1. Introduction

Multicore fiber (MCF) transmission has been intensively investigated as an attractive technology for considerable enlargement of the fiber capacity beyond the Shannon capacity limit of optical networks, [1]. In addition to high-capacity digital communications, MCFs constitute a promising medium to develop interesting applications such as multi-parameter optical fiber sensing [2] and Microwave Photonics (MWP) signal processing [3]. One of the major physical impairments that could compromise the MCF performance in the above-mentioned areas of application relates to the existence of possible bending and twisting effects that may affect deployed fiber links. In the context of weakly MCFs, these detrimental effects have been studied mainly in relation to the fiber inter-core crosstalk considering both homogeneous and heterogeneous core configurations [4,5].

However, a variety of applications where time-delay control and synchronization play an important role require in addition a thorough analysis of the group delay performance of the different fiber cores. This is the case of radio-over-fiber transmission and signal processing in 5G (and beyond) fiber-wireless communications and MWP scenarios. Here, we include timing synchronization [6] of wireless signals in MCF-based radio access networks feeding multiple input multiple-output (MIMO) antenna systems [7]. Another representative scenario is reconfigurable microwave signal processing, where the radiofrequency (RF) signal is generated, modified or controlled by optical solutions based on incoherent discrete-time architectures [3]. In this regard, we have previously proposed the exploitation of both homogeneous and multicore fibers as a compact medium to provide different signal processing functionalities [8], including optical beamforming for wireless phased-array antennas [9], reconfigurable RF signal filtering [9] or multicavity optoelectronic oscillation [10].

The influence of fiber bending on the intercore differential group delay (DGD) have been previously addressed in [11] for a 10-km 4-core fiber link. There, the authors realized that the measured bending-induced DGD was lower than the one predicted from simulation, what they attributed to possible fiber twisting one will expect in a 10-km link. Nevertheless, this statement has not been corroborated by providing either the pertinent theory or the experimental demonstration for different levels of twisting and bending.

In this paper, we provide the theoretical and experimental evaluation of the effects of fiber bending and twisting on the group delay performance of a commercial homogeneous 7-core fiber link. After providing the theoretical expressions for the computation of the DGD accumulated on a given fiber length for arbitrary bending and twisting conditions (as long as one assumes a constant twist rate along the fiber), we evaluate experimentally how the accumulated DGD is affected for different twisting scenarios applied to 25, 35 and 50-mm curvature radii. Finally, we experimentally demonstrate how these conditions reflect on the performance of a particular time-sensitive MWP signal processing functionality, microwave signal filtering, up to 50 GHz. All in all, we demonstrate how twisting actually counteracts the degradation introduced by the fiber curvature itself on the accumulated DGD.

2. Theoretical modeling

In general, a bent fiber core m can be described as the corresponding straight core with an equivalent refractive index distribution n_{eq, m}, [12]:

 

Fig. 1. Multicore fiber curvature with a bending radius R_b and local polar coordinates (r,θ) indicated in the MCF cross section. (b) Effect of the fiber curvature on the refractive index profile of cores 1, 2 and 5 as compared to the straight condition.

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Under the assumption of weak inter-core crosstalk, we can simplify the modal analysis and treat each core in isolation, without loss of generality. For a given core m, the propagation constant under bent condition (or equivalent propagation constant), β_eq,m, can be expressed as

From Eq. (1), we see that the highest variation on the equivalent refractive index due to fiber curvatures occurs when the core forms an angle of θ = kπ, k ε Z, to the curvature plane. In addition, as Fig. 1(b) shows, the equivalent refractive indices of cores located in the 2^nd and 3^rd section quadrants decrease their values with respect to the equivalent straight fiber, while those located in the 1^st and 4^th quadrants increase their refractive indices. In addition, Eqs. (2) and (3) show that this effect is also translated to the core phase propagation constant and the group delay. Actually, the phase propagation constant variation due to fiber curvatures has been widely studied in many works, [15,16], but there is a lack of investigation on how curvatures affect to the group delay.

The effect of fiber twisting with a constant twist rate γ can be understood as a linear rotation of the fiber cross section along a given fiber length L; in other words, as a linear increment of the angle θ_m for a given core m. If we applied an ideal curvature with a fixed bending radius over the whole fiber length, the angle θ_m of each core m would be preserved. The DGD between the outer cores and the central core would then accumulate linearly with the fiber length unless that core m is placed at an angle θ = π/2 + kπ, k ɛ Z (where the core belongs to the curvature tangential plane and the DGD is cancelled out). We can denote the longitudinal evolution of the angular coordinate of core m in the cross-sectional area of the fiber as θ_m = θ_m,i + γz, being z the longitudinal coordinate and θ_{m, i} the initial angular position (i.e., when z = 0). Thus, for a given core m, the DGD with respect to the corresponding straight condition τ_m is a function of z: DGD(z) = τ_eq,m(z) - τ_m. After a given propagation length L, the accumulated DGD is given by

 

Fig. 2. Computed differential group delay (DGD) as a function of the twist rate (rad/km) for a 1-km MCF link and different bending radii (R_b = 50, 75, 150 mm), comparing the results from Eqs. (4) and (9).

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3. Experimental evaluation

We have experimentally evaluated both bending and twisting effects on the DGD of a commercial homogeneous 7-core fiber following the experimental setup illustrated in Fig. 3. The fiber has step-index cores (core radius a = 4 µm and core-to-cladding relative index difference Δ = 0.31%) and a core pitch of 35 µm. We measured the DGDs between straight and bent fiber conditions over a small piece of fiber (L = 1 m). This way, we can carry out the straight fiber measurements and properly manage the applied twist while curving the fiber. Small bending radii of 25, 35 and 50 mm (below the typical fiber spool radius) are applied to force a representative variation in the core group delays when the fiber is bent. The fiber is locked with clamps at both ends: before fiber winding, a first clamp holds the fiber at one point, while the second clamp maintains the curvature and twisting conditions after winding.

 

Fig. 3. Experimental setup for the measure of the DGD between central and outer cores.

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As Fig. 3 shows, the optical signal coming from a tunable laser is injected into the central core (Core 1) and one of the outer cores (Cores 2-7) of the MCF. Prior to each measurement, we adjusted the optical paths for the central core to have a differential delay of around 20 ps greater than the outer cores. This was done to avoid zero-DGD situations in which the DGD measurement method is not accurate. At the fiber output, both signals are coupled together and injected into an Optical Spectrum Analyzer (OSA). Sweeping the optical wavelength of the laser reveals the interference pattern from which we measure the differential delay between both cores, [17]. As an example, Fig. 4(a) represents the interference pattern of cores 1-2 measured when the fiber is bent at a 25-mm radius within a 10-nm range (from 1545 to 1555 nm). This pattern is then Fourier transformed into the time domain to obtain the temporal representation of Fig. 4(b); the difference between these two peaks determines the DGD due to the curvature.

 

Fig. 4. (a) Interference pattern (optical power) measured by the OSA when the fiber is bent at a 25-mm radius (upper) and in straight condition (lower); and (b) temporal waveforms obtained from the inverse Fourier Transform of the interference patterns when the fiber is bent at a 25-mm radius (blue) and in straight condition (orange).

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Table 1 gathers the measured DGD values between the outer and central cores for the three bending radii when two different conditions are applied: (a) the fiber is bent among the cylinder carefully trying not to induce any fiber twisting; and (b) the fiber is bent while forcing an intentional twist. For the forced twist experiments, the fiber twisting was manually performed by introducing several intentional rotations distributed along the 1-m fiber while fiber winding. We can see as expected that the DGD is bigger as the bending radius is smaller, as well as how twisting the fiber produces an important reduction on the core DGDs. Slightly asymmetric behavior between the cores with opposite delay variation can be attributed to both fabrication mismatches on the core positions (radial, r_m, and angular, θ_m,i) with respect to the ideal configuration and also to measurement tolerances.

Table 1. Normalized inter-core DGD (ps/m) values measured between pair of cores for different bending and twisting conditions

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Figure 5(a) depicts the core DGDs located inside the fiber cross-section for the representative case of R_b = 35 mm with and without twist. The position of the tangential plane has been estimated from the core DGD measurements gathered in Table 1. As shown, cores located at one side of the tangential plane have DGDs with opposite signs than those located in the other side. Moreover, the DGD of those cores located in opposite radial positions (i.e., θ_m,i - θ_n,i = kπ) have similar absolute value (ideally identical) but opposite signs. The worst-case DGD variation occurs when the core forms an angle of ±π/2 with the tangential plane, but this case cannot actually be measured with precision since we do not have control over the curvature plane (which we assume random) during the measurement process. However, we can estimate the worst-case DGD from the measured values by doing a plane correction. For instance, in the case of the tangential planes of Fig. 5(a), we can apply the plane correction θ_c (in degrees) that satisfies:

 

Fig. 5. (a) Measured core differential group delays between bent and unbent fiber for a bending radius of 35 mm with and without twist. (b) Differential group delay dependence on the bending radius: Solid and dashed lines correspond to the worst-case computed theoretical responses without twist, while error bars represent the worst-case experimental values (red bars: no twist, blue bars: forced twist).

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4. Experimental impact of bending and twisting effects on microwave signal processing applications

Non-desired group delay variations between cores could affect applications where time-delay control and synchronization play a crucial role. This is the case, for instance, of fiber-distributed RF signal processing, where the MCF offers the required parallelism for the compact implementation of a sampled true time delay line architecture using a single optical fiber, [8]. Therefore, we have experimentally evaluated how bending and twisting affect a delay-sensitive application such as Microwave Photonics signal filtering. We implemented a 7-tap finite impulse response filter with a Free Spectral Range of 10 GHz by adjusting the group delay difference between adjacent filter samples (or basic differential delay) to Δτ = 100 ps. The filter frequency response was measured for two different fiber conditions: (1) bent without intentional twist and (2) bent with a forced twist, each one for three different bending radii: 25, 35 and 50 mm.

The experimental setup used to measure the filter electrical response is depicted in Fig. 6. This setup was also used in [18], where we presented some preliminary results about this topic. The optical signal is generated by a broadband source (BS) which is then filtered to obtain an optical carrier with a -3-dB bandwidth of 0.3 nm. The use of a BS makes the filter response insensitive to environmental changes such as temperature or vibrations. The optical signal is then modulated and equally launched to all cores using a 1 × 8 coupler and a fan-in device. The fiber condition is change for each curvature/twist case similarly to what we did with group delay measurement. At each fanout output, 2 cm of singlemode fiber (approximately 100-ps delay) were added incrementally from tap to tap to obtain the basic differential delay. A series of variable delay lines (VDLs) at the output of the MCF were used to finely tune the tap delays. To obtain a uniform amplitude distribution between the filter taps, we used Variable Optical Attenuators (VOAs). Output signals are combined before photodetection. The electrical response is measured by a Vectorial Network Analyzer (VNA), which also generates the RF input signal.

 

Fig. 6. Setup scheme for the experimental measure of MWP signal filtering response.

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Figure 7 illustrates the measured filter transfer function as a function of the RF frequency for the different bending/twisting cases stated before. The blue solid line represents the case in which the fiber is wound over the cylinder with a forced regular twist and the red dash-dotted line represents the case where the fiber is bent trying not to produce any twist. In the absence of twist, the filter response when the fiber is bent at 25 mm (Fig. 7(a)) experiences distortions even in the first resonance. When the curvature radius is 35 mm (Fig. 7(b)), the distortions are less important but still present at high frequency resonances. Finally, in the last case (Fig. 7(c)), distortions only appear importantly at the last resonance represented. In all cases, the filter response greatly improves when the intentional twist is applied due to the reduction in the DGDs between cores, as we have seen in section 3. This is particularly important for high-frequency resonances where bent-induced DGD deviations have greater effect on the RF response. These results show how applying a moderate twist to the fiber helps reducing the DGD variations introduced by the fiber itself and, thus, improve the performance of the implemented signal processing functionality.

 

Fig. 7. Measured filter response for: (solid blue) fiber bent with high twist, (dash-dotted red) fiber bent with low twist; and different radius: a) 25 mm b) 35 mm c) 50 mm.

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5. Conclusions

One of the most important sources of degradation that could compromise the performance of time-delay sensitive applications that incorporate MCFs relates to possible bending and twisting effects. In this paper, we have evaluated both theoretically and experimentally how fiber bending and twisting induce group delay variations between the central and outer cores of an MCF. We have obtained analytical expressions for the computation of the bend-induced DGD accumulated on a given fiber length for a constant twist rate. We must note that the effect of the fiber twist on the intercore DGD has been evaluated in terms of the linear rotation of the fiber cross-section area along the fiber. However, we have not considered the effect of twist-induced birefringence caused by possible deformations in the ellipticity of the cores as a consequence of the applied torsional strain, [19].

Experimental comparison of the effect of different twisting conditions was carried on a 1-m link of homogeneous 7-core MCF for three different bending radii of 25, 35 and 50 mm. These effects were also evaluated experimentally in the context of a particular time-sensitive scenario, time-discrete microwave signal filtering, where the 7 cores acted as the optical carriers for the 7 filter taps. Our results for RF frequencies up to 50 GHz show how even a tightly bent 1-m MCF can degrade the performance of this application and how fiber twisting is actually beneficial to counteract the group delay variations produced by the fiber curvature. These findings are generally applicable to time-delay sensitive application areas such as radio-over-fiber distribution and MWP signal processing in fiber-wireless access networks, as well as high-capacity long-haul digital communications where digital MIMO processing may be required.