applying index-matching gel to the outer surface of the bent fiber, and through the use of non-controlled bending.